+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/ynat/ynat_plus.ma".
-
-(* NATURAL NUMBERS WITH INFINITY ********************************************)
-
-(* right subtraction *)
-definition yminus_dx: nat → ynat → nat ≝ λx,y. match y with
-[ yinj n ⇒ x-n
-| Y ⇒ 0
-].
-
-interpretation "ynat right minus" 'minus x y = (yminus_dx x y).
-
-lemma yminus_dx_inj (m) (n): m - yinj n = m - n.
-// qed.
-
-lemma yminus_dx_Y2: ∀m. m - (∞) = 0.
-// qed.
-
-lemma yminus_dx_succ_bi (n:ynat) (m): ↑m - ↑n = m - n.
-* // qed.
-
-lemma yminus_dx_S2: ∀n:ynat. ∀m. m - ↑n = ↓(m - n).
-* // qed.
-
-lemma yplus_minus_dx (x) (y): x ≤ yinj y → yinj y = x + yinj (y-x).
-* [ #x #y #H >yplus_inj /4 width=1 by yle_inv_inj, minus_to_plus, eq_f/ ]
-#y #H lapply (yle_inv_Y1 … H) -H #H destruct
-qed-.
-
-(*
-
-
-(* Basic properties *********************************************************)
-
-lemma yminus_inj: ∀m,n. yinj m - yinj n = yinj (m - n).
-#m #n elim n -n //
-#n #IH >yminus_S2 >IH -IH >eq_minus_S_pred //
-qed.
-
-lemma yminus_Y_inj: ∀n. ∞ - yinj n = ∞.
-#n elim n -n //
-qed.
-
-lemma yminus_O1: ∀x:ynat. 0 - x = 0.
-* // qed.
-
-lemma yminus_refl: ∀x:ynat. x - x = 0.
-* // qed.
-
-lemma yminus_minus_comm: ∀y,z,x. x - y - z = x - z - y.
-* #y [ * #z [ * // ] ] >yminus_O1 //
-qed.
-
-(* Properties on predecessor ************************************************)
-
-lemma yminus_SO2: ∀m. m - 1 = ⫰m.
-* //
-qed.
-
-lemma yminus_pred1: ∀x,y. ⫰x - y = ⫰(x-y).
-#x * // #y elim y -y //
-qed.
-
-lemma yminus_pred: ∀n,m. 0 < m → 0 < n → ⫰m - ⫰n = m - n.
-* // #n *
-[ #m #Hm #Hn >yminus_inj >yminus_inj
- /4 width=1 by ylt_inv_inj, minus_pred_pred, eq_f/
-| >yminus_Y_inj //
-]
-qed-.
-
-(* Properties on successor **************************************************)
-
-lemma yminus_succ1_inj: ∀n:nat. ∀m:ynat. n ≤ m → ⫯m - n = ⫯(m - n).
-#n *
-[ #m #Hmn >yminus_inj >yminus_inj
- /4 width=1 by yle_inv_inj, plus_minus, eq_f/
-| >yminus_Y_inj //
-]
-qed-.
-
-lemma yminus_succ2: ∀y,x. x - ⫯y = ⫰(x-y).
-* //
-qed.
-
-(* Properties on order ******************************************************)
-
-lemma yle_minus_sn: ∀n,m. m - n ≤ m.
-* // #n * /2 width=1 by yle_inj/
-qed.
-
-lemma yle_to_minus: ∀m:ynat. ∀n:ynat. m ≤ n → m - n = 0.
-#m #n * -m -n /3 width=3 by eq_minus_O, eq_f/
-qed-.
-
-lemma yminus_to_le: ∀n:ynat. ∀m:ynat. m - n = 0 → m ≤ n.
-* // #n *
-[ #m >yminus_inj #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
- /2 width=1 by yle_inj/
-| >yminus_Y_inj #H destruct
-]
-qed.
-
-lemma monotonic_yle_minus_dx: ∀x,y. x ≤ y → ∀z. x - z ≤ y - z.
-#x #y #Hxy * //
-#z elim z -z /3 width=1 by yle_pred/
-qed.
-
-(* Properties on strict order ***********************************************)
-
-lemma ylt_to_minus: ∀x,y:ynat. x < y → 0 < y - x.
-#x #y #H elim H -x -y /3 width=1 by ylt_inj, lt_plus_to_minus_r/
-qed.
-
-lemma yminus_to_lt: ∀x,y:ynat. 0 < y - x → x < y.
-* [2: #y #H elim (ylt_yle_false … H) // ]
-#m * /4 width=1 by ylt_inv_inj, ylt_inj, lt_minus_to_plus_r/
-qed-.
-
-lemma monotonic_ylt_minus_dx: ∀x,y:ynat. x < y → ∀z:nat. z ≤ x → x - z < y - z.
-#x #y * -x -y
-/4 width=1 by ylt_inj, yle_inv_inj, monotonic_lt_minus_l/
-qed.
-
-(* Properties on minus ******************************************************)
-
-lemma yplus_minus_inj: ∀m:ynat. ∀n:nat. m + n - n = m.
-#m #n elim n -n //
-#n #IHn >(yplus_succ2 m n) >(yminus_succ … n) //
-qed.
-
-lemma yplus_minus: ∀m,n. m + n - n ≤ m.
-#m * //
-qed.
