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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 definition replace_2 (A) (B): relation3 (relation2 A B) (relation A) (relation B) ≝
21 λR,Sa,Sb. ∀a1,b1. R a1 b1 → ∀a2. Sa a1 a2 → ∀b2. Sb b1 b2 → R a2 b2.
23 (* Inclusion ****************************************************************)
25 definition subR2 (S1) (S2): relation (relation2 S1 S2) ≝
26 λR1,R2. (∀a1,a2. R1 a1 a2 → R2 a1 a2).
28 interpretation "2-relation inclusion"
29 'subseteq R1 R2 = (subR2 ?? R1 R2).
31 definition subR3 (S1) (S2) (S3): relation (relation3 S1 S2 S3) ≝
32 λR1,R2. (∀a1,a2,a3. R1 a1 a2 a3 → R2 a1 a2 a3).
34 interpretation "3-relation inclusion"
35 'subseteq R1 R2 = (subR3 ??? R1 R2).
37 (* Properties of relations **************************************************)
39 definition relation5: Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0]
40 ≝ λA,B,C,D,E.A→B→C→D→E→Prop.
42 definition relation6: Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0]
43 ≝ λA,B,C,D,E,F.A→B→C→D→E→F→Prop.
45 (**) (* we dont use "∀a. reflexive … (R a)" since auto seems to dislike repeatd δ-expansion *)
46 definition c_reflexive (A) (B): predicate (relation3 A B B) ≝
49 definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥).
51 definition Transitive: ∀A. ∀R: relation A. Prop ≝ λA,R.
52 ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2.
54 definition left_cancellable: ∀A. ∀R: relation A. Prop ≝ λA,R.
55 ∀a0,a1. R a0 a1 → ∀a2. R a0 a2 → R a1 a2.
57 definition right_cancellable: ∀A. ∀R: relation A. Prop ≝ λA,R.
58 ∀a1,a0. R a1 a0 → ∀a2. R a2 a0 → R a1 a2.
60 definition pw_confluent2: ∀A. relation A → relation A → predicate A ≝ λA,R1,R2,a0.
61 ∀a1. R1 a0 a1 → ∀a2. R2 a0 a2 →
62 ∃∃a. R2 a1 a & R1 a2 a.
64 definition confluent2: ∀A. relation (relation A) ≝ λA,R1,R2.
65 ∀a0. pw_confluent2 A R1 R2 a0.
67 definition transitive2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
68 ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 →
69 ∃∃a. R2 a1 a & R1 a a2.
71 definition bi_confluent: ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R.
72 ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. R a0 b0 a2 b2 →
73 ∃∃a,b. R a1 b1 a b & R a2 b2 a b.
75 definition lsub_trans: ∀A,B. relation2 (A→relation B) (relation A) ≝ λA,B,R1,R2.
76 ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 L1 L2 → R1 L1 T1 T2.
78 definition s_r_confluent1: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2.
79 ∀L1,T1,T2. R1 L1 T1 T2 → ∀L2. R2 T1 L1 L2 → R2 T2 L1 L2.
81 definition is_mono: ∀B:Type[0]. predicate (predicate B) ≝
82 λB,R. ∀b1. R b1 → ∀b2. R b2 → b1 = b2.
84 definition is_inj2: ∀A,B:Type[0]. predicate (relation2 A B) ≝
85 λA,B,R. ∀a1,b. R a1 b → ∀a2. R a2 b → a1 = a2.
87 (* Normal form and strong normalization *************************************)
89 definition NF: ∀A. relation A → relation A → predicate A ≝
90 λA,R,S,a1. ∀a2. R a1 a2 → S a1 a2.
92 definition NF_dec: ∀A. relation A → relation A → Prop ≝
93 λA,R,S. ∀a1. NF A R S a1 ∨
94 ∃∃a2. R … a1 a2 & (S a1 a2 → ⊥).
96 inductive SN (A) (R,S:relation A): predicate A ≝
97 | SN_intro: ∀a1. (∀a2. R a1 a2 → (S a1 a2 → ⊥) → SN A R S a2) → SN A R S a1
100 lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a.
102 @SN_intro #a2 #HRa12 #HSa12
103 elim HSa12 -HSa12 /2 width=1 by/
106 definition NF_sn: ∀A. relation A → relation A → predicate A ≝
107 λA,R,S,a2. ∀a1. R a1 a2 → S a1 a2.
109 inductive SN_sn (A) (R,S:relation A): predicate A ≝
110 | SN_sn_intro: ∀a2. (∀a1. R a1 a2 → (S a1 a2 → ⊥) → SN_sn A R S a1) → SN_sn A R S a2
113 lemma NF_to_SN_sn: ∀A,R,S,a. NF_sn A R S a → SN_sn A R S a.
115 @SN_sn_intro #a1 #HRa12 #HSa12
116 elim HSa12 -HSa12 /2 width=1 by/
119 (* Relations on unboxed triples *********************************************)
121 definition tri_RC: ∀A,B,C. tri_relation A B C → tri_relation A B C ≝
122 λA,B,C,R,a1,b1,c1,a2,b2,c2. R … a1 b1 c1 a2 b2 c2 ∨
123 ∧∧ a1 = a2 & b1 = b2 & c1 = c2.
125 lemma tri_RC_reflexive: ∀A,B,C,R. tri_reflexive A B C (tri_RC … R).
126 /3 width=1 by and3_intro, or_intror/ qed.