cofe.v 13 KB
 Robbert Krebbers committed Nov 16, 2015 1 ``````Require Export prelude.prelude. `````` Robbert Krebbers committed Nov 11, 2015 2 3 4 5 ``````Obligation Tactic := idtac. (** Unbundeled version *) Class Dist A := dist : nat → relation A. `````` Robbert Krebbers committed Nov 12, 2015 6 ``````Instance: Params (@dist) 3. `````` Robbert Krebbers committed Nov 11, 2015 7 8 9 10 ``````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. `````` Robbert Krebbers committed Nov 17, 2015 11 12 13 14 15 16 ``````Ltac cofe_subst := repeat match goal with | _ => progress simplify_equality' | H: @dist _ ?d ?n ?x _ |- _ => setoid_subst_aux (@dist _ d n) x | H: @dist _ ?d ?n _ ?x |- _ => symmetry in H;setoid_subst_aux (@dist _ d n) x end. `````` Robbert Krebbers committed Nov 11, 2015 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 `````` 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. `````` Robbert Krebbers committed Nov 19, 2015 48 49 50 51 52 ``````Arguments cofe_car _ : simpl never. Arguments cofe_equiv _ _ _ : simpl never. Arguments cofe_dist _ _ _ _ : simpl never. Arguments cofe_compl _ _ : simpl never. Arguments cofe_cofe _ : simpl never. `````` Robbert Krebbers committed Nov 11, 2015 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 `````` (** 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. `````` Robbert Krebbers committed Nov 12, 2015 79 `````` Instance ne_proper `{Cofe B} (f : A → B) `````` Robbert Krebbers committed Nov 11, 2015 80 81 `````` `{!∀ 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. `````` Robbert Krebbers committed Nov 12, 2015 82 `````` Instance ne_proper_2 `{Cofe B, Cofe C} (f : A → B → C) `````` Robbert Krebbers committed Nov 11, 2015 83 84 85 86 87 88 89 90 91 92 `````` `{!∀ 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. `````` Robbert Krebbers committed Nov 12, 2015 93 94 95 96 97 `````` 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 contractive_proper `{Cofe B} (f : A → B) `{!Contractive f} : Proper ((≡) ==> (≡)) f | 100 := _. `````` Robbert Krebbers committed Nov 11, 2015 98 99 ``````End cofe. `````` Robbert Krebbers committed Nov 22, 2015 100 101 102 103 104 105 ``````(** Mapping a chain *) Program Definition chain_map `{Dist A, Dist B} (f : A → B) `{!∀ n, Proper (dist n ==> dist n) f} (c : chain A) : chain B := {| chain_car n := f (c n) |}. Next Obligation. by intros A ? B ? f Hf c n i ?; apply Hf, chain_cauchy. Qed. `````` Robbert Krebbers committed Nov 18, 2015 106 107 108 109 ``````(** Timeless elements *) Class Timeless `{Dist A, Equiv A} (x : A) := timeless y : x ={1}= y → x ≡ y. Arguments timeless {_ _ _} _ {_} _ _. `````` Robbert Krebbers committed Nov 11, 2015 110 ``````(** Fixpoint *) `````` Robbert Krebbers committed Nov 17, 2015 111 112 ``````Program Definition fixpoint_chain `{Cofe A, Inhabited A} (f : A → A) `{!Contractive f} : chain A := {| chain_car i := Nat.iter i f inhabitant |}. `````` Robbert Krebbers committed Nov 11, 2015 113 114 115 116 ``````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. `````` Robbert Krebbers committed Nov 17, 2015 117 118 ``````Program Definition fixpoint `{Cofe A, Inhabited A} (f : A → A) `{!Contractive f} : A := compl (fixpoint_chain f). `````` Robbert Krebbers committed Nov 11, 2015 119 120 `````` Section fixpoint. `````` Robbert Krebbers committed Nov 17, 2015 121 122 `````` Context `{Cofe A, Inhabited A} (f : A → A) `{!Contractive f}. Lemma fixpoint_unfold : fixpoint f ≡ f (fixpoint f). `````` Robbert Krebbers