:: SCMPDS_6 semantic presentation

begin

             set A = NAT ;

set D = SCM-Data-Loc ;

Lm1:  (Stop SCMPDS) . 0 = halt SCMPDS
by AFINSQ_1:34;

Lm2:  0 in dom (Stop SCMPDS)
by COMPOS_1:3;


theorem :: SCMPDS_6:1
canceled;

theorem :: SCMPDS_6:2
canceled;

theorem :: SCMPDS_6:3
canceled;

theorem :: SCMPDS_6:4
canceled;

theorem :: SCMPDS_6:5
canceled;

theorem Th6: :: SCMPDS_6:6
for i being Instruction of SCMPDS
for I being Program of SCMPDS holds card (i ';' I) = (card I) + 1
proof
let i be Instruction of SCMPDS; ::_thesis: for I being Program of SCMPDS holds card (i ';' I) = (card I) + 1
let I be Program of SCMPDS; ::_thesis: card (i ';' I) = (card I) + 1
thus  card (i ';' I) = card ((Load i) ';' I)
.= (card (Load i)) + (card I) by AFINSQ_1:17
.= (card I) + 1 by COMPOS_1:54 ; ::_thesis: verum
end;

theorem Th7: :: SCMPDS_6:7
for i being Instruction of SCMPDS
for I being Program of SCMPDS holds (i ';' I) . 0 = i
proof
let i be Instruction of SCMPDS; ::_thesis: for I being Program of SCMPDS holds (i ';' I) . 0 = i
let I be Program of SCMPDS; ::_thesis: (i ';' I) . 0 = i
 ( i ';' I = (Load i) ';' I & 0 in dom (Load i) ) by COMPOS_1:50;
hence (i ';' I) . 0 = (Load i) . 0 by AFINSQ_1:def_3
.= i by COMPOS_1:52 ;
::_thesis: verum
end;

theorem :: SCMPDS_6:8
canceled;

theorem :: SCMPDS_6:9
canceled;

theorem :: SCMPDS_6:10
canceled;

theorem Th11: :: SCMPDS_6:11
for i being Instruction of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for P being Instruction-Sequence of SCMPDS holds CurInstr ((P +* (stop (i ';' I))),(Initialize s)) = i
proof
let i be Instruction of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for P being Instruction-Sequence of SCMPDS holds CurInstr ((P +* (stop (i ';' I))),(Initialize s)) = i
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for P being Instruction-Sequence of SCMPDS holds CurInstr ((P +* (stop (i ';' I))),(Initialize s)) = i
let I be Program of SCMPDS; ::_thesis: for P being Instruction-Sequence of SCMPDS holds CurInstr ((P +* (stop (i ';' I))),(Initialize s)) = i
let P be Instruction-Sequence of SCMPDS; ::_thesis: CurInstr ((P +* (stop (i ';' I))),(Initialize s)) = i
set iI = i ';' I;
set P3 = P +* (stop (i ';' I));
A1: 0 in dom (Load i) by COMPOS_1:50;
 card (i ';' I) = (card I) + 1 by Th6;
then A2: 0 in dom (i ';' I) by AFINSQ_1:66;
 i ';' I c= stop (i ';' I) by AFINSQ_1:74;
then  dom (i ';' I) c= dom (stop (i ';' I)) by RELAT_1:11;
then A3: 0 in dom (stop (i ';' I)) by A2;
A4: (P +* (stop (i ';' I))) /. (IC (Initialize s)) = (P +* (stop (i ';' I))) . (IC (Initialize s)) by PBOOLE:143;
(P +* (stop (i ';' I))) . 0 = (P +* (stop (i ';' I))) . 0
.= (stop (i ';' I)) . 0 by A3, FUNCT_4:13
.= (i ';' I) . 0 by A2, COMPOS_1:63
.= ((Load i) ';' I) . 0
.= (Load i) . 0 by A1, AFINSQ_1:def_3
.= i by COMPOS_1:52 ;
hence  CurInstr ((P +* (stop (i ';' I))),(Initialize s)) = i by A4, MEMSTR_0:47; ::_thesis: verum
end;

theorem Th12: :: SCMPDS_6:12
for s being State of SCMPDS
for m1, m2 being Element of NAT st IC s = m1 holds
ICplusConst (s,m2) = m1 + m2
proof
let s be State of SCMPDS; ::_thesis: for m1, m2 being Element of NAT st IC s = m1 holds
ICplusConst (s,m2) = m1 + m2
let m1, m2 be Element of NAT ; ::_thesis: ( IC s = m1 implies ICplusConst (s,m2) = m1 + m2 )
 ex m being Element of NAT st
( m = IC s & ICplusConst (s,m2) = abs (m + m2) ) by SCMPDS_2:def_18;
hence  ( IC s = m1 implies ICplusConst (s,m2) = m1 + m2 ) by ABSVALUE:def_1; ::_thesis: verum
end;

theorem Th13: :: SCMPDS_6:13
for I, J being Program of SCMPDS holds Shift ((stop J),(card I)) c= stop (I ';' J)
proof
let I, J be Program of SCMPDS; ::_thesis: Shift ((stop J),(card I)) c= stop (I ';' J)
stop (I ';' J) = (I ';' J) ';' (Stop SCMPDS)
.= I ';' (J ';' (Stop SCMPDS)) by AFINSQ_1:27
.= I ';' (stop J) ;
then  stop (I ';' J) = I +* (Shift ((stop J),(card I))) ;
hence  Shift ((stop J),(card I)) c= stop (I ';' J) by FUNCT_4:25; ::_thesis: verum
end;

theorem :: SCMPDS_6:14
canceled;

theorem :: SCMPDS_6:15
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for i being No-StopCode parahalting Instruction of SCMPDS
for J being parahalting shiftable Program of SCMPDS
for a being Int_position holds (IExec ((i ';' J),P,s)) . a = (IExec (J,P,(Initialize (Exec (i,s))))) . a
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for i being No-StopCode parahalting Instruction of SCMPDS
for J being parahalting shiftable Program of SCMPDS
for a being Int_position holds (IExec ((i ';' J),P,s)) . a = (IExec (J,P,(Initialize (Exec (i,s))))) . a
let s be 0 -started State of SCMPDS; ::_thesis: for i being No-StopCode parahalting Instruction of SCMPDS
for J being parahalting shiftable Program of SCMPDS
for a being Int_position holds (IExec ((i ';' J),P,s)) . a = (IExec (J,P,(Initialize (Exec (i,s))))) . a
let i be No-StopCode parahalting Instruction of SCMPDS; ::_thesis: for J being parahalting shiftable Program of SCMPDS
for a being Int_position holds (IExec ((i ';' J),P,s)) . a = (IExec (J,P,(Initialize (Exec (i,s))))) . a
let J be parahalting shiftable Program of SCMPDS; ::_thesis: for a being Int_position holds (IExec ((i ';' J),P,s)) . a = (IExec (J,P,(Initialize (Exec (i,s))))) . a
let a be Int_position; ::_thesis: (IExec ((i ';' J),P,s)) . a = (IExec (J,P,(Initialize (Exec (i,s))))) . a
thus (IExec ((i ';' J),P,s)) . a = (IExec (((Load i) ';' J),P,s)) . a
.= (IExec (J,P,(Initialize (IExec ((Load i),P,s))))) . a by SCMPDS_5:35
.= (IExec (J,P,(Initialize (Exec (i,s))))) . a by SCMPDS_5:40 ; ::_thesis: verum
end;

theorem :: SCMPDS_6:16
for a being Int_position
for k1, k2 being Integer holds (a,k1) <>0_goto k2 <> halt SCMPDS
proof
 InsCode (halt SCMPDS) = 0 by COMPOS_1:70;
hence  for a being Int_position
for k1, k2 being Integer holds (a,k1) <>0_goto k2 <> halt SCMPDS by SCMPDS_2:16; ::_thesis: verum
end;

theorem :: SCMPDS_6:17
for a being Int_position
for k1, k2 being Integer holds (a,k1) <=0_goto k2 <> halt SCMPDS
proof
 InsCode (halt SCMPDS) = 0 by COMPOS_1:70;
hence  for a being Int_position
for k1, k2 being Integer holds (a,k1) <=0_goto k2 <> halt SCMPDS by SCMPDS_2:17; ::_thesis: verum
end;

theorem :: SCMPDS_6:18
for a being Int_position
for k1, k2 being Integer holds (a,k1) >=0_goto k2 <> halt SCMPDS
proof
 InsCode (halt SCMPDS) = 0 by COMPOS_1:70;
hence  for a being Int_position
for k1, k2 being Integer holds (a,k1) >=0_goto k2 <> halt SCMPDS by SCMPDS_2:18; ::_thesis: verum
end;

definition
let k1 be Integer;
func Goto k1 ->  Program of SCMPDS equals :: SCMPDS_6:def 1
 Load (goto k1);
coherence
 Load (goto k1) is Program of SCMPDS ;
end;

:: deftheorem  defines Goto SCMPDS_6:def_1_:_
 for k1 being Integer holds Goto k1 = Load (goto k1);

registration
let n be Element of NAT ;
cluster goto (n + 1) ->  No-StopCode ;
correctness
coherence
goto (n + 1) is No-StopCode ;
by SCMPDS_5:21;
cluster goto (- (n + 1)) ->  No-StopCode ;
correctness
coherence
goto (- (n + 1)) is No-StopCode ;
by SCMPDS_5:21;
end;

registration
let n be Element of NAT ;
cluster Goto (n + 1) ->  halt-free ;
correctness
coherence
Goto (n + 1) is halt-free ;
;
cluster Goto (- (n + 1)) ->  halt-free ;
correctness
coherence
Goto (- (n + 1)) is halt-free ;
;
end;

theorem Th19: :: SCMPDS_6:19
for k1 being Integer holds
( 0 in dom (Goto k1) & (Goto k1) . 0 = goto k1 ) by AFINSQ_1:34, AFINSQ_1:65;

begin

definition
let I be Program of SCMPDS;
let s be State of SCMPDS;
let P be Instruction-Sequence of SCMPDS;
predI is_closed_on s,P means :Def2: :: SCMPDS_6:def 2
 for k being Element of NAT holds IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom (stop I);
predI is_halting_on s,P means :Def3: :: SCMPDS_6:def 3
P +* (stop I) halts_on Initialize s;
end;

:: deftheorem Def2 defines is_closed_on SCMPDS_6:def_2_:_
 for I being Program of SCMPDS
for s being State of SCMPDS
for P being Instruction-Sequence of SCMPDS holds
( I is_closed_on s,P iff for k being Element of NAT holds IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom (stop I) );

:: deftheorem Def3 defines is_halting_on SCMPDS_6:def_3_:_
 for I being Program of SCMPDS
for s being State of SCMPDS
for P being Instruction-Sequence of SCMPDS holds
( I is_halting_on s,P iff P +* (stop I) halts_on Initialize s );

theorem Th20: :: SCMPDS_6:20
for I being Program of SCMPDS holds
( I is paraclosed iff for s being State of SCMPDS
for P being Instruction-Sequence of SCMPDS holds I is_closed_on s,P )
proof
let I be Program of SCMPDS; ::_thesis: ( I is paraclosed iff for s being State of SCMPDS
for P being Instruction-Sequence of SCMPDS holds I is_closed_on s,P )
hereby  ::_thesis: ( ( for s being State of SCMPDS
for P being Instruction-Sequence of SCMPDS holds I is_closed_on s,P ) implies I is paraclosed )
assume A1: I is paraclosed ; ::_thesis: for s being State of SCMPDS
for P being Instruction-Sequence of SCMPDS holds I is_closed_on s,P
let s be State of SCMPDS; ::_thesis: for P being Instruction-Sequence of SCMPDS holds I is_closed_on s,P
let P be Instruction-Sequence of SCMPDS; ::_thesis: I is_closed_on s,P
 stop I c= P +* (stop I) by FUNCT_4:25;
then  for n being Element of NAT holds IC (Comput ((P +* (stop I)),(Initialize s),n)) in dom (stop I) by A1, SCMPDS_4:def_6;
hence  I is_closed_on s,P by Def2; ::_thesis: verum
end;
assume A2: for s being State of SCMPDS
for P being Instruction-Sequence of SCMPDS holds I is_closed_on s,P ; ::_thesis: I is paraclosed
let s be 0 -started State of SCMPDS; :: according to SCMPDS_4:def_6 ::_thesis: for b1 being Element of NAT
for b2 being set holds
( not stop I c= b2 or IC (Comput (b2,s,b1)) in K249((stop I)) )
let k be Element of NAT ; ::_thesis: for b1 being set holds
( not stop I c= b1 or IC (Comput (b1,s,k)) in K249((stop I)) )
let P be Instruction-Sequence of SCMPDS; ::_thesis: ( not stop I c= P or IC (Comput (P,s,k)) in K249((stop I)) )
A3: Initialize s = s by MEMSTR_0:44;
assume  stop I c= P ; ::_thesis: IC (Comput (P,s,k)) in K249((stop I))
then A4: P = P +* (stop I) by FUNCT_4:98;
 I is_closed_on s,P by A2;
hence  IC (Comput (P,s,k)) in dom (stop I) by A3, A4, Def2; ::_thesis: verum
end;

theorem Th21: :: SCMPDS_6:21
for I being Program of SCMPDS holds
( I is parahalting iff for s being State of SCMPDS
for P being Instruction-Sequence of SCMPDS holds I is_halting_on s,P )
proof
let I be Program of SCMPDS; ::_thesis: ( I is parahalting iff for s being State of SCMPDS
for P being Instruction-Sequence of SCMPDS holds I is_halting_on s,P )
hereby  ::_thesis: ( ( for s being State of SCMPDS
for P being Instruction-Sequence of SCMPDS holds I is_halting_on s,P ) implies I is parahalting )
assume A1: I is parahalting ; ::_thesis: for s being State of SCMPDS
for P being Instruction-Sequence of SCMPDS holds I is_halting_on s,P
let s be State of SCMPDS; ::_thesis: for P being Instruction-Sequence of SCMPDS holds I is_halting_on s,P
let P be Instruction-Sequence of SCMPDS; ::_thesis: I is_halting_on s,P
 stop I c= P +* (stop I) by FUNCT_4:25;
then  P +* (stop I) halts_on Initialize s by A1, SCMPDS_4:def_7;
hence  I is_halting_on s,P by Def3; ::_thesis: verum
end;
assume A2: for s being State of SCMPDS
for P being Instruction-Sequence of SCMPDS holds I is_halting_on s,P ; ::_thesis: I is parahalting
let s be 0 -started State of SCMPDS; :: according to SCMPDS_4:def_7 ::_thesis: for b1 being set holds
( not stop I c= b1 or b1 halts_on s )
let P be Instruction-Sequence of SCMPDS; ::_thesis: ( not stop I c= P or P halts_on s )
A3: Initialize s = s by MEMSTR_0:44;
assume  stop I c= P ; ::_thesis: P halts_on s
then A4: P = P +* (stop I) by FUNCT_4:98;
 I is_halting_on s,P by A2;
hence  P halts_on s by A4, Def3, A3; ::_thesis: verum
end;

theorem Th22: :: SCMPDS_6:22
for P1, P2 being Instruction-Sequence of SCMPDS
for s1, s2 being State of SCMPDS
for I being Program of SCMPDS 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 SCMPDS; ::_thesis: for s1, s2 being State of SCMPDS
for I being Program of SCMPDS st DataPart s1 = DataPart s2 & I is_closed_on s1,P1 holds
I is_closed_on s2,P2
let s1, s2 be State of SCMPDS; ::_thesis: for I being Program of SCMPDS st DataPart s1 = DataPart s2 & I is_closed_on s1,P1 holds
I is_closed_on s2,P2
let I be Program of SCMPDS; ::_thesis: ( DataPart s1 = DataPart s2 & I is_closed_on s1,P1 implies I is_closed_on s2,P2 )
set pI = stop I;
set S1 = Initialize s1;
set S2 = Initialize s2;
set E1 = P1 +* (stop I);
set E2 = P2 +* (stop I);
assume A1: DataPart s1 = DataPart s2 ; ::_thesis: ( not I is_closed_on s1,P1 or I is_closed_on s2,P2 )
A2: Comput ((P2 +* (stop I)),(Initialize s2),0) = Initialize s2 by EXTPRO_1:2;
A3: Comput ((P1 +* (stop I)),(Initialize s1),0) = Initialize s1 by EXTPRO_1:2;
then A4: DataPart (Comput ((P1 +* (stop I)),(Initialize s1),0)) = DataPart s1 by MEMSTR_0:45
.= DataPart (Comput ((P2 +* (stop I)),(Initialize s2),0)) by A1, A2, MEMSTR_0:45 ;
defpred S1[ Element of NAT ] means ( IC (Comput ((P1 +* (stop I)),(Initialize s1),$1)) = IC (Comput ((P2 +* (stop I)),(Initialize s2),$1)) & CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),$1))) = CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),$1))) & DataPart (Comput ((P1 +* (stop I)),(Initialize s1),$1)) = DataPart (Comput ((P2 +* (stop I)),(Initialize s2),$1)) );
A5: 0 in dom (stop I) by COMPOS_1:36;
assume A6: I is_closed_on s1,P1 ; ::_thesis: I is_closed_on s2,P2
A7: 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] )
A8: Comput ((P2 +* (stop I)),(Initialize s2),(k + 1)) = Following ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),k))) by EXTPRO_1:3;
assume A9: S1[k] ; ::_thesis: S1[k + 1]
then A10: for a being Int_position holds (Comput ((P1 +* (stop I)),(Initialize s1),k)) . a = (Comput ((P2 +* (stop I)),(Initialize s2),k)) . a by SCMPDS_4:8;
 stop I c= P2 +* (stop I) by FUNCT_4:25;
then A11: stop I c= P2 +* (stop I) ;
A12: IC (Comput ((P1 +* (stop I)),(Initialize s1),(k + 1))) in dom (stop I) by A6, Def2;
A13: Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)) = Following ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),k))) by EXTPRO_1:3;
 Comput ((P1 +* (stop I)),(Initialize s1),k) = Comput ((P2 +* (stop I)),(Initialize s2),k) by A9, A10, SCMPDS_4:2;
then A14: Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)) = Comput ((P2 +* (stop I)),(Initialize s2),(k + 1)) by A9, A8, A13;
then A15: Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)) = Comput ((P2 +* (stop I)),(Initialize s2),(k + 1)) ;
A16: IC (Comput ((P1 +* (stop I)),(Initialize s1),(k + 1))) = IC (Comput ((P2 +* (stop I)),(Initialize s2),(k + 1))) by A14;
A17: (P1 +* (stop I)) /. (IC (Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)))) = (P1 +* (stop I)) . (IC (Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)))) by PBOOLE:143;
A18: (P2 +* (stop I)) /. (IC (Comput ((P2 +* (stop I)),(Initialize s2),(k + 1)))) = (P2 +* (stop I)) . (IC (Comput ((P2 +* (stop I)),(Initialize s2),(k + 1)))) by PBOOLE:143;
 stop I c= P1 +* (stop I) by FUNCT_4:25;
then  stop I c= P1 +* (stop I) ;
then  CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)))) = (stop I) . (IC (Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)))) by A12, A17, GRFUNC_1:2
.= CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),(k + 1)))) by A11, A16, A12, A18, GRFUNC_1:2 ;
hence  S1[k + 1] by A15; ::_thesis: verum
end;
A19: IC (Comput ((P2 +* (stop I)),(Initialize s2),0)) = IC (Initialize s2) by A2
.= 0 by MEMSTR_0:def_11 ;
A20: (P1 +* (stop I)) /. (IC (Comput ((P1 +* (stop I)),(Initialize s1),0))) = (P1 +* (stop I)) . (IC (Comput ((P1 +* (stop I)),(Initialize s1),0))) by PBOOLE:143;
A21: (P2 +* (stop I)) /. (IC (Comput ((P2 +* (stop I)),(Initialize s2),0))) = (P2 +* (stop I)) . (IC (Comput ((P2 +* (stop I)),(Initialize s2),0))) by PBOOLE:143;
A22: IC (Comput ((P1 +* (stop I)),(Initialize s1),0)) = IC (Initialize s1) by A3
.= 0 by MEMSTR_0:def_11 ;
then  CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),0))) = (stop I) . 0 by A5, A20, FUNCT_4:13
.= CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),0))) by A19, A5, A21, FUNCT_4:13 ;
then A23: S1[ 0 ] by A22, A19, A4;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P2_+*_(stop_I)),(Initialize_s2),k))_in_dom_(stop_I)
let k be Element of NAT ; ::_thesis: IC (Comput ((P2 +* (stop I)),(Initialize s2),k)) in dom (stop I)
A24: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A23, A7);
 IC (Comput ((P1 +* (stop I)),(Initialize s1),k)) in dom (stop I) by A6, Def2;
hence  IC (Comput ((P2 +* (stop I)),(Initialize s2),k)) in dom (stop I) by A24; ::_thesis: verum
end;
hence  I is_closed_on s2,P2 by Def2; ::_thesis: verum
end;

theorem :: SCMPDS_6:23
for P1, P2 being Instruction-Sequence of SCMPDS
for s1, s2 being State of SCMPDS
for I being Program of SCMPDS 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 SCMPDS; ::_thesis: for s1, s2 being State of SCMPDS
for I being Program of SCMPDS 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 SCMPDS; ::_thesis: for I being Program of SCMPDS 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 SCMPDS; ::_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 pI = stop I;
set S1 = Initialize s1;
set S2 = Initialize s2;
set E1 = P1 +* (stop I);
set E2 = P2 +* (stop I);
defpred S1[ Element of NAT ] means ( IC (Comput ((P1 +* (stop I)),(Initialize s1),$1)) = IC (Comput ((P2 +* (stop I)),(Initialize s2),$1)) & CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),$1))) = CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),$1))) & DataPart (Comput ((P1 +* (stop I)),(Initialize s1),$1)) = DataPart (Comput ((P2 +* (stop I)),(Initialize s2),$1)) );
A1: Comput ((P1 +* (stop I)),(Initialize s1),0) = Initialize s1 by EXTPRO_1:2;
A2: Comput ((P2 +* (stop I)),(Initialize s2),0) = Initialize s2 by EXTPRO_1:2;
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 A3: Comput ((P1 +* (stop I)),(Initialize s1),0) = Comput ((P2 +* (stop I)),(Initialize s2),0) by A1, A2, MEMSTR_0:80;
A4: 0 in dom (stop I) by COMPOS_1:36;
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 +* (stop I)),(Initialize s2),(k + 1)) = Following ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),k))) by EXTPRO_1:3;
assume A8: S1[k] ; ::_thesis: S1[k + 1]
then  for a being Int_position holds (Comput ((P1 +* (stop I)),(Initialize s1),k)) . a = (Comput ((P2 +* (stop I)),(Initialize s2),k)) . a by SCMPDS_4:8;
then A9: Comput ((P1 +* (stop I)),(Initialize s1),k) = Comput ((P2 +* (stop I)),(Initialize s2),k) by A8, SCMPDS_4:2;
A10: stop I c= P2 +* (stop I) by FUNCT_4:25;
A11: IC (Comput ((P1 +* (stop I)),(Initialize s1),(k + 1))) in dom (stop I) by A5, Def2;
 Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)) = Following ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),k))) by EXTPRO_1:3;
then A12: Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)) = Comput ((P2 +* (stop I)),(Initialize s2),(k + 1)) by A8, A9, A7;
then A13: IC (Comput ((P1 +* (stop I)),(Initialize s1),(k + 1))) = IC (Comput ((P2 +* (stop I)),(Initialize s2),(k + 1))) ;
A14: (P1 +* (stop I)) /. (IC (Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)))) = (P1 +* (stop I)) . (IC (Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)))) by PBOOLE:143;
A15: (P2 +* (stop I)) /. (IC (Comput ((P2 +* (stop I)),(Initialize s2),(k + 1)))) = (P2 +* (stop I)) . (IC (Comput ((P2 +* (stop I)),(Initialize s2),(k + 1)))) by PBOOLE:143;
 stop I c= P1 +* (stop I) by FUNCT_4:25;
then  CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)))) = (stop I) . (IC (Comput ((P1 +* (stop I)),(Initialize s1),(k + 1)))) by A11, A14, GRFUNC_1:2
.= CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),(k + 1)))) by A10, A13, A11, A15, GRFUNC_1:2 ;
hence  S1[k + 1] by A12; ::_thesis: verum
end;
A16: IC (Comput ((P2 +* (stop I)),(Initialize s2),0)) = IC (Initialize s2) by A2
.= 0 by MEMSTR_0:def_11 ;
assume  I is_halting_on s1,P1 ; ::_thesis: ( I is_closed_on s2,P2 & I is_halting_on s2,P2 )
then  P1 +* (stop I) halts_on Initialize s1 by Def3;
then consider m being Element of NAT  such that
A17: CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),m))) = halt SCMPDS by EXTPRO_1:29;
A18: (P1 +* (stop I)) /. (IC (Comput ((P1 +* (stop I)),(Initialize s1),0))) = (P1 +* (stop I)) . (IC (Comput ((P1 +* (stop I)),(Initialize s1),0))) by PBOOLE:143;
A19: (P2 +* (stop I)) /. (IC (Comput ((P2 +* (stop I)),(Initialize s2),0))) = (P2 +* (stop I)) . (IC (Comput ((P2 +* (stop I)),(Initialize s2),0))) by PBOOLE:143;
IC (Comput ((P1 +* (stop I)),(Initialize s1),0)) = IC (Initialize s1) by A1
.= 0 by MEMSTR_0:def_11 ;
then  CurInstr ((P1 +* (stop I)),(Comput ((P1 +* (stop I)),(Initialize s1),0))) = (stop I) . 0 by A4, A18, FUNCT_4:13
.= CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),0))) by A16, A4, A19, FUNCT_4:13 ;
then A20: S1[ 0 ] by A3;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P2_+*_(stop_I)),(Initialize_s2),k))_in_dom_(stop_I)
let k be Element of NAT ; ::_thesis: IC (Comput ((P2 +* (stop I)),(Initialize s2),k)) in dom (stop I)
A21: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A20, A6);
 IC (Comput ((P1 +* (stop I)),(Initialize s1),k)) in dom (stop I) by A5, Def2;
hence  IC (Comput ((P2 +* (stop I)),(Initialize s2),k)) in dom (stop I) by A21; ::_thesis: verum
end;
hence  I is_closed_on s2,P2 by Def2; ::_thesis: I is_halting_on s2,P2
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A20, A6);
then  CurInstr ((P2 +* (stop I)),(Comput ((P2 +* (stop I)),(Initialize s2),m))) = halt SCMPDS by A17;
then  P2 +* (stop I) halts_on Initialize s2 by EXTPRO_1:29;
hence  I is_halting_on s2,P2 by Def3; ::_thesis: verum
end;

theorem Th24: :: SCMPDS_6:24
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I, J being Program of SCMPDS holds
( I is_closed_on s,P iff I is_closed_on Initialize s,P +* J )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I, J being Program of SCMPDS holds
( I is_closed_on s,P iff I is_closed_on Initialize s,P +* J )
let s be State of SCMPDS; ::_thesis: for I, J being Program of SCMPDS holds
( I is_closed_on s,P iff I is_closed_on Initialize s,P +* J )
let I, J be Program of SCMPDS; ::_thesis: ( I is_closed_on s,P iff I is_closed_on Initialize s,P +* J )
 DataPart s = DataPart (Initialize s) by MEMSTR_0:45;
hence  ( I is_closed_on s,P iff I is_closed_on Initialize s,P +* J ) by Th22; ::_thesis: verum
end;

theorem Th25: :: SCMPDS_6:25
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I, J being Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( ( for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),s) holds
IC (Comput ((P +* (stop I)),s,k)) = IC (Comput ((P +* (stop (I ';' J))),s,k)) ) & DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop (I ';' J))),s,(LifeSpan ((P +* (stop I)),s)))) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I, J being Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( ( for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),s) holds
IC (Comput ((P +* (stop I)),s,k)) = IC (Comput ((P +* (stop (I ';' J))),s,k)) ) & DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop (I ';' J))),s,(LifeSpan ((P +* (stop I)),s)))) )
let s be 0 -started State of SCMPDS; ::_thesis: for I, J being Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( ( for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),s) holds
IC (Comput ((P +* (stop I)),s,k)) = IC (Comput ((P +* (stop (I ';' J))),s,k)) ) & DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop (I ';' J))),s,(LifeSpan ((P +* (stop I)),s)))) )
let I, J be Program of SCMPDS; ::_thesis: ( I is_closed_on s,P & I is_halting_on s,P implies ( ( for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),s) holds
IC (Comput ((P +* (stop I)),s,k)) = IC (Comput ((P +* (stop (I ';' J))),s,k)) ) & DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop (I ';' J))),s,(LifeSpan ((P +* (stop I)),s)))) ) )
assume A1: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or ( ( for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),s) holds
IC (Comput ((P +* (stop I)),s,k)) = IC (Comput ((P +* (stop (I ';' J))),s,k)) ) & DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop (I ';' J))),s,(LifeSpan ((P +* (stop I)),s)))) ) )
A2: Initialize s = s by MEMSTR_0:44;
set pI = stop I;
set pIJ = stop (I ';' J);
set P1 = P +* (stop I);
defpred S1[ Element of NAT ] means ( $1 <= LifeSpan ((P +* (stop I)),s) implies Comput ((P +* (stop I)),s,$1) = Comput (((P +* (stop I)) +* (stop (I ';' J))),s,$1) );
assume  I is_halting_on s,P ; ::_thesis: ( ( for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),s) holds
IC (Comput ((P +* (stop I)),s,k)) = IC (Comput ((P +* (stop (I ';' J))),s,k)) ) & DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop (I ';' J))),s,(LifeSpan ((P +* (stop I)),s)))) )
then A3: P +* (stop I) halts_on s by Def3, A2;
A4: for m being Element of NAT st S1[m] holds
S1[m + 1]
proof
set JS = J ';' (Stop SCMPDS);
set E2 = (P +* (stop I)) +* (stop (I ';' J));
let m be Element of NAT ; ::_thesis: ( S1[m] implies S1[m + 1] )
assume A5: ( m <= LifeSpan ((P +* (stop I)),s) implies Comput ((P +* (stop I)),s,m) = Comput (((P +* (stop I)) +* (stop (I ';' J))),s,m) ) ; ::_thesis: S1[m + 1]
A6: stop (I ';' J) c= (P +* (stop I)) +* (stop (I ';' J)) by FUNCT_4:25;
A7: Comput ((P +* (stop I)),s,(m + 1)) = Following ((P +* (stop I)),(Comput ((P +* (stop I)),s,m))) by EXTPRO_1:3;
A8: stop (I ';' J) = (I ';' J) ';' (Stop SCMPDS)
.= I ';' (J ';' (Stop SCMPDS)) by AFINSQ_1:27 ;
dom (I ';' (J ';' (Stop SCMPDS))) = dom (I +* (Shift ((J ';' (Stop SCMPDS)),(card I))))
.= (dom I) \/ (dom (Shift ((J ';' (Stop SCMPDS)),(card I)))) by FUNCT_4:def_1 ;
then A9: dom I c= dom (I ';' (J ';' (Stop SCMPDS))) by XBOOLE_1:7;
A10: Comput (((P +* (stop I)) +* (stop (I ';' J))),s,(m + 1)) = Following (((P +* (stop I)) +* (stop (I ';' J))),(Comput (((P +* (stop I)) +* (stop (I ';' J))),s,m))) by EXTPRO_1:3;
A11: IC (Comput ((P +* (stop I)),s,m)) in dom (stop I) by A1, Def2, A2;
A12: (P +* (stop I)) /. (IC (Comput ((P +* (stop I)),s,m))) = (P +* (stop I)) . (IC (Comput ((P +* (stop I)),s,m))) by PBOOLE:143;
 stop I c= P +* (stop I) by FUNCT_4:25;
then A13: CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),s,m))) = (stop I) . (IC (Comput ((P +* (stop I)),s,m))) by A11, A12, GRFUNC_1:2;
assume A14: m + 1 <= LifeSpan ((P +* (stop I)),s) ; ::_thesis: Comput ((P +* (stop I)),s,(m + 1)) = Comput (((P +* (stop I)) +* (stop (I ';' J))),s,(m + 1))
then  m < LifeSpan ((P +* (stop I)),s) by NAT_1:13;
then  (stop I) . (IC (Comput ((P +* (stop I)),s,m))) <> halt SCMPDS by A3, A13, EXTPRO_1:def_15;
then A15: IC (Comput ((P +* (stop I)),s,m)) in dom I by A11, COMPOS_1:51;
A16: ((P +* (stop I)) +* (stop (I ';' J))) /. (IC (Comput (((P +* (stop I)) +* (stop (I ';' J))),s,m))) = ((P +* (stop I)) +* (stop (I ';' J))) . (IC (Comput (((P +* (stop I)) +* (stop (I ';' J))),s,m))) by PBOOLE:143;
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),s,m))) = (I ';' (Stop SCMPDS)) . (IC (Comput ((P +* (stop I)),s,m))) by A13
.= I . (IC (Comput ((P +* (stop I)),s,m))) by A15, AFINSQ_1:def_3
.= (stop (I ';' J)) . (IC (Comput ((P +* (stop I)),s,m))) by A15, A8, AFINSQ_1:def_3
.= ((P +* (stop I)) +* (stop (I ';' J))) . (IC (Comput ((P +* (stop I)),s,m))) by A6, A15, A8, A9, GRFUNC_1:2
.= CurInstr (((P +* (stop I)) +* (stop (I ';' J))),(Comput (((P +* (stop I)) +* (stop (I ';' J))),s,m))) by A5, A14, A16, NAT_1:13 ;
hence  Comput ((P +* (stop I)),s,(m + 1)) = Comput (((P +* (stop I)) +* (stop (I ';' J))),s,(m + 1)) by A5, A14, A7, A10, NAT_1:13; ::_thesis: verum
end;
 ( Comput ((P +* (stop I)),s,0) = s & Comput (((P +* (stop I)) +* (stop (I ';' J))),s,0) = s ) by EXTPRO_1:2;
then A17: S1[ 0 ] ;
A18: for m being Element of NAT holds S1[m] from NAT_1:sch_1(A17, A4);
A19: (P +* (stop I)) +* (stop (I ';' J)) = P +* ((stop I) +* (stop (I ';' J))) by FUNCT_4:14
.= P +* (stop (I ';' J)) by SCMPDS_5:14 ;
thus  for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),s) holds
IC (Comput ((P +* (stop I)),s,k)) = IC (Comput ((P +* (stop (I ';' J))),s,k)) by A18, A19; ::_thesis: DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop (I ';' J))),s,(LifeSpan ((P +* (stop I)),s))))
 Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s))) = Comput ((P +* (stop (I ';' J))),s,(LifeSpan ((P +* (stop I)),s))) by A19, A18;
hence  DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop (I ';' J))),s,(LifeSpan ((P +* (stop I)),s)))) ; ::_thesis: verum
end;

theorem Th26: :: SCMPDS_6:26
for s being State of SCMPDS
for P being Instruction-Sequence of SCMPDS
for I being Program of SCMPDS
for k being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),(Initialize s)) holds
IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom I
proof
let s be State of SCMPDS; ::_thesis: for P being Instruction-Sequence of SCMPDS
for I being Program of SCMPDS
for k being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),(Initialize s)) holds
IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom I
let P be Instruction-Sequence of SCMPDS; ::_thesis: for I being Program of SCMPDS
for k being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),(Initialize s)) holds
IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom I
let I be Program of SCMPDS; ::_thesis: for k being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),(Initialize s)) holds
IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom I
let k be Element of NAT ; ::_thesis: ( I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),(Initialize s)) implies IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom I )
set ss = Initialize s;
set PP = P +* (stop I);
set m = LifeSpan ((P +* (stop I)),(Initialize s));
set Sp = Stop SCMPDS;
assume that
A1: I is_closed_on s,P and
A2: I is_halting_on s,P and
A3: k < LifeSpan ((P +* (stop I)),(Initialize s)) ; ::_thesis: IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom I
set Sk = Comput ((P +* (stop I)),(Initialize s),k);
set Ik = IC (Comput ((P +* (stop I)),(Initialize s),k));
A4: IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom (stop I) by A1, Def2;
reconsider n = IC (Comput ((P +* (stop I)),(Initialize s),k)) as Element of NAT ;
A5: stop I c= P +* (stop I) by FUNCT_4:25;
A6: P +* (stop I) halts_on Initialize s by A2, Def3;
A7: now__::_thesis:_not_n_=_card_I
A8: (P +* (stop I)) /. (IC (Comput ((P +* (stop I)),(Initialize s),k))) = (P +* (stop I)) . (IC (Comput ((P +* (stop I)),(Initialize s),k))) by PBOOLE:143;
assume A9: n = card I ; ::_thesis: contradiction
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),k))) = (P +* (stop I)) . (IC (Comput ((P +* (stop I)),(Initialize s),k))) by A8
.= (stop I) . (0 + n) by A4, A5, GRFUNC_1:2
.= halt SCMPDS by A9, Lm1, Lm2, AFINSQ_1:def_3 ;
hence  contradiction by A3, A6, EXTPRO_1:def_15; ::_thesis: verum
end;
 card (stop I) = (card I) + 1 by COMPOS_1:55;
then  n < (card I) + 1 by A4, AFINSQ_1:66;
then  n <= card I by INT_1:7;
then  n < card I by A7, XXREAL_0:1;
hence  IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom I by AFINSQ_1:66; ::_thesis: verum
end;

theorem Th27: :: SCMPDS_6:27
for P being Instruction-Sequence of SCMPDS
for I, J being Program of SCMPDS
for s being 0 -started State of SCMPDS
for k being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),s) holds
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),s,k))) = CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k)))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for I, J being Program of SCMPDS
for s being 0 -started State of SCMPDS
for k being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),s) holds
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),s,k))) = CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k)))
let I, J be Program of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for k being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),s) holds
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),s,k))) = CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k)))
let s be 0 -started State of SCMPDS; ::_thesis: for k being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),s) holds
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),s,k))) = CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k)))
let k be Element of NAT ; ::_thesis: ( I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),s) implies CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),s,k))) = CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k))) )
set P1 = P +* (stop I);
set P2 = P +* (stop (I ';' J));
set s3 = Comput ((P +* (stop I)),s,k);
set s4 = Comput ((P +* (stop (I ';' J))),s,k);
set P3 = P +* (stop I);
set P4 = P +* (stop (I ';' J));
set SS = Stop SCMPDS;
assume that
A1: I is_closed_on s,P and
A2: ( I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),s) ) ; ::_thesis: CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),s,k))) = CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k)))
A3: Initialize s = s by MEMSTR_0:44;
then A4: IC (Comput ((P +* (stop I)),s,k)) in dom I by A1, A2, Th26;
A5: IC (Comput ((P +* (stop I)),s,k)) = IC (Comput ((P +* (stop (I ';' J))),s,k)) by A1, A2, Th25;
A6: IC (Comput ((P +* (stop I)),s,k)) in dom (stop I) by A1, Def2, A3;
A7: ( dom (stop I) c= dom (stop (I ';' J)) & stop (I ';' J) c= P +* (stop (I ';' J)) ) by FUNCT_4:25, SCMPDS_5:13;
A8: stop I c= P +* (stop I) by FUNCT_4:25;
A9: stop (I ';' J) = (I ';' J) ';' (Stop SCMPDS)
.= I ';' (J ';' (Stop SCMPDS)) by AFINSQ_1:27 ;
A10: (P +* (stop I)) /. (IC (Comput ((P +* (stop I)),s,k))) = (P +* (stop I)) . (IC (Comput ((P +* (stop I)),s,k))) by PBOOLE:143;
A11: (P +* (stop (I ';' J))) /. (IC (Comput ((P +* (stop (I ';' J))),s,k))) = (P +* (stop (I ';' J))) . (IC (Comput ((P +* (stop (I ';' J))),s,k))) by PBOOLE:143;
thus  CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),s,k))) = (P +* (stop I)) . (IC (Comput ((P +* (stop I)),s,k))) by A10
.= (stop I) . (IC (Comput ((P +* (stop I)),s,k))) by A6, A8, GRFUNC_1:2
.= I . (IC (Comput ((P +* (stop I)),s,k))) by A4, AFINSQ_1:def_3
.= (stop (I ';' J)) . (IC (Comput ((P +* (stop I)),s,k))) by A4, A9, AFINSQ_1:def_3
.= (P +* (stop (I ';' J))) . (IC (Comput ((P +* (stop (I ';' J))),s,k))) by A5, A6, A7, GRFUNC_1:2
.= CurInstr ((P +* (stop (I ';' J))),(Comput ((P +* (stop (I ';' J))),s,k))) by A11 ; ::_thesis: verum
end;

theorem Th28: :: SCMPDS_6:28
for P being Instruction-Sequence of SCMPDS
for I being halt-free Program of SCMPDS
for s being State of SCMPDS
for k being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),(Initialize s)) holds
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),k))) <> halt SCMPDS
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for I being halt-free Program of SCMPDS
for s being State of SCMPDS
for k being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),(Initialize s)) holds
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),k))) <> halt SCMPDS
let I be halt-free Program of SCMPDS; ::_thesis: for s being State of SCMPDS
for k being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),(Initialize s)) holds
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),k))) <> halt SCMPDS
let s be State of SCMPDS; ::_thesis: for k being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),(Initialize s)) holds
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),k))) <> halt SCMPDS
let k be Element of NAT ; ::_thesis: ( I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),(Initialize s)) implies CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),k))) <> halt SCMPDS )
set ss = Initialize s;
set PP = P +* (stop I);
set s2 = Comput ((P +* (stop I)),(Initialize s),k);
set P2 = P +* (stop I);
assume  ( I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),(Initialize s)) ) ; ::_thesis: CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),k))) <> halt SCMPDS
then A1: IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom I by Th26;
A2: (P +* (stop I)) /. (IC (Comput ((P +* (stop I)),(Initialize s),k))) = (P +* (stop I)) . (IC (Comput ((P +* (stop I)),(Initialize s),k))) by PBOOLE:143;
A3: stop I c= P +* (stop I) by FUNCT_4:25;
 I c= stop I by AFINSQ_1:74;
then  I c= P +* (stop I) by A3, XBOOLE_1:1;
then  CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),k))) = I . (IC (Comput ((P +* (stop I)),(Initialize s),k))) by A1, A2, GRFUNC_1:2;
hence  CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),k))) <> halt SCMPDS by A1, COMPOS_1:def_27; ::_thesis: verum
end;

theorem Th29: :: SCMPDS_6:29
for P being Instruction-Sequence of SCMPDS
for I being halt-free Program of SCMPDS
for s being State of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) = card I
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for I being halt-free Program of SCMPDS
for s being State of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) = card I
let I be halt-free Program of SCMPDS; ::_thesis: for s being State of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) = card I
let s be State of SCMPDS; ::_thesis: ( I is_closed_on s,P & I is_halting_on s,P implies IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) = card I )
set s1 = Initialize s;
set P1 = P +* (stop I);
assume that
A1: I is_closed_on s,P and
A2: I is_halting_on s,P ; ::_thesis: IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) = card I
set Css = Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))));
reconsider n = IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) as Element of NAT ;
A3: stop I c= P +* (stop I) by FUNCT_4:25;
 I c= stop I by AFINSQ_1:74;
then A4: I c= P +* (stop I) by A3, XBOOLE_1:1;
A5: P +* (stop I) halts_on Initialize s by A2, Def3;
now__::_thesis:_not_IC_(Comput_((P_+*_(stop_I)),(Initialize_s),(LifeSpan_((P_+*_(stop_I)),(Initialize_s)))))_in_dom_I
A6: (P +* (stop I)) /. (IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) = (P +* (stop I)) . (IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by PBOOLE:143;
assume A7: IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) in dom I ; ::_thesis: contradiction
then I . (IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) = (P +* (stop I)) . (IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A4, GRFUNC_1:2
.= CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A6
.= halt SCMPDS by A5, EXTPRO_1:def_15 ;
hence  contradiction by A7, COMPOS_1:def_27; ::_thesis: verum
end;
then A8: n >= card I by AFINSQ_1:66;
A9: card (stop I) = (card I) + 1 by COMPOS_1:55;
 IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) in dom (stop I) by A1, Def2;
then  n < (card I) + 1 by A9, AFINSQ_1:66;
then  n <= card I by NAT_1:13;
then  IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) = card I by A8, XXREAL_0:1;
hence  IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) = card I ; ::_thesis: verum
end;

