let p be non NAT -defined autonomic FinPartState of ; :: thesis: for s1, s2 being State of SCM+FSA st NPP p c= s1 & NPP p c= s2 holds
for P1, P2 being the Instructions of SCM+FSA -valued ManySortedSet of NAT st ProgramPart p c= P1 & ProgramPart p c= P2 holds
for i being Element of NAT
for da being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = f :=<0,...,0> da & f in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . da) & k2 = abs ((Comput (P2,s2,i)) . da) holds
k1 |-> 0 = k2 |-> 0
let s1, s2 be State of SCM+FSA; :: thesis: ( NPP p c= s1 & NPP p c= s2 implies for P1, P2 being the Instructions of SCM+FSA -valued ManySortedSet of NAT st ProgramPart p c= P1 & ProgramPart p c= P2 holds
for i being Element of NAT
for da being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = f :=<0,...,0> da & f in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . da) & k2 = abs ((Comput (P2,s2,i)) . da) holds
k1 |-> 0 = k2 |-> 0 )
assume B1: ( NPP p c= s1 & NPP p c= s2 ) ; :: thesis: for P1, P2 being the Instructions of SCM+FSA -valued ManySortedSet of NAT st ProgramPart p c= P1 & ProgramPart p c= P2 holds
for i being Element of NAT
for da being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = f :=<0,...,0> da & f in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . da) & k2 = abs ((Comput (P2,s2,i)) . da) holds
k1 |-> 0 = k2 |-> 0
let P1, P2 be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: ( ProgramPart p c= P1 & ProgramPart p c= P2 implies for i being Element of NAT
for da being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = f :=<0,...,0> da & f in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . da) & k2 = abs ((Comput (P2,s2,i)) . da) holds
k1 |-> 0 = k2 |-> 0 )
assume A2: ( ProgramPart p c= P1 & ProgramPart p c= P2 ) ; :: thesis: for i being Element of NAT
for da being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = f :=<0,...,0> da & f in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . da) & k2 = abs ((Comput (P2,s2,i)) . da) holds
k1 |-> 0 = k2 |-> 0
let i be Element of NAT ; :: thesis: for da being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = f :=<0,...,0> da & f in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . da) & k2 = abs ((Comput (P2,s2,i)) . da) holds
k1 |-> 0 = k2 |-> 0
let da be Int-Location ; :: thesis: for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = f :=<0,...,0> da & f in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . da) & k2 = abs ((Comput (P2,s2,i)) . da) holds
k1 |-> 0 = k2 |-> 0
let f be FinSeq-Location ; :: thesis: ( CurInstr (P1,(Comput (P1,s1,i))) = f :=<0,...,0> da & f in dom p implies for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . da) & k2 = abs ((Comput (P2,s2,i)) . da) holds
k1 |-> 0 = k2 |-> 0 )
set Cs1i1 = Comput (P1,s1,(i + 1));
set Cs2i1 = Comput (P2,s2,(i + 1));
A3: (Comput (P1,s1,(i + 1))) | (dom (NPP p)) = (Comput (P2,s2,(i + 1))) | (dom (NPP p)) by B1, EXTPRO_1:def 9, A2, B1;
set Cs2i = Comput (P2,s2,i);
set Cs1i = Comput (P1,s1,i);
set I = CurInstr (P1,(Comput (P1,s1,i)));
A4: Comput (P1,s1,(i + 1)) = Following (P1,(Comput (P1,s1,i))) by EXTPRO_1:4
.= Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i))) ;
A5: ( f in dom (NPP p) implies ( ((Comput (P1,s1,(i + 1))) | (dom (NPP p))) . f = (Comput (P1,s1,(i + 1))) . f & ((Comput (P2,s2,(i + 1))) | (dom (NPP p))) . f = (Comput (P2,s2,(i + 1))) . f ) ) by FUNCT_1:72;
A6: Comput (P2,s2,(i + 1)) = Following (P2,(Comput (P2,s2,i))) by EXTPRO_1:4
.= Exec ((CurInstr (P2,(Comput (P2,s2,i)))),(Comput (P2,s2,i))) ;
assume that
A7: CurInstr (P1,(Comput (P1,s1,i))) = f :=<0,...,0> da and
A8: f in dom p ; :: thesis: for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . da) & k2 = abs ((Comput (P2,s2,i)) . da) holds
k1 |-> 0 = k2 |-> 0
A9: f in dom (NPP p) by A8, Lm2;
A10: ( ex k1 being Element of NAT st
( k1 = abs ((Comput (P1,s1,i)) . da) & (Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i)))) . f = k1 |-> 0 ) & ex k2 being Element of NAT st
( k2 = abs ((Comput (P2,s2,i)) . da) & (Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P2,s2,i)))) . f = k2 |-> 0 ) ) by A7, SCMFSA_2:101;
let i1, i2 be Element of NAT ; :: thesis: ( i1 = abs ((Comput (P1,s1,i)) . da) & i2 = abs ((Comput (P2,s2,i)) . da) implies i1 |-> 0 = i2 |-> 0 )
assume ( i1 = abs ((Comput (P1,s1,i)) . da) & i2 = abs ((Comput (P2,s2,i)) . da) & i1 |-> 0 <> i2 |-> 0 ) ; :: thesis: contradiction
hence contradiction by B1, A4, A6, A5, A3, A10, AMISTD_5:7, A9, A2; :: thesis: verum