Commit 4ae55518 by Ralf Jung

### simplify stateful lifting lemmas; prove fork lemma

parent fa175d94
 ... ... @@ -280,22 +280,20 @@ Inductive prim_step : expr -> state -> expr -> state -> option expr -> Prop := prim_step (Cas (Loc l) e1 e2) σ LitTrue (<[l:=v2]>σ) None . Definition reducible e: Prop := exists σ e' σ' ef, prim_step e σ e' σ' ef. Definition reducible e σ : Prop := ∃ e' σ' ef, prim_step e σ e' σ' ef. Lemma reducible_not_value e: reducible e -> e2v e = None. Lemma reducible_not_value e σ : reducible e σ → e2v e = None. Proof. intros (σ' & e'' & σ'' & ef & Hstep). destruct Hstep; simpl in *; reflexivity. intros (e' & σ' & ef & Hstep). destruct Hstep; simpl in *; reflexivity. Qed. Definition stuck (e : expr) : Prop := forall K e', e = fill K e' -> ~reducible e'. Definition stuck (e : expr) σ : Prop := ∀ K e', e = fill K e' → ~reducible e' σ. Lemma values_stuck v : stuck (v2e v). Lemma values_stuck v σ : stuck (v2e v) σ. Proof. intros ?? Heq. edestruct (fill_value K) as [v' Hv']. ... ... @@ -309,9 +307,9 @@ Section step_by_value. expression has a non-value e in the hole, then K is a left sub-context of K' - in other words, e also contains the reducible expression *) Lemma step_by_value {K K' e e'} : Lemma step_by_value {K K' e e' σ} : fill K e = fill K' e' -> reducible e' -> reducible e' σ -> e2v e = None -> exists K'', K' = comp_ctx K K''. Proof. ... ... @@ -324,7 +322,7 @@ Proof. by erewrite ?v2v). Ltac bad_red Hfill e' Hred := exfalso; destruct e'; try discriminate Hfill; []; case: Hfill; intros; subst; destruct Hred as (σ' & e'' & σ'' & ef & Hstep); case: Hfill; intros; subst; destruct Hred as (e'' & σ'' & ef & Hstep); inversion Hstep; done || (clear Hstep; subst; eapply fill_not_value2; last ( try match goal with [ H : _ = fill _ _ |- _ ] => erewrite <-H end; simpl; ... ... @@ -451,14 +449,14 @@ Section Language. |}. Next Obligation. intros e1 σ1 e2 σ2 ef (K & e1' & e2' & He1 & He2 & Hstep). subst e1 e2. eapply fill_not_value. eapply reducible_not_value. do 4 eexists; eassumption. eapply fill_not_value. eapply reducible_not_value. do 3 eexists; eassumption. Qed. Next Obligation. intros ? ? ? ? ? Hatomic (K & e1' & e2' & Heq1 & Heq2 & Hstep). subst. move: (Hatomic). rewrite (atomic_fill e1' K). - rewrite !fill_empty. eauto using atomic_step. - assumption. - clear Hatomic. eapply reducible_not_value. do 4 eexists; eassumption. - clear Hatomic. eapply reducible_not_value. do 3 eexists; eassumption. Qed. (** We can have bind with arbitrary evaluation contexts **) ... ... @@ -470,8 +468,8 @@ Section Language. exists (comp_ctx K K'), e1', e2'. rewrite -!fill_comp Heq1 Heq2. split; last split; reflexivity || assumption. - intros ? ? ? ? ? Hnval (K'' & e1'' & e2'' & Heq1 & Heq2 & Hstep). destruct (step_by_value Heq1) as [K' HeqK]. + do 4 eexists. eassumption. destruct (step_by_value (σ:=σ1) Heq1) as [K' HeqK]. + do 3 eexists. eassumption. + assumption. + subst e2 K''. rewrite -fill_comp in Heq1. apply fill_inj_r in Heq1. subst e1'. exists (fill K' e2''). split; first by rewrite -fill_comp. ... ... @@ -479,15 +477,15 @@ Section Language. Qed. Lemma prim_ectx_step e1 σ1 e2 σ2 ef : reducible e1 → reducible e1 σ1 → ectx_step e1 σ1 e2 σ2 ef → prim_step e1 σ1 e2 σ2 ef. Proof. intros Hred (K' & e1' & e2' & Heq1 & Heq2 & Hstep). destruct (@step_by_value K' EmptyCtx e1' e1) as [K'' [HK' HK'']%comp_empty]. destruct (@step_by_value K' EmptyCtx e1' e1 σ1) as [K'' [HK' HK'']%comp_empty]. - by rewrite fill_empty. - done. - apply reducible_not_value. do 4 eexists; eassumption. - eapply reducible_not_value. do 3 eexists; eassumption. - subst K' K'' e1 e2. by rewrite !fill_empty. Qed. ... ...
