let P be Instruction-Sequence of SCMPDS; for s being 0 -started State of SCMPDS
for I, J 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,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P )
let s be 0 -started State of SCMPDS; for I, J 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,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P )
let I, J 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,J) is_closed_on s,P & if>0 (a,k1,I,J) 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,J) is_closed_on s,P & if>0 (a,k1,I,J) 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,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P ) )
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;
set P2 = P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J));
set P3 = P +* (stop (if>0 (a,k1,I,J)));
set s4 = Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1);
set P4 = P +* (stop (if>0 (a,k1,I,J)));
set i = (a,k1) <=0_goto ((card I) + 2);
A1:
0 in dom (stop (if>0 (a,k1,I,J)))
by COMPOS_1:36;
A2:
Initialize s = s
by MEMSTR_0:44;
then A3:
IC s = 0
by MEMSTR_0:47;
A4: if>0 (a,k1,I,J) =
(((a,k1) <=0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J
by SCMPDS_4:14
.=
((a,k1) <=0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J)
by SCMPDS_4:14
;
then A5:
Shift ((stop ((I ';' (Goto ((card J) + 1))) ';' J)),1) c= P +* (stop (if>0 (a,k1,I,J)))
by Lm6;
A6: Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,(0 + 1)) =
Following ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,0)))
by EXTPRO_1:3
.=
Following ((P +* (stop (if>0 (a,k1,I,J)))),s)
by EXTPRO_1:2
.=
Exec (((a,k1) <=0_goto ((card I) + 2)),s)
by A4, Th3, A2
;
for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)) . a
by A6, A2, SCMPDS_2:56;
then A7:
DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if>0 (a,k1,I,J)))),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,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P ) )
then A8: IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)) =
(IC s) + 1
by A6, SCMPDS_2:56
.=
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,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P ) )
assume A10:
I is_halting_on s,P
; ( if>0 (a,k1,I,J) is_closed_on s,P & if>0 (a,k1,I,J) is_halting_on s,P )
then A11:
(I ';' (Goto ((card J) + 1))) ';' J is_closed_on s,P
by A9, Th21;
then A12:
( Start-At (0,SCMPDS) c= Initialize s & (I ';' (Goto ((card J) + 1))) ';' J is_closed_on Initialize s,P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) )
by FUNCT_4:25;
A13:
stop ((I ';' (Goto ((card J) + 1))) ';' J) c= P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))
by FUNCT_4:25;
(I ';' (Goto ((card J) + 1))) ';' J is_halting_on s,P
by A9, A10, Th21;
then A14:
P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on Initialize s
;
A15: card (stop (if>0 (a,k1,I,J))) =
(card (if>0 (a,k1,I,J))) + 1
by COMPOS_1:55
.=
((card ((I ';' (Goto ((card J) + 1))) ';' J)) + 1) + 1
by A4, Th1
;
now for k being Nat holds IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,k)) in dom (stop (if>0 (a,k1,I,J)))let k be
Nat;
IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,b1)) in dom (stop (if>0 (a,k1,I,J)))per cases
( 0 < k or k = 0 )
;
suppose
0 < k
;
IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,b1)) in dom (stop (if>0 (a,k1,I,J)))then consider k1 being
Nat such that A16:
k1 + 1
= k
by NAT_1:6;
reconsider k1 =
k1 as
Nat ;
reconsider m =
IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Initialize s),k1)) as
Nat ;
A17:
card (stop (if>0 (a,k1,I,J))) = (card (stop ((I ';' (Goto ((card J) + 1))) ';' J))) + 1
by A15, COMPOS_1:55;
m in dom (stop ((I ';' (Goto ((card J) + 1))) ';' J))
by A11;
then
m < card (stop ((I ';' (Goto ((card J) + 1))) ';' J))
by AFINSQ_1:66;
then A18:
m + 1
< card (stop (if>0 (a,k1,I,J)))
by A17, XREAL_1:6;
IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,k)) =
IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)),k1))
by A16, EXTPRO_1:4
.=
m + 1
by A12, A5, A8, A7, Th22, A13
;
hence
IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,k)) in dom (stop (if>0 (a,k1,I,J)))
by A18, AFINSQ_1:66;
verum end; suppose
k = 0
;
IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,b1)) in dom (stop (if>0 (a,k1,I,J)))hence
IC (Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,k)) in dom (stop (if>0 (a,k1,I,J)))
by A1, A3, EXTPRO_1:2;
verum end; end; end;
hence
if>0 (a,k1,I,J) is_closed_on s,P
by A2; if>0 (a,k1,I,J) is_halting_on s,P
A19:
Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Initialize s))) + 1)) = Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Initialize s))))
by EXTPRO_1:4;
CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,((LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Initialize s))) + 1)))) =
CurInstr ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),(Comput ((P +* (stop (if>0 (a,k1,I,J)))),s,1)),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Initialize s))))))
by A19
.=
CurInstr ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Initialize s),(LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Initialize s))))))
by A12, A5, A8, A7, Th22, A13
.=
halt SCMPDS
by A14, EXTPRO_1:def 15
;
then
P +* (stop (if>0 (a,k1,I,J))) halts_on s
by EXTPRO_1:29;
hence
if>0 (a,k1,I,J) is_halting_on s,P
by A2; verum