ofe.v 41.6 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*". `````` Ralf Jung committed Jan 25, 2017 270 `````` Context `{Cofe A, Inhabited A} (f : A → A) k `{!Contractive (Nat.iter k f)}. `````` Ralf Jung committed Jan 25, 2017 271 272 273 `````` (* TODO: Can we get rid of this assumption, derive it from contractivity? *) Context `{!∀ n, Proper (dist n ==> dist n) f}. `````` Ralf Jung committed Jan 25, 2017 274 `````` Lemma fixpointK_unfold : fixpointK k f ≡ f (fixpointK k f). `````` Ralf Jung committed Jan 25, 2017 275 276 `````` Proof. apply equiv_dist=>n. `````` Ralf Jung committed Jan 25, 2017 277 `````` rewrite /fixpointK fixpoint_eq /fixpoint_def (conv_compl n (fixpoint_chain _)) //. `````` Ralf Jung committed Jan 25, 2017 278 `````` induction n as [|n IH]; simpl. `````` Ralf Jung committed Jan 25, 2017 279 280 `````` - rewrite -Nat_iter_S Nat_iter_S_r. eapply contractive_0; first done. - rewrite -[f _]Nat_iter_S Nat_iter_S_r. eapply contractive_S; first done. eauto. `````` Ralf Jung committed Jan 25, 2017 281 282 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 283 `````` Lemma fixpointK_unique (x : A) : x ≡ f x → x ≡ fixpointK k f. `````` Ralf Jung committed Jan 25, 2017 284 `````` Proof. `````` Ralf Jung committed Jan 25, 2017 285 286 287 288 `````` intros Hf. apply fixpoint_unique, equiv_dist=>n. (* Forward reasoning is so annoying... *) assert (x ≡{n}≡ f x) by by apply equiv_dist. clear Contractive0. induction k; first done. by rewrite {1}Hf {1}IHn0. `````` Ralf Jung committed Jan 25, 2017 289 290 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 291 292 `````` Section fixpointK_ne. Context (g : A → A) `{!Contractive (Nat.iter k g), !∀ 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. `````` Ralf Jung committed Jan 25, 2017 296 297 `````` rewrite /fixpointK=>Hne /=. apply fixpoint_ne=>? /=. clear Contractive0 Contractive1. induction k; first by auto. simpl. rewrite IHn0. apply Hne. `````` Ralf Jung committed Jan 25, 2017 298 299 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 300 301 302 303 `````` 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 304 `````` `````` Robbert Krebbers committed Dec 05, 2016 305 ``````(** Mutual fixpoints *) `````` Ralf Jung committed Jan 25, 2017 306 ``````Section fixpointAB. `````` 307 308 `````` Local Unset Default Proof Using. `````` Robbert Krebbers committed Dec 05, 2016 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 `````` 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 350 ``````End fixpointAB. `````` Robbert Krebbers committed Dec 05, 2016 351 `````` `````` Ralf Jung committed Jan 25, 2017 352 ``````Section fixpointAB_ne. `````` Robbert Krebbers committed Dec 05, 2016 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 `````` 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 384 ``````End fixpointAB_ne. `````` Robbert Krebbers committed Dec 05, 2016 385 `````` `````` Robbert Krebbers committed Jul 25, 2016 386 ``````(** Function space *) `````` Ralf Jung committed Nov 22, 2016 387 ``````(* We make [ofe_fun] a definition so that we can register it as a canonical `````` Robbert Krebbers committed Aug 05, 2016 388 ``````structure. *) `````` Ralf Jung committed Nov 22, 2016 389 ``````Definition ofe_fun (A : Type) (B : ofeT) := A → B. `````` Robbert Krebbers committed Jul 25, 2016 390 `````` `````` Ralf Jung committed Nov 22, 2016 391 392 393 394 395 ``````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 396 397 398 399 400 401 402 403 404 405 `````` 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 406 `````` Canonical Structure ofe_funC := OfeT (ofe_fun A B) ofe_fun_ofe_mixin. `````` Robbert Krebbers committed