map_reduce.v 18.3 KB
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 Robbert Krebbers committed Jul 05, 2019 1 From stdpp Require Import sorting. Robbert Krebbers committed Jul 07, 2019 2 From actris.channel Require Import proto_channel proofmode. Robbert Krebbers committed Jul 05, 2019 3 From iris.heap_lang Require Import proofmode notation. Robbert Krebbers committed Jul 08, 2019 4 From actris.utils Require Import llist compare contribution. Robbert Krebbers committed Jul 09, 2019 5 From actris.examples Require Import map sort_fg_client. Robbert Krebbers committed Jul 05, 2019 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 From iris.algebra Require Import gmultiset. From Coq Require Import SetoidPermutation. (** Functional version of map reduce (aka the specification) *) Fixpoint group_insert {A} `{EqDecision K} (i : K) (x : A) (ixss : list (K * list A)) : list (K * list A) := match ixss with | [] => [(i,[x])] | (j,xs) :: ixss => if decide (i = j) then (j,x::xs) :: ixss else (j,xs) :: group_insert i x ixss end. Fixpoint group {A} `{EqDecision K} (ixs : list (K * A)) : list (K * list A) := match ixs with | [] => [] | (i,x) :: ixs => group_insert i x (group ixs) end. Definition map_reduce {A B C} `{EqDecision K} (map : A → list (K * B)) (red : K → list B → list C) : list A → list C := mbind (curry red) ∘ group ∘ mbind map. Instance: Params (@group_insert) 5. Instance: Params (@group) 3. Instance: Params (@group) 7. Robbert Krebbers committed Jul 06, 2019 32 33 34 35 36 37 38 39 40 41 42 (** Distributed version *) Definition par_map_reduce_map_server : val := rec: "go" "n" "cmap" "csort" "xs" := if: "n" = #0 then #() else if: recv "cmap" then (* send item to mapper *) if: lisnil "xs" then send "cmap" #false;; "go" ("n" - #1) "cmap" "csort" "xs" else send "cmap" #true;; Robbert Krebbers committed Jul 08, 2019 43 44 send "cmap" (lpop "xs");; "go" "n" "cmap" "csort" "xs" Robbert Krebbers committed Jul 06, 2019 45 46 47 48 49 50 51 else (* receive item from mapper *) let: "zs" := recv "cmap" in send_all "csort" "zs";; "go" "n" "cmap" "csort" "xs". Definition par_map_reduce_collect : val := rec: "go" "csort" "i" "ys" := Robbert Krebbers committed Jul 08, 2019 52 if: ~recv "csort" then NONE else Robbert Krebbers committed Jul 06, 2019 53 54 let: "jy" := recv "csort" in let: "j" := Fst "jy" in let: "y" := Snd "jy" in Robbert Krebbers committed Jul 08, 2019 55 56 if: "i" = "j" then lcons "y" "ys";; "go" "csort" "j" "ys" else SOME ("j", "y"). Robbert Krebbers committed Jul 06, 2019 57 58 59 Definition par_map_reduce_reduce_server : val := rec: "go" "n" "csort" "cred" "acc" "zs" := Robbert Krebbers committed Jul 08, 2019 60 if: "n" = #0 then #() else Robbert Krebbers committed Jul 06, 2019 61 62 63 64 65 66 67 if: recv "cred" then (* Send item to mapper *) match: "acc" with NONE => (* nothing left *) send "cred" #false;; "go" ("n" - #1) "csort" "cred" NONE "zs" | SOME "acc" => (* Read subsequent items with the same key *) Robbert Krebbers committed Jul 08, 2019 68 let: "ys" := lnil #() in lcons (Snd "acc") "ys";; Robbert Krebbers committed Jul 08, 2019 69 let: "new_acc" := par_map_reduce_collect "csort" (Fst "acc") "ys" in Robbert Krebbers committed Jul 06, 2019 70 send "cred" #true;; Robbert Krebbers committed Jul 08, 2019 71 72 send "cred" (Fst "acc", "ys");; "go" "n" "csort" "cred" "new_acc" "zs" Robbert Krebbers committed Jul 06, 2019 73 74 75 end else (* receive item from mapper *) let: "zs'" := recv "cred" in Robbert Krebbers committed Jul 08, 2019 76 77 lprep "zs" "zs'";; "go" "n" "csort" "cred" "acc" "zs". Robbert Krebbers committed Jul 06, 2019 78 79 80 81 82 Definition cmpZfst : val := λ: "x" "y", Fst "x" ≤ Fst "y". Definition par_map_reduce : val := λ: "n" "map" "red" "xs", let: "cmap" := start_map_service "n" "map" in Robbert Krebbers committed Jul 09, 2019 83 let: "csort" := start_chan (λ: "c", sort_service_fg cmpZfst "c") in Robbert Krebbers committed Jul 06, 2019 84 85 86 87 88 89 par_map_reduce_map_server "n" "cmap" "csort" "xs";; send "csort" #stop;; let: "cred" := start_map_service "n" "red" in (* We need the first sorted element in the loop to compare subsequent elements *) if: ~recv "csort" then lnil #() else (* Handle the empty case *) let: "jy" := recv "csort" in Robbert Krebbers committed Jul 08, 2019 90 91 let: "zs" := lnil #() in par_map_reduce_reduce_server "n" "csort" "cred" (SOME "jy") "zs";; "zs". Robbert Krebbers committed Jul 06, 2019 92 93 Robbert Krebbers committed Jul 05, 2019 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 (** Properties about the functional version *) Local Infix "≡ₚₚ" := (PermutationA (prod_relation (=) (≡ₚ))) (at level 70, no associativity) : stdpp_scope. Notation "(≡ₚₚ)" := (PermutationA (prod_relation (=) (≡ₚ))) (only parsing) : stdpp_scope. Section group. Context {A : Type} `{EqDecision K}. Implicit Types i : K. Implicit Types xs : list A. Implicit Types ixs : list (K * A). Implicit Types ixss : list (K * list A). Lemma elem_of_group_insert j i x ixss : j ∈ (group_insert i x ixss).*1 → i = j ∨ j ∈ ixss.*1. Proof. induction ixss as [|[i' x'] ixss IH]; repeat (simplify_eq/= || case_decide); set_solver. Qed. Lemma group_insert_commute i1 i2 x1 x2 ixss : group_insert i1 x1 (group_insert i2 x2 ixss) ≡ₚₚ group_insert i2 x2 (group_insert i1 x1 ixss). Proof. induction ixss as [|[j x] ixss IH]; repeat (simplify_eq/= || case_decide); repeat constructor; done. Qed. Lemma group_insert_nodup i x ixss : NoDup ixss.*1 → NoDup (group_insert i x ixss).*1. Proof. pose proof @elem_of_group_insert. induction ixss as [|[j xs] ixss IH]; csimpl; inversion_clear 1; repeat (simplify_eq/= || case_decide); repeat constructor; set_solver. Qed. Lemma group_nodup ixs : NoDup (group ixs).