:: SCMFSA8B semantic presentation

begin

set A = NAT ;

set D = Data-Locations ;

set SA0 = Start-At (0,SCM+FSA);

   Lm1:  for I, J being Program of SCM+FSA holds Reloc (J,(card I)) c= I ";" J
by FUNCT_4:25;

theorem Th1: :: SCMFSA8B:1
for P being Instruction-Sequence of SCM+FSA
for I being Program of SCM+FSA
for s being State of SCM+FSA st I is_closed_on s,P holds
0 in dom I
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for I being Program of SCM+FSA
for s being State of SCM+FSA st I is_closed_on s,P holds
0 in dom I
let I be Program of SCM+FSA; ::_thesis: for s being State of SCM+FSA st I is_closed_on s,P holds
0 in dom I
let s be State of SCM+FSA; ::_thesis: ( I is_closed_on s,P implies 0 in dom I )
assume A1: I is_closed_on s,P ; ::_thesis: 0 in dom I
A2: IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
IC (Comput ((P +* I),(Initialize s),0)) = (Initialize s) . (IC )
.= IC (Start-At (0,SCM+FSA)) by A2, FUNCT_4:13
.= 0 by FUNCOP_1:72 ;
hence  0 in dom I by A1, SCMFSA7B:def_6; ::_thesis: verum
end;

theorem :: SCMFSA8B:2
canceled;

theorem Th3: :: SCMFSA8B:3
for P1, P2 being Instruction-Sequence of SCM+FSA
for s1, s2 being State of SCM+FSA
for I being Program of SCM+FSA st DataPart s1 = DataPart s2 & I is_closed_on s1,P1 holds
I is_closed_on s2,P2
proof
let P1, P2 be Instruction-Sequence of SCM+FSA; ::_thesis: for s1, s2 being State of SCM+FSA
for I being Program of SCM+FSA st DataPart s1 = DataPart s2 & I is_closed_on s1,P1 holds
I is_closed_on s2,P2
let s1, s2 be State of SCM+FSA; ::_thesis: for I being Program of SCM+FSA st DataPart s1 = DataPart s2 & I is_closed_on s1,P1 holds
I is_closed_on s2,P2
let I be Program of SCM+FSA; ::_thesis: ( DataPart s1 = DataPart s2 & I is_closed_on s1,P1 implies I is_closed_on s2,P2 )
set S1 = Initialize s1;
set S2 = Initialize s2;
assume A1: DataPart s1 = DataPart s2 ; ::_thesis: ( not I is_closed_on s1,P1 or I is_closed_on s2,P2 )
A2: DataPart (Comput ((P1 +* I),(Initialize s1),0)) = DataPart s1 by MEMSTR_0:79
.= DataPart (Comput ((P2 +* I),(Initialize s2),0)) by A1, MEMSTR_0:79 ;
assume A3: I is_closed_on s1,P1 ; ::_thesis: I is_closed_on s2,P2
then A4: 0 in dom I by Th1;
defpred S1[ Nat] means ( IC (Comput ((P1 +* I),(Initialize s1),$1)) = IC (Comput ((P2 +* I),(Initialize s2),$1)) & CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),$1))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),$1))) & DataPart (Comput ((P1 +* I),(Initialize s1),$1)) = DataPart (Comput ((P2 +* I),(Initialize s2),$1)) );
A5: now__::_thesis:_for_k_being_Element_of_NAT_st_S1[k]_holds_
S1[k_+_1]
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
A6: Comput ((P2 +* I),(Initialize s2),(k + 1)) = Following ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k))) by EXTPRO_1:3
.= Exec ((CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k)))),(Comput ((P2 +* I),(Initialize s2),k))) ;
assume A7: S1[k] ; ::_thesis: S1[k + 1]
then A8: for f being FinSeq-Location holds (Comput ((P1 +* I),(Initialize s1),k)) . f = (Comput ((P2 +* I),(Initialize s2),k)) . f by SCMFSA_M:2;
 for a being Int-Location holds (Comput ((P1 +* I),(Initialize s1),k)) . a = (Comput ((P2 +* I),(Initialize s2),k)) . a by A7, SCMFSA_M:2;
then A9: Comput ((P1 +* I),(Initialize s1),k) = Comput ((P2 +* I),(Initialize s2),k) by A7, A8, SCMFSA_2:61;
A10: IC (Comput ((P1 +* I),(Initialize s1),(k + 1))) in dom I by A3, SCMFSA7B:def_6;
Comput ((P1 +* I),(Initialize s1),(k + 1)) = Following ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k))) by EXTPRO_1:3
.= Exec ((CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k)))),(Comput ((P1 +* I),(Initialize s1),k))) ;
then A11: Comput ((P1 +* I),(Initialize s1),(k + 1)) = Comput ((P2 +* I),(Initialize s2),(k + 1)) by A7, A9, A6;
A12: IC (Comput ((P1 +* I),(Initialize s1),(k + 1))) = IC (Comput ((P2 +* I),(Initialize s2),(k + 1))) by A11;
A13: (P1 +* I) /. (IC (Comput ((P1 +* I),(Initialize s1),(k + 1)))) = (P1 +* I) . (IC (Comput ((P1 +* I),(Initialize s1),(k + 1)))) by PBOOLE:143;
A14: (P2 +* I) /. (IC (Comput ((P2 +* I),(Initialize s2),(k + 1)))) = (P2 +* I) . (IC (Comput ((P2 +* I),(Initialize s2),(k + 1)))) by PBOOLE:143;
A15: I c= P1 +* I by FUNCT_4:25;
A16: I c= P2 +* I by FUNCT_4:25;
CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),(k + 1)))) = I . (IC (Comput ((P1 +* I),(Initialize s1),(k + 1)))) by A10, A13, A15, GRFUNC_1:2
.= CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),(k + 1)))) by A12, A10, A14, A16, GRFUNC_1:2 ;
hence  S1[k + 1] by A11; ::_thesis: verum
end;
A17: (P1 +* I) /. (IC (Comput ((P1 +* I),(Initialize s1),0))) = (P1 +* I) . (IC (Comput ((P1 +* I),(Initialize s1),0))) by PBOOLE:143;
A18: (P2 +* I) /. (IC (Comput ((P2 +* I),(Initialize s2),0))) = (P2 +* I) . (IC (Comput ((P2 +* I),(Initialize s2),0))) by PBOOLE:143;
A19: IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
then A20: IC (Comput ((P2 +* I),(Initialize s2),0)) = IC (Start-At (0,SCM+FSA)) by FUNCT_4:13
.= 0 by FUNCOP_1:72 ;
A21: IC (Comput ((P1 +* I),(Initialize s1),0)) = IC (Start-At (0,SCM+FSA)) by A19, FUNCT_4:13
.= 0 by FUNCOP_1:72 ;
then  CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),0))) = I . 0 by A4, A17, FUNCT_4:13
.= CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),0))) by A20, A4, A18, FUNCT_4:13 ;
then A22: S1[ 0 ] by A21, A20, A2;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P2_+*_I),(Initialize_s2),k))_in_dom_I
let k be Element of NAT ; ::_thesis: IC (Comput ((P2 +* I),(Initialize s2),k)) in dom I
A23: IC (Comput ((P1 +* I),(Initialize s1),k)) in dom I by A3, SCMFSA7B:def_6;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A22, A5);
hence  IC (Comput ((P2 +* I),(Initialize s2),k)) in dom I by A23; ::_thesis: verum
end;
hence  I is_closed_on s2,P2 by SCMFSA7B:def_6; ::_thesis: verum
end;

theorem Th4: :: SCMFSA8B:4
for s1, s2 being State of SCM+FSA
for I, J being Program of SCM+FSA st DataPart s1 = DataPart s2 holds
Initialize s1 = Initialize s2 by MEMSTR_0:80;

theorem Th5: :: SCMFSA8B:5
for P1, P2 being Instruction-Sequence of SCM+FSA
for s1, s2 being State of SCM+FSA
for I being Program of SCM+FSA st DataPart s1 = DataPart s2 & I is_closed_on s1,P1 & I is_halting_on s1,P1 holds
( I is_closed_on s2,P2 & I is_halting_on s2,P2 )
proof
let P1, P2 be Instruction-Sequence of SCM+FSA; ::_thesis: for s1, s2 being State of SCM+FSA
for I being Program of SCM+FSA st DataPart s1 = DataPart s2 & I is_closed_on s1,P1 & I is_halting_on s1,P1 holds
( I is_closed_on s2,P2 & I is_halting_on s2,P2 )
let s1, s2 be State of SCM+FSA; ::_thesis: for I being Program of SCM+FSA st DataPart s1 = DataPart s2 & I is_closed_on s1,P1 & I is_halting_on s1,P1 holds
( I is_closed_on s2,P2 & I is_halting_on s2,P2 )
let I be Program of SCM+FSA; ::_thesis: ( DataPart s1 = DataPart s2 & I is_closed_on s1,P1 & I is_halting_on s1,P1 implies ( I is_closed_on s2,P2 & I is_halting_on s2,P2 ) )
set S1 = Initialize s1;
set S2 = Initialize s2;
defpred S1[ Nat] means ( IC (Comput ((P1 +* I),(Initialize s1),$1)) = IC (Comput ((P2 +* I),(Initialize s2),$1)) & CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),$1))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),$1))) & DataPart (Comput ((P1 +* I),(Initialize s1),$1)) = DataPart (Comput ((P2 +* I),(Initialize s2),$1)) );
A1: IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
A2: IC (Comput ((P1 +* I),(Initialize s1),0)) = IC (Start-At (0,SCM+FSA)) by A1, FUNCT_4:13
.= 0 by FUNCOP_1:72 ;
A3: IC (Comput ((P2 +* I),(Initialize s2),0)) = IC (Start-At (0,SCM+FSA)) by A1, FUNCT_4:13
.= 0 by FUNCOP_1:72 ;
assume  DataPart s1 = DataPart s2 ; ::_thesis: ( not I is_closed_on s1,P1 or not I is_halting_on s1,P1 or ( I is_closed_on s2,P2 & I is_halting_on s2,P2 ) )
then A4: Comput ((P1 +* I),(Initialize s1),0) = Comput ((P2 +* I),(Initialize s2),0) by Th4;
assume A5: I is_closed_on s1,P1 ; ::_thesis: ( not I is_halting_on s1,P1 or ( I is_closed_on s2,P2 & I is_halting_on s2,P2 ) )
A6: now__::_thesis:_for_k_being_Element_of_NAT_st_S1[k]_holds_
S1[k_+_1]
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
A7: Comput ((P2 +* I),(Initialize s2),(k + 1)) = Following ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k))) by EXTPRO_1:3
.= Exec ((CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k)))),(Comput ((P2 +* I),(Initialize s2),k))) ;
assume A8: S1[k] ; ::_thesis: S1[k + 1]
then A9: for f being FinSeq-Location holds (Comput ((P1 +* I),(Initialize s1),k)) . f = (Comput ((P2 +* I),(Initialize s2),k)) . f by SCMFSA_M:2;
 for a being Int-Location holds (Comput ((P1 +* I),(Initialize s1),k)) . a = (Comput ((P2 +* I),(Initialize s2),k)) . a by A8, SCMFSA_M:2;
then A10: Comput ((P1 +* I),(Initialize s1),k) = Comput ((P2 +* I),(Initialize s2),k) by A8, A9, SCMFSA_2:61;
A11: IC (Comput ((P1 +* I),(Initialize s1),(k + 1))) in dom I by A5, SCMFSA7B:def_6;
Comput ((P1 +* I),(Initialize s1),(k + 1)) = Following ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k))) by EXTPRO_1:3
.= Exec ((CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k)))),(Comput ((P1 +* I),(Initialize s1),k))) ;
then A12: Comput ((P1 +* I),(Initialize s1),(k + 1)) = Comput ((P2 +* I),(Initialize s2),(k + 1)) by A8, A10, A7;
A13: IC (Comput ((P1 +* I),(Initialize s1),(k + 1))) = IC (Comput ((P2 +* I),(Initialize s2),(k + 1))) by A12;
A14: (P1 +* I) /. (IC (Comput ((P1 +* I),(Initialize s1),(k + 1)))) = (P1 +* I) . (IC (Comput ((P1 +* I),(Initialize s1),(k + 1)))) by PBOOLE:143;
A15: (P2 +* I) /. (IC (Comput ((P2 +* I),(Initialize s2),(k + 1)))) = (P2 +* I) . (IC (Comput ((P2 +* I),(Initialize s2),(k + 1)))) by PBOOLE:143;
A16: I c= P1 +* I by FUNCT_4:25;
A17: I c= P2 +* I by FUNCT_4:25;
CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),(k + 1)))) = I . (IC (Comput ((P1 +* I),(Initialize s1),(k + 1)))) by A11, A14, A16, GRFUNC_1:2
.= CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),(k + 1)))) by A13, A11, A15, A17, GRFUNC_1:2 ;
hence  S1[k + 1] by A12; ::_thesis: verum
end;
assume  I is_halting_on s1,P1 ; ::_thesis: ( I is_closed_on s2,P2 & I is_halting_on s2,P2 )
then  P1 +* I halts_on Initialize s1 by SCMFSA7B:def_7;
then consider m being Element of NAT  such that
A18: CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),m))) = halt SCM+FSA by EXTPRO_1:29;
A19: (P1 +* I) /. (IC (Comput ((P1 +* I),(Initialize s1),0))) = (P1 +* I) . (IC (Comput ((P1 +* I),(Initialize s1),0))) by PBOOLE:143;
A20: (P2 +* I) /. (IC (Comput ((P2 +* I),(Initialize s2),0))) = (P2 +* I) . (IC (Comput ((P2 +* I),(Initialize s2),0))) by PBOOLE:143;
A21: 0 in dom I by A5, Th1;
then  CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),0))) = I . 0 by A2, A19, FUNCT_4:13
.= CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),0))) by A3, A21, A20, FUNCT_4:13 ;
then A22: S1[ 0 ] by A4;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P2_+*_I),(Initialize_s2),k))_in_dom_I
let k be Element of NAT ; ::_thesis: IC (Comput ((P2 +* I),(Initialize s2),k)) in dom I
A23: IC (Comput ((P1 +* I),(Initialize s1),k)) in dom I by A5, SCMFSA7B:def_6;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A22, A6);
hence  IC (Comput ((P2 +* I),(Initialize s2),k)) in dom I by A23; ::_thesis: verum
end;
hence  I is_closed_on s2,P2 by SCMFSA7B:def_6; ::_thesis: I is_halting_on s2,P2
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A22, A6);
then  CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),m))) = halt SCM+FSA by A18;
then  P2 +* I halts_on Initialize s2 by EXTPRO_1:29;
hence  I is_halting_on s2,P2 by SCMFSA7B:def_7; ::_thesis: verum
end;

theorem Th6: :: SCMFSA8B:6
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being Program of SCM+FSA holds
( I is_closed_on Initialized s,P iff I is_closed_on s +* (Initialize ((intloc 0) .--> 1)),P +* J )
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being Program of SCM+FSA holds
( I is_closed_on Initialized s,P iff I is_closed_on s +* (Initialize ((intloc 0) .--> 1)),P +* J )
let s be State of SCM+FSA; ::_thesis: for I, J being Program of SCM+FSA holds
( I is_closed_on Initialized s,P iff I is_closed_on s +* (Initialize ((intloc 0) .--> 1)),P +* J )
let I, J be Program of SCM+FSA; ::_thesis: ( I is_closed_on Initialized s,P iff I is_closed_on s +* (Initialize ((intloc 0) .--> 1)),P +* J )
 DataPart (Initialized s) = DataPart (s +* (Initialize ((intloc 0) .--> 1))) ;
hence  ( I is_closed_on Initialized s,P iff I is_closed_on s +* (Initialize ((intloc 0) .--> 1)),P +* J ) by Th3; ::_thesis: verum
end;

theorem Th7: :: SCMFSA8B:7
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for l being Element of NAT holds
( I is_closed_on s,P iff I is_closed_on s +* (Start-At (0,SCM+FSA)),P +* I )
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for l being Element of NAT holds
( I is_closed_on s,P iff I is_closed_on s +* (Start-At (0,SCM+FSA)),P +* I )
let s be State of SCM+FSA; ::_thesis: for I, J being Program of SCM+FSA
for l being Element of NAT holds
( I is_closed_on s,P iff I is_closed_on s +* (Start-At (0,SCM+FSA)),P +* I )
let I, J be Program of SCM+FSA; ::_thesis: for l being Element of NAT holds
( I is_closed_on s,P iff I is_closed_on s +* (Start-At (0,SCM+FSA)),P +* I )
let l be Element of NAT ; ::_thesis: ( I is_closed_on s,P iff I is_closed_on s +* (Start-At (0,SCM+FSA)),P +* I )
 DataPart s = DataPart (Initialize s) by MEMSTR_0:79;
hence  ( I is_closed_on s,P iff I is_closed_on s +* (Start-At (0,SCM+FSA)),P +* I ) by Th3; ::_thesis: verum
end;

theorem Th8: :: SCMFSA8B:8
for P1, P2 being Instruction-Sequence of SCM+FSA
for s1 being 0 -started State of SCM+FSA
for s2 being State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s1,P1 & I c= P1 holds
for n being Element of NAT st IC s2 = n & DataPart s1 = DataPart s2 & Reloc (I,n) c= P2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
proof
let P1, P2 be Instruction-Sequence of SCM+FSA; ::_thesis: for s1 being 0 -started State of SCM+FSA
for s2 being State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s1,P1 & I c= P1 holds
for n being Element of NAT st IC s2 = n & DataPart s1 = DataPart s2 & Reloc (I,n) c= P2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
let s1 be 0 -started State of SCM+FSA; ::_thesis: for s2 being State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s1,P1 & I c= P1 holds
for n being Element of NAT st IC s2 = n & DataPart s1 = DataPart s2 & Reloc (I,n) c= P2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
let s2 be State of SCM+FSA; ::_thesis: for I being Program of SCM+FSA st I is_closed_on s1,P1 & I c= P1 holds
for n being Element of NAT st IC s2 = n & DataPart s1 = DataPart s2 & Reloc (I,n) c= P2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
let I be Program of SCM+FSA; ::_thesis: ( I is_closed_on s1,P1 & I c= P1 implies for n being Element of NAT st IC s2 = n & DataPart s1 = DataPart s2 & Reloc (I,n) c= P2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) ) )
A1: Start-At (0,SCM+FSA) c= s1 by MEMSTR_0:29;
assume A2: I is_closed_on s1,P1 ; ::_thesis: ( not I c= P1 or for n being Element of NAT st IC s2 = n & DataPart s1 = DataPart s2 & Reloc (I,n) c= P2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) ) )
assume A3: I c= P1 ; ::_thesis: for n being Element of NAT st IC s2 = n & DataPart s1 = DataPart s2 & Reloc (I,n) c= P2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
let n be Element of NAT ; ::_thesis: ( IC s2 = n & DataPart s1 = DataPart s2 & Reloc (I,n) c= P2 implies for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) ) )
A4: IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
defpred S1[ Nat] means ( (IC (Comput (P1,s1,$1))) + n = IC (Comput (P2,s2,$1)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,$1)))),n) = CurInstr (P2,(Comput (P2,s2,$1))) & DataPart (Comput (P1,s1,$1)) = DataPart (Comput (P2,s2,$1)) );
A5: IC (Comput (P1,s1,0)) = IC s1
.= IC (Start-At (0,SCM+FSA)) by A1, A4, GRFUNC_1:2
.= 0 by FUNCOP_1:72 ;
assume A6: IC s2 = n ; ::_thesis: ( not DataPart s1 = DataPart s2 or not Reloc (I,n) c= P2 or for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) ) )
A7: 0 in dom I by A2, Th1;
then A8: 0 + n in dom (Reloc (I,n)) by COMPOS_1:46;
 IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
then A9: P1 . (IC s1) = P1 . (IC (Start-At (0,SCM+FSA))) by A1, GRFUNC_1:2
.= P1 . 0 by FUNCOP_1:72
.= I . 0 by A7, A3, GRFUNC_1:2 ;
assume  DataPart s1 = DataPart s2 ; ::_thesis: ( not Reloc (I,n) c= P2 or for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) ) )
then A10: DataPart (Comput (P1,s1,0)) = DataPart s2
.= DataPart (Comput (P2,s2,0)) ;
assume A11: Reloc (I,n) c= P2 ; ::_thesis: for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
let i be Element of NAT ; ::_thesis: ( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
A12: P2 /. (IC s2) = P2 . (IC s2) by PBOOLE:143;
A13: CurInstr (P1,s1) = I . 0 by A9, PBOOLE:143;
IncAddr ((CurInstr (P1,(Comput (P1,s1,0)))),n) = IncAddr ((CurInstr (P1,s1)),n)
.= (Reloc (I,n)) . (0 + n) by A13, A7, COMPOS_1:35
.= CurInstr (P2,s2) by A6, A8, A12, A11, GRFUNC_1:2
.= CurInstr (P2,(Comput (P2,s2,0))) ;
then A14: S1[ 0 ] by A6, A5, A10;
A15: 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] )
A16: Comput (P1,s1,(k + 1)) = Following (P1,(Comput (P1,s1,k))) by EXTPRO_1:3
.= Exec ((CurInstr (P1,(Comput (P1,s1,k)))),(Comput (P1,s1,k))) ;
reconsider l = IC (Comput (P1,s1,(k + 1))) as Element of NAT ;
reconsider j = CurInstr (P1,(Comput (P1,s1,(k + 1)))) as Instruction of SCM+FSA ;
A17: Comput (P2,s2,(k + 1)) = Following (P2,(Comput (P2,s2,k))) by EXTPRO_1:3
.= Exec ((CurInstr (P2,(Comput (P2,s2,k)))),(Comput (P2,s2,k))) ;
A18: Initialize s1 = s1 by A1, FUNCT_4:98;
A19: P1 = P1 +* I by A3, FUNCT_4:98;
then A20: IC (Comput (P1,s1,(k + 1))) in dom I by A2, A18, SCMFSA7B:def_6;
assume A21: S1[k] ; ::_thesis: S1[k + 1]
hence A22: (IC (Comput (P1,s1,(k + 1)))) + n = IC (Comput (P2,s2,(k + 1))) by A16, A17, SCMFSA6A:8; ::_thesis: ( IncAddr ((CurInstr (P1,(Comput (P1,s1,(k + 1))))),n) = CurInstr (P2,(Comput (P2,s2,(k + 1)))) & DataPart (Comput (P1,s1,(k + 1))) = DataPart (Comput (P2,s2,(k + 1))) )
then A23: IC (Comput (P2,s2,(k + 1))) in dom (Reloc (I,n)) by A20, COMPOS_1:46;
A24: l in dom I by A18, A2, A19, SCMFSA7B:def_6;
j = P1 . (IC (Comput (P1,s1,(k + 1)))) by PBOOLE:143
.= I . l by A20, A3, GRFUNC_1:2 ;
hence  IncAddr ((CurInstr (P1,(Comput (P1,s1,(k + 1))))),n) = (Reloc (I,n)) . (l + n) by A24, COMPOS_1:35
.= P2 . (IC (Comput (P2,s2,(k + 1)))) by A23, A22, A11, GRFUNC_1:2
.= CurInstr (P2,(Comput (P2,s2,(k + 1)))) by PBOOLE:143 ;
::_thesis: DataPart (Comput (P1,s1,(k + 1))) = DataPart (Comput (P2,s2,(k + 1)))
thus  DataPart (Comput (P1,s1,(k + 1))) = DataPart (Comput (P2,s2,(k + 1))) by A21, A16, A17, SCMFSA6A:8; ::_thesis: verum
end;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A14, A15);
hence  ( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) ) ; ::_thesis: verum
end;

theorem Th9: :: SCMFSA8B:9
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for i being parahalting keeping_0 Instruction of SCM+FSA
for J being parahalting Program of SCM+FSA
for a being Int-Location holds (IExec ((i ";" J),P,s)) . a = (IExec (J,P,(Exec (i,(Initialized s))))) . a
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for i being parahalting keeping_0 Instruction of SCM+FSA
for J being parahalting Program of SCM+FSA
for a being Int-Location holds (IExec ((i ";" J),P,s)) . a = (IExec (J,P,(Exec (i,(Initialized s))))) . a
let s be State of SCM+FSA; ::_thesis: for i being parahalting keeping_0 Instruction of SCM+FSA
for J being parahalting Program of SCM+FSA
for a being Int-Location holds (IExec ((i ";" J),P,s)) . a = (IExec (J,P,(Exec (i,(Initialized s))))) . a
let i be parahalting keeping_0 Instruction of SCM+FSA; ::_thesis: for J being parahalting Program of SCM+FSA
for a being Int-Location holds (IExec ((i ";" J),P,s)) . a = (IExec (J,P,(Exec (i,(Initialized s))))) . a
let J be parahalting Program of SCM+FSA; ::_thesis: for a being Int-Location holds (IExec ((i ";" J),P,s)) . a = (IExec (J,P,(Exec (i,(Initialized s))))) . a
let a be Int-Location; ::_thesis: (IExec ((i ";" J),P,s)) . a = (IExec (J,P,(Exec (i,(Initialized s))))) . a
A1: IExec ((Macro i),P,s) = Exec (i,(Initialized s)) by SCMFSA6C:5;
thus (IExec ((i ";" J),P,s)) . a = (IExec (J,P,(IExec ((Macro i),P,s)))) . a by SCMFSA6C:1
.= (IExec (J,P,(IExec ((Macro i),P,s)))) . a
.= (IExec (J,P,(Exec (i,(Initialized s))))) . a by A1
.= (IExec (J,P,(Exec (i,(Initialized s))))) . a ; ::_thesis: verum
end;

theorem Th10: :: SCMFSA8B:10
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for i being parahalting keeping_0 Instruction of SCM+FSA
for J being parahalting Program of SCM+FSA
for f being FinSeq-Location holds (IExec ((i ";" J),P,s)) . f = (IExec (J,P,(Exec (i,(Initialized s))))) . f
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for i being parahalting keeping_0 Instruction of SCM+FSA
for J being parahalting Program of SCM+FSA
for f being FinSeq-Location holds (IExec ((i ";" J),P,s)) . f = (IExec (J,P,(Exec (i,(Initialized s))))) . f
let s be State of SCM+FSA; ::_thesis: for i being parahalting keeping_0 Instruction of SCM+FSA
for J being parahalting Program of SCM+FSA
for f being FinSeq-Location holds (IExec ((i ";" J),P,s)) . f = (IExec (J,P,(Exec (i,(Initialized s))))) . f
let i be parahalting keeping_0 Instruction of SCM+FSA; ::_thesis: for J being parahalting Program of SCM+FSA
for f being FinSeq-Location holds (IExec ((i ";" J),P,s)) . f = (IExec (J,P,(Exec (i,(Initialized s))))) . f
let J be parahalting Program of SCM+FSA; ::_thesis: for f being FinSeq-Location holds (IExec ((i ";" J),P,s)) . f = (IExec (J,P,(Exec (i,(Initialized s))))) . f
let f be FinSeq-Location ; ::_thesis: (IExec ((i ";" J),P,s)) . f = (IExec (J,P,(Exec (i,(Initialized s))))) . f
A1: IExec ((Macro i),P,s) = Exec (i,(Initialized s)) by SCMFSA6C:5;
thus (IExec ((i ";" J),P,s)) . f = (IExec (J,P,(IExec ((Macro i),P,s)))) . f by SCMFSA6C:2
.= (IExec (J,P,(IExec ((Macro i),P,s)))) . f
.= (IExec (J,P,(Exec (i,(Initialized s))))) . f by A1
.= (IExec (J,P,(Exec (i,(Initialized s))))) . f ; ::_thesis: verum
end;

definition
let a be Int-Location;
let I, J be Program of SCM+FSA;
func if=0 (a,I,J) ->  Program of SCM+FSA equals :: SCMFSA8B:def 1
((((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA);
coherence
 ((((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA) is Program of SCM+FSA ;
func if>0 (a,I,J) ->  Program of SCM+FSA equals :: SCMFSA8B:def 2
((((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA);
coherence
 ((((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA) is Program of SCM+FSA ;
end;

