iprod.v 13.6 KB
 Robbert Krebbers committed Mar 10, 2016 1 2 ``````From iris.algebra Require Export cmra. From iris.algebra Require Import upred. `````` Robbert Krebbers committed Mar 11, 2016 3 ``````From iris.prelude Require Import finite. `````` Ralf Jung committed Feb 08, 2016 4 `````` `````` Robbert Krebbers committed Feb 08, 2016 5 ``````(** * Indexed product *) `````` Ralf Jung committed Feb 08, 2016 6 7 ``````(** Need to put this in a definition to make canonical structures to work. *) Definition iprod {A} (B : A → cofeT) := ∀ x, B x. `````` Robbert Krebbers committed Feb 08, 2016 8 ``````Definition iprod_insert {A} {B : A → cofeT} `{∀ x x' : A, Decision (x = x')} `````` Ralf Jung committed Feb 08, 2016 9 10 `````` (x : A) (y : B x) (f : iprod B) : iprod B := λ x', match decide (x = x') with left H => eq_rect _ B y _ H | right _ => f x' end. `````` Robbert Krebbers committed Feb 08, 2016 11 12 13 14 ``````Global Instance iprod_empty {A} {B : A → cofeT} `{∀ x, Empty (B x)} : Empty (iprod B) := λ x, ∅. Definition iprod_singleton {A} {B : A → cofeT} `{∀ x x' : A, Decision (x = x'), ∀ x : A, Empty (B x)} `````` Ralf Jung committed Feb 08, 2016 15 `````` (x : A) (y : B x) : iprod B := iprod_insert x y ∅. `````` Robbert Krebbers committed Feb 08, 2016 16 17 ``````Instance: Params (@iprod_insert) 4. Instance: Params (@iprod_singleton) 5. `````` Ralf Jung committed Feb 08, 2016 18 19 20 21 22 `````` Section iprod_cofe. Context {A} {B : A → cofeT}. Implicit Types x : A. Implicit Types f g : iprod B. `````` Robbert Krebbers committed Feb 08, 2016 23 `````` `````` Ralf Jung committed Feb 08, 2016 24 `````` Instance iprod_equiv : Equiv (iprod B) := λ f g, ∀ x, f x ≡ g x. `````` Ralf Jung committed Feb 10, 2016 25 `````` Instance iprod_dist : Dist (iprod B) := λ n f g, ∀ x, f x ≡{n}≡ g x. `````` Ralf Jung committed Feb 08, 2016 26 27 28 29 30 31 32 33 `````` Program Definition iprod_chain (c : chain (iprod B)) (x : A) : chain (B x) := {| chain_car n := c n x |}. Next Obligation. by intros c x n i ?; apply (chain_cauchy c). Qed. Program Instance iprod_compl : Compl (iprod B) := λ c x, compl (iprod_chain c x). Definition iprod_cofe_mixin : CofeMixin (iprod B). Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 34 `````` - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|]. `````` Ralf Jung committed Feb 08, 2016 35 `````` intros Hfg k; apply equiv_dist; intros n; apply Hfg. `````` Robbert Krebbers committed Feb 17, 2016 36 `````` - intros n; split. `````` Ralf Jung committed Feb 08, 2016 37 38 `````` + by intros f x. + by intros f g ? x. `````` Ralf Jung committed Feb 20, 2016 39 `````` + by intros f g h ?? x; trans (g x). `````` Robbert Krebbers committed Feb 17, 2016 40 `````` - intros n f g Hfg x; apply dist_S, Hfg. `````` Robbert Krebbers committed Feb 18, 2016 41 42 `````` - intros n c x. rewrite /compl /iprod_compl (conv_compl n (iprod_chain c x)). `````` Ralf Jung committed Feb 08, 2016 43 44 45 46 `````` apply (chain_cauchy c); lia. Qed. Canonical Structure iprodC : cofeT := CofeT iprod_cofe_mixin. `````` Robbert Krebbers committed Feb 08, 2016 47 48 49 50 `````` (** Properties of empty *) Section empty. Context `{∀ x, Empty (B