*1. Proof. induction ixs as [|[i x] ixs IH]; csimpl; auto using group_insert_nodup, NoDup_nil_2. Qed. Lemma grouped_permutation_elem_of ixss1 ixss2 i : ixss1 ≡ₚₚ ixss2 → i ∈ ixss1.*1 → i ∈ ixss2.*1. Proof. induction 1 as [|[i1 xs1] [i2 xs2] ixss1 ixss2 [??]|[i1 xs1] [i2 xs2] ixss|]; set_solver. Qed. Lemma grouped_permutation_nodup ixss1 ixss2 : ixss1 ≡ₚₚ ixss2 → NoDup ixss1.*1 → NoDup ixss2.*1. Proof. pose proof @grouped_permutation_elem_of. induction 1 as [|[i1 xs1] [i2 xs2] ixss1 ixss2 [??]|[i1 xs1] [i2 xs2] ixss|]; csimpl; rewrite ?NoDup_cons; try set_solver. Qed. Lemma group_insert_perm ixss1 ixss2 i x : NoDup ixss1.*1 → ixss1 ≡ₚₚ ixss2 → group_insert i x ixss1 ≡ₚₚ group_insert i x ixss2. Proof. induction 2 as [|[i1 xs1] [i2 xs2] ixss1 ixss2 [??]|[i1 xs1] [i2 xs2] ixss|]; repeat match goal with | _ => progress (simplify_eq/= || case_decide) | H : NoDup (_ :: _) |- _ => inversion_clear H end; first [repeat constructor; by auto |set_solver |etrans; eauto using grouped_permutation_nodup]. Qed. Global Instance group_perm : Proper ((≡ₚ) ==> (≡ₚₚ)) (@group A K _). Proof. induction 1; repeat (simplify_eq/= || case_decide || case_match); first [by etrans|auto using group_insert_perm, group_nodup, group_insert_commute]. Qed. Lemma group_fmap (i : K) xs : xs ≠ [] → group ((i,) <\$> xs) ≡ₚₚ [(i, xs)]. Proof. induction xs as [|x1 [|x2 xs] IH]; intros; simplify_eq/=; try done. etrans. { apply group_insert_perm, IH; auto using group_insert_nodup, group_nodup. } simpl; by case_decide. Qed. Lemma group_insert_snoc ixss i x j ys : i ≠ j → group_insert i x (ixss ++ [(j, ys)]) ≡ₚₚ group_insert i x ixss ++ [(j,ys)]. Proof. intros. induction ixss as [|[i' xs'] ixss IH]; repeat (simplify_eq/= || case_decide); repeat constructor; by auto. Qed. Lemma group_snoc ixs j ys : j ∉ ixs.*1 → ys ≠ [] → group (ixs ++ ((j,) <\$> ys)) ≡ₚₚ group ixs ++ [(j,ys)]. Proof. induction ixs as [|[i x] ixs IH]; csimpl; first by auto using group_fmap. rewrite ?not_elem_of_cons=> -[??]. etrans; last by apply group_insert_snoc. apply group_insert_perm, IH; auto using group_nodup. Qed. End group. Section map_reduce. Context {A B C} `{EqDecision K} (map : A → list (K * B)) (red : K → list B → list C). Context `{!∀ j, Proper ((≡ₚ) ==> (≡ₚ)) (red j)}. Global Instance bind_red_perm : Proper ((≡ₚₚ) ==> (≡ₚ)) (mbind (curry red)). Proof. induction 1 as [|[i1 xs1] [i2 xs2] ixss1 ixss2 [??]|[i1 xs1] [i2 xs2] ixss|]; simplify_eq/=; try done. - repeat (done || f_equiv). - by rewrite !assoc_L (comm _ (red i2 xs2)). - by etrans. Qed. Global Instance map_reduce_perm : Proper ((≡ₚ) ==> (≡ₚ)) (map_reduce map red). Proof. intros xs1 xs2 Hxs. by rewrite /map_reduce /= Hxs. Qed. End map_reduce. (** Correctness proofs of the distributed version *) Class map_reduceG Σ A B `{Countable A, Countable B} := { Robbert Krebbers committed Jul 06, 2019 209 210 map_reduce_mapG :> mapG Σ A; map_reduce_reduceG :> mapG Σ (Z * list B); Robbert Krebbers committed Jul 05, 2019 211 212 213 214 }. Section mapper. Context `{Countable A, Countable B} {C : Type}. Robbert Krebbers committed Jul 06, 2019 215 Context `{!heapG Σ, !proto_chanG Σ, !map_reduceG Σ A B} (N : namespace). Robbert Krebbers committed Jul 05, 2019 216 217 218 219 Context (IA : A → val → iProp Σ) (IB : Z → B → val → iProp Σ) (IC : C → val → iProp Σ). Context (map : A → list (Z * B)) (red : Z → list B → list C). Context `{!