Require Import Arith.

Theorem context_examples:
  forall (A B C D E F:Prop), A -> B -> C -> D -> E -> F -> A.
Proof.
  intro.
  intro P.
  intros Q R.
  intros until F.
  intros.
  revert H.
  revert F H4.
  revert R H2.
  rename E into HELLO.
  intros.
  clear Q HELLO H3 H1.

  assert (H1: 1+1=2).
    reflexivity.

  (* exact H. *)
  assumption.
Qed.

Theorem apply_example:
  forall (A B C:Prop), (A -> B) -> (B -> C) -> A -> C.
Proof.
  intros.

  apply H0.
  apply H.
  assumption.

(*
  apply H in H1.
  apply H0 in H1.
  assumption.
*)
Qed.

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

Theorem search_example:
  forall (n:nat), 0 < n -> n < n+n.
Proof.
  intros.
  Search nat.
  Search (_ < _+_).
  apply Nat.lt_add_pos_r.
  exact H.
Qed.

Theorem plus_0n_n0:
  forall (n:nat), 0+n = n+0.
Proof.
  intro.
  simpl.
  Print "+".
  unfold "+".
  fold Nat.add.
  rewrite Nat.add_comm.
  Check Nat.add_comm.
  reflexivity.
Qed.

Theorem equality_examples:
 forall (m n:nat), m+1 = n+1 -> m=n.
Proof.
  intros.
  symmetry.
  symmetry in H.
  transitivity (n+n-m).
  rewrite Nat.add_comm in H.
  simpl in H.
  rewrite Nat.add_comm in H.
  simpl in H.
  injection H.
  intro.
  subst.
  remember (m+m) as hello.
Admitted.