- basics: some support for abstract triangular confluence (which

[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 c1b9c77c03c85a8cc739b6542b81563eebc37b22..bff1aaeef80042a2ae2e838317d0c5ed5c858a58 100644 (file)
--- a/matita/matita/lib/basics/star.ma
+++ b/matita/matita/lib/basics/star.ma
@@ -107,7 +107,7 @@ 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)).
@@ -139,3 +139,62 @@ 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. relation A → relation A → Prop ≝ λA,S,R.
+                  ∀a,a1. S a1 a → ∀a2. R a1 a2 → S a2 a.
+
+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/
+qed.