(* File inspired from https://gitlab.mpi-sws.org/iris/stdpp/-/blob/8c98553ad0ca2029b30cf18b58e321ec3a79172b/stdpp/base.v *)
From Corelib Require Export Morphisms RelationClasses ListDef Setoid.
From Stdlib Require Export List Peano Utf8 Permutation.
From Stdlib Require Export Program.Basics Program.Syntax.

Export ListNotations.


Notation length := Datatypes.length.
Global Generalizable All Variables.
Global Unset Transparent Obligations.

Definition tc_opaque {A} (x : A) : A := x.
(* Typeclasses Opaque tc_opaque. *)

Global Arguments tc_opaque {_} _ /.
Declare Scope stdpp_scope.
Delimit Scope stdpp_scope with stdpp.
Global Open Scope stdpp_scope.
Notation "(=)" := eq (only parsing) : stdpp_scope.
Notation "( x =.)" := (eq x) (only parsing) : stdpp_scope.
Notation "(.= x )" := (λ y, eq y x) (only parsing) : stdpp_scope.
Notation "(≠)" := (λ x y, x ≠ y) (only parsing) : stdpp_scope.
Notation "( x ≠.)" := (λ y, x ≠ y) (only parsing) : stdpp_scope.
Notation "(.≠ x )" := (λ y, y ≠ x) (only parsing) : stdpp_scope.
Infix "=@{ A }" := (@eq A)
  (at level 70, only parsing, no associativity) : stdpp_scope.
Notation "(=@{ A } )" := (@eq A) (only parsing) : stdpp_scope.
Notation "(≠@{ A } )" := (λ X Y, ¬X =@{A} Y) (only parsing) : stdpp_scope.
Notation "X ≠@{ A } Y":= (¬X =@{ A } Y)
  (at level 70, only parsing, no associativity) : stdpp_scope.

Global Hint Extern 0 (_ = _) => reflexivity : core.
Global Hint Extern 100 (_ ≠ _) => discriminate : core.

Global Instance: ∀ A, PreOrder (=@{A}).
Proof. split; repeat intro; congruence. Qed.
Class Equiv A := equiv: relation A.
Global Instance equiv_rewrite_relation `{Equiv A} :
  RewriteRelation (@equiv A _) | 150 := {}.

Infix "≡" := equiv (at level 70, no associativity) : stdpp_scope.
Infix "≡@{ A }" := (@equiv A _)
  (at level 70, only parsing, no associativity) : stdpp_scope.

Notation "(≡)" := equiv (only parsing) : stdpp_scope.
Notation "( X ≡.)" := (equiv X) (only parsing) : stdpp_scope.
Notation "(.≡ X )" := (λ Y, Y ≡ X) (only parsing) : stdpp_scope.
Notation "(≢)" := (λ X Y, ¬X ≡ Y) (only parsing) : stdpp_scope.
Notation "X ≢ Y":= (¬X ≡ Y) (at level 70, no associativity) : stdpp_scope.
Notation "( X ≢.)" := (λ Y, X ≢ Y) (only parsing) : stdpp_scope.
Notation "(.≢ X )" := (λ Y, Y ≢ X) (only parsing) : stdpp_scope.

Notation "(≡@{ A } )" := (@equiv A _) (only parsing) : stdpp_scope.
Notation "(≢@{ A } )" := (λ X Y, ¬X ≡@{A} Y) (only parsing) : stdpp_scope.
Notation "X ≢@{ A } Y":= (¬X ≡@{ A } Y)
  (at level 70, only parsing, no associativity) : stdpp_scope.
Class LeibnizEquiv A `{Equiv A} :=
  leibniz_equiv (x y : A) : x ≡ y → x = y.
Global Hint Mode LeibnizEquiv ! - : typeclass_instances.

Global Instance: Params (@equiv) 2 := {}.
Global Instance equiv_default_relation `{Equiv A} :
  DefaultRelation (≡@{A}) | 3 := {}.
Global Hint Extern 0 (_ ≡ _) => reflexivity : core.
Global Hint Extern 0 (_ ≡ _) => symmetry; assumption : core.