∀ j, Proper ((≡ₚ) ==> (≡ₚ)) (red j)}. Local Open Scope nat_scope. Robbert Krebbers committed Jul 06, 2019 220 Implicit Types n : nat. Robbert Krebbers committed Jul 05, 2019 221 222 223 224 Definition IZB (iy : Z * B) (w : val) : iProp Σ := (∃ w', ⌜ w = (#iy.1, w')%V ⌝ ∧ IB iy.1 iy.2 w')%I. Definition IZBs (iys : Z * list B) (w : val) : iProp Σ := Robbert Krebbers committed Jul 08, 2019 225 (∃ (l : loc), ⌜ w = (#iys.1, #l)%V ⌝ ∗ llist (IB iys.1) l (iys.2))%I. Robbert Krebbers committed Jul 05, 2019 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 Definition RZB : relation (Z * B) := prod_relation (≤)%Z (λ _ _ : B, True). Instance RZB_dec : RelDecision RZB. Proof. solve_decision. Qed. Instance RZB_total : Total RZB. Proof. intros [i1 y1] [i2 y2]. destruct (total (≤)%Z i1 i2); [left|right]; done. Qed. Instance RZB_trans : Transitive RZB. Proof. by apply (prod_relation_trans _). Qed. Lemma RZB_cmp_spec : cmp_spec IZB RZB cmpZfst. Proof. iIntros ([i1 y1] [i2 y2] v1 v2) "!>"; iIntros (Φ) "[HI1 HI2] HΦ". iDestruct "HI1" as (w1 ->) "HI1". iDestruct "HI2" as (w2 ->) "HI2 /=". wp_lam; wp_pures. iSpecialize ("HΦ" with "[HI1 HI2]"). { iSplitL "HI1"; rewrite /IZB /=; eauto with iFrame. } repeat case_bool_decide=> //; unfold RZB, prod_relation in *; naive_solver. Qed. Robbert Krebbers committed Jul 08, 2019 243 Lemma par_map_reduce_map_server_spec n cmap csort l xs X ys : Robbert Krebbers committed Jul 05, 2019 244 245 (n = 0 → X = ∅ ∧ xs = []) → {{{ Robbert Krebbers committed Jul 08, 2019 246 llist IA l xs ∗ Robbert Krebbers committed Jul 06, 2019 247 cmap ↣ map_worker_protocol IA IZB map n (X : gmultiset A) @ N ∗ Robbert Krebbers committed Jul 09, 2019 248 csort ↣ sort_fg_head_protocol IZB RZB ys @ N Robbert Krebbers committed Jul 05, 2019 249 }}} Robbert Krebbers committed Jul 08, 2019 250 par_map_reduce_map_server #n cmap csort #l Robbert Krebbers committed Jul 05, 2019 251 252 {{{ ys', RET #(); ⌜ys' ≡ₚ (xs ++ elements X) ≫= map⌝ ∗ Robbert Krebbers committed Jul 09, 2019 253 csort ↣ sort_fg_head_protocol IZB RZB (ys ++ ys') @ N Robbert Krebbers committed Jul 05, 2019 254 255 }}}. Proof. Robbert Krebbers committed Jul 08, 2019 256 257 iIntros (Hn Φ) "(Hl & Hcmap & Hcsort) HΦ". iLöb as "IH" forall (n xs X ys Hn Φ); wp_rec; wp_pures; simpl. Robbert Krebbers committed Jul 05, 2019 258 259 260 261 case_bool_decide; wp_pures; simplify_eq/=. { destruct Hn as [-> ->]; first lia. iApply ("HΦ" \$! []). rewrite right_id_L. auto. } destruct n as [|n]=> //=. wp_branch as %?