ofe.v 31.7 KB
 Robbert Krebbers committed Mar 10, 2016 1 ``````From iris.algebra Require Export base. `````` Robbert Krebbers committed Nov 11, 2015 2 `````` `````` Ralf Jung committed Nov 22, 2016 3 ``````(** This files defines (a shallow embedding of) the category of OFEs: `````` Ralf Jung committed Feb 16, 2016 4 5 6 7 8 9 10 11 `````` Complete ordered families of equivalences. This is a cartesian closed category, and mathematically speaking, the entire development lives in this category. However, we will generally prefer to work with raw Coq functions plus some registered Proper instances for non-expansiveness. This makes writing such functions much easier. It turns out that it many cases, we do not even need non-expansiveness. *) `````` Robbert Krebbers committed Nov 11, 2015 12 13 ``````(** Unbundeled version *) Class Dist A := dist : nat → relation A. `````` Robbert Krebbers committed Nov 12, 2015 14 ``````Instance: Params (@dist) 3. `````` Ralf Jung committed Feb 10, 2016 15 16 ``````Notation "x ≡{ n }≡ y" := (dist n x y) (at level 70, n at next level, format "x ≡{ n }≡ y"). `````` Robbert Krebbers committed Feb 13, 2016 17 ``````Hint Extern 0 (_ ≡{_}≡ _) => reflexivity. `````` Ralf Jung committed Feb 10, 2016 18 ``````Hint Extern 0 (_ ≡{_}≡ _) => symmetry; assumption. `````` Robbert Krebbers committed Jan 13, 2016 19 20 21 `````` Tactic Notation "cofe_subst" ident(x) := repeat match goal with `````` Robbert Krebbers committed Feb 17, 2016 22 `````` | _ => progress simplify_eq/= `````` Robbert Krebbers committed Jan 13, 2016 23 24 25 26 `````` | H:@dist ?A ?d ?n x _ |- _ => setoid_subst_aux (@dist A d n) x | H:@dist ?A ?d ?n _ x |- _ => symmetry in H;setoid_subst_aux (@dist A d n) x end. Tactic Notation "cofe_subst" := `````` Robbert Krebbers committed Nov 17, 2015 27 `````` repeat match goal with `````` Robbert Krebbers committed Feb 17, 2016 28 `````` | _ => progress simplify_eq/= `````` Robbert Krebbers committed Dec 21, 2015 29 30 `````` | H:@dist ?A ?d ?n ?x _ |- _ => setoid_subst_aux (@dist A d n) x | H:@dist ?A ?d ?n _ ?x |- _ => symmetry in H;setoid_subst_aux (@dist A d n) x `````` Robbert Krebbers committed Nov 17, 2015 31 `````` end. `````` Robbert Krebbers committed Nov 11, 2015 32 `````` `````` Ralf Jung committed Nov 22, 2016 33 ``````Record OfeMixin A `{Equiv A, Dist A} := { `````` Ralf Jung committed Feb 10, 2016 34 `````` mixin_equiv_dist x y : x ≡ y ↔ ∀ n, x ≡{n}≡ y; `````` Robbert Krebbers committed Jan 14, 2016 35 `````` mixin_dist_equivalence n : Equivalence (dist n); `````` Ralf Jung committed Nov 22, 2016 36 `````` mixin_dist_S n x y : x ≡{S n}≡ y → x ≡{n}≡ y `````` Robbert Krebbers committed Nov 11, 2015 37 38 39 ``````}. (** Bundeled version *) `````` Ralf Jung committed Nov 22, 2016 40 41 42 43 44 ``````Structure ofeT := OfeT' { ofe_car :> Type; ofe_equiv : Equiv ofe_car; ofe_dist : Dist ofe_car; ofe_mixin : OfeMixin ofe_car; `````` Robbert Krebbers committed Jun 15, 2016 45 `````` _ : Type `````` Robbert Krebbers committed Nov 11, 2015 46 ``````}. `````` Ralf Jung committed Nov 22, 2016 47 48 49 50 51 52 53 54 55 ``````Arguments OfeT' _ {_ _} _ _. Notation OfeT A m := (OfeT' A m A). Add Printing Constructor ofeT. Hint Extern 0 (Equiv _) => eapply (@ofe_equiv _) : typeclass_instances. Hint Extern 0 (Dist _) => eapply (@ofe_dist _) : typeclass_instances. Arguments ofe_car : simpl never. Arguments ofe_equiv : simpl never. Arguments ofe_dist : simpl never. Arguments ofe_mixin : simpl never. `````` Robbert Krebbers committed Jan 14, 2016 56 57 `````` (** Lifting properties from the mixin *) `````` Ralf Jung committed Nov 22, 2016 58 59 ``````Section ofe_mixin. Context {A : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 60 `````` Implicit Types x y : A. `````` Ralf Jung committed Feb 10, 2016 61 `````` Lemma equiv_dist x y : x ≡ y ↔ ∀ n, x ≡{n}≡ y. `````` Ralf Jung committed Nov 22, 2016 62 `````` Proof. apply (mixin_equiv_dist _ (ofe_mixin A)). Qed. `````` Robbert Krebbers committed Jan 14, 2016 63 `````` Global Instance dist_equivalence n : Equivalence (@dist A _ n). `````` Ralf Jung committed Nov 22, 2016 64 `````` Proof. apply (mixin_dist_equivalence _ (ofe_mixin A)). Qed. `````` Ralf Jung committed Feb 10, 2016 65 `````` Lemma dist_S n x y : x ≡{S n}≡ y → x ≡{n}≡ y. `````` Ralf Jung committed Nov 22, 2016 66 67 `````` Proof. apply (mixin_dist_S _ (ofe_mixin A)). Qed. End ofe_mixin. `````` Robbert Krebbers committed Jan 14, 2016 68 `````` `````` Robbert Krebbers committed May 28, 2016 69 70 ``````Hint Extern 1 (_ ≡{_}≡ _) => apply equiv_dist; assumption. `````` Robbert Krebbers committed Feb 24, 2016 71 ``````(** Discrete COFEs and Timeless elements *) `````` Ralf Jung committed Mar 15, 2016 72 ``````(* TODO: On paper, We called these "discrete elements". I think that makes `````` Ralf Jung committed Mar 07, 2016 73 `````` more sense. *) `````` Robbert Krebbers committed May 27, 2016 74 75 ``````Class Timeless `{Equiv A, Dist A} (x : A) := timeless y : x ≡{0}≡ y → x ≡ y. Arguments timeless {_ _ _} _ {_} _ _. `````` Ralf Jung committed Nov 22, 2016 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 ``````Class Discrete (A : ofeT) := discrete_timeless (x : A) :> Timeless x. (** OFEs with a completion *) Record chain (A : ofeT) := { chain_car :> nat → A; chain_cauchy n i : n ≤ i → chain_car i ≡{n}≡ chain_car n }. Arguments chain_car {_} _ _. Arguments chain_cauchy {_} _ _ _ _. Notation Compl A := (chain A%type → A). Class Cofe (A : ofeT) := { compl : Compl A; conv_compl n c : compl c ≡{n}≡ c n; }. Arguments compl : simpl never. `````` Robbert Krebbers committed Feb 24, 2016 92 `````` `````` Robbert Krebbers committed Nov 11, 2015 93 94 ``````(** General properties *) Section cofe. `````` Ralf Jung committed Nov 22, 2016 95 `````` Context {A : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 96 `````` Implicit Types x y : A. `````` Robbert Krebbers committed Nov 11, 2015 97 98 99 `````` Global Instance cofe_equivalence : Equivalence ((≡) : relation A). Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 100 101 `````` - by intros x; rewrite equiv_dist. - by intros x y; rewrite !equiv_dist. `````` Ralf Jung committed Feb 20, 2016 102 `````` - by intros x y z; rewrite !equiv_dist; intros; trans y. `````` Robbert Krebbers committed Nov 11, 2015 103 `````` Qed. `````` Robbert Krebbers committed Jan 14, 2016 104 `````` Global Instance dist_ne n : Proper (dist n ==> dist n ==> iff) (@dist A _ n). `````` Robbert Krebbers committed Nov 11, 2015 105 106 `````` Proof. intros x1 x2 ? y1 y2 ?; split; intros. `````` Ralf Jung committed Feb 20, 2016 107 108 `````` - by trans x1; [|trans y1]. - by trans x2; [|trans y2]. `````` Robbert Krebbers committed Nov 11, 2015 109 `````` Qed. `````` Robbert Krebbers committed Jan 14, 2016 110 `````` Global Instance dist_proper n : Proper ((≡) ==> (≡) ==> iff) (@dist A _ n). `````` Robbert Krebbers committed Nov 11, 2015 111 `````` Proof. `````` Robbert Krebbers committed Jan 13, 2016 112 `````` by move => x1 x2 /equiv_dist Hx y1 y2 /equiv_dist Hy; rewrite (Hx n) (Hy n). `````` Robbert Krebbers committed Nov 11, 2015 113 114 115 `````` Qed. Global Instance dist_proper_2 n x : Proper ((≡) ==> iff) (dist n x). Proof. by apply dist_proper. Qed. `````` Robbert Krebbers committed Feb 18, 2016 116 `````` Lemma dist_le n n' x y : x ≡{n}≡ y → n' ≤ n → x ≡{n'}≡ y. `````` Robbert Krebbers committed Nov 11, 2015 117 `````` Proof. induction 2; eauto using dist_S. Qed. `````` Ralf Jung committed Feb 29, 2016 118 119 `````` Lemma dist_le' n n' x y : n' ≤ n → x ≡{n}≡ y → x ≡{n'}≡ y. Proof. intros; eauto using dist_le. Qed. `````` Ralf Jung committed Nov 22, 2016 120 `````` Instance ne_proper {B : ofeT} (f : A → B) `````` Robbert Krebbers committed Nov 11, 2015 121 122 `````` `{!∀ 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. `````` Ralf Jung committed Nov 22, 2016 123 `````` Instance ne_proper_2 {B C : ofeT} (f : A → B → C) `````` Robbert Krebbers committed Nov 11, 2015 124 125 126 127 `````` `{!∀ n, Proper (dist n ==> dist n ==> dist n) f} : Proper ((≡) ==> (≡) ==> (≡)) f | 100. Proof. unfold Proper, respectful; setoid_rewrite equiv_dist. `````` Robbert Krebbers committed Jan 13, 2016 128 `````` by intros x1 x2 Hx y1 y2 Hy n; rewrite (Hx n) (Hy n). `````` Robbert Krebbers committed Nov 11, 2015 129 `````` Qed. `````` Robbert Krebbers committed Feb 24, 2016 130 `````` `````` Ralf Jung committed Nov 22, 2016 131 `````` Lemma conv_compl' `{Cofe A} n (c : chain A) : compl c ≡{n}≡ c (S n). `````` Ralf Jung committed Feb 29, 2016 132 133 134 135 `````` Proof. transitivity (c n); first by apply conv_compl. symmetry. apply chain_cauchy. omega. Qed. `````` Robbert Krebbers committed Feb 24, 2016 136 137 `````` Lemma timeless_iff n (x : A) `{!Timeless x} y : x ≡ y ↔ x ≡{n}≡ y. Proof. `````` Robbert Krebbers committed May 28, 2016 138 `````` split; intros; auto. apply (timeless _), dist_le with n; auto with lia. `````` Robbert Krebbers committed Feb 24, 2016 139 `````` Qed. `````` Robbert Krebbers committed Nov 11, 2015 140 141 ``````End cofe. `````` Robbert Krebbers committed Dec 02, 2016 142 143 144 145 ``````(** Contractive functions *) Class Contractive {A B : ofeT} (f : A → B) := contractive n x y : (∀ i, i < n → x ≡{i}≡ y) → f x ≡{n}≡ f y. `````` Ralf Jung committed Nov 22, 2016 146 ``````Instance const_contractive {A B : ofeT} (x : A) : Contractive (@const A B x). `````` Robbert Krebbers committed Mar 06, 2016 147 148 ``````Proof. by intros n y1 y2. Qed. `````` Robbert Krebbers committed Dec 02, 2016 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 ``````Section contractive. Context {A B : ofeT} (f : A → B) `{!Contractive f}. Implicit Types x y : A. Lemma contractive_0 x y : f x ≡{0}≡ f y. Proof. eauto using contractive with omega. Qed. Lemma contractive_S n x y : x ≡{n}≡ y → f x ≡{S n}≡ f y. Proof. eauto using contractive, dist_le with omega. Qed. Global Instance contractive_ne n : Proper (dist n ==> dist n) f | 100. Proof. by intros x y ?; apply dist_S, contractive_S. Qed. Global Instance contractive_proper : Proper ((≡) ==> (≡)) f | 100. Proof. apply (ne_proper _). Qed. End contractive. `````` Robbert Krebbers committed Nov 22, 2015 164 ``````(** Mapping a chain *) `````` Ralf Jung committed Nov 22, 2016 165 ``````Program Definition chain_map {A B : ofeT} (f : A → B) `````` Robbert Krebbers committed Nov 22, 2015 166 167 `````` `{!∀ n, Proper (dist n ==> dist n) f} (c : chain A) : chain B := {| chain_car n := f (c n) |}. `````` Ralf Jung committed Nov 22, 2016 168 ``````Next Obligation. by intros A B f Hf c n i ?; apply Hf, chain_cauchy. Qed. `````` Robbert Krebbers committed Nov 22, 2015 169 `````` `````` Robbert Krebbers committed Nov 11, 2015 170 ``````(** Fixpoint *) `````` Ralf Jung committed Nov 22, 2016 171 ``````Program Definition fixpoint_chain {A : ofeT} `{Inhabited A} (f : A → A) `````` Robbert Krebbers committed Feb 10, 2016 172 `````` `{!Contractive f} : chain A := {| chain_car i := Nat.iter (S i) f inhabitant |}. `````` Robbert Krebbers committed Nov 11, 2015 173 ``````Next Obligation. `````` Robbert Krebbers committed Mar 06, 2016 174 175 `````` intros A ? f ? n. induction n as [|n IH]; intros [|i] ?; simpl in *; try reflexivity || omega. `````` Robbert Krebbers committed Feb 17, 2016 176 177 `````` - apply (contractive_0 f). - apply (contractive_S f), IH; auto with omega. `````` Robbert Krebbers committed Nov 11, 2015 178 ``````Qed. `````` Robbert Krebbers committed Mar 18, 2016 179 `````` `````` Ralf Jung committed Nov 22, 2016 180 ``````Program Definition fixpoint_def `{Cofe A, Inhabited A} (f : A → A) `````` Robbert Krebbers committed Nov 17, 2015 181 `````` `{!Contractive f} : A := compl (fixpoint_chain f). `````` Robbert Krebbers committed Mar 18, 2016 182 ``````Definition fixpoint_aux : { x | x = @fixpoint_def }. by eexists. Qed. `````` Ralf Jung committed Nov 22, 2016 183 ``````Definition fixpoint {A AC AiH} f {Hf} := proj1_sig fixpoint_aux A AC AiH f Hf. `````` Robbert Krebbers committed Mar 18, 2016 184 ``````Definition fixpoint_eq : @fixpoint = @fixpoint_def := proj2_sig fixpoint_aux. `````` Robbert Krebbers committed Nov 11, 2015 185 186 `````` Section fixpoint. `````` Ralf Jung committed Nov 22, 2016 