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
222
223
224
225
226
227
228
229
230
|
(* Bug report #2830 (evar defined twice) covers different bugs *)
(* 1- This was submitted by qb.h.agws *)
Module A.
Set Implicit Arguments.
Inductive Bit := O | I.
Inductive BitString: nat -> Set :=
| bit: Bit -> BitString 0
| bitStr: forall n: nat, Bit -> BitString n -> BitString (S n).
Definition BitOr (a b: Bit) :=
match a, b with
| O, O => O
| _, _ => I
end.
(* Should fail with an error; used to failed in 8.4 and trunk with
anomaly Evd.define: cannot define an evar twice *)
Fail Fixpoint StringOr (n m: nat) (a: BitString n) (b: BitString m) :=
match a with
| bit a' =>
match b with
| bit b' => bit (BitOr a' b')
| bitStr b' bT => bitStr b' (StringOr (bit a') bT)
end
| bitStr a' aT =>
match b with
| bit b' => bitStr a' (StringOr aT (bit b'))
| bitStr b' bT => bitStr (BitOr a' b') (StringOr aT bT)
end
end.
End A.
(* 2- This was submitted by Andrew Appel *)
Module B.
From Stdlib Require Import Program Relations.
Record ageable_facts (A:Type) (level: A -> nat) (age1:A -> option A) :=
{ af_unage : forall x x' y', level x' = level y' -> age1 x = Some x' -> exists y, age1 y = Some y'
; af_level1 : forall x, age1 x = None <-> level x = 0
; af_level2 : forall x y, age1 x = Some y -> level x = S (level y)
}.
Arguments af_unage {A level age1}.
Arguments af_level1 {A level age1}.
Arguments af_level2 {A level age1}.
Class ageable (A:Type) := mkAgeable
{ level : A -> nat
; age1 : A -> option A
; age_facts : ageable_facts A level age1
}.
Definition age {A} `{ageable A} (x y:A) := age1 x = Some y.
Definition necR {A} `{ageable A} : relation A := clos_refl_trans A age.
Delimit Scope pred with pred.
Local Open Scope pred.
Definition hereditary {A} (R:A->A->Prop) (p:A->Prop) :=
forall a a':A, R a a' -> p a -> p a'.
Definition pred (A:Type) {AG:ageable A} :=
{ p:A -> Prop | hereditary age p }.
Bind Scope pred with pred.
Definition app_pred {A} `{ageable A} (p:pred A) : A -> Prop := proj1_sig p.
Definition pred_hereditary `{ageable} (p:pred A) := proj2_sig p.
Coercion app_pred : pred >-> Funclass.
Global Opaque pred.
Definition derives {A} `{ageable A} (P Q:pred A) := forall a:A, P a -> Q a.
Arguments derives : default implicits.
Program Definition andp {A} `{ageable A} (P Q:pred A) : pred A :=
fun a:A => P a /\ Q a.
Next Obligation.
intros; intro; intuition; apply pred_hereditary with a; auto.
Qed.
Program Definition imp {A} `{ageable A} (P Q:pred A) : pred A :=
fun a:A => forall a':A, necR a a' -> P a' -> Q a'.
Next Obligation.
intros; intro; intuition.
apply H1; auto.
apply rt_trans with a'; auto.
apply rt_step; auto.
Qed.
Program Definition allp {A} `{ageable A} {B: Type} (f: B -> pred A) : pred A
:= fun a => forall b, f b a.
Next Obligation.
intros; intro; intuition.
apply pred_hereditary with a; auto.
apply H1.
Qed.
Notation "P '<-->' Q" := (andp (imp P Q) (imp Q P)) (at level 57, no associativity) : pred.
Notation "P '|--' Q" := (derives P Q) (at level 80, no associativity).
Notation "'All' x ':' T ',' P " := (allp (fun x:T => P%pred)) (at level 65, x at level 99) : pred.
Lemma forall_pred_ext {A} `{agA : ageable A}: forall B P Q,
(All x : B, (P x <--> Q x)) |-- (All x : B, P x) <--> (All x: B, Q x).
Abort.
End B.
(* 3. *)
(* This was submitted by Anthony Cowley *)
From Stdlib Require Import Morphisms.
From Stdlib Require Import Setoid.
Module C.
Reserved Notation "a ~> b" (at level 70, right associativity).
Reserved Notation "a ≈ b" (at level 54).
Reserved Notation "a ∘ b" (at level 50, left associativity).
Generalizable All Variables.
