Section EqDec.
Context {Σ_funcs : funcs_signature}.
Context {Σ_preds : preds_signature}.
Context {ops : operators}.
Hypothesis eq_dec_Funcs : eq_dec syms.
Hypothesis eq_dec_Preds : eq_dec preds.
Hypothesis eq_dec_binop : eq_dec binop.
Hypothesis eq_dec_quantop : eq_dec quantop.
Global Instance dec_term : eq_dec term.
Proof with subst; try (now left + (right; intros[=]; resolve_existT; congruence)).
intros t. induction t as [ | ]; intros [|? v']...
- decide (x = n)...
- decide (F = f)... destruct (dec_vec_in _ _ _ X v')...
Qed.
Instance dec_falsity : eq_dec falsity_flag.

Lemma eq_dep_falsity b phi psi :
eq_dep falsity_flag (@form Σ_funcs Σ_preds ops) b phi b psi <-> phi = psi.