rtmaps with finite colength

[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 325e07d387be02cc1c321016b426b5da69d03e63..d70ea1bfa74cc134edde18bd68c9ffdcc6a9dead 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
@@ -23,6 +23,10 @@ interpretation "nat successor" 'Successor m = (S m).
 
 interpretation "nat predecessor" 'Predecessor m = (pred m).
 
+interpretation "nat min" 'and x y = (min x y).
+
+interpretation "nat max" 'or x y = (max x y).
+
 (* Iota equations ***********************************************************)
 
 lemma pred_O: pred 0 = 0.
@@ -31,6 +35,17 @@ normalize // qed.
 lemma pred_S: ∀m. pred (S m) = m.
 // qed.
 
+lemma max_O1: ∀n. n = (0 ∨ n).
+// qed.
+
+lemma max_O2: ∀n. n = (n ∨ 0).
+// qed.
+
+lemma max_SS: ∀n1,n2. ⫯(n1∨n2) = (⫯n1 ∨ ⫯n2).
+#n1 #n2 elim (decidable_le n1 n2) #H normalize
+[ >(le_to_leb_true … H) | >(not_le_to_leb_false … H) ] -H //
+qed.
+
 (* Equations ****************************************************************)
 
 lemma minus_plus_m_m_commutative: ∀n,m:nat. n = m + n - m.
@@ -112,6 +127,12 @@ 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 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-.
+
+lemma max_S2_le_S: ∀n1,n2,n. (n1 ∨ n2) ≤ n → (n1 ∨ ⫯n2) ≤ ⫯n.
+/2 width=1 by max_S1_le_S/ qed-.
+
 lemma arith_j: ∀x,y,z. x-y-1 ≤ x-(y-z)-1.
 /3 width=1 by monotonic_le_minus_l, monotonic_le_minus_r/ qed.
 
@@ -220,7 +241,7 @@ qed-.
 (* Iterators ****************************************************************)
 
 (* Note: see also: lib/arithemetics/bigops.ma *)
-let rec iter (n:nat) (B:Type[0]) (op: B → B) (nil: B) ≝
+rec definition iter (n:nat) (B:Type[0]) (op: B → B) (nil: B) ≝
   match n with
    [ O   ⇒ nil
    | S k ⇒ op (iter k B op nil)
@@ -249,7 +270,7 @@ qed.
 (* Trichotomy operator ******************************************************)
 
 (* Note: this is "if eqb n1 n2 then a2 else if leb n1 n2 then a1 else a3" *)
-let rec tri (A:Type[0]) n1 n2 a1 a2 a3 on n1 : A ≝
+rec definition tri (A:Type[0]) n1 n2 a1 a2 a3 on n1 : A ≝
   match n1 with
   [ O    ⇒ match n2 with [ O ⇒ a2 | S n2 ⇒ a1 ]
   | S n1 ⇒ match n2 with [ O ⇒ a3 | S n2 ⇒ tri A n1 n2 a1 a2 a3 ]