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 a29899d451a6b8b10152012ced8c0b7ed252839b..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 **************************************************)
@@ -51,12 +54,18 @@ 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.
 // qed-.
 
+lemma plus_minus_m_m_commutative (n) (m): m ≤ n → n = m+(n-m).
+/2 width=1 by plus_minus_associative/ qed-.
+
 lemma plus_to_minus_2: ∀m1,m2,n1,n2. n1 ≤ m1 → n2 ≤ m2 →
                        m1+n2 = m2+n1 → m1-n1 = m2-n2.
 #m1 #m2 #n1 #n2 #H1 #H2 #H
@@ -127,6 +136,14 @@ qed-.
 lemma le_plus_to_minus_comm: ∀n,m,p. n ≤ p+m → n-p ≤ m.
 /2 width=1 by le_plus_to_minus/ qed-.
 
+lemma le_inv_S1: ∀m,n. ↑m ≤ n → ∃∃p. m ≤ p & ↑p = n.
+#m *
+[ #H lapply (le_n_O_to_eq … H) -H
+  #H destruct
+| /3 width=3 by monotonic_pred, ex2_intro/
+]
+qed-.
+
 (* Note: this might interfere with nat.ma *)
 lemma monotonic_lt_pred: ∀m,n. m < n → 0 < m → pred m < pred n.
 #m #n #Hmn #Hm whd >(S_pred … Hm)
@@ -139,6 +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-.
 
@@ -158,6 +183,15 @@ qed-.
 lemma lt_le_false: ∀x,y. x < y → y ≤ x → ⊥.
 /3 width=4 by lt_refl_false, lt_to_le_to_lt/ qed-.
 
+lemma le_dec (n) (m): Decidable (n≤m).
+#n elim n -n [ /2 width=1 by or_introl/ ]
+#n #IH * [ /3 width=2 by lt_zero_false, or_intror/ ]
+#m elim (IH m) -IH
+[ /3 width=1 by or_introl, le_S_S/
+| /4 width=1 by or_intror, le_S_S_to_le/
+]
+qed-.
+
 lemma succ_inv_refl_sn: ∀x. ↑x = x → ⊥.
 #x #H @(lt_le_false x (↑x)) //
 qed-.
@@ -271,6 +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 *)
@@ -322,3 +371,34 @@ lemma tri_gt: ∀A,a1,a2,a3,n1,n2. n2 < n1 → tri A n1 n2 a1 a2 a3 = a3.
 | #n1 #IH #n2 elim n2 -n2 /3 width=1 by monotonic_lt_pred/
 ]
 qed.
+
+(* Decidability of predicates ***********************************************)
+
+lemma dec_lt (R:predicate nat):
+      (∀n. Decidable … (R n)) →
+      ∀n. Decidable … (∃∃m. m < n & R m).
+#R #HR #n elim n -n [| #n * ]
+[ @or_intror * /2 width=2 by lt_zero_false/
+| * /4 width=3 by lt_S, or_introl, ex2_intro/
+| #H0 elim (HR n) -HR
+  [ /3 width=3 by or_introl, ex2_intro/
+  | #Hn @or_intror * #m #Hmn #Hm
+    elim (le_to_or_lt_eq … Hmn) -Hmn #H destruct [ -Hn | -H0 ]
+    /4 width=3 by lt_S_S_to_lt, ex2_intro/
+  ]
+]
+qed-.
+
+lemma dec_min (R:predicate nat):
+      (∀n. Decidable … (R n)) → ∀n. R n →
+      ∃∃m. m ≤ n & R m & (∀p. p < m → R p → ⊥).
+#R #HR #n
+@(nat_elim1 n) -n #n #IH #Hn
+elim (dec_lt … HR n) -HR [ -Hn | -IH ]
+[ * #p #Hpn #Hp
+  elim (IH … Hpn Hp) -IH -Hp #m #Hmp #Hm #HNm
+  @(ex3_intro … Hm HNm) -HNm
+  /3 width=3 by lt_to_le, le_to_lt_to_lt/
+| /4 width=4 by ex3_intro, ex2_intro/
+]
+qed-.