:: deftheorem  defines if=0 SCMFSA8B:def_1_:_
 for a being Int-Location
for I, J being Program of SCM+FSA holds if=0 (a,I,J) = ((((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA);

:: deftheorem  defines if>0 SCMFSA8B:def_2_:_
 for a being Int-Location
for I, J being Program of SCM+FSA holds if>0 (a,I,J) = ((((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA);

definition
let a be Int-Location;
let I, J be Program of SCM+FSA;
func if<0 (a,I,J) ->  Program of SCM+FSA equals :: SCMFSA8B:def 3
 if=0 (a,J,(if>0 (a,J,I)));
coherence
 if=0 (a,J,(if>0 (a,J,I))) is Program of SCM+FSA ;
end;

:: deftheorem  defines if<0 SCMFSA8B:def_3_:_
 for a being Int-Location
for I, J being Program of SCM+FSA holds if<0 (a,I,J) = if=0 (a,J,(if>0 (a,J,I)));

Lm2:  for a being Int-Location
for I, J being Program of SCM+FSA holds
( 0 in dom (if=0 (a,I,J)) & 1 in dom (if=0 (a,I,J)) & 0 in dom (if>0 (a,I,J)) & 1 in dom (if>0 (a,I,J)) )
proof
let a be Int-Location; ::_thesis: for I, J being Program of SCM+FSA holds
( 0 in dom (if=0 (a,I,J)) & 1 in dom (if=0 (a,I,J)) & 0 in dom (if>0 (a,I,J)) & 1 in dom (if>0 (a,I,J)) )
let I, J be Program of SCM+FSA; ::_thesis: ( 0 in dom (if=0 (a,I,J)) & 1 in dom (if=0 (a,I,J)) & 0 in dom (if>0 (a,I,J)) & 1 in dom (if>0 (a,I,J)) )
set i = a =0_goto ((card J) + 3);
if=0 (a,I,J) = (((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA)) by SCMFSA6A:25
.= ((a =0_goto ((card J) + 3)) ";" J) ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA))) by SCMFSA6A:25
.= (a =0_goto ((card J) + 3)) ";" (J ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA)))) by SCMFSA6A:29
.= (Macro (a =0_goto ((card J) + 3))) ";" (J ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA)))) ;
then A1: dom (Macro (a =0_goto ((card J) + 3))) c= dom (if=0 (a,I,J)) by SCMFSA6A:17;
A2: dom (Macro (a =0_goto ((card J) + 3))) = {0,1} by COMPOS_1:61;
then A3: 1 in dom (Macro (a =0_goto ((card J) + 3))) by TARSKI:def_2;
 0 in dom (Macro (a =0_goto ((card J) + 3))) by A2, TARSKI:def_2;
hence  ( 0 in dom (if=0 (a,I,J)) & 1 in dom (if=0 (a,I,J)) ) by A1, A3; ::_thesis: ( 0 in dom (if>0 (a,I,J)) & 1 in dom (if>0 (a,I,J)) )
set i = a >0_goto ((card J) + 3);
if>0 (a,I,J) = (((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA)) by SCMFSA6A:25
.= ((a >0_goto ((card J) + 3)) ";" J) ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA))) by SCMFSA6A:25
.= (a >0_goto ((card J) + 3)) ";" (J ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA)))) by SCMFSA6A:29
.= (Macro (a >0_goto ((card J) + 3))) ";" (J ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA)))) ;
then A4: dom (Macro (a >0_goto ((card J) + 3))) c= dom (if>0 (a,I,J)) by SCMFSA6A:17;
A5: dom (Macro (a >0_goto ((card J) + 3))) = {0,1} by COMPOS_1:61;
then A6: 1 in dom (Macro (a >0_goto ((card J) + 3))) by TARSKI:def_2;
 0 in dom (Macro (a >0_goto ((card J) + 3))) by A5, TARSKI:def_2;
hence  ( 0 in dom (if>0 (a,I,J)) & 1 in dom (if>0 (a,I,J)) ) by A4, A6; ::_thesis: verum
end;

Lm3:  for a being Int-Location
for I, J being Program of SCM+FSA holds
( (if=0 (a,I,J)) . 0 = a =0_goto ((card J) + 3) & (if=0 (a,I,J)) . 1 = goto 2 & (if>0 (a,I,J)) . 0 = a >0_goto ((card J) + 3) & (if>0 (a,I,J)) . 1 = goto 2 )
proof
let a be Int-Location; ::_thesis: for I, J being Program of SCM+FSA holds
( (if=0 (a,I,J)) . 0 = a =0_goto ((card J) + 3) & (if=0 (a,I,J)) . 1 = goto 2 & (if>0 (a,I,J)) . 0 = a >0_goto ((card J) + 3) & (if>0 (a,I,J)) . 1 = goto 2 )
let I, J be Program of SCM+FSA; ::_thesis: ( (if=0 (a,I,J)) . 0 = a =0_goto ((card J) + 3) & (if=0 (a,I,J)) . 1 = goto 2 & (if>0 (a,I,J)) . 0 = a >0_goto ((card J) + 3) & (if>0 (a,I,J)) . 1 = goto 2 )
set i = a =0_goto ((card J) + 3);
A1: if=0 (a,I,J) = (((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA)) by SCMFSA6A:25
.= ((a =0_goto ((card J) + 3)) ";" J) ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA))) by SCMFSA6A:25
.= (a =0_goto ((card J) + 3)) ";" (J ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA)))) by SCMFSA6A:29
.= (Macro (a =0_goto ((card J) + 3))) ";" (J ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA)))) ;
A2: dom (Macro (a =0_goto ((card J) + 3))) = {0,1} by COMPOS_1:61;
then  0 in dom (Macro (a =0_goto ((card J) + 3))) by TARSKI:def_2;
hence (if=0 (a,I,J)) . 0 = (Directed (Macro (a =0_goto ((card J) + 3)))) . 0 by A1, SCMFSA8A:14
.= a =0_goto ((card J) + 3) by SCMFSA7B:1 ;
::_thesis: ( (if=0 (a,I,J)) . 1 = goto 2 & (if>0 (a,I,J)) . 0 = a >0_goto ((card J) + 3) & (if>0 (a,I,J)) . 1 = goto 2 )
 1 in dom (Macro (a =0_goto ((card J) + 3))) by A2, TARSKI:def_2;
hence (if=0 (a,I,J)) . 1 = (Directed (Macro (a =0_goto ((card J) + 3)))) . 1 by A1, SCMFSA8A:14
.= goto 2 by SCMFSA7B:2 ;
::_thesis: ( (if>0 (a,I,J)) . 0 = a >0_goto ((card J) + 3) & (if>0 (a,I,J)) . 1 = goto 2 )
set i = a >0_goto ((card J) + 3);
A3: if>0 (a,I,J) = (((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA)) by SCMFSA6A:25
.= ((a >0_goto ((card J) + 3)) ";" J) ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA))) by SCMFSA6A:25
.= (a >0_goto ((card J) + 3)) ";" (J ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA)))) by SCMFSA6A:29
.= (Macro (a >0_goto ((card J) + 3))) ";" (J ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA)))) ;
A4: dom (Macro (a >0_goto ((card J) + 3))) = {0,1} by COMPOS_1:61;
then  0 in dom (Macro (a >0_goto ((card J) + 3))) by TARSKI:def_2;
hence (if>0 (a,I,J)) . 0 = (Directed (Macro (a >0_goto ((card J) + 3)))) . 0 by A3, SCMFSA8A:14
.= a >0_goto ((card J) + 3) by SCMFSA7B:1 ;
::_thesis: (if>0 (a,I,J)) . 1 = goto 2
 1 in dom (Macro (a >0_goto ((card J) + 3))) by A4, TARSKI:def_2;
hence (if>0 (a,I,J)) . 1 = (Directed (Macro (a >0_goto ((card J) + 3)))) . 1 by A3, SCMFSA8A:14
.= goto 2 by SCMFSA7B:2 ;
::_thesis: verum
end;

theorem Th11: :: SCMFSA8B:11
for I, J being Program of SCM+FSA
for a being Int-Location holds card (if=0 (a,I,J)) = ((card I) + (card J)) + 4
proof
let I, J be Program of SCM+FSA; ::_thesis: for a being Int-Location holds card (if=0 (a,I,J)) = ((card I) + (card J)) + 4
let a be Int-Location; ::_thesis: card (if=0 (a,I,J)) = ((card I) + (card J)) + 4
A1: card (Stop SCM+FSA) = 1 by COMPOS_1:4;
thus  card (if=0 (a,I,J)) = (card ((((Macro (a =0_goto ((card J) + 3))) ";" J) ";" (Goto ((card I) + 1))) ";" I)) + 1 by A1, SCMFSA6A:21
.= ((card (((Macro (a =0_goto ((card J) + 3))) ";" J) ";" (Goto ((card I) + 1)))) + (card I)) + 1 by SCMFSA6A:21
.= (((card ((Macro (a =0_goto ((card J) + 3))) ";" J)) + (card (Goto ((card I) + 1)))) + (card I)) + 1 by SCMFSA6A:21
.= (((card ((Macro (a =0_goto ((card J) + 3))) ";" J)) + 1) + (card I)) + 1 by SCMFSA8A:15
.= ((((card (Macro (a =0_goto ((card J) + 3)))) + (card J)) + 1) + (card I)) + 1 by SCMFSA6A:21
.= (((2 + (card J)) + 1) + (card I)) + 1 by COMPOS_1:56
.= ((card I) + (card J)) + 4 ; ::_thesis: verum
end;

theorem Th12: :: SCMFSA8B:12
for I, J being Program of SCM+FSA
for a being Int-Location holds card (if>0 (a,I,J)) = ((card I) + (card J)) + 4
proof
let I, J be Program of SCM+FSA; ::_thesis: for a being Int-Location holds card (if>0 (a,I,J)) = ((card I) + (card J)) + 4
let a be Int-Location; ::_thesis: card (if>0 (a,I,J)) = ((card I) + (card J)) + 4
A1: card (Stop SCM+FSA) = 1 by COMPOS_1:4;
thus  card (if>0 (a,I,J)) = (card ((((Macro (a >0_goto ((card J) + 3))) ";" J) ";" (Goto ((card I) + 1))) ";" I)) + 1 by A1, SCMFSA6A:21
.= ((card (((Macro (a >0_goto ((card J) + 3))) ";" J) ";" (Goto ((card I) + 1)))) + (card I)) + 1 by SCMFSA6A:21
.= (((card ((Macro (a >0_goto ((card J) + 3))) ";" J)) + (card (Goto ((card I) + 1)))) + (card I)) + 1 by SCMFSA6A:21
.= (((card ((Macro (a >0_goto ((card J) + 3))) ";" J)) + 1) + (card I)) + 1 by SCMFSA8A:15
.= ((((card (Macro (a >0_goto ((card J) + 3)))) + (card J)) + 1) + (card I)) + 1 by SCMFSA6A:21
.= (((2 + (card J)) + 1) + (card I)) + 1 by COMPOS_1:56
.= ((card I) + (card J)) + 4 ; ::_thesis: verum
end;

theorem Th13: :: SCMFSA8B:13
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a = 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if=0 (a,I,J) is_closed_on s,P & if=0 (a,I,J) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a = 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if=0 (a,I,J) is_closed_on s,P & if=0 (a,I,J) is_halting_on s,P )
let s be State of SCM+FSA; ::_thesis: for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a = 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if=0 (a,I,J) is_closed_on s,P & if=0 (a,I,J) is_halting_on s,P )
let I, J be Program of SCM+FSA; ::_thesis: for a being read-write Int-Location st s . a = 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if=0 (a,I,J) is_closed_on s,P & if=0 (a,I,J) is_halting_on s,P )
let a be read-write Int-Location; ::_thesis: ( s . a = 0 & I is_closed_on s,P & I is_halting_on s,P implies ( if=0 (a,I,J) is_closed_on s,P & if=0 (a,I,J) is_halting_on s,P ) )
set I1 = I ";" (Stop SCM+FSA);
set s1 = Initialize s;
set P1 = P +* (I ";" (Stop SCM+FSA));
set s3 = Initialize s;
set P3 = P +* (if=0 (a,I,J));
set s4 = Comput ((P +* (if=0 (a,I,J))),(Initialize s),1);
set i = a =0_goto ((card J) + 3);
A1: not a in dom (Start-At (0,SCM+FSA)) by SCMFSA_2:102;
A2: 0 in dom (if=0 (a,I,J)) by Lm2;
A3: (P +* (if=0 (a,I,J))) . 0 = (if=0 (a,I,J)) . 0 by A2, FUNCT_4:13
.= a =0_goto ((card J) + 3) by Lm3 ;
 IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
then A4: IC (Initialize s) = IC (Start-At (0,SCM+FSA)) by FUNCT_4:13
.= 0 by FUNCOP_1:72 ;
A5: if=0 (a,I,J) c= P +* (if=0 (a,I,J)) by FUNCT_4:25;
A6: if=0 (a,I,J) = (((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA)) by SCMFSA6A:25;
card (((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) = (card ((Macro (a =0_goto ((card J) + 3))) ";" J)) + (card (Goto ((card I) + 1))) by SCMFSA6A:21
.= (card ((Macro (a =0_goto ((card J) + 3))) ";" J)) + 1 by SCMFSA8A:15
.= ((card (Macro (a =0_goto ((card J) + 3)))) + (card J)) + 1 by SCMFSA6A:21
.= ((card J) + 2) + 1 by COMPOS_1:56
.= (card J) + (2 + 1) ;
then A7: Reloc ((I ";" (Stop SCM+FSA)),((card J) + 3)) c= if=0 (a,I,J) by A6, Lm1;
A8: Reloc ((I ";" (Stop SCM+FSA)),((card J) + 3)) c= P +* (if=0 (a,I,J)) by A7, A5, XBOOLE_1:1;
A9: Comput ((P +* (if=0 (a,I,J))),(Initialize s),(0 + 1)) = Following ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (if=0 (a,I,J))),(Initialize s))
.= Exec ((a =0_goto ((card J) + 3)),(Initialize s)) by A4, A3, PBOOLE:143 ;
A10: for f being FinSeq-Location holds (Initialize s) . f = (Comput ((P +* (if=0 (a,I,J))),(Initialize s),1)) . f by A9, SCMFSA_2:70;
 for a being Int-Location holds (Initialize s) . a = (Comput ((P +* (if=0 (a,I,J))),(Initialize s),1)) . a by A9, SCMFSA_2:70;
then A11: DataPart (Initialize s) = DataPart (Comput ((P +* (if=0 (a,I,J))),(Initialize s),1)) by A10, SCMFSA_M:2;
assume  s . a = 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or ( if=0 (a,I,J) is_closed_on s,P & if=0 (a,I,J) is_halting_on s,P ) )
then  (Initialize s) . a = 0 by A1, FUNCT_4:11;
then A12: IC (Comput ((P +* (if=0 (a,I,J))),(Initialize s),1)) = (card J) + 3 by A9, SCMFSA_2:70;
assume A13: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or ( if=0 (a,I,J) is_closed_on s,P & if=0 (a,I,J) is_halting_on s,P ) )
assume A14: I is_halting_on s,P ; ::_thesis: ( if=0 (a,I,J) is_closed_on s,P & if=0 (a,I,J) is_halting_on s,P )
then A15: I ";" (Stop SCM+FSA) is_closed_on s,P by A13, SCMFSA8A:30;
 I ";" (Stop SCM+FSA) is_halting_on s,P by A13, A14, SCMFSA8A:30;
then A16: P +* (I ";" (Stop SCM+FSA)) halts_on Initialize s by SCMFSA7B:def_7;
 DataPart s = DataPart (Initialize s) by MEMSTR_0:79;
then A17: I ";" (Stop SCM+FSA) is_closed_on Initialize s,P +* (I ";" (Stop SCM+FSA)) by A15, Th3;
A18: I ";" (Stop SCM+FSA) c= P +* (I ";" (Stop SCM+FSA)) by FUNCT_4:25;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(if=0_(a,I,J))),(Initialize_s),k))_in_dom_(if=0_(a,I,J))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (if=0 (a,I,J))),(Initialize s),b1)) in dom (if=0 (a,I,J))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (if=0 (a,I,J))),(Initialize s),b1)) in dom (if=0 (a,I,J))
then consider k1 being Nat such that
A19: k1 + 1 = k by NAT_1:6;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def_12;
reconsider m = IC (Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialize s),k1)) as Element of NAT ;
 m in dom (I ";" (Stop SCM+FSA)) by A15, SCMFSA7B:def_6;
then A20: m < card (I ";" (Stop SCM+FSA)) by AFINSQ_1:66;
 card (Stop SCM+FSA) = 1 by COMPOS_1:4;
then A21: card (I ";" (Stop SCM+FSA)) = (card I) + 1 by SCMFSA6A:21;
card (if=0 (a,I,J)) = ((card I) + (card J)) + 4 by Th11
.= ((card J) + 3) + (card (I ";" (Stop SCM+FSA))) by A21 ;
then A22: m + ((card J) + 3) < card (if=0 (a,I,J)) by A20, XREAL_1:6;
IC (Comput ((P +* (if=0 (a,I,J))),(Initialize s),k)) = IC (Comput ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialize s),1)),k1)) by A19, EXTPRO_1:4
.= m + ((card J) + 3) by A17, A12, A11, Th8, A18, A8 ;
hence  IC (Comput ((P +* (if=0 (a,I,J))),(Initialize s),k)) in dom (if=0 (a,I,J)) by A22, AFINSQ_1:66; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (if=0 (a,I,J))),(Initialize s),b1)) in dom (if=0 (a,I,J))
hence  IC (Comput ((P +* (if=0 (a,I,J))),(Initialize s),k)) in dom (if=0 (a,I,J)) by A2, A4, EXTPRO_1:2; ::_thesis: verum
end;
end;
end;
hence  if=0 (a,I,J) is_closed_on s,P by SCMFSA7B:def_6; ::_thesis: if=0 (a,I,J) is_halting_on s,P
CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialize s),((LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialize s))) + 1)))) = CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialize s),1)),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialize s)))))) by EXTPRO_1:4
.= IncAddr ((CurInstr ((P +* (I ";" (Stop SCM+FSA))),(Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialize s),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialize s))))))),((card J) + 3)) by A17, A12, A11, Th8, A8, A18
.= IncAddr ((halt SCM+FSA),((card J) + 3)) by A16, EXTPRO_1:def_15
.= halt SCM+FSA by COMPOS_0:4 ;
then  P +* (if=0 (a,I,J)) halts_on Initialize s by EXTPRO_1:29;
hence  if=0 (a,I,J) is_halting_on s,P by SCMFSA7B:def_7; ::_thesis: verum
end;

theorem Th14: :: SCMFSA8B:14
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a = 0 & I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a = 0 & I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
let s be State of SCM+FSA; ::_thesis: for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a = 0 & I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
let I, J be Program of SCM+FSA; ::_thesis: for a being read-write Int-Location st s . a = 0 & I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
let a be read-write Int-Location; ::_thesis: ( s . a = 0 & I is_closed_on Initialized s,P & I is_halting_on Initialized s,P implies IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
set I1 = I ";" (Stop SCM+FSA);
set s1 = Initialized s;
set P1 = P +* (I ";" (Stop SCM+FSA));
set P3 = P +* (if=0 (a,I,J));
A1: I ";" (Stop SCM+FSA) c= P +* (I ";" (Stop SCM+FSA)) by FUNCT_4:25;
set s4 = Comput ((P +* (if=0 (a,I,J))),(Initialized s),1);
set i = a =0_goto ((card J) + 3);
A2: if=0 (a,I,J) = (((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA)) by SCMFSA6A:25;
A3: 0 in dom (if=0 (a,I,J)) by Lm2;
A4: (P +* (if=0 (a,I,J))) . 0 = (if=0 (a,I,J)) . 0 by A3, FUNCT_4:13
.= a =0_goto ((card J) + 3) by Lm3 ;
A5: dom (Initialize ((intloc 0) .--> 1)) = {(intloc 0),(IC )} by SCMFSA_M:11;
 ( a <> intloc 0 & a <> IC ) by SCMFSA_2:56;
then A6: not a in dom (Initialize ((intloc 0) .--> 1)) by A5, TARSKI:def_2;
 IC in dom (Initialize ((intloc 0) .--> 1)) by MEMSTR_0:48;
then A7: IC (Initialized s) = IC (Initialize ((intloc 0) .--> 1)) by FUNCT_4:13
.= 0 by MEMSTR_0:def_11 ;
A8: Comput ((P +* (if=0 (a,I,J))),(Initialized s),(0 + 1)) = Following ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),0))) by EXTPRO_1:3
.= Following ((P +* (if=0 (a,I,J))),(Initialized s))
.= Exec ((a =0_goto ((card J) + 3)),(Initialized s)) by A7, A4, PBOOLE:143 ;
A9: if=0 (a,I,J) c= P +* (if=0 (a,I,J)) by FUNCT_4:25;
assume  s . a = 0 ; ::_thesis: ( not I is_closed_on Initialized s,P or not I is_halting_on Initialized s,P or IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
then  (Initialized s) . a = 0 by A6, FUNCT_4:11;
then A10: IC (Comput ((P +* (if=0 (a,I,J))),(Initialized s),1)) = (card J) + 3 by A8, SCMFSA_2:70;
A11: for f being FinSeq-Location holds (Initialized s) . f = (Comput ((P +* (if=0 (a,I,J))),(Initialized s),1)) . f by A8, SCMFSA_2:70;
 for a being Int-Location holds (Initialized s) . a = (Comput ((P +* (if=0 (a,I,J))),(Initialized s),1)) . a by A8, SCMFSA_2:70;
then A12: DataPart (Initialized s) = DataPart (Comput ((P +* (if=0 (a,I,J))),(Initialized s),1)) by A11, SCMFSA_M:2;
card (((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) = (card ((Macro (a =0_goto ((card J) + 3))) ";" J)) + (card (Goto ((card I) + 1))) by SCMFSA6A:21
.= (card ((Macro (a =0_goto ((card J) + 3))) ";" J)) + 1 by SCMFSA8A:15
.= ((card (Macro (a =0_goto ((card J) + 3)))) + (card J)) + 1 by SCMFSA6A:21
.= ((card J) + 2) + 1 by COMPOS_1:56
.= (card J) + (2 + 1) ;
then A13: Reloc ((I ";" (Stop SCM+FSA)),((card J) + 3)) c= if=0 (a,I,J) by A2, Lm1;
A14: Reloc ((I ";" (Stop SCM+FSA)),((card J) + 3)) c= P +* (if=0 (a,I,J)) by A13, A9, XBOOLE_1:1;
assume A15: I is_closed_on Initialized s,P ; ::_thesis: ( not I is_halting_on Initialized s,P or IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
assume A16: I is_halting_on Initialized s,P ; ::_thesis: IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
then A17: P +* (I ";" (Stop SCM+FSA)) halts_on Initialized s by A15, SCMFSA8A:34;
 I ";" (Stop SCM+FSA) is_closed_on Initialized s,P by A15, A16, SCMFSA8A:30;
then A18: I ";" (Stop SCM+FSA) is_closed_on Initialized s,P +* (I ";" (Stop SCM+FSA)) by Th6;
A19: CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),((LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) + 1)))) = CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),1)),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s)))))) by EXTPRO_1:4
.= IncAddr ((CurInstr ((P +* (I ";" (Stop SCM+FSA))),(Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialized s),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))))))),((card J) + 3)) by A18, A10, A12, Th8, A14, A1
.= IncAddr ((halt SCM+FSA),((card J) + 3)) by A17, EXTPRO_1:def_15
.= halt SCM+FSA by COMPOS_0:4 ;
then A20: P +* (if=0 (a,I,J)) halts_on Initialized s by EXTPRO_1:29;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_(LifeSpan_((P_+*_(I_";"_(Stop_SCM+FSA))),(Initialized_s)))_+_1_holds_
CurInstr_((P_+*_(if=0_(a,I,J))),(Comput_((P_+*_(if=0_(a,I,J))),(Initialized_s),l)))_<>_halt_SCM+FSA
let l be Element of NAT ; ::_thesis: ( l < (LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) + 1 implies CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),b1))) <> halt SCM+FSA )
assume A21: l < (LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) + 1 ; ::_thesis: CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),b1))) <> halt SCM+FSA
A22: Comput ((P +* (if=0 (a,I,J))),(Initialized s),0) = Initialized s ;
percases ( l = 0 or l <> 0 ) ;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),b1))) <> halt SCM+FSA
hence  CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),l))) <> halt SCM+FSA by A7, A4, A22, PBOOLE:143; ::_thesis: verum
end;
suppose l <> 0 ; ::_thesis: not CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),b1))) = halt SCM+FSA
then consider n being Nat such that
A23: l = n + 1 by NAT_1:6;
assume A24: CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),l))) = halt SCM+FSA ; ::_thesis: contradiction
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A25: InsCode (halt SCM+FSA) = 0 by COMPOS_1:70;
InsCode (CurInstr ((P +* (I ";" (Stop SCM+FSA))),(Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialized s),n)))) = InsCode (IncAddr ((CurInstr ((P +* (I ";" (Stop SCM+FSA))),(Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialized s),n)))),((card J) + 3))) by COMPOS_0:def_9
.= InsCode (CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),1)),n)))) by A18, A10, A12, Th8, A14, A1
.= 0 by A23, A24, EXTPRO_1:4, A25 ;
then A26: CurInstr ((P +* (I ";" (Stop SCM+FSA))),(Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialized s),n))) = halt SCM+FSA by SCMFSA_2:95;
 n < LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s)) by A21, A23, XREAL_1:6;
hence  contradiction by A17, A26, EXTPRO_1:def_15; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),l))) = halt SCM+FSA holds
(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) + 1 <= l ;
then A27: LifeSpan ((P +* (if=0 (a,I,J))),(Initialized s)) = (LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) + 1 by A19, A20, EXTPRO_1:def_15;
A28: DataPart (Result ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) = DataPart (Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialized s),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))))) by A15, A16, EXTPRO_1:23, SCMFSA8A:34
.= DataPart (Comput ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),1)),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))))) by A18, A10, A12, Th8, A1, A14
.= DataPart (Comput ((P +* (if=0 (a,I,J))),(Initialized s),((LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (if=0 (a,I,J))),(Initialized s))) by A20, A27, EXTPRO_1:23 ;
A29: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if=0_(a,I,J)),P,s))_holds_
(IExec_((if=0_(a,I,J)),P,s))_._x_=_((IExec_((I_";"_(Stop_SCM+FSA)),P,s))_+*_(Start-At_((((card_I)_+_(card_J))_+_3),SCM+FSA)))_._x
let x be set ; ::_thesis: ( x in dom (IExec ((if=0 (a,I,J)),P,s)) implies (IExec ((if=0 (a,I,J)),P,s)) . b1 = ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1 )
A30: IExec ((I ";" (Stop SCM+FSA)),P,s) = Result ((P +* (I ";" (Stop SCM+FSA))),(Initialized s)) by SCMFSA6B:def_1;
A31: dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) = {(IC )} by FUNCOP_1:13;
A32: IExec ((if=0 (a,I,J)),P,s) = Result ((P +* (if=0 (a,I,J))),(Initialized s)) by SCMFSA6B:def_1;
assume A33: x in dom (IExec ((if=0 (a,I,J)),P,s)) ; ::_thesis: (IExec ((if=0 (a,I,J)),P,s)) . b1 = ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1
percases ( x is Int-Location or x is FinSeq-Location or x = IC ) by A33, SCMFSA_M:1;
supposeA34: x is Int-Location ; ::_thesis: (IExec ((if=0 (a,I,J)),P,s)) . b1 = ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1
then  x <> IC by SCMFSA_2:56;
then A35: not x in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A31, TARSKI:def_1;
thus (IExec ((if=0 (a,I,J)),P,s)) . x = (Result ((P +* (if=0 (a,I,J))),(Initialized s))) . x by A32
.= (Result ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) . x by A28, A34, SCMFSA_M:2
.= (IExec ((I ";" (Stop SCM+FSA)),P,s)) . x by A30
.= ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . x by A35, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA36: x is FinSeq-Location ; ::_thesis: (IExec ((if=0 (a,I,J)),P,s)) . b1 = ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1
then  x <> IC by SCMFSA_2:57;
then A37: not x in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A31, TARSKI:def_1;
thus (IExec ((if=0 (a,I,J)),P,s)) . x = (Result ((P +* (if=0 (a,I,J))),(Initialized s))) . x by A32
.= (Result ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) . x by A28, A36, SCMFSA_M:2
.= (IExec ((I ";" (Stop SCM+FSA)),P,s)) . x by A30
.= ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . x by A37, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA38: x = IC ; ::_thesis: (IExec ((if=0 (a,I,J)),P,s)) . b1 = ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1
then A39: x in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A31, TARSKI:def_1;
A40: IC (Result ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) = (IExec ((I ";" (Stop SCM+FSA)),P,s)) . (IC ) by A30
.= IC ((IExec (I,P,s)) +* (Start-At ((card I),SCM+FSA))) by A15, A16, SCMFSA8A:36
.= card I by FUNCT_4:113 ;
thus (IExec ((if=0 (a,I,J)),P,s)) . x = (Result ((P +* (if=0 (a,I,J))),(Initialized s))) . x by A32
.= (Comput ((P +* (if=0 (a,I,J))),(Initialized s),((LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) + 1))) . x by A20, A27, EXTPRO_1:23
.= IC (Comput ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),1)),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))))) by A38, EXTPRO_1:4
.= (IC (Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialized s),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s)))))) + ((card J) + 3) by A18, A10, A12, Th8, A14, A1
.= (IC (Result ((P +* (I ";" (Stop SCM+FSA))),(Initialized s)))) + ((card J) + 3) by A15, A16, EXTPRO_1:23, SCMFSA8A:34
.= (Start-At (((card I) + ((card J) + 3)),SCM+FSA)) . (IC ) by A40, FUNCOP_1:72
.= ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . x by A38, A39, FUNCT_4:13 ; ::_thesis: verum
end;
end;
end;
dom (IExec ((if=0 (a,I,J)),P,s)) = the carrier of SCM+FSA by PARTFUN1:def_2
.= dom ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) by PARTFUN1:def_2 ;
hence  IExec ((if=0 (a,I,J)),P,s) = (IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A29, FUNCT_1:2
.= ((IExec (I,P,s)) +* (Start-At ((card I),SCM+FSA))) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A15, A16, SCMFSA8A:36
.= (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by FUNCT_4:114 ;
::_thesis: verum
end;

