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));
A1: s +* (Initialize (stop J)) = (Initialize s) +* (stop J) by COMPOS_1:125;
A2: s +* (Initialize (stop (if<0 (a,k1,I,J)))) = (Initialize s) +* (stop (if<0 (a,k1,I,J))) by COMPOS_1:125;
A3: 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 ;
A4: 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 EXTPRO_1:4
.= Following ((ProgramPart ((Initialize s) +* (stop (if<0 (a,k1,I,J))))),((Initialize s) +* (stop (if<0 (a,k1,I,J))))) by EXTPRO_1:3
.= Exec (((a,k1) >=0_goto ((card I) + 2)),((Initialize s) +* (stop (if<0 (a,k1,I,J))))) by A3, Th22, A2 ;
A5: 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 A6: ((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 A2, FUNCT_4:12
.= s . (DataLoc ((s . a),k1)) by A5, A2, FUNCT_4:12 ;
A7: IC ((Initialize s) +* (stop (if<0 (a,k1,I,J)))) = 0 by A2, FUNCT_4:26, SCMPDS_5:18;
assume s . (DataLoc ((s . a),k1)) >= 0 ; :: thesis:
then A8: 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 A4, A6, SCMPDS_2:69
.= 0 + ((card I) + 2) by A7, Th23 ;
assume A9: J is_closed_on s ; :: thesis:
then A10: ( Initialize (stop J) c= (Initialize s) +* (stop J) & J is_closed_on (Initialize s) +* (stop J) ) by Th38, A1, FUNCT_4:26;
A11: stop (if<0 (a,k1,I,J)) c= (Initialize s) +* (stop (if<0 (a,k1,I,J))) by FUNCT_4:26;
A12: 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_1:20
.= (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 A11, XBOOLE_1:1;
then A13: 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 A14: ProgramPart ((Initialize s) +* (stop J)) halts_on (Initialize s) +* (stop J) by Def3;
A15: DataPart ((Initialize s) +* (stop J)) = DataPart ((Initialize s) +* (stop (if<0 (a,k1,I,J)))) by COMPOS_1:138, FUNCT_7:134;

now

let a be Int_position ; :: thesis:
thus ((Initialize s) +* (stop J)) . a = ((Initialize s) +* (stop (if<0 (a,k1,I,J)))) . a by A15, 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 A4, SCMPDS_2:69 ; :: thesis:

end;

then A16: 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
A17: 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 A9, Def2;
then m < card (stop J) by AFINSQ_1:70;
then A18: m + ((card I) + 2) < (card (stop J)) + ((card I) + 2) by XREAL_1:8;
A19: card (stop J) = (card J) + 1 by SCMPDS_5:7;
A20: 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 A12, AFINSQ_1:20
.= ((card I) + 2) + (card (stop J)) by A19 ;
A21: 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 A17, EXTPRO_1:5
.= m + ((card I) + 2) by A10, A16, A13, A8, Th45, A21 ;
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 A20, A18, 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 EXTPRO_1:3;
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 A7, COMPOS_1:135; :: thesis:

end;