ofe.v 43.9 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. `````` Ralf Jung committed Jan 27, 2017 20 21 ``````Notation NonExpansive f := (∀ n, Proper (dist n ==> dist n) f). Notation NonExpansive2 f := (∀ n, Proper (dist n ==> dist n ==> dist n) f). `````` Robbert Krebbers committed Jan 13, 2016 22 `````` `````` Robbert Krebbers committed Feb 09, 2017 23 ``````Tactic Notation "ofe_subst" ident(x) := `````` Robbert Krebbers committed Jan 13, 2016 24 `````` repeat match goal with `````` Robbert Krebbers committed Feb 17, 2016 25 `````` | _ => progress simplify_eq/= `````` Robbert Krebbers committed Jan 13, 2016 26 27 28 `````` | 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. `````` Robbert Krebbers committed Feb 09, 2017 29 ``````Tactic Notation "ofe_subst" := `````` Robbert Krebbers committed Nov 17, 2015 30 `````` repeat match goal with `````` Robbert Krebbers committed Feb 17, 2016 31 `````` | _ => progress simplify_eq/= `````` Robbert Krebbers committed Dec 21, 2015 32 33 `````` | 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 34 `````` end. `````` Robbert Krebbers committed Nov 11, 2015 35 `````` `````` Ralf Jung committed Nov 22, 2016 36 ``````Record OfeMixin A `{Equiv A, Dist A} := { `````` Ralf Jung committed Feb 10, 2016 37 `````` mixin_equiv_dist x y : x ≡ y ↔ ∀ n, x ≡{n}≡ y; `````` Robbert Krebbers committed Jan 14, 2016 38 `````` mixin_dist_equivalence n : Equivalence (dist n); `````` Ralf Jung committed Nov 22, 2016 39 `````` mixin_dist_S n x y : x ≡{S n}≡ y → x ≡{n}≡ y `````` Robbert Krebbers committed Nov 11, 2015 40 41 42 ``````}. (** Bundeled version *) `````` Ralf Jung committed Nov 22, 2016 43 44 45 46 47 ``````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 48 `````` _ : Type `````` Robbert Krebbers committed Nov 11, 2015 49 ``````}. `````` Ralf Jung committed Nov 22, 2016 50 51 52 53 54 55 56 57 58 ``````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 59 `````` `````` Robbert Krebbers committed Feb 09, 2017 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 ``````(** When declaring instances of subclasses of OFE (like CMRAs and unital CMRAs) we need Coq to *infer* the canonical OFE instance of a given type and take the mixin out of it. This makes sure we do not use two different OFE instances in different places (see for example the constructors [CMRAT] and [UCMRAT] in the file [cmra.v].) In order to infer the OFE instance, we use the definition [ofe_mixin_of'] which is inspired by the [clone] trick in ssreflect. It works as follows, when type checking [@ofe_mixin_of' A ?Ac id] Coq faces a unification problem: ofe_car ?Ac ~ A which will resolve [?Ac] to the canonical OFE instance corresponding to [A]. The definition [@ofe_mixin_of' A ?Ac id] will then provide the corresponding mixin. Note that type checking of [ofe_mixin_of' A id] will fail when [A] does not have a canonical OFE instance. The notation [ofe_mixin_of A] that we define on top of [ofe_mixin_of' A id] hides the [id] and normalizes the mixin to head normal form. The latter is to ensure that we do not end up with redundant canonical projections to the mixin, i.e. them all being of the shape [ofe_mixin_of' A id]. *) Definition ofe_mixin_of' A {Ac : ofeT} (f : Ac → A) : OfeMixin Ac := ofe_mixin Ac. Notation ofe_mixin_of A := ltac:(let H := eval hnf in (ofe_mixin_of' A id) in exact H) (only parsing). `````` Robbert Krebbers committed Jan 14, 2016 85 ``````(** Lifting properties from the mixin *) `````` Ralf Jung committed Nov 22, 2016 86 87 ``````Section ofe_mixin. Context {A : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 88 `````` Implicit Types x y : A. `````` Ralf Jung committed Feb 10, 2016 89 `````` Lemma equiv_dist x y : x ≡ y ↔ ∀ n, x ≡{n}≡ y. `````` Ralf Jung committed Nov 22, 2016 90 `````` Proof. apply (mixin_equiv_dist _ (ofe_mixin A)). Qed. `````` Robbert Krebbers committed Jan 14, 2016 91 `````` Global Instance dist_equivalence n : Equivalence (@dist A _ n). `````` Ralf Jung committed Nov 22, 2016 92 `````` Proof. apply (mixin_dist_equivalence _ (ofe_mixin A)). Qed. `````` Ralf Jung committed Feb 10, 2016 93 `````` Lemma dist_S n x y : x ≡{S n}≡ y → x ≡{n}≡ y. `````` Ralf Jung committed Nov 22, 2016 94 95 `````` Proof. apply (mixin_dist_S _ (ofe_mixin A)). Qed. End ofe_mixin. `````` Robbert Krebbers committed Jan 14, 2016 96 `````` `````` Robbert Krebbers committed May 28, 2016 