let q be NAT -defined the InstructionsF of SCM -valued finite non halt-free Function; 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 = SubFrom (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; 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 = SubFrom (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; ( 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 = SubFrom (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 )
; 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 = SubFrom (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; ( 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 = SubFrom (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 )
; 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 = SubFrom (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; for da, db being Data-Location
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = SubFrom (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; for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = SubFrom (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; ( I = CurInstr (P1,(Comput (P1,s1,i))) & I = SubFrom (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)))
; ( not I = SubFrom (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 = SubFrom (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) )
; 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_3:4;
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_3:4;
hence
contradiction
by A8, A6, A7, A2, A1, EXTPRO_1:def 10; verum