:: SFMASTR3 semantic presentation

begin


theorem Th1: :: SFMASTR3:1
for s being State of SCM+FSA
for aa being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st I is_closed_on Initialized s,p & I is_halting_on Initialized s,p & not I destroys aa holds
(IExec (I,p,s)) . aa = (Initialized s) . aa
proof
let s be State of SCM+FSA; ::_thesis: for aa being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st I is_closed_on Initialized s,p & I is_halting_on Initialized s,p & not I destroys aa holds
(IExec (I,p,s)) . aa = (Initialized s) . aa
let aa be Int-Location; ::_thesis: for I being Program of
for p being Instruction-Sequence of SCM+FSA st I is_closed_on Initialized s,p & I is_halting_on Initialized s,p & not I destroys aa holds
(IExec (I,p,s)) . aa = (Initialized s) . aa
let I be Program of ; ::_thesis: for p being Instruction-Sequence of SCM+FSA st I is_closed_on Initialized s,p & I is_halting_on Initialized s,p & not I destroys aa holds
(IExec (I,p,s)) . aa = (Initialized s) . aa
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( I is_closed_on Initialized s,p & I is_halting_on Initialized s,p & not I destroys aa implies (IExec (I,p,s)) . aa = (Initialized s) . aa )
set a = aa;
A1: DataPart (Initialized s) = DataPart (Initialize (Initialized s)) by MEMSTR_0:79;
assume  ( I is_closed_on Initialized s,p & I is_halting_on Initialized s,p & not I destroys aa ) ; ::_thesis: (IExec (I,p,s)) . aa = (Initialized s) . aa
hence (IExec (I,p,s)) . aa = (Comput ((p +* I),(Initialize (Initialized s)),0)) . aa by SCMFSA8C:60
.= (Initialize (Initialized s)) . aa
.= (Initialized s) . aa by A1, SCMFSA_M:2 ;
::_thesis: verum
end;

theorem Th2: :: SFMASTR3:2
for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 holds
DataPart (IExec ((Stop SCM+FSA),p,s)) = DataPart s
proof
let s be State of SCM+FSA; ::_thesis: for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 holds
DataPart (IExec ((Stop SCM+FSA),p,s)) = DataPart s
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( s . (intloc 0) = 1 implies DataPart (IExec ((Stop SCM+FSA),p,s)) = DataPart s )
assume A1: s . (intloc 0) = 1 ; ::_thesis: DataPart (IExec ((Stop SCM+FSA),p,s)) = DataPart s
thus  DataPart (IExec ((Stop SCM+FSA),p,s)) = DataPart (Initialized s) by SCMFSA8C:14
.= DataPart s by A1, SCMFSA_M:19 ; ::_thesis: verum
end;

theorem Th3: :: SFMASTR3:3
for aa being Int-Location holds not Stop SCM+FSA refers aa
proof
let aa be Int-Location; ::_thesis: not Stop SCM+FSA refers aa
A1: rng (Stop SCM+FSA) = {(halt SCM+FSA)} by AFINSQ_1:33;
let i be Instruction of SCM+FSA; :: according to SCMFSA7B:def_2 ::_thesis: ( not i in rng (Stop SCM+FSA) or not i refers aa )
assume  i in rng (Stop SCM+FSA) ; ::_thesis: not i refers aa
then  i = halt SCM+FSA by A1, TARSKI:def_1;
hence  not i refers aa by SCMFSA8C:49; ::_thesis: verum
end;

theorem Th4: :: SFMASTR3:4
for aa, bb, cc being Int-Location st aa <> bb holds
not cc := bb refers aa
proof
let aa, bb, cc be Int-Location; ::_thesis: ( aa <> bb implies not cc := bb refers aa )
assume A1: aa <> bb ; ::_thesis: not cc := bb refers aa
now__::_thesis:_for_e_being_Int-Location
for_l_being_Element_of_NAT_
for_f_being_FinSeq-Location_holds_
(_e_:=_aa_<>_cc_:=_bb_&_AddTo_(e,aa)_<>_cc_:=_bb_&_SubFrom_(e,aa)_<>_cc_:=_bb_&_MultBy_(e,aa)_<>_cc_:=_bb_&_Divide_(aa,e)_<>_cc_:=_bb_&_Divide_(e,aa)_<>_cc_:=_bb_&_aa_=0_goto_l_<>_cc_:=_bb_&_aa_>0_goto_l_<>_cc_:=_bb_&_e_:=_(f,aa)_<>_cc_:=_bb_&_(f,e)_:=_aa_<>_cc_:=_bb_&_(f,aa)_:=_e_<>_cc_:=_bb_&_f_:=<0,...,0>_aa_<>_cc_:=_bb_)
let e be Int-Location; ::_thesis: for l being Element of NAT
for f being FinSeq-Location holds
( e := aa <> cc := bb & AddTo (e,aa) <> cc := bb & SubFrom (e,aa) <> cc := bb & MultBy (e,aa) <> cc := bb & Divide (aa,e) <> cc := bb & Divide (e,aa) <> cc := bb & aa =0_goto l <> cc := bb & aa >0_goto l <> cc := bb & e := (f,aa) <> cc := bb & (f,e) := aa <> cc := bb & (f,aa) := e <> cc := bb & f :=<0,...,0> aa <> cc := bb )
let l be Element of NAT ; ::_thesis: for f being FinSeq-Location holds
( e := aa <> cc := bb & AddTo (e,aa) <> cc := bb & SubFrom (e,aa) <> cc := bb & MultBy (e,aa) <> cc := bb & Divide (aa,e) <> cc := bb & Divide (e,aa) <> cc := bb & aa =0_goto l <> cc := bb & aa >0_goto l <> cc := bb & e := (f,aa) <> cc := bb & (f,e) := aa <> cc := bb & (f,aa) := e <> cc := bb & f :=<0,...,0> aa <> cc := bb )
let f be FinSeq-Location ; ::_thesis: ( e := aa <> cc := bb & AddTo (e,aa) <> cc := bb & SubFrom (e,aa) <> cc := bb & MultBy (e,aa) <> cc := bb & Divide (aa,e) <> cc := bb & Divide (e,aa) <> cc := bb & aa =0_goto l <> cc := bb & aa >0_goto l <> cc := bb & e := (f,aa) <> cc := bb & (f,e) := aa <> cc := bb & (f,aa) := e <> cc := bb & f :=<0,...,0> aa <> cc := bb )
thus  e := aa <> cc := bb by A1, SF_MASTR:1; ::_thesis: ( AddTo (e,aa) <> cc := bb & SubFrom (e,aa) <> cc := bb & MultBy (e,aa) <> cc := bb & Divide (aa,e) <> cc := bb & Divide (e,aa) <> cc := bb & aa =0_goto l <> cc := bb & aa >0_goto l <> cc := bb & e := (f,aa) <> cc := bb & (f,e) := aa <> cc := bb & (f,aa) := e <> cc := bb & f :=<0,...,0> aa <> cc := bb )
A2: InsCode (cc := bb) = 1 by SCMFSA_2:18;
hence  AddTo (e,aa) <> cc := bb by SCMFSA_2:19; ::_thesis: ( SubFrom (e,aa) <> cc := bb & MultBy (e,aa) <> cc := bb & Divide (aa,e) <> cc := bb & Divide (e,aa) <> cc := bb & aa =0_goto l <> cc := bb & aa >0_goto l <> cc := bb & e := (f,aa) <> cc := bb & (f,e) := aa <> cc := bb & (f,aa) := e <> cc := bb & f :=<0,...,0> aa <> cc := bb )
thus  SubFrom (e,aa) <> cc := bb by A2, SCMFSA_2:20; ::_thesis: ( MultBy (e,aa) <> cc := bb & Divide (aa,e) <> cc := bb & Divide (e,aa) <> cc := bb & aa =0_goto l <> cc := bb & aa >0_goto l <> cc := bb & e := (f,aa) <> cc := bb & (f,e) := aa <> cc := bb & (f,aa) := e <> cc := bb & f :=<0,...,0> aa <> cc := bb )
thus  MultBy (e,aa) <> cc := bb by A2, SCMFSA_2:21; ::_thesis: ( Divide (aa,e) <> cc := bb & Divide (e,aa) <> cc := bb & aa =0_goto l <> cc := bb & aa >0_goto l <> cc := bb & e := (f,aa) <> cc := bb & (f,e) := aa <> cc := bb & (f,aa) := e <> cc := bb & f :=<0,...,0> aa <> cc := bb )
thus  ( Divide (aa,e) <> cc := bb & Divide (e,aa) <> cc := bb ) by A2, SCMFSA_2:22; ::_thesis: ( aa =0_goto l <> cc := bb & aa >0_goto l <> cc := bb & e := (f,aa) <> cc := bb & (f,e) := aa <> cc := bb & (f,aa) := e <> cc := bb & f :=<0,...,0> aa <> cc := bb )
thus  aa =0_goto l <> cc := bb by A2, SCMFSA_2:24; ::_thesis: ( aa >0_goto l <> cc := bb & e := (f,aa) <> cc := bb & (f,e) := aa <> cc := bb & (f,aa) := e <> cc := bb & f :=<0,...,0> aa <> cc := bb )
thus  aa >0_goto l <> cc := bb by A2, SCMFSA_2:25; ::_thesis: ( e := (f,aa) <> cc := bb & (f,e) := aa <> cc := bb & (f,aa) := e <> cc := bb & f :=<0,...,0> aa <> cc := bb )
thus  e := (f,aa) <> cc := bb by A2, SCMFSA_2:26; ::_thesis: ( (f,e) := aa <> cc := bb & (f,aa) := e <> cc := bb & f :=<0,...,0> aa <> cc := bb )
thus  ( (f,e) := aa <> cc := bb & (f,aa) := e <> cc := bb ) by A2, SCMFSA_2:27; ::_thesis: f :=<0,...,0> aa <> cc := bb
thus  f :=<0,...,0> aa <> cc := bb by A2, SCMFSA_2:29; ::_thesis: verum
end;
hence  not cc := bb refers aa by SCMFSA7B:def_1; ::_thesis: verum
end;

theorem Th5: :: SFMASTR3:5
for s being State of SCM+FSA
for a being read-write Int-Location
for bb being Int-Location
for f being FinSeq-Location holds (Exec ((a := (f,bb)),s)) . a = (s . f) /. (abs (s . bb))
proof
let s be State of SCM+FSA; ::_thesis: for a being read-write Int-Location
for bb being Int-Location
for f being FinSeq-Location holds (Exec ((a := (f,bb)),s)) . a = (s . f) /. (abs (s . bb))
let a be read-write Int-Location; ::_thesis: for bb being Int-Location
for f being FinSeq-Location holds (Exec ((a := (f,bb)),s)) . a = (s . f) /. (abs (s . bb))
let bb be Int-Location; ::_thesis: for f being FinSeq-Location holds (Exec ((a := (f,bb)),s)) . a = (s . f) /. (abs (s . bb))
let f be FinSeq-Location ; ::_thesis: (Exec ((a := (f,bb)),s)) . a = (s . f) /. (abs (s . bb))
 ex k being Element of NAT st
( k = abs (s . bb) & (Exec ((a := (f,bb)),s)) . a = (s . f) /. k ) by SCMFSA_2:72;
hence  (Exec ((a := (f,bb)),s)) . a = (s . f) /. (abs (s . bb)) ; ::_thesis: verum
end;

theorem Th6: :: SFMASTR3:6
for s being State of SCM+FSA
for aa, bb being Int-Location
for f being FinSeq-Location holds (Exec (((f,aa) := bb),s)) . f = (s . f) +* ((abs (s . aa)),(s . bb))
proof
let s be State of SCM+FSA; ::_thesis: for aa, bb being Int-Location
for f being FinSeq-Location holds (Exec (((f,aa) := bb),s)) . f = (s . f) +* ((abs (s . aa)),(s . bb))
let aa, bb be Int-Location; ::_thesis: for f being FinSeq-Location holds (Exec (((f,aa) := bb),s)) . f = (s . f) +* ((abs (s . aa)),(s . bb))
let f be FinSeq-Location ; ::_thesis: (Exec (((f,aa) := bb),s)) . f = (s . f) +* ((abs (s . aa)),(s . bb))
 ex k being Element of NAT st
( k = abs (s . aa) & (Exec (((f,aa) := bb),s)) . f = (s . f) +* (k,(s . bb)) ) by SCMFSA_2:73;
hence  (Exec (((f,aa) := bb),s)) . f = (s . f) +* ((abs (s . aa)),(s . bb)) ; ::_thesis: verum
end;

registration
let a be read-write Int-Location;
let b be Int-Location;
let I, J be good Program of ;
cluster if>0 (a,b,I,J) ->  good ;
coherence
 if>0 (a,b,I,J) is good
proof
 if>0 (a,b,I,J) = (SubFrom (a,b)) ";" (if>0 (a,I,J)) by SCMFSA8B:def_5;
hence  if>0 (a,b,I,J) is good ; ::_thesis: verum
end;
end;

theorem Th7: :: SFMASTR3:7
for aa, bb being Int-Location
for I, J being Program of holds UsedIntLoc (if>0 (aa,bb,I,J)) = ({aa,bb} \/ (UsedIntLoc I)) \/ (UsedIntLoc J)
proof
let aa, bb be Int-Location; ::_thesis: for I, J being Program of holds UsedIntLoc (if>0 (aa,bb,I,J)) = ({aa,bb} \/ (UsedIntLoc I)) \/ (UsedIntLoc J)
let I, J be Program of ; ::_thesis: UsedIntLoc (if>0 (aa,bb,I,J)) = ({aa,bb} \/ (UsedIntLoc I)) \/ (UsedIntLoc J)
set a = aa;
thus  UsedIntLoc (if>0 (aa,bb,I,J)) = UsedIntLoc ((SubFrom (aa,bb)) ";" (if>0 (aa,I,J))) by SCMFSA8B:def_5
.= (UsedIntLoc (SubFrom (aa,bb))) \/ (UsedIntLoc (if>0 (aa,I,J))) by SF_MASTR:29
.= {aa,bb} \/ (UsedIntLoc (if>0 (aa,I,J))) by SF_MASTR:14
.= {aa,bb} \/ (({aa} \/ (UsedIntLoc I)) \/ (UsedIntLoc J)) by SCMFSA9A:9
.= {aa,bb} \/ ({aa} \/ ((UsedIntLoc I) \/ (UsedIntLoc J))) by XBOOLE_1:4
.= ({aa,bb} \/ {aa}) \/ ((UsedIntLoc I) \/ (UsedIntLoc J)) by XBOOLE_1:4
.= {aa,bb} \/ ((UsedIntLoc I) \/ (UsedIntLoc J)) by ZFMISC_1:9
.= ({aa,bb} \/ (UsedIntLoc I)) \/ (UsedIntLoc J) by XBOOLE_1:4 ; ::_thesis: verum
end;

theorem Th8: :: SFMASTR3:8
for aa, bb being Int-Location
for I being Program of st not I destroys aa holds
not while>0 (bb,I) destroys aa
proof
let aa, bb be Int-Location; ::_thesis: for I being Program of st not I destroys aa holds
not while>0 (bb,I) destroys aa
let I be Program of ; ::_thesis: ( not I destroys aa implies not while>0 (bb,I) destroys aa )
set J = ((card I) + 4) .--> (goto 0);
set F = if>0 (bb,(I ";" (Goto 0)),(Stop SCM+FSA));
A1: not Goto 0 destroys aa by SCMFSA8C:57;
A2: not Stop SCM+FSA destroys aa by SCMFSA8C:56;
assume  not I destroys aa ; ::_thesis: not while>0 (bb,I) destroys aa
then  not I ";" (Goto 0) destroys aa by A1, SCMFSA8C:52;
then A3: not if>0 (bb,(I ";" (Goto 0)),(Stop SCM+FSA)) destroys aa by A2, SCMFSA8C:88;
 ( not ((card I) + 4) .--> (goto 0) destroys aa & while>0 (bb,I) = (if>0 (bb,(I ";" (Goto 0)),(Stop SCM+FSA))) +* (((card I) + 4) .--> (goto 0)) ) by SCMFSA_9:30, SCMFSA_9:def_2;
hence  not while>0 (bb,I) destroys aa by A3, SCMFSA8A:11; ::_thesis: verum
end;

theorem Th9: :: SFMASTR3:9
for cc, aa, bb being Int-Location
for I, J being Program of st cc <> aa & not I destroys cc & not J destroys cc holds
not if>0 (aa,bb,I,J) destroys cc
proof
let cc, aa, bb be Int-Location; ::_thesis: for I, J being Program of st cc <> aa & not I destroys cc & not J destroys cc holds
not if>0 (aa,bb,I,J) destroys cc
let I, J be Program of ; ::_thesis: ( cc <> aa & not I destroys cc & not J destroys cc implies not if>0 (aa,bb,I,J) destroys cc )
assume that
A1: cc <> aa and
A2: ( not I destroys cc & not J destroys cc ) ; ::_thesis: not if>0 (aa,bb,I,J) destroys cc
 ( if>0 (aa,bb,I,J) = (SubFrom (aa,bb)) ";" (if>0 (aa,I,J)) & not if>0 (aa,I,J) destroys cc ) by A2, SCMFSA8B:def_5, SCMFSA8C:88;
hence  not if>0 (aa,bb,I,J) destroys cc by A1, SCMFSA7B:8, SCMFSA8C:53; ::_thesis: verum
end;

begin

definition
let p be Instruction-Sequence of SCM+FSA;
let a, b, c be Int-Location;
let I be Program of ;
let s be State of SCM+FSA;
func StepForUp (a,b,c,I,p,s) ->  Function of NAT,(product (the_Values_of SCM+FSA)) equals :: SFMASTR3:def 1
 StepWhile>0 ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),(((s . c) - (s . b)) + 1))) +* (a,(s . b))));
coherence
 StepWhile>0 ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),(((s . c) - (s . b)) + 1))) +* (a,(s . b)))) is Function of NAT,(product (the_Values_of SCM+FSA)) ;
end;

:: deftheorem  defines StepForUp SFMASTR3:def_1_:_
 for p being Instruction-Sequence of SCM+FSA
for a, b, c being Int-Location
for I being Program of
for s being State of SCM+FSA holds StepForUp (a,b,c,I,p,s) = StepWhile>0 ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),(((s . c) - (s . b)) + 1))) +* (a,(s . b))));

theorem Th10: :: SFMASTR3:10
for s being State of SCM+FSA
for a being read-write Int-Location
for bb, cc being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . (intloc 0) = 1
proof
let s be State of SCM+FSA; ::_thesis: for a being read-write Int-Location
for bb, cc being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . (intloc 0) = 1
let a be read-write Int-Location; ::_thesis: for bb, cc being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . (intloc 0) = 1
let bb, cc be Int-Location; ::_thesis: for I being Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . (intloc 0) = 1
let I be Program of ; ::_thesis: for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . (intloc 0) = 1
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( s . (intloc 0) = 1 implies ((StepForUp (a,bb,cc,I,p,s)) . 0) . (intloc 0) = 1 )
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I));
set S = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb));
A1: ((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . (intloc 0) = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) . (intloc 0) by FUNCT_7:32
.= s . (intloc 0) by FUNCT_7:32 ;
assume  s . (intloc 0) = 1 ; ::_thesis: ((StepForUp (a,bb,cc,I,p,s)) . 0) . (intloc 0) = 1
hence  ((StepForUp (a,bb,cc,I,p,s)) . 0) . (intloc 0) = 1 by A1, SCMFSA_9:def_5; ::_thesis: verum
end;

theorem Th11: :: SFMASTR3:11
for s being State of SCM+FSA
for a being read-write Int-Location
for bb, cc being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA holds ((StepForUp (a,bb,cc,I,p,s)) . 0) . a = s . bb
proof
let s be State of SCM+FSA; ::_thesis: for a being read-write Int-Location
for bb, cc being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA holds ((StepForUp (a,bb,cc,I,p,s)) . 0) . a = s . bb
let a be read-write Int-Location; ::_thesis: for bb, cc being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA holds ((StepForUp (a,bb,cc,I,p,s)) . 0) . a = s . bb
let bb, cc be Int-Location; ::_thesis: for I being Program of
for p being Instruction-Sequence of SCM+FSA holds ((StepForUp (a,bb,cc,I,p,s)) . 0) . a = s . bb
let I be Program of ; ::_thesis: for p being Instruction-Sequence of SCM+FSA holds ((StepForUp (a,bb,cc,I,p,s)) . 0) . a = s . bb
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ((StepForUp (a,bb,cc,I,p,s)) . 0) . a = s . bb
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I));
set S = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb));
 ( (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . 0 = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb)) & a in dom (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) ) by SCMFSA_2:42, SCMFSA_9:def_5;
hence  ((StepForUp (a,bb,cc,I,p,s)) . 0) . a = s . bb by FUNCT_7:31; ::_thesis: verum
end;

theorem Th12: :: SFMASTR3:12
for s being State of SCM+FSA
for a being read-write Int-Location
for bb, cc being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st a <> bb holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . bb = s . bb
proof
let s be State of SCM+FSA; ::_thesis: for a being read-write Int-Location
for bb, cc being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st a <> bb holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . bb = s . bb
let a be read-write Int-Location; ::_thesis: for bb, cc being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st a <> bb holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . bb = s . bb
let bb, cc be Int-Location; ::_thesis: for I being Program of
for p being Instruction-Sequence of SCM+FSA st a <> bb holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . bb = s . bb
let I be Program of ; ::_thesis: for p being Instruction-Sequence of SCM+FSA st a <> bb holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . bb = s . bb
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( a <> bb implies ((StepForUp (a,bb,cc,I,p,s)) . 0) . bb = s . bb )
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I));
set S = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb));
 bb in {a,bb,cc} by ENUMSET1:def_1;
then  bb in {a,bb,cc} \/ (UsedIntLoc I) by XBOOLE_0:def_3;
then A1: bb <> 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) by SCMFSA_M:25;
assume  a <> bb ; ::_thesis: ((StepForUp (a,bb,cc,I,p,s)) . 0) . bb = s . bb
then ((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . bb = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) . bb by FUNCT_7:32
.= s . bb by A1, FUNCT_7:32 ;
hence  ((StepForUp (a,bb,cc,I,p,s)) . 0) . bb = s . bb by SCMFSA_9:def_5; ::_thesis: verum
end;

theorem Th13: :: SFMASTR3:13
for s being State of SCM+FSA
for a being read-write Int-Location
for cc, bb being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st a <> cc holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . cc = s . cc
proof
let s be State of SCM+FSA; ::_thesis: for a being read-write Int-Location
for cc, bb being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st a <> cc holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . cc = s . cc
let a be read-write Int-Location; ::_thesis: for cc, bb being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st a <> cc holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . cc = s . cc
let cc, bb be Int-Location; ::_thesis: for I being Program of
for p being Instruction-Sequence of SCM+FSA st a <> cc holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . cc = s . cc
let I be Program of ; ::_thesis: for p being Instruction-Sequence of SCM+FSA st a <> cc holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . cc = s . cc
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( a <> cc implies ((StepForUp (a,bb,cc,I,p,s)) . 0) . cc = s . cc )
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I));
set S = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb));
 cc in {a,bb,cc} by ENUMSET1:def_1;
then  cc in {a,bb,cc} \/ (UsedIntLoc I) by XBOOLE_0:def_3;
then A1: cc <> 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) by SCMFSA_M:25;
assume  a <> cc ; ::_thesis: ((StepForUp (a,bb,cc,I,p,s)) . 0) . cc = s . cc
then ((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . cc = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) . cc by FUNCT_7:32
.= s . cc by A1, FUNCT_7:32 ;
hence  ((StepForUp (a,bb,cc,I,p,s)) . 0) . cc = s . cc by SCMFSA_9:def_5; ::_thesis: verum
end;

theorem Th14: :: SFMASTR3:14
for s being State of SCM+FSA
for a being read-write Int-Location
for dd, bb, cc being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st a <> dd & dd in UsedIntLoc I holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . dd = s . dd
proof
let s be State of SCM+FSA; ::_thesis: for a being read-write Int-Location
for dd, bb, cc being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st a <> dd & dd in UsedIntLoc I holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . dd = s . dd
let a be read-write Int-Location; ::_thesis: for dd, bb, cc being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st a <> dd & dd in UsedIntLoc I holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . dd = s . dd
let dd, bb, cc be Int-Location; ::_thesis: for I being Program of
for p being Instruction-Sequence of SCM+FSA st a <> dd & dd in UsedIntLoc I holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . dd = s . dd
let I be Program of ; ::_thesis: for p being Instruction-Sequence of SCM+FSA st a <> dd & dd in UsedIntLoc I holds
((StepForUp (a,bb,cc,I,p,s)) . 0) . dd = s . dd
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( a <> dd & dd in UsedIntLoc I implies ((StepForUp (a,bb,cc,I,p,s)) . 0) . dd = s . dd )
assume that
A1: a <> dd and
A2: dd in UsedIntLoc I ; ::_thesis: ((StepForUp (a,bb,cc,I,p,s)) . 0) . dd = s . dd
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I));
 dd in {a,bb,cc} \/ (UsedIntLoc I) by A2, XBOOLE_0:def_3;
then A3: dd <> 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) by SCMFSA_M:25;
set S = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb));
((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . dd = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) . dd by A1, FUNCT_7:32
.= s . dd by A3, FUNCT_7:32 ;
hence  ((StepForUp (a,bb,cc,I,p,s)) . 0) . dd = s . dd by SCMFSA_9:def_5; ::_thesis: verum
end;

theorem Th15: :: SFMASTR3:15
for s being State of SCM+FSA
for a being read-write Int-Location
for bb, cc being Int-Location
for f being FinSeq-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA holds ((StepForUp (a,bb,cc,I,p,s)) . 0) . f = s . f
proof
let s be State of SCM+FSA; ::_thesis: for a being read-write Int-Location
for bb, cc being Int-Location
for f being FinSeq-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA holds ((StepForUp (a,bb,cc,I,p,s)) . 0) . f = s . f
let a be read-write Int-Location; ::_thesis: for bb, cc being Int-Location
for f being FinSeq-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA holds ((StepForUp (a,bb,cc,I,p,s)) . 0) . f = s . f
let bb, cc be Int-Location; ::_thesis: for f being FinSeq-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA holds ((StepForUp (a,bb,cc,I,p,s)) . 0) . f = s . f
let f be FinSeq-Location ; ::_thesis: for I being Program of
for p being Instruction-Sequence of SCM+FSA holds ((StepForUp (a,bb,cc,I,p,s)) . 0) . f = s . f
let I be Program of ; ::_thesis: for p being Instruction-Sequence of SCM+FSA holds ((StepForUp (a,bb,cc,I,p,s)) . 0) . f = s . f
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ((StepForUp (a,bb,cc,I,p,s)) . 0) . f = s . f
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I));
set S = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb));
((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . f = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) . f by FUNCT_7:32, SCMFSA_2:58
.= s . f by FUNCT_7:32, SCMFSA_2:58 ;
hence  ((StepForUp (a,bb,cc,I,p,s)) . 0) . f = s . f by SCMFSA_9:def_5; ::_thesis: verum
end;