97 98 ``````Hint Extern 1 (_ ≡{_}≡ _) => apply equiv_dist; assumption. `````` Ralf Jung committed Dec 21, 2016 99 ``````(** Discrete OFEs and Timeless elements *) `````` Ralf Jung committed Mar 15, 2016 100 ``````(* TODO: On paper, We called these "discrete elements". I think that makes `````` Ralf Jung committed Mar 07, 2016 101 `````` more sense. *) `````` Robbert Krebbers committed Jan 22, 2017 102 103 104 105 ``````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 106 107 108 109 110 111 112 113 114 115 ``````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 116 ``````Program Definition chain_map {A B : ofeT} (f : A → B) `````` Ralf Jung committed Jan 27, 2017 117 `````` `{!NonExpansive f} (c : chain A) : chain B := `````` Robbert Krebbers committed Dec 05, 2016 118 119 120 `````` {| 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 121 122 123 124 125 126 ``````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 127 `````` `````` Robbert Krebbers committed Feb 09, 2017 128 ``````Lemma compl_chain_map `{Cofe A, Cofe B} (f : A → B) c `(NonExpansive f) : `````` Jacques-Henri Jourdan committed Jan 05, 2017 129 130 131 `````` compl (chain_map f c) ≡ f (compl c). Proof. apply equiv_dist=>n. by rewrite !conv_compl. Qed. `````` Robbert Krebbers committed Nov 11, 2015 132 ``````(** General properties *) `````` Robbert Krebbers committed Feb 09, 2017 133 ``````Section ofe. `````` Ralf Jung committed Nov 22, 2016 134 `````` Context {A : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 135 `````` Implicit Types x y : A. `````` Robbert Krebbers committed Feb 09, 2017 136 `````` Global Instance ofe_equivalence : Equivalence ((≡) : relation A). `````` Robbert Krebbers committed Nov 11, 2015 137 138 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 139 140 `````` - by intros x; rewrite equiv_dist. - by intros x y; rewrite !equiv_dist. `````` Ralf Jung committed Feb 20, 2016 141 `````` - by intros x y z; rewrite !equiv_dist; intros; trans y. `````` Robbert Krebbers committed Nov 11, 2015 142 `````` Qed. `````` Robbert Krebbers committed Jan 14, 2016 143 `````` Global Instance dist_ne n : Proper (dist n ==> dist n ==> iff) (@dist A _ n). `````` Robbert Krebbers committed Nov 11, 2015 144 145 `````` Proof. intros x1 x2 ? y1 y2 ?; split; intros. `````` Ralf Jung committed Feb 20, 2016 146 147 `````` - by trans x1; [|trans y1]. - by trans x2; [|trans y2]. `````` Robbert Krebbers committed Nov 11, 2015 148 `````` Qed. `````` Robbert Krebbers committed Jan 14, 2016 149 `````` Global Instance dist_proper n : Proper ((≡) ==> (≡) ==> iff) (@dist A _ n). `````` Robbert Krebbers committed Nov 11, 2015 150 `````` Proof. `````` Robbert Krebbers committed Jan 13, 2016 151 `````` by move => x1 x2 /equiv_dist Hx y1 y2 /equiv_dist Hy; rewrite (Hx n) (Hy n). `````` Robbert Krebbers committed Nov 11, 2015 152 153 154 `````` 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 155 `````` Lemma dist_le n n' x y : x ≡{n}≡ y → n' ≤ n → x ≡{n'}≡ y. `````` Robbert Krebbers committed Nov 11, 2015 156 `````` Proof. induction 2; eauto using dist_S. Qed. `````` Ralf Jung committed Feb 29, 2016 157 158 `````` 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 159 `````` Instance ne_proper {B : ofeT} (f : A → B) `````` Ralf Jung committed Jan 27, 2017 160 `````` `{!NonExpansive f} : Proper ((≡) ==> (≡)) f | 100. `````` Robbert Krebbers committed Nov 11, 2015 161 `````` Proof. by intros x1 x2; rewrite !equiv_dist; intros Hx n; rewrite (Hx n). Qed. `````` Ralf Jung committed Nov 22, 2016 162 `````` Instance ne_proper_2 {B C : ofeT} (f : A → B → C) `````` Ralf Jung committed Jan 27, 2017 163 `````` `{!NonExpansive2 f} : `````` Robbert Krebbers committed Nov 11, 2015 164 165 166 `````` Proper ((≡) ==> (≡) ==> (≡)) f | 100. Proof. unfold Proper, respectful; setoid_rewrite equiv_dist. `````` Robbert Krebbers committed Jan 13, 2016 167 `````` by intros x1 x2 Hx y1 y2 Hy n; rewrite (Hx n) (Hy n). `````` Robbert Krebbers committed Nov 11, 2015 168 `````` Qed. `````` Robbert Krebbers committed Feb 24, 2016 169 `````` `````` Ralf Jung committed Nov 22, 2016 170 `````` Lemma conv_compl' `{Cofe A} n (c : chain A) : compl c ≡{n}≡ c (S n). `````` Ralf Jung committed Feb 29, 2016 171 172 173 174 `````` Proof. transitivity (c n); first by apply conv_compl. symmetry. apply chain_cauchy. omega. Qed. `````` Robbert Krebbers committed Feb 24, 2016 