187 `````` Context `{Cofe A, Inhabited A} (f : A → A) `{!Contractive f}. `````` Robbert Krebbers committed Aug 21, 2016 188 `````` `````` Robbert Krebbers committed Nov 17, 2015 189 `````` Lemma fixpoint_unfold : fixpoint f ≡ f (fixpoint f). `````` Robbert Krebbers committed Nov 11, 2015 190 `````` Proof. `````` Robbert Krebbers committed Mar 18, 2016 191 192 `````` apply equiv_dist=>n. rewrite fixpoint_eq /fixpoint_def (conv_compl n (fixpoint_chain f)) //. `````` Robbert Krebbers committed Feb 12, 2016 193 `````` induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S. `````` Robbert Krebbers committed Nov 11, 2015 194 `````` Qed. `````` Robbert Krebbers committed Aug 21, 2016 195 196 197 `````` Lemma fixpoint_unique (x : A) : x ≡ f x → x ≡ fixpoint f. Proof. `````` Robbert Krebbers committed Aug 22, 2016 198 199 200 `````` rewrite !equiv_dist=> Hx n. induction n as [|n IH]; simpl in *. - rewrite Hx fixpoint_unfold; eauto using contractive_0. - rewrite Hx fixpoint_unfold. apply (contractive_S _), IH. `````` Robbert Krebbers committed Aug 21, 2016 201 202 `````` Qed. `````` Robbert Krebbers committed Nov 17, 2015 203 `````` Lemma fixpoint_ne (g : A → A) `{!Contractive g} n : `````` Ralf Jung committed Feb 10, 2016 204 `````` (∀ z, f z ≡{n}≡ g z) → fixpoint f ≡{n}≡ fixpoint g. `````` Robbert Krebbers committed Nov 11, 2015 205 `````` Proof. `````` Robbert Krebbers committed Mar 18, 2016 206 `````` intros Hfg. rewrite fixpoint_eq /fixpoint_def `````` Robbert Krebbers committed Feb 18, 2016 207 `````` (conv_compl n (fixpoint_chain f)) (conv_compl n (fixpoint_chain g)) /=. `````` Robbert Krebbers committed Feb 10, 2016 208 209 `````` induction n as [|n IH]; simpl in *; [by rewrite !Hfg|]. rewrite Hfg; apply contractive_S, IH; auto using dist_S. `````` Robbert Krebbers committed Nov 11, 2015 210 `````` Qed. `````` Robbert Krebbers committed Nov 17, 2015 211 212 `````` Lemma fixpoint_proper (g : A → A) `{!Contractive g} : (∀ x, f x ≡ g x) → fixpoint f ≡ fixpoint g. `````` Robbert Krebbers committed Nov 11, 2015 213 214 215 `````` Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_ne. Qed. End fixpoint. `````` Robbert Krebbers committed Jul 25, 2016 216 ``````(** Function space *) `````` Ralf Jung committed Nov 22, 2016 217 ``````(* We make [ofe_fun] a definition so that we can register it as a canonical `````` Robbert Krebbers committed Aug 05, 2016 218 ``````structure. *) `````` Ralf Jung committed Nov 22, 2016 219 ``````Definition ofe_fun (A : Type) (B : ofeT) := A → B. `````` Robbert Krebbers committed Jul 25, 2016 220 `````` `````` Ralf Jung committed Nov 22, 2016 221 222 223 224 225 ``````Section ofe_fun. Context {A : Type} {B : ofeT}. Instance ofe_fun_equiv : Equiv (ofe_fun A B) := λ f g, ∀ x, f x ≡ g x. Instance ofe_fun_dist : Dist (ofe_fun A B) := λ n f g, ∀ x, f x ≡{n}≡ g x. Definition ofe_fun_ofe_mixin : OfeMixin (ofe_fun A B). `````` Robbert Krebbers committed Jul 25, 2016 226 227 228 229 230 231 232 233 234 235 `````` Proof. split. - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|]. intros Hfg k; apply equiv_dist=> n; apply Hfg. - intros n; split. + by intros f x. + by intros f g ? x. + by intros f g h ?? x; trans (g x). - by intros n f g ? x; apply dist_S. Qed. `````` Ralf Jung committed Nov 22, 2016 236 `````` Canonical Structure ofe_funC := OfeT (ofe_fun A B) ofe_fun_ofe_mixin. `````` Robbert Krebbers committed Jul 25, 2016 237 `````` `````` Ralf Jung committed Nov 22, 2016 238 239 240 241 242 243 244 245 246 `````` Program Definition ofe_fun_chain `(c : chain ofe_funC) (x : A) : chain B := {| chain_car n := c n x |}. Next Obligation. intros c x n i ?. by apply (chain_cauchy c). Qed. Global Program Instance ofe_fun_cofe `{Cofe B} : Cofe ofe_funC := { compl c x := compl (ofe_fun_chain c x) }. Next Obligation. intros ? n c x. apply (conv_compl n (ofe_fun_chain c x)). Qed. End ofe_fun. Arguments ofe_funC : clear implicits. `````` Robbert Krebbers committed Jul 25, 2016 247 ``````Notation "A -c> B" := `````` Ralf Jung committed Nov 22, 2016 248 249 `````` (ofe_funC A B) (at level 99, B at level 200, right associativity). Instance ofe_fun_inhabited {A} {B : ofeT} `{Inhabited B} : `````` Robbert Krebbers committed Jul 25, 2016 250 251 `````` Inhabited (A -c> B) := populate (λ _, inhabitant). `````` Robbert Krebbers committed Jul 25, 2016 252 ``````(** Non-expansive function space *) `````` Ralf Jung committed Nov 22, 2016 253 254 255 ``````Record ofe_mor (A B : ofeT) : Type := CofeMor { ofe_mor_car :> A → B; ofe_mor_ne n : Proper (dist n ==> dist n) ofe_mor_car `````` Robbert Krebbers committed Nov 11, 2015 256 257 ``````}. Arguments CofeMor {_ _} _ {_}. `````` Ralf Jung committed Nov 22, 2016 258 259 ``````Add Printing Constructor ofe_mor. Existing Instance ofe_mor_ne. `````` Robbert Krebbers committed Nov 11, 2015 260 `````` `````` Robbert Krebbers committed Jun 17, 2016 261 262 263 264 ``````Notation "'λne' x .. y , t" := (@CofeMor _ _ (λ x, .. (@CofeMor _ _ (λ y, t) _) ..) _) (at level 200, x binder, y binder, right associativity). `````` Ralf Jung committed Nov 22, 2016 265 266 267 268 269 270 271 ``````Section ofe_mor. Context {A B : ofeT}. Global Instance ofe_mor_proper (f : ofe_mor A B) : Proper ((≡) ==> (≡)) f. Proof. apply ne_proper, ofe_mor_ne. Qed. Instance ofe_mor_equiv : Equiv (ofe_mor A B) := λ f g, ∀ x, f x ≡ g x. Instance ofe_mor_dist : Dist (ofe_mor A B) := λ n f g, ∀ x, f x ≡{n}≡ g x. Definition ofe_mor_ofe_mixin : OfeMixin (ofe_mor A B). `````` Robbert Krebbers committed Jan 14, 2016 272 273 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 274 `````` - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|]. `````` Robbert Krebbers committed Feb 18, 2016 275 `````` intros Hfg k; apply equiv_dist=> n; apply Hfg. `````` Robbert Krebbers committed Feb 17, 2016 276 `````` - intros n; split. `````` Robbert Krebbers committed Jan 14, 2016 277 278 `````` + by intros f x. + by intros f g ? x. `````` Ralf Jung committed Feb 20, 2016 279 `````` + by intros f g h ?? x; trans (g x). `````` Robbert Krebbers committed Feb 17, 2016 280 `````` - by intros n f g ? x; apply