theorem Th15: :: SCMFSA8B:15
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a <> 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if=0 (a,I,J) is_closed_on s,P & if=0 (a,I,J) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a <> 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if=0 (a,I,J) is_closed_on s,P & if=0 (a,I,J) is_halting_on s,P )
let s be State of SCM+FSA; ::_thesis: for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a <> 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if=0 (a,I,J) is_closed_on s,P & if=0 (a,I,J) is_halting_on s,P )
let I, J be Program of SCM+FSA; ::_thesis: for a being read-write Int-Location st s . a <> 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if=0 (a,I,J) is_closed_on s,P & if=0 (a,I,J) is_halting_on s,P )
let a be read-write Int-Location; ::_thesis: ( s . a <> 0 & J is_closed_on s,P & J is_halting_on s,P implies ( if=0 (a,I,J) is_closed_on s,P & if=0 (a,I,J) is_halting_on s,P ) )
set I1 = I ";" (Stop SCM+FSA);
set JI2 = ((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA);
set s2 = Initialize s;
set P2 = P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA));
A1: ((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA) c= P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) by FUNCT_4:25;
set s3 = Initialize s;
set P3 = P +* (if=0 (a,I,J));
set s4 = Comput ((P +* (if=0 (a,I,J))),(Initialize s),1);
set s5 = Comput ((P +* (if=0 (a,I,J))),(Initialize s),2);
set i = a =0_goto ((card J) + 3);
 IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
then A2: IC (Initialize s) = IC (Start-At (0,SCM+FSA)) by FUNCT_4:13
.= 0 by FUNCOP_1:72 ;
A3: if=0 (a,I,J) c= P +* (if=0 (a,I,J)) by FUNCT_4:25;
A4: if=0 (a,I,J) = (((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA)) by SCMFSA6A:25
.= ((a =0_goto ((card J) + 3)) ";" J) ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA))) by SCMFSA6A:25
.= (a =0_goto ((card J) + 3)) ";" (J ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA)))) by SCMFSA6A:29
.= (a =0_goto ((card J) + 3)) ";" ((J ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA))) by SCMFSA6A:25
.= (Macro (a =0_goto ((card J) + 3))) ";" (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) by SCMFSA6A:25 ;
then  Reloc ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),(card (Macro (a =0_goto ((card J) + 3))))) c= if=0 (a,I,J) by Lm1;
then A5: Reloc ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),2) c= if=0 (a,I,J) by COMPOS_1:56;
A6: Reloc ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),2) c= P +* (if=0 (a,I,J)) by A5, A3, XBOOLE_1:1;
A7: not a in dom (Start-At (0,SCM+FSA)) by SCMFSA_2:102;
A8: 0 in dom (if=0 (a,I,J)) by Lm2;
A9: if=0 (a,I,J) c= P +* (if=0 (a,I,J)) by FUNCT_4:25;
A10: (P +* (if=0 (a,I,J))) . 0 = (if=0 (a,I,J)) . 0 by A8, FUNCT_4:13
.= a =0_goto ((card J) + 3) by Lm3 ;
A11: 1 in dom (if=0 (a,I,J)) by Lm2;
A12: Comput ((P +* (if=0 (a,I,J))),(Initialize s),(0 + 1)) = Following ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (if=0 (a,I,J))),(Initialize s))
.= Exec ((a =0_goto ((card J) + 3)),(Initialize s)) by A2, A10, PBOOLE:143 ;
assume  s . a <> 0 ; ::_thesis: ( not J is_closed_on s,P or not J is_halting_on s,P or ( if=0 (a,I,J) is_closed_on s,P & if=0 (a,I,J) is_halting_on s,P ) )
then  (Initialize s) . a <> 0 by A7, FUNCT_4:11;
then A13: IC (Comput ((P +* (if=0 (a,I,J))),(Initialize s),1)) = succ 0 by A2, A12, SCMFSA_2:70
.= 0 + 1 ;
A14: (P +* (if=0 (a,I,J))) . 1 = (if=0 (a,I,J)) . 1 by A11, A9, GRFUNC_1:2
.= goto 2 by Lm3 ;
assume A15: J is_closed_on s,P ; ::_thesis: ( not J is_halting_on s,P or ( if=0 (a,I,J) is_closed_on s,P & if=0 (a,I,J) is_halting_on s,P ) )
A16: Comput ((P +* (if=0 (a,I,J))),(Initialize s),(1 + 1)) = Following ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialize s),1))) by EXTPRO_1:3
.= Exec ((goto 2),(Comput ((P +* (if=0 (a,I,J))),(Initialize s),1))) by A13, A14, PBOOLE:143 ;
A17: now__::_thesis:_for_f_being_FinSeq-Location_holds_(Initialize_s)_._f_=_(Comput_((P_+*_(if=0_(a,I,J))),(Initialize_s),2))_._f
let f be FinSeq-Location ; ::_thesis: (Initialize s) . f = (Comput ((P +* (if=0 (a,I,J))),(Initialize s),2)) . f
thus (Initialize s) . f = (Comput ((P +* (if=0 (a,I,J))),(Initialize s),1)) . f by A12, SCMFSA_2:70
.= (Comput ((P +* (if=0 (a,I,J))),(Initialize s),2)) . f by A16, SCMFSA_2:69 ; ::_thesis: verum
end;
now__::_thesis:_for_a_being_Int-Location_holds_(Initialize_s)_._a_=_(Comput_((P_+*_(if=0_(a,I,J))),(Initialize_s),2))_._a
let a be Int-Location; ::_thesis: (Initialize s) . a = (Comput ((P +* (if=0 (a,I,J))),(Initialize s),2)) . a
thus (Initialize s) . a = (Comput ((P +* (if=0 (a,I,J))),(Initialize s),1)) . a by A12, SCMFSA_2:70
.= (Comput ((P +* (if=0 (a,I,J))),(Initialize s),2)) . a by A16, SCMFSA_2:69 ; ::_thesis: verum
end;
then A18: DataPart (Initialize s) = DataPart (Comput ((P +* (if=0 (a,I,J))),(Initialize s),2)) by A17, SCMFSA_M:2;
assume A19: J is_halting_on s,P ; ::_thesis: ( if=0 (a,I,J) is_closed_on s,P & if=0 (a,I,J) is_halting_on s,P )
then A20: ((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA) is_closed_on s,P by A15, SCMFSA8A:37;
then A21: ((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA) is_closed_on Initialize s,P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) by Th7;
A22: P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) halts_on Initialize s by A15, A19, SCMFSA8A:38;
A23: IC (Comput ((P +* (if=0 (a,I,J))),(Initialize s),2)) = 2 by A16, SCMFSA_2:69;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(if=0_(a,I,J))),(Initialize_s),k))_in_dom_(if=0_(a,I,J))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (if=0 (a,I,J))),(Initialize s),b1)) in dom (if=0 (a,I,J))
 ( k = 0 or 0 + 1 < k + 1 ) by XREAL_1:6;
then A24: ( k = 0 or 1 <= k ) by NAT_1:13;
percases ( 1 < k or k = 0 or k = 1 ) by A24, XXREAL_0:1;
supposeA25: 1 < k ; ::_thesis: IC (Comput ((P +* (if=0 (a,I,J))),(Initialize s),b1)) in dom (if=0 (a,I,J))
then consider k1 being Nat such that
A26: k1 + 1 = k by NAT_1:6;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def_12;
 0 + 1 < k1 + 1 by A25, A26;
then consider k2 being Nat such that
A27: k2 + 1 = k1 by NAT_1:6;
reconsider k2 = k2 as Element of NAT by ORDINAL1:def_12;
reconsider m = IC (Comput ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialize s),k2)) as Element of NAT ;
A28: card (if=0 (a,I,J)) = (card (Macro (a =0_goto ((card J) + 3)))) + (card (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))) by A4, SCMFSA6A:21
.= 2 + (card (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))) by COMPOS_1:56 ;
 m in dom (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) by A20, SCMFSA7B:def_6;
then  m < card (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) by AFINSQ_1:66;
then A29: m + 2 < card (if=0 (a,I,J)) by A28, XREAL_1:6;
IC (Comput ((P +* (if=0 (a,I,J))),(Initialize s),k)) = IC (Comput ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialize s),1)),k1)) by A26, EXTPRO_1:4
.= IC (Comput ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialize s),1)),1)),k2)) by A27, EXTPRO_1:4
.= IC (Comput ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialize s),(1 + 1))),k2)) by EXTPRO_1:4
.= m + 2 by A21, A23, A18, Th8, A6, A1 ;
hence  IC (Comput ((P +* (if=0 (a,I,J))),(Initialize s),k)) in dom (if=0 (a,I,J)) by A29, AFINSQ_1:66; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (if=0 (a,I,J))),(Initialize s),b1)) in dom (if=0 (a,I,J))
hence  IC (Comput ((P +* (if=0 (a,I,J))),(Initialize s),k)) in dom (if=0 (a,I,J)) by A8, A2, EXTPRO_1:2; ::_thesis: verum
end;
suppose k = 1 ; ::_thesis: IC (Comput ((P +* (if=0 (a,I,J))),(Initialize s),b1)) in dom (if=0 (a,I,J))
hence  IC (Comput ((P +* (if=0 (a,I,J))),(Initialize s),k)) in dom (if=0 (a,I,J)) by A13, Lm2; ::_thesis: verum
end;
end;
end;
hence  if=0 (a,I,J) is_closed_on s,P by SCMFSA7B:def_6; ::_thesis: if=0 (a,I,J) is_halting_on s,P
CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialize s),((LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialize s))) + 2)))) = CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialize s),2)),(LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialize s)))))) by EXTPRO_1:4
.= IncAddr ((CurInstr ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Comput ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialize s),(LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialize s))))))),2) by A21, A23, A18, Th8, A6, A1
.= IncAddr ((halt SCM+FSA),2) by A22, EXTPRO_1:def_15
.= halt SCM+FSA by COMPOS_0:4 ;
then  P +* (if=0 (a,I,J)) halts_on Initialize s by EXTPRO_1:29;
hence  if=0 (a,I,J) is_halting_on s,P by SCMFSA7B:def_7; ::_thesis: verum
end;

theorem Th16: :: SCMFSA8B:16
for P being Instruction-Sequence of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st s . a <> 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P holds
IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for I, J being Program of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st s . a <> 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P holds
IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
let I, J be Program of SCM+FSA; ::_thesis: for a being read-write Int-Location
for s being State of SCM+FSA st s . a <> 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P holds
IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
let a be read-write Int-Location; ::_thesis: for s being State of SCM+FSA st s . a <> 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P holds
IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
let s be State of SCM+FSA; ::_thesis: ( s . a <> 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P implies IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
set I1 = I ";" (Stop SCM+FSA);
set JI2 = ((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA);
set s2 = Initialized s;
set P2 = P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA));
A1: ((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA) c= P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) by FUNCT_4:25;
set P3 = P +* (if=0 (a,I,J));
set s4 = Comput ((P +* (if=0 (a,I,J))),(Initialized s),1);
set s5 = Comput ((P +* (if=0 (a,I,J))),(Initialized s),2);
set i = a =0_goto ((card J) + 3);
 0 in dom (if=0 (a,I,J)) by Lm2;
then A2: (P +* (if=0 (a,I,J))) . 0 = (if=0 (a,I,J)) . 0 by FUNCT_4:13
.= a =0_goto ((card J) + 3) by Lm3 ;
A3: 1 in dom (if=0 (a,I,J)) by Lm2;
A4: (P +* (if=0 (a,I,J))) . 1 = (if=0 (a,I,J)) . 1 by A3, FUNCT_4:13
.= goto 2 by Lm3 ;
A5: dom (Initialize ((intloc 0) .--> 1)) = {(intloc 0),(IC )} by SCMFSA_M:11;
 ( a <> intloc 0 & a <> IC ) by SCMFSA_2:56;
then A6: not a in dom (Initialize ((intloc 0) .--> 1)) by A5, TARSKI:def_2;
 IC in dom (Initialize ((intloc 0) .--> 1)) by MEMSTR_0:48;
then A7: IC (Initialized s) = IC (Initialize ((intloc 0) .--> 1)) by FUNCT_4:13
.= 0 by MEMSTR_0:def_11 ;
if=0 (a,I,J) = (((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA)) by SCMFSA6A:25
.= ((a =0_goto ((card J) + 3)) ";" J) ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA))) by SCMFSA6A:25
.= (a =0_goto ((card J) + 3)) ";" (J ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA)))) by SCMFSA6A:29
.= (a =0_goto ((card J) + 3)) ";" ((J ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA))) by SCMFSA6A:25
.= (Macro (a =0_goto ((card J) + 3))) ";" (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) by SCMFSA6A:25 ;
then  Reloc ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),(card (Macro (a =0_goto ((card J) + 3))))) c= if=0 (a,I,J) by Lm1;
then A8: Reloc ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),2) c= if=0 (a,I,J) by COMPOS_1:56;
 if=0 (a,I,J) c= P +* (if=0 (a,I,J)) by FUNCT_4:25;
then A9: Reloc ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),2) c= P +* (if=0 (a,I,J)) by A8, XBOOLE_1:1;
A10: Comput ((P +* (if=0 (a,I,J))),(Initialized s),(0 + 1)) = Following ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),0))) by EXTPRO_1:3
.= Following ((P +* (if=0 (a,I,J))),(Initialized s))
.= Exec ((a =0_goto ((card J) + 3)),(Initialized s)) by A7, A2, PBOOLE:143 ;
assume  s . a <> 0 ; ::_thesis: ( not J is_closed_on Initialized s,P or not J is_halting_on Initialized s,P or IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
then  (Initialized s) . a <> 0 by A6, FUNCT_4:11;
then A11: IC (Comput ((P +* (if=0 (a,I,J))),(Initialized s),1)) = succ 0 by A7, A10, SCMFSA_2:70
.= 0 + 1 ;
assume A12: J is_closed_on Initialized s,P ; ::_thesis: ( not J is_halting_on Initialized s,P or IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
A13: Comput ((P +* (if=0 (a,I,J))),(Initialized s),(1 + 1)) = Following ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),1))) by EXTPRO_1:3
.= Exec ((goto 2),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),1))) by A11, A4, PBOOLE:143 ;
then A14: IC (Comput ((P +* (if=0 (a,I,J))),(Initialized s),2)) = 2 by SCMFSA_2:69;
A15: now__::_thesis:_for_f_being_FinSeq-Location_holds_(Initialized_s)_._f_=_(Comput_((P_+*_(if=0_(a,I,J))),(Initialized_s),2))_._f
let f be FinSeq-Location ; ::_thesis: (Initialized s) . f = (Comput ((P +* (if=0 (a,I,J))),(Initialized s),2)) . f
thus (Initialized s) . f = (Comput ((P +* (if=0 (a,I,J))),(Initialized s),1)) . f by A10, SCMFSA_2:70
.= (Comput ((P +* (if=0 (a,I,J))),(Initialized s),2)) . f by A13, SCMFSA_2:69 ; ::_thesis: verum
end;
now__::_thesis:_for_a_being_Int-Location_holds_(Initialized_s)_._a_=_(Comput_((P_+*_(if=0_(a,I,J))),(Initialized_s),2))_._a
let a be Int-Location; ::_thesis: (Initialized s) . a = (Comput ((P +* (if=0 (a,I,J))),(Initialized s),2)) . a
thus (Initialized s) . a = (Comput ((P +* (if=0 (a,I,J))),(Initialized s),1)) . a by A10, SCMFSA_2:70
.= (Comput ((P +* (if=0 (a,I,J))),(Initialized s),2)) . a by A13, SCMFSA_2:69 ; ::_thesis: verum
end;
then A16: DataPart (Initialized s) = DataPart (Comput ((P +* (if=0 (a,I,J))),(Initialized s),2)) by A15, SCMFSA_M:2;
assume A17: J is_halting_on Initialized s,P ; ::_thesis: IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
then A18: P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) halts_on Initialized s by A12, SCMFSA8A:39;
 ((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA) is_closed_on Initialized s,P by A12, A17, SCMFSA8A:37;
then A19: ((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA) is_closed_on Initialized s,P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) by Th6;
A20: CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),((LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) + 2)))) = CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),2)),(LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s)))))) by EXTPRO_1:4
.= IncAddr ((CurInstr ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Comput ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s),(LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))))))),2) by A19, A9, A14, A16, Th8, A1
.= IncAddr ((halt SCM+FSA),2) by A18, EXTPRO_1:def_15
.= halt SCM+FSA by COMPOS_0:4 ;
then A21: P +* (if=0 (a,I,J)) halts_on Initialized s by EXTPRO_1:29;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_(LifeSpan_((P_+*_(((J_";"_(Goto_((card_I)_+_1)))_";"_I)_";"_(Stop_SCM+FSA))),(Initialized_s)))_+_2_holds_
CurInstr_((P_+*_(if=0_(a,I,J))),(Comput_((P_+*_(if=0_(a,I,J))),(Initialized_s),l)))_<>_halt_SCM+FSA
let l be Element of NAT ; ::_thesis: ( l < (LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) + 2 implies CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),b1))) <> halt SCM+FSA )
assume A22: l < (LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) + 2 ; ::_thesis: CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),b1))) <> halt SCM+FSA
A23: Comput ((P +* (if=0 (a,I,J))),(Initialized s),0) = Initialized s ;
percases ( l = 0 or l = 1 or ( l <> 0 & l <> 1 ) ) ;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),b1))) <> halt SCM+FSA
hence  CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),l))) <> halt SCM+FSA by A7, A2, A23, PBOOLE:143; ::_thesis: verum
end;
suppose l = 1 ; ::_thesis: CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),b1))) <> halt SCM+FSA
hence  CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),l))) <> halt SCM+FSA by A11, A4, PBOOLE:143; ::_thesis: verum
end;
supposeA24: ( l <> 0 & l <> 1 ) ; ::_thesis: not CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),b1))) = halt SCM+FSA
assume A25: CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),l))) = halt SCM+FSA ; ::_thesis: contradiction
consider n being Nat such that
A26: l = n + 1 by A24, NAT_1:6;
 n <> 0 by A24, A26;
then consider l2 being Nat such that
A27: n = l2 + 1 by NAT_1:6;
reconsider l2 = l2 as Element of NAT by ORDINAL1:def_12;
A28: InsCode (halt SCM+FSA) = 0 by COMPOS_1:70;
A29: Comput ((P +* (if=0 (a,I,J))),(Initialized s),(l2 + (1 + 1))) = Comput ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),(1 + 1))),l2) by EXTPRO_1:4;
InsCode (CurInstr ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Comput ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s),l2)))) = InsCode (IncAddr ((CurInstr ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Comput ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s),l2)))),2)) by COMPOS_0:def_9
.= 0 by A26, A27, A25, A29, A19, A14, A16, Th8, A9, A1, A28 ;
then A30: CurInstr ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Comput ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s),l2))) = halt SCM+FSA by SCMFSA_2:95;
 n + 1 < ((LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) + 1) + 1 by A22, A26;
then  n < (LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) + 1 by XREAL_1:6;
then  l2 < LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s)) by A27, XREAL_1:6;
hence  contradiction by A18, A30, EXTPRO_1:def_15; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),l))) = halt SCM+FSA holds
(LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) + 2 <= l ;
then A31: LifeSpan ((P +* (if=0 (a,I,J))),(Initialized s)) = (LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) + 2 by A20, A21, EXTPRO_1:def_15;
A32: DataPart (Result ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) = DataPart (Comput ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s),(LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))))) by A17, A12, EXTPRO_1:23, SCMFSA8A:39
.= DataPart (Comput ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),2)),(LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))))) by A19, A14, A16, Th8, A9, A1
.= DataPart (Comput ((P +* (if=0 (a,I,J))),(Initialized s),((LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) + 2))) by EXTPRO_1:4
.= DataPart (Result ((P +* (if=0 (a,I,J))),(Initialized s))) by A21, A31, EXTPRO_1:23 ;
A33: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if=0_(a,I,J)),P,s))_holds_
(IExec_((if=0_(a,I,J)),P,s))_._x_=_((IExec_((((J_";"_(Goto_((card_I)_+_1)))_";"_I)_";"_(Stop_SCM+FSA)),P,s))_+*_(Start-At_((((card_I)_+_(card_J))_+_3),SCM+FSA)))_._x
let x be set ; ::_thesis: ( x in dom (IExec ((if=0 (a,I,J)),P,s)) implies (IExec ((if=0 (a,I,J)),P,s)) . b1 = ((IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1 )
A34: dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) = {(IC )} by FUNCOP_1:13;
A35: IExec ((if=0 (a,I,J)),P,s) = Result ((P +* (if=0 (a,I,J))),(Initialized s)) by SCMFSA6B:def_1;
A36: IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s) = Result ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s)) by SCMFSA6B:def_1;
assume A37: x in dom (IExec ((if=0 (a,I,J)),P,s)) ; ::_thesis: (IExec ((if=0 (a,I,J)),P,s)) . b1 = ((IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1
percases ( x is Int-Location or x is FinSeq-Location or x = IC ) by A37, SCMFSA_M:1;
supposeA38: x is Int-Location ; ::_thesis: (IExec ((if=0 (a,I,J)),P,s)) . b1 = ((IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1
then  x <> IC by SCMFSA_2:56;
then A39: not x in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A34, TARSKI:def_1;
thus (IExec ((if=0 (a,I,J)),P,s)) . x = (Result ((P +* (if=0 (a,I,J))),(Initialized s))) . x by A35
.= (Result ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) . x by A32, A38, SCMFSA_M:2
.= (IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) . x by A36
.= ((IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . x by A39, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA40: x is FinSeq-Location ; ::_thesis: (IExec ((if=0 (a,I,J)),P,s)) . b1 = ((IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1
then  x <> IC by SCMFSA_2:57;
then A41: not x in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A34, TARSKI:def_1;
thus (IExec ((if=0 (a,I,J)),P,s)) . x = (Result ((P +* (if=0 (a,I,J))),(Initialized s))) . x by A35
.= (Result ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) . x by A32, A40, SCMFSA_M:2
.= (IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) . x by A36
.= ((IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . x by A41, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA42: x = IC ; ::_thesis: (IExec ((if=0 (a,I,J)),P,s)) . b1 = ((IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1
then A43: x in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A34, TARSKI:def_1;
A44: IC (Result ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) = IC (IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) by A36
.= ((card I) + (card J)) + 1 by A12, A17, SCMFSA8A:40 ;
thus (IExec ((if=0 (a,I,J)),P,s)) . x = (Result ((P +* (if=0 (a,I,J))),(Initialized s))) . x by A35
.= (Comput ((P +* (if=0 (a,I,J))),(Initialized s),((LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) + 2))) . x by A21, A31, EXTPRO_1:23
.= IC (Comput ((P +* (if=0 (a,I,J))),(Comput ((P +* (if=0 (a,I,J))),(Initialized s),2)),(LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))))) by A42, EXTPRO_1:4
.= (IC (Comput ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s),(LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s)))))) + 2 by A19, A14, A16, Th8, A9, A1
.= (IC (Result ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s)))) + 2 by A17, A12, EXTPRO_1:23, SCMFSA8A:39
.= (Start-At (((((card I) + (card J)) + 1) + 2),SCM+FSA)) . (IC ) by A44, FUNCOP_1:72
.= ((IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . x by A42, A43, FUNCT_4:13 ; ::_thesis: verum
end;
end;
end;
dom (IExec ((if=0 (a,I,J)),P,s)) = the carrier of SCM+FSA by PARTFUN1:def_2
.= dom ((IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) by PARTFUN1:def_2 ;
hence  IExec ((if=0 (a,I,J)),P,s) = (IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A33, FUNCT_1:2
.= ((IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 1),SCM+FSA))) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A12, A17, SCMFSA8A:41
.= (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by FUNCT_4:114 ;
::_thesis: verum
end;

