X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fbasics%2Fstar.ma;h=36aee8fe45465c016a6ca3709046a0c6fefd11c7;hb=ea368a02a071bb99eeb84bf24ab4000acb314d60;hp=bff1aaeef80042a2ae2e838317d0c5ed5c858a58;hpb=e0827239f4b44f2af9c7f88c4c7c41f2a193ae37;p=helm.git

diff --git a/matita/matita/lib/basics/star.ma b/matita/matita/lib/basics/star.ma
index bff1aaeef..36aee8fe4 100644
--- a/matita/matita/lib/basics/star.ma
+++ b/matita/matita/lib/basics/star.ma
@@ -11,12 +11,6 @@
 
 include "basics/relations.ma".
 
-(********** relations **********)
-
-definition subR ≝ λA.λR,S:relation A.(∀a,b. R a b → S a b).
-
-definition inv ≝ λA.λR:relation A.λa,b.R b a.
-
 (* transitive closcure (plus) *)
 
 inductive TC (A:Type[0]) (R:relation A) (a:A): A → Prop ≝
@@ -52,8 +46,8 @@ qed.
   
 (* star *)
 inductive star (A:Type[0]) (R:relation A) (a:A): A → Prop ≝
-  |inj: ∀b,c.star A R a b → R b c → star A R a c
-  |refl: star A R a a.
+  |sstep: ∀b,c.star A R a b → R b c → star A R a c
+  |srefl: star A R a a.
 
 lemma R_to_star: ∀A,R,a,b. R a b → star A R a b.
 #A #R #a #b /2/
@@ -88,6 +82,66 @@ theorem star_inv: ∀A,R.
 #H (elim H) /2/ normalize #c #d #H1 #H2 #H3 @(trans_star … H3) /2/
 qed.
 
+lemma star_decomp_l : 
+  ∀A,R,x,y.star A R x y → x = y ∨ ∃z.R x z ∧ star A R z y.
+#A #R #x #y #Hstar elim Hstar
+[ #b #c #Hleft #Hright *
+  [ #H1 %2 @(ex_intro ?? c) % //
+  | * #x0 * #H1 #H2 %2 @(ex_intro ?? x0) % /2/ ]
+| /2/ ]
+qed.
+
+(* right associative version of star *)
+inductive starl (A:Type[0]) (R:relation A) : A → A → Prop ≝
+  |sstepl: ∀a,b,c.R a b → starl A R b c → starl A R a c
+  |refll: ∀a.starl A R a a.
+ 
+lemma starl_comp : ∀A,R,a,b,c.
+  starl A R a b → R b c → starl A R a c.
+#A #R #a #b #c #Hstar elim Hstar 
+  [#a1 #b1 #c1 #Rab #sbc #Hind #a1 @(sstepl … Rab) @Hind //
+  |#a1 #Rac @(sstepl … Rac) //
+  ]
+qed.
+
+lemma star_compl : ∀A,R,a,b,c.
+  R a b → star A R b c → star A R a c.
+#A #R #a #b #c #Rab #Hstar elim Hstar 
+  [#b1 #c1 #sbb1 #Rb1c1 #Hind @(sstep … Rb1c1) @Hind
+  |@(sstep … Rab) //
+  ]
+qed.
+
+lemma star_to_starl: ∀A,R,a,b.star A R a b → starl A R a b.
+#A #R #a #b #Hs elim Hs //
+#d #c #sad #Rdc #sad @(starl_comp … Rdc) //
+qed.
+
+lemma starl_to_star: ∀A,R,a,b.starl A R a b → star A R a b.
+#A #R #a #b #Hs elim Hs // -Hs -b -a
+#a #b #c #Rab #sbc #sbc @(star_compl … Rab) //
+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 *)
 
 lemma TC_to_star: ∀A,R,a,b.TC A R a b → star A R a b.
@@ -152,7 +206,7 @@ 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
@@ -173,28 +227,215 @@ lemma TC_dx_to_TC: ∀A. ∀R: relation A.
 #A #R #a1 #a2 #Ha12 elim Ha12 -a1 -a2 /2 width=3/
 qed.
 
-fact TC_star_ind_dx_aux: ∀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,a. TC … R a1 a → a = a2 → P a1.
-#A #R #HR #a2 #P #Ha2 #H #a1 #a #Ha1
+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 /3 width=4/
+[ #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_star_ind_dx_aux … HR … Ha2 H … Ha12) //
+@(TC_ind_dx … P ? H … Ha12) /3 width=4/
 qed-.
 
-definition Conf3: ∀A. relation A → relation A → Prop ≝ λA,S,R.
-                  ∀a,a1. S a1 a → ∀a2. R a1 a2 → S a2 a.
+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 TC_Conf3: ∀A,S,R. Conf3 A S R → Conf3 A S (TC … R).
-#A #S #R #HSR #a #a1 #Ha1 #a2 #H elim H -a2 /2 width=3/
+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-.