let n be Nat; for R being non trivial Ring
for a being Data-Location of R
for loc being Nat
for s1, s2 being State of (SCM R)
for P1, P2 being Instruction-Sequence of (SCM R)
for q being NAT -defined the InstructionsF of (SCM b1) -valued finite non halt-free Function
for p being non empty b8 -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds
( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )
let R be non trivial Ring; for a being Data-Location of R
for loc being Nat
for s1, s2 being State of (SCM R)
for P1, P2 being Instruction-Sequence of (SCM R)
for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function
for p being non empty b7 -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds
( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )
let a be Data-Location of R; for loc being Nat
for s1, s2 being State of (SCM R)
for P1, P2 being Instruction-Sequence of (SCM R)
for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function
for p being non empty b6 -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds
( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )
let loc be Nat; for s1, s2 being State of (SCM R)
for P1, P2 being Instruction-Sequence of (SCM R)
for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function
for p being non empty b5 -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds
( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )
let s1, s2 be State of (SCM R); for P1, P2 being Instruction-Sequence of (SCM R)
for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function
for p being non empty b3 -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds
( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )
let P1, P2 be Instruction-Sequence of (SCM R); for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function
for p being non empty b1 -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds
( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )
set Cs2i1 = Comput (P2,s2,(n + 1));
set Cs1i1 = Comput (P1,s1,(n + 1));
set I = CurInstr (P1,(Comput (P1,s1,n)));
let q be NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function; for p being non empty q -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds
( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )
let p be non empty q -autonomic FinPartState of (SCM R); ( p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 implies ( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R ) )
assume that
A1:
( p c= s1 & p c= s2 )
and
A2:
( q c= P1 & q c= P2 )
; ( not CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc or not loc <> (IC (Comput (P1,s1,n))) + 1 or ( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R ) )
A3:
CurInstr (P1,(Comput (P1,s1,n))) = CurInstr (P2,(Comput (P2,s2,n)))
by A1, A2, AMISTD_5:7;
set Cs2i = Comput (P2,s2,n);
set Cs1i = Comput (P1,s1,n);
A4: Comput (P1,s1,(n + 1)) =
Following (P1,(Comput (P1,s1,n)))
by EXTPRO_1:3
.=
Exec ((CurInstr (P1,(Comput (P1,s1,n)))),(Comput (P1,s1,n)))
;
A5: Comput (P2,s2,(n + 1)) =
Following (P2,(Comput (P2,s2,n)))
by EXTPRO_1:3
.=
Exec ((CurInstr (P2,(Comput (P2,s2,n)))),(Comput (P2,s2,n)))
;
IC in dom p
by AMISTD_5:6;
then A6:
( ((Comput (P1,s1,(n + 1))) | (dom p)) . (IC ) = (Comput (P1,s1,(n + 1))) . (IC ) & ((Comput (P2,s2,(n + 1))) | (dom p)) . (IC ) = (Comput (P2,s2,(n + 1))) . (IC ) )
by FUNCT_1:49;
assume that
A7:
CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc
and
A8:
loc <> (IC (Comput (P1,s1,n))) + 1
; ( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )
A9:
IC (Comput (P1,s1,n)) = IC (Comput (P2,s2,n))
by A1, A2, AMISTD_5:7;
hereby ( (Comput (P2,s2,n)) . a = 0. R implies (Comput (P1,s1,n)) . a = 0. R )
assume
(
(Comput (P1,s1,n)) . a = 0. R &
(Comput (P2,s2,n)) . a <> 0. R )
;
contradictionthen
(
(Comput (P1,s1,(n + 1))) . (IC ) = loc &
(Comput (P2,s2,(n + 1))) . (IC ) = (IC (Comput (P2,s2,n))) + 1 )
by A3, A4, A5, A7, SCMRING2:16;
hence
contradiction
by A1, A9, A6, A8, A2, EXTPRO_1:def 10;
verum
end;
assume that
A10:
(Comput (P2,s2,n)) . a = 0. R
and
A11:
(Comput (P1,s1,n)) . a <> 0. R
; contradiction
A12:
(Comput (P1,s1,(n + 1))) . (IC ) = (IC (Comput (P1,s1,n))) + 1
by A4, A7, A11, SCMRING2:16;
(Comput (P2,s2,(n + 1))) . (IC ) = loc
by A3, A5, A7, A10, SCMRING2:16;
hence
contradiction
by A1, A6, A8, A12, A2, EXTPRO_1:def 10; verum