ofe.v 49.3 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 Oct 10, 2017 36 37 38 39 40 41 42 43 ``````Section mixin. Local Set Primitive Projections. Record OfeMixin A `{Equiv A, Dist A} := { mixin_equiv_dist x y : x ≡ y ↔ ∀ n, x ≡{n}≡ y; mixin_dist_equivalence n : Equivalence (dist n); mixin_dist_S n x y : x ≡{S n}≡ y → x ≡{n}≡ y }. End mixin. `````` Robbert Krebbers committed Nov 11, 2015 44 45 `````` (** Bundeled version *) `````` Ralf Jung committed Nov 22, 2016 46 47 48 49 50 ``````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 51 `````` _ : Type `````` Robbert Krebbers committed Nov 11, 2015 52 ``````}. `````` Ralf Jung committed Nov 22, 2016 53 54 55 56 57 58 59 60 61 ``````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 62 `````` `````` Robbert Krebbers committed Feb 09, 2017 63 64 65 ``````(** 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 `````` Robbert Krebbers committed Oct 25, 2017 66 ``````different places (see for example the constructors [CmraT] and [UcmraT] in the `````` Robbert Krebbers committed Feb 09, 2017 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 ``````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 88 ``````(** Lifting properties from the mixin *) `````` Ralf Jung committed Nov 22, 2016 89 90 ``````Section ofe_mixin. Context {A : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 91 `````` Implicit Types x y : A. `````` Ralf Jung committed Feb 10, 2016 92 `````` Lemma equiv_dist x y : x ≡ y ↔ ∀ n, x ≡{n}≡ y. `````` Ralf Jung committed Nov 22, 2016 93 `````` Proof. apply (mixin_equiv_dist _ (ofe_mixin A)). Qed. `````` Robbert Krebbers committed Jan 14, 2016 94 `````` Global Instance dist_equivalence n : Equivalence (@dist A _ n). `````` Ralf Jung committed Nov 22, 2016 95 `````` Proof. apply (mixin_dist_equivalence _ (ofe_mixin A)). Qed. `````` Ralf Jung committed Feb 10, 2016 96 `````` Lemma dist_S n x y : x ≡{S n}≡ y → x ≡{n}≡ y. `````` Ralf Jung committed Nov 22, 2016 97 98 `````` Proof. apply (mixin_dist_S _ (ofe_mixin A)). Qed. End ofe_mixin. `````` Robbert Krebbers committed Jan 14, 2016 99 `````` `````` Robbert Krebbers committed May 28, 2016 100 101 ``````Hint Extern 1 (_ ≡{_}≡ _) => apply equiv_dist; assumption. `````` Robbert Krebbers committed Oct 25, 2017 102 103 104 105 106 107 ``````(** Discrete OFEs and discrete OFE elements *) Class Discrete {A : ofeT} (x : A) := discrete y : x ≡{0}≡ y → x ≡ y. Arguments discrete {_} _ {_} _ _. Hint Mode Discrete + ! : typeclass_instances. Instance: Params (@Discrete) 1. `````` Robbert Krebbers committed Oct 25, 2017 108 ``````Class OfeDiscrete (A : ofeT) := ofe_discrete_discrete (x : A) :> Discrete x. `````` Ralf Jung committed Nov 22, 2016 109 110 111 112 113 114 115 116 117 `````` (** 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 118 ``````Program Definition chain_map {A B : ofeT} (f : A → B) `````` Ralf Jung committed Jan 27, 2017 119 `````` `{!NonExpansive f} (c : chain A) : chain B := `````` Robbert Krebbers committed Dec 05, 2016 120 121 122 `````` {| 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 123 124 125 126 127 128 ``````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 129 `````` `````` Robbert Krebbers committed Feb 09, 2017 130 ``````Lemma compl_chain_map `{Cofe A, Cofe B} (f : A → B) c `(NonExpansive f) : `````` Jacques-Henri Jourdan committed Jan 05, 2017 131 132 133 `````` compl (chain_map f c) ≡ f (compl c). Proof. apply equiv_dist=>n. by rewrite !conv_compl. Qed. `````` Ralf Jung committed Mar 01, 2017 134 135 136 137 138 139 140 141 ``````Program Definition chain_const {A : ofeT} (a : A) : chain A := {| chain_car n := a |}. Next Obligation. by intros A a n i _. Qed. Lemma compl_chain_const {A : ofeT} `{!Cofe A} (a : A) : compl (chain_const a) ≡ a. Proof. apply equiv_dist=>n. by rewrite conv_compl. Qed. `````` Robbert Krebbers committed Nov 11, 2015 142 ``````(** General properties *) `````` Robbert Krebbers committed Feb 09, 2017 143 ``````Section ofe. `````` Ralf Jung committed Nov 22, 2016 144 `````` Context {A : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 145 `````` Implicit Types x y : A. `````` Robbert Krebbers committed Feb 09, 2017 146 `````` Global Instance ofe_equivalence : Equivalence ((≡) : relation A). `````` Robbert Krebbers committed Nov 11, 2015 147 148 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 149 150 `````` - by intros x; rewrite equiv_dist. - by intros x y; rewrite !equiv_dist. `````` Ralf Jung committed Feb 20, 2016 151 `````` - by intros x y z; rewrite !equiv_dist; intros; trans y. `````` Robbert Krebbers committed Nov 11, 2015 152 `````` Qed. `````` Robbert Krebbers committed Jan 14, 2016 153 `````` Global Instance dist_ne n : Proper (dist n ==> dist n ==> iff) (@dist A _ n). `````` Robbert Krebbers committed Nov 11, 2015 154 155 `````` Proof. intros x1 x2 ? y1 y2 ?; split; intros. `````` Ralf Jung committed Feb 20, 2016 156 157 `````` - by trans x1; [|trans y1]. - by trans x2; [|trans y2]. `````` Robbert Krebbers committed Nov 11, 2015 158 `````` Qed. `````` Robbert Krebbers committed Jan 14, 2016 159 `````` Global Instance dist_proper n : Proper ((≡) ==> (≡) ==> iff) (@dist A _ n). `````` Robbert Krebbers committed Nov 11, 2015 160 `````` Proof. `````` Robbert Krebbers committed Jan 13, 2016 161 `````` by move => x1 x2 /equiv_dist Hx y1 y2 /equiv_dist Hy; rewrite (Hx n) (Hy n). `````` Robbert Krebbers committed Nov 11, 2015 162 163 164 `````` Qed. Global Instance dist_proper_2 n x : Proper ((≡) ==> iff) (dist n x). Proof. by apply dist_proper. Qed. `````` Robbert Krebbers committed Oct 25, 2017 165 166 `````` Global Instance Discrete_proper : Proper ((≡) ==> iff) (@Discrete A). Proof. intros x y Hxy. rewrite /Discrete. by setoid_rewrite Hxy. Qed. `````` Robbert Krebbers committed Feb 11, 2017 167 `````` `````` Robbert Krebbers committed Feb 18, 2016 168 `````` Lemma dist_le n n' x y : x ≡{n}≡ y → n' ≤ n → x ≡{n'}≡ y. `````` Robbert Krebbers committed Nov 11, 2015 169 `````` Proof. induction 2; eauto using dist_S. Qed. `````` Ralf Jung committed Feb 29, 2016 170 171 `````` Lemma dist_le' n n' x y : n' ≤ n → x ≡{n}≡ y → x ≡{n'}≡ y. Proof. intros; eauto using dist_le. Qed. `````` Robbert Krebbers committed Feb 11, 2017 172 173 `````` Instance ne_proper {B : ofeT} (f : A → B) `{!NonExpansive f} : Proper ((≡) ==> (≡)) f | 100. `````` Robbert Krebbers committed Nov 11, 2015 174 `````` Proof. by intros x1 x2; rewrite !equiv_dist; intros Hx n; rewrite (Hx n). Qed. `````` Robbert Krebbers committed Feb 11, 2017 175 `````` Instance ne_proper_2 {B C : ofeT} (f : A → B → C) `{!NonExpansive2 f} : `````` Robbert Krebbers committed Nov 11, 2015 176 177 178 `````` Proper ((≡) ==> (≡) ==> (≡)) f | 100. Proof. unfold Proper, respectful; setoid_rewrite equiv_dist. `````` Robbert Krebbers committed Jan 13, 2016 179 `````` by intros x1 x2 Hx y1 y2 Hy n; rewrite (Hx n) (Hy n). `````` Robbert Krebbers committed Nov 11, 2015 180 `````` Qed. `````` Robbert Krebbers committed Feb 24, 2016 181 `````` `````` Ralf Jung committed Nov 22, 2016 182 `````` Lemma conv_compl' `{Cofe A} n (c : chain A) : compl c ≡{n}≡ c (S n). `````` Ralf Jung committed Feb 29, 2016 183 184 185 186 `````` Proof. transitivity (c n); first by apply conv_compl. symmetry. apply chain_cauchy. omega. Qed. `````` Robbert Krebbers committed Apr 13, 2017 187 `````` `````` Robbert Krebbers committed Oct 25, 2017 188 `````` Lemma discrete_iff n (x : A) `{!Discrete x} y : x ≡ y ↔ x ≡{n}≡ y. `````` Robbert Krebbers committed Feb 24, 2016 189 `````` Proof. `````` Robbert Krebbers committed Oct 25, 2017 190 `````` split; intros; auto. apply (discrete _), dist_le with n; auto with lia. `````` Robbert Krebbers committed Feb 24, 2016 191 `````` Qed. `````` Robbert Krebbers committed Oct 25, 2017 192 `````` Lemma discrete_iff_0 n (x : A) `{!Discrete x} y : x ≡{0}≡ y ↔ x ≡{n}≡ y. `````` Robbert Krebbers committed Nov 28, 2017 193 `````` Proof. by rewrite -!discrete_iff. Qed. `````` Robbert Krebbers committed Feb 09, 2017 194 ``````End ofe. `````` Robbert Krebbers committed Nov 11, 2015 195 `````` `````` Robbert Krebbers committed Dec 02, 2016 196 ``````(** Contractive functions *) `````` Robbert Krebbers committed Aug 17, 2017 197 ``````Definition dist_later `{Dist A} (n : nat) (x y : A) : Prop := `````` Robbert Krebbers committed Dec 05, 2016 198 `````` match n with 0 => True | S n => x ≡{n}≡ y end. `````` Robbert Krebbers committed Aug 17, 2017 199 ``````Arguments dist_later _ _ !