let P be Instruction-Sequence of SCMPDS; :: thesis: for s being State of SCMPDS
for I 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) 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
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 ; :: 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 Th3 ;
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)) = (IC (Initialize s)) + 1 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) ;
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 ;
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 Th1 ;

now :: thesis: for k being 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 Nat; :: thesis: IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),b₁)) in dom (stop (if=0 (a,k1,I)))

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