ofe.v 41.4 KB
 Robbert Krebbers committed Mar 10, 2016 1 ``````From iris.algebra Require Export base. `````` Ralf Jung committed Jan 05, 2017 2 ``````Set Default Proof Using "Type". `````` Robbert Krebbers committed Nov 11, 2015 3 `````` `````` Ralf Jung committed Nov 22, 2016 4 ``````(** This files defines (a shallow embedding of) the category of OFEs: `````` Ralf Jung committed Feb 16, 2016 5 6 7 8 9 10 11 12 `````` 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 13 14 ``````(** Unbundeled version *) Class Dist A := dist : nat → relation A. `````` Robbert Krebbers committed Nov 12, 2015 15 ``````Instance: Params (@dist) 3. `````` Ralf Jung committed Feb 10, 2016 16 17 ``````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 18 ``````Hint Extern 0 (_ ≡{_}≡ _) => reflexivity. `````` Ralf Jung committed Feb 10, 2016 19 ``````Hint Extern 0 (_ ≡{_}≡ _) => symmetry; assumption. `````` Robbert Krebbers committed Jan 13, 2016 20 21 22 `````` Tactic Notation "cofe_subst" ident(x) := repeat match goal with `````` Robbert Krebbers committed Feb 17, 2016 23 `````` | _ => progress simplify_eq/= `````` Robbert Krebbers committed Jan 13, 2016 24 25 26 27 `````` | 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 28 `````` repeat match goal with `````` Robbert Krebbers committed Feb 17, 2016 29 `````` | _ => progress simplify_eq/= `````` Robbert Krebbers committed Dec 21, 2015 30 31 `````` | 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 32 `````` end. `````` Robbert Krebbers committed Nov 11, 2015 33 `````` `````` Ralf Jung committed Nov 22, 2016 34 ``````Record OfeMixin A `{Equiv A, Dist A} := { `````` Ralf Jung committed Feb 10, 2016 35 `````` mixin_equiv_dist x y : x ≡ y ↔ ∀ n, x ≡{n}≡ y; `````` Robbert Krebbers committed Jan 14, 2016 36 `````` mixin_dist_equivalence n : Equivalence (dist n); `````` Ralf Jung committed Nov 22, 2016 37 `````` mixin_dist_S n x y : x ≡{S n}≡ y → x ≡{n}≡ y `````` Robbert Krebbers committed Nov 11, 2015 38 39 40 ``````}. (** Bundeled version *) `````` Ralf Jung committed Nov 22, 2016 41 42 43 44 45 ``````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 46 `````` _ : Type `````` Robbert Krebbers committed Nov 11, 2015 47 ``````}. `````` Ralf Jung committed Nov 22, 2016 48 49 50 51 52 53 54 55 56 ``````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 57 58 `````` (** Lifting properties from the mixin *) `````` Ralf Jung committed Nov 22, 2016 59 60 ``````Section ofe_mixin. Context {A : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 61 `````` Implicit Types x y : A. `````` Ralf Jung committed Feb 10, 2016 62 `````` Lemma equiv_dist x y : x ≡ y ↔ ∀ n, x ≡{n}≡ y. `````` Ralf Jung committed Nov 22, 2016 63 `````` Proof. apply (mixin_equiv_dist _ (ofe_mixin A)). Qed. `````` Robbert Krebbers committed Jan 14, 2016 64 `````` Global Instance dist_equivalence n : Equivalence (@dist A _ n). `````` Ralf Jung committed Nov 22, 2016 65 `````` Proof. apply (mixin_dist_equivalence _ (ofe_mixin A)). Qed. `````` Ralf Jung committed Feb 10, 2016 66 `````` Lemma dist_S n x y : x ≡{S n}≡ y → x ≡{n}≡ y. `````` Ralf Jung committed Nov 22, 2016 67 68 `````` Proof. apply (mixin_dist_S _ (ofe_mixin A)). Qed. End ofe_mixin. `````` Robbert Krebbers committed Jan 14, 2016 69 `````` `````` Robbert Krebbers committed May 28, 2016 70 71 ``````Hint Extern 1 (_ ≡{_}≡ _) => apply equiv_dist; assumption. `````` Ralf Jung committed Dec 21, 2016 72 ``````(** Discrete OFEs and Timeless elements *) `````` Ralf Jung committed Mar 15, 2016 73 ``````(* TODO: On paper, We called these "discrete elements". I think that makes `````` Ralf Jung committed Mar 07, 2016 74 `````` more sense. *) `````` Robbert Krebbers committed Jan 22, 2017 75 76 77 78 ``````Class Timeless {A : ofeT} (x : A) := timeless y : x ≡{0}≡ y → x ≡ y. Arguments timeless {_} _ {_} _ _. Hint Mode Timeless + ! : typeclass_instances. `````` Ralf Jung committed Nov 22, 2016 79 80 81 82 83 84 85 86 87 88 ``````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 {_} _ _ _ _. `````` Robbert Krebbers committed Dec 05, 2016 89 90 91 92 93 ``````Program Definition chain_map {A B : ofeT} (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. `````` Ralf Jung committed Nov 22, 2016 94 95 96 97 98 99 ``````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 100 `````` `````` Jacques-Henri Jourdan committed Jan 05, 2017 101 102 103 104 105 ``````Lemma compl_chain_map `{Cofe A, Cofe B} (f : A → B) c `(∀ n : nat, Proper (dist n ==> dist n) f) : compl (chain_map f c) ≡ f (compl c). Proof. apply equiv_dist=>n. by rewrite !conv_compl. Qed. `````` Robbert Krebbers committed Nov 11, 2015 106 107 ``````(** General properties *) Section cofe. `````` Ralf Jung committed Nov 22, 2016 108 `````` Context {A : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 109 `````` Implicit Types x y : A. `````` Robbert Krebbers committed Nov 11, 2015 110 111 112 `````` Global Instance cofe_equivalence : Equivalence ((≡) : relation A). Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 113 114 `````` - by intros x; rewrite equiv_dist. - by intros x y; rewrite !equiv_dist. `````` Ralf Jung committed Feb 20, 2016 115 `````` - by intros x y z; rewrite !equiv_dist; intros; trans y. `````` Robbert Krebbers committed Nov 11, 2015 116 `````` Qed. `````` Robbert Krebbers committed Jan 14, 2016 117 `````` Global Instance dist_ne n : Proper (dist n ==> dist n ==> iff) (@dist A _ n). `````` Robbert Krebbers committed Nov 11, 2015 118 119 `````` Proof. intros x1 x2 ? y1 y2 ?; split; intros. `````` Ralf Jung committed Feb 20, 2016 120 121 `````` - by trans x1; [|trans y1]. - by trans x2; [|trans y2]. `````` Robbert Krebbers committed Nov 11, 2015 122 `````` Qed. `````` Robbert Krebbers committed Jan 14, 2016 123 `````` Global Instance dist_proper n : Proper ((≡) ==> (≡) ==> iff) (@dist A _ n). `````` Robbert Krebbers committed Nov 11, 2015 124 `````` Proof. `````` Robbert Krebbers committed Jan 13, 2016 125 `````` by move => x1 x2 /equiv_dist Hx y1 y2 /equiv_dist Hy; rewrite (Hx n) (Hy n). `````` Robbert Krebbers committed Nov 11, 2015 126 127 128 `````` 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 129 `````` Lemma dist_le n n' x y : x ≡{n}≡ y → n' ≤ n → x ≡{n'}≡ y. `````` Robbert Krebbers committed Nov 11, 2015 130 `````` Proof. induction 2; eauto using dist_S. Qed. `````` Ralf Jung committed Feb 29, 2016 131 132 `````` 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 133 `````` Instance ne_proper {B : ofeT} (f : A → B) `````` Robbert Krebbers committed Nov 11, 2015 134 135 `````` `{!∀ 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 136 `````` Instance ne_proper_2 {B C : ofeT} (f : A → B → C) `````` Robbert Krebbers committed Nov 11, 2015 137 138 139 140 `````` `{!∀ 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 141 `````` by intros x1 x2 Hx y1 y2 Hy n; rewrite (Hx n) (Hy n). `````` Robbert Krebbers committed Nov 11, 2015 142 `````` Qed. `````` Robbert Krebbers committed Feb 24, 2016 143 `````` `````` Ralf Jung committed Nov 22, 2016 144 `````` Lemma conv_compl' `{Cofe A} n (c : chain A) : compl c ≡{n}≡ c (S n). `````` Ralf Jung committed Feb 29, 2016 145 146 147 148 `````` Proof. transitivity (c n); first by apply conv_compl. symmetry. apply chain_cauchy. omega. Qed. `````` Robbert Krebbers committed Feb 24, 2016 149 150 `````` Lemma timeless_iff n (x : A) `{!Timeless x} y : x ≡ y ↔ x ≡{n}≡ y. Proof. `````` Robbert Krebbers committed May 28, 2016 151 `````` split; intros; auto. apply (timeless _), dist_le with n; auto with lia. `````` Robbert Krebbers committed Feb 24, 2016 152 `````` Qed. `````` Robbert Krebbers committed Nov 11, 2015 153 154 ``````End cofe. `````` Robbert Krebbers committed Dec 02, 2016 155 ``````(** Contractive functions *) `````` Robbert Krebbers committed Dec 05, 2016 156 157 158 159 160 161 162 163 ``````Definition dist_later {A : ofeT} (n : nat) (x y : A) : Prop := match n with 0 => True | S n => x ≡{n}≡ y end. Arguments dist_later _ !_ _ _ /. Global Instance dist_later_equivalence A n : Equivalence (@dist_later A n). Proof. destruct n as [|n]. by split. apply dist_equivalence. Qed. Notation Contractive f := (∀ n, Proper (dist_later n ==> dist n) f). `````` Robbert Krebbers committed Dec 02, 2016 164 `````` `````` Ralf Jung committed Nov 22, 2016 165 ``````Instance const_contractive {A B : ofeT} (x : A) : Contractive (@const A B x). `````` Robbert Krebbers committed Mar 06, 2016 166 167 ``````Proof. by intros n y1 y2. Qed. `````` Robbert Krebbers committed Dec 02, 2016 168 ``````Section contractive. `````` Ralf Jung committed Jan 25, 2017 169 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Dec 02, 2016 170 171 172 173 `````` Context {A B : ofeT} (f : A → B) `{!Contractive f}. Implicit Types x y : A. Lemma contractive_0 x y : f x ≡{0}≡ f y. `````` Robbert Krebbers committed Dec 05, 2016 174 `````` Proof. by apply (_ : Contractive f). Qed. `````` Robbert Krebbers committed Dec 02, 2016 175 `````` Lemma contractive_S n x y : x ≡{n}≡ y → f x ≡{S n}≡ f y. `````` Robbert Krebbers committed Dec 05, 2016 176 `````` Proof. intros. by apply (_ : Contractive f). Qed. `````` Robbert Krebbers committed Dec 02, 2016 177 178 179 180 181 182 183 `````` 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 Dec 05, 2016 184 185 186 187 188 189 