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Axiom magic : False.
(* Submitted by Dachuan Yu (bug #220) *)
Fixpoint T (n : nat) : Type :=
match n with
| O => nat -> Prop
| S n' => T n'
end.
Inductive R : forall n : nat, T n -> nat -> Prop :=
| RO : forall (Psi : T 0) (l : nat), Psi l -> R 0 Psi l
| RS :
forall (n : nat) (Psi : T (S n)) (l : nat), R n Psi l -> R (S n) Psi l.
Definition Psi00 (n : nat) : Prop := False.
Definition Psi0 : T 0 := Psi00.
Lemma Inversion_RO : forall l : nat, R 0 Psi0 l -> Psi00 l.
inversion 1.
Abort.
(* Submitted by Pierre Casteran (bug #540) *)
Set Implicit Arguments.
Unset Strict Implicit.
Parameter rule : Set -> Type.
Inductive extension (I : Set) : Type :=
| NL : extension I
| add_rule : rule I -> extension I -> extension I.
Inductive in_extension (I : Set) (r : rule I) : extension I -> Type :=
| in_first : forall e, in_extension r (add_rule r e)
| in_rest : forall e r', in_extension r e -> in_extension r (add_rule r' e).
Implicit Arguments NL [I].
Inductive super_extension (I : Set) (e : extension I) :
extension I -> Type :=
| super_NL : super_extension e NL
| super_add :
forall r (e' : extension I),
in_extension r e ->
super_extension e e' -> super_extension e (add_rule r e').
Lemma super_def :
forall (I : Set) (e1 e2 : extension I),
super_extension e2 e1 -> forall ru, in_extension ru e1 -> in_extension ru e2.
Proof.
simple induction 1.
inversion 1; auto.
elim magic.
Qed.
(* Example from Norbert Schirmer on Coq-Club, Sep 2000 *)
Set Strict Implicit.
Unset Implicit Arguments.
Definition Q (n m : nat) (prf : n <= m) := True.
Goal forall (n m : nat) (H : S n <= m), Q (S n) m H = True.
intros.
dependent inversion_clear H.
elim magic.
elim magic.
Qed.
(* Submitted by Boris Yakobowski (bug #529) *)
(* Check that Inversion does not fail due to unnormalized evars *)
Set Implicit Arguments.
Unset Strict Implicit.
Require Import Bvector.
Inductive I : nat -> Set :=
| C1 : I 1
| C2 : forall k i : nat, vector (I i) k -> I i.
Inductive SI : forall k : nat, I k -> vector nat k -> nat -> Prop :=
SC2 :
forall (k i vf : nat) (v : vector (I i) k) (xi : vector nat i),
SI (C2 v) xi vf.
Theorem SUnique :
forall (k : nat) (f : I k) (c : vector nat k) v v',
SI f c v -> SI f c v' -> v = v'.
Proof.
induction 1.
intros H; inversion H.
Admitted.
(* Used to failed at some time *)
Set Strict Implicit.
Unset Implicit Arguments.
Parameter bar : forall p q : nat, p = q -> Prop.
Inductive foo : nat -> nat -> Prop :=
C : forall (a b : nat) (Heq : a = b), bar a b Heq -> foo a b.
Lemma depinv : forall a b, foo a b -> True.
intros a b H.
inversion H.
Abort.
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