X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fbasics%2Fstar.ma;h=2e3b775e4de1c21c79d4d9b4ecf856fb1ede8d40;hb=a04bfe6d381b281db15e8b432f6f221576aad439;hp=166ba5d7e3de10a6ad24a2525f259de44a1ccf1a;hpb=439b6ec33d749ba4e6ae0938e973a85bc23e306e;p=helm.git

diff --git a/matita/matita/lib/basics/star.ma b/matita/matita/lib/basics/star.ma
index 166ba5d7e..2e3b775e4 100644
--- a/matita/matita/lib/basics/star.ma
+++ b/matita/matita/lib/basics/star.ma
@@ -196,7 +196,7 @@ lemma TC_star_ind: ∀A,R. reflexive A R → ∀a1. ∀P:predicate A.
                    P a1 → (∀a,a2. TC … R a1 a → R a a2 → P a → P a2) →
                    ∀a2. TC … R a1 a2 → P a2.
 #A #R #H #a1 #P #Ha1 #IHa1 #a2 #Ha12 elim Ha12 -a2 /3 width=4/
-qed.
+qed-.
 
 inductive TC_dx (A:Type[0]) (R:relation A): A → A → Prop ≝
   |inj_dx: ∀a,c. R a c → TC_dx A R a c
@@ -254,3 +254,89 @@ definition Conf3: ∀A,B. relation2 A B → relation A → Prop ≝ λA,B,S,R.
 lemma TC_Conf3: ∀A,B,S,R. Conf3 A B S R → Conf3 A B S (TC … R).
 #A #B #S #R #HSR #b #a1 #Ha1 #a2 #H elim H -a2 /2 width=3/
 qed.
+
+inductive bi_TC (A,B:Type[0]) (R:bi_relation A B) (a:A) (b:B): relation2 A B ≝
+  |bi_inj : ∀c,d. R a b c d → bi_TC A B R a b c d
+  |bi_step: ∀c,d,e,f. bi_TC A B R a b c d → R c d e f → bi_TC A B R a b e f.
+
+lemma bi_TC_strap: ∀A,B. ∀R:bi_relation A B. ∀a1,a,a2,b1,b,b2.
+                   R a1 b1 a b → bi_TC … R a b a2 b2 → bi_TC … R a1 b1 a2 b2.
+#A #B #R #a1 #a #a2 #b1 #b #b2 #HR #H elim H -a2 -b2 /2 width=4/ /3 width=4/
+qed.
+
+lemma bi_TC_reflexive: ∀A,B,R. bi_reflexive A B R →
+                       bi_reflexive A B (bi_TC … R).
+/2 width=1/ qed.
+
+inductive bi_TC_dx (A,B:Type[0]) (R:bi_relation A B): bi_relation A B ≝
+  |bi_inj_dx  : ∀a1,a2,b1,b2. R a1 b1 a2 b2 → bi_TC_dx A B R a1 b1 a2 b2
+  |bi_step_dx : ∀a1,a,a2,b1,b,b2. R a1 b1 a b → bi_TC_dx A B R a b a2 b2 →
+                bi_TC_dx A B R a1 b1 a2 b2.
+
+lemma bi_TC_dx_strap: ∀A,B. ∀R: bi_relation A B.
+                      ∀a1,a,a2,b1,b,b2. bi_TC_dx A B R a1 b1 a b →
+                      R a b a2 b2 → bi_TC_dx A B R a1 b1 a2 b2.
+#A #B #R #a1 #a #a2 #b1 #b #b2 #H1 elim H1 -a -b /3 width=4/
+qed.
+
+lemma bi_TC_to_bi_TC_dx: ∀A,B. ∀R: bi_relation A B.
+                         ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 →
+                         bi_TC_dx … R a1 b1 a2 b2.
+#A #B #R #a1 #a2 #b1 #b2 #H12 elim H12 -a2 -b2 /2 width=4/
+qed.
+
+lemma bi_TC_dx_to_bi_TC: ∀A,B. ∀R: bi_relation A B.
