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 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 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 `````` 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. `````` Robbert Krebbers committed Nov 12, 2015 74 `````` Instance ne_proper `{Cofe B} (f : A → B) `````` Robbert Krebbers committed Nov 11, 2015 75 76 `````` `{!∀ 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 77 `````` Instance ne_proper_2 `{Cofe B, Cofe C} (f : A → B → C) `````` Robbert Krebbers committed Nov 11, 2015 78 79 80 81 82 83 84 85 86 87 `````` `{!∀ 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 88 89 90 91 92 `````` 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 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 ``````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. `````` Robbert Krebbers committed Nov 12, 2015 164 165 166 167 168 ``````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 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 ``````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). `````` Robbert Krebbers committed Nov 16, 2015 223 ``````Instance prod_map_ne `{Dist A, Dist A', Dist B, Dist B'} n : `````` Robbert Krebbers committed Nov 11, 2015 224 225 226 227 228 229 230 231 232 233 234 235 236 237 `````` 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. `````` Robbert Krebbers committed Nov 16, 2015 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 `````` (** 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. Definition discrete_cofeC : cofeT := CofeT A. End discrete_cofe. Arguments discrete_cofeC _ {_ _}. (** 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 299 `````` Instance laterC_map_contractive (A B : cofeT) : Contractive (@laterC_map A B). `````` Robbert Krebbers committed Nov 16, 2015 300 301 `````` Proof. intros n f g Hf n'; apply Hf. Qed. End later.``````