X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Farithmetics%2Fnat.ma;h=249bd274e74622e59d69ab7d77782a86c713bc4d;hb=80178d6cf86b78bb9fc47f397f4bcfb1fd15a24f;hp=67b7de50b7dfb5711e9679b502720862e49c6d03;hpb=6b87a3e9d6dd7c3abb922750587444ac3fd08e16;p=helm.git

diff --git a/matita/matita/lib/arithmetics/nat.ma b/matita/matita/lib/arithmetics/nat.ma
index 67b7de50b..249bd274e 100644
--- a/matita/matita/lib/arithmetics/nat.ma
+++ b/matita/matita/lib/arithmetics/nat.ma
@@ -164,7 +164,7 @@ lemma times_times: ∀x,y,z. x*(y*z) = y*(x*z).
 // qed. 
 
 theorem times_n_1 : ∀n:nat. n = n * 1.
-#n // qed.
+// qed.
 
 theorem minus_S_S: ∀n,m:nat.S n - S m = n -m.
 // qed.
@@ -187,6 +187,13 @@ theorem eq_minus_S_pred: ∀n,m. n - (S m) = pred(n -m).
 lemma plus_plus_comm_23: ∀x,y,z. x + y + z = x + z + y.
 // qed.
 
+lemma discr_plus_xy_minus_xz: ∀x,z,y. x + y = x - z → y = 0.
+#x elim x -x // #x #IHx * normalize
+[ #y #H @(IHx 0) <minus_n_O /2 width=1/
+| #z #y >plus_n_Sm #H lapply (IHx … H) -x -z #H destruct
+]
+qed-.
+
 (* Negated equalities *******************************************************)
 
 theorem not_eq_S: ∀n,m:nat. n ≠ m → S n ≠ S m.
@@ -439,6 +446,10 @@ 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)
@@ -495,13 +506,31 @@ 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
-cut (∀n,a. f a = n → P a) /2 width=3/ -a
-#n @(nat_elim1 … n) -n #n /3 width=3/ (**) (* auto very slow (274s) without #n *)
+@(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.
@@ -664,6 +693,10 @@ lapply (minus_le x y) <H -H #H
 elim (not_le_Sn_n x) #H0 elim (H0 ?) //
 qed-.
 
+lemma plus_le_0: ∀x,y. x + y ≤ 0 → x = 0 ∧ y = 0.
+#x #y #H elim (le_inv_plus_l … H) -H #H1 #H2 /3 width=1/
+qed-.
+
 (* Still more equalities ****************************************************)
 
 theorem eq_minus_O: ∀n,m:nat.