Lm3:  for P being Instruction-Sequence of SCMPDS
for I being halt-free Program of SCMPDS
for J being Program of SCMPDS
for s being 0 -started State of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),s)) + 1))) = ((card I) + (card J)) + 1 & DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),s)) + 1))) & ( for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),s) holds
CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,k))) <> halt SCMPDS ) & IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop I)),s)))) = card I & P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s & LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) = (LifeSpan ((P +* (stop I)),s)) + 1 )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for I being halt-free Program of SCMPDS
for J being Program of SCMPDS
for s being 0 -started State of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),s)) + 1))) = ((card I) + (card J)) + 1 & DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),s)) + 1))) & ( for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),s) holds
CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,k))) <> halt SCMPDS ) & IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop I)),s)))) = card I & P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s & LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) = (LifeSpan ((P +* (stop I)),s)) + 1 )
let I be halt-free Program of SCMPDS; ::_thesis: for J being Program of SCMPDS
for s being 0 -started State of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),s)) + 1))) = ((card I) + (card J)) + 1 & DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),s)) + 1))) & ( for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),s) holds
CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,k))) <> halt SCMPDS ) & IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop I)),s)))) = card I & P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s & LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) = (LifeSpan ((P +* (stop I)),s)) + 1 )
let J be Program of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),s)) + 1))) = ((card I) + (card J)) + 1 & DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),s)) + 1))) & ( for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),s) holds
CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,k))) <> halt SCMPDS ) & IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop I)),s)))) = card I & P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s & LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) = (LifeSpan ((P +* (stop I)),s)) + 1 )
let s be 0 -started State of SCMPDS; ::_thesis: ( I is_closed_on s,P & I is_halting_on s,P implies ( IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),s)) + 1))) = ((card I) + (card J)) + 1 & DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),s)) + 1))) & ( for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),s) holds
CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,k))) <> halt SCMPDS ) & IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop I)),s)))) = card I & P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s & LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) = (LifeSpan ((P +* (stop I)),s)) + 1 ) )
assume A1: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or ( IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),s)) + 1))) = ((card I) + (card J)) + 1 & DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),s)) + 1))) & ( for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),s) holds
CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,k))) <> halt SCMPDS ) & IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop I)),s)))) = card I & P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s & LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) = (LifeSpan ((P +* (stop I)),s)) + 1 ) )
set G1 = Goto ((card J) + 1);
set SS = Stop SCMPDS;
set J2 = ((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS);
set IJ = (I ';' (Goto ((card J) + 1))) ';' J;
set pJ = stop ((I ';' (Goto ((card J) + 1))) ';' J);
set s1 = s;
set P1 = P +* (stop I);
set s2 = s;
reconsider P2 = P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) as Instruction-Sequence of SCMPDS ;
assume A2: I is_halting_on s,P ; ::_thesis: ( IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),s)) + 1))) = ((card I) + (card J)) + 1 & DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),s)) + 1))) & ( for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),s) holds
CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,k))) <> halt SCMPDS ) & IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop I)),s)))) = card I & P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s & LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) = (LifeSpan ((P +* (stop I)),s)) + 1 )
A3: Initialize s = s by MEMSTR_0:44;
set sm = Comput (P2,s,(LifeSpan ((P +* (stop I)),s)));
A4: (I ';' (Goto ((card J) + 1))) ';' J = I ';' ((Goto ((card J) + 1)) ';' J) by AFINSQ_1:27;
then A5: IC (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = IC (Comput (P2,s,(LifeSpan ((P +* (stop I)),s)))) by A1, A2, Th25;
then A6: IC (Comput (P2,s,(LifeSpan ((P +* (stop I)),s)))) = card I by A1, A2, Th29, A3;
A7: 0 in dom (Goto ((card J) + 1)) by Th19;
A8: (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS)) . 0 = ((Goto ((card J) + 1)) ';' (J ';' (Stop SCMPDS))) . 0 by AFINSQ_1:27
.= (Goto ((card J) + 1)) . 0 by A7, AFINSQ_1:def_3
.= goto ((card J) + 1) by Th19 ;
card ((Goto ((card J) + 1)) ';' J) = (card (Goto ((card J) + 1))) + (card J) by AFINSQ_1:17
.= 1 + (card J) by COMPOS_1:54 ;
then A9: (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS)) . ((card J) + 1) = (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS)) . (0 + (card ((Goto ((card J) + 1)) ';' J)))
.= halt SCMPDS by Lm1, Lm2, AFINSQ_1:def_3 ;
A10: card (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS)) = card ((Goto ((card J) + 1)) ';' (J ';' (Stop SCMPDS))) by AFINSQ_1:27
.= (card (Goto ((card J) + 1))) + (card (J ';' (Stop SCMPDS))) by AFINSQ_1:17
.= 1 + (card (J ';' (Stop SCMPDS))) by COMPOS_1:54 ;
then A11: 0 in dom (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS)) by AFINSQ_1:66;
A12: stop ((I ';' (Goto ((card J) + 1))) ';' J) = ((I ';' (Goto ((card J) + 1))) ';' J) ';' (Stop SCMPDS)
.= (I ';' ((Goto ((card J) + 1)) ';' J)) ';' (Stop SCMPDS) by AFINSQ_1:27
.= I ';' (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS)) by AFINSQ_1:27 ;
then A13: card (stop ((I ';' (Goto ((card J) + 1))) ';' J)) = (card I) + (card (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS))) by AFINSQ_1:17;
then  (card I) + 0 < card (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by XREAL_1:6;
then A14: card I in dom (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by AFINSQ_1:66;
A15: card (Stop SCMPDS) = 1 by AFINSQ_1:33;
A16: card (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS)) = 1 + ((card J) + (card (Stop SCMPDS))) by A10, AFINSQ_1:17
.= (card J) + (1 + (card (Stop SCMPDS))) ;
then  (card J) + 1 < card (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS)) by A15, XREAL_1:6;
then A17: (card J) + 1 in dom (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS)) by AFINSQ_1:66;
 card (stop ((I ';' (Goto ((card J) + 1))) ';' J)) = (((card I) + (card J)) + 1) + 1 by A13, A16, A15;
then  ((card I) + (card J)) + 1 < card (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by NAT_1:13;
then A18: ((card I) + (card J)) + 1 in dom (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by AFINSQ_1:66;
A19: P2 /. (IC (Comput (P2,s,(LifeSpan ((P +* (stop I)),s))))) = P2 . (IC (Comput (P2,s,(LifeSpan ((P +* (stop I)),s))))) by PBOOLE:143;
A20: CurInstr (P2,(Comput (P2,s,(LifeSpan ((P +* (stop I)),s))))) = P2 . (card I) by A1, A2, A5, Th29, A19, A3
.= P2 . (card I)
.= (stop ((I ';' (Goto ((card J) + 1))) ';' J)) . (card I) by A14, FUNCT_4:13
.= (I ';' (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS))) . (0 + (card I)) by A12
.= goto ((card J) + 1) by A11, A8, AFINSQ_1:def_3 ;
A21: now__::_thesis:_for_a_being_Int_position_holds_(Comput_(P2,s,((LifeSpan_((P_+*_(stop_I)),s))_+_1)))_._a_=_(Comput_(P2,s,(LifeSpan_((P_+*_(stop_I)),s))))_._a
let a be Int_position; ::_thesis: (Comput (P2,s,((LifeSpan ((P +* (stop I)),s)) + 1))) . a = (Comput (P2,s,(LifeSpan ((P +* (stop I)),s)))) . a
thus (Comput (P2,s,((LifeSpan ((P +* (stop I)),s)) + 1))) . a = (Following (P2,(Comput (P2,s,(LifeSpan ((P +* (stop I)),s)))))) . a by EXTPRO_1:3
.= (Comput (P2,s,(LifeSpan ((P +* (stop I)),s)))) . a by A20, SCMPDS_2:54 ; ::_thesis: verum
end;
A22: P2 /. (IC (Comput (P2,s,((LifeSpan ((P +* (stop I)),s)) + 1)))) = P2 . (IC (Comput (P2,s,((LifeSpan ((P +* (stop I)),s)) + 1)))) by PBOOLE:143;
thus  IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),s)) + 1))) = IC (Following (P2,(Comput (P2,s,(LifeSpan ((P +* (stop I)),s)))))) by EXTPRO_1:3
.= ICplusConst ((Comput (P2,s,(LifeSpan ((P +* (stop I)),s)))),((card J) + 1)) by A20, SCMPDS_2:54
.= (card I) + ((card J) + 1) by A6, Th12
.= ((card I) + (card J)) + 1 ; ::_thesis: ( DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),s)) + 1))) & ( for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),s) holds
CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,k))) <> halt SCMPDS ) & IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop I)),s)))) = card I & P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s & LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) = (LifeSpan ((P +* (stop I)),s)) + 1 )
then A23: CurInstr (P2,(Comput (P2,s,((LifeSpan ((P +* (stop I)),s)) + 1)))) = P2 . (((card I) + (card J)) + 1) by A22
.= (stop ((I ';' (Goto ((card J) + 1))) ';' J)) . (((card I) + (card J)) + 1) by A18, FUNCT_4:13
.= (I ';' (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS))) . ((card I) + ((card J) + 1)) by A12
.= halt SCMPDS by A17, A9, AFINSQ_1:def_3 ;
 DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput (P2,s,(LifeSpan ((P +* (stop I)),s)))) by A1, A2, A4, Th25;
hence  DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = DataPart (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),s)) + 1))) by A21, SCMPDS_4:8; ::_thesis: ( ( for k being Element of NAT st k <= LifeSpan ((P +* (stop I)),s) holds
CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,k))) <> halt SCMPDS ) & IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop I)),s)))) = card I & P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s & LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) = (LifeSpan ((P +* (stop I)),s)) + 1 )
hereby  ::_thesis: ( IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop I)),s)))) = card I & P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s & LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) = (LifeSpan ((P +* (stop I)),s)) + 1 )
let k be Element of NAT ; ::_thesis: ( k <= LifeSpan ((P +* (stop I)),s) implies CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,b1))) <> halt SCMPDS )
assume A25: k <= LifeSpan ((P +* (stop I)),s) ; ::_thesis: CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,b1))) <> halt SCMPDS
percases ( k < LifeSpan ((P +* (stop I)),s) or LifeSpan ((P +* (stop I)),s) <= k ) ;
supposeA26: k < LifeSpan ((P +* (stop I)),s) ; ::_thesis: CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,b1))) <> halt SCMPDS
then  CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),s,k))) <> halt SCMPDS by A1, A2, Th28, A3;
hence  CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,k))) <> halt SCMPDS by A1, A2, A4, A26, Th27; ::_thesis: verum
end;
suppose LifeSpan ((P +* (stop I)),s) <= k ; ::_thesis: CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,b1))) <> halt SCMPDS
then  k = LifeSpan ((P +* (stop I)),s) by A25, XXREAL_0:1;
hence  CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,k))) <> halt SCMPDS by A20; ::_thesis: verum
end;
end;
end;
thus  IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop I)),s)))) = card I by A1, A2, A5, Th29, A3; ::_thesis: ( P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s & LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) = (LifeSpan ((P +* (stop I)),s)) + 1 )
thus A27: P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s by A23, EXTPRO_1:29; ::_thesis: LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) = (LifeSpan ((P +* (stop I)),s)) + 1
now__::_thesis:_for_k_being_Element_of_NAT_st_CurInstr_(P2,(Comput_(P2,s,k)))_=_halt_SCMPDS_holds_
(LifeSpan_((P_+*_(stop_I)),s))_+_1_<=_k
let k be Element of NAT ; ::_thesis: ( CurInstr (P2,(Comput (P2,s,k))) = halt SCMPDS implies (LifeSpan ((P +* (stop I)),s)) + 1 <= k )
assume  CurInstr (P2,(Comput (P2,s,k))) = halt SCMPDS ; ::_thesis: (LifeSpan ((P +* (stop I)),s)) + 1 <= k
then  LifeSpan ((P +* (stop I)),s) < k by A24;
hence  (LifeSpan ((P +* (stop I)),s)) + 1 <= k by INT_1:7; ::_thesis: verum
end;
hence  LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) = (LifeSpan ((P +* (stop I)),s)) + 1 by A23, A27, EXTPRO_1:def_15; ::_thesis: verum
end;

theorem Th30: :: SCMPDS_6:30
for P being Instruction-Sequence of SCMPDS
for I, J being Program of SCMPDS
for s being 0 -started State of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for I, J being Program of SCMPDS
for s being 0 -started State of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P )
let I, J be Program of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
( (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P )
let s be 0 -started State of SCMPDS; ::_thesis: ( I is_closed_on s,P & I is_halting_on s,P implies ( (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P ) )
set G = Goto ((card J) + 1);
set IJ = (I ';' (Goto ((card J) + 1))) ';' J;
set J2 = I ';' ((Goto ((card J) + 1)) ';' J);
set pJ = stop (I ';' ((Goto ((card J) + 1)) ';' J));
set pI = stop I;
set P1 = P +* (stop I);
set P2 = P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)));
set m = LifeSpan ((P +* (stop I)),s);
set SS = Stop SCMPDS;
set s3 = Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)));
set s4 = Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,(LifeSpan ((P +* (stop I)),s)));
set P3 = P +* (stop I);
set P4 = P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)));
A1: Initialize s = s by MEMSTR_0:44;
A2: (I ';' (Goto ((card J) + 1))) ';' J = I ';' ((Goto ((card J) + 1)) ';' J) by AFINSQ_1:27;
A3: I c= stop I by AFINSQ_1:74;
 stop I c= P +* (stop I) by FUNCT_4:25;
then A4: I c= P +* (stop I) by A3, XBOOLE_1:1;
A5: dom (stop I) c= dom (stop (I ';' ((Goto ((card J) + 1)) ';' J))) by SCMPDS_5:13;
set JS = ((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS);
reconsider n = IC (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) as Element of NAT ;
A6: card (stop I) = (card I) + 1 by COMPOS_1:55;
assume A7: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or ( (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P ) )
then  IC (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) in dom (stop I) by Def2, A1;
then  n < (card I) + 1 by A6, AFINSQ_1:66;
then A8: n <= card I by INT_1:7;
A9: stop (I ';' ((Goto ((card J) + 1)) ';' J)) c= P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J))) by FUNCT_4:25;
A10: stop (I ';' ((Goto ((card J) + 1)) ';' J)) = (I ';' ((Goto ((card J) + 1)) ';' J)) ';' (Stop SCMPDS)
.= I ';' (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS)) by AFINSQ_1:27 ;
then  I c= stop (I ';' ((Goto ((card J) + 1)) ';' J)) by AFINSQ_1:74;
then  I c= P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J))) by A9, XBOOLE_1:1;
then A11: I c= P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J))) ;
assume A12: I is_halting_on s,P ; ::_thesis: ( (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P )
then A13: P +* (stop I) halts_on s by Def3, A1;
A14: (P +* (stop I)) /. (IC (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s))))) = (P +* (stop I)) . (IC (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s))))) by PBOOLE:143;
A15: (P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))) /. (IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,(LifeSpan ((P +* (stop I)),s))))) = (P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))) . (IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,(LifeSpan ((P +* (stop I)),s))))) by PBOOLE:143;
percases ( IC (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) <> card I or IC (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = card I ) ;
suppose IC (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) <> card I ; ::_thesis: ( (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P )
then  n < card I by A8, XXREAL_0:1;
then A16: IC (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) in dom I by AFINSQ_1:66;
A17: halt SCMPDS = CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s))))) by A13, EXTPRO_1:def_15
.= I . (IC (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s))))) by A4, A16, A14, GRFUNC_1:2
.= (P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))) . (IC (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s))))) by A11, A16, GRFUNC_1:2
.= CurInstr ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),(Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,(LifeSpan ((P +* (stop I)),s))))) by A7, A12, Th25, A15 ;
then A18: P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J))) halts_on s by EXTPRO_1:29;
hence  (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P by A2, Def3, A1; ::_thesis: (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(I_';'_((Goto_((card_J)_+_1))_';'_J)))),s,k))_in_dom_(stop_(I_';'_((Goto_((card_J)_+_1))_';'_J)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,b1)) in dom (stop (I ';' ((Goto ((card J) + 1)) ';' J)))
set C1k = IC (Comput ((P +* (stop I)),s,k));
set C2k = IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,k));
percases ( k <= LifeSpan ((P +* (stop I)),s) or k > LifeSpan ((P +* (stop I)),s) ) ;
supposeA19: k <= LifeSpan ((P +* (stop I)),s) ; ::_thesis: IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,b1)) in dom (stop (I ';' ((Goto ((card J) + 1)) ';' J)))
 IC (Comput ((P +* (stop I)),s,k)) in dom (stop I) by A7, Def2, A1;
then  IC (Comput ((P +* (stop I)),s,k)) in dom (stop (I ';' ((Goto ((card J) + 1)) ';' J))) by A5;
hence  IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,k)) in dom (stop (I ';' ((Goto ((card J) + 1)) ';' J))) by A7, A12, A19, Th25; ::_thesis: verum
end;
supposeA20: k > LifeSpan ((P +* (stop I)),s) ; ::_thesis: IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,b1)) in dom (stop (I ';' ((Goto ((card J) + 1)) ';' J)))
set m2 = LifeSpan ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s);
A21: LifeSpan ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s) <= LifeSpan ((P +* (stop I)),s) by A17, A18, EXTPRO_1:def_15;
then  IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,k)) = IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,(LifeSpan ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s)))) by A18, A20, EXTPRO_1:25, XXREAL_0:2
.= IC (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s)))) by A7, A12, A21, Th25 ;
then  IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,k)) in dom (stop I) by A7, Def2, A1;
hence  IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,k)) in dom (stop (I ';' ((Goto ((card J) + 1)) ';' J))) by A5; ::_thesis: verum
end;
end;
end;
hence  (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P by A2, Def2, A1; ::_thesis: verum
end;
supposeA22: IC (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) = card I ; ::_thesis: ( (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P )
then A23: IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,(LifeSpan ((P +* (stop I)),s)))) = card I by A7, A12, Th25;
A24: 0 in dom (Goto ((card J) + 1)) by Th19;
A25: card (Stop SCMPDS) = 1 by AFINSQ_1:33;
A26: ((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS) = (Goto ((card J) + 1)) ';' (J ';' (Stop SCMPDS)) by AFINSQ_1:27;
then A27: card (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS)) = (card (Goto ((card J) + 1))) + (card (J ';' (Stop SCMPDS))) by AFINSQ_1:17
.= 1 + (card (J ';' (Stop SCMPDS))) by COMPOS_1:54
.= ((card J) + 1) + 1 by A25, AFINSQ_1:17 ;
then A28: 0 in dom (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS)) by AFINSQ_1:66;
 (card J) + 1 < card (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS)) by A27, NAT_1:13;
then A29: (card J) + 1 in dom (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS)) by AFINSQ_1:66;
card (stop (I ';' ((Goto ((card J) + 1)) ';' J))) = (card I) + ((card J) + (1 + 1)) by A10, A27, AFINSQ_1:17
.= (((card I) + (card J)) + 1) + 1 ;
then A30: ((card I) + (card J)) + 1 < card (stop (I ';' ((Goto ((card J) + 1)) ';' J))) by NAT_1:13;
then A31: ((card I) + (card J)) + 1 in dom (stop (I ';' ((Goto ((card J) + 1)) ';' J))) by AFINSQ_1:66;
A32: (P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))) /. (IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,(LifeSpan ((P +* (stop I)),s))))) = (P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))) . (IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,(LifeSpan ((P +* (stop I)),s))))) by PBOOLE:143;
 card (stop (I ';' ((Goto ((card J) + 1)) ';' J))) = (card I) + (card (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS))) by A10, AFINSQ_1:17;
then  (card I) + 0 < card (stop (I ';' ((Goto ((card J) + 1)) ';' J))) by XREAL_1:6;
then A33: card I in dom (stop (I ';' ((Goto ((card J) + 1)) ';' J))) by AFINSQ_1:66;
A34: CurInstr ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),(Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,(LifeSpan ((P +* (stop I)),s))))) = (P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))) . (card I) by A7, A12, A22, Th25, A32
.= (P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))) . (card I)
.= (I ';' (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS))) . (0 + (card I)) by A10, A9, A33, GRFUNC_1:2
.= (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS)) . 0 by A28, AFINSQ_1:def_3
.= (Goto ((card J) + 1)) . 0 by A26, A24, AFINSQ_1:def_3
.= goto ((card J) + 1) by Th19 ;
card ((Goto ((card J) + 1)) ';' J) = (card (Goto ((card J) + 1))) + (card J) by AFINSQ_1:17
.= 1 + (card J) by COMPOS_1:54 ;
then A35: (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS)) . ((card J) + 1) = (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS)) . (0 + (card ((Goto ((card J) + 1)) ';' J)))
.= halt SCMPDS by Lm1, Lm2, AFINSQ_1:def_3 ;
A36: (P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))) /. (IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) = (P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))) . (IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) by PBOOLE:143;
A37: IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,((LifeSpan ((P +* (stop I)),s)) + 1))) = IC (Following ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),(Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,(LifeSpan ((P +* (stop I)),s)))))) by EXTPRO_1:3
.= ICplusConst ((Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,(LifeSpan ((P +* (stop I)),s)))),((card J) + 1)) by A34, SCMPDS_2:54
.= (card I) + ((card J) + 1) by A23, Th12
.= ((card I) + (card J)) + 1 ;
then A38: CurInstr ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),(Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,((LifeSpan ((P +* (stop I)),s)) + 1)))) = (P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))) . (((card I) + (card J)) + 1) by A36
.= (I ';' (((Goto ((card J) + 1)) ';' J) ';' (Stop SCMPDS))) . ((card I) + ((card J) + 1)) by A10, A9, A31, GRFUNC_1:2
.= halt SCMPDS by A29, A35, AFINSQ_1:def_3 ;
then A39: P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J))) halts_on s by EXTPRO_1:29;
hence  (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P by A2, Def3, A1; ::_thesis: (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(I_';'_((Goto_((card_J)_+_1))_';'_J)))),s,k))_in_dom_(stop_(I_';'_((Goto_((card_J)_+_1))_';'_J)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,b1)) in dom (stop (I ';' ((Goto ((card J) + 1)) ';' J)))
set C1k = IC (Comput ((P +* (stop I)),s,k));
set C2k = IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,k));
percases ( k <= LifeSpan ((P +* (stop I)),s) or k > LifeSpan ((P +* (stop I)),s) ) ;
supposeA40: k <= LifeSpan ((P +* (stop I)),s) ; ::_thesis: IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,b1)) in dom (stop (I ';' ((Goto ((card J) + 1)) ';' J)))
 IC (Comput ((P +* (stop I)),s,k)) in dom (stop I) by A7, Def2, A1;
then  IC (Comput ((P +* (stop I)),s,k)) in dom (stop (I ';' ((Goto ((card J) + 1)) ';' J))) by A5;
hence  IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,k)) in dom (stop (I ';' ((Goto ((card J) + 1)) ';' J))) by A7, A12, A40, Th25; ::_thesis: verum
end;
supposeA41: k > LifeSpan ((P +* (stop I)),s) ; ::_thesis: IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,b1)) in dom (stop (I ';' ((Goto ((card J) + 1)) ';' J)))
set m2 = LifeSpan ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s);
A42: LifeSpan ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s) <= (LifeSpan ((P +* (stop I)),s)) + 1 by A38, A39, EXTPRO_1:def_15;
 k >= (LifeSpan ((P +* (stop I)),s)) + 1 by A41, INT_1:7;
then  IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,k)) = IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,(LifeSpan ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s)))) by A39, A42, EXTPRO_1:25, XXREAL_0:2
.= ((card I) + (card J)) + 1 by A37, A39, A42, EXTPRO_1:25 ;
hence  IC (Comput ((P +* (stop (I ';' ((Goto ((card J) + 1)) ';' J)))),s,k)) in dom (stop (I ';' ((Goto ((card J) + 1)) ';' J))) by A30, AFINSQ_1:66; ::_thesis: verum
end;
end;
end;
hence  (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P by A2, Def2, A1; ::_thesis: verum
end;
end;
end;

theorem Th31: :: SCMPDS_6:31
for s2 being State of SCMPDS
for P1, P2 being Instruction-Sequence of SCMPDS
for s1 being 0 -started State of SCMPDS
for I being shiftable Program of SCMPDS st stop I c= P1 & I is_closed_on s1,P1 holds
for n being Element of NAT st Shift ((stop I),n) c= P2 & IC s2 = n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & CurInstr (P1,(Comput (P1,s1,i))) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
proof
let s2 be State of SCMPDS; ::_thesis: for P1, P2 being Instruction-Sequence of SCMPDS
for s1 being 0 -started State of SCMPDS
for I being shiftable Program of SCMPDS st stop I c= P1 & I is_closed_on s1,P1 holds
for n being Element of NAT st Shift ((stop I),n) c= P2 & IC s2 = n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & CurInstr (P1,(Comput (P1,s1,i))) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
let P1, P2 be Instruction-Sequence of SCMPDS; ::_thesis: for s1 being 0 -started State of SCMPDS
for I being shiftable Program of SCMPDS st stop I c= P1 & I is_closed_on s1,P1 holds
for n being Element of NAT st Shift ((stop I),n) c= P2 & IC s2 = n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & CurInstr (P1,(Comput (P1,s1,i))) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
let s1 be 0 -started State of SCMPDS; ::_thesis: for I being shiftable Program of SCMPDS st stop I c= P1 & I is_closed_on s1,P1 holds
for n being Element of NAT st Shift ((stop I),n) c= P2 & IC s2 = n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & CurInstr (P1,(Comput (P1,s1,i))) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
let I be shiftable Program of SCMPDS; ::_thesis: ( stop I c= P1 & I is_closed_on s1,P1 implies for n being Element of NAT st Shift ((stop I),n) c= P2 & IC s2 = n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & CurInstr (P1,(Comput (P1,s1,i))) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) ) )
set SI = stop I;
assume that
A1: stop I c= P1 and
A2: I is_closed_on s1,P1 ; ::_thesis: for n being Element of NAT st Shift ((stop I),n) c= P2 & IC s2 = n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & CurInstr (P1,(Comput (P1,s1,i))) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
A3: Initialize s1 = s1 by MEMSTR_0:44;
let n be Element of NAT ; ::_thesis: ( Shift ((stop I),n) c= P2 & IC s2 = n & DataPart s1 = DataPart s2 implies for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & CurInstr (P1,(Comput (P1,s1,i))) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) ) )
defpred S1[ Element of NAT ] means ( (IC (Comput (P1,s1,$1))) + n = IC (Comput (P2,s2,$1)) & CurInstr (P1,(Comput (P1,s1,$1))) = CurInstr (P2,(Comput (P2,s2,$1))) & DataPart (Comput (P1,s1,$1)) = DataPart (Comput (P2,s2,$1)) );
assume that
A4: Shift ((stop I),n) c= P2 and
A5: IC s2 = n and
A6: DataPart s1 = DataPart s2 ; ::_thesis: for i being Element of NAT holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & CurInstr (P1,(Comput (P1,s1,i))) = 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)) & CurInstr (P1,(Comput (P1,s1,i))) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )
A7: DataPart (Comput (P1,s1,0)) = DataPart s2 by A6, EXTPRO_1:2
.= DataPart (Comput (P2,s2,0)) by EXTPRO_1:2 ;
A8: 0 in dom (stop I) by COMPOS_1:36;
then A9: 0 + n in dom (Shift ((stop I),n)) by VALUED_1:24;
A10: P1 . (IC s1) = P1 . (IC (Initialize s1)) by A3
.= P1 . 0 by MEMSTR_0:def_11
.= (stop I) . 0 by A1, A8, GRFUNC_1:2 ;
A11: P1 = P1 +* (stop I) by A1, FUNCT_4:98;
A12: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A13: S1[k] ; ::_thesis: S1[k + 1]
reconsider m = IC (Comput (P1,s1,k)) as Element of NAT ;
set i = CurInstr (P1,(Comput (P1,s1,k)));
A14: Comput (P1,s1,(k + 1)) = Following (P1,(Comput (P1,s1,k))) by EXTPRO_1:3;
reconsider l = IC (Comput (P1,s1,(k + 1))) as Element of NAT ;
A15: IC (Comput (P1,s1,(k + 1))) in dom (stop I) by A2, A11, Def2, A3;
then A16: l + n in dom (Shift ((stop I),n)) by VALUED_1:24;
A17: Comput (P2,s2,(k + 1)) = Following (P2,(Comput (P2,s2,k))) by EXTPRO_1:3;
A18: IC (Comput (P1,s1,k)) in dom (stop I) by A2, A11, Def2, A3;
A19: P1 /. (IC (Comput (P1,s1,k))) = P1 . (IC (Comput (P1,s1,k))) by PBOOLE:143;
A20: CurInstr (P1,(Comput (P1,s1,k))) = P1 . (IC (Comput (P1,s1,k))) by A19
.= (stop I) . (IC (Comput (P1,s1,k))) by A1, A18, GRFUNC_1:2 ;
then A21: ( InsCode (CurInstr (P1,(Comput (P1,s1,k)))) <> 1 & InsCode (CurInstr (P1,(Comput (P1,s1,k)))) <> 3 ) by A18, SCMPDS_4:def_9;
A22: CurInstr (P1,(Comput (P1,s1,k))) valid_at m by A18, A20, SCMPDS_4:def_9;
hence A23: (IC (Comput (P1,s1,(k + 1)))) + n = IC (Comput (P2,s2,(k + 1))) by A13, A14, A17, A21, SCMPDS_4:28; ::_thesis: ( CurInstr (P1,(Comput (P1,s1,(k + 1)))) = CurInstr (P2,(Comput (P2,s2,(k + 1)))) & DataPart (Comput (P1,s1,(k + 1))) = DataPart (Comput (P2,s2,(k + 1))) )
A24: P1 /. (IC (Comput (P1,s1,(k + 1)))) = P1 . (IC (Comput (P1,s1,(k + 1)))) by PBOOLE:143;
A25: P2 /. (IC (Comput (P2,s2,(k + 1)))) = P2 . (IC (Comput (P2,s2,(k + 1)))) by PBOOLE:143;
CurInstr (P1,(Comput (P1,s1,(k + 1)))) = P1 . l by A24
.= (stop I) . l by A1, A15, GRFUNC_1:2
.= (stop I) . l ;
hence  CurInstr (P1,(Comput (P1,s1,(k + 1)))) = (Shift ((stop I),n)) . (IC (Comput (P2,s2,(k + 1)))) by A23, A15, VALUED_1:def_12
.= P2 . (IC (Comput (P2,s2,(k + 1)))) by A4, A23, A16, GRFUNC_1:2
.= CurInstr (P2,(Comput (P2,s2,(k + 1)))) by A25 ;
::_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 A13, A14, A17, A21, A22, SCMPDS_4:28; ::_thesis: verum
end;
A26: IC (Comput (P1,s1,0)) = IC s1 by EXTPRO_1:2
.= IC (Initialize s1) by A3
.= 0 by MEMSTR_0:def_11 ;
A27: Comput (P1,s1,0) = s1 by EXTPRO_1:2;
A28: Comput (P2,s2,0) = s2 by EXTPRO_1:2;
A29: P2 /. (IC s2) = P2 . (IC s2) by PBOOLE:143;
A30: P1 /. (IC s1) = P1 . (IC s1) by PBOOLE:143;
CurInstr (P1,(Comput (P1,s1,0))) = CurInstr (P1,s1) by A27
.= (Shift ((stop I),n)) . (0 + n) by A8, A10, A30, VALUED_1:def_12
.= CurInstr (P2,s2) by A4, A5, A9, A29, GRFUNC_1:2
.= CurInstr (P2,(Comput (P2,s2,0))) by A28 ;
then A31: S1[ 0 ] by A5, A26, A7, EXTPRO_1:2;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A31, A12);
hence  ( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & CurInstr (P1,(Comput (P1,s1,i))) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) ) ; ::_thesis: verum
end;

theorem Th32: :: SCMPDS_6:32
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free Program of SCMPDS
for J being Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) = ((card I) + (card J)) + 1
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free Program of SCMPDS
for J being Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) = ((card I) + (card J)) + 1
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free Program of SCMPDS
for J being Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) = ((card I) + (card J)) + 1
let I be halt-free Program of SCMPDS; ::_thesis: for J being Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) = ((card I) + (card J)) + 1
let J be Program of SCMPDS; ::_thesis: ( I is_closed_on s,P & I is_halting_on s,P implies IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) = ((card I) + (card J)) + 1 )
set m = (LifeSpan ((P +* (stop I)),(Initialize s))) + 1;
set G = Goto ((card J) + 1);
set P2 = P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J));
A1: Initialize s = s by MEMSTR_0:44;
assume A2: ( I is_closed_on s,P & I is_halting_on s,P ) ; ::_thesis: IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) = ((card I) + (card J)) + 1
then  ( P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s & LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) = (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 ) by Lm3, A1;
then  IC (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) = IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) by EXTPRO_1:23
.= ((card I) + (card J)) + 1 by A2, Lm3, A1 ;
hence  IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) = ((card I) + (card J)) + 1 ; ::_thesis: verum
end;

theorem Th33: :: SCMPDS_6:33
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free Program of SCMPDS
for J being Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free Program of SCMPDS
for J being Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS))
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free Program of SCMPDS
for J being Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS))
let I be halt-free Program of SCMPDS; ::_thesis: for J being Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS))
let J be Program of SCMPDS; ::_thesis: ( I is_closed_on s,P & I is_halting_on s,P implies IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS)) )
set P1 = P +* (stop I);
set m = (LifeSpan ((P +* (stop I)),s)) + 1;
set G = Goto ((card J) + 1);
set P2 = P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J));
set l = ((card I) + (card J)) + 1;
assume that
A1: I is_closed_on s,P and
A2: I is_halting_on s,P ; ::_thesis: IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS))
 Initialize s = s by MEMSTR_0:44;
then A3: P +* (stop I) halts_on s by A2, Def3;
 ( P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s & LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) = (LifeSpan ((P +* (stop I)),s)) + 1 ) by A1, A2, Lm3;
then A4: Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) = Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,((LifeSpan ((P +* (stop I)),s)) + 1)) by EXTPRO_1:23;
then  DataPart (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) = DataPart (Comput ((P +* (stop I)),s,(LifeSpan ((P +* (stop I)),s)))) by A1, A2, Lm3;
then A5: DataPart (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) = DataPart (Result ((P +* (stop I)),s)) by A3, EXTPRO_1:23
.= DataPart ((Result ((P +* (stop I)),s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS))) by MEMSTR_0:79 ;
IC (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) = ((card I) + (card J)) + 1 by A1, A2, A4, Lm3
.= IC ((Result ((P +* (stop I)),s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS))) by FUNCT_4:113 ;
then A6: Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) = (Result ((P +* (stop I)),s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS)) by A5, MEMSTR_0:78;
thus  IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s) = Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)
.= (Result ((P +* (stop I)),s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS)) by A6
.= (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS)) ; ::_thesis: verum
end;

theorem Th34: :: SCMPDS_6:34
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being halt-free Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
IC (IExec (I,P,(Initialize s))) = card I
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being halt-free Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
IC (IExec (I,P,(Initialize s))) = card I
let s be State of SCMPDS; ::_thesis: for I being halt-free Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
IC (IExec (I,P,(Initialize s))) = card I
let I be halt-free Program of SCMPDS; ::_thesis: ( I is_closed_on s,P & I is_halting_on s,P implies IC (IExec (I,P,(Initialize s))) = card I )
set s1 = Initialize s;
set P1 = P +* (stop I);
assume that
A1: I is_closed_on s,P and
A2: I is_halting_on s,P ; ::_thesis: IC (IExec (I,P,(Initialize s))) = card I
A3: P +* (stop I) halts_on Initialize s by A2, Def3;
thus  IC (IExec (I,P,(Initialize s))) = IC (Result ((P +* (stop I)),(Initialize s)))
.= IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A3, EXTPRO_1:23
.= card I by A1, A2, Th29 ; ::_thesis: verum
end;

begin

definition
let a be Int_position;
let k be Integer;
let I, J be Program of SCMPDS;
func if=0 (a,k,I,J) ->  Program of SCMPDS equals :: SCMPDS_6:def 4
((((a,k) <>0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1))) ';' J;
coherence
 ((((a,k) <>0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1))) ';' J is Program of SCMPDS ;
func if>0 (a,k,I,J) ->  Program of SCMPDS equals :: SCMPDS_6:def 5
((((a,k) <=0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1))) ';' J;
coherence
 ((((a,k) <=0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1))) ';' J is Program of SCMPDS ;
func if<0 (a,k,I,J) ->  Program of SCMPDS equals :: SCMPDS_6:def 6
((((a,k) >=0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1))) ';' J;
coherence
 ((((a,k) >=0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1))) ';' J is Program of SCMPDS ;
end;

:: deftheorem  defines if=0 SCMPDS_6:def_4_:_
 for a being Int_position
for k being Integer
for I, J being Program of SCMPDS holds if=0 (a,k,I,J) = ((((a,k) <>0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1))) ';' J;

:: deftheorem  defines if>0 SCMPDS_6:def_5_:_
 for a being Int_position
for k being Integer
for I, J being Program of SCMPDS holds if>0 (a,k,I,J) = ((((a,k) <=0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1))) ';' J;

:: deftheorem  defines if<0 SCMPDS_6:def_6_:_
 for a being Int_position
for k being Integer
for I, J being Program of SCMPDS holds if<0 (a,k,I,J) = ((((a,k) >=0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1))) ';' J;

definition
let a be Int_position;
let k be Integer;
let I be Program of SCMPDS;
func if=0 (a,k,I) ->  Program of SCMPDS equals :: SCMPDS_6:def 7
((a,k) <>0_goto ((card I) + 1)) ';' I;
coherence
 ((a,k) <>0_goto ((card I) + 1)) ';' I is Program of SCMPDS ;
func if<>0 (a,k,I) ->  Program of SCMPDS equals :: SCMPDS_6:def 8
(((a,k) <>0_goto 2) ';' (goto ((card I) + 1))) ';' I;
coherence
 (((a,k) <>0_goto 2) ';' (goto ((card I) + 1))) ';' I is Program of SCMPDS ;
func if>0 (a,k,I) ->  Program of SCMPDS equals :: SCMPDS_6:def 9
((a,k) <=0_goto ((card I) + 1)) ';' I;
coherence
 ((a,k) <=0_goto ((card I) + 1)) ';' I is Program of SCMPDS ;
func if<=0 (a,k,I) ->  Program of SCMPDS equals :: SCMPDS_6:def 10
(((a,k) <=0_goto 2) ';' (goto ((card I) + 1))) ';' I;
coherence
 (((a,k) <=0_goto 2) ';' (goto ((card I) + 1))) ';' I is Program of SCMPDS ;
func if<0 (a,k,I) ->  Program of SCMPDS equals :: SCMPDS_6:def 11
((a,k) >=0_goto ((card I) + 1)) ';' I;
coherence
 ((a,k) >=0_goto ((card I) + 1)) ';' I is Program of SCMPDS ;
func if>=0 (a,k,I) ->  Program of SCMPDS equals :: SCMPDS_6:def 12
(((a,k) >=0_goto 2) ';' (goto ((card I) + 1))) ';' I;
coherence
 (((a,k) >=0_goto 2) ';' (goto ((card I) + 1))) ';' I is Program of SCMPDS ;
end;

:: deftheorem  defines if=0 SCMPDS_6:def_7_:_
 for a being Int_position
for k being Integer
for I being Program of SCMPDS holds if=0 (a,k,I) = ((a,k) <>0_goto ((card I) + 1)) ';' I;

:: deftheorem  defines if<>0 SCMPDS_6:def_8_:_
 for a being Int_position
for k being Integer
for I being Program of SCMPDS holds if<>0 (a,k,I) = (((a,k) <>0_goto 2) ';' (goto ((card I) + 1))) ';' I;

:: deftheorem  defines if>0 SCMPDS_6:def_9_:_
 for a being Int_position
for k being Integer
for I being Program of SCMPDS holds if>0 (a,k,I) = ((a,k) <=0_goto ((card I) + 1)) ';' I;