-
-lemma yminus_plus2: ∀z,y,x:ynat. x - (y + z) = x - y - z.
-* // #z * [2: >yplus_Y1 >yminus_O1 // ] #y *
-[ #x >yplus_inj >yminus_inj >yminus_inj >yminus_inj /2 width=1 by eq_f/
-| >yplus_inj >yminus_Y_inj //
-]
-qed.
-
-(* Forward lemmas on minus **************************************************)
-
-lemma yle_plus1_to_minus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y.
-#x #z #y #H lapply (monotonic_yle_minus_dx … H y) -H //
-qed-.
-
-lemma yle_plus1_to_minus_inj1: ∀x,z:ynat. ∀y:nat. y + x ≤ z → x ≤ z - y.
-/2 width=1 by yle_plus1_to_minus_inj2/ qed-.
-
-lemma yle_plus2_to_minus_inj2: ∀x,y:ynat. ∀z:nat. x ≤ y + z → x - z ≤ y.
-/2 width=1 by monotonic_yle_minus_dx/ qed-.
-
-lemma yle_plus2_to_minus_inj1: ∀x,y:ynat. ∀z:nat. x ≤ z + y → x - z ≤ y.
-/2 width=1 by yle_plus2_to_minus_inj2/ qed-.
-
-lemma yplus_minus_assoc_inj: ∀x:nat. ∀y,z:ynat. x ≤ y → z + (y - x) = z + y - x.
-#x *
-[ #y * // #z >yminus_inj >yplus_inj >yplus_inj
- /4 width=1 by yle_inv_inj, plus_minus, eq_f/
-| >yminus_Y_inj //
-]
-qed-.
-
-lemma yplus_minus_assoc_comm_inj: ∀x:nat. ∀y,z:ynat. x ≤ y → z - (y - x) = z + x - y.
-#x *
-[ #y *
- [ #z >yminus_inj >yminus_inj >yplus_inj >yminus_inj
- /4 width=1 by yle_inv_inj, minus_le_minus_minus_comm, eq_f/
- | >yminus_inj >yminus_Y_inj //
- ]
-| >yminus_Y_inj //
-]
-qed-.
-
-lemma yplus_minus_comm_inj: ∀y:nat. ∀x,z:ynat. y ≤ x → x + z - y = x - y + z.
-#y * // #x * //
-#z #Hxy >yplus_inj >yminus_inj <plus_minus
-/2 width=1 by yle_inv_inj/
-qed-.
-
-lemma ylt_plus1_to_minus_inj2: ∀x,z:ynat. ∀y:nat. x + y < z → x < z - y.
-#x #z #y #H lapply (monotonic_ylt_minus_dx … H y ?) -H //
-qed-.
-
-lemma ylt_plus1_to_minus_inj1: ∀x,z:ynat. ∀y:nat. y + x < z → x < z - y.
-/2 width=1 by ylt_plus1_to_minus_inj2/ qed-.
-
-lemma ylt_plus2_to_minus_inj2: ∀x,y:ynat. ∀z:nat. z ≤ x → x < y + z → x - z < y.
-/2 width=1 by monotonic_ylt_minus_dx/ qed-.
-
-lemma ylt_plus2_to_minus_inj1: ∀x,y:ynat. ∀z:nat. z ≤ x → x < z + y → x - z < y.
-/2 width=1 by ylt_plus2_to_minus_inj2/ qed-.
-
-lemma yplus_inv_minus: ∀x1,y1. y1 ≤ yinj x1 →
- ∀x2,y2. yinj x1 + x2 = yinj y2 + y1 →
- yinj x1 - y1 = yinj y2 - x2 ∧ x2 ≤ yinj y2.
-#x1 #y1 #Hyx1 #x2 #y2 #H0
-lapply (yle_fwd_plus_ge_inj … x2 y2 Hyx1 ?) // #Hxy2
-elim (yle_inv_inj2 … Hyx1) -Hyx1 #m #Hyx1 #H destruct
-elim (yle_inv_inj2 … Hxy2) #n #H1 #H destruct
->yplus_inj in H0; >yplus_inj >yminus_inj >yminus_inj #H0
-@conj // lapply (yinj_inj … H0) -H0 /3 width=1 by arith_b1, eq_f/
-qed-.
-
-(* Inversion lemmas on minus ************************************************)
-
-lemma yle_inv_plus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y ∧ y ≤ z.
-/3 width=3 by yle_plus1_to_minus_inj2, yle_trans, conj/ qed-.
-
-lemma yle_inv_plus_inj1: ∀x,z:ynat. ∀y:nat. y + x ≤ z → x ≤ z - y ∧ y ≤ z.
-/2 width=1 by yle_inv_plus_inj2/ qed-.
-
-lemma yle_inv_plus_inj_dx: ∀x,y:ynat. ∀z:nat. x + y ≤ z →
- ∃∃m,n. x = yinj m & y = yinj n & x ≤ z - y & y ≤ z.
-#x #y #z #Hz elim (yle_inv_inj2 … Hz)
-#z0 #_ #H elim (yplus_inv_inj … H) -H
-#m #n #H1 #H2 #H3 destruct
-elim (yle_inv_plus_inj2 … Hz) -Hz /2 width=2 by ex4_2_intro/
-qed-.
-*)
projects / helm.git / blobdiff