+                         ∀a1,a2,b1,b2. bi_TC_dx … R a1 b1 a2 b2 →
+                         bi_TC … R a1 b1 a2 b2.
+#A #b #R #a1 #a2 #b1 #b2 #H12 elim H12 -a1 -a2 -b1 -b2 /2 width=4/
+qed.
+
+fact bi_TC_ind_dx_aux: ∀A,B,R,a2,b2. ∀P:relation2 A B.
+                       (∀a1,b1. R a1 b1 a2 b2 → P a1 b1) →
+                       (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) →
+                       ∀a1,a,b1,b. bi_TC … R a1 b1 a b → a = a2 → b = b2 → P a1 b1.
+#A #B #R #a2 #b2 #P #H1 #H2 #a1 #a #b1 #b #H1
+elim (bi_TC_to_bi_TC_dx ??????? H1) -a1 -a -b1 -b
+[ #a1 #x #b1 #y #H1 #Hx #Hy destruct /2 width=1/
+| #a1 #a #x #b1 #b #y #H1 #H #IH #Hx #Hy destruct /3 width=5/
+]
+qed-.
+
+lemma bi_TC_ind_dx: ∀A,B,R,a2,b2. ∀P:relation2 A B.
+                    (∀a1,b1. R a1 b1 a2 b2 → P a1 b1) →
+                    (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) →
+                    ∀a1,b1. bi_TC … R a1 b1 a2 b2 → P a1 b1.
+#A #B #R #a2 #b2 #P #H1 #H2 #a1 #b1 #H12
+@(bi_TC_ind_dx_aux ?????? H1 H2 … H12) //
+qed-.
+
+lemma bi_TC_symmetric: ∀A,B,R. bi_symmetric A B R →
+                       bi_symmetric A B (bi_TC … R).
+#A #B #R #HR #a1 #a2 #b1 #b2 #H21
+@(bi_TC_ind_dx ?????????? H21) -a2 -b2 /3 width=1/ /3 width=4/
+qed.
+
+lemma bi_TC_transitive: ∀A,B,R. bi_transitive A B (bi_TC … R).
+#A #B #R #a1 #a #b1 #b #H elim H -a -b /2 width=4/ /3 width=4/
+qed.
+
+definition bi_Conf3: ∀A,B,C. relation3 A B C → bi_relation A B → Prop ≝ λA,B,C,S,R.
+                     ∀c,a1,b1. S a1 b1 c → ∀a2,b2. R a1 b1 a2 b2 → S a2 b2 c.
+
+lemma bi_TC_Conf3: ∀A,B,C,S,R. bi_Conf3 A B C S R → bi_Conf3 A B C S (bi_TC … R).
+#A #B #C #S #R #HSR #c #a1 #b1 #Hab1 #a2 #b2 #H elim H -a2 -b2 /2 width=4/
+qed.
+
+lemma bi_TC_star_ind: ∀A,B,R. bi_reflexive A B R → ∀a1,b1. ∀P:relation2 A B.
+                      P a1 b1 → (∀a,a2,b,b2. bi_TC … R a1 b1 a b → R a b a2 b2 → P a b → P a2 b2) →
+                      ∀a2,b2. bi_TC … R a1 b1 a2 b2 → P a2 b2.
+#A #B #R #HR #a1 #b1 #P #H1 #IH #a2 #b2 #H12 elim H12 -a2 -b2 /3 width=5/
+qed-.
+
+lemma bi_TC_star_ind_dx: ∀A,B,R. bi_reflexive A B R →
+                         ∀a2,b2. ∀P:relation2 A B. P a2 b2 →
+                         (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) →
+                         ∀a1,b1. bi_TC … R a1 b1 a2 b2 → P a1 b1.
+#A #B #R #HR #a2 #b2 #P #H2 #IH #a1 #b1 #H12
+@(bi_TC_ind_dx … P ? IH … H12) /3 width=5/
+qed-.