Jul 25, 2016 407 `````` `````` Ralf Jung committed Nov 22, 2016 408 409 410 411 412 413 414 415 416 `````` 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 417 ``````Notation "A -c> B" := `````` Ralf Jung committed Nov 22, 2016 418 419 `````` (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 420 421 `````` Inhabited (A -c> B) := populate (λ _, inhabitant). `````` Robbert Krebbers committed Jul 25, 2016 422 ``````(** Non-expansive function space *) `````` Ralf Jung committed Nov 22, 2016 423 424 425 ``````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 426 427 ``````}. Arguments CofeMor {_ _} _ {_}. `````` Ralf Jung committed Nov 22, 2016 428 429 ``````Add Printing Constructor ofe_mor. Existing Instance ofe_mor_ne. `````` Robbert Krebbers committed Nov 11, 2015 430 `````` `````` Robbert Krebbers committed Jun 17, 2016 431 432 433 434 ``````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 435 436 437 438 439 440 441 ``````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 442 443 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 444 `````` - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|]. `````` Robbert Krebbers committed Feb 18, 2016 445 `````` intros Hfg k; apply equiv_dist=> n; apply Hfg. `````` Robbert Krebbers committed Feb 17, 2016 446 `````` - intros n; split. `````` Robbert Krebbers committed Jan 14, 2016 447 448 `````` + by intros f x. + by intros f g ? x. `````` Ralf Jung committed Feb 20, 2016 449 `````` + by intros f g h ?? x; trans (g x). `````` Robbert Krebbers committed Feb 17, 2016 450 `````` - by intros n f g ? x; apply dist_S. `````` Robbert Krebbers committed Jan 14, 2016 451 `````` Qed. `````` Ralf Jung committed Nov 22, 2016 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 `````` 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 469 `````` `````` Ralf Jung committed Nov 22, 2016 470 471 `````` 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 472 `````` Proof. intros f g Hfg x y Hx; rewrite Hx; apply Hfg. Qed. `````` Ralf Jung committed Nov 22, 2016 473 474 475 `````` 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 476 `````` Proof. done. Qed. `````` Ralf Jung committed Nov 22, 2016 477 ``````End ofe_mor. `````` Robbert Krebbers committed Jan 14, 2016 478 `````` `````` Ralf Jung committed Nov 22, 2016 479 ``````Arguments ofe_morC : clear implicits. `````` Robbert Krebbers committed Jul 25, 2016 480 ``````Notation "A -n> B" := `````` Ralf Jung committed Nov 22, 2016 481 482 `````` (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 483 `````` Inhabited (A -n> B) := populate (λne _, inhabitant). `````` Robbert Krebbers committed Nov 11, 2015 484 `````` `````` Ralf Jung committed Mar 17, 2016 485 ``````(** Identity and composition and constant function *) `````` Robbert Krebbers committed Nov 11, 2015 486 487 ``````Definition cid {A} : A -n> A := CofeMor id. Instance: Params (@cid) 1. `````` Ralf Jung committed Nov 22, 2016 488 ``````Definition cconst {A B : ofeT} (x : B) : A -n> B := CofeMor (const x). `````` Ralf Jung committed Mar 17, 2016 489 ``````Instance: Params (@cconst) 2. `````` Robbert Krebbers committed Mar 02, 2016 490 `````` `````` Robbert Krebbers committed Nov 11, 2015 491 492 493 494 495 ``````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 496 `````` f1 ≡{n}≡ f2 → g1 ≡{n}≡ g2 → f1 ◎ g1 ≡{n}≡ f2 ◎ g2. `````` Robbert Krebbers committed Jan 13, 2016 497 ``````Proof. by intros Hf Hg x; rewrite /= (Hg x) (Hf (g2 x)). Qed. `````` Robbert Krebbers committed Nov 11, 2015 498 `````` `````` Ralf Jung committed Mar 02, 2016 499 ``````(* Function space maps *) `````` Ralf Jung committed Nov 22, 2016 500 ``````Definition