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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Class A (A : Type).
Instance an: A nat.
Class B (A : Type) (a : A).
Instance bn0: B nat 0.
Instance bn1: B nat 1.
Goal A nat.
Proof.
typeclasses eauto.
Qed.
Goal B nat 2.
Proof.
Fail typeclasses eauto.
Abort.
Goal exists T : Type, A T.
Proof.
eexists. typeclasses eauto.
Defined.
Hint Extern 0 (_ /\ _) => constructor : typeclass_instances.
Existing Class and.
Goal exists (T : Type) (t : T), A T /\ B T t.
Proof.
eexists. eexists. typeclasses eauto.
Defined.
Instance ab: A bool. (* Backtrack on A instance *)
Goal exists (T : Type) (t : T), A T /\ B T t.
Proof.
eexists. eexists. typeclasses eauto.
Defined.
Class C {T} `(a : A T) (t : T).
Require Import Classes.Init.
Hint Extern 0 { x : ?A & _ } =>
unshelve class_apply @existT : typeclass_instances.
Existing Class sigT.
Set Typeclasses Debug.
Instance can: C an 0.
(* Backtrack on instance implementation *)
Goal exists (T : Type) (t : T), { x : A T & C x t }.
Proof.
eexists. eexists. typeclasses eauto.
Defined.
Class D T `(a: A T).
Instance: D _ an.
Goal exists (T : Type), { x : A T & D T x }.
Proof.
eexists. typeclasses eauto.
Defined.
(* Example from Nicolas Magaud on coq-club - Jul 2000 *)
Definition Nat : Set := nat.
Parameter S' : Nat -> Nat.
Parameter plus' : Nat -> Nat -> Nat.
Lemma simpl_plus_l_rr1 :
(forall n0 : Nat,
(forall m p : Nat, plus' n0 m = plus' n0 p -> m = p) ->
forall m p : Nat, S' (plus' n0 m) = S' (plus' n0 p) -> m = p) ->
forall n : Nat,
(forall m p : Nat, plus' n m = plus' n p -> m = p) ->
forall m p : Nat, S' (plus' n m) = S' (plus' n p) -> m = p.
intros.
apply H0. apply f_equal_nat.
Time info_eauto.
Undo.
Set Typeclasses Debug.
Set Typeclasses Iterative Deepening.
Time typeclasses eauto 6 with nocore. Show Proof.
Undo.
Time eauto. (* does EApply H *)
Qed.
(* Example from Nicolas Tabareau on coq-club - Feb 2016.
Full backtracking on dependent subgoals.
*)
Require Import Coq.Classes.Init.
Module NTabareau.
Set Typeclasses Dependency Order.
Unset Typeclasses Iterative Deepening.
Notation "x .1" := (projT1 x) (at level 3).
Notation "x .2" := (projT2 x) (at level 3).
Parameter myType: Type.
Class Foo (a:myType) := {}.
Class Bar (a:myType) := {}.
Class Qux (a:myType) := {}.
Parameter fooTobar : forall a (H : Foo a), {b: myType & Bar b}.
Parameter barToqux : forall a (H : Bar a), {b: myType & Qux b}.
Hint Extern 5 (Bar ?D.1) =>
destruct D; simpl : typeclass_instances.
Hint Extern 5 (Qux ?D.1) =>
destruct D; simpl : typeclass_instances.
Hint Extern 1 myType =>
unshelve refine (fooTobar _ _).1 : typeclass_instances.
Hint Extern 1 myType => unshelve refine (barToqux _ _).1 : typeclass_instances.
Hint Extern 0 { x : _ & _ } => simple refine (existT _ _ _) : typeclass_instances.
Unset Typeclasses Debug.
Definition trivial a (H : Foo a) : {b : myType & Qux b}.
Proof.
Time typeclasses eauto 10 with typeclass_instances.
Undo. Set Typeclasses Iterative Deepening.
Time typeclasses eauto with typeclass_instances.
Defined.
End NTabareau.
Module NTabareauClasses.
Set Typeclasses Dependency Order.
Unset Typeclasses Iterative Deepening.
Notation "x .1" := (projT1 x) (at level 3).
Notation "x .2" := (projT2 x) (at level 3).
Parameter myType: Type.
Existing Class myType.
Class Foo (a:myType) := {}.
Class Bar (a:myType) := {}.
Class Qux (a:myType) := {}.
Parameter fooTobar : forall a (H : Foo a), {b: myType & Bar b}.
Parameter barToqux : forall a (H : Bar a), {b: myType & Qux b}.
Hint Extern 5 (Bar ?D.1) =>
destruct D; simpl : typeclass_instances.
Hint Extern 5 (Qux ?D.1) =>
destruct D; simpl : typeclass_instances.
Hint Extern 1 myType =>
unshelve notypeclasses refine (fooTobar _ _).1 : typeclass_instances.
Hint Extern 1 myType =>
unshelve notypeclasses refine (barToqux _ _).1 : typeclass_instances.
Hint Extern 0 { x : _ & _ } =>
unshelve notypeclasses refine (existT _ _ _) : typeclass_instances.
Unset Typeclasses Debug.
Definition trivial a (H : Foo a) : {b : myType & Qux b}.
Proof.
Time typeclasses eauto 10 with typeclass_instances.
Undo. Set Typeclasses Iterative Deepening.
(* Much faster in iteratove deepening mode *)
Time typeclasses eauto with typeclass_instances.
Defined.
End NTabareauClasses.
Require Import List.
Parameter in_list : list (nat * nat) -> nat -> Prop.
Definition not_in_list (l : list (nat * nat)) (n : nat) : Prop :=
~ in_list l n.
(* Hints Unfold not_in_list. *)
Axiom
lem1 :
forall (l1 l2 : list (nat * nat)) (n : nat),
not_in_list (l1 ++ l2) n -> not_in_list l1 n.
Axiom
lem2 :
forall (l1 l2 : list (nat * nat)) (n : nat),
not_in_list (l1 ++ l2) n -> not_in_list l2 n.
Axiom
lem3 :
forall (l : list (nat * nat)) (n p q : nat),
not_in_list ((p, q) :: l) n -> not_in_list l n.
Axiom
lem4 :
forall (l1 l2 : list (nat * nat)) (n : nat),
not_in_list l1 n -> not_in_list l2 n -> not_in_list (l1 ++ l2) n.
Hint Resolve lem1 lem2 lem3 lem4: essai.
Goal
forall (l : list (nat * nat)) (n p q : nat),
not_in_list ((p, q) :: l) n -> not_in_list l n.
intros.
eauto with essai.
Qed.
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