(* Version: 23.04. *)
(* * Preliminaries by Autosubst *)
(* ** Axiomatic Assumptions For our development, we have to extend Coq with _functional extensionality_. *)
(* ** Functional Extensionality We import the axiom from the Coq Standard Library and derive a utility tactic to make the assumption practically usable. *)
Tactic Notation "nointr" tactic(t) :=
let m := fresh "marker" in
pose (m := tt);
t; revert_until m; clear m.
Ltac fext := nointr repeat (
match goal with
[ |- ?x = ?y ] =>
(refine (@functional_extensionality_dep _ _ _ _ _) ||
refine (@forall_extensionality _ _ _ _) ||
refine (@forall_extensionalityP _ _ _ _) ||
refine (@forall_extensionalityS _ _ _ _)); intro
end).

(* ** Propositional Extensionality We state the axiom of _propositional extensionality_ directly and use it to prove _proof irrelevance_. *)
Axiom pext : forall P Q : Prop, (P <-> Q) -> (P = Q).