update in ground_2, static_2, basic_2

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / lib / arith.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma b/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
index 6ce878b44bf0d73d1ba13dfd8e23a19cf008441e..79bcc2548bca92b905b524c6adfda5ae919040cc 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
@@ -12,9 +12,12 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "arithmetics/nat.ma".
+include "ground_2/xoa/ex_3_1.ma".
+include "ground_2/xoa/or_3.ma".
 include "ground_2/notation/functions/uparrow_1.ma".
 include "ground_2/notation/functions/downarrow_1.ma".
-include "arithmetics/nat.ma".
+include "ground_2/pull/pull_2.ma".
 include "ground_2/lib/relations.ma".
 
 (* ARITHMETICAL PROPERTIES **************************************************)
@@ -51,7 +54,10 @@ qed.
 
 (* Equalities ***************************************************************)
 
-lemma plus_SO: ∀n. n + 1 = ↑n.
+lemma plus_SO_sn (n): 1 + n = ↑n.
+// qed-.
+
+lemma plus_SO_dx (n): n + 1 = ↑n.
 // qed.
 
 lemma minus_plus_m_m_commutative: ∀n,m:nat. n = m + n - m.
@@ -150,6 +156,14 @@ lemma lt_S_S: ∀x,y. x < y → ↑x < ↑y.
 lemma lt_S: ∀n,m. n < m → n < ↑m.
 /2 width=1 by le_S/ qed.
 
+lemma monotonic_lt_minus_r:
+∀p,q,n. q < n -> q < p → n-p < n-q.
+#p #q #n #Hn #H
+lapply (monotonic_le_minus_r … n H) -H #H
+@(le_to_lt_to_lt … H) -H
+/2 width=1 by lt_plus_to_minus/
+qed.
+
 lemma max_S1_le_S: ∀n1,n2,n. (n1 ∨ n2) ≤ n → (↑n1 ∨ n2) ≤ ↑n.
 /4 width=2 by to_max, le_maxr, le_S_S, le_S/ qed-.
 
@@ -291,6 +305,21 @@ lemma le_elim: ∀R:relation nat.
 #n1 #H elim (lt_le_false … H) -H //
 qed-.
 
+lemma nat_elim_le_sn (Q:relation …):
+      (∀m1,m2. (∀m. m < m2-m1 → Q (m2-m) m2) → m1 ≤ m2 → Q m1 m2) →
+      ∀n1,n2. n1 ≤ n2 → Q n1 n2.
+#Q #IH #n1 #n2 #Hn
+<(minus_minus_m_m … Hn) -Hn
+lapply (minus_le n2 n1)
+let d ≝ (n2-n1)
+@(nat_elim1 … d) -d -n1 #d
+@pull_2 #Hd
+<(minus_minus_m_m … Hd) in ⊢ (%→?); -Hd
+let n1 ≝ (n2-d) #IHd
+@IH -IH [| // ] #m #Hn
+/4 width=3 by lt_to_le, lt_to_le_to_lt/
+qed-.
+
 (* Iterators ****************************************************************)
 
 (* Note: see also: lib/arithemetics/bigops.ma *)