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(* First a simplification of the bug *)
Set Printing Universes.
Inductive enc (A:Type (*1*)) (* : Type.1 *) := C : A -> enc A.
Definition id (X:Type(*4*)) (x:X) := x.
Lemma test : let S := Type(*5 : 6*) in enc S -> S.
simpl; intros.
refine (enc _).
apply id.
apply Prop.
Defined.
(* Then the original bug *)
Require Import List.
Inductive a : Set := (* some dummy inductive *)
b : (list a) -> a. (* i don't know if this *)
(* happens for smaller *)
(* ones *)
Inductive sg : Type := Sg. (* single *)
Definition ipl2 (P : a -> Type) := (* in Prop, that means P is true forall *)
fold_right (fun x => fun A => prod (P x) A) sg. (* the elements of a given list *)
Definition ind
: forall S : a -> Type,
(forall ls : list a, ipl2 S ls -> S (b ls)) -> forall s : a, S s :=
fun (S : a -> Type)
(X : forall ls : list a, ipl2 S ls -> S (b ls)) =>
fix ind2 (s : a) :=
match s as a return (S a) with
| b l =>
X l
(list_rect (fun l0 : list a => ipl2 S l0) Sg
(fun (a0 : a) (l0 : list a) (IHl : ipl2 S l0) =>
pair (ind2 a0) IHl) l)
end. (* some induction principle *)
Implicit Arguments ind [S].
Lemma k : a -> Type. (* some ininteresting lemma *)
intro;pattern H;apply ind;intros.
assert (K : Type).
induction ls.
exact sg.
exact sg.
exact (prod K sg).
Defined.
Lemma k' : a -> Type. (* same lemma but with our bug *)
intro;pattern H;apply ind;intros.
refine (prod _ _).
induction ls.
exact sg.
exact sg.
exact sg. (* Proof complete *)
Defined. (* bug *)
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