:: deftheorem  defines if<=0 SCMPDS_6:def_10_:_
 for a being Int_position
for k being Integer
for I being Program of SCMPDS holds if<=0 (a,k,I) = (((a,k) <=0_goto 2) ';' (goto ((card I) + 1))) ';' I;

:: deftheorem  defines if<0 SCMPDS_6:def_11_:_
 for a being Int_position
for k being Integer
for I being Program of SCMPDS holds if<0 (a,k,I) = ((a,k) >=0_goto ((card I) + 1)) ';' I;

:: deftheorem  defines if>=0 SCMPDS_6:def_12_:_
 for a being Int_position
for k being Integer
for I being Program of SCMPDS holds if>=0 (a,k,I) = (((a,k) >=0_goto 2) ';' (goto ((card I) + 1))) ';' I;

Lm4:  for n being Element of NAT
for i being Instruction of SCMPDS
for I, J being Program of SCMPDS holds card (((i ';' I) ';' (Goto n)) ';' J) = ((card I) + (card J)) + 2
proof
let n be Element of NAT ; ::_thesis: for i being Instruction of SCMPDS
for I, J being Program of SCMPDS holds card (((i ';' I) ';' (Goto n)) ';' J) = ((card I) + (card J)) + 2
let i be Instruction of SCMPDS; ::_thesis: for I, J being Program of SCMPDS holds card (((i ';' I) ';' (Goto n)) ';' J) = ((card I) + (card J)) + 2
let I, J be Program of SCMPDS; ::_thesis: card (((i ';' I) ';' (Goto n)) ';' J) = ((card I) + (card J)) + 2
set G = Goto n;
thus  card (((i ';' I) ';' (Goto n)) ';' J) = (card ((i ';' I) ';' (Goto n))) + (card J) by AFINSQ_1:17
.= ((card (i ';' I)) + (card (Goto n))) + (card J) by AFINSQ_1:17
.= ((card (i ';' I)) + 1) + (card J) by COMPOS_1:54
.= (((card I) + 1) + 1) + (card J) by Th6
.= ((card I) + (card J)) + 2 ; ::_thesis: verum
end;

begin

theorem :: SCMPDS_6:35
for a being Int_position
for k1 being Integer
for I, J being Program of SCMPDS holds card (if=0 (a,k1,I,J)) = ((card I) + (card J)) + 2 by Lm4;

theorem :: SCMPDS_6:36
for a being Int_position
for k1 being Integer
for I, J being Program of SCMPDS holds
( 0 in dom (if=0 (a,k1,I,J)) & 1 in dom (if=0 (a,k1,I,J)) )
proof
let a be Int_position; ::_thesis: for k1 being Integer
for I, J being Program of SCMPDS holds
( 0 in dom (if=0 (a,k1,I,J)) & 1 in dom (if=0 (a,k1,I,J)) )
let k1 be Integer; ::_thesis: for I, J being Program of SCMPDS holds
( 0 in dom (if=0 (a,k1,I,J)) & 1 in dom (if=0 (a,k1,I,J)) )
let I, J be Program of SCMPDS; ::_thesis: ( 0 in dom (if=0 (a,k1,I,J)) & 1 in dom (if=0 (a,k1,I,J)) )
set ci = card (if=0 (a,k1,I,J));
 card (if=0 (a,k1,I,J)) = ((card I) + (card J)) + 2 by Lm4;
then  2 <= card (if=0 (a,k1,I,J)) by NAT_1:12;
then  1 < card (if=0 (a,k1,I,J)) by XXREAL_0:2;
hence  ( 0 in dom (if=0 (a,k1,I,J)) & 1 in dom (if=0 (a,k1,I,J)) ) by AFINSQ_1:66; ::_thesis: verum
end;

Lm5:  for i being Instruction of SCMPDS
for I, J, K being Program of SCMPDS holds (((i ';' I) ';' J) ';' K) . 0 = i
proof
let i be Instruction of SCMPDS; ::_thesis: for I, J, K being Program of SCMPDS holds (((i ';' I) ';' J) ';' K) . 0 = i
let I, J, K be Program of SCMPDS; ::_thesis: (((i ';' I) ';' J) ';' K) . 0 = i
A1: 0 in dom (Load i) by COMPOS_1:50;
((i ';' I) ';' J) ';' K = (i ';' (I ';' J)) ';' K by SCMPDS_4:14
.= i ';' ((I ';' J) ';' K) by SCMPDS_4:14
.= (Load i) ';' ((I ';' J) ';' K) ;
hence (((i ';' I) ';' J) ';' K) . 0 = (Load i) . 0 by A1, AFINSQ_1:def_3
.= i by COMPOS_1:52 ;
::_thesis: verum
end;

theorem :: SCMPDS_6:37
for a being Int_position
for k1 being Integer
for I, J being Program of SCMPDS holds (if=0 (a,k1,I,J)) . 0 = (a,k1) <>0_goto ((card I) + 2) by Lm5;

Lm6:  for i being Instruction of SCMPDS
for I being Program of SCMPDS
for P being Instruction-Sequence of SCMPDS holds Shift ((stop I),1) c= P +* (stop (i ';' I))
proof
let i be Instruction of SCMPDS; ::_thesis: for I being Program of SCMPDS
for P being Instruction-Sequence of SCMPDS holds Shift ((stop I),1) c= P +* (stop (i ';' I))
let I be Program of SCMPDS; ::_thesis: for P being Instruction-Sequence of SCMPDS holds Shift ((stop I),1) c= P +* (stop (i ';' I))
let P be Instruction-Sequence of SCMPDS; ::_thesis: Shift ((stop I),1) c= P +* (stop (i ';' I))
set pI = stop I;
set iI = i ';' I;
set piI = stop (i ';' I);
set P3 = P +* (stop (i ';' I));
 ( card (Load i) = 1 & i ';' I = (Load i) ';' I ) by COMPOS_1:54;
then A1: Shift ((stop I),1) c= stop (i ';' I) by Th13;
 stop (i ';' I) c= P +* (stop (i ';' I)) by FUNCT_4:25;
then  Shift ((stop I),1) c= P +* (stop (i ';' I)) by A1, XBOOLE_1:1;
hence  Shift ((stop I),1) c= P +* (stop (i ';' I)) ; ::_thesis: verum
end;

Lm7:  for i, j being Instruction of SCMPDS
for I being Program of SCMPDS
for P being Instruction-Sequence of SCMPDS holds Shift ((stop I),2) c= P +* (stop ((i ';' j) ';' I))
proof
let i, j be Instruction of SCMPDS; ::_thesis: for I being Program of SCMPDS
for P being Instruction-Sequence of SCMPDS holds Shift ((stop I),2) c= P +* (stop ((i ';' j) ';' I))
let I be Program of SCMPDS; ::_thesis: for P being Instruction-Sequence of SCMPDS holds Shift ((stop I),2) c= P +* (stop ((i ';' j) ';' I))
let P be Instruction-Sequence of SCMPDS; ::_thesis: Shift ((stop I),2) c= P +* (stop ((i ';' j) ';' I))
set pI = stop I;
set pjI = stop ((i ';' j) ';' I);
set P3 = P +* (stop ((i ';' j) ';' I));
card (i ';' j) = card ((Load i) ';' (Load j))
.= (card (Load i)) + (card (Load j)) by AFINSQ_1:17
.= 1 + (card (Load j)) by COMPOS_1:54
.= 1 + 1 by COMPOS_1:54 ;
then A1: Shift ((stop I),2) c= stop ((i ';' j) ';' I) by Th13;
 stop ((i ';' j) ';' I) c= P +* (stop ((i ';' j) ';' I)) by FUNCT_4:25;
then  Shift ((stop I),2) c= P +* (stop ((i ';' j) ';' I)) by A1, XBOOLE_1:1;
hence  Shift ((stop I),2) c= P +* (stop ((i ';' j) ';' I)) ; ::_thesis: verum
end;

theorem Th38: :: SCMPDS_6:38
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I, J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I, J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P )
let s be 0 -started State of SCMPDS; ::_thesis: for I, J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P )
let I, J be shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P )
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P )
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P implies ( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
assume A1: s . (DataLoc ((s . a),k1)) = 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or ( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P ) )
set i = (a,k1) <>0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
set I2 = (I ';' (Goto ((card J) + 1))) ';' J;
set IF = if=0 (a,k1,I,J);
set pIF = stop (if=0 (a,k1,I,J));
set pI2 = stop ((I ';' (Goto ((card J) + 1))) ';' J);
set P2 = P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J));
set P3 = P +* (stop (if=0 (a,k1,I,J)));
set s4 = Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1);
set P4 = P +* (stop (if=0 (a,k1,I,J)));
A2: Initialize s = s by MEMSTR_0:44;
then A3: IC s = 0 by MEMSTR_0:47;
A4: if=0 (a,k1,I,J) = (((a,k1) <>0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J by SCMPDS_4:14
.= ((a,k1) <>0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J) by SCMPDS_4:14 ;
then A5: Shift ((stop ((I ';' (Goto ((card J) + 1))) ';' J)),1) c= P +* (stop (if=0 (a,k1,I,J))) by Lm6;
A6: Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,(0 + 1)) = Following ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,0))) by EXTPRO_1:3
.= Following ((P +* (stop (if=0 (a,k1,I,J)))),s) by EXTPRO_1:2
.= Exec (((a,k1) <>0_goto ((card I) + 2)),s) by A4, Th11, A2 ;
 for a being Int_position holds s . a = (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)) . a by A6, SCMPDS_2:55;
then A7: DataPart s = DataPart (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)) by SCMPDS_4:8;
s . (DataLoc ((s . a),k1)) = s . (DataLoc ((s . a),k1))
.= 0 by A1 ;
then A8: IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)) = succ (IC s) by A6, SCMPDS_2:55
.= 0 + 1 by A3 ;
A9: 0 in dom (stop (if=0 (a,k1,I,J))) by COMPOS_1:36;
assume A10: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or ( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P ) )
assume A11: I is_halting_on s,P ; ::_thesis: ( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P )
then  (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P by A10, Th30;
then A12: P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s by Def3, A2;
A13: (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P by A10, A11, Th30;
then A14: ( Start-At (0,SCMPDS) c= s & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) ) by Th24, A2, FUNCT_4:25;
A15: card (stop (if=0 (a,k1,I,J))) = (card (if=0 (a,k1,I,J))) + 1 by COMPOS_1:55
.= ((card ((I ';' (Goto ((card J) + 1))) ';' J)) + 1) + 1 by A4, Th6 ;
A16: stop ((I ';' (Goto ((card J) + 1))) ';' J) c= P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by FUNCT_4:25;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(if=0_(a,k1,I,J)))),s,k))_in_dom_(stop_(if=0_(a,k1,I,J)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,b1)) in dom (stop (if=0 (a,k1,I,J)))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,b1)) in dom (stop (if=0 (a,k1,I,J)))
then consider k1 being Nat such that
A17: k1 + 1 = k by NAT_1:6;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def_12;
reconsider m = IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,k1)) as Element of NAT ;
A18: card (stop (if=0 (a,k1,I,J))) = (card (stop ((I ';' (Goto ((card J) + 1))) ';' J))) + 1 by A15, COMPOS_1:55;
 m in dom (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by A13, Def2, A2;
then  m < card (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by AFINSQ_1:66;
then A19: m + 1 < card (stop (if=0 (a,k1,I,J))) by A18, XREAL_1:6;
IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,k)) = IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),k1)) by A17, EXTPRO_1:4
.= m + 1 by A14, A5, A8, A7, Th31, A16 ;
hence  IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,k)) in dom (stop (if=0 (a,k1,I,J))) by A19, AFINSQ_1:66; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,b1)) in dom (stop (if=0 (a,k1,I,J)))
hence  IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,k)) in dom (stop (if=0 (a,k1,I,J))) by A9, A3, EXTPRO_1:2; ::_thesis: verum
end;
end;
end;
hence  if=0 (a,k1,I,J) is_closed_on s,P by Def2, A2; ::_thesis: if=0 (a,k1,I,J) is_halting_on s,P
A20: Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1)) = Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))) by EXTPRO_1:4;
CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1)))) = CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) by A20
.= CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) by A14, A5, A8, A7, Th31, A16
.= halt SCMPDS by A12, EXTPRO_1:def_15 ;
then  P +* (stop (if=0 (a,k1,I,J))) halts_on s by EXTPRO_1:29;
hence  if=0 (a,k1,I,J) is_halting_on s,P by Def3, A2; ::_thesis: verum
end;

theorem Th39: :: SCMPDS_6:39
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P )
let I be Program of SCMPDS; ::_thesis: for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P )
let J be shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P )
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P )
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) <> 0 & J is_closed_on s,P & J is_halting_on s,P implies ( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
set pJ = stop J;
set s1 = Initialize s;
set P1 = P +* (stop J);
set IF = if=0 (a,k1,I,J);
set pIF = stop (if=0 (a,k1,I,J));
set s3 = Initialize s;
set P3 = P +* (stop (if=0 (a,k1,I,J)));
set s4 = Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),1);
set P4 = P +* (stop (if=0 (a,k1,I,J)));
set i = (a,k1) <>0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
set iG = (((a,k1) <>0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1));
A1: if=0 (a,k1,I,J) = (((a,k1) <>0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J by SCMPDS_4:14
.= ((a,k1) <>0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J) by SCMPDS_4:14 ;
A2: Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <>0_goto ((card I) + 2)),(Initialize s)) by A1, Th11 ;
A3: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A4: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A3, FUNCT_4:11 ;
A5: IC (Initialize s) = 0 by MEMSTR_0:47;
assume  s . (DataLoc ((s . a),k1)) <> 0 ; ::_thesis: ( not J is_closed_on s,P or not J is_halting_on s,P or ( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P ) )
then A6: IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),1)) = ICplusConst ((Initialize s),((card I) + 2)) by A2, A4, SCMPDS_2:55
.= 0 + ((card I) + 2) by A5, Th12 ;
assume A7: J is_closed_on s,P ; ::_thesis: ( not J is_halting_on s,P or ( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P ) )
then A8: ( Start-At (0,SCMPDS) c= Initialize s & J is_closed_on Initialize s,P +* (stop J) ) by Th24, FUNCT_4:25;
A9: stop (if=0 (a,k1,I,J)) c= P +* (stop (if=0 (a,k1,I,J))) by FUNCT_4:25;
A10: card ((((a,k1) <>0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1))) = (card (((a,k1) <>0_goto ((card I) + 2)) ';' I)) + (card (Goto ((card J) + 1))) by AFINSQ_1:17
.= (card (((a,k1) <>0_goto ((card I) + 2)) ';' I)) + 1 by COMPOS_1:54
.= ((card I) + 1) + 1 by Th6
.= (card I) + (1 + 1) ;
then  Shift ((stop J),((card I) + 2)) c= stop (if=0 (a,k1,I,J)) by Th13;
then A11: Shift ((stop J),((card I) + 2)) c= P +* (stop (if=0 (a,k1,I,J))) by A9, XBOOLE_1:1;
assume  J is_halting_on s,P ; ::_thesis: ( if=0 (a,k1,I,J) is_closed_on s,P & if=0 (a,k1,I,J) is_halting_on s,P )
then A12: P +* (stop J) halts_on Initialize s by Def3;
 for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),1)) . a by A2, SCMPDS_2:55;
then A13: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),1)) by SCMPDS_4:8;
A14: stop J c= P +* (stop J) by FUNCT_4:25;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(if=0_(a,k1,I,J)))),(Initialize_s),k))_in_dom_(stop_(if=0_(a,k1,I,J)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),b1)) in dom (stop (if=0 (a,k1,I,J)))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),b1)) in dom (stop (if=0 (a,k1,I,J)))
then consider k1 being Nat such that
A15: k1 + 1 = k by NAT_1:6;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def_12;
reconsider m = IC (Comput ((P +* (stop J)),(Initialize s),k1)) as Element of NAT ;
 m in dom (stop J) by A7, Def2;
then  m < card (stop J) by AFINSQ_1:66;
then A16: m + ((card I) + 2) < (card (stop J)) + ((card I) + 2) by XREAL_1:6;
A17: card (stop J) = (card J) + 1 by COMPOS_1:55;
A18: card (stop (if=0 (a,k1,I,J))) = (card (if=0 (a,k1,I,J))) + 1 by COMPOS_1:55
.= (((card I) + 2) + (card J)) + 1 by A10, AFINSQ_1:17
.= ((card I) + 2) + (card (stop J)) by A17 ;
IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),k)) = IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),1)),k1)) by A15, EXTPRO_1:4
.= m + ((card I) + 2) by A8, A13, A11, A6, Th31, A14 ;
hence  IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),k)) in dom (stop (if=0 (a,k1,I,J))) by A18, A16, AFINSQ_1:66; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),b1)) in dom (stop (if=0 (a,k1,I,J)))
then  Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),k) = Initialize s by EXTPRO_1:2;
hence  IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),k)) in dom (stop (if=0 (a,k1,I,J))) by A5, COMPOS_1:36; ::_thesis: verum
end;
end;
end;
hence  if=0 (a,k1,I,J) is_closed_on s,P by Def2; ::_thesis: if=0 (a,k1,I,J) is_halting_on s,P
A19: Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1)) = Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),1)),(LifeSpan ((P +* (stop J)),(Initialize s)))) by EXTPRO_1:4;
CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),1)),(LifeSpan ((P +* (stop J)),(Initialize s)))))) by A19
.= CurInstr ((P +* (stop J)),(Comput ((P +* (stop J)),(Initialize s),(LifeSpan ((P +* (stop J)),(Initialize s)))))) by A8, A13, A11, A6, Th31, A14
.= halt SCMPDS by A12, EXTPRO_1:def_15 ;
then  P +* (stop (if=0 (a,k1,I,J))) halts_on Initialize s by EXTPRO_1:29;
hence  if=0 (a,k1,I,J) is_halting_on s,P by Def3; ::_thesis: verum
end;

theorem Th40: :: SCMPDS_6:40
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let J be shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P implies IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
set b = DataLoc ((s . a),k1);
assume A1: s . (DataLoc ((s . a),k1)) = 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
set i = (a,k1) <>0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
set I2 = (I ';' (Goto ((card J) + 1))) ';' J;
set IF = if=0 (a,k1,I,J);
set pI2 = stop ((I ';' (Goto ((card J) + 1))) ';' J);
set s2 = s;
set s3 = s;
set P2 = P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J));
set P3 = P +* (stop (if=0 (a,k1,I,J)));
set s4 = Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1);
set P4 = P +* (stop (if=0 (a,k1,I,J)));
A2: Initialize s = s by MEMSTR_0:44;
then A3: IC s = 0 by MEMSTR_0:47;
A4: if=0 (a,k1,I,J) = (((a,k1) <>0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J by SCMPDS_4:14
.= ((a,k1) <>0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J) by SCMPDS_4:14 ;
then A5: Shift ((stop ((I ';' (Goto ((card J) + 1))) ';' J)),1) c= P +* (stop (if=0 (a,k1,I,J))) by Lm6;
A6: Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,(0 + 1)) = Following ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,0))) by EXTPRO_1:3
.= Following ((P +* (stop (if=0 (a,k1,I,J)))),s) by EXTPRO_1:2
.= Exec (((a,k1) <>0_goto ((card I) + 2)),s) by A4, Th11, A2 ;
s . (DataLoc ((s . a),k1)) = s . (DataLoc ((s . a),k1))
.= 0 by A1 ;
then A7: IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)) = succ (IC s) by A6, SCMPDS_2:55
.= 0 + 1 by A3 ;
 for a being Int_position holds s . a = (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)) . a by A6, SCMPDS_2:55;
then A8: DataPart s = DataPart (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)) by SCMPDS_4:8;
set SAl = Start-At ((((card I) + (card J)) + 2),SCMPDS);
assume A9: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
assume A10: I is_halting_on s,P ; ::_thesis: IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
then  (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P by A9, Th30;
then A11: P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s by Def3, A2;
 (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P by A9, A10, Th30;
then A12: ( Start-At (0,SCMPDS) c= s & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) ) by Th24, A2, FUNCT_4:25;
A13: stop ((I ';' (Goto ((card J) + 1))) ';' J) c= P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by FUNCT_4:25;
A14: Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1)) = Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))) by EXTPRO_1:4;
A15: CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1)))) = CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) by A14
.= CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) by A12, A5, A7, A8, Th31, A13
.= halt SCMPDS by A11, EXTPRO_1:def_15 ;
then A16: P +* (stop (if=0 (a,k1,I,J))) halts_on s by EXTPRO_1:29;
A17: CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),s) = (a,k1) <>0_goto ((card I) + 2) by A4, Th11, A2;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_(LifeSpan_((P_+*_(stop_((I_';'_(Goto_((card_J)_+_1)))_';'_J))),s))_+_1_holds_
CurInstr_((P_+*_(stop_(if=0_(a,k1,I,J)))),(Comput_((P_+*_(stop_(if=0_(a,k1,I,J)))),s,l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 implies CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,b1))) <> halt SCMPDS )
assume A18: l < (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 ; ::_thesis: CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,b1))) <> halt SCMPDS
A19: Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,0) = s by EXTPRO_1:2;
percases ( l = 0 or l <> 0 ) ;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,b1))) <> halt SCMPDS
then  CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,l))) = CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),s) by A19;
hence  CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,l))) <> halt SCMPDS by A17; ::_thesis: verum
end;
suppose l <> 0 ; ::_thesis: not CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,b1))) = halt SCMPDS
then consider n being Nat such that
A20: l = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A21: n < LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) by A18, A20, XREAL_1:6;
assume A22: CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,l))) = halt SCMPDS ; ::_thesis: contradiction
A23: Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,(n + 1)) = Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),n) by EXTPRO_1:4;
CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,n))) = CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),n))) by A12, A5, A7, A8, Th31, A13
.= halt SCMPDS by A20, A22, A23 ;
hence  contradiction by A11, A21, EXTPRO_1:def_15; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,l))) = halt SCMPDS holds
(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 <= l ;
then A24: LifeSpan ((P +* (stop (if=0 (a,k1,I,J)))),s) = (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 by A15, A16, EXTPRO_1:def_15;
A25: DataPart (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) = DataPart (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)))) by A11, EXTPRO_1:23
.= DataPart (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)))) by A12, A5, A7, A8, Th31, A13
.= DataPart (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (stop (if=0 (a,k1,I,J)))),s)) by A16, A24, EXTPRO_1:23 ;
A26: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if=0_(a,k1,I,J)),P,s))_holds_
(IExec_((if=0_(a,k1,I,J)),P,s))_._x_=_((IExec_(((I_';'_(Goto_((card_J)_+_1)))_';'_J),P,s))_+*_(Start-At_((((card_I)_+_(card_J))_+_2),SCMPDS)))_._x
let x be set ; ::_thesis: ( x in dom (IExec ((if=0 (a,k1,I,J)),P,s)) implies (IExec ((if=0 (a,k1,I,J)),P,s)) . b1 = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1 )
A27: dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A28: x in dom (IExec ((if=0 (a,k1,I,J)),P,s)) ; ::_thesis: (IExec ((if=0 (a,k1,I,J)),P,s)) . b1 = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A28, SCMPDS_4:6;
supposeA29: x is Int_position ; ::_thesis: (IExec ((if=0 (a,k1,I,J)),P,s)) . b1 = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A30: not x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A27, TARSKI:def_1;
thus (IExec ((if=0 (a,k1,I,J)),P,s)) . x = (Result ((P +* (stop (if=0 (a,k1,I,J)))),s)) . x
.= (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) . x by A25, A29, SCMPDS_4:8
.= (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) . x
.= ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by A30, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA31: x = IC ; ::_thesis: (IExec ((if=0 (a,k1,I,J)),P,s)) . b1 = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1
A32: IC (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) = IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s))
.= ((card I) + (card J)) + 1 by A9, A10, Th32 ;
A33: x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A27, A31, TARSKI:def_1;
thus (IExec ((if=0 (a,k1,I,J)),P,s)) . x = (Result ((P +* (stop (if=0 (a,k1,I,J)))),s)) . x
.= (Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1))) . x by A16, A24, EXTPRO_1:23
.= IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)))) by A31, EXTPRO_1:4
.= (IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) + 1 by A12, A5, A7, A8, Th31, A13
.= (IC (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))) + 1 by A11, EXTPRO_1:23
.= IC (Start-At (((((card I) + (card J)) + 1) + 1),SCMPDS)) by A32, FUNCOP_1:72
.= ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by A31, A33, FUNCT_4:13 ; ::_thesis: verum
end;
end;
end;
dom (IExec ((if=0 (a,k1,I,J)),P,s)) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A26, FUNCT_1:2
.= ((IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A9, A10, Th33
.= (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by MEMSTR_0:36 ;
::_thesis: verum
end;

theorem Th41: :: SCMPDS_6:41
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for J being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if=0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for J being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if=0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for J being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if=0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let I be Program of SCMPDS; ::_thesis: for J being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if=0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let J be halt-free shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if=0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if=0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) <> 0 & J is_closed_on s,P & J is_halting_on s,P implies IExec ((if=0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set pJ = stop J;
set s1 = Initialize s;
set P1 = P +* (stop J);
set P2 = P +* (stop J);
set IF = if=0 (a,k1,I,J);
set pIF = stop (if=0 (a,k1,I,J));
set s3 = Initialize s;
set P3 = P +* (stop (if=0 (a,k1,I,J)));
set s4 = Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),1);
set P4 = P +* (stop (if=0 (a,k1,I,J)));
set i = (a,k1) <>0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
set iG = (((a,k1) <>0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1));
set SAl = Start-At ((((card I) + (card J)) + 2),SCMPDS);
A1: if=0 (a,k1,I,J) = (((a,k1) <>0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J by SCMPDS_4:14
.= ((a,k1) <>0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J) by SCMPDS_4:14 ;
A2: Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <>0_goto ((card I) + 2)),(Initialize s)) by A1, Th11 ;
A3: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A4: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A3, FUNCT_4:11 ;
A5: IC (Initialize s) = 0 by MEMSTR_0:47;
assume  s . (DataLoc ((s . a),k1)) <> 0 ; ::_thesis: ( not J is_closed_on s,P or not J is_halting_on s,P or IExec ((if=0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
then A6: IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),1)) = ICplusConst ((Initialize s),((card I) + 2)) by A2, A4, SCMPDS_2:55
.= 0 + ((card I) + 2) by A5, Th12 ;
 for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),1)) . a by A2, SCMPDS_2:55;
then A7: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),1)) by SCMPDS_4:8;
card ((((a,k1) <>0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1))) = (card (((a,k1) <>0_goto ((card I) + 2)) ';' I)) + (card (Goto ((card J) + 1))) by AFINSQ_1:17
.= (card (((a,k1) <>0_goto ((card I) + 2)) ';' I)) + 1 by COMPOS_1:54
.= ((card I) + 1) + 1 by Th6
.= (card I) + (1 + 1) ;
then A8: Shift ((stop J),((card I) + 2)) c= stop (if=0 (a,k1,I,J)) by Th13;
 stop (if=0 (a,k1,I,J)) c= P +* (stop (if=0 (a,k1,I,J))) by FUNCT_4:25;
then  Shift ((stop J),((card I) + 2)) c= P +* (stop (if=0 (a,k1,I,J))) by A8, XBOOLE_1:1;
then A9: Shift ((stop J),((card I) + 2)) c= P +* (stop (if=0 (a,k1,I,J))) ;
assume A10: J is_closed_on s,P ; ::_thesis: ( not J is_halting_on s,P or IExec ((if=0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
then A11: ( Start-At (0,SCMPDS) c= Initialize s & J is_closed_on Initialize s,P +* (stop J) ) by Th24, FUNCT_4:25;
A12: stop J c= P +* (stop J) by FUNCT_4:25;
assume A13: J is_halting_on s,P ; ::_thesis: IExec ((if=0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
then A14: P +* (stop J) halts_on Initialize s by Def3;
A15: Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1)) = Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),1)),(LifeSpan ((P +* (stop J)),(Initialize s)))) by EXTPRO_1:4;
A16: CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),1)),(LifeSpan ((P +* (stop J)),(Initialize s)))))) by A15
.= CurInstr ((P +* (stop J)),(Comput ((P +* (stop J)),(Initialize s),(LifeSpan ((P +* (stop J)),(Initialize s)))))) by A11, A9, A6, A7, Th31, A12
.= halt SCMPDS by A14, EXTPRO_1:def_15 ;
then A17: P +* (stop (if=0 (a,k1,I,J))) halts_on Initialize s by EXTPRO_1:29;
A18: CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s)) = (a,k1) <>0_goto ((card I) + 2) by A1, Th11;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_(LifeSpan_((P_+*_(stop_J)),(Initialize_s)))_+_1_holds_
CurInstr_((P_+*_(stop_(if=0_(a,k1,I,J)))),(Comput_((P_+*_(stop_(if=0_(a,k1,I,J)))),(Initialize_s),l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < (LifeSpan ((P +* (stop J)),(Initialize s))) + 1 implies CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),b1))) <> halt SCMPDS )
assume A19: l < (LifeSpan ((P +* (stop J)),(Initialize s))) + 1 ; ::_thesis: CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),b1))) <> halt SCMPDS
A20: Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),0) = Initialize s by EXTPRO_1:2;
percases ( l = 0 or l <> 0 ) ;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),b1))) <> halt SCMPDS
then  CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),l))) = CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s)) by A20;
hence  CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),l))) <> halt SCMPDS by A18; ::_thesis: verum
end;
suppose l <> 0 ; ::_thesis: not CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),b1))) = halt SCMPDS
then consider n being Nat such that
A21: l = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A22: n < LifeSpan ((P +* (stop J)),(Initialize s)) by A19, A21, XREAL_1:6;
assume A23: CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),l))) = halt SCMPDS ; ::_thesis: contradiction
A24: Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),(n + 1)) = Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),1)),n) by EXTPRO_1:4;
CurInstr ((P +* (stop J)),(Comput ((P +* (stop J)),(Initialize s),n))) = CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),1)),n))) by A11, A9, A6, A7, Th31, A12
.= halt SCMPDS by A21, A23, A24 ;
hence  contradiction by A14, A22, EXTPRO_1:def_15; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),l))) = halt SCMPDS holds
(LifeSpan ((P +* (stop J)),(Initialize s))) + 1 <= l ;
then A25: LifeSpan ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s)) = (LifeSpan ((P +* (stop J)),(Initialize s))) + 1 by A16, A17, EXTPRO_1:def_15;
A26: DataPart (Result ((P +* (stop J)),(Initialize s))) = DataPart (Comput ((P +* (stop J)),(Initialize s),(LifeSpan ((P +* (stop J)),(Initialize s))))) by A14, EXTPRO_1:23
.= DataPart (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),1)),(LifeSpan ((P +* (stop J)),(Initialize s))))) by A11, A9, A6, A7, Th31, A12
.= DataPart (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s))) by A17, A25, EXTPRO_1:23 ;
A27: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if=0_(a,k1,I,J)),P,(Initialize_s)))_holds_
(IExec_((if=0_(a,k1,I,J)),P,(Initialize_s)))_._x_=_((IExec_(J,P,(Initialize_s)))_+*_(Start-At_((((card_I)_+_(card_J))_+_2),SCMPDS)))_._x
let x be set ; ::_thesis: ( x in dom (IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) implies (IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b1 = ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1 )
A28: dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A29: x in dom (IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) ; ::_thesis: (IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b1 = ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A29, SCMPDS_4:6;
supposeA30: x is Int_position ; ::_thesis: (IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b1 = ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A31: not x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A28, TARSKI:def_1;
thus (IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . x = (Result ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s))) . x
.= (Result ((P +* (stop J)),(Initialize s))) . x by A26, A30, SCMPDS_4:8
.= (IExec (J,P,(Initialize s))) . x
.= ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by A31, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA32: x = IC ; ::_thesis: (IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b1 = ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1
A33: IC (Result ((P +* (stop J)),(Initialize s))) = IC (IExec (J,P,(Initialize s)))
.= card J by A10, A13, Th34 ;
A34: x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A28, A32, TARSKI:def_1;
thus (IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . x = (Result ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s))) . x
.= (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1))) . x by A17, A25, EXTPRO_1:23
.= IC (Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Comput ((P +* (stop (if=0 (a,k1,I,J)))),(Initialize s),1)),(LifeSpan ((P +* (stop J)),(Initialize s))))) by A32, EXTPRO_1:4
.= (IC (Comput ((P +* (stop J)),(Initialize s),(LifeSpan ((P +* (stop J)),(Initialize s)))))) + ((card I) + 2) by A11, A9, A6, A7, Th31, A12
.= (IC (Result ((P +* (stop J)),(Initialize s)))) + ((card I) + 2) by A14, EXTPRO_1:23
.= IC (Start-At (((card J) + ((card I) + 2)),SCMPDS)) by A33, FUNCOP_1:72
.= ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by A32, A34, FUNCT_4:13 ; ::_thesis: verum
end;
end;
end;
dom (IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((if=0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A27, FUNCT_1:2; ::_thesis: verum
end;

registration
let I, J be parahalting shiftable Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if=0 (a,k1,I,J) ->  parahalting shiftable ;
correctness
coherence
( if=0 (a,k1,I,J) is shiftable & if=0 (a,k1,I,J) is parahalting );
proof
set i = (a,k1) <>0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
set IF = if=0 (a,k1,I,J);
reconsider IJ = (I ';' (Goto ((card J) + 1))) ';' J as shiftable Program of SCMPDS ;
thus  if=0 (a,k1,I,J) is shiftable ; ::_thesis: if=0 (a,k1,I,J) is parahalting
let s be 0 -started State of SCMPDS; :: according to SCMPDS_4:def_7 ::_thesis: for b1 being set holds
( not stop (if=0 (a,k1,I,J)) c= b1 or b1 halts_on s )
let P be Instruction-Sequence of SCMPDS; ::_thesis: ( not stop (if=0 (a,k1,I,J)) c= P or P halts_on s )
A1: Initialize s = s by MEMSTR_0:44;
assume  stop (if=0 (a,k1,I,J)) c= P ; ::_thesis: P halts_on s
then A2: P = P +* (stop (if=0 (a,k1,I,J))) by FUNCT_4:98;
A3: ( J is_closed_on s,P & J is_halting_on s,P ) by Th20, Th21;
A4: ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
percases ( s . (DataLoc ((s . a),k1)) = 0 or s . (DataLoc ((s . a),k1)) <> 0 ) ;
suppose s . (DataLoc ((s . a),k1)) = 0 ; ::_thesis: P halts_on s
then  if=0 (a,k1,I,J) is_halting_on s,P by A4, Th38;
hence  P halts_on s by A1, A2, Def3; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),k1)) <> 0 ; ::_thesis: P halts_on s
then  if=0 (a,k1,I,J) is_halting_on s,P by A3, Th39;
hence  P halts_on s by A1, A2, Def3; ::_thesis: verum
end;
end;
end;
end;

registration
let I, J be halt-free Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if=0 (a,k1,I,J) ->  halt-free ;
coherence
 if=0 (a,k1,I,J) is halt-free ;
end;

theorem :: SCMPDS_6:42
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I, J being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if=0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I, J being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if=0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
let s be 0 -started State of SCMPDS; ::_thesis: for I, J being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if=0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
let I, J be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer holds IC (IExec ((if=0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
let a be Int_position; ::_thesis: for k1 being Integer holds IC (IExec ((if=0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
let k1 be Integer; ::_thesis: IC (IExec ((if=0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
set IF = if=0 (a,k1,I,J);
A1: ( J is_closed_on s,P & J is_halting_on s,P ) by Th20, Th21;
A2: ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
A3: Initialize s = s by MEMSTR_0:44;
hereby  ::_thesis: verum
percases ( s . (DataLoc ((s . a),k1)) = 0 or s . (DataLoc ((s . a),k1)) <> 0 ) ;
suppose s . (DataLoc ((s . a),k1)) = 0 ; ::_thesis: IC (IExec ((if=0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
then  IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A2, Th40;
hence  IC (IExec ((if=0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2 by FUNCT_4:113; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),k1)) <> 0 ; ::_thesis: IC (IExec ((if=0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
then  IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A1, Th41, A3;
hence  IC (IExec ((if=0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2 by FUNCT_4:113; ::_thesis: verum
end;
end;
end;
end;

theorem :: SCMPDS_6:43
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
(IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
(IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free parahalting shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
(IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let I be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
(IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let J be shiftable Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
(IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
(IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) = 0 implies (IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b )
assume A1: s . (DataLoc ((s . a),k1)) = 0 ; ::_thesis: (IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
A2: not b in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by SCMPDS_4:18;
 ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
then  IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A1, Th40;
hence  (IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b by A2, FUNCT_4:11; ::_thesis: verum
end;

theorem :: SCMPDS_6:44
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for J being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for J being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for J being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
let I be Program of SCMPDS; ::_thesis: for J being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
let J be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) <> 0 implies (IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b )
assume A1: s . (DataLoc ((s . a),k1)) <> 0 ; ::_thesis: (IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
A2: not b in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by SCMPDS_4:18;
 ( J is_closed_on s,P & J is_halting_on s,P ) by Th20, Th21;
then  IExec ((if=0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A1, Th41;
hence  (IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b by A2, FUNCT_4:11; ::_thesis: verum
end;

begin

theorem :: SCMPDS_6:45
for a being Int_position
for k1 being Integer
for I being Program of SCMPDS holds card (if=0 (a,k1,I)) = (card I) + 1 by Th6;

theorem :: SCMPDS_6:46
for a being Int_position
for k1 being Integer
for I being Program of SCMPDS holds 0 in dom (if=0 (a,k1,I))
proof
let a be Int_position; ::_thesis: for k1 being Integer
for I being Program of SCMPDS holds 0 in dom (if=0 (a,k1,I))
let k1 be Integer; ::_thesis: for I being Program of SCMPDS holds 0 in dom (if=0 (a,k1,I))
let I be Program of SCMPDS; ::_thesis: 0 in dom (if=0 (a,k1,I))
set ci = card (if=0 (a,k1,I));
 card (if=0 (a,k1,I)) = (card I) + 1 by Th6;
hence  0 in dom (if=0 (a,k1,I)) by AFINSQ_1:66; ::_thesis: verum
end;

theorem :: SCMPDS_6:47
for a being Int_position
for k1 being Integer
for I being Program of SCMPDS holds (if=0 (a,k1,I)) . 0 = (a,k1) <>0_goto ((card I) + 1) by Th7;

theorem Th48: :: SCMPDS_6:48
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if=0 (a,k1,I) is_closed_on s,P & if=0 (a,k1,I) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if=0 (a,k1,I) is_closed_on s,P & if=0 (a,k1,I) is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if=0 (a,k1,I) is_closed_on s,P & if=0 (a,k1,I) is_halting_on s,P )
let I be shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if=0 (a,k1,I) is_closed_on s,P & if=0 (a,k1,I) is_halting_on s,P )
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if=0 (a,k1,I) is_closed_on s,P & if=0 (a,k1,I) is_halting_on s,P )
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P implies ( if=0 (a,k1,I) is_closed_on s,P & if=0 (a,k1,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
assume A1: s . (DataLoc ((s . a),k1)) = 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or ( if=0 (a,k1,I) is_closed_on s,P & if=0 (a,k1,I) is_halting_on s,P ) )
set i = (a,k1) <>0_goto ((card I) + 1);
set IF = if=0 (a,k1,I);
set pIF = stop (if=0 (a,k1,I));
set pI = stop I;
set s2 = Initialize s;
set s3 = Initialize s;
set P2 = P +* (stop I);
set P3 = P +* (stop (if=0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if=0 (a,k1,I)));
A2: IC (Initialize s) = 0 by MEMSTR_0:47;
A3: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
A4: Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if=0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <>0_goto ((card I) + 1)),(Initialize s)) by Th11 ;
 for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)) . a by A4, SCMPDS_2:55;
then A5: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)) by SCMPDS_4:8;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= 0 by A1, A3, FUNCT_4:11 ;
then A6: IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)) = succ (IC (Initialize s)) by A4, SCMPDS_2:55
.= 0 + 1 by A2 ;
assume A7: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or ( if=0 (a,k1,I) is_closed_on s,P & if=0 (a,k1,I) is_halting_on s,P ) )
then A8: I is_closed_on Initialize s,P +* (stop I) by Th24;
assume  I is_halting_on s,P ; ::_thesis: ( if=0 (a,k1,I) is_closed_on s,P & if=0 (a,k1,I) is_halting_on s,P )
then A9: P +* (stop I) halts_on Initialize s by Def3;
A10: 0 in dom (stop (if=0 (a,k1,I))) by COMPOS_1:36;
A11: ( Start-At (0,SCMPDS) c= Initialize s & Shift ((stop I),1) c= P +* (stop (if=0 (a,k1,I))) ) by Lm6, FUNCT_4:25;
A12: stop I c= P +* (stop I) by FUNCT_4:25;
A13: card (stop (if=0 (a,k1,I))) = (card (if=0 (a,k1,I))) + 1 by COMPOS_1:55
.= ((card I) + 1) + 1 by Th6 ;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(if=0_(a,k1,I)))),(Initialize_s),k))_in_dom_(stop_(if=0_(a,k1,I)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if=0 (a,k1,I)))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if=0 (a,k1,I)))
then consider k1 being Nat such that
A14: k1 + 1 = k by NAT_1:6;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def_12;
reconsider m = IC (Comput ((P +* (stop I)),(Initialize s),k1)) as Element of NAT ;
A15: card (stop (if=0 (a,k1,I))) = (card (stop I)) + 1 by A13, COMPOS_1:55;
 m in dom (stop I) by A7, Def2;
then  m < card (stop I) by AFINSQ_1:66;
then A16: m + 1 < card (stop (if=0 (a,k1,I))) by A15, XREAL_1:6;
IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),k)) = IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)),k1)) by A14, EXTPRO_1:4
.= m + 1 by A8, A11, A6, A5, Th31, A12 ;
hence  IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if=0 (a,k1,I))) by A16, AFINSQ_1:66; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if=0 (a,k1,I)))
hence  IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if=0 (a,k1,I))) by A10, A2, EXTPRO_1:2; ::_thesis: verum
end;
end;
end;
hence  if=0 (a,k1,I) is_closed_on s,P by Def2; ::_thesis: if=0 (a,k1,I) is_halting_on s,P
A17: Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)) = Comput ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))) by EXTPRO_1:4;
CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A17
.= CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A8, A11, A6, A5, Th31, A12
.= halt SCMPDS by A9, EXTPRO_1:def_15 ;
then  P +* (stop (if=0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
hence  if=0 (a,k1,I) is_halting_on s,P by Def3; ::_thesis: verum
end;

