Require Import Arith.

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

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

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

Compute isred (Dark (Light (Dark (Dark Blue)))).
Compute isblue (Dark (Light (Dark (Dark Blue)))).

Compute 1 =? 1.

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

Print nat.
Print "=?".
Print "=".
Print true.
Check true.
Check True.
Print True.

(* Compute if 1 = 1 then true else false. <-- ERROR *)

Definition zilch (n:nat) := (n = 0).
Check zilch.
Compute zilch 0.
Compute zilch 1.

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

Theorem zilch_twice:
  forall x y, zilch (x+y) -> x+y = 0.
Proof.
  intros.
  destruct (x+y).
    reflexivity.
    discriminate H.
Qed.