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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 Transitive: ∀A. ∀R: relation A. Prop ≝ λA,R.
24 ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2.
26 definition confluent2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
27 ∀a0,a1. R1 a0 a1 → ∀a2. R2 a0 a2 →
28 ∃∃a. R2 a1 a & R1 a2 a.
30 definition transitive2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
31 ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 →
32 ∃∃a. R2 a1 a & R1 a a2.
34 definition bi_confluent: ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R.
35 ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. R a0 b0 a2 b2 →
36 ∃∃a,b. R a1 b1 a b & R a2 b2 a b.
38 definition LTC: ∀A:Type[0]. ∀B. (A→relation B) → (A→relation B) ≝
41 definition s_r_trans: ∀A,B. relation2 (A→relation B) (relation A) ≝ λA,B,R1,R2.
42 ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 L1 L2 → LTC … R1 L1 T1 T2.
44 definition s_rs_trans: ∀A,B. relation2 (A→relation B) (relation A) ≝ λA,B,R1,R2.
45 ∀L2,T1,T2. LTC … R1 L2 T1 T2 → ∀L1. R2 L1 L2 → LTC … R1 L1 T1 T2.
47 lemma TC_strip1: ∀A,R1,R2. confluent2 A R1 R2 →
48 ∀a0,a1. TC … R1 a0 a1 → ∀a2. R2 a0 a2 →
49 ∃∃a. R2 a1 a & TC … R1 a2 a.
50 #A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1
52 elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3/
53 | #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
54 elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
55 elim (HR12 … Ha1 … Ha0) -HR12 -a /4 width=5/
59 lemma TC_strip2: ∀A,R1,R2. confluent2 A R1 R2 →
60 ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a0 a1 →
61 ∃∃a. TC … R2 a1 a & R1 a2 a.
62 #A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2
64 elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3/
65 | #a #a2 #_ #Ha2 #IHa0 #a1 #Ha01
66 elim (IHa0 … Ha01) -a0 #a0 #Ha10 #Ha0
67 elim (HR12 … Ha0 … Ha2) -HR12 -a /4 width=3/
71 lemma TC_confluent2: ∀A,R1,R2.
72 confluent2 A R1 R2 → confluent2 A (TC … R1) (TC … R2).
73 #A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1
75 elim (TC_strip2 … HR12 … Ha02 … Ha01) -HR12 -a0 /3 width=3/
76 | #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
77 elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
78 elim (TC_strip2 … HR12 … Ha0 … Ha1) -HR12 -a /4 width=5/
82 lemma TC_strap1: ∀A,R1,R2. transitive2 A R1 R2 →
83 ∀a1,a0. TC … R1 a1 a0 → ∀a2. R2 a0 a2 →
84 ∃∃a. R2 a1 a & TC … R1 a a2.
85 #A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0
87 elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3/
88 | #a #a0 #_ #Ha0 #IHa #a2 #Ha02
89 elim (HR12 … Ha0 … Ha02) -HR12 -a0 #a0 #Ha0 #Ha02
90 elim (IHa … Ha0) -a /4 width=5/
94 lemma TC_strap2: ∀A,R1,R2. transitive2 A R1 R2 →
95 ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a1 a0 →
96 ∃∃a. TC … R2 a1 a & R1 a a2.
97 #A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2
99 elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3/
100 | #a #a2 #_ #Ha02 #IHa #a1 #Ha10
101 elim (IHa … Ha10) -a0 #a0 #Ha10 #Ha0
102 elim (HR12 … Ha0 … Ha02) -HR12 -a /4 width=3/
106 lemma TC_transitive2: ∀A,R1,R2.
107 transitive2 A R1 R2 → transitive2 A (TC … R1) (TC … R2).
108 #A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0
109 [ #a0 #Ha10 #a2 #Ha02
110 elim (TC_strap2 … HR12 … Ha02 … Ha10) -HR12 -a0 /3 width=3/
111 | #a #a0 #_ #Ha0 #IHa #a2 #Ha02
112 elim (TC_strap2 … HR12 … Ha02 … Ha0) -HR12 -a0 #a0 #Ha0 #Ha02
113 elim (IHa … Ha0) -a /4 width=5/
117 definition NF: ∀A. relation A → relation A → predicate A ≝
118 λA,R,S,a1. ∀a2. R a1 a2 → S a2 a1.