theorem Th49: :: SCMPDS_6:49
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
( if=0 (a,k1,I) is_closed_on s,P & if=0 (a,k1,I) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
( if=0 (a,k1,I) is_closed_on s,P & if=0 (a,k1,I) is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
( if=0 (a,k1,I) is_closed_on s,P & if=0 (a,k1,I) is_halting_on s,P )
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
( if=0 (a,k1,I) is_closed_on s,P & if=0 (a,k1,I) is_halting_on s,P )
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
( if=0 (a,k1,I) is_closed_on s,P & if=0 (a,k1,I) is_halting_on s,P )
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) <> 0 implies ( if=0 (a,k1,I) is_closed_on s,P & if=0 (a,k1,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
assume A1: s . (DataLoc ((s . a),k1)) <> 0 ; ::_thesis: ( if=0 (a,k1,I) is_closed_on s,P & if=0 (a,k1,I) is_halting_on s,P )
set i = (a,k1) <>0_goto ((card I) + 1);
set IF = if=0 (a,k1,I);
set pIF = stop (if=0 (a,k1,I));
set s3 = Initialize s;
set P3 = P +* (stop (if=0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if=0 (a,k1,I)));
A2: IC (Initialize s) = 0 by MEMSTR_0:47;
A3: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A4: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A3, FUNCT_4:11 ;
Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if=0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <>0_goto ((card I) + 1)),(Initialize s)) by Th11 ;
then A5: IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)) = ICplusConst ((Initialize s),((card I) + 1)) by A1, A4, SCMPDS_2:55
.= 0 + ((card I) + 1) by A2, Th12 ;
A6: card (if=0 (a,k1,I)) = (card I) + 1 by Th6;
then A7: (card I) + 1 in dom (stop (if=0 (a,k1,I))) by COMPOS_1:64;
A8: (P +* (stop (if=0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1))) = (P +* (stop (if=0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1))) by PBOOLE:143;
 stop (if=0 (a,k1,I)) c= P +* (stop (if=0 (a,k1,I))) by FUNCT_4:25;
then  stop (if=0 (a,k1,I)) c= P +* (stop (if=0 (a,k1,I))) ;
then (P +* (stop (if=0 (a,k1,I)))) . ((card I) + 1) = (stop (if=0 (a,k1,I))) . ((card I) + 1) by A7, GRFUNC_1:2
.= halt SCMPDS by A6, COMPOS_1:64 ;
then A9: CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1))) = halt SCMPDS by A5, A8;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(if=0_(a,k1,I)))),(Initialize_s),k))_in_dom_(stop_(if=0_(a,k1,I)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if=0 (a,k1,I)))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if=0 (a,k1,I)))
then  1 + 0 <= k by INT_1:7;
hence  IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if=0 (a,k1,I))) by A7, A5, A9, EXTPRO_1:5; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if=0 (a,k1,I)))
then  Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),k) = Initialize s by EXTPRO_1:2;
hence  IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if=0 (a,k1,I))) by A2, COMPOS_1:36; ::_thesis: verum
end;
end;
end;
hence  if=0 (a,k1,I) is_closed_on s,P by Def2; ::_thesis: if=0 (a,k1,I) is_halting_on s,P
 P +* (stop (if=0 (a,k1,I))) halts_on Initialize s by A9, EXTPRO_1:29;
hence  if=0 (a,k1,I) is_halting_on s,P by Def3; ::_thesis: verum
end;

theorem Th50: :: SCMPDS_6:50
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
let s be State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) = 0 & I is_closed_on s,P & I is_halting_on s,P implies IExec ((if=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) )
set b = DataLoc ((s . a),k1);
assume A1: s . (DataLoc ((s . a),k1)) = 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or IExec ((if=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) )
set i = (a,k1) <>0_goto ((card I) + 1);
set IF = if=0 (a,k1,I);
set pI = stop I;
set pIF = stop (if=0 (a,k1,I));
set s2 = Initialize s;
set P2 = P +* (stop I);
set s3 = Initialize s;
set P3 = P +* (stop (if=0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if=0 (a,k1,I)));
A2: IC (Initialize s) = 0 by MEMSTR_0:47;
A3: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
A4: Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if=0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <>0_goto ((card I) + 1)),(Initialize s)) by Th11 ;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= 0 by A1, A3, FUNCT_4:11 ;
then A5: IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)) = succ (IC (Initialize s)) by A4, SCMPDS_2:55
.= 0 + 1 by A2 ;
 for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)) . a by A4, SCMPDS_2:55;
then A6: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)) by SCMPDS_4:8;
assume A7: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or IExec ((if=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) )
then A8: I is_closed_on Initialize s,P +* (stop I) by Th24;
set SAl = Start-At (((card I) + 1),SCMPDS);
A9: ( Start-At (0,SCMPDS) c= Initialize s & Shift ((stop I),1) c= P +* (stop (if=0 (a,k1,I))) ) by Lm6, FUNCT_4:25;
A10: stop I c= P +* (stop I) by FUNCT_4:25;
assume A11: I is_halting_on s,P ; ::_thesis: IExec ((if=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
then A12: P +* (stop I) halts_on Initialize s by Def3;
A13: Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)) = Comput ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))) by EXTPRO_1:4;
A14: CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A13
.= CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A8, A9, A5, A6, Th31, A10
.= halt SCMPDS by A12, EXTPRO_1:def_15 ;
then A15: P +* (stop (if=0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
A16: CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Initialize s)) = (a,k1) <>0_goto ((card I) + 1) by Th11;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_(LifeSpan_((P_+*_(stop_I)),(Initialize_s)))_+_1_holds_
CurInstr_((P_+*_(stop_(if=0_(a,k1,I)))),(Comput_((P_+*_(stop_(if=0_(a,k1,I)))),(Initialize_s),l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 implies CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS )
assume A17: l < (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 ; ::_thesis: CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
A18: Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),0) = Initialize s by EXTPRO_1:2;
percases ( l = 0 or l <> 0 ) ;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
then  CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),l))) = CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Initialize s)) by A18;
hence  CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A16; ::_thesis: verum
end;
suppose l <> 0 ; ::_thesis: not CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),b1))) = halt SCMPDS
then consider n being Nat such that
A19: l = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A20: n < LifeSpan ((P +* (stop I)),(Initialize s)) by A17, A19, XREAL_1:6;
assume A21: CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS ; ::_thesis: contradiction
A22: Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),(n + 1)) = Comput ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)),n) by EXTPRO_1:4;
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),n))) = CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)),n))) by A8, A9, A5, A6, Th31, A10
.= halt SCMPDS by A19, A21, A22 ;
hence  contradiction by A12, A20, EXTPRO_1:def_15; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS holds
(LifeSpan ((P +* (stop I)),(Initialize s))) + 1 <= l ;
then A23: LifeSpan ((P +* (stop (if=0 (a,k1,I)))),(Initialize s)) = (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 by A14, A15, EXTPRO_1:def_15;
A24: DataPart (Result ((P +* (stop I)),(Initialize s))) = DataPart (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A12, EXTPRO_1:23
.= DataPart (Comput ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A8, A9, A5, A6, Th31, A10
.= DataPart (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (stop (if=0 (a,k1,I)))),(Initialize s))) by A15, A23, EXTPRO_1:23 ;
A25: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if=0_(a,k1,I)),P,(Initialize_s)))_holds_
(IExec_((if=0_(a,k1,I)),P,(Initialize_s)))_._x_=_((IExec_(I,P,(Initialize_s)))_+*_(Start-At_(((card_I)_+_1),SCMPDS)))_._x
let x be set ; ::_thesis: ( x in dom (IExec ((if=0 (a,k1,I)),P,(Initialize s))) implies (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . b1 )
A26: dom (Start-At (((card I) + 1),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A27: x in dom (IExec ((if=0 (a,k1,I)),P,(Initialize s))) ; ::_thesis: (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A27, SCMPDS_4:6;
supposeA28: x is Int_position ; ::_thesis: (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A29: not x in dom (Start-At (((card I) + 1),SCMPDS)) by A26, TARSKI:def_1;
thus (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . x = (Result ((P +* (stop (if=0 (a,k1,I)))),(Initialize s))) . x
.= (Result ((P +* (stop I)),(Initialize s))) . x by A24, A28, SCMPDS_4:8
.= (IExec (I,P,(Initialize s))) . x
.= ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . x by A29, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA30: x = IC ; ::_thesis: (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . b1
A31: IC (Result ((P +* (stop I)),(Initialize s))) = IC (IExec (I,P,(Initialize s)))
.= card I by A7, A11, Th34 ;
A32: x in dom (Start-At (((card I) + 1),SCMPDS)) by A26, A30, TARSKI:def_1;
thus (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . x = (Result ((P +* (stop (if=0 (a,k1,I)))),(Initialize s))) . x
.= (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) . x by A15, A23, EXTPRO_1:23
.= IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A30, EXTPRO_1:4
.= (IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) + 1 by A8, A9, A5, A6, Th31, A10
.= (IC (Result ((P +* (stop I)),(Initialize s)))) + 1 by A12, EXTPRO_1:23
.= IC (Start-At (((card I) + 1),SCMPDS)) by A31, FUNCOP_1:72
.= ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . x by A30, A32, FUNCT_4:13 ; ::_thesis: verum
end;
end;
end;
dom (IExec ((if=0 (a,k1,I)),P,(Initialize s))) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((if=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) by A25, FUNCT_1:2; ::_thesis: verum
end;

theorem Th51: :: SCMPDS_6:51
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
IExec ((if=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
IExec ((if=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
IExec ((if=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
IExec ((if=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
IExec ((if=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) <> 0 implies IExec ((if=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set IF = if=0 (a,k1,I);
set pIF = stop (if=0 (a,k1,I));
set s3 = Initialize s;
set P3 = P +* (stop (if=0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if=0 (a,k1,I)));
set i = (a,k1) <>0_goto ((card I) + 1);
set SAl = Start-At (((card I) + 1),SCMPDS);
A1: IC (Initialize s) = 0 by MEMSTR_0:47;
A2: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A3: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A2, FUNCT_4:11 ;
A4: Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if=0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <>0_goto ((card I) + 1)),(Initialize s)) by Th11 ;
assume  s . (DataLoc ((s . a),k1)) <> 0 ; ::_thesis: IExec ((if=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
then A5: IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)) = ICplusConst ((Initialize s),((card I) + 1)) by A4, A3, SCMPDS_2:55
.= 0 + ((card I) + 1) by A1, Th12 ;
A6: stop (if=0 (a,k1,I)) c= P +* (stop (if=0 (a,k1,I))) by FUNCT_4:25;
A7: (P +* (stop (if=0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1))) = (P +* (stop (if=0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1))) by PBOOLE:143;
A8: card (if=0 (a,k1,I)) = (card I) + 1 by Th6;
then  (card I) + 1 in dom (stop (if=0 (a,k1,I))) by COMPOS_1:64;
then (P +* (stop (if=0 (a,k1,I)))) . ((card I) + 1) = (stop (if=0 (a,k1,I))) . ((card I) + 1) by A6, GRFUNC_1:2
.= halt SCMPDS by A8, COMPOS_1:64 ;
then A9: CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1))) = halt SCMPDS by A5, A7;
then A10: P +* (stop (if=0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
A11: CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Initialize s)) = (a,k1) <>0_goto ((card I) + 1) by Th11;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_1_holds_
CurInstr_((P_+*_(stop_(if=0_(a,k1,I)))),(Comput_((P_+*_(stop_(if=0_(a,k1,I)))),(Initialize_s),l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < 1 implies CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS )
A12: Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),0) = Initialize s by EXTPRO_1:2;
assume  l < 1 ; ::_thesis: CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS
then  l < 1 + 0 ;
then  l = 0 by NAT_1:13;
then  CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),l))) = CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Initialize s)) by A12;
hence  CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A11; ::_thesis: verum
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS holds
1 <= l ;
then  LifeSpan ((P +* (stop (if=0 (a,k1,I)))),(Initialize s)) = 1 by A9, A10, EXTPRO_1:def_15;
then A13: Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1) = Result ((P +* (stop (if=0 (a,k1,I)))),(Initialize s)) by A10, EXTPRO_1:23;
A14: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if=0_(a,k1,I)),P,(Initialize_s)))_holds_
(IExec_((if=0_(a,k1,I)),P,(Initialize_s)))_._x_=_(s_+*_(Start-At_(((card_I)_+_1),SCMPDS)))_._x
A15: dom (Start-At (((card I) + 1),SCMPDS)) = {(IC )} by FUNCOP_1:13;
let x be set ; ::_thesis: ( x in dom (IExec ((if=0 (a,k1,I)),P,(Initialize s))) implies (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 1),SCMPDS))) . b1 )
assume A16: x in dom (IExec ((if=0 (a,k1,I)),P,(Initialize s))) ; ::_thesis: (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 1),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A16, SCMPDS_4:6;
supposeA17: x is Int_position ; ::_thesis: (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 1),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A18: not x in dom (Start-At (((card I) + 1),SCMPDS)) by A15, TARSKI:def_1;
A19: not x in dom (Start-At (0,SCMPDS)) by A17, SCMPDS_4:18;
thus (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . x = (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)) . x by A13
.= (Initialize s) . x by A4, A17, SCMPDS_2:55
.= s . x by A19, FUNCT_4:11
.= (s +* (Start-At (((card I) + 1),SCMPDS))) . x by A18, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA20: x = IC ; ::_thesis: (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 1),SCMPDS))) . b1
hence (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . x = (card I) + 1 by A5, A13
.= (s +* (Start-At (((card I) + 1),SCMPDS))) . x by A20, FUNCT_4:113 ;
::_thesis: verum
end;
end;
end;
dom (IExec ((if=0 (a,k1,I)),P,(Initialize s))) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom (s +* (Start-At (((card I) + 1),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((if=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS)) by A14, FUNCT_1:2; ::_thesis: verum
end;

registration
let I be parahalting shiftable Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if=0 (a,k1,I) ->  parahalting shiftable ;
correctness
coherence
( if=0 (a,k1,I) is shiftable & if=0 (a,k1,I) is parahalting );
proof
set i = (a,k1) <>0_goto ((card I) + 1);
set IF = if=0 (a,k1,I);
thus  if=0 (a,k1,I) is shiftable ; ::_thesis: if=0 (a,k1,I) is parahalting
let s be 0 -started State of SCMPDS; :: according to SCMPDS_4:def_7 ::_thesis: for b1 being set holds
( not stop (if=0 (a,k1,I)) c= b1 or b1 halts_on s )
let P be Instruction-Sequence of SCMPDS; ::_thesis: ( not stop (if=0 (a,k1,I)) c= P or P halts_on s )
A1: Initialize s = s by MEMSTR_0:44;
assume  stop (if=0 (a,k1,I)) c= P ; ::_thesis: P halts_on s
then A2: P = P +* (stop (if=0 (a,k1,I))) by FUNCT_4:98;
A3: ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
percases ( s . (DataLoc ((s . a),k1)) = 0 or s . (DataLoc ((s . a),k1)) <> 0 ) ;
suppose s . (DataLoc ((s . a),k1)) = 0 ; ::_thesis: P halts_on s
then  if=0 (a,k1,I) is_halting_on s,P by A3, Th48;
hence  P halts_on s by A1, A2, Def3; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),k1)) <> 0 ; ::_thesis: P halts_on s
then  if=0 (a,k1,I) is_halting_on s,P by Th49;
hence  P halts_on s by A1, A2, Def3; ::_thesis: verum
end;
end;
end;
end;

registration
let I be halt-free Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if=0 (a,k1,I) ->  halt-free ;
coherence
 if=0 (a,k1,I) is halt-free ;
end;

theorem :: SCMPDS_6:52
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if=0 (a,k1,I)),P,s)) = (card I) + 1
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if=0 (a,k1,I)),P,s)) = (card I) + 1
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if=0 (a,k1,I)),P,s)) = (card I) + 1
let I be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer holds IC (IExec ((if=0 (a,k1,I)),P,s)) = (card I) + 1
let a be Int_position; ::_thesis: for k1 being Integer holds IC (IExec ((if=0 (a,k1,I)),P,s)) = (card I) + 1
let k1 be Integer; ::_thesis: IC (IExec ((if=0 (a,k1,I)),P,s)) = (card I) + 1
set IF = if=0 (a,k1,I);
A1: ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
A2: Initialize s = s by MEMSTR_0:44;
percases ( s . (DataLoc ((s . a),k1)) = 0 or s . (DataLoc ((s . a),k1)) <> 0 ) ;
suppose s . (DataLoc ((s . a),k1)) = 0 ; ::_thesis: IC (IExec ((if=0 (a,k1,I)),P,s)) = (card I) + 1
then  IExec ((if=0 (a,k1,I)),P,s) = (IExec (I,P,s)) +* (Start-At (((card I) + 1),SCMPDS)) by A1, Th50, A2;
hence  IC (IExec ((if=0 (a,k1,I)),P,s)) = (card I) + 1 by FUNCT_4:113; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),k1)) <> 0 ; ::_thesis: IC (IExec ((if=0 (a,k1,I)),P,s)) = (card I) + 1
then  IExec ((if=0 (a,k1,I)),P,s) = s +* (Start-At (((card I) + 1),SCMPDS)) by Th51, A2;
hence  IC (IExec ((if=0 (a,k1,I)),P,s)) = (card I) + 1 by FUNCT_4:113; ::_thesis: verum
end;
end;
end;

theorem :: SCMPDS_6:53
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
(IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
(IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let s be State of SCMPDS; ::_thesis: for I being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
(IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let I be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
(IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
(IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) = 0 implies (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b )
assume A1: s . (DataLoc ((s . a),k1)) = 0 ; ::_thesis: (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
A2: not b in dom (Start-At (((card I) + 1),SCMPDS)) by SCMPDS_4:18;
 ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
then  IExec ((if=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) by A1, Th50;
hence  (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b by A2, FUNCT_4:11; ::_thesis: verum
end;

theorem :: SCMPDS_6:54
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b = s . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b = s . b
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b = s . b
let I be Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b = s . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b = s . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) <> 0 implies (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b = s . b )
assume  s . (DataLoc ((s . a),k1)) <> 0 ; ::_thesis: (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b = s . b
then A1: IExec ((if=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS)) by Th51;
 not b in dom (Start-At (((card I) + 1),SCMPDS)) by SCMPDS_4:18;
hence  (IExec ((if=0 (a,k1,I)),P,(Initialize s))) . b = s . b by A1, FUNCT_4:11; ::_thesis: verum
end;

Lm8:  for i, j being Instruction of SCMPDS
for I being Program of SCMPDS holds card ((i ';' j) ';' I) = (card I) + 2
proof
let i, j be Instruction of SCMPDS; ::_thesis: for I being Program of SCMPDS holds card ((i ';' j) ';' I) = (card I) + 2
let I be Program of SCMPDS; ::_thesis: card ((i ';' j) ';' I) = (card I) + 2
thus  card ((i ';' j) ';' I) = card (i ';' (j ';' I)) by SCMPDS_4:16
.= (card (j ';' I)) + 1 by Th6
.= ((card I) + 1) + 1 by Th6
.= (card I) + 2 ; ::_thesis: verum
end;

begin

theorem :: SCMPDS_6:55
for a being Int_position
for k1 being Integer
for I being Program of SCMPDS holds card (if<>0 (a,k1,I)) = (card I) + 2 by Lm8;

Lm9:  for i, j being Instruction of SCMPDS
for I being Program of SCMPDS holds
( 0 in dom ((i ';' j) ';' I) & 1 in dom ((i ';' j) ';' I) )
proof
let i, j be Instruction of SCMPDS; ::_thesis: for I being Program of SCMPDS holds
( 0 in dom ((i ';' j) ';' I) & 1 in dom ((i ';' j) ';' I) )
let I be Program of SCMPDS; ::_thesis: ( 0 in dom ((i ';' j) ';' I) & 1 in dom ((i ';' j) ';' I) )
set ci = card ((i ';' j) ';' I);
 card ((i ';' j) ';' I) = (card I) + 2 by Lm8;
then  2 <= card ((i ';' j) ';' I) by NAT_1:11;
then  1 < card ((i ';' j) ';' I) by XXREAL_0:2;
hence  ( 0 in dom ((i ';' j) ';' I) & 1 in dom ((i ';' j) ';' I) ) by AFINSQ_1:66; ::_thesis: verum
end;

theorem :: SCMPDS_6:56
for a being Int_position
for k1 being Integer
for I being Program of SCMPDS holds
( 0 in dom (if<>0 (a,k1,I)) & 1 in dom (if<>0 (a,k1,I)) ) by Lm9;

Lm10:  for i, j being Instruction of SCMPDS
for I being Program of SCMPDS holds
( ((i ';' j) ';' I) . 0 = i & ((i ';' j) ';' I) . 1 = j )
proof
let i, j be Instruction of SCMPDS; ::_thesis: for I being Program of SCMPDS holds
( ((i ';' j) ';' I) . 0 = i & ((i ';' j) ';' I) . 1 = j )
let I be Program of SCMPDS; ::_thesis: ( ((i ';' j) ';' I) . 0 = i & ((i ';' j) ';' I) . 1 = j )
set jI = j ';' I;
A1: (i ';' j) ';' I = i ';' (j ';' I) by SCMPDS_4:16
.= (Load i) ';' (j ';' I) ;
 0 in dom (Load i) by COMPOS_1:50;
hence ((i ';' j) ';' I) . 0 = (Load i) . 0 by A1, AFINSQ_1:def_3
.= i by COMPOS_1:52 ;
::_thesis: ((i ';' j) ';' I) . 1 = j
A2: 0 in dom (Load j) by COMPOS_1:50;
 card (j ';' I) = (card I) + 1 by Th6;
then A3: ( card (Load i) = 1 & 0 in dom (j ';' I) ) by AFINSQ_1:66, COMPOS_1:54;
thus ((i ';' j) ';' I) . 1 = ((Load i) ';' (j ';' I)) . (0 + 1) by A1
.= (j ';' I) . 0 by A3, AFINSQ_1:def_3
.= ((Load j) ';' I) . 0
.= (Load j) . 0 by A2, AFINSQ_1:def_3
.= j by COMPOS_1:52 ; ::_thesis: verum
end;

theorem :: SCMPDS_6:57
for a being Int_position
for k1 being Integer
for I being Program of SCMPDS holds
( (if<>0 (a,k1,I)) . 0 = (a,k1) <>0_goto 2 & (if<>0 (a,k1,I)) . 1 = goto ((card I) + 1) ) by Lm10;

theorem Th58: :: SCMPDS_6:58
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<>0 (a,k1,I) is_closed_on s,P & if<>0 (a,k1,I) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<>0 (a,k1,I) is_closed_on s,P & if<>0 (a,k1,I) is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<>0 (a,k1,I) is_closed_on s,P & if<>0 (a,k1,I) is_halting_on s,P )
let I be shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<>0 (a,k1,I) is_closed_on s,P & if<>0 (a,k1,I) is_halting_on s,P )
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<>0 (a,k1,I) is_closed_on s,P & if<>0 (a,k1,I) is_halting_on s,P )
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) <> 0 & I is_closed_on s,P & I is_halting_on s,P implies ( if<>0 (a,k1,I) is_closed_on s,P & if<>0 (a,k1,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
set IF = if<>0 (a,k1,I);
set pIF = stop (if<>0 (a,k1,I));
set pI = stop I;
set s2 = Initialize s;
set P2 = P +* (stop I);
set s3 = Initialize s;
set P3 = P +* (stop (if<>0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if<>0 (a,k1,I)));
set i = (a,k1) <>0_goto 2;
set j = goto ((card I) + 1);
A1: if<>0 (a,k1,I) = ((a,k1) <>0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:16;
A2: Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <>0_goto 2),(Initialize s)) by A1, Th11 ;
 for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)) . a by A2, SCMPDS_2:55;
then A3: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)) by SCMPDS_4:8;
A4: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A5: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A4, FUNCT_4:11 ;
A6: IC (Initialize s) = 0 by MEMSTR_0:47;
assume  s . (DataLoc ((s . a),k1)) <> 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or ( if<>0 (a,k1,I) is_closed_on s,P & if<>0 (a,k1,I) is_halting_on s,P ) )
then A7: IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)) = ICplusConst ((Initialize s),2) by A2, A5, SCMPDS_2:55
.= 0 + 2 by A6, Th12 ;
assume A8: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or ( if<>0 (a,k1,I) is_closed_on s,P & if<>0 (a,k1,I) is_halting_on s,P ) )
then A9: I is_closed_on Initialize s,P +* (stop I) by Th24;
assume  I is_halting_on s,P ; ::_thesis: ( if<>0 (a,k1,I) is_closed_on s,P & if<>0 (a,k1,I) is_halting_on s,P )
then A10: P +* (stop I) halts_on Initialize s by Def3;
A11: ( Start-At (0,SCMPDS) c= Initialize s & Shift ((stop I),2) c= P +* (stop (if<>0 (a,k1,I))) ) by Lm7, FUNCT_4:25;
A12: stop I c= P +* (stop I) by FUNCT_4:25;
A13: 0 in dom (stop (if<>0 (a,k1,I))) by COMPOS_1:36;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(if<>0_(a,k1,I)))),(Initialize_s),k))_in_dom_(stop_(if<>0_(a,k1,I)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<>0 (a,k1,I)))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<>0 (a,k1,I)))
then consider k1 being Nat such that
A14: k1 + 1 = k by NAT_1:6;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def_12;
reconsider m = IC (Comput ((P +* (stop I)),(Initialize s),k1)) as Element of NAT ;
A15: card (stop (if<>0 (a,k1,I))) = 1 + (card (if<>0 (a,k1,I))) by COMPOS_1:55
.= 1 + ((card I) + 2) by Lm8
.= (1 + (card I)) + 2
.= (card (stop I)) + 2 by COMPOS_1:55 ;
 m in dom (stop I) by A8, Def2;
then  m < card (stop I) by AFINSQ_1:66;
then A16: m + 2 < card (stop (if<>0 (a,k1,I))) by A15, XREAL_1:6;
IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),k)) = IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)),k1)) by A14, EXTPRO_1:4
.= m + 2 by A9, A11, A7, A3, Th31, A12 ;
hence  IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if<>0 (a,k1,I))) by A16, AFINSQ_1:66; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<>0 (a,k1,I)))
hence  IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if<>0 (a,k1,I))) by A13, A6, EXTPRO_1:2; ::_thesis: verum
end;
end;
end;
hence  if<>0 (a,k1,I) is_closed_on s,P by Def2; ::_thesis: if<>0 (a,k1,I) is_halting_on s,P
A17: Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)) = Comput ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))) by EXTPRO_1:4;
CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A17
.= CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A9, A11, A7, A3, Th31, A12
.= halt SCMPDS by A10, EXTPRO_1:def_15 ;
then  P +* (stop (if<>0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
hence  if<>0 (a,k1,I) is_halting_on s,P by Def3; ::_thesis: verum
end;

theorem Th59: :: SCMPDS_6:59
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
( if<>0 (a,k1,I) is_closed_on s,P & if<>0 (a,k1,I) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
( if<>0 (a,k1,I) is_closed_on s,P & if<>0 (a,k1,I) is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
( if<>0 (a,k1,I) is_closed_on s,P & if<>0 (a,k1,I) is_halting_on s,P )
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
( if<>0 (a,k1,I) is_closed_on s,P & if<>0 (a,k1,I) is_halting_on s,P )
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
( if<>0 (a,k1,I) is_closed_on s,P & if<>0 (a,k1,I) is_halting_on s,P )
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) = 0 implies ( if<>0 (a,k1,I) is_closed_on s,P & if<>0 (a,k1,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
assume A1: s . (DataLoc ((s . a),k1)) = 0 ; ::_thesis: ( if<>0 (a,k1,I) is_closed_on s,P & if<>0 (a,k1,I) is_halting_on s,P )
set IF = if<>0 (a,k1,I);
set pIF = stop (if<>0 (a,k1,I));
set s3 = Initialize s;
set P3 = P +* (stop (if<>0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1);
set s5 = Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2);
set P4 = P +* (stop (if<>0 (a,k1,I)));
set P5 = P +* (stop (if<>0 (a,k1,I)));
A2: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A3: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= 0 by A1, A2, FUNCT_4:11 ;
A4: stop (if<>0 (a,k1,I)) c= P +* (stop (if<>0 (a,k1,I))) by FUNCT_4:25;
set i = (a,k1) <>0_goto 2;
set j = goto ((card I) + 1);
A5: if<>0 (a,k1,I) = ((a,k1) <>0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:16;
A6: IC (Initialize s) = 0 by MEMSTR_0:47;
Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <>0_goto 2),(Initialize s)) by A5, Th11 ;
then A7: IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)) = succ (IC (Initialize s)) by A3, SCMPDS_2:55
.= 0 + 1 by A6 ;
A8: 1 in dom (if<>0 (a,k1,I)) by Lm9;
then  1 in dom (stop (if<>0 (a,k1,I))) by COMPOS_1:62;
then A9: (P +* (stop (if<>0 (a,k1,I)))) . 1 = (stop (if<>0 (a,k1,I))) . 1 by A4, GRFUNC_1:2
.= (if<>0 (a,k1,I)) . 1 by A8, COMPOS_1:63
.= goto ((card I) + 1) by Lm10 ;
A10: card (if<>0 (a,k1,I)) = (card I) + 2 by Lm8;
then A11: (card I) + 2 in dom (stop (if<>0 (a,k1,I))) by COMPOS_1:64;
A12: (P +* (stop (if<>0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1))) = (P +* (stop (if<>0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1))) by PBOOLE:143;
Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),(1 + 1)) = Following ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1))) by EXTPRO_1:3
.= Exec ((goto ((card I) + 1)),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1))) by A7, A9, A12 ;
then A13: IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2)) = ICplusConst ((Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)),((card I) + 1)) by SCMPDS_2:54
.= ((card I) + 1) + 1 by A7, Th12
.= (card I) + (1 + 1) ;
A14: (P +* (stop (if<>0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2))) = (P +* (stop (if<>0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2))) by PBOOLE:143;
 stop (if<>0 (a,k1,I)) c= P +* (stop (if<>0 (a,k1,I))) by FUNCT_4:25;
then (P +* (stop (if<>0 (a,k1,I)))) . ((card I) + 2) = (stop (if<>0 (a,k1,I))) . ((card I) + 2) by A11, GRFUNC_1:2
.= halt SCMPDS by A10, COMPOS_1:64 ;
then A15: CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2))) = halt SCMPDS by A13, A14;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(if<>0_(a,k1,I)))),(Initialize_s),k))_in_dom_(stop_(if<>0_(a,k1,I)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<>0 (a,k1,I)))
A16: ( k = 0 or 0 + 1 <= k ) by INT_1:7;
percases ( k = 0 or k = 1 or 1 < k ) by A16, XXREAL_0:1;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<>0 (a,k1,I)))
then  Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),k) = Initialize s by EXTPRO_1:2;
hence  IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if<>0 (a,k1,I))) by A6, COMPOS_1:36; ::_thesis: verum
end;
suppose k = 1 ; ::_thesis: IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<>0 (a,k1,I)))
hence  IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if<>0 (a,k1,I))) by A8, A7, COMPOS_1:62; ::_thesis: verum
end;
suppose 1 < k ; ::_thesis: IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<>0 (a,k1,I)))
then  1 + 1 <= k by INT_1:7;
hence  IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if<>0 (a,k1,I))) by A11, A13, A15, EXTPRO_1:5; ::_thesis: verum
end;
end;
end;
hence  if<>0 (a,k1,I) is_closed_on s,P by Def2; ::_thesis: if<>0 (a,k1,I) is_halting_on s,P
 P +* (stop (if<>0 (a,k1,I))) halts_on Initialize s by A15, EXTPRO_1:29;
hence  if<>0 (a,k1,I) is_halting_on s,P by Def3; ::_thesis: verum
end;

theorem Th60: :: SCMPDS_6:60
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))
let s be State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) <> 0 & I is_closed_on s,P & I is_halting_on s,P implies IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set IF = if<>0 (a,k1,I);
set pI = stop I;
set pIF = stop (if<>0 (a,k1,I));
set s2 = Initialize s;
set P2 = P +* (stop I);
set s3 = Initialize s;
set P3 = P +* (stop (if<>0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if<>0 (a,k1,I)));
set i = (a,k1) <>0_goto 2;
set j = goto ((card I) + 1);
set SAl = Start-At (((card I) + 2),SCMPDS);
A1: if<>0 (a,k1,I) = ((a,k1) <>0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:16;
A2: Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <>0_goto 2),(Initialize s)) by A1, Th11 ;
A3: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A4: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A3, FUNCT_4:11 ;
A5: IC (Initialize s) = 0 by MEMSTR_0:47;
A6: ( Start-At (0,SCMPDS) c= Initialize s & Shift ((stop I),2) c= P +* (stop (if<>0 (a,k1,I))) ) by Lm7, FUNCT_4:25;
A7: stop I c= P +* (stop I) by FUNCT_4:25;
 for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)) . a by A2, SCMPDS_2:55;
then A8: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)) by SCMPDS_4:8;
assume  s . (DataLoc ((s . a),k1)) <> 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS)) )
then A9: IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)) = ICplusConst ((Initialize s),2) by A2, A4, SCMPDS_2:55
.= 0 + 2 by A5, Th12 ;
assume A10: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS)) )
then A11: I is_closed_on Initialize s,P +* (stop I) by Th24;
assume A12: I is_halting_on s,P ; ::_thesis: IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))
then A13: P +* (stop I) halts_on Initialize s by Def3;
A14: Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)) = Comput ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))) by EXTPRO_1:4;
A15: CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A14
.= CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A11, A6, A9, A8, Th31, A7
.= halt SCMPDS by A13, EXTPRO_1:def_15 ;
then A16: P +* (stop (if<>0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
A17: CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s)) = (a,k1) <>0_goto 2 by A1, Th11;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_(LifeSpan_((P_+*_(stop_I)),(Initialize_s)))_+_1_holds_
CurInstr_((P_+*_(stop_(if<>0_(a,k1,I)))),(Comput_((P_+*_(stop_(if<>0_(a,k1,I)))),(Initialize_s),l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 implies CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS )
assume A18: l < (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 ; ::_thesis: CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
A19: Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),0) = Initialize s by EXTPRO_1:2;
percases ( l = 0 or l <> 0 ) ;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
then  CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),l))) = CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s)) by A19;
hence  CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A17; ::_thesis: verum
end;
suppose l <> 0 ; ::_thesis: not CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b1))) = halt SCMPDS
then consider n being Nat such that
A20: l = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A21: n < LifeSpan ((P +* (stop I)),(Initialize s)) by A18, A20, XREAL_1:6;
assume A22: CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS ; ::_thesis: contradiction
A23: Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),(n + 1)) = Comput ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)),n) by EXTPRO_1:4;
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),n))) = CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)),n))) by A11, A6, A9, A8, Th31, A7
.= halt SCMPDS by A20, A22, A23 ;
hence  contradiction by A13, A21, EXTPRO_1:def_15; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS holds
(LifeSpan ((P +* (stop I)),(Initialize s))) + 1 <= l ;
then A24: LifeSpan ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s)) = (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 by A15, A16, EXTPRO_1:def_15;
A25: DataPart (Result ((P +* (stop I)),(Initialize s))) = DataPart (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A13, EXTPRO_1:23
.= DataPart (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A11, A6, A9, A8, Th31, A7
.= DataPart (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s))) by A16, A24, EXTPRO_1:23 ;
A26: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if<>0_(a,k1,I)),P,(Initialize_s)))_holds_
(IExec_((if<>0_(a,k1,I)),P,(Initialize_s)))_._x_=_((IExec_(I,P,(Initialize_s)))_+*_(Start-At_(((card_I)_+_2),SCMPDS)))_._x
let x be set ; ::_thesis: ( x in dom (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) implies (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) . b1 )
A27: dom (Start-At (((card I) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A28: x in dom (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) ; ::_thesis: (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A28, SCMPDS_4:6;
supposeA29: x is Int_position ; ::_thesis: (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A30: not x in dom (Start-At (((card I) + 2),SCMPDS)) by A27, TARSKI:def_1;
thus (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . x = (Result ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s))) . x
.= (Result ((P +* (stop I)),(Initialize s))) . x by A25, A29, SCMPDS_4:8
.= (IExec (I,P,(Initialize s))) . x
.= ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) . x by A30, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA31: x = IC ; ::_thesis: (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) . b1
A32: IC (Result ((P +* (stop I)),(Initialize s))) = IC (IExec (I,P,(Initialize s)))
.= card I by A10, A12, Th34 ;
A33: x in dom (Start-At (((card I) + 2),SCMPDS)) by A27, A31, TARSKI:def_1;
thus (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . x = (Result ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s))) . x
.= (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) . x by A16, A24, EXTPRO_1:23
.= IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A31, EXTPRO_1:4
.= (IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) + 2 by A11, A6, A9, A8, Th31, A7
.= (IC (Result ((P +* (stop I)),(Initialize s)))) + 2 by A13, EXTPRO_1:23
.= IC (Start-At (((card I) + 2),SCMPDS)) by A32, FUNCOP_1:72
.= ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) . x by A31, A33, FUNCT_4:13 ; ::_thesis: verum
end;
end;
end;
dom (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS)) by A26, FUNCT_1:2; ::_thesis: verum
end;

theorem Th61: :: SCMPDS_6:61
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) = 0 implies IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS)) )
set b = DataLoc ((s . a),k1);
assume A1: s . (DataLoc ((s . a),k1)) = 0 ; ::_thesis: IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
set SAl = Start-At (((card I) + 2),SCMPDS);
set i = (a,k1) <>0_goto 2;
set j = goto ((card I) + 1);
set IF = if<>0 (a,k1,I);
set pIF = stop (if<>0 (a,k1,I));
set s3 = Initialize s;
set P3 = P +* (stop (if<>0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1);
set s5 = Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2);
set P4 = P +* (stop (if<>0 (a,k1,I)));
set P5 = P +* (stop (if<>0 (a,k1,I)));
A2: if<>0 (a,k1,I) = ((a,k1) <>0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:16;
A3: stop (if<>0 (a,k1,I)) c= P +* (stop (if<>0 (a,k1,I))) by FUNCT_4:25;
then A4: stop (if<>0 (a,k1,I)) c= P +* (stop (if<>0 (a,k1,I))) ;
A5: stop (if<>0 (a,k1,I)) c= P +* (stop (if<>0 (a,k1,I))) by A3;
A6: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
A7: IC (Initialize s) = 0 by MEMSTR_0:47;
A8: Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <>0_goto 2),(Initialize s)) by A2, Th11 ;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= 0 by A1, A6, FUNCT_4:11 ;
then A9: IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)) = succ (IC (Initialize s)) by A8, SCMPDS_2:55
.= 0 + 1 by A7 ;
A10: 1 in dom (if<>0 (a,k1,I)) by Lm9;
then  1 in dom (stop (if<>0 (a,k1,I))) by COMPOS_1:62;
then A11: (P +* (stop (if<>0 (a,k1,I)))) . 1 = (stop (if<>0 (a,k1,I))) . 1 by A4, GRFUNC_1:2
.= (if<>0 (a,k1,I)) . 1 by A10, COMPOS_1:63
.= goto ((card I) + 1) by Lm10 ;
A12: (P +* (stop (if<>0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1))) = (P +* (stop (if<>0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1))) by PBOOLE:143;
A13: Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),(1 + 1)) = Following ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1))) by EXTPRO_1:3
.= Exec ((goto ((card I) + 1)),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1))) by A9, A11, A12 ;
then A14: IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2)) = ICplusConst ((Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)),((card I) + 1)) by SCMPDS_2:54
.= ((card I) + 1) + 1 by A9, Th12
.= (card I) + (1 + 1) ;
A15: (P +* (stop (if<>0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2))) = (P +* (stop (if<>0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2))) by PBOOLE:143;
A16: card (if<>0 (a,k1,I)) = (card I) + 2 by Lm8;
then  (card I) + 2 in dom (stop (if<>0 (a,k1,I))) by COMPOS_1:64;
then (P +* (stop (if<>0 (a,k1,I)))) . ((card I) + 2) = (stop (if<>0 (a,k1,I))) . ((card I) + 2) by A5, GRFUNC_1:2
.= halt SCMPDS by A16, COMPOS_1:64 ;
then A17: CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2))) = halt SCMPDS by A14, A15;
then A18: P +* (stop (if<>0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
A19: CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s)) = (a,k1) <>0_goto 2 by A2, Th11;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_1_+_1_holds_
CurInstr_((P_+*_(stop_(if<>0_(a,k1,I)))),(Comput_((P_+*_(stop_(if<>0_(a,k1,I)))),(Initialize_s),l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < 1 + 1 implies CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS )
assume  l < 1 + 1 ; ::_thesis: CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
then A20: l <= 1 by NAT_1:13;
A21: Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),0) = Initialize s by EXTPRO_1:2;
A22: (P +* (stop (if<>0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),l))) = (P +* (stop (if<>0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),l))) by PBOOLE:143;
percases ( l = 0 or l = 1 ) by A20, NAT_1:25;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
then  CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),l))) = CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s)) by A21;
hence  CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A19; ::_thesis: verum
end;
suppose l = 1 ; ::_thesis: CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
hence  CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A9, A11, A22; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (if<>0 (a,k1,I)))),(Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS holds
2 <= l ;
then  LifeSpan ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s)) = 2 by A17, A18, EXTPRO_1:def_15;
then A23: Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2) = Result ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s)) by A18, EXTPRO_1:23;
A24: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if<>0_(a,k1,I)),P,(Initialize_s)))_holds_
(IExec_((if<>0_(a,k1,I)),P,(Initialize_s)))_._x_=_(s_+*_(Start-At_(((card_I)_+_2),SCMPDS)))_._x
A25: dom (Start-At (((card I) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
let x be set ; ::_thesis: ( x in dom (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) implies (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1 )
assume A26: x in dom (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) ; ::_thesis: (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A26, SCMPDS_4:6;
supposeA27: x is Int_position ; ::_thesis: (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A28: not x in dom (Start-At (((card I) + 2),SCMPDS)) by A25, TARSKI:def_1;
A29: not x in dom (Start-At (0,SCMPDS)) by A27, SCMPDS_4:18;
thus (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . x = (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),2)) . x by A23
.= (Comput ((P +* (stop (if<>0 (a,k1,I)))),(Initialize s),1)) . x by A13, A27, SCMPDS_2:54
.= (Initialize s) . x by A8, A27, SCMPDS_2:55
.= s . x by A29, FUNCT_4:11
.= (s +* (Start-At (((card I) + 2),SCMPDS))) . x by A28, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA30: x = IC ; ::_thesis: (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1
hence (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . x = (card I) + 2 by A14, A23
.= (s +* (Start-At (((card I) + 2),SCMPDS))) . x by A30, FUNCT_4:113 ;
::_thesis: verum
end;
end;
end;
dom (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom (s +* (Start-At (((card I) + 2),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS)) by A24, FUNCT_1:2; ::_thesis: verum
end;