_ _ _ /. `````` Robbert Krebbers committed Dec 05, 2016 200 `````` `````` Robbert Krebbers committed Aug 17, 2017 201 ``````Global Instance dist_later_equivalence (A : ofeT) n : Equivalence (@dist_later A _ n). `````` Robbert Krebbers committed Dec 05, 2016 202 203 ``````Proof. destruct n as [|n]. by split. apply dist_equivalence. Qed. `````` Ralf Jung committed Feb 22, 2017 204 205 206 ``````Lemma dist_dist_later {A : ofeT} n (x y : A) : dist n x y → dist_later n x y. Proof. intros Heq. destruct n; first done. exact: dist_S. Qed. `````` Ralf Jung committed Mar 01, 2017 207 208 209 210 211 212 213 214 215 216 217 ``````Lemma dist_later_dist {A : ofeT} n (x y : A) : dist_later (S n) x y → dist n x y. Proof. done. Qed. (* We don't actually need this lemma (as our tactics deal with this through other means), but technically speaking, this is the reason why pre-composing a non-expansive function to a contractive function preserves contractivity. *) Lemma ne_dist_later {A B : ofeT} (f : A → B) : NonExpansive f → ∀ n, Proper (dist_later n ==> dist_later n) f. Proof. intros Hf [|n]; last exact: Hf. hnf. by intros. Qed. `````` Robbert Krebbers committed Dec 05, 2016 218 ``````Notation Contractive f := (∀ n, Proper (dist_later n ==> dist n) f). `````` Robbert Krebbers committed Dec 02, 2016 219 `````` `````` Ralf Jung committed Nov 22, 2016 220 ``````Instance const_contractive {A B : ofeT} (x : A) : Contractive (@const A B x). `````` Robbert Krebbers committed Mar 06, 2016 221 222 ``````Proof. by intros n y1 y2. Qed. `````` Robbert Krebbers committed Dec 02, 2016 223 ``````Section contractive. `````` Ralf Jung committed Jan 25, 2017 224 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Dec 02, 2016 225 226 227 228 `````` 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 229 `````` Proof. by apply (_ : Contractive f). Qed. `````` Robbert Krebbers committed Dec 02, 2016 230 `````` Lemma contractive_S n x y : x ≡{n}≡ y → f x ≡{S n}≡ f y. `````` Robbert Krebbers committed Dec 05, 2016 231 `````` Proof. intros. by apply (_ : Contractive f). Qed. `````` Robbert Krebbers committed Dec 02, 2016 232 `````` `````` Ralf Jung committed Jan 27, 2017 233 234 `````` 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 235 236 237 238 `````` Global Instance contractive_proper : Proper ((≡) ==> (≡)) f | 100. Proof. apply (ne_proper _). Qed. End contractive. `````` Robbert Krebbers committed Dec 05, 2016 239 240 ``````Ltac f_contractive := match goal with `````` Robbert Krebbers committed Aug 17, 2017 241 242 243 `````` | |- ?f _ ≡{_}≡ ?f _ => simple apply (_ : Proper (dist_later _ ==> _) f) | |- ?f _ _ ≡{_}≡ ?f _ _ => simple apply (_ : Proper (dist_later _ ==> _ ==> _) f) | |- ?f _ _ ≡{_}≡ ?f _ _ => simple apply (_ : Proper (_ ==> dist_later _ ==> _) f) `````` Robbert Krebbers committed Dec 05, 2016 244 245 `````` end; try match goal with `````` Robbert Krebbers committed Aug 17, 2017 246 `````` | |- @dist_later ?A _ ?n ?x ?y => `````` Ralf Jung committed Mar 01, 2017 247 `````` destruct n as [|n]; [exact I|change (@dist A _ n x y)] `````` Robbert Krebbers committed Dec 05, 2016 248 `````` end; `````` Robbert Krebbers committed Aug 17, 2017 249 `````` try simple apply reflexivity. `````` Robbert Krebbers committed Dec 05, 2016 250 `````` `````` Robbert Krebbers committed Aug 17, 2017 251 252 ``````Ltac solve_contractive := solve_proper_core ltac:(fun _ => first [f_contractive | f_equiv]). `````` Robbert Krebbers committed Nov 22, 2015 253 `````` `````` Robbert Krebbers committed Mar 09, 2017 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 ``````(** Limit preserving predicates *) Class LimitPreserving `{!Cofe A} (P : A → Prop) : Prop := limit_preserving (c : chain A) : (∀ n, P (c n)) → P (compl c). Hint Mode LimitPreserving + + ! : typeclass_instances. Section limit_preserving. Context `{Cofe A}. (* These are not instances as they will never fire automatically... but they can still be helpful in proving things to be limit preserving. *) Lemma limit_preserving_ext (P Q : A → Prop) : (∀ x, P x ↔ Q x) → LimitPreserving P → LimitPreserving Q. Proof. intros HP Hlimit c ?. apply HP, Hlimit=> n; by apply HP. Qed. Global Instance limit_preserving_const (P : Prop) : LimitPreserving (λ _, P). Proof. intros c HP. apply (HP 0). Qed. `````` Robbert Krebbers committed Oct 25, 2017 271 `````` Lemma limit_preserving_discrete (P : A → Prop) : `````` Robbert Krebbers committed Mar 09, 2017 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 `````` Proper (dist 0 ==> impl) P → LimitPreserving P. Proof. intros PH c Hc. by rewrite (conv_compl 0). Qed. Lemma limit_preserving_and (P1 P2 : A → Prop) : LimitPreserving P1 → LimitPreserving P2 → LimitPreserving (λ x, P1 x ∧ P2 x). Proof. intros Hlim1 Hlim2 c Hc. split. apply Hlim1, Hc. apply Hlim2, Hc. Qed. Lemma limit_preserving_impl (P1 P2 : A → Prop) : Proper (dist 0 ==> impl) P1 → LimitPreserving P2 → LimitPreserving (λ x, P1 x → P2 x). Proof. intros Hlim1 Hlim2 c Hc HP1. apply Hlim2=> n; apply Hc. eapply Hlim1, HP1. apply dist_le with n; last lia. apply (conv_compl n). Qed. Lemma limit_preserving_forall {B} (P : B → A → Prop) : (∀ y, LimitPreserving (P y)) → LimitPreserving (λ x, ∀ y, P y x). Proof. intros Hlim c Hc y. by apply Hlim. Qed. End limit_preserving. `````` Robbert Krebbers committed Nov 11, 2015 294 ``````(** Fixpoint *) `````` Ralf Jung committed Nov 22, 2016 295 ``````Program Definition fixpoint_chain {A : ofeT} `{Inhabited A} (f : A → A) `````` Robbert Krebbers committed Feb 10, 2016 296 `````` `{!Contractive f} : chain A := {| chain_car i := Nat.iter (S i) f inhabitant |}. `````` Robbert Krebbers committed Nov 11, 2015 297 ``````Next Obligation. `````` Robbert Krebbers committed Mar 06, 2016 298 `````` intros A ? f ? n. `````` Robbert Krebbers committed Dec 05, 2016 299 `````` induction n as [|n IH]=> -[|i] //= ?; try omega. `````` Robbert Krebbers committed Feb 17, 2016 300 301 `````` - apply (contractive_0 f). - apply (contractive_S f), IH; auto with omega. `````` Robbert Krebbers committed Nov 11, 2015 302 ``````Qed. `````` Robbert Krebbers committed Mar 18, 2016 303 `````` `````` Ralf Jung committed Nov 22, 2016 304 ``````Program Definition fixpoint_def `{Cofe A, Inhabited A} (f : A → A) `````` Robbert Krebbers committed Nov 17, 2015 305 `````` `{!Contractive f} : A := compl (fixpoint_chain f). `````` Ralf Jung committed Jan 11, 2017 306 307 308 ``````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 309 310 `````` Section fixpoint. `````` Ralf Jung committed Nov 22, 2016 311 `````` Context `{Cofe A, Inhabited A} (f : A → A) `{!Contractive f}. `````` Robbert Krebbers committed Aug 21, 2016 312 `````` `````` Robbert Krebbers committed Nov 17, 2015 313 `````` Lemma fixpoint_unfold : fixpoint f ≡ f (fixpoint f). `````` Robbert Krebbers committed Nov 11, 2015 314 `````` Proof. `````` Robbert Krebbers committed Mar 18, 2016 315 316 `````` apply equiv_dist=>n. rewrite fixpoint_eq /fixpoint_def (conv_compl n (fixpoint_chain f)) //. `````` Robbert Krebbers committed Feb 12, 2016 317 `````` induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S. `````` Robbert Krebbers committed Nov 11, 2015 318 `````` Qed. `````` Robbert Krebbers committed Aug 21, 2016 319 320 321 `````` Lemma fixpoint_unique (x : A) : x ≡ f x → x ≡ fixpoint f. Proof. `````` Robbert Krebbers committed Aug 22, 2016 322 323 324 `````` 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 325 326 `````` Qed. `````` Robbert Krebbers committed Nov 17, 2015 327 `````` Lemma fixpoint_ne (g : A → A) `{!Contractive g} n : `````` Ralf Jung committed Feb 10, 2016 328 `````` (∀ z, f z ≡{n}≡ g z) → fixpoint f ≡{n}≡ fixpoint g. `````` Robbert Krebbers committed Nov 11, 2015 329 `````` Proof. `````` Robbert Krebbers committed Mar 18, 2016 330 `````` intros Hfg. rewrite fixpoint_eq /fixpoint_def `````` Robbert Krebbers committed Feb 18, 2016 331 `````` (conv_compl n (fixpoint_chain f)) (conv_compl n (fixpoint_chain g)) /=. `````` Robbert Krebbers committed Feb 10, 2016 332 333 `````` 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 334 `````` Qed. `````` Robbert Krebbers committed Nov 17, 2015 335 336 `````` Lemma fixpoint_proper (g : A → A) `{!Contractive g} : (∀ x, f x ≡ g x) → fixpoint f ≡ fixpoint g. `````` Robbert Krebbers committed Nov 11, 2015 337 `````` Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_ne. Qed. `````` Jacques-Henri Jourdan committed Dec 23, 2016 338 339 `````` Lemma fixpoint_ind (P : A → Prop) : `````` Jacques-Henri Jourdan committed Dec 23, 2016 340 `````` Proper ((≡) ==> impl) P → `````` Jacques-Henri Jourdan committed Dec 23, 2016 341 `````` (∃ x, P x) → (∀ x, P x → P (f x)) → `````` Robbert Krebbers committed Mar 09, 2017 342 `````` LimitPreserving P → `````` Jacques-Henri Jourdan committed Dec 23, 2016 343 344 345 346 `````` 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). `````` Robbert Krebbers committed Mar 09, 2017 347 348 `````` { intros n. rewrite /chcar. induction n as [|n IH]=> -[|i] //=; eauto using contractive_0, contractive_S with omega. } `````` Jacques-Henri Jourdan committed Dec 23, 2016 349 `````` set (fp2 := compl {| chain_cauchy := Hcauch |}). `````` Robbert Krebbers committed Mar 09, 2017 350 351 352 353 `````` assert (f fp2 ≡ fp2). { apply equiv_dist=>n. rewrite /fp2 (conv_compl n) /= /chcar. induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S. } rewrite -(fixpoint_unique fp2) //. `````` Robbert Krebbers committed Mar 11, 2017 354 `````` apply Hlim=> n /=. by apply Nat_iter_ind. `````` Jacques-Henri Jourdan committed Dec 23, 2016 355 `````` Qed. `````` Robbert Krebbers committed Nov 11, 2015 356 357 ``````End fixpoint. `````` Robbert Krebbers committed Mar 09, 2017 