proof.v 16.6 KB
 Robbert Krebbers committed Feb 24, 2016 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 ``````From prelude Require Import functions. From algebra Require Import upred_big_op upred_tactics. From program_logic Require Import sts saved_prop. From heap_lang Require Export heap wp_tactics. From barrier Require Import protocol. From barrier Require Export barrier. Import uPred. (** The monoids we need. *) (* Not bundling heapG, as it may be shared with other users. *) Class barrierG Σ := BarrierG { barrier_stsG :> stsG heap_lang Σ sts; barrier_savedPropG :> savedPropG heap_lang Σ; }. Definition barrierGF : iFunctors := [stsGF sts; agreeF]. Instance inGF_barrierG `{inGF heap_lang Σ (stsGF sts), inGF heap_lang Σ agreeF} : barrierG Σ. Proof. split; apply _. Qed. (** Now we come to the Iris part of the proof. *) Section proof. Context {Σ : iFunctorG} `{!heapG Σ, !barrierG Σ}. Context (heapN N : namespace). Local Notation iProp := (iPropG heap_lang Σ). Definition waiting (P : iProp) (I : gset gname) : iProp := (∃ Ψ : gname → iProp, ▷ (P -★ Π★{set I} Ψ) ★ Π★{set I} (λ i, saved_prop_own i (Ψ i)))%I. Definition ress (I : gset gname) : iProp := (Π★{set I} (λ i, ∃ R, saved_prop_own i R ★ ▷ R))%I. Coercion state_to_val (s : state) : val := match s with State Low _ => '0 | State High _ => '1 end. Arguments state_to_val !_ /. Definition barrier_inv (l : loc) (P : iProp) (s : state) : iProp := (l ↦ s ★ match s with State Low I' => waiting P I' | State High I' => ress I' end )%I. Definition barrier_ctx (γ : gname) (l : loc) (P : iProp) : iProp := (■ (heapN ⊥ N) ★ heap_ctx heapN ★ sts_ctx γ N (barrier_inv l P))%I. Definition send (l : loc) (P : iProp) : iProp := (∃ γ, barrier_ctx γ l P ★ sts_ownS γ low_states {[ Send ]})%I. Definition recv (l : loc) (R : iProp) : iProp := (∃ γ P Q i, barrier_ctx γ l P ★ sts_ownS γ (i_states i) {[ Change i ]} ★ saved_prop_own i Q ★ ▷ (Q -★ R))%I. (** Setoids *) Global Instance waiting_ne n : Proper (dist n ==> (=) ==> dist n) waiting. `````` Ralf Jung committed Feb 25, 2016 56 ``````Proof. solve_proper. Qed. `````` Robbert Krebbers committed Feb 24, 2016 57 ``````Global Instance barrier_inv_ne n l : `````` Ralf Jung committed Feb 25, 2016 58 59 `````` Proper (dist n ==> eq ==> dist n) (barrier_inv l). Proof. solve_proper. Qed. `````` Robbert Krebbers committed Feb 24, 2016 60 ``````Global Instance barrier_ctx_ne n γ l : Proper (dist n ==> dist n) (barrier_ctx γ l). `````` Ralf Jung committed Feb 25, 2016 61 ``````Proof. solve_proper. Qed. `````` Robbert Krebbers committed Feb 24, 2016 62 ``````Global Instance send_ne n l : Proper (dist n ==> dist n) (send l). `````` Ralf Jung committed Feb 25, 2016 63 ``````Proof. solve_proper. Qed. `````` Robbert Krebbers committed Feb 24, 2016 64 ``````Global Instance recv_ne n l : Proper (dist n ==> dist n) (recv l). `````` Ralf Jung committed Feb 25, 2016 65 ``````Proof. solve_proper. Qed. `````` Robbert Krebbers committed Feb 24, 2016 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 `````` (** Helper lemmas *) Lemma waiting_split i i1 i2 Q R1 R2 P I : i ∈ I → i1 ∉ I → i2 ∉ I → i1 ≠ i2 → (saved_prop_own i2 R2 ★ saved_prop_own i1 R1 ★ saved_prop_own i Q ★ (Q -★ R1 ★ R2) ★ waiting P I) ⊑ waiting P ({[i1]} ∪ ({[i2]} ∪ (I ∖ {[i]}))). Proof. intros. rewrite /waiting !sep_exist_l. apply exist_elim=>Ψ. rewrite -(exist_intro (<[i1:=R1]> (<[i2:=R2]> Ψ))). rewrite [(Π★{set _} (λ _, saved_prop_own _ _))%I](big_sepS_delete _ I i) //. rewrite !assoc [(_ ★ (_ -★ _))%I]comm !assoc [(_ ★ ▷ _)%I]comm. rewrite !assoc [(_ ★ _ i _)%I]comm !assoc [(_ ★ _ i _)%I]comm -!assoc. do 4 (rewrite big_sepS_insert; last set_solver). rewrite !fn_lookup_insert fn_lookup_insert_ne // !fn_lookup_insert. rewrite 3!assoc. apply sep_mono. `````` Robbert Krebbers committed Feb 25, 2016 82 `````` - rewrite saved_prop_agree. strip_later. `````` Robbert Krebbers committed Feb 24, 2016 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 `````` apply wand_intro_l. rewrite [(_ ★ (_ -★ Π★{set _} _))%I]comm !assoc wand_elim_r. rewrite (big_sepS_delete _ I i) //. rewrite [(_ ★ Π★{set _} _)%I]comm [(_ ★ Π★{set _} _)%I]comm -!assoc. apply sep_mono. + apply big_sepS_mono; [done|] => j. rewrite elem_of_difference not_elem_of_singleton=> -[??]. by do 2 (rewrite fn_lookup_insert_ne; last naive_solver). + rewrite !assoc. eapply wand_apply_r'; first done. apply: (eq_rewrite (Ψ i) Q (λ x, x)%I); last by eauto with I. rewrite eq_sym. eauto with I. - rewrite !assoc [(saved_prop_own i2 _ ★ _)%I]comm; apply sep_mono_r. apply big_sepS_mono; [done|]=> j. rewrite elem_of_difference not_elem_of_singleton=> -[??]. by do 2 (rewrite fn_lookup_insert_ne; last naive_solver). Qed. Lemma ress_split i i1 i2 Q R1 R2 I : i ∈ I → i1 ∉ I → i2 ∉ I → i1 ≠ i2 → (saved_prop_own i2 R2 ★ saved_prop_own i1 R1 ★ saved_prop_own i Q ★ (Q -★ R1 ★ R2) ★ ress I) ⊑ ress ({[i1]} ∪ ({[i2]} ∪ (I ∖ {[i]}))). Proof. intros. rewrite /ress. rewrite [(Π★{set _} _)%I](big_sepS_delete _ I i) // !assoc !sep_exist_l !sep_exist_r. apply exist_elim=>R. do 2 (rewrite big_sepS_insert; last set_solver). rewrite -(exist_intro R1) -(exist_intro R2) [(_ i2 _ ★ _)%I]comm -!assoc. apply sep_mono_r. rewrite !assoc. apply sep_mono_l. rewrite [(▷ _ ★ _ i2 _)%I]comm -!assoc. apply sep_mono_r. rewrite !assoc [(_ ★ _ i R)%I]comm !assoc saved_prop_agree. rewrite [(▷ _ ★ _)%I]comm -!assoc. eapply wand_apply_l. { by rewrite <-later_wand, <-later_intro. } { by rewrite later_sep. } `````` Robbert Krebbers committed Feb 25, 2016 117 `````` strip_later. apply: (eq_rewrite R Q (λ x, x)%I); eauto with I. `````` Robbert Krebbers committed Feb 24, 2016 118 119 120 ``````Qed. (** Actual proofs *) `````` Ralf Jung committed Feb 28, 2016 121 ``````Lemma newbarrier_spec (P : iProp) (Φ : val → iProp) : `````` Robbert Krebbers committed Feb 24, 2016 122 123 `````` heapN ⊥ N → (heap_ctx heapN ★ ∀ l, recv l P ★ send l P -★ Φ (LocV l)) `````` Ralf Jung committed Feb 28, 2016 124 `````` ⊑ || newbarrier '() {{ Φ }}. `````` Robbert Krebbers committed Feb 24, 2016 125 ``````Proof. `````` Ralf Jung committed Feb 28, 2016 126 `````` intros HN. rewrite /newbarrier. wp_seq. `````` Robbert Krebbers committed Feb 24, 2016 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 `````` rewrite -wp_pvs. wp eapply wp_alloc; eauto with I ndisj. apply forall_intro=>l. rewrite (forall_elim l). apply wand_intro_l. rewrite !assoc. apply pvs_wand_r. (* The core of this proof: Allocating the STS and the saved prop. *) eapply sep_elim_True_r; first by eapply (saved_prop_alloc _ P). rewrite