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Require Import Program.Tactics.
Obligation Tactic := idtac.
Set Universe Polymorphism.
Inductive paths {A : Type} (a : A) : A -> Type :=
idpath : paths a a.
Inductive Empty : Type :=.
Inductive Unit : Type := tt.
Definition not (A : Type) := A -> Empty.
Lemma nat_path_O_S (n : nat) (H : paths O (S n)) : Empty.
Proof. refine (
match H in paths _ i return
match i with
| O => Unit
| S _ => Empty
end
with
| idpath _ => tt
end
). Defined.
Lemma symmetry {A} (x y : A) (p : paths x y) : paths y x.
Proof.
destruct p. apply idpath.
Defined.
Lemma nat_path_S_O (n : nat) (H : paths (S n) O) : Empty.
Proof. eapply nat_path_O_S. exact (symmetry _ _ H). Defined.
Set Printing Universes.
Program Fixpoint succ_not_zero (n:nat) : not (paths (S n) 0) :=
match n as n return not (paths (S n) 0) with
| 0 => nat_path_S_O _
| S n' => let dummy := succ_not_zero n' in _
end.
Next Obligation.
intros f _ n dummy H. exact (nat_path_S_O _ H).
Show Universes.
Defined.
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