Require Import Arith.
Require Import List.

Theorem apply_example2:
  forall (A:Prop), (forall (B:Prop), B -> A) -> A.
Proof.
  intros.
  apply H with (B := True).
  exact I. (* apply I, reflexivity also work *)
Qed.

Theorem search_example:
  forall (n:nat), 0 < n -> n < n+n.
Proof.
  intros.
  Search nat.
  Search (_ < _ + _).
Abort.

Print "+".
Print "<".
Print "<=".

Theorem and_example:
  forall (A B:Prop), A /\ B -> B /\ A.
Proof.
  intros.
  destruct H.
  split.
    assumption.
    assumption.
Qed.

Print "/\".
Print "\/".

Theorem or_example:
  forall (A B:Prop), A \/ B -> B \/ A.
Proof.
  intros.
  destruct H.
    right. assumption.
    left. assumption.
Qed.

Theorem exists_example:
  forall (f:nat->nat), (exists n, f(n)=0) ->
                       (exists n, f(n)+n=n).
Proof.
  intros.
  destruct H.
  exists x.
  rewrite H. reflexivity.
Qed.

Theorem specialize_example:
  forall (A:Prop), (forall (B:Prop), B -> A) -> A.
Proof.
  intros.
  specialize H with (B := True).
  apply H.
  reflexivity.
Qed.

Print True.
Print eq.
Print False.