pvs_frame_l. apply pvs_strip_pvs. rewrite sep_exist_l. apply exist_elim=>i. trans (pvs ⊤ ⊤ (heap_ctx heapN ★ ▷ (barrier_inv l P (State Low {[ i ]})) ★ saved_prop_own i P)). - rewrite -pvs_intro. cancel [heap_ctx heapN]. rewrite {1}[saved_prop_own _ _]always_sep_dup. cancel [saved_prop_own i P]. rewrite /barrier_inv /waiting -later_intro. cancel [l ↦ '0]%I. rewrite -(exist_intro (const P)) /=. rewrite -[saved_prop_own _ _](left_id True%I (★)%I). by rewrite !big_sepS_singleton /= wand_diag -later_intro. - rewrite (sts_alloc (barrier_inv l P) ⊤ N); last by eauto. rewrite !pvs_frame_r !pvs_frame_l. rewrite pvs_trans'. apply pvs_strip_pvs. rewrite sep_exist_r sep_exist_l. apply exist_elim=>γ. rewrite /recv /send. rewrite -(exist_intro γ) -(exist_intro P). rewrite -(exist_intro P) -(exist_intro i) -(exist_intro γ). (* This is even more annoying than usually, since rewrite sometimes unfolds stuff... *) rewrite [barrier_ctx _ _ _]lock !assoc [(_ ★ locked (barrier_ctx _ _ _))%I]comm !assoc -lock. rewrite -always_sep_dup. (* TODO: This is cancelling below a pvs. *) rewrite [barrier_ctx _ _ _]lock always_and_sep_l -!assoc assoc -lock. rewrite -pvs_frame_l. rewrite /barrier_ctx const_equiv // left_id. apply sep_mono_r. rewrite [(saved_prop_own _ _ ★ _)%I]comm !assoc. rewrite -pvs_frame_r. apply sep_mono_l. rewrite -assoc [(▷ _ ★ _)%I]comm assoc -pvs_frame_r. eapply sep_elim_True_r; last eapply sep_mono_l. { rewrite -later_intro. apply wand_intro_l. by rewrite right_id. } rewrite (sts_own_weaken ⊤ _ _ (i_states i ∩ low_states) _ ({[ Change i ]} ∪ {[ Send ]})). + apply pvs_mono. rewrite -sts_ownS_op; eauto using i_states_closed, low_states_closed. set_solver. `````` Robbert Krebbers committed Feb 24, 2016 163 164 `````` + intros []; set_solver. + set_solver. `````` Robbert Krebbers committed Feb 24, 2016 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 `````` + auto using sts.closed_op, i_states_closed, low_states_closed. Qed. Lemma signal_spec l P (Φ : val → iProp) : (send l P ★ P ★ Φ '()) ⊑ || signal (LocV l) {{ Φ }}. Proof. rewrite /signal /send /barrier_ctx. rewrite sep_exist_r. apply exist_elim=>γ. rewrite -!assoc. apply const_elim_sep_l=>?. wp_let. (* I think some evars here are better than repeating *everything* *) eapply (sts_fsaS _ (wp_fsa _)) with (N0:=N) (γ0:=γ); simpl; eauto with I ndisj. rewrite !assoc [(_ ★ sts_ownS _ _ _)%I]comm -!assoc. apply sep_mono_r. apply forall_intro=>-[p I]. apply wand_intro_l. rewrite -!assoc. apply const_elim_sep_l=>Hs. destruct p; last done. rewrite {1}/barrier_inv =>/={Hs}. rewrite later_sep. eapply wp_store with (v' := '0); eauto with I ndisj. `````` Robbert Krebbers committed Feb 25, 2016 181 `````` strip_later. cancel [l ↦ '0]%I. `````` Robbert Krebbers committed Feb 24, 2016 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 `````` apply wand_intro_l. rewrite -(exist_intro (State High I)). rewrite -(exist_intro ∅). rewrite const_equiv /=; last by eauto using signal_step. rewrite left_id -later_intro {2}/barrier_inv -!assoc. apply sep_mono_r. rewrite !assoc [(_ ★ P)%I]comm !assoc -2!assoc. apply sep_mono; last first. { apply wand_intro_l. eauto with I. } (* Now we come to the core of the proof: Updating from waiting to ress. *) rewrite /waiting /ress sep_exist_l. apply