(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_3_2.ma".
include "ground_2/ynat/ynat_lt.ma".
(* NATURAL NUMBERS WITH INFINITY ********************************************)
(* addition *)
-definition yplus: ynat → ynat → ynat ≝ λx,y. match x with
-[ yinj m ⇒ ysucc^m y
+definition yplus: ynat → ynat → ynat ≝ λx,y. match y with
+[ yinj n ⇒ ysucc^n x
| Y ⇒ Y
].
interpretation "ynat plus" 'plus x y = (yplus x y).
+lemma yplus_O2: ∀m:ynat. m + 0 = m.
+// qed.
+
+lemma yplus_S2: ∀m:ynat. ∀n. m + S n = ↑(m + n).
+// qed.
+
+lemma yplus_Y2: ∀m:ynat. m + (∞) = ∞.
+// qed.
+
(* Properties on successor **************************************************)
-lemma yplus_succ1: ∀m,n. (⫯m) + n = ⫯(m + n).
-* //
+lemma yplus_succ2: ∀m,n. m + ↑n = ↑(m + n).
+#m * //
+qed.
+
+lemma yplus_succ1: ∀m,n. ↑m + n = ↑(m + n).
+#m * // #n elim n -n //
qed.
-lemma yplus_SO1: ∀m. 1 + m = ⫯m.
+lemma yplus_succ_swap: ∀m,n. m + ↑n = ↑m + n.
+// qed.
+
+lemma yplus_SO2: ∀m. m + 1 = ↑m.
* //
-qed.
+qed.
(* Basic properties *********************************************************)
-lemma yplus_inj: ∀m,n. yinj m + yinj n = yinj (m + n).
-#m elim m -m //
-#m #IHm #n >(yplus_succ1 m) >IHm -IHm //
+lemma yplus_inj: ∀n,m. yinj m + yinj n = yinj (m + n).
+#n elim n -n //
+#n #IHn #m >(yplus_succ2 ? n) >IHn -IHn
+<plus_n_Sm //
qed.
-lemma yplus_Y2: ∀m. (m + ∞) = ∞.
-* normalize //
+lemma yplus_Y1: ∀m. ∞ + m = ∞.
+* // #m elim m -m //
qed.
lemma yplus_comm: commutative … yplus.
* [ #m ] * [1,3: #n ] //
-normalize >ysucc_iter_Y //
qed.
lemma yplus_assoc: associative … yplus.
-* // #x * //
-#y #z >yplus_inj whd in ⊢ (??%%); >iter_plus //
+#x #y * // #z cases y -y
+[ #y >yplus_inj whd in ⊢ (??%%); <iter_plus //
+| >yplus_Y1 //
+]
+qed.
+
+lemma yplus_O1: ∀n:ynat. 0 + n = n.
+#n >yplus_comm // qed.
+
+lemma yplus_comm_23: ∀x,y,z. x + z + y = x + y + z.
+#x #y #z >yplus_assoc //
+qed.
+
+lemma yplus_comm_24: ∀x1,x2,x3,x4. x1 + x4 + x3 + x2 = x1 + x2 + x3 + x4.
+#x1 #x2 #x3 #x4
+>yplus_assoc >yplus_assoc >yplus_assoc >yplus_assoc
+/2 width=1 by eq_f2/
+qed.
+
+lemma yplus_assoc_23: ∀x1,x2,x3,x4. x1 + x2 + (x3 + x4) = x1 + (x2 + x3) + x4.
+#x1 #x2 #x3 #x4 >yplus_assoc >yplus_assoc
+/2 width=1 by eq_f2/
+qed.
+
+(* Inversion lemmas on successor *********************************************)
+
+lemma yplus_inv_succ_lt_dx: ∀x,y,z:ynat. 0 < y → x + y = ↑z → x + ↓y = z.
+#x #y #z #H <(ylt_inv_O1 y) // >yplus_succ2
+/2 width=1 by ysucc_inv_inj/
+qed-.
+
+lemma yplus_inv_succ_lt_sn: ∀x,y,z:ynat. 0 < x → x + y = ↑z → ↓x + y = z.
