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
|
Require Import Setoid.
Require Import RelationClasses.
Require Import Morphisms.
Set Implicit Arguments.
Set Strict Implicit.
(** The purpose of this tactic is to try to automatically derive morphisms
for functions
**)
Theorem Proper_red : forall T U (rT : relation T) (rU : relation U) (f : T -> U),
(forall x x', rT x x' -> rU (f x) (f x')) ->
Proper (rT ==> rU) f.
intuition.
Qed.
Theorem respectful_red : forall T U (rT : relation T) (rU : relation U) (f g : T -> U),
(forall x x', rT x x' -> rU (f x) (g x')) ->
respectful rT rU f g.
intuition.
Qed.
Theorem respectful_if_bool T : forall (x x' : bool) (t t' f f' : T) eqT,
x = x' ->
eqT t t' -> eqT f f' ->
eqT (if x then t else f) (if x' then t' else f') .
intros; subst; auto; destruct x'; auto.
Qed.
Ltac derive_morph :=
repeat
first [ lazymatch goal with
| |- Proper _ _ => red; intros
| |- (_ ==> _)%signature _ _ => red; intros
end
| apply respectful_red; intros
| apply respectful_if_bool; intros
| match goal with
| [ H : (_ ==> ?EQ)%signature ?F ?F' |- ?EQ (?F _) (?F' _) ] =>
apply H
| [ |- ?EQ (?F _) (?F _) ] =>
let inst := constr:(_ : Proper (_ ==> EQ) F) in
apply inst
| [ H : (_ ==> _ ==> ?EQ)%signature ?F ?F' |- ?EQ (?F _ _) (?F' _ _) ] =>
apply H
| [ |- ?EQ (?F _ _) (?F' _ _) ] =>
let inst := constr:(_ : Proper (_ ==> _ ==> EQ) F) in
apply inst
| [ |- ?EQ (?F _ _ _) (?F _ _ _) ] =>
let inst := constr:(_ : Proper (_ ==> _ ==> _ ==> EQ) F) in
apply inst
| [ |- ?EQ (?F _) (?F _) ] => unfold F
| [ |- ?EQ (?F _ _) (?F _ _) ] => unfold F
| [ |- ?EQ (?F _ _ _) (?F _ _ _) ] => unfold F
end ].
Global Instance Proper_andb : Proper (@eq bool ==> @eq bool ==> @eq bool) andb.
derive_morph; auto.
Qed.
Section K.
Variable F : bool -> bool -> bool.
Hypothesis Fproper : Proper (@eq bool ==> @eq bool ==> @eq bool) F.
Existing Instance Fproper.
Definition food (x y z : bool) : bool :=
F x (F y z).
Global Instance Proper_food : Proper (@eq bool ==> @eq bool ==> @eq bool ==> @eq bool) food.
Proof.
derive_morph; auto.
Qed.
Global Instance Proper_S : Proper (@eq nat ==> @eq nat) S.
Proof.
derive_morph; auto.
Qed.
End K.
Require Import List.
Section Map.
Variable T : Type.
Variable eqT : relation T.
Inductive listEq {T} (eqT : relation T) : relation (list T) :=
| listEq_nil : listEq eqT nil nil
| listEq_cons : forall x x' y y', eqT x x' -> listEq eqT y y' ->listEq eqT (x :: y) (x' :: y').
Theorem listEq_match V U (eqV : relation V) (eqU : relation U) : forall x x' : list V,
forall X X' Y Y',
eqU X X' ->
(eqV ==> listEq eqV ==> eqU)%signature Y Y' ->
listEq eqV x x' ->
eqU (match x with
| nil => X
| x :: xs => Y x xs
end)
(match x' with
| nil => X'
| x :: xs => Y' x xs
end).
Proof.
intros. induction H1; auto. derive_morph; auto.
Qed.
Variable U : Type.
Variable eqU : relation U.
Variable f : T -> U.
Variable fproper : Proper (eqT ==> eqU) f.
Definition hd (l : list T) : option T :=
match l with
| nil => None
| l :: _ => Some l
end.
(*
Global Instance Proper_hd : Proper (listEq eqT ==> optionEq eqT) hd.
Proof.
foo. (** This has binders in the match... **)
Abort.
*)
Fixpoint map' (l : list T) : list U :=
match l with
| nil => nil
| l :: ls => f l :: map' ls
end.
Global Instance Proper_map' : Proper (listEq eqT ==> listEq eqU) map'.
Proof.
derive_morph. induction H; econstructor; derive_morph; auto.
Qed.
End Map.
|
