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, db being Data-Location
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = MultBy (da,db) & da in dom p holds
((Comput (P1,s1,i)) . da) * ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) * ((Comput (P2,s2,i)) . db)
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, db being Data-Location
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = MultBy (da,db) & da in dom p holds
((Comput (P1,s1,i)) . da) * ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) * ((Comput (P2,s2,i)) . db)
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, db being Data-Location
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = MultBy (da,db) & da in dom p holds
((Comput (P1,s1,i)) . da) * ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) * ((Comput (P2,s2,i)) . db) )
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, db being Data-Location
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = MultBy (da,db) & da in dom p holds
((Comput (P1,s1,i)) . da) * ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) * ((Comput (P2,s2,i)) . db)
let P1, P2 be Instruction-Sequence of SCM; :: thesis: ( q c= P1 & q c= P2 implies for i being Nat
for da, db being Data-Location
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = MultBy (da,db) & da in dom p holds
((Comput (P1,s1,i)) . da) * ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) * ((Comput (P2,s2,i)) . db) )
assume A2: ( q c= P1 & q c= P2 ) ; :: thesis: for i being Nat
for da, db being Data-Location
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = MultBy (da,db) & da in dom p holds
((Comput (P1,s1,i)) . da) * ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) * ((Comput (P2,s2,i)) . db)
let i be Nat; :: thesis: for da, db being Data-Location
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = MultBy (da,db) & da in dom p holds
((Comput (P1,s1,i)) . da) * ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) * ((Comput (P2,s2,i)) . db)
let da, db be Data-Location; :: thesis: for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = MultBy (da,db) & da in dom p holds
((Comput (P1,s1,i)) . da) * ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) * ((Comput (P2,s2,i)) . db)
let I be Instruction of SCM; :: thesis: ( I = CurInstr (P1,(Comput (P1,s1,i))) & I = MultBy (da,db) & da in dom p implies ((Comput (P1,s1,i)) . da) * ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) * ((Comput (P2,s2,i)) . db) )
assume A3: I = CurInstr (P1,(Comput (P1,s1,i))) ; :: thesis: ( not I = MultBy (da,db) or not da in dom p or ((Comput (P1,s1,i)) . da) * ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) * ((Comput (P2,s2,i)) . db) )
set Cs2i1 = Comput (P2,s2,(i + 1));
set Cs2i = Comput (P2,s2,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))) ;
assume that
A5: I = MultBy (da,db) and
A6: ( da in dom p & ((Comput (P1,s1,i)) . da) * ((Comput (P1,s1,i)) . db) <> ((Comput (P2,s2,i)) . da) * ((Comput (P2,s2,i)) . db) ) ; :: thesis: contradiction
I = CurInstr (P2,(Comput (P2,s2,i))) by A3, A2, A1, AMISTD_5:7;
then A7: (Comput (P2,s2,(i + 1))) . da = ((Comput (P2,s2,i)) . da) * ((Comput (P2,s2,i)) . db) by A4, A5, AMI_3:5;
set Cs1i1 = Comput (P1,s1,(i + 1));
set Cs1i = Comput (P1,s1,i);
A8: ( da in dom p implies ( ((Comput (P1,s1,(i + 1))) | (dom p)) . da = (Comput (P1,s1,(i + 1))) . da & ((Comput (P2,s2,(i + 1))) | (dom p)) . da = (Comput (P2,s2,(i + 1))) . da ) ) by FUNCT_1:49;
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))) ;
then (Comput (P1,s1,(i + 1))) . da = ((Comput (P1,s1,i)) . da) * ((Comput (P1,s1,i)) . db) by A3, A5, AMI_3:5;
hence contradiction by A8, A6, A7, A2, A1, EXTPRO_1:def 10; :: thesis: verum