(* Some useful tactics *)
Tactic Notation "spec" hyp(H) :=
  match type of H with ?a -> _ => 
    let H1 := fresh in (assert (H1: a); [|generalize (H H1); clear H H1; intro H]) end.

Tactic Notation "spec" hyp(H) constr(a) :=
  (generalize (H a); clear H; intro H). 

Tactic Notation "spec" hyp(H) constr(a) constr(b) :=
  (generalize (H a b); clear H; intro H).

 Tactic Notation "spec" hyp(H) constr(a) constr(b) constr(c) :=
  (generalize (H a b c); clear H; intro H).

Tactic Notation "spec" hyp(H) constr(a) constr(b) constr(c) constr(d) :=
  (generalize (H a b c d); clear H; intro H).

Tactic Notation "spec" hyp(H) constr(a) constr(b) constr(c) constr(d) constr(e) :=
  (generalize (H a b c d e); clear H; intro H).

Ltac remembertac2 x a :=
  let x := fresh x in
  let H := fresh "Heq" x in
  (set (x:=a) in *; assert (H: a=x) by reflexivity; clearbody x).

Tactic Notation "cases" constr(a) :=
  let x := fresh in (remembertac2 x a; destruct x).

Require ClassicalFacts.  (* no axioms in this line *)
Require Export Coq.Logic.ClassicalChoice. 
     (* Axioms dependent_unique_choice, relational_choice *)

Tactic Notation "LEM" constr(a) :=
  destruct (classic a).








Require Import Omega.

Inductive Tree : Type :=
  | Leaf : Tree
  | Node : Tree -> Tree -> Tree.

Definition tree1 := Leaf.

Definition tree2 := Node (Node Leaf Leaf) Leaf.

Fixpoint leaves(t : Tree) : nat :=
  match t with
   | Leaf => 1
   | Node tl tr => leaves tl + leaves tr
  end.

Fixpoint nodes(t : Tree) : nat :=
  match t with
   | Leaf => 0
   | Node tl tr => 1 + nodes tl + nodes tr
  end.

Lemma leaves_gt_nodes: forall t,
  leaves t > nodes t.
Proof.
intro.
  induction t.
  simpl.
  omega.
  simpl.
  omega.
Qed.



(*
Lemma leaves_ge_nodes: forall t,
  leaves t >= nodes t.
Proof.
  induction t.
  simpl.
  omega.
  simpl.
  omega.
Qed.
*)

Lemma leaves_ge_nodes: forall t,
  leaves t >= nodes t.
Proof.
  intro.
  generalize (leaves_gt_nodes t); intro.
  omega.
Qed.

Fixpoint size(t : Tree) : nat :=
  match t with
   | Leaf => 1
   | Node tl tr => 1 + size tl + size tr
  end.

Lemma size_eq_leavesnodes: forall t,
  size t = leaves t + nodes t.
Proof.
  induction t.
  simpl.
  trivial.
  simpl.
  omega.
Qed.

Lemma leaves_ge_nodes': forall t,
  leaves t >= nodes t.
Proof.
  assert (forall t, leaves t > nodes t).
    induction t; simpl; omega.
  intro.
  spec H t.
  omega.
Qed.

Inductive Nat : Type :=
  | Z : Nat
  | S : Nat -> Nat.

Fixpoint Add (n1 n2 : Nat) : Nat :=
  match n1 with
   | Z => n2
   | S n1' => S (Add n1' n2)
  end.

Lemma Add_Comm : forall n1 n2,
  Add n1 n2 = Add n2 n1.
Proof.
  intros.
  induction n2.
    induction n1.
    trivial.
    simpl.
    rewrite IHn1.
    simpl.
    trivial.
  simpl.
  rewrite <- IHn2.
(*  clear IHn2. *)
  induction n1.
  simpl.
  trivial.
  simpl.
  rewrite IHn1.
  trivial.
(* uh oh... :-) *)
Qed.

Print Add_Comm.

Print leaves_gt_nodes.

(* revert, clear, etc. are very useful... as well as assert *)