 Require Export barrier.parameter. Require Import prelude.gmap iris.lifting. Require Import prelude.gmap iris.lifting barrier.heap_lang. Import uPred. (* TODO RJ: Figure out a way to to always use our Σ. *) ... ... @@ -11,7 +11,35 @@ Proof. by apply (wp_bind (Σ:=Σ) (K := fill K)), fill_is_ctx. Qed. (** Base axioms for core primitives of the language. *) (** Base axioms for core primitives of the language: Stateful reductions *) Lemma wp_lift_step E1 E2 (φ : expr → state → Prop) Q e1 σ1 : E1 ⊆ E2 → to_val e1 = None → reducible e1 σ1 → (∀ e2 σ2 ef, prim_step e1 σ1 e2 σ2 ef → ef = None ∧ φ e2 σ2) → pvs E2 E1 (ownP (Σ:=Σ) σ1 ★ ▷ ∀ e2 σ2, (■ φ e2 σ2 ∧ ownP (Σ:=Σ) σ2) -★ pvs E1 E2 (wp (Σ:=Σ) E2 e2 Q)) ⊑ wp (Σ:=Σ) E2 e1 Q. Proof. (* RJ FIXME WTF the bound names of wp_lift_step *changed*?!?? *) intros ? He Hsafe Hstep. etransitivity; last eapply wp_lift_step with (σ2 := σ1) (φ0 := λ e' σ' ef, ef = None ∧ φ e' σ'); last first. - intros e2 σ2 ef Hstep'%prim_ectx_step; last done. by apply Hstep. - destruct Hsafe as (e' & σ' & ? & ?). do 3 eexists. exists EmptyCtx. do 2 eexists. split_ands; try (by rewrite fill_empty); eassumption. - done. - eassumption. - apply pvs_mono. apply sep_mono; first done. apply later_mono. apply forall_mono=>e2. apply forall_mono=>σ2. apply forall_intro=>ef. apply wand_intro_l. rewrite always_and_sep_l' -associative -always_and_sep_l'. apply const_elim_l; move=>[-> Hφ]. eapply const_intro_l; first eexact Hφ. rewrite always_and_sep_l' associative -always_and_sep_l' wand_elim_r. apply pvs_mono. rewrite right_id. done. Qed. (* TODO RJ: Figure out some better way to make the postcondition a predicate over a *location* *) ... ... @@ -23,11 +51,9 @@ Lemma wp_alloc E σ v: Proof. (* RJ FIXME: rewrite would be nicer... *) etransitivity; last eapply wp_lift_step with (σ1 := σ) (φ := λ e' σ' ef, ∃ l, e' = Loc l ∧ σ' = <[l:=v]>σ ∧ σ !! l = None ∧ ef = None); (φ := λ e' σ', ∃ l, e' = Loc l ∧ σ' = <[l:=v]>σ ∧ σ !! l = None); last first. - intros e2 σ2 ef Hstep%prim_ectx_step; last first. { exists ∅. do 3 eexists. eapply AllocS with (l:=0); by rewrite ?v2v. } inversion_clear Hstep. - intros e2 σ2 ef Hstep. inversion_clear Hstep. split; first done. rewrite v2v in Hv. inversion_clear Hv. eexists; split_ands; done. - (* RJ FIXME: Need to find a fresh location. *) admit. ... ... @@ -36,9 +62,9 @@ Proof. - (* RJ FIXME I am sure there is a better way to invoke right_id, but I could not find it. *) rewrite -pvs_intro. rewrite -{1}[ownP σ](@right_id _ _ _ _ uPred.sep_True). apply sep_mono; first done. rewrite -later_intro. apply forall_intro=>e2. apply forall_intro=>σ2. apply forall_intro=>ef. apply forall_intro=>e2. apply forall_intro=>σ2. apply wand_intro_l. rewrite right_id. rewrite -pvs_intro. apply const_elim_l. intros [l [-> [-> [Hl ->]]]]. rewrite right_id. apply const_elim_l. intros [l [-> [-> Hl]]]. rewrite -wp_value'; last reflexivity. erewrite <-exist_intro with (a := l). apply and_intro. + by apply const_intro. ... ... @@ -50,21 +76,17 @@ Lemma wp_load E σ l v: ownP (Σ:=Σ) σ ⊑ wp (Σ:=Σ) E (Load (Loc l)) (λ v', ■(v' = v) ∧ ownP (Σ:=Σ) σ). Proof. intros Hl. etransitivity; last eapply wp_lift_step with (σ1 := σ) (φ := λ e' σ' ef, e' = v2e v ∧ σ' = σ ∧ ef = None); last first. - intros e2 σ2 ef Hstep%prim_ectx_step; last first. { exists σ. do 3 eexists. eapply LoadS; eassumption. } move: Hl. inversion_clear Hstep=>{σ}. rewrite Hlookup. case=>->. done. - do 3 eexists. exists EmptyCtx. do 2 eexists. split_ands; try (by rewrite fill_empty); []. eapply LoadS; eassumption. (φ := λ e' σ', e' = v2e v ∧ σ' = σ); last first. - intros e2 σ2 ef Hstep. move: Hl. inversion_clear Hstep=>{σ}. rewrite