+#x #y #z #H <(ylt_inv_O1 x) // >yplus_succ1
+/2 width=1 by ysucc_inv_inj/
+
+qed-.
+
+(* Inversion lemmas on order ************************************************)
+
+lemma yle_inv_plus_dx: ∀x,y. x ≤ y → ∃z. x + z = y.
+#x #y #H elim H -x -y /2 width=2 by ex_intro/
+#m #n #H @(ex_intro … (yinj (n-m))) (**) (* explicit constructor *)
+/3 width=1 by plus_minus, eq_f/
+qed-.
+
+lemma yle_inv_plus_sn: ∀x,y. x ≤ y → ∃z. z + x = y.
+#x #y #H elim (yle_inv_plus_dx … H) -H
+/2 width=2 by ex_intro/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma yplus_inv_inj: ∀z,y,x. x + y = yinj z →
+ ∃∃m,n. m + n = z & x = yinj m & y = yinj n.
+#z * [2: normalize #x #H destruct ]
+#y * [2: >yplus_Y1 #H destruct ]
+/3 width=5 by yinj_inj, ex3_2_intro/
+qed-.
+
+lemma yplus_inv_O: ∀x,y:ynat. x + y = 0 → x = 0 ∧ y = 0.
+#x #y #H elim (yplus_inv_inj … H) -H
+#m * /2 width=1 by conj/ #n <plus_n_Sm #H destruct
+qed-.
+
+lemma discr_yplus_xy_x: ∀x,y. x + y = x → x = ∞ ∨ y = 0.
+* /2 width=1 by or_introl/
+#x elim x -x /2 width=1 by or_intror/
+#x #IHx * [2: >yplus_Y2 #H destruct ]
+#y <ysucc_inj >yplus_succ1 #H
+lapply (ysucc_inv_inj … H) -H #H
+elim (IHx … H) -IHx -H /2 width=1 by or_introl, or_intror/
+qed-.
+
+lemma discr_yplus_x_xy: ∀x,y. x = x + y → x = ∞ ∨ y = 0.
+/2 width=1 by discr_yplus_xy_x/ qed-.
+
+lemma yplus_inv_monotonic_dx_inj: ∀z,x,y. x + yinj z = y + yinj z → x = y.
+#z @(nat_ind_plus … z) -z /3 width=2 by ysucc_inv_inj/
+qed-.
+
+lemma yplus_inv_monotonic_dx: ∀z,x,y. z < ∞ → x + z = y + z → x = y.
+#z #x #y #H elim (ylt_inv_Y2 … H) -H /2 width=2 by yplus_inv_monotonic_dx_inj/
+qed-.
+
+lemma yplus_inv_Y2: ∀x,y. x + y = ∞ → x = ∞ ∨ y = ∞.
+* /2 width=1 by or_introl/ #x * // #y >yplus_inj #H destruct
+qed-.
+
+lemma yplus_inv_monotonic_23: ∀z,x1,x2,y1,y2. z < ∞ →
+ x1 + z + y1 = x2 + z + y2 → x1 + y1 = x2 + y2.
+#z #x1 #x2 #y1 #y2 #Hz #H @(yplus_inv_monotonic_dx z) //
+>yplus_comm_23 >H -H //
+qed-.
+
+(* Inversion lemmas on strict_order *****************************************)
+
+lemma ylt_inv_plus_Y: ∀x,y. x + y < ∞ → x < ∞ ∧ y < ∞.
+#x #y #H elim (ylt_inv_Y2 … H) -H
+#z #H elim (yplus_inv_inj … H) -H /2 width=1 by conj/
+qed-.
+
+lemma ylt_inv_plus_sn: ∀x,y. x < y → ∃∃z. ↑z + x = y & x < ∞.
+#x #y #H elim (ylt_inv_le … H) -H
+#Hx #H elim (yle_inv_plus_sn … H) -H
+/2 width=2 by ex2_intro/
+qed-.
+
+lemma ylt_inv_plus_dx: ∀x,y. x < y → ∃∃z. x + ↑z = y & x < ∞.
+#x #y #H elim (ylt_inv_plus_sn … H) -H
+#z >yplus_comm /2 width=2 by ex2_intro/
+qed-.
