let s be State of SCMPDS ; :: thesis: for I 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 is_closed_on s & if<=0 a,k1,I is_halting_on s )
let I 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 is_closed_on s & if<=0 a,k1,I 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 is_closed_on s & if<=0 a,k1,I 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 is_closed_on s & if<=0 a,k1,I is_halting_on s ) )
set b = DataLoc (s . a),k1;
set IF = if<=0 a,k1,I;
set pIF = stop (if<=0 a,k1,I);
set IsIF = Initialized (stop (if<=0 a,k1,I));
set pI = stop I;
set IsI = Initialized (stop I);
set s2 = s +* (Initialized (stop I));
set s3 = s +* (Initialized (stop (if<=0 a,k1,I)));
set s4 = Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),1;
set i = a,k1 <=0_goto 2;
set j = goto ((card I) + 1);
A1: if<=0 a,k1,I = (a,k1 <=0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:52;
A2: Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),(0 + 1) = Following (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),0 ) by AMI_1:14
.= Following (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))) by AMI_1:13
.= Exec (a,k1 <=0_goto 2),(s +* (Initialized (stop (if<=0 a,k1,I)))) by A1, Th22 ;
A3: DataPart (s +* (Initialized (stop I))) = DataPart (s +* (Initialized (stop (if<=0 a,k1,I)))) by SCMPDS_4:24, SCMPDS_4:36;
now
let a be Int_position ; :: thesis: (s +* (Initialized (stop I))) . a = (Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),1) . a
thus (s +* (Initialized (stop I))) . a = (s +* (Initialized (stop (if<=0 a,k1,I)))) . a by A3, SCMPDS_4:23
.= (Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),1) . a by A2, SCMPDS_2:68 ; :: thesis: verum
end;
then A4: DataPart (s +* (Initialized (stop I))) = DataPart (Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),1) by SCMPDS_4:23;
A5: not DataLoc (s . a),k1 in dom (Initialized (stop (if<=0 a,k1,I))) by SCMPDS_4:31;
not a in dom (Initialized (stop (if<=0 a,k1,I))) by SCMPDS_4:31;
then A6: (s +* (Initialized (stop (if<=0 a,k1,I)))) . (DataLoc ((s +* (Initialized (stop (if<=0 a,k1,I)))) . a),k1) = (s +* (Initialized (stop (if<=0 a,k1,I)))) . (DataLoc (s . a),k1) by FUNCT_4:12
.= s . (DataLoc (s . a),k1) by A5, FUNCT_4:12 ;
A7: IC (s +* (Initialized (stop (if<=0 a,k1,I)))) = 0 by FUNCT_4:26, SCMPDS_5:18;
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 is_closed_on s & if<=0 a,k1,I is_halting_on s ) )
then A8: IC (Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),1) = ICplusConst (s +* (Initialized (stop (if<=0 a,k1,I)))),2 by A2, A6, SCMPDS_2:68
.= 0 + 2 by A7, Th23 ;
assume A9: I is_closed_on s ; :: thesis: ( not I is_halting_on s or ( if<=0 a,k1,I is_closed_on s & if<=0 a,k1,I is_halting_on s ) )
then A10: I is_closed_on s +* (Initialized (stop I)) by Th38;
assume I is_halting_on s ; :: thesis: ( if<=0 a,k1,I is_closed_on s & if<=0 a,k1,I is_halting_on s )
then A11: ProgramPart (s +* (Initialized (stop I))) halts_on s +* (Initialized (stop I)) by Def3;
A12: ( Initialized (stop I) c= s +* (Initialized (stop I)) & Shift (stop I),2 c= Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),1 ) by Lm6, FUNCT_4:26;
A13: 0 in dom (stop (if<=0 a,k1,I)) by SCMPDS_4:75;
now
let k be Element of NAT ; :: thesis: IC (Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),b1) in dom (stop (if<=0 a,k1,I))
per cases ( 0 < k or k = 0 ) ;
suppose 0 < k ; :: thesis: IC (Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),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 Element of NAT by ORDINAL1:def 13;
reconsider m = IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k1) as Element of NAT ;
A15: card (stop (if<=0 a,k1,I)) = 1 + (card (if<=0 a,k1,I)) by SCMPDS_5:7
.= 1 + ((card I) + 2) by Lm8
.= (1 + (card I)) + 2
.= (card (stop I)) + 2 by SCMPDS_5:7 ;
m in dom (stop I) by A9, Def2;
then m < card (stop I) by SCMPDS_4:1;
then A16: m + 2 < card (stop (if<=0 a,k1,I)) by A15, XREAL_1:8;
T: ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I)))) = ProgramPart (Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),1) by AMI_1:144;
IC (Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),k) = IC (Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),1),k1) by A14, AMI_1:51
.= m + 2 by A10, A12, A8, A4, Th45, T ;
hence IC (Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),k) in dom (stop (if<=0 a,k1,I)) by A16, SCMPDS_4:1; :: thesis: verum
end;
suppose k = 0 ; :: thesis: IC (Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),b1) in dom (stop (if<=0 a,k1,I))
hence IC (Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),k) in dom (stop (if<=0 a,k1,I)) by A13, A7, AMI_1:13; :: thesis: verum
end;
end;
end;
hence if<=0 a,k1,I is_closed_on s by Def2; :: thesis: if<=0 a,k1,I is_halting_on s
T: ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I)))) = ProgramPart (Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),1) by AMI_1:144;
x: Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),((LifeSpan (s +* (Initialized (stop I)))) + 1) = Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),1),(LifeSpan (s +* (Initialized (stop I)))) by AMI_1:51;
CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),((LifeSpan (s +* (Initialized (stop I)))) + 1))),(Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),((LifeSpan (s +* (Initialized (stop I)))) + 1)) = CurInstr (ProgramPart (Comput (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),1)),(Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),1),(LifeSpan (s +* (Initialized (stop I)))))),(Comput (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),1)),(Comput (ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I))))),(s +* (Initialized (stop (if<=0 a,k1,I)))),1),(LifeSpan (s +* (Initialized (stop I))))) by x, T
.= CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (s +* (Initialized (stop I)))))),(Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),(LifeSpan (s +* (Initialized (stop I))))) by A10, A12, A8, A4, Th45
.= halt SCMPDS by A11, AMI_1:def 46 ;
then ProgramPart (s +* (Initialized (stop (if<=0 a,k1,I)))) halts_on s +* (Initialized (stop (if<=0 a,k1,I))) by AMI_1:146;
hence if<=0 a,k1,I is_halting_on s by Def3; :: thesis: verum