theorem Th16: :: SFMASTR3:16
for s being State of SCM+FSA
for a being read-write Int-Location
for bb, cc being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 holds
for aux being read-write Int-Location st aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) holds
DataPart (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) = DataPart ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb)))
proof
let s be State of SCM+FSA; ::_thesis: for a being read-write Int-Location
for bb, cc being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 holds
for aux being read-write Int-Location st aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) holds
DataPart (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) = DataPart ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb)))
let a be read-write Int-Location; ::_thesis: for bb, cc being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 holds
for aux being read-write Int-Location st aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) holds
DataPart (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) = DataPart ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb)))
let bb, cc be Int-Location; ::_thesis: for I being Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 holds
for aux being read-write Int-Location st aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) holds
DataPart (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) = DataPart ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb)))
let I be Program of ; ::_thesis: for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 holds
for aux being read-write Int-Location st aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) holds
DataPart (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) = DataPart ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb)))
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( s . (intloc 0) = 1 implies for aux being read-write Int-Location st aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) holds
DataPart (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) = DataPart ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) )
assume A1: s . (intloc 0) = 1 ; ::_thesis: for aux being read-write Int-Location st aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) holds
DataPart (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) = DataPart ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb)))
 ( cc = intloc 0 or cc is read-write ) by SCMFSA_M:def_2;
then A2: (Initialized s) . cc = s . cc by A1, SCMFSA_M:9, SCMFSA_M:37;
set i3 = a := bb;
let aux be read-write Int-Location; ::_thesis: ( aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) implies DataPart (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) = DataPart ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) )
assume A3: aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) ; ::_thesis: DataPart (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) = DataPart ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb)))
 bb in {a,bb,cc} by ENUMSET1:def_1;
then  bb in {a,bb,cc} \/ (UsedIntLoc I) by XBOOLE_0:def_3;
then A4: bb <> aux by A3, SCMFSA_M:25;
set s2 = (s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb));
A5: aux in dom s by SCMFSA_2:42;
 a in {a,bb,cc} by ENUMSET1:def_1;
then  a in {a,bb,cc} \/ (UsedIntLoc I) by XBOOLE_0:def_3;
then A6: a <> aux by A3, SCMFSA_M:25;
then A7: ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . aux = (s +* (aux,(((s . cc) - (s . bb)) + 1))) . aux by FUNCT_7:32
.= ((s . cc) - (s . bb)) + 1 by A5, FUNCT_7:31 ;
set i2 = AddTo (aux,(intloc 0));
set i1 = SubFrom (aux,bb);
set i0 = aux := cc;
set s1 = IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s);
A8: (IExec (((aux := cc) ";" (SubFrom (aux,bb))),p,s)) . (intloc 0) = (Exec ((SubFrom (aux,bb)),(Exec ((aux := cc),(Initialized s))))) . (intloc 0) by SCMFSA6C:8
.= (Exec ((aux := cc),(Initialized s))) . (intloc 0) by SCMFSA_2:65
.= (Initialized s) . (intloc 0) by SCMFSA_2:63
.= 1 by SCMFSA_M:9 ;
A9: a in dom (s +* (aux,(((s . cc) - (s . bb)) + 1))) by SCMFSA_2:42;
 ( bb = intloc 0 or bb is read-write ) by SCMFSA_M:def_2;
then A10: (Initialized s) . bb = s . bb by A1, SCMFSA_M:9, SCMFSA_M:37;
A11: (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) . a = (Exec ((a := bb),(IExec ((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))),p,s)))) . a by SCMFSA6C:6
.= (IExec ((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))),p,s)) . bb by SCMFSA_2:63
.= (Exec ((AddTo (aux,(intloc 0))),(IExec (((aux := cc) ";" (SubFrom (aux,bb))),p,s)))) . bb by SCMFSA6C:6
.= (IExec (((aux := cc) ";" (SubFrom (aux,bb))),p,s)) . bb by A4, SCMFSA_2:64
.= (Exec ((SubFrom (aux,bb)),(Exec ((aux := cc),(Initialized s))))) . bb by SCMFSA6C:8
.= (Exec ((aux := cc),(Initialized s))) . bb by A4, SCMFSA_2:65
.= s . bb by A4, A10, SCMFSA_2:63 ;
A12: (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) . aux = (Exec ((a := bb),(IExec ((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))),p,s)))) . aux by SCMFSA6C:6
.= (IExec ((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))),p,s)) . aux by A6, SCMFSA_2:63
.= (Exec ((AddTo (aux,(intloc 0))),(IExec (((aux := cc) ";" (SubFrom (aux,bb))),p,s)))) . aux by SCMFSA6C:6
.= ((IExec (((aux := cc) ";" (SubFrom (aux,bb))),p,s)) . aux) + 1 by A8, SCMFSA_2:64
.= ((Exec ((SubFrom (aux,bb)),(Exec ((aux := cc),(Initialized s))))) . aux) + 1 by SCMFSA6C:8
.= (((Exec ((aux := cc),(Initialized s))) . aux) - ((Exec ((aux := cc),(Initialized s))) . bb)) + 1 by SCMFSA_2:65
.= (((Initialized s) . cc) - ((Exec ((aux := cc),(Initialized s))) . bb)) + 1 by SCMFSA_2:63
.= ((s . cc) - (s . bb)) + 1 by A4, A2, A10, SCMFSA_2:63 ;
now__::_thesis:_(_(_for_x_being_Int-Location_holds_(IExec_(((((aux_:=_cc)_";"_(SubFrom_(aux,bb)))_";"_(AddTo_(aux,(intloc_0))))_";"_(a_:=_bb)),p,s))_._x_=_((s_+*_(aux,(((s_._cc)_-_(s_._bb))_+_1)))_+*_(a,(s_._bb)))_._x_)_&_(_for_x_being_FinSeq-Location_holds_(IExec_(((((aux_:=_cc)_";"_(SubFrom_(aux,bb)))_";"_(AddTo_(aux,(intloc_0))))_";"_(a_:=_bb)),p,s))_._x_=_((s_+*_(aux,(((s_._cc)_-_(s_._bb))_+_1)))_+*_(a,(s_._bb)))_._x_)_)
hereby  ::_thesis: for x being FinSeq-Location holds (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) . x = ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . x
let x be Int-Location; ::_thesis: (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) . b1 = ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . b1
percases ( x = a or x = aux or ( x <> aux & x <> a ) ) ;
suppose x = a ; ::_thesis: (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) . b1 = ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . b1
hence  (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) . x = ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . x by A11, A9, FUNCT_7:31; ::_thesis: verum
end;
suppose x = aux ; ::_thesis: (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) . b1 = ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . b1
hence  (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) . x = ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . x by A12, A7; ::_thesis: verum
end;
supposeA13: ( x <> aux & x <> a ) ; ::_thesis: (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) . b1 = ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . b1
then A14: ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . x = (s +* (aux,(((s . cc) - (s . bb)) + 1))) . x by FUNCT_7:32
.= s . x by A13, FUNCT_7:32 ;
A15: ( x = intloc 0 or x is read-write ) by SCMFSA_M:def_2;
(IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) . x = (Exec ((a := bb),(IExec ((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))),p,s)))) . x by SCMFSA6C:6
.= (IExec ((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))),p,s)) . x by A13, SCMFSA_2:63
.= (Exec ((AddTo (aux,(intloc 0))),(IExec (((aux := cc) ";" (SubFrom (aux,bb))),p,s)))) . x by SCMFSA6C:6
.= (IExec (((aux := cc) ";" (SubFrom (aux,bb))),p,s)) . x by A13, SCMFSA_2:64
.= (Exec ((SubFrom (aux,bb)),(Exec ((aux := cc),(Initialized s))))) . x by SCMFSA6C:8
.= (Exec ((aux := cc),(Initialized s))) . x by A13, SCMFSA_2:65
.= (Initialized s) . x by A13, SCMFSA_2:63
.= s . x by A1, A15, SCMFSA_M:9, SCMFSA_M:37 ;
hence  (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) . x = ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . x by A14; ::_thesis: verum
end;
end;
end;
let x be FinSeq-Location ; ::_thesis: (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) . x = ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . x
thus (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) . x = (Exec ((a := bb),(IExec ((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))),p,s)))) . x by SCMFSA6C:7
.= (IExec ((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))),p,s)) . x by SCMFSA_2:63
.= (Exec ((AddTo (aux,(intloc 0))),(IExec (((aux := cc) ";" (SubFrom (aux,bb))),p,s)))) . x by SCMFSA6C:7
.= (IExec (((aux := cc) ";" (SubFrom (aux,bb))),p,s)) . x by SCMFSA_2:64
.= (Exec ((SubFrom (aux,bb)),(Exec ((aux := cc),(Initialized s))))) . x by SCMFSA6C:9
.= (Exec ((aux := cc),(Initialized s))) . x by SCMFSA_2:65
.= (Initialized s) . x by SCMFSA_2:63
.= s . x by SCMFSA_M:37
.= (s +* (aux,(((s . cc) - (s . bb)) + 1))) . x by FUNCT_7:32, SCMFSA_2:58
.= ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . x by FUNCT_7:32, SCMFSA_2:58 ; ::_thesis: verum
end;
hence  DataPart (IExec (((((aux := cc) ";" (SubFrom (aux,bb))) ";" (AddTo (aux,(intloc 0)))) ";" (a := bb)),p,s)) = DataPart ((s +* (aux,(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) by SCMFSA_M:2; ::_thesis: verum
end;

definition
let p be Instruction-Sequence of SCM+FSA;
let a, b, c be Int-Location;
let I be Program of ;
let s be State of SCM+FSA;
pred ProperForUpBody a,b,c,I,s,p means :Def2: :: SFMASTR3:def 2
 for i being Element of NAT st i < ((s . c) - (s . b)) + 1 holds
( I is_closed_on (StepForUp (a,b,c,I,p,s)) . i,p +* (while>0 ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),(intloc 0)))))) & I is_halting_on (StepForUp (a,b,c,I,p,s)) . i,p +* (while>0 ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),(intloc 0)))))) );
end;

:: deftheorem Def2 defines ProperForUpBody SFMASTR3:def_2_:_
 for p being Instruction-Sequence of SCM+FSA
for a, b, c being Int-Location
for I being Program of
for s being State of SCM+FSA holds
( ProperForUpBody a,b,c,I,s,p iff for i being Element of NAT st i < ((s . c) - (s . b)) + 1 holds
( I is_closed_on (StepForUp (a,b,c,I,p,s)) . i,p +* (while>0 ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),(intloc 0)))))) & I is_halting_on (StepForUp (a,b,c,I,p,s)) . i,p +* (while>0 ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),(intloc 0)))))) ) );

theorem Th17: :: SFMASTR3:17
for s being State of SCM+FSA
for aa, bb, cc being Int-Location
for p being Instruction-Sequence of SCM+FSA
for I being parahalting Program of holds ProperForUpBody aa,bb,cc,I,s,p
proof
let s be State of SCM+FSA; ::_thesis: for aa, bb, cc being Int-Location
for p being Instruction-Sequence of SCM+FSA
for I being parahalting Program of holds ProperForUpBody aa,bb,cc,I,s,p
let aa, bb, cc be Int-Location; ::_thesis: for p being Instruction-Sequence of SCM+FSA
for I being parahalting Program of holds ProperForUpBody aa,bb,cc,I,s,p
let p be Instruction-Sequence of SCM+FSA; ::_thesis: for I being parahalting Program of holds ProperForUpBody aa,bb,cc,I,s,p
let I be parahalting Program of ; ::_thesis: ProperForUpBody aa,bb,cc,I,s,p
let i be Element of NAT ; :: according to SFMASTR3:def_2 ::_thesis: ( i < ((s . cc) - (s . bb)) + 1 implies ( I is_closed_on (StepForUp (aa,bb,cc,I,p,s)) . i,p +* (while>0 ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (aa,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) & I is_halting_on (StepForUp (aa,bb,cc,I,p,s)) . i,p +* (while>0 ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (aa,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) ) )
assume  i < ((s . cc) - (s . bb)) + 1 ; ::_thesis: ( I is_closed_on (StepForUp (aa,bb,cc,I,p,s)) . i,p +* (while>0 ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (aa,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) & I is_halting_on (StepForUp (aa,bb,cc,I,p,s)) . i,p +* (while>0 ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (aa,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) )
thus  I is_closed_on (StepForUp (aa,bb,cc,I,p,s)) . i,p +* (while>0 ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (aa,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by SCMFSA7B:18; ::_thesis: I is_halting_on (StepForUp (aa,bb,cc,I,p,s)) . i,p +* (while>0 ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (aa,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0))))))
thus  I is_halting_on (StepForUp (aa,bb,cc,I,p,s)) . i,p +* (while>0 ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (aa,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by SCMFSA7B:19; ::_thesis: verum
end;

theorem Th18: :: SFMASTR3:18
for s being State of SCM+FSA
for a being read-write Int-Location
for bb, cc being Int-Location
for Ig being good Program of
for k being Element of NAT
for p being Instruction-Sequence of SCM+FSA st ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (intloc 0) = 1 & Ig is_closed_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) & Ig is_halting_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) holds
((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (intloc 0) = 1
proof
let s be State of SCM+FSA; ::_thesis: for a being read-write Int-Location
for bb, cc being Int-Location
for Ig being good Program of
for k being Element of NAT
for p being Instruction-Sequence of SCM+FSA st ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (intloc 0) = 1 & Ig is_closed_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) & Ig is_halting_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) holds
((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (intloc 0) = 1
let a be read-write Int-Location; ::_thesis: for bb, cc being Int-Location
for Ig being good Program of
for k being Element of NAT
for p being Instruction-Sequence of SCM+FSA st ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (intloc 0) = 1 & Ig is_closed_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) & Ig is_halting_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) holds
((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (intloc 0) = 1
let bb, cc be Int-Location; ::_thesis: for Ig being good Program of
for k being Element of NAT
for p being Instruction-Sequence of SCM+FSA st ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (intloc 0) = 1 & Ig is_closed_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) & Ig is_halting_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) holds
((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (intloc 0) = 1
let Ig be good Program of ; ::_thesis: for k being Element of NAT
for p being Instruction-Sequence of SCM+FSA st ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (intloc 0) = 1 & Ig is_closed_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) & Ig is_halting_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) holds
((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (intloc 0) = 1
let k be Element of NAT ; ::_thesis: for p being Instruction-Sequence of SCM+FSA st ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (intloc 0) = 1 & Ig is_closed_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) & Ig is_halting_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) holds
((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (intloc 0) = 1
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (intloc 0) = 1 & Ig is_closed_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) & Ig is_halting_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) implies ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (intloc 0) = 1 )
set I = Ig;
assume that
A1: ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (intloc 0) = 1 and
A2: ( Ig is_closed_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) & Ig is_halting_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) ) ; ::_thesis: ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (intloc 0) = 1
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig));
set IB = (Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)));
set SW2 = StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))));
A3: (Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))) = Ig ";" ((AddTo (a,(intloc 0))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) by SCMFSA6A:28;
percases ( ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) <= 0 or ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0 ) ;
suppose ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) <= 0 ; ::_thesis: ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (intloc 0) = 1
then  DataPart ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . (k + 1)) = DataPart ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) by SCMFSA9A:31;
hence  ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (intloc 0) = 1 by A1, SCMFSA_M:2; ::_thesis: verum
end;
supposeA4: ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0 ; ::_thesis: ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (intloc 0) = 1
A5: ( Ig is_closed_on Initialized ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) & Ig is_halting_on Initialized ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) ) by A1, A2, SFMASTR2:5;
A6: (AddTo (a,(intloc 0))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))) is_closed_on IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) by SCMFSA7B:18;
then  ( (AddTo (a,(intloc 0))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))) is_halting_on IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) & (Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))) is_closed_on Initialized ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) ) by A3, A5, SCMFSA7B:19, SFMASTR1:2;
then A7: DataPart ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . (k + 1)) = DataPart (IExec (((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) by A1, A3, A4, A5, A6, SCMFSA9A:32, SFMASTR1:3;
(IExec (((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . (intloc 0) = (IExec (((AddTo (a,(intloc 0))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),(IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))))) . (intloc 0) by A3, A5, SFMASTR1:7
.= (Exec ((SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))),(Exec ((AddTo (a,(intloc 0))),(Initialized (IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)))))))) . (intloc 0) by SCMFSA6C:8
.= (Exec ((AddTo (a,(intloc 0))),(Initialized (IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)))))) . (intloc 0) by SCMFSA_2:65
.= (Initialized (IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)))) . (intloc 0) by SCMFSA_2:64
.= 1 by SCMFSA_M:9 ;
hence  ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (intloc 0) = 1 by A7, SCMFSA_M:2; ::_thesis: verum
end;
end;
end;

theorem Th19: :: SFMASTR3:19
for s being State of SCM+FSA
for a being read-write Int-Location
for bb, cc being Int-Location
for Ig being good Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p holds
for k being Element of NAT st k <= ((s . cc) - (s . bb)) + 1 holds
( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (intloc 0) = 1 & ( not Ig destroys a implies ( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . a = k + (s . bb) & ((StepForUp (a,bb,cc,Ig,p,s)) . k) . a <= (s . cc) + 1 ) ) & (((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + k = ((s . cc) - (s . bb)) + 1 )
proof
let s be State of SCM+FSA; ::_thesis: for a being read-write Int-Location
for bb, cc being Int-Location
for Ig being good Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p holds
for k being Element of NAT st k <= ((s . cc) - (s . bb)) + 1 holds
( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (intloc 0) = 1 & ( not Ig destroys a implies ( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . a = k + (s . bb) & ((StepForUp (a,bb,cc,Ig,p,s)) . k) . a <= (s . cc) + 1 ) ) & (((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + k = ((s . cc) - (s . bb)) + 1 )
let a be read-write Int-Location; ::_thesis: for bb, cc being Int-Location
for Ig being good Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p holds
for k being Element of NAT st k <= ((s . cc) - (s . bb)) + 1 holds
( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (intloc 0) = 1 & ( not Ig destroys a implies ( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . a = k + (s . bb) & ((StepForUp (a,bb,cc,Ig,p,s)) . k) . a <= (s . cc) + 1 ) ) & (((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + k = ((s . cc) - (s . bb)) + 1 )
let bb, cc be Int-Location; ::_thesis: for Ig being good Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p holds
for k being Element of NAT st k <= ((s . cc) - (s . bb)) + 1 holds
( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (intloc 0) = 1 & ( not Ig destroys a implies ( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . a = k + (s . bb) & ((StepForUp (a,bb,cc,Ig,p,s)) . k) . a <= (s . cc) + 1 ) ) & (((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + k = ((s . cc) - (s . bb)) + 1 )
let Ig be good Program of ; ::_thesis: for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p holds
for k being Element of NAT st k <= ((s . cc) - (s . bb)) + 1 holds
( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (intloc 0) = 1 & ( not Ig destroys a implies ( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . a = k + (s . bb) & ((StepForUp (a,bb,cc,Ig,p,s)) . k) . a <= (s . cc) + 1 ) ) & (((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + k = ((s . cc) - (s . bb)) + 1 )
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p implies for k being Element of NAT st k <= ((s . cc) - (s . bb)) + 1 holds
( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (intloc 0) = 1 & ( not Ig destroys a implies ( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . a = k + (s . bb) & ((StepForUp (a,bb,cc,Ig,p,s)) . k) . a <= (s . cc) + 1 ) ) & (((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + k = ((s . cc) - (s . bb)) + 1 ) )
set I = Ig;
assume that
A1: s . (intloc 0) = 1 and
A2: ProperForUpBody a,bb,cc,Ig,s,p ; ::_thesis: for k being Element of NAT st k <= ((s . cc) - (s . bb)) + 1 holds
( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (intloc 0) = 1 & ( not Ig destroys a implies ( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . a = k + (s . bb) & ((StepForUp (a,bb,cc,Ig,p,s)) . k) . a <= (s . cc) + 1 ) ) & (((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + k = ((s . cc) - (s . bb)) + 1 )
set scb1 = ((s . cc) - (s . bb)) + 1;
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig));
set SF = StepForUp (a,bb,cc,Ig,p,s);
set IB = (Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)));
set s2 = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb));
set p2 = p;
set SW2 = StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))));
A3: (Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))) = Ig ";" ((AddTo (a,(intloc 0))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) by SCMFSA6A:28;
defpred S1[ Nat] means ( $1 <= ((s . cc) - (s . bb)) + 1 implies ( ((StepForUp (a,bb,cc,Ig,p,s)) . $1) . (intloc 0) = 1 & ( not Ig destroys a implies ( ((StepForUp (a,bb,cc,Ig,p,s)) . $1) . a = $1 + (s . bb) & ((StepForUp (a,bb,cc,Ig,p,s)) . $1) . a <= (s . cc) + 1 ) ) & (((StepForUp (a,bb,cc,Ig,p,s)) . $1) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + $1 = ((s . cc) - (s . bb)) + 1 ) );
 a in {a,bb,cc} by ENUMSET1:def_1;
then  a in {a,bb,cc} \/ (UsedIntLoc Ig) by XBOOLE_0:def_3;
then A4: 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)) <> a by SCMFSA_M:25;
A5: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A6: S1[k] ; ::_thesis: S1[k + 1]
thus  S1[k + 1] ::_thesis: verum
proof
A7: not 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)) in UsedIntLoc Ig
proof
assume  1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)) in UsedIntLoc Ig ; ::_thesis: contradiction
then  1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)) in {a,bb,cc} \/ (UsedIntLoc Ig) by XBOOLE_0:def_3;
hence  contradiction by SCMFSA_M:25; ::_thesis: verum
end;
set k1 = k + 1;
assume A8: k + 1 <= ((s . cc) - (s . bb)) + 1 ; ::_thesis: ( ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (intloc 0) = 1 & ( not Ig destroys a implies ( ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . a = (k + 1) + (s . bb) & ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . a <= (s . cc) + 1 ) ) & (((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + (k + 1) = ((s . cc) - (s . bb)) + 1 )
A9: k < k + 1 by XREAL_1:29;
then A10: ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0 by A6, A8, XREAL_1:8, XXREAL_0:2;
A11: k < ((s . cc) - (s . bb)) + 1 by A8, A9, XXREAL_0:2;
then A12: Ig is_closed_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) by A2, Def2;
then A13: Ig is_closed_on Initialized ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) by A6, A8, A9, SFMASTR2:4, XXREAL_0:2;
A14: Ig is_halting_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) by A2, A11, Def2;
then A15: Ig is_halting_on Initialized ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) by A6, A8, A9, A12, SFMASTR2:5, XXREAL_0:2;
thus  ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (intloc 0) = 1 by A6, A8, A9, A12, A14, Th18, XXREAL_0:2; ::_thesis: ( ( not Ig destroys a implies ( ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . a = (k + 1) + (s . bb) & ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . a <= (s . cc) + 1 ) ) & (((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + (k + 1) = ((s . cc) - (s . bb)) + 1 )
A16: (AddTo (a,(intloc 0))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))) is_closed_on IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) by SCMFSA7B:18;
then A17: (Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))) is_closed_on Initialized ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) by A3, A13, A15, SFMASTR1:2;
 (AddTo (a,(intloc 0))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))) is_halting_on IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) by SCMFSA7B:19;
then  (Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))) is_halting_on Initialized ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) by A3, A13, A15, A16, SFMASTR1:3;
then A18: DataPart ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . (k + 1)) = DataPart (IExec (((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) by A6, A8, A9, A10, A17, SCMFSA9A:32, XXREAL_0:2;
hereby  ::_thesis: (((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + (k + 1) = ((s . cc) - (s . bb)) + 1
assume A19: not Ig destroys a ; ::_thesis: ( ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . a = (k + 1) + (s . bb) & ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . a <= (s . cc) + 1 )
A20: (IExec (((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . a = (IExec (((AddTo (a,(intloc 0))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),(IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))))) . a by A3, A13, A15, SFMASTR1:7
.= (Exec ((SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))),(Exec ((AddTo (a,(intloc 0))),(Initialized (IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)))))))) . a by SCMFSA6C:8
.= (Exec ((AddTo (a,(intloc 0))),(Initialized (IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)))))) . a by A4, SCMFSA_2:65
.= ((Initialized (IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)))) . a) + ((Initialized (IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)))) . (intloc 0)) by SCMFSA_2:64
.= ((Initialized (IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)))) . a) + 1 by SCMFSA_M:9
.= ((IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . a) + 1 by SCMFSA_M:37
.= ((Initialized ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)) . a) + 1 by A13, A15, A19, Th1
.= (((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . a) + 1 by SCMFSA_M:37 ;
hence  ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . a = (k + 1) + (s . bb) by A6, A8, A9, A18, A19, SCMFSA_M:2, XXREAL_0:2; ::_thesis: ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . a <= (s . cc) + 1
 (k + 1) + (s . bb) <= (((s . cc) + 1) - (s . bb)) + (s . bb) by A8, XREAL_1:6;
hence  ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . a <= (s . cc) + 1 by A6, A8, A9, A18, A19, A20, SCMFSA_M:2, XXREAL_0:2; ::_thesis: verum
end;
(IExec (((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) = (IExec (((AddTo (a,(intloc 0))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),(IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))))) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) by A3, A13, A15, SFMASTR1:7
.= (Exec ((SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))),(Exec ((AddTo (a,(intloc 0))),(Initialized (IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)))))))) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) by SCMFSA6C:8
.= ((Exec ((AddTo (a,(intloc 0))),(Initialized (IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)))))) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) - ((Exec ((AddTo (a,(intloc 0))),(Initialized (IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)))))) . (intloc 0)) by SCMFSA_2:65
.= ((Exec ((AddTo (a,(intloc 0))),(Initialized (IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)))))) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) - ((Initialized (IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)))) . (intloc 0)) by SCMFSA_2:64
.= ((Exec ((AddTo (a,(intloc 0))),(Initialized (IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)))))) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) - 1 by SCMFSA_M:9
.= ((Initialized (IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)))) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) - 1 by A4, SCMFSA_2:64
.= ((IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) - 1 by SCMFSA_M:37
.= ((Initialized ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) - 1 by A13, A15, A7, Th1, SFMASTR1:1
.= (((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) - 1 by SCMFSA_M:37 ;
hence (((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + (k + 1) = ((((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) - 1) + (k + 1) by A18, SCMFSA_M:2
.= ((s . cc) - (s . bb)) + 1 by A6, A8, A9, XXREAL_0:2 ;
::_thesis: verum
end;
end;
A21: a in dom (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) by SCMFSA_2:42;
A22: 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)) in dom s by SCMFSA_2:42;
A23: S1[ 0 ]
proof
assume A24: 0 <= ((s . cc) - (s . bb)) + 1 ; ::_thesis: ( ((StepForUp (a,bb,cc,Ig,p,s)) . 0) . (intloc 0) = 1 & ( not Ig destroys a implies ( ((StepForUp (a,bb,cc,Ig,p,s)) . 0) . a = 0 + (s . bb) & ((StepForUp (a,bb,cc,Ig,p,s)) . 0) . a <= (s . cc) + 1 ) ) & (((StepForUp (a,bb,cc,Ig,p,s)) . 0) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + 0 = ((s . cc) - (s . bb)) + 1 )
A25: (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . 0 = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb)) by SCMFSA_9:def_5;
hence ((StepForUp (a,bb,cc,Ig,p,s)) . 0) . (intloc 0) = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) . (intloc 0) by FUNCT_7:32
.= 1 by A1, FUNCT_7:32 ;
::_thesis: ( ( not Ig destroys a implies ( ((StepForUp (a,bb,cc,Ig,p,s)) . 0) . a = 0 + (s . bb) & ((StepForUp (a,bb,cc,Ig,p,s)) . 0) . a <= (s . cc) + 1 ) ) & (((StepForUp (a,bb,cc,Ig,p,s)) . 0) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + 0 = ((s . cc) - (s . bb)) + 1 )
hereby  ::_thesis: (((StepForUp (a,bb,cc,Ig,p,s)) . 0) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + 0 = ((s . cc) - (s . bb)) + 1
assume  not Ig destroys a ; ::_thesis: ( ((StepForUp (a,bb,cc,Ig,p,s)) . 0) . a = 0 + (s . bb) & ((StepForUp (a,bb,cc,Ig,p,s)) . 0) . a <= (s . cc) + 1 )
thus  ((StepForUp (a,bb,cc,Ig,p,s)) . 0) . a = 0 + (s . bb) by A21, A25, FUNCT_7:31; ::_thesis: ((StepForUp (a,bb,cc,Ig,p,s)) . 0) . a <= (s . cc) + 1
 0 + (s . bb) <= (((s . cc) + 1) - (s . bb)) + (s . bb) by A24, XREAL_1:6;
hence  ((StepForUp (a,bb,cc,Ig,p,s)) . 0) . a <= (s . cc) + 1 by A21, A25, FUNCT_7:31; ::_thesis: verum
end;
thus (((StepForUp (a,bb,cc,Ig,p,s)) . 0) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + 0 = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) by A4, A25, FUNCT_7:32
.= ((s . cc) - (s . bb)) + 1 by A22, FUNCT_7:31 ; ::_thesis: verum
end;
thus  for k being Element of NAT holds S1[k] from NAT_1:sch_1(A23, A5); ::_thesis: verum
end;

