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);
A3: 0 in dom (stop (if>0 (a,k1,I,J))) by COMPOS_1:36;
I: Initialize s = s by MEMSTR_0:44;
then A4: IC s = 0 by MEMSTR_0:47;
A7: 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 A8: Shift ((stop ((I ';' (Goto ((card J) + 1))) ';' J)),1) c= P +* (stop (if>0 (a,k1,I,J))) by Lm6;
A9: 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 A7, Th22, I ;
for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)) . a by A9, SCMPDS_2:56, I;
then A11: 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 A12: IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)) = succ (IC s) by A9, SCMPDS_2:56
.= 0 + 1 by A4 ;
assume A13: 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 A14: 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 A15: (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P by A13, Th44;
then A16: ( 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 Th38, FUNCT_4:25;
UU: 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 A13, A14, Th44;
then A17: P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on Initialize s by Def3;
A18: 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 A7, Th15 ;

now

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)))

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
A19: 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 ;
A20: card (stop (if>0 (a,k1,I,J))) = (card (stop ((I ';' (Goto ((card J) + 1))) ';' J))) + 1 by A18, COMPOS_1:55;
m in dom (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by A15, Def2;
then m < card (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by AFINSQ_1:66;
then A21: m + 1 < card (stop (if>0 (a,k1,I,J))) by A20, 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 A19, EXTPRO_1:4
.= m + 1 by A16, A8, A12, A11, Th45, UU ;
hence IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,k)) in dom (stop (if>0 (a,k1,I,J))) by A21, 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 A3, A4, EXTPRO_1:2; :: thesis: verum

end;

end;

end;

hence if>0 (a,k1,I,J) is_closed_on s,P by Def2, I; :: thesis: if>0 (a,k1,I,J) is_halting_on s,P
A24: 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 A24
.= 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 A16, A8, A12, A11, Th45, UU
.= halt SCMPDS by A17, 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, I; :: thesis: verum