358 `````` `````` Ralf Jung committed Jan 25, 2017 359 360 361 ``````(** 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 362 `````` `````` Ralf Jung committed Jan 25, 2017 363 ``````Section fixpointK. `````` Ralf Jung committed Jan 25, 2017 364 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Jan 25, 2017 365 `````` Context `{Cofe A, Inhabited A} (f : A → A) (k : nat). `````` Ralf Jung committed Feb 23, 2017 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 `````` Context {f_contractive : Contractive (Nat.iter k f)} {f_ne : NonExpansive f}. (* Note than f_ne is crucial here: there are functions f such that f^2 is contractive, but f is not non-expansive. Consider for example f: SPred → SPred (where SPred is "downclosed sets of natural numbers"). Define f (using informative excluded middle) as follows: f(N) = N (where N is the set of all natural numbers) f({0, ..., n}) = {0, ... n-1} if n is even (so n-1 is at least -1, in which case we return the empty set) f({0, ..., n}) = {0, ..., n+2} if n is odd In other words, if we consider elements of SPred as ordinals, then we decreaste odd finite ordinals by 1 and increase even finite ordinals by 2. f is not non-expansive: Consider f({0}) = ∅ and f({0,1}) = f({0,1,2,3}). The arguments are clearly 0-equal, but the results are not. Now consider g := f^2. We have g(N) = N g({0, ..., n}) = {0, ... n+1} if n is even g({0, ..., n}) = {0, ..., n+4} if n is odd g is contractive. All outputs contain 0, so they are all 0-equal. Now consider two n-equal inputs. We have to show that the outputs are n+1-equal. Either they both do not contain n in which case they have to be fully equal and hence so are the results. Or else they both contain n, so the results will both contain n+1, so the results are n+1-equal. *) `````` Robbert Krebbers committed Jan 25, 2017 389 390 `````` Let f_proper : Proper ((≡) ==> (≡)) f := ne_proper f. `````` Ralf Jung committed Feb 23, 2017 391 `````` Local Existing Instance f_proper. `````` Ralf Jung committed Jan 25, 2017 392 `````` `````` Ralf Jung committed Jan 25, 2017 393 `````` Lemma fixpointK_unfold : fixpointK k f ≡ f (fixpointK k f). `````` Ralf Jung committed Jan 25, 2017 394 `````` Proof. `````` Robbert Krebbers committed Jan 25, 2017 395 396 `````` symmetry. rewrite /fixpointK. apply fixpoint_unique. by rewrite -Nat_iter_S_r Nat_iter_S -fixpoint_unfold. `````` Ralf Jung committed Jan 25, 2017 397 398 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 399 `````` Lemma fixpointK_unique (x : A) : x ≡ f x → x ≡ fixpointK k f. `````` Ralf Jung committed Jan 25, 2017 400 `````` Proof. `````` Robbert Krebbers committed Jan 25, 2017 401 402 `````` intros Hf. apply fixpoint_unique. clear f_contractive. induction k as [|k' IH]=> //=. by rewrite -IH. `````` Ralf Jung committed Jan 25, 2017 403 404 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 405 `````` Section fixpointK_ne. `````` Robbert Krebbers committed Jan 25, 2017 406 `````` Context (g : A → A) `{g_contractive : !Contractive (Nat.iter k g)}. `````` Ralf Jung committed Jan 27, 2017 407 `````` Context {g_ne : NonExpansive g}. `````` Ralf Jung committed Jan 25, 2017 408 `````` `````` Ralf Jung committed Jan 25, 2017 409 `````` Lemma fixpointK_ne n : (∀ z, f z ≡{n}≡ g z) → fixpointK k f ≡{n}≡ fixpointK k g. `````` Ralf Jung committed Jan 25, 2017 410 `````` Proof. `````` Robbert Krebbers committed Jan 25, 2017 411 412 413 `````` 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 414 415 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 416 417 418 `````` 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. `````` Ralf Jung committed Feb 21, 2017 419 420 421 422 `````` Lemma fixpointK_ind (P : A → Prop) : Proper ((≡) ==> impl) P → (∃ x, P x) → (∀ x, P x → P (f x)) → `````` Robbert Krebbers committed Mar 09, 2017 423 `````` LimitPreserving P → `````` Ralf Jung committed Feb 21, 2017 424 425 `````` P (fixpointK k f). Proof. `````` Robbert Krebbers committed Mar 09, 2017 426 `````` intros. rewrite /fixpointK. apply fixpoint_ind; eauto. `````` Robbert Krebbers committed Mar 11, 2017 427 `````` intros; apply Nat_iter_ind; auto. `````` Ralf Jung committed Feb 21, 2017 428 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 429 ``````End fixpointK. `````` Ralf Jung committed Jan 25, 2017 430 `````` `````` Robbert Krebbers committed Dec 05, 2016 431 ``````(** Mutual fixpoints *) `````` Ralf Jung committed Jan 25, 2017 432 ``````Section fixpointAB. `````` 433 434 `````` Local Unset Default Proof Using. `````` Robbert Krebbers