theorem Th20: :: SFMASTR3:20
for s being State of SCM+FSA
for a being read-write Int-Location
for bb, cc being Int-Location
for Ig being good Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p holds
for k being Element of NAT holds
( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0 iff k < ((s . cc) - (s . bb)) + 1 )
proof
let s be State of SCM+FSA; ::_thesis: for a being read-write Int-Location
for bb, cc being Int-Location
for Ig being good Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p holds
for k being Element of NAT holds
( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0 iff k < ((s . cc) - (s . bb)) + 1 )
let a be read-write Int-Location; ::_thesis: for bb, cc being Int-Location
for Ig being good Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p holds
for k being Element of NAT holds
( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0 iff k < ((s . cc) - (s . bb)) + 1 )
let bb, cc be Int-Location; ::_thesis: for Ig being good Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p holds
for k being Element of NAT holds
( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0 iff k < ((s . cc) - (s . bb)) + 1 )
let Ig be good Program of ; ::_thesis: for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p holds
for k being Element of NAT holds
( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0 iff k < ((s . cc) - (s . bb)) + 1 )
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p implies for k being Element of NAT holds
( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0 iff k < ((s . cc) - (s . bb)) + 1 ) )
set I = Ig;
set SF = StepForUp (a,bb,cc,Ig,p,s);
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig));
set s2 = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb));
set p2 = p;
set IB = (Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)));
set SW2 = StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))));
set scb1 = ((s . cc) - (s . bb)) + 1;
defpred S1[ Nat] means ( ((StepForUp (a,bb,cc,Ig,p,s)) . $1) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0 implies $1 < ((s . cc) - (s . bb)) + 1 );
assume A1: ( s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p ) ; ::_thesis: for k being Element of NAT holds
( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0 iff k < ((s . cc) - (s . bb)) + 1 )
A2: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume that
A3: S1[k] and
A4: ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0 ; ::_thesis: k + 1 < ((s . cc) - (s . bb)) + 1
A5: ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0
proof
assume A6: ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) <= 0 ; ::_thesis: contradiction
then  DataPart ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) = DataPart ((StepForUp (a,bb,cc,Ig,p,s)) . k) by SCMFSA9A:31;
hence  contradiction by A4, A6, SCMFSA_M:2; ::_thesis: verum
end;
then reconsider scb1 = ((s . cc) - (s . bb)) + 1 as Element of NAT by A3, INT_1:3;
assume  k + 1 >= ((s . cc) - (s . bb)) + 1 ; ::_thesis: contradiction
then A7: (((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + (k + 1) > 0 + scb1 by A4, XREAL_1:8;
 k + 1 <= scb1 by A3, A5, NAT_1:13;
hence  contradiction by A1, A7, Th19; ::_thesis: verum
end;
let k be Element of NAT ; ::_thesis: ( ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0 iff k < ((s . cc) - (s . bb)) + 1 )
 a in {a,bb,cc} by ENUMSET1:def_1;
then  a in {a,bb,cc} \/ (UsedIntLoc Ig) by XBOOLE_0:def_3;
then A8: 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)) <> a by SCMFSA_M:25;
A9: 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)) in dom s by SCMFSA_2:42;
A10: S1[ 0 ]
proof
 (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . 0 = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb)) by SCMFSA_9:def_5;
then A11: ((StepForUp (a,bb,cc,Ig,p,s)) . 0) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) by A8, FUNCT_7:32
.= ((s . cc) - (s . bb)) + 1 by A9, FUNCT_7:31 ;
assume  ((StepForUp (a,bb,cc,Ig,p,s)) . 0) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0 ; ::_thesis: 0 < ((s . cc) - (s . bb)) + 1
hence  0 < ((s . cc) - (s . bb)) + 1 by A11; ::_thesis: verum
end;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A10, A2);
hence  S1[k] ; ::_thesis: ( k < ((s . cc) - (s . bb)) + 1 implies ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0 )
assume A12: k < ((s . cc) - (s . bb)) + 1 ; ::_thesis: ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0
then A13: k - k < (((s . cc) - (s . bb)) + 1) - k by XREAL_1:9;
 (((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + k = ((s . cc) - (s . bb)) + 1 by A1, A12, Th19;
hence  ((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0 by A13; ::_thesis: verum
end;

theorem Th21: :: SFMASTR3:21
for s being State of SCM+FSA
for a being read-write Int-Location
for bb, cc being Int-Location
for Ig being good Program of
for k being Element of NAT
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p & k < ((s . cc) - (s . bb)) + 1 holds
((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations) = (IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepForUp (a,bb,cc,Ig,p,s)) . k))) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations)
proof
let s be State of SCM+FSA; ::_thesis: for a being read-write Int-Location
for bb, cc being Int-Location
for Ig being good Program of
for k being Element of NAT
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p & k < ((s . cc) - (s . bb)) + 1 holds
((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations) = (IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepForUp (a,bb,cc,Ig,p,s)) . k))) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations)
let a be read-write Int-Location; ::_thesis: for bb, cc being Int-Location
for Ig being good Program of
for k being Element of NAT
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p & k < ((s . cc) - (s . bb)) + 1 holds
((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations) = (IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepForUp (a,bb,cc,Ig,p,s)) . k))) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations)
let bb, cc be Int-Location; ::_thesis: for Ig being good Program of
for k being Element of NAT
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p & k < ((s . cc) - (s . bb)) + 1 holds
((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations) = (IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepForUp (a,bb,cc,Ig,p,s)) . k))) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations)
let Ig be good Program of ; ::_thesis: for k being Element of NAT
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p & k < ((s . cc) - (s . bb)) + 1 holds
((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations) = (IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepForUp (a,bb,cc,Ig,p,s)) . k))) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations)
let k be Element of NAT ; ::_thesis: for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p & k < ((s . cc) - (s . bb)) + 1 holds
((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations) = (IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepForUp (a,bb,cc,Ig,p,s)) . k))) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations)
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( s . (intloc 0) = 1 & ProperForUpBody a,bb,cc,Ig,s,p & k < ((s . cc) - (s . bb)) + 1 implies ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations) = (IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepForUp (a,bb,cc,Ig,p,s)) . k))) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations) )
assume that
A1: s . (intloc 0) = 1 and
A2: ProperForUpBody a,bb,cc,Ig,s,p and
A3: k < ((s . cc) - (s . bb)) + 1 ; ::_thesis: ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations) = (IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepForUp (a,bb,cc,Ig,p,s)) . k))) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations)
set FL = FinSeq-Locations ;
set I = Ig;
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig));
set SF = StepForUp (a,bb,cc,Ig,p,s);
set IB = (Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)));
set s2 = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb));
set p2 = p;
set SW2 = StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))));
 StepForUp (a,bb,cc,Ig,p,s) = StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb)))) ;
then A4: ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . (intloc 0) = 1 by A1, A2, A3, Th19;
set scb1 = ((s . cc) - (s . bb)) + 1;
A5: (((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + k = ((s . cc) - (s . bb)) + 1 by A1, A2, A3, Th19;
A6: ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) > 0
proof
assume  ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) <= 0 ; ::_thesis: contradiction
then  (((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + k < 0 + (((s . cc) - (s . bb)) + 1) by A3, XREAL_1:8;
hence  contradiction by A5; ::_thesis: verum
end;
set S2 = IExec (((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k));
set IB1 = Ig ";" (AddTo (a,(intloc 0)));
set Iloc = {a,bb,cc} \/ (UsedIntLoc Ig);
set S1 = IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k));
A7: Macro (AddTo (a,(intloc 0))) is_closed_on IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) by SCMFSA7B:18;
A8: Ig is_closed_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) by A2, A3, Def2;
then A9: Ig is_closed_on Initialized ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) by A4, SFMASTR2:4;
 Ig is_halting_on (StepForUp (a,bb,cc,Ig,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) by A2, A3, Def2;
then A10: Ig is_halting_on Initialized ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) by A8, A4, SFMASTR2:5;
 Macro (AddTo (a,(intloc 0))) is_halting_on IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) by SCMFSA7B:19;
then A11: Ig ";" (AddTo (a,(intloc 0))) is_halting_on Initialized ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) by A9, A10, A7, SFMASTR1:3;
now__::_thesis:_(_(_for_x_being_Int-Location_st_x_in_{a,bb,cc}_\/_(UsedIntLoc_Ig)_holds_
(IExec_((Ig_";"_(AddTo_(a,(intloc_0)))),(p_+*_(while>0_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),((Ig_";"_(AddTo_(a,(intloc_0))))_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),(intloc_0))))))),((StepWhile>0_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),((Ig_";"_(AddTo_(a,(intloc_0))))_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),(intloc_0)))),p,((s_+*_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),(((s_._cc)_-_(s_._bb))_+_1)))_+*_(a,(s_._bb)))))_._k)))_._x_=_(IExec_(((Ig_";"_(AddTo_(a,(intloc_0))))_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),(intloc_0)))),(p_+*_(while>0_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),((Ig_";"_(AddTo_(a,(intloc_0))))_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),(intloc_0))))))),((StepWhile>0_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),((Ig_";"_(AddTo_(a,(intloc_0))))_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),(intloc_0)))),p,((s_+*_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),(((s_._cc)_-_(s_._bb))_+_1)))_+*_(a,(s_._bb)))))_._k)))_._x_)_&_(_for_x_being_FinSeq-Location_holds_(IExec_((Ig_";"_(AddTo_(a,(intloc_0)))),(p_+*_(while>0_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),((Ig_";"_(AddTo_(a,(intloc_0))))_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),(intloc_0))))))),((StepWhile>0_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),((Ig_";"_(AddTo_(a,(intloc_0))))_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),(intloc_0)))),p,((s_+*_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),(((s_._cc)_-_(s_._bb))_+_1)))_+*_(a,(s_._bb)))))_._k)))_._x_=_(IExec_(((Ig_";"_(AddTo_(a,(intloc_0))))_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),(intloc_0)))),(p_+*_(while>0_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),((Ig_";"_(AddTo_(a,(intloc_0))))_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),(intloc_0))))))),((StepWhile>0_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),((Ig_";"_(AddTo_(a,(intloc_0))))_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),(intloc_0)))),p,((s_+*_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_Ig))),(((s_._cc)_-_(s_._bb))_+_1)))_+*_(a,(s_._bb)))))_._k)))_._x_)_)
hereby  ::_thesis: for x being FinSeq-Location holds (IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . x = (IExec (((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . x
let x be Int-Location; ::_thesis: ( x in {a,bb,cc} \/ (UsedIntLoc Ig) implies (IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . x = (IExec (((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . x )
assume  x in {a,bb,cc} \/ (UsedIntLoc Ig) ; ::_thesis: (IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . x = (IExec (((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . x
then A12: x <> 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)) by SCMFSA_M:25;
(IExec (((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . x = (Exec ((SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))),(IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))))) . x by A9, A10, A7, A11, SFMASTR1:2, SFMASTR1:11
.= (IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . x by A12, SCMFSA_2:65 ;
hence  (IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . x = (IExec (((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . x ; ::_thesis: verum
end;
let x be FinSeq-Location ; ::_thesis: (IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . x = (IExec (((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . x
(IExec (((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . x = (Exec ((SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))),(IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))))) . x by A9, A10, A7, A11, SFMASTR1:2, SFMASTR1:12
.= (IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . x by SCMFSA_2:65 ;
hence  (IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . x = (IExec (((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) . x ; ::_thesis: verum
end;
then A13: (IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations) = (IExec (((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations) by SCMFSA_M:28;
A14: ( (Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))) = Ig ";" ((AddTo (a,(intloc 0))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) & (AddTo (a,(intloc 0))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))) is_closed_on IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) ) by SCMFSA6A:28, SCMFSA7B:18;
then  ( (AddTo (a,(intloc 0))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))) is_halting_on IExec (Ig,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k)),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) & (Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))) is_closed_on Initialized ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))) ) by A9, A10, SCMFSA7B:19, SFMASTR1:2;
then  DataPart ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . (k + 1)) = DataPart (IExec (((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k))) by A4, A6, A9, A10, A14, SCMFSA9A:32, SFMASTR1:3;
hence  ((StepForUp (a,bb,cc,Ig,p,s)) . (k + 1)) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations) = (IExec ((Ig ";" (AddTo (a,(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),((StepForUp (a,bb,cc,Ig,p,s)) . k))) | (({a,bb,cc} \/ (UsedIntLoc Ig)) \/ FinSeq-Locations) by A13, RELAT_1:153, SCMFSA_2:100, XBOOLE_1:9; ::_thesis: verum
end;

definition
let a, b, c be Int-Location;
let I be Program of ;
func for-up (a,b,c,I) ->  Program of  equals :: SFMASTR3:def 3
(((((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))) := c) ";" (SubFrom ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),b))) ";" (AddTo ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := b)) ";" (while>0 ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),(intloc 0))))));
coherence
 (((((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))) := c) ";" (SubFrom ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),b))) ";" (AddTo ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := b)) ";" (while>0 ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),(intloc 0)))))) is Program of ;
end;

:: deftheorem  defines for-up SFMASTR3:def_3_:_
 for a, b, c being Int-Location
for I being Program of holds for-up (a,b,c,I) = (((((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))) := c) ";" (SubFrom ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),b))) ";" (AddTo ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := b)) ";" (while>0 ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,b,c} \/ (UsedIntLoc I))),(intloc 0))))));

theorem Th22: :: SFMASTR3:22
for aa, bb, cc being Int-Location
for I being Program of holds {aa,bb,cc} \/ (UsedIntLoc I) c= UsedIntLoc (for-up (aa,bb,cc,I))
proof
let aa, bb, cc be Int-Location; ::_thesis: for I being Program of holds {aa,bb,cc} \/ (UsedIntLoc I) c= UsedIntLoc (for-up (aa,bb,cc,I))
let I be Program of ; ::_thesis: {aa,bb,cc} \/ (UsedIntLoc I) c= UsedIntLoc (for-up (aa,bb,cc,I))
set aux = 1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I));
set i0 = (1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))) := cc;
set i1 = SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),bb);
set i2 = AddTo ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0));
set i3 = aa := bb;
set I4 = while>0 ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (aa,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))));
A1: UsedIntLoc (((((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (aa := bb)) = (UsedIntLoc ((((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0))))) \/ (UsedIntLoc (aa := bb)) by SF_MASTR:30
.= ((UsedIntLoc (((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),bb)))) \/ (UsedIntLoc (AddTo ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0))))) \/ (UsedIntLoc (aa := bb)) by SF_MASTR:30
.= (((UsedIntLoc ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))) := cc)) \/ (UsedIntLoc (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),bb)))) \/ (UsedIntLoc (AddTo ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0))))) \/ (UsedIntLoc (aa := bb)) by SF_MASTR:31
.= (({(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),cc} \/ (UsedIntLoc (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),bb)))) \/ (UsedIntLoc (AddTo ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0))))) \/ (UsedIntLoc (aa := bb)) by SF_MASTR:14
.= (({(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),cc} \/ {(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),bb}) \/ (UsedIntLoc (AddTo ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0))))) \/ (UsedIntLoc (aa := bb)) by SF_MASTR:14
.= (({(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),cc} \/ {(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),bb}) \/ {(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)}) \/ (UsedIntLoc (aa := bb)) by SF_MASTR:14
.= (({(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),cc} \/ {(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),bb}) \/ {(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)}) \/ {aa,bb} by SF_MASTR:14 ;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {aa,bb,cc} \/ (UsedIntLoc I) or x in UsedIntLoc (for-up (aa,bb,cc,I)) )
A2: UsedIntLoc (while>0 ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (aa,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) = {(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I)))} \/ (UsedIntLoc ((I ";" (AddTo (aa,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0))))) by SCMFSA9A:24
.= {(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I)))} \/ ((UsedIntLoc (I ";" (AddTo (aa,(intloc 0))))) \/ (UsedIntLoc (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0))))) by SF_MASTR:30
.= {(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I)))} \/ (((UsedIntLoc I) \/ (UsedIntLoc (AddTo (aa,(intloc 0))))) \/ (UsedIntLoc (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0))))) by SF_MASTR:30
.= {(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I)))} \/ ((UsedIntLoc I) \/ ((UsedIntLoc (AddTo (aa,(intloc 0)))) \/ (UsedIntLoc (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by XBOOLE_1:4
.= (UsedIntLoc I) \/ ({(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I)))} \/ ((UsedIntLoc (AddTo (aa,(intloc 0)))) \/ (UsedIntLoc (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by XBOOLE_1:4 ;
assume  x in {aa,bb,cc} \/ (UsedIntLoc I) ; ::_thesis: x in UsedIntLoc (for-up (aa,bb,cc,I))
then A3: ( x in {aa,bb,cc} or x in UsedIntLoc I ) by XBOOLE_0:def_3;
A4: UsedIntLoc (for-up (aa,bb,cc,I)) = (UsedIntLoc (((((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (aa := bb))) \/ (UsedIntLoc (while>0 ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (aa,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0))))))) by SF_MASTR:27;
percases ( x = aa or x = bb or x = cc or x in UsedIntLoc I ) by A3, ENUMSET1:def_1;
suppose x = aa ; ::_thesis: x in UsedIntLoc (for-up (aa,bb,cc,I))
then  x in {aa,bb} by TARSKI:def_2;
then  x in UsedIntLoc (((((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (aa := bb)) by A1, XBOOLE_0:def_3;
hence  x in UsedIntLoc (for-up (aa,bb,cc,I)) by A4, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose x = bb ; ::_thesis: x in UsedIntLoc (for-up (aa,bb,cc,I))
then  x in {aa,bb} by TARSKI:def_2;
then  x in UsedIntLoc (((((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (aa := bb)) by A1, XBOOLE_0:def_3;
hence  x in UsedIntLoc (for-up (aa,bb,cc,I)) by A4, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose x = cc ; ::_thesis: x in UsedIntLoc (for-up (aa,bb,cc,I))
then  x in {(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),cc} by TARSKI:def_2;
then  x in {(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),cc} \/ {(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),bb} by XBOOLE_0:def_3;
then  x in ({(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),cc} \/ {(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),bb}) \/ {(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)} by XBOOLE_0:def_3;
then  x in (({(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),cc} \/ {(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),bb}) \/ {(1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)}) \/ {aa,bb} by XBOOLE_0:def_3;
hence  x in UsedIntLoc (for-up (aa,bb,cc,I)) by A1, A4, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose x in UsedIntLoc I ; ::_thesis: x in UsedIntLoc (for-up (aa,bb,cc,I))
then  x in UsedIntLoc (while>0 ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (aa,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({aa,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by A2, XBOOLE_0:def_3;
hence  x in UsedIntLoc (for-up (aa,bb,cc,I)) by A4, XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;

registration
let a be read-write Int-Location;
let b, c be Int-Location;
let I be good Program of ;
cluster for-up (a,b,c,I) ->  good ;
coherence
 for-up (a,b,c,I) is good ;
end;

theorem Th23: :: SFMASTR3:23
for a being read-write Int-Location
for aa, bb, cc being Int-Location
for I being Program of st a <> aa & aa <> 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) & not I destroys aa holds
not for-up (a,bb,cc,I) destroys aa
proof
let a be read-write Int-Location; ::_thesis: for aa, bb, cc being Int-Location
for I being Program of st a <> aa & aa <> 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) & not I destroys aa holds
not for-up (a,bb,cc,I) destroys aa
let aa, bb, cc be Int-Location; ::_thesis: for I being Program of st a <> aa & aa <> 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) & not I destroys aa holds
not for-up (a,bb,cc,I) destroys aa
let I be Program of ; ::_thesis: ( a <> aa & aa <> 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) & not I destroys aa implies not for-up (a,bb,cc,I) destroys aa )
assume that
A1: a <> aa and
A2: aa <> 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) and
A3: not I destroys aa ; ::_thesis: not for-up (a,bb,cc,I) destroys aa
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I));
set i2 = AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0));
A4: not AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)) destroys aa by A2, SCMFSA7B:7;
set i3 = a := bb;
set i1 = SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb);
set i0 = (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc;
set I03 = ((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb);
 ( not (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc destroys aa & not SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb) destroys aa ) by A2, SCMFSA7B:6, SCMFSA7B:8;
then  not (((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) destroys aa by A4, SCMFSA8C:54, SCMFSA8C:55;
then A5: not ((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb) destroys aa by A1, SCMFSA7B:6, SCMFSA8C:54;
set IB = (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)));
set I4 = while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))));
 not I ";" (AddTo (a,(intloc 0))) destroys aa by A1, A3, SCMFSA7B:7, SCMFSA8C:54;
then  not (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) destroys aa by A2, SCMFSA7B:8, SCMFSA8C:54;
then  not while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))))) destroys aa by Th8;
hence  not for-up (a,bb,cc,I) destroys aa by A5, SCMFSA8C:52; ::_thesis: verum
end;

theorem Th24: :: SFMASTR3:24
for s being State of SCM+FSA
for a being read-write Int-Location
for bb, cc being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & s . bb > s . cc holds
( ( for x being Int-Location st x <> a & x in {bb,cc} \/ (UsedIntLoc I) holds
(IExec ((for-up (a,bb,cc,I)),p,s)) . x = s . x ) & ( for f being FinSeq-Location holds (IExec ((for-up (a,bb,cc,I)),p,s)) . f = s . f ) )
proof
let s be State of SCM+FSA; ::_thesis: for a being read-write Int-Location
for bb, cc being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & s . bb > s . cc holds
( ( for x being Int-Location st x <> a & x in {bb,cc} \/ (UsedIntLoc I) holds
(IExec ((for-up (a,bb,cc,I)),p,s)) . x = s . x ) & ( for f being FinSeq-Location holds (IExec ((for-up (a,bb,cc,I)),p,s)) . f = s . f ) )
let a be read-write Int-Location; ::_thesis: for bb, cc being Int-Location
for I being Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & s . bb > s . cc holds
( ( for x being Int-Location st x <> a & x in {bb,cc} \/ (UsedIntLoc I) holds
(IExec ((for-up (a,bb,cc,I)),p,s)) . x = s . x ) & ( for f being FinSeq-Location holds (IExec ((for-up (a,bb,cc,I)),p,s)) . f = s . f ) )
let bb, cc be Int-Location; ::_thesis: for I being Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & s . bb > s . cc holds
( ( for x being Int-Location st x <> a & x in {bb,cc} \/ (UsedIntLoc I) holds
(IExec ((for-up (a,bb,cc,I)),p,s)) . x = s . x ) & ( for f being FinSeq-Location holds (IExec ((for-up (a,bb,cc,I)),p,s)) . f = s . f ) )
let I be Program of ; ::_thesis: for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & s . bb > s . cc holds
( ( for x being Int-Location st x <> a & x in {bb,cc} \/ (UsedIntLoc I) holds
(IExec ((for-up (a,bb,cc,I)),p,s)) . x = s . x ) & ( for f being FinSeq-Location holds (IExec ((for-up (a,bb,cc,I)),p,s)) . f = s . f ) )
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( s . (intloc 0) = 1 & s . bb > s . cc implies ( ( for x being Int-Location st x <> a & x in {bb,cc} \/ (UsedIntLoc I) holds
(IExec ((for-up (a,bb,cc,I)),p,s)) . x = s . x ) & ( for f being FinSeq-Location holds (IExec ((for-up (a,bb,cc,I)),p,s)) . f = s . f ) ) )
assume that
A1: s . (intloc 0) = 1 and
A2: s . bb > s . cc ; ::_thesis: ( ( for x being Int-Location st x <> a & x in {bb,cc} \/ (UsedIntLoc I) holds
(IExec ((for-up (a,bb,cc,I)),p,s)) . x = s . x ) & ( for f being FinSeq-Location holds (IExec ((for-up (a,bb,cc,I)),p,s)) . f = s . f ) )
 ( cc = intloc 0 or cc is read-write ) by SCMFSA_M:def_2;
then A3: (Initialized s) . cc = s . cc by A1, SCMFSA_M:9, SCMFSA_M:37;
set MI = for-up (a,bb,cc,I);
set i3 = a := bb;
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I));
set i0 = (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc;
set i1 = SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb);
set i2 = AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0));
set IB = (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)));
set I4 = while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))));
set I03 = ((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb);
set s1 = IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s);
set p1 = p;
set s2 = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb));
A4: (IExec ((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))),p,s)) . (intloc 0) = (Exec ((SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb)),(Exec (((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc),(Initialized s))))) . (intloc 0) by SCMFSA6C:8
.= (Exec (((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc),(Initialized s))) . (intloc 0) by SCMFSA_2:65
.= (Initialized s) . (intloc 0) by SCMFSA_2:63
.= 1 by SCMFSA_M:9 ;
 ( bb = intloc 0 or bb is read-write ) by SCMFSA_M:def_2;
then A5: (Initialized s) . bb = s . bb by A1, SCMFSA_M:9, SCMFSA_M:37;
A6: (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)) . (intloc 0) = (Exec ((a := bb),(IExec (((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,s)))) . (intloc 0) by SCMFSA6C:6
.= (IExec (((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,s)) . (intloc 0) by SCMFSA_2:63
.= (Exec ((AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))),(IExec ((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))),p,s)))) . (intloc 0) by SCMFSA6C:6
.= 1 by A4, SCMFSA_2:64 ;
 (s . bb) - (s . bb) > (s . cc) - (s . bb) by A2, XREAL_1:9;
then  (s . cc) - (s . bb) <= - 1 by INT_1:8;
then A7: ((s . cc) - (s . bb)) + 1 <= (- 1) + 1 by XREAL_1:6;
set s3 = IExec ((for-up (a,bb,cc,I)),p,s);
 a in {a,bb,cc} by ENUMSET1:def_1;
then  a in {a,bb,cc} \/ (UsedIntLoc I) by XBOOLE_0:def_3;
then A8: a <> 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) by SCMFSA_M:25;
 bb in {a,bb,cc} by ENUMSET1:def_1;
then  bb in {a,bb,cc} \/ (UsedIntLoc I) by XBOOLE_0:def_3;
then A9: bb <> 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) by SCMFSA_M:25;
A10: (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) = (Exec ((a := bb),(IExec (((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,s)))) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) by SCMFSA6C:6
.= (IExec (((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,s)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) by A8, SCMFSA_2:63
.= (Exec ((AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))),(IExec ((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))),p,s)))) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) by SCMFSA6C:6
.= ((IExec ((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))),p,s)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)))) + 1 by A4, SCMFSA_2:64
.= ((Exec ((SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb)),(Exec (((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc),(Initialized s))))) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)))) + 1 by SCMFSA6C:8
.= (((Exec (((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc),(Initialized s))) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)))) - ((Exec (((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc),(Initialized s))) . bb)) + 1 by SCMFSA_2:65
.= (((Initialized s) . cc) - ((Exec (((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc),(Initialized s))) . bb)) + 1 by SCMFSA_2:63
.= ((s . cc) - (s . bb)) + 1 by A9, A3, A5, SCMFSA_2:63 ;
then  ( while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))))) is_halting_on IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s),p & while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))))) is_closed_on IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s),p ) by A7, SCMFSA_9:38;
then A11: DataPart (IExec ((for-up (a,bb,cc,I)),p,s)) = DataPart (IExec ((while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) by SFMASTR1:9
.= DataPart (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)) by A10, A7, A6, SCMFSA9A:35
.= DataPart ((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) by A1, Th16 ;
hereby  ::_thesis: for f being FinSeq-Location holds (IExec ((for-up (a,bb,cc,I)),p,s)) . f = s . f
let x be Int-Location; ::_thesis: ( x <> a & x in {bb,cc} \/ (UsedIntLoc I) implies (IExec ((for-up (a,bb,cc,I)),p,s)) . x = s . x )
assume that
A12: x <> a and
A13: x in {bb,cc} \/ (UsedIntLoc I) ; ::_thesis: (IExec ((for-up (a,bb,cc,I)),p,s)) . x = s . x
 x in {a,bb,cc} \/ (UsedIntLoc I)
proof
percases ( x in {bb,cc} or x in UsedIntLoc I ) by A13, XBOOLE_0:def_3;
suppose x in {bb,cc} ; ::_thesis: x in {a,bb,cc} \/ (UsedIntLoc I)
then  ( x = bb or x = cc ) by TARSKI:def_2;
then  x in {a,bb,cc} by ENUMSET1:def_1;
hence  x in {a,bb,cc} \/ (UsedIntLoc I) by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose x in UsedIntLoc I ; ::_thesis: x in {a,bb,cc} \/ (UsedIntLoc I)
hence  x in {a,bb,cc} \/ (UsedIntLoc I) by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
then A14: x <> 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) by SCMFSA_M:25;
thus (IExec ((for-up (a,bb,cc,I)),p,s)) . x = ((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . x by A11, SCMFSA_M:2
.= (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) . x by A12, FUNCT_7:32
.= s . x by A14, FUNCT_7:32 ; ::_thesis: verum
end;
let x be FinSeq-Location ; ::_thesis: (IExec ((for-up (a,bb,cc,I)),p,s)) . x = s . x
thus (IExec ((for-up (a,bb,cc,I)),p,s)) . x = ((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . x by A11, SCMFSA_M:2
.= (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) . x by FUNCT_7:32, SCMFSA_2:58
.= s . x by FUNCT_7:32, SCMFSA_2:58 ; ::_thesis: verum
end;

