propagating the arithmetics library, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / nat_le.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_le.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_le.ma
index 347ea4f890be465ae9941a6c9a20b86f05a51695..5d99e4c55a1606fca21959c80055157a9c83d9fe 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_le.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_le.ma
@@ -12,13 +12,12 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground/insert_eq/insert_eq_0.ma".
+include "ground/generated/insert_eq_1.ma".
 include "ground/arith/nat_succ.ma".
 
 (* ORDER FOR NON-NEGATIVE INTEGERS ******************************************)
 
 (*** le *)
-(*** le_ind *)
 inductive nle (m:nat): predicate nat ≝
 | nle_refl   : nle m m
 | nle_succ_dx: ∀n. nle m n → nle m (↑n)
@@ -45,7 +44,7 @@ lemma nle_succ_bi (m) (n): m ≤ n → ↑m ≤ ↑n.
 qed.
 
 (*** le_or_ge *)
-lemma nle_ge_dis (m) (n): ∨∨ m ≤ n | n ≤ m.
+lemma nat_split_le_ge (m) (n): ∨∨ m ≤ n | n ≤ m.
 #m #n @(nat_ind_2_succ … m n) -m -n
 [ /2 width=1 by or_introl/
 | /2 width=1 by or_intror/
@@ -63,7 +62,7 @@ qed-.
 
 (*** le_S_S_to_le *)
 lemma nle_inv_succ_bi (m) (n): ↑m ≤ ↑n → m ≤ n.
-#m #n @(insert_eq_0 … (↑n))
+#m #n @(insert_eq_1 … (↑n))
 #x * -x
 [ #H >(eq_inv_nsucc_bi … H) -n //
 | #o #Ho #H >(eq_inv_nsucc_bi … H) -n
@@ -73,7 +72,7 @@ qed-.
 
 (*** le_n_O_to_eq *)
 lemma nle_inv_zero_dx (m): m ≤ 𝟎 → 𝟎 = m.
-#m @(insert_eq_0 … (𝟎))
+#m @(insert_eq_1 … (𝟎))
 #y * -y
 [ #H destruct //
 | #y #_ #H elim (eq_inv_zero_nsucc … H)
@@ -88,7 +87,7 @@ lemma nle_inv_succ_zero (m): ↑m ≤ 𝟎 → ⊥.
 
 lemma nle_inv_succ_sn_refl (m): ↑m ≤ m → ⊥.
 #m @(nat_ind_succ … m) -m [| #m #IH ] #H
-[ /3 width=2 by nle_inv_zero_dx, eq_inv_zero_nsucc/
+[ /2 width=2 by nle_inv_succ_zero/
 | /3 width=1 by nle_inv_succ_bi/
 ]
 qed-.
@@ -110,7 +109,7 @@ lemma nle_ind_alt (Q: relation2 nat nat):
       (∀m,n. m ≤ n → Q m n → Q (↑m) (↑n)) →
       ∀m,n. m ≤ n → Q m n.
 #Q #IH1 #IH2 #m #n @(nat_ind_2_succ … m n) -m -n //
-[ #m #H elim (nle_inv_succ_zero … H)
+[ #m #_ #H elim (nle_inv_succ_zero … H)
 | /4 width=1 by nle_inv_succ_bi/
 ]
 qed-.
@@ -124,7 +123,7 @@ qed-.
 
 (*** decidable_le le_dec *)
 lemma nle_dec (m) (n): Decidable … (m ≤ n).
-#m #n elim (nle_ge_dis m n) [ /2 width=1 by or_introl/ ]
+#m #n elim (nat_split_le_ge m n) [ /2 width=1 by or_introl/ ]
 #Hnm elim (eq_nat_dec m n) [ #H destruct /2 width=1 by nle_refl, or_introl/ ]
 /4 width=1 by nle_antisym, or_intror/
 qed-.