committed Nov 11, 2015 123 124 `````` Proof. apply equiv_dist; intros n; unfold fixpoint. `````` Robbert Krebbers committed Nov 17, 2015 125 126 `````` rewrite (conv_compl (fixpoint_chain f) n). by rewrite (chain_cauchy (fixpoint_chain f) n (S n)) at 1 by lia. `````` Robbert Krebbers committed Nov 11, 2015 127 `````` Qed. `````` Robbert Krebbers committed Nov 17, 2015 128 129 `````` Lemma fixpoint_ne (g : A → A) `{!Contractive g} n : (∀ z, f z ={n}= g z) → fixpoint f ={n}= fixpoint g. `````` Robbert Krebbers committed Nov 11, 2015 130 `````` Proof. `````` Robbert Krebbers committed Nov 17, 2015 131 132 `````` intros Hfg; unfold fixpoint. rewrite (conv_compl (fixpoint_chain f) n),(conv_compl (fixpoint_chain g) n). `````` Robbert Krebbers committed Nov 11, 2015 133 134 135 `````` induction n as [|n IH]; simpl in *; [done|]. rewrite Hfg; apply contractive, IH; auto using dist_S. Qed. `````` Robbert Krebbers committed Nov 17, 2015 136 137 `````` Lemma fixpoint_proper (g : A → A) `{!Contractive g} : (∀ x, f x ≡ g x) → fixpoint f ≡ fixpoint g. `````` Robbert Krebbers committed Nov 11, 2015 138 139 `````` Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_ne. Qed. End fixpoint. `````` Robbert Krebbers committed Nov 17, 2015 140 ``````Global Opaque fixpoint. `````` Robbert Krebbers committed Nov 11, 2015 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 `````` (** 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. `````` Robbert Krebbers committed Nov 12, 2015 180 181 182 183 184 ``````Instance cofe_mor_car_ne A B n : Proper (dist n ==> dist n ==> dist n) (@cofe_mor_car A B). Proof. intros f g Hfg x y Hx; rewrite Hx; apply Hfg. Qed. Instance cofe_mor_car_proper A B : Proper ((≡) ==> (≡) ==> (≡)) (@cofe_mor_car A B) := ne_proper_2 _. `````` Robbert Krebbers committed Nov 11, 2015 185 186 187 188 ``````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). `````` Robbert Krebbers committed Nov 17, 2015 189 190 ``````Instance cofe_more_inhabited (A B : cofeT) `{Inhabited B} : Inhabited (A -n> B) := populate (CofeMor (λ _, inhabitant)). `````` Robbert Krebbers committed Nov 11, 2015 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 `````` (** 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). `````` Robbert Krebbers committed Nov 22, 2015 221 222 223 224 ``````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) := _. `````` Robbert Krebbers committed Nov 11, 2015 225 ``````Instance prod_compl `{Compl A, Compl B} : Compl (A * B) := λ c, `````` Robbert Krebbers committed Nov 22, 2015 226 `````` (compl (chain_map fst c), compl (chain_map snd c)). `````` Robbert Krebbers committed Nov 11, 2015 227 228 229 230 231 232 233 234 ``````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. `````` Robbert Krebbers committed Nov 22, 2015 235 236 `````` * intros c n; split. apply (conv_compl (chain_map fst c) n). apply (conv_compl (chain_map snd c) n). `````` Robbert Krebbers committed Nov 11, 2015 237 ``````Qed. `````` Robbert Krebbers committed Nov 18, 2015 238 239 240 ``````Instance pair_timeless `{Dist A, Equiv A, Dist B, Equiv B} (x : A) (y : B) : Timeless x → Timeless y → Timeless (x,y). Proof. by intros ?? [x' y'] [??]; split; apply (timeless _). Qed. `````` Robbert Krebbers committed Nov 11, 2015 241 ``````Canonical Structure prodC (A B : cofeT) : cofeT := CofeT (A * B). `````` Robbert Krebbers committed Nov 16, 2015 242 ``````Instance prod_map_ne `{Dist A, Dist A', Dist B, Dist B'} n : `````` Robbert Krebbers committed Nov 11, 2015 243 244 