190 ``````Ltac f_contractive := match goal with | |- ?f _ ≡{_}≡ ?f _ => apply (_ : Proper (dist_later _ ==> _) f) | |- ?f _ _ ≡{_}≡ ?f _ _ => apply (_ : Proper (dist_later _ ==> _ ==> _) f) | |- ?f _ _ ≡{_}≡ ?f _ _ => apply (_ : Proper (_ ==> dist_later _ ==> _) f) end; try match goal with `````` Jacques-Henri Jourdan committed Dec 26, 2016 191 192 `````` | |- @dist_later ?A ?n ?x ?y => destruct n as [|n]; [done|change (@dist A _ n x y)] `````` Robbert Krebbers committed Dec 05, 2016 193 194 195 196 197 198 `````` end; try reflexivity. Ltac solve_contractive := preprocess_solve_proper; solve [repeat (first [f_contractive|f_equiv]; try eassumption)]. `````` Robbert Krebbers committed Nov 22, 2015 199 `````` `````` Robbert Krebbers committed Nov 11, 2015 200 ``````(** Fixpoint *) `````` Ralf Jung committed Nov 22, 2016 201 ``````Program Definition fixpoint_chain {A : ofeT} `{Inhabited A} (f : A → A) `````` Robbert Krebbers committed Feb 10, 2016 202 `````` `{!Contractive f} : chain A := {| chain_car i := Nat.iter (S i) f inhabitant |}. `````` Robbert Krebbers committed Nov 11, 2015 203 ``````Next Obligation. `````` Robbert Krebbers committed Mar 06, 2016 204 `````` intros A ? f ? n. `````` Robbert Krebbers committed Dec 05, 2016 205 `````` induction n as [|n IH]=> -[|i] //= ?; try omega. `````` Robbert Krebbers committed Feb 17, 2016 206 207 `````` - apply (contractive_0 f). - apply (contractive_S f), IH; auto with omega. `````` Robbert Krebbers committed Nov 11, 2015 208 ``````Qed. `````` Robbert Krebbers committed Mar 18, 2016 209 `````` `````` Ralf Jung committed Nov 22, 2016 210 ``````Program Definition fixpoint_def `{Cofe A, Inhabited A} (f : A → A) `````` Robbert Krebbers committed Nov 17, 2015 211 `````` `{!Contractive f} : A := compl (fixpoint_chain f). `````` Ralf Jung committed Jan 11, 2017 212 213 214 ``````Definition fixpoint_aux : seal (@fixpoint_def). by eexists. Qed. Definition fixpoint {A AC AiH} f {Hf} := unseal fixpoint_aux A AC AiH f Hf. Definition fixpoint_eq : @fixpoint = @fixpoint_def := seal_eq fixpoint_aux. `````` Robbert Krebbers committed Nov 11, 2015 215 216 `````` Section fixpoint. `````` Ralf Jung committed Nov 22, 2016 217 `````` Context `{Cofe A, Inhabited A} (f : A → A) `{!Contractive f}. `````` Robbert Krebbers committed Aug 21, 2016 218 `````` `````` Robbert Krebbers committed Nov 17, 2015 219 `````` Lemma fixpoint_unfold : fixpoint f ≡ f (fixpoint f). `````` Robbert Krebbers committed Nov 11, 2015 220 `````` Proof. `````` Robbert Krebbers committed Mar 18, 2016 221 222 `````` apply equiv_dist=>n. rewrite fixpoint_eq /fixpoint_def (conv_compl n (fixpoint_chain f)) //. `````` Robbert Krebbers committed Feb 12, 2016 223 `````` induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S. `````` Robbert Krebbers committed Nov 11, 2015 224 `````` Qed. `````` Robbert Krebbers committed Aug 21, 2016 225 226 227 `````` Lemma fixpoint_unique (x : A) : x ≡ f x → x ≡ fixpoint f. Proof. `````` Robbert Krebbers committed Aug 22, 2016 228 229 230 `````` 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 231 232 `````` Qed. `````` Robbert Krebbers committed Nov 17, 2015 233 `````` Lemma fixpoint_ne (g : A → A) `{!Contractive g} n : `````` Ralf Jung committed Feb 10, 2016 234 `````` (∀ z, f z ≡{n}≡ g z) → fixpoint f ≡{n}≡ fixpoint g. `````` Robbert Krebbers committed Nov 11, 2015 235 `````` Proof. `````` Robbert Krebbers committed Mar 18, 2016 236 `````` intros Hfg. rewrite fixpoint_eq /fixpoint_def `````` Robbert Krebbers committed Feb 18, 2016 237 `````` (conv_compl n (fixpoint_chain f)) (conv_compl n (fixpoint_chain g)) /=. `````` Robbert Krebbers committed Feb 10, 2016 238 239 `````` 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 240 `````` Qed. `````` Robbert Krebbers committed Nov 17, 2015 241 242 `````` Lemma fixpoint_proper (g : A → A) `{!Contractive g} : (∀ x, f x ≡ g x) → fixpoint f ≡ fixpoint g. `````` Robbert Krebbers committed Nov 11, 2015 243 `````` Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_ne. Qed. `````` Jacques-Henri Jourdan committed Dec 23, 2016 244 245 `````` Lemma fixpoint_ind (P : A → Prop) : `````` Jacques-Henri Jourdan committed Dec 23, 2016 246 `````` Proper ((≡) ==> impl) P → `````` Jacques-Henri Jourdan committed Dec 23, 2016 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 `````` (∃ x, P x) → (∀ x, P x → P (f x)) → (∀ (c : chain A), (∀ n, P (c n)) → P (compl c)) → P (fixpoint f). Proof. intros ? [x Hx] Hincr Hlim. set (chcar i := Nat.iter (S i) f x). assert (Hcauch : ∀ n i : nat, n ≤ i → chcar i ≡{n}≡ chcar n). { intros n. induction n as [|n IH]=> -[|i] //= ?; try omega. - apply (contractive_0 