Commit d011f232 by Robbert Krebbers

### Valid STS elements should be non-empty.

parent 4847b5c1
 ... ... @@ -29,6 +29,7 @@ Inductive frame_step (T : set B) (s1 s2 : A) : Prop := T1 ∩ (tok s1 ∪ T) ≡ ∅ → step (s1,T1) (s2,T2) → frame_step T s1 s2. Hint Resolve Frame_step. Record closed (T : set B) (S : set A) : Prop := Closed { closed_ne : S ≢ ∅; closed_disjoint s : s ∈ S → tok s ∩ T ≡ ∅; closed_step s1 s2 : s1 ∈ S → frame_step T s1 s2 → s2 ∈ S }. ... ... @@ -44,7 +45,8 @@ Global Instance sts_unit : Unit (t R tok) := λ x, | frag S' _ => frag (up_set ∅ S') ∅ | auth s _ => frag (up ∅ s) ∅ end. Inductive sts_disjoint : Disjoint (t R tok) := | frag_frag_disjoint S1 S2 T1 T2 : T1 ∩ T2 ≡ ∅ → frag S1 T1 ⊥ frag S2 T2 | frag_frag_disjoint S1 S2 T1 T2 : S1 ∩ S2 ≢ ∅ → T1 ∩ T2 ≡ ∅ → frag S1 T1 ⊥ frag S2 T2 | auth_frag_disjoint s S T1 T2 : s ∈ S → T1 ∩ T2 ≡ ∅ → auth s T1 ⊥ frag S T2 | frag_auth_disjoint s S T1 T2 : s ∈ S → T1 ∩ T2 ≡ ∅ → frag S T1 ⊥ auth s T2. Global Existing Instance sts_disjoint. ... ... @@ -64,6 +66,7 @@ Global Instance sts_minus : Minus (t R tok) := λ x1 x2, end. Hint Extern 10 (equiv (A:=set _) _ _) => esolve_elem_of : sts. Hint Extern 10 (¬(equiv (A:=set _) _ _)) => esolve_elem_of : sts. Hint Extern 10 (_ ∈ _) => esolve_elem_of : sts. Hint Extern 10 (_ ⊆ _) => esolve_elem_of : sts. Instance: Equivalence ((≡) : relation (t R tok)). ... ... @@ -83,16 +86,14 @@ Qed. Instance closed_proper : Proper ((≡) ==> (≡) ==> iff) closed. Proof. by split; apply closed_proper'. Qed. Lemma closed_op T1 T2 S1 S2 : closed T1 S1 → closed T2 S2 → T1 ∩ T2 ≡ ∅ → closed (T1 ∪ T2) (S1 ∩ S2). closed T1 S1 → closed T2 S2 → T1 ∩ T2 ≡ ∅ → S1 ∩ S2 ≢ ∅ → closed (T1 ∪ T2) (S1 ∩ S2). Proof. intros [? Hstep1] [? Hstep2] ?; split; [esolve_elem_of|]. intros s3 s4; rewrite !elem_of_intersection; intros [??] [T ??]; split. * apply Hstep1 with s3; eauto with sts. * apply Hstep2 with s3; eauto with sts. intros [_ ? Hstep1] [_ ? Hstep2] ?; split; [done|esolve_elem_of|]. intros s3 s4; rewrite !elem_of_intersection; intros [??] [T3 T4 ?]; split. * apply Hstep1 with s3, Frame_step with T3 T4; auto with sts. * apply Hstep2 with s3, Frame_step with T3 T4; auto with sts. Qed. Lemma closed_all : closed ∅ set_all. Proof. split; auto with sts. Qed. Hint Resolve closed_all : sts. Instance up_preserving : Proper (flip (⊆) ==> (=) ==> (⊆)) up. Proof. intros T T' HT s ? <-; apply elem_of_subseteq. ... ... @@ -105,30 +106,32 @@ Instance up_set_proper : Proper ((≡) ==> (≡) ==> (≡)) up_set. Proof. by intros T1 T2 HT S1 S2 HS; unfold up_set; rewrite HS, HT. Qed. Lemma elem_of_up s T : s ∈ up T s. Proof. constructor. Qed. Lemma suseteq_up_set S T : S ⊆ up_set T S. Lemma subseteq_up_set S T : S ⊆ up_set T S. Proof. intros s ?; apply elem_of_bind; eauto using elem_of_up. Qed. Lemma closed_up_set S T : (∀ s, s ∈ S → tok s ∩ T ≡ ∅) → closed T (up_set T S). Lemma closed_up_set S T : (∀ s, s ∈ S → tok s ∩ T ≡ ∅) → S ≢ ∅ → closed T (up_set T S). Proof. intros HS; unfold up_set; split. intros HS Hne; unfold up_set; split. * assert (∀ s, s ∈ up T s) by eauto using elem_of_up. esolve_elem_of. * intros s; rewrite !elem_of_bind; intros (s'&Hstep&Hs'). specialize (HS s' Hs'); clear Hs' S. specialize (HS s' Hs'); clear Hs' Hne S. induction Hstep as [s|s1 s2 s3 [T1 T2 ? Hstep] ? IH]; auto. inversion_clear Hstep; apply IH; clear IH; auto with sts. * intros s1 s2; rewrite !elem_of_bind; intros (s&?