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 = da := db & da in dom p holds
(Comput (P1,s1,i)) . db = (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 = da := db & da in dom p holds
(Comput (P1,s1,i)) . db = (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 = da := db & da in dom p holds
(Comput (P1,s1,i)) . db = (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 = da := db & da in dom p holds
(Comput (P1,s1,i)) . db = (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 = da := db & da in dom p holds
(Comput (P1,s1,i)) . db = (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 = da := db & da in dom p holds
(Comput (P1,s1,i)) . db = (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 = da := db & da in dom p holds
(Comput (P1,s1,i)) . db = (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 = da := db & da in dom p holds
(Comput (P1,s1,i)) . db = (Comput (P2,s2,i)) . db
let I be Instruction of SCM; :: thesis: ( I = CurInstr (P1,(Comput (P1,s1,i))) & I = da := db & da in dom p implies (Comput (P1,s1,i)) . db = (Comput (P2,s2,i)) . db )
assume A3: I = CurInstr (P1,(Comput (P1,s1,i))) ; :: thesis: ( not I = da := db or not da in dom p or (Comput (P1,s1,i)) . db = (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 = da := db and
A6: ( da in dom p & (Comput (P1,s1,i)) . db <> (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)) . db by A4, A5, AMI_3:2;
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)) . db by A3, A5, AMI_3:2;
hence contradiction by A8, A6, A7, A2, A1, EXTPRO_1:def 10; :: thesis: verum