cofe.v 9.24 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
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
Require Export prelude.
Obligation Tactic := idtac.

(** Unbundeled version *)
Class Dist A := dist : nat  relation A.
Instance: Params (@dist) 2.
Notation "x ={ n }= y" := (dist n x y)
  (at level 70, n at next level, format "x  ={ n }=  y").
Hint Extern 0 (?x ={_}= ?x) => reflexivity.
Hint Extern 0 (_ ={_}= _) => symmetry; assumption.

Record chain (A : Type) `{Dist A} := {
  chain_car :> nat  A;
  chain_cauchy n i : n  i  chain_car n ={n}= chain_car i
}.
Arguments chain_car {_ _} _ _.
Arguments chain_cauchy {_ _} _ _ _ _.
Class Compl A `{Dist A} := compl : chain A  A.

Class Cofe A `{Equiv A, Compl A} := {
  equiv_dist x y : x  y   n, x ={n}= y;
  dist_equivalence n :> Equivalence (dist n);
  dist_S n x y : x ={S n}= y  x ={n}= y;
  dist_0 x y : x ={0}= y;
  conv_compl (c : chain A) n : compl c ={n}= c n
}.
Hint Extern 0 (_ ={0}= _) => apply dist_0.
Class Contractive `{Dist A, Dist B} (f : A -> B) :=
  contractive n : Proper (dist n ==> dist (S n)) f.

(** Bundeled version *)
Structure cofeT := CofeT {
  cofe_car :> Type;
  cofe_equiv : Equiv cofe_car;
  cofe_dist : Dist cofe_car;
  cofe_compl : Compl cofe_car;
  cofe_cofe : Cofe cofe_car
}.
Arguments CofeT _ {_ _ _ _}.
Add Printing Constructor cofeT.
Existing Instances cofe_equiv cofe_dist cofe_compl cofe_cofe.

(** General properties *)
Section cofe.
  Context `{Cofe A}.
  Global Instance cofe_equivalence : Equivalence (() : relation A).
  Proof.
    split.
    * by intros x; rewrite equiv_dist.
    * by intros x y; rewrite !equiv_dist.
    * by intros x y z; rewrite !equiv_dist; intros; transitivity y.
  Qed.
  Global Instance dist_ne n : Proper (dist n ==> dist n ==> iff) (dist n).
  Proof.
    intros x1 x2 ? y1 y2 ?; split; intros.
    * by transitivity x1; [done|]; transitivity y1.
    * by transitivity x2; [done|]; transitivity y2.
  Qed.
  Global Instance dist_proper n : Proper (() ==> () ==> iff) (dist n).
  Proof.
    intros x1 x2 Hx y1 y2 Hy.
    by rewrite equiv_dist in Hx, Hy; rewrite (Hx n), (Hy n).
  Qed.
  Global Instance dist_proper_2 n x : Proper (() ==> iff) (dist n x).
  Proof. by apply dist_proper. Qed.
  Lemma dist_le x y n n' : x ={n}= y  n'  n  x ={n'}= y.
  Proof. induction 2; eauto using dist_S. Qed.
  Global Instance contractive_ne `{Cofe B} (f : A  B) `{!Contractive f} n :
    Proper (dist n ==> dist n) f | 100.
  Proof. by intros x1 x2 ?; apply dist_S, contractive. Qed.
  Global Instance ne_proper `{Cofe B} (f : A  B)
    `{! n, Proper (dist n ==> dist n) f} : Proper (() ==> ()) f | 100.
  Proof. by intros x1 x2; rewrite !equiv_dist; intros Hx n; rewrite (Hx n). Qed.
  Global Instance ne_proper_2 `{Cofe B, Cofe C} (f : A  B  C)
    `{! n, Proper (dist n ==> dist n ==> dist n) f} :
    Proper (() ==> () ==> ()) f | 100.
  Proof.
     unfold Proper, respectful; setoid_rewrite equiv_dist.
     by intros x1 x2 Hx y1 y2 Hy n; rewrite Hx, Hy.
  Qed.
  Lemma compl_ne (c1 c2: chain A) n : c1 n ={n}= c2 n  compl c1 ={n}= compl c2.
  Proof. intros. by rewrite (conv_compl c1 n), (conv_compl c2 n). Qed.
  Lemma compl_ext (c1 c2 : chain A) : ( i, c1 i  c2 i)  compl c1  compl c2.
  Proof. setoid_rewrite equiv_dist; naive_solver eauto using compl_ne. Qed.
End cofe.

