### Simplify barrier protocol proofs.

parent 2d9c5f33
 ... @@ -17,12 +17,9 @@ Inductive prim_step : relation state := ... @@ -17,12 +17,9 @@ Inductive prim_step : relation state := | ChangeI p I2 I1 : prim_step (State p I1) (State p I2) | ChangeI p I2 I1 : prim_step (State p I1) (State p I2) | ChangePhase I : prim_step (State Low I) (State High I). | ChangePhase I : prim_step (State Low I) (State High I). Definition change_tok (I : gset gname) : set token := {[ t | match t with Change i => i ∉ I | Send => False end ]}. Definition send_tok (p : phase) : set token := match p with Low => ∅ | High => {[ Send ]} end. Definition tok (s : state) : set token := Definition tok (s : state) : set token := change_tok (state_I s) ∪ send_tok (state_phase s). {[ t | ∃ i, t = Change i ∧ i ∉ state_I s ]} ∪ (if state_phase s is High then {[ Send ]} else ∅). Global Arguments tok !_ /. Global Arguments tok !_ /. Canonical Structure sts := sts.STS prim_step tok. Canonical Structure sts := sts.STS prim_step tok. ... @@ -35,30 +32,22 @@ Definition low_states : set state := {[ s | state_phase s = Low ]}. ... @@ -35,30 +32,22 @@ Definition low_states : set state := {[ s | state_phase s = Low ]}. Lemma i_states_closed i : sts.closed (i_states i) {[ Change i ]}. Lemma i_states_closed i : sts.closed (i_states i) {[ Change i ]}. Proof. Proof. split. split; first (intros [[] I]; set_solver). - intros [p I]. rewrite /= /i_states /change_tok. destruct p; set_solver. (* If we do the destruct of the states early, and then inversion - (* If we do the destruct of the states early, and then inversion on the proof of a transition, it doesn't work - we do not obtain on the proof of a transition, it doesn't work - we do not obtain the equalities we need. So we destruct the states late, because this the equalities we need. So we destruct the states late, because this means we can use "destruct" instead of "inversion". *) means we can use "destruct" instead of "inversion". *) intros s1 s2 Hs1 [T1 T2 Hdisj Hstep']. intros s1 s2 Hs1 [T1 T2 Hdisj Hstep']. inversion_clear Hstep' as [? ? ? ? Htrans _ _ Htok]. inversion_clear Hstep' as [? ? ? ? Htrans _ _ Htok]. destruct Htrans as [[] ??|]; done || set_solver. destruct Htrans; simpl in *; last done. move: Hs1 Hdisj Htok. rewrite elem_of_equiv_empty elem_of_equiv. move=> ? /(_ (Change i)) Hdisj /(_ (Change i)); move: Hdisj. rewrite elem_of_intersection elem_of_union !elem_of_mkSet. intros; apply dec_stable. destruct p; set_solver. Qed. Qed. Lemma low_states_closed : sts.closed low_states {[ Send ]}. Lemma low_states_closed : sts.closed low_states {[ Send ]}. Proof. Proof. split. split; first (intros [??]; set_solver). - intros [p I]. rewrite /low_states. set_solver. intros s1 s2 Hs1 [T1 T2 Hdisj Hstep']. - intros s1 s2 Hs1 [T1 T2 Hdisj Hstep']. inversion_clear Hstep' as [? ? ? ? Htrans _ _ Htok]. inversion_clear Hstep' as [? ? ? ? Htrans _ _ Htok]. destruct Htrans as [[] ??|]; done || set_solver. destruct Htrans; simpl in *; first by destruct p. exfalso; apply dec_stable; set_solver. Qed. Qed. (* Proof that we can take the steps we need. *) (* Proof that we can take the steps we need. *) ... @@ -70,12 +59,8 @@ Lemma wait_step i I : ... @@ -70,12 +59,8 @@ Lemma wait_step i I : sts.steps (State High I, {[ Change i ]}) (State High (I ∖ {[ i ]}), ∅). sts.steps (State High I, {[ Change i ]}) (State High (I ∖ {[ i ]}), ∅). Proof. Proof. intros. apply rtc_once. intros. apply rtc_once. constructor; first constructor; rewrite /= /change_tok; [set_solver by eauto..|]. constructor; first constructor; [set_solver..|]. (* TODO this proof is rather annoying. *) apply elem_of_equiv=>-[j|]; last set_solver. apply elem_of_equiv=>t. rewrite !elem_of_union. rewrite !elem_of_mkSet /change_tok /=. destruct t as [j|]; last set_solver. rewrite elem_of_difference elem_of_singleton. destruct (decide (i = j)); set_solver. destruct (decide (i = j)); set_solver. Qed. Qed. ... @@ -85,17 +70,14 @@ Lemma split_step p i i1 i2 I : ... @@ -85,17 +70,14 @@ Lemma split_step p i i1 i2 I : (State p I, {[ Change i ]}) (State p I, {[ Change i ]}) (State p ({[i1]} ∪ ({[i2]} ∪ (I ∖ {[i]}))), {[ Change i1; Change i2 ]}). (State p ({[i1]} ∪ ({[i2]} ∪ (I ∖ {[i]}))), {[ Change i1; Change i2 ]}). Proof. Proof. intros. apply rtc_once. intros. apply rtc_once. constructor; first constructor. constructor; first constructor; simpl. - destruct p; set_solver. - destruct p; set_solver. - destruct p; set_solver. (* This gets annoying... and I think I can see a pattern with all these proofs. Automatable? *) - apply elem_of_equiv=> /= -[j|]; last set_solver. - apply elem_of_equiv=>t. destruct t; last set_solver. set_unfold; rewrite !(inj_iff Change). rewrite !elem_of_mkSet !not_elem_of_union !not_elem_of_singleton assert (Change j ∈ match p with Low => ∅ | High => {[Send]} end ↔ False) not_elem_of_difference elem_of_singleton !(inj_iff Change). as -> by (destruct p; set_solver). destruct p; naive_solver. destruct (decide (i1 = j)) as [->|]; first naive_solver. - apply elem_of_equiv=>t. destruct t as [j|]; last set_solver. destruct (decide (i2 = j)) as [->|]; first naive_solver. rewrite !elem_of_mkSet !not_elem_of_union !not_elem_of_singleton destruct (decide (i = j)) as [->|]; naive_solver. not_elem_of_difference elem_of_singleton !(inj_iff Change). destruct (decide (i1 = j)) as [->|]; first tauto. destruct (decide (i2 = j)) as [->|]; intuition. Qed. Qed.
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