theorem Th17: :: SCMFSA8B:17
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( if=0 (a,I,J) is parahalting & ( s . a = 0 implies IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <> 0 implies IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( if=0 (a,I,J) is parahalting & ( s . a = 0 implies IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <> 0 implies IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
let s be State of SCM+FSA; ::_thesis: for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( if=0 (a,I,J) is parahalting & ( s . a = 0 implies IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <> 0 implies IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
let I, J be parahalting Program of SCM+FSA; ::_thesis: for a being read-write Int-Location holds
( if=0 (a,I,J) is parahalting & ( s . a = 0 implies IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <> 0 implies IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
let a be read-write Int-Location; ::_thesis: ( if=0 (a,I,J) is parahalting & ( s . a = 0 implies IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <> 0 implies IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
A1: I is_halting_on Initialized s,P by SCMFSA7B:19;
 for s being 0 -started State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA st if=0 (a,I,J) c= P holds
P halts_on s
proof
let s be 0 -started State of SCM+FSA; ::_thesis: for P being Instruction-Sequence of SCM+FSA st if=0 (a,I,J) c= P holds
P halts_on s
 Start-At (0,SCM+FSA) c= s by MEMSTR_0:29;
then A2: s = Initialize s by FUNCT_4:98;
let Q be Instruction-Sequence of SCM+FSA; ::_thesis: ( if=0 (a,I,J) c= Q implies Q halts_on s )
assume A3: if=0 (a,I,J) c= Q ; ::_thesis: Q halts_on s
A4: if=0 (a,I,J) c= Q by A3;
A5: I is_closed_on s,Q by SCMFSA7B:18;
A6: I is_halting_on s,Q by SCMFSA7B:19;
A7: J is_halting_on s,Q by SCMFSA7B:19;
A8: J is_closed_on s,Q by SCMFSA7B:18;
A9: Q +* (if=0 (a,I,J)) = Q by A4, FUNCT_4:98;
percases ( s . a = 0 or s . a <> 0 ) ;
suppose s . a = 0 ; ::_thesis: Q halts_on s
then  if=0 (a,I,J) is_halting_on s,Q by A5, A6, Th13;
hence  Q halts_on s by A2, A9, SCMFSA7B:def_7; ::_thesis: verum
end;
suppose s . a <> 0 ; ::_thesis: Q halts_on s
then  if=0 (a,I,J) is_halting_on s,Q by A8, A7, Th15;
hence  Q halts_on s by A2, A9, SCMFSA7B:def_7; ::_thesis: verum
end;
end;
end;
hence  if=0 (a,I,J) is parahalting by AMISTD_1:def_11; ::_thesis: ( ( s . a = 0 implies IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <> 0 implies IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
A10: J is_closed_on Initialized s,P by SCMFSA7B:18;
 I is_closed_on Initialized s,P by SCMFSA7B:18;
hence  ( s . a = 0 implies IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) by A1, Th14; ::_thesis: ( s . a <> 0 implies IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
 J is_halting_on Initialized s,P by SCMFSA7B:19;
hence  ( s . a <> 0 implies IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) by Th16, A10; ::_thesis: verum
end;

theorem Th18: :: SCMFSA8B:18
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( IC (IExec ((if=0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3 & ( s . a = 0 implies ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <> 0 implies ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( IC (IExec ((if=0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3 & ( s . a = 0 implies ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <> 0 implies ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
let s be State of SCM+FSA; ::_thesis: for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( IC (IExec ((if=0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3 & ( s . a = 0 implies ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <> 0 implies ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
let I, J be parahalting Program of SCM+FSA; ::_thesis: for a being read-write Int-Location holds
( IC (IExec ((if=0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3 & ( s . a = 0 implies ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <> 0 implies ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
let a be read-write Int-Location; ::_thesis: ( IC (IExec ((if=0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3 & ( s . a = 0 implies ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <> 0 implies ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
hereby  ::_thesis: ( ( s . a = 0 implies ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <> 0 implies ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
percases ( s . a = 0 or s . a <> 0 ) ;
suppose s . a = 0 ; ::_thesis: IC (IExec ((if=0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3
then  IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by Th17;
hence  IC (IExec ((if=0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3 by FUNCT_4:113; ::_thesis: verum
end;
suppose s . a <> 0 ; ::_thesis: IC (IExec ((if=0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3
then  IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by Th17;
hence  IC (IExec ((if=0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3 by FUNCT_4:113; ::_thesis: verum
end;
end;
end;
hereby  ::_thesis: ( s . a <> 0 implies ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) )
assume  s . a = 0 ; ::_thesis: ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) )
then A1: IExec ((if=0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by Th17;
hereby  ::_thesis: for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f
let d be Int-Location; ::_thesis: (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d
 not d in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by SCMFSA_2:102;
hence  (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d by A1, FUNCT_4:11; ::_thesis: verum
end;
let f be FinSeq-Location ; ::_thesis: (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f
 not f in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by SCMFSA_2:103;
hence  (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f by A1, FUNCT_4:11; ::_thesis: verum
end;
assume  s . a <> 0 ; ::_thesis: ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) )
then A2: IExec ((if=0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by Th17;
hereby  ::_thesis: for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f
let d be Int-Location; ::_thesis: (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d
 not d in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by SCMFSA_2:102;
hence  (IExec ((if=0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d by A2, FUNCT_4:11; ::_thesis: verum
end;
let f be FinSeq-Location ; ::_thesis: (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f
 not f in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by SCMFSA_2:103;
hence  (IExec ((if=0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f by A2, FUNCT_4:11; ::_thesis: verum
end;

theorem Th19: :: SCMFSA8B:19
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a > 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>0 (a,I,J) is_closed_on s,P & if>0 (a,I,J) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a > 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>0 (a,I,J) is_closed_on s,P & if>0 (a,I,J) is_halting_on s,P )
let s be State of SCM+FSA; ::_thesis: for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a > 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>0 (a,I,J) is_closed_on s,P & if>0 (a,I,J) is_halting_on s,P )
let I, J be Program of SCM+FSA; ::_thesis: for a being read-write Int-Location st s . a > 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>0 (a,I,J) is_closed_on s,P & if>0 (a,I,J) is_halting_on s,P )
let a be read-write Int-Location; ::_thesis: ( s . a > 0 & I is_closed_on s,P & I is_halting_on s,P implies ( if>0 (a,I,J) is_closed_on s,P & if>0 (a,I,J) is_halting_on s,P ) )
set I1 = I ";" (Stop SCM+FSA);
set s1 = Initialize s;
set P1 = P +* (I ";" (Stop SCM+FSA));
set s3 = Initialize s;
set P3 = P +* (if>0 (a,I,J));
set s4 = Comput ((P +* (if>0 (a,I,J))),(Initialize s),1);
set i = a >0_goto ((card J) + 3);
A1: I ";" (Stop SCM+FSA) c= P +* (I ";" (Stop SCM+FSA)) by FUNCT_4:25;
A2: not a in dom (Start-At (0,SCM+FSA)) by SCMFSA_2:102;
A3: 0 in dom (if>0 (a,I,J)) by Lm2;
A4: (P +* (if>0 (a,I,J))) . 0 = (if>0 (a,I,J)) . 0 by A3, FUNCT_4:13
.= a >0_goto ((card J) + 3) by Lm3 ;
 IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
then A5: IC (Initialize s) = IC (Start-At (0,SCM+FSA)) by FUNCT_4:13
.= 0 by FUNCOP_1:72 ;
A6: if>0 (a,I,J) c= P +* (if>0 (a,I,J)) by FUNCT_4:25;
A7: if>0 (a,I,J) = (((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA)) by SCMFSA6A:25;
card (((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) = (card ((Macro (a >0_goto ((card J) + 3))) ";" J)) + (card (Goto ((card I) + 1))) by SCMFSA6A:21
.= (card ((Macro (a >0_goto ((card J) + 3))) ";" J)) + 1 by SCMFSA8A:15
.= ((card (Macro (a >0_goto ((card J) + 3)))) + (card J)) + 1 by SCMFSA6A:21
.= ((card J) + 2) + 1 by COMPOS_1:56
.= (card J) + (2 + 1) ;
then A8: Reloc ((I ";" (Stop SCM+FSA)),((card J) + 3)) c= if>0 (a,I,J) by A7, Lm1;
A9: Reloc ((I ";" (Stop SCM+FSA)),((card J) + 3)) c= P +* (if>0 (a,I,J)) by A8, A6, XBOOLE_1:1;
A10: Comput ((P +* (if>0 (a,I,J))),(Initialize s),(0 + 1)) = Following ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (if>0 (a,I,J))),(Initialize s))
.= Exec ((a >0_goto ((card J) + 3)),(Initialize s)) by A5, A4, PBOOLE:143 ;
A11: for f being FinSeq-Location holds (Initialize s) . f = (Comput ((P +* (if>0 (a,I,J))),(Initialize s),1)) . f by A10, SCMFSA_2:71;
 for a being Int-Location holds (Initialize s) . a = (Comput ((P +* (if>0 (a,I,J))),(Initialize s),1)) . a by A10, SCMFSA_2:71;
then A12: DataPart (Initialize s) = DataPart (Comput ((P +* (if>0 (a,I,J))),(Initialize s),1)) by A11, SCMFSA_M:2;
assume  s . a > 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or ( if>0 (a,I,J) is_closed_on s,P & if>0 (a,I,J) is_halting_on s,P ) )
then  (Initialize s) . a > 0 by A2, FUNCT_4:11;
then A13: IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),1)) = (card J) + 3 by A10, SCMFSA_2:71;
assume A14: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or ( if>0 (a,I,J) is_closed_on s,P & if>0 (a,I,J) is_halting_on s,P ) )
assume A15: I is_halting_on s,P ; ::_thesis: ( if>0 (a,I,J) is_closed_on s,P & if>0 (a,I,J) is_halting_on s,P )
then A16: I ";" (Stop SCM+FSA) is_closed_on s,P by A14, SCMFSA8A:30;
 I ";" (Stop SCM+FSA) is_halting_on s,P by A14, A15, SCMFSA8A:30;
then A17: P +* (I ";" (Stop SCM+FSA)) halts_on Initialize s by SCMFSA7B:def_7;
 DataPart s = DataPart (Initialize s) by MEMSTR_0:79;
then A18: I ";" (Stop SCM+FSA) is_closed_on Initialize s,P +* (I ";" (Stop SCM+FSA)) by A16, Th3;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(if>0_(a,I,J))),(Initialize_s),k))_in_dom_(if>0_(a,I,J))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),b1)) in dom (if>0 (a,I,J))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),b1)) in dom (if>0 (a,I,J))
then consider k1 being Nat such that
A19: k1 + 1 = k by NAT_1:6;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def_12;
reconsider m = IC (Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialize s),k1)) as Element of NAT ;
 m in dom (I ";" (Stop SCM+FSA)) by A16, SCMFSA7B:def_6;
then A20: m < card (I ";" (Stop SCM+FSA)) by AFINSQ_1:66;
 card (Stop SCM+FSA) = 1 by COMPOS_1:4;
then A21: card (I ";" (Stop SCM+FSA)) = (card I) + 1 by SCMFSA6A:21;
card (if>0 (a,I,J)) = ((card I) + (card J)) + 4 by Th12
.= ((card J) + 3) + (card (I ";" (Stop SCM+FSA))) by A21 ;
then A22: m + ((card J) + 3) < card (if>0 (a,I,J)) by A20, XREAL_1:6;
IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),k)) = IC (Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),1)),k1)) by A19, EXTPRO_1:4
.= m + ((card J) + 3) by A18, A13, A12, Th8, A9, A1 ;
hence  IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),k)) in dom (if>0 (a,I,J)) by A22, AFINSQ_1:66; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),b1)) in dom (if>0 (a,I,J))
hence  IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),k)) in dom (if>0 (a,I,J)) by A3, A5, EXTPRO_1:2; ::_thesis: verum
end;
end;
end;
hence  if>0 (a,I,J) is_closed_on s,P by SCMFSA7B:def_6; ::_thesis: if>0 (a,I,J) is_halting_on s,P
CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),((LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialize s))) + 1)))) = CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),1)),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialize s)))))) by EXTPRO_1:4
.= IncAddr ((CurInstr ((P +* (I ";" (Stop SCM+FSA))),(Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialize s),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialize s))))))),((card J) + 3)) by A18, A13, A12, Th8, A9, A1
.= IncAddr ((halt SCM+FSA),((card J) + 3)) by A17, EXTPRO_1:def_15
.= halt SCM+FSA by COMPOS_0:4 ;
then  P +* (if>0 (a,I,J)) halts_on Initialize s by EXTPRO_1:29;
hence  if>0 (a,I,J) is_halting_on s,P by SCMFSA7B:def_7; ::_thesis: verum
end;

theorem Th20: :: SCMFSA8B:20
for P being Instruction-Sequence of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st s . a > 0 & I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for I, J being Program of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st s . a > 0 & I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
let I, J be Program of SCM+FSA; ::_thesis: for a being read-write Int-Location
for s being State of SCM+FSA st s . a > 0 & I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
let a be read-write Int-Location; ::_thesis: for s being State of SCM+FSA st s . a > 0 & I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
let s be State of SCM+FSA; ::_thesis: ( s . a > 0 & I is_closed_on Initialized s,P & I is_halting_on Initialized s,P implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
set I1 = I ";" (Stop SCM+FSA);
set s1 = Initialized s;
set P1 = P +* (I ";" (Stop SCM+FSA));
set P3 = P +* (if>0 (a,I,J));
set s4 = Comput ((P +* (if>0 (a,I,J))),(Initialized s),1);
set i = a >0_goto ((card J) + 3);
A1: I ";" (Stop SCM+FSA) c= P +* (I ";" (Stop SCM+FSA)) by FUNCT_4:25;
A2: if>0 (a,I,J) = (((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA)) by SCMFSA6A:25;
A3: 0 in dom (if>0 (a,I,J)) by Lm2;
A4: (P +* (if>0 (a,I,J))) . 0 = (if>0 (a,I,J)) . 0 by A3, FUNCT_4:13
.= a >0_goto ((card J) + 3) by Lm3 ;
A5: dom (Initialize ((intloc 0) .--> 1)) = {(intloc 0),(IC )} by SCMFSA_M:11;
 ( a <> intloc 0 & a <> IC ) by SCMFSA_2:56;
then A6: not a in dom (Initialize ((intloc 0) .--> 1)) by A5, TARSKI:def_2;
 IC in dom (Initialize ((intloc 0) .--> 1)) by MEMSTR_0:48;
then A7: IC (Initialized s) = IC (Initialize ((intloc 0) .--> 1)) by FUNCT_4:13
.= 0 by MEMSTR_0:def_11 ;
A8: Comput ((P +* (if>0 (a,I,J))),(Initialized s),(0 + 1)) = Following ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),0))) by EXTPRO_1:3
.= Following ((P +* (if>0 (a,I,J))),(Initialized s))
.= Exec ((a >0_goto ((card J) + 3)),(Initialized s)) by A7, A4, PBOOLE:143 ;
A9: if>0 (a,I,J) c= P +* (if>0 (a,I,J)) by FUNCT_4:25;
assume  s . a > 0 ; ::_thesis: ( not I is_closed_on Initialized s,P or not I is_halting_on Initialized s,P or IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
then  (Initialized s) . a > 0 by A6, FUNCT_4:11;
then A10: IC (Comput ((P +* (if>0 (a,I,J))),(Initialized s),1)) = (card J) + 3 by A8, SCMFSA_2:71;
A11: for f being FinSeq-Location holds (Initialized s) . f = (Comput ((P +* (if>0 (a,I,J))),(Initialized s),1)) . f by A8, SCMFSA_2:71;
 for a being Int-Location holds (Initialized s) . a = (Comput ((P +* (if>0 (a,I,J))),(Initialized s),1)) . a by A8, SCMFSA_2:71;
then A12: DataPart (Initialized s) = DataPart (Comput ((P +* (if>0 (a,I,J))),(Initialized s),1)) by A11, SCMFSA_M:2;
card (((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) = (card ((Macro (a >0_goto ((card J) + 3))) ";" J)) + (card (Goto ((card I) + 1))) by SCMFSA6A:21
.= (card ((Macro (a >0_goto ((card J) + 3))) ";" J)) + 1 by SCMFSA8A:15
.= ((card (Macro (a >0_goto ((card J) + 3)))) + (card J)) + 1 by SCMFSA6A:21
.= ((card J) + 2) + 1 by COMPOS_1:56
.= (card J) + (2 + 1) ;
then A13: Reloc ((I ";" (Stop SCM+FSA)),((card J) + 3)) c= if>0 (a,I,J) by A2, Lm1;
A14: Reloc ((I ";" (Stop SCM+FSA)),((card J) + 3)) c= P +* (if>0 (a,I,J)) by A13, A9, XBOOLE_1:1;
assume A15: I is_closed_on Initialized s,P ; ::_thesis: ( not I is_halting_on Initialized s,P or IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
assume A16: I is_halting_on Initialized s,P ; ::_thesis: IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
then A17: P +* (I ";" (Stop SCM+FSA)) halts_on Initialized s by A15, SCMFSA8A:34;
 I ";" (Stop SCM+FSA) is_closed_on Initialized s,P by A15, A16, SCMFSA8A:30;
then A18: I ";" (Stop SCM+FSA) is_closed_on Initialized s,P +* (I ";" (Stop SCM+FSA)) by Th6;
A19: CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),((LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) + 1)))) = CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),1)),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s)))))) by EXTPRO_1:4
.= IncAddr ((CurInstr ((P +* (I ";" (Stop SCM+FSA))),(Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialized s),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))))))),((card J) + 3)) by A18, A10, A12, Th8, A14, A1
.= IncAddr ((halt SCM+FSA),((card J) + 3)) by A17, EXTPRO_1:def_15
.= halt SCM+FSA by COMPOS_0:4 ;
then A20: P +* (if>0 (a,I,J)) halts_on Initialized s by EXTPRO_1:29;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_(LifeSpan_((P_+*_(I_";"_(Stop_SCM+FSA))),(Initialized_s)))_+_1_holds_
CurInstr_((P_+*_(if>0_(a,I,J))),(Comput_((P_+*_(if>0_(a,I,J))),(Initialized_s),l)))_<>_halt_SCM+FSA
let l be Element of NAT ; ::_thesis: ( l < (LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) + 1 implies CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),b1))) <> halt SCM+FSA )
assume A21: l < (LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) + 1 ; ::_thesis: CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),b1))) <> halt SCM+FSA
A22: Comput ((P +* (if>0 (a,I,J))),(Initialized s),0) = Initialized s ;
percases ( l = 0 or l <> 0 ) ;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),b1))) <> halt SCM+FSA
hence  CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),l))) <> halt SCM+FSA by A7, A4, A22, PBOOLE:143; ::_thesis: verum
end;
suppose l <> 0 ; ::_thesis: not CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),b1))) = halt SCM+FSA
then consider n being Nat such that
A23: l = n + 1 by NAT_1:6;
assume A24: CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),l))) = halt SCM+FSA ; ::_thesis: contradiction
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A25: InsCode (halt SCM+FSA) = 0 by COMPOS_1:70;
InsCode (CurInstr ((P +* (I ";" (Stop SCM+FSA))),(Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialized s),n)))) = InsCode (IncAddr ((CurInstr ((P +* (I ";" (Stop SCM+FSA))),(Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialized s),n)))),((card J) + 3))) by COMPOS_0:def_9
.= InsCode (CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),1)),n)))) by A18, A10, A12, Th8, A14, A1
.= 0 by A23, A24, EXTPRO_1:4, A25 ;
then A26: CurInstr ((P +* (I ";" (Stop SCM+FSA))),(Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialized s),n))) = halt SCM+FSA by SCMFSA_2:95;
 n < LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s)) by A21, A23, XREAL_1:6;
hence  contradiction by A17, A26, EXTPRO_1:def_15; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),l))) = halt SCM+FSA holds
(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) + 1 <= l ;
then A27: LifeSpan ((P +* (if>0 (a,I,J))),(Initialized s)) = (LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) + 1 by A19, A20, EXTPRO_1:def_15;
A28: DataPart (Result ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) = DataPart (Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialized s),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))))) by A15, A16, EXTPRO_1:23, SCMFSA8A:34
.= DataPart (Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),1)),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))))) by A18, A10, A12, Th8, A14, A1
.= DataPart (Comput ((P +* (if>0 (a,I,J))),(Initialized s),((LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (if>0 (a,I,J))),(Initialized s))) by A20, A27, EXTPRO_1:23 ;
A29: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if>0_(a,I,J)),P,s))_holds_
(IExec_((if>0_(a,I,J)),P,s))_._x_=_((IExec_((I_";"_(Stop_SCM+FSA)),P,s))_+*_(Start-At_((((card_I)_+_(card_J))_+_3),SCM+FSA)))_._x
let x be set ; ::_thesis: ( x in dom (IExec ((if>0 (a,I,J)),P,s)) implies (IExec ((if>0 (a,I,J)),P,s)) . b1 = ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1 )
A30: dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) = {(IC )} by FUNCOP_1:13;
A31: IExec ((if>0 (a,I,J)),P,s) = Result ((P +* (if>0 (a,I,J))),(Initialized s)) by SCMFSA6B:def_1;
A32: IExec ((I ";" (Stop SCM+FSA)),P,s) = Result ((P +* (I ";" (Stop SCM+FSA))),(Initialized s)) by SCMFSA6B:def_1;
assume A33: x in dom (IExec ((if>0 (a,I,J)),P,s)) ; ::_thesis: (IExec ((if>0 (a,I,J)),P,s)) . b1 = ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1
percases ( x is Int-Location or x is FinSeq-Location or x = IC ) by A33, SCMFSA_M:1;
supposeA34: x is Int-Location ; ::_thesis: (IExec ((if>0 (a,I,J)),P,s)) . b1 = ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1
then  x <> IC by SCMFSA_2:56;
then A35: not x in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A30, TARSKI:def_1;
thus (IExec ((if>0 (a,I,J)),P,s)) . x = (Result ((P +* (if>0 (a,I,J))),(Initialized s))) . x by A31
.= (Result ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) . x by A28, A34, SCMFSA_M:2
.= (IExec ((I ";" (Stop SCM+FSA)),P,s)) . x by A32
.= ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . x by A35, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA36: x is FinSeq-Location ; ::_thesis: (IExec ((if>0 (a,I,J)),P,s)) . b1 = ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1
then  x <> IC by SCMFSA_2:57;
then A37: not x in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A30, TARSKI:def_1;
thus (IExec ((if>0 (a,I,J)),P,s)) . x = (Result ((P +* (if>0 (a,I,J))),(Initialized s))) . x by A31
.= (Result ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) . x by A28, A36, SCMFSA_M:2
.= (IExec ((I ";" (Stop SCM+FSA)),P,s)) . x by A32
.= ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . x by A37, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA38: x = IC ; ::_thesis: (IExec ((if>0 (a,I,J)),P,s)) . b1 = ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1
then A39: x in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A30, TARSKI:def_1;
A40: IC (Result ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) = (IExec ((I ";" (Stop SCM+FSA)),P,s)) . (IC ) by A32
.= IC ((IExec (I,P,s)) +* (Start-At ((card I),SCM+FSA))) by A15, A16, SCMFSA8A:36
.= card I by FUNCT_4:113 ;
thus (IExec ((if>0 (a,I,J)),P,s)) . x = (Result ((P +* (if>0 (a,I,J))),(Initialized s))) . x by A31
.= (Comput ((P +* (if>0 (a,I,J))),(Initialized s),((LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))) + 1))) . x by A20, A27, EXTPRO_1:23
.= IC (Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),1)),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s))))) by A38, EXTPRO_1:4
.= (IC (Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialized s),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialized s)))))) + ((card J) + 3) by A18, A10, A12, Th8, A1, A14
.= (IC (Result ((P +* (I ";" (Stop SCM+FSA))),(Initialized s)))) + ((card J) + 3) by A15, A16, EXTPRO_1:23, SCMFSA8A:34
.= (Start-At (((card I) + ((card J) + 3)),SCM+FSA)) . (IC ) by A40, FUNCOP_1:72
.= ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . x by A38, A39, FUNCT_4:13 ; ::_thesis: verum
end;
end;
end;
dom (IExec ((if>0 (a,I,J)),P,s)) = the carrier of SCM+FSA by PARTFUN1:def_2
.= dom ((IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) by PARTFUN1:def_2 ;
hence  IExec ((if>0 (a,I,J)),P,s) = (IExec ((I ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A29, FUNCT_1:2
.= ((IExec (I,P,s)) +* (Start-At ((card I),SCM+FSA))) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A15, A16, SCMFSA8A:36
.= (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by FUNCT_4:114 ;
::_thesis: verum
end;

