- revision of ground_2 and basic_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 3eec591c692269b16eae8bff92bb6efbfb7bc4cf..b23fb9de8280f4fe9efd714c1c4462a07956c4dc 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
@@ -17,8 +17,27 @@ include "ground_2/lib/star.ma".
 
 (* ARITHMETICAL PROPERTIES **************************************************)
 
+(* 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.
+// qed-.
+
+(* Note: uses minus_minus_comm, minus_plus_m_m, commutative_plus, plus_minus *)
+lemma plus_minus_minus_be: ∀x,y,z. y ≤ z → z ≤ x → (x - z) + (z - y) = x - y.
+#x #z #y #Hzy #Hyx >plus_minus // >commutative_plus >plus_minus //
+qed-.
+
+fact plus_minus_minus_be_aux: ∀i,x,y,z. y ≤ z → z ≤ x → i = z - y → x - z + i = x - y.
+/2 width=1 by plus_minus_minus_be/ qed-.
+
 lemma plus_n_2: ∀n. n + 2 = n + 1 + 1.
 // qed.
 
@@ -52,6 +71,20 @@ lemma arith_i: ∀x,y,z. y < x → x+z-y-1 = x-y-1+z.
 
 (* Properties ***************************************************************)
 
+lemma eq_nat_dec: ∀n1,n2:nat. Decidable (n1 = n2).
+#n1 elim n1 -n1 [| #n1 #IHn1 ] * [2,4: #n2 ]
+[1,4: @or_intror #H destruct
+| elim (IHn1 n2) -IHn1 /3 width=1 by or_intror, or_introl/
+| /2 width=1 by or_introl/
+]
+qed-.
+
+lemma lt_or_eq_or_gt: ∀m,n. ∨∨ m < n | n = m | n < m.
+#m #n elim (lt_or_ge m n) /2 width=1 by or3_intro0/
+#H elim H -m /2 width=1 by or3_intro1/
+#m #Hm * /3 width=1 by not_le_to_lt, le_S_S, or3_intro2/
+qed-.
+
 fact le_repl_sn_conf_aux: ∀x,y,z:nat. x ≤ z → x = y → y ≤ z.
 // qed-.
 
@@ -86,15 +119,11 @@ qed.
 
 (* Inversion & forward lemmas ***********************************************)
 
-axiom eq_nat_dec: ∀n1,n2:nat. Decidable (n1 = n2).
+lemma discr_plus_xy_y: ∀x,y. x + y = y → x = 0.
+// qed-.
 
-axiom lt_dec: ∀n1,n2. Decidable (n1 < n2).
-
-lemma lt_or_eq_or_gt: ∀m,n. ∨∨ m < n | n = m | n < m.
-#m #n elim (lt_or_ge m n) /2 width=1 by or3_intro0/
-#H elim H -m /2 width=1 by or3_intro1/
-#m #Hm * /3 width=1 by not_le_to_lt, le_S_S, or3_intro2/
-qed-.
+lemma lt_plus_SO_to_le: ∀x,y. x < y + 1 → x ≤ y.
+/2 width=1 by monotonic_pred/ qed-.
 
 lemma lt_refl_false: ∀n. n < n → ⊥.
 #n #H elim (lt_to_not_eq … H) -H /2 width=1 by/
@@ -104,11 +133,6 @@ lemma lt_zero_false: ∀n. n < 0 → ⊥.
 #n #H elim (lt_to_not_le … H) -H /2 width=1 by/
 qed-.
 
-lemma false_lt_to_le: ∀x,y. (x < y → ⊥) → y ≤ x.
-#x #y #H elim (decidable_lt x y) /2 width=1 by not_lt_to_le/
-#Hxy elim (H Hxy)
-qed-.
-
 lemma pred_inv_refl: ∀m. pred m = m → m = 0.
 * // normalize #m #H elim (lt_refl_false m) //
 qed-.
@@ -128,9 +152,22 @@ lemma plus_xySz_x_false: ∀z,x,y. x + y + S z = x → ⊥.
 lemma plus_xSy_x_false: ∀y,x. x + S y = x → ⊥.
 /2 width=4 by plus_xySz_x_false/ qed-.
 
+(* Note this should go in nat.ma *)
+lemma discr_x_minus_xy: ∀x,y. x = x - y → x = 0 ∨ y = 0.
+#x @(nat_ind_plus … x) -x /2 width=1 by or_introl/
+#x #_ #y @(nat_ind_plus … y) -y /2 width=1 by or_intror/
+#y #_ >minus_plus_plus_l
+#H lapply (discr_plus_xy_minus_xz … H) -H
+#H destruct
+qed-.
+
+lemma zero_eq_plus: ∀x,y. 0 = x + y → 0 = x ∧ 0 = y.
+* /2 width=1 by conj/ #x #y normalize #H destruct
+qed-.
+
 (* Iterators ****************************************************************)
 
-(* Note: see also: lib/arithemetcs/bigops.ma *)
+(* Note: see also: lib/arithemetics/bigops.ma *)
 let rec iter (n:nat) (B:Type[0]) (op: B → B) (nil: B) ≝
   match n with
    [ O   ⇒ nil
@@ -139,6 +176,12 @@ let rec iter (n:nat) (B:Type[0]) (op: B → B) (nil: B) ≝
 
 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.