committed Dec 05, 2016 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 `````` 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 476 ``````End fixpointAB. `````` Robbert Krebbers committed Dec 05, 2016 477 `````` `````` Ralf Jung committed Jan 25, 2017 478 ``````Section fixpointAB_ne. `````` Robbert Krebbers committed Dec 05, 2016 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 `````` 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 510 ``````End fixpointAB_ne. `````` Robbert Krebbers committed Dec 05, 2016 511 `````` `````` Robbert Krebbers committed Jul 25, 2016 512 ``````(** Non-expansive function space *) `````` Ralf Jung committed Nov 22, 2016 513 514 ``````Record ofe_mor (A B : ofeT) : Type := CofeMor { ofe_mor_car :> A → B; `````` Ralf Jung committed Jan 27, 2017 515 `````` ofe_mor_ne : NonExpansive ofe_mor_car `````` Robbert Krebbers committed Nov 11, 2015 516 517 ``````}. Arguments CofeMor {_ _} _ {_}. `````` Ralf Jung committed Nov 22, 2016 518 519 ``````Add Printing Constructor ofe_mor. Existing Instance ofe_mor_ne. `````` Robbert Krebbers committed Nov 11, 2015 520 `````` `````` Robbert Krebbers committed Jun 17, 2016 521 522 523 524 ``````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 525 526 527 528 529 530 531 ``````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 532 533 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 534 `````` - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|]. `````` Robbert Krebbers committed Feb 18, 2016 535 `````` intros Hfg k; apply equiv_dist=> n; apply Hfg. `````` Robbert Krebbers committed Feb 17, 2016 536 `````` - intros n; split. `````` Robbert Krebbers committed Jan 14, 2016 537 538 `````` + by intros f x. + by intros f g ? x. `````` Ralf Jung committed Feb 20, 2016 539 `````` + by intros f g h ?? x; trans (g x). `````` Robbert Krebbers committed Feb 17, 2016 540 `````` - by intros n f g ? x; apply dist_S. `````` Robbert Krebbers committed Jan 14, 2016 541 `````` Qed. `````` Ralf Jung committed Nov 22, 2016 542 543 544 545 546 547 548 549 550 551 552 `````` 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. `````` Jacques-Henri Jourdan committed Jul 28, 2017 553 `````` Global Program Instance ofe_mor_cofe `{Cofe B} : Cofe ofe_morC := `````` Ralf Jung committed Nov 22, 2016 554 555 556 557 558 `````` {| 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 559 `````` `````` Ralf Jung committed Jan 27, 2017 560 561 562 `````` 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 563 564 565 `````` 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 566 `````` Proof. done. Qed. `````` Ralf Jung committed Nov 22, 2016 567 ``````End ofe_mor. `````` Robbert Krebbers committed Jan 14, 2016 568 `````` `````` Ralf Jung committed Nov 22, 2016 569 ``````Arguments ofe_morC : clear implicits. `````` Robbert Krebbers committed Jul 25, 2016 570 ``````Notation "A -n> B" := `````` Ralf Jung committed Nov 22, 2016 571 572 `````` (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 573 `````` Inhabited (A -n> B) := populate (λne _, inhabitant). `````` Robbert Krebbers committed Nov 11, 2015 574 `````` `````` Ralf Jung committed Mar 17, 2016 575 ``````(** Identity and composition and constant function *) `````` Robbert Krebbers committed Nov 11, 2015 576 577 ``````Definition cid {A} : A -n> A := CofeMor id. Instance: Params (@cid) 1. `````` Ralf Jung committed Nov 22, 2016 578 ``````Definition cconst {A B : ofeT} (x : B) : A -n> B := CofeMor (const x). `````` Ralf Jung committed Mar 17, 2016 579 ``````Instance: Params (@cconst) 2. `````` Robbert Krebbers committed Mar 02, 2016 580 `````` `````` Robbert Krebbers committed Nov 11, 2015 581 582 583 584 ``````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). `````` Ralf Jung committed Nov 16, 2017 585 586 587 ``````Global Instance ccompose_ne {A B C} : NonExpansive2 (@ccompose A B C). Proof. intros n ?? Hf g1 g2 Hg x. rewrite /= (Hg x) (Hf (g2 x)) //. Qed. `````` Robbert Krebbers committed Nov 11, 2015 588 `````` `````` Ralf Jung committed Mar 02, 2016 589 ``````(* Function space maps *) `````` Ralf Jung committed Nov 22, 2016 590 ``````Definition ofe_mor_map {A A' B B'} (f : A' -n> A) (g : B -n> B') `````` Ralf Jung committed Mar 02, 2016 591 `````` (h : A -n> B) : A' -n> B' := g ◎ h ◎ f. `````` Ralf Jung committed Nov 22, 2016 592 593 ``````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 594 ``````Proof. intros ??? ??? ???. by repeat apply ccompose_ne. Qed. `````` Ralf Jung committed Mar 02, 2016 595 `````` `````` Ralf Jung