sts.v 6.78 KB
 Ralf Jung committed Feb 23, 2016 1 ``````From algebra Require Export sts upred_tactics. `````` Robbert Krebbers committed Feb 23, 2016 2 ``````From program_logic Require Export invariants global_functor. `````` Ralf Jung committed Feb 15, 2016 3 4 ``````Import uPred. `````` Ralf Jung committed Mar 06, 2016 5 ``````(** The CMRA we need. *) `````` Robbert Krebbers committed Feb 17, 2016 6 ``````Class stsG Λ Σ (sts : stsT) := StsG { `````` Robbert Krebbers committed Mar 01, 2016 7 `````` sts_inG :> inG Λ Σ (stsR sts); `````` Robbert Krebbers committed Feb 16, 2016 8 `````` sts_inhabited :> Inhabited (sts.state sts); `````` Ralf Jung committed Feb 15, 2016 9 ``````}. `````` Robbert Krebbers committed Feb 17, 2016 10 ``````Coercion sts_inG : stsG >-> inG. `````` Ralf Jung committed Mar 06, 2016 11 ``````(** The Functor we need. *) `````` Robbert Krebbers committed Mar 02, 2016 12 ``````Definition stsGF (sts : stsT) : rFunctor := constRF (stsR sts). `````` Ralf Jung committed Mar 06, 2016 13 ``````(* Show and register that they match. *) `````` Ralf Jung committed Feb 23, 2016 14 ``````Instance inGF_stsG sts `{inGF Λ Σ (stsGF sts)} `````` Robbert Krebbers committed Feb 22, 2016 15 16 `````` `{Inhabited (sts.state sts)} : stsG Λ Σ sts. Proof. split; try apply _. apply: inGF_inG. Qed. `````` Ralf Jung committed Feb 22, 2016 17 `````` `````` Ralf Jung committed Mar 06, 2016 18 19 20 21 22 23 24 25 26 27 ``````Section definitions. Context `{i : stsG Λ Σ sts} (γ : gname). Definition sts_ownS (S : sts.states sts) (T : sts.tokens sts) : iPropG Λ Σ:= own γ (sts_frag S T). Definition sts_own (s : sts.state sts) (T : sts.tokens sts) : iPropG Λ Σ := own γ (sts_frag_up s T). Definition sts_inv (φ : sts.state sts → iPropG Λ Σ) : iPropG Λ Σ := (∃ s, own γ (sts_auth s ∅) ★ φ s)%I. Definition sts_ctx (N : namespace) (φ: sts.state sts → iPropG Λ Σ) : iPropG Λ Σ := inv N (sts_inv φ). `````` Ralf Jung committed Feb 24, 2016 28 `````` `````` Ralf Jung committed Mar 06, 2016 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 `````` Global Instance sts_inv_ne n : Proper (pointwise_relation _ (dist n) ==> dist n) sts_inv. Proof. solve_proper. Qed. Global Instance sts_inv_proper : Proper (pointwise_relation _ (≡) ==> (≡)) sts_inv. Proof. solve_proper. Qed. Global Instance sts_ownS_proper : Proper ((≡) ==> (≡) ==> (≡)) sts_ownS. Proof. solve_proper. Qed. Global Instance sts_own_proper s : Proper ((≡) ==> (≡)) (sts_own s). Proof. solve_proper. Qed. Global Instance sts_ctx_ne n N : Proper (pointwise_relation _ (dist n) ==> dist n) (sts_ctx N). Proof. solve_proper. Qed. Global Instance sts_ctx_proper N : Proper (pointwise_relation _ (≡) ==> (≡)) (sts_ctx N). Proof. solve_proper. Qed. Global Instance sts_ctx_always_stable N φ : AlwaysStable (sts_ctx N φ). Proof. apply _. Qed. End definitions. Typeclasses Opaque sts_own sts_ownS sts_ctx. `````` Robbert Krebbers committed Feb 17, 2016 49 50 51 52 ``````Instance: Params (@sts_inv) 