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(* This requires cumulativity *)
Definition Type2 := Type.
Definition Type1 : Type2 := Type.
Lemma lem1 : (True -> Type1) -> Type2.
intro H.
apply H.
exact I.
Qed.
Lemma lem2 :
forall (A : Type) (P : A -> Type) (x : A),
(forall y : A, x = y -> P y) -> P x.
auto.
Qed.
Lemma lem3 : forall P : Prop, P.
intro P; pattern P.
apply lem2.
Abort.
(* Check managing of universe constraints in inversion (BZ#855) *)
Inductive dep_eq : forall X : Type, X -> X -> Prop :=
| intro_eq : forall (X : Type) (f : X), dep_eq X f f
| intro_feq :
forall (A : Type) (B : A -> Type),
let T := forall x : A, B x in
forall (f g : T) (x : A), dep_eq (B x) (f x) (g x) -> dep_eq T f g.
Require Import Relations.
Theorem dep_eq_trans : forall X : Type, transitive X (dep_eq X).
Proof.
unfold transitive.
intros X f g h H1 H2.
inversion H1.
Abort.
(* Submitted by Bas Spitters (BZ#935) *)
(* This is a problem with the status of the type in LetIn: is it a
user-provided one or an inferred one? At the current time, the
kernel type-check the type in LetIn, which means that it must be
considered as user-provided when calling the kernel. However, in
practice it is inferred so that a universe refresh is needed to set
its status as "user-provided".
Especially, universe refreshing was not done for "set/pose" *)
Lemma ind_unsec : forall Q : nat -> Type, True.
intro.
set (C := forall m, Q m -> Q m).
exact I.
Qed.
(* Submitted by Danko Ilik (bug report #1507); related to LetIn *)
Record U : Type := { A:=Type; a:A }.
(** Check assignment of sorts to inductives and records. *)
Variable sh : list nat.
Definition is_box_in_shape (b :nat * nat) := True.
Definition myType := Type.
Module Ind.
Inductive box_in : myType :=
myBox (coord : nat * nat) (_ : is_box_in_shape coord) : box_in.
End Ind.
Module Rec.
Record box_in : myType :=
BoxIn { coord :> nat * nat; _ : is_box_in_shape coord }.
End Rec.
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