exist_elim=>{Φ} Φ. rewrite later_wand {1}(later_intro P) !assoc wand_elim_r. rewrite big_sepS_later -big_sepS_sepS. apply big_sepS_mono'=>i. by rewrite -(exist_intro (Φ i)) comm. Qed. Lemma wait_spec l P (Φ : val → iProp) : (recv l P ★ (P -★ Φ '())) ⊑ || wait (LocV l) {{ Φ }}. Proof. rename P into R. wp_rec. rewrite {1}/recv /barrier_ctx. rewrite !sep_exist_r. apply exist_elim=>γ. rewrite !sep_exist_r. apply exist_elim=>P. rewrite !sep_exist_r. apply exist_elim=>Q. rewrite !sep_exist_r. apply exist_elim=>i. rewrite -!assoc. apply const_elim_sep_l=>?. wp_focus (! _)%L. (* I think some evars here are better than repeating *everything* *) eapply (sts_fsaS _ (wp_fsa _)) with (N0:=N) (γ0:=γ); simpl; eauto with I ndisj. rewrite !assoc [(_ ★ sts_ownS _ _ _)%I]comm -!assoc. apply sep_mono_r. apply forall_intro=>-[p I]. apply wand_intro_l. rewrite -!assoc. apply const_elim_sep_l=>Hs. rewrite {1}/barrier_inv =>/=. rewrite later_sep. eapply wp_load; eauto with I ndisj. `````` Robbert Krebbers committed Feb 25, 2016 212 `````` rewrite -!assoc. apply sep_mono_r. strip_later. `````` Robbert Krebbers committed Feb 24, 2016 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 `````` apply wand_intro_l. destruct p. { (* a Low state. The comparison fails, and we recurse. *) rewrite -(exist_intro (State Low I)) -(exist_intro {[ Change i ]}). rewrite [(■ sts.steps _ _ )%I]const_equiv /=; last by apply rtc_refl. rewrite left_id -[(▷ barrier_inv _ _ _)%I]later_intro {3}/barrier_inv. rewrite -!assoc. apply sep_mono_r, sep_mono_r, wand_intro_l. wp_op; first done. intros _. wp_if. rewrite !assoc. rewrite -always_wand_impl always_elim. rewrite -{2}pvs_wp. apply pvs_wand_r. rewrite -(exist_intro γ) -(exist_intro P) -(exist_intro Q) -(exist_intro i). rewrite !assoc. do 3 (rewrite -pvs_frame_r; apply sep_mono; last (try apply later_intro; reflexivity)). rewrite [(_ ★ heap_ctx _)%I]comm -!assoc. rewrite const_equiv // left_id -pvs_frame_l. apply sep_mono_r. rewrite comm -pvs_frame_l. apply sep_mono_r. apply sts_own_weaken; eauto using i_states_closed. } (* a High state: the comparison succeeds, and we perform a transition and return to the client *) rewrite [(_ ★ □ (_ → _ ))%I]sep_elim_l. rewrite -(exist_intro (State High (I ∖ {[ i ]}))) -(exist_intro ∅). change (i ∈ I) in Hs. rewrite const_equiv /=; last by eauto using wait_step. rewrite left_id -[(▷ barrier_inv _ _ _)%I]later_intro {2}/barrier_inv. rewrite -!assoc. apply sep_mono_r. rewrite /ress. rewrite (big_sepS_delete _ I i) // [(_ ★ Π★{set _} _)%I]comm -!assoc. apply sep_mono_r. rewrite !sep_exist_r. apply exist_elim=>Q'. apply wand_intro_l. rewrite [(heap_ctx _ ★ _)%I]sep_elim_r. rewrite [(sts_own _ _ _ ★ _)%I]sep_elim_r [(sts_ctx _ _ _ ★ _)%I]sep_elim_r. rewrite !assoc [(_ ★ saved_prop_own i Q)%I]comm !assoc saved_prop_agree. `````` Robbert Krebbers committed Feb 25, 2016 242 `````` wp_op; [|done]=> _. wp_if. `````` Robbert Krebbers committed Feb 24, 2016 243 244 245 246 247 `````` eapply wand_apply_r; [done..|]. eapply wand_apply_r; [done..