the decentralization of core notation continues ...

[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 04d2d7317e4416b7031d18f5125f2ce78eeb72bc..391e085309e3ee2cbc9f535ee07af7f2179f34a1 100644 (file)
--- 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/
@@ -99,22 +93,22 @@ qed.
 
 (* right associative version of star *)
 inductive starl (A:Type[0]) (R:relation A) : A → A → Prop ≝
-  |injl: ∀a,b,c.R a b → starl A R b c → starl A R a c
+  |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 @(injl … Rab) @Hind //
-  |#a1 #Rac @(injl … Rac) //
+  [#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 @(inj … Rb1c1) @Hind
-  |@(inj … Rab) //
+  [#b1 #c1 #sbb1 #Rb1c1 #Hind @(sstep … Rb1c1) @Hind
+  |@(sstep … Rab) //
   ]
 qed.
 
@@ -128,17 +122,27 @@ 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 *)
+(* 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/
@@ -189,62 +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_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
-elim (TC_to_TC_dx ???? Ha1) -a1 -a
-[ #a #c #Hac #H destruct /3 width=4/
-| #a #b #c #Hab #Hbc #IH #H destruct /3 width=4/
-]
-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) //
-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.