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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/xoa/and_3.ma".
17 include "ground/xoa/ex_2_2.ma".
18 include "ground/lib/logic.ma".
20 (* 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 replace_2 (A) (B): relation3 (relation2 A B) (relation A) (relation B) ≝
29 λR,Sa,Sb. ∀a1,b1. R a1 b1 → ∀a2. Sa a1 a2 → ∀b2. Sb b1 b2 → R a2 b2.
31 (* * we don't use "∀a. reflexive … (R a)" since auto seems to dislike repeatd δ-expansion *)
32 definition c_reflexive (A) (B): predicate (relation3 A B B) ≝
35 definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥).
37 definition Transitive (A) (R:relation A): Prop ≝
38 ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2.
40 definition left_cancellable (A) (R:relation A): Prop ≝
41 ∀a0,a1. R a0 a1 → ∀a2. R a0 a2 → R a1 a2.
43 definition right_cancellable (A) (R:relation A): Prop ≝
44 ∀a1,a0. R a1 a0 → ∀a2. R a2 a0 → R a1 a2.
46 definition pw_confluent2 (A) (R1,R2:relation A): predicate A ≝
48 ∀a1. R1 a0 a1 → ∀a2. R2 a0 a2 →
49 ∃∃a. R2 a1 a & R1 a2 a.
51 definition confluent2 (A): relation (relation A) ≝
53 ∀a0. pw_confluent2 A R1 R2 a0.
55 definition transitive2 (A) (R1,R2:relation A): Prop ≝
56 ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 →
57 ∃∃a. R2 a1 a & R1 a a2.
59 definition confluent1 (A) (B): relation2 (relation2 A B) (relation A) ≝
60 λR1,R2. ∀a1,b. R1 a1 b → ∀a2. R2 a1 a2 → R1 a2 b.
62 definition bi_confluent (A) (B) (R: bi_relation A B): Prop ≝
63 ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. R a0 b0 a2 b2 →
64 ∃∃a,b. R a1 b1 a b & R a2 b2 a b.
66 definition lsub_trans (A) (B): relation2 (A→relation B) (relation A) ≝
68 ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 L1 L2 → R1 L1 T1 T2.
70 definition s_r_confluent1 (A) (B): relation2 (A→relation B) (B→relation A) ≝
72 ∀L1,T1,T2. R1 L1 T1 T2 → ∀L2. R2 T1 L1 L2 → R2 T2 L1 L2.
74 definition is_mono (B:Type[0]): predicate (predicate B) ≝
75 λR. ∀b1. R b1 → ∀b2. R b2 → b1 = b2.
77 definition is_inj2 (A,B:Type[0]): predicate (relation2 A B) ≝
78 λR. ∀a1,b. R a1 b → ∀a2. R a2 b → a1 = a2.
80 (* NOTE: Normal form and strong normalization *******************************)
82 definition NF (A): relation A → relation A → predicate A ≝
83 λR,S,a1. ∀a2. R a1 a2 → S a1 a2.
85 definition NF_dec (A): relation A → relation A → Prop ≝
86 λR,S. ∀a1. NF … R S a1 ∨
87 ∃∃a2. R … a1 a2 & (S a1 a2 → ⊥).
89 inductive SN (A) (R,S:relation A): predicate A ≝
90 | SN_intro: ∀a1. (∀a2. R a1 a2 → (S a1 a2 → ⊥) → SN … R S a2) → SN … R S a1
93 lemma NF_to_SN (A) (R) (S): ∀a. NF A R S a → SN A R S a.
95 @SN_intro #a2 #HRa12 #HSa12
96 elim HSa12 -HSa12 /2 width=1 by/
99 definition NF_sn (A): relation A → relation A → predicate A ≝
100 λR,S,a2. ∀a1. R a1 a2 → S a1 a2.
102 inductive SN_sn (A) (R,S:relation A): predicate A ≝
103 | SN_sn_intro: ∀a2. (∀a1. R a1 a2 → (S a1 a2 → ⊥) → SN_sn … R S a1) → SN_sn … R S a2
106 lemma NF_to_SN_sn (A) (R) (S): ∀a. NF_sn A R S a → SN_sn A R S a.
108 @SN_sn_intro #a1 #HRa12 #HSa12
109 elim HSa12 -HSa12 /2 width=1 by/
112 (* NOTE: Normal form and strong normalization with unboxed triples **********)
114 inductive SN3 (A) (B) (C) (R,S:relation6 A B C A B C): relation3 A B C ≝
115 | SN3_intro: ∀a1,b1,c1. (∀a2,b2,c2. R a1 b1 c1 a2 b2 c2 → (S a1 b1 c1 a2 b2 c2 → ⊥) → SN3 … R S a2 b2 c2) → SN3 … R S a1 b1 c1
118 (* NOTE: Reflexive closure with unboxed triples *****************************)
120 definition tri_RC (A,B,C): tri_relation A B C → tri_relation A B C ≝
121 λR,a1,b1,c1,a2,b2,c2.
122 ∨∨ 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.
128 (* NOTE: Inclusion for relations ********************************************)
130 definition subR2 (S1) (S2): relation (relation2 S1 S2) ≝
131 λR1,R2. (∀a1,a2. R1 a1 a2 → R2 a1 a2).
134 "2-relation inclusion"
135 'subseteq R1 R2 = (subR2 ?? R1 R2).
137 definition subR3 (S1) (S2) (S3): relation (relation3 S1 S2 S3) ≝
138 λR1,R2. (∀a1,a2,a3. R1 a1 a2 a3 → R2 a1 a2 a3).
141 "3-relation inclusion"
142 'subseteq R1 R2 = (subR3 ??? R1 R2).
144 (* Main constructions with eq ***********************************************)
146 theorem canc_sn_eq (A):
147 left_cancellable A (eq …).
150 theorem canc_dx_eq (A):
151 right_cancellable A (eq …).