registration
let I be parahalting shiftable Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if<>0 (a,k1,I) ->  parahalting shiftable ;
correctness
coherence
( if<>0 (a,k1,I) is shiftable & if<>0 (a,k1,I) is parahalting );
proof
set i = (a,k1) <>0_goto 2;
set j = goto ((card I) + 1);
set IF = if<>0 (a,k1,I);
set pIF = stop (if<>0 (a,k1,I));
thus  if<>0 (a,k1,I) is shiftable ; ::_thesis: if<>0 (a,k1,I) is parahalting
let s be 0 -started State of SCMPDS; :: according to SCMPDS_4:def_7 ::_thesis: for b1 being set holds
( not stop (if<>0 (a,k1,I)) c= b1 or b1 halts_on s )
let P be Instruction-Sequence of SCMPDS; ::_thesis: ( not stop (if<>0 (a,k1,I)) c= P or P halts_on s )
A1: Initialize s = s by MEMSTR_0:44;
assume  stop (if<>0 (a,k1,I)) c= P ; ::_thesis: P halts_on s
then A2: P = P +* (stop (if<>0 (a,k1,I))) by FUNCT_4:98;
A3: ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
percases ( s . (DataLoc ((s . a),k1)) <> 0 or s . (DataLoc ((s . a),k1)) = 0 ) ;
suppose s . (DataLoc ((s . a),k1)) <> 0 ; ::_thesis: P halts_on s
then  if<>0 (a,k1,I) is_halting_on s,P by A3, Th58;
hence  P halts_on s by A1, A2, Def3; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),k1)) = 0 ; ::_thesis: P halts_on s
then  if<>0 (a,k1,I) is_halting_on s,P by Th59;
hence  P halts_on s by A1, A2, Def3; ::_thesis: verum
end;
end;
end;
end;

registration
let I be halt-free Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if<>0 (a,k1,I) ->  halt-free ;
coherence
 if<>0 (a,k1,I) is halt-free ;
end;

theorem :: SCMPDS_6:62
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) = (card I) + 2
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) = (card I) + 2
let s be State of SCMPDS; ::_thesis: for I being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) = (card I) + 2
let I be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer holds IC (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) = (card I) + 2
let a be Int_position; ::_thesis: for k1 being Integer holds IC (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) = (card I) + 2
let k1 be Integer; ::_thesis: IC (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) = (card I) + 2
set IF = if<>0 (a,k1,I);
A1: ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
percases ( s . (DataLoc ((s . a),k1)) <> 0 or s . (DataLoc ((s . a),k1)) = 0 ) ;
suppose s . (DataLoc ((s . a),k1)) <> 0 ; ::_thesis: IC (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) = (card I) + 2
then  IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS)) by A1, Th60;
hence  IC (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) = (card I) + 2 by FUNCT_4:113; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),k1)) = 0 ; ::_thesis: IC (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) = (card I) + 2
then  IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS)) by Th61;
hence  IC (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) = (card I) + 2 by FUNCT_4:113; ::_thesis: verum
end;
end;
end;

theorem :: SCMPDS_6:63
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let s be State of SCMPDS; ::_thesis: for I being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let I be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) <> 0 implies (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b )
assume A1: s . (DataLoc ((s . a),k1)) <> 0 ; ::_thesis: (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
A2: not b in dom (Start-At (((card I) + 2),SCMPDS)) by SCMPDS_4:18;
 ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
then  IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS)) by A1, Th60;
hence  (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b by A2, FUNCT_4:11; ::_thesis: verum
end;

theorem :: SCMPDS_6:64
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
(IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b = s . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
(IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b = s . b
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
(IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b = s . b
let I be Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
(IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b = s . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) = 0 holds
(IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b = s . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) = 0 implies (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b = s . b )
assume  s . (DataLoc ((s . a),k1)) = 0 ; ::_thesis: (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b = s . b
then A1: IExec ((if<>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS)) by Th61;
 not b in dom (Start-At (((card I) + 2),SCMPDS)) by SCMPDS_4:18;
hence  (IExec ((if<>0 (a,k1,I)),P,(Initialize s))) . b = s . b by A1, FUNCT_4:11; ::_thesis: verum
end;

begin

theorem :: SCMPDS_6:65
for a being Int_position
for k1 being Integer
for I, J being Program of SCMPDS holds card (if>0 (a,k1,I,J)) = ((card I) + (card J)) + 2 by Lm4;

theorem :: SCMPDS_6:66
for a being Int_position
for k1 being Integer
for I, J being Program of SCMPDS holds
( 0 in dom (if>0 (a,k1,I,J)) & 1 in dom (if>0 (a,k1,I,J)) )
proof
let a be Int_position; ::_thesis: for k1 being Integer
for I, J being Program of SCMPDS holds
( 0 in dom (if>0 (a,k1,I,J)) & 1 in dom (if>0 (a,k1,I,J)) )
let k1 be Integer; ::_thesis: for I, J being Program of SCMPDS holds
( 0 in dom (if>0 (a,k1,I,J)) & 1 in dom (if>0 (a,k1,I,J)) )
let I, J be Program of SCMPDS; ::_thesis: ( 0 in dom (if>0 (a,k1,I,J)) & 1 in dom (if>0 (a,k1,I,J)) )
set ci = card (if>0 (a,k1,I,J));
 card (if>0 (a,k1,I,J)) = ((card I) + (card J)) + 2 by Lm4;
then  2 <= card (if>0 (a,k1,I,J)) by NAT_1:12;
then  1 < card (if>0 (a,k1,I,J)) by XXREAL_0:2;
hence  ( 0 in dom (if>0 (a,k1,I,J)) & 1 in dom (if>0 (a,k1,I,J)) ) by AFINSQ_1:66; ::_thesis: verum
end;

theorem :: SCMPDS_6:67
for a being Int_position
for k1 being Integer
for I, J being Program of SCMPDS holds (if>0 (a,k1,I,J)) . 0 = (a,k1) <=0_goto ((card I) + 2) by Lm5;

theorem Th68: :: SCMPDS_6:68
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I, J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I, J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P )
let s be 0 -started State of SCMPDS; ::_thesis: for I, J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P )
let I, J be shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P )
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P )
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P implies ( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
set G = Goto ((card J) + 1);
set I2 = (I ';' (Goto ((card J) + 1))) ';' J;
set IF = if>0 (a,k1,I,J);
set pIF = stop (if>0 (a,k1,I,J));
set pI2 = stop ((I ';' (Goto ((card J) + 1))) ';' J);
set s2 = Initialize s;
set P2 = P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J));
set P3 = P +* (stop (if>0 (a,k1,I,J)));
set s4 = Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1);
set P4 = P +* (stop (if>0 (a,k1,I,J)));
set i = (a,k1) <=0_goto ((card I) + 2);
A1: 0 in dom (stop (if>0 (a,k1,I,J))) by COMPOS_1:36;
A2: Initialize s = s by MEMSTR_0:44;
then A3: IC s = 0 by MEMSTR_0:47;
A4: if>0 (a,k1,I,J) = (((a,k1) <=0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J by SCMPDS_4:14
.= ((a,k1) <=0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J) by SCMPDS_4:14 ;
then A5: Shift ((stop ((I ';' (Goto ((card J) + 1))) ';' J)),1) c= P +* (stop (if>0 (a,k1,I,J))) by Lm6;
A6: Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,(0 + 1)) = Following ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,0))) by EXTPRO_1:3
.= Following ((P +* (stop (if>0 (a,k1,I,J)))),s) by EXTPRO_1:2
.= Exec (((a,k1) <=0_goto ((card I) + 2)),s) by A4, Th11, A2 ;
 for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)) . a by A6, A2, SCMPDS_2:56;
then A7: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)) by SCMPDS_4:8;
assume  s . (DataLoc ((s . a),k1)) > 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or ( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P ) )
then A8: IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)) = succ (IC s) by A6, SCMPDS_2:56
.= 0 + 1 by A3 ;
assume A9: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or ( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P ) )
assume A10: I is_halting_on s,P ; ::_thesis: ( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P )
then A11: (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P by A9, Th30;
then A12: ( Start-At (0,SCMPDS) c= Initialize s & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on Initialize s,P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) ) by Th24, FUNCT_4:25;
A13: stop ((I ';' (Goto ((card J) + 1))) ';' J) c= P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by FUNCT_4:25;
 (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P by A9, A10, Th30;
then A14: P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on Initialize s by Def3;
A15: card (stop (if>0 (a,k1,I,J))) = (card (if>0 (a,k1,I,J))) + 1 by COMPOS_1:55
.= ((card ((I ';' (Goto ((card J) + 1))) ';' J)) + 1) + 1 by A4, Th6 ;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(if>0_(a,k1,I,J)))),s,k))_in_dom_(stop_(if>0_(a,k1,I,J)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,b1)) in dom (stop (if>0 (a,k1,I,J)))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,b1)) in dom (stop (if>0 (a,k1,I,J)))
then consider k1 being Nat such that
A16: k1 + 1 = k by NAT_1:6;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def_12;
reconsider m = IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Initialize s),k1)) as Element of NAT ;
A17: card (stop (if>0 (a,k1,I,J))) = (card (stop ((I ';' (Goto ((card J) + 1))) ';' J))) + 1 by A15, COMPOS_1:55;
 m in dom (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by A11, Def2;
then  m < card (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by AFINSQ_1:66;
then A18: m + 1 < card (stop (if>0 (a,k1,I,J))) by A17, XREAL_1:6;
IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,k)) = IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)),k1)) by A16, EXTPRO_1:4
.= m + 1 by A12, A5, A8, A7, Th31, A13 ;
hence  IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,k)) in dom (stop (if>0 (a,k1,I,J))) by A18, AFINSQ_1:66; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,b1)) in dom (stop (if>0 (a,k1,I,J)))
hence  IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,k)) in dom (stop (if>0 (a,k1,I,J))) by A1, A3, EXTPRO_1:2; ::_thesis: verum
end;
end;
end;
hence  if>0 (a,k1,I,J) is_closed_on s,P by Def2, A2; ::_thesis: if>0 (a,k1,I,J) is_halting_on s,P
A19: Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Initialize s))) + 1)) = Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Initialize s)))) by EXTPRO_1:4;
CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Initialize s)))))) by A19
.= CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Initialize s),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Initialize s)))))) by A12, A5, A8, A7, Th31, A13
.= halt SCMPDS by A14, EXTPRO_1:def_15 ;
then  P +* (stop (if>0 (a,k1,I,J))) halts_on s by EXTPRO_1:29;
hence  if>0 (a,k1,I,J) is_halting_on s,P by Def3, A2; ::_thesis: verum
end;

theorem Th69: :: SCMPDS_6:69
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P )
let I be Program of SCMPDS; ::_thesis: for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P )
let J be shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P )
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P )
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) <= 0 & J is_closed_on s,P & J is_halting_on s,P implies ( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
set pJ = stop J;
set s1 = Initialize s;
set P1 = P +* (stop J);
set IF = if>0 (a,k1,I,J);
set pIF = stop (if>0 (a,k1,I,J));
set s3 = Initialize s;
set P3 = P +* (stop (if>0 (a,k1,I,J)));
set s4 = Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s),1);
set P4 = P +* (stop (if>0 (a,k1,I,J)));
set i = (a,k1) <=0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
set iG = (((a,k1) <=0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1));
A1: if>0 (a,k1,I,J) = (((a,k1) <=0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J by SCMPDS_4:14
.= ((a,k1) <=0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J) by SCMPDS_4:14 ;
A2: Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <=0_goto ((card I) + 2)),(Initialize s)) by A1, Th11 ;
A3: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A4: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A3, FUNCT_4:11 ;
A5: IC (Initialize s) = 0 by MEMSTR_0:47;
assume  s . (DataLoc ((s . a),k1)) <= 0 ; ::_thesis: ( not J is_closed_on s,P or not J is_halting_on s,P or ( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P ) )
then A6: IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s),1)) = ICplusConst ((Initialize s),((card I) + 2)) by A2, A4, SCMPDS_2:56
.= 0 + ((card I) + 2) by A5, Th12 ;
assume A7: J is_closed_on s,P ; ::_thesis: ( not J is_halting_on s,P or ( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P ) )
then A8: ( Start-At (0,SCMPDS) c= Initialize s & J is_closed_on Initialize s,P +* (stop J) ) by Th24, FUNCT_4:25;
A9: stop J c= P +* (stop J) by FUNCT_4:25;
A10: stop (if>0 (a,k1,I,J)) c= P +* (stop (if>0 (a,k1,I,J))) by FUNCT_4:25;
A11: card ((((a,k1) <=0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1))) = (card (((a,k1) <=0_goto ((card I) + 2)) ';' I)) + (card (Goto ((card J) + 1))) by AFINSQ_1:17
.= (card (((a,k1) <=0_goto ((card I) + 2)) ';' I)) + 1 by COMPOS_1:54
.= ((card I) + 1) + 1 by Th6
.= (card I) + (1 + 1) ;
then  Shift ((stop J),((card I) + 2)) c= stop (if>0 (a,k1,I,J)) by Th13;
then A12: Shift ((stop J),((card I) + 2)) c= P +* (stop (if>0 (a,k1,I,J))) by A10, XBOOLE_1:1;
assume  J is_halting_on s,P ; ::_thesis: ( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P )
then A13: P +* (stop J) halts_on Initialize s by Def3;
 for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s),1)) . a by A2, SCMPDS_2:56;
then A14: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s),1)) by SCMPDS_4:8;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(if>0_(a,k1,I,J)))),(Initialize_s),k))_in_dom_(stop_(if>0_(a,k1,I,J)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s),b1)) in dom (stop (if>0 (a,k1,I,J)))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s),b1)) in dom (stop (if>0 (a,k1,I,J)))
then consider k1 being Nat such that
A15: k1 + 1 = k by NAT_1:6;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def_12;
reconsider m = IC (Comput ((P +* (stop J)),(Initialize s),k1)) as Element of NAT ;
 m in dom (stop J) by A7, Def2;
then  m < card (stop J) by AFINSQ_1:66;
then A16: m + ((card I) + 2) < (card (stop J)) + ((card I) + 2) by XREAL_1:6;
A17: card (stop J) = (card J) + 1 by COMPOS_1:55;
A18: card (stop (if>0 (a,k1,I,J))) = (card (if>0 (a,k1,I,J))) + 1 by COMPOS_1:55
.= (((card I) + 2) + (card J)) + 1 by A11, AFINSQ_1:17
.= ((card I) + 2) + (card (stop J)) by A17 ;
IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s),k)) = IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s),1)),k1)) by A15, EXTPRO_1:4
.= m + ((card I) + 2) by A8, A14, A12, A6, Th31, A9 ;
hence  IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s),k)) in dom (stop (if>0 (a,k1,I,J))) by A18, A16, AFINSQ_1:66; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s),b1)) in dom (stop (if>0 (a,k1,I,J)))
then  Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s),k) = Initialize s by EXTPRO_1:2;
hence  IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s),k)) in dom (stop (if>0 (a,k1,I,J))) by A5, COMPOS_1:36; ::_thesis: verum
end;
end;
end;
hence  if>0 (a,k1,I,J) is_closed_on s,P by Def2; ::_thesis: if>0 (a,k1,I,J) is_halting_on s,P
A19: Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1)) = Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s),1)),(LifeSpan ((P +* (stop J)),(Initialize s)))) by EXTPRO_1:4;
CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Initialize s),1)),(LifeSpan ((P +* (stop J)),(Initialize s)))))) by A19
.= CurInstr ((P +* (stop J)),(Comput ((P +* (stop J)),(Initialize s),(LifeSpan ((P +* (stop J)),(Initialize s)))))) by A8, A14, A12, A6, Th31, A9
.= halt SCMPDS by A13, EXTPRO_1:def_15 ;
then  P +* (stop (if>0 (a,k1,I,J))) halts_on Initialize s by EXTPRO_1:29;
hence  if>0 (a,k1,I,J) is_halting_on s,P by Def3; ::_thesis: verum
end;

theorem Th70: :: SCMPDS_6:70
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let J be shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P implies IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set G = Goto ((card J) + 1);
set I2 = (I ';' (Goto ((card J) + 1))) ';' J;
set IF = if>0 (a,k1,I,J);
set pIF = stop (if>0 (a,k1,I,J));
set pI2 = stop ((I ';' (Goto ((card J) + 1))) ';' J);
set P2 = P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J));
set P3 = P +* (stop (if>0 (a,k1,I,J)));
set s4 = Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1);
set P4 = P +* (stop (if>0 (a,k1,I,J)));
set i = (a,k1) <=0_goto ((card I) + 2);
set SAl = Start-At ((((card I) + (card J)) + 2),SCMPDS);
A1: Initialize s = s by MEMSTR_0:44;
then A2: IC s = 0 by MEMSTR_0:47;
A3: if>0 (a,k1,I,J) = (((a,k1) <=0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J by SCMPDS_4:14
.= ((a,k1) <=0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J) by SCMPDS_4:14 ;
then A4: Shift ((stop ((I ';' (Goto ((card J) + 1))) ';' J)),1) c= P +* (stop (if>0 (a,k1,I,J))) by Lm6;
A5: Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,(0 + 1)) = Following ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,0))) by EXTPRO_1:3
.= Following ((P +* (stop (if>0 (a,k1,I,J)))),s) by EXTPRO_1:2
.= Exec (((a,k1) <=0_goto ((card I) + 2)),s) by A3, Th11, A1 ;
assume  s . (DataLoc ((s . a),k1)) > 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
then A6: IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)) = succ (IC s) by A5, SCMPDS_2:56
.= 0 + 1 by A2 ;
 for a being Int_position holds s . a = (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)) . a by A5, SCMPDS_2:56;
then A7: DataPart s = DataPart (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)) by SCMPDS_4:8;
assume A8: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
assume A9: I is_halting_on s,P ; ::_thesis: IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
then  (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P by A8, Th30;
then A10: P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s by Def3, A1;
 (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P by A8, A9, Th30;
then A11: ( Start-At (0,SCMPDS) c= s & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) ) by Th24, A1, FUNCT_4:25;
A12: stop ((I ';' (Goto ((card J) + 1))) ';' J) c= P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by FUNCT_4:25;
A13: Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1)) = Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))) by EXTPRO_1:4;
A14: CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1)))) = CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) by A13
.= CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) by A11, A4, A6, A7, Th31, A12
.= halt SCMPDS by A10, EXTPRO_1:def_15 ;
then A15: P +* (stop (if>0 (a,k1,I,J))) halts_on s by EXTPRO_1:29;
A16: CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),s) = (a,k1) <=0_goto ((card I) + 2) by A3, Th11, A1;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_(LifeSpan_((P_+*_(stop_((I_';'_(Goto_((card_J)_+_1)))_';'_J))),s))_+_1_holds_
CurInstr_((P_+*_(stop_(if>0_(a,k1,I,J)))),(Comput_((P_+*_(stop_(if>0_(a,k1,I,J)))),s,l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 implies CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,b1))) <> halt SCMPDS )
assume A17: l < (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 ; ::_thesis: CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,b1))) <> halt SCMPDS
A18: Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,0) = s by EXTPRO_1:2;
percases ( l = 0 or l <> 0 ) ;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,b1))) <> halt SCMPDS
then  CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,l))) = CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),s) by A18;
hence  CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,l))) <> halt SCMPDS by A16; ::_thesis: verum
end;
suppose l <> 0 ; ::_thesis: not CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,b1))) = halt SCMPDS
then consider n being Nat such that
A19: l = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A20: n < LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) by A17, A19, XREAL_1:6;
assume A21: CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,l))) = halt SCMPDS ; ::_thesis: contradiction
A22: Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,(n + 1)) = Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)),n) by EXTPRO_1:4;
CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,n))) = CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)),n))) by A11, A4, A6, A7, Th31, A12
.= halt SCMPDS by A19, A21, A22 ;
hence  contradiction by A10, A20, EXTPRO_1:def_15; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,l))) = halt SCMPDS holds
(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 <= l ;
then A23: LifeSpan ((P +* (stop (if>0 (a,k1,I,J)))),s) = (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 by A14, A15, EXTPRO_1:def_15;
A24: DataPart (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) = DataPart (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)))) by A10, EXTPRO_1:23
.= DataPart (Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)))) by A11, A4, A6, A7, Th31, A12
.= DataPart (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (stop (if>0 (a,k1,I,J)))),s)) by A15, A23, EXTPRO_1:23 ;
A25: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if>0_(a,k1,I,J)),P,s))_holds_
(IExec_((if>0_(a,k1,I,J)),P,s))_._x_=_((IExec_(((I_';'_(Goto_((card_J)_+_1)))_';'_J),P,s))_+*_(Start-At_((((card_I)_+_(card_J))_+_2),SCMPDS)))_._x
let x be set ; ::_thesis: ( x in dom (IExec ((if>0 (a,k1,I,J)),P,s)) implies (IExec ((if>0 (a,k1,I,J)),P,s)) . b1 = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1 )
A26: dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A27: x in dom (IExec ((if>0 (a,k1,I,J)),P,s)) ; ::_thesis: (IExec ((if>0 (a,k1,I,J)),P,s)) . b1 = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A27, SCMPDS_4:6;
supposeA28: x is Int_position ; ::_thesis: (IExec ((if>0 (a,k1,I,J)),P,s)) . b1 = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A29: not x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A26, TARSKI:def_1;
thus (IExec ((if>0 (a,k1,I,J)),P,s)) . x = (Result ((P +* (stop (if>0 (a,k1,I,J)))),s)) . x
.= (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) . x by A24, A28, SCMPDS_4:8
.= (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) . x
.= ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by A29, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA30: x = IC ; ::_thesis: (IExec ((if>0 (a,k1,I,J)),P,s)) . b1 = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1
A31: IC (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) = IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s))
.= ((card I) + (card J)) + 1 by A8, A9, Th32 ;
A32: x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A26, A30, TARSKI:def_1;
thus (IExec ((if>0 (a,k1,I,J)),P,s)) . x = (Result ((P +* (stop (if>0 (a,k1,I,J)))),s)) . x
.= (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1))) . x by A15, A23, EXTPRO_1:23
.= IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)))) by A30, EXTPRO_1:4
.= (IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) + 1 by A11, A4, A6, A7, Th31, A12
.= (IC (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))) + 1 by A10, EXTPRO_1:23
.= IC (Start-At (((((card I) + (card J)) + 1) + 1),SCMPDS)) by A31, FUNCOP_1:72
.= ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by A30, A32, FUNCT_4:13 ; ::_thesis: verum
end;
end;
end;
dom (IExec ((if>0 (a,k1,I,J)),P,s)) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A25, FUNCT_1:2
.= ((IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A8, A9, Th33
.= (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by MEMSTR_0:36 ;
::_thesis: verum
end;

theorem Th71: :: SCMPDS_6:71
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for J being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for J being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let s be 0 -started State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for J being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let I be Program of SCMPDS; ::_thesis: for J being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let J be halt-free shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) <= 0 & J is_closed_on s,P & J is_halting_on s,P implies IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
A1: Initialize s = s by MEMSTR_0:44;
set b = DataLoc ((s . a),k1);
set pJ = stop J;
set s1 = s;
set P1 = P +* (stop J);
set IF = if>0 (a,k1,I,J);
set pIF = stop (if>0 (a,k1,I,J));
set s3 = s;
set P3 = P +* (stop (if>0 (a,k1,I,J)));
set s4 = Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1);
set P4 = P +* (stop (if>0 (a,k1,I,J)));
set i = (a,k1) <=0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
set iG = (((a,k1) <=0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1));
set SAl = Start-At ((((card I) + (card J)) + 2),SCMPDS);
A2: if>0 (a,k1,I,J) = (((a,k1) <=0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J by SCMPDS_4:14
.= ((a,k1) <=0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J) by SCMPDS_4:14 ;
A3: Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,(0 + 1)) = Following ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,0))) by EXTPRO_1:3
.= Following ((P +* (stop (if>0 (a,k1,I,J)))),s) by EXTPRO_1:2
.= Exec (((a,k1) <=0_goto ((card I) + 2)),s) by A2, Th11, A1 ;
A4: IC s = 0 by A1, MEMSTR_0:47;
assume  s . (DataLoc ((s . a),k1)) <= 0 ; ::_thesis: ( not J is_closed_on s,P or not J is_halting_on s,P or IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
then A5: IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)) = ICplusConst (s,((card I) + 2)) by A3, SCMPDS_2:56
.= 0 + ((card I) + 2) by A4, Th12 ;
 for a being Int_position holds s . a = (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)) . a by A3, SCMPDS_2:56;
then A6: DataPart s = DataPart (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)) by SCMPDS_4:8;
card ((((a,k1) <=0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1))) = (card (((a,k1) <=0_goto ((card I) + 2)) ';' I)) + (card (Goto ((card J) + 1))) by AFINSQ_1:17
.= (card (((a,k1) <=0_goto ((card I) + 2)) ';' I)) + 1 by COMPOS_1:54
.= ((card I) + 1) + 1 by Th6
.= (card I) + (1 + 1) ;
then A7: Shift ((stop J),((card I) + 2)) c= stop (if>0 (a,k1,I,J)) by Th13;
 stop (if>0 (a,k1,I,J)) c= P +* (stop (if>0 (a,k1,I,J))) by FUNCT_4:25;
then A8: Shift ((stop J),((card I) + 2)) c= P +* (stop (if>0 (a,k1,I,J))) by A7, XBOOLE_1:1;
assume A9: J is_closed_on s,P ; ::_thesis: ( not J is_halting_on s,P or IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
then A10: ( Start-At (0,SCMPDS) c= s & J is_closed_on s,P +* (stop J) ) by Th24, A1, FUNCT_4:25;
A11: stop J c= P +* (stop J) by FUNCT_4:25;
assume A12: J is_halting_on s,P ; ::_thesis: IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
then A13: P +* (stop J) halts_on s by Def3, A1;
A14: Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop J)),s)) + 1)) = Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop J)),s))) by EXTPRO_1:4;
A15: CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop J)),s)) + 1)))) = CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop J)),s))))) by A14
.= CurInstr ((P +* (stop J)),(Comput ((P +* (stop J)),s,(LifeSpan ((P +* (stop J)),s))))) by A10, A8, A5, A6, Th31, A11
.= halt SCMPDS by A13, EXTPRO_1:def_15 ;
then A16: P +* (stop (if>0 (a,k1,I,J))) halts_on s by EXTPRO_1:29;
A17: CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),s) = (a,k1) <=0_goto ((card I) + 2) by A2, Th11, A1;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_(LifeSpan_((P_+*_(stop_J)),s))_+_1_holds_
CurInstr_((P_+*_(stop_(if>0_(a,k1,I,J)))),(Comput_((P_+*_(stop_(if>0_(a,k1,I,J)))),s,l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < (LifeSpan ((P +* (stop J)),s)) + 1 implies CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,b1))) <> halt SCMPDS )
assume A18: l < (LifeSpan ((P +* (stop J)),s)) + 1 ; ::_thesis: CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,b1))) <> halt SCMPDS
A19: Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,0) = s by EXTPRO_1:2;
percases ( l = 0 or l <> 0 ) ;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,b1))) <> halt SCMPDS
then  CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,l))) = CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),s) by A19;
hence  CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,l))) <> halt SCMPDS by A17; ::_thesis: verum
end;
suppose l <> 0 ; ::_thesis: not CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,b1))) = halt SCMPDS
then consider n being Nat such that
A20: l = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A21: n < LifeSpan ((P +* (stop J)),s) by A18, A20, XREAL_1:6;
assume A22: CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,l))) = halt SCMPDS ; ::_thesis: contradiction
A23: Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,(n + 1)) = Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)),n) by EXTPRO_1:4;
CurInstr ((P +* (stop J)),(Comput ((P +* (stop J)),s,n))) = CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)),n))) by A10, A8, A5, A6, Th31, A11
.= halt SCMPDS by A20, A22, A23 ;
hence  contradiction by A13, A21, EXTPRO_1:def_15; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,l))) = halt SCMPDS holds
(LifeSpan ((P +* (stop J)),s)) + 1 <= l ;
then A24: LifeSpan ((P +* (stop (if>0 (a,k1,I,J)))),s) = (LifeSpan ((P +* (stop J)),s)) + 1 by A15, A16, EXTPRO_1:def_15;
A25: DataPart (Result ((P +* (stop J)),s)) = DataPart (Comput ((P +* (stop J)),s,(LifeSpan ((P +* (stop J)),s)))) by A13, EXTPRO_1:23
.= DataPart (Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop J)),s)))) by A10, A8, A5, A6, Th31, A11
.= DataPart (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop J)),s)) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (stop (if>0 (a,k1,I,J)))),s)) by A16, A24, EXTPRO_1:23 ;
A26: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if>0_(a,k1,I,J)),P,s))_holds_
(IExec_((if>0_(a,k1,I,J)),P,s))_._x_=_((IExec_(J,P,s))_+*_(Start-At_((((card_I)_+_(card_J))_+_2),SCMPDS)))_._x
let x be set ; ::_thesis: ( x in dom (IExec ((if>0 (a,k1,I,J)),P,s)) implies (IExec ((if>0 (a,k1,I,J)),P,s)) . b1 = ((IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1 )
A27: dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A28: x in dom (IExec ((if>0 (a,k1,I,J)),P,s)) ; ::_thesis: (IExec ((if>0 (a,k1,I,J)),P,s)) . b1 = ((IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A28, SCMPDS_4:6;
supposeA29: x is Int_position ; ::_thesis: (IExec ((if>0 (a,k1,I,J)),P,s)) . b1 = ((IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A30: not x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A27, TARSKI:def_1;
thus (IExec ((if>0 (a,k1,I,J)),P,s)) . x = (Result ((P +* (stop (if>0 (a,k1,I,J)))),s)) . x
.= (Result ((P +* (stop J)),s)) . x by A25, A29, SCMPDS_4:8
.= (IExec (J,P,s)) . x
.= ((IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by A30, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA31: x = IC ; ::_thesis: (IExec ((if>0 (a,k1,I,J)),P,s)) . b1 = ((IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1
A32: IC (Result ((P +* (stop J)),s)) = IC (IExec (J,P,s))
.= card J by A9, A12, Th34, A1 ;
A33: x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A27, A31, TARSKI:def_1;
thus (IExec ((if>0 (a,k1,I,J)),P,s)) . x = (Result ((P +* (stop (if>0 (a,k1,I,J)))),s)) . x
.= (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop J)),s)) + 1))) . x by A16, A24, EXTPRO_1:23
.= IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop J)),s)))) by A31, EXTPRO_1:4
.= (IC (Comput ((P +* (stop J)),s,(LifeSpan ((P +* (stop J)),s))))) + ((card I) + 2) by A10, A8, A5, A6, Th31, A11
.= (IC (Result ((P +* (stop J)),s))) + ((card I) + 2) by A13, EXTPRO_1:23
.= IC (Start-At (((card J) + ((card I) + 2)),SCMPDS)) by A32, FUNCOP_1:72
.= ((IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by A31, A33, FUNCT_4:13 ; ::_thesis: verum
end;
end;
end;
dom (IExec ((if>0 (a,k1,I,J)),P,s)) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom ((IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A26, FUNCT_1:2; ::_thesis: verum
end;

registration
let I, J be parahalting shiftable Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if>0 (a,k1,I,J) ->  parahalting shiftable ;
correctness
coherence
( if>0 (a,k1,I,J) is shiftable & if>0 (a,k1,I,J) is parahalting );
proof
set i = (a,k1) <=0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
set IF = if>0 (a,k1,I,J);
set pIF = stop (if>0 (a,k1,I,J));
reconsider IJ = (I ';' (Goto ((card J) + 1))) ';' J as shiftable Program of SCMPDS ;
thus  if>0 (a,k1,I,J) is shiftable ; ::_thesis: if>0 (a,k1,I,J) is parahalting
let s be 0 -started State of SCMPDS; :: according to SCMPDS_4:def_7 ::_thesis: for b1 being set holds
( not stop (if>0 (a,k1,I,J)) c= b1 or b1 halts_on s )
let P be Instruction-Sequence of SCMPDS; ::_thesis: ( not stop (if>0 (a,k1,I,J)) c= P or P halts_on s )
A1: Initialize s = s by MEMSTR_0:44;
assume  stop (if>0 (a,k1,I,J)) c= P ; ::_thesis: P halts_on s
then A2: P = P +* (stop (if>0 (a,k1,I,J))) by FUNCT_4:98;
A3: ( J is_closed_on s,P & J is_halting_on s,P ) by Th20, Th21;
A4: ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
percases ( s . (DataLoc ((s . a),k1)) > 0 or s . (DataLoc ((s . a),k1)) <= 0 ) ;
suppose s . (DataLoc ((s . a),k1)) > 0 ; ::_thesis: P halts_on s
then  if>0 (a,k1,I,J) is_halting_on s,P by A4, Th68;
hence  P halts_on s by A1, A2, Def3; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),k1)) <= 0 ; ::_thesis: P halts_on s
then  if>0 (a,k1,I,J) is_halting_on s,P by A3, Th69;
hence  P halts_on s by A1, A2, Def3; ::_thesis: verum
end;
end;
end;
end;

registration
let I, J be halt-free Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if>0 (a,k1,I,J) ->  halt-free ;
coherence
 if>0 (a,k1,I,J) is halt-free ;
end;

theorem :: SCMPDS_6:72
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I, J being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if>0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I, J being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if>0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
let s be 0 -started State of SCMPDS; ::_thesis: for I, J being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if>0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
let I, J be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer holds IC (IExec ((if>0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
let a be Int_position; ::_thesis: for k1 being Integer holds IC (IExec ((if>0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
let k1 be Integer; ::_thesis: IC (IExec ((if>0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
set IF = if>0 (a,k1,I,J);
A1: ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
A2: ( J is_closed_on s,P & J is_halting_on s,P ) by Th20, Th21;
percases ( s . (DataLoc ((s . a),k1)) > 0 or s . (DataLoc ((s . a),k1)) <= 0 ) ;
suppose s . (DataLoc ((s . a),k1)) > 0 ; ::_thesis: IC (IExec ((if>0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
then  IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A1, Th70;
hence  IC (IExec ((if>0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2 by FUNCT_4:113; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),k1)) <= 0 ; ::_thesis: IC (IExec ((if>0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
then  IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A2, Th71;
hence  IC (IExec ((if>0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2 by FUNCT_4:113; ::_thesis: verum
end;
end;
end;

theorem :: SCMPDS_6:73
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free parahalting shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let I be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let J be shiftable Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) > 0 implies (IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b )
assume A1: s . (DataLoc ((s . a),k1)) > 0 ; ::_thesis: (IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
A2: not b in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by SCMPDS_4:18;
 ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
then  IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A1, Th70;
hence  (IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b by A2, FUNCT_4:11; ::_thesis: verum
end;

theorem :: SCMPDS_6:74
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for J being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (J,P,s)) . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being Program of SCMPDS
for J being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (J,P,s)) . b
let s be 0 -started State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for J being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (J,P,s)) . b
let I be Program of SCMPDS; ::_thesis: for J being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (J,P,s)) . b
let J be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (J,P,s)) . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (J,P,s)) . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) <= 0 implies (IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (J,P,s)) . b )
set IF = if>0 (a,k1,I,J);
assume A1: s . (DataLoc ((s . a),k1)) <= 0 ; ::_thesis: (IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (J,P,s)) . b
A2: not b in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by SCMPDS_4:18;
 ( J is_closed_on s,P & J is_halting_on s,P ) by Th20, Th21;
then  IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A1, Th71;
hence  (IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (J,P,s)) . b by A2, FUNCT_4:11; ::_thesis: verum
end;

begin

theorem :: SCMPDS_6:75
for a being Int_position
for k1 being Integer
for I being Program of SCMPDS holds card (if>0 (a,k1,I)) = (card I) + 1 by Th6;

theorem :: SCMPDS_6:76
for a being Int_position
for k1 being Integer
for I being Program of SCMPDS holds 0 in dom (if>0 (a,k1,I))
proof
let a be Int_position; ::_thesis: for k1 being Integer
for I being Program of SCMPDS holds 0 in dom (if>0 (a,k1,I))
let k1 be Integer; ::_thesis: for I being Program of SCMPDS holds 0 in dom (if>0 (a,k1,I))
let I be Program of SCMPDS; ::_thesis: 0 in dom (if>0 (a,k1,I))
set ci = card (if>0 (a,k1,I));
 card (if>0 (a,k1,I)) = (card I) + 1 by Th6;
hence  0 in dom (if>0 (a,k1,I)) by AFINSQ_1:66; ::_thesis: verum
end;

theorem :: SCMPDS_6:77
for a being Int_position
for k1 being Integer
for I being Program of SCMPDS holds (if>0 (a,k1,I)) . 0 = (a,k1) <=0_goto ((card I) + 1) by Th7;

theorem Th78: :: SCMPDS_6:78
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>0 (a,k1,I) is_closed_on s,P & if>0 (a,k1,I) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>0 (a,k1,I) is_closed_on s,P & if>0 (a,k1,I) is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>0 (a,k1,I) is_closed_on s,P & if>0 (a,k1,I) is_halting_on s,P )
let I be shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>0 (a,k1,I) is_closed_on s,P & if>0 (a,k1,I) is_halting_on s,P )
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>0 (a,k1,I) is_closed_on s,P & if>0 (a,k1,I) is_halting_on s,P )
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P implies ( if>0 (a,k1,I) is_closed_on s,P & if>0 (a,k1,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
set IF = if>0 (a,k1,I);
set pIF = stop (if>0 (a,k1,I));
set pI = stop I;
set s2 = Initialize s;
set s3 = Initialize s;
set P2 = P +* (stop I);
set P3 = P +* (stop (if>0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if>0 (a,k1,I)));
set i = (a,k1) <=0_goto ((card I) + 1);
A1: 0 in dom (stop (if>0 (a,k1,I))) by COMPOS_1:36;
A2: IC (Initialize s) = 0 by MEMSTR_0:47;
A3: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A4: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A3, FUNCT_4:11 ;
A5: Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <=0_goto ((card I) + 1)),(Initialize s)) by Th11 ;
 for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)) . a by A5, SCMPDS_2:56;
then A6: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)) by SCMPDS_4:8;
assume  s . (DataLoc ((s . a),k1)) > 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or ( if>0 (a,k1,I) is_closed_on s,P & if>0 (a,k1,I) is_halting_on s,P ) )
then A7: IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)) = succ (IC (Initialize s)) by A5, A4, SCMPDS_2:56
.= 0 + 1 by A2 ;
assume A8: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or ( if>0 (a,k1,I) is_closed_on s,P & if>0 (a,k1,I) is_halting_on s,P ) )
then A9: I is_closed_on Initialize s,P +* (stop I) by Th24;
assume  I is_halting_on s,P ; ::_thesis: ( if>0 (a,k1,I) is_closed_on s,P & if>0 (a,k1,I) is_halting_on s,P )
then A10: P +* (stop I) halts_on Initialize s by Def3;
A11: ( Start-At (0,SCMPDS) c= Initialize s & Shift ((stop I),1) c= P +* (stop (if>0 (a,k1,I))) ) by Lm6, FUNCT_4:25;
A12: stop I c= P +* (stop I) by FUNCT_4:25;
A13: card (stop (if>0 (a,k1,I))) = (card (if>0 (a,k1,I))) + 1 by COMPOS_1:55
.= ((card I) + 1) + 1 by Th6 ;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(if>0_(a,k1,I)))),(Initialize_s),k))_in_dom_(stop_(if>0_(a,k1,I)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if>0 (a,k1,I)))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if>0 (a,k1,I)))
then consider k1 being Nat such that
A14: k1 + 1 = k by NAT_1:6;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def_12;
reconsider m = IC (Comput ((P +* (stop I)),(Initialize s),k1)) as Element of NAT ;
A15: card (stop (if>0 (a,k1,I))) = (card (stop I)) + 1 by A13, COMPOS_1:55;
 m in dom (stop I) by A8, Def2;
then  m < card (stop I) by AFINSQ_1:66;
then A16: m + 1 < card (stop (if>0 (a,k1,I))) by A15, XREAL_1:6;
IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),k)) = IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)),k1)) by A14, EXTPRO_1:4
.= m + 1 by A9, A11, A7, A6, Th31, A12 ;
hence  IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if>0 (a,k1,I))) by A16, AFINSQ_1:66; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if>0 (a,k1,I)))
hence  IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if>0 (a,k1,I))) by A1, A2, EXTPRO_1:2; ::_thesis: verum
end;
end;
end;
hence  if>0 (a,k1,I) is_closed_on s,P by Def2; ::_thesis: if>0 (a,k1,I) is_halting_on s,P
A17: Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)) = Comput ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))) by EXTPRO_1:4;
CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A17
.= CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A9, A11, A7, A6, Th31, A12
.= halt SCMPDS by A10, EXTPRO_1:def_15 ;
then  P +* (stop (if>0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
hence  if>0 (a,k1,I) is_halting_on s,P by Def3; ::_thesis: verum
end;