Lm1: now__::_thesis:_for_s_being_State_of_SCM+FSA
for_a_being_read-write_Int-Location
for_bb,_cc_being_Int-Location
for_p_being_Instruction-Sequence_of_SCM+FSA
for_I_being_good_Program_of_st_s_._(intloc_0)_=_1_&_(_ProperForUpBody_a,bb,cc,I,s,p_or_I_is_parahalting_)_holds_
(_ProperBodyWhile>0_1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I)),(I_";"_(AddTo_(a,(intloc_0))))_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I))),(intloc_0))),_IExec_((((((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I)))_:=_cc)_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I))),bb)))_";"_(AddTo_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I))),(intloc_0))))_";"_(a_:=_bb)),p,s),p_&_WithVariantWhile>0_1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I)),(I_";"_(AddTo_(a,(intloc_0))))_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I))),(intloc_0))),_IExec_((((((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I)))_:=_cc)_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I))),bb)))_";"_(AddTo_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I))),(intloc_0))))_";"_(a_:=_bb)),p,s),p_)
let s be State of SCM+FSA; ::_thesis: for a being read-write Int-Location
for bb, cc being Int-Location
for p being Instruction-Sequence of SCM+FSA
for I being good Program of st s . (intloc 0) = 1 & ( ProperForUpBody a,bb,cc,I,s,p or I is parahalting ) holds
( ProperBodyWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)),(I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s),p & WithVariantWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)),(I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s),p )
let a be read-write Int-Location; ::_thesis: for bb, cc being Int-Location
for p being Instruction-Sequence of SCM+FSA
for I being good Program of st s . (intloc 0) = 1 & ( ProperForUpBody a,bb,cc,I,s,p or I is parahalting ) holds
( ProperBodyWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)),(I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s),p & WithVariantWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)),(I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s),p )
let bb, cc be Int-Location; ::_thesis: for p being Instruction-Sequence of SCM+FSA
for I being good Program of st s . (intloc 0) = 1 & ( ProperForUpBody a,bb,cc,I,s,p or I is parahalting ) holds
( ProperBodyWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)),(I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s),p & WithVariantWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)),(I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s),p )
let p be Instruction-Sequence of SCM+FSA; ::_thesis: for I being good Program of st s . (intloc 0) = 1 & ( ProperForUpBody a,bb,cc,I,s,p or I is parahalting ) holds
( ProperBodyWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)),(I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s),p & WithVariantWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)),(I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s),p )
let I be good Program of ; ::_thesis: ( s . (intloc 0) = 1 & ( ProperForUpBody a,bb,cc,I,s,p or I is parahalting ) implies ( ProperBodyWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)),(I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s),p & WithVariantWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)),(I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s),p ) )
assume that
A1: s . (intloc 0) = 1 and
A2: ( ProperForUpBody a,bb,cc,I,s,p or I is parahalting ) ; ::_thesis: ( ProperBodyWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)),(I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s),p & WithVariantWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)),(I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s),p )
A3: ProperForUpBody a,bb,cc,I,s,p by A2, Th17;
set scb1 = ((s . cc) - (s . bb)) + 1;
set SF = StepForUp (a,bb,cc,I,p,s);
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I));
set IB = (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)));
set s2 = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb));
set p2 = p;
set IB2 = (AddTo (a,(intloc 0))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)));
set SW2 = StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))));
A4: StepForUp (a,bb,cc,I,p,s) = StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb)))) ;
A5: (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) = I ";" ((AddTo (a,(intloc 0))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) by SCMFSA6A:28;
A6: ProperBodyWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)),(I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))),(s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb)),p
proof
let k be Element of NAT ; :: according to SCMFSA9A:def_4 ::_thesis: ( ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) <= 0 or ( (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_closed_on (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) & (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_halting_on (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) ) )
A7: (AddTo (a,(intloc 0))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_closed_on IExec (I,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))))))),((StepForUp (a,bb,cc,I,p,s)) . k)),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by SCMFSA7B:18;
assume  ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) > 0 ; ::_thesis: ( (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_closed_on (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) & (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_halting_on (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) )
then A8: k < ((s . cc) - (s . bb)) + 1 by A1, A3, A4, Th20;
then A9: I is_closed_on (StepForUp (a,bb,cc,I,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by A3, Def2;
A10: ((StepForUp (a,bb,cc,I,p,s)) . k) . (intloc 0) = 1 by A1, A3, A8, Th19;
 I is_halting_on (StepForUp (a,bb,cc,I,p,s)) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by A3, A8, Def2;
then A11: I is_halting_on Initialized ((StepForUp (a,bb,cc,I,p,s)) . k),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by A10, A9, SFMASTR2:5;
A12: I is_closed_on Initialized ((StepForUp (a,bb,cc,I,p,s)) . k),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by A10, A9, SFMASTR2:4;
then A13: (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_closed_on Initialized ((StepForUp (a,bb,cc,I,p,s)) . k),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by A5, A11, A7, SFMASTR1:2;
hence  (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_closed_on (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by A10, SFMASTR2:4; ::_thesis: (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_halting_on (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))))))
 (AddTo (a,(intloc 0))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_halting_on IExec (I,(p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))))))),((StepForUp (a,bb,cc,I,p,s)) . k)),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by SCMFSA7B:19;
then  (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_halting_on Initialized ((StepForUp (a,bb,cc,I,p,s)) . k),p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by A5, A12, A11, A7, SFMASTR1:3;
hence  (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_halting_on (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by A10, A13, SFMASTR2:5; ::_thesis: verum
end;
set i3 = a := bb;
set i2 = AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0));
set i1 = SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb);
set i0 = (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc;
set s1 = IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s);
set p1 = p;
set SW1 = StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)));
deffunc H1( State of SCM+FSA) ->  Element of NAT = abs ($1 . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))));
consider f being Function of (product (the_Values_of SCM+FSA)),NAT such that
A14: for x being Element of product (the_Values_of SCM+FSA) holds f . x = H1(x) from FUNCT_2:sch_4();
A15: for x being State of SCM+FSA holds f . x = H1(x)
proof
let x be State of SCM+FSA; ::_thesis: f . x = H1(x)
reconsider x = x as Element of product (the_Values_of SCM+FSA) by CARD_3:107;
 f . x = H1(x) by A14;
hence  f . x = H1(x) ; ::_thesis: verum
end;
A16: DataPart (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)) = DataPart ((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) by A1, Th16;
thus  ProperBodyWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)),(I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s),p ::_thesis: WithVariantWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)),(I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s),p
proof
let k be Element of NAT ; :: according to SCMFSA9A:def_4 ::_thesis: ( ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) <= 0 or ( (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_closed_on (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) & (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_halting_on (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) ) )
assume A17: ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) > 0 ; ::_thesis: ( (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_closed_on (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) & (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_halting_on (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) )
A18: DataPart ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) = DataPart ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k) by A16, A6, SCMFSA9A:34;
then A19: ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) = ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) by SCMFSA_M:2;
then A20: (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_closed_on (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by A6, A17, SCMFSA9A:def_4;
hence  (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_closed_on (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by A18, SCMFSA8B:3; ::_thesis: (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_halting_on (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))))))
 (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_halting_on (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by A6, A17, A19, SCMFSA9A:def_4;
hence  (I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))) is_halting_on (StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k,p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))))) by A18, A20, SCMFSA8B:5; ::_thesis: verum
end;
A21: for k being Element of NAT holds
( f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) < f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k) or ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) <= 0 )
proof
let k be Element of NAT ; ::_thesis: ( f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) < f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k) or ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) <= 0 )
A22: DataPart ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k) = DataPart ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) by A16, A6, SCMFSA9A:34;
then A23: ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) = ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) by SCMFSA_M:2;
 DataPart ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . (k + 1)) = DataPart ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) by A16, A6, SCMFSA9A:34;
then A24: ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) = ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . (k + 1)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) by SCMFSA_M:2;
now__::_thesis:_(_((StepWhile>0_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I))),((I_";"_(AddTo_(a,(intloc_0))))_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I))),(intloc_0)))),p,(IExec_((((((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I)))_:=_cc)_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I))),bb)))_";"_(AddTo_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I))),(intloc_0))))_";"_(a_:=_bb)),p,s))))_._k)_._(1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I)))_>_0_implies_f_._((StepWhile>0_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I))),((I_";"_(AddTo_(a,(intloc_0))))_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I))),(intloc_0)))),p,(IExec_((((((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I)))_:=_cc)_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I))),bb)))_";"_(AddTo_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I))),(intloc_0))))_";"_(a_:=_bb)),p,s))))_._(k_+_1))_<_f_._((StepWhile>0_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I))),((I_";"_(AddTo_(a,(intloc_0))))_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I))),(intloc_0)))),p,(IExec_((((((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I)))_:=_cc)_";"_(SubFrom_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I))),bb)))_";"_(AddTo_((1_-stRWNotIn_({a,bb,cc}_\/_(UsedIntLoc_I))),(intloc_0))))_";"_(a_:=_bb)),p,s))))_._k)_)
assume A25: ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) > 0 ; ::_thesis: f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) < f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k)
then A26: k < ((s . cc) - (s . bb)) + 1 by A1, A3, A4, A23, Th20;
 k < ((s . cc) - (s . bb)) + 1 by A1, A3, A4, A23, A25, Th20;
then A27: (((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)))) + k = ((s . cc) - (s . bb)) + 1 by A1, A3, A4, Th19;
reconsider scb1 = ((s . cc) - (s . bb)) + 1 as Element of NAT by A26, INT_1:3;
A28: k + 1 <= scb1 by A26, NAT_1:13;
then A29: (((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . (k + 1)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)))) + (k + 1) = ((s . cc) - (s . bb)) + 1 by A1, A3, A4, Th19;
A30: f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k) = abs (((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)))) by A15
.= ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) by A23, A25, ABSVALUE:def_1 ;
percases ( ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) > 0 or ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) <= 0 ) ;
supposeA31: ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) > 0 ; ::_thesis: f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) < f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k)
f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) = abs (((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)))) by A15
.= (scb1 - k) - 1 by A24, A29, A31, ABSVALUE:def_1 ;
hence  f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) < f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k) by A30, A27, XREAL_1:146; ::_thesis: verum
end;
supposeA32: ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) <= 0 ; ::_thesis: f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) < f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k)
 ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . (k + 1)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) = scb1 - (k + 1) by A29;
then A33: ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) = 0 by A24, A28, A32, XREAL_1:48;
f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) = abs (((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)))) by A15
.= 0 by A33, ABSVALUE:def_1 ;
hence  f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) < f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k) by A22, A25, A30, SCMFSA_M:2; ::_thesis: verum
end;
end;
end;
hence  ( f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (k + 1)) < f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k) or ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) <= 0 ) ; ::_thesis: verum
end;
thus  WithVariantWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)),(I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s),p ::_thesis: verum
proof
take  f ; :: according to SCMFSA9A:def_5 ::_thesis: for b1 being Element of NAT holds
( not f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . b1) <= f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (b1 + 1)) or ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . b1) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) <= 0 )
thus  for b1 being Element of NAT holds
( not f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . b1) <= f . ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . (b1 + 1)) or ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),((I ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(intloc 0)))) ";" (a := bb)),p,s)))) . b1) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))) <= 0 ) by A21; ::_thesis: verum
end;
end;

theorem Th25: :: SFMASTR3:25
for s being State of SCM+FSA
for a being read-write Int-Location
for cc, bb being Int-Location
for Ig being good Program of
for k being Element of NAT
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & k = ((s . cc) - (s . bb)) + 1 & ( ProperForUpBody a,bb,cc,Ig,s,p or Ig is parahalting ) holds
DataPart (IExec ((for-up (a,bb,cc,Ig)),p,s)) = DataPart ((StepForUp (a,bb,cc,Ig,p,s)) . k)
proof
let s be State of SCM+FSA; ::_thesis: for a being read-write Int-Location
for cc, bb being Int-Location
for Ig being good Program of
for k being Element of NAT
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & k = ((s . cc) - (s . bb)) + 1 & ( ProperForUpBody a,bb,cc,Ig,s,p or Ig is parahalting ) holds
DataPart (IExec ((for-up (a,bb,cc,Ig)),p,s)) = DataPart ((StepForUp (a,bb,cc,Ig,p,s)) . k)
let a be read-write Int-Location; ::_thesis: for cc, bb being Int-Location
for Ig being good Program of
for k being Element of NAT
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & k = ((s . cc) - (s . bb)) + 1 & ( ProperForUpBody a,bb,cc,Ig,s,p or Ig is parahalting ) holds
DataPart (IExec ((for-up (a,bb,cc,Ig)),p,s)) = DataPart ((StepForUp (a,bb,cc,Ig,p,s)) . k)
let cc, bb be Int-Location; ::_thesis: for Ig being good Program of
for k being Element of NAT
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & k = ((s . cc) - (s . bb)) + 1 & ( ProperForUpBody a,bb,cc,Ig,s,p or Ig is parahalting ) holds
DataPart (IExec ((for-up (a,bb,cc,Ig)),p,s)) = DataPart ((StepForUp (a,bb,cc,Ig,p,s)) . k)
let Ig be good Program of ; ::_thesis: for k being Element of NAT
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & k = ((s . cc) - (s . bb)) + 1 & ( ProperForUpBody a,bb,cc,Ig,s,p or Ig is parahalting ) holds
DataPart (IExec ((for-up (a,bb,cc,Ig)),p,s)) = DataPart ((StepForUp (a,bb,cc,Ig,p,s)) . k)
let k be Element of NAT ; ::_thesis: for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & k = ((s . cc) - (s . bb)) + 1 & ( ProperForUpBody a,bb,cc,Ig,s,p or Ig is parahalting ) holds
DataPart (IExec ((for-up (a,bb,cc,Ig)),p,s)) = DataPart ((StepForUp (a,bb,cc,Ig,p,s)) . k)
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( s . (intloc 0) = 1 & k = ((s . cc) - (s . bb)) + 1 & ( ProperForUpBody a,bb,cc,Ig,s,p or Ig is parahalting ) implies DataPart (IExec ((for-up (a,bb,cc,Ig)),p,s)) = DataPart ((StepForUp (a,bb,cc,Ig,p,s)) . k) )
set I = Ig;
assume that
A1: s . (intloc 0) = 1 and
A2: k = ((s . cc) - (s . bb)) + 1 and
A3: ( ProperForUpBody a,bb,cc,Ig,s,p or Ig is parahalting ) ; ::_thesis: DataPart (IExec ((for-up (a,bb,cc,Ig)),p,s)) = DataPart ((StepForUp (a,bb,cc,Ig,p,s)) . k)
set scb1 = ((s . cc) - (s . bb)) + 1;
set SF = StepForUp (a,bb,cc,Ig,p,s);
A4: ProperForUpBody a,bb,cc,Ig,s,p by A3, Th17;
set i3 = a := bb;
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig));
set i0 = (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc;
set i1 = SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb);
set i2 = AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0));
set IB = (Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)));
set I03 = ((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb);
set s1 = IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s);
set p1 = p;
A5: (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s)) . (intloc 0) = (Exec ((a := bb),(IExec (((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,s)))) . (intloc 0) by SCMFSA6C:6
.= (IExec (((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,s)) . (intloc 0) by SCMFSA_2:63
.= (Exec ((AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))),(IExec ((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))),p,s)))) . (intloc 0) by SCMFSA6C:6
.= (IExec ((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))),p,s)) . (intloc 0) by SCMFSA_2:64
.= (Exec ((SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb)),(Exec (((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc),(Initialized s))))) . (intloc 0) by SCMFSA6C:8
.= (Exec (((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc),(Initialized s))) . (intloc 0) by SCMFSA_2:65
.= (Initialized s) . (intloc 0) by SCMFSA_2:63
.= 1 by SCMFSA_M:9 ;
then A6: DataPart (Initialized (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s))) = DataPart (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s)) by SCMFSA_M:19;
set Ex1 = ExitsAtWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,(Initialized (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s))));
set SW1 = StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,(Initialized (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s))));
A7: ProperBodyWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)),(Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s),p by A1, A3, Lm1;
then A8: ProperBodyWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)),(Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))), Initialized (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s)),p by A6, SCMFSA9A:38;
set s2 = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb));
set p2 = p;
set SW2 = StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))));
A9: DataPart (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s)) = DataPart ((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) by A1, Th16;
then  DataPart ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,(Initialized (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s))))) . k) = DataPart ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) by A6, A8, SCMFSA9A:34;
then A10: ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,(Initialized (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s))))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) = ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) by SCMFSA_M:2;
A11: WithVariantWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)),(Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s),p by A1, A3, Lm1;
then A12: WithVariantWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)),(Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))), Initialized (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s)),p by A5, A6, A7, SCMFSA9A:41;
consider K being Element of NAT  such that
A13: ExitsAtWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,(Initialized (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s)))) = K and
A14: ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,(Initialized (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s))))) . K) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) <= 0 and
A15: ( ( for i being Element of NAT st ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,(Initialized (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s))))) . i) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) <= 0 holds
K <= i ) & DataPart (Comput ((p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),(Initialize (Initialized (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s)))),(LifeSpan ((p +* (while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))))),(Initialize (Initialized (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s)))))))) = DataPart ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,(Initialized (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s))))) . K) ) by A12, A8, SCMFSA9A:def_6;
 DataPart ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,(Initialized (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s))))) . K) = DataPart ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . K) by A6, A8, A9, SCMFSA9A:34;
then A16: ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,(Initialized (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s))))) . K) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) = ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))))) . K) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) by SCMFSA_M:2;
A17: now__::_thesis:_not_K_<_((s_._cc)_-_(s_._bb))_+_1
assume A18: K < ((s . cc) - (s . bb)) + 1 ; ::_thesis: contradiction
then  (((StepForUp (a,bb,cc,Ig,p,s)) . K) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + K < 0 + (((s . cc) - (s . bb)) + 1) by A14, A16, XREAL_1:8;
hence  contradiction by A1, A4, A18, Th19; ::_thesis: verum
end;
 (((StepForUp (a,bb,cc,Ig,p,s)) . k) . (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)))) + k = ((s . cc) - (s . bb)) + 1 by A1, A2, A4, Th19;
then  K <= k by A2, A15, A10;
then A19: ExitsAtWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,(Initialized (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s)))) = k by A2, A13, A17, XXREAL_0:1;
set MI = for-up (a,bb,cc,Ig);
set I4 = while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))));
 ( while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))) is_halting_on IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s),p & while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))) is_closed_on IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s),p ) by A7, A11, SCMFSA9A:27;
hence  DataPart (IExec ((for-up (a,bb,cc,Ig)),p,s)) = DataPart (IExec ((while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))))),p,(IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s)))) by SFMASTR1:9
.= DataPart ((StepWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,(Initialized (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s))))) . (ExitsAtWhile>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))),p,(Initialized (IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s)))))) by A8, A12, SCMFSA9A:36
.= DataPart ((StepForUp (a,bb,cc,Ig,p,s)) . k) by A6, A8, A9, A19, SCMFSA9A:34 ;
::_thesis: verum
end;

theorem Th26: :: SFMASTR3:26
for s being State of SCM+FSA
for a being read-write Int-Location
for bb, cc being Int-Location
for Ig being good Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ( ProperForUpBody a,bb,cc,Ig,s,p or Ig is parahalting ) holds
( for-up (a,bb,cc,Ig) is_closed_on s,p & for-up (a,bb,cc,Ig) is_halting_on s,p )
proof
let s be State of SCM+FSA; ::_thesis: for a being read-write Int-Location
for bb, cc being Int-Location
for Ig being good Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ( ProperForUpBody a,bb,cc,Ig,s,p or Ig is parahalting ) holds
( for-up (a,bb,cc,Ig) is_closed_on s,p & for-up (a,bb,cc,Ig) is_halting_on s,p )
let a be read-write Int-Location; ::_thesis: for bb, cc being Int-Location
for Ig being good Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ( ProperForUpBody a,bb,cc,Ig,s,p or Ig is parahalting ) holds
( for-up (a,bb,cc,Ig) is_closed_on s,p & for-up (a,bb,cc,Ig) is_halting_on s,p )
let bb, cc be Int-Location; ::_thesis: for Ig being good Program of
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ( ProperForUpBody a,bb,cc,Ig,s,p or Ig is parahalting ) holds
( for-up (a,bb,cc,Ig) is_closed_on s,p & for-up (a,bb,cc,Ig) is_halting_on s,p )
let Ig be good Program of ; ::_thesis: for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 & ( ProperForUpBody a,bb,cc,Ig,s,p or Ig is parahalting ) holds
( for-up (a,bb,cc,Ig) is_closed_on s,p & for-up (a,bb,cc,Ig) is_halting_on s,p )
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( s . (intloc 0) = 1 & ( ProperForUpBody a,bb,cc,Ig,s,p or Ig is parahalting ) implies ( for-up (a,bb,cc,Ig) is_closed_on s,p & for-up (a,bb,cc,Ig) is_halting_on s,p ) )
set I = Ig;
assume that
A1: s . (intloc 0) = 1 and
A2: ( ProperForUpBody a,bb,cc,Ig,s,p or Ig is parahalting ) ; ::_thesis: ( for-up (a,bb,cc,Ig) is_closed_on s,p & for-up (a,bb,cc,Ig) is_halting_on s,p )
set i3 = a := bb;
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig));
set i0 = (1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc;
set i1 = SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb);
set i2 = AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0));
set IB = (Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)));
set I4 = while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))));
set I03 = ((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb);
set s1 = IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s);
set p1 = p;
A3: ( ProperBodyWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)),(Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s),p & WithVariantWhile>0 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig)),(Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))), IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s),p ) by A1, A2, Lm1;
then A4: while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))) is_closed_on IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s),p by SCMFSA9A:27;
reconsider I03 = ((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb) as parahalting Program of ;
set MI = for-up (a,bb,cc,Ig);
A5: ( I03 is_closed_on Initialized s,p & I03 is_halting_on Initialized s,p ) by SCMFSA7B:18, SCMFSA7B:19;
then A6: for-up (a,bb,cc,Ig) is_closed_on Initialized s,p by A4, SFMASTR1:2;
hence  for-up (a,bb,cc,Ig) is_closed_on s,p by A1, SFMASTR2:4; ::_thesis: for-up (a,bb,cc,Ig) is_halting_on s,p
 while>0 ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),((Ig ";" (AddTo (a,(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0))))) is_halting_on IExec ((((((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))) := cc) ";" (SubFrom ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),bb))) ";" (AddTo ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc Ig))),(intloc 0)))) ";" (a := bb)),p,s),p by A3, SCMFSA9A:27;
then  for-up (a,bb,cc,Ig) is_halting_on Initialized s,p by A5, A4, SFMASTR1:3;
hence  for-up (a,bb,cc,Ig) is_halting_on s,p by A1, A6, SFMASTR2:5; ::_thesis: verum
end;

begin

definition
let start, finish, minpos be Int-Location;
let f be FinSeq-Location ;
func FinSeqMin (f,start,finish,minpos) ->  Program of  equals :: SFMASTR3:def 4
(minpos := start) ";" (for-up ((3 -rdRWNotIn {start,finish,minpos}),start,finish,((((1 -stRWNotIn {start,finish,minpos}) := (f,(3 -rdRWNotIn {start,finish,minpos}))) ";" ((2 -ndRWNotIn {start,finish,minpos}) := (f,minpos))) ";" (if>0 ((2 -ndRWNotIn {start,finish,minpos}),(1 -stRWNotIn {start,finish,minpos}),(Macro (minpos := (3 -rdRWNotIn {start,finish,minpos}))),(Stop SCM+FSA))))));
coherence
 (minpos := start) ";" (for-up ((3 -rdRWNotIn {start,finish,minpos}),start,finish,((((1 -stRWNotIn {start,finish,minpos}) := (f,(3 -rdRWNotIn {start,finish,minpos}))) ";" ((2 -ndRWNotIn {start,finish,minpos}) := (f,minpos))) ";" (if>0 ((2 -ndRWNotIn {start,finish,minpos}),(1 -stRWNotIn {start,finish,minpos}),(Macro (minpos := (3 -rdRWNotIn {start,finish,minpos}))),(Stop SCM+FSA)))))) is Program of ;
end;

