1 (**************************************************************************)
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/xoa/xoa_props.ma".
18 (* GENERIC RELATIONS ********************************************************)
20 (* PROPERTIES OF RELATIONS **************************************************)
22 definition relation5 : Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0]
23 ≝ λA,B,C,D,E.A→B→C→D→E→Prop.
25 definition relation6 : Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0]
26 ≝ λA,B,C,D,E,F.A→B→C→D→E→F→Prop.
28 definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥).
30 definition Transitive: ∀A. ∀R: relation A. Prop ≝ λA,R.
31 ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2.
33 definition left_cancellable: ∀A. ∀R: relation A. Prop ≝ λA,R.
34 ∀a0,a1. R a0 a1 → ∀a2. R a0 a2 → R a1 a2.
36 definition right_cancellable: ∀A. ∀R: relation A. Prop ≝ λA,R.
37 ∀a1,a0. R a1 a0 → ∀a2. R a2 a0 → R a1 a2.
39 definition pw_confluent2: ∀A. relation A → relation A → predicate A ≝ λA,R1,R2,a0.
40 ∀a1. R1 a0 a1 → ∀a2. R2 a0 a2 →
41 ∃∃a. R2 a1 a & R1 a2 a.
43 definition confluent2: ∀A. relation (relation A) ≝ λA,R1,R2.
44 ∀a0. pw_confluent2 A R1 R2 a0.
46 definition transitive2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
47 ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 →
48 ∃∃a. R2 a1 a & R1 a a2.
50 definition bi_confluent: ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R.
51 ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. R a0 b0 a2 b2 →
52 ∃∃a,b. R a1 b1 a b & R a2 b2 a b.
54 definition lsub_trans: ∀A,B. relation2 (A→relation B) (relation A) ≝ λA,B,R1,R2.
55 ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 L1 L2 → R1 L1 T1 T2.
57 definition s_r_confluent1: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2.
58 ∀L1,T1,T2. R1 L1 T1 T2 → ∀L2. R2 T1 L1 L2 → R2 T2 L1 L2.
60 definition is_mono: ∀B:Type[0]. predicate (predicate B) ≝
61 λB,R. ∀b1. R b1 → ∀b2. R b2 → b1 = b2.
63 definition is_inj2: ∀A,B:Type[0]. predicate (relation2 A B) ≝
64 λA,B,R. ∀a1,b. R a1 b → ∀a2. R a2 b → a1 = a2.
66 (* Normal form and strong normalization *************************************)
68 definition NF: ∀A. relation A → relation A → predicate A ≝
69 λA,R,S,a1. ∀a2. R a1 a2 → S a1 a2.
71 definition NF_dec: ∀A. relation A → relation A → Prop ≝
72 λA,R,S. ∀a1. NF A R S a1 ∨
73 ∃∃a2. R … a1 a2 & (S a1 a2 → ⊥).
75 inductive SN (A) (R,S:relation A): predicate A ≝
76 | SN_intro: ∀a1. (∀a2. R a1 a2 → (S a1 a2 → ⊥) → SN A R S a2) → SN A R S a1
79 lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a.
81 @SN_intro #a2 #HRa12 #HSa12
82 elim HSa12 -HSa12 /2 width=1 by/
85 definition NF_sn: ∀A. relation A → relation A → predicate A ≝
86 λA,R,S,a2. ∀a1. R a1 a2 → S a1 a2.
88 inductive SN_sn (A) (R,S:relation A): predicate A ≝
89 | SN_sn_intro: ∀a2. (∀a1. R a1 a2 → (S a1 a2 → ⊥) → SN_sn A R S a1) → SN_sn A R S a2
92 lemma NF_to_SN_sn: ∀A,R,S,a. NF_sn A R S a → SN_sn A R S a.
94 @SN_sn_intro #a1 #HRa12 #HSa12
95 elim HSa12 -HSa12 /2 width=1 by/
98 (* Relations on unboxed triples *********************************************)
100 definition tri_RC: ∀A,B,C. tri_relation A B C → tri_relation A B C ≝
101 λA,B,C,R,a1,b1,c1,a2,b2,c2. R … a1 b1 c1 a2 b2 c2 ∨
102 ∧∧ a1 = a2 & b1 = b2 & c1 = c2.
104 lemma tri_RC_reflexive: ∀A,B,C,R. tri_reflexive A B C (tri_RC … R).
105 /3 width=1 by and3_intro, or_intror/ qed.