x)}. Definition iprod_lookup_empty x : ∅ x = ∅ := eq_refl. `````` Ralf Jung committed Feb 09, 2016 51 `````` Global Instance iprod_empty_timeless : `````` Robbert Krebbers committed Feb 08, 2016 52 `````` (∀ x : A, Timeless (∅ : B x)) → Timeless (∅ : iprod B). `````` Ralf Jung committed Feb 18, 2016 53 `````` Proof. intros ? f Hf x. by apply: timeless. Qed. `````` Robbert Krebbers committed Feb 08, 2016 54 55 `````` End empty. `````` Ralf Jung committed Feb 08, 2016 56 `````` (** Properties of iprod_insert. *) `````` Ralf Jung committed Feb 08, 2016 57 `````` Context `{∀ x x' : A, Decision (x = x')}. `````` Ralf Jung committed Feb 08, 2016 58 `````` `````` Robbert Krebbers committed Feb 18, 2016 59 `````` Global Instance iprod_insert_ne n x : `````` Ralf Jung committed Feb 08, 2016 60 61 62 63 64 65 66 `````` Proper (dist n ==> dist n ==> dist n) (iprod_insert x). Proof. intros y1 y2 ? f1 f2 ? x'; rewrite /iprod_insert. by destruct (decide _) as [[]|]. Qed. Global Instance iprod_insert_proper x : Proper ((≡) ==> (≡) ==> (≡)) (iprod_insert x) := ne_proper_2 _. `````` Ralf Jung committed Feb 08, 2016 67 `````` `````` Ralf Jung committed Feb 08, 2016 68 69 70 71 72 73 74 75 76 `````` Lemma iprod_lookup_insert f x y : (iprod_insert x y f) x = y. Proof. rewrite /iprod_insert; destruct (decide _) as [Hx|]; last done. by rewrite (proof_irrel Hx eq_refl). Qed. Lemma iprod_lookup_insert_ne f x x' y : x ≠ x' → (iprod_insert x y f) x' = f x'. Proof. by rewrite /iprod_insert; destruct (decide _). Qed. `````` Robbert Krebbers committed Feb 08, 2016 77 `````` Global Instance iprod_lookup_timeless f x : Timeless f → Timeless (f x). `````` Ralf Jung committed Feb 08, 2016 78 `````` Proof. `````` Robbert Krebbers committed Feb 08, 2016 79 `````` intros ? y ?. `````` Ralf Jung committed Feb 08, 2016 80 `````` cut (f ≡ iprod_insert x y f). `````` Robbert Krebbers committed Feb 08, 2016 81 `````` { by move=> /(_ x)->; rewrite iprod_lookup_insert. } `````` Ralf Jung committed Feb 18, 2016 82 `````` by apply: timeless=>x'; destruct (decide (x = x')) as [->|]; `````` Robbert Krebbers committed Feb 08, 2016 83 `````` rewrite ?iprod_lookup_insert ?iprod_lookup_insert_ne. `````` Ralf Jung committed Feb 08, 2016 84 85 86 87 `````` Qed. Global Instance iprod_insert_timeless f x y : Timeless f → Timeless y → Timeless (iprod_insert x y f). Proof. `````` Robbert Krebbers committed Feb 08, 2016 88 `````` intros ?? g Heq x'; destruct (decide (x = x')) as [->|]. `````` Robbert Krebbers committed Feb 17, 2016 89 `````` - rewrite iprod_lookup_insert. `````` Ralf Jung committed Feb 18, 2016 90 `````` apply: timeless. by rewrite -(Heq x') iprod_lookup_insert. `````` Robbert Krebbers committed Feb 17, 2016 91 `````` - rewrite iprod_lookup_insert_ne //. `````` Ralf Jung committed Feb 18, 2016 92 `````` apply: timeless. by rewrite -(Heq x') iprod_lookup_insert_ne. `````` Ralf Jung committed Feb 08, 2016 93 94 95 `````` Qed. (** Properties of iprod_singletom. *) `````` Ralf Jung committed Feb 08, 2016 96 `````` Context `{∀ x : A, Empty (B x)}. `````` Robbert Krebbers committed Feb 08, 2016 97 `````` `````` Robbert Krebbers committed Feb 18, 2016 98 `````` Global Instance iprod_singleton_ne