- new syntax for let rec/corec with flavor specifier (tested on lambdadelta/ground_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 b23fb9de8280f4fe9efd714c1c4462a07956c4dc..1ba525475622dfe3fb6777c4cbe59847a4584a1d 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
@@ -12,11 +12,17 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground_2/notation/functions/successor_1.ma".
+include "ground_2/notation/functions/predecessor_1.ma".
 include "arithmetics/nat.ma".
 include "ground_2/lib/star.ma".
 
 (* ARITHMETICAL PROPERTIES **************************************************)
 
+interpretation "nat successor" 'Successor m = (S m).
+
+interpretation "nat predecessor" 'Predecessor m = (pred m).
+
 (* Iota equations ***********************************************************)
 
 lemma pred_O: pred 0 = 0.
@@ -100,6 +106,12 @@ lemma monotonic_lt_pred: ∀m,n. m < n → O < m → pred m < pred n.
 @le_S_S_to_le >S_pred /2 width=3 by transitive_lt/
 qed.
 
+lemma lt_S_S: ∀x,y. x < y → ⫯x < ⫯y.
+/2 width=1 by le_S_S/ qed.
+
+lemma lt_S: ∀n,m. n < m → n < ⫯m.
+/2 width=1 by 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.
 
@@ -133,6 +145,24 @@ lemma lt_zero_false: ∀n. n < 0 → ⊥.
 #n #H elim (lt_to_not_le … H) -H /2 width=1 by/
 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 lt_inv_O1: ∀n. 0 < n → ∃m. ⫯m = n.
+* /2 width=2 by ex_intro/
+#H cases (lt_le_false … H) -H //
+qed-.
+
+lemma lt_inv_S1: ∀m,n. ⫯m < n → ∃∃p. m < p & ⫯p = n.
+#m * /3 width=3 by lt_S_S_to_lt, ex2_intro/
+#H cases (lt_le_false … H) -H //
+qed-.
+
+lemma lt_inv_gen: ∀y,x. x < y → ∃∃z. x ≤ z & ⫯z = y.
+* /3 width=3 by le_S_S_to_le, ex2_intro/
+#x #H elim (lt_le_false … H) -H //
+qed-.
+
 lemma pred_inv_refl: ∀m. pred m = m → m = 0.
 * // normalize #m #H elim (lt_refl_false m) //
 qed-.
@@ -165,10 +195,32 @@ lemma zero_eq_plus: ∀x,y. 0 = x + y → 0 = x ∧ 0 = y.
 * /2 width=1 by conj/ #x #y normalize #H destruct
 qed-.
 
+lemma lt_S_S_to_lt: ∀x,y. ⫯x < ⫯y → x < y.
+/2 width=1 by le_S_S_to_le/ qed-.
+
+lemma lt_elim: ∀R:relation nat.
+               (∀n2. R O (⫯n2)) →
+               (∀n1,n2. R n1 n2 → R (⫯n1) (⫯n2)) →
+               ∀n2,n1. n1 < n2 → R n1 n2.
+#R #IH1 #IH2 #n2 elim n2 -n2
+[ #n1 #H elim (lt_le_false … H) -H //
+| #n2 #IH * /4 width=1 by lt_S_S_to_lt/
+]
+qed-.
+
+lemma le_elim: ∀R:relation nat.
+               (∀n2. R O (n2)) →
+               (∀n1,n2. R n1 n2 → R (⫯n1) (⫯n2)) →
+               ∀n1,n2. n1 ≤ n2 → R n1 n2.
+#R #IH1 #IH2 #n1 #n2 @(nat_elim2 … n1 n2) -n1 -n2
+/4 width=1 by monotonic_pred/ -IH1 -IH2
+#n1 #H elim (lt_le_false … H) -H //
+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)
@@ -197,7 +249,7 @@ 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 ]