committed Nov 22, 2016 596 597 ``````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 598 599 ``````Instance ofe_morC_map_ne {A A' B B'} : NonExpansive2 (@ofe_morC_map A A' B B'). `````` Ralf Jung committed Mar 02, 2016 600 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 601 `````` intros n f f' Hf g g' Hg ?. rewrite /= /ofe_mor_map. `````` Robbert Krebbers committed Mar 02, 2016 602 `````` by repeat apply ccompose_ne. `````` Ralf Jung committed Mar 02, 2016 603 604 ``````Qed. `````` Robbert Krebbers committed Nov 11, 2015 605 ``````(** unit *) `````` Robbert Krebbers committed Jan 14, 2016 606 607 ``````Section unit. Instance unit_dist : Dist unit := λ _ _ _, True. `````` Ralf Jung committed Nov 22, 2016 608 `````` Definition unit_ofe_mixin : OfeMixin unit. `````` Robbert Krebbers committed Jan 14, 2016 609 `````` Proof. by repeat split; try exists 0. Qed. `````` Ralf Jung committed Nov 22, 2016 610 `````` Canonical Structure unitC : ofeT := OfeT unit unit_ofe_mixin. `````` Robbert Krebbers committed Nov 28, 2016 611 `````` `````` Ralf Jung committed Nov 22, 2016 612 613 `````` Global Program Instance unit_cofe : Cofe unitC := { compl x := () }. Next Obligation. by repeat split; try exists 0. Qed. `````` Robbert Krebbers committed Nov 28, 2016 614 `````` `````` Robbert Krebbers committed Oct 25, 2017 615 `````` Global Instance unit_ofe_discrete : OfeDiscrete unitC. `````` Robbert Krebbers committed Jan 31, 2016 616 `````` Proof. done. Qed. `````` Robbert Krebbers committed Jan 14, 2016 617 ``````End unit. `````` Robbert Krebbers committed Nov 11, 2015 618 619 `````` (** Product *) `````` Robbert Krebbers committed Jan 14, 2016 620 ``````Section product. `````` Ralf Jung committed Nov 22, 2016 621 `````` Context {A B : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 622 623 624 `````` Instance prod_dist : Dist (A * B) := λ n, prod_relation (dist n) (dist n). Global Instance pair_ne : `````` Ralf Jung committed Jan 27, 2017 625 626 627 `````` 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 628 `````` Definition prod_ofe_mixin : OfeMixin (A * B). `````` Robbert Krebbers committed Jan 14, 2016 629 630 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 631 `````` - intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation. `````` Robbert Krebbers committed Jan 14, 2016 632 `````` rewrite !equiv_dist; naive_solver. `````` Robbert Krebbers committed Feb 17, 2016 633 634 `````` - apply _. - by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S. `````` Robbert Krebbers committed Jan 14, 2016 635 `````` Qed. `````` Ralf Jung committed Nov 22, 2016 636 637 638 639 640 641 642 643 644 `````` 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. `````` Robbert Krebbers committed Oct 25, 2017 645 646 647 `````` Global Instance prod_discrete (x : A * B) : Discrete (x.1) → Discrete (x.2) → Discrete x. Proof. by intros ???[??]; split; apply (discrete _). Qed. `````` Robbert Krebbers committed Oct 25, 2017 648 649 `````` Global Instance prod_ofe_discrete : OfeDiscrete A → OfeDiscrete B → OfeDiscrete prodC. `````` Robbert Krebbers committed Feb 24, 2016 650 `````` Proof. intros ?? [??]; apply _. Qed. `````` Robbert Krebbers committed Jan 14, 2016 651 652 653 654 655 ``````End product. Arguments prodC : clear implicits. Typeclasses Opaque prod_dist. `````` Ralf Jung committed Nov 22, 2016 656 ``````Instance prod_map_ne {A A' B B' : ofeT} n : `````` Robbert Krebbers committed Nov 11, 2015 657 658 659 660 661 `````` 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 662 663 664 ``````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 665 `````` `````` Robbert Krebbers committed Mar 02, 2016 666 667 ``````(** Functors *) Structure cFunctor := CFunctor { `````` Ralf Jung committed Nov 22, 2016 668 `````` cFunctor_car : ofeT → ofeT → ofeT; `````` Robbert Krebbers committed Mar 02, 2016 669 670 `````` 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 671 672 `````` cFunctor_ne {A1 A2 B1 B2} : NonExpansive (@cFunctor_map A1 A2 B1 B2); `````` Ralf Jung committed Nov 22, 2016 673 `````` cFunctor_id {A B : ofeT} (x : cFunctor_car A B) : `````` Robbert Krebbers committed Mar 02, 2016 674 675 676 677 678 `````` 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 679 ``````Existing Instance cFunctor_ne. `````` Robbert Krebbers committed Mar 02, 2016 680 681 ``````Instance: Params (@cFunctor_map) 5. `````` Ralf Jung committed Mar 07, 2016 682 683 684 ``````Delimit Scope cFunctor_scope with CF. Bind Scope cFunctor_scope with cFunctor. `````` Ralf