dist_S. `````` Robbert Krebbers committed Jan 14, 2016 281 `````` Qed. `````` Ralf Jung committed Nov 22, 2016 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 `````` Canonical Structure ofe_morC := OfeT (ofe_mor A B) ofe_mor_ofe_mixin. Program Definition ofe_mor_chain (c : chain ofe_morC) (x : A) : chain B := {| chain_car n := c n x |}. Next Obligation. intros c x n i ?. by apply (chain_cauchy c). Qed. Program Definition ofe_mor_compl `{Cofe B} : Compl ofe_morC := λ c, {| ofe_mor_car x := compl (ofe_mor_chain c x) |}. Next Obligation. intros ? c n x y Hx. by rewrite (conv_compl n (ofe_mor_chain c x)) (conv_compl n (ofe_mor_chain c y)) /= Hx. Qed. Global Program Instance ofe_more_cofe `{Cofe B} : Cofe ofe_morC := {| compl := ofe_mor_compl |}. Next Obligation. intros ? n c x; simpl. by rewrite (conv_compl n (ofe_mor_chain c x)) /=. Qed. `````` Robbert Krebbers committed Jan 14, 2016 299 `````` `````` Ralf Jung committed Nov 22, 2016 300 301 `````` Global Instance ofe_mor_car_ne n : Proper (dist n ==> dist n ==> dist n) (@ofe_mor_car A B). `````` Robbert Krebbers committed Jan 14, 2016 302 `````` Proof. intros f g Hfg x y Hx; rewrite Hx; apply Hfg. Qed. `````` Ralf Jung committed Nov 22, 2016 303 304 305 `````` Global Instance ofe_mor_car_proper : Proper ((≡) ==> (≡) ==> (≡)) (@ofe_mor_car A B) := ne_proper_2 _. Lemma ofe_mor_ext (f g : ofe_mor A B) : f ≡ g ↔ ∀ x, f x ≡ g x. `````` Robbert Krebbers committed Jan 14, 2016 306 `````` Proof. done. Qed. `````` Ralf Jung committed Nov 22, 2016 307 ``````End ofe_mor. `````` Robbert Krebbers committed Jan 14, 2016 308 `````` `````` Ralf Jung committed Nov 22, 2016 309 ``````Arguments ofe_morC : clear implicits. `````` Robbert Krebbers committed Jul 25, 2016 310 ``````Notation "A -n> B" := `````` Ralf Jung committed Nov 22, 2016 311 312 `````` (ofe_morC A B) (at level 99, B at level 200, right associativity). Instance ofe_mor_inhabited {A B : ofeT} `{Inhabited B} : `````` Robbert Krebbers committed Jul 25, 2016 313 `````` Inhabited (A -n> B) := populate (λne _, inhabitant). `````` Robbert Krebbers committed Nov 11, 2015 314 `````` `````` Ralf Jung committed Mar 17, 2016 315 ``````(** Identity and composition and constant function *) `````` Robbert Krebbers committed Nov 11, 2015 316 317 ``````Definition cid {A} : A -n> A := CofeMor id. Instance: Params (@cid) 1. `````` Ralf Jung committed Nov 22, 2016 318 ``````Definition cconst {A B : ofeT} (x : B) : A -n> B := CofeMor (const x). `````` Ralf Jung committed Mar 17, 2016 319 ``````Instance: Params (@cconst) 2. `````` Robbert Krebbers committed Mar 02, 2016 320 `````` `````` Robbert Krebbers committed Nov 11, 2015 321 322 323 324 325 ``````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 : `````` Ralf Jung committed Feb 10, 2016 326 `````` f1 ≡{n}≡ f2 → g1 ≡{n}≡ g2 → f1 ◎ g1 ≡{n}≡ f2 ◎ g2. `````` Robbert Krebbers committed Jan 13, 2016 327 ``````Proof. by intros Hf Hg x; rewrite /= (Hg x) (Hf (g2 x)). Qed. `````` Robbert Krebbers committed Nov 11, 2015 328 `````` `````` Ralf Jung committed Mar 02, 2016 329 ``````(* Function space maps *) `````` Ralf Jung committed Nov 22, 2016 330 ``````Definition ofe_mor_map {A A' B B'} (f : A' -n> A) (g : B -n> B') `````` Ralf Jung committed Mar 02, 2016 331 `````` (h : A -n> B) : A' -n> B' := g ◎ h ◎ f. `````` Ralf Jung committed Nov 22, 2016 332 333 ``````Instance ofe_mor_map_ne {A A' B B'} n : Proper (dist n ==> dist n ==> dist n ==> dist n) (@ofe_mor_map A A' B B'). `````` Robbert Krebbers committed Mar 02, 2016 334 ``````Proof. intros ??? ??? ???. by repeat apply ccompose_ne. Qed. `````` Ralf Jung committed Mar 02, 2016 335 `````` `````` Ralf Jung committed Nov 22, 2016 336 337 338 339 ``````Definition ofe_morC_map {A A' B B'} (f : A' -n> A) (g : B -n> B') : (A -n> B) -n> (A' -n> B') := CofeMor (ofe_mor_map f g). Instance ofe_morC_map_ne {A A' B B'} n : Proper (dist n ==> dist n ==> dist n) (@ofe_morC_map A A' B B'). `````` Ralf Jung committed Mar 02, 2016 340 ``````Proof. `````` Ralf Jung committed Nov 22, 2016 341 `````` intros f f' Hf g g' Hg ?. rewrite /= /ofe_mor_map. `````` Robbert Krebbers committed Mar 02, 2016 342 `````` by repeat apply ccompose_ne. `````` Ralf Jung committed Mar 02, 2016 343 344 ``````Qed. `````` Robbert Krebbers committed Nov 11, 2015 345 ``````(** unit *) `````` Robbert Krebbers committed Jan 14, 2016 346 347 ``````Section unit. Instance unit_dist : Dist unit := λ _ _ _, True. `````` Ralf Jung committed Nov 22, 2016 348 `````` Definition unit_ofe_mixin : OfeMixin unit. `````` Robbert Krebbers committed Jan 14, 2016 349 `````` Proof. by repeat split; try exists 0. Qed. `````` Ralf Jung committed Nov 22, 2016 350 `````` Canonical Structure unitC : ofeT := OfeT unit unit_ofe_mixin. `````` Robbert Krebbers committed Nov 28, 2016 351 `````` `````` Ralf Jung committed Nov 22, 2016 352 353 `````` Global Program Instance unit_cofe : Cofe unitC := { compl x := () }. Next Obligation. by repeat split; try exists 0. Qed. `````` Robbert Krebbers committed Nov 28, 2016 354 355 `````` Global Instance unit_discrete_cofe : Discrete unitC. `````` Robbert Krebbers committed Jan 31, 2016 356 `````` Proof. done. Qed. `````` Robbert Krebbers committed Jan 14, 2016 357 ``````End unit. `````` Robbert Krebbers committed Nov 11, 2015 358 359 `````` (** Product *) `````` Robbert Krebbers committed Jan 14, 2016 360 ``````Section product. `````` Ralf Jung committed Nov 22, 2016 361 `````` Context {A B : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 362 363 364 365 366 367 `````` Instance prod_dist : Dist (A * B) := λ n, prod_relation (dist n) (dist n). Global Instance pair_ne : Proper (dist n ==> dist n ==> dist n) (@pair A B) := _. Global Instance fst_ne : Proper (dist n ==> dist n) (@fst A B) := _. Global Instance snd_ne : Proper (dist n ==> dist n) (@snd A B) := _. `````` Ralf Jung committed Nov 22, 2016 368 `````` Definition prod_ofe_mixin : OfeMixin (A * B). `````` Robbert Krebbers committed Jan 14, 2016 369 370 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 371 `````` - intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation. `````` Robbert Krebbers committed Jan 14, 2016 372 `````` rewrite !equiv_dist; naive_solver. `````` Robbert Krebbers committed Feb 17, 2016 373 374 `````` - apply _. - by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S. `````` Robbert Krebbers committed Jan 14, 2016 375 `````` Qed. `````` Ralf Jung committed Nov 22, 2016 376 377 378 379 380 381 382 383 384 `````` Canonical Structure prodC : ofeT := OfeT (A * B) prod_ofe_mixin. Global Program Instance prod_cofe `{Cofe A, Cofe B} : Cofe prodC := { compl c := (compl (chain_map fst c), compl (chain_map snd c)) }. Next Obligation. intros ?? n c; split. apply (conv_compl n (chain_map fst c)). apply (conv_compl n (chain_map snd c)). Qed. `````` Jacques-Henri Jourdan committed Jun 15, 2016 385 386 387 `````` Global Instance prod_timeless (x : A * B) : Timeless (x.1) → Timeless (x.2) → Timeless x. Proof. by intros ???