ipm_paper.v 9.06 KB
 Robbert Krebbers committed Jan 17, 2017 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 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 ``````From iris.base_logic Require Import base_logic. From iris.proofmode Require Import tactics. From iris.program_logic Require Export hoare. From iris.heap_lang Require Import proofmode notation. Set Default Proof Using "Type". (** The proofs from Section 3.1 *) Section demo. Context {M : ucmraT}. Notation iProp := (uPred M). (* The version in Coq *) Lemma and_exist A (P R: Prop) (Ψ: A → Prop) : P ∧ (∃ a, Ψ a) ∧ R → ∃ a, P ∧ Ψ a. Proof. intros [HP [HΨ HR]]. destruct HΨ as [x HΨ]. exists x. split. assumption. assumption. Qed. (* The version in IPM *) Lemma sep_exist A (P R: iProp) (Ψ: A → iProp) : P ∗ (∃ a, Ψ a) ∗ R ⊢ ∃ a, Ψ a ∗ P. Proof. iIntros "[HP [HΨ HR]]". iDestruct "HΨ" as (x) "HΨ". iExists x. iSplitL "HΨ". iAssumption. iAssumption. Qed. (* The short version in IPM, as in the paper *) Lemma sep_exist_short A (P R: iProp) (Ψ: A → iProp) : P ∗ (∃ a, Ψ a) ∗ R ⊢ ∃ a, Ψ a ∗ P. Proof. iIntros "[HP [HΨ HR]]". iFrame "HP". iAssumption. Qed. (* An even shorter version in IPM, using the frame introduction pattern `\$` *) Lemma sep_exist_shorter A (P R: iProp) (Ψ: A → iProp) : P ∗ (∃ a, Ψ a) ∗ R ⊢ ∃ a, Ψ a ∗ P. Proof. by iIntros "[\$ [??]]". Qed. End demo. (** The proofs from Section 3.2 *) (** In the Iris development we often write specifications directly using weakest preconditions, in sort of `CPS` style, so that they can be applied easier when proving client code. A version of [list_reverse] in that style can be found in the file [theories/tests/list_reverse.v]. *) Section list_reverse. Context `{!heapG Σ}. Notation iProp := (iProp Σ). Implicit Types l : loc. Fixpoint is_list (hd : val) (xs : list val) : iProp := match xs with | [] => ⌜hd = NONEV⌝ | x :: xs => ∃ l hd', ⌜hd = SOMEV #l⌝ ∗ l ↦ (x,hd') ∗ is_list hd' xs end%I. Definition rev : val := rec: "rev" "hd" "acc" := match: "hd" with NONE => "acc" | SOME "l" => let: "tmp1" := Fst !"l" in let: "tmp2" := Snd !"l" in "l" <- ("tmp1", "acc");; "rev" "tmp2" "hd" end. Lemma rev_acc_ht hd acc xs ys : {{ is_list hd xs ∗ is_list acc ys }} rev hd acc {{ w, is_list w (reverse xs ++ ys) }}. Proof. iIntros "!# [Hxs Hys]". iLöb as "IH" forall (hd acc xs ys). wp_rec; wp_let. destruct xs as [|x xs]; iSimplifyEq. - (* nil *) by wp_match. - (* cons *) iDestruct "Hxs" as (l hd') "(% & Hx & Hxs)"; iSimplifyEq. wp_match. wp_load. wp_proj. wp_let. wp_load. wp_proj. wp_let. wp_store. rewrite reverse_cons -assoc. iApply ("IH" \$! hd' (InjRV #l) xs (x :: ys) with "Hxs [Hx Hys]"). iExists l, acc; by iFrame. Qed. Lemma rev_ht hd xs : {{ is_list hd xs }} rev hd NONE {{ w, is_list w (reverse xs) }}. Proof. iIntros "!