(* *)
(**************************************************************************)
+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.
+normalize // qed.
+
+lemma pred_S: ∀m. pred (S m) = m.
+// qed.
+
(* Equations ****************************************************************)
lemma minus_plus_m_m_commutative: ∀n,m:nat. n = m + n - m.
@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 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.
#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 pred_inv_refl: ∀m. pred m = m → m = 0.
* // normalize #m #H elim (lt_refl_false m) //
qed-.
* /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 *)
interpretation "iterated function" 'exp op n = (iter n ? op).
+lemma iter_O: ∀B:Type[0]. ∀f:B→B.∀b. f^0 b = b.
+// qed.
+
+lemma iter_S: ∀B:Type[0]. ∀f:B→B.∀b,l. f^(S l) b = f (f^l b).
+// qed.
+
lemma iter_SO: ∀B:Type[0]. ∀f:B→B. ∀b,l. f^(l+1) b = f (f^l b).
#B #f #b #l >commutative_plus //
qed.