let p be autonomic non programmed FinPartState of SCM+FSA ; :: thesis: for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for i being Element of NAT
for da, db being Int-Location
for f being FinSeq-Location st CurInstr (Computation s1,i) = f,db := da & f in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Computation s1,i) . db) & k2 = abs ((Computation s2,i) . db) holds
((Computation s1,i) . f) +* k1,((Computation s1,i) . da) = ((Computation s2,i) . f) +* k2,((Computation s2,i) . da)
let s1, s2 be State of SCM+FSA ; :: thesis: ( p c= s1 & p c= s2 implies for i being Element of NAT
for da, db being Int-Location
for f being FinSeq-Location st CurInstr (Computation s1,i) = f,db := da & f in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Computation s1,i) . db) & k2 = abs ((Computation s2,i) . db) holds
((Computation s1,i) . f) +* k1,((Computation s1,i) . da) = ((Computation s2,i) . f) +* k2,((Computation s2,i) . da) )
assume A1: ( p c= s1 & p c= s2 ) ; :: thesis: for i being Element of NAT
for da, db being Int-Location
for f being FinSeq-Location st CurInstr (Computation s1,i) = f,db := da & f in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Computation s1,i) . db) & k2 = abs ((Computation s2,i) . db) holds
((Computation s1,i) . f) +* k1,((Computation s1,i) . da) = ((Computation s2,i) . f) +* k2,((Computation s2,i) . da)
let i be Element of NAT ; :: thesis: for da, db being Int-Location
for f being FinSeq-Location st CurInstr (Computation s1,i) = f,db := da & f in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Computation s1,i) . db) & k2 = abs ((Computation s2,i) . db) holds
((Computation s1,i) . f) +* k1,((Computation s1,i) . da) = ((Computation s2,i) . f) +* k2,((Computation s2,i) . da)
let da, db be Int-Location ; :: thesis: for f being FinSeq-Location st CurInstr (Computation s1,i) = f,db := da & f in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Computation s1,i) . db) & k2 = abs ((Computation s2,i) . db) holds
((Computation s1,i) . f) +* k1,((Computation s1,i) . da) = ((Computation s2,i) . f) +* k2,((Computation s2,i) . da)
let f be FinSeq-Location ; :: thesis: ( CurInstr (Computation s1,i) = f,db := da & f in dom p implies for k1, k2 being Element of NAT st k1 = abs ((Computation s1,i) . db) & k2 = abs ((Computation s2,i) . db) holds
((Computation s1,i) . f) +* k1,((Computation s1,i) . da) = ((Computation s2,i) . f) +* k2,((Computation s2,i) . da) )
set I = CurInstr (Computation s1,i);
set Cs1i = Computation s1,i;
set Cs2i = Computation s2,i;
set Cs1i1 = Computation s1,(i + 1);
set Cs2i1 = Computation s2,(i + 1);
A2: Computation s1,(i + 1) = Following (Computation s1,i) by AMI_1:14
.= Exec (CurInstr (Computation s1,i)),(Computation s1,i) ;
A3: Computation s2,(i + 1) = Following (Computation s2,i) by AMI_1:14
.= Exec (CurInstr (Computation s2,i)),(Computation s2,i) ;
A4: ( f in dom p implies ( ((Computation s1,(i + 1)) | (dom p)) . f = (Computation s1,(i + 1)) . f & ((Computation s2,(i + 1)) | (dom p)) . f = (Computation s2,(i + 1)) . f ) ) by FUNCT_1:72;
A5: (Computation s1,(i + 1)) | (dom p) = (Computation s2,(i + 1)) | (dom p) by A1, AMI_1:def 25;
assume A6: ( CurInstr (Computation s1,i) = f,db := da & f in dom p ) ; :: thesis: for k1, k2 being Element of NAT st k1 = abs ((Computation s1,i) . db) & k2 = abs ((Computation s2,i) . db) holds
((Computation s1,i) . f) +* k1,((Computation s1,i) . da) = ((Computation s2,i) . f) +* k2,((Computation s2,i) . da)
let i1, i2 be Element of NAT ; :: thesis: ( i1 = abs ((Computation s1,i) . db) & i2 = abs ((Computation s2,i) . db) implies ((Computation s1,i) . f) +* i1,((Computation s1,i) . da) = ((Computation s2,i) . f) +* i2,((Computation s2,i) . da) )
assume that
A7: ( i1 = abs ((Computation s1,i) . db) & i2 = abs ((Computation s2,i) . db) ) and
A8: ((Computation s1,i) . f) +* i1,((Computation s1,i) . da) <> ((Computation s2,i) . f) +* i2,((Computation s2,i) . da) ; :: thesis: contradiction
consider k1 being Element of NAT such that
A9: ( k1 = abs ((Computation s1,i) . db) & (Exec (CurInstr (Computation s1,i)),(Computation s1,i)) . f = ((Computation s1,i) . f) +* k1,((Computation s1,i) . da) ) by A6, SCMFSA_2:99;
consider k2 being Element of NAT such that
A10: ( k2 = abs ((Computation s2,i) . db) & (Exec (CurInstr (Computation s1,i)),(Computation s2,i)) . f = ((Computation s2,i) . f) +* k2,((Computation s2,i) . da) ) by A6, SCMFSA_2:99;
thus contradiction by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, Th18; :: thesis: verum