Require Import Arith.

Example and_exercise :
  forall n m : nat, n + m = 0 -> n = 0 /\ m = 0.
Proof.
  intros n m H. 
  destruct n.
  - auto.
  - inversion H.
Qed.

Example and_exercise':
  forall n m : nat, n + m = 0 -> n = 0 /\ m = 0.
Proof.
  (* https://github.com/yanhick/coq-exercises/blob/master/LF/Logic.v *)
  intros n m H. 
  split.
  - destruct n.
    + reflexivity.
    + inversion H.
  - destruct m.
    + reflexivity.
    + rewrite plus_comm in H.
      inversion H. (* XXX: Why rewrite helped for inversion? *)
Qed.

Lemma proj2 : forall P Q : Prop, P /\ Q -> Q.
Proof.
  intros p q H.
  destruct H as [Hp Hq].
  exact Hq.
Qed.

Theorem and_assoc : forall P Q R : Prop,
  P /\ (Q /\ R) -> (P /\ Q) /\ R.
Proof.
    intros p q r H.
    destruct H as [Hp [Hq Hr]].
    split.
    - split.
      + exact Hp.
      + exact Hq.
    - exact Hr.
Qed.

Fact not_implies_our_not : forall (P:Prop),
  ~ P -> (forall (Q:Prop), P -> Q).
Proof.
  intros p Hp' q Hp.
  contradiction.
Qed.

Theorem double_not: forall P:Prop, P -> ~~P. 
Proof.
  intros p Hp.
  (* Or just [auto] *)
  unfold not.
  intro pF.
  apply pF.
  exact Hp.
Qed.

Theorem contrapositive : forall (P Q : Prop),
    (P -> Q) -> (~Q -> ~P).
Proof.
  intros p q Hpq.
  intro Hq'.
  unfold not.
  intro Hp.
  destruct Hq'.  (* XXX: What happened here? *)
  apply Hpq.
  exact Hp.
Qed.

Theorem not_both_true_and_false : forall P : Prop,
  ~ (P /\ ~P).
Proof.
  intros p.
  intro H.
  destruct H as [H'].
  destruct H.
  exact H'.
Qed.

Theorem not_true_is_false : forall b : bool,
  b <> true -> b = false.
Proof.
  intros b Hb'.
  unfold not in Hb'.
  destruct b.
  - destruct Hb'.
    reflexivity.
  - reflexivity.
Qed.

Theorem iff_sym : forall P Q : Prop,
  (P <-> Q) -> (Q <-> P).
Proof.
  intros p q Hpqiff.
	destruct Hpqiff as [Hpq Hqp].
  split.
  - exact Hqp.
  - exact Hpq.
Qed.

Lemma not_true_iff_false : forall b,
  b <> true <-> b = false.
Proof.
	intros b.
	split.
  - intro HbnqT.
	  apply not_true_is_false.
		exact HbnqT.
  - intro HbeqF. 
		rewrite HbeqF.
	  auto.
Qed.

Theorem or_distributes_over_and : forall P Q R: Prop,
  P \/ (Q /\ R) <-> (P \/ Q) /\ (P \/ R).
Proof.
	intros p q r.
	split.
  - intro H.
		split.
    + destruct H.
      * auto.
      * destruct H; auto.
    + destruct H.
      * auto.
      * destruct H; auto.
  - intro H.
    destruct H.
		left.
		destruct H0.
    + exact H0.
    + destruct H.
		  * exact H.
      * 
Abort.

Lemma mul_eq_0 : forall n m, n * m = 0 <-> n = 0 \/ m = 0.
Proof.
  intros n m.
  split.
  - intro H.
Abort.

Theorem or_assoc :
  forall P Q R : Prop, P \/ (Q \/ R) <-> (P \/ Q) \/ R.
Proof.
  intros p q r.
  split.
  - intro H.
    destruct H.
    + left.
      left.
      exact H.
    + destruct H.
      * left.
        right.
        exact H.
      * right.
        exact H.
  - intros [[Hp | Hq] | Hr].
    + left.
      exact Hp.
    + right.
      left.
      exact Hq.
    + right.
      right.
      exact Hr.
Qed.

Lemma mul_eq_0_ternary :
  forall n m p, n * m * p = 0 <-> n = 0 \/ m = 0 \/ p = 0.
Proof.
  intros n m p.
  split.
  - intro H.
    Search (_ * _ = 0).
    (*rewrite mult_is_O.*)
Abort.

Lemma apply_iff_example :
  forall n m : nat, n * m = 0 -> n = 0 \/ m = 0.
Proof.
  intros n m H.
  induction n.
  - auto.
  - induction m.
    + auto.
Abort.


Require Import ssreflect ssrbool.

Theorem eqb_eq: forall (n1 n2: nat),
  Nat.eqb n1 n2 = true <-> n1 = n2.
Proof.
  move => n1 n2.
  case (Nat.eqb n1 n2) eqn:H.
  - by rewrite PeanoNat.Nat.eqb_eq in H.
  - by rewrite PeanoNat.Nat.eqb_neq in H.
Qed.