let P be Instruction-Sequence of SCMPDS; :: thesis: for s being State of SCMPDS
for I being Program of
for J being shiftable Program of
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
for J being shiftable Program of
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 ; :: thesis: for J being shiftable Program of
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 ; :: 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, Th3 ;
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, Th4 ;
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 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 Th1
.= (card I) + (1 + 1) ;
then Shift ((stop J),((card I) + 2)) c= stop (if<0 (a,k1,I,J)) by Th5;
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 ;
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 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 Nat; :: thesis: IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),(Initialize s),b₁)) in dom (stop (if<0 (a,k1,I,J)))

per cases ( 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 Nat ;
reconsider m = IC (Comput ((P +* (stop J)),(Initialize s),k1)) as Nat ;
m in dom (stop J) by A7;
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, Th22, 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 ; :: 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, Th22, 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 ; :: thesis: verum