175 176 `````` Lemma timeless_iff n (x : A) `{!Timeless x} y : x ≡ y ↔ x ≡{n}≡ y. Proof. `````` Robbert Krebbers committed May 28, 2016 177 `````` split; intros; auto. apply (timeless _), dist_le with n; auto with lia. `````` Robbert Krebbers committed Feb 24, 2016 178 `````` Qed. `````` Robbert Krebbers committed Feb 09, 2017 179 ``````End ofe. `````` Robbert Krebbers committed Nov 11, 2015 180 `````` `````` Robbert Krebbers committed Dec 02, 2016 181 ``````(** Contractive functions *) `````` Robbert Krebbers committed Dec 05, 2016 182 183 184 185 186 187 188 189 ``````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 190 `````` `````` Ralf Jung committed Nov 22, 2016 191 ``````Instance const_contractive {A B : ofeT} (x : A) : Contractive (@const A B x). `````` Robbert Krebbers committed Mar 06, 2016 192 193 ``````Proof. by intros n y1 y2. Qed. `````` Robbert Krebbers committed Dec 02, 2016 194 ``````Section contractive. `````` Ralf Jung committed Jan 25, 2017 195 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Dec 02, 2016 196 197 198 199 `````` 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 200 `````` Proof. by apply (_ : Contractive f). Qed. `````` Robbert Krebbers committed Dec 02, 2016 201 `````` Lemma contractive_S n x y : x ≡{n}≡ y → f x ≡{S n}≡ f y. `````` Robbert Krebbers committed Dec 05, 2016 202 `````` Proof. intros. by apply (_ : Contractive f). Qed. `````` Robbert Krebbers committed Dec 02, 2016 203 `````` `````` Ralf Jung committed Jan 27, 2017 204 205 `````` Global Instance contractive_ne : NonExpansive f | 100. Proof. by intros n x y ?; apply dist_S, contractive_S. Qed. `````` Robbert Krebbers committed Dec 02, 2016 206 207 208 209 `````` Global Instance contractive_proper : Proper ((≡) ==> (≡)) f | 100. Proof. apply (ne_proper _). Qed. End contractive. `````` Robbert Krebbers committed Dec 05, 2016 210 211 212 213 214 215 216 ``````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 217 218 `````` | |- @dist_later ?A ?n ?x ?y => destruct n as [|n]; [done|change (@dist A _ n x y)] `````` Robbert Krebbers committed Dec 05, 2016 219 220 221 222 223 224 `````` end; try reflexivity. Ltac solve_contractive := preprocess_solve_proper; solve [repeat (first [f_contractive|f_equiv]; try eassumption)]. `````` Robbert Krebbers committed Nov 22, 2015 225 `````` `````` Robbert Krebbers committed Nov 11, 2015 226 ``````(** Fixpoint *) `````` Ralf Jung committed Nov 22, 2016 227 ``````Program Definition fixpoint_chain {A : ofeT} `{Inhabited A} (f : A → A) `````` Robbert Krebbers committed Feb 10, 2016 228 `````` `{!Contractive f} : chain A := {| chain_car i := Nat.iter (S i) f inhabitant |}. `````` Robbert Krebbers committed Nov 11, 2015 229 ``````Next Obligation. `````` Robbert Krebbers committed Mar 06, 2016 230 `````` intros A ? f ? n. `````` Robbert Krebbers committed Dec 05, 2016 231 `````` induction n as [|n IH]=> -[|i] //= ?; try omega. `````` Robbert Krebbers committed Feb 17, 2016 232 233 `````` - apply (contractive_0 f). - apply (contractive_S f), IH; auto with omega. `````` Robbert Krebbers committed Nov 11, 2015 234 ``````Qed. `````` Robbert Krebbers committed Mar 18, 2016 235 `````` `````` Ralf Jung committed Nov 22, 2016 236 ``````Program Definition fixpoint_def `{Cofe A, Inhabited A} (f : A → A) `````` Robbert Krebbers committed Nov 17, 2015 237 `````` `{!Contractive f} : A := compl (fixpoint_chain f). `````` Ralf Jung committed Jan 11, 2017 238 239 240 ``````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 241 242 `````` Section fixpoint. `````` Ralf Jung committed Nov 22, 2016 243 `````` Context `{Cofe A, Inhabited A} (f : A → A) `{!Contractive f}. `````` Robbert Krebbers committed Aug 21, 2016 244 `````` `````` Robbert Krebbers committed Nov 17, 2015 245 `````` Lemma fixpoint_unfold : fixpoint f ≡ f (fixpoint f). `````` Robbert Krebbers committed Nov 11, 2015 246 `````` Proof. `````` Robbert Krebbers committed Mar 18, 2016 247 248 `````` apply equiv_dist=>n. rewrite fixpoint_eq /fixpoint_def (conv_compl n (fixpoint_chain f)) //. `````` Robbert Krebbers committed Feb 12, 2016 249 `````` induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S. `````` Robbert Krebbers committed Nov 11, 2015 250 `````` Qed. `````` Robbert Krebbers committed Aug 21, 2016 251 252 253 `````` Lemma fixpoint_unique (x : A) : x ≡ f x → x ≡ fixpoint f. Proof. `````` Robbert Krebbers committed Aug 