# Hxs". rewrite -(right_id_L [] (++) (reverse xs)). iApply (rev_acc_ht hd NONEV with "[Hxs]"); simpl; by iFrame. Qed. End list_reverse. (** The proofs from Section 5 *) (** This part contains a formalization of the monotone counter, but with an explicit contruction of the monoid, as we have also done in the proof mode paper. This should simplify explaining and understanding what is happening. A version that uses the authoritative monoid and natural number monoid under max can be found in [theories/heap_lang/lib/counter.v]. *) (** The invariant rule in the paper is in fact derived from mask changing update modalities (which we did not cover in the paper). Normally we use these mask changing update modalities directly in our proofs, but in this file we use the first prove the rule as a lemma, and then use that. *) Lemma wp_inv_open `{irisG Λ Σ} N E P e Φ : `````` Robbert Krebbers committed Nov 04, 2017 104 `````` nclose N ⊆ E → Atomic e → `````` Robbert Krebbers committed Jan 17, 2017 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 `````` inv N P ∗ (▷ P -∗ WP e @ E ∖ ↑N {{ v, ▷ P ∗ Φ v }}) ⊢ WP e @ E {{ Φ }}. Proof. iIntros (??) "[#Hinv Hwp]". iMod (inv_open E N P with "Hinv") as "[HP Hclose]"=>//. iApply wp_wand_r; iSplitL "HP Hwp"; [by iApply "Hwp"|]. iIntros (v) "[HP \$]". by iApply "Hclose". Qed. Definition newcounter : val := λ: <>, ref #0. Definition incr : val := rec: "incr" "l" := let: "n" := !"l" in if: CAS "l" "n" (#1 + "n") then #() else "incr" "l". Definition read : val := λ: "l", !"l". (** The CMRA we need. *) Inductive M := Auth : nat → M | Frag : nat → M | Bot. Section M. Arguments cmra_op _ !_ !_/. Arguments op _ _ !_ !_/. Arguments core _ _ !_/. Canonical Structure M_C : ofeT := leibnizC M. Instance M_valid : Valid M := λ x, x ≠ Bot. Instance M_op : Op M := λ x y, match x, y with | Auth n, Frag j | Frag j, Auth n => if decide (j ≤ n)%nat then Auth n else Bot | Frag i, Frag j => Frag (max i j) | _, _ => Bot end. Instance M_pcore : PCore M := λ x, Some match x with Auth j | Frag j => Frag j | _ => Bot end. `````` Robbert Krebbers committed Sep 17, 2017 139 `````` Instance M_unit : Unit M := Frag 0. `````` Robbert Krebbers committed Jan 17, 2017 140 141 142 143 144 145 146 147 148 149 150 151 152 153 `````` Definition M_ra_mixin : RAMixin M. Proof. apply ra_total_mixin; try solve_proper || eauto. - intros [n1|i1|] [n2|i2|] [n3|i3|]; repeat (simpl; case_decide); f_equal/=; lia. - intros [n1|i1|] [n2|i2|]; repeat (simpl; case_decide); f_equal/=; lia. - intros [n|i|]; repeat (simpl; case_decide); f_equal/=; lia. - by intros [n|i|]. - intros [n1|i1|] y [[n2|i2|] ?]; exists (core y); simplify_eq/=; repeat (simpl; case_decide); f_equal/=; lia. - intros [n1|i1|] [n2|i2|]; simpl; by try case_decide. Qed. Canonical Structure M_R : cmraT := discreteR M M_ra_mixin. `````` Robbert Krebbers committed Feb 09, 2017 154 `````` `````` Robbert Krebbers committed Oct 25, 2017 155 `````` Global Instance M_discrete : CmraDiscrete M_R. `````` Robbert Krebbers committed Feb 09, 2017 156 157 `````` Proof. apply discrete_cmra_discrete. Qed. `````` Robbert Krebbers committed Oct 25, 2017 158 `````` Definition M_ucmra_mixin : UcmraMixin M. `````` Robbert Krebbers committed Jan 17, 2017 159 160 161 162 `````` Proof. split; try (done || apply _). intros [?|?