|%_; wp_pures. Robbert Krebbers committed Jul 08, 2019 262 - wp_apply (lisnil_spec with "Hl"); iIntros "Hl". Robbert Krebbers committed Jul 08, 2019 263 destruct xs as [|x xs]; csimpl; wp_pures. Robbert Krebbers committed Jul 05, 2019 264 + wp_select. wp_pures. rewrite Nat2Z.inj_succ Z.sub_1_r Z.pred_succ. Robbert Krebbers committed Jul 08, 2019 265 266 267 268 iApply ("IH" \$! _ [] with "[%] Hl Hcmap Hcsort HΦ"); naive_solver. + wp_select. wp_apply (lpop_spec with "Hl"); iIntros (v) "[HIx Hl]". wp_send with "[\$HIx]". wp_apply ("IH" with "[%] Hl Hcmap Hcsort"); first done. iIntros (ys'). Robbert Krebbers committed Jul 05, 2019 269 270 rewrite gmultiset_elements_disj_union gmultiset_elements_singleton. rewrite assoc_L -(comm _ [x]). iApply "HΦ". Robbert Krebbers committed Jul 08, 2019 271 - wp_recv (x k) as (Hx) "Hk". Robbert Krebbers committed Jul 09, 2019 272 rewrite -(right_id END%proto _ (sort_fg_head_protocol _ _ _)). Robbert Krebbers committed Jul 08, 2019 273 wp_apply (send_all_spec with "[\$Hk \$Hcsort]"); iIntros "Hcsort". Robbert Krebbers committed Jul 05, 2019 274 rewrite right_id. Robbert Krebbers committed Jul 08, 2019 275 wp_apply ("IH" with "[] Hl Hcmap Hcsort"); first done. Robbert Krebbers committed Jul 05, 2019 276 277 278 279 280 281 282 283 iIntros (ys'). iDestruct 1 as (Hys) "Hcsort"; simplify_eq/=. rewrite -assoc_L. iApply ("HΦ" \$! (map x ++ ys') with "[\$Hcsort]"). iPureIntro. rewrite (gmultiset_disj_union_difference {[ x ]} X) -?gmultiset_elem_of_singleton_subseteq //. rewrite (comm disj_union) gmultiset_elements_disj_union. by rewrite gmultiset_elements_singleton assoc_L bind_app -Hys /= right_id_L comm. Qed. Robbert Krebbers committed Jul 08, 2019 284 Lemma par_map_reduce_collect_spec csort iys iys_sorted i l ys : Robbert Krebbers committed Jul 05, 2019 285 286 287 288 289 290 let acc := from_option (λ '(i,y,w), [(i,y)]) [] in let accv := from_option (λ '(i,y,w), SOMEV (#(i:Z),w)) NONEV in ys ≠ [] → Sorted RZB (iys_sorted ++ ((i,) <\$> ys)) → i ∉ iys_sorted.*1 → {{{ Robbert Krebbers committed Jul 08, 2019 291 llist (IB i) l (reverse ys) ∗ Robbert Krebbers committed Jul 09, 2019 292 csort ↣ sort_fg_tail_protocol IZB RZB iys (iys_sorted ++ ((i,) <\$> ys)) @ N Robbert Krebbers committed Jul 05, 2019 293 }}} Robbert Krebbers committed Jul 08, 2019 294 par_map_reduce_collect csort #i #l Robbert Krebbers committed Jul 08, 2019 295 {{{ ys' miy, RET accv miy; Robbert Krebbers committed Jul 05, 2019 296 297 298 ⌜ Sorted RZB ((iys_sorted ++ ((i,) <\$> ys ++ ys')) ++ acc miy) ⌝ ∗ ⌜ from_option (λ '(j,_,_), i ≠ j ∧ j ∉ iys_sorted.