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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "ground_2/ynat/ynat_lt.ma".
17 (* NATURAL NUMBERS WITH INFINITY ********************************************)
20 definition yminus: ynat → ynat → ynat ≝ λx,y. match y with
25 interpretation "ynat minus" 'minus x y = (yminus x y).
27 (* Basic properties *********************************************************)
29 lemma yminus_inj: ∀n,m. yinj m - yinj n = yinj (m - n).
30 #n elim n -n /2 width=3 by trans_eq/
33 lemma yminus_Y_inj: ∀n. ∞ - yinj n = ∞.
34 #n elim n -n // normalize
38 lemma yminus_O1: ∀x:ynat. 0 - x = 0.
41 lemma yminus_refl: ∀x:ynat. x - x = 0.
44 lemma yminus_minus_comm: ∀y,z,x. x - y - z = x - z - y.
45 * #y [ * #z [ * // ] ] >yminus_O1 //
48 (* Properties on predecessor ************************************************)
50 lemma yminus_SO2: ∀m. m - 1 = ⫰m.
54 lemma yminus_pred: ∀n,m. 0 < m → 0 < n → ⫰m - ⫰n = m - n.
56 [ #m #Hm #Hn >yminus_inj >yminus_inj
57 /4 width=1 by ylt_inv_inj, minus_pred_pred, eq_f/
62 (* Properties on successor **************************************************)
64 lemma yminus_succ: ∀n,m. ⫯m - ⫯n = m - n.
65 * // #n * [2: >yminus_Y_inj // ]
69 lemma yminus_succ1_inj: ∀n:nat. ∀m:ynat. n ≤ m → ⫯m - n = ⫯(m - n).
71 [ #m #Hmn >yminus_inj >yminus_inj
72 /4 width=1 by yle_inv_inj, plus_minus, eq_f/
77 lemma yminus_succ2: ∀y,x. x - ⫯y = ⫰(x-y).
81 (* Properties on order ******************************************************)
83 lemma yle_minus_sn: ∀n,m. m - n ≤ m.
84 * // #n * /2 width=1 by yle_inj/
87 lemma yle_to_minus: ∀m:ynat. ∀n:ynat. m ≤ n → m - n = 0.
88 #m #n * -m -n /3 width=3 by eq_minus_O, eq_f/
91 lemma yminus_to_le: ∀n:ynat. ∀m:ynat. m - n = 0 → m ≤ n.
93 [ #m >yminus_inj #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
94 /2 width=1 by yle_inj/
95 | >yminus_Y_inj #H destruct
99 lemma monotonic_yle_minus_dx: ∀x,y. x ≤ y → ∀z. x - z ≤ y - z.
101 #z elim z -z /3 width=1 by yle_pred/
104 (* Properties on strict order ***********************************************)
106 lemma monotonic_ylt_minus_dx: ∀x,y:ynat. x < y → ∀z:nat. z ≤ x → x - z < y - z.
108 /4 width=1 by ylt_inj, yle_inv_inj, monotonic_lt_minus_l/