propagating the arithmetics library, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / lib / ltc.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/lib/ltc.ma b/matita/matita/contribs/lambdadelta/ground/lib/ltc.ma
index 4f3962671e212fb675d969686f101c41c5d24702..1d08fb15d98b9bd7af24e7d55f3ed133c8eea363 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/lib/ltc.ma
+++ b/matita/matita/contribs/lambdadelta/ground/lib/ltc.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground/insert_eq/insert_eq_0.ma".
+include "ground/insert_eq/insert_eq_1.ma".
 include "ground/lib/functions.ma".
 
 (* LABELLED TRANSITIVE CLOSURE **********************************************)
@@ -22,7 +22,7 @@ inductive ltc (A:Type[0]) (f) (B) (R:relation3 A B B): relation3 A B B ≝
 | ltc_trans: ∀a1,a2,b1,b,b2. ltc … a1 b1 b → ltc … a2 b b2 → ltc … (f a1 a2) b1 b2
 .
 
-(* Basic properties *********************************************************)
+(* Basic constructions ******************************************************)
 
 lemma ltc_sn (A) (f) (B) (R): ∀a1,b1,b. R a1 b1 b →
                               ∀a2,b2. ltc A f B R a2 b b2 → ltc … f … R (f a1 a2) b1 b2.
@@ -32,13 +32,13 @@ lemma ltc_dx (A) (f) (B) (R): ∀a1,b1,b. ltc A f B R a1 b1 b →
                               ∀a2,b2. R a2 b b2 → ltc … f … R (f a1 a2) b1 b2.
 /3 width=3 by ltc_rc, ltc_trans/ qed.
 
-(* Basic eliminators ********************************************************)
+(* Basic eliminations *******************************************************)
 
 lemma ltc_ind_sn (A) (f) (B) (R) (Q:relation2 A B) (b2): associative … f →
                  (∀a,b1. R a b1 b2 → Q a b1) →
                  (∀a1,a2,b1,b. R a1 b1 b → ltc … f … R a2 b b2 → Q a2 b → Q (f a1 a2) b1) →
                  ∀a,b1. ltc … f … R a b1 b2 → Q a b1.
-#A #f #B #R #Q #b2 #Hf #IH1 #IH2 #a #b1 @(insert_eq_0 … b2)
+#A #f #B #R #Q #b2 #Hf #IH1 #IH2 #a #b1 @(insert_eq_1 … b2)
 #b0 #H elim H -a -b1 -b0 /2 width=2 by/
 #a1 #a2 #b1 #b #b0 #H #Hb2 #_
 generalize in match Hb2; generalize in match a2; -Hb2 -a2
@@ -49,14 +49,14 @@ lemma ltc_ind_dx (A) (f) (B) (R) (Q:A→predicate B) (b1): associative … f →
                  (∀a,b2. R a b1 b2 → Q a b2) →
                  (∀a1,a2,b,b2. ltc … f … R a1 b1 b → Q a1 b → R a2 b b2 → Q (f a1 a2) b2) →
                  ∀a,b2. ltc … f … R a b1 b2 → Q a b2.
-#A #f #B #R #Q #b1 #Hf #IH1 #IH2 #a #b2 @(insert_eq_0 … b1)
+#A #f #B #R #Q #b1 #Hf #IH1 #IH2 #a #b2 @(insert_eq_1 … b1)
 #b0 #H elim H -a -b0 -b2 /2 width=2 by/
 #a1 #a2 #b0 #b #b2 #Hb0 #H #IHb0 #_
 generalize in match IHb0; generalize in match Hb0; generalize in match a1; -IHb0 -Hb0 -a1
 elim H -a2 -b -b2 /4 width=4 by ltc_trans/
 qed-.
 
-(* Advanced elimiators with reflexivity *************************************)
+(* Advanced elimiations with reflexivity ************************************)
 
 lemma ltc_ind_sn_refl (A) (i) (f) (B) (R) (Q:relation2 A B) (b2):
                       associative … f → right_identity … f i → reflexive B (R i) →
@@ -78,7 +78,7 @@ lemma ltc_ind_dx_refl (A) (i) (f) (B) (R) (Q:A→predicate B) (b1):
 >(H2f a) -H2f /3 width=4 by ltc_rc/
 qed-.
 
-(* Properties with lsub *****************************************************)
+(* Constructions with lsub **************************************************)
 
 lemma ltc_lsub_trans: ∀A,f. associative … f →
                       ∀B,C,R,S. (∀n. lsub_trans B C (λL. R L n) S) →