theorem Th21: :: SCMFSA8B:21
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a <= 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if>0 (a,I,J) is_closed_on s,P & if>0 (a,I,J) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a <= 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if>0 (a,I,J) is_closed_on s,P & if>0 (a,I,J) is_halting_on s,P )
let s be State of SCM+FSA; ::_thesis: for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a <= 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if>0 (a,I,J) is_closed_on s,P & if>0 (a,I,J) is_halting_on s,P )
let I, J be Program of SCM+FSA; ::_thesis: for a being read-write Int-Location st s . a <= 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if>0 (a,I,J) is_closed_on s,P & if>0 (a,I,J) is_halting_on s,P )
let a be read-write Int-Location; ::_thesis: ( s . a <= 0 & J is_closed_on s,P & J is_halting_on s,P implies ( if>0 (a,I,J) is_closed_on s,P & if>0 (a,I,J) is_halting_on s,P ) )
set I1 = I ";" (Stop SCM+FSA);
set JI2 = ((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA);
set s2 = Initialize s;
set P2 = P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA));
A1: ((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA) c= P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) by FUNCT_4:25;
set s3 = Initialize s;
set P3 = P +* (if>0 (a,I,J));
set s4 = Comput ((P +* (if>0 (a,I,J))),(Initialize s),1);
set s5 = Comput ((P +* (if>0 (a,I,J))),(Initialize s),2);
set i = a >0_goto ((card J) + 3);
 IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
then A2: IC (Initialize s) = IC (Start-At (0,SCM+FSA)) by FUNCT_4:13
.= 0 by FUNCOP_1:72 ;
A3: if>0 (a,I,J) c= P +* (if>0 (a,I,J)) by FUNCT_4:25;
A4: if>0 (a,I,J) = (((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA)) by SCMFSA6A:25
.= ((a >0_goto ((card J) + 3)) ";" J) ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA))) by SCMFSA6A:25
.= (a >0_goto ((card J) + 3)) ";" (J ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA)))) by SCMFSA6A:29
.= (a >0_goto ((card J) + 3)) ";" ((J ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA))) by SCMFSA6A:25
.= (Macro (a >0_goto ((card J) + 3))) ";" (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) by SCMFSA6A:25 ;
then A5: Reloc ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),(card (Macro (a >0_goto ((card J) + 3))))) c= if>0 (a,I,J) by Lm1;
 card (Macro (a >0_goto ((card J) + 3))) = 2 by COMPOS_1:56;
then A6: Reloc ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),2) c= P +* (if>0 (a,I,J)) by A3, A5, XBOOLE_1:1;
A7: not a in dom (Start-At (0,SCM+FSA)) by SCMFSA_2:102;
A8: 0 in dom (if>0 (a,I,J)) by Lm2;
A9: (P +* (if>0 (a,I,J))) . 0 = (if>0 (a,I,J)) . 0 by A8, FUNCT_4:13
.= a >0_goto ((card J) + 3) by Lm3 ;
A10: 1 in dom (if>0 (a,I,J)) by Lm2;
A11: Comput ((P +* (if>0 (a,I,J))),(Initialize s),(0 + 1)) = Following ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (if>0 (a,I,J))),(Initialize s))
.= Exec ((a >0_goto ((card J) + 3)),(Initialize s)) by A2, A9, PBOOLE:143 ;
assume  s . a <= 0 ; ::_thesis: ( not J is_closed_on s,P or not J is_halting_on s,P or ( if>0 (a,I,J) is_closed_on s,P & if>0 (a,I,J) is_halting_on s,P ) )
then  (Initialize s) . a <= 0 by A7, FUNCT_4:11;
then A12: IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),1)) = succ 0 by A2, A11, SCMFSA_2:71
.= 0 + 1 ;
A13: (P +* (if>0 (a,I,J))) . 1 = (if>0 (a,I,J)) . 1 by A10, FUNCT_4:13
.= goto 2 by Lm3 ;
assume A14: J is_closed_on s,P ; ::_thesis: ( not J is_halting_on s,P or ( if>0 (a,I,J) is_closed_on s,P & if>0 (a,I,J) is_halting_on s,P ) )
A15: Comput ((P +* (if>0 (a,I,J))),(Initialize s),(1 + 1)) = Following ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),1))) by EXTPRO_1:3
.= Exec ((goto 2),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),1))) by A12, A13, PBOOLE:143 ;
A16: now__::_thesis:_for_f_being_FinSeq-Location_holds_(Initialize_s)_._f_=_(Comput_((P_+*_(if>0_(a,I,J))),(Initialize_s),2))_._f
let f be FinSeq-Location ; ::_thesis: (Initialize s) . f = (Comput ((P +* (if>0 (a,I,J))),(Initialize s),2)) . f
thus (Initialize s) . f = (Comput ((P +* (if>0 (a,I,J))),(Initialize s),1)) . f by A11, SCMFSA_2:71
.= (Comput ((P +* (if>0 (a,I,J))),(Initialize s),2)) . f by A15, SCMFSA_2:69 ; ::_thesis: verum
end;
now__::_thesis:_for_a_being_Int-Location_holds_(Initialize_s)_._a_=_(Comput_((P_+*_(if>0_(a,I,J))),(Initialize_s),2))_._a
let a be Int-Location; ::_thesis: (Initialize s) . a = (Comput ((P +* (if>0 (a,I,J))),(Initialize s),2)) . a
thus (Initialize s) . a = (Comput ((P +* (if>0 (a,I,J))),(Initialize s),1)) . a by A11, SCMFSA_2:71
.= (Comput ((P +* (if>0 (a,I,J))),(Initialize s),2)) . a by A15, SCMFSA_2:69 ; ::_thesis: verum
end;
then A17: DataPart (Initialize s) = DataPart (Comput ((P +* (if>0 (a,I,J))),(Initialize s),2)) by A16, SCMFSA_M:2;
assume A18: J is_halting_on s,P ; ::_thesis: ( if>0 (a,I,J) is_closed_on s,P & if>0 (a,I,J) is_halting_on s,P )
then A19: ((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA) is_closed_on s,P by A14, SCMFSA8A:37;
then A20: ((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA) is_closed_on Initialize s,P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) by Th7;
A21: P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) halts_on Initialize s by A14, A18, SCMFSA8A:38;
A22: IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),2)) = 2 by A15, SCMFSA_2:69;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(if>0_(a,I,J))),(Initialize_s),k))_in_dom_(if>0_(a,I,J))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),b1)) in dom (if>0 (a,I,J))
 ( k = 0 or 0 + 1 < k + 1 ) by XREAL_1:6;
then A23: ( k = 0 or 1 <= k ) by NAT_1:13;
percases ( 1 < k or k = 0 or k = 1 ) by A23, XXREAL_0:1;
supposeA24: 1 < k ; ::_thesis: IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),b1)) in dom (if>0 (a,I,J))
then consider k1 being Nat such that
A25: k1 + 1 = k by NAT_1:6;
 0 + 1 < k1 + 1 by A24, A25;
then consider k2 being Nat such that
A26: k2 + 1 = k1 by NAT_1:6;
reconsider k1 = k1, k2 = k2 as Element of NAT by ORDINAL1:def_12;
reconsider m = IC (Comput ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialize s),k2)) as Element of NAT ;
A27: card (if>0 (a,I,J)) = (card (Macro (a >0_goto ((card J) + 3)))) + (card (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))) by A4, SCMFSA6A:21
.= 2 + (card (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))) by COMPOS_1:56 ;
 m in dom (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) by A19, SCMFSA7B:def_6;
then  m < card (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) by AFINSQ_1:66;
then A28: m + 2 < card (if>0 (a,I,J)) by A27, XREAL_1:6;
IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),k)) = IC (Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),1)),k1)) by A25, EXTPRO_1:4
.= IC (Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),1)),1)),k2)) by A26, EXTPRO_1:4
.= IC (Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),(1 + 1))),k2)) by EXTPRO_1:4
.= m + 2 by A20, A22, A17, Th8, A6, A1 ;
hence  IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),k)) in dom (if>0 (a,I,J)) by A28, AFINSQ_1:66; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),b1)) in dom (if>0 (a,I,J))
hence  IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),k)) in dom (if>0 (a,I,J)) by A8, A2, EXTPRO_1:2; ::_thesis: verum
end;
suppose k = 1 ; ::_thesis: IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),b1)) in dom (if>0 (a,I,J))
hence  IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),k)) in dom (if>0 (a,I,J)) by A12, Lm2; ::_thesis: verum
end;
end;
end;
hence  if>0 (a,I,J) is_closed_on s,P by SCMFSA7B:def_6; ::_thesis: if>0 (a,I,J) is_halting_on s,P
CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),((LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialize s))) + 2)))) = CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),2)),(LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialize s)))))) by EXTPRO_1:4
.= IncAddr ((CurInstr ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Comput ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialize s),(LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialize s))))))),2) by A20, A6, A22, A17, Th8, A1
.= IncAddr ((halt SCM+FSA),2) by A21, EXTPRO_1:def_15
.= halt SCM+FSA by COMPOS_0:4 ;
then  P +* (if>0 (a,I,J)) halts_on Initialize s by EXTPRO_1:29;
hence  if>0 (a,I,J) is_halting_on s,P by SCMFSA7B:def_7; ::_thesis: verum
end;

theorem Th22: :: SCMFSA8B:22
for P being Instruction-Sequence of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st s . a <= 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P holds
IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for I, J being Program of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st s . a <= 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P holds
IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
let I, J be Program of SCM+FSA; ::_thesis: for a being read-write Int-Location
for s being State of SCM+FSA st s . a <= 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P holds
IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
let a be read-write Int-Location; ::_thesis: for s being State of SCM+FSA st s . a <= 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P holds
IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
let s be State of SCM+FSA; ::_thesis: ( s . a <= 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
set I1 = I ";" (Stop SCM+FSA);
set JI2 = ((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA);
set s2 = Initialized s;
set P2 = P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA));
A1: ((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA) c= P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) by FUNCT_4:25;
set P3 = P +* (if>0 (a,I,J));
set s4 = Comput ((P +* (if>0 (a,I,J))),(Initialized s),1);
set s5 = Comput ((P +* (if>0 (a,I,J))),(Initialized s),2);
set i = a >0_goto ((card J) + 3);
 0 in dom (if>0 (a,I,J)) by Lm2;
then A2: (P +* (if>0 (a,I,J))) . 0 = (if>0 (a,I,J)) . 0 by FUNCT_4:13
.= a >0_goto ((card J) + 3) by Lm3 ;
A3: 1 in dom (if>0 (a,I,J)) by Lm2;
A4: (P +* (if>0 (a,I,J))) . 1 = (if>0 (a,I,J)) . 1 by A3, FUNCT_4:13
.= goto 2 by Lm3 ;
A5: dom (Initialize ((intloc 0) .--> 1)) = {(intloc 0),(IC )} by SCMFSA_M:11;
 ( a <> intloc 0 & a <> IC ) by SCMFSA_2:56;
then A6: not a in dom (Initialize ((intloc 0) .--> 1)) by A5, TARSKI:def_2;
 IC in dom (Initialize ((intloc 0) .--> 1)) by MEMSTR_0:48;
then A7: IC (Initialized s) = IC (Initialize ((intloc 0) .--> 1)) by FUNCT_4:13
.= 0 by MEMSTR_0:def_11 ;
if>0 (a,I,J) = (((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA)) by SCMFSA6A:25
.= ((a >0_goto ((card J) + 3)) ";" J) ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA))) by SCMFSA6A:25
.= (a >0_goto ((card J) + 3)) ";" (J ";" ((Goto ((card I) + 1)) ";" (I ";" (Stop SCM+FSA)))) by SCMFSA6A:29
.= (a >0_goto ((card J) + 3)) ";" ((J ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA))) by SCMFSA6A:25
.= (Macro (a >0_goto ((card J) + 3))) ";" (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) by SCMFSA6A:25 ;
then  Reloc ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),(card (Macro (a >0_goto ((card J) + 3))))) c= if>0 (a,I,J) by Lm1;
then A8: Reloc ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),2) c= if>0 (a,I,J) by COMPOS_1:56;
 if>0 (a,I,J) c= P +* (if>0 (a,I,J)) by FUNCT_4:25;
then A9: Reloc ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),2) c= P +* (if>0 (a,I,J)) by A8, XBOOLE_1:1;
A10: Comput ((P +* (if>0 (a,I,J))),(Initialized s),(0 + 1)) = Following ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),0))) by EXTPRO_1:3
.= Following ((P +* (if>0 (a,I,J))),(Initialized s))
.= Exec ((a >0_goto ((card J) + 3)),(Initialized s)) by A7, A2, PBOOLE:143 ;
assume  s . a <= 0 ; ::_thesis: ( not J is_closed_on Initialized s,P or not J is_halting_on Initialized s,P or IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
then  (Initialized s) . a <= 0 by A6, FUNCT_4:11;
then A11: IC (Comput ((P +* (if>0 (a,I,J))),(Initialized s),1)) = succ 0 by A7, A10, SCMFSA_2:71
.= 0 + 1 ;
assume A12: J is_closed_on Initialized s,P ; ::_thesis: ( not J is_halting_on Initialized s,P or IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
A13: Comput ((P +* (if>0 (a,I,J))),(Initialized s),(1 + 1)) = Following ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),1))) by EXTPRO_1:3
.= Exec ((goto 2),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),1))) by A11, A4, PBOOLE:143 ;
then A14: IC (Comput ((P +* (if>0 (a,I,J))),(Initialized s),2)) = 2 by SCMFSA_2:69;
A15: now__::_thesis:_for_f_being_FinSeq-Location_holds_(Initialized_s)_._f_=_(Comput_((P_+*_(if>0_(a,I,J))),(Initialized_s),2))_._f
let f be FinSeq-Location ; ::_thesis: (Initialized s) . f = (Comput ((P +* (if>0 (a,I,J))),(Initialized s),2)) . f
thus (Initialized s) . f = (Comput ((P +* (if>0 (a,I,J))),(Initialized s),1)) . f by A10, SCMFSA_2:71
.= (Comput ((P +* (if>0 (a,I,J))),(Initialized s),2)) . f by A13, SCMFSA_2:69 ; ::_thesis: verum
end;
now__::_thesis:_for_a_being_Int-Location_holds_(Initialized_s)_._a_=_(Comput_((P_+*_(if>0_(a,I,J))),(Initialized_s),2))_._a
let a be Int-Location; ::_thesis: (Initialized s) . a = (Comput ((P +* (if>0 (a,I,J))),(Initialized s),2)) . a
thus (Initialized s) . a = (Comput ((P +* (if>0 (a,I,J))),(Initialized s),1)) . a by A10, SCMFSA_2:71
.= (Comput ((P +* (if>0 (a,I,J))),(Initialized s),2)) . a by A13, SCMFSA_2:69 ; ::_thesis: verum
end;
then A16: DataPart (Initialized s) = DataPart (Comput ((P +* (if>0 (a,I,J))),(Initialized s),2)) by A15, SCMFSA_M:2;
assume A17: J is_halting_on Initialized s,P ; ::_thesis: IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))
then A18: P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) halts_on Initialized s by A12, SCMFSA8A:39;
 ((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA) is_closed_on Initialized s,P by A12, A17, SCMFSA8A:37;
then A19: ((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA) is_closed_on Initialized s,P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) by Th6;
A20: CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),((LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) + 2)))) = CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),2)),(LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s)))))) by EXTPRO_1:4
.= IncAddr ((CurInstr ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Comput ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s),(LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))))))),2) by A19, A14, A16, Th8, A9, A1
.= IncAddr ((halt SCM+FSA),2) by A18, EXTPRO_1:def_15
.= halt SCM+FSA by COMPOS_0:4 ;
then A21: P +* (if>0 (a,I,J)) halts_on Initialized s by EXTPRO_1:29;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_(LifeSpan_((P_+*_(((J_";"_(Goto_((card_I)_+_1)))_";"_I)_";"_(Stop_SCM+FSA))),(Initialized_s)))_+_2_holds_
CurInstr_((P_+*_(if>0_(a,I,J))),(Comput_((P_+*_(if>0_(a,I,J))),(Initialized_s),l)))_<>_halt_SCM+FSA
let l be Element of NAT ; ::_thesis: ( l < (LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) + 2 implies CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),b1))) <> halt SCM+FSA )
assume A22: l < (LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) + 2 ; ::_thesis: CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),b1))) <> halt SCM+FSA
A23: Comput ((P +* (if>0 (a,I,J))),(Initialized s),0) = Initialized s ;
percases ( l = 0 or l = 1 or ( l <> 0 & l <> 1 ) ) ;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),b1))) <> halt SCM+FSA
hence  CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),l))) <> halt SCM+FSA by A7, A2, A23, PBOOLE:143; ::_thesis: verum
end;
suppose l = 1 ; ::_thesis: CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),b1))) <> halt SCM+FSA
hence  CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),l))) <> halt SCM+FSA by A11, A4, PBOOLE:143; ::_thesis: verum
end;
supposeA24: ( l <> 0 & l <> 1 ) ; ::_thesis: not CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),b1))) = halt SCM+FSA
assume A25: CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),l))) = halt SCM+FSA ; ::_thesis: contradiction
consider n being Nat such that
A26: l = n + 1 by A24, NAT_1:6;
 n <> 0 by A24, A26;
then consider l2 being Nat such that
A27: n = l2 + 1 by NAT_1:6;
reconsider l2 = l2 as Element of NAT by ORDINAL1:def_12;
InsCode (CurInstr ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Comput ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s),l2)))) = InsCode (IncAddr ((CurInstr ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Comput ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s),l2)))),2)) by COMPOS_0:def_9
.= InsCode (CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),2)),l2)))) by A19, A14, A16, Th8, A9, A1
.= InsCode (CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),(l2 + (1 + 1)))))) by EXTPRO_1:4
.= 0 by A26, A27, A25, COMPOS_1:70 ;
then A28: CurInstr ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Comput ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s),l2))) = halt SCM+FSA by SCMFSA_2:95;
 n + 1 < ((LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) + 1) + 1 by A22, A26;
then  n < (LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) + 1 by XREAL_1:6;
then  l2 < LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s)) by A27, XREAL_1:6;
hence  contradiction by A18, A28, EXTPRO_1:def_15; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),l))) = halt SCM+FSA holds
(LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) + 2 <= l ;
then A29: LifeSpan ((P +* (if>0 (a,I,J))),(Initialized s)) = (LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) + 2 by A20, A21, EXTPRO_1:def_15;
A30: DataPart (Result ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) = DataPart (Comput ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s),(LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))))) by A17, A12, EXTPRO_1:23, SCMFSA8A:39
.= DataPart (Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),2)),(LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))))) by A19, A14, A16, Th8, A9, A1
.= DataPart (Comput ((P +* (if>0 (a,I,J))),(Initialized s),((LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) + 2))) by EXTPRO_1:4
.= DataPart (Result ((P +* (if>0 (a,I,J))),(Initialized s))) by A21, A29, EXTPRO_1:23 ;
A31: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if>0_(a,I,J)),P,s))_holds_
(IExec_((if>0_(a,I,J)),P,s))_._x_=_((IExec_((((J_";"_(Goto_((card_I)_+_1)))_";"_I)_";"_(Stop_SCM+FSA)),P,s))_+*_(Start-At_((((card_I)_+_(card_J))_+_3),SCM+FSA)))_._x
let x be set ; ::_thesis: ( x in dom (IExec ((if>0 (a,I,J)),P,s)) implies (IExec ((if>0 (a,I,J)),P,s)) . b1 = ((IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1 )
A32: dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) = {(IC )} by FUNCOP_1:13;
A33: IExec ((if>0 (a,I,J)),P,s) = Result ((P +* (if>0 (a,I,J))),(Initialized s)) by SCMFSA6B:def_1;
A34: IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s) = Result ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s)) by SCMFSA6B:def_1;
assume A35: x in dom (IExec ((if>0 (a,I,J)),P,s)) ; ::_thesis: (IExec ((if>0 (a,I,J)),P,s)) . b1 = ((IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1
percases ( x is Int-Location or x is FinSeq-Location or x = IC ) by A35, SCMFSA_M:1;
supposeA36: x is Int-Location ; ::_thesis: (IExec ((if>0 (a,I,J)),P,s)) . b1 = ((IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1
then  x <> IC by SCMFSA_2:56;
then A37: not x in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A32, TARSKI:def_1;
thus (IExec ((if>0 (a,I,J)),P,s)) . x = (Result ((P +* (if>0 (a,I,J))),(Initialized s))) . x by A33
.= (Result ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) . x by A30, A36, SCMFSA_M:2
.= (IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) . x by A34
.= ((IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . x by A37, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA38: x is FinSeq-Location ; ::_thesis: (IExec ((if>0 (a,I,J)),P,s)) . b1 = ((IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1
then  x <> IC by SCMFSA_2:57;
then A39: not x in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A32, TARSKI:def_1;
thus (IExec ((if>0 (a,I,J)),P,s)) . x = (Result ((P +* (if>0 (a,I,J))),(Initialized s))) . x by A33
.= (Result ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) . x by A30, A38, SCMFSA_M:2
.= (IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) . x by A34
.= ((IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . x by A39, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA40: x = IC ; ::_thesis: (IExec ((if>0 (a,I,J)),P,s)) . b1 = ((IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . b1
then A41: x in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A32, TARSKI:def_1;
A42: IC (Result ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) = IC (IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) by A34
.= ((card I) + (card J)) + 1 by A12, A17, SCMFSA8A:40 ;
thus (IExec ((if>0 (a,I,J)),P,s)) . x = (Result ((P +* (if>0 (a,I,J))),(Initialized s))) . x by A33
.= (Comput ((P +* (if>0 (a,I,J))),(Initialized s),((LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))) + 2))) . x by A21, A29, EXTPRO_1:23
.= IC (Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialized s),2)),(LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s))))) by A40, EXTPRO_1:4
.= (IC (Comput ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s),(LifeSpan ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s)))))) + 2 by A19, A14, A16, Th8, A9, A1
.= (IC (Result ((P +* (((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA))),(Initialized s)))) + 2 by A17, A12, EXTPRO_1:23, SCMFSA8A:39
.= (Start-At (((((card I) + (card J)) + 1) + 2),SCM+FSA)) . (IC ) by A42, FUNCOP_1:72
.= ((IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) . x by A40, A41, FUNCT_4:13 ; ::_thesis: verum
end;
end;
end;
dom (IExec ((if>0 (a,I,J)),P,s)) = the carrier of SCM+FSA by PARTFUN1:def_2
.= dom ((IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) by PARTFUN1:def_2 ;
hence  IExec ((if>0 (a,I,J)),P,s) = (IExec ((((J ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)),P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A31, FUNCT_1:2
.= ((IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 1),SCM+FSA))) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by A12, A17, SCMFSA8A:41
.= (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by FUNCT_4:114 ;
::_thesis: verum
end;

theorem Th23: :: SCMFSA8B:23
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( if>0 (a,I,J) is parahalting & ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( if>0 (a,I,J) is parahalting & ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
let s be State of SCM+FSA; ::_thesis: for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( if>0 (a,I,J) is parahalting & ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
let I, J be parahalting Program of SCM+FSA; ::_thesis: for a being read-write Int-Location holds
( if>0 (a,I,J) is parahalting & ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
let a be read-write Int-Location; ::_thesis: ( if>0 (a,I,J) is parahalting & ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
A1: I is_halting_on Initialized s,P by SCMFSA7B:19;
 for s being 0 -started State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA st if>0 (a,I,J) c= P holds
P halts_on s
proof
let s be 0 -started State of SCM+FSA; ::_thesis: for P being Instruction-Sequence of SCM+FSA st if>0 (a,I,J) c= P holds
P halts_on s
 Start-At (0,SCM+FSA) c= s by MEMSTR_0:29;
then A2: s = Initialize s by FUNCT_4:98;
let Q be Instruction-Sequence of SCM+FSA; ::_thesis: ( if>0 (a,I,J) c= Q implies Q halts_on s )
assume A3: if>0 (a,I,J) c= Q ; ::_thesis: Q halts_on s
A4: if>0 (a,I,J) c= Q by A3;
A5: I is_closed_on s,Q by SCMFSA7B:18;
A6: I is_halting_on s,Q by SCMFSA7B:19;
A7: J is_halting_on s,Q by SCMFSA7B:19;
A8: J is_closed_on s,Q by SCMFSA7B:18;
A9: Q +* (if>0 (a,I,J)) = Q by A4, FUNCT_4:98;
percases ( s . a > 0 or s . a <= 0 ) ;
suppose s . a > 0 ; ::_thesis: Q halts_on s
then  if>0 (a,I,J) is_halting_on s,Q by A5, A6, Th19;
hence  Q halts_on s by A2, A9, SCMFSA7B:def_7; ::_thesis: verum
end;
suppose s . a <= 0 ; ::_thesis: Q halts_on s
then  if>0 (a,I,J) is_halting_on s,Q by A8, A7, Th21;
hence  Q halts_on s by A2, A9, SCMFSA7B:def_7; ::_thesis: verum
end;
end;
end;
hence  if>0 (a,I,J) is parahalting by AMISTD_1:def_11; ::_thesis: ( ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )
A10: J is_closed_on Initialized s,P by SCMFSA7B:18;
 I is_closed_on Initialized s,P by SCMFSA7B:18;
hence  ( s . a > 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) by A1, Th20; ::_thesis: ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) )
 J is_halting_on Initialized s,P by SCMFSA7B:19;
hence  ( s . a <= 0 implies IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) by Th22, A10; ::_thesis: verum
end;

theorem Th24: :: SCMFSA8B:24
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( IC (IExec ((if>0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3 & ( s . a > 0 implies ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <= 0 implies ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( IC (IExec ((if>0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3 & ( s . a > 0 implies ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <= 0 implies ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
let s be State of SCM+FSA; ::_thesis: for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( IC (IExec ((if>0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3 & ( s . a > 0 implies ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <= 0 implies ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
let I, J be parahalting Program of SCM+FSA; ::_thesis: for a being read-write Int-Location holds
( IC (IExec ((if>0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3 & ( s . a > 0 implies ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <= 0 implies ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
let a be read-write Int-Location; ::_thesis: ( IC (IExec ((if>0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3 & ( s . a > 0 implies ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <= 0 implies ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
hereby  ::_thesis: ( ( s . a > 0 implies ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <= 0 implies ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
percases ( s . a > 0 or s . a <= 0 ) ;
suppose s . a > 0 ; ::_thesis: IC (IExec ((if>0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3
then  IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by Th23;
hence  IC (IExec ((if>0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3 by FUNCT_4:113; ::_thesis: verum
end;
suppose s . a <= 0 ; ::_thesis: IC (IExec ((if>0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3
then  IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by Th23;
hence  IC (IExec ((if>0 (a,I,J)),P,s)) = ((card I) + (card J)) + 3 by FUNCT_4:113; ::_thesis: verum
end;
end;
end;
hereby  ::_thesis: ( s . a <= 0 implies ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) )
assume  s . a > 0 ; ::_thesis: ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) )
then A1: IExec ((if>0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by Th23;
hereby  ::_thesis: for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f
let d be Int-Location; ::_thesis: (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d
 not d in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by SCMFSA_2:102;
hence  (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (I,P,s)) . d by A1, FUNCT_4:11; ::_thesis: verum
end;
let f be FinSeq-Location ; ::_thesis: (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f
 not f in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by SCMFSA_2:103;
hence  (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (I,P,s)) . f by A1, FUNCT_4:11; ::_thesis: verum
end;
assume  s . a <= 0 ; ::_thesis: ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) )
then A2: IExec ((if>0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by Th23;
hereby  ::_thesis: for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f
let d be Int-Location; ::_thesis: (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d
 not d in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by SCMFSA_2:102;
hence  (IExec ((if>0 (a,I,J)),P,s)) . d = (IExec (J,P,s)) . d by A2, FUNCT_4:11; ::_thesis: verum
end;
let f be FinSeq-Location ; ::_thesis: (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f
 not f in dom (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) by SCMFSA_2:103;
hence  (IExec ((if>0 (a,I,J)),P,s)) . f = (IExec (J,P,s)) . f by A2, FUNCT_4:11; ::_thesis: verum
end;

theorem :: SCMFSA8B:25
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a < 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a < 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P )
let s be State of SCM+FSA; ::_thesis: for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a < 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P )
let I, J be Program of SCM+FSA; ::_thesis: for a being read-write Int-Location st s . a < 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P )
let a be read-write Int-Location; ::_thesis: ( s . a < 0 & I is_closed_on s,P & I is_halting_on s,P implies ( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P ) )
assume A1: s . a < 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or ( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P ) )
assume that
A2: I is_closed_on s,P and
A3: I is_halting_on s,P ; ::_thesis: ( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P )
A4: if>0 (a,J,I) is_halting_on s,P by A1, A2, A3, Th21;
 if>0 (a,J,I) is_closed_on s,P by A1, A2, A3, Th21;
hence  ( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P ) by A1, A4, Th15; ::_thesis: verum
end;

theorem Th26: :: SCMFSA8B:26
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a < 0 & I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a < 0 & I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
let s be State of SCM+FSA; ::_thesis: for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a < 0 & I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
let I, J be Program of SCM+FSA; ::_thesis: for a being read-write Int-Location st s . a < 0 & I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
let a be read-write Int-Location; ::_thesis: ( s . a < 0 & I is_closed_on Initialized s,P & I is_halting_on Initialized s,P implies IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) )
assume A1: s . a < 0 ; ::_thesis: ( not I is_closed_on Initialized s,P or not I is_halting_on Initialized s,P or IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) )
assume that
A2: I is_closed_on Initialized s,P and
A3: I is_halting_on Initialized s,P ; ::_thesis: IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
A4: (Initialized s) . a <= 0 by A1, SCMFSA_M:37;
then A5: if>0 (a,J,I) is_halting_on Initialized s,P by A2, A3, Th21;
 if>0 (a,J,I) is_closed_on Initialized s,P by A2, A3, A4, Th21;
hence  IExec ((if<0 (a,I,J)),P,s) = (IExec ((if>0 (a,J,I)),P,s)) +* (Start-At ((((card (if>0 (a,J,I))) + (card J)) + 3),SCM+FSA)) by A1, A5, Th16
.= (IExec ((if>0 (a,J,I)),P,s)) +* (Start-At ((((((card I) + (card J)) + 4) + (card J)) + 3),SCM+FSA)) by Th12
.= ((IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) +* (Start-At ((((((card I) + (card J)) + 4) + (card J)) + 3),SCM+FSA)) by A1, A2, A3, Th22
.= (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) by FUNCT_4:114 ;
::_thesis: verum
end;