:: deftheorem  defines FinSeqMin SFMASTR3:def_4_:_
 for start, finish, minpos being Int-Location
for f being FinSeq-Location holds FinSeqMin (f,start,finish,minpos) = (minpos := start) ";" (for-up ((3 -rdRWNotIn {start,finish,minpos}),start,finish,((((1 -stRWNotIn {start,finish,minpos}) := (f,(3 -rdRWNotIn {start,finish,minpos}))) ";" ((2 -ndRWNotIn {start,finish,minpos}) := (f,minpos))) ";" (if>0 ((2 -ndRWNotIn {start,finish,minpos}),(1 -stRWNotIn {start,finish,minpos}),(Macro (minpos := (3 -rdRWNotIn {start,finish,minpos}))),(Stop SCM+FSA))))));

registration
let start, finish be Int-Location;
let minpos be read-write Int-Location;
let f be FinSeq-Location ;
cluster FinSeqMin (f,start,finish,minpos) ->  good ;
coherence
 FinSeqMin (f,start,finish,minpos) is good ;
end;

theorem Th27: :: SFMASTR3:27
for c being read-write Int-Location
for aa, bb being Int-Location
for f being FinSeq-Location st c <> aa holds
not FinSeqMin (f,aa,bb,c) destroys aa
proof
let c be read-write Int-Location; ::_thesis: for aa, bb being Int-Location
for f being FinSeq-Location st c <> aa holds
not FinSeqMin (f,aa,bb,c) destroys aa
let aa, bb be Int-Location; ::_thesis: for f being FinSeq-Location st c <> aa holds
not FinSeqMin (f,aa,bb,c) destroys aa
let f be FinSeq-Location ; ::_thesis: ( c <> aa implies not FinSeqMin (f,aa,bb,c) destroys aa )
set a = aa;
set b = bb;
set aux1 = 1 -stRWNotIn {aa,bb,c};
set aux2 = 2 -ndRWNotIn {aa,bb,c};
set cv = 3 -rdRWNotIn {aa,bb,c};
set i10 = (1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}));
set i11 = (2 -ndRWNotIn {aa,bb,c}) := (f,c);
set I12 = if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA));
set I1B = (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)));
set I1 = for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))));
A1: not Stop SCM+FSA destroys aa by SCMFSA8C:56;
A2: aa in {aa,bb,c} by ENUMSET1:def_1;
then  1 -stRWNotIn {aa,bb,c} <> aa by SCMFSA_M:25;
then A3: not (1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c})) destroys aa by SCMFSA7B:14;
A4: 2 -ndRWNotIn {aa,bb,c} <> aa by A2, SCMFSA_M:25;
then  not (2 -ndRWNotIn {aa,bb,c}) := (f,c) destroys aa by SCMFSA7B:14;
then A5: not ((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c)) destroys aa by A3, SCMFSA8C:55;
assume A6: c <> aa ; ::_thesis: not FinSeqMin (f,aa,bb,c) destroys aa
then  not Macro (c := (3 -rdRWNotIn {aa,bb,c})) destroys aa by SCMFSA7B:6, SCMFSA8C:48;
then  not if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)) destroys aa by A4, A1, Th9;
then A7: not (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))) destroys aa by A5, SCMFSA8C:52;
 aa in {(3 -rdRWNotIn {aa,bb,c}),aa,bb} by ENUMSET1:def_1;
then  aa in {(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))) by XBOOLE_0:def_3;
then A8: aa <> 1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))))) by SCMFSA_M:25;
 3 -rdRWNotIn {aa,bb,c} <> aa by A2, SCMFSA_M:25;
then  not for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))) destroys aa by A7, A8, Th23;
hence  not FinSeqMin (f,aa,bb,c) destroys aa by A6, SCMFSA7B:6, SCMFSA8C:53; ::_thesis: verum
end;

theorem Th28: :: SFMASTR3:28
for c being read-write Int-Location
for aa, bb being Int-Location
for f being FinSeq-Location holds {aa,bb,c} c= UsedIntLoc (FinSeqMin (f,aa,bb,c))
proof
let c be read-write Int-Location; ::_thesis: for aa, bb being Int-Location
for f being FinSeq-Location holds {aa,bb,c} c= UsedIntLoc (FinSeqMin (f,aa,bb,c))
let aa, bb be Int-Location; ::_thesis: for f being FinSeq-Location holds {aa,bb,c} c= UsedIntLoc (FinSeqMin (f,aa,bb,c))
let f be FinSeq-Location ; ::_thesis: {aa,bb,c} c= UsedIntLoc (FinSeqMin (f,aa,bb,c))
set a = aa;
set b = bb;
set aux1 = 1 -stRWNotIn {aa,bb,c};
set aux2 = 2 -ndRWNotIn {aa,bb,c};
set cv = 3 -rdRWNotIn {aa,bb,c};
set i0 = c := aa;
set i10 = (1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}));
set i11 = (2 -ndRWNotIn {aa,bb,c}) := (f,c);
set I12 = if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA));
set I1B = (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)));
set I1 = for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))));
A1: UsedIntLoc ((c := aa) ";" (for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))) = (UsedIntLoc (c := aa)) \/ (UsedIntLoc (for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))) by SF_MASTR:29;
A2: {(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))) c= UsedIntLoc (for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))))) by Th22;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {aa,bb,c} or x in UsedIntLoc (FinSeqMin (f,aa,bb,c)) )
A3: UsedIntLoc (c := aa) = {c,aa} by SF_MASTR:14;
assume A4: x in {aa,bb,c} ; ::_thesis: x in UsedIntLoc (FinSeqMin (f,aa,bb,c))
percases ( x = aa or x = bb or x = c ) by A4, ENUMSET1:def_1;
suppose x = aa ; ::_thesis: x in UsedIntLoc (FinSeqMin (f,aa,bb,c))
then  x in {c,aa} by TARSKI:def_2;
hence  x in UsedIntLoc (FinSeqMin (f,aa,bb,c)) by A1, A3, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose x = bb ; ::_thesis: x in UsedIntLoc (FinSeqMin (f,aa,bb,c))
then  x in {(3 -rdRWNotIn {aa,bb,c}),aa,bb} by ENUMSET1:def_1;
then  x in {(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))) by XBOOLE_0:def_3;
hence  x in UsedIntLoc (FinSeqMin (f,aa,bb,c)) by A1, A2, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose x = c ; ::_thesis: x in UsedIntLoc (FinSeqMin (f,aa,bb,c))
then  x in {c,aa} by TARSKI:def_2;
hence  x in UsedIntLoc (FinSeqMin (f,aa,bb,c)) by A1, A3, XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;

theorem Th29: :: SFMASTR3:29
for s being State of SCM+FSA
for c being read-write Int-Location
for aa, bb being Int-Location
for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 holds
( FinSeqMin (f,aa,bb,c) is_closed_on s,p & FinSeqMin (f,aa,bb,c) is_halting_on s,p )
proof
let s be State of SCM+FSA; ::_thesis: for c being read-write Int-Location
for aa, bb being Int-Location
for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 holds
( FinSeqMin (f,aa,bb,c) is_closed_on s,p & FinSeqMin (f,aa,bb,c) is_halting_on s,p )
let c be read-write Int-Location; ::_thesis: for aa, bb being Int-Location
for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 holds
( FinSeqMin (f,aa,bb,c) is_closed_on s,p & FinSeqMin (f,aa,bb,c) is_halting_on s,p )
let aa, bb be Int-Location; ::_thesis: for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 holds
( FinSeqMin (f,aa,bb,c) is_closed_on s,p & FinSeqMin (f,aa,bb,c) is_halting_on s,p )
let f be FinSeq-Location ; ::_thesis: for p being Instruction-Sequence of SCM+FSA st s . (intloc 0) = 1 holds
( FinSeqMin (f,aa,bb,c) is_closed_on s,p & FinSeqMin (f,aa,bb,c) is_halting_on s,p )
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( s . (intloc 0) = 1 implies ( FinSeqMin (f,aa,bb,c) is_closed_on s,p & FinSeqMin (f,aa,bb,c) is_halting_on s,p ) )
assume A1: s . (intloc 0) = 1 ; ::_thesis: ( FinSeqMin (f,aa,bb,c) is_closed_on s,p & FinSeqMin (f,aa,bb,c) is_halting_on s,p )
set a = aa;
set b = bb;
set aux1 = 1 -stRWNotIn {aa,bb,c};
set aux2 = 2 -ndRWNotIn {aa,bb,c};
set cv = 3 -rdRWNotIn {aa,bb,c};
set i0 = c := aa;
set i10 = (1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}));
set i11 = (2 -ndRWNotIn {aa,bb,c}) := (f,c);
set I12 = if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA));
set I1B = (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)));
set I1 = for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))));
set s1 = IExec ((Macro (c := aa)),p,s);
set p1 = p;
A2: (IExec ((Macro (c := aa)),p,s)) . (intloc 0) = (IExec ((Macro (c := aa)),p,s)) . (intloc 0)
.= (Exec ((c := aa),(Initialized s))) . (intloc 0) by SCMFSA6C:5
.= (Exec ((c := aa),(Initialized s))) . (intloc 0)
.= (Initialized s) . (intloc 0) by SCMFSA_2:63
.= 1 by SCMFSA_M:9 ;
then A3: for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))) is_closed_on IExec ((Macro (c := aa)),p,s),p by Th26;
A4: ( Macro (c := aa) is_closed_on Initialized s,p & Macro (c := aa) is_halting_on Initialized s,p ) by SCMFSA7B:18, SCMFSA7B:19;
then A5: FinSeqMin (f,aa,bb,c) is_closed_on Initialized s,p by A3, SFMASTR1:2;
hence  FinSeqMin (f,aa,bb,c) is_closed_on s,p by A1, SFMASTR2:4; ::_thesis: FinSeqMin (f,aa,bb,c) is_halting_on s,p
 for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))) is_halting_on IExec ((Macro (c := aa)),p,s),p by A2, Th26;
then  FinSeqMin (f,aa,bb,c) is_halting_on Initialized s,p by A3, A4, SFMASTR1:3;
hence  FinSeqMin (f,aa,bb,c) is_halting_on s,p by A1, A5, SFMASTR2:5; ::_thesis: verum
end;

theorem Th30: :: SFMASTR3:30
for s being State of SCM+FSA
for c being read-write Int-Location
for aa, bb being Int-Location
for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA st aa <> c & bb <> c & s . (intloc 0) = 1 holds
( (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . f = s . f & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . aa = s . aa & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . bb = s . bb )
proof
let s be State of SCM+FSA; ::_thesis: for c being read-write Int-Location
for aa, bb being Int-Location
for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA st aa <> c & bb <> c & s . (intloc 0) = 1 holds
( (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . f = s . f & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . aa = s . aa & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . bb = s . bb )
let c be read-write Int-Location; ::_thesis: for aa, bb being Int-Location
for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA st aa <> c & bb <> c & s . (intloc 0) = 1 holds
( (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . f = s . f & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . aa = s . aa & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . bb = s . bb )
let aa, bb be Int-Location; ::_thesis: for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA st aa <> c & bb <> c & s . (intloc 0) = 1 holds
( (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . f = s . f & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . aa = s . aa & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . bb = s . bb )
let f be FinSeq-Location ; ::_thesis: for p being Instruction-Sequence of SCM+FSA st aa <> c & bb <> c & s . (intloc 0) = 1 holds
( (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . f = s . f & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . aa = s . aa & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . bb = s . bb )
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( aa <> c & bb <> c & s . (intloc 0) = 1 implies ( (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . f = s . f & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . aa = s . aa & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . bb = s . bb ) )
assume that
A1: aa <> c and
A2: bb <> c and
A3: s . (intloc 0) = 1 ; ::_thesis: ( (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . f = s . f & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . aa = s . aa & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . bb = s . bb )
set a = aa;
set b = bb;
set i0 = c := aa;
set s1 = Exec ((c := aa),(Initialized s));
set p1 = p;
A4: ( aa = intloc 0 or aa is read-write ) by SCMFSA_M:def_2;
A5: ( bb = intloc 0 or bb is read-write ) by SCMFSA_M:def_2;
A6: (Exec ((c := aa),(Initialized s))) . bb = (Initialized s) . bb by A2, SCMFSA_2:63
.= s . bb by A3, A5, SCMFSA_M:9, SCMFSA_M:37 ;
set cv = 3 -rdRWNotIn {aa,bb,c};
set aux2 = 2 -ndRWNotIn {aa,bb,c};
set aux1 = 1 -stRWNotIn {aa,bb,c};
set i10 = (1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}));
set i11 = (2 -ndRWNotIn {aa,bb,c}) := (f,c);
set I12 = if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA));
set I1B = (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)));
set I1 = for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))));
A7: 2 -ndRWNotIn {aa,bb,c} <> 3 -rdRWNotIn {aa,bb,c} by SCMFSA_M:26;
 3 -rdRWNotIn {aa,bb,c} in {(3 -rdRWNotIn {aa,bb,c}),aa,bb} by ENUMSET1:def_1;
then A8: 3 -rdRWNotIn {aa,bb,c} in {(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))) by XBOOLE_0:def_3;
A9: 1 -stRWNotIn {aa,bb,c} <> 3 -rdRWNotIn {aa,bb,c} by SCMFSA_M:26;
A10: bb in {aa,bb,c} by ENUMSET1:def_1;
then A11: 3 -rdRWNotIn {aa,bb,c} <> bb by SCMFSA_M:25;
A12: 1 -stRWNotIn {aa,bb,c} <> bb by A10, SCMFSA_M:25;
A13: 2 -ndRWNotIn {aa,bb,c} <> bb by A10, SCMFSA_M:25;
A14: (Exec ((c := aa),(Initialized s))) . aa = (Initialized s) . aa by A1, SCMFSA_2:63
.= s . aa by A3, A4, SCMFSA_M:9, SCMFSA_M:37 ;
 bb in {(3 -rdRWNotIn {aa,bb,c}),aa,bb} by ENUMSET1:def_1;
then A15: bb in {(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))) by XBOOLE_0:def_3;
A16: aa in {aa,bb,c} by ENUMSET1:def_1;
then A17: 3 -rdRWNotIn {aa,bb,c} <> aa by SCMFSA_M:25;
A18: 1 -stRWNotIn {aa,bb,c} <> aa by A16, SCMFSA_M:25;
A19: 2 -ndRWNotIn {aa,bb,c} <> aa by A16, SCMFSA_M:25;
A20: (Exec ((c := aa),(Initialized s))) . f = (Initialized s) . f by SCMFSA_2:63
.= s . f by SCMFSA_M:37 ;
 aa in {(3 -rdRWNotIn {aa,bb,c}),aa,bb} by ENUMSET1:def_1;
then A21: aa in {(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))) by XBOOLE_0:def_3;
 c in {aa,bb,c} by ENUMSET1:def_1;
then A22: 3 -rdRWNotIn {aa,bb,c} <> c by SCMFSA_M:25;
A23: (Exec ((c := aa),(Initialized s))) . (intloc 0) = (Initialized s) . (intloc 0) by SCMFSA_2:63
.= 1 by SCMFSA_M:9 ;
then A24: ( for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))) is_closed_on Exec ((c := aa),(Initialized s)),p & for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))) is_halting_on Exec ((c := aa),(Initialized s)),p ) by Th26;
percases ( s . aa > s . bb or s . aa <= s . bb ) ;
supposeA25: s . aa > s . bb ; ::_thesis: ( (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . f = s . f & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . aa = s . aa & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . bb = s . bb )
thus (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . f = (IExec ((for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))))),p,(Exec ((c := aa),(Initialized s))))) . f by A24, SFMASTR1:15
.= s . f by A23, A14, A6, A20, A25, Th24 ; ::_thesis: ( (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . aa = s . aa & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . bb = s . bb )
 aa in {aa,bb} by TARSKI:def_2;
then A26: aa in {aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))) by XBOOLE_0:def_3;
thus (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . aa = (IExec ((for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))))),p,(Exec ((c := aa),(Initialized s))))) . aa by A24, SFMASTR1:14
.= s . aa by A17, A23, A14, A6, A25, A26, Th24 ; ::_thesis: (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . bb = s . bb
 bb in {aa,bb} by TARSKI:def_2;
then A27: bb in {aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))) by XBOOLE_0:def_3;
thus (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . bb = (IExec ((for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))))),p,(Exec ((c := aa),(Initialized s))))) . bb by A24, SFMASTR1:14
.= s . bb by A11, A23, A14, A6, A25, A27, Th24 ; ::_thesis: verum
end;
supposeA28: s . aa <= s . bb ; ::_thesis: ( (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . f = s . f & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . aa = s . aa & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . bb = s . bb )
set SF = StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))));
A29: (s . aa) - (s . aa) <= (s . bb) - (s . aa) by A28, XREAL_1:9;
then reconsider k = ((s . bb) - (s . aa)) + 1 as Element of NAT by INT_1:3;
defpred S1[ Nat] means ( 0 < $1 & $1 <= k implies ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . $1) . (intloc 0) = 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . $1) . (3 -rdRWNotIn {aa,bb,c}) = $1 + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . $1) . f = (Exec ((c := aa),(Initialized s))) . f & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . $1) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . $1) . bb = (Exec ((c := aa),(Initialized s))) . bb ) );
A30: ProperForUpBody 3 -rdRWNotIn {aa,bb,c},aa,bb,(((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))), Exec ((c := aa),(Initialized s)),p by Th17;
A31: for n being Element of NAT st S1[n] holds
S1[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] )
assume that
A32: S1[n] and
 0 < n + 1 and
A33: n + 1 <= k ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (intloc 0) = 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb )
A34: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (intloc 0) = 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) = n + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) <= (Exec ((c := aa),(Initialized s))) . bb )
proof
percases ( 0 = n or 0 < n ) ;
supposeA35: 0 = n ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (intloc 0) = 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) = n + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) <= (Exec ((c := aa),(Initialized s))) . bb )
hence  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (intloc 0) = 1 by A23, Th10; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) = n + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) <= (Exec ((c := aa),(Initialized s))) . bb )
thus  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) = n + ((Exec ((c := aa),(Initialized s))) . aa) by A35, Th11; ::_thesis: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) <= (Exec ((c := aa),(Initialized s))) . bb
thus  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) <= (Exec ((c := aa),(Initialized s))) . bb by A14, A6, A28, A35, Th11; ::_thesis: verum
end;
supposeA36: 0 < n ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (intloc 0) = 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) = n + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) <= (Exec ((c := aa),(Initialized s))) . bb )
hence  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (intloc 0) = 1 by A32, A33, NAT_1:13; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) = n + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) <= (Exec ((c := aa),(Initialized s))) . bb )
thus  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) = n + ((Exec ((c := aa),(Initialized s))) . aa) by A32, A33, A36, NAT_1:13; ::_thesis: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) <= (Exec ((c := aa),(Initialized s))) . bb
 (n + 1) - 1 <= (((s . bb) - (s . aa)) + 1) - 1 by A33, XREAL_1:9;
hence  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) <= (Exec ((c := aa),(Initialized s))) . bb by A14, A6, A32, A33, A36, NAT_1:13, XREAL_1:19; ::_thesis: verum
end;
end;
end;
 n < n + 1 by XREAL_1:29;
then  n < k by A33, XXREAL_0:2;
then A37: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) | (({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))))) \/ FinSeq-Locations) = (IExec ((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) | (({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))))) \/ FinSeq-Locations) by A23, A14, A6, A30, Th21;
then A38: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (IExec ((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . f by SCMFSA_M:28
.= (Exec ((AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0))),(IExec (((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . f by SFMASTR1:12
.= (IExec (((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . f by SCMFSA_2:64
.= (IExec ((if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . f by SFMASTR1:8 ;
A39: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (IExec ((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . bb by A15, A37, SCMFSA_M:28
.= (Exec ((AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0))),(IExec (((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . bb by SFMASTR1:11
.= (IExec (((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . bb by A11, SCMFSA_2:64
.= (IExec ((if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . bb by SFMASTR1:7 ;
A40: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (IExec ((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . aa by A21, A37, SCMFSA_M:28
.= (Exec ((AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0))),(IExec (((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . aa by SFMASTR1:11
.= (IExec (((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . aa by A17, SCMFSA_2:64
.= (IExec ((if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . aa by SFMASTR1:7 ;
A41: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (IExec ((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (3 -rdRWNotIn {aa,bb,c}) by A8, A37, SCMFSA_M:28
.= (Exec ((AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0))),(IExec (((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . (3 -rdRWNotIn {aa,bb,c}) by SFMASTR1:11
.= ((IExec (((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (3 -rdRWNotIn {aa,bb,c})) + ((IExec (((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (intloc 0)) by SCMFSA_2:64
.= ((IExec (((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (3 -rdRWNotIn {aa,bb,c})) + 1 by SCMFSA6B:11
.= ((IExec ((if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . (3 -rdRWNotIn {aa,bb,c})) + 1 by SFMASTR1:7 ;
set ss = IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n));
set S0 = Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n);
set S1 = Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)));
set S2 = Exec (((2 -ndRWNotIn {aa,bb,c}) := (f,c)),(Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))));
A42: (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . f = (Exec (((2 -ndRWNotIn {aa,bb,c}) := (f,c)),(Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . f by SCMFSA6C:9
.= (Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . f by SCMFSA_2:72
.= (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . f by SCMFSA_2:72 ;
A43: (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (intloc 0) = (Exec (((2 -ndRWNotIn {aa,bb,c}) := (f,c)),(Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . (intloc 0) by SCMFSA6C:8
.= (Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . (intloc 0) by SCMFSA_2:72
.= (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . (intloc 0) by SCMFSA_2:72
.= 1 by SCMFSA_M:9 ;
then A44: DataPart (IExec ((Stop SCM+FSA),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) = DataPart (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) by Th2;
A45: (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . bb = (Exec (((2 -ndRWNotIn {aa,bb,c}) := (f,c)),(Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . bb by SCMFSA6C:8
.= (Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . bb by A13, SCMFSA_2:72
.= (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . bb by A12, SCMFSA_2:72 ;
A46: (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . aa = (Exec (((2 -ndRWNotIn {aa,bb,c}) := (f,c)),(Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . aa by SCMFSA6C:8
.= (Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . aa by A19, SCMFSA_2:72
.= (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . aa by A18, SCMFSA_2:72 ;
A47: (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (3 -rdRWNotIn {aa,bb,c}) = (Exec (((2 -ndRWNotIn {aa,bb,c}) := (f,c)),(Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . (3 -rdRWNotIn {aa,bb,c}) by SCMFSA6C:8
.= (Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . (3 -rdRWNotIn {aa,bb,c}) by A7, SCMFSA_2:72
.= (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . (3 -rdRWNotIn {aa,bb,c}) by A9, SCMFSA_2:72 ;
A48: ( not Macro (c := (3 -rdRWNotIn {aa,bb,c})) refers 2 -ndRWNotIn {aa,bb,c} & not Stop SCM+FSA refers 2 -ndRWNotIn {aa,bb,c} ) by A7, Th3, Th4, SCMFSA8C:51;
percases ( 0 = n or 0 < n ) ;
supposeA49: 0 = n ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (intloc 0) = 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb )
thus  ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (intloc 0) = 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb ) ::_thesis: verum
proof
thus  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (intloc 0) = 1 by A23, A14, A6, A30, A33, Th19; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb )
A50: (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . f = ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . 0) . f by A49, SCMFSA_M:37
.= s . f by A20, Th15 ;
A51: (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . (3 -rdRWNotIn {aa,bb,c}) = ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . 0) . (3 -rdRWNotIn {aa,bb,c}) by A49, SCMFSA_M:37
.= s . aa by A14, Th11 ;
A52: (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . bb = ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . 0) . bb by A5, A34, A49, SCMFSA_M:9, SCMFSA_M:37
.= (Exec ((c := aa),(Initialized s))) . bb by A11, Th13 ;
A53: (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . aa = ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . 0) . aa by A4, A34, A49, SCMFSA_M:9, SCMFSA_M:37
.= (Exec ((c := aa),(Initialized s))) . aa by A17, Th12 ;
thus  ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb ) ::_thesis: verum
proof
percases ( (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (2 -ndRWNotIn {aa,bb,c}) <= (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (1 -stRWNotIn {aa,bb,c}) or (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (2 -ndRWNotIn {aa,bb,c}) > (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (1 -stRWNotIn {aa,bb,c}) ) ;
supposeA54: (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (2 -ndRWNotIn {aa,bb,c}) <= (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (1 -stRWNotIn {aa,bb,c}) ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb )
hence ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = ((IExec ((Stop SCM+FSA),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . (3 -rdRWNotIn {aa,bb,c})) + 1 by A7, A48, A41, SCMFSA8B:40
.= (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) by A14, A44, A47, A49, A51, SCMFSA_M:2 ;
::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb )
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (IExec ((Stop SCM+FSA),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . f by A48, A38, A54, SCMFSA8B:40
.= (Exec ((c := aa),(Initialized s))) . f by A20, A44, A42, A50, SCMFSA_M:2 ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb )
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (IExec ((Stop SCM+FSA),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . aa by A19, A48, A40, A54, SCMFSA8B:40
.= (Exec ((c := aa),(Initialized s))) . aa by A44, A46, A53, SCMFSA_M:2 ; ::_thesis: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (IExec ((Stop SCM+FSA),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . bb by A13, A48, A39, A54, SCMFSA8B:40
.= (Exec ((c := aa),(Initialized s))) . bb by A44, A45, A52, SCMFSA_M:2 ; ::_thesis: verum
end;
supposeA55: (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (2 -ndRWNotIn {aa,bb,c}) > (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (1 -stRWNotIn {aa,bb,c}) ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb )
A56: (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . (3 -rdRWNotIn {aa,bb,c}) = (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . (3 -rdRWNotIn {aa,bb,c})
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . (3 -rdRWNotIn {aa,bb,c}) by SCMFSA6C:5
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . (3 -rdRWNotIn {aa,bb,c})
.= (Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . (3 -rdRWNotIn {aa,bb,c}) by A22, SCMFSA_2:63 ;
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = ((IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . (3 -rdRWNotIn {aa,bb,c})) + 1 by A7, A48, A41, A55, SCMFSA8B:40
.= ((Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . (3 -rdRWNotIn {aa,bb,c})) + 1 by A56
.= (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) by A14, A47, A49, A51, SCMFSA_M:37 ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb )
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . f by A48, A38, A55, SCMFSA8B:40
.= (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . f
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . f by SCMFSA6C:5
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . f
.= (Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . f by SCMFSA_2:63
.= (Exec ((c := aa),(Initialized s))) . f by A20, A42, A50, SCMFSA_M:37 ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb )
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . aa by A19, A48, A40, A55, SCMFSA8B:40
.= (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . aa
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . aa by SCMFSA6C:5
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . aa
.= (Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . aa by A1, SCMFSA_2:63
.= (Exec ((c := aa),(Initialized s))) . aa by A4, A43, A46, A53, SCMFSA_M:9, SCMFSA_M:37 ; ::_thesis: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . bb by A13, A48, A39, A55, SCMFSA8B:40
.= (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . bb
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . bb by SCMFSA6C:5
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . bb
.= (Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . bb by A2, SCMFSA_2:63
.= (Exec ((c := aa),(Initialized s))) . bb by A5, A43, A45, A52, SCMFSA_M:9, SCMFSA_M:37 ; ::_thesis: verum
end;
end;
end;
end;
end;
supposeA57: 0 < n ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (intloc 0) = 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb )
thus  ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (intloc 0) = 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb ) ::_thesis: verum
proof
thus  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (intloc 0) = 1 by A23, A14, A6, A30, A33, Th19; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb )
A58: (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . (3 -rdRWNotIn {aa,bb,c}) = ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) by SCMFSA_M:37;
A59: (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . aa = (Exec ((c := aa),(Initialized s))) . aa by A4, A32, A33, A57, NAT_1:13, SCMFSA_M:9, SCMFSA_M:37;
A60: (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . bb = (Exec ((c := aa),(Initialized s))) . bb by A5, A32, A33, A57, NAT_1:13, SCMFSA_M:9, SCMFSA_M:37;
A61: (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . f = s . f by A20, A32, A33, A57, NAT_1:13, SCMFSA_M:37;
thus  ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb ) ::_thesis: verum
proof
percases ( (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (2 -ndRWNotIn {aa,bb,c}) <= (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (1 -stRWNotIn {aa,bb,c}) or (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (2 -ndRWNotIn {aa,bb,c}) > (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (1 -stRWNotIn {aa,bb,c}) ) ;
supposeA62: (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (2 -ndRWNotIn {aa,bb,c}) <= (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (1 -stRWNotIn {aa,bb,c}) ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb )
hence ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = ((IExec ((Stop SCM+FSA),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . (3 -rdRWNotIn {aa,bb,c})) + 1 by A7, A48, A41, SCMFSA8B:40
.= ((IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (3 -rdRWNotIn {aa,bb,c})) + 1 by A44, SCMFSA_M:2
.= (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) by A34, A47, A58 ;
::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb )
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (IExec ((Stop SCM+FSA),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . f by A48, A38, A62, SCMFSA8B:40
.= (Exec ((c := aa),(Initialized s))) . f by A20, A44, A42, A61, SCMFSA_M:2 ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb )
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (IExec ((Stop SCM+FSA),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . aa by A19, A48, A40, A62, SCMFSA8B:40
.= (Exec ((c := aa),(Initialized s))) . aa by A44, A46, A59, SCMFSA_M:2 ; ::_thesis: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (IExec ((Stop SCM+FSA),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . bb by A13, A48, A39, A62, SCMFSA8B:40
.= (Exec ((c := aa),(Initialized s))) . bb by A44, A45, A60, SCMFSA_M:2 ; ::_thesis: verum
end;
supposeA63: (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (2 -ndRWNotIn {aa,bb,c}) > (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (1 -stRWNotIn {aa,bb,c}) ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb )
A64: (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . (3 -rdRWNotIn {aa,bb,c}) = (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . (3 -rdRWNotIn {aa,bb,c})
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . (3 -rdRWNotIn {aa,bb,c}) by SCMFSA6C:5
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . (3 -rdRWNotIn {aa,bb,c})
.= (Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . (3 -rdRWNotIn {aa,bb,c}) by A22, SCMFSA_2:63
.= (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (3 -rdRWNotIn {aa,bb,c}) by SCMFSA_M:37 ;
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = ((IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . (3 -rdRWNotIn {aa,bb,c})) + 1 by A7, A48, A41, A63, SCMFSA8B:40
.= ((IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (3 -rdRWNotIn {aa,bb,c})) + 1 by A64
.= (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) by A34, A47, A58 ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb )
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . f by A48, A38, A63, SCMFSA8B:40
.= (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . f
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . f by SCMFSA6C:5
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . f
.= (Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . f by SCMFSA_2:63
.= (Exec ((c := aa),(Initialized s))) . f by A20, A42, A61, SCMFSA_M:37 ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (Exec ((c := aa),(Initialized s))) . aa & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb )
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . aa = (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . aa by A19, A48, A40, A63, SCMFSA8B:40
.= (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . aa
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . aa by SCMFSA6C:5
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . aa
.= (Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . aa by A1, SCMFSA_2:63
.= (Exec ((c := aa),(Initialized s))) . aa by A4, A43, A46, A59, SCMFSA_M:9, SCMFSA_M:37 ; ::_thesis: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (Exec ((c := aa),(Initialized s))) . bb
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . bb = (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . bb by A13, A48, A39, A63, SCMFSA8B:40
.= (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . bb
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . bb by SCMFSA6C:5
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . bb
.= (Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . bb by A2, SCMFSA_2:63
.= (Exec ((c := aa),(Initialized s))) . bb by A5, A43, A45, A60, SCMFSA_M:9, SCMFSA_M:37 ; ::_thesis: verum
end;
end;
end;
end;
end;
end;
end;
A65: S1[ 0 ] ;
A66: for n being Element of NAT holds S1[n] from NAT_1:sch_1(A65, A31);
A67: DataPart (IExec ((for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))))),p,(Exec ((c := aa),(Initialized s))))) = DataPart ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . k) by A23, A14, A6, Th25;
thus (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . f = (IExec ((for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))))),p,(Exec ((c := aa),(Initialized s))))) . f by A24, SFMASTR1:15
.= ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . k) . f by A67, SCMFSA_M:2
.= s . f by A20, A29, A66 ; ::_thesis: ( (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . aa = s . aa & (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . bb = s . bb )
thus (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . aa = (IExec ((for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))))),p,(Exec ((c := aa),(Initialized s))))) . aa by A24, SFMASTR1:14
.= ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . k) . aa by A67, SCMFSA_M:2
.= s . aa by A14, A29, A66 ; ::_thesis: (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . bb = s . bb
thus (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . bb = (IExec ((for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))))),p,(Exec ((c := aa),(Initialized s))))) . bb by A24, SFMASTR1:14
.= ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . k) . bb by A67, SCMFSA_M:2
.= s . bb by A6, A29, A66 ; ::_thesis: verum
end;
end;
end;