*1) (iys_sorted ++ ((i,) <\$> ys ++ ys') ≡ₚ iys) miy ⌝ ∗ Robbert Krebbers committed Jul 08, 2019 299 llist (IB i) l (reverse (ys ++ ys')) ∗ Robbert Krebbers committed Jul 09, 2019 300 csort ↣ from_option (λ _, sort_fg_tail_protocol IZB RZB iys Robbert Krebbers committed Jul 05, 2019 301 302 303 304 ((iys_sorted ++ ((i,) <\$> ys ++ ys')) ++ acc miy)) END%proto miy @ N ∗ from_option (λ '(i,y,w), IB i y w) True miy }}}. Proof. Robbert Krebbers committed Jul 08, 2019 305 306 iIntros (acc accv Hys Hsort Hi Φ) "[Hl Hcsort] HΦ". iLöb as "IH" forall (ys Hys Hsort Hi Φ); wp_rec; wp_pures; simpl. Robbert Krebbers committed Jul 05, 2019 307 wp_branch as %_|%Hperm; last first; wp_pures. Robbert Krebbers committed Jul 08, 2019 308 { iApply ("HΦ" \$! [] None with "[Hl \$Hcsort]"); simpl. Robbert Krebbers committed Jul 08, 2019 309 iEval (rewrite !right_id_L); auto with iFrame. } Robbert Krebbers committed Jul 08, 2019 310 wp_recv ([j y] ?) as (Htl w ->) "HIy /=". wp_pures. rewrite -assoc_L. Robbert Krebbers committed Jul 05, 2019 311 case_bool_decide as Hij; simplify_eq/=; wp_pures. Robbert Krebbers committed Jul 08, 2019 312 - wp_apply (lcons_spec with "[\$Hl \$HIy]"); iIntros "Hl". Robbert Krebbers committed Jul 05, 2019 313 rewrite -reverse_snoc. wp_apply ("IH" \$! (ys ++ [y]) Robbert Krebbers committed Jul 08, 2019 314 with "[%] [%] [//] Hl [Hcsort] [HΦ]"); try iClear "IH". Robbert Krebbers committed Jul 05, 2019 315 316 317 + intros ?; discriminate_list. + rewrite fmap_app /= assoc_L. by apply Sorted_snoc. + by rewrite fmap_app /= assoc_L. Robbert Krebbers committed Jul 08, 2019 318 319 + iIntros "!>" (ys' miy). rewrite -!(assoc_L _ ys) /=. iApply "HΦ". - iApply ("HΦ" \$! [] (Some (j,y,w))). Robbert Krebbers committed Jul 05, 2019 320 321 322 323 324 325 326 327 328 329 330 331 332 rewrite /= !right_id_L assoc_L. iFrame. iPureIntro; split. { by apply Sorted_snoc. } split; first congruence. intros [[j' y'] [-> Hj]]%elem_of_list_fmap. destruct Hij; do 2 f_equal. destruct ys as [|y'' ys _] using rev_ind; first done. move: Htl. rewrite fmap_app assoc_L /=. inversion 1 as [|?? [? _]]; discriminate_list || simplify_list_eq. assert (RZB (j',y') (i,y'')) as [??]; last (simpl in *; lia). apply (Sorted_StronglySorted _) in Hsort. eapply elem_of_StronglySorted_app; set_solver. Qed. Robbert Krebbers committed Jul 08, 2019 333 Lemma par_map_reduce_reduce_server_spec n iys iys_sorted miy zs l Y csort cred : Robbert Krebbers committed Jul 05, 2019 334 335 336 337 338 339 let acc := from_option (λ '(i,y,w), [(i,y)]) [] in let accv := from_option (λ '(i,y,w), SOMEV (#(i:Z),w)) NONEV in (n = 0 → miy = None ∧ Y = ∅) → from_option (λ '(i,_,_), i ∉ iys_sorted.*1) (iys_sorted ≡ₚ iys) miy → Sorted RZB (iys_sorted ++ acc miy) → {{{ Robbert Krebbers committed Jul 08, 2019 340 llist IC l zs ∗ Robbert Krebbers committed Jul 09, 2019 341 csort ↣ from_option (λ _, sort_fg_tail_protocol IZB RZB iys Robbert Krebbers committed Jul 05, 2019 342 (iys_sorted ++ acc miy)) END%proto miy @ N ∗ Robbert Krebbers committed Jul 06, 2019 343 cred ↣ map_worker_protocol IZBs IC (curry red) n (Y : gmultiset (Z * list B)) @ N ∗ Robbert Krebbers committed Jul 08, 2019 344 from_option (λ '(i,y,w), IB i y w) True miy Robbert