f). - apply (contractive_S f), IH; auto with omega. } set (fp2 := compl {| chain_cauchy := Hcauch |}). rewrite -(fixpoint_unique fp2); first by apply Hlim; induction n; apply Hincr. apply equiv_dist=>n. rewrite /fp2 (conv_compl n) /= /chcar. induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S. Qed. `````` Robbert Krebbers committed Nov 11, 2015 262 263 ``````End fixpoint. `````` Ralf Jung committed Jan 25, 2017 264 265 266 ``````(** Fixpoint of f when f^k is contractive. **) Definition fixpointK `{Cofe A, Inhabited A} k (f : A → A) `{!Contractive (Nat.iter k f)} := fixpoint (Nat.iter k f). `````` Ralf Jung committed Jan 25, 2017 267 `````` `````` Ralf Jung committed Jan 25, 2017 268 ``````Section fixpointK. `````` Ralf Jung committed Jan 25, 2017 269 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Jan 25, 2017 270 271 `````` Context `{Cofe A, Inhabited A} (f : A → A) (k : nat). Context `{f_contractive : !Contractive (Nat.iter k f)}. `````` Ralf Jung committed Jan 25, 2017 272 `````` (* TODO: Can we get rid of this assumption, derive it from contractivity? *) `````` Robbert Krebbers committed Jan 25, 2017 273 274 275 276 `````` Context `{f_ne : !∀ n, Proper (dist n ==> dist n) f}. Let f_proper : Proper ((≡) ==> (≡)) f := ne_proper f. Existing Instance f_proper. `````` Ralf Jung committed Jan 25, 2017 277 `````` `````` Ralf Jung committed Jan 25, 2017 278 `````` Lemma fixpointK_unfold : fixpointK k f ≡ f (fixpointK k f). `````` Ralf Jung committed Jan 25, 2017 279 `````` Proof. `````` Robbert Krebbers committed Jan 25, 2017 280 281 `````` symmetry. rewrite /fixpointK. apply fixpoint_unique. by rewrite -Nat_iter_S_r Nat_iter_S -fixpoint_unfold. `````` Ralf Jung committed Jan 25, 2017 282 283 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 284 `````` Lemma fixpointK_unique (x : A) : x ≡ f x → x ≡ fixpointK k f. `````` Ralf Jung committed Jan 25, 2017 285 `````` Proof. `````` Robbert Krebbers committed Jan 25, 2017 286 287 `````` intros Hf. apply fixpoint_unique. clear f_contractive. induction k as [|k' IH]=> //=. by rewrite -IH. `````` Ralf Jung committed Jan 25, 2017 288 289 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 290 `````` Section fixpointK_ne. `````` Robbert Krebbers committed Jan 25, 2017 291 292 `````` Context (g : A → A) `{g_contractive : !Contractive (Nat.iter k g)}. Context {g_ne : ∀ n, Proper (dist n ==> dist n) g}. `````` Ralf Jung committed Jan 25, 2017 293 `````` `````` Ralf Jung committed Jan 25, 2017 294 `````` Lemma fixpointK_ne n : (∀ z, f z ≡{n}≡ g z) → fixpointK k f ≡{n}≡ fixpointK k g. `````` Ralf Jung committed Jan 25, 2017 295 `````` Proof. `````` Robbert Krebbers committed Jan 25, 2017 296 297 298 `````` rewrite /fixpointK=> Hfg /=. apply fixpoint_ne=> z. clear f_contractive g_contractive. induction k as [|k' IH]=> //=. by rewrite IH Hfg. `````` Ralf Jung committed Jan 25, 2017 299 300 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 301 302 303 304 `````` Lemma fixpointK_proper : (∀ z, f z ≡ g z) → fixpointK k f ≡ fixpointK k g. Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpointK_ne. Qed. End fixpointK_ne. End fixpointK. `````` Ralf Jung committed Jan 25, 2017 305 `````` `````` Robbert Krebbers committed Dec 05, 2016 306 ``````(** Mutual fixpoints *) `````` Ralf Jung committed Jan 25, 2017 307 ``````Section fixpointAB. `````` 308 309 `````` Local Unset Default Proof Using. `````` Robbert Krebbers committed Dec 05, 2016 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 `````` Context `{Cofe A, Cofe B, !Inhabited A, !Inhabited B}. Context (fA : A → B → A). Context (fB : A → B → B). Context `{∀ n, Proper (dist_later n ==> dist n ==> dist n) fA}. Context `{∀ n, Proper (dist_later n ==> dist_later n ==> dist n) fB}. Local Definition fixpoint_AB (x : A) : B := fixpoint (fB x). Local Instance fixpoint_AB_contractive : Contractive fixpoint_AB. Proof. intros n x x' Hx; rewrite /fixpoint_AB. apply fixpoint_ne=> y. by f_contractive. Qed. Local Definition fixpoint_AA (x : A) : A := fA x (fixpoint_AB x). Local Instance fixpoint_AA_contractive : Contractive fixpoint_AA. Proof. solve_contractive. Qed. Definition fixpoint_A : A := fixpoint fixpoint_AA. Definition fixpoint_B : B := fixpoint_AB fixpoint_A. Lemma fixpoint_A_unfold : fA fixpoint_A fixpoint_B ≡ fixpoint_A. Proof. by rewrite {2}/fixpoint_A (fixpoint_unfold _). Qed. Lemma fixpoint_B_unfold : fB fixpoint_A fixpoint_B ≡ fixpoint_B. Proof. by rewrite {2}/fixpoint_B /fixpoint_AB (fixpoint_unfold _). Qed. Instance: Proper ((≡) ==> (≡) ==> (≡)) fA. Proof. apply ne_proper_2=> n x x' ? y y' ?. f_contractive; auto using dist_S. Qed. Instance: Proper ((≡) ==> (≡) ==> (≡)) fB. Proof. apply ne_proper_2=> n x x' ? y y' ?. f_contractive; auto using dist_S. Qed. Lemma fixpoint_A_unique p q : fA p q ≡ p → fB p q ≡ q → p ≡ fixpoint_A. Proof. intros HfA HfB. rewrite -HfA. apply fixpoint_unique. rewrite /fixpoint_AA. f_equiv=> //. apply fixpoint_unique. by rewrite HfA HfB. Qed. Lemma fixpoint_B_unique p q : fA p q ≡ p → fB p q ≡ q → q ≡ fixpoint_B. Proof. intros. apply fixpoint_unique. by rewrite -fixpoint_A_unique. Qed. `````` Ralf Jung committed Jan 25, 2017 351 ``````End fixpointAB. `````` Robbert Krebbers committed Dec 05, 2016 352 `````` `````` Ralf Jung committed Jan 25, 2017 353 ``````Section fixpointAB_ne. `````` Robbert Krebbers committed Dec 05, 2016 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 `````` Context `{Cofe A, Cofe B, !Inhabited A, !Inhabited B}. Context (fA fA' : A → B → A). Context (fB fB' : A → B → B). Context `{∀ n, Proper (dist_later n ==> dist n ==> dist n) fA}. Context `{∀ n, Proper (dist_later n ==> dist n ==> dist n) fA'}. Context `{∀ n, Proper (dist_later n ==> dist_later n ==> dist n) fB}. Context `{∀ n, Proper (dist_later n ==> dist_later n ==> dist n) fB'}. Lemma fixpoint_A_ne n : (∀ x y, fA x y ≡{n}≡ fA' x y) → (∀ x y, fB x y ≡{n}≡ fB' x y) → fixpoint_A fA fB ≡{n}≡ fixpoint_A fA' fB'. Proof. intros HfA HfB. apply fixpoint_ne=> z. rewrite /fixpoint_AA /fixpoint_AB HfA. f_equiv. by apply fixpoint_ne. Qed. Lemma fixpoint_B_ne n : (∀ x y, fA x y ≡{n}≡ fA' x y) → (∀ x y, fB x y ≡{n}≡ fB' x y) → fixpoint_B fA fB ≡{n}≡ fixpoint_B fA' fB'. Proof. intros HfA HfB. apply fixpoint_ne=> z. rewrite HfB. f_contractive. apply fixpoint_A_ne; auto using dist_S. Qed. Lemma fixpoint_A_proper : (∀ x y, fA x y ≡ fA' x y) → (∀ x y, fB x y ≡ fB' x y) → fixpoint_A fA fB ≡ fixpoint_A fA' fB'. Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_A_ne. Qed. Lemma fixpoint_B_proper : (∀ x y, fA x y ≡ fA' x y) → (∀ x y, fB x y ≡ fB' x y) → fixpoint_B fA fB ≡ fixpoint_B fA' fB'. Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_B_ne. Qed. `````` Ralf Jung committed Jan 25, 2017 385 ``````End fixpointAB_ne. `````` Robbert Krebbers committed Dec 05, 2016 386 `````` `````` Robbert Krebbers committed Jul 25, 2016 387 ``````(** Function space *) `````` Ralf Jung committed Nov 22, 2016 388 ``````(* We make [ofe_fun] a definition so that we can register it as a canonical `````` Robbert Krebbers committed Aug 05, 2016 389 ``````structure. *) `````` Ralf Jung committed Nov 22, 2016 390 ``````Definition ofe_fun (A : Type) (B : ofeT) := A → B. `````` Robbert Krebbers committed Jul 25, 2016 391 `````` `````` Ralf Jung committed Nov 22, 2016 392 393 394 395 396 ``````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 397 398 399 400 401 402 403 404 405 406 `````` 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 407 `````` Canonical Structure ofe_funC := OfeT (ofe_fun A B) ofe_fun_ofe_mixin. `````` Robbert Krebbers committed Jul 25, 2016 408 `````` `````` Ralf Jung committed Nov 22, 2016 409 410 411 412 413 414 415 416 417 `````` 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 418 ``````Notation "A -c> B" := `````` Ralf Jung committed Nov 22, 2016 419 420 `````` (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 421 422 `````` Inhabited (A -c> B) := populate (λ _, inhabitant). `````` Robbert Krebbers committed Jul 25, 2016 423 ``````(** Non-expansive function space *) `````` Ralf Jung committed Nov 22, 2016 424 425 426 ``````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 427 428 ``````}. Arguments CofeMor {_ _} _ {_}. `````` Ralf Jung committed Nov 22, 2016 429 430 ``````Add Printing Constructor ofe_mor. Existing Instance ofe_mor_ne. `````` Robbert Krebbers committed Nov 11, 2015 431 `````` `````` Robbert Krebbers committed Jun 17, 2016 432 433 434 435 ``````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 436 437 438 439 440 441 442 ``````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 443 444 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 445 `````` - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|]. `````` Robbert Krebbers committed Feb 18, 2016 446 `````` intros Hfg k; apply equiv_dist=> n; apply Hfg. `````` Robbert Krebbers committed Feb 17, 2016 447 `````` - intros n; split. `````` Robbert Krebbers committed Jan 14, 2016 448 449 `````` + by intros f x. + by intros f g ? x. `````` Ralf Jung committed Feb 20, 2016 450 `````` + by intros f g h ?? x; trans (g x). `````` Robbert Krebbers committed Feb 17, 2016 451 `````` - by intros n f g ? x; apply dist_S. `````` Robbert Krebbers committed Jan 14, 2016 452 `````` Qed. `````` Ralf Jung committed Nov 22, 2016 