### The different notions of confluence and properties about them.

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 ... ... @@ -62,6 +62,19 @@ Definition rtsc {A} (R : relation A) := rtc (sc R). (** Strongly normalizing elements. *) Notation sn R := (Acc (flip R)). (** The various kinds of "confluence" properties. Any relation that has the diamond property is confluent, and any confluent relation is locally confluent. The naming convention are taken from "Term Rewriting and All That" by Baader and Nipkow. *) Definition diamond {A} (R : relation A) := ∀ x y1 y2, R x y1 → R x y2 → ∃ z, R y1 z ∧ R y2 z. Definition confluent {A} (R : relation A) := diamond (rtc R). Definition locally_confluent {A} (R : relation A) := ∀ x y1 y2, R x y1 → R x y2 → ∃ z, rtc R y1 z ∧ rtc R y2 z. Hint Unfold nf red : core. (** * General theorems *) ... ... @@ -234,7 +247,60 @@ Section properties. intros H. cut (∀ z, rtc R x z → all_loop R z); [eauto|]. cofix FIX. constructor; eauto using rtc_r. Qed. End rtc. (** An alternative definition of confluence; also known as the Church-Rosser property. *) Lemma confluent_alt : confluent R ↔ (∀ x y, rtsc R x y → ∃ z, rtc R x z ∧ rtc R y z). Proof. split. - intros Hcr. induction 1 as [x|x y1 y1' [Hy1|Hy1] Hy1' (z&IH1&IH2)]; eauto. destruct (Hcr y1 x z) as (z'&?&?); eauto using rtc_transitive. - intros Hcr x y1 y2 Hy1 Hy2. apply Hcr; trans x; eauto using rtc_rtsc_rl, rtc_rtsc_lr. Qed. Lemma confluent_nf_r x y : confluent R → rtsc R x y → nf R y → rtc R x y. Proof. rewrite confluent_alt. intros Hcr ??. destruct (Hcr x y) as (z&Hx&Hy); auto. by apply rtc_nf in Hy as ->. Qed. Lemma confluent_nf_l x y : confluent R → rtsc R x y → nf R x → rtc R y x. Proof. intros. by apply (confluent_nf_r y x). Qed. Lemma diamond_confluent : diamond R → confluent R. Proof. intros Hdiam. assert (∀ x y1 y2, rtc R x y1 → R x y2 → ∃ z, rtc R y1 z ∧ rtc R y2 z) as Hstrip. { intros x y1 y2 Hy1; revert y2. induction Hy1 as [x|x y1 y1' Hy1 Hy1' IH]; [by eauto|]; intros y2 Hy2. destruct (Hdiam x y1 y2) as (z&Hy1z&Hy2z); auto. destruct (IH z) as (z'&?&?); eauto. } intros x y1 y2 Hy1; revert y2. induction Hy1 as [x|x y1 y1' Hy1 Hy1' IH]; [by eauto|]; intros y2 Hy2. destruct (Hstrip x y2 y1) as (z&?&?); eauto. destruct (IH z) as (z'&?&?); eauto using rtc_transitive. Qed. Lemma confluent_locally_confluent : confluent R → locally_confluent R. Proof. unfold confluent, locally_confluent; eauto. Qed. (** The following is also known as Newman's lemma *) Lemma locally_confluent_confluent : (∀ x, sn R x) → locally_confluent R → confluent R. Proof. intros Hsn Hcr x. induction (Hsn x) as [x _ IH]. intros y1 y2 Hy1 Hy2. destruct Hy1 as [x|x y1 y1' Hy1 Hy1']; [by eauto|]. destruct Hy2 as [x|x y2 y2' Hy2 Hy2']; [by eauto|]. destruct (Hcr x y1 y2) as (z&Hy1z&Hy2z); auto. destruct (IH _ Hy1 y1' z) as (z1&?&?); auto. destruct (IH _ Hy2 y2' z1) as (z2&?&?); eauto using rtc_transitive. Qed. End properties. (** * Theorems on sub relations *) Section subrel. ... ...
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