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1From stdpp Require Import relations.
2
3Definition deterministic {A} (R : relation A) :=
4 ∀ x y1 y2, R x y1 → R x y2 → y1 = y2.
5
6(* As in Baader and Nipkow (1998): *)
7(* y is a normal form of x if x -->* y and y is in normal form. *)
8Definition is_nf_of `(R : relation A) x y := (rtc R x y) ∧ nf R y.
9
10(** The reflexive closure. *)
11Inductive rc {A} (R : relation A) : relation A :=
12 | rc_refl x : rc R x x
13 | rc_once x y : R x y → rc R x y.
14
15Hint Constructors rc : core.
16
17(* Baader and Nipkow, Definition 2.7.3. *)
18Definition strongly_confluent {A} (R : relation A) :=
19 ∀ x y1 y2, R x y1 → R x y2 → ∃ z, rtc R y1 z ∧ rc R y2 z.
20
21(* Baader and Nipkow, Definition 2.1.4. *)
22Definition semi_confluent {A} (R : relation A) :=
23 ∀ x y1 y2, R x y1 → rtc R x y2 → ∃ z, rtc R y1 z ∧ rtc R y2 z.
24
25(* Lemma 2.7.4 from Baader and Nipkow *)
26Lemma strong__semi_confluence {A} (R : relation A) :
27 strongly_confluent R → semi_confluent R.
28Proof.
29 intros Hstrc.
30 unfold strongly_confluent in Hstrc.
31 unfold semi_confluent.
32 intros x1 y1 xn Hxy1 [steps Hsteps] % rtc_nsteps.
33 revert x1 y1 xn Hxy1 Hsteps.
34 induction steps; intros x1 y1 xn Hxy1 Hsteps.
35 - inv Hsteps. exists y1.
36 split.
37 + apply rtc_refl.
38 + by apply rtc_once.
39 - inv Hsteps as [|? ? x2 ? Hx1x2 Hsteps'].
40 destruct (Hstrc x1 y1 x2 Hxy1 Hx1x2) as [y2 [Hy21 Hy22]].
41 inv Hy22.
42 + exists xn. split; [|done].
43 apply rtc_transitive with (y := y2); [done|].
44 rewrite rtc_nsteps. by exists steps.
45 + destruct (IHsteps x2 y2 xn H Hsteps') as [yn [Hy2yn Hxnyn]].
46 exists yn. split; [|done].
47 by apply rtc_transitive with (y := y2).
48Qed.
49
50(* Copied from stdpp, originates from Baader and Nipkow (1998) *)
51Definition cr {A} (R : relation A) :=
52 ∀ x y, rtsc R x y → ∃ z, rtc R x z ∧ rtc R y z.
53
54(* Part of Lemma 2.1.5 from Baader and Nipkow (1998) *)
55Lemma semi_confluence__cr {A} (R : relation A) :
56 semi_confluent R → cr R.
57Proof.
58 intros Hsc.
59 unfold semi_confluent in Hsc.
60 unfold cr.
61 intros x y [steps Hsteps] % rtc_nsteps.
62 revert x y Hsteps.
63 induction steps; intros x y Hsteps.
64 - inv Hsteps. by exists y.
65 - inv Hsteps. rename x into y', y into x, y0 into y.
66 apply IHsteps in H1 as H1'. destruct H1' as [z [Hz1 Hz2]].
67 destruct H0.
68 + exists z. split; [|done].
69 by apply rtc_l with (y := y).
70 + destruct (Hsc y y' z H Hz1) as [z' [Hz'1 Hz'2]].
71 exists z'. split; [done|].
72 by apply rtc_transitive with (y := z).
73Qed.