let q be NAT -defined the InstructionsF of SCM+FSA -valued finite non halt-free Function; for p being non empty q -autonomic FinPartState of SCM+FSA
for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for P1, P2 being Instruction-Sequence of SCM+FSA st q c= P1 & q c= P2 holds
for i being Nat
for da being Int-Location
for loc being Nat st CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )
let p be non empty q -autonomic FinPartState of SCM+FSA; for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for P1, P2 being Instruction-Sequence of SCM+FSA st q c= P1 & q c= P2 holds
for i being Nat
for da being Int-Location
for loc being Nat st CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )
let s1, s2 be State of SCM+FSA; ( p c= s1 & p c= s2 implies for P1, P2 being Instruction-Sequence of SCM+FSA st q c= P1 & q c= P2 holds
for i being Nat
for da being Int-Location
for loc being Nat st CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 ) )
assume A1:
( p c= s1 & p c= s2 )
; for P1, P2 being Instruction-Sequence of SCM+FSA st q c= P1 & q c= P2 holds
for i being Nat
for da being Int-Location
for loc being Nat st CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )
let P1, P2 be Instruction-Sequence of SCM+FSA; ( q c= P1 & q c= P2 implies for i being Nat
for da being Int-Location
for loc being Nat st CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 ) )
assume A2:
( q c= P1 & q c= P2 )
; for i being Nat
for da being Int-Location
for loc being Nat st CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )
let i be Nat; for da being Int-Location
for loc being Nat st CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )
let da be Int-Location; for loc being Nat st CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )
let loc be Nat; ( CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 implies ( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 ) )
set I = CurInstr (P1,(Comput (P1,s1,i)));
set Cs1i = Comput (P1,s1,i);
set Cs2i = Comput (P2,s2,i);
set Cs1i1 = Comput (P1,s1,(i + 1));
set Cs2i1 = Comput (P2,s2,(i + 1));
A3: Comput (P1,s1,(i + 1)) =
Following (P1,(Comput (P1,s1,i)))
by EXTPRO_1:3
.=
Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i)))
;
A4: Comput (P2,s2,(i + 1)) =
Following (P2,(Comput (P2,s2,i)))
by EXTPRO_1:3
.=
Exec ((CurInstr (P2,(Comput (P2,s2,i)))),(Comput (P2,s2,i)))
;
IC in dom p
by AMISTD_5:6;
then A5:
( ((Comput (P1,s1,(i + 1))) | (dom p)) . (IC ) = (Comput (P1,s1,(i + 1))) . (IC ) & ((Comput (P2,s2,(i + 1))) | (dom p)) . (IC ) = (Comput (P2,s2,(i + 1))) . (IC ) )
by FUNCT_1:49;
assume that
A6:
CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc
and
A7:
loc <> (IC (Comput (P1,s1,i))) + 1
; ( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )
A8:
CurInstr (P1,(Comput (P1,s1,i))) = CurInstr (P2,(Comput (P2,s2,i)))
by A1, A2, AMISTD_5:7;
A9:
now ( (Comput (P2,s2,i)) . da = 0 implies not (Comput (P1,s1,i)) . da <> 0 )assume
(
(Comput (P2,s2,i)) . da = 0 &
(Comput (P1,s1,i)) . da <> 0 )
;
contradictionthen
(
(Comput (P2,s2,(i + 1))) . (IC ) = loc &
(Comput (P1,s1,(i + 1))) . (IC ) = (IC (Comput (P1,s1,i))) + 1 )
by A8, A3, A4, A6, SCMFSA_2:70;
hence
contradiction
by A1, A5, A7, A2, EXTPRO_1:def 10;
verum end;
A10:
(Comput (P1,s1,(i + 1))) | (dom p) = (Comput (P2,s2,(i + 1))) | (dom p)
by A1, A2, EXTPRO_1:def 10;
now ( (Comput (P1,s1,i)) . da = 0 implies not (Comput (P2,s2,i)) . da <> 0 )assume
(
(Comput (P1,s1,i)) . da = 0 &
(Comput (P2,s2,i)) . da <> 0 )
;
contradictionthen
(
(Comput (P1,s1,(i + 1))) . (IC ) = loc &
(Comput (P2,s2,(i + 1))) . (IC ) = (IC (Comput (P2,s2,i))) + 1 )
by A8, A3, A4, A6, SCMFSA_2:70;
hence
contradiction
by A1, A5, A10, A7, A2, AMISTD_5:7;
verum end;
hence
( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )
by A9; verum