let P be Instruction-Sequence of SCM+FSA; for s being State of SCM+FSA
for I, J being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location st s . a > 0 & I is_halting_on s,P holds
if>0 (a,I,J) is_halting_on s,P
let s be State of SCM+FSA; for I, J being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location st s . a > 0 & I is_halting_on s,P holds
if>0 (a,I,J) is_halting_on s,P
let I, J be really-closed MacroInstruction of SCM+FSA ; for a being read-write Int-Location st s . a > 0 & I is_halting_on s,P holds
if>0 (a,I,J) is_halting_on s,P
let a be read-write Int-Location; ( s . a > 0 & I is_halting_on s,P implies if>0 (a,I,J) is_halting_on s,P )
set I1 = I ";" (Stop SCM+FSA);
set s1 = Initialize s;
set P1 = P +* (I ";" (Stop SCM+FSA));
set s3 = Initialize s;
set P3 = P +* (if>0 (a,I,J));
set s4 = Comput ((P +* (if>0 (a,I,J))),(Initialize s),1);
set i = a >0_goto ((card J) + 3);
A1:
I ";" (Stop SCM+FSA) c= P +* (I ";" (Stop SCM+FSA))
by FUNCT_4:25;
A2:
not a in dom (Start-At (0,SCM+FSA))
by SCMFSA_2:102;
A3:
0 in dom (if>0 (a,I,J))
by AFINSQ_1:65;
A4: (P +* (if>0 (a,I,J))) . 0 =
(if>0 (a,I,J)) . 0
by A3, FUNCT_4:13
.=
a >0_goto ((card J) + 3)
by Lm2
;
IC in dom (Start-At (0,SCM+FSA))
by MEMSTR_0:15;
then A5: IC (Initialize s) =
IC (Start-At (0,SCM+FSA))
by FUNCT_4:13
.=
0
by FUNCOP_1:72
;
A6:
if>0 (a,I,J) c= P +* (if>0 (a,I,J))
by FUNCT_4:25;
A7:
if>0 (a,I,J) = (((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" (I ";" (Stop SCM+FSA))
by SCMFSA6A:25;
card (((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) =
(card ((Macro (a >0_goto ((card J) + 3))) ";" J)) + (card (Goto ((card I) + 1)))
by SCMFSA6A:21
.=
(card ((Macro (a >0_goto ((card J) + 3))) ";" J)) + 1
by SCMFSA8A:15
.=
((card (Macro (a >0_goto ((card J) + 3)))) + (card J)) + 1
by SCMFSA6A:21
.=
((card J) + 2) + 1
by COMPOS_1:56
.=
(card J) + (2 + 1)
;
then A8:
Reloc ((I ";" (Stop SCM+FSA)),((card J) + 3)) c= if>0 (a,I,J)
by A7, SCMFSA6A:38;
A9:
Reloc ((I ";" (Stop SCM+FSA)),((card J) + 3)) c= P +* (if>0 (a,I,J))
by A8, A6, XBOOLE_1:1;
A10: Comput ((P +* (if>0 (a,I,J))),(Initialize s),(0 + 1)) =
Following ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),0)))
by EXTPRO_1:3
.=
Following ((P +* (if>0 (a,I,J))),(Initialize s))
.=
Exec ((a >0_goto ((card J) + 3)),(Initialize s))
by A5, A4, PBOOLE:143
;
A11:
for f being FinSeq-Location holds (Initialize s) . f = (Comput ((P +* (if>0 (a,I,J))),(Initialize s),1)) . f
by A10, SCMFSA_2:71;
for a being Int-Location holds (Initialize s) . a = (Comput ((P +* (if>0 (a,I,J))),(Initialize s),1)) . a
by A10, SCMFSA_2:71;
then A12:
DataPart (Initialize s) = DataPart (Comput ((P +* (if>0 (a,I,J))),(Initialize s),1))
by A11, SCMFSA_M:2;
assume
s . a > 0
; ( not I is_halting_on s,P or if>0 (a,I,J) is_halting_on s,P )
then
(Initialize s) . a > 0
by A2, FUNCT_4:11;
then A13:
IC (Comput ((P +* (if>0 (a,I,J))),(Initialize s),1)) = (card J) + 3
by A10, SCMFSA_2:71;
assume A14:
I is_halting_on s,P
; if>0 (a,I,J) is_halting_on s,P
I ";" (Stop SCM+FSA) is_halting_on s,P
by A14, SCMFSA8A:30;
then A15:
P +* (I ";" (Stop SCM+FSA)) halts_on Initialize s
by SCMFSA7B:def 7;
CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),((LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialize s))) + 1)))) =
CurInstr ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Comput ((P +* (if>0 (a,I,J))),(Initialize s),1)),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialize s))))))
by EXTPRO_1:4
.=
IncAddr ((CurInstr ((P +* (I ";" (Stop SCM+FSA))),(Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialize s),(LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialize s))))))),((card J) + 3))
by A13, A12, Th2, A9, A1
.=
IncAddr ((halt SCM+FSA),((card J) + 3))
by A15, EXTPRO_1:def 15
.=
halt SCM+FSA
by COMPOS_0:4
;
then
P +* (if>0 (a,I,J)) halts_on Initialize s
by EXTPRO_1:29;
hence
if>0 (a,I,J) is_halting_on s,P
by SCMFSA7B:def 7; verum