120 inductive SN (A) (R,S:relation A): predicate A ≝
121 | SN_intro: ∀a1. (∀a2. R a1 a2 → (S a2 a1 → ⊥) → SN A R S a2) → SN A R S a1
124 lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a.
126 @SN_intro #a2 #HRa12 #HSa12
127 elim HSa12 -HSa12 /2 width=1/
130 definition NF_sn: ∀A. relation A → relation A → predicate A ≝
131 λA,R,S,a2. ∀a1. R a1 a2 → S a2 a1.
133 inductive SN_sn (A) (R,S:relation A): predicate A ≝
134 | SN_sn_intro: ∀a2. (∀a1. R a1 a2 → (S a2 a1 → ⊥) → SN_sn A R S a1) → SN_sn A R S a2
137 lemma NF_to_SN_sn: ∀A,R,S,a. NF_sn A R S a → SN_sn A R S a.
139 @SN_sn_intro #a1 #HRa12 #HSa12
140 elim HSa12 -HSa12 /2 width=1/
143 lemma bi_TC_strip: ∀A,B,R. bi_confluent A B R →
144 ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. bi_TC … R a0 b0 a2 b2 →
145 ∃∃a,b. bi_TC … R a1 b1 a b & R a2 b2 a b.
146 #A #B #R #HR #a0 #a1 #b0 #b1 #H01 #a2 #b2 #H elim H -a2 -b2
148 elim (HR … H01 … H02) -HR -a0 -b0 /3 width=4/
149 | #a2 #b2 #a3 #b3 #_ #H23 * #a #b #H1 #H2
150 elim (HR … H23 … H2) -HR -a0 -b0 -a2 -b2 /3 width=4/
154 lemma bi_TC_confluent: ∀A,B,R. bi_confluent A B R →
155 bi_confluent A B (bi_TC … R).
156 #A #B #R #HR #a0 #a1 #b0 #b1 #H elim H -a1 -b1
157 [ #a1 #b1 #H01 #a2 #b2 #H02
158 elim (bi_TC_strip … HR … H01 … H02) -a0 -b0 /3 width=4/
159 | #a1 #b1 #a3 #b3 #_ #H13 #IH #a2 #b2 #H02
160 elim (IH … H02) -a0 -b0 #a0 #b0 #H10 #H20
161 elim (bi_TC_strip … HR … H13 … H10) -a1 -b1 /3 width=7/
165 lemma bi_TC_decomp_r: ∀A,B. ∀R:bi_relation A B.
166 ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 →
168 ∃∃a,b. bi_TC … R a1 b1 a b & R a b a2 b2.
169 #A #B #R #a1 #a2 #b1 #b2 * -a2 -b2 /2 width=1/ /3 width=4/
172 lemma bi_TC_decomp_l: ∀A,B. ∀R:bi_relation A B.
173 ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 →
175 ∃∃a,b. R a1 b1 a b & bi_TC … R a b a2 b2.
176 #A #B #R #a1 #a2 #b1 #b2 #H @(bi_TC_ind_dx … a1 b1 H) -a1 -b1
178 | #a1 #a #b1 #b #Hab1 #Hab2 #_ /3 width=4/
182 lemma s_r_trans_TC1: ∀A,B,R,S. s_r_trans A B R S → s_rs_trans A B R S.
183 #A #B #R #S #HRS #L2 #T1 #T2 #H elim H -T2 [ /3 width=3/ ]
184 #T #T2 #_ #HT2 #IHT1 #L1 #HL12
185 lapply (HRS … HT2 … HL12) -HRS -HT2 /3 width=3/
188 lemma s_r_trans_TC2: ∀A,B,R,S. s_rs_trans A B R S → s_r_trans A B R (TC … S).
189 #A #B #R #S #HRS #L2 #T1 #T2 #HT12 #L1 #H @(TC_ind_dx … L1 H) -L1 /2 width=3/ /3 width=3/