|]. apply: (eq_rewrite Q' Q (λ x, x)%I); last by eauto with I. rewrite eq_sym. eauto with I. Qed. `````` Ralf Jung committed Mar 01, 2016 248 249 250 ``````Lemma recv_split E l P1 P2 : nclose N ⊆ E → recv l (P1 ★ P2) ⊑ |={E}=> recv l P1 ★ recv l P2. `````` Robbert Krebbers committed Feb 24, 2016 251 ``````Proof. `````` Ralf Jung committed Mar 01, 2016 252 253 `````` rename P1 into R1. rename P2 into R2. intros HN. rewrite {1}/recv /barrier_ctx. `````` Robbert Krebbers committed Feb 24, 2016 254 `````` apply exist_elim=>γ. rewrite sep_exist_r. apply exist_elim=>P. `````` Ralf Jung committed Mar 01, 2016 255 256 `````` apply exist_elim=>Q. apply exist_elim=>i. rewrite -!assoc. apply const_elim_sep_l=>?. rewrite -pvs_trans'. `````` Robbert Krebbers committed Feb 24, 2016 257 `````` (* I think some evars here are better than repeating *everything* *) `````` Ralf Jung committed Mar 01, 2016 258 `````` eapply pvs_mk_fsa, (sts_fsaS _ pvs_fsa) with (N0:=N) (γ0:=γ); simpl; `````` Robbert Krebbers committed Feb 24, 2016 259 260 261 `````` eauto with I ndisj. rewrite !assoc [(_ ★ sts_ownS _ _ _)%I]comm -!assoc. apply sep_mono_r. apply forall_intro=>-[p I]. apply wand_intro_l. rewrite -!assoc. `````` Ralf Jung committed Mar 01, 2016 262 `````` apply const_elim_sep_l=>Hs. rewrite /pvs_fsa. `````` Robbert Krebbers committed Feb 24, 2016 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 `````` eapply sep_elim_True_l. { eapply saved_prop_alloc_strong with (P0 := R1) (G := I). } rewrite pvs_frame_r. apply pvs_strip_pvs. rewrite sep_exist_r. apply exist_elim=>i1. rewrite always_and_sep_l. rewrite -assoc. apply const_elim_sep_l=>Hi1. eapply sep_elim_True_l. { eapply saved_prop_alloc_strong with (P0 := R2) (G := I ∪ {[ i1 ]}). } rewrite pvs_frame_r. apply pvs_mono. rewrite sep_exist_r. apply exist_elim=>i2. rewrite always_and_sep_l. rewrite -assoc. apply const_elim_sep_l=>Hi2. rewrite ->not_elem_of_union, elem_of_singleton in Hi2. destruct Hi2 as [Hi2 Hi12]. change (i ∈ I) in Hs. destruct p. (* Case I: Low state. *) - rewrite -(exist_intro (State Low ({[i1]} ∪ ({[i2]} ∪ (I ∖ {[i]}))))). rewrite -(exist_intro ({[Change i1 ]} ∪ {[ Change i2 ]})). rewrite [(■ sts.steps _ _)%I]const_equiv; last by eauto using split_step. `````` Ralf Jung committed Mar 01, 2016 278 `````` rewrite left_id {1 3}/barrier_inv. `````` Robbert Krebbers committed Feb 24, 2016 279 280 281 `````` (* FIXME ssreflect rewrite fails if there are evars around. Also, this is very slow because we don't have a proof mode. *) rewrite -(waiting_split _ _ _ Q R1 R2); [|done..]. rewrite {1}[saved_prop_own i1 _]always_sep_dup. `````` Ralf Jung committed Mar 01, 2016 282 283 284 285 286 `````` rewrite {1}[saved_prop_own i2 _]always_sep_dup !later_sep. rewrite -![(▷ saved_prop_own _ _)%I]later_intro. ecancel [▷ l ↦ _; saved_prop_own i1 _; saved_prop_own i2 _ ; ▷ waiting _ _ ; ▷ (Q -★ _) ; saved_prop_own i _]%I. apply wand_intro_l. rewrite !assoc. rewrite /recv. `````` Robbert Krebbers committed Feb 24, 2016 287 288 289 `````` rewrite -(exist_intro γ) -(exist_intro P) -(exist_intro R1) -(exist_intro i1). rewrite -(exist_intro γ) -(exist_intro P) -(exist_intro R2) -(exist_intro i2). do 2 rewrite !