update in ground_2, static_2, basic_2

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / lib / relations.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/lib/relations.ma b/matita/matita/contribs/lambdadelta/ground_2/lib/relations.ma
index fa8bae34356880363acc0a6369876121b7bbe6d9..47524fd84d4669ebe1b4d425074aece7642941eb 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/lib/relations.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/lib/relations.ma
@@ -13,12 +13,14 @@
 (**************************************************************************)
 
 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.
+           λR,Sa,Sb. ∀a1,b1. R a1 b1 → ∀a2. Sa a1 a2 → ∀b2. Sb b1 b2 → R a2 b2.
 
 (* Inclusion ****************************************************************)
 
@@ -36,81 +38,93 @@ interpretation "3-relation inclusion"
 
 (* 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 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.
+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 dont use "∀a. reflexive … (R a)" since auto seems to dislike repeatd δ-expansion *)  
+(**) (* 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.
+           λR. ∀a,b. R a b b.
 
 definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥).
 
-definition Transitive: ∀A. ∀R: relation A. Prop ≝ λA,R.
-                       ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2.
+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 ≝ λA,R.
-                             ∀a0,a1. R a0 a1 → ∀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 ≝ λA,R.
-                              ∀a1,a0. R a1 a0 → ∀a2. R a2 a0 → 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. relation A → relation A → predicate A ≝ λA,R1,R2,a0.
-                          ∀a1. R1 a0 a1 → ∀a2. R2 a0 a2 →
-                          ∃∃a. R2 a1 a & R1 a2 a.
+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) ≝ λA,R1,R2.
-                       ∀a0. pw_confluent2 A R1 R2 a0.
+definition confluent2 (A): relation (relation A) ≝
+           λR1,R2.
+           ∀a0. pw_confluent2 A R1 R2 a0.
 
-definition transitive2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
-                        ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 →
-                        ∃∃a. R2 a1 a & R1 a a2.
+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 ≝ λA,B,R.
-                         ∀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 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) ≝ λA,B,R1,R2.
-                       ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 L1 L2 → R1 L1 T1 T2.
+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) ≝ λA,B,R1,R2.
-                           ∀L1,T1,T2. R1 L1 T1 T2 → ∀L2. R2 T1 L1 L2 → R2 T2 L1 L2.
+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) ≝ 
-                    λB,R. ∀b1. R b1 → ∀b2. R b2 → b1 = b2.
+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) ≝ 
-                    λA,B,R. ∀a1,b. R a1 b → ∀a2. R a2 b → a1 = a2.
+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 ≝
-   λA,R,S,a1. ∀a2. R a1 a2 → S a1 a2.
+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 ≝
-                   λA,R,S. ∀a1. NF A R 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.
+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 ≝
-   λA,R,S,a2. ∀a1. R a1 a2 → S a1 a2.
+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.
+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/
@@ -118,9 +132,10 @@ qed.
 
 (* Relations on unboxed triples *********************************************)
 
-definition tri_RC: ∀A,B,C. tri_relation A B C → tri_relation A B C ≝
-                   λA,B,C,R,a1,b1,c1,a2,b2,c2. R … a1 b1 c1 a2 b2 c2 ∨
-                   ∧∧ a1 = a2 & b1 = b2 & c1 = c2.
+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).
+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.