1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
(* \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.
|
