Library Coq.Logic.FunctionalExtensionality


This module states the axiom of (dependent) functional extensionality and (dependent) eta-expansion. It introduces a tactic extensionality to apply the axiom of extensionality to an equality goal.
The converse of functional extensionality.

Lemma equal_f : forall {A B : Type} {f g : A -> B},
  f = g -> forall x, f x = g x.

Lemma equal_f_dep : forall {A B} {f g : forall (x : A), B x},
  f = g -> forall x, f x = g x.

Statements of functional extensionality for simple and dependent functions.

Axiom functional_extensionality_dep : forall {A} {B : A -> Type},
  forall (f g : forall x : A, B x),
  (forall x, f x = g x) -> f = g.

Lemma functional_extensionality {A B} (f g : A -> B) :
  (forall x, f x = g x) -> f = g.

Extensionality of foralls follows from functional extensionality.
Lemma forall_extensionality {A} {B C : A -> Type} (H : forall x : A, B x = C x)
: (forall x, B x) = (forall x, C x).

Lemma forall_extensionalityP {A} {B C : A -> Prop} (H : forall x : A, B x = C x)
: (forall x, B x) = (forall x, C x).

Lemma forall_extensionalityS {A} {B C : A -> Set} (H : forall x : A, B x = C x)
: (forall x, B x) = (forall x, C x).

A version of functional_extensionality_dep which is provably equal to eq_refl on fun _ => eq_refl
Definition functional_extensionality_dep_good
           {A} {B : A -> Type}
           (f g : forall x : A, B x)
           (H : forall x, f x = g x)
  : f = g
  := eq_trans (eq_sym (functional_extensionality_dep f f (fun _ => eq_refl)))
              (functional_extensionality_dep f g H).

Lemma functional_extensionality_dep_good_refl {A B} f
  : @functional_extensionality_dep_good A B f f (fun _ => eq_refl) = eq_refl.

Opaque functional_extensionality_dep_good.

Lemma forall_sig_eq_rect
      {A B} (f : forall a : A, B a)
      (P : { g : _ | (forall a, f a = g a) } -> Type)
      (k : P (exist (fun g => forall a, f a = g a) f (fun a => eq_refl)))
      g
: P g.
Definition forall_eq_rect
      {A B} (f : forall a : A, B a)
      (P : forall g, (forall a, f a = g a) -> Type)
      (k : P f (fun a => eq_refl))
      g H
  : P g H
  := @forall_sig_eq_rect A B f (fun g => P (proj1_sig g) (proj2_sig g)) k (exist _ g H).

Definition forall_eq_rect_comp {A B} f P k
  : @forall_eq_rect A B f P k f (fun _ => eq_refl) = k.

Definition f_equal__functional_extensionality_dep_good
           {A B f g} H a
  : f_equal (fun h => h a) (@functional_extensionality_dep_good A B f g H) = H a.

Definition f_equal__functional_extensionality_dep_good__fun
           {A B f g} H
  : (fun a => f_equal (fun h => h a) (@functional_extensionality_dep_good A B f g H)) = H.

Apply functional_extensionality, introducing variable x.

Tactic Notation "extensionality" ident(x) :=
  match goal with
    [ |- ?X = ?Y ] =>
    (apply (@functional_extensionality _ _ X Y) ||
     apply (@functional_extensionality_dep _ _ X Y) ||
     apply forall_extensionalityP ||
     apply forall_extensionalityS ||
     apply forall_extensionality) ; intro x
  end.

Iteratively apply functional_extensionality on an hypothesis until finding an equality statement
Ltac extensionality_in_checker tac :=
  first [ tac tt | fail 1 "Anomaly: Unexpected error in extensionality tactic. Please report." ].
Tactic Notation "extensionality" "in" hyp(H) :=
  let rec check_is_extensional_equality H :=
      lazymatch type of H with
      | _ = _ => constr:(Prop)
      | forall a : ?A, ?T
        => let Ha := fresh in
           constr:(forall a : A, match H a with Ha => ltac:(let v := check_is_extensional_equality Ha in exact v) end)
      end in
  let assert_is_extensional_equality H :=
      first [ let dummy := check_is_extensional_equality H in idtac
            | fail 1 "Not an extensional equality" ] in
  let assert_not_intensional_equality H :=
      lazymatch type of H with
      | _ = _ => fail "Already an intensional equality"
      | _ => idtac
      end in
  let enforce_no_body H :=
      (tryif (let dummy := (eval unfold H in H) in idtac)
        then clearbody H
        else idtac) in
  let rec extensionality_step_make_type H :=
      lazymatch type of H with
      | forall a : ?A, ?f = ?g
        => constr:({ H' | (fun a => f_equal (fun h => h a) H') = H })
      | forall a : ?A, _
        => let H' := fresh in
           constr:(forall a : A, match H a with H' => ltac:(let ret := extensionality_step_make_type H' in exact ret) end)
      end in
  let rec eta_contract T :=
      lazymatch (eval cbv beta in T) with
      | context T'[fun a : ?A => ?f a]
        => let T'' := context T'[f] in
           eta_contract T''
      | ?T => T
      end in
  let rec lift_sig_extensionality H :=
      lazymatch type of H with
      | sig _ => H
      | forall a : ?A, _
        => let Ha := fresh in
           let ret := constr:(fun a : A => match H a with Ha => ltac:(let v := lift_sig_extensionality Ha in exact v) end) in
           lazymatch type of ret with
           | forall a : ?A, sig (fun b : ?B => @?f a b = @?g a b)
             => eta_contract (exist (fun b : (forall a : A, B) => (fun a : A => f a (b a)) = (fun a : A => g a (b a)))
                                    (fun a : A => proj1_sig (ret a))
                                    (@functional_extensionality_dep_good _ _ _ _ (fun a : A => proj2_sig (ret a))))
           end
      end in
  let extensionality_pre_step H H_out Heq :=
      let T := extensionality_step_make_type H in
      let H' := fresh in
      assert (H' : T) by (intros; eexists; apply f_equal__functional_extensionality_dep_good__fun);
      let H''b := lift_sig_extensionality H' in
      case H''b; clear H';
      intros H_out Heq in
  let rec extensionality_rec H H_out Heq :=
      lazymatch type of H with
      | forall a, _ = _
        => extensionality_pre_step H H_out Heq
      | _
        => let pre_H_out' := fresh H_out in
           let H_out' := fresh pre_H_out' in
           extensionality_pre_step H H_out' Heq;
           let Heq' := fresh Heq in
           extensionality_rec H_out' H_out Heq';
           subst H_out'
      end in
  first [ assert_is_extensional_equality H | fail 1 "Not an extensional equality" ];
  first [ assert_not_intensional_equality H | fail 1 "Already an intensional equality" ];
  (tryif enforce_no_body H then idtac else clearbody H);
  let H_out := fresh in
  let Heq := fresh "Heq" in
  extensionality_in_checker ltac:(fun tt => extensionality_rec H H_out Heq);
  
  destruct Heq; rename H_out into H.

Eta expansion follows from extensionality.

Lemma eta_expansion_dep {A} {B : A -> Type} (f : forall x : A, B x) :
  f = fun x => f x.

Lemma eta_expansion {A B} (f : A -> B) : f = fun x => f x.