theorem Th79: :: SCMPDS_6:79
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
( if>0 (a,k1,I) is_closed_on s,P & if>0 (a,k1,I) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
( if>0 (a,k1,I) is_closed_on s,P & if>0 (a,k1,I) is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
( if>0 (a,k1,I) is_closed_on s,P & if>0 (a,k1,I) is_halting_on s,P )
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
( if>0 (a,k1,I) is_closed_on s,P & if>0 (a,k1,I) is_halting_on s,P )
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
( if>0 (a,k1,I) is_closed_on s,P & if>0 (a,k1,I) is_halting_on s,P )
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) <= 0 implies ( if>0 (a,k1,I) is_closed_on s,P & if>0 (a,k1,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
assume A1: s . (DataLoc ((s . a),k1)) <= 0 ; ::_thesis: ( if>0 (a,k1,I) is_closed_on s,P & if>0 (a,k1,I) is_halting_on s,P )
set i = (a,k1) <=0_goto ((card I) + 1);
set IF = if>0 (a,k1,I);
set pIF = stop (if>0 (a,k1,I));
set s3 = Initialize s;
set P3 = P +* (stop (if>0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if>0 (a,k1,I)));
A2: IC (Initialize s) = 0 by MEMSTR_0:47;
A3: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A4: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A3, FUNCT_4:11 ;
Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <=0_goto ((card I) + 1)),(Initialize s)) by Th11 ;
then A5: IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)) = ICplusConst ((Initialize s),((card I) + 1)) by A1, A4, SCMPDS_2:56
.= 0 + ((card I) + 1) by A2, Th12 ;
A6: card (if>0 (a,k1,I)) = (card I) + 1 by Th6;
then A7: (card I) + 1 in dom (stop (if>0 (a,k1,I))) by COMPOS_1:64;
A8: (P +* (stop (if>0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1))) = (P +* (stop (if>0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1))) by PBOOLE:143;
 stop (if>0 (a,k1,I)) c= P +* (stop (if>0 (a,k1,I))) by FUNCT_4:25;
then  stop (if>0 (a,k1,I)) c= P +* (stop (if>0 (a,k1,I))) ;
then (P +* (stop (if>0 (a,k1,I)))) . ((card I) + 1) = (stop (if>0 (a,k1,I))) . ((card I) + 1) by A7, GRFUNC_1:2
.= halt SCMPDS by A6, COMPOS_1:64 ;
then A9: CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1))) = halt SCMPDS by A5, A8;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(if>0_(a,k1,I)))),(Initialize_s),k))_in_dom_(stop_(if>0_(a,k1,I)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if>0 (a,k1,I)))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if>0 (a,k1,I)))
then  1 + 0 <= k by INT_1:7;
hence  IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if>0 (a,k1,I))) by A7, A5, A9, EXTPRO_1:5; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if>0 (a,k1,I)))
then  Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),k) = Initialize s by EXTPRO_1:2;
hence  IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if>0 (a,k1,I))) by A2, COMPOS_1:36; ::_thesis: verum
end;
end;
end;
hence  if>0 (a,k1,I) is_closed_on s,P by Def2; ::_thesis: if>0 (a,k1,I) is_halting_on s,P
 P +* (stop (if>0 (a,k1,I))) halts_on Initialize s by A9, EXTPRO_1:29;
hence  if>0 (a,k1,I) is_halting_on s,P by Def3; ::_thesis: verum
end;

theorem Th80: :: SCMPDS_6:80
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
let s be State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P implies IExec ((if>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set IF = if>0 (a,k1,I);
set pI = stop I;
set pIF = stop (if>0 (a,k1,I));
set s2 = Initialize s;
set s3 = Initialize s;
set P2 = P +* (stop I);
set P3 = P +* (stop (if>0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if>0 (a,k1,I)));
set i = (a,k1) <=0_goto ((card I) + 1);
set SAl = Start-At (((card I) + 1),SCMPDS);
A1: IC (Initialize s) = 0 by MEMSTR_0:47;
A2: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A3: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A2, FUNCT_4:11 ;
A4: Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <=0_goto ((card I) + 1)),(Initialize s)) by Th11 ;
assume  s . (DataLoc ((s . a),k1)) > 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or IExec ((if>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) )
then A5: IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)) = succ (IC (Initialize s)) by A4, A3, SCMPDS_2:56
.= 0 + 1 by A1 ;
 for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)) . a by A4, SCMPDS_2:56;
then A6: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)) by SCMPDS_4:8;
assume A7: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or IExec ((if>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) )
then A8: I is_closed_on Initialize s,P +* (stop I) by Th24;
A9: ( Start-At (0,SCMPDS) c= Initialize s & Shift ((stop I),1) c= P +* (stop (if>0 (a,k1,I))) ) by Lm6, FUNCT_4:25;
A10: stop I c= P +* (stop I) by FUNCT_4:25;
assume A11: I is_halting_on s,P ; ::_thesis: IExec ((if>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
then A12: P +* (stop I) halts_on Initialize s by Def3;
A13: Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)) = Comput ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))) by EXTPRO_1:4;
A14: CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A13
.= CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A8, A9, A5, A6, Th31, A10
.= halt SCMPDS by A12, EXTPRO_1:def_15 ;
then A15: P +* (stop (if>0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
A16: CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) = (a,k1) <=0_goto ((card I) + 1) by Th11;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_(LifeSpan_((P_+*_(stop_I)),(Initialize_s)))_+_1_holds_
CurInstr_((P_+*_(stop_(if>0_(a,k1,I)))),(Comput_((P_+*_(stop_(if>0_(a,k1,I)))),(Initialize_s),l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 implies CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS )
assume A17: l < (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 ; ::_thesis: CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
A18: Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),0) = Initialize s by EXTPRO_1:2;
percases ( l = 0 or l <> 0 ) ;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
then  CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) = CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) by A18;
hence  CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A16; ::_thesis: verum
end;
suppose l <> 0 ; ::_thesis: not CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),b1))) = halt SCMPDS
then consider n being Nat such that
A19: l = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A20: n < LifeSpan ((P +* (stop I)),(Initialize s)) by A17, A19, XREAL_1:6;
assume A21: CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS ; ::_thesis: contradiction
A22: Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),(n + 1)) = Comput ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)),n) by EXTPRO_1:4;
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),n))) = CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)),n))) by A8, A9, A5, A6, Th31, A10
.= halt SCMPDS by A19, A21, A22 ;
hence  contradiction by A12, A20, EXTPRO_1:def_15; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS holds
(LifeSpan ((P +* (stop I)),(Initialize s))) + 1 <= l ;
then A23: LifeSpan ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) = (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 by A14, A15, EXTPRO_1:def_15;
A24: DataPart (Result ((P +* (stop I)),(Initialize s))) = DataPart (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A12, EXTPRO_1:23
.= DataPart (Comput ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A8, A9, A5, A6, Th31, A10
.= DataPart (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (stop (if>0 (a,k1,I)))),(Initialize s))) by A15, A23, EXTPRO_1:23 ;
A25: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if>0_(a,k1,I)),P,(Initialize_s)))_holds_
(IExec_((if>0_(a,k1,I)),P,(Initialize_s)))_._x_=_((IExec_(I,P,(Initialize_s)))_+*_(Start-At_(((card_I)_+_1),SCMPDS)))_._x
let x be set ; ::_thesis: ( x in dom (IExec ((if>0 (a,k1,I)),P,(Initialize s))) implies (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . b1 )
A26: dom (Start-At (((card I) + 1),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A27: x in dom (IExec ((if>0 (a,k1,I)),P,(Initialize s))) ; ::_thesis: (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A27, SCMPDS_4:6;
supposeA28: x is Int_position ; ::_thesis: (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A29: not x in dom (Start-At (((card I) + 1),SCMPDS)) by A26, TARSKI:def_1;
thus (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . x = (Result ((P +* (stop (if>0 (a,k1,I)))),(Initialize s))) . x
.= (Result ((P +* (stop I)),(Initialize s))) . x by A24, A28, SCMPDS_4:8
.= (IExec (I,P,(Initialize s))) . x
.= ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . x by A29, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA30: x = IC ; ::_thesis: (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . b1
A31: IC (Result ((P +* (stop I)),(Initialize s))) = IC (IExec (I,P,(Initialize s)))
.= card I by A7, A11, Th34 ;
A32: x in dom (Start-At (((card I) + 1),SCMPDS)) by A26, A30, TARSKI:def_1;
thus (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . x = (Result ((P +* (stop (if>0 (a,k1,I)))),(Initialize s))) . x
.= (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) . x by A15, A23, EXTPRO_1:23
.= IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A30, EXTPRO_1:4
.= (IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) + 1 by A8, A9, A5, A6, Th31, A10
.= (IC (Result ((P +* (stop I)),(Initialize s)))) + 1 by A12, EXTPRO_1:23
.= IC (Start-At (((card I) + 1),SCMPDS)) by A31, FUNCOP_1:72
.= ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . x by A30, A32, FUNCT_4:13 ; ::_thesis: verum
end;
end;
end;
dom (IExec ((if>0 (a,k1,I)),P,(Initialize s))) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((if>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) by A25, FUNCT_1:2; ::_thesis: verum
end;

theorem Th81: :: SCMPDS_6:81
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) <= 0 implies IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set IF = if>0 (a,k1,I);
set pIF = stop (if>0 (a,k1,I));
set s3 = Initialize s;
set P3 = P +* (stop (if>0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if>0 (a,k1,I)));
set i = (a,k1) <=0_goto ((card I) + 1);
set SAl = Start-At (((card I) + 1),SCMPDS);
A1: IC (Initialize s) = 0 by MEMSTR_0:47;
A2: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A3: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A2, FUNCT_4:11 ;
A4: Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <=0_goto ((card I) + 1)),(Initialize s)) by Th11 ;
assume  s . (DataLoc ((s . a),k1)) <= 0 ; ::_thesis: IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
then A5: IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)) = ICplusConst ((Initialize s),((card I) + 1)) by A4, A3, SCMPDS_2:56
.= 0 + ((card I) + 1) by A1, Th12 ;
 stop (if>0 (a,k1,I)) c= P +* (stop (if>0 (a,k1,I))) by FUNCT_4:25;
then A6: stop (if>0 (a,k1,I)) c= P +* (stop (if>0 (a,k1,I))) ;
A7: (P +* (stop (if>0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1))) = (P +* (stop (if>0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1))) by PBOOLE:143;
A8: card (if>0 (a,k1,I)) = (card I) + 1 by Th6;
then  (card I) + 1 in dom (stop (if>0 (a,k1,I))) by COMPOS_1:64;
then (P +* (stop (if>0 (a,k1,I)))) . ((card I) + 1) = (stop (if>0 (a,k1,I))) . ((card I) + 1) by A6, GRFUNC_1:2
.= halt SCMPDS by A8, COMPOS_1:64 ;
then A9: CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1))) = halt SCMPDS by A5, A7;
then A10: P +* (stop (if>0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
A11: CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) = (a,k1) <=0_goto ((card I) + 1) by Th11;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_1_holds_
CurInstr_((P_+*_(stop_(if>0_(a,k1,I)))),(Comput_((P_+*_(stop_(if>0_(a,k1,I)))),(Initialize_s),l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < 1 implies CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS )
A12: Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),0) = Initialize s by EXTPRO_1:2;
assume  l < 1 ; ::_thesis: CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS
then  l < 1 + 0 ;
then  l = 0 by NAT_1:13;
then  CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) = CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) by A12;
hence  CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A11; ::_thesis: verum
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS holds
1 <= l ;
then  LifeSpan ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) = 1 by A9, A10, EXTPRO_1:def_15;
then A13: Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1) = Result ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) by A10, EXTPRO_1:23;
A14: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if>0_(a,k1,I)),P,(Initialize_s)))_holds_
(IExec_((if>0_(a,k1,I)),P,(Initialize_s)))_._x_=_(s_+*_(Start-At_(((card_I)_+_1),SCMPDS)))_._x
A15: dom (Start-At (((card I) + 1),SCMPDS)) = {(IC )} by FUNCOP_1:13;
let x be set ; ::_thesis: ( x in dom (IExec ((if>0 (a,k1,I)),P,(Initialize s))) implies (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 1),SCMPDS))) . b1 )
assume A16: x in dom (IExec ((if>0 (a,k1,I)),P,(Initialize s))) ; ::_thesis: (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 1),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A16, SCMPDS_4:6;
supposeA17: x is Int_position ; ::_thesis: (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 1),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A18: not x in dom (Start-At (((card I) + 1),SCMPDS)) by A15, TARSKI:def_1;
A19: not x in dom (Start-At (0,SCMPDS)) by A17, SCMPDS_4:18;
thus (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . x = (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)) . x by A13
.= (Initialize s) . x by A4, A17, SCMPDS_2:56
.= s . x by A19, FUNCT_4:11
.= (s +* (Start-At (((card I) + 1),SCMPDS))) . x by A18, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA20: x = IC ; ::_thesis: (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 1),SCMPDS))) . b1
hence (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . x = (card I) + 1 by A5, A13
.= (s +* (Start-At (((card I) + 1),SCMPDS))) . x by A20, FUNCT_4:113 ;
::_thesis: verum
end;
end;
end;
dom (IExec ((if>0 (a,k1,I)),P,(Initialize s))) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom (s +* (Start-At (((card I) + 1),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS)) by A14, FUNCT_1:2; ::_thesis: verum
end;

registration
let I be parahalting shiftable Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if>0 (a,k1,I) ->  parahalting shiftable ;
correctness
coherence
( if>0 (a,k1,I) is shiftable & if>0 (a,k1,I) is parahalting );
proof
set i = (a,k1) <=0_goto ((card I) + 1);
set IF = if>0 (a,k1,I);
set pIF = stop (if>0 (a,k1,I));
thus  if>0 (a,k1,I) is shiftable ; ::_thesis: if>0 (a,k1,I) is parahalting
let s be 0 -started State of SCMPDS; :: according to SCMPDS_4:def_7 ::_thesis: for b1 being set holds
( not stop (if>0 (a,k1,I)) c= b1 or b1 halts_on s )
let P be Instruction-Sequence of SCMPDS; ::_thesis: ( not stop (if>0 (a,k1,I)) c= P or P halts_on s )
A1: Initialize s = s by MEMSTR_0:44;
assume  stop (if>0 (a,k1,I)) c= P ; ::_thesis: P halts_on s
then A2: P = P +* (stop (if>0 (a,k1,I))) by FUNCT_4:98;
A3: ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
percases ( s . (DataLoc ((s . a),k1)) > 0 or s . (DataLoc ((s . a),k1)) <= 0 ) ;
suppose s . (DataLoc ((s . a),k1)) > 0 ; ::_thesis: P halts_on s
then  if>0 (a,k1,I) is_halting_on s,P by A3, Th78;
hence  P halts_on s by A2, Def3, A1; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),k1)) <= 0 ; ::_thesis: P halts_on s
then  if>0 (a,k1,I) is_halting_on s,P by Th79;
hence  P halts_on s by A2, Def3, A1; ::_thesis: verum
end;
end;
end;
end;

registration
let I be halt-free Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if>0 (a,k1,I) ->  halt-free ;
coherence
 if>0 (a,k1,I) is halt-free ;
end;

theorem :: SCMPDS_6:82
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if>0 (a,k1,I)),P,s)) = (card I) + 1
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if>0 (a,k1,I)),P,s)) = (card I) + 1
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if>0 (a,k1,I)),P,s)) = (card I) + 1
let I be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer holds IC (IExec ((if>0 (a,k1,I)),P,s)) = (card I) + 1
let a be Int_position; ::_thesis: for k1 being Integer holds IC (IExec ((if>0 (a,k1,I)),P,s)) = (card I) + 1
let k1 be Integer; ::_thesis: IC (IExec ((if>0 (a,k1,I)),P,s)) = (card I) + 1
set IF = if>0 (a,k1,I);
A1: ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
A2: Initialize s = s by MEMSTR_0:44;
percases ( s . (DataLoc ((s . a),k1)) > 0 or s . (DataLoc ((s . a),k1)) <= 0 ) ;
suppose s . (DataLoc ((s . a),k1)) > 0 ; ::_thesis: IC (IExec ((if>0 (a,k1,I)),P,s)) = (card I) + 1
then  IExec ((if>0 (a,k1,I)),P,s) = (IExec (I,P,s)) +* (Start-At (((card I) + 1),SCMPDS)) by A1, Th80, A2;
hence  IC (IExec ((if>0 (a,k1,I)),P,s)) = (card I) + 1 by FUNCT_4:113; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),k1)) <= 0 ; ::_thesis: IC (IExec ((if>0 (a,k1,I)),P,s)) = (card I) + 1
then  IExec ((if>0 (a,k1,I)),P,s) = s +* (Start-At (((card I) + 1),SCMPDS)) by Th81, A2;
hence  IC (IExec ((if>0 (a,k1,I)),P,s)) = (card I) + 1 by FUNCT_4:113; ::_thesis: verum
end;
end;
end;

theorem :: SCMPDS_6:83
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
(IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
(IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let s be State of SCMPDS; ::_thesis: for I being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
(IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let I be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
(IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
(IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) > 0 implies (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b )
assume A1: s . (DataLoc ((s . a),k1)) > 0 ; ::_thesis: (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
A2: not b in dom (Start-At (((card I) + 1),SCMPDS)) by SCMPDS_4:18;
 ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
then  IExec ((if>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) by A1, Th80;
hence  (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b by A2, FUNCT_4:11; ::_thesis: verum
end;

theorem :: SCMPDS_6:84
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
(IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = s . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
(IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = s . b
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
(IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = s . b
let I be Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
(IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = s . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
(IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = s . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) <= 0 implies (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = s . b )
assume  s . (DataLoc ((s . a),k1)) <= 0 ; ::_thesis: (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = s . b
then A1: IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS)) by Th81;
 not b in dom (Start-At (((card I) + 1),SCMPDS)) by SCMPDS_4:18;
hence  (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = s . b by A1, FUNCT_4:11; ::_thesis: verum
end;

begin

theorem :: SCMPDS_6:85
for a being Int_position
for k1 being Integer
for I being Program of SCMPDS holds card (if<=0 (a,k1,I)) = (card I) + 2 by Lm8;

theorem :: SCMPDS_6:86
for a being Int_position
for k1 being Integer
for I being Program of SCMPDS holds
( 0 in dom (if<=0 (a,k1,I)) & 1 in dom (if<=0 (a,k1,I)) ) by Lm9;

theorem :: SCMPDS_6:87
for a being Int_position
for k1 being Integer
for I being Program of SCMPDS holds
( (if<=0 (a,k1,I)) . 0 = (a,k1) <=0_goto 2 & (if<=0 (a,k1,I)) . 1 = goto ((card I) + 1) ) by Lm10;

theorem Th88: :: SCMPDS_6:88
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<=0 (a,k1,I) is_closed_on s,P & if<=0 (a,k1,I) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<=0 (a,k1,I) is_closed_on s,P & if<=0 (a,k1,I) is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<=0 (a,k1,I) is_closed_on s,P & if<=0 (a,k1,I) is_halting_on s,P )
let I be shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<=0 (a,k1,I) is_closed_on s,P & if<=0 (a,k1,I) is_halting_on s,P )
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<=0 (a,k1,I) is_closed_on s,P & if<=0 (a,k1,I) is_halting_on s,P )
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s,P & I is_halting_on s,P implies ( if<=0 (a,k1,I) is_closed_on s,P & if<=0 (a,k1,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
set IF = if<=0 (a,k1,I);
set pIF = stop (if<=0 (a,k1,I));
set pI = stop I;
set s2 = Initialize s;
set s3 = Initialize s;
set P2 = P +* (stop I);
set P3 = P +* (stop (if<=0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if<=0 (a,k1,I)));
set i = (a,k1) <=0_goto 2;
set j = goto ((card I) + 1);
A1: stop I c= P +* (stop I) by FUNCT_4:25;
A2: if<=0 (a,k1,I) = ((a,k1) <=0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:16;
A3: Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <=0_goto 2),(Initialize s)) by A2, Th11 ;
 for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)) . a by A3, SCMPDS_2:56;
then A4: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)) by SCMPDS_4:8;
A5: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A6: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A5, FUNCT_4:11 ;
A7: IC (Initialize s) = 0 by MEMSTR_0:47;
assume  s . (DataLoc ((s . a),k1)) <= 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or ( if<=0 (a,k1,I) is_closed_on s,P & if<=0 (a,k1,I) is_halting_on s,P ) )
then A8: IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)) = ICplusConst ((Initialize s),2) by A3, A6, SCMPDS_2:56
.= 0 + 2 by A7, Th12 ;
assume A9: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or ( if<=0 (a,k1,I) is_closed_on s,P & if<=0 (a,k1,I) is_halting_on s,P ) )
then A10: I is_closed_on Initialize s,P +* (stop I) by Th24;
assume  I is_halting_on s,P ; ::_thesis: ( if<=0 (a,k1,I) is_closed_on s,P & if<=0 (a,k1,I) is_halting_on s,P )
then A11: P +* (stop I) halts_on Initialize s by Def3;
A12: ( Start-At (0,SCMPDS) c= Initialize s & Shift ((stop I),2) c= P +* (stop (if<=0 (a,k1,I))) ) by Lm7, FUNCT_4:25;
A13: 0 in dom (stop (if<=0 (a,k1,I))) by COMPOS_1:36;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(if<=0_(a,k1,I)))),(Initialize_s),k))_in_dom_(stop_(if<=0_(a,k1,I)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<=0 (a,k1,I)))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<=0 (a,k1,I)))
then consider k1 being Nat such that
A14: k1 + 1 = k by NAT_1:6;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def_12;
reconsider m = IC (Comput ((P +* (stop I)),(Initialize s),k1)) as Element of NAT ;
A15: card (stop (if<=0 (a,k1,I))) = 1 + (card (if<=0 (a,k1,I))) by COMPOS_1:55
.= 1 + ((card I) + 2) by Lm8
.= (1 + (card I)) + 2
.= (card (stop I)) + 2 by COMPOS_1:55 ;
 m in dom (stop I) by A9, Def2;
then  m < card (stop I) by AFINSQ_1:66;
then A16: m + 2 < card (stop (if<=0 (a,k1,I))) by A15, XREAL_1:6;
IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),k)) = IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)),k1)) by A14, EXTPRO_1:4
.= m + 2 by A10, A12, A8, A4, Th31, A1 ;
hence  IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if<=0 (a,k1,I))) by A16, AFINSQ_1:66; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<=0 (a,k1,I)))
hence  IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if<=0 (a,k1,I))) by A13, A7, EXTPRO_1:2; ::_thesis: verum
end;
end;
end;
hence  if<=0 (a,k1,I) is_closed_on s,P by Def2; ::_thesis: if<=0 (a,k1,I) is_halting_on s,P
A17: Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)) = Comput ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))) by EXTPRO_1:4;
CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A17
.= CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A10, A12, A8, A4, Th31, A1
.= halt SCMPDS by A11, EXTPRO_1:def_15 ;
then  P +* (stop (if<=0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
hence  if<=0 (a,k1,I) is_halting_on s,P by Def3; ::_thesis: verum
end;

theorem Th89: :: SCMPDS_6:89
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
( if<=0 (a,k1,I) is_closed_on s,P & if<=0 (a,k1,I) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
( if<=0 (a,k1,I) is_closed_on s,P & if<=0 (a,k1,I) is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
( if<=0 (a,k1,I) is_closed_on s,P & if<=0 (a,k1,I) is_halting_on s,P )
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
( if<=0 (a,k1,I) is_closed_on s,P & if<=0 (a,k1,I) is_halting_on s,P )
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
( if<=0 (a,k1,I) is_closed_on s,P & if<=0 (a,k1,I) is_halting_on s,P )
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) > 0 implies ( if<=0 (a,k1,I) is_closed_on s,P & if<=0 (a,k1,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
assume A1: s . (DataLoc ((s . a),k1)) > 0 ; ::_thesis: ( if<=0 (a,k1,I) is_closed_on s,P & if<=0 (a,k1,I) is_halting_on s,P )
set i = (a,k1) <=0_goto 2;
set j = goto ((card I) + 1);
set IF = if<=0 (a,k1,I);
set pIF = stop (if<=0 (a,k1,I));
set s3 = Initialize s;
set P3 = P +* (stop (if<=0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1);
set s5 = Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),2);
set P4 = P +* (stop (if<=0 (a,k1,I)));
set P5 = P +* (stop (if<=0 (a,k1,I)));
A2: if<=0 (a,k1,I) = ((a,k1) <=0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:16;
A3: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A4: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A3, FUNCT_4:11 ;
A5: IC (Initialize s) = 0 by MEMSTR_0:47;
Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <=0_goto 2),(Initialize s)) by A2, Th11 ;
then A6: IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)) = succ (IC (Initialize s)) by A1, A4, SCMPDS_2:56
.= 0 + 1 by A5 ;
A7: stop (if<=0 (a,k1,I)) c= P +* (stop (if<=0 (a,k1,I))) by FUNCT_4:25;
then A8: stop (if<=0 (a,k1,I)) c= P +* (stop (if<=0 (a,k1,I))) ;
A9: 1 in dom (if<=0 (a,k1,I)) by Lm9;
then  1 in dom (stop (if<=0 (a,k1,I))) by COMPOS_1:62;
then A10: (P +* (stop (if<=0 (a,k1,I)))) . 1 = (stop (if<=0 (a,k1,I))) . 1 by A8, GRFUNC_1:2
.= (if<=0 (a,k1,I)) . 1 by A9, COMPOS_1:63
.= goto ((card I) + 1) by Lm10 ;
A11: card (if<=0 (a,k1,I)) = (card I) + 2 by Lm8;
then A12: (card I) + 2 in dom (stop (if<=0 (a,k1,I))) by COMPOS_1:64;
A13: (P +* (stop (if<=0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1))) = (P +* (stop (if<=0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1))) by PBOOLE:143;
Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),(1 + 1)) = Following ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1))) by EXTPRO_1:3
.= Exec ((goto ((card I) + 1)),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1))) by A6, A10, A13 ;
then A14: IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),2)) = ICplusConst ((Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)),((card I) + 1)) by SCMPDS_2:54
.= ((card I) + 1) + 1 by A6, Th12
.= (card I) + (1 + 1) ;
A15: (P +* (stop (if<=0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),2))) = (P +* (stop (if<=0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),2))) by PBOOLE:143;
 stop (if<=0 (a,k1,I)) c= P +* (stop (if<=0 (a,k1,I))) by A7;
then (P +* (stop (if<=0 (a,k1,I)))) . ((card I) + 2) = (stop (if<=0 (a,k1,I))) . ((card I) + 2) by A12, GRFUNC_1:2
.= halt SCMPDS by A11, COMPOS_1:64 ;
then A16: CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),2))) = halt SCMPDS by A14, A15;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(if<=0_(a,k1,I)))),(Initialize_s),k))_in_dom_(stop_(if<=0_(a,k1,I)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<=0 (a,k1,I)))
A17: ( k = 0 or 0 + 1 <= k ) by INT_1:7;
percases ( k = 0 or k = 1 or 1 < k ) by A17, XXREAL_0:1;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<=0 (a,k1,I)))
then  Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),k) = Initialize s by EXTPRO_1:2;
hence  IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if<=0 (a,k1,I))) by A5, COMPOS_1:36; ::_thesis: verum
end;
suppose k = 1 ; ::_thesis: IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<=0 (a,k1,I)))
hence  IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if<=0 (a,k1,I))) by A9, A6, COMPOS_1:62; ::_thesis: verum
end;
suppose 1 < k ; ::_thesis: IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<=0 (a,k1,I)))
then  1 + 1 <= k by INT_1:7;
hence  IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if<=0 (a,k1,I))) by A12, A14, A16, EXTPRO_1:5; ::_thesis: verum
end;
end;
end;
hence  if<=0 (a,k1,I) is_closed_on s,P by Def2; ::_thesis: if<=0 (a,k1,I) is_halting_on s,P
 P +* (stop (if<=0 (a,k1,I))) halts_on Initialize s by A16, EXTPRO_1:29;
hence  if<=0 (a,k1,I) is_halting_on s,P by Def3; ::_thesis: verum
end;

theorem Th90: :: SCMPDS_6:90
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))
let s be State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s,P & I is_halting_on s,P implies IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set IF = if<=0 (a,k1,I);
set pIF = stop (if<=0 (a,k1,I));
set pI = stop I;
set s2 = Initialize s;
set s3 = Initialize s;
set P2 = P +* (stop I);
set P3 = P +* (stop (if<=0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if<=0 (a,k1,I)));
set i = (a,k1) <=0_goto 2;
set j = goto ((card I) + 1);
set SAl = Start-At (((card I) + 2),SCMPDS);
A1: stop I c= P +* (stop I) by FUNCT_4:25;
A2: if<=0 (a,k1,I) = ((a,k1) <=0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:16;
A3: Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <=0_goto 2),(Initialize s)) by A2, Th11 ;
A4: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A5: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A4, FUNCT_4:11 ;
A6: IC (Initialize s) = 0 by MEMSTR_0:47;
A7: ( Start-At (0,SCMPDS) c= Initialize s & Shift ((stop I),2) c= P +* (stop (if<=0 (a,k1,I))) ) by Lm7, FUNCT_4:25;
 for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)) . a by A3, SCMPDS_2:56;
then A8: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)) by SCMPDS_4:8;
assume  s . (DataLoc ((s . a),k1)) <= 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS)) )
then A9: IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)) = ICplusConst ((Initialize s),2) by A3, A5, SCMPDS_2:56
.= 0 + 2 by A6, Th12 ;
assume A10: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS)) )
then A11: I is_closed_on Initialize s,P +* (stop I) by Th24;
assume A12: I is_halting_on s,P ; ::_thesis: IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))
then A13: P +* (stop I) halts_on Initialize s by Def3;
A14: Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)) = Comput ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))) by EXTPRO_1:4;
A15: CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A14
.= CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A11, A7, A9, A8, Th31, A1
.= halt SCMPDS by A13, EXTPRO_1:def_15 ;
then A16: P +* (stop (if<=0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
A17: CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s)) = (a,k1) <=0_goto 2 by A2, Th11;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_(LifeSpan_((P_+*_(stop_I)),(Initialize_s)))_+_1_holds_
CurInstr_((P_+*_(stop_(if<=0_(a,k1,I)))),(Comput_((P_+*_(stop_(if<=0_(a,k1,I)))),(Initialize_s),l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 implies CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS )
assume A18: l < (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 ; ::_thesis: CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
A19: Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),0) = Initialize s by EXTPRO_1:2;
percases ( l = 0 or l <> 0 ) ;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
then  CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),l))) = CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s)) by A19;
hence  CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A17; ::_thesis: verum
end;
suppose l <> 0 ; ::_thesis: not CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),b1))) = halt SCMPDS
then consider n being Nat such that
A20: l = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A21: n < LifeSpan ((P +* (stop I)),(Initialize s)) by A18, A20, XREAL_1:6;
assume A22: CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS ; ::_thesis: contradiction
A23: Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),(n + 1)) = Comput ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)),n) by EXTPRO_1:4;
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),n))) = CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)),n))) by A11, A7, A9, A8, Th31, A1
.= halt SCMPDS by A20, A22, A23 ;
hence  contradiction by A13, A21, EXTPRO_1:def_15; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS holds
(LifeSpan ((P +* (stop I)),(Initialize s))) + 1 <= l ;
then A24: LifeSpan ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s)) = (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 by A15, A16, EXTPRO_1:def_15;
A25: DataPart (Result ((P +* (stop I)),(Initialize s))) = DataPart (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A13, EXTPRO_1:23
.= DataPart (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A11, A7, A9, A8, Th31, A1
.= DataPart (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s))) by A16, A24, EXTPRO_1:23 ;
A26: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if<=0_(a,k1,I)),P,(Initialize_s)))_holds_
(IExec_((if<=0_(a,k1,I)),P,(Initialize_s)))_._x_=_((IExec_(I,P,(Initialize_s)))_+*_(Start-At_(((card_I)_+_2),SCMPDS)))_._x
let x be set ; ::_thesis: ( x in dom (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) implies (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) . b1 )
A27: dom (Start-At (((card I) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A28: x in dom (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) ; ::_thesis: (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A28, SCMPDS_4:6;
supposeA29: x is Int_position ; ::_thesis: (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A30: not x in dom (Start-At (((card I) + 2),SCMPDS)) by A27, TARSKI:def_1;
thus (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . x = (Result ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s))) . x
.= (Result ((P +* (stop I)),(Initialize s))) . x by A25, A29, SCMPDS_4:8
.= (IExec (I,P,(Initialize s))) . x
.= ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) . x by A30, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA31: x = IC ; ::_thesis: (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) . b1
A32: IC (Result ((P +* (stop I)),(Initialize s))) = IC (IExec (I,P,(Initialize s)))
.= card I by A10, A12, Th34 ;
A33: x in dom (Start-At (((card I) + 2),SCMPDS)) by A27, A31, TARSKI:def_1;
thus (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . x = (Result ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s))) . x
.= (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) . x by A16, A24, EXTPRO_1:23
.= IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A31, EXTPRO_1:4
.= (IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) + 2 by A11, A7, A9, A8, Th31, A1
.= (IC (Result ((P +* (stop I)),(Initialize s)))) + 2 by A13, EXTPRO_1:23
.= IC (Start-At (((card I) + 2),SCMPDS)) by A32, FUNCOP_1:72
.= ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) . x by A31, A33, FUNCT_4:13 ; ::_thesis: verum
end;
end;
end;
dom (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS)) by A26, FUNCT_1:2; ::_thesis: verum
end;

theorem Th91: :: SCMPDS_6:91
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) > 0 implies IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set IF = if<=0 (a,k1,I);
set pIF = stop (if<=0 (a,k1,I));
set s3 = Initialize s;
set P3 = P +* (stop (if<=0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1);
set s5 = Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),2);
set P4 = P +* (stop (if<=0 (a,k1,I)));
set P5 = P +* (stop (if<=0 (a,k1,I)));
set i = (a,k1) <=0_goto 2;
set j = goto ((card I) + 1);
set SAl = Start-At (((card I) + 2),SCMPDS);
A1: if<=0 (a,k1,I) = ((a,k1) <=0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:16;
A2: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A3: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A2, FUNCT_4:11 ;
A4: IC (Initialize s) = 0 by MEMSTR_0:47;
A5: stop (if<=0 (a,k1,I)) c= P +* (stop (if<=0 (a,k1,I))) by FUNCT_4:25;
then A6: stop (if<=0 (a,k1,I)) c= P +* (stop (if<=0 (a,k1,I))) ;
A7: stop (if<=0 (a,k1,I)) c= P +* (stop (if<=0 (a,k1,I))) by A5;
A8: Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <=0_goto 2),(Initialize s)) by A1, Th11 ;
assume  s . (DataLoc ((s . a),k1)) > 0 ; ::_thesis: IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
then A9: IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)) = succ (IC (Initialize s)) by A8, A3, SCMPDS_2:56
.= 0 + 1 by A4 ;
A10: 1 in dom (if<=0 (a,k1,I)) by Lm9;
then  1 in dom (stop (if<=0 (a,k1,I))) by COMPOS_1:62;
then A11: (P +* (stop (if<=0 (a,k1,I)))) . 1 = (stop (if<=0 (a,k1,I))) . 1 by A6, GRFUNC_1:2
.= (if<=0 (a,k1,I)) . 1 by A10, COMPOS_1:63
.= goto ((card I) + 1) by Lm10 ;
A12: (P +* (stop (if<=0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1))) = (P +* (stop (if<=0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1))) by PBOOLE:143;
A13: Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),(1 + 1)) = Following ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1))) by EXTPRO_1:3
.= Exec ((goto ((card I) + 1)),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1))) by A9, A11, A12 ;
then A14: IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),2)) = ICplusConst ((Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)),((card I) + 1)) by SCMPDS_2:54
.= ((card I) + 1) + 1 by A9, Th12
.= (card I) + (1 + 1) ;
A15: (P +* (stop (if<=0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),2))) = (P +* (stop (if<=0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),2))) by PBOOLE:143;
A16: card (if<=0 (a,k1,I)) = (card I) + 2 by Lm8;
then  (card I) + 2 in dom (stop (if<=0 (a,k1,I))) by COMPOS_1:64;
then (P +* (stop (if<=0 (a,k1,I)))) . ((card I) + 2) = (stop (if<=0 (a,k1,I))) . ((card I) + 2) by A7, GRFUNC_1:2
.= halt SCMPDS by A16, COMPOS_1:64 ;
then A17: CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),2))) = halt SCMPDS by A14, A15;
then A18: P +* (stop (if<=0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
A19: CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s)) = (a,k1) <=0_goto 2 by A1, Th11;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_1_+_1_holds_
CurInstr_((P_+*_(stop_(if<=0_(a,k1,I)))),(Comput_((P_+*_(stop_(if<=0_(a,k1,I)))),(Initialize_s),l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < 1 + 1 implies CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS )
assume  l < 1 + 1 ; ::_thesis: CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
then A20: l <= 1 by NAT_1:13;
A21: Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),0) = Initialize s by EXTPRO_1:2;
A22: (P +* (stop (if<=0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),l))) = (P +* (stop (if<=0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),l))) by PBOOLE:143;
percases ( l = 0 or l = 1 ) by A20, NAT_1:25;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
then  CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),l))) = CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s)) by A21;
hence  CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A19; ::_thesis: verum
end;
suppose l = 1 ; ::_thesis: CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
hence  CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A9, A11, A22; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (if<=0 (a,k1,I)))),(Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS holds
2 <= l ;
then  LifeSpan ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s)) = 2 by A17, A18, EXTPRO_1:def_15;
then A23: Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),2) = Result ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s)) by A18, EXTPRO_1:23;
A24: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if<=0_(a,k1,I)),P,(Initialize_s)))_holds_
(IExec_((if<=0_(a,k1,I)),P,(Initialize_s)))_._x_=_(s_+*_(Start-At_(((card_I)_+_2),SCMPDS)))_._x
A25: dom (Start-At (((card I) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
let x be set ; ::_thesis: ( x in dom (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) implies (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1 )
assume A26: x in dom (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) ; ::_thesis: (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A26, SCMPDS_4:6;
supposeA27: x is Int_position ; ::_thesis: (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A28: not x in dom (Start-At (((card I) + 2),SCMPDS)) by A25, TARSKI:def_1;
A29: not x in dom (Start-At (0,SCMPDS)) by A27, SCMPDS_4:18;
thus (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . x = (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),2)) . x by A23
.= (Comput ((P +* (stop (if<=0 (a,k1,I)))),(Initialize s),1)) . x by A13, A27, SCMPDS_2:54
.= (Initialize s) . x by A8, A27, SCMPDS_2:56
.= s . x by A29, FUNCT_4:11
.= (s +* (Start-At (((card I) + 2),SCMPDS))) . x by A28, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA30: x = IC ; ::_thesis: (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1
hence (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . x = (card I) + 2 by A14, A23
.= (s +* (Start-At (((card I) + 2),SCMPDS))) . x by A30, FUNCT_4:113 ;
::_thesis: verum
end;
end;
end;
dom (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom (s +* (Start-At (((card I) + 2),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS)) by A24, FUNCT_1:2; ::_thesis: verum
end;

registration
let I be parahalting shiftable Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if<=0 (a,k1,I) ->  parahalting shiftable ;
correctness
coherence
( if<=0 (a,k1,I) is shiftable & if<=0 (a,k1,I) is parahalting );
proof
set i = (a,k1) <=0_goto 2;
set j = goto ((card I) + 1);
set IF = if<=0 (a,k1,I);
set pIF = stop (if<=0 (a,k1,I));
thus  if<=0 (a,k1,I) is shiftable ; ::_thesis: if<=0 (a,k1,I) is parahalting
let s be 0 -started State of SCMPDS; :: according to SCMPDS_4:def_7 ::_thesis: for b1 being set holds
( not stop (if<=0 (a,k1,I)) c= b1 or b1 halts_on s )
let P be Instruction-Sequence of SCMPDS; ::_thesis: ( not stop (if<=0 (a,k1,I)) c= P or P halts_on s )
A1: Initialize s = s by MEMSTR_0:44;
assume  stop (if<=0 (a,k1,I)) c= P ; ::_thesis: P halts_on s
then A2: P = P +* (stop (if<=0 (a,k1,I))) by FUNCT_4:98;
A3: ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
percases ( s . (DataLoc ((s . a),k1)) <= 0 or s . (DataLoc ((s . a),k1)) > 0 ) ;
suppose s . (DataLoc ((s . a),k1)) <= 0 ; ::_thesis: P halts_on s
then  if<=0 (a,k1,I) is_halting_on s,P by A3, Th88;
hence  P halts_on s by A2, Def3, A1; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),k1)) > 0 ; ::_thesis: P halts_on s
then  if<=0 (a,k1,I) is_halting_on s,P by Th89;
hence  P halts_on s by A2, Def3, A1; ::_thesis: verum
end;
end;
end;
end;

registration
let I be halt-free Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if<=0 (a,k1,I) ->  halt-free ;
coherence
 if<=0 (a,k1,I) is halt-free ;
end;

theorem :: SCMPDS_6:92
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if<=0 (a,k1,I)),P,s)) = (card I) + 2
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if<=0 (a,k1,I)),P,s)) = (card I) + 2
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if<=0 (a,k1,I)),P,s)) = (card I) + 2
let I be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer holds IC (IExec ((if<=0 (a,k1,I)),P,s)) = (card I) + 2
let a be Int_position; ::_thesis: for k1 being Integer holds IC (IExec ((if<=0 (a,k1,I)),P,s)) = (card I) + 2
let k1 be Integer; ::_thesis: IC (IExec ((if<=0 (a,k1,I)),P,s)) = (card I) + 2
set IF = if<=0 (a,k1,I);
A1: ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
A2: Initialize s = s by MEMSTR_0:44;
percases ( s . (DataLoc ((s . a),k1)) <= 0 or s . (DataLoc ((s . a),k1)) > 0 ) ;
suppose s . (DataLoc ((s . a),k1)) <= 0 ; ::_thesis: IC (IExec ((if<=0 (a,k1,I)),P,s)) = (card I) + 2
then  IExec ((if<=0 (a,k1,I)),P,s) = (IExec (I,P,s)) +* (Start-At (((card I) + 2),SCMPDS)) by A1, Th90, A2;
hence  IC (IExec ((if<=0 (a,k1,I)),P,s)) = (card I) + 2 by FUNCT_4:113; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),k1)) > 0 ; ::_thesis: IC (IExec ((if<=0 (a,k1,I)),P,s)) = (card I) + 2
then  IExec ((if<=0 (a,k1,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS)) by Th91, A2;
hence  IC (IExec ((if<=0 (a,k1,I)),P,s)) = (card I) + 2 by FUNCT_4:113; ::_thesis: verum
end;
end;
end;