Jung committed Mar 07, 2016 685 686 687 ``````Class cFunctorContractive (F : cFunctor) := cFunctor_contractive A1 A2 B1 B2 :> Contractive (@cFunctor_map F A1 A2 B1 B2). `````` Ralf Jung committed Nov 22, 2016 688 ``````Definition cFunctor_diag (F: cFunctor) (A: ofeT) : ofeT := cFunctor_car F A A. `````` Robbert Krebbers committed Mar 02, 2016 689 690 ``````Coercion cFunctor_diag : cFunctor >-> Funclass. `````` Ralf Jung committed Nov 22, 2016 691 ``````Program Definition constCF (B : ofeT) : cFunctor := `````` Robbert Krebbers committed Mar 02, 2016 692 693 `````` {| cFunctor_car A1 A2 := B; cFunctor_map A1 A2 B1 B2 f := cid |}. Solve Obligations with done. `````` Ralf Jung committed Jan 06, 2017 694 ``````Coercion constCF : ofeT >-> cFunctor. `````` Robbert Krebbers committed Mar 02, 2016 695 `````` `````` Ralf Jung committed Mar 07, 2016 696 ``````Instance constCF_contractive B : cFunctorContractive (constCF B). `````` Robbert Krebbers committed Mar 07, 2016 697 ``````Proof. rewrite /cFunctorContractive; apply _. Qed. `````` Ralf Jung committed Mar 07, 2016 698 699 700 701 `````` 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 702 ``````Notation "∙" := idCF : cFunctor_scope. `````` Ralf Jung committed Mar 07, 2016 703 `````` `````` Robbert Krebbers committed Mar 02, 2016 704 705 706 707 708 ``````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 709 710 711 ``````Next Obligation. intros ?? A1 A2 B1 B2 n ???; by apply prodC_map_ne; apply cFunctor_ne. Qed. `````` Robbert Krebbers committed Mar 02, 2016 712 713 714 715 716 ``````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 717 ``````Notation "F1 * F2" := (prodCF F1%CF F2%CF) : cFunctor_scope. `````` Robbert Krebbers committed Mar 02, 2016 718 `````` `````` Ralf Jung committed Mar 07, 2016 719 720 721 722 723 724 725 726 ``````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 727 ``````Program Definition ofe_morCF (F1 F2 : cFunctor) : cFunctor := {| `````` Robbert Krebbers committed Jul 25, 2016 728 `````` cFunctor_car A B := cFunctor_car F1 B A -n> cFunctor_car F2 A B; `````` Ralf Jung committed Mar 02, 2016 729 `````` cFunctor_map A1 A2 B1 B2 fg := `````` Ralf Jung committed Nov 22, 2016 730 `````` ofe_morC_map (cFunctor_map F1 (fg.2, fg.1)) (cFunctor_map F2 fg) `````` Ralf Jung committed Mar 02, 2016 731 ``````|}. `````` Robbert Krebbers committed Mar 07, 2016 732 733 ``````Next Obligation. intros F1 F2 A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *. `````` Ralf Jung committed Nov 22, 2016 734 `````` apply ofe_morC_map_ne; apply cFunctor_ne; split; by apply Hfg. `````` Robbert Krebbers committed Mar 07, 2016 735 ``````Qed. `````` Ralf Jung committed Mar 02, 2016 736 ``````Next Obligation. `````` Robbert Krebbers committed Mar 02, 2016 737 738 `````` intros F1 F2 A B [f ?] ?; simpl. rewrite /= !cFunctor_id. apply (ne_proper f). apply cFunctor_id. `````` Ralf Jung committed Mar 02, 2016 739 740 ``````Qed. Next Obligation. `````` Robbert Krebbers committed Mar 02, 2016 741 742 `````` 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 743 ``````Qed. `````` Ralf Jung committed Jan 06, 2017 744 ``````Notation "F1 -n> F2" := (ofe_morCF F1%CF F2%CF) : cFunctor_scope. `````` Ralf Jung committed Mar 02, 2016 745 `````` `````` Ralf Jung committed Nov 22, 2016 746 ``````Instance ofe_morCF_contractive F1 F2 : `````` Ralf Jung committed Mar 07, 2016 747 `````` cFunctorContractive F1 → cFunctorContractive F2 → `````` Ralf Jung committed Nov 22, 2016 748 `````` cFunctorContractive (ofe_morCF F1 F2). `````` Ralf Jung committed Mar 07, 2016 749 750 ``````Proof. intros ?? A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *. `````` Robbert Krebbers committed Dec 05, 2016 751 `````` apply ofe_morC_map_ne; apply cFunctor_contractive; destruct n, Hfg; by split. `````` Ralf Jung committed Mar 07, 2016 752 753 ``````Qed. `````` Robbert Krebbers committed May 27, 2016 754 755 ``````(** Sum *) Section sum. `````` Ralf Jung committed Nov 22, 2016 756 `````` Context {A B : ofeT}. `````` Robbert Krebbers committed May 27, 2016 757 758 `````` Instance sum_dist : Dist (A + B) := λ n, sum_relation (dist n) (dist n). `````` Ralf Jung committed Jan 27, 2017 759 760 `````` Global Instance inl_ne : NonExpansive (@inl A B) := _. Global Instance inr_ne : NonExpansive (@inr A B) := _. `````` Robbert Krebbers committed May 27, 2016 761 762 763 `````` 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 764 765 766 767 768 769 770 771 772 773 774 775 `````` Definition sum_ofe_mixin : OfeMixin (A + B). Proof``````