let q be NAT -defined the InstructionsF of SCM+FSA -valued finite non halt-free Function; :: thesis: 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, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = Divide (da,db) & db in dom p holds
((Comput (P1,s1,i)) . da) mod ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) mod ((Comput (P2,s2,i)) . db)
let p be non empty q -autonomic FinPartState of SCM+FSA; :: thesis: 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, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = Divide (da,db) & db in dom p holds
((Comput (P1,s1,i)) . da) mod ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) mod ((Comput (P2,s2,i)) . db)
let s1, s2 be State of SCM+FSA; :: thesis: ( 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, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = Divide (da,db) & db in dom p holds
((Comput (P1,s1,i)) . da) mod ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) mod ((Comput (P2,s2,i)) . db) )
assume A1: ( p c= s1 & p c= s2 ) ; :: thesis: for P1, P2 being Instruction-Sequence of SCM+FSA st q c= P1 & q c= P2 holds
for i being Nat
for da, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = Divide (da,db) & db in dom p holds
((Comput (P1,s1,i)) . da) mod ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) mod ((Comput (P2,s2,i)) . db)
let P1, P2 be Instruction-Sequence of SCM+FSA; :: thesis: ( q c= P1 & q c= P2 implies for i being Nat
for da, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = Divide (da,db) & db in dom p holds
((Comput (P1,s1,i)) . da) mod ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) mod ((Comput (P2,s2,i)) . db) )
assume A2: ( q c= P1 & q c= P2 ) ; :: thesis: for i being Nat
for da, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = Divide (da,db) & db in dom p holds
((Comput (P1,s1,i)) . da) mod ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) mod ((Comput (P2,s2,i)) . db)
let i be Nat; :: thesis: for da, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = Divide (da,db) & db in dom p holds
((Comput (P1,s1,i)) . da) mod ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) mod ((Comput (P2,s2,i)) . db)
let da, db be Int-Location; :: thesis: ( CurInstr (P1,(Comput (P1,s1,i))) = Divide (da,db) & db in dom p implies ((Comput (P1,s1,i)) . da) mod ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) mod ((Comput (P2,s2,i)) . db) )
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 (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
A4: CurInstr (P1,(Comput (P1,s1,i))) = Divide (da,db) and
A5: db in dom p and
A6: ((Comput (P1,s1,i)) . da) mod ((Comput (P1,s1,i)) . db) <> ((Comput (P2,s2,i)) . da) mod ((Comput (P2,s2,i)) . db) ; :: thesis: contradiction
A7: ( ((Comput (P1,s1,(i + 1))) | (dom p)) . db = (Comput (P1,s1,(i + 1))) . db & ((Comput (P2,s2,(i + 1))) | (dom p)) . db = (Comput (P2,s2,(i + 1))) . db ) by A5, 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 A8: (Comput (P1,s1,(i + 1))) . db = ((Comput (P1,s1,i)) . da) mod ((Comput (P1,s1,i)) . db) by A4, SCMFSA_2:67;
CurInstr (P1,(Comput (P1,s1,i))) = CurInstr (P2,(Comput (P2,s2,i))) by A1, A2, AMISTD_5:7;
then (Comput (P2,s2,(i + 1))) . db = ((Comput (P2,s2,i)) . da) mod ((Comput (P2,s2,i)) . db) by A3, A4, SCMFSA_2:67;
hence contradiction by A1, A6, A7, A8, A2, EXTPRO_1:def 10; :: thesis: verum