[??]; split; apply (timeless _). Qed. `````` Robbert Krebbers committed Feb 24, 2016 388 389 `````` Global Instance prod_discrete_cofe : Discrete A → Discrete B → Discrete prodC. Proof. intros ?? [??]; apply _. Qed. `````` Robbert Krebbers committed Jan 14, 2016 390 391 392 393 394 ``````End product. Arguments prodC : clear implicits. Typeclasses Opaque prod_dist. `````` Ralf Jung committed Nov 22, 2016 395 ``````Instance prod_map_ne {A A' B B' : ofeT} n : `````` Robbert Krebbers committed Nov 11, 2015 396 397 398 399 400 401 402 403 404 `````` 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. `````` Robbert Krebbers committed Mar 02, 2016 405 406 ``````(** Functors *) Structure cFunctor := CFunctor { `````` Ralf Jung committed Nov 22, 2016 407 `````` cFunctor_car : ofeT → ofeT → ofeT; `````` Robbert Krebbers committed Mar 02, 2016 408 409 `````` cFunctor_map {A1 A2 B1 B2} : ((A2 -n> A1) * (B1 -n> B2)) → cFunctor_car A1 B1 -n> cFunctor_car A2 B2; `````` Robbert Krebbers committed Mar 07, 2016 410 411 `````` cFunctor_ne {A1 A2 B1 B2} n : Proper (dist n ==> dist n) (@cFunctor_map A1 A2 B1 B2); `````` Ralf Jung committed Nov 22, 2016 412 `````` cFunctor_id {A B : ofeT} (x : cFunctor_car A B) : `````` Robbert Krebbers committed Mar 02, 2016 413 414 415 416 417 `````` cFunctor_map (cid,cid) x ≡ x; cFunctor_compose {A1 A2 A3 B1 B2 B3} (f : A2 -n> A1) (g : A3 -n> A2) (f' : B1 -n> B2) (g' : B2 -n> B3) x : cFunctor_map (f◎g, g'◎f') x ≡ cFunctor_map (g,g') (cFunctor_map (f,f') x) }. `````` Robbert Krebbers committed Mar 07, 2016 418 ``````Existing Instance cFunctor_ne. `````` Robbert Krebbers committed Mar 02, 2016 419 420 ``````Instance: Params (@cFunctor_map) 5. `````` Ralf Jung committed Mar 07, 2016 421 422 423 ``````Delimit Scope cFunctor_scope with CF. Bind Scope cFunctor_scope with cFunctor. `````` Ralf Jung committed Mar 07, 2016 424 425 426 ``````Class cFunctorContractive (F : cFunctor) := cFunctor_contractive A1 A2 B1 B2 :> Contractive (@cFunctor_map F A1 A2 B1 B2). `````` Ralf Jung committed Nov 22, 2016 427 ``````Definition cFunctor_diag (F: cFunctor) (A: ofeT) : ofeT := cFunctor_car F A A. `````` Robbert Krebbers committed Mar 02, 2016 428 429 ``````Coercion cFunctor_diag : cFunctor >-> Funclass. `````` Ralf Jung committed Nov 22, 2016 430 ``````Program Definition constCF (B : ofeT) : cFunctor := `````` Robbert Krebbers committed Mar 02, 2016 431 432 433 `````` {| cFunctor_car A1 A2 := B; cFunctor_map A1 A2 B1 B2 f := cid |}. Solve Obligations with done. `````` Ralf Jung committed Mar 07, 2016 434 ``````Instance constCF_contractive B : cFunctorContractive (constCF B). `````` Robbert Krebbers committed Mar 07, 2016 435 ``````Proof. rewrite /cFunctorContractive; apply _. Qed. `````` Ralf Jung committed Mar 07, 2016 436 437 438 439 440 `````` Program Definition idCF : cFunctor := {| cFunctor_car A1 A2 := A2; cFunctor_map A1 A2 B1 B2 f := f.2 |}. Solve Obligations with done. `````` Robbert Krebbers committed Mar 02, 2016 441 442 443 444 445 ``````Program Definition prodCF (F1 F2 : cFunctor) : cFunctor := {| cFunctor_car A B := prodC (cFunctor_car F1 A B) (cFunctor_car F2 A B); cFunctor_map A1 A2 B1 B2 fg := prodC_map (cFunctor_map F1 fg) (cFunctor_map F2 fg) |}. `````` Robbert Krebbers committed Mar 07, 2016 446 447 448 ``````Next Obligation. intros ?? A1 A2 B1 B2 n ???; by apply prodC_map_ne; apply cFunctor_ne. Qed. `````` Robbert Krebbers committed Mar 02, 2016 449 450 451 452 453 454 ``````Next Obligation. by intros F1 F2 A B [??]; rewrite /= !cFunctor_id. Qed. Next Obligation. intros F1 F2 A1 A2 A3 B1 B2 B3 f g f' g' [??]; simpl. by rewrite !cFunctor_compose. Qed. `````` Ralf Jung committed Mar 07, 2016 455 456 457 458 459 460 461 462 ``````Instance prodCF_contractive F1 F2 : cFunctorContractive F1 → cFunctorContractive F2 → cFunctorContractive (prodCF F1 F2). Proof. intros ?? A1 A2 B1 B2 n ???; by apply prodC_map_ne; apply cFunctor_contractive. Qed. `````` Ralf Jung committed Nov 22, 2016 463 ``````Instance compose_ne {A} {B B' : ofeT} (f : B -n> B') n : `````` Jacques-Henri Jourdan committed Oct 05, 2016 464 465 466 `````` Proper (dist n ==> dist n) (compose f : (A -c> B) → A -c> B'). Proof. intros g g' Hf x; simpl. by rewrite (Hf x). Qed. `````` Ralf Jung committed Nov 22, 2016 467 ``````Definition ofe_funC_map {A B B'} (f : B -n> B') : (A -c> B) -n> (A -c> B') := `````` Jacques-Henri Jourdan committed Oct 05, 2016 468 `````` @CofeMor (_ -c> _) (_ -c> _) (compose f) _. `````` Ralf Jung committed Nov 22, 2016 469 470 ``````Instance ofe_funC_map_ne {A B B'} n : Proper (dist n ==> dist n) (@ofe_funC_map A B B'). `````` Jacques-Henri Jourdan committed Oct 05, 2016 471 472 ``````Proof. intros f f' Hf g x. apply Hf. Qed. `````` Ralf Jung committed Nov 22, 2016 473 474 475 ``````Program Definition ofe_funCF (T : Type) (F : cFunctor) : cFunctor := {| cFunctor_car A B := ofe_funC T (cFunctor_car F A B); cFunctor_map A1 A2 B1 B2 fg := ofe_funC_map (cFunctor_map F fg) `````` Jacques-Henri Jourdan committed Oct 05, 2016 476 477 ``````|}. Next Obligation. `````` Ralf Jung committed Nov 22, 2016 478 `````` intros ?? A1 A2 B1 B2 n ???; by apply ofe_funC_map_ne; apply cFunctor_ne. `````` Jacques-Henri Jourdan committed Oct 05, 2016 479 480 481 482 483 484 