- list.ma: improved notation for constant lists (a "term 19" was missing)

[helm.git] / matita / matita / lib / basics / star.ma

diff --git a/matita/matita/lib/basics/star.ma b/matita/matita/lib/basics/star.ma
index 2e3b775e4de1c21c79d4d9b4ecf856fb1ede8d40..892cff583c9b85d7deb844b8c53a5051c0a93d83 100644 (file)
--- a/matita/matita/lib/basics/star.ma
+++ b/matita/matita/lib/basics/star.ma
@@ -122,15 +122,25 @@ lemma starl_to_star: ∀A,R,a,b.starl A R a b → star A R a b.
 #a #b #c #Rab #sbc #sbc @(star_compl … Rab) //
 qed.
 
-lemma star_ind_l : 
-  ∀A:Type[0].∀R:relation A.∀Q:A → A → Prop.
-  (∀a.Q a a) → 
-  (∀a,b,c.R a b → star A R b c → Q b c → Q a c) → 
-  ∀a,b.star A R a b → Q a b.
-#A #R #Q #H1 #H2 #a #b #H0 
-elim (star_to_starl ???? H0) // -H0 -b -a 
-#a #b #c #Rab #slbc @H2 // @starl_to_star //
-qed.  
+fact star_ind_l_aux: ∀A,R,a2. ∀P:predicate A.
+                     P a2 →
+                     (∀a1,a. R a1 a → star … R a a2 → P a → P a1) →
+                     ∀a1,a. star … R a1 a → a = a2 → P a1.
+#A #R #a2 #P #H1 #H2 #a1 #a #Ha1
+elim (star_to_starl ???? Ha1) -a1 -a
+[ #a #b #c #Hab #Hbc #IH #H destruct /3 width=4/
+| #a #H destruct /2 width=1/
+]
+qed-.
+
+(* imporeved version of star_ind_l with "left_parameter" *)
+lemma star_ind_l: ∀A,R,a2. ∀P:predicate A.
+                  P a2 →
+                  (∀a1,a. R a1 a → star … R a a2 → P a → P a1) →
+                  ∀a1. star … R a1 a2 → P a1.
+#A #R #a2 #P #H1 #H2 #a1 #Ha12
+@(star_ind_l_aux … H1 H2 … Ha12) //
+qed.
 
 (* RC and star *)
 
@@ -340,3 +350,74 @@ lemma bi_TC_star_ind_dx: ∀A,B,R. bi_reflexive A B R →
 #A #B #R #HR #a2 #b2 #P #H2 #IH #a1 #b1 #H12
 @(bi_TC_ind_dx … P ? IH … H12) /3 width=5/
 qed-.
+
+definition bi_star: ∀A,B,R. bi_relation A B ≝ λA,B,R,a1,b1,a2,b2.
+                    (a1 = a2 ∧ b1 = b2) ∨ bi_TC A B R a1 b1 a2 b2.
+
+lemma bi_star_bi_reflexive: ∀A,B,R. bi_reflexive A B (bi_star … R).
+/3 width=1/ qed.
+
+lemma bi_TC_to_bi_star: ∀A,B,R,a1,b1,a2,b2.
+                        bi_TC A B R a1 b1 a2 b2 → bi_star A B R a1 b1 a2 b2.
+/2 width=1/ qed.
+
+lemma bi_R_to_bi_star: ∀A,B,R,a1,b1,a2,b2.
+                       R a1 b1 a2 b2 → bi_star A B R a1 b1 a2 b2.
+/3 width=1/ qed.
+
+lemma bi_star_strap1: ∀A,B,R,a1,a,a2,b1,b,b2. bi_star A B R a1 b1 a b →
+                      R a b a2 b2 → bi_star A B R a1 b1 a2 b2.
+#A #B #R #a1 #a #a2 #b1 #b #b2 *
+[ * #H1 #H2 destruct /2 width=1/
+| /3 width=4/
+]
+qed.
+
+lemma bi_star_strap2: ∀A,B,R,a1,a,a2,b1,b,b2. R a1 b1 a b →
+                      bi_star A B R a b a2 b2 → bi_star A B R a1 b1 a2 b2.
+#A #B #R #a1 #a #a2 #b1 #b #b2 #H *
+[ * #H1 #H2 destruct /2 width=1/
+| /3 width=4/
+]
+qed.
+
+lemma bi_star_to_bi_TC_to_bi_TC: ∀A,B,R,a1,a,a2,b1,b,b2. bi_star A B R a1 b1 a b →
+                                 bi_TC A B R a b a2 b2 → bi_TC A B R a1 b1 a2 b2.
+#A #B #R #a1 #a #a2 #b1 #b #b2 *
+[ * #H1 #H2 destruct /2 width=1/
+| /2 width=4/
+]
+qed.
+
+lemma bi_TC_to_bi_star_to_bi_TC: ∀A,B,R,a1,a,a2,b1,b,b2. bi_TC A B R a1 b1 a b →
+                                 bi_star A B R a b a2 b2 → bi_TC A B R a1 b1 a2 b2.
+#A #B #R #a1 #a #a2 #b1 #b #b2 #H *
+[ * #H1 #H2 destruct /2 width=1/
+| /2 width=4/
+]
+qed.
+
+lemma bi_tansitive_bi_star: ∀A,B,R. bi_transitive A B (bi_star … R).
+#A #B #R #a1 #a #b1 #b #H #a2 #b2 *
+[ * #H1 #H2 destruct /2 width=1/
+| /3 width=4/
+]
+qed.
+
+lemma bi_star_ind: ∀A,B,R,a1,b1. ∀P:relation2 A B. P a1 b1 →
+                   (∀a,a2,b,b2. bi_star … R a1 b1 a b → R a b a2 b2 → P a b → P a2 b2) →
+                   ∀a2,b2. bi_star … R a1 b1 a2 b2 → P a2 b2.
+#A #B #R #a1 #b1 #P #H #IH #a2 #b2 *
+[ * #H1 #H2 destruct //
+| #H12 elim H12 -a2 -b2 /2 width=5/ -H /3 width=5/
+]
+qed-.
+
+lemma bi_star_ind_dx: ∀A,B,R,a2,b2. ∀P:relation2 A B. P a2 b2 →
+                      (∀a1,a,b1,b. R a1 b1 a b → bi_star … R a b a2 b2 → P a b → P a1 b1) →
+                      ∀a1,b1. bi_star … R a1 b1 a2 b2 → P a1 b1.
+#A #B #R #a2 #b2 #P #H #IH #a1 #b1 *
+[ * #H1 #H2 destruct //
+| #H12 @(bi_TC_ind_dx ?????????? H12) -a1 -b1 /2 width=5/ -H /3 width=5/
+]
+qed-.