let f be FinSeq-Location ; :: thesis: for i being Instruction of SCM+FSA
for s being State of SCM+FSA st not f in UsedInt*Loc i holds
(Exec (i,s)) . f = s . f

let i be Instruction of SCM+FSA; :: thesis: for s being State of SCM+FSA st not f in UsedInt*Loc i holds
(Exec (i,s)) . f = s . f

let s be State of SCM+FSA; :: thesis: ( not f in UsedInt*Loc i implies (Exec (i,s)) . f = s . f )
assume A1: not f in UsedInt*Loc i ; :: thesis: (Exec (i,s)) . f = s . f
InsCode i <= 12 by SCMFSA_2:16;
then not not InsCode i = 0 & ... & not InsCode i = 12 ;
per cases then ( InsCode i = 0 or InsCode i = 1 or InsCode i = 2 or InsCode i = 3 or InsCode i = 4 or InsCode i = 5 or InsCode i = 6 or InsCode i = 7 or InsCode i = 8 or InsCode i = 9 or InsCode i = 10 or InsCode i = 11 or InsCode i = 12 ) ;
suppose InsCode i = 0 ; :: thesis: (Exec (i,s)) . f = s . f
end;
suppose InsCode i = 1 ; :: thesis: (Exec (i,s)) . f = s . f
end;
suppose InsCode i = 2 ; :: thesis: (Exec (i,s)) . f = s . f
end;
suppose InsCode i = 3 ; :: thesis: (Exec (i,s)) . f = s . f
end;
suppose InsCode i = 4 ; :: thesis: (Exec (i,s)) . f = s . f
end;
suppose InsCode i = 5 ; :: thesis: (Exec (i,s)) . f = s . f
end;
suppose InsCode i = 6 ; :: thesis: (Exec (i,s)) . f = s . f
then ex l being Nat st i = goto l by SCMFSA_2:35;
hence (Exec (i,s)) . f = s . f by SCMFSA_2:69; :: thesis: verum
end;
suppose InsCode i = 7 ; :: thesis: (Exec (i,s)) . f = s . f
then ex l being Nat ex a being Int-Location st i = a =0_goto l by SCMFSA_2:36;
hence (Exec (i,s)) . f = s . f by SCMFSA_2:70; :: thesis: verum
end;
suppose InsCode i = 8 ; :: thesis: (Exec (i,s)) . f = s . f
then ex l being Nat ex a being Int-Location st i = a >0_goto l by SCMFSA_2:37;
hence (Exec (i,s)) . f = s . f by SCMFSA_2:71; :: thesis: verum
end;
suppose InsCode i = 9 ; :: thesis: (Exec (i,s)) . f = s . f
then ex a, b being Int-Location ex g being FinSeq-Location st i = b := (g,a) by SCMFSA_2:38;
hence (Exec (i,s)) . f = s . f by SCMFSA_2:72; :: thesis: verum
end;
suppose InsCode i = 10 ; :: thesis: (Exec (i,s)) . f = s . f
then consider a, b being Int-Location, g being FinSeq-Location such that
A2: i = (g,a) := b by SCMFSA_2:39;
UsedInt*Loc i = {g} by ;
then f <> g by ;
hence (Exec (i,s)) . f = s . f by ; :: thesis: verum
end;
suppose InsCode i = 11 ; :: thesis: (Exec (i,s)) . f = s . f
end;
suppose InsCode i = 12 ; :: thesis: (Exec (i,s)) . f = s . f
then consider a being Int-Location, g being FinSeq-Location such that
A3: i = g :=<0,...,0> a by SCMFSA_2:41;
UsedInt*Loc i = {g} by ;
then f <> g by ;
hence (Exec (i,s)) . f = s . f by ; :: thesis: verum
end;
end;