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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * Copyright INRIA, CNRS and contributors *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(* \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) *)
(************************************************************************)
(************************************************************************)
Lemma essai : forall x : nat, x = x.
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.
(************************************************************************)
Lemma T : nat.
refine (S _).
Abort.
(************************************************************************)
Lemma essai2 : forall x : nat, x = x.
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.
Lemma essai : nat.
refine (f _ ((fun x : nat => _:nat) 0)).
Restart.
refine (f _ 0).
Abort.
(************************************************************************)
Parameter P : nat -> Prop.
Lemma essai : {x : nat | x = 1}.
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.
(************************************************************************)
Lemma essai : forall n : nat, {x : nat | x = S n}.
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 *)
Require Import Init.Wf.
Require Import Wf_nat.
Lemma essai_wf : nat -> nat.
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.
Require Import Arith_base.
Lemma fibo : nat -> nat.
refine
(well_founded_induction _ (fun _ : nat => nat)
(fun (x0 : nat) (fib : forall x : nat, x < x0 -> nat) =>
match zerop x0 with
| left _ => 1
| right h1 =>
match zerop (pred x0) with
| left _ => 1
| right h2 => fib (pred x0) _ + fib (pred (pred x0)) _
end
end)).
exact lt_wf.
auto with arith.
apply Nat.lt_trans with (m := pred x0); auto with arith.
Qed.
|