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 `````` 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 470 `````` `````` Ralf Jung committed Nov 22, 2016 471 472 `````` 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 473 `````` Proof. intros f g Hfg x y Hx; rewrite Hx; apply Hfg. Qed. `````` Ralf Jung committed Nov 22, 2016 474 475 476 `````` 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 477 `````` Proof. done. Qed. `````` Ralf Jung committed Nov 22, 2016 478 ``````End ofe_mor. `````` Robbert Krebbers committed Jan 14, 2016 479 `````` `````` Ralf Jung committed Nov 22, 2016 480 ``````Arguments ofe_morC : clear implicits. `````` Robbert Krebbers committed Jul 25, 2016 481 ``````Notation "A -n> B" := `````` Ralf Jung committed Nov 22, 2016 482 483 `````` (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 484 `````` Inhabited (A -n> B) := populate (λne _, inhabitant). `````` Robbert Krebbers committed Nov 11, 2015 485 `````` `````` Ralf Jung committed Mar 17, 2016 486 ``````(** Identity and composition and constant function *) `````` Robbert Krebbers committed Nov 11, 2015 487 488 ``````Definition cid {A} : A -n> A := CofeMor id. Instance: Params (@cid) 1. `````` Ralf Jung committed Nov 22, 2016 489 ``````Definition cconst {A B : ofeT} (x : B) : A -n> B := CofeMor (const x). `````` Ralf Jung committed Mar 17, 2016 490 ``````Instance: Params (@cconst) 2. `````` Robbert Krebbers committed Mar 02, 2016 491 `````` `````` Robbert Krebbers committed Nov 11, 2015 492 493 494 495 496 ``````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 497 `````` f1 ≡{n}≡ f2 → g1 ≡{n}≡ g2 → f1 ◎ g1 ≡{n}≡ f2 ◎ g2. `````` Robbert Krebbers committed Jan 13, 2016 498 ``````Proof. by intros Hf Hg x; rewrite /= (Hg x) (Hf (g2 x)). Qed. `````` Robbert Krebbers committed Nov 11, 2015 499 `````` `````` Ralf Jung committed Mar 02, 2016 500 ``````(* Function space maps *) `````` Ralf Jung committed Nov 22, 2016 501 ``````Definition ofe_mor_map {A A' B B'} (f : A' -n> A) (g : B -n> B') `````` Ralf Jung committed Mar 02, 2016 502 `````` (h : A -n> B) : A' -n> B' := g ◎ h ◎ f. `````` Ralf Jung committed Nov 22, 2016 503 504 ``````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 505 ``````Proof. intros ??? ??? ???. by repeat apply ccompose_ne. Qed. `````` Ralf Jung committed Mar 02, 2016 506 `````` `````` Ralf Jung committed Nov 22, 2016 507 508 509 510 ``````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 511 ``````Proof. `````` Ralf Jung committed Nov 22, 2016 512 `````` intros f f' Hf g g' Hg ?. rewrite /= /ofe_mor_map. `````` Robbert Krebbers committed Mar 02, 2016 513 `````` by repeat apply ccompose_ne. `````` Ralf Jung committed Mar 02, 2016 514 515 ``````Qed. `````` Robbert Krebbers committed Nov 11, 2015 516 ``````(** unit *) `````` Robbert Krebbers committed Jan 14, 2016 517 518 ``````Section unit. Instance unit_dist : Dist unit := λ _ _ _, True. `````` Ralf Jung committed Nov 22, 2016 519 `````` Definition unit_ofe_mixin : OfeMixin unit. `````` Robbert Krebbers committed Jan 14, 2016 520 `````` Proof. by repeat split; try exists 0. Qed. `````` Ralf Jung committed Nov 22, 2016 521 `````` Canonical Structure unitC : ofeT := OfeT unit unit_ofe_mixin. `````` Robbert Krebbers committed Nov 28, 2016 522 `````` `````` Ralf Jung committed Nov 22, 2016 523 524 `````` Global Program Instance unit_cofe : Cofe unitC := { compl x := () }. Next Obligation. by repeat split; try exists 0. Qed. `````` Robbert Krebbers committed Nov 28, 2016 525 526 `````` Global Instance unit_discrete_cofe : Discrete unitC. `````` Robbert Krebbers committed Jan 31, 2016 527 `````` Proof. done. Qed. `````` Robbert Krebbers committed Jan 14, 2016 528 ``````End unit. `````` Robbert Krebbers committed Nov 11, 2015 529 530 `````` (** Product *) `````` Robbert Krebbers committed Jan 14, 2016 531 ``````Section product. `````` Ralf Jung committed Nov 22, 2016 532 `````` Context {A B : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 533 534 535 536 537 538 `````` 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 539 `````` Definition prod_ofe_mixin : OfeMixin (A * B). `````` Robbert Krebbers committed Jan 14, 2016 540 541 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 542 `````` - intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation. `````` Robbert Krebbers committed Jan 14, 2016 543 `````` rewrite !equiv_dist; naive_solver. `````` Robbert Krebbers committed Feb 17, 2016 544 545 `````` - apply _. - by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S. `````` Robbert Krebbers committed Jan 14, 2016 546 `````` Qed. `````` Ralf Jung committed Nov 22, 2016 547 548 549 550 551 552 553 