context-free parallel reduction on closures is confluent!

[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 09730b6f22e586a145bca785f77dba56f30d80c4..166ba5d7e3de10a6ad24a2525f259de44a1ccf1a 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/
@@ -97,24 +91,46 @@ lemma star_decomp_l :
 | /2/ ]
 qed.
 
-axiom star_ind_l : 
+(* 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.
+
+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) → 
-  ∀x,y.star A R x y → Q x y.
-(* #A #R #Q #H1 #H2 #x #y #H0 elim H0
-[ #b #c #Hleft #Hright #IH
-  cases (star_decomp_l ???? Hleft)
-  [ #Hx @H2 //
-  | * #z * #H3 #H4 @(H2 … H3) /2/
-[
-|
-generalize in match (λb.H2 x b y); elim H0
-[#b #c #Hleft #Hright #H2' #H3 @H3
- @(H3 b)
- elim H0
-[ #b #c #Hleft #Hright #IH //
-| *)
+  ∀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.  
 
 (* RC and star *)
 
@@ -201,23 +217,35 @@ 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,B. relation2 A B → relation A → Prop ≝ λA,B,S,R.