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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 lsub_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 → R1 L1 T1 T2.
44 definition s_r_trans: ∀A,B. relation2 (A→relation B) (relation A) ≝ λA,B,R1,R2.
45 ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 L1 L2 → LTC … R1 L1 T1 T2.
47 definition s_rs_trans: ∀A,B. relation2 (A→relation B) (relation A) ≝ λA,B,R1,R2.
48 ∀L2,T1,T2. LTC … R1 L2 T1 T2 → ∀L1. R2 L1 L2 → LTC … R1 L1 T1 T2.
50 lemma TC_strip1: ∀A,R1,R2. confluent2 A R1 R2 →
51 ∀a0,a1. TC … R1 a0 a1 → ∀a2. R2 a0 a2 →
52 ∃∃a. R2 a1 a & TC … R1 a2 a.
53 #A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1
55 elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3/
56 | #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
57 elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
58 elim (HR12 … Ha1 … Ha0) -HR12 -a /4 width=5/
62 lemma TC_strip2: ∀A,R1,R2. confluent2 A R1 R2 →
63 ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a0 a1 →
64 ∃∃a. TC … R2 a1 a & R1 a2 a.
65 #A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2
67 elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3/
68 | #a #a2 #_ #Ha2 #IHa0 #a1 #Ha01
69 elim (IHa0 … Ha01) -a0 #a0 #Ha10 #Ha0
70 elim (HR12 … Ha0 … Ha2) -HR12 -a /4 width=3/
74 lemma TC_confluent2: ∀A,R1,R2.
75 confluent2 A R1 R2 → confluent2 A (TC … R1) (TC … R2).
76 #A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1
78 elim (TC_strip2 … HR12 … Ha02 … Ha01) -HR12 -a0 /3 width=3/
79 | #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
80 elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
81 elim (TC_strip2 … HR12 … Ha0 … Ha1) -HR12 -a /4 width=5/
85 lemma TC_strap1: ∀A,R1,R2. transitive2 A R1 R2 →
86 ∀a1,a0. TC … R1 a1 a0 → ∀a2. R2 a0 a2 →
87 ∃∃a. R2 a1 a & TC … R1 a a2.
88 #A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0
90 elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3/
91 | #a #a0 #_ #Ha0 #IHa #a2 #Ha02
92 elim (HR12 … Ha0 … Ha02) -HR12 -a0 #a0 #Ha0 #Ha02
93 elim (IHa … Ha0) -a /4 width=5/
97 lemma TC_strap2: ∀A,R1,R2. transitive2 A R1 R2 →
98 ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a1 a0 →
99 ∃∃a. TC … R2 a1 a & R1 a a2.
100 #A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2
101 [ #a2 #Ha02 #a1 #Ha10
102 elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3/
103 | #a #a2 #_ #Ha02 #IHa #a1 #Ha10
104 elim (IHa … Ha10) -a0 #a0 #Ha10 #Ha0
105 elim (HR12 … Ha0 … Ha02) -HR12 -a /4 width=3/
109 lemma TC_transitive2: ∀A,R1,R2.
110 transitive2 A R1 R2 → transitive2 A (TC … R1) (TC … R2).
111 #A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0
112 [ #a0 #Ha10 #a2 #Ha02
113 elim (TC_strap2 … HR12 … Ha02 … Ha10) -HR12 -a0 /3 width=3/
114 | #a #a0 #_ #Ha0 #IHa #a2 #Ha02
115 elim (TC_strap2 … HR12 … Ha02 … Ha0) -HR12 -a0 #a0 #Ha0 #Ha02
116 elim (IHa … Ha0) -a /4 width=5/
120 definition NF: ∀A. relation A → relation A → predicate A ≝
121 λA,R,S,a1. ∀a2. R a1 a2 → S a2 a1.
123 definition NF_dec: ∀A. relation A → relation A → Prop ≝
124 λA,R,S. ∀a1. NF A R S a1 ∨
125 ∃∃a2. R … a1 a2 & (S a2 a1 → ⊥).
127 inductive SN (A) (R,S:relation A): predicate A ≝
128 | SN_intro: ∀a1. (∀a2. R a1 a2 → (S a2 a1 → ⊥) → SN A R S a2) → SN A R S a1
131 lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a.
133 @SN_intro #a2 #HRa12 #HSa12
134 elim HSa12 -HSa12 /2 width=1/
137 lemma SN_to_NF: ∀A,R,S. NF_dec A R S →
139 ∃∃a2. star … R a1 a2 & NF A R S a2.
