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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Test des definitions inductives imbriquees *)
Require Import List.
Inductive X : Set :=
cons1 : list X -> X.
Inductive Y : Set :=
cons2 : list (Y * Y) -> Y.
(* Test inductive types with local definitions *)
Inductive eq1 : forall A:Type, let B:=A in A -> Prop :=
refl1 : eq1 True I.
Check
fun (P : forall A : Type, let B := A in A -> Type) (f : P True I) (A : Type) =>
let B := A in
fun (a : A) (e : eq1 A a) =>
match e in (eq1 A0 B0 a0) return (P A0 a0) with
| refl1 => f
end.
Inductive eq2 (A:Type) (a:A)
: forall B C:Type, let D:=(A*B*C)%type in D -> Prop :=
refl2 : eq2 A a unit bool (a,tt,true).
(* Check that induction variables are cleared even with in clause *)
Lemma foo : forall n m : nat, n + m = n + m.
Proof.
intros; induction m as [|m] in n |- *.
auto.
auto.
Qed.
(* Check selection of occurrences by pattern *)
Goal forall x, S x = S (S x).
intros.
induction (S _) in |- * at -2.
now_show (0=1).
Undo 2.
induction (S _) in |- * at 1 3.
now_show (0=1).
Undo 2.
induction (S _) in |- * at 1.
now_show (0=S (S x)).
Undo 2.
induction (S _) in |- * at 2.
now_show (S x=0).
Undo 2.
induction (S _) in |- * at 3.
now_show (S x=1).
Undo 2.
Fail induction (S _) in |- * at 4.
Abort.
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