let q be NAT -defined the InstructionsF of SCM -valued finite non halt-free Function; :: thesis: for p being non empty q -autonomic FinPartState of SCM
for s1, s2 being State of SCM st p c= s1 & p c= s2 holds
for P1, P2 being Instruction-Sequence of SCM st q c= P1 & q c= P2 holds
for i being Nat
for da being Data-Location
for loc being Nat
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & 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; :: thesis: for s1, s2 being State of SCM st p c= s1 & p c= s2 holds
for P1, P2 being Instruction-Sequence of SCM st q c= P1 & q c= P2 holds
for i being Nat
for da being Data-Location
for loc being Nat
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & 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; :: thesis: ( p c= s1 & p c= s2 implies for P1, P2 being Instruction-Sequence of SCM st q c= P1 & q c= P2 holds
for i being Nat
for da being Data-Location
for loc being Nat
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & 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 ) ; :: thesis: for P1, P2 being Instruction-Sequence of SCM st q c= P1 & q c= P2 holds
for i being Nat
for da being Data-Location
for loc being Nat
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & 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; :: thesis: ( q c= P1 & q c= P2 implies for i being Nat
for da being Data-Location
for loc being Nat
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & 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 ) ; :: thesis: for i being Nat
for da being Data-Location
for loc being Nat
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & 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; :: thesis: for da being Data-Location
for loc being Nat
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & 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 Data-Location; :: thesis: for loc being Nat
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & 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; :: thesis: for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & 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 Instruction of SCM; :: thesis: ( I = CurInstr (P1,(Comput (P1,s1,i))) & 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 ) )
assume A3: I = CurInstr (P1,(Comput (P1,s1,i))) ; :: thesis: ( not I = da =0_goto loc or not loc <> (IC (Comput (P1,s1,i))) + 1 or ( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 ) )
set Cs2i1 = Comput (P2,s2,(i + 1));
set Cs1i1 = Comput (P1,s1,(i + 1));
set Cs2i = Comput (P2,s2,i);
set Cs1i = Comput (P1,s1,i);
A4: 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))) ;
A5: 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 A6: ( ((Comput (P1,s1,(i + 1))) | (dom p)) . (IC ) = IC (Comput (P1,s1,(i + 1))) & ((Comput (P2,s2,(i + 1))) | (dom p)) . (IC ) = IC (Comput (P2,s2,(i + 1))) ) by FUNCT_1:49;
assume that
A7: I = da =0_goto loc and
A8: loc <> (IC (Comput (P1,s1,i))) + 1 ; :: thesis: ( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )
A9: I = CurInstr (P2,(Comput (P2,s2,i))) by A3, A2, A1, AMISTD_5:7;

A10: now :: thesis: ( (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 ) ; :: thesis: contradiction
then ( (Comput (P2,s2,(i + 1))) . (IC ) = loc & (Comput (P1,s1,(i + 1))) . (IC ) = (IC (Comput (P1,s1,i))) + 1 ) by A3, A9, A4, A5, A7, AMI_3:8;
hence contradiction by A6, A8, A2, A1, EXTPRO_1:def 10; :: thesis: verum

end;

A11: (Comput (P1,s1,(i + 1))) | (dom p) = (Comput (P2,s2,(i + 1))) | (dom p) by A2, A1, EXTPRO_1:def 10;

now :: thesis: ( (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 ) ; :: thesis: contradiction
then ( (Comput (P1,s1,(i + 1))) . (IC ) = loc & (Comput (P2,s2,(i + 1))) . (IC ) = (IC (Comput (P2,s2,i))) + 1 ) by A3, A9, A4, A5, A7, AMI_3:8;
hence contradiction by A6, A11, A8, A2, A1, AMISTD_5:7; :: thesis: verum

end;

hence ( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 ) by A10; :: thesis: verum