Class Category (Object:Type) (Hom:Object -> Object -> Type) := {
hom := Hom where "a ~> b" := (hom a b) : category_scope
; ob := Object
; id : forall a, hom a a
; comp : forall c b a, hom b c -> hom a b -> hom a c
where "g ∘ f" := (comp _ _ _ g f) : category_scope
; eqv : forall a b, hom a b -> hom a b -> Prop
where "f ≈ g" := (eqv _ _ f g) : category_scope
; eqv_equivalence : forall a b, Equivalence (eqv a b)
; comp_respects : forall a b c,
Proper (eqv b c ==> eqv a b ==> eqv a c) (comp c b a)
; left_identity : forall `(f:a ~> b), id b ∘ f ≈ f
; right_identity : forall `(f:a ~> b), f ∘ id a ≈ f
; associativity : forall `(f:a~>b) `(g:b~>c) `(h:c~>d),
h ∘ (g ∘ f) ≈ (h ∘ g) ∘ f
}.
Notation "a ~> b" := (@hom _ _ _ a b) : category_scope.
Notation "g ∘ f" := (@comp _ _ _ _ _ _ g f) : category_scope.
Notation "a ≈ b" := (@eqv _ _ _ _ _ a b) : category_scope.
Notation "a ~{ C }~> b" := (@hom _ _ C a b) (at level 100) : category_scope.
Coercion ob : Category >-> Sortclass.
Open Scope category_scope.
Add Parametric Relation `(C:Category Ob Hom) (a b : Ob) : (hom a b) (eqv a b)
reflexivity proved by (@Equivalence_Reflexive _ _ (eqv_equivalence a b))
symmetry proved by (@Equivalence_Symmetric _ _ (eqv_equivalence a b))
transitivity proved by (@Equivalence_Transitive _ _ (eqv_equivalence a b))
as parametric_relation_eqv.
Add Parametric Morphism `(C:Category Ob Hom) (c b a : Ob) : (comp c b a)
with signature (eqv _ _ ==> eqv _ _ ==> eqv _ _) as parametric_morphism_comp.
intros x y Heq x' y'. apply comp_respects. exact Heq.
Defined.
Class Functor `(C:Category) `(D:Category) (im : C -> D) := {
functor_im := im
; fmap : forall {a b}, `(a ~> b) -> im a ~> im b
; fmap_respects : forall a b (f f' : a ~> b), f ≈ f' -> fmap f ≈ fmap f'
; fmap_preserves_id : forall a, fmap (id a) ≈ id (im a)
; fmap_preserves_comp : forall `(f:a~>b) `(g:b~>c),
fmap g ∘ fmap f ≈ fmap (g ∘ f)
}.
Coercion functor_im : Functor >-> Funclass.
Arguments fmap [Object Hom C Object0 Hom0 D im] _ [a b].
Add Parametric Morphism `(C:Category) `(D:Category)
(Im:C->D) (F:Functor C D Im) (a b:C) : (@fmap _ _ C _ _ D Im F a b)
with signature (@eqv C _ C a b ==> @eqv D _ D (Im a) (Im b))
as parametric_morphism_fmap.
intros. apply fmap_respects. assumption. Qed.
(* HERE IS THE PROBLEMATIC INSTANCE. If we change this to a regular Definition,
then the problem goes away. *)
#[export] Instance functor_comp `{C:Category} `{D:Category} `{E:Category}
{Gim} (G:Functor D E Gim) {Fim} (F:Functor C D Fim)
: Functor C E (Basics.compose Gim Fim).
intros. apply Build_Functor with (fmap := fun a b f => fmap G (fmap F f)).
abstract (intros; rewrite H; reflexivity).
abstract (intros; repeat (rewrite fmap_preserves_id); reflexivity).
abstract (intros; repeat (rewrite fmap_preserves_comp); reflexivity).
Defined.
Definition skel {A:Type} : relation A := @eq A.
#[export] Instance skel_equiv A : Equivalence (@skel A).
Admitted.
Import FunctionalExtensionality.
#[export] Instance set_cat : Category Type (fun A B => A -> B).
refine {|
id := fun A => fun x => x
; comp c b a f g := fun x => f (g x)
; eqv := fun A B => @skel (A -> B)
|}.
intros. compute. symmetry. apply eta_expansion.
intros. compute. symmetry. apply eta_expansion.
intros. compute. reflexivity.
Defined.
(* The [list] type constructor is a Functor. *)
From Stdlib Require Import List.
Definition setList (A:set_cat) := list A.
#[export] Instance list_functor : Functor set_cat set_cat setList.
apply Build_Functor with (fmap := @map).
intros. rewrite H. reflexivity.
intros; simpl; apply functional_extensionality.
induction x; [auto|simpl]. rewrite IHx. reflexivity.
intros; simpl; apply functional_extensionality.
induction x; [auto|simpl]. rewrite IHx. reflexivity.
Defined.
Local Notation "[ a , .. , b ]" := (a :: .. (b :: nil) ..) : list_scope.
Definition setFmap {Fim} {F:Functor set_cat set_cat Fim} `(f:A~>B) (xs:Fim A) := fmap F f xs.
(* We want to infer the [Functor] instance based on the value's
structure, but the [functor_comp] instance throws things awry. *)
Eval cbv in setFmap (fun x => x * 3) [67,8].
End C.
|