(assoc (★)%I) [(_ ★ sts_ownS _ _ _)%I]comm. `````` Ralf Jung committed Mar 01, 2016 290 291 `````` rewrite -!assoc. rewrite [(sts_ownS _ _ _ ★ _ ★ _)%I]assoc. rewrite -pvs_frame_r. apply sep_mono. `````` Robbert Krebbers committed Feb 24, 2016 292 `````` * rewrite -sts_ownS_op; eauto using i_states_closed. `````` Ralf Jung committed Mar 01, 2016 293 294 `````` + apply sts_own_weaken; eauto using sts.closed_op, i_states_closed. set_solver. `````` Robbert Krebbers committed Feb 24, 2016 295 296 297 298 299 300 301 302 303 304 305 `````` + set_solver. * rewrite const_equiv // !left_id. rewrite {1}[heap_ctx _]always_sep_dup {1}[sts_ctx _ _ _]always_sep_dup. rewrite !wand_diag -!later_intro. solve_sep_entails. (* Case II: High state. TODO: Lots of this script is just copy-n-paste of the previous one. Most of that is because the goals are fairly similar in structure, and the proof scripts are mostly concerned with manually managaing the structure (assoc, comm, dup) of the context. *) - rewrite -(exist_intro (State High ({[i1]} ∪ ({[i2]} ∪ (I ∖ {[i]}))))). rewrite -(exist_intro ({[Change i1 ]} ∪ {[ Change i2 ]})). rewrite const_equiv; last by eauto using split_step. `````` Ralf Jung committed Mar 01, 2016 306 `````` rewrite left_id {1 3}/barrier_inv. `````` Robbert Krebbers committed Feb 24, 2016 307 308 `````` rewrite -(ress_split _ _ _ Q R1 R2); [|done..]. rewrite {1}[saved_prop_own i1 _]always_sep_dup. `````` Ralf Jung committed Mar 01, 2016 309 310 311 312 313 `````` rewrite {1}[saved_prop_own i2 _]always_sep_dup !later_sep. rewrite -![(▷ saved_prop_own _ _)%I]later_intro. ecancel [▷ l ↦ _; saved_prop_own i1 _; saved_prop_own i2 _ ; ▷ ress _ ; ▷ (Q -★ _) ; saved_prop_own i _]%I. apply wand_intro_l. rewrite !assoc. rewrite /recv. `````` Robbert Krebbers committed Feb 24, 2016 314 315 316 317 318 319 320 `````` rewrite -(exist_intro γ) -(exist_intro P) -(exist_intro R1) -(exist_intro i1). rewrite -(exist_intro γ) -(exist_intro P) -(exist_intro R2) -(exist_intro i2). do 2 rewrite !(assoc (★)%I) [(_ ★ sts_ownS _ _ _)%I]comm. rewrite -!assoc. rewrite [(sts_ownS _ _ _ ★ _ ★ _)%I]assoc -pvs_frame_r. apply sep_mono. * rewrite -sts_ownS_op; eauto using i_states_closed. + apply sts_own_weaken; eauto using sts.closed_op, i_states_closed. `````` Robbert Krebbers committed Feb 24, 2016 321 `````` set_solver. `````` Robbert Krebbers committed Feb 24, 2016 322 323 324 325 326 327 `````` + set_solver. * rewrite const_equiv // !left_id. rewrite {1}[heap_ctx _]always_sep_dup {1}[sts_ctx _ _ _]always_sep_dup. rewrite !wand_diag -!later_intro. solve_sep_entails. Qed. `````` Ralf Jung committed Mar 01, 2016 328 ``````Lemma recv_weaken l P1 P2 : `````` Robbert Krebbers committed Feb 24, 2016 329 330 331 332 333 334 335 336 337 `````` (P1 -★ P2) ⊑ (recv l P1 -★ recv l P2). Proof. apply wand_intro_l. rewrite /recv. rewrite sep_exist_r. apply exist_mono=>γ. rewrite sep_exist_r. apply exist_mono=>P. rewrite sep_exist_r. apply exist_mono=>Q. rewrite sep_exist_r. apply exist_mono=>i. rewrite -!assoc. apply sep_mono_r, sep_mono_r, sep_mono_r, sep_mono_r, sep_mono_r. rewrite (later_intro (P1 -★ _)%I) -later_sep. apply later_mono. apply wand_intro_l. by rewrite !assoc wand_elim_r wand_elim_r. Qed. `````` Ralf Jung committed Feb 29, 2016 338 339 340 341 `````` Lemma recv_mono l P1 P2 : P1 ⊑ P2 → recv l P1 ⊑ recv l P2. Proof. `````` Ralf Jung committed Mar 01, 2016 342 `````` intros HP%entails_wand. apply wand_entails. rewrite HP. apply recv_weaken. `````` Ralf Jung committed Feb 29, 2016 343 344 ``````Qed. `````` Robbert Krebbers committed Feb 24, 2016 345 ``End proof.``