Quelle absevarprop.v Sprache: Coq

(************************************************************************)
(*         *      The Rocq Prover / The Rocq 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)         *)
(************************************************************************)

(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)

Require Import ssreflect ssrbool ssrfun.
Require Import TestSuite.ssr_mini_mathcomp.

Lemma test15: forall (y : nat) (x : 'I_2), y < 1 -> val x = y -> Some x = insub y.
move=> y x le_1 defx; rewrite insubT ?(leq_trans le_1) // => ?.
by congr (Some _); apply: val_inj=> /=; exact: defx.
Qed.

Axiom P : nat -> Prop.
Axiom Q : forall n, P n -> Prop.
Definition R := fun (x : nat) (p : P x) m (q : P (x+1)) => m > 0.

Inductive myEx : Type := ExI : forall n (pn : P n) pn', Q n pn -> R n pn n pn' -> myEx.

Parameter P1 : P 1.
Parameter P11 : P (1 + 1).
Parameter Q1 : forall P1, Q 1 P1.

Lemma testmE1 : myEx.
Proof.
apply: ExI 1 _ _ _ _.
  match goal with |- P 1 => exact: P1 | _ => fail end.
  match goal with |- P (1+1) => exact: P11 | _ => fail end.
  match goal with |- forall p : P 1, Q 1 p => move=> *; exact: Q1 | _ => fail end.
match goal with |- forall (p : P 1) (q : P (1+1)), is_true (R 1 p 1 q) => done | _ => fail end.
Qed.

Lemma testE2 : exists y : { x | P x }, sval y = 1.
Proof.
apply: ex_intro (exist _ 1 _) _.
  match goal with |- P 1 => exact: P1 | _ => fail end.
match goal with |- forall p : P 1, @sval _ _ (@exist _ _ 1 p) = 1 => done | _ => fail end.
Qed.

Lemma testE3 : exists y : { x | P x }, sval y = 1.
Proof.
have := (ex_intro _ (exist _ 1 _) _); apply.
  match goal with |- P 1 => exact: P1 | _ => fail end.
match goal with |- forall p : P 1, @sval _ _ (@exist _ _ 1 p) = 1 => done | _ => fail end.
Qed.

Lemma testE4 : P 2 -> exists y : { x | P x }, sval y = 2.
Proof.
move=> P2; apply: ex_intro (exist _ 2 _) _.
match goal with |- @sval _ _ (@exist _ _ 2 P2) = 2 => done | _ => fail end.
Qed.

#[export] Hint Resolve P1.

Lemma testmE12 : myEx.
Proof.
apply: ExI 1 _ _ _ _.
  match goal with |- P (1+1) => exact: P11 | _ => fail end.
  match goal with |- Q 1 P1 => exact: Q1 | _ => fail end.
match goal with |- forall (q : P (1+1)), is_true (R 1 P1 1 q) => done | _ => fail end.
Qed.

Create HintDb SSR.

#[export] Hint Resolve P11 : SSR.

Ltac ssrautoprop := trivial with SSR.

Lemma testmE13 : myEx.
Proof.
apply: ExI 1 _ _ _ _.
  match goal with |- Q 1 P1 => exact: Q1 | _ => fail end.
match goal with |- is_true (R 1 P1 1 P11) => done | _ => fail end.
Qed.

Definition R1 := fun (x : nat) (p : P x) m (q : P (x+1)) (r : Q x p) => m > 0.

Inductive myEx1 : Type :=
  ExI1 : forall n (pn : P n) pn' (q : Q n pn), R1 n pn n pn' q -> myEx1.

#[export] Hint Extern 0 (Q 1 P1) => apply (Q1 P1) : SSR.

(* tests that goals in prop are solved in the right order, propagating instantiations,
   thus the goal Q 1 ?p1 is faced by trivial after ?p1, and is thus evar free *)
Lemma testmE14 : myEx1.
Proof.
apply: ExI1 1 _ _ _ _.
match goal with |- is_true (R1 1 P1 1 P11 (Q1 P1)) => done | _ => fail end.
Qed.

quality99%

¤ Dauer der Verarbeitung: 0.3 Sekunden ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.