- improved fqu allows to prove fqu_cpx_trans and its derivatives

[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 8ce8b954c1bd353ce1d1e6f7ddf8e2411a863337..2c6ad2a5623d947abd287fa3e1ceccddb880087b 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
@@ -51,6 +51,9 @@ qed.
 
 (* Equations ****************************************************************)
 
+lemma plus_SO: ∀n. n + 1 = ⫯n.
+// qed.
+
 lemma minus_plus_m_m_commutative: ∀n,m:nat. n = m + n - m.
 // qed-.
 
@@ -93,6 +96,20 @@ qed.
 lemma arith_i: ∀x,y,z. y < x → x+z-y-1 = x-y-1+z.
 /2 width=1 by plus_minus/ qed-.
 
+lemma idempotent_max: ∀n:nat. n = (n ∨ n).
+#n normalize >le_to_leb_true //
+qed.
+
+lemma associative_max: associative … max.
+#x #y #z normalize
+@(leb_elim x y) normalize #Hxy
+@(leb_elim y z) normalize #Hyz //
+[1,2: >le_to_leb_true /2 width=3 by transitive_le/
+| >not_le_to_leb_false /4 width=3 by lt_to_not_le, not_le_to_lt, transitive_lt/
+  >not_le_to_leb_false //
+]
+qed.
+
 (* Properties ***************************************************************)
 
 lemma eq_nat_dec: ∀n1,n2:nat. Decidable (n1 = n2).
@@ -155,6 +172,17 @@ qed.
 
 (* Inversion & forward lemmas ***********************************************)
 
+lemma nat_split: ∀x. x = 0 ∨ ∃y. ⫯y = x.
+* /3 width=2 by ex_intro, or_introl, or_intror/
+qed-.
+
+lemma max_inv_O3: ∀x,y. (x ∨ y) = 0 → 0 = x ∧ 0 = y.
+/4 width=2 by le_maxr, le_maxl, le_n_O_to_eq, conj/
+qed-.
+
+lemma plus_inv_O3: ∀x,y. x + y = 0 → x = 0 ∧ y = 0.
+/2 width=1 by plus_le_0/ qed-.
+
 lemma discr_plus_xy_y: ∀x,y. x + y = y → x = 0.
 // qed-.