554 555 `````` 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 556 557 558 `````` 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 559 560 `````` Global Instance prod_discrete_cofe : Discrete A → Discrete B → Discrete prodC. Proof. intros ?? [??]; apply _. Qed. `````` Robbert Krebbers committed Jan 14, 2016 561 562 563 564 565 ``````End product. Arguments prodC : clear implicits. Typeclasses Opaque prod_dist. `````` Ralf Jung committed Nov 22, 2016 566 ``````Instance prod_map_ne {A A' B B' : ofeT} n : `````` Robbert Krebbers committed Nov 11, 2015 567 568 569 570 571 572 573 574 575 `````` 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 576 577 ``````(** Functors *) Structure cFunctor := CFunctor { `````` Ralf Jung committed Nov 22, 2016 578 `````` cFunctor_car : ofeT → ofeT → ofeT; `````` Robbert Krebbers committed Mar 02, 2016 579 580 `````` 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 581 582 `````` cFunctor_ne {A1 A2 B1 B2} n : Proper (dist n ==> dist n) (@cFunctor_map A1 A2 B1 B2); `````` Ralf Jung committed Nov 22, 2016 583 `````` cFunctor_id {A B : ofeT} (x : cFunctor_car A B) : `````` Robbert Krebbers committed Mar 02, 2016 584 585 586 587 588 `````` 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 589 ``````Existing Instance cFunctor_ne. `````` Robbert Krebbers committed Mar 02, 2016 590 591 ``````Instance: Params (@cFunctor_map) 5. `````` Ralf Jung committed Mar 07, 2016 592 593 594 ``````Delimit Scope cFunctor_scope with CF. Bind Scope cFunctor_scope with cFunctor. `````` Ralf Jung committed Mar 07, 2016 595 596 597 ``````Class cFunctorContractive (F : cFunctor) := cFunctor_contractive A1 A2 B1 B2 :> Contractive (@cFunctor_map F A1 A2 B1 B2). `````` Ralf Jung committed Nov 22, 2016 598 ``````Definition cFunctor_diag (F: cFunctor) (A: ofeT) : ofeT := cFunctor_car F A A. `````` Robbert Krebbers committed Mar 02, 2016 599 600 ``````Coercion cFunctor_diag : cFunctor >-> Funclass. `````` Ralf Jung committed Nov 22, 2016 601 ``````Program Definition constCF (B : ofeT) : cFunctor := `````` Robbert Krebbers committed Mar 02, 2016 602 603 `````` {| cFunctor_car A1 A2 := B; cFunctor_map A1 A2 B1 B2 f := cid |}. Solve Obligations with done. `````` Ralf Jung committed Jan 06, 2017 604 ``````Coercion constCF : ofeT >-> cFunctor. `````` Robbert Krebbers committed Mar 02, 2016 605 `````` `````` Ralf Jung committed Mar 07, 2016 606 ``````Instance constCF_contractive B : cFunctorContractive (constCF B). `````` Robbert Krebbers committed Mar 07, 2016 607 ``````Proof. rewrite /cFunctorContractive; apply _. Qed. `````` Ralf Jung committed Mar 07, 2016 608 609 610 611 `````` Program Definition idCF : cFunctor := {| cFunctor_car A1 A2 := A2; cFunctor_map A1 A2 B1 B2 f := f.2 |}. Solve Obligations with done. `````` Ralf Jung committed Jan 06, 2017 612 ``````Notation "∙" := idCF : cFunctor_scope. `````` Ralf Jung committed Mar 07, 2016 613 `````` `````` Robbert Krebbers committed Mar 02, 2016 614 615 616 617 618 ``````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 619 620 621 ``````Next Obligation. intros ?? A1 A2 B1 B2 n ???; by apply prodC_map_ne; apply cFunctor_ne. Qed. `````` Robbert Krebbers committed Mar 02, 2016 622 623 624 625 626 ``````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 Jan 06, 2017 627 ``````Notation "F1 * F2" := (prodCF F1%CF F2%CF) : cFunctor_scope. `````` Robbert Krebbers committed Mar 02, 2016 628 `````` `````` Ralf Jung committed Mar 07, 2016 629 630 631 632 633 634 635 636 ``````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 637 ``````Instance compose_ne {A} {B B' : ofeT} (f : B -n> B') n : `````` Jacques-Henri Jourdan committed Oct 05, 2016 638 639 640 `````` 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 641 ``````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 642 `````` @CofeMor (_ -c> _) (_ -c> _) (compose f) _. `````` Ralf Jung committed Nov 22, 2016 643 644 ``````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 645 646 ``````Proof. intros f f' Hf g x. apply Hf. Qed. `````` Ralf Jung committed Nov 22, 2016 647 648 649 ``````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 650 651 ``````|}. Next Obligation. `````` Ralf Jung committed Nov 22, 2016 652 `````` intros ?? A1 A2 B1 B2 n ???; by apply ofe_funC_map_ne; apply cFunctor_ne. `````` Jacques-Henri Jourdan committed Oct 05, 2016 653 654 655 656 657 658 ``````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 Jan 06, 2017 659 ``````Notation "T -c> F" := (ofe_funCF T%type F%CF) : cFunctor_scope. `````` Jacques-Henri Jourdan committed Oct 05, 2016 660 `````` `````` Ralf Jung committed Nov 22, 2016 661 662 ``````Instance