485 ``````Qed. Next Obligation. intros F1 F2 A B ??. by rewrite /= /compose /= !cFunctor_id. Qed. Next Obligation. intros T F A1 A2 A3 B1 B2 B3 f g f' g' ??; simpl. by rewrite !cFunctor_compose. Qed. `````` Ralf Jung committed Nov 22, 2016 486 487 ``````Instance ofe_funCF_contractive (T : Type) (F : cFunctor) : cFunctorContractive F → cFunctorContractive (ofe_funCF T F). `````` Jacques-Henri Jourdan committed Oct 05, 2016 488 489 ``````Proof. intros ?? A1 A2 B1 B2 n ???; `````` Ralf Jung committed Nov 22, 2016 490 `````` by apply ofe_funC_map_ne; apply cFunctor_contractive. `````` Jacques-Henri Jourdan committed Oct 05, 2016 491 492 ``````Qed. `````` Ralf Jung committed Nov 22, 2016 493 ``````Program Definition ofe_morCF (F1 F2 : cFunctor) : cFunctor := {| `````` Robbert Krebbers committed Jul 25, 2016 494 `````` cFunctor_car A B := cFunctor_car F1 B A -n> cFunctor_car F2 A B; `````` Ralf Jung committed Mar 02, 2016 495 `````` cFunctor_map A1 A2 B1 B2 fg := `````` Ralf Jung committed Nov 22, 2016 496 `````` ofe_morC_map (cFunctor_map F1 (fg.2, fg.1)) (cFunctor_map F2 fg) `````` Ralf Jung committed Mar 02, 2016 497 ``````|}. `````` Robbert Krebbers committed Mar 07, 2016 498 499 ``````Next Obligation. intros F1 F2 A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *. `````` Ralf Jung committed Nov 22, 2016 500 `````` apply ofe_morC_map_ne; apply cFunctor_ne; split; by apply Hfg. `````` Robbert Krebbers committed Mar 07, 2016 501 ``````Qed. `````` Ralf Jung committed Mar 02, 2016 502 ``````Next Obligation. `````` Robbert Krebbers committed Mar 02, 2016 503 504 `````` intros F1 F2 A B [f ?] ?; simpl. rewrite /= !cFunctor_id. apply (ne_proper f). apply cFunctor_id. `````` Ralf Jung committed Mar 02, 2016 505 506 ``````Qed. Next Obligation. `````` Robbert Krebbers committed Mar 02, 2016 507 508 `````` intros F1 F2 A1 A2 A3 B1 B2 B3 f g f' g' [h ?] ?; simpl in *. rewrite -!cFunctor_compose. do 2 apply (ne_proper _). apply cFunctor_compose. `````` Ralf Jung committed Mar 02, 2016 509 510 ``````Qed. `````` Ralf Jung committed Nov 22, 2016 511 ``````Instance ofe_morCF_contractive F1 F2 : `````` Ralf Jung committed Mar 07, 2016 512 `````` cFunctorContractive F1 → cFunctorContractive F2 → `````` Ralf Jung committed Nov 22, 2016 513 `````` cFunctorContractive (ofe_morCF F1 F2). `````` Ralf Jung committed Mar 07, 2016 514 515 ``````Proof. intros ?? A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *. `````` Ralf Jung committed Nov 22, 2016 516 `````` apply ofe_morC_map_ne; apply cFunctor_contractive=>i ?; split; by apply Hfg. `````` Ralf Jung committed Mar 07, 2016 517 518 ``````Qed. `````` Robbert Krebbers committed May 27, 2016 519 520 ``````(** Sum *) Section sum. `````` Ralf Jung committed Nov 22, 2016 521 `````` Context {A B : ofeT}. `````` Robbert Krebbers committed May 27, 2016 522 523 524 525 526 527 528 `````` Instance sum_dist : Dist (A + B) := λ n, sum_relation (dist n) (dist n). Global Instance inl_ne : Proper (dist n ==> dist n) (@inl A B) := _. Global Instance inr_ne : Proper (dist n ==> dist n) (@inr A B) := _. Global Instance inl_ne_inj : Inj (dist n) (dist n) (@inl A B) := _. Global Instance inr_ne_inj : Inj (dist n) (dist n) (@inr A B) := _. `````` Ralf Jung committed Nov 22, 2016 529 530 531 532 533 534 535 536 537 538 539 540 `````` Definition sum_ofe_mixin : OfeMixin (A + B). Proof. split. - intros x y; split=> Hx. + destruct Hx=> n; constructor; by apply equiv_dist. + destruct (Hx 0); constructor; apply equiv_dist=> n; by apply (inj _). - apply _. - destruct 1; constructor; by apply dist_S. Qed. Canonical Structure sumC : ofeT := OfeT (A + B) sum_ofe_mixin. Program Definition inl_chain (c : chain sumC) (a : A) : chain A := `````` Robbert Krebbers committed May 27, 2016 541 542 `````` {| chain_car n := match c n return _ with inl a' => a' | _ => a end |}. Next Obligation. intros c a n i ?; simpl. by destruct (chain_cauchy c n i). Qed. `````` Ralf Jung committed Nov 22, 2016 543 `````` Program Definition inr_chain (c : chain sumC) (b : B) : chain B := `````` Robbert Krebbers committed May 27, 2016 544 545 546 `````` {| chain_car n := match c n return _ with inr b' => b' | _ => b end |}. Next Obligation. intros c b n i ?; simpl. by destruct (chain_cauchy c n i). Qed. `````` Ralf Jung committed Nov 22, 2016 547 `````` Definition sum_compl `{Cofe A, Cofe B} : Compl sumC := λ c, `````` Robbert Krebbers committed May 27, 2016 548 549 550 551 `````` match c 0 with | inl a => inl (compl (inl_chain c a)) | inr b => inr (compl (inr_chain c b)) end. `````` Ralf Jung committed Nov 22, 2016 552 553 554 555 556 557 558 `````` Global Program Instance sum_cofe `{Cofe A, Cofe B} : Cofe sumC := { compl := sum_compl }. Next Obligation. intros ?? n c; rewrite /compl /sum_compl. feed inversion (chain_cauchy c 0 n); first by auto with lia; constructor. - rewrite (conv_compl n (inl_chain c _)) /=. destruct (c n); naive_solver. - rewrite (conv_compl n (inr_chain c _)) /=. destruct (c n); naive_solver. `````` Robbert Krebbers committed May 27, 2016 559 560 561 562 563 564 565 566 567 568 569 570 571 `````` Qed. Global Instance inl_timeless (x : A) : Timeless x → Timeless (inl x). Proof. inversion_clear 2; constructor; by apply (timeless _). Qed. Global Instance inr_timeless (y : B) : Timeless y → Timeless (inr y). Proof. inversion_clear 2; constructor; by apply (timeless _). Qed. Global Instance sum_discrete_cofe : Discrete A → Discrete B → Discrete sumC. Proof. intros ?? [?|?]; apply _. Qed. End sum. Arguments sumC : clear implicits. Typeclasses Opaque sum_dist. `````` Ralf Jung committed Nov 22, 2016 572 ``````Instance sum_map_ne {A A' B B' : ofeT} n : `````` Robbert Krebbers committed May 27, 2016 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 `````` Proper ((dist n ==> dist n) ==> (dist n ==> dist n) ==> dist n ==> dist n) (@sum_map A A' B B'). Proof. intros f f' Hf g g' Hg ??; destruct 1; constructor; [by apply Hf|by apply Hg]. Qed. Definition sumC_map {A A' B B'} (f : A -n> A') (g : B -n> B') : sumC A B -n> sumC A' B' := CofeMor (sum_map f g). Instance sumC_map_ne {A A' B B'} n : Proper (dist n ==> dist n ==> dist n) (@sumC_map A A' B B'). Proof. intros f f' Hf g g' Hg [?