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 8e850c697dc21b058684a6756207470cb84ad3ae..391e085309e3ee2cbc9f535ee07af7f2179f34a1 100644 (file)
--- a/matita/matita/lib/basics/star.ma
+++ b/matita/matita/lib/basics/star.ma
@@ -11,11 +11,47 @@
 
 include "basics/relations.ma".
 
-(********** relations **********)
+(* transitive closcure (plus) *)
 
+inductive TC (A:Type[0]) (R:relation A) (a:A): A → Prop ≝
+  |inj: ∀c. R a c → TC A R a c
+  |step : ∀b,c.TC A R a b → R b c → TC A R a c.
+
+theorem trans_TC: ∀A,R,a,b,c. 
+  TC A R a b → TC A R b c → TC A R a c.
+#A #R #a #b #c #Hab #Hbc (elim Hbc) /2/
+qed.
+
+theorem TC_idem: ∀A,R. exteqR … (TC A R) (TC A (TC A R)).
+#A #R #a #b % /2/ #H (elim H) /2/
+qed.
+
+lemma monotonic_TC: ∀A,R,S. subR A R S → subR A (TC A R) (TC A S).
+#A #R #S #subRS #a #b #H (elim H) /3/
+qed.
+
+lemma sub_TC: ∀A,R,S. subR A R (TC A S) → subR A (TC A R) (TC A S).
+#A #R #S #Hsub #a #b #H (elim H) /3/
+qed.
+
+theorem sub_TC_to_eq: ∀A,R,S. subR A R S → subR A S (TC A R) → 
+  exteqR … (TC A R) (TC A S).
+#A #R #S #sub1 #sub2 #a #b % /2/
+qed.
+
+theorem TC_inv: ∀A,R. exteqR ?? (TC A (inv A R)) (inv A (TC A R)).
+#A #R #a #b %
+#H (elim H) /2/ normalize #c #d #H1 #H2 #H3 @(trans_TC … H3) /2/
+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/
+qed.
 
 theorem trans_star: ∀A,R,a,b,c. 
   star A R a b → star A R b c → star A R a c.
@@ -26,8 +62,6 @@ theorem star_star: ∀A,R. exteqR … (star A R) (star A (star A R)).
 #A #R #a #b % /2/ #H (elim H) /2/
 qed.
 
-definition subR ≝ λA.λR,S:relation A.(∀a,b. R a b → S a b).
-
 lemma monotonic_star: ∀A,R,S. subR A R S → subR A (star A R) (star A S).
 #A #R #S #subRS #a #b #H (elim H) /3/
 qed.
@@ -42,6 +76,82 @@ theorem sub_star_to_eq: ∀A,R,S. subR A R S → subR A S (star A R) →
 #A #R #S #sub1 #sub2 #a #b % /2/
 qed.
 
+theorem star_inv: ∀A,R. 
+  exteqR ?? (star A (inv A R)) (inv A (star A R)).
+#A #R #a #b %
+#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.
+
+(* 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/
+qed.
+
+lemma star_case: ∀A,R,a,b. star A R a b → a = b ∨ TC A R a b.
+#A #R #a #b #H (elim H) /2/ #c #d #star_ac #Rcd * #H1 %2 /2/.
+qed.
+
 (* equiv -- smallest equivalence relation containing R *)
 
 inductive equiv (A:Type[0]) (R:relation A) : A → A → Prop ≝
@@ -51,7 +161,7 @@ inductive equiv (A:Type[0]) (R:relation A) : A → A → Prop ≝
   
 theorem trans_equiv: ∀A,R,a,b,c. 
   equiv A R a b → equiv A R b c → equiv A R a c.
-#A #R #a #b #c #Hab #Hbc (inversion Hbc) /2/
+#A #R #a #b #c #Hab #Hbc (elim Hbc) /2/
 qed.
  
 theorem equiv_equiv: ∀A,R. exteqR … (equiv A R) (equiv A (equiv A R)).