5. Instance: Params (@sts_ownS) 5. Instance: Params (@sts_own) 6. Instance: Params (@sts_ctx) 6. `````` Ralf Jung committed Feb 15, 2016 53 54 `````` Section sts. `````` Robbert Krebbers committed Feb 17, 2016 55 `````` Context `{stsG Λ Σ sts} (φ : sts.state sts → iPropG Λ Σ). `````` Ralf Jung committed Feb 15, 2016 56 57 58 `````` Implicit Types N : namespace. Implicit Types P Q R : iPropG Λ Σ. Implicit Types γ : gname. `````` Robbert Krebbers committed Feb 16, 2016 59 60 61 62 `````` Implicit Types S : sts.states sts. Implicit Types T : sts.tokens sts. (** Setoids *) `````` Ralf Jung committed Feb 15, 2016 63 `````` `````` Ralf Jung committed Feb 15, 2016 64 65 `````` (* The same rule as implication does *not* hold, as could be shown using sts_frag_included. *) `````` Ralf Jung committed Feb 17, 2016 66 `````` Lemma sts_ownS_weaken E γ S1 S2 T1 T2 : `````` 67 `````` T2 ⊆ T1 → S1 ⊆ S2 → sts.closed S2 T2 → `````` Robbert Krebbers committed Feb 19, 2016 68 `````` sts_ownS γ S1 T1 ⊑ (|={E}=> sts_ownS γ S2 T2). `````` Ralf Jung committed Feb 25, 2016 69 `````` Proof. intros ? ? ?. by apply own_update, sts_update_frag. Qed. `````` Ralf Jung committed Feb 15, 2016 70 `````` `````` Ralf Jung committed Feb 17, 2016 71 `````` Lemma sts_own_weaken E γ s S T1 T2 : `````` 72 `````` T2 ⊆ T1 → s ∈ S → sts.closed S T2 → `````` Robbert Krebbers committed Feb 19, 2016 73 `````` sts_own γ s T1 ⊑ (|={E}=> sts_ownS γ S T2). `````` Ralf Jung committed Feb 25, 2016 74 `````` Proof. intros ???. by apply own_update, sts_update_frag_up. Qed. `````` Ralf Jung committed Feb 15, 2016 75 `````` `````` Ralf Jung committed Feb 17, 2016 76 `````` Lemma sts_ownS_op γ S1 S2 T1 T2 : `````` Ralf Jung committed Feb 17, 2016 77 `````` T1 ∩ T2 ⊆ ∅ → sts.closed S1 T1 → sts.closed S2 T2 → `````` Ralf Jung committed Feb 17, 2016 78 `````` sts_ownS γ (S1 ∩ S2) (T1 ∪ T2) ≡ (sts_ownS γ S1 T1 ★ sts_ownS γ S2 T2)%I. `````` Ralf Jung committed Feb 25, 2016 79 `````` Proof. intros. by rewrite /sts_ownS -own_op sts_op_frag. Qed. `````` Ralf Jung committed Feb 17, 2016 80 `````` `````` Ralf Jung committed Feb 17, 2016 81 82 `````` Lemma sts_alloc E N s : nclose N ⊆ E → `````` Robbert Krebbers committed Feb 19, 2016 83 `````` ▷ φ s ⊑ (|={E}=> ∃ γ, sts_ctx γ N φ ∧ sts_own γ s (⊤ ∖ sts.tok s)). `````` Ralf Jung committed Feb 15, 2016 84 `````` Proof. `````` Ralf Jung committed Feb 17, 2016 85 `````` intros HN. eapply sep_elim_True_r. `````` Ralf Jung committed Mar 05, 2016 86 `````` { apply (own_alloc (sts_auth s (⊤ ∖ sts.tok s)) E). `````` Robbert Krebbers committed Feb 17, 2016 87 `````` apply sts_auth_valid; set_solver. } `````` Ralf Jung committed Mar 05, 2016 88 `````` rewrite pvs_frame_l. apply pvs_strip_pvs. `````` Ralf Jung committed Feb 15, 2016 89 `````` rewrite sep_exist_l. apply exist_elim=>γ. rewrite -(exist_intro γ). `````` Ralf Jung committed Feb 20, 2016 90 `````` trans (▷ sts_inv γ φ ★ sts_own γ s (⊤ ∖ sts.tok s))%I. `````` Ralf Jung