(* Contribution to the Coq Library   V6.3 (July 1999)                    *)

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(* This contribution was updated for Coq V5.10 by the COQ workgroup.        *)
(* January 1995                                                             *)
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(* Goal: forall (A B : Prop) (_ : forall _ : forall _ : forall _ : forall _ : A, B, A, A, B), B *)
(*                                ex1_auto.v                                *)
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Theorem trivial : forall A : Prop, A -> A.
(* Goal: forall (A B : Prop) (_ : forall _ : forall _ : forall _ : forall _ : A, B, A, A, B), B *)
auto.
Qed.

Theorem and_commutative : forall A B : Prop, A /\ B -> B /\ A.
(* Goal: forall (A B : Prop) (_ : forall _ : forall _ : forall _ : forall _ : A, B, A, A, B), B *)
intros A B H'; elim H'; auto.
Qed.

Theorem or_commutative : forall A B : Prop, A \/ B -> B \/ A.
(* Goal: forall (A B : Prop) (_ : forall _ : forall _ : forall _ : forall _ : A, B, A, A, B), B *)
intros A B H'; elim H'; auto.
Qed.

Theorem mp : forall A B : Prop, A -> (A -> B) -> B.
(* Goal: forall (A B : Prop) (_ : forall _ : forall _ : forall _ : forall _ : A, B, A, A, B), B *)
auto.
Qed.

Theorem S : forall A B C : Prop, (A -> B -> C) -> (A -> B) -> A -> C.
(* Goal: forall (A B : Prop) (_ : forall _ : forall _ : forall _ : forall _ : A, B, A, A, B), B *)
auto.
Qed.

Theorem Praeclarum :
 forall x y z t : Prop, (x -> z) /\ (y -> t) -> x /\ y -> z /\ t.
(* Goal: forall (x y z t : Prop) (_ : and (forall _ : x, z) (forall _ : y, t)) (_ : and x y), and z t *)
intros x y z t H'; elim H'.
(* Goal: forall (_ : forall _ : x, z) (_ : forall _ : y, t) (_ : and x y), and z t *)
intros H'0 H'1 H'2; elim H'2.
(* Goal: forall (A B : Prop) (_ : forall _ : forall _ : forall _ : forall _ : A, B, A, A, B), B *)
auto.
Qed.

Theorem resolution :
 forall (p q : Type -> Prop) (a : Type),
 p a -> (forall x : Type, p x -> q x) -> q a.
(* Goal: forall (A B : Prop) (_ : forall _ : forall _ : forall _ : forall _ : A, B, A, A, B), B *)
auto.
Qed.

Theorem Witnesses :
 forall (a b : Type) (p : Type -> Prop), p a \/ p b -> exists x : Type, p x.
intros a b p h; elim h;
 [ intro H'; clear h; try exact H' | clear h; intro H' ].
(* Goal: @ex Type (fun x : Type => p x) *)
(* Goal: @ex Type (fun x : Type => p x) *)
exists a; try assumption.
(* Goal: @ex Type (fun x : Type => p x) *)
exists b; try assumption.
Qed.

Theorem Simple :
 forall (A : Set) (R : A -> A -> Prop),
 (forall x y z : A, R x y /\ R y z -> R x z) ->
 (forall x y : A, R x y -> R y x) ->
 forall x : A, (exists y : A, R x y) -> R x x.
(* Goal: forall (A : Set) (R : forall (_ : A) (_ : A), Prop) (_ : forall (x y z : A) (_ : and (R x y) (R y z)), R x z) (_ : forall (x y : A) (_ : R x y), R y x) (x : A) (_ : @ex A (fun y : A => R x y)), R x x *)
intros A R H' H'0 x H'1; try assumption.
(* Goal: R x x *)
elim H'1; intros y E; clear H'1; try exact E.
(* Goal: forall (A B : Prop) (_ : forall _ : forall _ : forall _ : forall _ : A, B, A, A, B), B *)
apply H' with y; auto.
Qed.

Theorem not_not : forall a : Prop, a -> ~ ~ a.
(* Goal: forall (A B : Prop) (_ : forall _ : forall _ : forall _ : forall _ : A, B, A, A, B), B *)
unfold not in |- *; auto.
Qed.

Theorem mini_cases : forall x y : Prop, (x \/ ~ y) /\ y -> x.
intros x y h; elim h; intros h0 H'; elim h0;
 [ clear h h0; intro H'0 | intro H'0; elim H'0; clear h h0; try assumption ];
(* Goal: forall (A B : Prop) (_ : forall _ : forall _ : forall _ : forall _ : A, B, A, A, B), B *)
 auto.
Qed.

Require Import Classical.
(*This theorem needs classical logic*)

Theorem not_not_converse : forall a : Prop, ~ ~ a -> a.
(* Goal: forall (a : Prop) (_ : not (not a)), a *)
intros a H'; try assumption; auto 10.
generalize (classic a); intro h; elim h;
 [ intro H'0; clear h; try exact H'0 | clear h; intro H'0 ].
(* Goal: a *)
elim H'; assumption.
Qed.

Theorem not_quite_classic : forall a : Prop, ~ ~ (a \/ ~ a).
(* Goal: forall (A B : Prop) (_ : forall _ : forall _ : forall _ : forall _ : A, B, A, A, B), B *)
unfold not in |- *; auto.
Qed.

Theorem Peirce : forall A B : Prop, ((((A -> B) -> A) -> A) -> B) -> B.
(* Goal: forall (A B : Prop) (_ : forall _ : forall _ : forall _ : forall _ : A, B, A, A, B), B *)
auto.
Qed.

