let P be Instruction-Sequence of SCMPDS; :: thesis: for s being 0 -started State of SCMPDS
for I, J being shiftable Program of
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
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 ; :: 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, Th3, 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)) = (IC s) + 1 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, Th21;
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 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, Th21;
then A14: P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on s by 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, Th1 ;

now :: thesis: for k being 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 Nat; :: thesis: IC (Comput ((P +* (stop (if<0 (a,k1,I,J)))),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)))),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 Nat ;
reconsider m = IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),s,k1)) as 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, 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, Th22, 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 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, Th22, 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 A2; :: thesis: verum