let P be Instruction-Sequence of SCMPDS; for s being State of SCMPDS
for I being shiftable Program of
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) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
let s be State of SCMPDS; for I being shiftable Program of
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) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
let I be shiftable Program of ; 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) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
let a be 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) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
let k1 be Integer; ( s . (DataLoc ((s . a),k1)) < 0 & I is_closed_on s,P & I is_halting_on s,P implies ( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P ) )
set b = DataLoc ((s . a),k1);
set IF = if<0 (a,k1,I);
set pIF = stop (if<0 (a,k1,I));
set pI = stop I;
set s2 = Initialize s;
set s3 = Initialize s;
set P2 = P +* (stop I);
set P3 = P +* (stop (if<0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if<0 (a,k1,I)));
set i = (a,k1) >=0_goto ((card I) + 1);
A1:
stop I c= P +* (stop I)
by FUNCT_4:25;
A2:
0 in dom (stop (if<0 (a,k1,I)))
by COMPOS_1:36;
A3:
IC (Initialize s) = 0
by MEMSTR_0:47;
A4:
not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS))
by SCMPDS_4:18;
not a in dom (Start-At (0,SCMPDS))
by SCMPDS_4:18;
then A5: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) =
(Initialize s) . (DataLoc ((s . a),k1))
by FUNCT_4:11
.=
s . (DataLoc ((s . a),k1))
by A4, FUNCT_4:11
;
A6: Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),(0 + 1)) =
Following ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),0)))
by EXTPRO_1:3
.=
Following ((P +* (stop (if<0 (a,k1,I)))),(Initialize s))
by EXTPRO_1:2
.=
Exec (((a,k1) >=0_goto ((card I) + 1)),(Initialize s))
by Th3
;
for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)) . a
by A6, SCMPDS_2:57;
then A7:
DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1))
by SCMPDS_4:8;
assume
s . (DataLoc ((s . a),k1)) < 0
; ( not I is_closed_on s,P or not I is_halting_on s,P or ( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P ) )
then A8: IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)) =
(IC (Initialize s)) + 1
by A6, A5, SCMPDS_2:57
.=
0 + 1
by A3
;
assume A9:
I is_closed_on s,P
; ( not I is_halting_on s,P or ( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P ) )
then A10:
I is_closed_on Initialize s,P +* (stop I)
;
assume
I is_halting_on s,P
; ( if<0 (a,k1,I) is_closed_on s,P & if<0 (a,k1,I) is_halting_on s,P )
then A11:
P +* (stop I) halts_on Initialize s
;
A12:
( Start-At (0,SCMPDS) c= Initialize s & Shift ((stop I),1) c= P +* (stop (if<0 (a,k1,I))) )
by Lm6, FUNCT_4:25;
A13: card (stop (if<0 (a,k1,I))) =
(card (if<0 (a,k1,I))) + 1
by COMPOS_1:55
.=
((card I) + 1) + 1
by Th1
;
now for k being Nat holds IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if<0 (a,k1,I)))let k be
Nat;
IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),b1)) in dom (stop (if<0 (a,k1,I)))per cases
( 0 < k or k = 0 )
;
suppose
0 < k
;
IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),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
Nat ;
reconsider m =
IC (Comput ((P +* (stop I)),(Initialize s),k1)) as
Nat ;
A15:
card (stop (if<0 (a,k1,I))) = (card (stop I)) + 1
by A13, COMPOS_1:55;
m in dom (stop I)
by A9;
then
m < card (stop I)
by AFINSQ_1:66;
then A16:
m + 1
< card (stop (if<0 (a,k1,I)))
by A15, XREAL_1:6;
IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),k)) =
IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)),k1))
by A14, EXTPRO_1:4
.=
m + 1
by A10, A12, A8, A7, Th22, A1
;
hence
IC (Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if<0 (a,k1,I)))
by A16, AFINSQ_1:66;
verum end; end;
hence
if<0 (a,k1,I) is_closed_on s,P
; if<0 (a,k1,I) is_halting_on s,P
A17:
Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)) = Comput ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))
by EXTPRO_1:4;
CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) =
CurInstr ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Comput ((P +* (stop (if<0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))))
by A17
.=
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))))
by A10, A12, A8, A7, Th22, A1
.=
halt SCMPDS
by A11, EXTPRO_1:def 15
;
then
P +* (stop (if<0 (a,k1,I))) halts_on Initialize s
by EXTPRO_1:29;
hence
if<0 (a,k1,I) is_halting_on s,P
; verum