theorem Th31: :: SFMASTR3:31
for s being State of SCM+FSA
for c being read-write Int-Location
for aa, bb being Int-Location
for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA st 1 <= s . aa & s . aa <= s . bb & s . bb <= len (s . f) & aa <> c & bb <> c & s . (intloc 0) = 1 holds
(IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . c = min_at ((s . f),(abs (s . aa)),(abs (s . bb)))
proof
let s be State of SCM+FSA; ::_thesis: for c being read-write Int-Location
for aa, bb being Int-Location
for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA st 1 <= s . aa & s . aa <= s . bb & s . bb <= len (s . f) & aa <> c & bb <> c & s . (intloc 0) = 1 holds
(IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . c = min_at ((s . f),(abs (s . aa)),(abs (s . bb)))
let c be read-write Int-Location; ::_thesis: for aa, bb being Int-Location
for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA st 1 <= s . aa & s . aa <= s . bb & s . bb <= len (s . f) & aa <> c & bb <> c & s . (intloc 0) = 1 holds
(IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . c = min_at ((s . f),(abs (s . aa)),(abs (s . bb)))
let aa, bb be Int-Location; ::_thesis: for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA st 1 <= s . aa & s . aa <= s . bb & s . bb <= len (s . f) & aa <> c & bb <> c & s . (intloc 0) = 1 holds
(IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . c = min_at ((s . f),(abs (s . aa)),(abs (s . bb)))
let f be FinSeq-Location ; ::_thesis: for p being Instruction-Sequence of SCM+FSA st 1 <= s . aa & s . aa <= s . bb & s . bb <= len (s . f) & aa <> c & bb <> c & s . (intloc 0) = 1 holds
(IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . c = min_at ((s . f),(abs (s . aa)),(abs (s . bb)))
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( 1 <= s . aa & s . aa <= s . bb & s . bb <= len (s . f) & aa <> c & bb <> c & s . (intloc 0) = 1 implies (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . c = min_at ((s . f),(abs (s . aa)),(abs (s . bb))) )
set a = aa;
set b = bb;
assume that
A1: 1 <= s . aa and
A2: s . aa <= s . bb and
A3: s . bb <= len (s . f) and
A4: aa <> c and
A5: bb <> c and
A6: s . (intloc 0) = 1 ; ::_thesis: (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . c = min_at ((s . f),(abs (s . aa)),(abs (s . bb)))
A7: ( bb = intloc 0 or bb is read-write ) by SCMFSA_M:def_2;
set i0 = c := aa;
set s1 = Exec ((c := aa),(Initialized s));
set p1 = p;
A8: ( aa = intloc 0 or aa is read-write ) by SCMFSA_M:def_2;
reconsider sa = abs (s . aa) as Element of NAT ;
A9: s . aa = sa by A1, ABSVALUE:def_1;
 (s . aa) - (s . aa) <= (s . bb) - (s . aa) by A2, XREAL_1:9;
then reconsider sba = (s . bb) - (s . aa) as Element of NAT by INT_1:3;
A10: (Exec ((c := aa),(Initialized s))) . f = (Initialized s) . f by SCMFSA_2:63
.= s . f by SCMFSA_M:37 ;
set k = sba + 1;
set cv = 3 -rdRWNotIn {aa,bb,c};
set aux2 = 2 -ndRWNotIn {aa,bb,c};
set aux1 = 1 -stRWNotIn {aa,bb,c};
A11: 1 -stRWNotIn {aa,bb,c} <> 2 -ndRWNotIn {aa,bb,c} by SCMFSA_M:26;
set I12 = if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA));
set i10 = (1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}));
A12: 2 -ndRWNotIn {aa,bb,c} <> 3 -rdRWNotIn {aa,bb,c} by SCMFSA_M:26;
set i11 = (2 -ndRWNotIn {aa,bb,c}) := (f,c);
A13: 1 -stRWNotIn {aa,bb,c} <> 3 -rdRWNotIn {aa,bb,c} by SCMFSA_M:26;
A14: c in {aa,bb,c} by ENUMSET1:def_1;
then A15: 3 -rdRWNotIn {aa,bb,c} <> c by SCMFSA_M:25;
set I1B = (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)));
set I1 = for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))));
set SF = StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))));
defpred S1[ Nat] means ( 0 < $1 & $1 <= sba + 1 implies ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . $1) . (intloc 0) = 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . $1) . (3 -rdRWNotIn {aa,bb,c}) = $1 + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . $1) . f = (Exec ((c := aa),(Initialized s))) . f & ex sa1 being Element of NAT st
( sa1 = ($1 + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . $1) . c = min_at ((s . f),sa,sa1) ) ) );
 3 -rdRWNotIn {aa,bb,c} in {(3 -rdRWNotIn {aa,bb,c}),aa,bb} by ENUMSET1:def_1;
then A16: 3 -rdRWNotIn {aa,bb,c} in {(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))) by XBOOLE_0:def_3;
A17: ProperForUpBody 3 -rdRWNotIn {aa,bb,c},aa,bb,(((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))), Exec ((c := aa),(Initialized s)),p by Th17;
A18: 1 -stRWNotIn {aa,bb,c} <> c by A14, SCMFSA_M:25;
A19: 2 -ndRWNotIn {aa,bb,c} <> c by A14, SCMFSA_M:25;
A20: (Exec ((c := aa),(Initialized s))) . c = (Initialized s) . aa by SCMFSA_2:63
.= s . aa by A6, A8, SCMFSA_M:9, SCMFSA_M:37 ;
A21: (Exec ((c := aa),(Initialized s))) . aa = (Initialized s) . aa by A4, SCMFSA_2:63
.= s . aa by A6, A8, SCMFSA_M:9, SCMFSA_M:37 ;
A22: s . aa <= len (s . f) by A2, A3, XXREAL_0:2;
then A23: sa in dom (s . f) by A1, A9, FINSEQ_3:25;
A24: (Exec ((c := aa),(Initialized s))) . bb = (Initialized s) . bb by A5, SCMFSA_2:63
.= s . bb by A6, A7, SCMFSA_M:9, SCMFSA_M:37 ;
A25: (Exec ((c := aa),(Initialized s))) . (intloc 0) = (Initialized s) . (intloc 0) by SCMFSA_2:63
.= 1 by SCMFSA_M:9 ;
then A26: DataPart (IExec ((for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))))),p,(Exec ((c := aa),(Initialized s))))) = DataPart ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (sba + 1)) by A21, A24, Th25;
 c in {c,(3 -rdRWNotIn {aa,bb,c})} by TARSKI:def_2;
then  c in UsedIntLoc (c := (3 -rdRWNotIn {aa,bb,c})) by SF_MASTR:14;
then  c in UsedIntLoc (Macro (c := (3 -rdRWNotIn {aa,bb,c}))) by SF_MASTR:28;
then  c in {(2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c})} \/ (UsedIntLoc (Macro (c := (3 -rdRWNotIn {aa,bb,c})))) by XBOOLE_0:def_3;
then  c in ({(2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c})} \/ (UsedIntLoc (Macro (c := (3 -rdRWNotIn {aa,bb,c}))))) \/ (UsedIntLoc (Stop SCM+FSA)) by XBOOLE_0:def_3;
then  c in UsedIntLoc (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))) by Th7;
then  c in (UsedIntLoc (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c)))) \/ (UsedIntLoc (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) by XBOOLE_0:def_3;
then A27: c in UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) by SF_MASTR:27;
then A28: c in {(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))) by XBOOLE_0:def_3;
A29: for n being Element of NAT st S1[n] holds
S1[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] )
assume that
A30: S1[n] and
 0 < n + 1 and
A31: n + 1 <= sba + 1 ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (intloc 0) = 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ex sa1 being Element of NAT st
( sa1 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) ) )
A32: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (intloc 0) = 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) = n + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) <= (Exec ((c := aa),(Initialized s))) . bb )
proof
percases ( 0 = n or 0 < n ) ;
supposeA33: 0 = n ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (intloc 0) = 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) = n + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) <= (Exec ((c := aa),(Initialized s))) . bb )
hence  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (intloc 0) = 1 by A25, Th10; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) = n + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) <= (Exec ((c := aa),(Initialized s))) . bb )
thus  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) = n + ((Exec ((c := aa),(Initialized s))) . aa) by A33, Th11; ::_thesis: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) <= (Exec ((c := aa),(Initialized s))) . bb
thus  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) <= (Exec ((c := aa),(Initialized s))) . bb by A2, A21, A24, A33, Th11; ::_thesis: verum
end;
supposeA34: 0 < n ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (intloc 0) = 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) = n + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) <= (Exec ((c := aa),(Initialized s))) . bb )
hence  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (intloc 0) = 1 by A30, A31, NAT_1:13; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) = n + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) <= (Exec ((c := aa),(Initialized s))) . bb )
thus  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) = n + ((Exec ((c := aa),(Initialized s))) . aa) by A30, A31, A34, NAT_1:13; ::_thesis: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) <= (Exec ((c := aa),(Initialized s))) . bb
 (n + 1) - 1 <= (((s . bb) - (s . aa)) + 1) - 1 by A31, XREAL_1:9;
hence  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) <= (Exec ((c := aa),(Initialized s))) . bb by A21, A24, A30, A34, NAT_1:13, XREAL_1:19; ::_thesis: verum
end;
end;
end;
set S0 = Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n);
set S1 = Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)));
set S2 = Exec (((2 -ndRWNotIn {aa,bb,c}) := (f,c)),(Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))));
A35: (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . f = (Exec (((2 -ndRWNotIn {aa,bb,c}) := (f,c)),(Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . f by SCMFSA6C:9
.= (Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . f by SCMFSA_2:72
.= (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . f by SCMFSA_2:72 ;
(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (intloc 0) = (Exec (((2 -ndRWNotIn {aa,bb,c}) := (f,c)),(Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . (intloc 0) by SCMFSA6C:8
.= (Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . (intloc 0) by SCMFSA_2:72
.= (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . (intloc 0) by SCMFSA_2:72
.= 1 by SCMFSA_M:9 ;
then A36: DataPart (IExec ((Stop SCM+FSA),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) = DataPart (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) by Th2;
 n < n + 1 by XREAL_1:29;
then  n < sba + 1 by A31, XXREAL_0:2;
then A37: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) | (({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))))) \/ FinSeq-Locations) = (IExec ((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) | (({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))))) \/ FinSeq-Locations) by A25, A21, A24, A17, Th21;
then A38: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (IExec ((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . f by SCMFSA_M:28
.= (Exec ((AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0))),(IExec (((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . f by SFMASTR1:12
.= (IExec (((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . f by SCMFSA_2:64
.= (IExec ((if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . f by SFMASTR1:8 ;
A39: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = (IExec ((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . c by A28, A37, SCMFSA_M:28
.= (Exec ((AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0))),(IExec (((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . c by SFMASTR1:11
.= (IExec (((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . c by A15, SCMFSA_2:64
.= (IExec ((if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . c by SFMASTR1:7 ;
A40: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (IExec ((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (3 -rdRWNotIn {aa,bb,c}) by A16, A37, SCMFSA_M:28
.= (Exec ((AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0))),(IExec (((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . (3 -rdRWNotIn {aa,bb,c}) by SFMASTR1:11
.= ((IExec (((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (3 -rdRWNotIn {aa,bb,c})) + ((IExec (((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (intloc 0)) by SCMFSA_2:64
.= ((IExec (((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (3 -rdRWNotIn {aa,bb,c})) + 1 by SCMFSA6B:11
.= ((IExec ((if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . (3 -rdRWNotIn {aa,bb,c})) + 1 by SFMASTR1:7 ;
A41: ( not Macro (c := (3 -rdRWNotIn {aa,bb,c})) refers 2 -ndRWNotIn {aa,bb,c} & not Stop SCM+FSA refers 2 -ndRWNotIn {aa,bb,c} ) by A12, Th3, Th4, SCMFSA8C:51;
A42: (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . c = (Exec (((2 -ndRWNotIn {aa,bb,c}) := (f,c)),(Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . c by SCMFSA6C:8
.= (Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . c by A19, SCMFSA_2:72
.= (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . c by A18, SCMFSA_2:72
.= ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . c by SCMFSA_M:37 ;
A43: (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (3 -rdRWNotIn {aa,bb,c}) = (Exec (((2 -ndRWNotIn {aa,bb,c}) := (f,c)),(Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . (3 -rdRWNotIn {aa,bb,c}) by SCMFSA6C:8
.= (Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . (3 -rdRWNotIn {aa,bb,c}) by A12, SCMFSA_2:72
.= (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . (3 -rdRWNotIn {aa,bb,c}) by A13, SCMFSA_2:72 ;
percases ( 0 = n or 0 < n ) ;
supposeA44: 0 = n ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (intloc 0) = 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ex sa1 being Element of NAT st
( sa1 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) ) )
thus  ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (intloc 0) = 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ex sa1 being Element of NAT st
( sa1 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) ) ) ::_thesis: verum
proof
reconsider sa1 = ((n + 1) + sa) - 1 as Element of NAT by A44;
A45: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . 0) . c = (Exec ((c := aa),(Initialized s))) . c by A15, A27, Th14;
A46: (Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . c = (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . c by A18, SCMFSA_2:72
.= s . aa by A20, A44, A45, SCMFSA_M:37 ;
thus  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (intloc 0) = 1 by A25, A21, A24, A17, A31, Th19; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ex sa1 being Element of NAT st
( sa1 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) ) )
A47: (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . f = ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . 0) . f by A44, SCMFSA_M:37
.= s . f by A10, Th15 ;
then A48: (Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . f = s . f by SCMFSA_2:72;
A49: (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . (3 -rdRWNotIn {aa,bb,c}) = ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . 0) . (3 -rdRWNotIn {aa,bb,c}) by A44, SCMFSA_M:37
.= s . aa by A21, Th11 ;
then reconsider S0cv = (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . (3 -rdRWNotIn {aa,bb,c}) as Element of NAT by A1, INT_1:3;
A50: (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (1 -stRWNotIn {aa,bb,c}) = (Exec (((2 -ndRWNotIn {aa,bb,c}) := (f,c)),(Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . (1 -stRWNotIn {aa,bb,c}) by SCMFSA6C:8
.= (Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . (1 -stRWNotIn {aa,bb,c}) by A11, SCMFSA_2:72
.= ((Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . f) /. (abs ((Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . (3 -rdRWNotIn {aa,bb,c}))) by Th5
.= (s . f) . S0cv by A9, A23, A47, A49, PARTFUN1:def_6 ;
A51: (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (2 -ndRWNotIn {aa,bb,c}) = (Exec (((2 -ndRWNotIn {aa,bb,c}) := (f,c)),(Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . (2 -ndRWNotIn {aa,bb,c}) by SCMFSA6C:8
.= ((Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . f) /. (abs ((Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . c)) by Th5
.= (s . f) . S0cv by A9, A23, A49, A48, A46, PARTFUN1:def_6 ;
hence ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = ((IExec ((Stop SCM+FSA),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . (3 -rdRWNotIn {aa,bb,c})) + 1 by A12, A41, A40, A50, SCMFSA8B:40
.= (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) by A21, A36, A43, A44, A49, SCMFSA_M:2 ;
::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ex sa1 being Element of NAT st
( sa1 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) ) )
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (IExec ((Stop SCM+FSA),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . f by A41, A38, A50, A51, SCMFSA8B:40
.= (Exec ((c := aa),(Initialized s))) . f by A10, A36, A35, A47, SCMFSA_M:2 ; ::_thesis: ex sa1 being Element of NAT st
( sa1 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) )
take  sa1 ; ::_thesis: ( sa1 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) )
thus  sa1 = ((n + 1) + sa) - 1 ; ::_thesis: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1)
((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = (IExec ((Stop SCM+FSA),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . c by A19, A41, A39, A50, A51, SCMFSA8B:40
.= s . aa by A20, A36, A42, A44, A45, SCMFSA_M:2 ;
hence  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) by A1, A22, A9, A44, GRAPH_2:60; ::_thesis: verum
end;
end;
supposeA52: 0 < n ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (intloc 0) = 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ex sa1 being Element of NAT st
( sa1 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) ) )
thus  ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (intloc 0) = 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ex sa1 being Element of NAT st
( sa1 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) ) ) ::_thesis: verum
proof
A53: (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . (3 -rdRWNotIn {aa,bb,c}) = ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . (3 -rdRWNotIn {aa,bb,c}) by SCMFSA_M:37;
then reconsider S0cv = (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . (3 -rdRWNotIn {aa,bb,c}) as Element of NAT by A1, A21, A32, INT_1:3;
 ( 1 <= S0cv & S0cv <= len (s . f) ) by A1, A3, A9, A21, A24, A32, A53, NAT_1:12, XXREAL_0:2;
then A54: S0cv in dom (s . f) by FINSEQ_3:25;
A55: (Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . f = s . f by A10, A30, A31, A52, NAT_1:13, SCMFSA_M:37;
then A56: (Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . f = s . f by SCMFSA_2:72;
A57: (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (1 -stRWNotIn {aa,bb,c}) = (Exec (((2 -ndRWNotIn {aa,bb,c}) := (f,c)),(Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . (1 -stRWNotIn {aa,bb,c}) by SCMFSA6C:8
.= (Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . (1 -stRWNotIn {aa,bb,c}) by A11, SCMFSA_2:72
.= ((Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . f) /. (abs ((Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . (3 -rdRWNotIn {aa,bb,c}))) by Th5
.= ((Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . f) /. S0cv by ABSVALUE:def_1
.= (s . f) . S0cv by A55, A54, PARTFUN1:def_6 ;
 n + (s . aa) <= len (s . f) by A3, A21, A24, A32, XXREAL_0:2;
then  (n + (s . aa)) - 1 <= (len (s . f)) - 1 by XREAL_1:9;
then A58: ((n + (s . aa)) - 1) + 0 <= ((len (s . f)) - 1) + 1 by XREAL_1:7;
thus  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (intloc 0) = 1 by A25, A21, A24, A17, A31, Th19; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ex sa1 being Element of NAT st
( sa1 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) ) )
consider sa1 being Element of NAT  such that
A59: sa1 = (n + sa) - 1 and
A60: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . c = min_at ((s . f),sa,sa1) by A30, A31, A52, NAT_1:13;
reconsider SFnc = ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . c as Element of NAT by A60;
 0 + 1 <= n by A52, NAT_1:13;
then  1 - 1 <= n - 1 by XREAL_1:9;
then A61: 0 + (s . aa) <= (n - 1) + (s . aa) by XREAL_1:6;
then A62: sa <= SFnc by A1, A9, A59, A60, A58, GRAPH_2:59;
then A63: 1 <= SFnc by A1, A9, XXREAL_0:2;
A64: SFnc <= sa1 by A1, A9, A59, A60, A61, A58, GRAPH_2:59;
then  SFnc <= len (s . f) by A9, A59, A58, XXREAL_0:2;
then A65: SFnc in dom (s . f) by A63, FINSEQ_3:25;
A66: (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (2 -ndRWNotIn {aa,bb,c}) = (Exec (((2 -ndRWNotIn {aa,bb,c}) := (f,c)),(Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . (2 -ndRWNotIn {aa,bb,c}) by SCMFSA6C:8
.= ((Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . f) /. (abs ((Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . c)) by Th5
.= ((Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . f) /. (abs ((Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)) . c)) by A18, SCMFSA_2:72
.= ((Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . f) /. (abs (((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . c)) by SCMFSA_M:37
.= ((Exec (((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))),(Initialized ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . f) /. SFnc by ABSVALUE:def_1
.= (s . f) . SFnc by A56, A65, PARTFUN1:def_6 ;
A67: for i being Element of NAT st sa <= i & i < ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . c holds
(s . f) . i > (s . f) . SFnc by A1, A9, A59, A60, A61, A58, GRAPH_2:59;
thus  ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ex sa1 being Element of NAT st
( sa1 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) ) ) ::_thesis: verum
proof
A68: ((n + 1) + (s . aa)) - 1 <= len (s . f) by A3, A21, A24, A32, XXREAL_0:2;
A69: ((n + 1) + (s . aa)) - 1 = ((n + (s . aa)) + 1) - 1
.= n + sa by A1, ABSVALUE:def_1 ;
then A70: s . aa <= ((n + 1) + (s . aa)) - 1 by NAT_1:12;
percases ( (s . f) . S0cv < (s . f) . SFnc or (s . f) . SFnc <= (s . f) . S0cv ) ;
supposeA71: (s . f) . S0cv < (s . f) . SFnc ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ex sa1 being Element of NAT st
( sa1 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) ) )
A72: (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . (3 -rdRWNotIn {aa,bb,c}) = (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . (3 -rdRWNotIn {aa,bb,c})
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . (3 -rdRWNotIn {aa,bb,c}) by SCMFSA6C:5
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . (3 -rdRWNotIn {aa,bb,c})
.= (Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . (3 -rdRWNotIn {aa,bb,c}) by A15, SCMFSA_2:63 ;
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = ((IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . (3 -rdRWNotIn {aa,bb,c})) + 1 by A12, A41, A40, A57, A66, A71, SCMFSA8B:40
.= ((Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . (3 -rdRWNotIn {aa,bb,c})) + 1 by A72
.= ((IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (3 -rdRWNotIn {aa,bb,c})) + 1 by SCMFSA_M:37
.= (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) by A32, A43, A53 ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ex sa1 being Element of NAT st
( sa1 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) ) )
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . f by A41, A38, A57, A66, A71, SCMFSA8B:40
.= (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . f
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . f by SCMFSA6C:5
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . f
.= (Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . f by SCMFSA_2:63
.= (Exec ((c := aa),(Initialized s))) . f by A10, A35, A55, SCMFSA_M:37 ; ::_thesis: ex sa1 being Element of NAT st
( sa1 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) )
reconsider sa11 = ((n + 1) + sa) - 1 as Element of NAT by A69;
take  sa11 ; ::_thesis: ( sa11 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa11) )
thus  sa11 = ((n + 1) + sa) - 1 ; ::_thesis: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa11)
A73: for i being Element of NAT st s . aa <= i & i <= ((n + 1) + (s . aa)) - 1 holds
(s . f) . S0cv <= (s . f) . i
proof
let i be Element of NAT ; ::_thesis: ( s . aa <= i & i <= ((n + 1) + (s . aa)) - 1 implies (s . f) . S0cv <= (s . f) . i )
assume that
A74: s . aa <= i and
A75: i <= ((n + 1) + (s . aa)) - 1 ; ::_thesis: (s . f) . S0cv <= (s . f) . i
percases ( i < ((n + 1) + (s . aa)) - 1 or i = ((n + 1) + (s . aa)) - 1 ) by A75, XXREAL_0:1;
suppose i < ((n + 1) + (s . aa)) - 1 ; ::_thesis: (s . f) . S0cv <= (s . f) . i
then  i + 1 <= n + (s . aa) by A69, NAT_1:13;
then  (i + 1) - 1 <= (n + (s . aa)) - 1 by XREAL_1:9;
then  (s . f) . SFnc <= (s . f) . i by A1, A9, A59, A60, A61, A58, A74, GRAPH_2:59;
hence  (s . f) . S0cv <= (s . f) . i by A71, XXREAL_0:2; ::_thesis: verum
end;
suppose i = ((n + 1) + (s . aa)) - 1 ; ::_thesis: (s . f) . S0cv <= (s . f) . i
hence  (s . f) . S0cv <= (s . f) . i by A21, A32, SCMFSA_M:37; ::_thesis: verum
end;
end;
end;
A76: for i being Element of NAT st s . aa <= i & i < S0cv holds
(s . f) . i > (s . f) . S0cv
proof
let i be Element of NAT ; ::_thesis: ( s . aa <= i & i < S0cv implies (s . f) . i > (s . f) . S0cv )
assume that
A77: s . aa <= i and
A78: i < S0cv ; ::_thesis: (s . f) . i > (s . f) . S0cv
 i + 1 <= S0cv by A78, NAT_1:13;
then  (i + 1) - 1 <= S0cv - 1 by XREAL_1:9;
then  (s . f) . SFnc <= (s . f) . i by A1, A9, A21, A32, A53, A59, A60, A61, A58, A77, GRAPH_2:59;
hence  (s . f) . i > (s . f) . S0cv by A71, XXREAL_0:2; ::_thesis: verum
end;
((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . c by A19, A41, A39, A57, A66, A71, SCMFSA8B:40
.= (IExec ((Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . c
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . c by SCMFSA6C:5
.= (Exec ((c := (3 -rdRWNotIn {aa,bb,c})),(Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))))) . c
.= (Initialized (IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n)))) . (3 -rdRWNotIn {aa,bb,c}) by SCMFSA_2:63
.= S0cv by A43, SCMFSA_M:37 ;
hence  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa11) by A1, A9, A21, A32, A53, A70, A68, A73, A76, GRAPH_2:59; ::_thesis: verum
end;
supposeA79: (s . f) . SFnc <= (s . f) . S0cv ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ex sa1 being Element of NAT st
( sa1 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) ) )
thus  ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ex sa1 being Element of NAT st
( sa1 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) ) ) ::_thesis: verum
proof
A80: for i being Element of NAT st s . aa <= i & i <= ((n + 1) + (s . aa)) - 1 holds
(s . f) . SFnc <= (s . f) . i
proof
let i be Element of NAT ; ::_thesis: ( s . aa <= i & i <= ((n + 1) + (s . aa)) - 1 implies (s . f) . SFnc <= (s . f) . i )
assume that
A81: s . aa <= i and
A82: i <= ((n + 1) + (s . aa)) - 1 ; ::_thesis: (s . f) . SFnc <= (s . f) . i
percases ( i < ((n + 1) + (s . aa)) - 1 or i = ((n + 1) + (s . aa)) - 1 ) by A82, XXREAL_0:1;
suppose i < ((n + 1) + (s . aa)) - 1 ; ::_thesis: (s . f) . SFnc <= (s . f) . i
then  i + 1 <= n + (s . aa) by A69, NAT_1:13;
then  (i + 1) - 1 <= (n + (s . aa)) - 1 by XREAL_1:9;
hence  (s . f) . SFnc <= (s . f) . i by A1, A9, A59, A60, A61, A58, A81, GRAPH_2:59; ::_thesis: verum
end;
suppose i = ((n + 1) + (s . aa)) - 1 ; ::_thesis: (s . f) . SFnc <= (s . f) . i
hence  (s . f) . SFnc <= (s . f) . i by A21, A32, A79, SCMFSA_M:37; ::_thesis: verum
end;
end;
end;
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . (3 -rdRWNotIn {aa,bb,c}) = ((IExec ((Stop SCM+FSA),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . (3 -rdRWNotIn {aa,bb,c})) + 1 by A12, A41, A40, A57, A66, A79, SCMFSA8B:40
.= ((IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))) . (3 -rdRWNotIn {aa,bb,c})) + 1 by A36, SCMFSA_M:2
.= (n + 1) + ((Exec ((c := aa),(Initialized s))) . aa) by A32, A43, A53 ; ::_thesis: ( ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (Exec ((c := aa),(Initialized s))) . f & ex sa1 being Element of NAT st
( sa1 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) ) )
thus ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . f = (IExec ((Stop SCM+FSA),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . f by A41, A38, A57, A66, A79, SCMFSA8B:40
.= (Exec ((c := aa),(Initialized s))) . f by A10, A36, A35, A55, SCMFSA_M:2 ; ::_thesis: ex sa1 being Element of NAT st
( sa1 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa1) )
reconsider sa11 = ((n + 1) + sa) - 1 as Element of NAT by A69;
take  sa11 ; ::_thesis: ( sa11 = ((n + 1) + sa) - 1 & ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa11) )
thus  sa11 = ((n + 1) + sa) - 1 ; ::_thesis: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa11)
 (n + (s . aa)) - 1 <= ((n + (s . aa)) - 1) + 1 by XREAL_1:29;
then A83: SFnc <= ((n + 1) + (s . aa)) - 1 by A9, A59, A64, XXREAL_0:2;
((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = (IExec ((Stop SCM+FSA),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),(IExec ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))),(p +* (while>0 ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),((((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))) ";" (AddTo ((3 -rdRWNotIn {aa,bb,c}),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(3 -rdRWNotIn {aa,bb,c}),aa,bb} \/ (UsedIntLoc ((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))))),(intloc 0))))))),((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n))))) . c by A19, A41, A39, A57, A66, A79, SCMFSA8B:40
.= ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . n) . c by A36, A42, SCMFSA_M:2 ;
hence  ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (n + 1)) . c = min_at ((s . f),sa,sa11) by A1, A9, A62, A67, A70, A68, A83, A80, GRAPH_2:59; ::_thesis: verum
end;
end;
end;
end;
end;
end;
end;
end;
reconsider sb = abs (s . bb) as Element of NAT ;
A84: S1[ 0 ] ;
 for n being Element of NAT holds S1[n] from NAT_1:sch_1(A84, A29);
then consider sab being Element of NAT  such that
A85: sab = ((sba + 1) + sa) - 1 and
A86: ((StepForUp ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))),p,(Exec ((c := aa),(Initialized s))))) . (sba + 1)) . c = min_at ((s . f),sa,sab) ;
A87: sab = sb by A9, A85, ABSVALUE:def_1;
 ( for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))) is_closed_on Exec ((c := aa),(Initialized s)),p & for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA))))) is_halting_on Exec ((c := aa),(Initialized s)),p ) by A25, Th26;
hence (IExec ((FinSeqMin (f,aa,bb,c)),p,s)) . c = (IExec ((for-up ((3 -rdRWNotIn {aa,bb,c}),aa,bb,((((1 -stRWNotIn {aa,bb,c}) := (f,(3 -rdRWNotIn {aa,bb,c}))) ";" ((2 -ndRWNotIn {aa,bb,c}) := (f,c))) ";" (if>0 ((2 -ndRWNotIn {aa,bb,c}),(1 -stRWNotIn {aa,bb,c}),(Macro (c := (3 -rdRWNotIn {aa,bb,c}))),(Stop SCM+FSA)))))),p,(Exec ((c := aa),(Initialized s))))) . c by SFMASTR1:14
.= min_at ((s . f),(abs (s . aa)),(abs (s . bb))) by A26, A86, A87, SCMFSA_M:2 ;
::_thesis: verum
end;

