let s be State of SCMPDS ; :: thesis:
let I be Program of SCMPDS ; :: thesis:
let J be shiftable Program of SCMPDS ; :: thesis:
let a be Int_position ; :: thesis:
let k1 be Integer; :: thesis:
set b = DataLoc (s . a),k1;
set pJ = stop J;
set s1 = (Initialize s) +* (stop J);
set IF = if<0 a,k1,I,J;
set pIF = stop (if<0 a,k1,I,J);
set s3 = (Initialize s) +* (stop (if<0 a,k1,I,J));
set s4 = Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),1;
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));
I1: s +* (Initialize (stop J)) = (Initialize s) +* (stop J) by SCMPDS_4:5;
I2: s +* (Initialize (stop (if<0 a,k1,I,J))) = (Initialize s) +* (stop (if<0 a,k1,I,J)) by SCMPDS_4:5;
A1: if<0 a,k1,I,J = ((a,k1 >=0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J by SCMPDS_4:50
.= (a,k1 >=0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J) by SCMPDS_4:50 ;
A2: Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),(0 + 1) = Following (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),(Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),0 ) by AMI_1:14
.= Following (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))) by AMI_1:13
.= Exec (a,k1 >=0_goto ((card I) + 2)),((Initialize s) +* (stop (if<0 a,k1,I,J))) by A1, Th22, I2 ;
A3: not DataLoc (s . a),k1 in dom (Initialize (stop (if<0 a,k1,I,J))) by SCMPDS_4:31;
not a in dom (Initialize (stop (if<0 a,k1,I,J))) by SCMPDS_4:31;
then A4: ((Initialize s) +* (stop (if<0 a,k1,I,J))) . (DataLoc (((Initialize s) +* (stop (if<0 a,k1,I,J))) . a),k1) = ((Initialize s) +* (stop (if<0 a,k1,I,J))) . (DataLoc (s . a),k1) by I2, FUNCT_4:12
.= s . (DataLoc (s . a),k1) by A3, I2, FUNCT_4:12 ;
A5: IC ((Initialize s) +* (stop (if<0 a,k1,I,J))) = 0 by I2, FUNCT_4:26, SCMPDS_5:18;
assume s . (DataLoc (s . a),k1) >= 0 ; :: thesis:
then A6: IC (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),1) = ICplusConst ((Initialize s) +* (stop (if<0 a,k1,I,J))),((card I) + 2) by A2, A4, SCMPDS_2:69
.= 0 + ((card I) + 2) by A5, Th23 ;
assume A7: J is_closed_on s ; :: thesis:
then A8: ( Initialize (stop J) c= (Initialize s) +* (stop J) & J is_closed_on (Initialize s) +* (stop J) ) by Th38, I1, FUNCT_4:26;
A9: stop (if<0 a,k1,I,J) c= (Initialize s) +* (stop (if<0 a,k1,I,J)) by FUNCT_4:26;
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 SCMPDS_4:45
.= (card ((a,k1 >=0_goto ((card I) + 2)) ';' I)) + 1 by SCMPDS_5:6
.= ((card I) + 1) + 1 by Th15
.= (card I) + (1 + 1) ;
then Shift (stop J),((card I) + 2) c= stop (if<0 a,k1,I,J) by Th24;
then Shift (stop J),((card I) + 2) c= (Initialize s) +* (stop (if<0 a,k1,I,J)) by A9, XBOOLE_1:1;
then A11: Shift (stop J),((card I) + 2) c= Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),1 by AMI_1:81;
assume J is_halting_on s ; :: thesis:
then A12: ProgramPart ((Initialize s) +* (stop J)) halts_on (Initialize s) +* (stop J) by Def3;
A13: DataPart ((Initialize s) +* (stop J)) = DataPart ((Initialize s) +* (stop (if<0 a,k1,I,J))) by FUNCT_7:134, SCMPDS_4:24;

now

let a be Int_position ; :: thesis:
thus ((Initialize s) +* (stop J)) . a = ((Initialize s) +* (stop (if<0 a,k1,I,J))) . a by A13, SCMPDS_4:23
.= (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),1) . a by A2, SCMPDS_2:69 ; :: thesis:

end;

then A14: DataPart ((Initialize s) +* (stop J)) = DataPart (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),1) by SCMPDS_4:23;

now

let k be Element of NAT ; :: thesis:

per cases ;

suppose 0 < k ; :: thesis:

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 13;
reconsider m = IC (Comput (ProgramPart ((Initialize s) +* (stop J))),((Initialize s) +* (stop J)),k1) as Element of NAT ;
m in dom (stop J) by A7, Def2;
then m < card (stop J) by AFINSQ_1:70;
then A16: m + ((card I) + 2) < (card (stop J)) + ((card I) + 2) by XREAL_1:8;
A17: card (stop J) = (card J) + 1 by SCMPDS_5:7;
A18: card (stop (if<0 a,k1,I,J)) = (card (if<0 a,k1,I,J)) + 1 by SCMPDS_5:7
.= (((card I) + 2) + (card J)) + 1 by A10, SCMPDS_4:45
.= ((card I) + 2) + (card (stop J)) by A17 ;
T: ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J))) = ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),1) by AMI_1:123;
IC (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),k) = IC (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),(Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),1),k1) by A15, AMI_1:51
.= m + ((card I) + 2) by A8, A14, A11, A6, Th45, T ;
hence IC (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),k) in dom (stop (if<0 a,k1,I,J)) by A18, A16, AFINSQ_1:70; :: thesis:

end;

suppose k = 0 ; :: thesis:

then Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),k = (Initialize s) +* (stop (if<0 a,k1,I,J)) by AMI_1:13;
hence IC (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),k) in dom (stop (if<0 a,k1,I,J)) by A5, SCMPDS_4:75; :: thesis:

end;