Closed some daemons

[helm.git] / matita / matita / lib / basics / lists / lstar.ma

diff --git a/matita/matita/lib/basics/lists/lstar.ma b/matita/matita/lib/basics/lists/lstar.ma
index aafe8f5b7055da6393c2f463500c54395059f46c..780f7759dd31f2d8d43217e391abad68d946250e 100644 (file)
--- a/matita/matita/lib/basics/lists/lstar.ma
+++ b/matita/matita/lib/basics/lists/lstar.ma
@@ -13,6 +13,9 @@ include "basics/lists/list.ma".
 
 (* labelled reflexive and transitive closure ********************************)
 
+definition ltransitive: ∀A,B:Type[0]. predicate (list A → relation B) ≝ λA,B,R.
+                        ∀l1,b1,b. R l1 b1 b → ∀l2,b2. R l2 b b2 → R (l1@l2) b1 b2. 
+
 inductive lstar (A:Type[0]) (B:Type[0]) (R: A→relation B): list A → relation B ≝
 | lstar_nil : ∀b. lstar A B R ([]) b b
 | lstar_cons: ∀a,b1,b. R a b1 b →
@@ -72,7 +75,51 @@ theorem lstar_singlevalued: ∀A,B,R. (∀a. singlevalued ?? (R a)) →
 ]
 qed-.
 
-theorem lstar_trans: ∀A,B,R,l1,b1,b. lstar A B R l1 b1 b →
-                     ∀l2,b2. lstar A B R l2 b b2 → lstar A B R (l1@l2) b1 b2.
+theorem lstar_ltransitive: ∀A,B,R. ltransitive … (lstar A B R).
 #A #B #R #l1 #b1 #b #H @(lstar_ind_l ????????? H) -l1 -b1 normalize // /3 width=3/
 qed-.
+
+lemma lstar_app: ∀A,B,R,l,b1,b. lstar A B R l b1 b → ∀a,b2. R a b b2 →
+                 lstar A B R (l@[a]) b1 b2.
+#A #B #R #l #b1 #b #H @(lstar_ind_l ????????? H) -l -b1 /2 width=1/
+normalize /3 width=3/
+qed.
+
+inductive lstar_r (A:Type[0]) (B:Type[0]) (R: A→relation B): list A → relation B ≝
+| lstar_r_nil: ∀b. lstar_r A B R ([]) b b
+| lstar_r_app: ∀l,b1,b. lstar_r A B R l b1 b → ∀a,b2. R a b b2 →
+               lstar_r A B R (l@[a]) b1 b2
+.
+
+lemma lstar_r_cons: ∀A,B,R,l,b,b2. lstar_r A B R l b b2 → ∀a,b1. R a b1 b →
+                    lstar_r A B R (a::l) b1 b2.
+#A #B #R #l #b #b2 #H elim H -l -b2 /2 width=3/
+#l #b1 #b #_ #a #b2 #Hb2 #IHb1 #a0 #b0 #Hb01
+@(lstar_r_app … (a0::l) … Hb2) -b2 /2 width=1/
+qed.
+
+lemma lstar_lstar_r: ∀A,B,R,l,b1,b2. lstar A B R l b1 b2 → lstar_r A B R l b1 b2.
+#A #B #R #l #b1 #b2 #H @(lstar_ind_l ????????? H) -l -b1 // /2 width=3/
+qed.
+
+lemma lstar_r_inv_lstar: ∀A,B,R,l,b1,b2. lstar_r A B R l b1 b2 → lstar A B R l b1 b2.
+#A #B #R #l #b1 #b2 #H elim H -l -b1 -b2 // /2 width=3/
+qed-.
+
+fact lstar_ind_r_aux: ∀A,B,R,b1. ∀P:relation2 (list A) B.
+                      P ([]) b1 →
+                      (∀a,l,b,b2. lstar … R l b1 b → R a b b2 → P l b → P (l@[a]) b2) →
+                      ∀l,b,b2. lstar … R l b b2 → b = b1 → P l b2.
+#A #B #R #b1 #P #H1 #H2 #l #b #b2 #H elim (lstar_lstar_r ?????? H) -l -b -b2
+[ #b #H destruct //
+| #l #b #b0 #Hb0 #a #b2 #Hb02 #IH #H destruct /3 width=4 by lstar_r_inv_lstar/
+]
+qed-.
+
+lemma lstar_ind_r: ∀A,B,R,b1. ∀P:relation2 (list A) B.
+                   P ([]) b1 →
+                   (∀a,l,b,b2. lstar … R l b1 b → R a b b2 → P l b → P (l@[a]) b2) →
+                   ∀l,b2. lstar … R l b1 b2 → P l b2.
+#A #B #R #b1 #P #H1 #H2 #l #b2 #Hb12
+@(lstar_ind_r_aux … H1 H2 … Hb12) //
+qed-.