n x : `````` Ralf Jung committed Feb 08, 2016 99 100 101 102 `````` Proper (dist n ==> dist n) (iprod_singleton x). Proof. by intros y1 y2 Hy; rewrite /iprod_singleton Hy. Qed. Global Instance iprod_singleton_proper x : Proper ((≡) ==> (≡)) (iprod_singleton x) := ne_proper _. `````` Robbert Krebbers committed Feb 08, 2016 103 `````` `````` Ralf Jung committed Feb 08, 2016 104 105 `````` Lemma iprod_lookup_singleton x y : (iprod_singleton x y) x = y. Proof. by rewrite /iprod_singleton iprod_lookup_insert. Qed. `````` Robbert Krebbers committed Feb 08, 2016 106 `````` Lemma iprod_lookup_singleton_ne x x' y: x ≠ x' → (iprod_singleton x y) x' = ∅. `````` Ralf Jung committed Feb 08, 2016 107 `````` Proof. intros; by rewrite /iprod_singleton iprod_lookup_insert_ne. Qed. `````` Ralf Jung committed Feb 08, 2016 108 109 `````` Global Instance iprod_singleton_timeless x (y : B x) : `````` Robbert Krebbers committed Feb 08, 2016 110 111 `````` (∀ x : A, Timeless (∅ : B x)) → Timeless y → Timeless (iprod_singleton x y). Proof. eauto using iprod_insert_timeless, iprod_empty_timeless. Qed. `````` Ralf Jung committed Feb 08, 2016 112 113 114 115 116 ``````End iprod_cofe. Arguments iprodC {_} _. Section iprod_cmra. `````` Robbert Krebbers committed Mar 11, 2016 117 `````` Context `{Finite A} {B : A → cmraT}. `````` Ralf Jung committed Feb 08, 2016 118 `````` Implicit Types f g : iprod B. `````` Robbert Krebbers committed Feb 08, 2016 119 `````` `````` Ralf Jung committed Feb 08, 2016 120 `````` Instance iprod_op : Op (iprod B) := λ f g x, f x ⋅ g x. `````` Ralf Jung committed Mar 08, 2016 121 `````` Instance iprod_core : Core (iprod B) := λ f x, core (f x). `````` Robbert Krebbers committed Feb 24, 2016 122 `````` Instance iprod_valid : Valid (iprod B) := λ f, ∀ x, ✓ f x. `````` Robbert Krebbers committed Feb 11, 2016 123 `````` Instance iprod_validN : ValidN (iprod B) := λ n f, ∀ x, ✓{n} f x. `````` Robbert Krebbers committed Feb 08, 2016 124 125 `````` Definition iprod_lookup_op f g x : (f ⋅ g) x = f x ⋅ g x := eq_refl. `````` Ralf Jung committed Mar 08, 2016 126 `````` Definition iprod_lookup_core f x : (core f) x = core (f x) := eq_refl. `````` Robbert Krebbers committed Feb 08, 2016 127 `````` `````` Robbert Krebbers committed Feb 26, 2016 128 `````` Lemma iprod_included_spec (f g : iprod B) : f ≼ g ↔ ∀ x, f x ≼ g x. `````` Ralf Jung committed Feb 08, 2016 129 `````` Proof. `````` Robbert Krebbers committed Mar 11, 2016 130 131 `````` split; [by intros [h Hh] x; exists (h x); rewrite /op /iprod_op (Hh x)|]. intros [h ?]%finite_choice. by exists h. `````` Ralf Jung committed Feb 08, 2016 132 `````` Qed. `````` Robbert Krebbers committed Feb 08, 2016 133 `````` `````` Ralf Jung committed Feb 08, 2016 134 135 136 `````` Definition iprod_cmra_mixin : CMRAMixin (iprod B). Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 137 `````` - by intros n f1 f2 f3 Hf x; rewrite iprod_lookup_op (Hf x). `````` Ralf Jung committed Mar 08, 2016 138 `````` - by intros n f1 f2 Hf x; rewrite iprod_lookup_core (Hf x). `````` Robbert Krebbers committed Feb 17, 2016 139 `````` - by intros n f1 f2 Hf ? x; rewrite -(Hf x). `````` Robbert Krebbers committed Feb 24, 2016 140 141 142 `````` - intros g; split. + intros Hg n i; apply cmra_valid_validN, Hg. + intros Hg i; apply cmra_valid_validN=> n; apply Hg. `````` Robbert Krebbers committed Feb 17, 2016 143 144 145 `````` - intros n f Hf x; apply cmra_validN_S, Hf. - by intros f1 f2 f3 x; rewrite iprod_lookup_op assoc. - by intros f1 f2 x; rewrite iprod_lookup_op comm. `````` Ralf Jung committed Mar 08, 2016 146 147 `````` - by intros f x; rewrite iprod_lookup_op iprod_lookup_core cmra_core_l. - by intros f x; rewrite iprod_lookup_core cmra_core_idemp. `````` Robbert Krebbers committed Feb 26, 2016 148 `````` - intros f1 f2; rewrite !iprod_included_spec=> Hf x. `````` Ralf Jung committed Mar 08, 2016 149 `````` by rewrite iprod_lookup_core; apply cmra_core_preserving, Hf. `````` Robbert Krebbers committed Feb 17, 2016 150 `````` - intros n f1 f2 Hf x; apply cmra_validN_op_l with (f2 x), Hf. `````` Robbert Krebbers committed Feb 24, 2016 151 152 153 154 `````` - intros n f f1 f2 Hf Hf12. set (g x := cmra_extend n (f x) (f1 x) (f2 x) (Hf x) (Hf12 x)). exists ((λ x, (proj1_sig (g x)).1), (λ x, (proj1_sig (g x)).2)). split_and?; intros x; apply (proj2_sig (g x)). `````` Ralf Jung committed Feb 08, 2016 155 `````` Qed. `````` Robbert Krebbers committed Mar 01, 2016 156 `````` Canonical Structure iprodR : cmraT := CMRAT iprod_cofe_mixin iprod_cmra_mixin. `````` Ralf Jung committed Mar 08, 2016 157 158 `````` Global Instance iprod_cmra_unit `{∀ x, Empty (B x)} : (∀ x, CMRAUnit (B x)) → CMRAUnit iprodR. `````` Ralf Jung committed Feb 08, 2016 159 160 `````` Proof. intros ?; split. `````` Ralf Jung committed Mar 08, 2016 161 `````` - intros x; apply cmra_unit_valid. `````` Robbert Krebbers committed Feb 17, 2016 162 163 `````` - by intros f x; rewrite iprod_lookup_op left_id. - by apply _. `````` Ralf Jung committed Feb 08, 2016 164 165 `````` Qed. `````` Robbert Krebbers committed Feb 13, 2016 166 `````` (** Internalized properties *) `````` Ralf Jung committed Mar 10, 2016 167 `````` Lemma iprod_equivI {M} g1 g2 : (g1 ≡ g2) ⊣⊢ (∀ i, g1 i ≡ g2 i : uPred M). `````` Robbert Krebbers committed Feb 25, 2016 168 `````` Proof. by uPred.unseal. Qed. `````` Ralf Jung committed Mar 10, 2016 169 `````` Lemma iprod_validI {M} g : (✓ g) ⊣⊢ (∀ i, ✓ g i : uPred M). `````` Robbert Krebbers committed Feb 25, 2016 170 `````` Proof. by uPred.unseal. Qed. `````` Robbert Krebbers committed Feb 13, 2016 171 `````` `````` Ralf Jung committed Feb 08, 2016 172 `````` (** Properties of iprod_insert. *) `````` Ralf Jung committed Feb 08, 2016 173 `````` Context `{∀ x x' : A, Decision (x = x')}. `````` Ralf Jung committed Feb 08, 2016 174 `````` `````` Ralf Jung committed Feb 08, 2016 175 176 177 178 `````` Lemma iprod_insert_updateP x (P : B x → Prop) (Q : iprod B → Prop) g y1 : y1 ~~>: P → (∀ y2, P y2 → Q (iprod_insert x y2 g)) → iprod_insert x y1 g ~~>: Q. Proof. `````` Robbert Krebbers committed Feb 18, 2016 179 `````` intros Hy1 HP n gf Hg. destruct (Hy1 n (gf x)) as (y2&?