theorem :: SCMFSA8B:27
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a = 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P ) by Th13;

theorem Th28: :: SCMFSA8B:28
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a = 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P holds
IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a = 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P holds
IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
let s be State of SCM+FSA; ::_thesis: for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a = 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P holds
IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
let I, J be Program of SCM+FSA; ::_thesis: for a being read-write Int-Location st s . a = 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P holds
IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
let a be read-write Int-Location; ::_thesis: ( s . a = 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P implies IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) )
assume A1: s . a = 0 ; ::_thesis: ( not J is_closed_on Initialized s,P or not J is_halting_on Initialized s,P or IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) )
assume that
A2: J is_closed_on Initialized s,P and
A3: J is_halting_on Initialized s,P ; ::_thesis: IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
thus  IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card (if>0 (a,J,I))) + (card J)) + 3),SCM+FSA)) by A1, A2, A3, Th14
.= (IExec (J,P,s)) +* (Start-At ((((((card I) + (card J)) + 4) + (card J)) + 3),SCM+FSA)) by Th12
.= (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ; ::_thesis: verum
end;

theorem :: SCMFSA8B:29
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a > 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a > 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P )
let s be State of SCM+FSA; ::_thesis: for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a > 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P )
let I, J be Program of SCM+FSA; ::_thesis: for a being read-write Int-Location st s . a > 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P )
let a be read-write Int-Location; ::_thesis: ( s . a > 0 & J is_closed_on s,P & J is_halting_on s,P implies ( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P ) )
assume A1: s . a > 0 ; ::_thesis: ( not J is_closed_on s,P or not J is_halting_on s,P or ( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P ) )
assume that
A2: J is_closed_on s,P and
A3: J is_halting_on s,P ; ::_thesis: ( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P )
A4: if>0 (a,J,I) is_halting_on s,P by A1, A2, A3, Th19;
 if>0 (a,J,I) is_closed_on s,P by A1, A2, A3, Th19;
hence  ( if<0 (a,I,J) is_closed_on s,P & if<0 (a,I,J) is_halting_on s,P ) by A1, A4, Th15; ::_thesis: verum
end;

theorem Th30: :: SCMFSA8B:30
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a > 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P holds
IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a > 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P holds
IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
let s be State of SCM+FSA; ::_thesis: for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a > 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P holds
IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
let I, J be Program of SCM+FSA; ::_thesis: for a being read-write Int-Location st s . a > 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P holds
IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
let a be read-write Int-Location; ::_thesis: ( s . a > 0 & J is_closed_on Initialized s,P & J is_halting_on Initialized s,P implies IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) )
assume A1: s . a > 0 ; ::_thesis: ( not J is_closed_on Initialized s,P or not J is_halting_on Initialized s,P or IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) )
assume that
A2: J is_closed_on Initialized s,P and
A3: J is_halting_on Initialized s,P ; ::_thesis: IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
A4: (Initialized s) . a > 0 by A1, SCMFSA_M:37;
then A5: if>0 (a,J,I) is_halting_on Initialized s,P by A2, A3, Th19;
 if>0 (a,J,I) is_closed_on Initialized s,P by A2, A3, A4, Th19;
hence  IExec ((if<0 (a,I,J)),P,s) = (IExec ((if>0 (a,J,I)),P,s)) +* (Start-At ((((card (if>0 (a,J,I))) + (card J)) + 3),SCM+FSA)) by A1, A5, Th16
.= (IExec ((if>0 (a,J,I)),P,s)) +* (Start-At ((((((card I) + (card J)) + 4) + (card J)) + 3),SCM+FSA)) by Th12
.= ((IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))) +* (Start-At ((((((card I) + (card J)) + 4) + (card J)) + 3),SCM+FSA)) by A1, A2, A3, Th20
.= (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) by FUNCT_4:114 ;
::_thesis: verum
end;

theorem :: SCMFSA8B:31
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( if<0 (a,I,J) is parahalting & ( s . a < 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) & ( s . a >= 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) )
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( if<0 (a,I,J) is parahalting & ( s . a < 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) & ( s . a >= 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) )
let s be State of SCM+FSA; ::_thesis: for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( if<0 (a,I,J) is parahalting & ( s . a < 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) & ( s . a >= 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) )
let I, J be parahalting Program of SCM+FSA; ::_thesis: for a being read-write Int-Location holds
( if<0 (a,I,J) is parahalting & ( s . a < 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) & ( s . a >= 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) )
let a be read-write Int-Location; ::_thesis: ( if<0 (a,I,J) is parahalting & ( s . a < 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) & ( s . a >= 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) )
 if>0 (a,J,I) is parahalting by Th23;
hence  if<0 (a,I,J) is parahalting by Th17; ::_thesis: ( ( s . a < 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) & ( s . a >= 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) )
A1: I is_closed_on Initialized s,P by SCMFSA7B:18;
hereby  ::_thesis: ( s . a >= 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) )
A2: I is_halting_on Initialized s,P by SCMFSA7B:19;
assume  s . a < 0 ; ::_thesis: IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
hence  IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) by A2, Th26, A1; ::_thesis: verum
end;
hereby  ::_thesis: verum
assume A3: s . a >= 0 ; ::_thesis: IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
A4: J is_halting_on Initialized s,P by SCMFSA7B:19;
A5: J is_closed_on Initialized s,P by SCMFSA7B:18;
percases ( s . a = 0 or s . a <> 0 ) ;
suppose s . a = 0 ; ::_thesis: IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
hence  IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) by A4, Th28, A5; ::_thesis: verum
end;
suppose s . a <> 0 ; ::_thesis: IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
hence  IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) by A3, A4, Th30, A5; ::_thesis: verum
end;
end;
end;
end;

registration
let I, J be parahalting Program of SCM+FSA;
let a be read-write Int-Location;
cluster if=0 (a,I,J) ->  parahalting ;
correctness
coherence
if=0 (a,I,J) is parahalting ;
by Th17;
cluster if>0 (a,I,J) ->  parahalting ;
correctness
coherence
if>0 (a,I,J) is parahalting ;
by Th23;
end;

definition
let a, b be Int-Location;
let I, J be Program of SCM+FSA;
func if=0 (a,b,I,J) ->  Program of SCM+FSA equals :: SCMFSA8B:def 4
(SubFrom (a,b)) ";" (if=0 (a,I,J));
coherence
 (SubFrom (a,b)) ";" (if=0 (a,I,J)) is Program of SCM+FSA ;
func if>0 (a,b,I,J) ->  Program of SCM+FSA equals :: SCMFSA8B:def 5
(SubFrom (a,b)) ";" (if>0 (a,I,J));
coherence
 (SubFrom (a,b)) ";" (if>0 (a,I,J)) is Program of SCM+FSA ;
end;

:: deftheorem  defines if=0 SCMFSA8B:def_4_:_
 for a, b being Int-Location
for I, J being Program of SCM+FSA holds if=0 (a,b,I,J) = (SubFrom (a,b)) ";" (if=0 (a,I,J));

:: deftheorem  defines if>0 SCMFSA8B:def_5_:_
 for a, b being Int-Location
for I, J being Program of SCM+FSA holds if>0 (a,b,I,J) = (SubFrom (a,b)) ";" (if>0 (a,I,J));

registration
let a be Int-Location;
let I, J be Program of SCM+FSA;
cluster if=0 (a,I,J) ->  non halt-free ;
coherence
 not if=0 (a,I,J) is halt-free ;
cluster if>0 (a,I,J) ->  non halt-free ;
coherence
 not if>0 (a,I,J) is halt-free ;
cluster if<0 (a,I,J) ->  non halt-free ;
coherence
 not if<0 (a,I,J) is halt-free ;
end;

registration
let a, b be Int-Location;
let I, J be Program of SCM+FSA;
cluster if=0 (a,b,I,J) ->  non halt-free ;
coherence
 not if=0 (a,b,I,J) is halt-free ;
cluster if>0 (a,b,I,J) ->  non halt-free ;
coherence
 not if>0 (a,b,I,J) is halt-free ;
end;

notation
let a, b be Int-Location;
let I, J be Program of SCM+FSA;
synonym  if<0 (b,a,I,J) for  if>0 (a,b,I,J);
end;

registration
let I, J be parahalting Program of SCM+FSA;
let a, b be read-write Int-Location;
cluster if=0 (a,b,I,J) ->  parahalting ;
correctness
coherence
if=0 (a,b,I,J) is parahalting ;
;
cluster if>0 (a,b,I,J) ->  parahalting ;
correctness
coherence
if>0 (a,b,I,J) is parahalting ;
;
end;

theorem :: SCMFSA8B:32
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being Program of SCM+FSA holds DataPart (Result ((P +* I),(Initialized s))) = DataPart (IExec (I,P,s)) by SCMFSA6B:def_1;

theorem Th33: :: SCMFSA8B:33
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being Program of SCM+FSA holds Result ((P +* I),(Initialized s)) = IExec (I,P,s) by SCMFSA6B:def_1;

theorem Th34: :: SCMFSA8B:34
for s1, s2 being State of SCM+FSA
for i being Instruction of SCM+FSA
for a being Int-Location st ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & not i refers a & IC s1 = IC s2 holds
( ( for b being Int-Location st a <> b holds
(Exec (i,s1)) . b = (Exec (i,s2)) . b ) & ( for f being FinSeq-Location holds (Exec (i,s1)) . f = (Exec (i,s2)) . f ) & IC (Exec (i,s1)) = IC (Exec (i,s2)) )
proof
let s1, s2 be State of SCM+FSA; ::_thesis: for i being Instruction of SCM+FSA
for a being Int-Location st ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & not i refers a & IC s1 = IC s2 holds
( ( for b being Int-Location st a <> b holds
(Exec (i,s1)) . b = (Exec (i,s2)) . b ) & ( for f being FinSeq-Location holds (Exec (i,s1)) . f = (Exec (i,s2)) . f ) & IC (Exec (i,s1)) = IC (Exec (i,s2)) )
let i be Instruction of SCM+FSA; ::_thesis: for a being Int-Location st ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & not i refers a & IC s1 = IC s2 holds
( ( for b being Int-Location st a <> b holds
(Exec (i,s1)) . b = (Exec (i,s2)) . b ) & ( for f being FinSeq-Location holds (Exec (i,s1)) . f = (Exec (i,s2)) . f ) & IC (Exec (i,s1)) = IC (Exec (i,s2)) )
let a be Int-Location; ::_thesis: ( ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & not i refers a & IC s1 = IC s2 implies ( ( for b being Int-Location st a <> b holds
(Exec (i,s1)) . b = (Exec (i,s2)) . b ) & ( for f being FinSeq-Location holds (Exec (i,s1)) . f = (Exec (i,s2)) . f ) & IC (Exec (i,s1)) = IC (Exec (i,s2)) ) )
defpred S1[ State of SCM+FSA, State of SCM+FSA] means ( ( for b being Int-Location st a <> b holds
$1 . b = $2 . b ) & ( for f being FinSeq-Location holds $1 . f = $2 . f ) );
assume A1: S1[s1,s2] ; ::_thesis: ( i refers a or not IC s1 = IC s2 or ( ( for b being Int-Location st a <> b holds
(Exec (i,s1)) . b = (Exec (i,s2)) . b ) & ( for f being FinSeq-Location holds (Exec (i,s1)) . f = (Exec (i,s2)) . f ) & IC (Exec (i,s1)) = IC (Exec (i,s2)) ) )
assume A2: not i refers a ; ::_thesis: ( not IC s1 = IC s2 or ( ( for b being Int-Location st a <> b holds
(Exec (i,s1)) . b = (Exec (i,s2)) . b ) & ( for f being FinSeq-Location holds (Exec (i,s1)) . f = (Exec (i,s2)) . f ) & IC (Exec (i,s1)) = IC (Exec (i,s2)) ) )
A3: InsCode i <= 12 by SCMFSA_2:16;
A4: now__::_thesis:_for_b_being_Int-Location_st_a_<>_b_holds_
(Exec_(i,s1))_._b_=_(Exec_(i,s2))_._b
let b be Int-Location; ::_thesis: ( a <> b implies (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1 )
assume A5: a <> b ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
percases ( InsCode i = 0 or InsCode i = 1 or InsCode i = 2 or InsCode i = 3 or InsCode i = 4 or InsCode i = 5 or InsCode i = 6 or InsCode i = 7 or InsCode i = 8 or InsCode i = 9 or InsCode i = 10 or InsCode i = 11 or InsCode i = 12 ) by A3, NAT_1:36;
suppose InsCode i = 0 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then A6: i = halt SCM+FSA by SCMFSA_2:95;
hence (Exec (i,s1)) . b = s1 . b by EXTPRO_1:def_3
.= s2 . b by A1, A5
.= (Exec (i,s2)) . b by A6, EXTPRO_1:def_3 ;
::_thesis: verum
end;
suppose InsCode i = 1 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then consider da, db being Int-Location such that
A7: i = da := db by SCMFSA_2:30;
A8: a <> db by A2, A7, SCMFSA7B:def_1;
hereby  ::_thesis: verum
percases ( b = da or b <> da ) ;
supposeA9: b = da ; ::_thesis: (Exec (i,s1)) . b = (Exec (i,s2)) . b
hence (Exec (i,s1)) . b = s1 . db by A7, SCMFSA_2:63
.= s2 . db by A1, A8
.= (Exec (i,s2)) . b by A7, A9, SCMFSA_2:63 ;
::_thesis: verum
end;
supposeA10: b <> da ; ::_thesis: (Exec (i,s1)) . b = (Exec (i,s2)) . b
hence (Exec (i,s1)) . b = s1 . b by A7, SCMFSA_2:63
.= s2 . b by A1, A5
.= (Exec (i,s2)) . b by A7, A10, SCMFSA_2:63 ;
::_thesis: verum
end;
end;
end;
end;
suppose InsCode i = 2 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then consider da, db being Int-Location such that
A11: i = AddTo (da,db) by SCMFSA_2:31;
A12: a <> db by A2, A11, SCMFSA7B:def_1;
hereby  ::_thesis: verum
percases ( b = da or b <> da ) ;
supposeA13: b = da ; ::_thesis: (Exec (i,s1)) . b = (Exec (i,s2)) . b
hence (Exec (i,s1)) . b = (s1 . b) + (s1 . db) by A11, SCMFSA_2:64
.= (s2 . b) + (s1 . db) by A1, A5
.= (s2 . b) + (s2 . db) by A1, A12
.= (Exec (i,s2)) . b by A11, A13, SCMFSA_2:64 ;
::_thesis: verum
end;
supposeA14: b <> da ; ::_thesis: (Exec (i,s1)) . b = (Exec (i,s2)) . b
hence (Exec (i,s1)) . b = s1 . b by A11, SCMFSA_2:64
.= s2 . b by A1, A5
.= (Exec (i,s2)) . b by A11, A14, SCMFSA_2:64 ;
::_thesis: verum
end;
end;
end;
end;
suppose InsCode i = 3 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then consider da, db being Int-Location such that
A15: i = SubFrom (da,db) by SCMFSA_2:32;
A16: a <> db by A2, A15, SCMFSA7B:def_1;
hereby  ::_thesis: verum
percases ( b = da or b <> da ) ;
supposeA17: b = da ; ::_thesis: (Exec (i,s1)) . b = (Exec (i,s2)) . b
hence (Exec (i,s1)) . b = (s1 . b) - (s1 . db) by A15, SCMFSA_2:65
.= (s2 . b) - (s1 . db) by A1, A5
.= (s2 . b) - (s2 . db) by A1, A16
.= (Exec (i,s2)) . b by A15, A17, SCMFSA_2:65 ;
::_thesis: verum
end;
supposeA18: b <> da ; ::_thesis: (Exec (i,s1)) . b = (Exec (i,s2)) . b
hence (Exec (i,s1)) . b = s1 . b by A15, SCMFSA_2:65
.= s2 . b by A1, A5
.= (Exec (i,s2)) . b by A15, A18, SCMFSA_2:65 ;
::_thesis: verum
end;
end;
end;
end;
suppose InsCode i = 4 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then consider da, db being Int-Location such that
A19: i = MultBy (da,db) by SCMFSA_2:33;
A20: a <> db by A2, A19, SCMFSA7B:def_1;
hereby  ::_thesis: verum
percases ( b = da or b <> da ) ;
supposeA21: b = da ; ::_thesis: (Exec (i,s1)) . b = (Exec (i,s2)) . b
hence (Exec (i,s1)) . b = (s1 . b) * (s1 . db) by A19, SCMFSA_2:66
.= (s2 . b) * (s1 . db) by A1, A5
.= (s2 . b) * (s2 . db) by A1, A20
.= (Exec (i,s2)) . b by A19, A21, SCMFSA_2:66 ;
::_thesis: verum
end;
supposeA22: b <> da ; ::_thesis: (Exec (i,s1)) . b = (Exec (i,s2)) . b
hence (Exec (i,s1)) . b = s1 . b by A19, SCMFSA_2:66
.= s2 . b by A1, A5
.= (Exec (i,s2)) . b by A19, A22, SCMFSA_2:66 ;
::_thesis: verum
end;
end;
end;
end;
suppose InsCode i = 5 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then consider da, db being Int-Location such that
A23: i = Divide (da,db) by SCMFSA_2:34;
A24: a <> db by A2, A23, SCMFSA7B:def_1;
A25: a <> da by A2, A23, SCMFSA7B:def_1;
hereby  ::_thesis: verum
percases ( b = db or ( b = da & b <> db ) or ( b <> da & b <> db ) ) ;
supposeA26: b = db ; ::_thesis: (Exec (i,s1)) . b = (Exec (i,s2)) . b
hence (Exec (i,s1)) . b = (s1 . da) mod (s1 . db) by A23, SCMFSA_2:67
.= (s2 . da) mod (s1 . db) by A1, A25
.= (s2 . da) mod (s2 . db) by A1, A24
.= (Exec (i,s2)) . b by A23, A26, SCMFSA_2:67 ;
::_thesis: verum
end;
supposeA27: ( b = da & b <> db ) ; ::_thesis: (Exec (i,s1)) . b = (Exec (i,s2)) . b
hence (Exec (i,s1)) . b = (s1 . da) div (s1 . db) by A23, SCMFSA_2:67
.= (s1 . da) div (s2 . db) by A1, A24
.= (s2 . da) div (s2 . db) by A1, A25
.= (Exec (i,s2)) . b by A23, A27, SCMFSA_2:67 ;
::_thesis: verum
end;
supposeA28: ( b <> da & b <> db ) ; ::_thesis: (Exec (i,s1)) . b = (Exec (i,s2)) . b
hence (Exec (i,s1)) . b = s1 . b by A23, SCMFSA_2:67
.= s2 . b by A1, A5
.= (Exec (i,s2)) . b by A23, A28, SCMFSA_2:67 ;
::_thesis: verum
end;
end;
end;
end;
suppose InsCode i = 6 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then A29: ex loc being Element of NAT st i = goto loc by SCMFSA_2:35;
hence (Exec (i,s1)) . b = s1 . b by SCMFSA_2:69
.= s2 . b by A1, A5
.= (Exec (i,s2)) . b by A29, SCMFSA_2:69 ;
::_thesis: verum
end;
suppose InsCode i = 7 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then A30: ex loc being Element of NAT ex da being Int-Location st i = da =0_goto loc by SCMFSA_2:36;
hence (Exec (i,s1)) . b = s1 . b by SCMFSA_2:70
.= s2 . b by A1, A5
.= (Exec (i,s2)) . b by A30, SCMFSA_2:70 ;
::_thesis: verum
end;
suppose InsCode i = 8 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then A31: ex loc being Element of NAT ex da being Int-Location st i = da >0_goto loc by SCMFSA_2:37;
hence (Exec (i,s1)) . b = s1 . b by SCMFSA_2:71
.= s2 . b by A1, A5
.= (Exec (i,s2)) . b by A31, SCMFSA_2:71 ;
::_thesis: verum
end;
suppose InsCode i = 9 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then consider db, da being Int-Location, g being FinSeq-Location  such that
A32: i = da := (g,db) by SCMFSA_2:38;
A33: a <> db by A2, A32, SCMFSA7B:def_1;
hereby  ::_thesis: verum
percases ( b = da or b <> da ) ;
supposeA34: b = da ; ::_thesis: (Exec (i,s1)) . b = (Exec (i,s2)) . b
then consider m2 being Element of NAT  such that
A35: m2 = abs (s2 . db) and
A36: (Exec ((da := (g,db)),s2)) . b = (s2 . g) /. m2 by SCMFSA_2:72;
consider m1 being Element of NAT  such that
A37: m1 = abs (s1 . db) and
A38: (Exec ((da := (g,db)),s1)) . b = (s1 . g) /. m1 by A34, SCMFSA_2:72;
 m1 = m2 by A1, A33, A37, A35;
hence  (Exec (i,s1)) . b = (Exec (i,s2)) . b by A1, A32, A38, A36; ::_thesis: verum
end;
supposeA39: b <> da ; ::_thesis: (Exec (i,s1)) . b = (Exec (i,s2)) . b
hence (Exec (i,s1)) . b = s1 . b by A32, SCMFSA_2:72
.= s2 . b by A1, A5
.= (Exec (i,s2)) . b by A32, A39, SCMFSA_2:72 ;
::_thesis: verum
end;
end;
end;
end;
suppose InsCode i = 10 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then A40: ex db, da being Int-Location ex g being FinSeq-Location st i = (g,db) := da by SCMFSA_2:39;
hence (Exec (i,s1)) . b = s1 . b by SCMFSA_2:73
.= s2 . b by A1, A5
.= (Exec (i,s2)) . b by A40, SCMFSA_2:73 ;
::_thesis: verum
end;
suppose InsCode i = 11 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then consider da being Int-Location, g being FinSeq-Location  such that
A41: i = da :=len g by SCMFSA_2:40;
hereby  ::_thesis: verum
percases ( b = da or b <> da ) ;
supposeA42: b = da ; ::_thesis: (Exec (i,s1)) . b = (Exec (i,s2)) . b
hence (Exec (i,s1)) . b = len (s1 . g) by A41, SCMFSA_2:74
.= len (s2 . g) by A1
.= (Exec (i,s2)) . b by A41, A42, SCMFSA_2:74 ;
::_thesis: verum
end;
supposeA43: b <> da ; ::_thesis: (Exec (i,s1)) . b = (Exec (i,s2)) . b
hence (Exec (i,s1)) . b = s1 . b by A41, SCMFSA_2:74
.= s2 . b by A1, A5
.= (Exec (i,s2)) . b by A41, A43, SCMFSA_2:74 ;
::_thesis: verum
end;
end;
end;
end;
suppose InsCode i = 12 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then A44: ex da being Int-Location ex g being FinSeq-Location st i = g :=<0,...,0> da by SCMFSA_2:41;
hence (Exec (i,s1)) . b = s1 . b by SCMFSA_2:75
.= s2 . b by A1, A5
.= (Exec (i,s2)) . b by A44, SCMFSA_2:75 ;
::_thesis: verum
end;
end;
end;
assume A45: IC s1 = IC s2 ; ::_thesis: ( ( for b being Int-Location st a <> b holds
(Exec (i,s1)) . b = (Exec (i,s2)) . b ) & ( for f being FinSeq-Location holds (Exec (i,s1)) . f = (Exec (i,s2)) . f ) & IC (Exec (i,s1)) = IC (Exec (i,s2)) )
now__::_thesis:_for_f_being_FinSeq-Location_holds_(Exec_(i,s1))_._f_=_(Exec_(i,s2))_._f
let f be FinSeq-Location ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
percases ( InsCode i = 0 or InsCode i = 1 or InsCode i = 2 or InsCode i = 3 or InsCode i = 4 or InsCode i = 5 or InsCode i = 6 or InsCode i = 7 or InsCode i = 8 or InsCode i = 9 or InsCode i = 10 or InsCode i = 11 or InsCode i = 12 ) by A3, NAT_1:36;
suppose InsCode i = 0 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then A46: i = halt SCM+FSA by SCMFSA_2:95;
hence (Exec (i,s1)) . f = s1 . f by EXTPRO_1:def_3
.= s2 . f by A1
.= (Exec (i,s2)) . f by A46, EXTPRO_1:def_3 ;
::_thesis: verum
end;
suppose InsCode i = 1 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then A47: ex da, db being Int-Location st i = da := db by SCMFSA_2:30;
hence (Exec (i,s1)) . f = s1 . f by SCMFSA_2:63
.= s2 . f by A1
.= (Exec (i,s2)) . f by A47, SCMFSA_2:63 ;
::_thesis: verum
end;
suppose InsCode i = 2 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then A48: ex da, db being Int-Location st i = AddTo (da,db) by SCMFSA_2:31;
hence (Exec (i,s1)) . f = s1 . f by SCMFSA_2:64
.= s2 . f by A1
.= (Exec (i,s2)) . f by A48, SCMFSA_2:64 ;
::_thesis: verum
end;
suppose InsCode i = 3 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then A49: ex da, db being Int-Location st i = SubFrom (da,db) by SCMFSA_2:32;
hence (Exec (i,s1)) . f = s1 . f by SCMFSA_2:65
.= s2 . f by A1
.= (Exec (i,s2)) . f by A49, SCMFSA_2:65 ;
::_thesis: verum
end;
suppose InsCode i = 4 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then A50: ex da, db being Int-Location st i = MultBy (da,db) by SCMFSA_2:33;
hence (Exec (i,s1)) . f = s1 . f by SCMFSA_2:66
.= s2 . f by A1
.= (Exec (i,s2)) . f by A50, SCMFSA_2:66 ;
::_thesis: verum
end;
suppose InsCode i = 5 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then A51: ex da, db being Int-Location st i = Divide (da,db) by SCMFSA_2:34;
hence (Exec (i,s1)) . f = s1 . f by SCMFSA_2:67
.= s2 . f by A1
.= (Exec (i,s2)) . f by A51, SCMFSA_2:67 ;
::_thesis: verum
end;
suppose InsCode i = 6 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then A52: ex loc being Element of NAT st i = goto loc by SCMFSA_2:35;
hence (Exec (i,s1)) . f = s1 . f by SCMFSA_2:69
.= s2 . f by A1
.= (Exec (i,s2)) . f by A52, SCMFSA_2:69 ;
::_thesis: verum
end;
suppose InsCode i = 7 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then A53: ex loc being Element of NAT ex da being Int-Location st i = da =0_goto loc by SCMFSA_2:36;
hence (Exec (i,s1)) . f = s1 . f by SCMFSA_2:70
.= s2 . f by A1
.= (Exec (i,s2)) . f by A53, SCMFSA_2:70 ;
::_thesis: verum
end;
suppose InsCode i = 8 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then A54: ex loc being Element of NAT ex da being Int-Location st i = da >0_goto loc by SCMFSA_2:37;
hence (Exec (i,s1)) . f = s1 . f by SCMFSA_2:71
.= s2 . f by A1
.= (Exec (i,s2)) . f by A54, SCMFSA_2:71 ;
::_thesis: verum
end;
suppose InsCode i = 9 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then A55: ex db, da being Int-Location ex g being FinSeq-Location st i = da := (g,db) by SCMFSA_2:38;
hence (Exec (i,s1)) . f = s1 . f by SCMFSA_2:72
.= s2 . f by A1
.= (Exec (i,s2)) . f by A55, SCMFSA_2:72 ;
::_thesis: verum
end;
suppose InsCode i = 10 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then consider db, da being Int-Location, g being FinSeq-Location  such that
A56: i = (g,db) := da by SCMFSA_2:39;
A57: a <> db by A2, A56, SCMFSA7B:def_1;
A58: a <> da by A2, A56, SCMFSA7B:def_1;
hereby  ::_thesis: verum
percases ( f = g or f <> g ) ;
supposeA59: f = g ; ::_thesis: (Exec (i,s1)) . f = (Exec (i,s2)) . f
A60: s1 . da = s2 . da by A1, A58;
consider m2 being Element of NAT  such that
A61: m2 = abs (s2 . db) and
A62: (Exec (((g,db) := da),s2)) . g = (s2 . g) +* (m2,(s2 . da)) by SCMFSA_2:73;
consider m1 being Element of NAT  such that
A63: m1 = abs (s1 . db) and
A64: (Exec (((g,db) := da),s1)) . g = (s1 . g) +* (m1,(s1 . da)) by SCMFSA_2:73;
 m1 = m2 by A1, A57, A63, A61;
hence  (Exec (i,s1)) . f = (Exec (i,s2)) . f by A1, A56, A59, A64, A62, A60; ::_thesis: verum
end;
supposeA65: f <> g ; ::_thesis: (Exec (i,s1)) . f = (Exec (i,s2)) . f
hence (Exec (i,s1)) . f = s1 . f by A56, SCMFSA_2:73
.= s2 . f by A1
.= (Exec (i,s2)) . f by A56, A65, SCMFSA_2:73 ;
::_thesis: verum
end;
end;
end;
end;
suppose InsCode i = 11 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then A66: ex da being Int-Location ex g being FinSeq-Location st i = da :=len g by SCMFSA_2:40;
hence (Exec (i,s1)) . f = s1 . f by SCMFSA_2:74
.= s2 . f by A1
.= (Exec (i,s2)) . f by A66, SCMFSA_2:74 ;
::_thesis: verum
end;
suppose InsCode i = 12 ; ::_thesis: (Exec (i,s1)) . b1 = (Exec (i,s2)) . b1
then consider da being Int-Location, g being FinSeq-Location  such that
A67: i = g :=<0,...,0> da by SCMFSA_2:41;
A68: a <> da by A2, A67, SCMFSA7B:def_1;
hereby  ::_thesis: verum
percases ( f = g or f <> g ) ;
supposeA69: f = g ; ::_thesis: (Exec (i,s1)) . f = (Exec (i,s2)) . f
A70: ex m2 being Element of NAT st
( m2 = abs (s2 . da) & (Exec ((g :=<0,...,0> da),s2)) . g = m2 |-> 0 ) by SCMFSA_2:75;
 ex m1 being Element of NAT st
( m1 = abs (s1 . da) & (Exec ((g :=<0,...,0> da),s1)) . g = m1 |-> 0 ) by SCMFSA_2:75;
hence  (Exec (i,s1)) . f = (Exec (i,s2)) . f by A1, A67, A68, A69, A70; ::_thesis: verum
end;
supposeA71: f <> g ; ::_thesis: (Exec (i,s1)) . f = (Exec (i,s2)) . f
hence (Exec (i,s1)) . f = s1 . f by A67, SCMFSA_2:75
.= s2 . f by A1
.= (Exec (i,s2)) . f by A67, A71, SCMFSA_2:75 ;
::_thesis: verum
end;
end;
end;
end;
end;
end;
hence  S1[ Exec (i,s1), Exec (i,s2)] by A4; ::_thesis: IC (Exec (i,s1)) = IC (Exec (i,s2))
now__::_thesis:_(Exec_(i,s1))_._(IC_)_=_(Exec_(i,s2))_._(IC_)
percases ( InsCode i = 0 or InsCode i = 1 or InsCode i = 2 or InsCode i = 3 or InsCode i = 4 or InsCode i = 5 or InsCode i = 6 or InsCode i = 7 or InsCode i = 8 or InsCode i = 9 or InsCode i = 10 or InsCode i = 11 or InsCode i = 12 ) by A3, NAT_1:36;
suppose InsCode i = 0 ; ::_thesis: (Exec (i,s1)) . (IC ) = (Exec (i,s2)) . (IC )
then A72: i = halt SCM+FSA by SCMFSA_2:95;
hence (Exec (i,s1)) . (IC ) = IC s1 by EXTPRO_1:def_3
.= (Exec (i,s2)) . (IC ) by A45, A72, EXTPRO_1:def_3 ;
::_thesis: verum
end;
suppose InsCode i = 1 ; ::_thesis: (Exec (i,s1)) . (IC ) = (Exec (i,s2)) . (IC )
then A73: ex da, db being Int-Location st i = da := db by SCMFSA_2:30;
hence (Exec (i,s1)) . (IC ) = succ (IC s1) by SCMFSA_2:63
.= (Exec (i,s2)) . (IC ) by A45, A73, SCMFSA_2:63 ;
::_thesis: verum
end;
suppose InsCode i = 2 ; ::_thesis: (Exec (i,s1)) . (IC ) = (Exec (i,s2)) . (IC )
then A74: ex da, db being Int-Location st i = AddTo (da,db) by SCMFSA_2:31;
hence (Exec (i,s1)) . (IC ) = succ (IC s1) by SCMFSA_2:64
.= (Exec (i,s2)) . (IC ) by A45, A74, SCMFSA_2:64 ;
::_thesis: verum
end;
suppose InsCode i = 3 ; ::_thesis: (Exec (i,s1)) . (IC ) = (Exec (i,s2)) . (IC )
then A75: ex da, db being Int-Location st i = SubFrom (da,db) by SCMFSA_2:32;
hence (Exec (i,s1)) . (IC ) = succ (IC s1) by SCMFSA_2:65
.= (Exec (i,s2)) . (IC ) by A45, A75, SCMFSA_2:65 ;
::_thesis: verum
end;
suppose InsCode i = 4 ; ::_thesis: (Exec (i,s1)) . (IC ) = (Exec (i,s2)) . (IC )
then A76: ex da, db being Int-Location st i = MultBy (da,db) by SCMFSA_2:33;
hence (Exec (i,s1)) . (IC ) = succ (IC s1) by SCMFSA_2:66
.= (Exec (i,s2)) . (IC ) by A45, A76, SCMFSA_2:66 ;
::_thesis: verum
end;
suppose InsCode i = 5 ; ::_thesis: (Exec (i,s1)) . (IC ) = (Exec (i,s2)) . (IC )
then A77: ex da, db being Int-Location st i = Divide (da,db) by SCMFSA_2:34;
hence (Exec (i,s1)) . (IC ) = succ (IC s1) by SCMFSA_2:67
.= (Exec (i,s2)) . (IC ) by A45, A77, SCMFSA_2:67 ;
::_thesis: verum
end;
suppose InsCode i = 6 ; ::_thesis: (Exec (i,s1)) . (IC ) = (Exec (i,s2)) . (IC )
then consider loc being Element of NAT  such that
A78: i = goto loc by SCMFSA_2:35;
thus (Exec (i,s1)) . (IC ) = loc by A78, SCMFSA_2:69
.= (Exec (i,s2)) . (IC ) by A78, SCMFSA_2:69 ; ::_thesis: verum
end;
suppose InsCode i = 7 ; ::_thesis: (Exec (i,s1)) . (IC ) = (Exec (i,s2)) . (IC )
then consider loc being Element of NAT , da being Int-Location such that
A79: i = da =0_goto loc by SCMFSA_2:36;
 a <> da by A2, A79, SCMFSA7B:def_1;
then A80: s1 . da = s2 . da by A1;
hereby  ::_thesis: verum
percases ( s1 . da = 0 or s1 . da <> 0 ) ;
supposeA81: s1 . da = 0 ; ::_thesis: (Exec (i,s1)) . (IC ) = (Exec (i,s2)) . (IC )
hence (Exec (i,s1)) . (IC ) = loc by A79, SCMFSA_2:70
.= (Exec (i,s2)) . (IC ) by A79, A80, A81, SCMFSA_2:70 ;
::_thesis: verum
end;
supposeA82: s1 . da <> 0 ; ::_thesis: (Exec (i,s1)) . (IC ) = (Exec (i,s2)) . (IC )
hence (Exec (i,s1)) . (IC ) = succ (IC s1) by A79, SCMFSA_2:70
.= (Exec (i,s2)) . (IC ) by A45, A79, A80, A82, SCMFSA_2:70 ;
::_thesis: verum
end;
end;
end;
end;
suppose InsCode i = 8 ; ::_thesis: (Exec (i,s1)) . (IC ) = (Exec (i,s2)) . (IC )
then consider loc being Element of NAT , da being Int-Location such that
A83: i = da >0_goto loc by SCMFSA_2:37;
 a <> da by A2, A83, SCMFSA7B:def_1;
then A84: s1 . da = s2 . da by A1;
hereby  ::_thesis: verum
percases ( s1 . da > 0 or s1 . da <= 0 ) ;
supposeA85: s1 . da > 0 ; ::_thesis: (Exec (i,s1)) . (IC ) = (Exec (i,s2)) . (IC )
hence (Exec (i,s1)) . (IC ) = loc by A83, SCMFSA_2:71
.= (Exec (i,s2)) . (IC ) by A83, A84, A85, SCMFSA_2:71 ;
::_thesis: verum
end;
supposeA86: s1 . da <= 0 ; ::_thesis: (Exec (i,s1)) . (IC ) = (Exec (i,s2)) . (IC )
hence (Exec (i,s1)) . (IC ) = succ (IC s1) by A83, SCMFSA_2:71
.= (Exec (i,s2)) . (IC ) by A45, A83, A84, A86, SCMFSA_2:71 ;
::_thesis: verum
end;
end;
end;
end;
suppose InsCode i = 9 ; ::_thesis: (Exec (i,s1)) . (IC ) = (Exec (i,s2)) . (IC )
then A87: ex db, da being Int-Location ex g being FinSeq-Location st i = da := (g,db) by SCMFSA_2:38;
hence (Exec (i,s1)) . (IC ) = succ (IC s1) by SCMFSA_2:72
.= (Exec (i,s2)) . (IC ) by A45, A87, SCMFSA_2:72 ;
::_thesis: verum
end;
suppose InsCode i = 10 ; ::_thesis: (Exec (i,s1)) . (IC ) = (Exec (i,s2)) . (IC )
then A88: ex db, da being Int-Location ex g being FinSeq-Location st i = (g,db) := da by SCMFSA_2:39;
hence (Exec (i,s1)) . (IC ) = succ (IC s1) by SCMFSA_2:73
.= (Exec (i,s2)) . (IC ) by A45, A88, SCMFSA_2:73 ;
::_thesis: verum
end;
suppose InsCode i = 11 ; ::_thesis: (Exec (i,s1)) . (IC ) = (Exec (i,s2)) . (IC )
then A89: ex da being Int-Location ex g being FinSeq-Location st i = da :=len g by SCMFSA_2:40;
hence (Exec (i,s1)) . (IC ) = succ (IC s1) by SCMFSA_2:74
.= (Exec (i,s2)) . (IC ) by A45, A89, SCMFSA_2:74 ;
::_thesis: verum
end;
suppose InsCode i = 12 ; ::_thesis: (Exec (i,s1)) . (IC ) = (Exec (i,s2)) . (IC )
then A90: ex da being Int-Location ex g being FinSeq-Location st i = g :=<0,...,0> da by SCMFSA_2:41;
hence (Exec (i,s1)) . (IC ) = succ (IC s1) by SCMFSA_2:75
.= (Exec (i,s2)) . (IC ) by A45, A90, SCMFSA_2:75 ;
::_thesis: verum
end;
end;
end;
hence  IC (Exec (i,s1)) = IC (Exec (i,s2)) ; ::_thesis: verum
end;