|?]; constructor; [apply Hf|apply Hg]. Qed. Program Definition sumCF (F1 F2 : cFunctor) : cFunctor := {| cFunctor_car A B := sumC (cFunctor_car F1 A B) (cFunctor_car F2 A B); cFunctor_map A1 A2 B1 B2 fg := sumC_map (cFunctor_map F1 fg) (cFunctor_map F2 fg) |}. Next Obligation. intros ?? A1 A2 B1 B2 n ???; by apply sumC_map_ne; apply cFunctor_ne. Qed. Next Obligation. by intros F1 F2 A B [?|?]; rewrite /= !cFunctor_id. Qed. Next Obligation. intros F1 F2 A1 A2 A3 B1 B2 B3 f g f' g' [?|?]; simpl; by rewrite !cFunctor_compose. Qed. Instance sumCF_contractive F1 F2 : cFunctorContractive F1 → cFunctorContractive F2 → cFunctorContractive (sumCF F1 F2). Proof. intros ?? A1 A2 B1 B2 n ???; by apply sumC_map_ne; apply cFunctor_contractive. Qed. `````` Robbert Krebbers committed Nov 16, 2015 606 607 608 ``````(** Discrete cofe *) Section discrete_cofe. Context `{Equiv A, @Equivalence A (≡)}. `````` Robbert Krebbers committed Feb 10, 2016 609 `````` Instance discrete_dist : Dist A := λ n x y, x ≡ y. `````` Ralf Jung committed Nov 22, 2016 610 `````` Definition discrete_ofe_mixin : OfeMixin A. `````` Robbert Krebbers committed Nov 16, 2015 611 612 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 613 614 615 `````` - intros x y; split; [done|intros Hn; apply (Hn 0)]. - done. - done. `````` Ralf Jung committed Nov 22, 2016 616 617 618 619 620 621 622 `````` Qed. Global Program Instance discrete_cofe : Cofe (OfeT A discrete_ofe_mixin) := { compl c := c 0 }. Next Obligation. intros n c. rewrite /compl /=; symmetry; apply (chain_cauchy c 0 n). omega. `````` Robbert Krebbers committed Nov 16, 2015 623 624 625 `````` Qed. End discrete_cofe. `````` Ralf Jung committed Nov 22, 2016 626 627 ``````Notation discreteC A := (OfeT A discrete_ofe_mixin). Notation leibnizC A := (OfeT A (@discrete_ofe_mixin _ equivL _)). `````` Robbert Krebbers committed May 25, 2016 628 629 630 631 632 633 `````` Instance discrete_discrete_cofe `{Equiv A, @Equivalence A (≡)} : Discrete (discreteC A). Proof. by intros x y. Qed. Instance leibnizC_leibniz A : LeibnizEquiv (leibnizC A). Proof. by intros x y. Qed. `````` Robbert Krebbers committed Jan 16, 2016 634 `````` `````` Robbert Krebbers committed Dec 11, 2015 635 636 ``````Canonical Structure natC := leibnizC nat. Canonical Structure boolC := leibnizC bool. `````` Robbert Krebbers committed Nov 19, 2015 637 `````` `````` Robbert Krebbers committed May 25, 2016 638 639 ``````(* Option *) Section option. `````` Ralf Jung committed Nov 22, 2016 640 `````` Context {A : ofeT}. `````` Robbert Krebbers committed May 25, 2016 641 `````` `````` Robbert Krebbers committed May 27, 2016 642 `````` Instance option_dist : Dist (option A) := λ n, option_Forall2 (dist n). `````` Robbert Krebbers committed May 25, 2016 643 `````` Lemma dist_option_Forall2 n mx my : mx ≡{n}≡ my ↔ option_Forall2 (dist n) mx my. `````` Robbert Krebbers committed May 27, 2016 644 `````` Proof. done. Qed. `````` Robbert Krebbers committed May 25, 2016 645 `````` `````` Ralf Jung committed Nov 22, 2016 646 `````` Definition option_ofe_mixin : OfeMixin (option A). `````` Robbert Krebbers committed May 25, 2016 647 648 649 650 651 `````` Proof. split. - intros mx my; split; [by destruct 1; constructor; apply equiv_dist|]. intros Hxy; destruct (Hxy 0); constructor; apply equiv_dist. by intros n; feed inversion (Hxy n). `````` Robbert Krebbers committed May 27, 2016 652 `````` - apply _. `````` Robbert Krebbers committed May 25, 2016 653 654 `````` - destruct 1; constructor; by apply dist_S. Qed. `````` Ralf Jung committed Nov 22, 2016 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 `````` Canonical Structure optionC := OfeT (option A) option_ofe_mixin. Program Definition option_chain (c : chain optionC) (x : A) : chain A := {| chain_car n := from_option id x (c n) |}. Next Obligation. intros c x n i ?; simpl. by destruct (chain_cauchy c n i). Qed. Definition option_compl `{Cofe A} : Compl optionC := λ c, match c 0 with Some x => Some (compl (option_chain c x)) | None => None end. Global Program Instance option_cofe `{Cofe A} : Cofe optionC := { compl := option_compl }. Next Obligation. intros ? n c; rewrite /compl /option_compl. feed inversion (chain_cauchy c 0 n); auto with lia; []. constructor. rewrite (conv_compl n (option_chain c _)) /=. destruct (c n); naive_solver. Qed. `````` Robbert Krebbers committed May 25, 2016 671 672 673 674 675 676 677 678 679 `````` Global Instance option_discrete : Discrete A → Discrete optionC. Proof. destruct 2; constructor; by apply (timeless _). Qed. Global Instance Some_ne : Proper (dist n ==> dist n) (@Some A). Proof. by constructor. Qed. Global Instance is_Some_ne n : Proper (dist n ==> iff) (@is_Some A). Proof. destruct 1; split; eauto. Qed. Global Instance Some_dist_inj : Inj (dist n) (dist n) (@Some A). Proof. by inversion_clear 1. Qed. `````` Robbert Krebbers committed May 28, 2016 680 681 682 `````` Global Instance from_option_ne {B} (R : relation B) (f : A → B) n : Proper (dist n ==> R) f → Proper (R ==> dist n ==> R) (from_option f). Proof. destruct 3; simpl; auto. Qed. `````` Robbert Krebbers committed May 25, 2016 683 684 685 686 687 `````` Global Instance None_timeless : Timeless (@None A). Proof. inversion_clear 1; constructor. Qed. Global Instance Some_timeless x : Timeless x → Timeless (Some x). Proof. by intros ?; inversion_clear 1; constructor; apply timeless. Qed. `````` Robbert Krebbers committed May 27, 2016 688 689 690 691 692 693 694 695 696 697 698 699 700 `````` Lemma dist_None n mx : mx ≡{n}≡ None ↔ mx = None. Proof. split; [by inversion_clear 1|by intros ->]. Qed. Lemma dist_Some_inv_l n mx my x : mx ≡{n}≡ my → mx = Some x → ∃ y, my = Some y ∧ x ≡{n}≡ y. Proof. destruct 1; naive_solver. Qed. Lemma dist_Some_inv_r n mx my y : mx ≡{n}≡ my → my = Some y → ∃ x, mx = Some x ∧ x ≡{n}≡ y. Proof. destruct 1; naive_solver. Qed. Lemma dist_Some_inv_l' n my x : Some x ≡{n}≡ my → ∃ x', Some x' = my ∧ x ≡{n}≡ x'. Proof. intros ?