Krebbers committed Jul 05, 2019 345 }}} Robbert Krebbers committed Jul 08, 2019 346 par_map_reduce_reduce_server #n csort cred (accv miy) #l Robbert Krebbers committed Jul 08, 2019 347 {{{ zs', RET #(); Robbert Krebbers committed Jul 05, 2019 348 ⌜ (group iys_sorted ≫= curry red) ++ zs' ≡ₚ (group iys ++ elements Y) ≫= curry red ⌝ ∗ Robbert Krebbers committed Jul 08, 2019 349 llist IC l (zs' ++ zs) Robbert Krebbers committed Jul 05, 2019 350 351 }}}. Proof. Robbert Krebbers committed Jul 08, 2019 352 353 iIntros (acc accv Hn Hmiy Hsort Φ) "(Hl & Hcsort & Hcred & HImiy) HΦ". iLöb as "IH" forall (n iys_sorted miy zs Y Hn Hmiy Hsort Φ); wp_rec; wp_pures; simpl. Robbert Krebbers committed Jul 05, 2019 354 355 case_bool_decide; wp_pures; simplify_eq/=. { destruct Hn as [-> ->]; first lia. Robbert Krebbers committed Jul 08, 2019 356 iApply ("HΦ" \$! [] with "[\$Hl]"); iPureIntro; simpl. Robbert Krebbers committed Jul 05, 2019 357 358 359 360 361 by rewrite gmultiset_elements_empty !right_id_L Hmiy. } destruct n as [|n]=> //=. wp_branch as %?|%_; wp_pures. - destruct miy as [[[i y] w]|]; simplify_eq/=; wp_pures; last first. + wp_select. wp_pures. rewrite Nat2Z.inj_succ Z.sub_1_r Z.pred_succ. iApply ("IH" \$! _ _ None Robbert Krebbers committed Jul 08, 2019 362 363 364 365 366 367 368 369 with "[%] [%] [%] Hl Hcsort Hcred [] HΦ"); naive_solver. + wp_apply (lnil_spec (IB i) with "[//]"); iIntros (k) "Hk". wp_apply (lcons_spec with "[\$Hk \$HImiy]"); iIntros "Hk". wp_apply (par_map_reduce_collect_spec _ _ _ _ _ [_] with "[\$Hk \$Hcsort]"); try done. iIntros (ys' miy). iDestruct 1 as (? Hmiy') "(Hk & Hcsort & HImiy)"; csimpl. wp_select; wp_pures. wp_send ((i, reverse (y :: ys'))) with "[Hk]". { iExists k; simpl; auto. } Robbert Krebbers committed Jul 05, 2019 370 wp_pures. iApply ("IH" \$! _ (_ ++ _) miy Robbert Krebbers committed Jul 08, 2019 371 with "[%] [%] [//] Hl Hcsort Hcred HImiy"); first done. Robbert Krebbers committed Jul 05, 2019 372 { destruct miy as [[[i' y'] w']|]; set_solver +Hmiy'. } Robbert Krebbers committed Jul 08, 2019 373 iIntros "!>" (zs'). iDestruct 1 as (Hperm) "HIC". Robbert Krebbers committed Jul 05, 2019 374 375 376 377 378 iApply ("HΦ" with "[\$HIC]"); iPureIntro; move: Hperm. rewrite gmultiset_elements_disj_union gmultiset_elements_singleton. rewrite group_snoc // reverse_Permutation. rewrite !bind_app /= right_id_L -!assoc_L -(comm _ zs') !assoc_L. apply (inj (++ _)). Robbert Krebbers committed Jul 08, 2019 379 - wp_recv ([i ys] k) as (Hy) "Hk". Robbert Krebbers committed Jul 08, 2019 380 wp_apply (lprep_spec with "[\$Hl \$Hk]"); iIntros "[Hl _]". Robbert Krebbers committed Jul 08, 2019 381 382 wp_apply ("IH" with "[ ] [//] [//] Hl Hcsort Hcred HImiy"); first done. iIntros (zs'); iDestruct 1 as (Hzs) "HIC"; simplify_eq/=. Robbert Krebbers committed Jul 05, 2019 383 384 385 386 387 388 389 390 iApply ("HΦ" \$! (zs' ++ red i ys)). iSplit; last