let n be Element of NAT ; :: thesis: for R being non trivial good Ring
for a, b being Data-Location of R
for s1, s2 being State of (SCM R)
for P1, P2 being the Instructions of (SCM b₁) -valued ManySortedSet of NAT st not R is trivial holds
for p being non NAT -defined autonomic FinPartState of st NPP p c= s1 & NPP p c= s2 & ProgramPart p c= P1 & ProgramPart p c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = SubFrom (a,b) & a in dom p holds
((Comput (P1,s1,n)) . a) - ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) - ((Comput (P2,s2,n)) . b)
let R be non trivial good Ring; :: thesis: for a, b being Data-Location of R
for s1, s2 being State of (SCM R)
for P1, P2 being the Instructions of (SCM R) -valued ManySortedSet of NAT st not R is trivial holds
for p being non NAT -defined autonomic FinPartState of st NPP p c= s1 & NPP p c= s2 & ProgramPart p c= P1 & ProgramPart p c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = SubFrom (a,b) & a in dom p holds
((Comput (P1,s1,n)) . a) - ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) - ((Comput (P2,s2,n)) . b)
let a, b be Data-Location of R; :: thesis: for s1, s2 being State of (SCM R)
for P1, P2 being the Instructions of (SCM R) -valued ManySortedSet of NAT st not R is trivial holds
for p being non NAT -defined autonomic FinPartState of st NPP p c= s1 & NPP p c= s2 & ProgramPart p c= P1 & ProgramPart p c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = SubFrom (a,b) & a in dom p holds
((Comput (P1,s1,n)) . a) - ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) - ((Comput (P2,s2,n)) . b)
let s1, s2 be State of (SCM R); :: thesis: for P1, P2 being the Instructions of (SCM R) -valued ManySortedSet of NAT st not R is trivial holds
for p being non NAT -defined autonomic FinPartState of st NPP p c= s1 & NPP p c= s2 & ProgramPart p c= P1 & ProgramPart p c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = SubFrom (a,b) & a in dom p holds
((Comput (P1,s1,n)) . a) - ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) - ((Comput (P2,s2,n)) . b)
let P1, P2 be the Instructions of (SCM R) -valued ManySortedSet of NAT ; :: thesis: ( not R is trivial implies for p being non NAT -defined autonomic FinPartState of st NPP p c= s1 & NPP p c= s2 & ProgramPart p c= P1 & ProgramPart p c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = SubFrom (a,b) & a in dom p holds
((Comput (P1,s1,n)) . a) - ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) - ((Comput (P2,s2,n)) . b) )
assume not R is trivial ; :: thesis: for p being non NAT -defined autonomic FinPartState of st NPP p c= s1 & NPP p c= s2 & ProgramPart p c= P1 & ProgramPart p c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = SubFrom (a,b) & a in dom p holds
((Comput (P1,s1,n)) . a) - ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) - ((Comput (P2,s2,n)) . b)
set Cs2i1 = Comput (P2,s2,(n + 1));
set Cs1i1 = Comput (P1,s1,(n + 1));
set Cs2i = Comput (P2,s2,n);
set Cs1i = Comput (P1,s1,n);
set I = CurInstr (P1,(Comput (P1,s1,n)));
let p be non NAT -defined autonomic FinPartState of ; :: thesis: ( NPP p c= s1 & NPP p c= s2 & ProgramPart p c= P1 & ProgramPart p c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = SubFrom (a,b) & a in dom p implies ((Comput (P1,s1,n)) . a) - ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) - ((Comput (P2,s2,n)) . b) )
assume that
A1: ( NPP p c= s1 & NPP p c= s2 ) and
A2: ( ProgramPart p c= P1 & ProgramPart p c= P2 ) ; :: thesis: ( not CurInstr (P1,(Comput (P1,s1,n))) = SubFrom (a,b) or not a in dom p or ((Comput (P1,s1,n)) . a) - ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) - ((Comput (P2,s2,n)) . b) )
A3: ( a in dom (NPP p) implies ( ((Comput (P1,s1,(n + 1))) | (dom (NPP p))) . a = (Comput (P1,s1,(n + 1))) . a & ((Comput (P2,s2,(n + 1))) | (dom (NPP p))) . a = (Comput (P2,s2,(n + 1))) . a ) ) by FUNCT_1:72;
A4: Comput (P2,s2,(n + 1)) = Following (P2,(Comput (P2,s2,n))) by EXTPRO_1:4
.= Exec ((CurInstr (P2,(Comput (P2,s2,n)))),(Comput (P2,s2,n))) ;
assume that
A5: CurInstr (P1,(Comput (P1,s1,n))) = SubFrom (a,b) and
A6: ( a in dom p & ((Comput (P1,s1,n)) . a) - ((Comput (P1,s1,n)) . b) <> ((Comput (P2,s2,n)) . a) - ((Comput (P2,s2,n)) . b) ) ; :: thesis: contradiction
Comput (P1,s1,(n + 1)) = Following (P1,(Comput (P1,s1,n))) by EXTPRO_1:4
.= Exec ((CurInstr (P1,(Comput (P1,s1,n)))),(Comput (P1,s1,n))) ;
then A7: (Comput (P1,s1,(n + 1))) . a = ((Comput (P1,s1,n)) . a) - ((Comput (P1,s1,n)) . b) by A5, SCMRING2:15;
CurInstr (P1,(Comput (P1,s1,n))) = CurInstr (P2,(Comput (P2,s2,n))) by A1, AMISTD_5:7, A2;
then (Comput (P2,s2,(n + 1))) . a = ((Comput (P2,s2,n)) . a) - ((Comput (P2,s2,n)) . b) by A4, A5, SCMRING2:15;
hence contradiction by A1, A3, A6, A7, EXTPRO_1:def 9, Lm2, A2; :: thesis: verum