let s be 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 & I is_halting_on s holds
( if<0 a,k1,I,J is_closed_on s & if<0 a,k1,I,J is_halting_on s )
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 & I is_halting_on s holds
( if<0 a,k1,I,J is_closed_on s & if<0 a,k1,I,J is_halting_on s )
let a be Int_position ; :: thesis: for k1 being Integer st s . (DataLoc (s . a),k1) < 0 & I is_closed_on s & I is_halting_on s holds
( if<0 a,k1,I,J is_closed_on s & if<0 a,k1,I,J is_halting_on s )
let k1 be Integer; :: thesis: ( s . (DataLoc (s . a),k1) < 0 & I is_closed_on s & I is_halting_on s implies ( if<0 a,k1,I,J is_closed_on s & if<0 a,k1,I,J is_halting_on s ) )
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) +* (stop ((I ';' (Goto ((card J) + 1))) ';' 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);
I1: s +* (Initialize (stop ((I ';' (Goto ((card J) + 1))) ';' J))) = (Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' 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: 0 in dom (stop (if<0 a,k1,I,J)) by SCMPDS_4:75;
A2: IC ((Initialize s) +* (stop (if<0 a,k1,I,J))) = 0 by I2, FUNCT_4:26, SCMPDS_5:18;
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: 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 ;
then A6: Shift (stop ((I ';' (Goto ((card J) + 1))) ';' J)),1 c= Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),1 by Lm5;
A7: 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 A5, Th22, I2 ;
A8: DataPart ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' 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: ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))) . a = (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),1) . a
thus ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))) . a = ((Initialize s) +* (stop (if<0 a,k1,I,J))) . a by A8, 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 A7, SCMPDS_2:69 ; :: thesis: verum

end;

then A9: DataPart ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' 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;
assume s . (DataLoc (s . a),k1) < 0 ; :: thesis: ( not I is_closed_on s or not I is_halting_on s or ( if<0 a,k1,I,J is_closed_on s & if<0 a,k1,I,J is_halting_on s ) )
then A10: IC (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),1) = succ (IC ((Initialize s) +* (stop (if<0 a,k1,I,J)))) by A7, A4, SCMPDS_2:69
.= 0 + 1 by A2 ;
assume A11: I is_closed_on s ; :: thesis: ( not I is_halting_on s or ( if<0 a,k1,I,J is_closed_on s & if<0 a,k1,I,J is_halting_on s ) )
assume A12: I is_halting_on s ; :: thesis: ( if<0 a,k1,I,J is_closed_on s & if<0 a,k1,I,J is_halting_on s )
then A13: (I ';' (Goto ((card J) + 1))) ';' J is_closed_on s by A11, Th44;
then A14: ( Initialize (stop ((I ';' (Goto ((card J) + 1))) ';' J)) c= (Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on (Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) ) by Th38, I1, FUNCT_4:26;
(I ';' (Goto ((card J) + 1))) ';' J is_halting_on s by A11, A12, Th44;
then A15: ProgramPart ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))) halts_on (Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by Def3;
A16: card (stop (if<0 a,k1,I,J)) = (card (if<0 a,k1,I,J)) + 1 by SCMPDS_5:7
.= ((card ((I ';' (Goto ((card J) + 1))) ';' J)) + 1) + 1 by A5, Th15 ;

now

let k be Element of NAT ; :: thesis: IC (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),b₁) in dom (stop (if<0 a,k1,I,J))

per cases ( 0 < k or k = 0 ) ;

suppose 0 < k ; :: thesis: IC (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),b₁) in dom (stop (if<0 a,k1,I,J))

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 ((I ';' (Goto ((card J) + 1))) ';' J)))),((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),k1) as Element of NAT ;
A18: card (stop (if<0 a,k1,I,J)) = (card (stop ((I ';' (Goto ((card J) + 1))) ';' J))) + 1 by A16, SCMPDS_5:7;
m in dom (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by A13, Def2;
then m < card (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by AFINSQ_1:70;
then A19: m + 1 < card (stop (if<0 a,k1,I,J)) by A18, XREAL_1:8;
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 A17, AMI_1:51
.= m + 1 by A14, A6, A10, A9, 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 A19, AFINSQ_1:70; :: thesis: verum

end;

suppose k = 0 ; :: thesis: IC (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),b₁) in dom (stop (if<0 a,k1,I,J))

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 A1, A2, AMI_1:13; :: thesis: verum

end;

end;

end;

hence if<0 a,k1,I,J is_closed_on s by Def2; :: thesis: if<0 a,k1,I,J is_halting_on s
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;
x: Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),((LifeSpan (ProgramPart ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)))),((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)))) + 1) = 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),(LifeSpan (ProgramPart ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)))),((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)))) by AMI_1:51;
TX4: ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),1) = ProgramPart (Comput (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),1)),(Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),1),(LifeSpan (ProgramPart ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)))),((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))))) by AMI_1:123;
TX2: ProgramPart ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))) = ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)))),((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(LifeSpan (ProgramPart ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)))),((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))))) by AMI_1:123;
CurInstr (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))),((LifeSpan (ProgramPart ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)))),((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)))) + 1)) = CurInstr (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),1)),(Comput (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),1)),(Comput (ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J)))),((Initialize s) +* (stop (if<0 a,k1,I,J))),1),(LifeSpan (ProgramPart ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)))),((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))))) by x, T
.= CurInstr (ProgramPart ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)))),(Comput (ProgramPart ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)))),((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(LifeSpan (ProgramPart ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)))),((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))))) by A14, A6, A10, A9, Th45, TX2, TX4
.= halt SCMPDS by A15, AMI_1:def 46 ;
then ProgramPart ((Initialize s) +* (stop (if<0 a,k1,I,J))) halts_on (Initialize s) +* (stop (if<0 a,k1,I,J)) by AMI_1:146;
hence if<0 a,k1,I,J is_halting_on s by Def3; :: thesis: verum