(** Fixpoint *)
Program Definition fixpoint_chain `{Cofe A} (f : A  A) `{!Contractive f}
  (x : A) : chain A := {| chain_car i := Nat.iter i f x |}.
Next Obligation.
  intros A ???? f ? x n; induction n as [|n IH]; intros i ?; [done|].
  destruct i as [|i]; simpl; try lia; apply contractive, IH; auto with lia.
Qed.
Program Definition fixpoint `{Cofe A} (f : A  A) `{!Contractive f}
  (x : A) : A := compl (fixpoint_chain f x).

Section fixpoint.
  Context `{Cofe A} (f : A  A) `{!Contractive f}.
  Lemma fixpoint_unfold x : fixpoint f x  f (fixpoint f x).
  Proof.
    apply equiv_dist; intros n; unfold fixpoint.
    rewrite (conv_compl (fixpoint_chain f x) n).
    by rewrite (chain_cauchy (fixpoint_chain f x) n (S n)) at 1 by lia.
  Qed.
  Lemma fixpoint_ne (g : A  A) `{!Contractive g} x y n :
    ( z, f z ={n}= g z)  fixpoint f x ={n}= fixpoint g y.
  Proof.
    intros Hfg; unfold fixpoint; rewrite (conv_compl (fixpoint_chain f x) n),
      (conv_compl (fixpoint_chain g y) n).
    induction n as [|n IH]; simpl in *; [done|].
    rewrite Hfg; apply contractive, IH; auto using dist_S.
  Qed.
  Lemma fixpoint_proper (g : A  A) `{!Contractive g} x :
    ( x, f x  g x)  fixpoint f x  fixpoint g x.
  Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_ne. Qed.
End fixpoint.

(** Function space *)
Structure cofeMor (A B : cofeT) : Type := CofeMor {
  cofe_mor_car :> A  B;
  cofe_mor_ne n : Proper (dist n ==> dist n) cofe_mor_car
}.
Arguments CofeMor {_ _} _ {_}.
Add Printing Constructor cofeMor.
Existing Instance cofe_mor_ne.

Instance cofe_mor_proper `(f : cofeMor A B) : Proper (() ==> ()) f := _.
Instance cofe_mor_equiv {A B : cofeT} : Equiv (cofeMor A B) := λ f g,
   x, f x  g x.
Instance cofe_mor_dist (A B : cofeT) : Dist (cofeMor A B) := λ n f g,
   x, f x ={n}= g x.
Program Definition fun_chain `(c : chain (cofeMor A B)) (x : A) : chain B :=
  {| chain_car n := c n x |}.
Next Obligation. intros A B c x n i ?. by apply (chain_cauchy c). Qed.
Program Instance cofe_mor_compl (A B : cofeT) : Compl (cofeMor A B) := λ c,
  {| cofe_mor_car x := compl (fun_chain c x) |}.
Next Obligation.
  intros A B c n x y Hxy.
  rewrite (conv_compl (fun_chain c x) n), (conv_compl (fun_chain c y) n).
  simpl; rewrite Hxy; apply (chain_cauchy c); lia.
Qed.
Instance cofe_mor_cofe (A B : cofeT) : Cofe (cofeMor A B).
Proof.
  split.
  * intros X Y; split; [intros HXY n k; apply equiv_dist, HXY|].
    intros HXY k; apply equiv_dist; intros n; apply HXY.
  * intros n; split.
    + by intros f x.
    + by intros f g ? x.
    + by intros f g h ?? x; transitivity (g x).
  * by intros n f g ? x; apply dist_S.
  * by intros f g x.
  * intros c n x; simpl.
    rewrite (conv_compl (fun_chain c x) n); apply (chain_cauchy c); lia.
Qed.
Instance cofe_mor_car_proper :
  Proper (() ==> () ==> ()) (@cofe_mor_car A B).
Proof. intros A B f g Hfg x y Hx; rewrite Hx; apply Hfg. Qed.
Lemma cofe_mor_ext {A B} (f g : cofeMor A B) : f  g   x, f x  g x.
Proof. done. Qed.
Canonical Structure cofe_mor (A B : cofeT) : cofeT := CofeT (cofeMor A B).
Infix "-n>" := cofe_mor (at level 45, right associativity).