+
+(* Properties on order ******************************************************)
+
+lemma yle_plus_dx2: ∀n,m. n ≤ m + n.
+* //
+#n elim n -n //
+#n #IHn #m >(yplus_succ2 ? n) @(yle_succ n) // (**) (* full auto fails *)
+qed.
+
+lemma yle_plus_dx1: ∀n,m. m ≤ m + n.
+// qed.
+
+lemma yle_plus_dx1_trans: ∀x,z. z ≤ x → ∀y. z ≤ x + y.
+/2 width=3 by yle_trans/ qed.
+
+lemma yle_plus_dx2_trans: ∀y,z. z ≤ y → ∀x. z ≤ x + y.
+/2 width=3 by yle_trans/ qed.
+
+lemma monotonic_yle_plus_dx: ∀x,y. x ≤ y → ∀z. x + z ≤ y + z.
+#x #y #Hxy * //
+#z elim z -z /3 width=1 by yle_succ/
+qed.
+
+lemma monotonic_yle_plus_sn: ∀x,y. x ≤ y → ∀z. z + x ≤ z + y.
+/2 width=1 by monotonic_yle_plus_dx/ qed.
+
+lemma monotonic_yle_plus: ∀x1,y1. x1 ≤ y1 → ∀x2,y2. x2 ≤ y2 →
+ x1 + x2 ≤ y1 + y2.
+/3 width=3 by monotonic_yle_plus_dx, monotonic_yle_plus_sn, yle_trans/ qed.
+
+lemma ylt_plus_Y: ∀x,y. x < ∞ → y < ∞ → x + y < ∞.
+#x #y #Hx elim (ylt_inv_Y2 … Hx) -Hx
+#m #Hm #Hy elim (ylt_inv_Y2 … Hy) -Hy //
+qed.
+
+(* Forward lemmas on order **************************************************)
+
+lemma yle_fwd_plus_sn2: ∀x,y,z. x + y ≤ z → y ≤ z.
+/2 width=3 by yle_trans/ qed-.
+
+lemma yle_fwd_plus_sn1: ∀x,y,z. x + y ≤ z → x ≤ z.
+/2 width=3 by yle_trans/ qed-.
+
+lemma yle_inv_monotonic_plus_dx_inj: ∀x,y:ynat.∀z:nat. x + z ≤ y + z → x ≤ y.
+#x #y #z elim z -z /3 width=1 by yle_inv_succ/
+qed-.
+
+lemma yle_inv_monotonic_plus_sn_inj: ∀x,y:ynat.∀z:nat. z + x ≤ z + y → x ≤ y.
+/2 width=2 by yle_inv_monotonic_plus_dx_inj/ qed-.
+
+lemma yle_inv_monotonic_plus_dx: ∀x,y,z. z < ∞ → x + z ≤ y + z → x ≤ y.
+#x #y #z #Hz elim (ylt_inv_Y2 … Hz) -Hz #m #H destruct
+/2 width=2 by yle_inv_monotonic_plus_sn_inj/
+qed-.
+
+lemma yle_inv_monotonic_plus_sn: ∀x,y,z. z < ∞ → z + x ≤ z + y → x ≤ y.
+/2 width=3 by yle_inv_monotonic_plus_dx/ qed-.
+
+lemma yle_fwd_plus_ge: ∀m1,m2:nat. m2 ≤ m1 → ∀n1,n2:ynat. m1 + n1 ≤ m2 + n2 → n1 ≤ n2.
+#m1 #m2 #Hm12 #n1 #n2 #H
+lapply (monotonic_yle_plus … Hm12 … H) -Hm12 -H
+/2 width=2 by yle_inv_monotonic_plus_sn_inj/
+qed-.
+
+lemma yle_fwd_plus_ge_inj: ∀m1:nat. ∀m2,n1,n2:ynat. m2 ≤ m1 → m1 + n1 ≤ m2 + n2 → n1 ≤ n2.
+#m2 #m1 #n1 #n2 #H elim (yle_inv_inj2 … H) -H
+#x #H0 #H destruct /3 width=4 by yle_fwd_plus_ge, yle_inj/
+qed-.