ofe_funCF_contractive (T : Type) (F : cFunctor) : cFunctorContractive F → cFunctorContractive (ofe_funCF T F). `````` Jacques-Henri Jourdan committed Oct 05, 2016 663 664 ``````Proof. intros ?? A1 A2 B1 B2 n ???; `````` Ralf Jung committed Nov 22, 2016 665 `````` by apply ofe_funC_map_ne; apply cFunctor_contractive. `````` Jacques-Henri Jourdan committed Oct 05, 2016 666 667 ``````Qed. `````` Ralf Jung committed Nov 22, 2016 668 ``````Program Definition ofe_morCF (F1 F2 : cFunctor) : cFunctor := {| `````` Robbert Krebbers committed Jul 25, 2016 669 `````` cFunctor_car A B := cFunctor_car F1 B A -n> cFunctor_car F2 A B; `````` Ralf Jung committed Mar 02, 2016 670 `````` cFunctor_map A1 A2 B1 B2 fg := `````` Ralf Jung committed Nov 22, 2016 671 `````` ofe_morC_map (cFunctor_map F1 (fg.2, fg.1)) (cFunctor_map F2 fg) `````` Ralf Jung committed Mar 02, 2016 672 ``````|}. `````` Robbert Krebbers committed Mar 07, 2016 673 674 ``````Next Obligation. intros F1 F2 A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *. `````` Ralf Jung committed Nov 22, 2016 675 `````` apply ofe_morC_map_ne; apply cFunctor_ne; split; by apply Hfg. `````` Robbert Krebbers committed Mar 07, 2016 676 ``````Qed. `````` Ralf Jung committed Mar 02, 2016 677 ``````Next Obligation. `````` Robbert Krebbers committed Mar 02, 2016 678 679 `````` intros F1 F2 A B [f ?] ?; simpl. rewrite /= !cFunctor_id. apply (ne_proper f). apply cFunctor_id. `````` Ralf Jung committed Mar 02, 2016 680 681 ``````Qed. Next Obligation. `````` Robbert Krebbers committed Mar 02, 2016 682 683 `````` 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 684 ``````Qed. `````` Ralf Jung committed Jan 06, 2017 685 ``````Notation "F1 -n> F2" := (ofe_morCF F1%CF F2%CF) : cFunctor_scope. `````` Ralf Jung committed Mar 02, 2016 686 `````` `````` Ralf Jung committed Nov 22, 2016 687 ``````Instance ofe_morCF_contractive F1 F2 : `````` Ralf Jung committed Mar 07, 2016 688 `````` cFunctorContractive F1 → cFunctorContractive F2 → `````` Ralf Jung committed Nov 22, 2016 689 `````` cFunctorContractive (ofe_morCF F1 F2). `````` Ralf Jung committed Mar 07, 2016 690 691 ``````Proof. intros ?? A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *. `````` Robbert Krebbers committed Dec 05, 2016 692 `````` apply ofe_morC_map_ne; apply cFunctor_contractive; destruct n, Hfg; by split. `````` Ralf Jung committed Mar 07, 2016 693 694 ``````Qed. `````` Robbert Krebbers committed May 27, 2016 695 696 ``````(** Sum *) Section sum. `````` Ralf Jung committed Nov 22, 2016 697 `````` Context {A B : ofeT}. `````` Robbert Krebbers committed May 27, 2016 698 699 700 701 702 703 704 `````` 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 705 706 707 708 709 710 711 712 713 714 715 716 `````` 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 717 718 `````` {| 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 719 `````` Program Definition inr_chain (c : chain sumC) (b : B) : chain B := `````` Robbert Krebbers committed May 27, 2016 720 721 722 `````` {| 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 723 `````` Definition sum_compl `{Cofe A, Cofe B} : Compl sumC := λ c, `````` Robbert Krebbers committed May 27, 2016 724 725 726 727 `````` 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 728 729 730 731 732 733 734 `````` 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 735 736 737 738 739 740 741 742 743 744 745 746 747 `````` 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 748 ``````Instance sum_map_ne {A A' B B' : ofeT} n : `````` Robbert Krebbers committed May 27, 2016 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 `````` 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. `````` Ralf Jung committed Jan 06, 2017 773 ``````Notation "F1 + F2" := (sumCF F1%CF F2%CF) : cFunctor_scope. `````` Robbert Krebbers committed May 27, 2016 774 775 776 777 778 779 780 781 782 `````` 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 783 784 785 ``````(** Discrete cofe *) Section discrete_cofe. Context `{Equiv A, @Equivalence A (≡)}. `````` Robbert Krebbers committed Dec 05, 2016 786 `````` `````` Robbert Krebbers committed Feb 10, 2016 787 `````` Instance discrete_dist : Dist A := λ n x y, x ≡ y. `````` Ralf Jung committed Nov 22, 2016 788 `````` Definition discrete_ofe_mixin : OfeMixin A. `````` Ralf Jung committed Jan 25, 2017 789 `````` Proof using Type*. `````` Robbert Krebbers committed Nov 16, 2015 790 `````` split. `````` Robbert Krebbers committed Feb 17, 2016 791 792 793 `````` - intros x y; split; [done|intros Hn; apply (Hn 0)]. - done. - done. `````` Ralf Jung committed Nov 22, 2016 794 `````` Qed. ``````