let I be set ; :: thesis: ( I is Instruction of SCM+FSA iff ( I = [0,{},{}] or ex a, b being Int-Location st I = a := b or ex a, b being Int-Location st I = AddTo (a,b) or ex a, b being Int-Location st I = SubFrom (a,b) or ex a, b being Int-Location st I = MultBy (a,b) or ex a, b being Int-Location st I = Divide (a,b) or ex la being Nat st I = goto la or ex lb being Nat ex da being Int-Location st I = da =0_goto lb or ex lb being Nat ex da being Int-Location st I = da >0_goto lb or ex b, a being Int-Location ex fa being FinSeq-Location st I = a := (fa,b) or ex a, b being Int-Location ex fa being FinSeq-Location st I = (fa,a) := b or ex a being Int-Location ex f being FinSeq-Location st I = a :=len f or ex a being Int-Location ex f being FinSeq-Location st I = f :=<0,...,0> a ) )
thus ( not I is Instruction of SCM+FSA or I = [0,{},{}] or ex a, b being Int-Location st I = a := b or ex a, b being Int-Location st I = AddTo (a,b) or ex a, b being Int-Location st I = SubFrom (a,b) or ex a, b being Int-Location st I = MultBy (a,b) or ex a, b being Int-Location st I = Divide (a,b) or ex la being Nat st I = goto la or ex lb being Nat ex da being Int-Location st I = da =0_goto lb or ex lb being Nat ex da being Int-Location st I = da >0_goto lb or ex b, a being Int-Location ex fa being FinSeq-Location st I = a := (fa,b) or ex a, b being Int-Location ex fa being FinSeq-Location st I = (fa,a) := b or ex a being Int-Location ex f being FinSeq-Location st I = a :=len f or ex a being Int-Location ex f being FinSeq-Location st I = f :=<0,...,0> a ) :: thesis: ( ( I = [0,{},{}] or ex a, b being Int-Location st I = a := b or ex a, b being Int-Location st I = AddTo (a,b) or ex a, b being Int-Location st I = SubFrom (a,b) or ex a, b being Int-Location st I = MultBy (a,b) or ex a, b being Int-Location st I = Divide (a,b) or ex la being Nat st I = goto la or ex lb being Nat ex da being Int-Location st I = da =0_goto lb or ex lb being Nat ex da being Int-Location st I = da >0_goto lb or ex b, a being Int-Location ex fa being FinSeq-Location st I = a := (fa,b) or ex a, b being Int-Location ex fa being FinSeq-Location st I = (fa,a) := b or ex a being Int-Location ex f being FinSeq-Location st I = a :=len f or ex a being Int-Location ex f being FinSeq-Location st I = f :=<0,...,0> a ) implies I is Instruction of SCM+FSA )
proof
assume I is Instruction of SCM+FSA ; :: thesis: ( I = [0,{},{}] or ex a, b being Int-Location st I = a := b or ex a, b being Int-Location st I = AddTo (a,b) or ex a, b being Int-Location st I = SubFrom (a,b) or ex a, b being Int-Location st I = MultBy (a,b) or ex a, b being Int-Location st I = Divide (a,b) or ex la being Nat st I = goto la or ex lb being Nat ex da being Int-Location st I = da =0_goto lb or ex lb being Nat ex da being Int-Location st I = da >0_goto lb or ex b, a being Int-Location ex fa being FinSeq-Location st I = a := (fa,b) or ex a, b being Int-Location ex fa being FinSeq-Location st I = (fa,a) := b or ex a being Int-Location ex f being FinSeq-Location st I = a :=len f or ex a being Int-Location ex f being FinSeq-Location st I = f :=<0,...,0> a )
then reconsider J = I as Instruction of SCM+FSA ;
set n = InsCode J;
not not InsCode J = 0 & ... & not InsCode J = 12 by Th9;
hence ( I = [0,{},{}] or ex a, b being Int-Location st I = a := b or ex a, b being Int-Location st I = AddTo (a,b) or ex a, b being Int-Location st I = SubFrom (a,b) or ex a, b being Int-Location st I = MultBy (a,b) or ex a, b being Int-Location st I = Divide (a,b) or ex la being Nat st I = goto la or ex lb being Nat ex da being Int-Location st I = da =0_goto lb or ex lb being Nat ex da being Int-Location st I = da >0_goto lb or ex b, a being Int-Location ex fa being FinSeq-Location st I = a := (fa,b) or ex a, b being Int-Location ex fa being FinSeq-Location st I = (fa,a) := b or ex a being Int-Location ex f being FinSeq-Location st I = a :=len f or ex a being Int-Location ex f being FinSeq-Location st I = f :=<0,...,0> a ) by Th23, Th24, Th25, Th26, Th27, Th28, Th29, Th30, Th31, Th32, Th33, Th34, Th85; :: thesis: verum
end;
thus ( ( I = [0,{},{}] or ex a, b being Int-Location st I = a := b or ex a, b being Int-Location st I = AddTo (a,b) or ex a, b being Int-Location st I = SubFrom (a,b) or ex a, b being Int-Location st I = MultBy (a,b) or ex a, b being Int-Location st I = Divide (a,b) or ex la being Nat st I = goto la or ex lb being Nat ex da being Int-Location st I = da =0_goto lb or ex lb being Nat ex da being Int-Location st I = da >0_goto lb or ex b, a being Int-Location ex fa being FinSeq-Location st I = a := (fa,b) or ex a, b being Int-Location ex fa being FinSeq-Location st I = (fa,a) := b or ex a being Int-Location ex f being FinSeq-Location st I = a :=len f or ex a being Int-Location ex f being FinSeq-Location st I = f :=<0,...,0> a ) implies I is Instruction of SCM+FSA ) by SCMFSA_I:3; :: thesis: verum