&?). `````` Ralf Jung committed Feb 08, 2016 180 181 182 183 184 185 186 187 188 189 190 191 192 193 `````` { move: (Hg x). by rewrite iprod_lookup_op iprod_lookup_insert. } exists (iprod_insert x y2 g); split; [auto|]. intros x'; destruct (decide (x' = x)) as [->|]; rewrite iprod_lookup_op ?iprod_lookup_insert //; []. move: (Hg x'). by rewrite iprod_lookup_op !iprod_lookup_insert_ne. Qed. Lemma iprod_insert_updateP' x (P : B x → Prop) g y1 : y1 ~~>: P → iprod_insert x y1 g ~~>: λ g', ∃ y2, g' = iprod_insert x y2 g ∧ P y2. Proof. eauto using iprod_insert_updateP. Qed. Lemma iprod_insert_update g x y1 y2 : y1 ~~> y2 → iprod_insert x y1 g ~~> iprod_insert x y2 g. Proof. `````` Robbert Krebbers committed Feb 08, 2016 194 `````` rewrite !cmra_update_updateP; eauto using iprod_insert_updateP with subst. `````` Ralf Jung committed Feb 08, 2016 195 196 `````` Qed. `````` Ralf Jung committed Feb 08, 2016 197 `````` (** Properties of iprod_singleton. *) `````` Ralf Jung committed Mar 08, 2016 198 `````` Context `{∀ x, Empty (B x), ∀ x, CMRAUnit (B x)}. `````` Ralf Jung committed Feb 08, 2016 199 `````` `````` Robbert Krebbers committed Feb 11, 2016 200 `````` Lemma iprod_singleton_validN n x (y: B x) : ✓{n} iprod_singleton x y ↔ ✓{n} y. `````` Ralf Jung committed Feb 08, 2016 201 `````` Proof. `````` Robbert Krebbers committed Feb 08, 2016 202 203 204 `````` split; [by move=>/(_ x); rewrite iprod_lookup_singleton|]. move=>Hx x'; destruct (decide (x = x')) as [->|]; rewrite ?iprod_lookup_singleton ?iprod_lookup_singleton_ne //. `````` Ralf Jung committed Mar 08, 2016 205 `````` by apply cmra_unit_validN. `````` Ralf Jung committed Feb 08, 2016 206 207 `````` Qed. `````` Ralf Jung committed Mar 08, 2016 208 209 `````` Lemma iprod_core_singleton x (y : B x) : core (iprod_singleton x y) ≡ iprod_singleton x (core y). `````` Ralf Jung committed Feb 08, 2016 210 `````` Proof. `````` Robbert Krebbers committed Mar 15, 2016 211 212 213 `````` move=>x'; destruct (decide (x = x')) as [->|]; by rewrite iprod_lookup_core ?iprod_lookup_singleton ?iprod_lookup_singleton_ne // (persistent ∅). `````` Ralf Jung committed Feb 08, 2016 214 215 `````` Qed. `````` Robbert Krebbers committed Mar 15, 2016 216 217 218 219 `````` Global Instance iprod_singleton_persistent x (y : B x) : Persistent y → Persistent (iprod_singleton x y). Proof. intros. rewrite /Persistent iprod_core_singleton. by f_equiv. Qed. `````` Ralf Jung committed Feb 08, 2016 220 221 222 223 `````` Lemma iprod_op_singleton (x : A) (y1 y2 : B x) : iprod_singleton x y1 ⋅ iprod_singleton x y2 ≡ iprod_singleton x (y1 ⋅ y2). Proof. intros x'; destruct (decide (x' = x)) as [->|]. `````` Robbert Krebbers committed Feb 17, 2016 224 225 `````` - by rewrite iprod_lookup_op !iprod_lookup_singleton. - by rewrite iprod_lookup_op !iprod_lookup_singleton_ne // left_id. `````` Ralf Jung committed Feb 08, 2016 226 227 228 229 230 231 232 233 `````` Qed. Lemma iprod_singleton_updateP x (P : B x → Prop) (Q : iprod B → Prop) y1 : y1 ~~>: P → (∀ y2, P y2 → Q (iprod_singleton x y2)) → iprod_singleton x y1 ~~>: Q. Proof. rewrite /iprod_singleton; eauto using iprod_insert_updateP. Qed. Lemma iprod_singleton_updateP' x (P : B x → Prop) y1 : y1 ~~>: P → `````` Ralf Jung committed Feb 09, 2016 234 `````` iprod_singleton x y1 ~~>: λ g, ∃ y2, g = iprod_singleton x y2 ∧ P y2. `````` Ralf Jung committed Feb 08, 2016 235 `````` Proof. eauto using