begin

definition
let f be FinSeq-Location ;
let a, b be Int-Location;
func swap (f,a,b) ->  Program of  equals :: SFMASTR3:def 5
((((1 -stRWNotIn {a,b}) := (f,a)) ";" ((2 -ndRWNotIn {a,b}) := (f,b))) ";" ((f,a) := (2 -ndRWNotIn {a,b}))) ";" ((f,b) := (1 -stRWNotIn {a,b}));
coherence
 ((((1 -stRWNotIn {a,b}) := (f,a)) ";" ((2 -ndRWNotIn {a,b}) := (f,b))) ";" ((f,a) := (2 -ndRWNotIn {a,b}))) ";" ((f,b) := (1 -stRWNotIn {a,b})) is Program of ;
end;

:: deftheorem  defines swap SFMASTR3:def_5_:_
 for f being FinSeq-Location
for a, b being Int-Location holds swap (f,a,b) = ((((1 -stRWNotIn {a,b}) := (f,a)) ";" ((2 -ndRWNotIn {a,b}) := (f,b))) ";" ((f,a) := (2 -ndRWNotIn {a,b}))) ";" ((f,b) := (1 -stRWNotIn {a,b}));

registration
let f be FinSeq-Location ;
let a, b be Int-Location;
cluster swap (f,a,b) ->  good parahalting ;
coherence
 ( swap (f,a,b) is good & swap (f,a,b) is parahalting ) ;
end;

theorem Th32: :: SFMASTR3:32
for cc, aa, bb being Int-Location
for f being FinSeq-Location st cc <> 1 -stRWNotIn {aa,bb} & cc <> 2 -ndRWNotIn {aa,bb} holds
not swap (f,aa,bb) destroys cc
proof
let cc, aa, bb be Int-Location; ::_thesis: for f being FinSeq-Location st cc <> 1 -stRWNotIn {aa,bb} & cc <> 2 -ndRWNotIn {aa,bb} holds
not swap (f,aa,bb) destroys cc
let f be FinSeq-Location ; ::_thesis: ( cc <> 1 -stRWNotIn {aa,bb} & cc <> 2 -ndRWNotIn {aa,bb} implies not swap (f,aa,bb) destroys cc )
set a = aa;
set b = bb;
set aux1 = 1 -stRWNotIn {aa,bb};
set aux2 = 2 -ndRWNotIn {aa,bb};
assume  ( cc <> 1 -stRWNotIn {aa,bb} & cc <> 2 -ndRWNotIn {aa,bb} ) ; ::_thesis: not swap (f,aa,bb) destroys cc
then A1: ( not (1 -stRWNotIn {aa,bb}) := (f,aa) destroys cc & not (2 -ndRWNotIn {aa,bb}) := (f,bb) destroys cc ) by SCMFSA7B:14;
 not (f,aa) := (2 -ndRWNotIn {aa,bb}) destroys cc by SCMFSA7B:15;
then  not (((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))) ";" ((f,aa) := (2 -ndRWNotIn {aa,bb})) destroys cc by A1, SCMFSA8C:54, SCMFSA8C:55;
hence  not swap (f,aa,bb) destroys cc by SCMFSA7B:15, SCMFSA8C:54; ::_thesis: verum
end;

theorem Th33: :: SFMASTR3:33
for s being State of SCM+FSA
for aa, bb being Int-Location
for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA st 1 <= s . aa & s . aa <= len (s . f) & 1 <= s . bb & s . bb <= len (s . f) & s . (intloc 0) = 1 holds
(IExec ((swap (f,aa,bb)),p,s)) . f = ((s . f) +* ((s . aa),((s . f) . (s . bb)))) +* ((s . bb),((s . f) . (s . aa)))
proof
let s be State of SCM+FSA; ::_thesis: for aa, bb being Int-Location
for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA st 1 <= s . aa & s . aa <= len (s . f) & 1 <= s . bb & s . bb <= len (s . f) & s . (intloc 0) = 1 holds
(IExec ((swap (f,aa,bb)),p,s)) . f = ((s . f) +* ((s . aa),((s . f) . (s . bb)))) +* ((s . bb),((s . f) . (s . aa)))
let aa, bb be Int-Location; ::_thesis: for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA st 1 <= s . aa & s . aa <= len (s . f) & 1 <= s . bb & s . bb <= len (s . f) & s . (intloc 0) = 1 holds
(IExec ((swap (f,aa,bb)),p,s)) . f = ((s . f) +* ((s . aa),((s . f) . (s . bb)))) +* ((s . bb),((s . f) . (s . aa)))
let f be FinSeq-Location ; ::_thesis: for p being Instruction-Sequence of SCM+FSA st 1 <= s . aa & s . aa <= len (s . f) & 1 <= s . bb & s . bb <= len (s . f) & s . (intloc 0) = 1 holds
(IExec ((swap (f,aa,bb)),p,s)) . f = ((s . f) +* ((s . aa),((s . f) . (s . bb)))) +* ((s . bb),((s . f) . (s . aa)))
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( 1 <= s . aa & s . aa <= len (s . f) & 1 <= s . bb & s . bb <= len (s . f) & s . (intloc 0) = 1 implies (IExec ((swap (f,aa,bb)),p,s)) . f = ((s . f) +* ((s . aa),((s . f) . (s . bb)))) +* ((s . bb),((s . f) . (s . aa))) )
set a = aa;
set b = bb;
assume that
A1: 1 <= s . aa and
A2: s . aa <= len (s . f) and
A3: 1 <= s . bb and
A4: s . bb <= len (s . f) and
A5: s . (intloc 0) = 1 ; ::_thesis: (IExec ((swap (f,aa,bb)),p,s)) . f = ((s . f) +* ((s . aa),((s . f) . (s . bb)))) +* ((s . bb),((s . f) . (s . aa)))
set aux1 = 1 -stRWNotIn {aa,bb};
set aux2 = 2 -ndRWNotIn {aa,bb};
set i0 = (1 -stRWNotIn {aa,bb}) := (f,aa);
set i1 = (2 -ndRWNotIn {aa,bb}) := (f,bb);
set i2 = (f,aa) := (2 -ndRWNotIn {aa,bb});
set s0 = Initialized s;
set s1 = Exec (((1 -stRWNotIn {aa,bb}) := (f,aa)),(Initialized s));
set s2 = IExec ((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))),p,s);
A6: ( bb = intloc 0 or bb is read-write ) by SCMFSA_M:def_2;
reconsider sa = s . aa as Element of NAT by A1, INT_1:3;
set s0a = abs ((Initialized s) . aa);
set s2a = abs ((IExec ((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))),p,s)) . aa);
A7: ( aa = intloc 0 or aa is read-write ) by SCMFSA_M:def_2;
A8: sa = abs (s . aa) by ABSVALUE:def_1;
then A9: ( (Initialized s) . f = s . f & sa = abs ((Initialized s) . aa) ) by A5, A7, SCMFSA_M:9, SCMFSA_M:37;
reconsider sb = s . bb as Element of NAT by A3, INT_1:3;
A10: sb = abs (s . bb) by ABSVALUE:def_1;
set s3 = IExec (((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))) ";" ((f,aa) := (2 -ndRWNotIn {aa,bb}))),p,s);
A11: 1 -stRWNotIn {aa,bb} <> 2 -ndRWNotIn {aa,bb} by SCMFSA_M:26;
A12: (IExec (((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))) ";" ((f,aa) := (2 -ndRWNotIn {aa,bb}))),p,s)) . (1 -stRWNotIn {aa,bb}) = (Exec (((f,aa) := (2 -ndRWNotIn {aa,bb})),(IExec ((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))),p,s)))) . (1 -stRWNotIn {aa,bb}) by SCMFSA6C:6
.= (IExec ((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))),p,s)) . (1 -stRWNotIn {aa,bb}) by SCMFSA_2:73
.= (Exec (((2 -ndRWNotIn {aa,bb}) := (f,bb)),(Exec (((1 -stRWNotIn {aa,bb}) := (f,aa)),(Initialized s))))) . (1 -stRWNotIn {aa,bb}) by SCMFSA6C:8
.= (Exec (((1 -stRWNotIn {aa,bb}) := (f,aa)),(Initialized s))) . (1 -stRWNotIn {aa,bb}) by A11, SCMFSA_2:72
.= ((Initialized s) . f) /. (abs ((Initialized s) . aa)) by Th5
.= (s . f) . sa by A1, A2, A9, FINSEQ_4:15 ;
set i3 = (f,bb) := (1 -stRWNotIn {aa,bb});
A13: (IExec ((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))),p,s)) . f = (Exec (((2 -ndRWNotIn {aa,bb}) := (f,bb)),(Exec (((1 -stRWNotIn {aa,bb}) := (f,aa)),(Initialized s))))) . f by SCMFSA6C:9
.= (Exec (((1 -stRWNotIn {aa,bb}) := (f,aa)),(Initialized s))) . f by SCMFSA_2:72 ;
A14: (IExec (((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))) ";" ((f,aa) := (2 -ndRWNotIn {aa,bb}))),p,s)) . f = (Exec (((f,aa) := (2 -ndRWNotIn {aa,bb})),(IExec ((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))),p,s)))) . f by SCMFSA6C:7
.= ((IExec ((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))),p,s)) . f) +* ((abs ((IExec ((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))),p,s)) . aa)),((IExec ((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))),p,s)) . (2 -ndRWNotIn {aa,bb}))) by Th6 ;
A15: bb in {aa,bb} by TARSKI:def_2;
then A16: bb <> 2 -ndRWNotIn {aa,bb} by SCMFSA_M:25;
 bb <> 1 -stRWNotIn {aa,bb} by A15, SCMFSA_M:25;
then A17: (Exec (((1 -stRWNotIn {aa,bb}) := (f,aa)),(Initialized s))) . bb = (Initialized s) . bb by SCMFSA_2:72
.= s . bb by A5, A6, SCMFSA_M:9, SCMFSA_M:37 ;
A18: aa in {aa,bb} by TARSKI:def_2;
then A19: aa <> 2 -ndRWNotIn {aa,bb} by SCMFSA_M:25;
A20: aa <> 1 -stRWNotIn {aa,bb} by A18, SCMFSA_M:25;
(IExec ((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))),p,s)) . aa = (Exec (((2 -ndRWNotIn {aa,bb}) := (f,bb)),(Exec (((1 -stRWNotIn {aa,bb}) := (f,aa)),(Initialized s))))) . aa by SCMFSA6C:8
.= (Exec (((1 -stRWNotIn {aa,bb}) := (f,aa)),(Initialized s))) . aa by A19, SCMFSA_2:72
.= (Initialized s) . aa by A20, SCMFSA_2:72 ;
then A21: sa = abs ((IExec ((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))),p,s)) . aa) by A5, A8, A7, SCMFSA_M:9, SCMFSA_M:37;
set s1b = abs ((Exec (((1 -stRWNotIn {aa,bb}) := (f,aa)),(Initialized s))) . bb);
A22: (Exec (((1 -stRWNotIn {aa,bb}) := (f,aa)),(Initialized s))) . f = (Initialized s) . f by SCMFSA_2:72
.= s . f by SCMFSA_M:37 ;
A23: (IExec (((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))) ";" ((f,aa) := (2 -ndRWNotIn {aa,bb}))),p,s)) . bb = (Exec (((f,aa) := (2 -ndRWNotIn {aa,bb})),(IExec ((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))),p,s)))) . bb by SCMFSA6C:6
.= (IExec ((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))),p,s)) . bb by SCMFSA_2:73
.= (Exec (((2 -ndRWNotIn {aa,bb}) := (f,bb)),(Exec (((1 -stRWNotIn {aa,bb}) := (f,aa)),(Initialized s))))) . bb by SCMFSA6C:8
.= (Exec (((1 -stRWNotIn {aa,bb}) := (f,aa)),(Initialized s))) . bb by A16, SCMFSA_2:72 ;
A24: (IExec ((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))),p,s)) . (2 -ndRWNotIn {aa,bb}) = (Exec (((2 -ndRWNotIn {aa,bb}) := (f,bb)),(Exec (((1 -stRWNotIn {aa,bb}) := (f,aa)),(Initialized s))))) . (2 -ndRWNotIn {aa,bb}) by SCMFSA6C:8
.= ((Exec (((1 -stRWNotIn {aa,bb}) := (f,aa)),(Initialized s))) . f) /. (abs ((Exec (((1 -stRWNotIn {aa,bb}) := (f,aa)),(Initialized s))) . bb)) by Th5
.= (s . f) . sb by A3, A4, A10, A22, A17, FINSEQ_4:15 ;
thus (IExec ((swap (f,aa,bb)),p,s)) . f = (Exec (((f,bb) := (1 -stRWNotIn {aa,bb})),(IExec (((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))) ";" ((f,aa) := (2 -ndRWNotIn {aa,bb}))),p,s)))) . f by SCMFSA6C:7
.= ((s . f) +* ((s . aa),((s . f) . (s . bb)))) +* ((s . bb),((s . f) . (s . aa))) by A10, A22, A13, A14, A21, A17, A23, A24, A12, Th6 ; ::_thesis: verum
end;

theorem :: SFMASTR3:34
for s being State of SCM+FSA
for aa, bb being Int-Location
for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA st 1 <= s . aa & s . aa <= len (s . f) & 1 <= s . bb & s . bb <= len (s . f) & s . (intloc 0) = 1 holds
( ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . aa) = (s . f) . (s . bb) & ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . bb) = (s . f) . (s . aa) )
proof
let s be State of SCM+FSA; ::_thesis: for aa, bb being Int-Location
for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA st 1 <= s . aa & s . aa <= len (s . f) & 1 <= s . bb & s . bb <= len (s . f) & s . (intloc 0) = 1 holds
( ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . aa) = (s . f) . (s . bb) & ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . bb) = (s . f) . (s . aa) )
let aa, bb be Int-Location; ::_thesis: for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA st 1 <= s . aa & s . aa <= len (s . f) & 1 <= s . bb & s . bb <= len (s . f) & s . (intloc 0) = 1 holds
( ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . aa) = (s . f) . (s . bb) & ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . bb) = (s . f) . (s . aa) )
let f be FinSeq-Location ; ::_thesis: for p being Instruction-Sequence of SCM+FSA st 1 <= s . aa & s . aa <= len (s . f) & 1 <= s . bb & s . bb <= len (s . f) & s . (intloc 0) = 1 holds
( ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . aa) = (s . f) . (s . bb) & ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . bb) = (s . f) . (s . aa) )
let p be Instruction-Sequence of SCM+FSA; ::_thesis: ( 1 <= s . aa & s . aa <= len (s . f) & 1 <= s . bb & s . bb <= len (s . f) & s . (intloc 0) = 1 implies ( ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . aa) = (s . f) . (s . bb) & ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . bb) = (s . f) . (s . aa) ) )
set a = aa;
set b = bb;
assume that
A1: 1 <= s . aa and
A2: s . aa <= len (s . f) and
A3: 1 <= s . bb and
A4: s . bb <= len (s . f) and
A5: s . (intloc 0) = 1 ; ::_thesis: ( ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . aa) = (s . f) . (s . bb) & ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . bb) = (s . f) . (s . aa) )
A6: (IExec ((swap (f,aa,bb)),p,s)) . f = ((s . f) +* ((s . aa),((s . f) . (s . bb)))) +* ((s . bb),((s . f) . (s . aa))) by A1, A2, A3, A4, A5, Th33;
reconsider sa = s . aa as Element of NAT by A1, INT_1:3;
A7: sa in dom (s . f) by A1, A2, FINSEQ_3:25;
A8: dom ((s . f) +* ((s . aa),((s . f) . (s . bb)))) = dom (s . f) by FUNCT_7:30;
reconsider sb = s . bb as Element of NAT by A3, INT_1:3;
A9: sb in dom (s . f) by A3, A4, FINSEQ_3:25;
percases ( sa <> sb or sa = sb ) ;
suppose sa <> sb ; ::_thesis: ( ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . aa) = (s . f) . (s . bb) & ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . bb) = (s . f) . (s . aa) )
hence ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . aa) = ((s . f) +* ((s . aa),((s . f) . (s . bb)))) . (s . aa) by A6, FUNCT_7:32
.= (s . f) . (s . bb) by A7, FUNCT_7:31 ;
::_thesis: ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . bb) = (s . f) . (s . aa)
thus  ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . bb) = (s . f) . (s . aa) by A9, A6, A8, FUNCT_7:31; ::_thesis: verum
end;
suppose sa = sb ; ::_thesis: ( ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . aa) = (s . f) . (s . bb) & ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . bb) = (s . f) . (s . aa) )
hence  ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . aa) = (s . f) . (s . bb) by A7, A6, A8, FUNCT_7:31; ::_thesis: ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . bb) = (s . f) . (s . aa)
thus  ((IExec ((swap (f,aa,bb)),p,s)) . f) . (s . bb) = (s . f) . (s . aa) by A9, A6, A8, FUNCT_7:31; ::_thesis: verum
end;
end;
end;

theorem Th35: :: SFMASTR3:35
for aa, bb being Int-Location
for f being FinSeq-Location holds {aa,bb} c= UsedIntLoc (swap (f,aa,bb))
proof
let aa, bb be Int-Location; ::_thesis: for f being FinSeq-Location holds {aa,bb} c= UsedIntLoc (swap (f,aa,bb))
let f be FinSeq-Location ; ::_thesis: {aa,bb} c= UsedIntLoc (swap (f,aa,bb))
set a = aa;
set b = bb;
set aux1 = 1 -stRWNotIn {aa,bb};
set aux2 = 2 -ndRWNotIn {aa,bb};
set i0 = (1 -stRWNotIn {aa,bb}) := (f,aa);
set i1 = (2 -ndRWNotIn {aa,bb}) := (f,bb);
set i2 = (f,aa) := (2 -ndRWNotIn {aa,bb});
set i3 = (f,bb) := (1 -stRWNotIn {aa,bb});
A1: UsedIntLoc (swap (f,aa,bb)) = (UsedIntLoc ((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))) ";" ((f,aa) := (2 -ndRWNotIn {aa,bb})))) \/ (UsedIntLoc ((f,bb) := (1 -stRWNotIn {aa,bb}))) by SF_MASTR:30
.= ((UsedIntLoc (((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb)))) \/ (UsedIntLoc ((f,aa) := (2 -ndRWNotIn {aa,bb})))) \/ (UsedIntLoc ((f,bb) := (1 -stRWNotIn {aa,bb}))) by SF_MASTR:30
.= (((UsedIntLoc ((1 -stRWNotIn {aa,bb}) := (f,aa))) \/ (UsedIntLoc ((2 -ndRWNotIn {aa,bb}) := (f,bb)))) \/ (UsedIntLoc ((f,aa) := (2 -ndRWNotIn {aa,bb})))) \/ (UsedIntLoc ((f,bb) := (1 -stRWNotIn {aa,bb}))) by SF_MASTR:31
.= (({(1 -stRWNotIn {aa,bb}),aa} \/ (UsedIntLoc ((2 -ndRWNotIn {aa,bb}) := (f,bb)))) \/ (UsedIntLoc ((f,aa) := (2 -ndRWNotIn {aa,bb})))) \/ (UsedIntLoc ((f,bb) := (1 -stRWNotIn {aa,bb}))) by SF_MASTR:17
.= (({(1 -stRWNotIn {aa,bb}),aa} \/ {(2 -ndRWNotIn {aa,bb}),bb}) \/ (UsedIntLoc ((f,aa) := (2 -ndRWNotIn {aa,bb})))) \/ (UsedIntLoc ((f,bb) := (1 -stRWNotIn {aa,bb}))) by SF_MASTR:17
.= (({(1 -stRWNotIn {aa,bb}),aa} \/ {(2 -ndRWNotIn {aa,bb}),bb}) \/ {(2 -ndRWNotIn {aa,bb}),aa}) \/ (UsedIntLoc ((f,bb) := (1 -stRWNotIn {aa,bb}))) by SF_MASTR:17
.= (({(1 -stRWNotIn {aa,bb}),aa} \/ {(2 -ndRWNotIn {aa,bb}),bb}) \/ {(2 -ndRWNotIn {aa,bb}),aa}) \/ {(1 -stRWNotIn {aa,bb}),bb} by SF_MASTR:17 ;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {aa,bb} or x in UsedIntLoc (swap (f,aa,bb)) )
assume A2: x in {aa,bb} ; ::_thesis: x in UsedIntLoc (swap (f,aa,bb))
percases ( x = aa or x = bb ) by A2, TARSKI:def_2;
suppose x = aa ; ::_thesis: x in UsedIntLoc (swap (f,aa,bb))
then  x in {(2 -ndRWNotIn {aa,bb}),aa} by TARSKI:def_2;
then  x in ({(1 -stRWNotIn {aa,bb}),aa} \/ {(2 -ndRWNotIn {aa,bb}),bb}) \/ {(2 -ndRWNotIn {aa,bb}),aa} by XBOOLE_0:def_3;
hence  x in UsedIntLoc (swap (f,aa,bb)) by A1, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose x = bb ; ::_thesis: x in UsedIntLoc (swap (f,aa,bb))
then  x in {(1 -stRWNotIn {aa,bb}),bb} by TARSKI:def_2;
hence  x in UsedIntLoc (swap (f,aa,bb)) by A1, XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;

theorem :: SFMASTR3:36
for aa, bb being Int-Location
for f being FinSeq-Location holds UsedInt*Loc (swap (f,aa,bb)) = {f}
proof
let aa, bb be Int-Location; ::_thesis: for f being FinSeq-Location holds UsedInt*Loc (swap (f,aa,bb)) = {f}
let f be FinSeq-Location ; ::_thesis: UsedInt*Loc (swap (f,aa,bb)) = {f}
set a = aa;
set b = bb;
set aux1 = 1 -stRWNotIn {aa,bb};
set aux2 = 2 -ndRWNotIn {aa,bb};
thus  UsedInt*Loc (swap (f,aa,bb)) = (UsedInt*Loc ((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))) ";" ((f,aa) := (2 -ndRWNotIn {aa,bb})))) \/ (UsedInt*Loc ((f,bb) := (1 -stRWNotIn {aa,bb}))) by SF_MASTR:46
.= (UsedInt*Loc ((((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb))) ";" ((f,aa) := (2 -ndRWNotIn {aa,bb})))) \/ {f} by SF_MASTR:33
.= ((UsedInt*Loc (((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb)))) \/ (UsedInt*Loc ((f,aa) := (2 -ndRWNotIn {aa,bb})))) \/ {f} by SF_MASTR:46
.= ((UsedInt*Loc (((1 -stRWNotIn {aa,bb}) := (f,aa)) ";" ((2 -ndRWNotIn {aa,bb}) := (f,bb)))) \/ {f}) \/ {f} by SF_MASTR:33
.= (((UsedInt*Loc ((1 -stRWNotIn {aa,bb}) := (f,aa))) \/ (UsedInt*Loc ((2 -ndRWNotIn {aa,bb}) := (f,bb)))) \/ {f}) \/ {f} by SF_MASTR:47
.= (({f} \/ (UsedInt*Loc ((2 -ndRWNotIn {aa,bb}) := (f,bb)))) \/ {f}) \/ {f} by SF_MASTR:33
.= ({f} \/ {f}) \/ {f} by SF_MASTR:33
.= {f} ; ::_thesis: verum
end;

begin

definition
let f be FinSeq-Location ;
func Selection-sort f ->  Program of  equals :: SFMASTR3:def 6
((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f) ";" (for-up ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))));
coherence
 ((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f) ";" (for-up ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))) is Program of ;
end;

