X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fbasics%2Fstar.ma;h=391e085309e3ee2cbc9f535ee07af7f2179f34a1;hb=4adf9860cd26175c4d73b73e8adbb3c6ceaa19c9;hp=36aee8fe45465c016a6ca3709046a0c6fefd11c7;hpb=380ceb6b6552fd9ebd48d710ab12931d5d97e465;p=helm.git

diff --git a/matita/matita/lib/basics/star.ma b/matita/matita/lib/basics/star.ma
index 36aee8fe4..391e08530 100644
--- a/matita/matita/lib/basics/star.ma
+++ b/matita/matita/lib/basics/star.ma
@@ -142,7 +142,7 @@ lemma star_ind_l: ∀A,R,a2. ∀P:predicate A.
 @(star_ind_l_aux … H1 H2 … Ha12) //
 qed.
 
-(* RC and star *)
+(* TC and star *)
 
 lemma TC_to_star: ∀A,R,a,b.TC A R a b → star A R a b.
 #R #A #a #b #TCH (elim TCH) /2/
@@ -193,249 +193,3 @@ lemma WF_antimonotonic: ∀A,R,S. subR A R S →
 #H #Hind % #c #Rcb @Hind @subRS //
 qed.
 
-(* added from lambda_delta *)
-
-lemma TC_strap: ∀A. ∀R:relation A. ∀a1,a,a2.
-                R a1 a → TC … R a a2 → TC … R a1 a2.
-/3 width=3/ qed.
-
-lemma TC_reflexive: ∀A,R. reflexive A R → reflexive A (TC … R).
-/2 width=1/ qed.
-
-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-.
-
-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
-  |step_dx : ∀a,b,c. R a b → TC_dx A R b c → TC_dx A R a c.
-
-lemma TC_dx_strap: ∀A. ∀R: relation A.
-                   ∀a,b,c. TC_dx A R a b → R b c → TC_dx A R a c.
-#A #R #a #b #c #Hab elim Hab -a -b /3 width=3/
-qed.
-
-lemma TC_to_TC_dx: ∀A. ∀R: relation A.
-                   ∀a1,a2. TC … R a1 a2 → TC_dx … R a1 a2.
-#A #R #a1 #a2 #Ha12 elim Ha12 -a2 /2 width=3/
-qed.
-
-lemma TC_dx_to_TC: ∀A. ∀R: relation A.
-                   ∀a1,a2. TC_dx … R a1 a2 → TC … R a1 a2.
-#A #R #a1 #a2 #Ha12 elim Ha12 -a1 -a2 /2 width=3/
-qed.
-
-fact TC_ind_dx_aux: ∀A,R,a2. ∀P:predicate A.
-                    (∀a1. R a1 a2 → P a1) →
-                    (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
-                    ∀a1,a. TC … R a1 a → a = a2 → P a1.
-#A #R #a2 #P #H1 #H2 #a1 #a #Ha1
-elim (TC_to_TC_dx ???? Ha1) -a1 -a
-[ #a #c #Hac #H destruct /2 width=1/
-| #a #b #c #Hab #Hbc #IH #H destruct /3 width=4/
-]
-qed-.
-
-lemma TC_ind_dx: ∀A,R,a2. ∀P:predicate A.
-                 (∀a1. R a1 a2 → P a1) →
-                 (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
-                 ∀a1. TC … R a1 a2 → P a1.
-#A #R #a2 #P #H1 #H2 #a1 #Ha12
-@(TC_ind_dx_aux … H1 H2 … Ha12) //
-qed-.
-
-lemma TC_symmetric: ∀A,R. symmetric A R → symmetric A (TC … R).
-#A #R #HR #x #y #H @(TC_ind_dx ??????? H) -x /3 width=1/ /3 width=3/
-qed.
-
-lemma TC_star_ind_dx: ∀A,R. reflexive A R →
-                      ∀a2. ∀P:predicate A. P a2 →
-                      (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
-                      ∀a1. TC … R a1 a2 → P a1.
-#A #R #HR #a2 #P #Ha2 #H #a1 #Ha12
-@(TC_ind_dx … P ? H … Ha12) /3 width=4/
-qed-.
-
-definition Conf3: ∀A,B. relation2 A B → relation A → Prop ≝ λA,B,S,R.
-                  ∀b,a1. S a1 b → ∀a2. R a1 a2 → S a2 b.
-
-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-.
-
-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-.
-
-(* ************ confluence of star *****************)
-
-lemma star_strip: ∀A,R. confluent A R →
-                  ∀a0,a1. star … R a0 a1 → ∀a2. R a0 a2 →
-                  ∃∃a. R a1 a & star … R a2 a.
-#A #R #HR #a0 #a1 #H elim H -a1 /2 width=3/
-#a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
-elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
-elim (HR … Ha1 … Ha0) -a /3 width=5/
-qed-.
-
-lemma star_confluent: ∀A,R. confluent A R → confluent A (star … R).
-#A #R #HR #a0 #a1 #H elim H -a1 /2 width=3/
-#a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
-elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
-elim (star_strip … HR … Ha0 … Ha1) -a /3 width=5/
-qed-.