iprod_singleton_updateP. Qed. `````` Robbert Krebbers committed Feb 08, 2016 236 237 238 `````` Lemma iprod_singleton_update x y1 y2 : y1 ~~> y2 → iprod_singleton x y1 ~~> iprod_singleton x y2. Proof. eauto using iprod_insert_update. Qed. `````` Ralf Jung committed Feb 08, 2016 239 240 `````` Lemma iprod_singleton_updateP_empty x (P : B x → Prop) (Q : iprod B → Prop) : `````` Robbert Krebbers committed Feb 08, 2016 241 `````` ∅ ~~>: P → (∀ y2, P y2 → Q (iprod_singleton x y2)) → ∅ ~~>: Q. `````` Ralf Jung committed Feb 08, 2016 242 `````` Proof. `````` Robbert Krebbers committed Feb 18, 2016 243 `````` intros Hx HQ n gf Hg. destruct (Hx n (gf x)) as (y2&?&?); first apply Hg. `````` Robbert Krebbers committed Feb 08, 2016 244 245 `````` exists (iprod_singleton x y2); split; [by apply HQ|]. intros x'; destruct (decide (x' = x)) as [->|]. `````` Robbert Krebbers committed Feb 17, 2016 246 247 `````` - by rewrite iprod_lookup_op iprod_lookup_singleton. - rewrite iprod_lookup_op iprod_lookup_singleton_ne //. apply Hg. `````` Ralf Jung committed Feb 08, 2016 248 `````` Qed. `````` Ralf Jung committed Feb 09, 2016 249 250 251 `````` Lemma iprod_singleton_updateP_empty' x (P : B x → Prop) : ∅ ~~>: P → ∅ ~~>: λ g, ∃ y2, g = iprod_singleton x y2 ∧ P y2. Proof. eauto using iprod_singleton_updateP_empty. Qed. `````` Ralf Jung committed Feb 08, 2016 252 253 ``````End iprod_cmra. `````` Robbert Krebbers committed Mar 11, 2016 254 ``````Arguments iprodR {_ _ _} _. `````` Ralf Jung committed Feb 08, 2016 255 `````` `````` Robbert Krebbers committed Feb 08, 2016 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 ``````(** * Functor *) Definition iprod_map {A} {B1 B2 : A → cofeT} (f : ∀ x, B1 x → B2 x) (g : iprod B1) : iprod B2 := λ x, f _ (g x). Lemma iprod_map_ext {A} {B1 B2 : A → cofeT} (f1 f2 : ∀ x, B1 x → B2 x) g : (∀ x, f1 x (g x) ≡ f2 x (g x)) → iprod_map f1 g ≡ iprod_map f2 g. Proof. done. Qed. Lemma iprod_map_id {A} {B: A → cofeT} (g : iprod B) : iprod_map (λ _, id) g = g. Proof. done. Qed. Lemma iprod_map_compose {A} {B1 B2 B3 : A → cofeT} (f1 : ∀ x, B1 x → B2 x) (f2 : ∀ x, B2 x → B3 x) (g : iprod B1) : iprod_map (λ x, f2 x ∘ f1 x) g = iprod_map f2 (iprod_map f1 g). Proof. done. Qed. Instance iprod_map_ne {A} {B1 B2 : A → cofeT} (f : ∀ x, B1 x → B2 x) n : (∀ x, Proper (dist n ==> dist n) (f x)) → Proper (dist n ==> dist n) (iprod_map f). Proof. by intros ? y1 y2 Hy x; rewrite /iprod_map (Hy x). Qed. `````` Robbert Krebbers committed Mar 11, 2016 274 275 ``````Instance iprod_map_cmra_monotone `{Finite A} {B1 B2: A → cmraT} (f : ∀ x, B1 x → B2 x) : `````` Ralf Jung committed Feb 08, 2016 276 277 `````` (∀ x, CMRAMonotone (f x)) → CMRAMonotone (iprod_map f). Proof. `````` Robbert Krebbers committed Feb 26, 2016 278 `````` split; first apply _. `````` Robbert Krebbers committed Feb 26, 2016 279 `````` - intros n g Hg x; rewrite /iprod_map; apply (validN_preserving (f _)), Hg. `````` Robbert Krebbers committed Feb 26, 2016 280 `````` - intros g1 g2; rewrite !iprod_included_spec=> Hf x. `````` Robbert Krebbers committed Feb 26, 2016 281 `````` rewrite /iprod_map; apply (included_preserving _), Hf. `````` Ralf Jung committed Feb 08, 2016 282 283 ``````Qed. `````` Robbert