(** Identity and composition *)
Definition cid {A} : A -n> A := CofeMor id.
Instance: Params (@cid) 1.
Definition ccompose {A B C}
  (f : B -n> C) (g : A -n> B) : A -n> C := CofeMor (f  g).
Instance: Params (@ccompose) 3.
Infix "◎" := ccompose (at level 40, left associativity).
Lemma ccompose_ne {A B C} (f1 f2 : B -n> C) (g1 g2 : A -n> B) n :
  f1 ={n}= f2  g1 ={n}= g2  f1  g1 ={n}= f2  g2.
Proof. by intros Hf Hg x; simpl; rewrite (Hg x), (Hf (g2 x)). Qed.

(** Pre-composition as a functor *)
Local Instance ccompose_l_ne' {A B C} (f : B -n> A) n :
  Proper (dist n ==> dist n) (λ g : A -n> C, g  f).
Proof. by intros g1 g2 ?; apply ccompose_ne. Qed.
Definition ccompose_l {A B C} (f : B -n> A) : (A -n> C) -n> (B -n> C) :=
  CofeMor (λ g : A -n> C, g  f).
Instance ccompose_l_ne {A B C} : Proper (dist n ==> dist n) (@ccompose_l A B C).
Proof. by intros n f1 f2 Hf g x; apply ccompose_ne. Qed.

(** unit *)
Instance unit_dist : Dist unit := λ _ _ _, True.
Instance unit_compl : Compl unit := λ _, ().
Instance unit_cofe : Cofe unit.
Proof. by repeat split; try exists 0. Qed.

(** Product *)
Instance prod_dist `{Dist A, Dist B} : Dist (A * B) := λ n,
  prod_relation (dist n) (dist n).
Program Definition fst_chain `{Dist A, Dist B} (c : chain (A * B)) : chain A :=
  {| chain_car n := fst (c n) |}.
Next Obligation. by intros A ? B ? c n i ?; apply (chain_cauchy c n). Qed.
Program Definition snd_chain `{Dist A, Dist B} (c : chain (A * B)) : chain B :=
  {| chain_car n := snd (c n) |}.
Next Obligation. by intros A ? B ? c n i ?; apply (chain_cauchy c n). Qed.
Instance prod_compl `{Compl A, Compl B} : Compl (A * B) := λ c,
  (compl (fst_chain c), compl (snd_chain c)).
Instance prod_cofe `{Cofe A, Cofe B} : Cofe (A * B).
Proof.
  split.
  * intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation.
    rewrite !equiv_dist; naive_solver.
  * apply _.
  * by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S.
  * by split.
  * intros c n; split. apply (conv_compl (fst_chain c) n).
    apply (conv_compl (snd_chain c) n).
Qed.
Canonical Structure prodC (A B : cofeT) : cofeT := CofeT (A * B).
Local Instance prod_map_ne `{Dist A, Dist A', Dist B, Dist B'} n :
  Proper ((dist n ==> dist n) ==> (dist n ==> dist n) ==>
           dist n ==> dist n) (@prod_map A A' B B').
Proof. by intros f f' Hf g g' Hg ?? [??]; split; [apply Hf|apply Hg]. Qed.
Definition prodC_map {A A' B B'} (f : A -n> A') (g : B -n> B') :
  prodC A B -n> prodC A' B' := CofeMor (prod_map f g).
Instance prodC_map_ne {A A' B B'} n :
  Proper (dist n ==> dist n ==> dist n) (@prodC_map A A' B B').
Proof. intros f f' Hf g g' Hg [??]; split; [apply Hf|apply Hg]. Qed.

Instance pair_ne `{Dist A, Dist B} :
  Proper (dist n ==> dist n ==> dist n) (@pair A B) := _.
Instance fst_ne `{Dist A, Dist B} : Proper (dist n ==> dist n) (@fst A B) := _.
Instance snd_ne `{Dist A, Dist B} : Proper (dist n ==> dist n) (@snd A B) := _.
Typeclasses Opaque prod_dist.