22, 2016 254 255 256 `````` 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 257 258 `````` Qed. `````` Robbert Krebbers committed Nov 17, 2015 259 `````` Lemma fixpoint_ne (g : A → A) `{!Contractive g} n : `````` Ralf Jung committed Feb 10, 2016 260 `````` (∀ z, f z ≡{n}≡ g z) → fixpoint f ≡{n}≡ fixpoint g. `````` Robbert Krebbers committed Nov 11, 2015 261 `````` Proof. `````` Robbert Krebbers committed Mar 18, 2016 262 `````` intros Hfg. rewrite fixpoint_eq /fixpoint_def `````` Robbert Krebbers committed Feb 18, 2016 263 `````` (conv_compl n (fixpoint_chain f)) (conv_compl n (fixpoint_chain g)) /=. `````` Robbert Krebbers committed Feb 10, 2016 264 265 `````` 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 266 `````` Qed. `````` Robbert Krebbers committed Nov 17, 2015 267 268 `````` Lemma fixpoint_proper (g : A → A) `{!Contractive g} : (∀ x, f x ≡ g x) → fixpoint f ≡ fixpoint g. `````` Robbert Krebbers committed Nov 11, 2015 269 `````` Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_ne. Qed. `````` Jacques-Henri Jourdan committed Dec 23, 2016 270 271 `````` Lemma fixpoint_ind (P : A → Prop) : `````` Jacques-Henri Jourdan committed Dec 23, 2016 272 `````` Proper ((≡) ==> impl) P → `````` Jacques-Henri Jourdan committed Dec 23, 2016 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 `````` (∃ 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 288 289 ``````End fixpoint. `````` Ralf Jung committed Jan 25, 2017 290 291 292 ``````(** 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 293 `````` `````` Ralf Jung committed Jan 25, 2017 294 ``````Section fixpointK. `````` Ralf Jung committed Jan 25, 2017 295 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Jan 25, 2017 296 297 `````` Context `{Cofe A, Inhabited A} (f : A → A) (k : nat). Context `{f_contractive : !Contractive (Nat.iter k f)}. `````` Ralf Jung committed Jan 25, 2017 298 `````` (* TODO: Can we get rid of this assumption, derive it from contractivity? *) `````` Ralf Jung committed Jan 27, 2017 299 `````` Context {f_ne : NonExpansive f}. `````` Robbert Krebbers committed Jan 25, 2017 300 301 302 `````` Let f_proper : Proper ((≡) ==> (≡)) f := ne_proper f. Existing Instance f_proper. `````` Ralf Jung committed Jan 25, 2017 303 `````` `````` Ralf Jung committed Jan 25, 2017 304 `````` Lemma fixpointK_unfold : fixpointK k f ≡ f (fixpointK k f). `````` Ralf Jung committed Jan 25, 2017 305 `````` Proof. `````` Robbert Krebbers committed Jan 25, 2017 306 307 `````` symmetry. rewrite /fixpointK. apply fixpoint_unique. by rewrite -Nat_iter_S_r Nat_iter_S -fixpoint_unfold. `````` Ralf Jung committed Jan 25, 2017 308 309 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 310 `````` Lemma fixpointK_unique (x : A) : x ≡ f x → x ≡ fixpointK k f. `````` Ralf Jung committed Jan 25, 2017 311 `````` Proof. `````` Robbert Krebbers committed Jan 25, 2017 312 313 `````` intros Hf. apply fixpoint_unique. clear f_contractive. induction k as [|k' IH]=> //=. by rewrite -IH. `````` Ralf Jung committed Jan 25, 2017 314 315 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 316 `````` Section fixpointK_ne. `````` Robbert Krebbers committed Jan 25, 2017 317 `````` Context (g : A → A) `{g_contractive : !Contractive (Nat.iter k g)}. `````` Ralf Jung committed Jan 27, 2017 318 `````` Context {g_ne : NonExpansive g}. `````` Ralf Jung committed Jan 25, 2017 319 `````` `````` Ralf Jung committed Jan 25, 2017 320 `````` Lemma fixpointK_ne n : (∀ z, f z ≡{n}≡ g z) → fixpointK k f ≡{n}≡ fixpointK k g. `````` Ralf Jung committed Jan 25, 2017 321 `````` Proof. `````` Robbert Krebbers committed Jan 25, 2017 322 323 324 `````` 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 325 326 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 327 328 329 330 `````` 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 331 `````` `````` Robbert Krebbers committed Dec 05, 2016 332 ``````(** Mutual fixpoints *) `````` Ralf Jung committed Jan 25, 2017 333 ``````Section fixpointAB. `````` 334 335 `````` Local Unset Default Proof Using. `````` Robbert Krebbers committed Dec 05, 2016 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 `````` 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 377 ``````End fixpointAB. `````` Robbert Krebbers committed Dec 05, 2016 378 `````` `````` Ralf Jung committed Jan 25, 2017 379 ``````Section fixpointAB_ne. `````` Robbert