Require Import Arith.

Inductive color : Type :=
| Red : color
| Blue : color
| Dark (c:color) : color
| Light (c:color) : color.

Definition isdark (c:color) : bool :=
  match c with
  | Dark _ => true
  | _ => false
  end.

Fixpoint isred (c:color) : bool :=
  match c with
  | Red => true
  | Blue => false
  | Dark x => isred x
  | Light x => isred x
  end.

Print nat.

(*
Fixpoint factorial (n:nat) : nat :=
  if n <=? 1 then 1 else n*(factorial (n-1)).
*)

Fixpoint factorial (n:nat) : nat :=
  match n with
  | O => 1
  | S m => n * factorial m
  end.

Fixpoint isblue (c:color) : bool :=
  match c with
  | Red => false
  | Blue => true
  | Dark x => isblue x
  | Light x => isblue x
  end.

Theorem redblue_dichotomy:
  forall c, isblue c = negb (isred c).
Proof.
  intro.
  induction c.
    simpl. reflexivity.
    simpl. reflexivity.
    simpl. assumption.
    simpl. assumption.
Qed.

Check redblue_dichotomy.

Check true.
Check false.
Check Nat.eqb.
Check True.
Check False.
Check eq.

Compute if (1 =? 1) then 1 else 0.
(* Compute if (1 = 1) then 1 else 0. *)

Definition zilch (n:nat) : Prop := (n = 0).

Compute zilch 0.
Compute zilch 1.

Theorem zilch_onlyzero:
  forall n, zilch n -> n = 0.
Proof.
  intro.
  destruct n.
    intro. reflexivity.
    intro. discriminate.
Qed.

Theorem rewrite_example:
  forall x, x=1 -> x+x=2.
Proof.
  intros.
  rewrite H.
  simpl. reflexivity.
Qed.