(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
(************************************************************************)
refine
((fun x0 : nat => match x0 with
| O => _
| S p => _
end)).
Restart.
refine
(fun x0 : nat => match x0 as n return (n = n) with
| O => _
| S p => _
end). (* OK *)
Restart.
refine
(fun x0 : nat => match x0 as n return (n = n) with
| O => _
| S p => _
end). (* OK *)
Restart.
(** Refine [x0:nat]Cases x0 of O => ? | (S p) => ? end. (* cannot be executed *)
**)
Abort.
(************************************************************************)
refine (S _).
Abort.
(************************************************************************)
refine (fix f (x : nat) : x = x := _).
Restart.
refine
(fix f (x : nat) : x = x :=
match x as n return (n = n :>nat) with
| O => _
| S p => _
end).
Restart.
refine
(fix f (x : nat) : x = x :=
match x as n return (n = n) with
| O => _
| S p => _
end).
Restart.
refine
(fix f (x : nat) : x = x :=
match x as n return (n = n :>nat) with
| O => _
| S p => f_equal S _
end).
Restart.
refine
(fix f (x : nat) : x = x :=
match x as n return (n = n :>nat) with
| O => _
| S p => f_equal S _
end).
Abort.
(************************************************************************)
Parameter f : nat * nat -> nat -> nat.
refine (f _ ((fun x : nat => _:nat) 0)).
Restart.
refine (f _ 0).
Abort.
(************************************************************************)
Parameter P : nat -> Prop.
refine (exist _ 1 _).  (* ECHEC *)
Restart.
(* mais si on contraint par le but alors ca marche : *)
(* Remarque : on peut toujours faire ça *)
refine (exist _ 1 _:{x : nat | x = 1}).
Restart.
refine (exist (fun x : nat => x = 1) 1 _).
Abort.
(************************************************************************)
refine
(fun n : nat =>
match n return {x : nat | x = S n} with
| O => _
| S p => _
end).
Restart.
refine
(fun n : nat => match n with
| O => _
| S p => _
end).
Restart.
refine
(fun n : nat =>
match n return {x : nat | x = S n} with
| O => _
| S p => _
end).
Restart.
refine
(fix f (n : nat) : {x : nat | x = S n} :=
match n return {x : nat | x = S n} with
| O => _
| S p => _
end).
Restart.
refine
(fix f (n : nat) : {x : nat | x = S n} :=
match n return {x : nat | x = S n} with
| O => _
| S p => _
end).
exists 1. trivial.
elim (f p).
refine
(fun (x : nat) (h : x = S p) => exist (fun x : nat => x = S (S p)) (S x) _).
rewrite h. auto.
Qed.
(* Quelques essais de recurrence bien fondée *)
refine
(fun x : nat =>
well_founded_induction _ (fun _ : nat => nat -> nat)
(fun (phi0 : nat) (w : forall phi : nat, phi < phi0 -> nat -> nat) =>
w x _) x x).
exact lt_wf.
Abort.

Lemma fibo : nat -> nat.