|]; simpl; try case_decide; f_equal/=; lia. Qed. `````` Robbert Krebbers committed Oct 25, 2017 163 `````` Canonical Structure M_UR : ucmraT := UcmraT M M_ucmra_mixin. `````` Robbert Krebbers committed Jan 17, 2017 164 `````` `````` Robbert Krebbers committed Oct 25, 2017 165 `````` Global Instance frag_core_id n : CoreId (Frag n). `````` Robbert Krebbers committed Jan 17, 2017 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 `````` Proof. by constructor. Qed. Lemma auth_frag_valid j n : ✓ (Auth n ⋅ Frag j) → (j ≤ n)%nat. Proof. simpl. case_decide. done. by intros []. Qed. Lemma auth_frag_op (j n : nat) : (j ≤ n)%nat → Auth n = Auth n ⋅ Frag j. Proof. intros. by rewrite /= decide_True. Qed. Lemma M_update n : Auth n ~~> Auth (S n). Proof. apply cmra_discrete_update=>-[m|j|] /= ?; repeat case_decide; done || lia. Qed. End M. Class counterG Σ := CounterG { counter_tokG :> inG Σ M_UR }. Definition counterΣ : gFunctors := #[GFunctor (constRF M_UR)]. Instance subG_counterΣ {Σ} : subG counterΣ Σ → counterG Σ. Proof. intros [?%subG_inG _]%subG_inv. split; apply _. Qed. Section counter_proof. Context `{!heapG Σ, !counterG Σ}. Implicit Types l : loc. Definition I (γ : gname) (l : loc) : iProp Σ := (∃ c : nat, l ↦ #c ∗ own γ (Auth c))%I. Definition C (l : loc) (n : nat) : iProp Σ := (∃ N γ, inv N (I γ l) ∧ own γ (Frag n))%I. (** The main proofs. *) `````` Robbert Krebbers committed Oct 25, 2017 194 `````` Global Instance C_persistent l n : Persistent (C l n). `````` Robbert Krebbers committed Jan 17, 2017 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 `````` Proof. apply _. Qed. Lemma newcounter_spec : {{ True }} newcounter #() {{ v, ∃ l, ⌜v = #l⌝ ∧ C l 0 }}. Proof. iIntros "!# _ /=". rewrite -wp_fupd /newcounter /=. wp_seq. wp_alloc l as "Hl". iMod (own_alloc (Auth 0)) as (γ) "Hγ"; first done. rewrite (auth_frag_op 0 0) //; iDestruct "Hγ" as "[Hγ Hγf]". set (N:= nroot .@ "counter"). iMod (inv_alloc N _ (I γ l) with "[Hl Hγ]") as "#?". { iIntros "!>". iExists 0%nat. by iFrame. } iModIntro. rewrite /C; eauto 10. Qed. Lemma incr_spec l n : {{ C l n }} incr #l {{ v, ⌜v = #()⌝ ∧ C l (S n) }}. Proof. iIntros "!# Hl /=". iLöb as "IH". wp_rec. iDestruct "Hl" as (N γ) "[#Hinv Hγf]". wp_bind (! _)%E. iApply wp_inv_open; last iFrame "Hinv"; auto. iDestruct 1 as (c) "[Hl Hγ]". wp_load. iSplitL "Hl Hγ"; [iNext; iExists c; by iFrame|]. wp_let. wp_op. wp_bind (CAS _ _ _). iApply wp_inv_open; last iFrame "Hinv"; auto. iDestruct 1 as (c') ">[Hl Hγ]". destruct (decide (c' = c)) as [->|]. - iCombine "Hγ" "Hγf" as "Hγ". iDestruct (own_valid with "Hγ") as %?%auth_frag_valid; rewrite -auth_frag_op //. iMod (own_update with "Hγ") as "Hγ"; first apply M_update. rewrite (auth_frag_op (S n) (S c)); last lia; iDestruct "Hγ" as "[Hγ Hγf]". wp_cas_suc. iSplitL "Hl Hγ". { iNext. iExists (S c). rewrite Nat2Z.inj_succ Z.add_1_l. by iFrame. } wp_if. rewrite {3}/C; eauto 10. - wp_cas_fail; first (intros [=]; abstract omega). iSplitL "Hl Hγ"; [iNext; iExists c'; by iFrame|]. wp_if. iApply ("IH" with "[Hγf]"). rewrite {3}/C; eauto 10. Qed. Lemma read_spec l n : {{ C l n }} read #l {{ v, ∃ m : nat, ⌜v = #m ∧ n ≤ m⌝ ∧ C l m }}. Proof. iIntros "!# Hl /=". iDestruct "Hl" as (N γ) "[#Hinv Hγf]". rewrite /read /=. wp_let. iApply wp_inv_open; last iFrame "Hinv"; auto. iDestruct 1 as (c) "[Hl Hγ]". wp_load. iDestruct (own_valid γ (Frag n ⋅ Auth c) with "[-]") as % ?%auth_frag_valid. { iApply own_op. by iFrame. } rewrite (auth_frag_op c c); last lia; iDestruct "Hγ" as "[Hγ Hγf']". iSplitL "Hl Hγ"; [iNext; iExists c; by iFrame|]. rewrite /C; eauto 10 with omega. Qed. End counter_proof.``````