245 246 247 248 249 250 251 252 `````` 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. Typeclasses Opaque prod_dist. `````` Robbert Krebbers committed Nov 16, 2015 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 `````` (** Discrete cofe *) Section discrete_cofe. Context `{Equiv A, @Equivalence A (≡)}. Instance discrete_dist : Dist A := λ n x y, match n with 0 => True | S n => x ≡ y end. Instance discrete_compl : Compl A := λ c, c 1. Instance discrete_cofe : Cofe A. Proof. split. * intros x y; split; [by intros ? []|intros Hn; apply (Hn 1)]. * intros [|n]; [done|apply _]. * by intros [|n]. * done. * intros c [|n]; [done|apply (chain_cauchy c 1 (S n)); lia]. Qed. `````` Robbert Krebbers committed Nov 18, 2015 269 270 `````` Global Instance discrete_timeless (x : A) : Timeless x. Proof. by intros y. Qed. `````` Robbert Krebbers committed Nov 22, 2015 271 `````` Definition discreteC : cofeT := CofeT A. `````` Robbert Krebbers committed Nov 16, 2015 272 ``````End discrete_cofe. `````` Robbert Krebbers committed Nov 22, 2015 273 ``````Arguments discreteC _ {_ _}. `````` Robbert Krebbers committed Nov 16, 2015 274 `````` `````` Robbert Krebbers committed Nov 22, 2015 275 ``````Definition leibnizC (A : Type) : cofeT := @discreteC A equivL _. `````` Robbert Krebbers committed Nov 19, 2015 276 `````` `````` Robbert Krebbers committed Nov 16, 2015 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 ``````(** Later *) Inductive later (A : Type) : Type := Later { later_car : A }. Arguments Later {_} _. Arguments later_car {_} _. Section later. Instance later_equiv `{Equiv A} : Equiv (later A) := λ x y, later_car x ≡ later_car y. Instance later_dist `{Dist A} : Dist (later A) := λ n x y, match n with 0 => True | S n => later_car x ={n}= later_car y end. Program Definition later_chain `{Dist A} (c : chain (later A)) : chain A := {| chain_car n := later_car (c (S n)) |}. Next Obligation. intros A ? c n i ?; apply (chain_cauchy c (S n)); lia. Qed. Instance later_compl `{Compl A} : Compl (later A) := λ c, Later (compl (later_chain c)). Instance later_cofe `{Cofe A} : Cofe (later A). Proof. split. * intros x y; unfold equiv, later_equiv; rewrite !equiv_dist. split. intros Hxy [|n]; [done|apply Hxy]. intros Hxy n; apply (Hxy (S n)). * intros [|n]; [by split|split]; unfold dist, later_dist. + by intros [x]. + by intros [x] [y]. + by intros [x] [y] [z] ??; transitivity y. * intros [|n] [x] [y] ?; [done|]; unfold dist, later_dist; by apply dist_S. * done. * intros c [|n]; [done|by apply (conv_compl (later_chain c) n)]. Qed. Canonical Structure laterC (A : cofeT) : cofeT := CofeT (later A). Instance later_fmap : FMap later := λ A B f x, Later (f (later_car x)). Instance later_fmap_ne `{Cofe A, Cofe B} (f : A → B) : (∀ n, Proper (dist n ==> dist n) f) → ∀ n, Proper (dist n ==> dist n) (fmap f : later A → later B). Proof. intros Hf [|n] [x] [y] ?; do 2 red; simpl. done. by apply Hf. Qed. Lemma later_fmap_id {A} (x : later A) : id <\$> x = x. Proof. by destruct x. Qed. Lemma later_fmap_compose {A B C} (f : A → B) (g : B → C) (x : later A) : g ∘ f <\$> x = g <\$> f <\$> x. Proof. by destruct x. Qed. Definition laterC_map {A B} (f : A -n> B) : laterC A -n> laterC B := CofeMor (fmap f : laterC A → laterC B). `````` Robbert Krebbers committed Nov 16, 2015 318 `````` Instance laterC_map_contractive (A B : cofeT) : Contractive (@laterC_map A B). `````` Robbert Krebbers committed Nov 16, 2015 319 320 `````` Proof. intros n f g Hf n'; apply Hf. Qed. End later.``````