Class Inj {A B} (R : relation A) (S : relation B) (f : A → B) : Prop :=
  inj x y : S (f x) (f y) → R x y.

Class Inj2 {A B C} (R1 : relation A) (R2 : relation B)
    (S : relation C) (f : A → B → C) : Prop :=
  inj2 x1 x2 y1 y2 : S (f x1 x2) (f y1 y2) → R1 x1 y1 ∧ R2 x2 y2.

Global Arguments irreflexivity {_} _ {_} _ _ : assert.
Global Arguments inj {_ _ _ _} _ {_} _ _ _ : assert.
Global Arguments inj2 {_ _ _ _ _ _} _ {_} _ _ _ _ _: assert.

Global Instance inj2_inj_1 `{Inj2 A B C R1 R2 R3 f} y : Inj R1 R3 (λ x, f x y).
Proof. repeat intro; edestruct (inj2 f); eauto. Qed.
Global Instance inj2_inj_2 `{Inj2 A B C R1 R2 R3 f} x : Inj R2 R3 (f x).
Proof. repeat intro; edestruct (inj2 f); eauto. Qed.

Notation "(∧)" := and (only parsing) : stdpp_scope.
Notation "( A ∧.)" := (and A) (only parsing) : stdpp_scope.
Notation "(.∧ B )" := (λ A, A ∧ B) (only parsing) : stdpp_scope.

Notation "(∨)" := or (only parsing) : stdpp_scope.
Notation "( A ∨.)" := (or A) (only parsing) : stdpp_scope.
Notation "(.∨ B )" := (λ A, A ∨ B) (only parsing) : stdpp_scope.

Notation "(↔)" := iff (only parsing) : stdpp_scope.
Notation "( A ↔.)" := (iff A) (only parsing) : stdpp_scope.
Notation "(.↔ B )" := (λ A, A ↔ B) (only parsing) : stdpp_scope.

Global Hint Extern 0 (_ ↔ _) => reflexivity : core.
Global Hint Extern 0 (_ ↔ _) => symmetry; assumption : core.

Notation "(→)" := (λ A B, A → B) (only parsing) : stdpp_scope.
Notation "( A →.)" := (λ B, A → B) (only parsing) : stdpp_scope.
Notation "(.→ B )" := (λ A, A → B) (only parsing) : stdpp_scope.

Notation "t $ r" := (t r)
  (at level 65, right associativity, only parsing) : stdpp_scope.
Notation "($)" := (λ f x, f x) (only parsing) : stdpp_scope.
Notation "(.$ x )" := (λ f, f x) (only parsing) : stdpp_scope.

Infix "∘" := compose : stdpp_scope.
Notation "(∘)" := compose (only parsing) : stdpp_scope.
Notation "( f ∘.)" := (compose f) (only parsing) : stdpp_scope.
Notation "(.∘ f )" := (λ g, compose g f) (only parsing) : stdpp_scope.
(** Ensure that [simpl] unfolds [id], [compose], and [flip] when fully
applied. *)
Global Arguments id _ _ / : assert.
Global Arguments compose _ _ _ _ _ _ / : assert.
Global Arguments flip _ _ _ _ _ _ / : assert.
Global Arguments const _ _ _ _ / : assert.

Definition fun_map {A A' B B'} (f: A' → A) (g: B → B') (h : A → B) : A' → B' :=
  g ∘ h ∘ f.

Global Instance id_inj {A} : Inj (=) (=) (@id A).
Proof. intros ??; auto. Qed.
Global Instance compose_inj {A B C} R1 R2 R3 (f : A → B) (g : B → C) :
  Inj R1 R2 f → Inj R2 R3 g → Inj R1 R3 (g ∘ f).
Proof. red; intuition. Qed.

(** ** Products *)
Notation "( x ,.)" := (pair x) (only parsing) : stdpp_scope.
Notation "(., y )" := (λ x, (x,y)) (only parsing) : stdpp_scope.

Notation "p .1" := (fst p) (at level 1, left associativity, format "p .1").
Notation "p .2" := (snd p) (at level 1, left associativity, format "p .2").

