xoa: change in naming convenctions for existential quantifies

[helm.git] / matita / matita / lib / arithmetics / nat.ma

diff --git a/matita/matita/lib/arithmetics/nat.ma b/matita/matita/lib/arithmetics/nat.ma
index edb99b8c0c9dac85517e253e9282d7f34597bd89..c707c3207c985267d76ff35ed37cde111a104bb4 100644 (file)
--- a/matita/matita/lib/arithmetics/nat.ma
+++ b/matita/matita/lib/arithmetics/nat.ma
@@ -305,6 +305,10 @@ lemma lt_to_le: ∀x,y. x < y → x ≤ y.
 lemma inv_eq_minus_O: ∀x,y. x - y = 0 → x ≤ y.
 // qed-.
 
+lemma le_x_times_x: ∀x. x ≤ x * x.
+#x elim x -x //
+qed.
+
 (* lt *)
 
 theorem transitive_lt: transitive nat lt.
@@ -435,6 +439,10 @@ theorem decidable_lt: ∀n,m. decidable (n < m).
 theorem le_to_or_lt_eq: ∀n,m:nat. n ≤ m → n < m ∨ n = m.
 #n #m #lenm (elim lenm) /3/ qed.
 
+theorem eq_or_gt: ∀n. 0 = n ∨ 0 < n.
+#n elim (le_to_or_lt_eq 0 n ?) // /2 width=1/
+qed-.
+
 theorem increasing_to_le2: ∀f:nat → nat. increasing f → 
   ∀m:nat. f 0 ≤ m → ∃i. f i ≤ m ∧ m < f (S i).
 #f #incr #m #lem (elim lem)
@@ -491,6 +499,31 @@ cut (∀q:nat. q ≤ n → P q) /2/
  ]
 qed.
 
+fact f_ind_aux: ∀A. ∀f:A→ℕ. ∀P:predicate A.
+                (∀n. (∀a. f a < n → P a) → ∀a. f a = n → P a) →
+                ∀n,a. f a = n → P a.
+#A #f #P #H #n @(nat_elim1 … n) -n #n /3 width=3/ (**) (* auto slow (34s) without #n *)
+qed-.
+
+lemma f_ind: ∀A. ∀f:A→ℕ. ∀P:predicate A.
+             (∀n. (∀a. f a < n → P a) → ∀a. f a = n → P a) → ∀a. P a.
+#A #f #P #H #a
+@(f_ind_aux … H) -H [2: // | skip ]
+qed-.
+
+fact f2_ind_aux: ∀A1,A2. ∀f:A1→A2→ℕ. ∀P:relation2 A1 A2.
+                 (∀n. (∀a1,a2. f a1 a2 < n → P a1 a2) → ∀a1,a2. f a1 a2 = n → P a1 a2) →
+                 ∀n,a1,a2. f a1 a2 = n → P a1 a2.
+#A1 #A2 #f #P #H #n @(nat_elim1 … n) -n #n /3 width=3/ (**) (* auto slow (34s) without #n *)
+qed-.
+
+lemma f2_ind: ∀A1,A2. ∀f:A1→A2→ℕ. ∀P:relation2 A1 A2.
+              (∀n. (∀a1,a2. f a1 a2 < n → P a1 a2) → ∀a1,a2. f a1 a2 = n → P a1 a2) →
+              ∀a1,a2. P a1 a2.
+#A1 #A2 #f #P #H #a1 #a2
+@(f2_ind_aux … H) -H [2: // | skip ]
+qed-. 
+
 (* More negated equalities **************************************************)
 
 theorem lt_to_not_eq : ∀n,m:nat. n < m → n ≠ m.