+
+lemma yle_fwd_plus_yge: ∀n2,m1:ynat. ∀n1,m2:nat. m2 ≤ m1 → m1 + n1 ≤ m2 + n2 → n1 ≤ n2.
+* // #n2 * /2 width=4 by yle_fwd_plus_ge_inj/
+qed-.
+
+(* Properties on strict order ***********************************************)
+
+lemma ylt_plus_dx1_trans: ∀x,z. z < x → ∀y. z < x + y.
+/2 width=3 by ylt_yle_trans/ qed.
+
+lemma ylt_plus_dx2_trans: ∀y,z. z < y → ∀x. z < x + y.
+/2 width=3 by ylt_yle_trans/ qed.
+
+lemma monotonic_ylt_plus_dx_inj: ∀x,y. x < y → ∀z:nat. x + yinj z < y + yinj z.
+#x #y #Hxy #z elim z -z /3 width=1 by ylt_succ/
qed.
+lemma monotonic_ylt_plus_sn_inj: ∀x,y. x < y → ∀z:nat. yinj z + x < yinj z + y.
+/2 width=1 by monotonic_ylt_plus_dx_inj/ qed.
+
+lemma monotonic_ylt_plus_dx: ∀x,y. x < y → ∀z. z < ∞ → x + z < y + z.
+#x #y #Hxy #z #Hz elim (ylt_inv_Y2 … Hz) -Hz
+#m #H destruct /2 width=1 by monotonic_ylt_plus_dx_inj/
+qed.
+
+lemma monotonic_ylt_plus_sn: ∀x,y. x < y → ∀z. z < ∞ → z + x < z + y.
+/2 width=1 by monotonic_ylt_plus_dx/ qed.
+
+lemma monotonic_ylt_plus_inj: ∀m1,m2. m1 < m2 → ∀n1,n2. yinj n1 ≤ n2 → m1 + n1 < m2 + n2.
+/3 width=3 by monotonic_ylt_plus_sn_inj, monotonic_yle_plus_sn, ylt_yle_trans/
+qed.
+
+lemma monotonic_ylt_plus: ∀m1,m2. m1 < m2 → ∀n1. n1 < ∞ → ∀n2. n1 ≤ n2 → m1 + n1 < m2 + n2.
+#m1 #m2 #Hm12 #n1 #H elim (ylt_inv_Y2 … H) -H #m #H destruct /2 width=1 by monotonic_ylt_plus_inj/
+qed.
+
+(* Forward lemmas on strict order *******************************************)
+
+lemma ylt_inv_monotonic_plus_dx: ∀x,y,z. x + z < y + z → x < y.
+* [2: #y #z >yplus_comm #H elim (ylt_inv_Y1 … H) ]
+#x * // #y * [2: #H elim (ylt_inv_Y1 … H) ]
+/4 width=3 by ylt_inv_inj, ylt_inj, lt_plus_to_lt_l/
+qed-.
+
+lemma ylt_fwd_plus_ge: ∀m1,m2. m2 ≤ m1 → ∀n1,n2. m1 + n1 < m2 + n2 → n1 < n2.
+#m1 #m2 #Hm12 #n1 #n2 #H elim (ylt_fwd_gen … H)
+#x #H0 elim (yplus_inv_inj … H0) -H0
+#y #z #_ #H2 #H3 destruct -x
+elim (yle_inv_inj2 … Hm12)
+#x #_ #H0 destruct
+lapply (monotonic_ylt_plus … H … Hm12) -H -Hm12
+/2 width=2 by ylt_inv_monotonic_plus_dx/
+qed-.
+
+(* Properties on predeccessor ***********************************************)
+
+lemma yplus_pred1: ∀x,y:ynat. 0 < x → ↓x + y = ↓(x+y).
+#x * // #y elim y -y // #y #IH #Hx
+>yplus_S2 >yplus_S2 >IH -IH // >ylt_inv_O1
+/2 width=1 by ylt_plus_dx1_trans/
+qed-.
+
+lemma yplus_pred2: ∀x,y:ynat. 0 < y → x + ