(* Formalizing and Proving Theorems in Coq --- Homework 3, part b.
   curated by Tobias Kappé, May 2022.

   There are no exercises in this file; it just serves to show that all of
   the exercises in Lecture 1 can be automated. *)

Require Import Coq.Setoids.Setoid.

Lemma conj_commutes (P Q: Prop):
  P /\ Q -> Q /\ P
.
Proof.
  intuition.
Qed.

Lemma conj_commutes_alternative (P Q: Prop):
  P /\ Q -> Q /\ P
.
Proof.
  intuition.
Qed.

Lemma disj_commutes (P Q: Prop):
  P \/ Q -> Q \/ P
.
Proof.
  intuition.
Qed.

Lemma conj_commutes_iff (P Q: Prop):
  P /\ Q <-> Q /\ P
.
Proof.
  intuition.
Qed.

Lemma conj_distributes (P Q R: Prop):
  (P /\ Q) \/ (P /\ R) -> P /\ (Q \/ R)
.
Proof.
  intuition.
Qed.

Lemma conj_distributes' (P Q R: Prop):
  (P /\ R) \/ (Q /\ R) -> (P \/ Q) /\ R
.
Proof.
  intuition.
Qed.

Lemma conj_distributes_compl (P Q R: Prop):
  P /\ (Q \/ R) -> (P /\ Q) \/ (P /\ R)
.
Proof.
  intuition.
Qed.

Lemma conj_distributes_compl' (P Q R: Prop):
  (Q \/ R) /\ P -> (Q /\ P) \/ (R /\ P)
.
Proof.
  intuition.
Qed.

Lemma conj_distributes_compl'_alternative (P Q R: Prop):
  (Q \/ R) /\ P -> (Q /\ P) \/ (R /\ P)
.
Proof.
  intuition.
Qed.

Lemma disj_commutes_iff (P Q: Prop):
  P \/ Q <-> Q \/ P
.
Proof.
  intuition.
Qed.

Lemma disj_distributes (P Q R: Prop):
  P \/ (Q /\ R) -> (P \/ Q) /\ (P \/ R)
.
Proof.
  intuition.
Qed.

Lemma disj_distributes_compl (P Q R: Prop):
  (P \/ Q) /\ (P \/ R) -> P \/ (Q /\ R)
.
Proof.
  intuition.
Qed.

Lemma disj_associative (P Q R: Prop):
  (P \/ Q) \/ R -> P \/ (Q \/ R)
.
Proof.
  intuition.
Qed.

Lemma disj_associative_compl (P Q R: Prop):
  P \/ (Q \/ R) -> (P \/ Q) \/ R
.
Proof.
  intuition.
Qed.

Lemma conj_associative (P Q R: Prop):
  (P /\ Q) /\ R -> P /\ (Q /\ R)
.
Proof.
  intuition.
Qed.

Lemma conj_associative_compl (P Q R: Prop):
  P /\ (Q /\ R) -> (P /\ Q) /\ R
.
Proof.
  intuition.
Qed.

Lemma introduce_double_negation (P: Prop):
  P -> ~~P
.
Proof.
  intuition.
Qed.

Lemma demorgan_first (P Q: Prop):
  ~(P \/ Q) -> ~P /\ ~Q
.
Proof.
  intuition.
Qed.

Lemma demorgan_first_compl (P Q: Prop):
  ~P /\ ~Q -> ~(P \/ Q)
.
Proof.
  intuition.
Qed.

Lemma demorgan_second (P Q: Prop):
  ~P \/ ~Q -> ~(P /\ Q)
.
Proof.
  intuition.
Qed.

Require Import Coq.Logic.Classical.

Lemma demorgan_second_compl (P Q: Prop):
  ~(P /\ Q) -> ~P \/ ~Q
.
Proof.
  (* The intuition tactic does not help here, even with the extra axiom. *)
  intuition.
Abort.