Definition prod_map {A A' B B'} (f: A → A') (g: B → B') (p : A * B) : A' * B' :=
  (f (p.1), g (p.2)).
Global Arguments prod_map {_ _ _ _} _ _ !_ / : assert.

Global Instance pair_inj {A B} : Inj2 (=) (=) (=) (@pair A B).
Proof. injection 1; auto. Qed.
Global Instance prod_map_inj {A A' B B'} (f : A → A') (g : B → B') :
  Inj (=) (=) f → Inj (=) (=) g → Inj (=) (=) (prod_map f g).
Proof.
  intros ?? [??] [??] ?; simpl in *; f_equal;
    [apply (inj f)|apply (inj g)]; congruence.
Qed.

Definition prod_relation {A B} (R1 : relation A) (R2 : relation B) :
  relation (A * B) := λ x y, R1 (x.1) (y.1) ∧ R2 (x.2) (y.2).

Section prod_relation.
  Context `{RA : relation A, RB : relation B}.
  Global Instance pair_inj' : Inj2 RA RB (prod_relation RA RB) pair.
  Proof. inversion_clear 1; eauto. Qed.
End prod_relation.

Global Instance prod_equiv `{Equiv A,Equiv B} : Equiv (A * B) :=
  prod_relation (≡) (≡).

Section prod_setoid.
  Context `{Equiv A, Equiv B}.
  Global Instance pair_equiv_inj : Inj2 (≡) (≡) (≡@{A*B}) pair := _.
End prod_setoid.

(* Typeclasses Opaque prod_equiv. *)

(** ** Sums *)
Definition sum_map {A A' B B'} (f: A → A') (g: B → B') (xy : A + B) : A' + B' :=
  match xy with inl x => inl (f x) | inr y => inr (g y) end.
Global Arguments sum_map {_ _ _ _} _ _ !_ / : assert.

Global Instance inl_inj {A B} : Inj (=) (=) (@inl A B).
Proof. injection 1; auto. Qed.
Global Instance inr_inj {A B} : Inj (=) (=) (@inr A B).
Proof. injection 1; auto. Qed.

Global Instance sum_map_inj {A A' B B'} (f : A → A') (g : B → B') :
  Inj (=) (=) f → Inj (=) (=) g → Inj (=) (=) (sum_map f g).
Proof. intros ?? [?|?] [?|?] [=]; f_equal; apply (inj _); auto. Qed.

Inductive sum_relation {A B}
     (RA : relation A) (RB : relation B) : relation (A + B) :=
  | inl_related x1 x2 : RA x1 x2 → sum_relation RA RB (inl x1) (inl x2)
  | inr_related y1 y2 : RB y1 y2 → sum_relation RA RB (inr y1) (inr y2).

Section sum_relation.
  Context `{RA : relation A, RB : relation B}.
  Global Instance inl_inj' : Inj RA (sum_relation RA RB) inl.
  Proof. inversion_clear 1; auto. Qed.
  Global Instance inr_inj' : Inj RB (sum_relation RA RB) inr.
  Proof. inversion_clear 1; auto. Qed.
End sum_relation.

Global Instance sum_equiv `{Equiv A, Equiv B} : Equiv (A + B) := sum_relation (≡) (≡).

Global Instance inl_equiv_inj `{Equiv A, Equiv B} : Inj (≡) (≡) (@inl A B) := _.
Global Instance inr_equiv_inj `{Equiv A, Equiv B} : Inj (≡) (≡) (@inr A B) := _.

Notation "` x" := (proj1_sig x) (at level 10, format "` x") : stdpp_scope.

Definition f := Nat.add 0.
Global Instance h: Inj eq eq f. 
  unfold f. simpl. easy.
Qed.

Goal Inj eq eq (compose (@id nat) id).
apply _.
Qed.

Goal Inj eq eq (compose (compose (@id nat) id) id).
apply _.
Qed.

(* Goal Inj eq eq (fun (x:nat) => id (id x)).
apply _.
Qed. *)

Goal Inj eq eq (fun (x: nat) => (compose id id) (id x)).
apply (compose_inj eq eq); apply _.
Qed.