theorem :: SCMPDS_6:93
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
(IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
(IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let s be State of SCMPDS; ::_thesis: for I being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
(IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let I be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
(IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 holds
(IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) <= 0 implies (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b )
assume A1: s . (DataLoc ((s . a),k1)) <= 0 ; ::_thesis: (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
A2: not b in dom (Start-At (((card I) + 2),SCMPDS)) by SCMPDS_4:18;
 ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
then  IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS)) by A1, Th90;
hence  (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b by A2, FUNCT_4:11; ::_thesis: verum
end;

theorem :: SCMPDS_6:94
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
(IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b = s . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
(IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b = s . b
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
(IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b = s . b
let I be Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
(IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b = s . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 holds
(IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b = s . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) > 0 implies (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b = s . b )
assume  s . (DataLoc ((s . a),k1)) > 0 ; ::_thesis: (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b = s . b
then A1: IExec ((if<=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS)) by Th91;
 not b in dom (Start-At (((card I) + 2),SCMPDS)) by SCMPDS_4:18;
hence  (IExec ((if<=0 (a,k1,I)),P,(Initialize s))) . b = s . b by A1, FUNCT_4:11; ::_thesis: verum
end;

begin

theorem :: SCMPDS_6:95
for a being Int_position
for k1 being Integer
for I, J being Program of SCMPDS holds card (if<0 (a,k1,I,J)) = ((card I) + (card J)) + 2 by Lm4;

theorem :: SCMPDS_6:96
for a being Int_position
for k1 being Integer
for I, J being Program of SCMPDS holds
( 0 in dom (if<0 (a,k1,I,J)) & 1 in dom (if<0 (a,k1,I,J)) )
proof
let a be Int_position; ::_thesis: for k1 being Integer
for I, J being Program of SCMPDS holds
( 0 in dom (if<0 (a,k1,I,J)) & 1 in dom (if<0 (a,k1,I,J)) )
let k1 be Integer; ::_thesis: for I, J being Program of SCMPDS holds
( 0 in dom (if<0 (a,k1,I,J)) & 1 in dom (if<0 (a,k1,I,J)) )
let I, J be Program of SCMPDS; ::_thesis: ( 0 in dom (if<0 (a,k1,I,J)) & 1 in dom (if<0 (a,k1,I,J)) )
set ci = card (if<0 (a,k1,I,J));
 card (if<0 (a,k1,I,J)) = ((card I) + (card J)) + 2 by Lm4;
then  2 <= card (if<0 (a,k1,I,J)) by NAT_1:12;
then  1 < card (if<0 (a,k1,I,J)) by XXREAL_0:2;
hence  ( 0 in dom (if<0 (a,k1,I,J)) & 1 in dom (if<0 (a,k1,I,J)) ) by AFINSQ_1:66; ::_thesis: verum
end;

theorem :: SCMPDS_6:97
for a being Int_position
for k1 being Integer
for I, J being Program of SCMPDS holds (if<0 (a,k1,I,J)) . 0 = (a,k1) >=0_goto ((card I) + 2) by Lm5;

theorem Th98: :: SCMPDS_6:98
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I, J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I, J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P )
let s be 0 -started State of SCMPDS; ::_thesis: for I, J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P )
let I, J be shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P )
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P )
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P implies ( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
set G = Goto ((card J) + 1);
set I2 = (I ';' (Goto ((card J) + 1))) ';' J;
set IF = if<0 (a,k1,I,J);
set pIF = stop (if<0 (a,k1,I,J));
set pI2 = stop ((I ';' (Goto ((card J) + 1))) ';' J);
set P2 = P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J));
set P3 = P +* (stop (if<0 (a,k1,I,J)));
set s4 = Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1);
set P4 = P +* (stop (if<0 (a,k1,I,J)));
set i = (a,k1) >=0_goto ((card I) + 2);
A1: 0 in dom (stop (if<0 (a,k1,I,J))) by COMPOS_1:36;
A2: Initialize s = s by MEMSTR_0:44;
then A3: IC s = 0 by MEMSTR_0:47;
A4: if<0 (a,k1,I,J) = (((a,k1) >=0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J by SCMPDS_4:14
.= ((a,k1) >=0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J) by SCMPDS_4:14 ;
then A5: Shift ((stop ((I ';' (Goto ((card J) + 1))) ';' J)),1) c= P +* (stop (if<0 (a,k1,I,J))) by Lm6;
A6: Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,(0 + 1)) = Following ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,0))) by EXTPRO_1:3
.= Following ((P +* (stop (if<0 (a,k1,I,J)))),s) by EXTPRO_1:2
.= Exec (((a,k1) >=0_goto ((card I) + 2)),s) by A4, Th11, A2 ;
 for a being Int_position holds s . a = (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)) . a by A6, SCMPDS_2:57;
then A7: DataPart s = DataPart (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)) by SCMPDS_4:8;
assume  s . (DataLoc ((s . a),k1)) < 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or ( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P ) )
then A8: IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)) = succ (IC s) by A6, SCMPDS_2:57
.= 0 + 1 by A3 ;
assume A9: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or ( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P ) )
assume A10: I is_halting_on s,P ; ::_thesis: ( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P )
then A11: (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P by A9, Th30;
then A12: ( Start-At (0,SCMPDS) c= s & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) ) by Th24, A2, FUNCT_4:25;
A13: stop ((I ';' (Goto ((card J) + 1))) ';' J) c= P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by FUNCT_4:25;
 (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P by A9, A10, Th30;
then A14: P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s by Def3, A2;
A15: card (stop (if<0 (a,k1,I,J))) = (card (if<0 (a,k1,I,J))) + 1 by COMPOS_1:55
.= ((card ((I ';' (Goto ((card J) + 1))) ';' J)) + 1) + 1 by A4, Th6 ;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(if<0_(a,k1,I,J)))),s,k))_in_dom_(stop_(if<0_(a,k1,I,J)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,b1)) in dom (stop (if<0 (a,k1,I,J)))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,b1)) in dom (stop (if<0 (a,k1,I,J)))
then consider k1 being Nat such that
A16: k1 + 1 = k by NAT_1:6;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def_12;
reconsider m = IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,k1)) as Element of NAT ;
A17: card (stop (if<0 (a,k1,I,J))) = (card (stop ((I ';' (Goto ((card J) + 1))) ';' J))) + 1 by A15, COMPOS_1:55;
 m in dom (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by A11, Def2, A2;
then  m < card (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by AFINSQ_1:66;
then A18: m + 1 < card (stop (if<0 (a,k1,I,J))) by A17, XREAL_1:6;
IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,k)) = IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)),k1)) by A16, EXTPRO_1:4
.= m + 1 by A12, A5, A8, A7, Th31, A13 ;
hence  IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,k)) in dom (stop (if<0 (a,k1,I,J))) by A18, AFINSQ_1:66; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,b1)) in dom (stop (if<0 (a,k1,I,J)))
hence  IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,k)) in dom (stop (if<0 (a,k1,I,J))) by A1, A3, EXTPRO_1:2; ::_thesis: verum
end;
end;
end;
hence  if<0 (a,k1,I,J) is_closed_on s,P by Def2, A2; ::_thesis: if<0 (a,k1,I,J) is_halting_on s,P
A19: Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1)) = Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))) by EXTPRO_1:4;
CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1)))) = CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) by A19
.= CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) by A12, A5, A8, A7, Th31, A13
.= halt SCMPDS by A14, EXTPRO_1:def_15 ;
then  P +* (stop (if<0 (a,k1,I,J))) halts_on s by EXTPRO_1:29;
hence  if<0 (a,k1,I,J) is_halting_on s,P by Def3, A2; ::_thesis: verum
end;

theorem Th99: :: SCMPDS_6:99
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P )
let I be Program of SCMPDS; ::_thesis: for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P )
let J be shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P )
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P holds
( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P )
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P implies ( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
set pJ = stop J;
set s1 = Initialize s;
set P1 = P +* (stop J);
set IF = if<0 (a,k1,I,J);
set pIF = stop (if<0 (a,k1,I,J));
set s3 = Initialize s;
set P3 = P +* (stop (if<0 (a,k1,I,J)));
set s4 = Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1);
set P4 = P +* (stop (if<0 (a,k1,I,J)));
set i = (a,k1) >=0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
set iG = (((a,k1) >=0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1));
A1: if<0 (a,k1,I,J) = (((a,k1) >=0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J by SCMPDS_4:14
.= ((a,k1) >=0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J) by SCMPDS_4:14 ;
A2: Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) >=0_goto ((card I) + 2)),(Initialize s)) by A1, Th11 ;
A3: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A4: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A3, FUNCT_4:11 ;
A5: IC (Initialize s) = 0 by MEMSTR_0:47;
assume  s . (DataLoc ((s . a),k1)) >= 0 ; ::_thesis: ( not J is_closed_on s,P or not J is_halting_on s,P or ( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P ) )
then A6: IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)) = ICplusConst ((Initialize s),((card I) + 2)) by A2, A4, SCMPDS_2:57
.= 0 + ((card I) + 2) by A5, Th12 ;
assume A7: J is_closed_on s,P ; ::_thesis: ( not J is_halting_on s,P or ( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P ) )
then A8: ( Start-At (0,SCMPDS) c= Initialize s & J is_closed_on Initialize s,P +* (stop J) ) by Th24, FUNCT_4:25;
A9: stop J c= P +* (stop J) by FUNCT_4:25;
A10: stop (if<0 (a,k1,I,J)) c= P +* (stop (if<0 (a,k1,I,J))) by FUNCT_4:25;
A11: card ((((a,k1) >=0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1))) = (card (((a,k1) >=0_goto ((card I) + 2)) ';' I)) + (card (Goto ((card J) + 1))) by AFINSQ_1:17
.= (card (((a,k1) >=0_goto ((card I) + 2)) ';' I)) + 1 by COMPOS_1:54
.= ((card I) + 1) + 1 by Th6
.= (card I) + (1 + 1) ;
then  Shift ((stop J),((card I) + 2)) c= stop (if<0 (a,k1,I,J)) by Th13;
then  Shift ((stop J),((card I) + 2)) c= P +* (stop (if<0 (a,k1,I,J))) by A10, XBOOLE_1:1;
then A12: Shift ((stop J),((card I) + 2)) c= P +* (stop (if<0 (a,k1,I,J))) ;
assume  J is_halting_on s,P ; ::_thesis: ( if<0 (a,k1,I,J) is_closed_on s,P & if<0 (a,k1,I,J) is_halting_on s,P )
then A13: P +* (stop J) halts_on Initialize s by Def3;
 for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)) . a by A2, SCMPDS_2:57;
then A14: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)) by SCMPDS_4:8;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(if<0_(a,k1,I,J)))),(Initialize_s),k))_in_dom_(stop_(if<0_(a,k1,I,J)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),b1)) in dom (stop (if<0 (a,k1,I,J)))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),b1)) in dom (stop (if<0 (a,k1,I,J)))
then consider k1 being Nat such that
A15: k1 + 1 = k by NAT_1:6;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def_12;
reconsider m = IC (Comput ((P +* (stop J)),(Initialize s),k1)) as Element of NAT ;
 m in dom (stop J) by A7, Def2;
then  m < card (stop J) by AFINSQ_1:66;
then A16: m + ((card I) + 2) < (card (stop J)) + ((card I) + 2) by XREAL_1:6;
A17: card (stop J) = (card J) + 1 by COMPOS_1:55;
A18: card (stop (if<0 (a,k1,I,J))) = (card (if<0 (a,k1,I,J))) + 1 by COMPOS_1:55
.= (((card I) + 2) + (card J)) + 1 by A11, AFINSQ_1:17
.= ((card I) + 2) + (card (stop J)) by A17 ;
IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),k)) = IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)),k1)) by A15, EXTPRO_1:4
.= m + ((card I) + 2) by A8, A14, A12, A6, Th31, A9 ;
hence  IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),k)) in dom (stop (if<0 (a,k1,I,J))) by A18, A16, AFINSQ_1:66; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),b1)) in dom (stop (if<0 (a,k1,I,J)))
then  Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),k) = Initialize s by EXTPRO_1:2;
hence  IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),k)) in dom (stop (if<0 (a,k1,I,J))) by A5, COMPOS_1:36; ::_thesis: verum
end;
end;
end;
hence  if<0 (a,k1,I,J) is_closed_on s,P by Def2; ::_thesis: if<0 (a,k1,I,J) is_halting_on s,P
A19: Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1)) = Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)),(LifeSpan ((P +* (stop J)),(Initialize s)))) by EXTPRO_1:4;
CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)),(LifeSpan ((P +* (stop J)),(Initialize s)))))) by A19
.= CurInstr ((P +* (stop J)),(Comput ((P +* (stop J)),(Initialize s),(LifeSpan ((P +* (stop J)),(Initialize s)))))) by A8, A14, A12, A6, Th31, A9
.= halt SCMPDS by A13, EXTPRO_1:def_15 ;
then  P +* (stop (if<0 (a,k1,I,J))) halts_on Initialize s by EXTPRO_1:29;
hence  if<0 (a,k1,I,J) is_halting_on s,P by Def3; ::_thesis: verum
end;

theorem Th100: :: SCMPDS_6:100
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for J being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let J be shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P implies IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set G = Goto ((card J) + 1);
set I2 = (I ';' (Goto ((card J) + 1))) ';' J;
set IF = if<0 (a,k1,I,J);
set pIF = stop (if<0 (a,k1,I,J));
set pI2 = stop ((I ';' (Goto ((card J) + 1))) ';' J);
set P2 = P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J));
set P3 = P +* (stop (if<0 (a,k1,I,J)));
set s4 = Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1);
set P4 = P +* (stop (if<0 (a,k1,I,J)));
set i = (a,k1) >=0_goto ((card I) + 2);
set SAl = Start-At ((((card I) + (card J)) + 2),SCMPDS);
A1: Initialize s = s by MEMSTR_0:44;
then A2: IC s = 0 by MEMSTR_0:47;
A3: if<0 (a,k1,I,J) = (((a,k1) >=0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J by SCMPDS_4:14
.= ((a,k1) >=0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J) by SCMPDS_4:14 ;
then A4: Shift ((stop ((I ';' (Goto ((card J) + 1))) ';' J)),1) c= P +* (stop (if<0 (a,k1,I,J))) by Lm6;
A5: Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,(0 + 1)) = Following ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,0))) by EXTPRO_1:3
.= Following ((P +* (stop (if<0 (a,k1,I,J)))),s) by EXTPRO_1:2
.= Exec (((a,k1) >=0_goto ((card I) + 2)),s) by A3, Th11, A1 ;
assume  s . (DataLoc ((s . a),k1)) < 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
then A6: IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)) = succ (IC s) by A5, SCMPDS_2:57
.= 0 + 1 by A2 ;
 for a being Int_position holds s . a = (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)) . a by A5, SCMPDS_2:57;
then A7: DataPart s = DataPart (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)) by SCMPDS_4:8;
assume A8: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
assume A9: I is_halting_on s,P ; ::_thesis: IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
then  (I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P by A8, Th30;
then A10: P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s by Def3, A1;
 (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P by A8, A9, Th30;
then A11: ( Start-At (0,SCMPDS) c= s & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) ) by Th24, A1, FUNCT_4:25;
A12: stop ((I ';' (Goto ((card J) + 1))) ';' J) c= P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by FUNCT_4:25;
A13: Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1)) = Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))) by EXTPRO_1:4;
A14: CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1)))) = CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) by A13
.= CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) by A11, A4, A6, A7, Th31, A12
.= halt SCMPDS by A10, EXTPRO_1:def_15 ;
then A15: P +* (stop (if<0 (a,k1,I,J))) halts_on s by EXTPRO_1:29;
A16: CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),s) = (a,k1) >=0_goto ((card I) + 2) by A3, Th11, A1;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_(LifeSpan_((P_+*_(stop_((I_';'_(Goto_((card_J)_+_1)))_';'_J))),s))_+_1_holds_
CurInstr_((P_+*_(stop_(if<0_(a,k1,I,J)))),(Comput_((P_+*_(stop_(if<0_(a,k1,I,J)))),s,l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 implies CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,b1))) <> halt SCMPDS )
assume A17: l < (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 ; ::_thesis: CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,b1))) <> halt SCMPDS
A18: Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,0) = s by EXTPRO_1:2;
percases ( l = 0 or l <> 0 ) ;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,b1))) <> halt SCMPDS
then  CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,l))) = CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),s) by A18;
hence  CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,l))) <> halt SCMPDS by A16; ::_thesis: verum
end;
suppose l <> 0 ; ::_thesis: not CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,b1))) = halt SCMPDS
then consider n being Nat such that
A19: l = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A20: n < LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s) by A17, A19, XREAL_1:6;
assume A21: CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,l))) = halt SCMPDS ; ::_thesis: contradiction
A22: Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,(n + 1)) = Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)),n) by EXTPRO_1:4;
CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,n))) = CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)),n))) by A11, A4, A6, A7, Th31, A12
.= halt SCMPDS by A19, A21, A22 ;
hence  contradiction by A10, A20, EXTPRO_1:def_15; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,l))) = halt SCMPDS holds
(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 <= l ;
then A23: LifeSpan ((P +* (stop (if<0 (a,k1,I,J)))),s) = (LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1 by A14, A15, EXTPRO_1:def_15;
A24: DataPart (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) = DataPart (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)))) by A10, EXTPRO_1:23
.= DataPart (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)))) by A11, A4, A6, A7, Th31, A12
.= DataPart (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (stop (if<0 (a,k1,I,J)))),s)) by A15, A23, EXTPRO_1:23 ;
A25: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if<0_(a,k1,I,J)),P,s))_holds_
(IExec_((if<0_(a,k1,I,J)),P,s))_._x_=_((IExec_(((I_';'_(Goto_((card_J)_+_1)))_';'_J),P,s))_+*_(Start-At_((((card_I)_+_(card_J))_+_2),SCMPDS)))_._x
let x be set ; ::_thesis: ( x in dom (IExec ((if<0 (a,k1,I,J)),P,s)) implies (IExec ((if<0 (a,k1,I,J)),P,s)) . b1 = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1 )
A26: dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A27: x in dom (IExec ((if<0 (a,k1,I,J)),P,s)) ; ::_thesis: (IExec ((if<0 (a,k1,I,J)),P,s)) . b1 = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A27, SCMPDS_4:6;
supposeA28: x is Int_position ; ::_thesis: (IExec ((if<0 (a,k1,I,J)),P,s)) . b1 = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A29: not x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A26, TARSKI:def_1;
thus (IExec ((if<0 (a,k1,I,J)),P,s)) . x = (Result ((P +* (stop (if<0 (a,k1,I,J)))),s)) . x
.= (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) . x by A24, A28, SCMPDS_4:8
.= (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) . x
.= ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by A29, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA30: x = IC ; ::_thesis: (IExec ((if<0 (a,k1,I,J)),P,s)) . b1 = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1
A31: IC (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) = IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s))
.= ((card I) + (card J)) + 1 by A8, A9, Th32 ;
A32: x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A26, A30, TARSKI:def_1;
thus (IExec ((if<0 (a,k1,I,J)),P,s)) . x = (Result ((P +* (stop (if<0 (a,k1,I,J)))),s)) . x
.= (Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)) + 1))) . x by A15, A23, EXTPRO_1:23
.= IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s)))) by A30, EXTPRO_1:4
.= (IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))))) + 1 by A11, A4, A6, A7, Th31, A12
.= (IC (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s))) + 1 by A10, EXTPRO_1:23
.= IC (Start-At (((((card I) + (card J)) + 1) + 1),SCMPDS)) by A31, FUNCOP_1:72
.= ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by A30, A32, FUNCT_4:13 ; ::_thesis: verum
end;
end;
end;
dom (IExec ((if<0 (a,k1,I,J)),P,s)) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A25, FUNCT_1:2
.= ((IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A8, A9, Th33
.= (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by MEMSTR_0:36 ;
::_thesis: verum
end;

theorem Th101: :: SCMPDS_6:101
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for J being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for J being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for J being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let I be Program of SCMPDS; ::_thesis: for J being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let J be halt-free shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P holds
IExec ((if<0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) >= 0 & J is_closed_on s,P & J is_halting_on s,P implies IExec ((if<0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set pJ = stop J;
set s1 = Initialize s;
set P1 = P +* (stop J);
set IF = if<0 (a,k1,I,J);
set pIF = stop (if<0 (a,k1,I,J));
set s3 = Initialize s;
set P3 = P +* (stop (if<0 (a,k1,I,J)));
set s4 = Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1);
set P4 = P +* (stop (if<0 (a,k1,I,J)));
set i = (a,k1) >=0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
set iG = (((a,k1) >=0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1));
set SAl = Start-At ((((card I) + (card J)) + 2),SCMPDS);
A1: if<0 (a,k1,I,J) = (((a,k1) >=0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J by SCMPDS_4:14
.= ((a,k1) >=0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J) by SCMPDS_4:14 ;
A2: Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) >=0_goto ((card I) + 2)),(Initialize s)) by A1, Th11 ;
A3: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A4: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A3, FUNCT_4:11 ;
A5: IC (Initialize s) = 0 by MEMSTR_0:47;
assume  s . (DataLoc ((s . a),k1)) >= 0 ; ::_thesis: ( not J is_closed_on s,P or not J is_halting_on s,P or IExec ((if<0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
then A6: IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)) = ICplusConst ((Initialize s),((card I) + 2)) by A2, A4, SCMPDS_2:57
.= 0 + ((card I) + 2) by A5, Th12 ;
 for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)) . a by A2, SCMPDS_2:57;
then A7: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)) by SCMPDS_4:8;
card ((((a,k1) >=0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1))) = (card (((a,k1) >=0_goto ((card I) + 2)) ';' I)) + (card (Goto ((card J) + 1))) by AFINSQ_1:17
.= (card (((a,k1) >=0_goto ((card I) + 2)) ';' I)) + 1 by COMPOS_1:54
.= ((card I) + 1) + 1 by Th6
.= (card I) + (1 + 1) ;
then A8: Shift ((stop J),((card I) + 2)) c= stop (if<0 (a,k1,I,J)) by Th13;
 stop (if<0 (a,k1,I,J)) c= P +* (stop (if<0 (a,k1,I,J))) by FUNCT_4:25;
then  Shift ((stop J),((card I) + 2)) c= P +* (stop (if<0 (a,k1,I,J))) by A8, XBOOLE_1:1;
then A9: Shift ((stop J),((card I) + 2)) c= P +* (stop (if<0 (a,k1,I,J))) ;
assume A10: J is_closed_on s,P ; ::_thesis: ( not J is_halting_on s,P or IExec ((if<0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) )
then A11: ( Start-At (0,SCMPDS) c= Initialize s & J is_closed_on Initialize s,P +* (stop J) ) by Th24, FUNCT_4:25;
A12: stop J c= P +* (stop J) by FUNCT_4:25;
assume A13: J is_halting_on s,P ; ::_thesis: IExec ((if<0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))
then A14: P +* (stop J) halts_on Initialize s by Def3;
A15: Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1)) = Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)),(LifeSpan ((P +* (stop J)),(Initialize s)))) by EXTPRO_1:4;
A16: CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)),(LifeSpan ((P +* (stop J)),(Initialize s)))))) by A15
.= CurInstr ((P +* (stop J)),(Comput ((P +* (stop J)),(Initialize s),(LifeSpan ((P +* (stop J)),(Initialize s)))))) by A11, A9, A6, A7, Th31, A12
.= halt SCMPDS by A14, EXTPRO_1:def_15 ;
then A17: P +* (stop (if<0 (a,k1,I,J))) halts_on Initialize s by EXTPRO_1:29;
A18: CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s)) = (a,k1) >=0_goto ((card I) + 2) by A1, Th11;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_(LifeSpan_((P_+*_(stop_J)),(Initialize_s)))_+_1_holds_
CurInstr_((P_+*_(stop_(if<0_(a,k1,I,J)))),(Comput_((P_+*_(stop_(if<0_(a,k1,I,J)))),(Initialize_s),l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < (LifeSpan ((P +* (stop J)),(Initialize s))) + 1 implies CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),b1))) <> halt SCMPDS )
assume A19: l < (LifeSpan ((P +* (stop J)),(Initialize s))) + 1 ; ::_thesis: CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),b1))) <> halt SCMPDS
A20: Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),0) = Initialize s by EXTPRO_1:2;
percases ( l = 0 or l <> 0 ) ;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),b1))) <> halt SCMPDS
then  CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),l))) = CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s)) by A20;
hence  CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),l))) <> halt SCMPDS by A18; ::_thesis: verum
end;
suppose l <> 0 ; ::_thesis: not CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),b1))) = halt SCMPDS
then consider n being Nat such that
A21: l = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A22: n < LifeSpan ((P +* (stop J)),(Initialize s)) by A19, A21, XREAL_1:6;
assume A23: CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),l))) = halt SCMPDS ; ::_thesis: contradiction
A24: Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),(n + 1)) = Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)),n) by EXTPRO_1:4;
CurInstr ((P +* (stop J)),(Comput ((P +* (stop J)),(Initialize s),n))) = CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)),n))) by A11, A9, A6, A7, Th31, A12
.= halt SCMPDS by A21, A23, A24 ;
hence  contradiction by A14, A22, EXTPRO_1:def_15; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),l))) = halt SCMPDS holds
(LifeSpan ((P +* (stop J)),(Initialize s))) + 1 <= l ;
then A25: LifeSpan ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s)) = (LifeSpan ((P +* (stop J)),(Initialize s))) + 1 by A16, A17, EXTPRO_1:def_15;
A26: DataPart (Result ((P +* (stop J)),(Initialize s))) = DataPart (Comput ((P +* (stop J)),(Initialize s),(LifeSpan ((P +* (stop J)),(Initialize s))))) by A14, EXTPRO_1:23
.= DataPart (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)),(LifeSpan ((P +* (stop J)),(Initialize s))))) by A11, A9, A6, A7, Th31, A12
.= DataPart (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s))) by A17, A25, EXTPRO_1:23 ;
A27: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if<0_(a,k1,I,J)),P,(Initialize_s)))_holds_
(IExec_((if<0_(a,k1,I,J)),P,(Initialize_s)))_._x_=_((IExec_(J,P,(Initialize_s)))_+*_(Start-At_((((card_I)_+_(card_J))_+_2),SCMPDS)))_._x
let x be set ; ::_thesis: ( x in dom (IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) implies (IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . b1 = ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1 )
A28: dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A29: x in dom (IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) ; ::_thesis: (IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . b1 = ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A29, SCMPDS_4:6;
supposeA30: x is Int_position ; ::_thesis: (IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . b1 = ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A31: not x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A28, TARSKI:def_1;
thus (IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . x = (Result ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s))) . x
.= (Result ((P +* (stop J)),(Initialize s))) . x by A26, A30, SCMPDS_4:8
.= (IExec (J,P,(Initialize s))) . x
.= ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by A31, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA32: x = IC ; ::_thesis: (IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . b1 = ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . b1
A33: IC (Result ((P +* (stop J)),(Initialize s))) = IC (IExec (J,P,(Initialize s)))
.= card J by A10, A13, Th34 ;
A34: x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A28, A32, TARSKI:def_1;
thus (IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . x = (Result ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s))) . x
.= (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),((LifeSpan ((P +* (stop J)),(Initialize s))) + 1))) . x by A17, A25, EXTPRO_1:23
.= IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),1)),(LifeSpan ((P +* (stop J)),(Initialize s))))) by A32, EXTPRO_1:4
.= (IC (Comput ((P +* (stop J)),(Initialize s),(LifeSpan ((P +* (stop J)),(Initialize s)))))) + ((card I) + 2) by A11, A9, A6, A7, Th31, A12
.= (IC (Result ((P +* (stop J)),(Initialize s)))) + ((card I) + 2) by A14, EXTPRO_1:23
.= IC (Start-At (((card J) + ((card I) + 2)),SCMPDS)) by A33, FUNCOP_1:72
.= ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by A32, A34, FUNCT_4:13 ; ::_thesis: verum
end;
end;
end;
dom (IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom ((IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((if<0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A27, FUNCT_1:2; ::_thesis: verum
end;

registration
let I, J be parahalting shiftable Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if<0 (a,k1,I,J) ->  parahalting shiftable ;
correctness
coherence
( if<0 (a,k1,I,J) is shiftable & if<0 (a,k1,I,J) is parahalting );
proof
set i = (a,k1) >=0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
set IF = if<0 (a,k1,I,J);
set pIF = stop (if<0 (a,k1,I,J));
reconsider IJ = (I ';' (Goto ((card J) + 1))) ';' J as shiftable Program of SCMPDS ;
thus  if<0 (a,k1,I,J) is shiftable ; ::_thesis: if<0 (a,k1,I,J) is parahalting
let s be 0 -started State of SCMPDS; :: according to SCMPDS_4:def_7 ::_thesis: for b1 being set holds
( not stop (if<0 (a,k1,I,J)) c= b1 or b1 halts_on s )
let P be Instruction-Sequence of SCMPDS; ::_thesis: ( not stop (if<0 (a,k1,I,J)) c= P or P halts_on s )
A1: Initialize s = s by MEMSTR_0:44;
assume  stop (if<0 (a,k1,I,J)) c= P ; ::_thesis: P halts_on s
then A2: P = P +* (stop (if<0 (a,k1,I,J))) by FUNCT_4:98;
A3: ( J is_closed_on s,P & J is_halting_on s,P ) by Th20, Th21;
A4: ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
percases ( s . (DataLoc ((s . a),k1)) < 0 or s . (DataLoc ((s . a),k1)) >= 0 ) ;
suppose s . (DataLoc ((s . a),k1)) < 0 ; ::_thesis: P halts_on s
then  if<0 (a,k1,I,J) is_halting_on s,P by A4, Th98;
hence  P halts_on s by A2, Def3, A1; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),k1)) >= 0 ; ::_thesis: P halts_on s
then  if<0 (a,k1,I,J) is_halting_on s,P by A3, Th99;
hence  P halts_on s by A2, Def3, A1; ::_thesis: verum
end;
end;
end;
end;

registration
let I, J be halt-free Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if<0 (a,k1,I,J) ->  halt-free ;
coherence
 if<0 (a,k1,I,J) is halt-free ;
end;

theorem :: SCMPDS_6:102
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I, J being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if<0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I, J being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if<0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
let s be 0 -started State of SCMPDS; ::_thesis: for I, J being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if<0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
let I, J be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer holds IC (IExec ((if<0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
let a be Int_position; ::_thesis: for k1 being Integer holds IC (IExec ((if<0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
let k1 be Integer; ::_thesis: IC (IExec ((if<0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
set IF = if<0 (a,k1,I,J);
A1: ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
A2: ( J is_closed_on s,P & J is_halting_on s,P ) by Th20, Th21;
A3: Initialize s = s by MEMSTR_0:44;
percases ( s . (DataLoc ((s . a),k1)) < 0 or s . (DataLoc ((s . a),k1)) >= 0 ) ;
suppose s . (DataLoc ((s . a),k1)) < 0 ; ::_thesis: IC (IExec ((if<0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
then  IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A1, Th100;
hence  IC (IExec ((if<0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2 by FUNCT_4:113; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),k1)) >= 0 ; ::_thesis: IC (IExec ((if<0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
then  IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A2, Th101, A3;
hence  IC (IExec ((if<0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2 by FUNCT_4:113; ::_thesis: verum
end;
end;
end;

theorem :: SCMPDS_6:103
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if<0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if<0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free parahalting shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if<0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let I be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if<0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let J be shiftable Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if<0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if<0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) < 0 implies (IExec ((if<0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b )
assume A1: s . (DataLoc ((s . a),k1)) < 0 ; ::_thesis: (IExec ((if<0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
A2: not b in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by SCMPDS_4:18;
 ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
then  IExec ((if<0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A1, Th100;
hence  (IExec ((if<0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b by A2, FUNCT_4:11; ::_thesis: verum
end;

theorem :: SCMPDS_6:104
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for J being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
(IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for J being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
(IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for J being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
(IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
let I be Program of SCMPDS; ::_thesis: for J being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
(IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
let J be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
(IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
(IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) >= 0 implies (IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b )
set IF = if<0 (a,k1,I,J);
assume A1: s . (DataLoc ((s . a),k1)) >= 0 ; ::_thesis: (IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
A2: not b in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by SCMPDS_4:18;
 ( J is_closed_on s,P & J is_halting_on s,P ) by Th20, Th21;
then  IExec ((if<0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A1, Th101;
hence  (IExec ((if<0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b by A2, FUNCT_4:11; ::_thesis: verum
end;

begin

theorem :: SCMPDS_6:105
for a being Int_position
for k1 being Integer
for I being Program of SCMPDS holds card (if<0 (a,k1,I)) = (card I) + 1 by Th6;

theorem :: SCMPDS_6:106
for a being Int_position
for k1 being Integer
for I being Program of SCMPDS holds 0 in dom (if<0 (a,k1,I))
proof
let a be Int_position; ::_thesis: for k1 being Integer
for I being Program of SCMPDS holds 0 in dom (if<0 (a,k1,I))
let k1 be Integer; ::_thesis: for I being Program of SCMPDS holds 0 in dom (if<0 (a,k1,I))
let I be Program of SCMPDS; ::_thesis: 0 in dom (if<0 (a,k1,I))
set ci = card (if<0 (a,k1,I));
 card (if<0 (a,k1,I)) = (card I) + 1 by Th6;
hence  0 in dom (if<0 (a,k1,I)) by AFINSQ_1:66; ::_thesis: verum
end;

theorem :: SCMPDS_6:107
for a being Int_position
for k1 being Integer
for I being Program of SCMPDS holds (if<0 (a,k1,I)) . 0 = (a,k1) >=0_goto ((card I) + 1) by Th7;

theorem Th108: :: SCMPDS_6:108
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
let I be shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P implies ( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
set IF = if<0 (a,k1,I);
set pIF = stop (if<0 (a,k1,I));
set pI = stop I;
set s2 = Initialize s;
set s3 = Initialize s;
set P2 = P +* (stop I);
set P3 = P +* (stop (if<0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if<0 (a,k1,I)));
set i = (a,k1) >=0_goto ((card I) + 1);
A1: stop I c= P +* (stop I) by FUNCT_4:25;
A2: 0 in dom (stop (if<0 (a,k1,I))) by COMPOS_1:36;
A3: IC (Initialize s) = 0 by MEMSTR_0:47;
A4: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A5: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A4, FUNCT_4:11 ;
A6: Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if<0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) >=0_goto ((card I) + 1)),(Initialize s)) by Th11 ;
 for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)) . a by A6, SCMPDS_2:57;
then A7: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)) by SCMPDS_4:8;
assume  s . (DataLoc ((s . a),k1)) < 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or ( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P ) )
then A8: IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)) = succ (IC (Initialize s)) by A6, A5, SCMPDS_2:57
.= 0 + 1 by A3 ;
assume A9: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or ( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P ) )
then A10: I is_closed_on Initialize s,P +* (stop I) by Th24;
assume  I is_halting_on s,P ; ::_thesis: ( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
then A11: P +* (stop I) halts_on Initialize s by Def3;
A12: ( Start-At (0,SCMPDS) c= Initialize s & Shift ((stop I),1) c= P +* (stop (if<0 (a,k1,I))) ) by Lm6, FUNCT_4:25;
A13: card (stop (if<0 (a,k1,I))) = (card (if<0 (a,k1,I))) + 1 by COMPOS_1:55
.= ((card I) + 1) + 1 by Th6 ;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(if<0_(a,k1,I)))),(Initialize_s),k))_in_dom_(stop_(if<0_(a,k1,I)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<0 (a,k1,I)))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<0 (a,k1,I)))
then consider k1 being Nat such that
A14: k1 + 1 = k by NAT_1:6;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def_12;
reconsider m = IC (Comput ((P +* (stop I)),(Initialize s),k1)) as Element of NAT ;
A15: card (stop (if<0 (a,k1,I))) = (card (stop I)) + 1 by A13, COMPOS_1:55;
 m in dom (stop I) by A9, Def2;
then  m < card (stop I) by AFINSQ_1:66;
then A16: m + 1 < card (stop (if<0 (a,k1,I))) by A15, XREAL_1:6;
IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),k)) = IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)),k1)) by A14, EXTPRO_1:4
.= m + 1 by A10, A12, A8, A7, Th31, A1 ;
hence  IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if<0 (a,k1,I))) by A16, AFINSQ_1:66; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<0 (a,k1,I)))
hence  IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if<0 (a,k1,I))) by A2, A3, EXTPRO_1:2; ::_thesis: verum
end;
end;
end;
hence  if<0 (a,k1,I) is_closed_on s,P by Def2; ::_thesis: if<0 (a,k1,I) is_halting_on s,P
A17: Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)) = Comput ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))) by EXTPRO_1:4;
CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A17
.= CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A10, A12, A8, A7, Th31, A1
.= halt SCMPDS by A11, EXTPRO_1:def_15 ;
then  P +* (stop (if<0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
hence  if<0 (a,k1,I) is_halting_on s,P by Def3; ::_thesis: verum
end;

theorem Th109: :: SCMPDS_6:109
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) >= 0 implies ( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
assume A1: s . (DataLoc ((s . a),k1)) >= 0 ; ::_thesis: ( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
set i = (a,k1) >=0_goto ((card I) + 1);
set IF = if<0 (a,k1,I);
set pIF = stop (if<0 (a,k1,I));
set s3 = Initialize s;
set P3 = P +* (stop (if<0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if<0 (a,k1,I)));
A2: IC (Initialize s) = 0 by MEMSTR_0:47;
A3: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A4: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A3, FUNCT_4:11 ;
Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if<0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) >=0_goto ((card I) + 1)),(Initialize s)) by Th11 ;
then A5: IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)) = ICplusConst ((Initialize s),((card I) + 1)) by A1, A4, SCMPDS_2:57
.= 0 + ((card I) + 1) by A2, Th12 ;
A6: card (if<0 (a,k1,I)) = (card I) + 1 by Th6;
then A7: (card I) + 1 in dom (stop (if<0 (a,k1,I))) by COMPOS_1:64;
A8: (P +* (stop (if<0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1))) = (P +* (stop (if<0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1))) by PBOOLE:143;
 stop (if<0 (a,k1,I)) c= P +* (stop (if<0 (a,k1,I))) by FUNCT_4:25;
then  stop (if<0 (a,k1,I)) c= P +* (stop (if<0 (a,k1,I))) ;
then (P +* (stop (if<0 (a,k1,I)))) . ((card I) + 1) = (stop (if<0 (a,k1,I))) . ((card I) + 1) by A7, GRFUNC_1:2
.= halt SCMPDS by A6, COMPOS_1:64 ;
then A9: CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1))) = halt SCMPDS by A5, A8;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(if<0_(a,k1,I)))),(Initialize_s),k))_in_dom_(stop_(if<0_(a,k1,I)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<0 (a,k1,I)))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<0 (a,k1,I)))
then  1 + 0 <= k by INT_1:7;
hence  IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if<0 (a,k1,I))) by A7, A5, A9, EXTPRO_1:5; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<0 (a,k1,I)))
then  Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),k) = Initialize s by EXTPRO_1:2;
hence  IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if<0 (a,k1,I))) by A2, COMPOS_1:36; ::_thesis: verum
end;
end;
end;
hence  if<0 (a,k1,I) is_closed_on s,P by Def2; ::_thesis: if<0 (a,k1,I) is_halting_on s,P
 P +* (stop (if<0 (a,k1,I))) halts_on Initialize s by A9, EXTPRO_1:29;
hence  if<0 (a,k1,I) is_halting_on s,P by Def3; ::_thesis: verum
end;

theorem Th110: :: SCMPDS_6:110
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
let s be State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if<0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P implies IExec ((if<0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set IF = if<0 (a,k1,I);
set pIF = stop (if<0 (a,k1,I));
set pI = stop I;
set s2 = Initialize s;
set s3 = Initialize s;
set P2 = P +* (stop I);
set P3 = P +* (stop (if<0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if<0 (a,k1,I)));
set i = (a,k1) >=0_goto ((card I) + 1);
set SAl = Start-At (((card I) + 1),SCMPDS);
A1: stop I c= P +* (stop I) by FUNCT_4:25;
A2: IC (Initialize s) = 0 by MEMSTR_0:47;
A3: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A4: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A3, FUNCT_4:11 ;
A5: Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if<0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) >=0_goto ((card I) + 1)),(Initialize s)) by Th11 ;
assume  s . (DataLoc ((s . a),k1)) < 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or IExec ((if<0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) )
then A6: IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)) = succ (IC (Initialize s)) by A5, A4, SCMPDS_2:57
.= 0 + 1 by A2 ;
 for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)) . a by A5, SCMPDS_2:57;
then A7: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)) by SCMPDS_4:8;
assume A8: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or IExec ((if<0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) )
then A9: I is_closed_on Initialize s,P +* (stop I) by Th24;
A10: ( Start-At (0,SCMPDS) c= Initialize s & Shift ((stop I),1) c= P +* (stop (if<0 (a,k1,I))) ) by Lm6, FUNCT_4:25;
assume A11: I is_halting_on s,P ; ::_thesis: IExec ((if<0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
then A12: P +* (stop I) halts_on Initialize s by Def3;
A13: Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)) = Comput ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))) by EXTPRO_1:4;
A14: CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A13
.= CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A9, A10, A6, A7, Th31, A1
.= halt SCMPDS by A12, EXTPRO_1:def_15 ;
then A15: P +* (stop (if<0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
A16: CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Initialize s)) = (a,k1) >=0_goto ((card I) + 1) by Th11;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_(LifeSpan_((P_+*_(stop_I)),(Initialize_s)))_+_1_holds_
CurInstr_((P_+*_(stop_(if<0_(a,k1,I)))),(Comput_((P_+*_(stop_(if<0_(a,k1,I)))),(Initialize_s),l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 implies CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS )
assume A17: l < (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 ; ::_thesis: CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
A18: Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),0) = Initialize s by EXTPRO_1:2;
percases ( l = 0 or l <> 0 ) ;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
then  CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),l))) = CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Initialize s)) by A18;
hence  CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A16; ::_thesis: verum
end;
suppose l <> 0 ; ::_thesis: not CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),b1))) = halt SCMPDS
then consider n being Nat such that
A19: l = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A20: n < LifeSpan ((P +* (stop I)),(Initialize s)) by A17, A19, XREAL_1:6;
assume A21: CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS ; ::_thesis: contradiction
A22: Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),(n + 1)) = Comput ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)),n) by EXTPRO_1:4;
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),n))) = CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)),n))) by A9, A10, A6, A7, Th31, A1
.= halt SCMPDS by A19, A21, A22 ;
hence  contradiction by A12, A20, EXTPRO_1:def_15; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS holds
(LifeSpan ((P +* (stop I)),(Initialize s))) + 1 <= l ;
then A23: LifeSpan ((P +* (stop (if<0 (a,k1,I)))),(Initialize s)) = (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 by A14, A15, EXTPRO_1:def_15;
A24: DataPart (Result ((P +* (stop I)),(Initialize s))) = DataPart (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A12, EXTPRO_1:23
.= DataPart (Comput ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A9, A10, A6, A7, Th31, A1
.= DataPart (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (stop (if<0 (a,k1,I)))),(Initialize s))) by A15, A23, EXTPRO_1:23 ;
A25: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if<0_(a,k1,I)),P,(Initialize_s)))_holds_
(IExec_((if<0_(a,k1,I)),P,(Initialize_s)))_._x_=_((IExec_(I,P,(Initialize_s)))_+*_(Start-At_(((card_I)_+_1),SCMPDS)))_._x
let x be set ; ::_thesis: ( x in dom (IExec ((if<0 (a,k1,I)),P,(Initialize s))) implies (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . b1 )
A26: dom (Start-At (((card I) + 1),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A27: x in dom (IExec ((if<0 (a,k1,I)),P,(Initialize s))) ; ::_thesis: (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A27, SCMPDS_4:6;
supposeA28: x is Int_position ; ::_thesis: (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A29: not x in dom (Start-At (((card I) + 1),SCMPDS)) by A26, TARSKI:def_1;
thus (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . x = (Result ((P +* (stop (if<0 (a,k1,I)))),(Initialize s))) . x
.= (Result ((P +* (stop I)),(Initialize s))) . x by A24, A28, SCMPDS_4:8
.= (IExec (I,P,(Initialize s))) . x
.= ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . x by A29, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA30: x = IC ; ::_thesis: (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . b1
A31: IC (Result ((P +* (stop I)),(Initialize s))) = IC (IExec (I,P,(Initialize s)))
.= card I by A8, A11, Th34 ;
A32: x in dom (Start-At (((card I) + 1),SCMPDS)) by A26, A30, TARSKI:def_1;
thus (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . x = (Result ((P +* (stop (if<0 (a,k1,I)))),(Initialize s))) . x
.= (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) . x by A15, A23, EXTPRO_1:23
.= IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A30, EXTPRO_1:4
.= (IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) + 1 by A9, A10, A6, A7, Th31, A1
.= (IC (Result ((P +* (stop I)),(Initialize s)))) + 1 by A12, EXTPRO_1:23
.= IC (Start-At (((card I) + 1),SCMPDS)) by A31, FUNCOP_1:72
.= ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . x by A30, A32, FUNCT_4:13 ; ::_thesis: verum
end;
end;
end;
dom (IExec ((if<0 (a,k1,I)),P,(Initialize s))) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((if<0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) by A25, FUNCT_1:2; ::_thesis: verum
end;