140 #A #R #S #HRS #a1 #H elim H -a1
141 #a1 #_ #IHa1 elim (HRS a1) -HRS /2 width=3/
142 * #a0 #Ha10 #Ha01 elim (IHa1 … Ha10 Ha01) -IHa1 -Ha01 /3 width=3/
145 definition NF_sn: ∀A. relation A → relation A → predicate A ≝
146 λA,R,S,a2. ∀a1. R a1 a2 → S a2 a1.
148 inductive SN_sn (A) (R,S:relation A): predicate A ≝
149 | SN_sn_intro: ∀a2. (∀a1. R a1 a2 → (S a2 a1 → ⊥) → SN_sn A R S a1) → SN_sn A R S a2
152 lemma NF_to_SN_sn: ∀A,R,S,a. NF_sn A R S a → SN_sn A R S a.
154 @SN_sn_intro #a1 #HRa12 #HSa12
155 elim HSa12 -HSa12 /2 width=1/
158 lemma bi_TC_strip: ∀A,B,R. bi_confluent A B R →
159 ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. bi_TC … R a0 b0 a2 b2 →
160 ∃∃a,b. bi_TC … R a1 b1 a b & R a2 b2 a b.
161 #A #B #R #HR #a0 #a1 #b0 #b1 #H01 #a2 #b2 #H elim H -a2 -b2
163 elim (HR … H01 … H02) -HR -a0 -b0 /3 width=4/
164 | #a2 #b2 #a3 #b3 #_ #H23 * #a #b #H1 #H2
165 elim (HR … H23 … H2) -HR -a0 -b0 -a2 -b2 /3 width=4/
169 lemma bi_TC_confluent: ∀A,B,R. bi_confluent A B R →
170 bi_confluent A B (bi_TC … R).
171 #A #B #R #HR #a0 #a1 #b0 #b1 #H elim H -a1 -b1
172 [ #a1 #b1 #H01 #a2 #b2 #H02
173 elim (bi_TC_strip … HR … H01 … H02) -a0 -b0 /3 width=4/
174 | #a1 #b1 #a3 #b3 #_ #H13 #IH #a2 #b2 #H02
175 elim (IH … H02) -a0 -b0 #a0 #b0 #H10 #H20
176 elim (bi_TC_strip … HR … H13 … H10) -a1 -b1 /3 width=7/
180 lemma bi_TC_decomp_r: ∀A,B. ∀R:bi_relation A B.
181 ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 →
183 ∃∃a,b. bi_TC … R a1 b1 a b & R a b a2 b2.
184 #A #B #R #a1 #a2 #b1 #b2 * -a2 -b2 /2 width=1/ /3 width=4/
187 lemma bi_TC_decomp_l: ∀A,B. ∀R:bi_relation A B.
188 ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 →
190 ∃∃a,b. R a1 b1 a b & bi_TC … R a b a2 b2.
191 #A #B #R #a1 #a2 #b1 #b2 #H @(bi_TC_ind_dx … a1 b1 H) -a1 -b1
193 | #a1 #a #b1 #b #Hab1 #Hab2 #_ /3 width=4/
197 lemma TC_lsub_trans: ∀A,B,R,S. lsub_trans A B R S → lsub_trans A B (LTC … R) S.
198 #A #B #R #S #HRS #L2 #T1 #T2 #H elim H -T2 [ /3 width=3/ ]
199 #T #T2 #_ #HT2 #IHT1 #L1 #HL12
200 lapply (HRS … HT2 … HL12) -HRS -HT2 /3 width=3/
203 lemma s_r_trans_TC1: ∀A,B,R,S. s_r_trans A B R S → s_rs_trans A B R S.
204 #A #B #R #S #HRS #L2 #T1 #T2 #H elim H -T2 [ /3 width=3/ ]
205 #T #T2 #_ #HT2 #IHT1 #L1 #HL12
206 lapply (HRS … HT2 … HL12) -HRS -HT2 /3 width=3/
209 lemma s_r_trans_TC2: ∀A,B,R,S. s_rs_trans A B R S → s_r_trans A B R (TC … S).
210 #A #B #R #S #HRS #L2 #T1 #T2 #HT12 #L1 #H @(TC_ind_dx … L1 H) -L1 /2 width=3/ /3 width=3/