Krebbers committed Dec 05, 2016 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 `````` 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 411 ``````End fixpointAB_ne. `````` Robbert Krebbers committed Dec 05, 2016 412 `````` `````` Robbert Krebbers committed Jul 25, 2016 413 ``````(** Function space *) `````` Ralf Jung committed Nov 22, 2016 414 ``````(* We make [ofe_fun] a definition so that we can register it as a canonical `````` Robbert Krebbers committed Aug 05, 2016 415 ``````structure. *) `````` Ralf Jung committed Nov 22, 2016 416 ``````Definition ofe_fun (A : Type) (B : ofeT) := A → B. `````` Robbert Krebbers committed Jul 25, 2016 417 `````` `````` Ralf Jung committed Nov 22, 2016 418 419 420 421 422 ``````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 423 424 425 426 427 428 429 430 431 432 `````` 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 433 `````` Canonical Structure ofe_funC := OfeT (ofe_fun A B) ofe_fun_ofe_mixin. `````` Robbert Krebbers committed Jul 25, 2016 434 `````` `````` Ralf Jung committed Nov 22, 2016 435 436 437 438 439 440 441 442 443 `````` 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 444 ``````Notation "A -c> B" := `````` Ralf Jung committed Nov 22, 2016 445 446 `````` (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 447 448 `````` Inhabited (A -c> B) := populate (λ _, inhabitant). `````` Robbert Krebbers committed Jul 25, 2016 449 ``````(** Non-expansive function space *) `````` Ralf Jung committed Nov 22, 2016 450 451 ``````Record ofe_mor (A B : ofeT) : Type := CofeMor { ofe_mor_car :> A → B; `````` Ralf Jung committed Jan 27, 2017 452 `````` ofe_mor_ne : NonExpansive ofe_mor_car `````` Robbert Krebbers committed Nov 11, 2015 453 454 ``````}. Arguments CofeMor {_ _} _ {_}. `````` Ralf Jung committed Nov 22, 2016 455 456 ``````Add Printing Constructor ofe_mor. Existing Instance ofe_mor_ne. `````` Robbert Krebbers committed Nov 11, 2015 457 `````` `````` Robbert Krebbers committed Jun 17, 2016 458 459 460 461 ``````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 462 463 464 465 466 467 468 ``````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 469 470 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 471 `````` - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|]. `````` Robbert Krebbers committed Feb 18, 2016 472 `````` intros Hfg k; apply equiv_dist=> n; apply Hfg. `````` Robbert Krebbers committed Feb 17, 2016 473 `````` - intros n; split. `````` Robbert Krebbers committed Jan 14, 2016 474 475 `````` + by intros f x. + by intros f g ? x. `````` Ralf Jung committed Feb 20, 2016 476 `````` + by intros f g h ?? x; trans (g x). `````` Robbert Krebbers committed Feb 17, 2016 477 `````` - by intros n f g ? x; apply dist_S. `````` Robbert Krebbers committed Jan 14, 2016 478 `````` Qed. `````` Ralf Jung committed Nov 22, 2016 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 `````` 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 496 `````` `````` Ralf Jung committed Jan 27, 2017 497 498 499 `````` Global Instance ofe_mor_car_ne : NonExpansive2 (@ofe_mor_car A B). Proof. intros n f g Hfg x y Hx; rewrite Hx; apply Hfg. Qed. `````` Ralf Jung committed Nov 22, 2016 500 501 502 `````` 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 503 `````` Proof. done. Qed. `````` Ralf Jung committed Nov 22, 2016 504 ``````End ofe_mor. `````` Robbert Krebbers committed Jan 14, 2016 505 `````` `````` Ralf Jung committed Nov 22, 2016 506 ``````Arguments ofe_morC : clear implicits. `````` Robbert Krebbers committed Jul 25, 2016 507 ``````Notation "A -n> B" := `````` Ralf Jung committed Nov 22, 2016 508 509 `````` (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 510 `````` Inhabited (A -n> B) := populate (λne _, inhabitant). `````` Robbert Krebbers committed Nov 11, 2015 511 `````` `````` Ralf Jung committed Mar 17, 2016 512 ``````(** Identity and composition and constant function *) `````` Robbert Krebbers committed Nov 11, 2015 513 514 ``````Definition cid {A} : A -n> A := CofeMor id. Instance: Params (@cid) 1. `````` Ralf Jung committed Nov 22, 2016 515 ``````Definition cconst {A B : ofeT} (x : B) : A -n> B := CofeMor (const x). `````` Ralf Jung committed Mar 17, 2016 516 ``````Instance: Params (@cconst) 2. `````` Robbert Krebbers committed Mar 02, 2016 517 `````` `````` Robbert Krebbers committed Nov 11, 2015 518 519 520 521 522 ``````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 523 `````` f1 ≡{n}≡ f2 → g1 ≡{n}≡ g2 → f1 ◎ g1 ≡{n}≡ f2 ◎ g2. `````` Robbert Krebbers committed Jan 13, 2016 524 ``````Proof. by intros Hf Hg x; rewrite /= (Hg x) (Hf (g2 x)). Qed. `````` Robbert Krebbers committed Nov 11, 2015 525 `````` `````` Ralf Jung committed Mar 02, 2016 526 ``````(* Function space maps *) `````` Ralf Jung committed Nov 22, 2016 527 ``````Definition ofe_mor_map {A A' B B'} (f : A' -n> A) (g : B -n> B') `````` Ralf Jung committed Mar 02, 2016 528 `````` (h : A -n> B) : A' -n> B' := g ◎ h ◎ f. `````` Ralf Jung committed Nov 22, 2016 529 530 ``````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 531 ``````Proof. intros ??? ??? ???. by repeat apply ccompose_ne. Qed. `````` Ralf Jung committed Mar 02, 2016 532 `````` `````` Ralf Jung committed Nov 22, 2016 533 534 ``````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). `````` Ralf Jung committed Jan 27, 2017 535 536 ``````Instance ofe_morC_map_ne {A A' B B'} : NonExpansive2 (@ofe_morC_map A A' B B'). `````` Ralf Jung committed Mar 02, 2016 537 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 538 `````` intros n f f' Hf g g' Hg ?. rewrite /= /ofe_mor_map. `````` Robbert Krebbers committed Mar 02, 2016 539 `````` by repeat apply ccompose_ne. `````` Ralf Jung committed Mar 02, 2016 540 541 ``````Qed. `````` Robbert Krebbers committed Nov 11, 2015 542 ``````(** unit *) `````` Robbert Krebbers committed Jan 14, 2016 543 544 ``````Section unit. Instance unit_dist : Dist unit := λ _ _ _, True. `````` Ralf Jung committed Nov 22, 2016 545 `````` Definition unit_ofe_mixin : OfeMixin unit. `````` Robbert Krebbers committed Jan 14, 2016 546 `````` Proof. by repeat split; try exists 0. Qed. `````` Ralf Jung committed Nov 22, 2016 547 `````` Canonical Structure unitC : ofeT := OfeT unit unit_ofe_mixin. `````` Robbert Krebbers committed Nov 28, 2016 548 `````` `````` Ralf Jung committed Nov 22, 2016 549 550 `````` Global Program Instance unit_cofe : Cofe unitC := { compl x := () }. Next Obligation. by repeat split; try exists 0. Qed. `````` Robbert Krebbers committed Nov 28, 2016 551 552 `````` Global Instance unit_discrete_cofe : Discrete unitC. `````` Robbert Krebbers committed Jan 31, 2016 553 `````` Proof. done. Qed. `````` Robbert Krebbers committed Jan 14, 2016 554 ``````End unit. `````` Robbert Krebbers committed Nov 11, 2015 555 `````` `````` Robbert Krebbers committed Feb 10, 2017 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 ``````Lemma iso_ofe_mixin {A : ofeT} `{Equiv B, Dist B} (g : B → A) (g_equiv : ∀ y1 y2, y1 ≡ y2 ↔ g y1 ≡ g y2) (g_dist : ∀ n y1 y2, y1 ≡{n}≡ y2 ↔ g y1 ≡{n}≡ g y2) : OfeMixin B. Proof. split. - intros y1 y2. rewrite g_equiv. setoid_rewrite g_dist. apply equiv_dist. - split. + intros y. by apply g_dist. + intros y1 y2. by rewrite !g_dist. + intros y1 y2 y3. rewrite !g_dist. intros ??; etrans; eauto. - intros n y1 y2. rewrite !g_dist. apply dist_S. Qed. Program Definition iso_cofe {A B : ofeT} `{Cofe A} (f : A → B) (g : B → A) `(!NonExpansive g, !NonExpansive f) (fg : ∀ y, f (g y) ≡ y) : Cofe B := {| compl c := f (compl (chain_map g c)) |}. Next Obligation. intros A B ? f g ?? fg n c. by rewrite /= conv_compl /= fg. Qed. `````` Robbert Krebbers committed Nov 11, 2015 576 ``````(** Product *) `````` Robbert Krebbers committed Jan 14, 2016 577 ``````Section product. `````` Ralf Jung committed Nov 22, 2016 578 `````` Context {A B : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 579 580 581 `````` Instance prod_dist : Dist (A * B) := λ n, prod_relation (dist n) (dist n). Global Instance pair_ne : `````` Ralf Jung committed Jan 27, 2017 582 583 584 `````` NonExpansive2 (@pair A B) := _. Global Instance fst_ne : NonExpansive (@fst A B) := _. Global Instance snd_ne : NonExpansive (@snd A B) := _. `````` Ralf Jung committed Nov 22, 2016 585 `````` Definition prod_ofe_mixin : OfeMixin (A * B). `````` Robbert Krebbers committed Jan 14, 2016 586 587 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 588 `````` - intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation. `````` Robbert Krebbers committed Jan 14, 2016 589 `````` rewrite !equiv_dist; naive_solver. `````` Robbert Krebbers committed Feb 17, 2016 590 591 `````` - apply _. - by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S. `````` Robbert Krebbers committed Jan 14, 2016 592 `````` Qed. `````` Ralf Jung committed Nov 22, 2016 593 594 595 596 597 598 599 600 601 `````` 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 602 603 604 `````` 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 605 606 `````` Global Instance prod_discrete_cofe : Discrete A → Discrete B → Discrete prodC. Proof. intros ?? [??]; apply _. Qed. `````` Robbert Krebbers committed Jan 14, 2016 607 608 609 610 611 ``````End product. Arguments prodC : clear implicits. Typeclasses Opaque prod_dist. `````` Ralf Jung committed Nov 22, 2016 612 ``````Instance prod_map_ne {A A' B B' : ofeT} n : `````` Robbert Krebbers committed Nov 11, 2015 613 614 615 616 617 `````` 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). `````` Ralf Jung committed Jan 27, 2017 618 619 620 ``````Instance prodC_map_ne {A A' B B'} : NonExpansive2 (@prodC_map A A' B B'). Proof. intros n f f' Hf g g' Hg [??]; split; [apply Hf|apply Hg]. Qed. `````` Robbert Krebbers committed Nov 11, 2015 621 `````` `````` Robbert Krebbers committed Mar 02, 2016 622 623 ``````(** Functors *) Structure cFunctor := CFunctor { `````` Ralf Jung committed Nov 22, 2016 624 `````` cFunctor_car : ofeT → ofeT → ofeT; `````` Robbert Krebbers committed Mar 02, 2016 625 626 `````` cFunctor_map {A1 A2 B1 B2} : ((A2 -n> A1) * (B1 -n> B2)) → cFunctor_car A1 B1 -n> cFunctor_car A2 B2; `````` Ralf Jung committed Jan 27, 2017 627 628 `````` cFunctor_ne {A1 A2 B1 B2} : NonExpansive (@cFunctor_map A1 A2 B1 B2); `````` Ralf Jung committed Nov 22, 2016 629 `````` cFunctor_id {A B : ofeT} (x : cFunctor_car A B) : `````` Robbert Krebbers committed Mar 02, 2016 630 631 632 633 634 `````` 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 635 ``````Existing Instance cFunctor_ne. `````` Robbert Krebbers committed Mar 02, 2016 636 637 ``````Instance: Params (@cFunctor_map) 5. `````` Ralf Jung committed Mar 07, 2016 638 639 640 ``````Delimit Scope cFunctor_scope with CF. Bind Scope cFunctor_scope with cFunctor. `````` Ralf Jung committed Mar 07, 2016 641 642 643 ``````Class cFunctorContractive (F : cFunctor) := cFunctor_contractive A1 A2 B1 B2 :> Contractive (@cFunctor_map F A1 A2 B1 B2). `````` Ralf Jung committed Nov 22, 2016 644 ``````Definition cFunctor_diag (F: cFunctor) (A: ofeT) : ofeT := cFunctor_car F A A. `````` Robbert Krebbers committed Mar 02, 2016 645 646 ``````Coercion cFunctor_diag : cFunctor >-> Funclass. `````` Ralf Jung committed Nov 22, 2016 647 ``````Program Definition constCF (B : ofeT) : cFunctor := `````` Robbert Krebbers committed Mar 02, 2016 648 649 `````` {| cFunctor_car A1 A2 := B; cFunctor_map A1 A2 B1 B2 f := cid |}. Solve Obligations with done. `````` Ralf Jung committed Jan 06, 2017 650 ``````Coercion constCF : ofeT >-> cFunctor. `````` Robbert Krebbers committed Mar 02, 2016 651 `````` `````` Ralf Jung committed Mar 07, 2016 652 ``````Instance constCF_contractive B : cFunctorContractive (constCF B). `````` Robbert Krebbers committed Mar 07, 2016 653 ``````Proof. rewrite /cFunctorContractive; apply _. Qed. `````` Ralf Jung committed Mar 07, 2016 654 655 656 657 `````` 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 658 ``````Notation "∙" := idCF : cFunctor_scope. `````` Ralf Jung committed Mar 07, 2016 659 `````` `````` Robbert Krebbers committed Mar 02, 2016 660 661 662 663 664 ``````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 665 666 667 ``````Next Obligation. intros ?? A1 A2 B1 B2 n ???; by apply prodC_map_ne; apply cFunctor_ne. Qed. `````` Robbert Krebbers committed Mar 02, 2016 668 669 670 671 672 ``````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 673 ``````Notation "F1 * F2" := (prodCF F1%CF F2%CF) : cFunctor_scope. `````` Robbert Krebbers committed Mar 02, 2016 674 `````` `````` Ralf Jung committed Mar 07, 2016 675 676 677 678 679 680 681 682 ``````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 Jan 27, 2017 683 684 685 ``````Instance compose_ne {A} {B B' : ofeT} (f : B -n> B') : NonExpansive (compose f : (A -c> B) → A -c> B'). Proof. intros n g g' Hf x; simpl. by rewrite (Hf x). Qed. `````` Jacques-Henri Jourdan committed Oct 05, 2016 686 `````` `````` Ralf Jung committed Nov 22, 2016 687 ``````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 688 `````` @CofeMor (_ -c> _) (_ -c> _) (compose f) _. `````` Ralf Jung committed Jan 27, 2017 689 690 691 ``````Instance ofe_funC_map_ne {A B B'} : NonExpansive (@ofe_funC_map A B B'). Proof. intros n f f' Hf g x. apply Hf. Qed. `````` Jacques-Henri Jourdan committed Oct 05, 2016 692 `````` `````` Ralf Jung committed Nov 22, 2016 693 694 695 ``````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 696 697 ``````|}. Next Obligation. `````` Ralf Jung committed Nov 22, 2016 698 `````` intros ?? A1 A2 B1 B2 n ???; by apply ofe_funC_map_ne; apply cFunctor_ne. `````` Jacques-Henri Jourdan committed Oct 05, 2016 699 700 701 702 703 704 ``````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 705 ``````Notation "T -c> F" := (ofe_funCF T%type F%CF) : cFunctor_scope. `````` Jacques-Henri Jourdan committed Oct 05, 2016 706 `````` `````` Ralf Jung committed Nov 22, 2016 707 708 ``````Instance ofe_funCF_contractive (T : Type) (F : cFunctor) : cFunctorContractive F → cFunctorContractive (ofe_funCF T F). `````` Jacques-Henri Jourdan committed Oct 05, 2016 709 710 ``````Proof. intros ?? A1 A2 B1 B2 n ???; `````` Ralf Jung committed Nov 22, 2016 711 `````` by apply ofe_funC_map_ne; apply cFunctor_contractive. `````` Jacques-Henri Jourdan committed Oct 05, 2016 712 713 ``````Qed. `````` Ralf Jung committed Nov 22, 2016 714 ``````Program Definition ofe_morCF (F1 F2 : cFunctor) : cFunctor := {| `````` Robbert Krebbers committed Jul 25, 2016 715 `````` cFunctor_car A B := cFunctor_car F1 B A -n> cFunctor_car F2 A B; `````` Ralf Jung committed Mar 02, 2016 716 `````` cFunctor_map A1 A2 B1 B2 fg := `````` Ralf Jung committed Nov 22, 2016 717 `````` ofe_morC_map (cFunctor_map F1 (fg.2, fg.1)) (cFunctor_map F2 fg) `````` Ralf Jung committed Mar 02, 2016 718 ``````|}. `````` Robbert Krebbers committed Mar 07, 2016 719 720 ``````Next Obligation. intros F1 F2 A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *. `````` Ralf Jung committed Nov 22, 2016 721 `````` apply ofe_morC_map_ne; apply cFunctor_ne; split; by apply Hfg. `````` Robbert Krebbers committed Mar 07, 2016 722 ``````Qed. `````` Ralf Jung committed Mar 02, 2016 723 ``````Next Obligation. `````` Robbert Krebbers committed Mar 02, 2016 724 725 `````` intros F1 F2 A B [f ?] ?; simpl. rewrite /= !cFunctor_id. apply (ne_proper f). apply cFunctor_id. `````` Ralf Jung committed Mar 02, 2016 726 727 ``````Qed. Next Obligation. `````` Robbert Krebbers committed Mar 02, 2016 728 729 `````` 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 730 ``````Qed. `````` Ralf Jung committed Jan 06, 2017 731 ``````Notation "F1 -n> F2" := (ofe_morCF F1%CF F2%CF) : cFunctor_scope. `````` Ralf Jung committed Mar 02, 2016 732 `````` `````` Ralf Jung committed Nov 22, 2016 733 ``````Instance ofe_morCF_contractive F1 F2 : `````` Ralf Jung committed Mar 07, 2016 734 `````` cFunctorContractive F1 → cFunctorContractive F2 → `````` Ralf Jung committed Nov 22, 2016 735 `````` cFunctorContractive (ofe_morCF F1 F2). `````` Ralf Jung committed Mar 07, 2016 736 737 ``````Proof. intros ?? A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *. `````` Robbert Krebbers committed Dec 05, 2016 738 `````` apply ofe_morC_map_ne; apply cFunctor_contractive; destruct n, Hfg; by split. `````` Ralf Jung committed Mar 07, 2016 739 740 ``````Qed. `````` Robbert Krebbers committed May 27, 2016 741 742 ``````(** Sum *) Section sum. `````` Ralf Jung committed Nov 22, 2016 743 `````` Context {A B : ofeT}. `````` Robbert Krebbers committed May 27, 2016 744 745 `````` Instance sum_dist : Dist (A + B) := λ n, sum_relation (dist n) (dist n). `````` Ralf Jung committed Jan 27, 2017 746 747 `````` Global Instance inl_ne : NonExpansive (@inl A B) := _. Global Instance inr_ne : NonExpansive (@inr A B) := _. `````` Robbert Krebbers committed May 27, 2016 748 749 750 `````` 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 751 752 753 754 755 756 757 758 759 760 761 762 `````` 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 763 764 `````` {| 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 765 `````` Program Definition inr_chain (c : chain sumC) (b : B) : chain B := `````` Robbert Krebbers committed May 27, 2016 766 767 768 `````` {| 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 769 `````` Definition sum_compl `{Cofe A, Cofe B} : Compl sumC := λ c, `````` Robbert Krebbers committed May 27, 2016 770 771 772 773 `````` 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 774 775 776 777 778 779 780 `````` 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 _)) ``````