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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "basics/star.ma".
16 include "ground_2/xoa_props.ma".
17 include "ground_2/notation.ma".
19 (* PROPERTIES OF RELATIONS **************************************************)
21 definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥).
23 definition confluent2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
24 ∀a0,a1. R1 a0 a1 → ∀a2. R2 a0 a2 →
25 ∃∃a. R2 a1 a & R1 a2 a.
27 definition transitive2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
28 ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 →
29 ∃∃a. R2 a1 a & R1 a a2.
31 lemma TC_strip1: ∀A,R1,R2. confluent2 A R1 R2 →
32 ∀a0,a1. TC … R1 a0 a1 → ∀a2. R2 a0 a2 →
33 ∃∃a. R2 a1 a & TC … R1 a2 a.
34 #A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1
36 elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3/
37 | #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
38 elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
39 elim (HR12 … Ha1 … Ha0) -HR12 -a /4 width=3/
43 lemma TC_strip2: ∀A,R1,R2. confluent2 A R1 R2 →
44 ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a0 a1 →
45 ∃∃a. TC … R2 a1 a & R1 a2 a.
46 #A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2
48 elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3/
49 | #a #a2 #_ #Ha2 #IHa0 #a1 #Ha01
50 elim (IHa0 … Ha01) -a0 #a0 #Ha10 #Ha0
51 elim (HR12 … Ha0 … Ha2) -HR12 -a /4 width=3/
55 lemma TC_confluent2: ∀A,R1,R2.
56 confluent2 A R1 R2 → confluent2 A (TC … R1) (TC … R2).
57 #A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1
59 elim (TC_strip2 … HR12 … Ha02 … Ha01) -HR12 -a0 /3 width=3/
60 | #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
61 elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
62 elim (TC_strip2 … HR12 … Ha0 … Ha1) -HR12 -a /4 width=3/
66 lemma TC_strap1: ∀A,R1,R2. transitive2 A R1 R2 →
67 ∀a1,a0. TC … R1 a1 a0 → ∀a2. R2 a0 a2 →
68 ∃∃a. R2 a1 a & TC … R1 a a2.
69 #A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0
71 elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3/
72 | #a #a0 #_ #Ha0 #IHa #a2 #Ha02
73 elim (HR12 … Ha0 … Ha02) -HR12 -a0 #a0 #Ha0 #Ha02
74 elim (IHa … Ha0) -a /4 width=3/
78 lemma TC_strap2: ∀A,R1,R2. transitive2 A R1 R2 →
79 ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a1 a0 →
80 ∃∃a. TC … R2 a1 a & R1 a a2.
81 #A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2
83 elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3/
84 | #a #a2 #_ #Ha02 #IHa #a1 #Ha10
85 elim (IHa … Ha10) -a0 #a0 #Ha10 #Ha0
86 elim (HR12 … Ha0 … Ha02) -HR12 -a /4 width=3/
90 lemma TC_transitive2: ∀A,R1,R2.
91 transitive2 A R1 R2 → transitive2 A (TC … R1) (TC … R2).
92 #A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0
94 elim (TC_strap2 … HR12 … Ha02 … Ha10) -HR12 -a0 /3 width=3/
95 | #a #a0 #_ #Ha0 #IHa #a2 #Ha02
96 elim (TC_strap2 … HR12 … Ha02 … Ha0) -HR12 -a0 #a0 #Ha0 #Ha02
97 elim (IHa … Ha0) -a /4 width=3/
101 definition NF: ∀A. relation A → relation A → predicate A ≝
102 λA,R,S,a1. ∀a2. R a1 a2 → S a2 a1.
104 inductive SN (A) (R,S:relation A): predicate A ≝
105 | SN_intro: ∀a1. (∀a2. R a1 a2 → (S a2 a1 → ⊥) → SN A R S a2) → SN A R S a1
108 lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a.
110 @SN_intro #a2 #HRa12 #HSa12
111 elim (HSa12 ?) -HSa12 /2 width=1/