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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basics/relations.ma".
16 include "ground_2/lib/logic.ma".
18 (* GENERIC RELATIONS ********************************************************)
20 lemma insert_eq: ∀A,a. ∀Q1,Q2:predicate A. (∀a0. Q1 a0 → a = a0 → Q2 a0) → Q1 a → Q2 a.
23 definition replace_2 (A) (B): relation3 (relation2 A B) (relation A) (relation B) ≝
24 λR,Sa,Sb. ∀a1,b1. R a1 b1 → ∀a2. Sa a1 a2 → ∀b2. Sb b1 b2 → R a2 b2.
26 (* Inclusion ****************************************************************)
28 definition subR2 (S1) (S2): relation (relation2 S1 S2) ≝
29 λR1,R2. (∀a1,a2. R1 a1 a2 → R2 a1 a2).
31 interpretation "2-relation inclusion"
32 'subseteq R1 R2 = (subR2 ?? R1 R2).
34 definition subR3 (S1) (S2) (S3): relation (relation3 S1 S2 S3) ≝
35 λR1,R2. (∀a1,a2,a3. R1 a1 a2 a3 → R2 a1 a2 a3).
37 interpretation "3-relation inclusion"
38 'subseteq R1 R2 = (subR3 ??? R1 R2).
40 (* Properties of relations **************************************************)
42 definition relation5: Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0]
43 ≝ λA,B,C,D,E.A→B→C→D→E→Prop.
45 definition relation6: Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0]
46 ≝ λA,B,C,D,E,F.A→B→C→D→E→F→Prop.
48 (**) (* we dont use "∀a. reflexive … (R a)" since auto seems to dislike repeatd δ-expansion *)
49 definition c_reflexive (A) (B): predicate (relation3 A B B) ≝
52 definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥).
54 definition Transitive: ∀A. ∀R: relation A. Prop ≝ λA,R.
55 ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2.
57 definition left_cancellable: ∀A. ∀R: relation A. Prop ≝ λA,R.
58 ∀a0,a1. R a0 a1 → ∀a2. R a0 a2 → R a1 a2.
60 definition right_cancellable: ∀A. ∀R: relation A. Prop ≝ λA,R.
61 ∀a1,a0. R a1 a0 → ∀a2. R a2 a0 → R a1 a2.
63 definition pw_confluent2: ∀A. relation A → relation A → predicate A ≝ λA,R1,R2,a0.
64 ∀a1. R1 a0 a1 → ∀a2. R2 a0 a2 →
65 ∃∃a. R2 a1 a & R1 a2 a.
67 definition confluent2: ∀A. relation (relation A) ≝ λA,R1,R2.
68 ∀a0. pw_confluent2 A R1 R2 a0.
70 definition transitive2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
71 ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 →
72 ∃∃a. R2 a1 a & R1 a a2.
74 definition bi_confluent: ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R.
75 ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. R a0 b0 a2 b2 →
76 ∃∃a,b. R a1 b1 a b & R a2 b2 a b.
78 definition lsub_trans: ∀A,B. relation2 (A→relation B) (relation A) ≝ λA,B,R1,R2.
79 ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 L1 L2 → R1 L1 T1 T2.
81 definition s_r_confluent1: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2.
82 ∀L1,T1,T2. R1 L1 T1 T2 → ∀L2. R2 T1 L1 L2 → R2 T2 L1 L2.
84 definition is_mono: ∀B:Type[0]. predicate (predicate B) ≝
85 λB,R. ∀b1. R b1 → ∀b2. R b2 → b1 = b2.
87 definition is_inj2: ∀A,B:Type[0]. predicate (relation2 A B) ≝
88 λA,B,R. ∀a1,b. R a1 b → ∀a2. R a2 b → a1 = a2.
90 (* Normal form and strong normalization *************************************)
92 definition NF: ∀A. relation A → relation A → predicate A ≝
93 λA,R,S,a1. ∀a2. R a1 a2 → S a1 a2.
95 definition NF_dec: ∀A. relation A → relation A → Prop ≝
96 λA,R,S. ∀a1. NF A R S a1 ∨
97 ∃∃a2. R … a1 a2 & (S a1 a2 → ⊥).
99 inductive SN (A) (R,S:relation A): predicate A ≝
100 | SN_intro: ∀a1. (∀a2. R a1 a2 → (S a1 a2 → ⊥) → SN A R S a2) → SN A R S a1
103 lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a.
105 @SN_intro #a2 #HRa12 #HSa12
106 elim HSa12 -HSa12 /2 width=1 by/
109 definition NF_sn: ∀A. relation A → relation A → predicate A ≝
110 λA,R,S,a2. ∀a1. R a1 a2 → S a1 a2.
112 inductive SN_sn (A) (R,S:relation A): predicate A ≝
113 | SN_sn_intro: ∀a2. (∀a1. R a1 a2 → (S a1 a2 → ⊥) → SN_sn A R S a1) → SN_sn A R S a2
116 lemma NF_to_SN_sn: ∀A,R,S,a. NF_sn A R S a → SN_sn A R S a.
118 @SN_sn_intro #a1 #HRa12 #HSa12
119 elim HSa12 -HSa12 /2 width=1 by/
122 (* Relations on unboxed triples *********************************************)
124 definition tri_RC: ∀A,B,C. tri_relation A B C → tri_relation A B C ≝
125 λA,B,C,R,a1,b1,c1,a2,b2,c2. R … a1 b1 c1 a2 b2 c2 ∨
126 ∧∧ a1 = a2 & b1 = b2 & c1 = c2.
128 lemma tri_RC_reflexive: ∀A,B,C,R. tri_reflexive A B C (tri_RC … R).
129 /3 width=1 by and3_intro, or_intror/ qed.