theorem Th111: :: SCMPDS_6:111
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
IExec ((if<0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
IExec ((if<0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
IExec ((if<0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
IExec ((if<0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
IExec ((if<0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) >= 0 implies IExec ((if<0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set IF = if<0 (a,k1,I);
set pIF = stop (if<0 (a,k1,I));
set s3 = Initialize s;
set P3 = P +* (stop (if<0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if<0 (a,k1,I)));
set i = (a,k1) >=0_goto ((card I) + 1);
set SAl = Start-At (((card I) + 1),SCMPDS);
A1: IC (Initialize s) = 0 by MEMSTR_0:47;
A2: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A3: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A2, FUNCT_4:11 ;
A4: Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if<0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) >=0_goto ((card I) + 1)),(Initialize s)) by Th11 ;
assume  s . (DataLoc ((s . a),k1)) >= 0 ; ::_thesis: IExec ((if<0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
then A5: IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)) = ICplusConst ((Initialize s),((card I) + 1)) by A4, A3, SCMPDS_2:57
.= 0 + ((card I) + 1) by A1, Th12 ;
 stop (if<0 (a,k1,I)) c= P +* (stop (if<0 (a,k1,I))) by FUNCT_4:25;
then A6: stop (if<0 (a,k1,I)) c= P +* (stop (if<0 (a,k1,I))) ;
A7: (P +* (stop (if<0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1))) = (P +* (stop (if<0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1))) by PBOOLE:143;
A8: card (if<0 (a,k1,I)) = (card I) + 1 by Th6;
then  (card I) + 1 in dom (stop (if<0 (a,k1,I))) by COMPOS_1:64;
then (P +* (stop (if<0 (a,k1,I)))) . ((card I) + 1) = (stop (if<0 (a,k1,I))) . ((card I) + 1) by A6, GRFUNC_1:2
.= halt SCMPDS by A8, COMPOS_1:64 ;
then A9: CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1))) = halt SCMPDS by A5, A7;
then A10: P +* (stop (if<0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
A11: CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Initialize s)) = (a,k1) >=0_goto ((card I) + 1) by Th11;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_1_holds_
CurInstr_((P_+*_(stop_(if<0_(a,k1,I)))),(Comput_((P_+*_(stop_(if<0_(a,k1,I)))),(Initialize_s),l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < 1 implies CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS )
A12: Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),0) = Initialize s by EXTPRO_1:2;
assume  l < 1 ; ::_thesis: CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS
then  l < 1 + 0 ;
then  l = 0 by NAT_1:13;
then  CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),l))) = CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Initialize s)) by A12;
hence  CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A11; ::_thesis: verum
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS holds
1 <= l ;
then  LifeSpan ((P +* (stop (if<0 (a,k1,I)))),(Initialize s)) = 1 by A9, A10, EXTPRO_1:def_15;
then A13: Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1) = Result ((P +* (stop (if<0 (a,k1,I)))),(Initialize s)) by A10, EXTPRO_1:23;
A14: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if<0_(a,k1,I)),P,(Initialize_s)))_holds_
(IExec_((if<0_(a,k1,I)),P,(Initialize_s)))_._x_=_(s_+*_(Start-At_(((card_I)_+_1),SCMPDS)))_._x
A15: dom (Start-At (((card I) + 1),SCMPDS)) = {(IC )} by FUNCOP_1:13;
let x be set ; ::_thesis: ( x in dom (IExec ((if<0 (a,k1,I)),P,(Initialize s))) implies (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 1),SCMPDS))) . b1 )
assume A16: x in dom (IExec ((if<0 (a,k1,I)),P,(Initialize s))) ; ::_thesis: (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 1),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A16, SCMPDS_4:6;
supposeA17: x is Int_position ; ::_thesis: (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 1),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A18: not x in dom (Start-At (((card I) + 1),SCMPDS)) by A15, TARSKI:def_1;
A19: not x in dom (Start-At (0,SCMPDS)) by A17, SCMPDS_4:18;
thus (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . x = (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)) . x by A13
.= (Initialize s) . x by A4, A17, SCMPDS_2:57
.= s . x by A19, FUNCT_4:11
.= (s +* (Start-At (((card I) + 1),SCMPDS))) . x by A18, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA20: x = IC ; ::_thesis: (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 1),SCMPDS))) . b1
hence (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . x = (card I) + 1 by A5, A13
.= (s +* (Start-At (((card I) + 1),SCMPDS))) . x by A20, FUNCT_4:113 ;
::_thesis: verum
end;
end;
end;
dom (IExec ((if<0 (a,k1,I)),P,(Initialize s))) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom (s +* (Start-At (((card I) + 1),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((if<0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS)) by A14, FUNCT_1:2; ::_thesis: verum
end;

registration
let I be parahalting shiftable Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if<0 (a,k1,I) ->  parahalting shiftable ;
correctness
coherence
( if<0 (a,k1,I) is shiftable & if<0 (a,k1,I) is parahalting );
proof
set i = (a,k1) >=0_goto ((card I) + 1);
set IF = if<0 (a,k1,I);
set pIF = stop (if<0 (a,k1,I));
thus  if<0 (a,k1,I) is shiftable ; ::_thesis: if<0 (a,k1,I) is parahalting
let s be 0 -started State of SCMPDS; :: according to SCMPDS_4:def_7 ::_thesis: for b1 being set holds
( not stop (if<0 (a,k1,I)) c= b1 or b1 halts_on s )
let P be Instruction-Sequence of SCMPDS; ::_thesis: ( not stop (if<0 (a,k1,I)) c= P or P halts_on s )
A1: Initialize s = s by MEMSTR_0:44;
assume  stop (if<0 (a,k1,I)) c= P ; ::_thesis: P halts_on s
then A2: P = P +* (stop (if<0 (a,k1,I))) by FUNCT_4:98;
A3: ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
percases ( s . (DataLoc ((s . a),k1)) < 0 or s . (DataLoc ((s . a),k1)) >= 0 ) ;
suppose s . (DataLoc ((s . a),k1)) < 0 ; ::_thesis: P halts_on s
then  if<0 (a,k1,I) is_halting_on s,P by A3, Th108;
hence  P halts_on s by A2, Def3, A1; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),k1)) >= 0 ; ::_thesis: P halts_on s
then  if<0 (a,k1,I) is_halting_on s,P by Th109;
hence  P halts_on s by A2, Def3, A1; ::_thesis: verum
end;
end;
end;
end;

registration
let I be halt-free Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if<0 (a,k1,I) ->  halt-free ;
coherence
 if<0 (a,k1,I) is halt-free ;
end;

theorem :: SCMPDS_6:112
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if<0 (a,k1,I)),P,(Initialize s))) = (card I) + 1
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if<0 (a,k1,I)),P,(Initialize s))) = (card I) + 1
let s be State of SCMPDS; ::_thesis: for I being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if<0 (a,k1,I)),P,(Initialize s))) = (card I) + 1
let I be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer holds IC (IExec ((if<0 (a,k1,I)),P,(Initialize s))) = (card I) + 1
let a be Int_position; ::_thesis: for k1 being Integer holds IC (IExec ((if<0 (a,k1,I)),P,(Initialize s))) = (card I) + 1
let k1 be Integer; ::_thesis: IC (IExec ((if<0 (a,k1,I)),P,(Initialize s))) = (card I) + 1
set IF = if<0 (a,k1,I);
A1: ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
percases ( s . (DataLoc ((s . a),k1)) < 0 or s . (DataLoc ((s . a),k1)) >= 0 ) ;
suppose s . (DataLoc ((s . a),k1)) < 0 ; ::_thesis: IC (IExec ((if<0 (a,k1,I)),P,(Initialize s))) = (card I) + 1
then  IExec ((if<0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) by A1, Th110;
hence  IC (IExec ((if<0 (a,k1,I)),P,(Initialize s))) = (card I) + 1 by FUNCT_4:113; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),k1)) >= 0 ; ::_thesis: IC (IExec ((if<0 (a,k1,I)),P,(Initialize s))) = (card I) + 1
then  IExec ((if<0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS)) by Th111;
hence  IC (IExec ((if<0 (a,k1,I)),P,(Initialize s))) = (card I) + 1 by FUNCT_4:113; ::_thesis: verum
end;
end;
end;

theorem :: SCMPDS_6:113
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let s be State of SCMPDS; ::_thesis: for I being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let I be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) < 0 implies (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b )
assume A1: s . (DataLoc ((s . a),k1)) < 0 ; ::_thesis: (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
A2: not b in dom (Start-At (((card I) + 1),SCMPDS)) by SCMPDS_4:18;
 ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
then  IExec ((if<0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) by A1, Th110;
hence  (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b by A2, FUNCT_4:11; ::_thesis: verum
end;

theorem :: SCMPDS_6:114
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
(IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = s . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
(IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = s . b
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
(IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = s . b
let I be Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
(IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = s . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
(IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = s . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) >= 0 implies (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = s . b )
assume  s . (DataLoc ((s . a),k1)) >= 0 ; ::_thesis: (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = s . b
then A1: IExec ((if<0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS)) by Th111;
 not b in dom (Start-At (((card I) + 1),SCMPDS)) by SCMPDS_4:18;
hence  (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = s . b by A1, FUNCT_4:11; ::_thesis: verum
end;

begin

theorem :: SCMPDS_6:115
for a being Int_position
for k1 being Integer
for I being Program of SCMPDS holds card (if>=0 (a,k1,I)) = (card I) + 2 by Lm8;

theorem :: SCMPDS_6:116
for a being Int_position
for k1 being Integer
for I being Program of SCMPDS holds
( 0 in dom (if>=0 (a,k1,I)) & 1 in dom (if>=0 (a,k1,I)) ) by Lm9;

theorem :: SCMPDS_6:117
for a being Int_position
for k1 being Integer
for I being Program of SCMPDS holds
( (if>=0 (a,k1,I)) . 0 = (a,k1) >=0_goto 2 & (if>=0 (a,k1,I)) . 1 = goto ((card I) + 1) ) by Lm10;

theorem Th118: :: SCMPDS_6:118
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>=0 (a,k1,I) is_closed_on s,P & if>=0 (a,k1,I) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>=0 (a,k1,I) is_closed_on s,P & if>=0 (a,k1,I) is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>=0 (a,k1,I) is_closed_on s,P & if>=0 (a,k1,I) is_halting_on s,P )
let I be shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>=0 (a,k1,I) is_closed_on s,P & if>=0 (a,k1,I) is_halting_on s,P )
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & I is_closed_on s,P & I is_halting_on s,P holds
( if>=0 (a,k1,I) is_closed_on s,P & if>=0 (a,k1,I) is_halting_on s,P )
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) >= 0 & I is_closed_on s,P & I is_halting_on s,P implies ( if>=0 (a,k1,I) is_closed_on s,P & if>=0 (a,k1,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
set IF = if>=0 (a,k1,I);
set pIF = stop (if>=0 (a,k1,I));
set pI = stop I;
set s2 = Initialize s;
set s3 = Initialize s;
set P2 = P +* (stop I);
set P3 = P +* (stop (if>=0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if>=0 (a,k1,I)));
set i = (a,k1) >=0_goto 2;
set j = goto ((card I) + 1);
A1: stop I c= P +* (stop I) by FUNCT_4:25;
A2: if>=0 (a,k1,I) = ((a,k1) >=0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:16;
A3: Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) >=0_goto 2),(Initialize s)) by A2, Th11 ;
 for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)) . a by A3, SCMPDS_2:57;
then A4: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)) by SCMPDS_4:8;
A5: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A6: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A5, FUNCT_4:11 ;
A7: IC (Initialize s) = 0 by MEMSTR_0:47;
assume  s . (DataLoc ((s . a),k1)) >= 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or ( if>=0 (a,k1,I) is_closed_on s,P & if>=0 (a,k1,I) is_halting_on s,P ) )
then A8: IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)) = ICplusConst ((Initialize s),2) by A3, A6, SCMPDS_2:57
.= 0 + 2 by A7, Th12 ;
assume A9: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or ( if>=0 (a,k1,I) is_closed_on s,P & if>=0 (a,k1,I) is_halting_on s,P ) )
then A10: I is_closed_on Initialize s,P +* (stop I) by Th24;
assume  I is_halting_on s,P ; ::_thesis: ( if>=0 (a,k1,I) is_closed_on s,P & if>=0 (a,k1,I) is_halting_on s,P )
then A11: P +* (stop I) halts_on Initialize s by Def3;
A12: ( Start-At (0,SCMPDS) c= Initialize s & Shift ((stop I),2) c= P +* (stop (if>=0 (a,k1,I))) ) by Lm7, FUNCT_4:25;
A13: 0 in dom (stop (if>=0 (a,k1,I))) by COMPOS_1:36;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(if>=0_(a,k1,I)))),(Initialize_s),k))_in_dom_(stop_(if>=0_(a,k1,I)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if>=0 (a,k1,I)))
percases ( 0 < k or k = 0 ) ;
suppose 0 < k ; ::_thesis: IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if>=0 (a,k1,I)))
then consider k1 being Nat such that
A14: k1 + 1 = k by NAT_1:6;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def_12;
reconsider m = IC (Comput ((P +* (stop I)),(Initialize s),k1)) as Element of NAT ;
A15: card (stop (if>=0 (a,k1,I))) = 1 + (card (if>=0 (a,k1,I))) by COMPOS_1:55
.= 1 + ((card I) + 2) by Lm8
.= (1 + (card I)) + 2
.= (card (stop I)) + 2 by COMPOS_1:55 ;
 m in dom (stop I) by A9, Def2;
then  m < card (stop I) by AFINSQ_1:66;
then A16: m + 2 < card (stop (if>=0 (a,k1,I))) by A15, XREAL_1:6;
IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),k)) = IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)),k1)) by A14, EXTPRO_1:4
.= m + 2 by A10, A12, A8, A4, Th31, A1 ;
hence  IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if>=0 (a,k1,I))) by A16, AFINSQ_1:66; ::_thesis: verum
end;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if>=0 (a,k1,I)))
hence  IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if>=0 (a,k1,I))) by A13, A7, EXTPRO_1:2; ::_thesis: verum
end;
end;
end;
hence  if>=0 (a,k1,I) is_closed_on s,P by Def2; ::_thesis: if>=0 (a,k1,I) is_halting_on s,P
A17: Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)) = Comput ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))) by EXTPRO_1:4;
CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A17
.= CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A10, A12, A8, A4, Th31, A1
.= halt SCMPDS by A11, EXTPRO_1:def_15 ;
then  P +* (stop (if>=0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
hence  if>=0 (a,k1,I) is_halting_on s,P by Def3; ::_thesis: verum
end;

theorem Th119: :: SCMPDS_6:119
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
( if>=0 (a,k1,I) is_closed_on s,P & if>=0 (a,k1,I) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
( if>=0 (a,k1,I) is_closed_on s,P & if>=0 (a,k1,I) is_halting_on s,P )
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
( if>=0 (a,k1,I) is_closed_on s,P & if>=0 (a,k1,I) is_halting_on s,P )
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
( if>=0 (a,k1,I) is_closed_on s,P & if>=0 (a,k1,I) is_halting_on s,P )
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
( if>=0 (a,k1,I) is_closed_on s,P & if>=0 (a,k1,I) is_halting_on s,P )
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) < 0 implies ( if>=0 (a,k1,I) is_closed_on s,P & if>=0 (a,k1,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
assume A1: s . (DataLoc ((s . a),k1)) < 0 ; ::_thesis: ( if>=0 (a,k1,I) is_closed_on s,P & if>=0 (a,k1,I) is_halting_on s,P )
set i = (a,k1) >=0_goto 2;
set j = goto ((card I) + 1);
set IF = if>=0 (a,k1,I);
set pIF = stop (if>=0 (a,k1,I));
set s3 = Initialize s;
set P3 = P +* (stop (if>=0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1);
set s5 = Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),2);
set P4 = P +* (stop (if>=0 (a,k1,I)));
set P5 = P +* (stop (if>=0 (a,k1,I)));
A2: if>=0 (a,k1,I) = ((a,k1) >=0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:16;
A3: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A4: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A3, FUNCT_4:11 ;
A5: IC (Initialize s) = 0 by MEMSTR_0:47;
Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) >=0_goto 2),(Initialize s)) by A2, Th11 ;
then A6: IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)) = succ (IC (Initialize s)) by A1, A4, SCMPDS_2:57
.= 0 + 1 by A5 ;
A7: stop (if>=0 (a,k1,I)) c= P +* (stop (if>=0 (a,k1,I))) by FUNCT_4:25;
then A8: stop (if>=0 (a,k1,I)) c= P +* (stop (if>=0 (a,k1,I))) ;
A9: 1 in dom (if>=0 (a,k1,I)) by Lm9;
then  1 in dom (stop (if>=0 (a,k1,I))) by COMPOS_1:62;
then A10: (P +* (stop (if>=0 (a,k1,I)))) . 1 = (stop (if>=0 (a,k1,I))) . 1 by A8, GRFUNC_1:2
.= (if>=0 (a,k1,I)) . 1 by A9, COMPOS_1:63
.= goto ((card I) + 1) by Lm10 ;
A11: card (if>=0 (a,k1,I)) = (card I) + 2 by Lm8;
then A12: (card I) + 2 in dom (stop (if>=0 (a,k1,I))) by COMPOS_1:64;
A13: (P +* (stop (if>=0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1))) = (P +* (stop (if>=0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1))) by PBOOLE:143;
Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),(1 + 1)) = Following ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1))) by EXTPRO_1:3
.= Exec ((goto ((card I) + 1)),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1))) by A6, A10, A13 ;
then A14: IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),2)) = ICplusConst ((Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)),((card I) + 1)) by SCMPDS_2:54
.= ((card I) + 1) + 1 by A6, Th12
.= (card I) + (1 + 1) ;
A15: (P +* (stop (if>=0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),2))) = (P +* (stop (if>=0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),2))) by PBOOLE:143;
 stop (if>=0 (a,k1,I)) c= P +* (stop (if>=0 (a,k1,I))) by A7;
then (P +* (stop (if>=0 (a,k1,I)))) . ((card I) + 2) = (stop (if>=0 (a,k1,I))) . ((card I) + 2) by A12, GRFUNC_1:2
.= halt SCMPDS by A11, COMPOS_1:64 ;
then A16: CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),2))) = halt SCMPDS by A14, A15;
now__::_thesis:_for_k_being_Element_of_NAT_holds_IC_(Comput_((P_+*_(stop_(if>=0_(a,k1,I)))),(Initialize_s),k))_in_dom_(stop_(if>=0_(a,k1,I)))
let k be Element of NAT ; ::_thesis: IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if>=0 (a,k1,I)))
A17: ( k = 0 or 0 + 1 <= k ) by INT_1:7;
percases ( k = 0 or k = 1 or 1 < k ) by A17, XXREAL_0:1;
suppose k = 0 ; ::_thesis: IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if>=0 (a,k1,I)))
then  Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),k) = Initialize s by EXTPRO_1:2;
hence  IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if>=0 (a,k1,I))) by A5, COMPOS_1:36; ::_thesis: verum
end;
suppose k = 1 ; ::_thesis: IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if>=0 (a,k1,I)))
hence  IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if>=0 (a,k1,I))) by A9, A6, COMPOS_1:62; ::_thesis: verum
end;
suppose 1 < k ; ::_thesis: IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if>=0 (a,k1,I)))
then  1 + 1 <= k by INT_1:7;
hence  IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if>=0 (a,k1,I))) by A12, A14, A16, EXTPRO_1:5; ::_thesis: verum
end;
end;
end;
hence  if>=0 (a,k1,I) is_closed_on s,P by Def2; ::_thesis: if>=0 (a,k1,I) is_halting_on s,P
 P +* (stop (if>=0 (a,k1,I))) halts_on Initialize s by A16, EXTPRO_1:29;
hence  if>=0 (a,k1,I) is_halting_on s,P by Def3; ::_thesis: verum
end;

theorem Th120: :: SCMPDS_6:120
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))
let s be State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 & I is_closed_on s,P & I is_halting_on s,P holds
IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) >= 0 & I is_closed_on s,P & I is_halting_on s,P implies IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set IF = if>=0 (a,k1,I);
set pIF = stop (if>=0 (a,k1,I));
set pI = stop I;
set s2 = Initialize s;
set s3 = Initialize s;
set P2 = P +* (stop I);
set P3 = P +* (stop (if>=0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if>=0 (a,k1,I)));
set i = (a,k1) >=0_goto 2;
set j = goto ((card I) + 1);
set SAl = Start-At (((card I) + 2),SCMPDS);
A1: if>=0 (a,k1,I) = ((a,k1) >=0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:16;
A2: stop I c= P +* (stop I) by FUNCT_4:25;
A3: Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) >=0_goto 2),(Initialize s)) by A1, Th11 ;
A4: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A5: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A4, FUNCT_4:11 ;
A6: IC (Initialize s) = 0 by MEMSTR_0:47;
A7: ( Start-At (0,SCMPDS) c= Initialize s & Shift ((stop I),2) c= P +* (stop (if>=0 (a,k1,I))) ) by Lm7, FUNCT_4:25;
 for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)) . a by A3, SCMPDS_2:57;
then A8: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)) by SCMPDS_4:8;
assume  s . (DataLoc ((s . a),k1)) >= 0 ; ::_thesis: ( not I is_closed_on s,P or not I is_halting_on s,P or IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS)) )
then A9: IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)) = ICplusConst ((Initialize s),2) by A3, A5, SCMPDS_2:57
.= 0 + 2 by A6, Th12 ;
assume A10: I is_closed_on s,P ; ::_thesis: ( not I is_halting_on s,P or IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS)) )
then A11: I is_closed_on Initialize s,P +* (stop I) by Th24;
assume A12: I is_halting_on s,P ; ::_thesis: IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))
then A13: P +* (stop I) halts_on Initialize s by Def3;
A14: Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)) = Comput ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))) by EXTPRO_1:4;
A15: CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A14
.= CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A11, A7, A9, A8, Th31, A2
.= halt SCMPDS by A13, EXTPRO_1:def_15 ;
then A16: P +* (stop (if>=0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
A17: CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s)) = (a,k1) >=0_goto 2 by A1, Th11;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_(LifeSpan_((P_+*_(stop_I)),(Initialize_s)))_+_1_holds_
CurInstr_((P_+*_(stop_(if>=0_(a,k1,I)))),(Comput_((P_+*_(stop_(if>=0_(a,k1,I)))),(Initialize_s),l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 implies CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS )
assume A18: l < (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 ; ::_thesis: CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
A19: Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),0) = Initialize s by EXTPRO_1:2;
percases ( l = 0 or l <> 0 ) ;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
then  CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),l))) = CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s)) by A19;
hence  CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A17; ::_thesis: verum
end;
suppose l <> 0 ; ::_thesis: not CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),b1))) = halt SCMPDS
then consider n being Nat such that
A20: l = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A21: n < LifeSpan ((P +* (stop I)),(Initialize s)) by A18, A20, XREAL_1:6;
assume A22: CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS ; ::_thesis: contradiction
A23: Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),(n + 1)) = Comput ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)),n) by EXTPRO_1:4;
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),n))) = CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)),n))) by A11, A7, A9, A8, Th31, A2
.= halt SCMPDS by A20, A22, A23 ;
hence  contradiction by A13, A21, EXTPRO_1:def_15; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS holds
(LifeSpan ((P +* (stop I)),(Initialize s))) + 1 <= l ;
then A24: LifeSpan ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s)) = (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 by A15, A16, EXTPRO_1:def_15;
A25: DataPart (Result ((P +* (stop I)),(Initialize s))) = DataPart (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A13, EXTPRO_1:23
.= DataPart (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A11, A7, A9, A8, Th31, A2
.= DataPart (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s))) by A16, A24, EXTPRO_1:23 ;
A26: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if>=0_(a,k1,I)),P,(Initialize_s)))_holds_
(IExec_((if>=0_(a,k1,I)),P,(Initialize_s)))_._x_=_((IExec_(I,P,(Initialize_s)))_+*_(Start-At_(((card_I)_+_2),SCMPDS)))_._x
let x be set ; ::_thesis: ( x in dom (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) implies (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) . b1 )
A27: dom (Start-At (((card I) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A28: x in dom (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) ; ::_thesis: (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A28, SCMPDS_4:6;
supposeA29: x is Int_position ; ::_thesis: (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A30: not x in dom (Start-At (((card I) + 2),SCMPDS)) by A27, TARSKI:def_1;
thus (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . x = (Result ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s))) . x
.= (Result ((P +* (stop I)),(Initialize s))) . x by A25, A29, SCMPDS_4:8
.= (IExec (I,P,(Initialize s))) . x
.= ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) . x by A30, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA31: x = IC ; ::_thesis: (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b1 = ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) . b1
A32: IC (Result ((P +* (stop I)),(Initialize s))) = IC (IExec (I,P,(Initialize s)))
.= card I by A10, A12, Th34 ;
A33: x in dom (Start-At (((card I) + 2),SCMPDS)) by A27, A31, TARSKI:def_1;
thus (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . x = (Result ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s))) . x
.= (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) . x by A16, A24, EXTPRO_1:23
.= IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A31, EXTPRO_1:4
.= (IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) + 2 by A11, A7, A9, A8, Th31, A2
.= (IC (Result ((P +* (stop I)),(Initialize s)))) + 2 by A13, EXTPRO_1:23
.= IC (Start-At (((card I) + 2),SCMPDS)) by A32, FUNCOP_1:72
.= ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) . x by A31, A33, FUNCT_4:13 ; ::_thesis: verum
end;
end;
end;
dom (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS)) by A26, FUNCT_1:2; ::_thesis: verum
end;

theorem Th121: :: SCMPDS_6:121
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
let a be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) < 0 implies IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set IF = if>=0 (a,k1,I);
set pIF = stop (if>=0 (a,k1,I));
set s3 = Initialize s;
set P3 = P +* (stop (if>=0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1);
set s5 = Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),2);
set P4 = P +* (stop (if>=0 (a,k1,I)));
set P5 = P +* (stop (if>=0 (a,k1,I)));
set i = (a,k1) >=0_goto 2;
set j = goto ((card I) + 1);
set SAl = Start-At (((card I) + 2),SCMPDS);
A1: if>=0 (a,k1,I) = ((a,k1) >=0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:16;
A2: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A3: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A2, FUNCT_4:11 ;
A4: IC (Initialize s) = 0 by MEMSTR_0:47;
A5: stop (if>=0 (a,k1,I)) c= P +* (stop (if>=0 (a,k1,I))) by FUNCT_4:25;
then A6: stop (if>=0 (a,k1,I)) c= P +* (stop (if>=0 (a,k1,I))) ;
A7: stop (if>=0 (a,k1,I)) c= P +* (stop (if>=0 (a,k1,I))) by A5;
A8: Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) >=0_goto 2),(Initialize s)) by A1, Th11 ;
assume  s . (DataLoc ((s . a),k1)) < 0 ; ::_thesis: IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS))
then A9: IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)) = succ (IC (Initialize s)) by A8, A3, SCMPDS_2:57
.= 0 + 1 by A4 ;
A10: 1 in dom (if>=0 (a,k1,I)) by Lm9;
then  1 in dom (stop (if>=0 (a,k1,I))) by COMPOS_1:62;
then A11: (P +* (stop (if>=0 (a,k1,I)))) . 1 = (stop (if>=0 (a,k1,I))) . 1 by A6, GRFUNC_1:2
.= (if>=0 (a,k1,I)) . 1 by A10, COMPOS_1:63
.= goto ((card I) + 1) by Lm10 ;
A12: (P +* (stop (if>=0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1))) = (P +* (stop (if>=0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1))) by PBOOLE:143;
A13: Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),(1 + 1)) = Following ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1))) by EXTPRO_1:3
.= Exec ((goto ((card I) + 1)),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1))) by A9, A11, A12 ;
then A14: IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),2)) = ICplusConst ((Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)),((card I) + 1)) by SCMPDS_2:54
.= ((card I) + 1) + 1 by A9, Th12
.= (card I) + (1 + 1) ;
A15: (P +* (stop (if>=0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),2))) = (P +* (stop (if>=0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),2))) by PBOOLE:143;
A16: card (if>=0 (a,k1,I)) = (card I) + 2 by Lm8;
then  (card I) + 2 in dom (stop (if>=0 (a,k1,I))) by COMPOS_1:64;
then (P +* (stop (if>=0 (a,k1,I)))) . ((card I) + 2) = (stop (if>=0 (a,k1,I))) . ((card I) + 2) by A7, GRFUNC_1:2
.= halt SCMPDS by A16, COMPOS_1:64 ;
then A17: CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),2))) = halt SCMPDS by A14, A15;
then A18: P +* (stop (if>=0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
A19: CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s)) = (a,k1) >=0_goto 2 by A1, Th11;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_<_1_+_1_holds_
CurInstr_((P_+*_(stop_(if>=0_(a,k1,I)))),(Comput_((P_+*_(stop_(if>=0_(a,k1,I)))),(Initialize_s),l)))_<>_halt_SCMPDS
let l be Element of NAT ; ::_thesis: ( l < 1 + 1 implies CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS )
assume  l < 1 + 1 ; ::_thesis: CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
then A20: l <= 1 by NAT_1:13;
A21: Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),0) = Initialize s by EXTPRO_1:2;
A22: (P +* (stop (if>=0 (a,k1,I)))) /. (IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),l))) = (P +* (stop (if>=0 (a,k1,I)))) . (IC (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),l))) by PBOOLE:143;
percases ( l = 0 or l = 1 ) by A20, NAT_1:25;
suppose l = 0 ; ::_thesis: CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
then  CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),l))) = CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s)) by A21;
hence  CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A19; ::_thesis: verum
end;
suppose l = 1 ; ::_thesis: CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),b1))) <> halt SCMPDS
hence  CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A9, A11, A22; ::_thesis: verum
end;
end;
end;
then  for l being Element of NAT st CurInstr ((P +* (stop (if>=0 (a,k1,I)))),(Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS holds
2 <= l ;
then  LifeSpan ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s)) = 2 by A17, A18, EXTPRO_1:def_15;
then A23: Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),2) = Result ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s)) by A18, EXTPRO_1:23;
A24: now__::_thesis:_for_x_being_set_st_x_in_dom_(IExec_((if>=0_(a,k1,I)),P,(Initialize_s)))_holds_
(IExec_((if>=0_(a,k1,I)),P,(Initialize_s)))_._x_=_(s_+*_(Start-At_(((card_I)_+_2),SCMPDS)))_._x
A25: dom (Start-At (((card I) + 2),SCMPDS)) = {(IC )} by FUNCOP_1:13;
let x be set ; ::_thesis: ( x in dom (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) implies (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1 )
assume A26: x in dom (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) ; ::_thesis: (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1
percases ( x is Int_position or x = IC ) by A26, SCMPDS_4:6;
supposeA27: x is Int_position ; ::_thesis: (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1
then  x <> IC by SCMPDS_2:43;
then A28: not x in dom (Start-At (((card I) + 2),SCMPDS)) by A25, TARSKI:def_1;
A29: not x in dom (Start-At (0,SCMPDS)) by A27, SCMPDS_4:18;
thus (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . x = (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),2)) . x by A23
.= (Comput ((P +* (stop (if>=0 (a,k1,I)))),(Initialize s),1)) . x by A13, A27, SCMPDS_2:54
.= (Initialize s) . x by A8, A27, SCMPDS_2:57
.= s . x by A29, FUNCT_4:11
.= (s +* (Start-At (((card I) + 2),SCMPDS))) . x by A28, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA30: x = IC ; ::_thesis: (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b1 = (s +* (Start-At (((card I) + 2),SCMPDS))) . b1
hence (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . x = (card I) + 2 by A14, A23
.= (s +* (Start-At (((card I) + 2),SCMPDS))) . x by A30, FUNCT_4:113 ;
::_thesis: verum
end;
end;
end;
dom (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) = the carrier of SCMPDS by PARTFUN1:def_2
.= dom (s +* (Start-At (((card I) + 2),SCMPDS))) by PARTFUN1:def_2 ;
hence  IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS)) by A24, FUNCT_1:2; ::_thesis: verum
end;

registration
let I be parahalting shiftable Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if>=0 (a,k1,I) ->  parahalting shiftable ;
correctness
coherence
( if>=0 (a,k1,I) is shiftable & if>=0 (a,k1,I) is parahalting );
proof
set i = (a,k1) >=0_goto 2;
set j = goto ((card I) + 1);
set IF = if>=0 (a,k1,I);
set pIF = stop (if>=0 (a,k1,I));
thus  if>=0 (a,k1,I) is shiftable ; ::_thesis: if>=0 (a,k1,I) is parahalting
let s be 0 -started State of SCMPDS; :: according to SCMPDS_4:def_7 ::_thesis: for b1 being set holds
( not stop (if>=0 (a,k1,I)) c= b1 or b1 halts_on s )
let P be Instruction-Sequence of SCMPDS; ::_thesis: ( not stop (if>=0 (a,k1,I)) c= P or P halts_on s )
A1: Initialize s = s by MEMSTR_0:44;
assume  stop (if>=0 (a,k1,I)) c= P ; ::_thesis: P halts_on s
then A2: P = P +* (stop (if>=0 (a,k1,I))) by FUNCT_4:98;
A3: ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
percases ( s . (DataLoc ((s . a),k1)) >= 0 or s . (DataLoc ((s . a),k1)) < 0 ) ;
suppose s . (DataLoc ((s . a),k1)) >= 0 ; ::_thesis: P halts_on s
then  if>=0 (a,k1,I) is_halting_on s,P by A3, Th118;
hence  P halts_on s by A2, Def3, A1; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),k1)) < 0 ; ::_thesis: P halts_on s
then  if>=0 (a,k1,I) is_halting_on s,P by Th119;
hence  P halts_on s by A2, Def3, A1; ::_thesis: verum
end;
end;
end;
end;

registration
let I be halt-free Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if>=0 (a,k1,I) ->  halt-free ;
coherence
 if>=0 (a,k1,I) is halt-free ;
end;

theorem :: SCMPDS_6:122
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if>=0 (a,k1,I)),P,s)) = (card I) + 2
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if>=0 (a,k1,I)),P,s)) = (card I) + 2
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if>=0 (a,k1,I)),P,s)) = (card I) + 2
let I be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for k1 being Integer holds IC (IExec ((if>=0 (a,k1,I)),P,s)) = (card I) + 2
let a be Int_position; ::_thesis: for k1 being Integer holds IC (IExec ((if>=0 (a,k1,I)),P,s)) = (card I) + 2
let k1 be Integer; ::_thesis: IC (IExec ((if>=0 (a,k1,I)),P,s)) = (card I) + 2
set IF = if>=0 (a,k1,I);
A1: ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
A2: Initialize s = s by MEMSTR_0:44;
percases ( s . (DataLoc ((s . a),k1)) >= 0 or s . (DataLoc ((s . a),k1)) < 0 ) ;
suppose s . (DataLoc ((s . a),k1)) >= 0 ; ::_thesis: IC (IExec ((if>=0 (a,k1,I)),P,s)) = (card I) + 2
then  IExec ((if>=0 (a,k1,I)),P,s) = (IExec (I,P,s)) +* (Start-At (((card I) + 2),SCMPDS)) by A1, Th120, A2;
hence  IC (IExec ((if>=0 (a,k1,I)),P,s)) = (card I) + 2 by FUNCT_4:113; ::_thesis: verum
end;
suppose s . (DataLoc ((s . a),k1)) < 0 ; ::_thesis: IC (IExec ((if>=0 (a,k1,I)),P,s)) = (card I) + 2
then  IExec ((if>=0 (a,k1,I)),P,s) = s +* (Start-At (((card I) + 2),SCMPDS)) by Th121, A2;
hence  IC (IExec ((if>=0 (a,k1,I)),P,s)) = (card I) + 2 by FUNCT_4:113; ::_thesis: verum
end;
end;
end;

theorem :: SCMPDS_6:123
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
(IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
(IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let s be State of SCMPDS; ::_thesis: for I being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
(IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let I be halt-free parahalting shiftable Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
(IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) >= 0 holds
(IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) >= 0 implies (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b )
assume A1: s . (DataLoc ((s . a),k1)) >= 0 ; ::_thesis: (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
A2: not b in dom (Start-At (((card I) + 2),SCMPDS)) by SCMPDS_4:18;
 ( I is_closed_on s,P & I is_halting_on s,P ) by Th20, Th21;
then  IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 2),SCMPDS)) by A1, Th120;
hence  (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b by A2, FUNCT_4:11; ::_thesis: verum
end;

theorem :: SCMPDS_6:124
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b = s . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b = s . b
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b = s . b
let I be Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b = s . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b = s . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) < 0 implies (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b = s . b )
assume  s . (DataLoc ((s . a),k1)) < 0 ; ::_thesis: (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b = s . b
then A1: IExec ((if>=0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 2),SCMPDS)) by Th121;
 not b in dom (Start-At (((card I) + 2),SCMPDS)) by SCMPDS_4:18;
hence  (IExec ((if>=0 (a,k1,I)),P,(Initialize s))) . b = s . b by A1, FUNCT_4:11; ::_thesis: verum
end;

theorem :: SCMPDS_6:125
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS holds
( I is_closed_on s,P iff I is_closed_on Initialize s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS holds
( I is_closed_on s,P iff I is_closed_on Initialize s,P )
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS holds
( I is_closed_on s,P iff I is_closed_on Initialize s,P )
let I be Program of SCMPDS; ::_thesis: ( I is_closed_on s,P iff I is_closed_on Initialize s,P )
thus  ( I is_closed_on s,P implies I is_closed_on Initialize s,P ) ::_thesis: ( I is_closed_on Initialize s,P implies I is_closed_on s,P )
proof
assume  for k being Element of NAT holds IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom (stop I) ; :: according to SCMPDS_6:def_2 ::_thesis: I is_closed_on Initialize s,P
hence  for k being Element of NAT holds IC (Comput ((P +* (stop I)),(Initialize (Initialize s)),k)) in dom (stop I) ; :: according to SCMPDS_6:def_2 ::_thesis: verum
end;
assume  for k being Element of NAT holds IC (Comput ((P +* (stop I)),(Initialize (Initialize s)),k)) in dom (stop I) ; :: according to SCMPDS_6:def_2 ::_thesis: I is_closed_on s,P
hence  for k being Element of NAT holds IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom (stop I) ; :: according to SCMPDS_6:def_2 ::_thesis: verum
end;

theorem :: SCMPDS_6:126
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS holds
( I is_halting_on s,P iff I is_halting_on Initialize s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS holds
( I is_halting_on s,P iff I is_halting_on Initialize s,P )
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS holds
( I is_halting_on s,P iff I is_halting_on Initialize s,P )
let I be Program of SCMPDS; ::_thesis: ( I is_halting_on s,P iff I is_halting_on Initialize s,P )
thus  ( I is_halting_on s,P implies I is_halting_on Initialize s,P ) ::_thesis: ( I is_halting_on Initialize s,P implies I is_halting_on s,P )
proof
assume  P +* (stop I) halts_on Initialize s ; :: according to SCMPDS_6:def_3 ::_thesis: I is_halting_on Initialize s,P
hence  P +* (stop I) halts_on Initialize (Initialize s) ; :: according to SCMPDS_6:def_3 ::_thesis: verum
end;
assume  P +* (stop I) halts_on Initialize (Initialize s) ; :: according to SCMPDS_6:def_3 ::_thesis: I is_halting_on s,P
hence  P +* (stop I) halts_on Initialize s ; :: according to SCMPDS_6:def_3 ::_thesis: verum
end;