committed Feb 17, 2016 91 `````` { rewrite /sts_inv -(exist_intro s) later_sep. `````` Ralf Jung committed Feb 25, 2016 92 `````` ecancel [▷ φ _]%I. `````` Robbert Krebbers committed Feb 17, 2016 93 `````` by rewrite -later_intro -own_op sts_op_auth_frag_up; last set_solver. } `````` Ralf Jung committed Mar 05, 2016 94 `````` rewrite (inv_alloc N E) // /sts_ctx pvs_frame_r. `````` Ralf Jung committed Feb 15, 2016 95 96 97 `````` by rewrite always_and_sep_l. Qed. `````` Robbert Krebbers committed Feb 16, 2016 98 `````` Lemma sts_opened E γ S T : `````` Robbert Krebbers committed Feb 17, 2016 99 `````` (▷ sts_inv γ φ ★ sts_ownS γ S T) `````` Robbert Krebbers committed Feb 19, 2016 100 `````` ⊑ (|={E}=> ∃ s, ■ (s ∈ S) ★ ▷ φ s ★ own γ (sts_auth s T)). `````` Ralf Jung committed Feb 15, 2016 101 `````` Proof. `````` Ralf Jung committed Feb 25, 2016 102 `````` rewrite /sts_inv later_exist sep_exist_r. apply exist_elim=>s. `````` Ralf Jung committed Feb 15, 2016 103 `````` rewrite later_sep pvs_timeless !pvs_frame_r. apply pvs_mono. `````` Ralf Jung committed Mar 05, 2016 104 105 106 `````` rewrite -(exist_intro s). ecancel [▷ φ _]%I. rewrite -own_op own_valid_l discrete_valid. apply const_elim_sep_l=> Hvalid. `````` Robbert Krebbers committed Feb 24, 2016 107 `````` assert (s ∈ S) by eauto using sts_auth_frag_valid_inv. `````` Ralf Jung committed Mar 05, 2016 108 `````` rewrite const_equiv // left_id sts_op_auth_frag //. `````` Robbert Krebbers committed Feb 24, 2016 109 `````` by assert (✓ sts_frag S T) as [??] by eauto using cmra_valid_op_r. `````` Ralf Jung committed Feb 15, 2016 110 111 `````` Qed. `````` Robbert Krebbers committed Feb 16, 2016 112 `````` Lemma sts_closing E γ s T s' T' : `````` Ralf Jung committed Feb 20, 2016 113 `````` sts.steps (s, T) (s', T') → `````` Robbert Krebbers committed Feb 19, 2016 114 `````` (▷ φ s' ★ own γ (sts_auth s T)) ⊑ (|={E}=> ▷ sts_inv γ φ ★ sts_own γ s' T'). `````` Ralf Jung committed Feb 15, 2016 115 `````` Proof. `````` Ralf Jung committed Feb 25, 2016 116 `````` intros Hstep. rewrite /sts_inv -(exist_intro s') later_sep. `````` Ralf Jung committed Feb 23, 2016 117 `````` (* TODO it would be really nice to use cancel here *) `````` 118 `````` rewrite [(_ ★ ▷ φ _)%I]comm -assoc. `````` Robbert Krebbers committed Feb 16, 2016 119 `````` rewrite -pvs_frame_l. apply sep_mono_r. rewrite -later_intro. `````` Robbert Krebbers committed Feb 24, 2016 120 `````` rewrite own_valid_l discrete_valid. apply const_elim_sep_l=>Hval. `````` Ralf Jung committed Feb 20, 2016 121 `````` trans (|={E}=> own γ (sts_auth s' T'))%I. `````` Robbert Krebbers committed Feb 16, 2016 122 `````` { by apply own_update, sts_update_auth. } `````` Ralf Jung committed Feb 20, 2016 123 `````` by rewrite -own_op sts_op_auth_frag_up. `````` Ralf Jung committed Feb 15, 2016 124 `````` Qed. `````` Ralf Jung committed Feb 15, 2016 125 `````` `````` Ralf Jung committed Feb 15, 2016 126 127 `````` Context {V} (fsa : FSA Λ (globalF Σ) V) `{!FrameShiftAssertion fsaV fsa}. `````` Robbert Krebbers committed Feb 18, 2016 128 `````` Lemma sts_fsaS E N P (Ψ : V → iPropG Λ Σ) γ S T : `````` Ralf Jung committed Feb 15, 2016 129 `````` fsaV → nclose N ⊆ E → `````` Robbert Krebbers committed Feb 17, 2016 130 131 `````` P ⊑ sts_ctx γ N φ → P ⊑ (sts_ownS γ S T ★ ∀ s, `````` Ralf Jung committed Feb 15, 2016 132 `````` ■ (s ∈ S) ★ ▷ φ s -★ `````` Ralf Jung committed Feb 15, 2016 133 `````` fsa (E ∖ nclose N) (λ x, ∃ s' T', `````` Ralf Jung committed Feb 20, 2016 134 `````` ■ sts.steps (s, T) (s', T') ★ ▷ φ s' ★ `````` Robbert Krebbers committed Feb 18, 2016 135 136 `````` (sts_own γ s' T' -★ Ψ x))) → P ⊑ fsa E Ψ. `````` Ralf Jung committed Feb 15, 2016 137 `````` Proof. `````` Robbert Krebbers committed Feb 16, 2016 138 `````` rewrite /sts_ctx=>? HN Hinv Hinner. `````` Ralf Jung committed Feb 15, 2016 139 140 `````` eapply (inv_fsa fsa); eauto. rewrite Hinner=>{Hinner Hinv P HN}. apply wand_intro_l. rewrite assoc. `````` Robbert Krebbers committed Feb 16, 2016 141 `````` rewrite (sts_opened (E ∖ N)) !pvs_frame_r !sep_exist_r. `````` Ralf Jung committed Feb 15, 2016 142 143 `````` apply (fsa_strip_pvs fsa). apply exist_elim=>s. rewrite (forall_elim s). rewrite [(▷_ ★ _)%I]comm. `````` Ralf Jung committed Feb 20, 2016 144 145 146 `````` eapply wand_apply_r; first (by eapply (wand_frame_l (own γ _))); last first. { rewrite assoc [(_ ★ own _ _)%I]comm -assoc. done. } rewrite fsa_frame_l. `````` Ralf Jung committed Feb 15, 2016 147 `````` apply (fsa_mono_pvs fsa)=> x. `````` Ralf Jung committed Feb 15, 2016 148 `````` rewrite sep_exist_l; apply exist_elim=> s'. `````` Ralf Jung committed Feb 15, 2016 149 150 `````` rewrite sep_exist_l; apply exist_elim=>T'. rewrite comm -!assoc. apply const_elim_sep_l=>-Hstep. `````` Ralf Jung committed Feb 15, 2016 151 `````` rewrite assoc [(_ ★ (_ -★ _))%I]comm -assoc. `````` Robbert Krebbers committed Feb 16, 2016 152 `````` rewrite (sts_closing (E ∖ N)) //; []. `````` Ralf Jung committed Feb 15, 2016 153 154 155 156 `````` rewrite pvs_frame_l. apply pvs_mono. by rewrite assoc [(_ ★ ▷_)%I]comm -assoc wand_elim_l. Qed. `````` Robbert Krebbers committed Feb 18, 2016 157 `````` Lemma sts_fsa E N P (Ψ : V → iPropG Λ Σ) γ s0 T : `````` Ralf Jung committed Feb 15, 2016 158 `````` fsaV → nclose N ⊆ E → `````` Robbert Krebbers committed Feb 17, 2016 159 160 `````` P ⊑ sts_ctx γ N φ → P ⊑ (sts_own γ s0 T ★ ∀ s, `````` Robbert Krebbers committed Feb 16, 2016 161 `````` ■ (s ∈ sts.up s0 T) ★ ▷ φ s -★ `````` Ralf Jung committed Feb 15, 2016 162 `````` fsa (E ∖ nclose N) (λ x, ∃ s' T', `````` Ralf Jung committed Feb 20, 2016 163 `````` ■ (sts.steps (s, T) (s', T')) ★ ▷ φ s' ★ `````` Robbert Krebbers committed Feb 18, 2016 164 165 `````` (sts_own γ s' T' -★ Ψ x))) → P ⊑ fsa E Ψ. `````` Ralf Jung committed Feb 25, 2016 166 `````` Proof. by apply sts_fsaS. Qed. `````` Ralf Jung committed Feb 15, 2016 167 ``End sts.``