:: deftheorem  defines Selection-sort SFMASTR3:def_6_:_
 for f being FinSeq-Location holds Selection-sort f = ((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f) ";" (for-up ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))));

theorem :: SFMASTR3:37
for s being State of SCM+FSA
for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA
for S being State of SCM+FSA st S = IExec ((Selection-sort f),p,s) holds
( S . f is_non_decreasing_on 1, len (S . f) & ex p being Permutation of (dom (s . f)) st S . f = (s . f) * p )
proof
let s be State of SCM+FSA; ::_thesis: for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA
for S being State of SCM+FSA st S = IExec ((Selection-sort f),p,s) holds
( S . f is_non_decreasing_on 1, len (S . f) & ex p being Permutation of (dom (s . f)) st S . f = (s . f) * p )
let f be FinSeq-Location ; ::_thesis: for p being Instruction-Sequence of SCM+FSA
for S being State of SCM+FSA st S = IExec ((Selection-sort f),p,s) holds
( S . f is_non_decreasing_on 1, len (S . f) & ex p being Permutation of (dom (s . f)) st S . f = (s . f) * p )
let p be Instruction-Sequence of SCM+FSA; ::_thesis: for S being State of SCM+FSA st S = IExec ((Selection-sort f),p,s) holds
( S . f is_non_decreasing_on 1, len (S . f) & ex p being Permutation of (dom (s . f)) st S . f = (s . f) * p )
set minpos = 2 -ndRWNotIn ({} Int-Locations);
set cv = 1 -stRWNotIn ({} Int-Locations);
let S be State of SCM+FSA; ::_thesis: ( S = IExec ((Selection-sort f),p,s) implies ( S . f is_non_decreasing_on 1, len (S . f) & ex p being Permutation of (dom (s . f)) st S . f = (s . f) * p ) )
assume A1: S = IExec ((Selection-sort f),p,s) ; ::_thesis: ( S . f is_non_decreasing_on 1, len (S . f) & ex p being Permutation of (dom (s . f)) st S . f = (s . f) * p )
set I22 = swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)));
set finish = 1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))));
set i1 = (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f;
set I21 = FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)));
set I2B = (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))));
set I2 = for-up ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))));
set s1 = Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s));
set p1 = p;
A2: (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (intloc 0) = (Initialized s) . (intloc 0) by SCMFSA_2:74
.= 1 by SCMFSA_M:9 ;
 1 -stRWNotIn ({} Int-Locations) in {(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))} by TARSKI:def_2;
then  ( 1 -stRWNotIn ({} Int-Locations) <> 1 -stRWNotIn {(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))} & 1 -stRWNotIn ({} Int-Locations) <> 2 -ndRWNotIn {(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))} ) by SCMFSA_M:25;
then A3: not swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))) destroys 1 -stRWNotIn ({} Int-Locations) by Th32;
set SF = StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))));
A4: (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) = len ((Initialized s) . f) by SCMFSA_2:74
.= len (s . f) by SCMFSA_M:37 ;
then reconsider n = (((Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))) - ((Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (intloc 0))) + 1 as Element of NAT by A2;
defpred S1[ Element of NAT ] means ( $1 <= n implies ( ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . $1) . (1 -stRWNotIn ({} Int-Locations)) = $1 + ((Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (intloc 0)) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . $1) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) = (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . $1) . f is_split_at $1 & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . $1) . f is_non_decreasing_on 1,$1 & ex p being Permutation of (dom (s . f)) st ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . $1) . f = (s . f) * p ) );
defpred S2[ Nat] means ( $1 < n implies ( ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . $1) . (intloc 0) = 1 & (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_closed_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . $1,p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) & (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_halting_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . $1,p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) ) );
A5: for k being Element of NAT st S2[k] holds
S2[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S2[k] implies S2[k + 1] )
assume A6: S2[k] ; ::_thesis: S2[k + 1]
assume  k + 1 < n ; ::_thesis: ( ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . (intloc 0) = 1 & (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_closed_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) & (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_halting_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) )
hence A7: ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . (intloc 0) = 1 by A6, Th18, NAT_1:13; ::_thesis: ( (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_closed_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) & (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_halting_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) )
 (Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1))) . (intloc 0) = 1 by SCMFSA_M:9;
then A8: ( FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations))) is_closed_on Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) & FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations))) is_halting_on Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) ) by Th29;
A9: swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))) is_closed_on IExec ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1))),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) by SCMFSA7B:18;
then A10: (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_closed_on Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) by A8, SFMASTR1:2;
hence  (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_closed_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) by A7, SFMASTR2:4; ::_thesis: (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_halting_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))
 swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))) is_halting_on IExec ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1))),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) by SCMFSA7B:19;
then  (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_halting_on Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) by A8, A9, SFMASTR1:3;
hence  (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_halting_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) by A7, A10, SFMASTR2:5; ::_thesis: verum
end;
A11: S2[ 0 ]
proof
 (Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0)) . (intloc 0) = 1 by SCMFSA_M:9;
then A12: ( FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations))) is_closed_on Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) & FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations))) is_halting_on Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) ) by Th29;
assume  0 < n ; ::_thesis: ( ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . (intloc 0) = 1 & (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_closed_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0,p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) & (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_halting_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0,p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) )
thus A13: ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . (intloc 0) = 1 by A2, Th10; ::_thesis: ( (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_closed_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0,p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) & (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_halting_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0,p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) )
A14: swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))) is_closed_on IExec ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0)),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) by SCMFSA7B:18;
then A15: (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_closed_on Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) by A12, SFMASTR1:2;
hence  (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_closed_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0,p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) by A13, SFMASTR2:4; ::_thesis: (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_halting_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0,p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))
 swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))) is_halting_on IExec ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0)),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) by SCMFSA7B:19;
then  (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_halting_on Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) by A12, A14, SFMASTR1:3;
hence  (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_halting_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0,p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) by A13, A15, SFMASTR2:5; ::_thesis: verum
end;
A16: for k being Element of NAT holds S2[k] from NAT_1:sch_1(A11, A5);
A17: ProperForUpBody 1 -stRWNotIn ({} Int-Locations), intloc 0,1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))),(FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))), Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s)),p
proof
let i be Element of NAT ; :: according to SFMASTR3:def_2 ::_thesis: ( i < (((Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))) - ((Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (intloc 0))) + 1 implies ( (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_closed_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . i,p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) & (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_halting_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . i,p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) ) )
thus  ( i < (((Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))) - ((Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (intloc 0))) + 1 implies ( (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_closed_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . i,p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) & (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_halting_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . i,p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) ) ) by A16; ::_thesis: verum
end;
then A18: DataPart (IExec ((for-up ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) = DataPart ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . n) by A2, Th25;
 ( for-up ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))) is_halting_on Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s)),p & for-up ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))) is_closed_on Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s)),p ) by A2, A17, Th26;
then A19: S . f = (IExec ((for-up ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . f by A1, SFMASTR1:15
.= ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . n) . f by A18, SCMFSA_M:2 ;
 not FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations))) destroys 1 -stRWNotIn ({} Int-Locations) by Th27, SCMFSA_M:26;
then A20: not (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) destroys 1 -stRWNotIn ({} Int-Locations) by A3, SCMFSA8C:52;
A21: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A22: S1[k] ; ::_thesis: S1[k + 1]
A23: now__::_thesis:_(_k_<_n_implies_(_((StepForUp_((1_-stRWNotIn_({}_Int-Locations)),(intloc_0),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),p,(Exec_(((1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))_:=len_f),(Initialized_s)))))_._k)_._(intloc_0)_=_1_&_(FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))_is_closed_on_Initialized_((StepForUp_((1_-stRWNotIn_({}_Int-Locations)),(intloc_0),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),p,(Exec_(((1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))_:=len_f),(Initialized_s)))))_._k),p_+*_(while>0_((1_-stRWNotIn_({(1_-stRWNotIn_({}_Int-Locations)),(intloc_0),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))}_\/_(UsedIntLoc_((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))))),((((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))_";"_(AddTo_((1_-stRWNotIn_({}_Int-Locations)),(intloc_0))))_";"_(SubFrom_((1_-stRWNotIn_({(1_-stRWNotIn_({}_Int-Locations)),(intloc_0),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))}_\/_(UsedIntLoc_((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))))),(intloc_0))))))_&_(FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))_is_halting_on_Initialized_((StepForUp_((1_-stRWNotIn_({}_Int-Locations)),(intloc_0),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),p,(Exec_(((1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))_:=len_f),(Initialized_s)))))_._k),p_+*_(while>0_((1_-stRWNotIn_({(1_-stRWNotIn_({}_Int-Locations)),(intloc_0),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))}_\/_(UsedIntLoc_((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))))),((((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))_";"_(AddTo_((1_-stRWNotIn_({}_Int-Locations)),(intloc_0))))_";"_(SubFrom_((1_-stRWNotIn_({(1_-stRWNotIn_({}_Int-Locations)),(intloc_0),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))}_\/_(UsedIntLoc_((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))))),(intloc_0))))))_&_((StepForUp_((1_-stRWNotIn_({}_Int-Locations)),(intloc_0),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),p,(Exec_(((1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))_:=len_f),(Initialized_s)))))_._k)_._(1_-stRWNotIn_({}_Int-Locations))_=_k_+_((Exec_(((1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))_:=len_f),(Initialized_s)))_._(intloc_0))_&_((StepForUp_((1_-stRWNotIn_({}_Int-Locations)),(intloc_0),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),p,(Exec_(((1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))_:=len_f),(Initialized_s)))))_._k)_._(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))_=_(Exec_(((1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))_:=len_f),(Initialized_s)))_._(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))_&_((StepForUp_((1_-stRWNotIn_({}_Int-Locations)),(intloc_0),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),p,(Exec_(((1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))_:=len_f),(Initialized_s)))))_._k)_._(1_-stRWNotIn_({}_Int-Locations))_<=_(Exec_(((1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))_:=len_f),(Initialized_s)))_._(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))_&_((StepForUp_((1_-stRWNotIn_({}_Int-Locations)),(intloc_0),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),p,(Exec_(((1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))_:=len_f),(Initialized_s)))))_._(k_+_1))_|_(({(1_-stRWNotIn_({}_Int-Locations)),(intloc_0),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))}_\/_(UsedIntLoc_((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))))_\/_FinSeq-Locations)_=_(IExec_((((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))_";"_(AddTo_((1_-stRWNotIn_({}_Int-Locations)),(intloc_0)))),(p_+*_(while>0_((1_-stRWNotIn_({(1_-stRWNotIn_({}_Int-Locations)),(intloc_0),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))}_\/_(UsedIntLoc_((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))))),((((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))_";"_(AddTo_((1_-stRWNotIn_({}_Int-Locations)),(intloc_0))))_";"_(SubFrom_((1_-stRWNotIn_({(1_-stRWNotIn_({}_Int-Locations)),(intloc_0),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))}_\/_(UsedIntLoc_((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))))),(intloc_0))))))),((StepForUp_((1_-stRWNotIn_({}_Int-Locations)),(intloc_0),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),p,(Exec_(((1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))_:=len_f),(Initialized_s)))))_._k)))_|_(({(1_-stRWNotIn_({}_Int-Locations)),(intloc_0),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))}_\/_(UsedIntLoc_((FinSeqMin_(f,(1_-stRWNotIn_({}_Int-Locations)),(1_-stNotUsed_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations))))),(2_-ndRWNotIn_({}_Int-Locations))))_";"_(swap_(f,(1_-stRWNotIn_({}_Int-Locations)),(2_-ndRWNotIn_({}_Int-Locations)))))))_\/_FinSeq-Locations)_)_)
assume A24: k < n ; ::_thesis: ( ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (intloc 0) = 1 & (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_closed_on Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) & (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_halting_on Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stRWNotIn ({} Int-Locations)) = k + ((Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (intloc 0)) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) = (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stRWNotIn ({} Int-Locations)) <= (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) | (({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))) \/ FinSeq-Locations) = (IExec ((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) | (({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))) \/ FinSeq-Locations) )
hence A25: ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (intloc 0) = 1 by A16; ::_thesis: ( (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_closed_on Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) & (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_halting_on Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stRWNotIn ({} Int-Locations)) = k + ((Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (intloc 0)) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) = (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stRWNotIn ({} Int-Locations)) <= (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) | (({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))) \/ FinSeq-Locations) = (IExec ((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) | (({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))) \/ FinSeq-Locations) )
A26: (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_closed_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k,p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) by A16, A24;
hence  (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_closed_on Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) by A25, SFMASTR2:4; ::_thesis: ( (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_halting_on Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stRWNotIn ({} Int-Locations)) = k + ((Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (intloc 0)) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) = (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stRWNotIn ({} Int-Locations)) <= (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) | (({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))) \/ FinSeq-Locations) = (IExec ((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) | (({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))) \/ FinSeq-Locations) )
 (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_halting_on (StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k,p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) by A16, A24;
hence  (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) is_halting_on Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) by A25, A26, SFMASTR2:5; ::_thesis: ( ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stRWNotIn ({} Int-Locations)) = k + ((Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (intloc 0)) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) = (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stRWNotIn ({} Int-Locations)) <= (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) | (({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))) \/ FinSeq-Locations) = (IExec ((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) | (({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))) \/ FinSeq-Locations) )
thus  ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stRWNotIn ({} Int-Locations)) = k + ((Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (intloc 0)) by A22, A24; ::_thesis: ( ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) = (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stRWNotIn ({} Int-Locations)) <= (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) | (({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))) \/ FinSeq-Locations) = (IExec ((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) | (({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))) \/ FinSeq-Locations) )
thus  ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) = (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) by A22, A24; ::_thesis: ( ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stRWNotIn ({} Int-Locations)) <= (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) | (({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))) \/ FinSeq-Locations) = (IExec ((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) | (({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))) \/ FinSeq-Locations) )
thus  ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stRWNotIn ({} Int-Locations)) <= (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) by A2, A22, A24, NAT_1:13; ::_thesis: ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) | (({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))) \/ FinSeq-Locations) = (IExec ((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) | (({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))) \/ FinSeq-Locations)
thus  ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) | (({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))) \/ FinSeq-Locations) = (IExec ((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) | (({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))))) \/ FinSeq-Locations) by A2, A17, A24, Th21; ::_thesis: verum
end;
set F = ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f;
set F1 = ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f;
assume A27: k + 1 <= n ; ::_thesis: ( ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . (1 -stRWNotIn ({} Int-Locations)) = (k + 1) + ((Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (intloc 0)) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) = (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f is_split_at k + 1 & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f is_non_decreasing_on 1,k + 1 & ex p being Permutation of (dom (s . f)) st ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f = (s . f) * p )
then consider pp being Permutation of (dom (s . f)) such that
A28: ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f = (s . f) * pp by A22, NAT_1:13;
thus  ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . (1 -stRWNotIn ({} Int-Locations)) = (k + 1) + ((Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (intloc 0)) by A20, A2, A17, A27, Th19; ::_thesis: ( ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) = (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f is_split_at k + 1 & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f is_non_decreasing_on 1,k + 1 & ex p being Permutation of (dom (s . f)) st ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f = (s . f) * p )
A29: ( swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))) is_closed_on Initialized (IExec ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) & swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))) is_halting_on Initialized (IExec ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) ) by SCMFSA7B:18, SCMFSA7B:19;
A30: 1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) = 1 -stRWNotIn (UsedIntLoc (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) by SFMASTR1:def_4;
set ma = min_at ((((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f),(k + 1),(len (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f)));
A31: dom (s . f) = Seg (len (s . f)) by FINSEQ_1:def_3;
then A32: len (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f) = len (s . f) by A28, FINSEQ_2:43;
A33: 1 <= k + 1 by NAT_1:12;
then A34: k + 1 <= min_at ((((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f),(k + 1),(len (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f))) by A2, A4, A27, A32, GRAPH_2:59;
then A35: 1 <= min_at ((((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f),(k + 1),(len (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f))) by A33, XXREAL_0:2;
 min_at ((((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f),(k + 1),(len (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f))) <= len (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f) by A2, A4, A27, A32, A33, GRAPH_2:59;
then A36: min_at ((((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f),(k + 1),(len (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f))) in dom (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f) by A35, FINSEQ_3:25;
A37: {(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))} c= UsedIntLoc (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) by Th35;
 2 -ndRWNotIn ({} Int-Locations) in {(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))} by TARSKI:def_2;
then A38: 1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) <> 2 -ndRWNotIn ({} Int-Locations) by A30, A37, SCMFSA_M:25;
 1 -stRWNotIn ({} Int-Locations) in {(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))} by TARSKI:def_2;
then A39: 1 -stRWNotIn ({} Int-Locations) <> 1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) by A30, A37, SCMFSA_M:25;
A40: 1 -stRWNotIn ({} Int-Locations) <> 2 -ndRWNotIn ({} Int-Locations) by SCMFSA_M:26;
 (Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k)) . (intloc 0) = 1 by SCMFSA_M:9;
then A41: ( FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations))) is_closed_on Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) & FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations))) is_halting_on Initialized ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k),p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0)))))) ) by Th29;
A42: ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f = ((((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f) +* ((k + 1),((((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f) . (min_at ((((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f),(k + 1),(len (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f))))))) +* ((min_at ((((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f),(k + 1),(len (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f)))),((((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f) . (k + 1)))
proof
set S2 = IExec ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k));
A43: len (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f) = abs (len (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f)) by ABSVALUE:def_1;
 ( ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) = len (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f) & k + 1 = abs (k + 1) ) by A4, A23, A27, A28, A31, ABSVALUE:def_1, FINSEQ_2:43, NAT_1:13;
then A44: (IExec ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) . (2 -ndRWNotIn ({} Int-Locations)) = min_at ((((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f),(k + 1),(len (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f))) by A2, A23, A27, A38, A40, A33, A43, Th31, NAT_1:13;
then A45: 1 <= (IExec ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) . (2 -ndRWNotIn ({} Int-Locations)) by A33, A34, XXREAL_0:2;
A46: (IExec ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) . f = ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f by A23, A27, A38, A40, Th30, NAT_1:13;
then A47: (IExec ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) . (2 -ndRWNotIn ({} Int-Locations)) <= len ((IExec ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) . f) by A2, A4, A27, A32, A33, A44, GRAPH_2:59;
A48: ( (IExec ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) . (1 -stRWNotIn ({} Int-Locations)) = k + 1 & (IExec ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) . (intloc 0) = 1 ) by A2, A23, A27, A41, A38, A40, Th30, NAT_1:13, SCMFSA8C:67;
thus ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f = (IExec ((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) . f by A23, A27, NAT_1:13, SCMFSA_M:28
.= (Exec ((AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0))),(IExec (((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))))) . f by A23, A27, NAT_1:13, SFMASTR1:12
.= (IExec (((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) . f by SCMFSA_2:64
.= (IExec ((swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),(IExec ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))))) . f by A41, SFMASTR1:8
.= ((((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f) +* ((k + 1),((((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f) . (min_at ((((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f),(k + 1),(len (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f))))))) +* ((min_at ((((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f),(k + 1),(len (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f)))),((((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f) . (k + 1))) by A2, A4, A27, A32, A33, A46, A44, A45, A47, A48, Th33 ; ::_thesis: verum
end;
 k + 1 in dom (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f) by A2, A4, A27, A32, A33, FINSEQ_3:25;
then consider p1 being Permutation of (dom (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f)) such that
A49: ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f = (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f) * p1 by A36, A42, FUNCT_7:111;
 ( {(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations))} c= UsedIntLoc (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) & 1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) in {(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations))} ) by Th28, ENUMSET1:def_1;
then  1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) in (UsedIntLoc (FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations))))) \/ (UsedIntLoc (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) by XBOOLE_0:def_3;
then  1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) in UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) by SF_MASTR:27;
then  1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) in {(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))) by XBOOLE_0:def_3;
hence ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) = (IExec ((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) by A23, A27, NAT_1:13, SCMFSA_M:28
.= (Exec ((AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0))),(IExec (((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) by A23, A27, NAT_1:13, SFMASTR1:11
.= (IExec (((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) by A39, SCMFSA_2:64
.= (IExec ((swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),(IExec ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) by A41, SFMASTR1:7
.= (Initialized (IExec ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k)))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) by A30, A29, SCMFSA_M:25, SFMASTR2:1
.= (IExec ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))),(p +* (while>0 ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),((((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) ";" (AddTo ((1 -stRWNotIn ({} Int-Locations)),(intloc 0)))) ";" (SubFrom ((1 -stRWNotIn ({(1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))} \/ (UsedIntLoc ((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))))))),(intloc 0))))))),((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) by SCMFSA_M:37
.= (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) by A23, A27, A38, A40, Th30, NAT_1:13 ;
::_thesis: ( ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f is_split_at k + 1 & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f is_non_decreasing_on 1,k + 1 & ex p being Permutation of (dom (s . f)) st ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f = (s . f) * p )
thus  ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f is_split_at k + 1 by A2, A4, A22, A27, A32, A42, GRAPH_2:62, NAT_1:13; ::_thesis: ( ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f is_non_decreasing_on 1,k + 1 & ex p being Permutation of (dom (s . f)) st ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f = (s . f) * p )
thus  ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f is_non_decreasing_on 1,k + 1 by A2, A4, A22, A27, A32, A42, GRAPH_2:61, NAT_1:13; ::_thesis: ex p being Permutation of (dom (s . f)) st ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f = (s . f) * p
 dom (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . k) . f) = dom (s . f) by A32, FINSEQ_3:29;
then reconsider p1 = p1 as Permutation of (dom (s . f)) ;
reconsider ppp = pp * p1 as Permutation of (dom (s . f)) ;
take  ppp ; ::_thesis: ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f = (s . f) * ppp
thus  ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . (k + 1)) . f = (s . f) * ppp by A28, A49, RELAT_1:36; ::_thesis: verum
end;
A50: dom (s . f) = Seg (len (s . f)) by FINSEQ_1:def_3;
A51: 1 -stRWNotIn ({} Int-Locations) in {(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))} by TARSKI:def_2;
 ( 1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) = 1 -stRWNotIn (UsedIntLoc (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) & {(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))} c= UsedIntLoc (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) ) by Th35, SFMASTR1:def_4;
then A52: 1 -stRWNotIn ({} Int-Locations) <> 1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations)))) by A51, SCMFSA_M:25;
A53: S1[ 0 ]
proof
assume  0 <= n ; ::_thesis: ( ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . (1 -stRWNotIn ({} Int-Locations)) = 0 + ((Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (intloc 0)) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) = (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f is_split_at 0 & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f is_non_decreasing_on 1, 0 & ex p being Permutation of (dom (s . f)) st ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f = (s . f) * p )
thus  ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . (1 -stRWNotIn ({} Int-Locations)) = 0 + ((Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (intloc 0)) by Th11; ::_thesis: ( ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) = (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f is_split_at 0 & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f is_non_decreasing_on 1, 0 & ex p being Permutation of (dom (s . f)) st ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f = (s . f) * p )
thus  ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) = (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . (1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) by A52, Th13; ::_thesis: ( ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f is_split_at 0 & ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f is_non_decreasing_on 1, 0 & ex p being Permutation of (dom (s . f)) st ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f = (s . f) * p )
thus  ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f is_split_at 0 ::_thesis: ( ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f is_non_decreasing_on 1, 0 & ex p being Permutation of (dom (s . f)) st ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f = (s . f) * p )
proof
let i, j be Element of NAT ; :: according to GRAPH_2:def_13 ::_thesis: ( not 1 <= i or not i <= 0 or j <= 0 or not j <= len (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f) or (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f) . i <= (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f) . j )
assume that
A54: ( 1 <= i & i <= 0 ) and
 0 < j and
 j <= len (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f) ; ::_thesis: (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f) . i <= (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f) . j
thus  (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f) . i <= (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f) . j by A54; ::_thesis: verum
end;
thus  ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f is_non_decreasing_on 1, 0 ::_thesis: ex p being Permutation of (dom (s . f)) st ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f = (s . f) * p
proof
let i, j be Element of NAT ; :: according to GRAPH_2:def_12 ::_thesis: ( not 1 <= i or not i <= j or not j <= 0 or (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f) . i <= (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f) . j )
assume that
 1 <= i and
A55: ( i <= j & j <= 0 ) ; ::_thesis: (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f) . i <= (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f) . j
thus  (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f) . i <= (((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f) . j by A55; ::_thesis: verum
end;
 dom (s . f) = Seg (len (s . f)) by FINSEQ_1:def_3;
then reconsider p = idseq (len (s . f)) as Permutation of (dom (s . f)) by FINSEQ_2:55;
take  p ; ::_thesis: ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f = (s . f) * p
((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f = (Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))) . f by Th15
.= (Initialized s) . f by SCMFSA_2:74
.= s . f by SCMFSA_M:37 ;
hence  ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . 0) . f = (s . f) * p by FINSEQ_2:54; ::_thesis: verum
end;
A56: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A53, A21);
then  ex p being Permutation of (dom (s . f)) st ((StepForUp ((1 -stRWNotIn ({} Int-Locations)),(intloc 0),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),((FinSeqMin (f,(1 -stRWNotIn ({} Int-Locations)),(1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),(2 -ndRWNotIn ({} Int-Locations)))) ";" (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))),p,(Exec (((1 -stNotUsed (swap (f,(1 -stRWNotIn ({} Int-Locations)),(2 -ndRWNotIn ({} Int-Locations))))) :=len f),(Initialized s))))) . n) . f = (s . f) * p ;
then  len (S . f) = n by A2, A4, A19, A50, FINSEQ_2:43;
hence  S . f is_non_decreasing_on 1, len (S . f) by A56, A19; ::_thesis: ex p being Permutation of (dom (s . f)) st S . f = (s . f) * p
thus  ex p being Permutation of (dom (s . f)) st S . f = (s . f) * p by A56, A19; ::_thesis: verum
end;