Krebbers committed Feb 08, 2016 284 285 286 287 288 289 ``````Definition iprodC_map {A} {B1 B2 : A → cofeT} (f : iprod (λ x, B1 x -n> B2 x)) : iprodC B1 -n> iprodC B2 := CofeMor (iprod_map f). Instance iprodC_map_ne {A} {B1 B2 : A → cofeT} n : Proper (dist n ==> dist n) (@iprodC_map A B1 B2). Proof. intros f1 f2 Hf g x; apply Hf. Qed. `````` Robbert Krebbers committed Mar 02, 2016 290 291 292 ``````Program Definition iprodCF {C} (F : C → cFunctor) : cFunctor := {| cFunctor_car A B := iprodC (λ c, cFunctor_car (F c) A B); cFunctor_map A1 A2 B1 B2 fg := iprodC_map (λ c, cFunctor_map (F c) fg) `````` Ralf Jung committed Feb 08, 2016 293 ``````|}. `````` Robbert Krebbers committed Mar 07, 2016 294 295 296 ``````Next Obligation. intros C F A1 A2 B1 B2 n ?? g. by apply iprodC_map_ne=>?; apply cFunctor_ne. Qed. `````` Ralf Jung committed Feb 08, 2016 297 ``````Next Obligation. `````` Robbert Krebbers committed Mar 02, 2016 298 299 `````` intros C F A B g; simpl. rewrite -{2}(iprod_map_id g). apply iprod_map_ext=> y; apply cFunctor_id. `````` Ralf Jung committed Feb 08, 2016 300 301 ``````Qed. Next Obligation. `````` Robbert Krebbers committed Mar 02, 2016 302 303 304 `````` intros C F A1 A2 A3 B1 B2 B3 f1 f2 f1' f2' g. rewrite /= -iprod_map_compose. apply iprod_map_ext=>y; apply cFunctor_compose. Qed. `````` Ralf Jung committed Mar 07, 2016 305 306 307 308 309 310 311 ``````Instance iprodCF_contractive {C} (F : C → cFunctor) : (∀ c, cFunctorContractive (F c)) → cFunctorContractive (iprodCF F). Proof. intros ? A1 A2 B1 B2 n ?? g. by apply iprodC_map_ne=>c; apply cFunctor_contractive. Qed. `````` Robbert Krebbers committed Mar 11, 2016 312 ``````Program Definition iprodRF `{Finite C} (F : C → rFunctor) : rFunctor := {| `````` Robbert Krebbers committed Mar 02, 2016 313 314 315 `````` rFunctor_car A B := iprodR (λ c, rFunctor_car (F c) A B); rFunctor_map A1 A2 B1 B2 fg := iprodC_map (λ c, rFunctor_map (F c) fg) |}. `````` Robbert Krebbers committed Mar 07, 2016 316 ``````Next Obligation. `````` Robbert Krebbers committed Mar 11, 2016 317 318 `````` intros C ?? F A1 A2 B1 B2 n ?? g. by apply iprodC_map_ne=>?; apply rFunctor_ne. `````` Robbert Krebbers committed Mar 07, 2016 319 ``````Qed. `````` Robbert Krebbers committed Mar 02, 2016 320 ``````Next Obligation. `````` Robbert Krebbers committed Mar 11, 2016 321 `````` intros C ?? F A B g; simpl. rewrite -{2}(iprod_map_id g). `````` Robbert Krebbers committed Mar 02, 2016 322 323 324 `````` apply iprod_map_ext=> y; apply rFunctor_id. Qed. Next Obligation. `````` Robbert Krebbers committed Mar 11, 2016 325 `````` intros C ?? F A1 A2 A3 B1 B2 B3 f1 f2 f1' f2' g. rewrite /=-iprod_map_compose. `````` Robbert Krebbers committed Mar 02, 2016 326 `````` apply iprod_map_ext=>y; apply rFunctor_compose. `````` Ralf Jung committed Feb 08, 2016 327 ``````Qed. `````` Robbert Krebbers committed Mar 11, 2016 328 ``````Instance iprodRF_contractive `{Finite C} (F : C → rFunctor) : `````` Ralf Jung committed Mar 07, 2016 329 330 331 332 333 `````` (∀ c, rFunctorContractive (F c)) → rFunctorContractive (iprodRF F). Proof. intros ? A1 A2 B1 B2 n ?? g. by apply iprodC_map_ne=>c; apply rFunctor_contractive. Qed.``````