&?) ?; exists s. split; [eapply rtc_r|]; eauto. Qed. Lemma closed_up_set_empty S : closed ∅ (up_set ∅ S). Lemma closed_up_set_empty S : S ≢ ∅ → closed ∅ (up_set ∅ S). Proof. eauto using closed_up_set with sts. Qed. Lemma closed_up s T : tok s ∩ T ≡ ∅ → closed T (up T s). Proof. intros; rewrite <-(collection_bind_singleton (up T) s). apply closed_up_set; auto with sts. apply closed_up_set; esolve_elem_of. Qed. Lemma closed_up_empty s : closed ∅ (up ∅ s). Proof. eauto using closed_up with sts. Qed. Lemma up_closed S T : closed T S → up_set T S ≡ S. Proof. intros; split; auto using suseteq_up_set; intros s. intros; split; auto using subseteq_up_set; intros s. unfold up_set; rewrite elem_of_bind; intros (s'&Hstep&?). induction Hstep; eauto using closed_step. Qed. ... ... @@ -144,7 +147,7 @@ Proof. closed T S → s ∈ S → tok s ∩ T' ≡ ∅ → tok s ∩ (T ∪ T') ≡ ∅). { intros S T T' s [??]; esolve_elem_of. } destruct 3; simpl in *; auto using closed_op with sts. * intros []; simpl; eauto using closed_up, closed_up_set with sts. * intros []; simpl; eauto using closed_up, closed_up_set, closed_ne with sts. * intros ???? (z&Hy&?&Hxz); destruct Hxz; inversion Hy;clear Hy; setoid_subst; rewrite ?disjoint_union_difference; auto using closed_up with sts. eapply closed_up_set; eauto 2 using closed_disjoint with sts. ... ... @@ -153,22 +156,37 @@ Proof. * destruct 4; inversion_clear 1; constructor; auto with sts. * destruct 1; constructor; auto with sts. * destruct 3; constructor; auto with sts. * intros []; constructor; auto using elem_of_up with sts. * intros [|S T]; constructor; auto using elem_of_up with sts. assert (S ⊆ up_set ∅ S ∧ S ≢ ∅) by eauto using subseteq_up_set, closed_ne. esolve_elem_of. * intros [|S T]; constructor; auto with sts. assert (S ⊆ up_set ∅ S); auto using suseteq_up_set with sts. assert (S ⊆ up_set ∅ S); auto using subseteq_up_set with sts. * intros [s T|S T]; constructor; auto with sts. + by rewrite (up_closed (up _ _)) by auto using closed_up with sts. + by rewrite (up_closed (up_set _ _)) by auto using closed_up_set with sts. * intros x y ?? (z&Hy&?&Hxz); exists (unit (x ⋅ y)). destruct Hxz; inversion_clear Hy; simpl; split_ands; auto using closed_up_set_empty, closed_up_empty; constructor; unfold up_set; auto with sts. * intros x y ?? (z&Hy&_&Hxz); destruct Hxz; inversion_clear Hy; constructor; eauto using elem_of_up; auto with sts. + by rewrite (up_closed (up_set _ _)) by eauto using closed_up_set, closed_ne with sts. * intros x y ?? (z&Hy&?&Hxz); exists (unit (x ⋅ y)); split_ands. + destruct Hxz;inversion_clear Hy;constructor;unfold up_set;esolve_elem_of. + destruct Hxz; inversion_clear Hy; simpl; auto using closed_up_set_empty, closed_up_empty with sts. + destruct Hxz; inversion_clear Hy; constructor; repeat match goal with | |- context [ up_set ?T ?S ] => unless (S ⊆ up_set T S) by done; pose proof (subseteq_up_set S T) | |- context [ up ?T ?s ] => unless (s ∈ up T s) by done; pose proof (elem_of_up s T) end; auto with sts. * intros x y ?? (z&Hy&_&Hxz); destruct Hxz; inversion_clear Hy; constructor; repeat match goal with | |- context [ up_set ?T ?S ] => unless (S ⊆ up_set T S) by done; pose proof (subseteq_up_set S T) | |- context [ up ?T ?s ] => unless (s ∈ up T s) by done; pose proof (elem_of_up s T) end; auto with sts. * intros x y ?? (z&Hy&?&Hxz); destruct Hxz as [S1 S2 T1 T2| |]; inversion Hy; clear Hy; constructor; setoid_subst; rewrite ?disjoint_union_difference by done; auto. split; [|apply intersection_greatest; auto using suseteq_up_set with sts]. split; [|apply intersection_greatest; auto using subseteq_up_set with sts]. apply intersection_greatest; [auto with sts|]. intros s2; rewrite elem_of_intersection. unfold up_set; rewrite elem_of_bind; intros (?&s1&?&?&?). ... ... @@ -178,7 +196,7 @@ Lemma step_closed s1 s2 T1 T2 S Tf : step (s1,T1) (s2,T2) → closed Tf S → s1 ∈ S → T1 ∩ Tf ≡ ∅ → s2 ∈ S ∧ T2 ∩ Tf ≡ ∅ ∧ tok s2 ∩ T2 ≡ ∅. Proof. inversion_clear 1 as [???? HR Hs1 Hs2]; intros [? Hstep] ??; split_ands; auto. inversion_clear 1 as [???? HR Hs1 Hs2]; intros [?? Hstep]??; split_ands; auto. * eapply Hstep with s1, Frame_step with T1 T2; auto with sts. * clear Hstep Hs1 Hs2; esolve_elem_of. Qed. ... ...
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