ofe_mor_map {A A' B B'} (f : A' -n> A) (g : B -n> B') `````` Ralf Jung committed Mar 02, 2016 501 `````` (h : A -n> B) : A' -n> B' := g ◎ h ◎ f. `````` Ralf Jung committed Nov 22, 2016 502 503 ``````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 504 ``````Proof. intros ??? ??? ???. by repeat apply ccompose_ne. Qed. `````` Ralf Jung committed Mar 02, 2016 505 `````` `````` Ralf Jung committed Nov 22, 2016 506 507 508 509 ``````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 510 ``````Proof. `````` Ralf Jung committed Nov 22, 2016 511 `````` intros f f' Hf g g' Hg ?. rewrite /= /ofe_mor_map. `````` Robbert Krebbers committed Mar 02, 2016 512 `````` by repeat apply ccompose_ne. `````` Ralf Jung committed Mar 02, 2016 513 514 ``````Qed. `````` Robbert Krebbers committed Nov 11, 2015 515 ``````(** unit *) `````` Robbert Krebbers committed Jan 14, 2016 516 517 ``````Section unit. Instance unit_dist : Dist unit := λ _ _ _, True. `````` Ralf Jung committed Nov 22, 2016 518 `````` Definition unit_ofe_mixin : OfeMixin unit. `````` Robbert Krebbers committed Jan 14, 2016 519 `````` Proof. by repeat split; try exists 0. Qed. `````` Ralf Jung committed Nov 22, 2016 520 `````` Canonical Structure unitC : ofeT := OfeT unit unit_ofe_mixin. `````` Robbert Krebbers committed Nov 28, 2016 521 `````` `````` Ralf Jung committed Nov 22, 2016 522 523 `````` Global Program Instance unit_cofe : Cofe unitC := { compl x := () }. Next Obligation. by repeat split; try exists 0. Qed. `````` Robbert Krebbers committed Nov 28, 2016 524 525 `````` Global Instance unit_discrete_cofe : Discrete unitC. `````` Robbert Krebbers committed Jan 31, 2016 526 `````` Proof. done. Qed. `````` Robbert Krebbers committed Jan 14, 2016 527 ``````End unit. `````` Robbert Krebbers committed Nov 11, 2015 528 529 `````` (** Product *) `````` Robbert Krebbers committed Jan 14, 2016 530 ``````Section product. `````` Ralf Jung committed Nov 22, 2016 531 `````` Context {A B : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 532 533 534 535 536 537 `````` 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 538 `````` Definition prod_ofe_mixin : OfeMixin (A * B). `````` Robbert Krebbers committed Jan 14, 2016 539 540 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 541 `````` - intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation. `````` Robbert Krebbers committed Jan 14, 2016 542 `````` rewrite !equiv_dist; naive_solver. `````` Robbert Krebbers committed Feb 17, 2016 543 544 `````` - apply _. - by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S. `````` Robbert Krebbers committed Jan 14, 2016 545 `````` Qed. `````` Ralf Jung committed Nov 22, 2016 546 547 548 549 550 551 552 553 554 `````` 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 555 556 557 `````` 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 558 559 `````` Global Instance prod_discrete_cofe : Discrete A → Discrete B → Discrete prodC. Proof. intros ?? [??]; apply _. Qed. `````` Robbert Krebbers committed Jan 14, 2016 560 561 562 563 564 ``````End product. Arguments prodC : clear implicits. Typeclasses Opaque prod_dist. `````` Ralf Jung committed Nov 22, 2016 565 ``````Instance prod_map_ne {A A' B B' : ofeT} n : `````` Robbert Krebbers committed Nov 11, 2015 566 567 568 569 570 571 572 573 574 `````` 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 575 576 ``````(** Functors *) Structure cFunctor := CFunctor { `````` Ralf Jung committed Nov 22, 2016 577 `````` cFunctor_car : ofeT → ofeT → ofeT; `````` Robbert Krebbers committed Mar 02, 2016 578 579 `````` 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 580 581 `````` cFunctor_ne {A1 A2 B1 B2} n : Proper (dist n ==> dist n) (@cFunctor_map A1 A2 B1 B2); `````` Ralf Jung committed Nov 22, 2016 582 `````` cFunctor_id {A B : ofeT} (x : cFunctor_car A B) : `````` Robbert Krebbers committed Mar 02, 2016 583 584 585 586 587 `````` 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 588 ``````Existing Instance cFunctor_ne. `````` Robbert Krebbers committed Mar 02, 2016 589 590 ``````Instance: Params (@cFunctor_map) 5. `````` Ralf Jung committed Mar 07, 2016 591 592 593 ``````Delimit Scope cFunctor_scope with CF. Bind Scope cFunctor_scope with cFunctor. `````` Ralf Jung committed Mar 07, 2016 594 595 596 ``````Class cFunctorContractive (F : cFunctor) := cFunctor_contractive A1 A2 B1 B2 :> Contractive (@cFunctor_map F A1 A2 B1 B2). `````` Ralf Jung committed Nov 22, 2016 597 ``````Definition cFunctor_diag (F: cFunctor) (A: ofeT) : ofeT := cFunctor_car F A A. `````` Robbert Krebbers committed Mar 02, 2016 598 599 ``````Coercion cFunctor_diag : cFunctor >-> Funclass. `````` Ralf Jung committed Nov 22, 2016 600 ``````Program Definition constCF (B : ofeT) : cFunctor := `````` Robbert Krebbers committed Mar 02, 2016 601 602 `````` {| cFunctor_car A1 A2 := B; cFunctor_map A1 A2 B1 B2 f := cid |}. Solve Obligations with done. `````` Ralf Jung committed Jan 06, 2017 603 ``````Coercion constCF : ofeT >-> cFunctor. `````` Robbert Krebbers committed Mar 02, 2016 604 `````` `````` Ralf Jung committed Mar 07, 2016 605 ``````Instance constCF_contractive B : cFunctorContractive (constCF B). `````` Robbert Krebbers committed Mar 07, 2016 606 ``````Proof. rewrite /cFunctorContractive; apply _. Qed. `````` Ralf Jung committed Mar 07, 2016 607 608 609 610 `````` 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 611 ``````Notation "∙" := idCF : cFunctor_scope. `````` Ralf Jung committed Mar 07, 2016 612 `````` `````` Robbert Krebbers committed Mar 02, 2016 613 614 615 616 617 ``````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 618 619 620 ``````Next Obligation. intros ?? A1 A2 B1 B2 n ???; by apply prodC_map_ne; apply cFunctor_ne. Qed. `````` Robbert Krebbers committed Mar 02, 2016 621 622 623 624 625 ``````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 626 ``````Notation "F1 * F2" := (prodCF F1%CF F2%CF) : cFunctor_scope. `````` Robbert Krebbers committed Mar 02, 2016 627 `````` `````` Ralf Jung committed Mar 07, 2016 628 629 630 631 632 633 634 635 ``````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 636 ``````Instance compose_ne {A} {B B' : ofeT} (f : B -n> B') n : `````` Jacques-Henri Jourdan committed Oct 05, 2016 637 638 639 `````` 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 640 ``````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 641 `````` @CofeMor (_ -c> _) (_ -c> _) (compose f) _. `````` Ralf Jung committed Nov 22, 2016 642 643 ``````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 644 645 ``````Proof. intros f f' Hf g x. apply Hf. Qed. `````` Ralf Jung committed Nov 22, 2016 646 647 648 ``````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 649 650 ``````|}. Next Obligation. `````` Ralf Jung committed Nov 22, 2016 651 `````` intros ?? A1 A2 B1 B2 n ???; by apply ofe_funC_map_ne; apply cFunctor_ne. `````` Jacques-Henri Jourdan committed Oct 05, 2016 652 653 654 655 656 657 ``````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 658 ``````Notation "T -c> F" := (ofe_funCF T%type F%CF) : cFunctor_scope. `````` Jacques-Henri Jourdan committed Oct 05, 2016 659 `````` `````` Ralf Jung committed Nov 22, 2016 660 661 ``````Instance ofe_funCF_contractive (T : Type) (F : cFunctor) : cFunctorContractive F → cFunctorContractive (ofe_funCF T F). `````` Jacques-Henri Jourdan committed