Inductive match_specs: forall (p: program) (G: funspecs), Prop :=
| match_specs_nil: match_specs nil nil
| match_specs_cons: forall i vars f p' P G',
         not (In i (map (@fst _ _) p')) ->
         match_specs p' G' ->
         typecheck vars f = true ->
         match_specs ((i,(vars,f))::p') ((i,P)::G').Inductive match_specs: forall (p: program) (G: funspecs), Prop :=
| match_specs_nil: match_specs nil nil
| match_specs_cons: forall i vars f p' P G',
         not (In i (map (@fst _ _) p')) ->
         match_specs p' G' ->
         typecheck vars f = true ->
         match_specs ((i,(vars,f))::p') ((i,P)::G').Inductive match_specs: forall (p: program) (G: funspecs), Prop :=
| match_specs_nil: match_specs nil nil
| match_specs_cons: forall i vars f p' P G',
         not (In i (map (@fst _ _) p')) ->
         match_specs p' G' ->
         typecheck vars f = true ->
         match_specs ((i,(vars,f))::p') ((i,P)::G').

Lemma match_specs_boundary: forall p G i, match_specs p G -> i >= boundary p -> table_get G i = None.