theorem Th35: :: SCMFSA8B:35
for P1, P2 being Instruction-Sequence of SCM+FSA
for s1, s2 being State of SCM+FSA
for I being Program of SCM+FSA
for a being Int-Location st not I refers a & ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & I is_closed_on s1,P1 holds
for k being Element of NAT holds
( ( for b being Int-Location st a <> b holds
(Comput ((P1 +* I),(Initialize s1),k)) . b = (Comput ((P2 +* I),(Initialize s2),k)) . b ) & ( for f being FinSeq-Location holds (Comput ((P1 +* I),(Initialize s1),k)) . f = (Comput ((P2 +* I),(Initialize s2),k)) . f ) & IC (Comput ((P1 +* I),(Initialize s1),k)) = IC (Comput ((P2 +* I),(Initialize s2),k)) & CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k))) )
proof
let P1, P2 be Instruction-Sequence of SCM+FSA; ::_thesis: for s1, s2 being State of SCM+FSA
for I being Program of SCM+FSA
for a being Int-Location st not I refers a & ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & I is_closed_on s1,P1 holds
for k being Element of NAT holds
( ( for b being Int-Location st a <> b holds
(Comput ((P1 +* I),(Initialize s1),k)) . b = (Comput ((P2 +* I),(Initialize s2),k)) . b ) & ( for f being FinSeq-Location holds (Comput ((P1 +* I),(Initialize s1),k)) . f = (Comput ((P2 +* I),(Initialize s2),k)) . f ) & IC (Comput ((P1 +* I),(Initialize s1),k)) = IC (Comput ((P2 +* I),(Initialize s2),k)) & CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k))) )
let s1, s2 be State of SCM+FSA; ::_thesis: for I being Program of SCM+FSA
for a being Int-Location st not I refers a & ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & I is_closed_on s1,P1 holds
for k being Element of NAT holds
( ( for b being Int-Location st a <> b holds
(Comput ((P1 +* I),(Initialize s1),k)) . b = (Comput ((P2 +* I),(Initialize s2),k)) . b ) & ( for f being FinSeq-Location holds (Comput ((P1 +* I),(Initialize s1),k)) . f = (Comput ((P2 +* I),(Initialize s2),k)) . f ) & IC (Comput ((P1 +* I),(Initialize s1),k)) = IC (Comput ((P2 +* I),(Initialize s2),k)) & CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k))) )
let I be Program of SCM+FSA; ::_thesis: for a being Int-Location st not I refers a & ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & I is_closed_on s1,P1 holds
for k being Element of NAT holds
( ( for b being Int-Location st a <> b holds
(Comput ((P1 +* I),(Initialize s1),k)) . b = (Comput ((P2 +* I),(Initialize s2),k)) . b ) & ( for f being FinSeq-Location holds (Comput ((P1 +* I),(Initialize s1),k)) . f = (Comput ((P2 +* I),(Initialize s2),k)) . f ) & IC (Comput ((P1 +* I),(Initialize s1),k)) = IC (Comput ((P2 +* I),(Initialize s2),k)) & CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k))) )
let a be Int-Location; ::_thesis: ( not I refers a & ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & I is_closed_on s1,P1 implies for k being Element of NAT holds
( ( for b being Int-Location st a <> b holds
(Comput ((P1 +* I),(Initialize s1),k)) . b = (Comput ((P2 +* I),(Initialize s2),k)) . b ) & ( for f being FinSeq-Location holds (Comput ((P1 +* I),(Initialize s1),k)) . f = (Comput ((P2 +* I),(Initialize s2),k)) . f ) & IC (Comput ((P1 +* I),(Initialize s1),k)) = IC (Comput ((P2 +* I),(Initialize s2),k)) & CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k))) ) )
assume A1: not I refers a ; ::_thesis: ( ex b being Int-Location st
( a <> b & not s1 . b = s2 . b ) or ex f being FinSeq-Location st not s1 . f = s2 . f or not I is_closed_on s1,P1 or for k being Element of NAT holds
( ( for b being Int-Location st a <> b holds
(Comput ((P1 +* I),(Initialize s1),k)) . b = (Comput ((P2 +* I),(Initialize s2),k)) . b ) & ( for f being FinSeq-Location holds (Comput ((P1 +* I),(Initialize s1),k)) . f = (Comput ((P2 +* I),(Initialize s2),k)) . f ) & IC (Comput ((P1 +* I),(Initialize s1),k)) = IC (Comput ((P2 +* I),(Initialize s2),k)) & CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k))) ) )
set S2 = Initialize s2;
set Q2 = P2 +* I;
set S1 = Initialize s1;
set Q1 = P1 +* I;
A2: I c= P1 +* I by FUNCT_4:25;
A3: I c= P2 +* I by FUNCT_4:25;
defpred S1[ State of SCM+FSA, State of SCM+FSA] means ( ( for b being Int-Location st a <> b holds
$1 . b = $2 . b ) & ( for f being FinSeq-Location holds $1 . f = $2 . f ) );
assume that
A4: for b being Int-Location st a <> b holds
s1 . b = s2 . b and
A5: for f being FinSeq-Location holds s1 . f = s2 . f ; ::_thesis: ( not I is_closed_on s1,P1 or for k being Element of NAT holds
( ( for b being Int-Location st a <> b holds
(Comput ((P1 +* I),(Initialize s1),k)) . b = (Comput ((P2 +* I),(Initialize s2),k)) . b ) & ( for f being FinSeq-Location holds (Comput ((P1 +* I),(Initialize s1),k)) . f = (Comput ((P2 +* I),(Initialize s2),k)) . f ) & IC (Comput ((P1 +* I),(Initialize s1),k)) = IC (Comput ((P2 +* I),(Initialize s2),k)) & CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k))) ) )
A6: IC in dom (Start-At (0,SCM+FSA)) by MEMSTR_0:15;
A7: now__::_thesis:_for_f_being_FinSeq-Location_holds_(Initialize_s1)_._f_=_(Initialize_s2)_._f
let f be FinSeq-Location ; ::_thesis: (Initialize s1) . f = (Initialize s2) . f
A8: not f in dom (Start-At (0,SCM+FSA)) by SCMFSA_2:103;
hence (Initialize s1) . f = s1 . f by FUNCT_4:11
.= s2 . f by A5
.= (Initialize s2) . f by A8, FUNCT_4:11 ;
::_thesis: verum
end;
defpred S2[ Nat] means ( S1[ Comput ((P1 +* I),(Initialize s1),$1), Comput ((P2 +* I),(Initialize s2),$1)] & IC (Comput ((P1 +* I),(Initialize s1),$1)) = IC (Comput ((P2 +* I),(Initialize s2),$1)) & CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),$1))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),$1))) );
A9: IC (Comput ((P1 +* I),(Initialize s1),0)) = IC (Initialize s1)
.= IC (Start-At (0,SCM+FSA)) by A6, FUNCT_4:13
.= IC (Initialize s2) by A6, FUNCT_4:13
.= IC (Comput ((P2 +* I),(Initialize s2),0)) ;
A10: now__::_thesis:_for_b_being_Int-Location_st_a_<>_b_holds_
(Initialize_s1)_._b_=_(Initialize_s2)_._b
let b be Int-Location; ::_thesis: ( a <> b implies (Initialize s1) . b = (Initialize s2) . b )
assume A11: a <> b ; ::_thesis: (Initialize s1) . b = (Initialize s2) . b
A12: not b in dom (Start-At (0,SCM+FSA)) by SCMFSA_2:102;
hence (Initialize s1) . b = s1 . b by FUNCT_4:11
.= s2 . b by A4, A11
.= (Initialize s2) . b by A12, FUNCT_4:11 ;
::_thesis: verum
end;
assume A13: I is_closed_on s1,P1 ; ::_thesis: for k being Element of NAT holds
( ( for b being Int-Location st a <> b holds
(Comput ((P1 +* I),(Initialize s1),k)) . b = (Comput ((P2 +* I),(Initialize s2),k)) . b ) & ( for f being FinSeq-Location holds (Comput ((P1 +* I),(Initialize s1),k)) . f = (Comput ((P2 +* I),(Initialize s2),k)) . f ) & IC (Comput ((P1 +* I),(Initialize s1),k)) = IC (Comput ((P2 +* I),(Initialize s2),k)) & CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k))) )
A14: IC (Comput ((P1 +* I),(Initialize s1),0)) in dom I by A13, SCMFSA7B:def_6;
A15: 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] )
A16: Comput ((P1 +* I),(Initialize s1),(k + 1)) = Following ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k))) by EXTPRO_1:3
.= Exec ((CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k)))),(Comput ((P1 +* I),(Initialize s1),k))) ;
A17: IC (Comput ((P1 +* I),(Initialize s1),k)) in dom I by A13, SCMFSA7B:def_6;
A18: Comput ((P2 +* I),(Initialize s2),(k + 1)) = Following ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k))) by EXTPRO_1:3
.= Exec ((CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k)))),(Comput ((P2 +* I),(Initialize s2),k))) ;
A19: (P1 +* I) /. (IC (Comput ((P1 +* I),(Initialize s1),k))) = (P1 +* I) . (IC (Comput ((P1 +* I),(Initialize s1),k))) by PBOOLE:143;
 CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k))) = I . (IC (Comput ((P1 +* I),(Initialize s1),k))) by A17, A19, A2, GRFUNC_1:2;
then  CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k))) in rng I by A17, FUNCT_1:def_3;
then A20: not CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k))) refers a by A1, SCMFSA7B:def_2;
assume A21: S2[k] ; ::_thesis: S2[k + 1]
hence  S1[ Comput ((P1 +* I),(Initialize s1),(k + 1)), Comput ((P2 +* I),(Initialize s2),(k + 1))] by A16, A18, A20, Th34; ::_thesis: ( IC (Comput ((P1 +* I),(Initialize s1),(k + 1))) = IC (Comput ((P2 +* I),(Initialize s2),(k + 1))) & CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),(k + 1)))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),(k + 1)))) )
thus A22: IC (Comput ((P1 +* I),(Initialize s1),(k + 1))) = IC (Comput ((P2 +* I),(Initialize s2),(k + 1))) by A21, A16, A18, A20, Th34; ::_thesis: CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),(k + 1)))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),(k + 1))))
A23: IC (Comput ((P1 +* I),(Initialize s1),(k + 1))) in dom I by A13, SCMFSA7B:def_6;
A24: (P1 +* I) /. (IC (Comput ((P1 +* I),(Initialize s1),(k + 1)))) = (P1 +* I) . (IC (Comput ((P1 +* I),(Initialize s1),(k + 1)))) by PBOOLE:143;
A25: (P2 +* I) /. (IC (Comput ((P2 +* I),(Initialize s2),(k + 1)))) = (P2 +* I) . (IC (Comput ((P2 +* I),(Initialize s2),(k + 1)))) by PBOOLE:143;
thus  CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),(k + 1)))) = I . (IC (Comput ((P1 +* I),(Initialize s1),(k + 1)))) by A23, A24, A2, GRFUNC_1:2
.= CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),(k + 1)))) by A22, A23, A25, A3, GRFUNC_1:2 ; ::_thesis: verum
end;
CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),0))) = (P1 +* I) . (IC (Comput ((P1 +* I),(Initialize s1),0))) by PBOOLE:143
.= I . (IC (Comput ((P1 +* I),(Initialize s1),0))) by A14, A2, GRFUNC_1:2
.= (P2 +* I) . (IC (Comput ((P2 +* I),(Initialize s2),0))) by A9, A14, A3, GRFUNC_1:2
.= CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),0))) by PBOOLE:143 ;
then A26: S2[ 0 ] by A10, A7, A9;
 for k being Element of NAT holds S2[k] from NAT_1:sch_1(A26, A15);
hence  for k being Element of NAT holds
( ( for b being Int-Location st a <> b holds
(Comput ((P1 +* I),(Initialize s1),k)) . b = (Comput ((P2 +* I),(Initialize s2),k)) . b ) & ( for f being FinSeq-Location holds (Comput ((P1 +* I),(Initialize s1),k)) . f = (Comput ((P2 +* I),(Initialize s2),k)) . f ) & IC (Comput ((P1 +* I),(Initialize s1),k)) = IC (Comput ((P2 +* I),(Initialize s2),k)) & CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k))) ) ; ::_thesis: verum
end;

theorem :: SCMFSA8B:36
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for l being Element of NAT holds
( I is_closed_on s,P & I is_halting_on s,P iff ( I is_closed_on s +* (Start-At (l,SCM+FSA)),P +* I & I is_halting_on s +* (Start-At (l,SCM+FSA)),P +* I ) )
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for l being Element of NAT holds
( I is_closed_on s,P & I is_halting_on s,P iff ( I is_closed_on s +* (Start-At (l,SCM+FSA)),P +* I & I is_halting_on s +* (Start-At (l,SCM+FSA)),P +* I ) )
let s be State of SCM+FSA; ::_thesis: for I, J being Program of SCM+FSA
for l being Element of NAT holds
( I is_closed_on s,P & I is_halting_on s,P iff ( I is_closed_on s +* (Start-At (l,SCM+FSA)),P +* I & I is_halting_on s +* (Start-At (l,SCM+FSA)),P +* I ) )
let I, J be Program of SCM+FSA; ::_thesis: for l being Element of NAT holds
( I is_closed_on s,P & I is_halting_on s,P iff ( I is_closed_on s +* (Start-At (l,SCM+FSA)),P +* I & I is_halting_on s +* (Start-At (l,SCM+FSA)),P +* I ) )
let l be Element of NAT ; ::_thesis: ( I is_closed_on s,P & I is_halting_on s,P iff ( I is_closed_on s +* (Start-At (l,SCM+FSA)),P +* I & I is_halting_on s +* (Start-At (l,SCM+FSA)),P +* I ) )
 DataPart s = DataPart (s +* (Start-At (l,SCM+FSA))) by MEMSTR_0:79;
hence  ( I is_closed_on s,P & I is_halting_on s,P iff ( I is_closed_on s +* (Start-At (l,SCM+FSA)),P +* I & I is_halting_on s +* (Start-At (l,SCM+FSA)),P +* I ) ) by Th5; ::_thesis: verum
end;