by rewrite -assoc_L. iPureIntro. rewrite (gmultiset_disj_union_difference {[ i,ys ]} Y) -?gmultiset_elem_of_singleton_subseteq //. rewrite (comm disj_union) gmultiset_elements_disj_union. rewrite gmultiset_elements_singleton assoc_L Hzs !bind_app /=. by rewrite right_id_L !assoc_L. Qed. Robbert Krebbers committed Jul 08, 2019 391 Lemma par_map_reduce_spec n vmap vred l xs : Robbert Krebbers committed Jul 05, 2019 392 0 < n → Robbert Krebbers committed Jul 06, 2019 393 394 map_spec IA IZB map vmap -∗ map_spec IZBs IC (curry red) vred -∗ Robbert Krebbers committed Jul 08, 2019 395 {{{ llist IA l xs }}} Robbert Krebbers committed Jul 08, 2019 396 par_map_reduce #n vmap vred #l Robbert Krebbers committed Jul 08, 2019 397 {{{ k zs, RET #k; ⌜zs ≡ₚ map_reduce map red xs⌝ ∗ llist IC k zs }}}. Robbert Krebbers committed Jul 05, 2019 398 Proof. Robbert Krebbers committed Jul 08, 2019 399 iIntros (?) "#Hmap #Hred !>"; iIntros (Φ) "Hl HΦ". wp_lam; wp_pures. Robbert Krebbers committed Jul 06, 2019 400 wp_apply (start_map_service_spec with "Hmap [//]"); iIntros (cmap) "Hcmap". Robbert Krebbers committed Jul 09, 2019 401 wp_apply (start_chan_proto_spec N (sort_fg_protocol IZB RZB <++> END)%proto); Robbert Krebbers committed Jul 05, 2019 402 iIntros (csort) "Hcsort". Robbert Krebbers committed Jul 09, 2019 403 { wp_apply (sort_service_fg_spec N with "[] Hcsort"); last by auto. Robbert Krebbers committed Jul 05, 2019 404 405 iApply RZB_cmp_spec. } rewrite right_id. Robbert Krebbers committed Jul 08, 2019 406 wp_apply (par_map_reduce_map_server_spec with "[\$Hl \$Hcmap \$Hcsort]"); first lia. Robbert Krebbers committed Jul 05, 2019 407 iIntros (iys). rewrite gmultiset_elements_empty right_id_L. Robbert Krebbers committed Jul 06, 2019 408 409 iDestruct 1 as (Hiys) "Hcsort /=". wp_select; wp_pures; simpl. wp_apply (start_map_service_spec with "Hred [//]"); iIntros (cred) "Hcred". Robbert Krebbers committed Jul 05, 2019 410 wp_branch as %_|%Hnil; last first. Robbert Krebbers committed Jul 08, 2019 411 { wp_pures. wp_apply (lnil_spec with "[//]"); iIntros (k) "Hk". Robbert Krebbers committed Jul 08, 2019 412 iApply ("HΦ" \$! k [] with "[\$Hk]"); simpl. Robbert Krebbers committed Jul 05, 2019 413 414 by rewrite /map_reduce /= -Hiys -Hnil. } wp_recv ([i y] ?) as (_ w ->) "HIB /="; wp_pures. Robbert Krebbers committed Jul 08, 2019 415 wp_apply (lnil_spec with "[//]"); iIntros (k) "Hk". Robbert Krebbers committed Jul 08, 2019 416 wp_apply (par_map_reduce_reduce_server_spec _ _ [] (Some (i, y, w)) Robbert Krebbers committed Jul 08, 2019 417 with "[\$Hk \$Hcsort \$Hcred \$HIB]"); simpl; auto; [lia|set_solver|]. Robbert Krebbers committed Jul 08, 2019 418 419 420 iIntros (zs). rewrite /= gmultiset_elements_empty !right_id. iDestruct 1 as (Hzs) "Hk". wp_pures. iApply ("HΦ" with "[\$Hk]"). by rewrite Hzs Hiys. Robbert Krebbers committed Jul 05, 2019 421 422 Qed. End mapper.