+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basics/relations.ma".
-include "ground_2/xoa/and_3.ma".
-include "ground_2/xoa/ex_2_2.ma".
-include "ground_2/lib/logic.ma".
-
-(* GENERIC RELATIONS ********************************************************)
-
-definition replace_2 (A) (B): relation3 (relation2 A B) (relation A) (relation B) ≝
- λR,Sa,Sb. ∀a1,b1. R a1 b1 → ∀a2. Sa a1 a2 → ∀b2. Sb b1 b2 → R a2 b2.
-
-(* Inclusion ****************************************************************)
-
-definition subR2 (S1) (S2): relation (relation2 S1 S2) ≝
- λR1,R2. (∀a1,a2. R1 a1 a2 → R2 a1 a2).
-
-interpretation "2-relation inclusion"
- 'subseteq R1 R2 = (subR2 ?? R1 R2).
-
-definition subR3 (S1) (S2) (S3): relation (relation3 S1 S2 S3) ≝
- λR1,R2. (∀a1,a2,a3. R1 a1 a2 a3 → R2 a1 a2 a3).
-
-interpretation "3-relation inclusion"
- 'subseteq R1 R2 = (subR3 ??? R1 R2).
-
-(* Properties of relations **************************************************)
-
-definition relation5: Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] ≝
- λA,B,C,D,E.A→B→C→D→E→Prop.
-
-definition relation6: Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] ≝
- λA,B,C,D,E,F.A→B→C→D→E→F→Prop.
-
-(**) (* we don't use "∀a. reflexive … (R a)" since auto seems to dislike repeatd δ-expansion *)
-definition c_reflexive (A) (B): predicate (relation3 A B B) ≝
- λR. ∀a,b. R a b b.
-
-definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥).
-
-definition Transitive (A) (R:relation A): Prop ≝
- ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2.
-
-definition left_cancellable (A) (R:relation A): Prop ≝
- ∀a0,a1. R a0 a1 → ∀a2. R a0 a2 → R a1 a2.
-
-definition right_cancellable (A) (R:relation A): Prop ≝
- ∀a1,a0. R a1 a0 → ∀a2. R a2 a0 → R a1 a2.
-
-definition pw_confluent2 (A) (R1,R2:relation A): predicate A ≝
- λa0.
- ∀a1. R1 a0 a1 → ∀a2. R2 a0 a2 →
- ∃∃a. R2 a1 a & R1 a2 a.
-
-definition confluent2 (A): relation (relation A) ≝
- λR1,R2.
- ∀a0. pw_confluent2 A R1 R2 a0.
-
-definition transitive2 (A) (R1,R2:relation A): Prop ≝
- ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 →
- ∃∃a. R2 a1 a & R1 a a2.
-
-definition bi_confluent (A) (B) (R: bi_relation A B): Prop ≝
- ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. R a0 b0 a2 b2 →
- ∃∃a,b. R a1 b1 a b & R a2 b2 a b.
-
-definition lsub_trans (A) (B): relation2 (A→relation B) (relation A) ≝
- λR1,R2.
- ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 L1 L2 → R1 L1 T1 T2.
-
-definition s_r_confluent1 (A) (B): relation2 (A→relation B) (B→relation A) ≝
- λR1,R2.
- ∀L1,T1,T2. R1 L1 T1 T2 → ∀L2. R2 T1 L1 L2 → R2 T2 L1 L2.
-
-definition is_mono (B:Type[0]): predicate (predicate B) ≝
- λR. ∀b1. R b1 → ∀b2. R b2 → b1 = b2.
-
-definition is_inj2 (A,B:Type[0]): predicate (relation2 A B) ≝
- λR. ∀a1,b. R a1 b → ∀a2. R a2 b → a1 = a2.
-
-(* Main properties of equality **********************************************)
-
-theorem canc_sn_eq (A): left_cancellable A (eq …).
-// qed-.
-
-theorem canc_dx_eq (A): right_cancellable A (eq …).
-// qed-.
-
-(* Normal form and strong normalization *************************************)
-
-definition NF (A): relation A → relation A → predicate A ≝
- λR,S,a1. ∀a2. R a1 a2 → S a1 a2.
-
-definition NF_dec (A): relation A → relation A → Prop ≝
- λR,S. ∀a1. NF A R S a1 ∨
- ∃∃a2. R … a1 a2 & (S a1 a2 → ⊥).
-
-inductive SN (A) (R,S:relation A): predicate A ≝
-| SN_intro: ∀a1. (∀a2. R a1 a2 → (S a1 a2 → ⊥) → SN A R S a2) → SN A R S a1
-.
-
-lemma NF_to_SN (A) (R) (S): ∀a. NF A R S a → SN A R S a.
-#A #R #S #a1 #Ha1
-@SN_intro #a2 #HRa12 #HSa12
-elim HSa12 -HSa12 /2 width=1 by/
-qed.
-
-definition NF_sn (A): relation A → relation A → predicate A ≝
- λR,S,a2. ∀a1. R a1 a2 → S a1 a2.
-
-inductive SN_sn (A) (R,S:relation A): predicate A ≝
-| SN_sn_intro: ∀a2. (∀a1. R a1 a2 → (S a1 a2 → ⊥) → SN_sn A R S a1) → SN_sn A R S a2
-.
-
-lemma NF_to_SN_sn (A) (R) (S): ∀a. NF_sn A R S a → SN_sn A R S a.
-#A #R #S #a2 #Ha2
-@SN_sn_intro #a1 #HRa12 #HSa12
-elim HSa12 -HSa12 /2 width=1 by/
-qed.
-
-(* Relations on unboxed triples *********************************************)
-
-definition tri_RC (A,B,C): tri_relation A B C → tri_relation A B C ≝
- λR,a1,b1,c1,a2,b2,c2.
- ∨∨ R … a1 b1 c1 a2 b2 c2
- | ∧∧ a1 = a2 & b1 = b2 & c1 = c2.
-
-lemma tri_RC_reflexive (A) (B) (C): ∀R. tri_reflexive A B C (tri_RC … R).
-/3 width=1 by and3_intro, or_intror/ qed.
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