Hlookup. case=>->. done. - do 3 eexists. eapply LoadS; eassumption. - reflexivity. - reflexivity. - rewrite -pvs_intro. rewrite -{1}[ownP σ](@right_id _ _ _ _ uPred.sep_True). apply sep_mono; first done. rewrite -later_intro. apply forall_intro=>e2. apply forall_intro=>σ2. apply forall_intro=>ef. apply forall_intro=>e2. apply forall_intro=>σ2. apply wand_intro_l. rewrite right_id. rewrite -pvs_intro. apply const_elim_l. intros [-> [-> ->]]. rewrite right_id. apply const_elim_l. intros [-> ->]. rewrite -wp_value. apply and_intro. + by apply const_intro. + done. ... ... @@ -72,25 +94,47 @@ Qed. Lemma wp_store E σ l v v': σ !! l = Some v' → ownP (Σ:=Σ) σ ⊑ wp (Σ:=Σ) E (Store (Loc l) (v2e v)) (λ v', ■(v' = LitVUnit) ∧ ownP (Σ:=Σ) (<[l:=v]>σ)). ownP (Σ:=Σ) σ ⊑ wp (Σ:=Σ) E (Store (Loc l) (v2e v)) (λ v', ■(v' = LitVUnit) ∧ ownP (Σ:=Σ) (<[l:=v]>σ)). Proof. intros Hl. etransitivity; last eapply wp_lift_step with (σ1 := σ) (φ := λ e' σ' ef, e' = LitUnit ∧ σ' = <[l:=v]>σ ∧ ef = None); last first. - intros e2 σ2 ef Hstep%prim_ectx_step; last first. { exists σ. do 3 eexists. eapply StoreS; last (eexists; eassumption). by rewrite v2v. } move: Hl. inversion_clear Hstep=>{σ2}. rewrite v2v in Hv. inversion_clear Hv. done. - do 3 eexists. exists EmptyCtx. do 2 eexists. split_ands; try (by rewrite fill_empty); []. eapply StoreS; last (eexists; eassumption). by rewrite v2v. (φ := λ e' σ', e' = LitUnit ∧ σ' = <[l:=v]>σ); last first. - intros e2 σ2 ef Hstep. move: Hl. inversion_clear Hstep=>{σ2}. rewrite v2v in Hv. inversion_clear Hv. done. - do 3 eexists. eapply StoreS; last (eexists; eassumption). by rewrite v2v. - reflexivity. - reflexivity. - rewrite -pvs_intro. rewrite -{1}[ownP σ](@right_id _ _ _ _ uPred.sep_True). apply sep_mono; first done. rewrite -later_intro. apply forall_intro=>e2. apply forall_intro=>σ2. apply forall_intro=>ef. apply forall_intro=>e2. apply forall_intro=>σ2. apply wand_intro_l. rewrite right_id. rewrite -pvs_intro. apply const_elim_l. intros [-> [-> ->]]. rewrite right_id. apply const_elim_l. intros [-> ->]. rewrite -wp_value'; last reflexivity. apply and_intro. + by apply const_intro. + done. Qed. (** Base axioms for core primitives of the language: Stateless reductions *) Lemma wp_fork E e : ▷ wp (Σ:=Σ) coPset_all e (λ _, True) ⊑ wp (Σ:=Σ) E (Fork e) (λ _, True). Proof. etransitivity; last eapply wp_lift_pure_step with (φ := λ e' ef, e' = LitUnit ∧ ef = Some e); last first. - intros σ1 e2 σ2 ef Hstep%prim_ectx_step; last first. { do 3 eexists. eapply ForkS. } inversion_clear Hstep. done. - intros ?. do 3 eexists. exists EmptyCtx. do 2 eexists. split_ands; try (by rewrite fill_empty); []. eapply ForkS. - reflexivity. - apply later_mono. apply forall_intro=>e2. apply forall_intro=>ef. apply impl_intro_l. apply const_elim_l. intros [-> ->]. (* FIXME RJ This is ridicolous. *) transitivity (True ★ wp coPset_all e (λ _ : ival Σ, True))%I; first by rewrite left_id. apply sep_mono; last reflexivity. rewrite -wp_value'; reflexivity. Qed.
 ... ... @@ -454,6 +454,18 @@ Proof. intros H; apply impl_intro_l. by rewrite -H and_elim_l. Qed. Lemma const_intro_l φ Q R : φ → (■φ ∧ Q) ⊑ R → Q ⊑ R. Proof. intros ? <-. apply and_intro; last done. by apply const_intro. Qed. Lemma const_intro_r φ Q R : φ → (Q ∧ ■φ) ⊑ R → Q ⊑ R. Proof. (* FIXME RJ: Why does this not work? rewrite (commutative uPred_and Q (■φ)%I). *) intros ? <-. apply and_intro; first done. by apply const_intro. Qed. Lemma const_elim_l φ Q R : (φ → Q ⊑ R) → (■ φ ∧ Q) ⊑ R. Proof. intros; apply const_elim with φ; eauto. Qed. Lemma const_elim_r φ Q R : (φ → Q ⊑ R) → (Q ∧ ■ φ) ⊑ R. ... ...
Supports Markdown
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!