%(dist_Some_inv_l _ _ _ x); naive_solver. Qed. Lemma dist_Some_inv_r' n mx y : mx ≡{n}≡ Some y → ∃ y', mx = Some y' ∧ y ≡{n}≡ y'. Proof. intros ?%(dist_Some_inv_r _ _ _ y); naive_solver. Qed. `````` Robbert Krebbers committed May 25, 2016 701 702 ``````End option. `````` Robbert Krebbers committed May 27, 2016 703 ``````Typeclasses Opaque option_dist. `````` Robbert Krebbers committed May 25, 2016 704 705 ``````Arguments optionC : clear implicits. `````` Ralf Jung committed Nov 22, 2016 706 ``````Instance option_fmap_ne {A B : ofeT} n: `````` Robbert Krebbers committed May 28, 2016 707 708 `````` Proper ((dist n ==> dist n) ==> dist n ==> dist n) (@fmap option _ A B). Proof. intros f f' Hf ?? []; constructor; auto. Qed. `````` Robbert Krebbers committed May 25, 2016 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 ``````Definition optionC_map {A B} (f : A -n> B) : optionC A -n> optionC B := CofeMor (fmap f : optionC A → optionC B). Instance optionC_map_ne A B n : Proper (dist n ==> dist n) (@optionC_map A B). Proof. by intros f f' Hf []; constructor; apply Hf. Qed. Program Definition optionCF (F : cFunctor) : cFunctor := {| cFunctor_car A B := optionC (cFunctor_car F A B); cFunctor_map A1 A2 B1 B2 fg := optionC_map (cFunctor_map F fg) |}. Next Obligation. by intros F A1 A2 B1 B2 n f g Hfg; apply optionC_map_ne, cFunctor_ne. Qed. Next Obligation. intros F A B x. rewrite /= -{2}(option_fmap_id x). apply option_fmap_setoid_ext=>y; apply cFunctor_id. Qed. Next Obligation. intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -option_fmap_compose. apply option_fmap_setoid_ext=>y; apply cFunctor_compose. Qed. Instance optionCF_contractive F : cFunctorContractive F → cFunctorContractive (optionCF F). Proof. by intros ? A1 A2 B1 B2 n f g Hfg; apply optionC_map_ne, cFunctor_contractive. Qed. `````` Robbert Krebbers committed Nov 16, 2015 736 ``````(** Later *) `````` Robbert Krebbers committed Feb 10, 2016 737 ``````Inductive later (A : Type) : Type := Next { later_car : A }. `````` Robbert Krebbers committed Dec 21, 2015 738 ``````Add Printing Constructor later. `````` Robbert Krebbers committed Feb 10, 2016 739 ``````Arguments Next {_} _. `````` Robbert Krebbers committed Nov 16, 2015 740 ``````Arguments later_car {_} _. `````` Robbert Krebbers committed Dec 21, 2015 741 `````` `````` Robbert Krebbers committed Nov 16, 2015 742 ``````Section later. `````` Ralf Jung committed Nov 22, 2016 743 `````` Context {A : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 744 745 `````` Instance later_equiv : Equiv (later A) := λ x y, later_car x ≡ later_car y. Instance later_dist : Dist (later A) := λ n x y, `````` Ralf Jung committed Feb 10, 2016 746 `````` match n with 0 => True | S n => later_car x ≡{n}≡ later_car y end. `````` Ralf Jung committed Nov 22, 2016 747 `````` Definition later_ofe_mixin : OfeMixin (later A). `````` Robbert Krebbers committed Nov 16, 2015 748 749 `````` Proof. split. `````` Ralf Jung committed Apr 24, 2016 750 751 `````` - intros x y; unfold equiv, later_equiv; rewrite !equiv_dist. split. intros Hxy [|n]; [done|apply Hxy]. intros Hxy n; apply (Hxy (S n)). `````` Robbert Krebbers committed Feb 17, 2016 752 `````` - intros [|n]; [by split|split]; unfold dist, later_dist. `````` Robbert Krebbers committed Nov 16, 2015 753 754 `````` + by intros [x]. + by intros [x] [y]. `````` Ralf Jung committed Feb 20, 2016 755 `````` + by intros [x] [y] [z] ??; trans y. `````` Robbert Krebbers committed Feb 17, 2016 756 `````` - intros [|n] [x] [y] ?; [done|]; unfold dist, later_dist; by apply dist_S. `````` Robbert Krebbers committed Nov 16, 2015 757 `````` Qed. `````` Ralf Jung committed Nov 22, 2016 758 759 760 761 762 763 764 765 766 767 768 `````` Canonical Structure laterC : ofeT := OfeT (later A) later_ofe_mixin. Program Definition later_chain (c : chain laterC) : chain A := {| chain_car n := later_car (c (S n)) |}. Next Obligation. intros c n i ?; apply (chain_cauchy c (S n)); lia. Qed. Global Program Instance later_cofe `{Cofe A} : Cofe laterC := { compl c := Next (compl (later_chain c)) }. Next Obligation. intros ? [|n] c; [done|by apply (conv_compl n (later_chain c))]. Qed. `````` Robbert Krebbers committed Feb 10, 2016 769 770 `````` Global Instance Next_contractive : Contractive (@Next A). Proof. intros [|n] x y Hxy; [done|]; apply Hxy; lia. Qed. `````` Robbert Krebbers committed Feb 11, 2016 771 `````` Global Instance Later_inj n : Inj (dist n) (dist (S n)) (@Next A). `````` Robbert Krebbers committed Jan 16, 2016 772 `````` Proof. by intros x y. Qed. `````` Robbert Krebbers committed Nov 16, 2015 773 ``````End later. `````` Robbert Krebbers committed Jan 14, 2016 774 775 776 777 `````` Arguments laterC : clear implicits. Definition later_map {A B} (f : A → B) (x : later A) : later B := `````` Robbert Krebbers committed Feb 10, 2016 778 `````` Next (f (later_car x)). `````` Ralf Jung committed Nov 22, 2016 779 ``````Instance later_map_ne {A B : ofeT} (f : A → B) n : `````` Robbert Krebbers committed Jan 14, 2016 780 781 782 783 784 785 786 787 `````` Proper (dist (pred n) ==> dist (pred n)) f → Proper (dist n ==> dist n) (later_map f) | 0. Proof. destruct n as [|n]; intros Hf [x] [y] ?; do 2 red; simpl; auto. Qed. Lemma later_map_id {A} (x : later A) : later_map id x = x. Proof. by destruct x. Qed. Lemma later_map_compose {A B C} (f : A → B) (g : B → C) (x : later A) : later_map (g ∘ f) x = later_map g (later_map f x). Proof. by destruct x. Qed. `````` Ralf Jung committed Nov 22, 2016 788 ``````Lemma later_map_ext {A B : ofeT} (f g : A → B) x : `````` Robbert Krebbers committed Mar 02, 2016 789 790 `````` (∀ x, f x ≡ g x) → later_map f x ≡ later_map g x. Proof. destruct x; intros Hf; apply Hf. Qed. `````` Robbert Krebbers committed Jan 14, 2016 791 792 ``````Definition laterC_map {A B} (f : A -n> B) : laterC A -n> laterC B := CofeMor (later_map f). `````` Ralf Jung committed Nov 22, 2016 793 ``````Instance laterC_map_contractive (A B : ofeT) : Contractive (@laterC_map A B). `````` Robbert Krebbers committed Feb 10, 2016 794 ``````Proof. intros [|n] f g Hf n'; [done|]; apply Hf; lia. Qed. `````` Robbert Krebbers committed Mar 02, 2016 795 `````` ``````