From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.

Section RewriteFacilities.

  Lemma diseq:
    ∀ {X : Type} (p : X → Prop) (x y : X),
      ¬ p x → p y → x ≠ y.
  Proof. intros ? ? ? ? NP P EQ; subst; auto. Qed.


  Lemma eqprop_to_eqbool {X : eqType} {a b : X}: a = b → a == b.
  Proof. by intros; apply/eqP. Qed.

  Lemma eqbool_true {X : eqType} {a b : X}: a == b → a == b = true.
  Proof. by move =>/eqP EQ; subst b; rewrite eq_refl. Qed.

  Lemma eqbool_false {X : eqType} {a b : X}: a != b → a == b = false.
  Proof. by apply negbTE. Qed.


  Lemma eqbool_to_eqprop {X : eqType} {a b : X}: a == b → a = b.
  Proof. by intros; apply/eqP. Qed.

  Lemma neqprop_to_neqbool {X : eqType} {a b : X}: a ≠ b → a != b.
  Proof. by intros; apply/eqP. Qed.

  Lemma neqbool_to_neqprop {X : eqType} {a b : X}: a != b → a ≠ b.
  Proof. by intros; apply/eqP. Qed.

  Lemma neq_sym {X : eqType} {a b : X}:
    a != b → b != a.
  Proof.
    intros NEQ; apply/eqP; intros EQ;
      subst b; move: NEQ ⇒ /eqP NEQ; auto. Qed.

  Lemma neq_antirefl {X : eqType} {a : X}:
    (a != a) = false.
  Proof. by apply/eqP. Qed.


  Lemma option_inj_eq {X : eqType} {a b : X}:
    a == b → Some a == Some b.
  Proof. by move ⇒ /eqP EQ; apply/eqP; rewrite EQ. Qed.

  Lemma option_inj_neq {X : eqType} {a b : X}:
    a != b → Some a != Some b.
  Proof.
    by move ⇒ /eqP NEQ;
     apply/eqP; intros CONTR;
       apply: NEQ; inversion_clear CONTR. Qed.

  (* As a motivation for this file, we consider the following example. *)
  Section Example.

    (* Let X  be an arbitrary type ... *)
    Context {X : eqType}.

    (* ... f be an arbitrary function bool → bool ... *)
    Variable f : bool → bool.

    (* ... p be an arbitrary predicate on X ... *)
    Variable p : X → Prop.

    (* ... and let a and b be two elements of X such that ... *)
    Variables a b : X.

    (* ... p holds for a and doesn't hold for b. *)
    Hypothesis H_pa : p a.
    Hypothesis H_npb : ¬ p b.

    (* The following examples are commented out
       to expose the insides of the proofs. *)

    (*
    (* Simplifying some relatively sophisticated 
       expressions can be quite tedious. *)
    Goal f ((a == b) && f false) = f false.
    Proof.
      (* Things like simpl/compute make no sense here. *)
      (* One can use replace to generate a new goal. *)
      replace (a == b) with false; last first.
      (* However, this leads to a "loss of focus". Moreover, 
         the resulting goal is not so trivial to prove. *)
      { apply/eqP; rewrite eq_sym eqbF_neg.
       by apply/eqP; intros EQ; subst b; apply H_npb. }
       by rewrite Bool.andb_false_l.
    Abort.
     *)

    (*
    (* The second attempt. *)
    Goal f ((a == b) && f false) = f false.
      (* With the lemmas above one can compose multiple 
         transformations in a single rewrite. *)
       by rewrite (eqbool_false (neq_sym (neqprop_to_neqbool (diseq _ _ _ H_npb H_pa))))
       Bool.andb_false_l.
    Qed.
    *)

  End Example.

End RewriteFacilities.