Oct 05, 2016 662 663 ``````Proof. intros ?? A1 A2 B1 B2 n ???; `````` Ralf Jung committed Nov 22, 2016 664 `````` by apply ofe_funC_map_ne; apply cFunctor_contractive. `````` Jacques-Henri Jourdan committed Oct 05, 2016 665 666 ``````Qed. `````` Ralf Jung committed Nov 22, 2016 667 ``````Program Definition ofe_morCF (F1 F2 : cFunctor) : cFunctor := {| `````` Robbert Krebbers committed Jul 25, 2016 668 `````` cFunctor_car A B := cFunctor_car F1 B A -n> cFunctor_car F2 A B; `````` Ralf Jung committed Mar 02, 2016 669 `````` cFunctor_map A1 A2 B1 B2 fg := `````` Ralf Jung committed Nov 22, 2016 670 `````` ofe_morC_map (cFunctor_map F1 (fg.2, fg.1)) (cFunctor_map F2 fg) `````` Ralf Jung committed Mar 02, 2016 671 ``````|}. `````` Robbert Krebbers committed Mar 07, 2016 672 673 ``````Next Obligation. intros F1 F2 A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *. `````` Ralf Jung committed Nov 22, 2016 674 `````` apply ofe_morC_map_ne; apply cFunctor_ne; split; by apply Hfg. `````` Robbert Krebbers committed Mar 07, 2016 675 ``````Qed. `````` Ralf Jung committed Mar 02, 2016 676 ``````Next Obligation. `````` Robbert Krebbers committed Mar 02, 2016 677 678 `````` intros F1 F2 A B [f ?] ?; simpl. rewrite /= !cFunctor_id. apply (ne_proper f). apply cFunctor_id. `````` Ralf Jung committed Mar 02, 2016 679 680 ``````Qed. Next Obligation. `````` Robbert Krebbers committed Mar 02, 2016 681 682 `````` 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 683 ``````Qed. `````` Ralf Jung committed Jan 06, 2017 684 ``````Notation "F1 -n> F2" := (ofe_morCF F1%CF F2%CF) : cFunctor_scope. `````` Ralf Jung committed Mar 02, 2016 685 `````` `````` Ralf Jung committed Nov 22, 2016 686 ``````Instance ofe_morCF_contractive F1 F2 : `````` Ralf Jung committed Mar 07, 2016 687 `````` cFunctorContractive F1 → cFunctorContractive F2 → `````` Ralf Jung committed Nov 22, 2016 688 `````` cFunctorContractive (ofe_morCF F1 F2). `````` Ralf Jung committed Mar 07, 2016 689 690 ``````Proof. intros ?? A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *. `````` Robbert Krebbers committed Dec 05, 2016 691 `````` apply ofe_morC_map_ne; apply cFunctor_contractive; destruct n, Hfg; by split. `````` Ralf Jung committed Mar 07, 2016 692 693 ``````Qed. `````` Robbert Krebbers committed May 27, 2016 694 695 ``````(** Sum *) Section sum. `````` Ralf Jung committed Nov 22, 2016 696 `````` Context {A B : ofeT}. `````` Robbert Krebbers committed May 27, 2016 697 698 699 700 701 702 703 `````` 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 704 705 706 707 708 709 710 711 712 713 714 715 `````` 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 716 717 `````` {| 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 718 `````` Program Definition inr_chain (c : chain sumC) (b : B) : chain B := `````` Robbert Krebbers committed May 27, 2016 719 720 721 `````` {| 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 722 `````` Definition sum_compl `{Cofe A, Cofe B} : Compl sumC := λ c, `````` Robbert Krebbers committed May 27, 2016 723 724 725 726 `````` 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 727 728 729 730 731 732 733 `````` 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 734 735 736 737 738 739 740 741 742 743 744 745 746 `````` 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 747 ``````Instance sum_map_ne {A A' B B' : ofeT} n : `````` Robbert Krebbers committed May 27, 2016 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 `````` 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 772 ``````Notation "F1 + F2" := (sumCF F1%CF F2%CF) : cFunctor_scope. `````` Robbert Krebbers committed May 27, 2016 773 774 775 776 777