theorem Th37: :: SCMFSA8B:37
for P1, P2 being Instruction-Sequence of SCM+FSA
for s1, s2 being State of SCM+FSA
for I being Program of SCM+FSA
for a being Int-Location st not I refers a & ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & I is_closed_on s1,P1 & I is_halting_on s1,P1 holds
( I is_closed_on s2,P2 & I is_halting_on s2,P2 )
proof
let P1, P2 be Instruction-Sequence of SCM+FSA; ::_thesis: for s1, s2 being State of SCM+FSA
for I being Program of SCM+FSA
for a being Int-Location st not I refers a & ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & I is_closed_on s1,P1 & I is_halting_on s1,P1 holds
( I is_closed_on s2,P2 & I is_halting_on s2,P2 )
let s1, s2 be State of SCM+FSA; ::_thesis: for I being Program of SCM+FSA
for a being Int-Location st not I refers a & ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & I is_closed_on s1,P1 & I is_halting_on s1,P1 holds
( I is_closed_on s2,P2 & I is_halting_on s2,P2 )
let I be Program of SCM+FSA; ::_thesis: for a being Int-Location st not I refers a & ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & I is_closed_on s1,P1 & I is_halting_on s1,P1 holds
( I is_closed_on s2,P2 & I is_halting_on s2,P2 )
let a be Int-Location; ::_thesis: ( not I refers a & ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & I is_closed_on s1,P1 & I is_halting_on s1,P1 implies ( I is_closed_on s2,P2 & I is_halting_on s2,P2 ) )
assume A1: not I refers a ; ::_thesis: ( ex b being Int-Location st
( a <> b & not s1 . b = s2 . b ) or ex f being FinSeq-Location st not s1 . f = s2 . f or not I is_closed_on s1,P1 or not I is_halting_on s1,P1 or ( I is_closed_on s2,P2 & I is_halting_on s2,P2 ) )
set S2 = Initialize s2;
set Q2 = P2 +* I;
set S1 = Initialize s1;
set Q1 = P1 +* I;
assume A2: for b being Int-Location st a <> b holds
s1 . b = s2 . b ; ::_thesis: ( ex f being FinSeq-Location st not s1 . f = s2 . f or not I is_closed_on s1,P1 or not I is_halting_on s1,P1 or ( I is_closed_on s2,P2 & I is_halting_on s2,P2 ) )
assume A3: for f being FinSeq-Location holds s1 . f = s2 . f ; ::_thesis: ( not I is_closed_on s1,P1 or not I is_halting_on s1,P1 or ( I is_closed_on s2,P2 & I is_halting_on s2,P2 ) )
assume that
A4: I is_closed_on s1,P1 and
A5: I is_halting_on s1,P1 ; ::_thesis: ( I is_closed_on s2,P2 & I is_halting_on s2,P2 )
A6: now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P2_+*_I),(Initialize_s2),k))_in_dom_I
let k be Element of NAT ; ::_thesis: IC (Comput ((P2 +* I),(Initialize s2),k)) in dom I
 IC (Comput ((P1 +* I),(Initialize s1),k)) in dom I by A4, SCMFSA7B:def_6;
hence  IC (Comput ((P2 +* I),(Initialize s2),k)) in dom I by A1, Th35, A4, A3, A2; ::_thesis: verum
end;
 P1 +* I halts_on Initialize s1 by A5, SCMFSA7B:def_7;
then consider n being Element of NAT  such that
A7: CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),n))) = halt SCM+FSA by EXTPRO_1:29;
 CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),n))) = halt SCM+FSA by A1, A7, Th35, A4, A3, A2;
then  P2 +* I halts_on Initialize s2 by EXTPRO_1:29;
hence  ( I is_closed_on s2,P2 & I is_halting_on s2,P2 ) by A6, SCMFSA7B:def_6, SCMFSA7B:def_7; ::_thesis: verum
end;

theorem Th38: :: SCMFSA8B:38
for P1, P2 being Instruction-Sequence of SCM+FSA
for s1, s2 being State of SCM+FSA
for I being Program of SCM+FSA
for a being Int-Location st ( for d being read-write Int-Location st a <> d holds
s1 . d = s2 . d ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & not I refers a & I is_closed_on Initialized s1,P1 & I is_halting_on Initialized s1,P1 holds
( ( for d being Int-Location st a <> d holds
(IExec (I,P1,s1)) . d = (IExec (I,P2,s2)) . d ) & ( for f being FinSeq-Location holds (IExec (I,P1,s1)) . f = (IExec (I,P2,s2)) . f ) & IC (IExec (I,P1,s1)) = IC (IExec (I,P2,s2)) )
proof
let P1, P2 be Instruction-Sequence of SCM+FSA; ::_thesis: for s1, s2 being State of SCM+FSA
for I being Program of SCM+FSA
for a being Int-Location st ( for d being read-write Int-Location st a <> d holds
s1 . d = s2 . d ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & not I refers a & I is_closed_on Initialized s1,P1 & I is_halting_on Initialized s1,P1 holds
( ( for d being Int-Location st a <> d holds
(IExec (I,P1,s1)) . d = (IExec (I,P2,s2)) . d ) & ( for f being FinSeq-Location holds (IExec (I,P1,s1)) . f = (IExec (I,P2,s2)) . f ) & IC (IExec (I,P1,s1)) = IC (IExec (I,P2,s2)) )
let s1, s2 be State of SCM+FSA; ::_thesis: for I being Program of SCM+FSA
for a being Int-Location st ( for d being read-write Int-Location st a <> d holds
s1 . d = s2 . d ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & not I refers a & I is_closed_on Initialized s1,P1 & I is_halting_on Initialized s1,P1 holds
( ( for d being Int-Location st a <> d holds
(IExec (I,P1,s1)) . d = (IExec (I,P2,s2)) . d ) & ( for f being FinSeq-Location holds (IExec (I,P1,s1)) . f = (IExec (I,P2,s2)) . f ) & IC (IExec (I,P1,s1)) = IC (IExec (I,P2,s2)) )
let I be Program of SCM+FSA; ::_thesis: for a being Int-Location st ( for d being read-write Int-Location st a <> d holds
s1 . d = s2 . d ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & not I refers a & I is_closed_on Initialized s1,P1 & I is_halting_on Initialized s1,P1 holds
( ( for d being Int-Location st a <> d holds
(IExec (I,P1,s1)) . d = (IExec (I,P2,s2)) . d ) & ( for f being FinSeq-Location holds (IExec (I,P1,s1)) . f = (IExec (I,P2,s2)) . f ) & IC (IExec (I,P1,s1)) = IC (IExec (I,P2,s2)) )
let a be Int-Location; ::_thesis: ( ( for d being read-write Int-Location st a <> d holds
s1 . d = s2 . d ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & not I refers a & I is_closed_on Initialized s1,P1 & I is_halting_on Initialized s1,P1 implies ( ( for d being Int-Location st a <> d holds
(IExec (I,P1,s1)) . d = (IExec (I,P2,s2)) . d ) & ( for f being FinSeq-Location holds (IExec (I,P1,s1)) . f = (IExec (I,P2,s2)) . f ) & IC (IExec (I,P1,s1)) = IC (IExec (I,P2,s2)) ) )
assume that
A1: for d being read-write Int-Location st a <> d holds
s1 . d = s2 . d and
A2: for f being FinSeq-Location holds s1 . f = s2 . f ; ::_thesis: ( I refers a or not I is_closed_on Initialized s1,P1 or not I is_halting_on Initialized s1,P1 or ( ( for d being Int-Location st a <> d holds
(IExec (I,P1,s1)) . d = (IExec (I,P2,s2)) . d ) & ( for f being FinSeq-Location holds (IExec (I,P1,s1)) . f = (IExec (I,P2,s2)) . f ) & IC (IExec (I,P1,s1)) = IC (IExec (I,P2,s2)) ) )
A3: now__::_thesis:_for_d_being_Int-Location_st_a_<>_d_holds_
(Initialized_s1)_._d_=_(Initialized_s2)_._d
let d be Int-Location; ::_thesis: ( a <> d implies (Initialized s1) . b1 = (Initialized s2) . b1 )
assume A4: a <> d ; ::_thesis: (Initialized s1) . b1 = (Initialized s2) . b1
percases ( d = intloc 0 or d <> intloc 0 ) ;
supposeA5: d = intloc 0 ; ::_thesis: (Initialized s1) . b1 = (Initialized s2) . b1
hence (Initialized s1) . d = 1 by SCMFSA_M:9
.= (Initialized s2) . d by A5, SCMFSA_M:9 ;
::_thesis: verum
end;
suppose d <> intloc 0 ; ::_thesis: (Initialized s1) . b1 = (Initialized s2) . b1
then A6: d is read-write by SCMFSA_M:def_2;
hence (Initialized s1) . d = s1 . d by SCMFSA_M:37
.= s2 . d by A1, A4, A6
.= (Initialized s2) . d by A6, SCMFSA_M:37 ;
::_thesis: verum
end;
end;
end;
set S1 = Initialized s1;
set Q1 = P1 +* I;
set S2 = Initialized s2;
set Q2 = P2 +* I;
assume A7: not I refers a ; ::_thesis: ( not I is_closed_on Initialized s1,P1 or not I is_halting_on Initialized s1,P1 or ( ( for d being Int-Location st a <> d holds
(IExec (I,P1,s1)) . d = (IExec (I,P2,s2)) . d ) & ( for f being FinSeq-Location holds (IExec (I,P1,s1)) . f = (IExec (I,P2,s2)) . f ) & IC (IExec (I,P1,s1)) = IC (IExec (I,P2,s2)) ) )
A8: Initialized s2 = Initialize (Initialized s2) by MEMSTR_0:44;
assume that
A9: I is_closed_on Initialized s1,P1 and
A10: I is_halting_on Initialized s1,P1 ; ::_thesis: ( ( for d being Int-Location st a <> d holds
(IExec (I,P1,s1)) . d = (IExec (I,P2,s2)) . d ) & ( for f being FinSeq-Location holds (IExec (I,P1,s1)) . f = (IExec (I,P2,s2)) . f ) & IC (IExec (I,P1,s1)) = IC (IExec (I,P2,s2)) )
A11: now__::_thesis:_for_f_being_FinSeq-Location_holds_(Initialized_s1)_._f_=_(Initialized_s2)_._f
let f be FinSeq-Location ; ::_thesis: (Initialized s1) . f = (Initialized s2) . f
thus (Initialized s1) . f = s1 . f by SCMFSA_M:37
.= s2 . f by A2
.= (Initialized s2) . f by SCMFSA_M:37 ; ::_thesis: verum
end;
then  I is_halting_on Initialized s2,P2 by A7, A9, A10, A3, Th37;
then A12: P2 +* I halts_on Initialized s2 by A8, SCMFSA7B:def_7;
A13: Initialized s1 = Initialize (Initialized s1) by MEMSTR_0:44;
then A14: P1 +* I halts_on Initialized s1 by A10, SCMFSA7B:def_7;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_LifeSpan_((P1_+*_I),(Initialized_s1))_holds_
CurInstr_((P2_+*_I),(Comput_((P2_+*_I),(Initialized_s2),l)))_<>_halt_SCM+FSA
let l be Element of NAT ; ::_thesis: ( l < LifeSpan ((P1 +* I),(Initialized s1)) implies CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialized s2),l))) <> halt SCM+FSA )
assume  l < LifeSpan ((P1 +* I),(Initialized s1)) ; ::_thesis: CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialized s2),l))) <> halt SCM+FSA
then  CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialized s1),l))) <> halt SCM+FSA by A14, EXTPRO_1:def_15;
hence  CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialized s2),l))) <> halt SCM+FSA by A7, A9, A3, A11, A13, A8, Th35; ::_thesis: verum
end;
then A15: for l being Element of NAT st CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialized s2),l))) = halt SCM+FSA holds
LifeSpan ((P1 +* I),(Initialized s1)) <= l ;
CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialized s2),(LifeSpan ((P1 +* I),(Initialized s1)))))) = CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialized s1),(LifeSpan ((P1 +* I),(Initialized s1)))))) by A7, A9, A3, A11, A13, A8, Th35
.= halt SCM+FSA by A14, EXTPRO_1:def_15 ;
then A16: LifeSpan ((P1 +* I),(Initialized s1)) = LifeSpan ((P2 +* I),(Initialized s2)) by A12, A15, EXTPRO_1:def_15;
then A17: Result ((P2 +* I),(Initialized s2)) = Comput ((P2 +* I),(Initialized s2),(LifeSpan ((P1 +* I),(Initialized s1)))) by A12, EXTPRO_1:23;
A18: Result ((P1 +* I),(Initialized s1)) = Comput ((P1 +* I),(Initialized s1),(LifeSpan ((P1 +* I),(Initialized s1)))) by A14, EXTPRO_1:23;
A19: Result ((P1 +* I),(Initialized s1)) = IExec (I,P1,s1) by Th33;
A20: Result ((P2 +* I),(Initialized s2)) = IExec (I,P2,s2) by Th33;
thus  for d being Int-Location st a <> d holds
(IExec (I,P1,s1)) . d = (IExec (I,P2,s2)) . d by A20, A19, A7, A9, A3, A11, A13, A8, A17, A18, Th35; ::_thesis: ( ( for f being FinSeq-Location holds (IExec (I,P1,s1)) . f = (IExec (I,P2,s2)) . f ) & IC (IExec (I,P1,s1)) = IC (IExec (I,P2,s2)) )
thus  for f being FinSeq-Location holds (IExec (I,P1,s1)) . f = (IExec (I,P2,s2)) . f by A20, A19, A7, A9, A3, A11, A13, A8, A17, A18, Th35; ::_thesis: IC (IExec (I,P1,s1)) = IC (IExec (I,P2,s2))
thus  IC (IExec (I,P1,s1)) = IC (Result ((P1 +* I),(Initialized s1))) by SCMFSA6B:def_1
.= IC (Comput ((P1 +* I),(Initialized s1),(LifeSpan ((P1 +* I),(Initialized s1))))) by A14, EXTPRO_1:23
.= IC (Comput ((P2 +* I),(Initialized s2),(LifeSpan ((P2 +* I),(Initialized s2))))) by A7, A9, A3, A11, A13, A8, A16, Th35
.= IC (Result ((P2 +* I),(Initialized s2))) by A12, EXTPRO_1:23
.= IC (IExec (I,P2,s2)) by SCMFSA6B:def_1 ; ::_thesis: verum
end;

theorem :: SCMFSA8B:39
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being parahalting Program of SCM+FSA
for a, b being read-write Int-Location st not I refers a & not J refers a holds
( IC (IExec ((if=0 (a,b,I,J)),P,s)) = ((card I) + (card J)) + 5 & ( s . a = s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <> s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being parahalting Program of SCM+FSA
for a, b being read-write Int-Location st not I refers a & not J refers a holds
( IC (IExec ((if=0 (a,b,I,J)),P,s)) = ((card I) + (card J)) + 5 & ( s . a = s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <> s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
let s be State of SCM+FSA; ::_thesis: for I, J being parahalting Program of SCM+FSA
for a, b being read-write Int-Location st not I refers a & not J refers a holds
( IC (IExec ((if=0 (a,b,I,J)),P,s)) = ((card I) + (card J)) + 5 & ( s . a = s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <> s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
let I, J be parahalting Program of SCM+FSA; ::_thesis: for a, b being read-write Int-Location st not I refers a & not J refers a holds
( IC (IExec ((if=0 (a,b,I,J)),P,s)) = ((card I) + (card J)) + 5 & ( s . a = s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <> s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
let a, b be read-write Int-Location; ::_thesis: ( not I refers a & not J refers a implies ( IC (IExec ((if=0 (a,b,I,J)),P,s)) = ((card I) + (card J)) + 5 & ( s . a = s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <> s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) ) )
assume that
A1: not I refers a and
A2: not J refers a ; ::_thesis: ( IC (IExec ((if=0 (a,b,I,J)),P,s)) = ((card I) + (card J)) + 5 & ( s . a = s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <> s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
reconsider JJ = if=0 (a,I,J) as parahalting Program of SCM+FSA ;
reconsider II = Macro (SubFrom (a,b)) as keeping_0 parahalting Program of SCM+FSA ;
set i = SubFrom (a,b);
set s1 = Exec ((SubFrom (a,b)),(Initialized s));
A3: now__::_thesis:_for_c_being_read-write_Int-Location_st_a_<>_c_holds_
(Exec_((SubFrom_(a,b)),(Initialized_s)))_._c_=_s_._c
let c be read-write Int-Location; ::_thesis: ( a <> c implies (Exec ((SubFrom (a,b)),(Initialized s))) . c = s . c )
assume  a <> c ; ::_thesis: (Exec ((SubFrom (a,b)),(Initialized s))) . c = s . c
hence (Exec ((SubFrom (a,b)),(Initialized s))) . c = (Initialized s) . c by SCMFSA_2:65
.= s . c by SCMFSA_M:37 ;
::_thesis: verum
end;
 IExec ((if=0 (a,b,I,J)),P,s) = IncIC ((IExec (JJ,P,(IExec (II,P,s)))),(card II)) by SCMFSA6B:20;
hence  IC (IExec ((if=0 (a,b,I,J)),P,s)) = (IC (IExec (JJ,P,(IExec (II,P,s))))) + (card II) by FUNCT_4:113
.= (((card I) + (card J)) + 3) + (card II) by Th18
.= (((card I) + (card J)) + 3) + 2 by COMPOS_1:56
.= ((card I) + (card J)) + 5 ;
::_thesis: ( ( s . a = s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <> s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
A4: now__::_thesis:_for_f_being_FinSeq-Location_holds_(Exec_((SubFrom_(a,b)),(Initialized_s)))_._f_=_s_._f
let f be FinSeq-Location ; ::_thesis: (Exec ((SubFrom (a,b)),(Initialized s))) . f = s . f
thus (Exec ((SubFrom (a,b)),(Initialized s))) . f = (Initialized s) . f by SCMFSA_2:65
.= s . f by SCMFSA_M:37 ; ::_thesis: verum
end;
hereby  ::_thesis: ( s . a <> s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) )
assume A5: s . a = s . b ; ::_thesis: ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) )
A6: I is_closed_on Initialized s,P by SCMFSA7B:18;
A7: I is_halting_on Initialized s,P by SCMFSA7B:19;
A8: (Exec ((SubFrom (a,b)),(Initialized s))) . a = ((Initialized s) . a) - ((Initialized s) . b) by SCMFSA_2:65
.= (s . a) - ((Initialized s) . b) by SCMFSA_M:37
.= (s . a) - (s . b) by SCMFSA_M:37
.= 0 by A5 ;
hereby  ::_thesis: for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f
let d be Int-Location; ::_thesis: ( a <> d implies (IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d )
assume A9: a <> d ; ::_thesis: (IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d
thus (IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (JJ,P,(Exec ((SubFrom (a,b)),(Initialized s))))) . d by Th9
.= (IExec (I,P,(Exec ((SubFrom (a,b)),(Initialized s))))) . d by A8, Th18
.= (IExec (I,P,s)) . d by A1, A3, A4, A6, A7, A9, Th38 ; ::_thesis: verum
end;
let f be FinSeq-Location ; ::_thesis: (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f
thus (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (JJ,P,(Exec ((SubFrom (a,b)),(Initialized s))))) . f by Th10
.= (IExec (I,P,(Exec ((SubFrom (a,b)),(Initialized s))))) . f by A8, Th18
.= (IExec (I,P,s)) . f by A1, A3, A4, A6, A7, Th38 ; ::_thesis: verum
end;
A10: (Exec ((SubFrom (a,b)),(Initialized s))) . a = ((Initialized s) . a) - ((Initialized s) . b) by SCMFSA_2:65
.= (s . a) - ((Initialized s) . b) by SCMFSA_M:37
.= (s . a) - (s . b) by SCMFSA_M:37 ;
A11: J is_halting_on Initialized s,P by SCMFSA7B:19;
A12: J is_closed_on Initialized s,P by SCMFSA7B:18;
assume  s . a <> s . b ; ::_thesis: ( ( for d being Int-Location st a <> d holds
(IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) )
then A13: (s . a) + (- (s . b)) <> (s . b) + (- (s . b)) ;
hereby  ::_thesis: for f being FinSeq-Location holds (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f
let d be Int-Location; ::_thesis: ( a <> d implies (IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d )
assume A14: a <> d ; ::_thesis: (IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d
thus (IExec ((if=0 (a,b,I,J)),P,s)) . d = (IExec (JJ,P,(Exec ((SubFrom (a,b)),(Initialized s))))) . d by Th9
.= (IExec (J,P,(Exec ((SubFrom (a,b)),(Initialized s))))) . d by A10, A13, Th18
.= (IExec (J,P,s)) . d by A2, A3, A4, A12, A11, A14, Th38 ; ::_thesis: verum
end;
let f be FinSeq-Location ; ::_thesis: (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f
thus (IExec ((if=0 (a,b,I,J)),P,s)) . f = (IExec (JJ,P,(Exec ((SubFrom (a,b)),(Initialized s))))) . f by Th10
.= (IExec (J,P,(Exec ((SubFrom (a,b)),(Initialized s))))) . f by A10, A13, Th18
.= (IExec (J,P,s)) . f by A2, A3, A4, A12, A11, Th38 ; ::_thesis: verum
end;

theorem :: SCMFSA8B:40
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I, J being parahalting Program of SCM+FSA
for a, b being read-write Int-Location st not I refers a & not J refers a holds
( IC (IExec ((if>0 (a,b,I,J)),P,s)) = ((card I) + (card J)) + 5 & ( s . a > s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <= s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I, J being parahalting Program of SCM+FSA
for a, b being read-write Int-Location st not I refers a & not J refers a holds
( IC (IExec ((if>0 (a,b,I,J)),P,s)) = ((card I) + (card J)) + 5 & ( s . a > s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <= s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
let s be State of SCM+FSA; ::_thesis: for I, J being parahalting Program of SCM+FSA
for a, b being read-write Int-Location st not I refers a & not J refers a holds
( IC (IExec ((if>0 (a,b,I,J)),P,s)) = ((card I) + (card J)) + 5 & ( s . a > s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <= s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
let I, J be parahalting Program of SCM+FSA; ::_thesis: for a, b being read-write Int-Location st not I refers a & not J refers a holds
( IC (IExec ((if>0 (a,b,I,J)),P,s)) = ((card I) + (card J)) + 5 & ( s . a > s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <= s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
let a, b be read-write Int-Location; ::_thesis: ( not I refers a & not J refers a implies ( IC (IExec ((if>0 (a,b,I,J)),P,s)) = ((card I) + (card J)) + 5 & ( s . a > s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <= s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) ) )
assume that
A1: not I refers a and
A2: not J refers a ; ::_thesis: ( IC (IExec ((if>0 (a,b,I,J)),P,s)) = ((card I) + (card J)) + 5 & ( s . a > s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <= s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
reconsider JJ = if>0 (a,I,J) as parahalting Program of SCM+FSA ;
reconsider II = Macro (SubFrom (a,b)) as keeping_0 parahalting Program of SCM+FSA ;
set i = SubFrom (a,b);
set s1 = Exec ((SubFrom (a,b)),(Initialized s));
A3: now__::_thesis:_for_c_being_read-write_Int-Location_st_a_<>_c_holds_
(Exec_((SubFrom_(a,b)),(Initialized_s)))_._c_=_s_._c
let c be read-write Int-Location; ::_thesis: ( a <> c implies (Exec ((SubFrom (a,b)),(Initialized s))) . c = s . c )
assume  a <> c ; ::_thesis: (Exec ((SubFrom (a,b)),(Initialized s))) . c = s . c
hence (Exec ((SubFrom (a,b)),(Initialized s))) . c = (Initialized s) . c by SCMFSA_2:65
.= s . c by SCMFSA_M:37 ;
::_thesis: verum
end;
 IExec ((if>0 (a,b,I,J)),P,s) = IncIC ((IExec (JJ,P,(IExec (II,P,s)))),(card II)) by SCMFSA6B:20;
hence  IC (IExec ((if>0 (a,b,I,J)),P,s)) = (IC (IExec (JJ,P,(IExec (II,P,s))))) + (card II) by FUNCT_4:113
.= (((card I) + (card J)) + 3) + (card II) by Th24
.= (((card I) + (card J)) + 3) + 2 by COMPOS_1:56
.= ((card I) + (card J)) + 5 ;
::_thesis: ( ( s . a > s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) ) ) & ( s . a <= s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) ) )
A4: now__::_thesis:_for_f_being_FinSeq-Location_holds_(Exec_((SubFrom_(a,b)),(Initialized_s)))_._f_=_s_._f
let f be FinSeq-Location ; ::_thesis: (Exec ((SubFrom (a,b)),(Initialized s))) . f = s . f
thus (Exec ((SubFrom (a,b)),(Initialized s))) . f = (Initialized s) . f by SCMFSA_2:65
.= s . f by SCMFSA_M:37 ; ::_thesis: verum
end;
hereby  ::_thesis: ( s . a <= s . b implies ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) ) )
A5: (Exec ((SubFrom (a,b)),(Initialized s))) . a = ((Initialized s) . a) - ((Initialized s) . b) by SCMFSA_2:65
.= (s . a) - ((Initialized s) . b) by SCMFSA_M:37
.= (s . a) - (s . b) by SCMFSA_M:37 ;
assume  s . a > s . b ; ::_thesis: ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f ) )
then A6: (Exec ((SubFrom (a,b)),(Initialized s))) . a > 0 by A5, XREAL_1:50;
A7: I is_halting_on Initialized s,P by SCMFSA7B:19;
A8: I is_closed_on Initialized s,P by SCMFSA7B:18;
hereby  ::_thesis: for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f
let d be Int-Location; ::_thesis: ( a <> d implies (IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d )
assume A9: a <> d ; ::_thesis: (IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (I,P,s)) . d
thus (IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (JJ,P,(Exec ((SubFrom (a,b)),(Initialized s))))) . d by Th9
.= (IExec (I,P,(Exec ((SubFrom (a,b)),(Initialized s))))) . d by A6, Th24
.= (IExec (I,P,s)) . d by A1, A3, A4, A8, A7, A9, Th38 ; ::_thesis: verum
end;
let f be FinSeq-Location ; ::_thesis: (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (I,P,s)) . f
thus (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (JJ,P,(Exec ((SubFrom (a,b)),(Initialized s))))) . f by Th10
.= (IExec (I,P,(Exec ((SubFrom (a,b)),(Initialized s))))) . f by A6, Th24
.= (IExec (I,P,s)) . f by A1, A3, A4, A8, A7, Th38 ; ::_thesis: verum
end;
A10: (Exec ((SubFrom (a,b)),(Initialized s))) . a = ((Initialized s) . a) - ((Initialized s) . b) by SCMFSA_2:65
.= (s . a) - ((Initialized s) . b) by SCMFSA_M:37
.= (s . a) - (s . b) by SCMFSA_M:37 ;
A11: J is_closed_on Initialized s,P by SCMFSA7B:18;
A12: J is_halting_on Initialized s,P by SCMFSA7B:19;
assume  s . a <= s . b ; ::_thesis: ( ( for d being Int-Location st a <> d holds
(IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f ) )
then A13: (Exec ((SubFrom (a,b)),(Initialized s))) . a <= 0 by A10, XREAL_1:47;
hereby  ::_thesis: for f being FinSeq-Location holds (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f
let d be Int-Location; ::_thesis: ( a <> d implies (IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d )
assume A14: a <> d ; ::_thesis: (IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (J,P,s)) . d
thus (IExec ((if>0 (a,b,I,J)),P,s)) . d = (IExec (JJ,P,(Exec ((SubFrom (a,b)),(Initialized s))))) . d by Th9
.= (IExec (J,P,(Exec ((SubFrom (a,b)),(Initialized s))))) . d by A13, Th24
.= (IExec (J,P,s)) . d by A2, A3, A4, A11, A12, A14, Th38 ; ::_thesis: verum
end;
let f be FinSeq-Location ; ::_thesis: (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (J,P,s)) . f
thus (IExec ((if>0 (a,b,I,J)),P,s)) . f = (IExec (JJ,P,(Exec ((SubFrom (a,b)),(Initialized s))))) . f by Th10
.= (IExec (J,P,(Exec ((SubFrom (a,b)),(Initialized s))))) . f by A13, Th24
.= (IExec (J,P,s)) . f by A2, A3, A4, A11, A12, Th38 ; ::_thesis: verum
end;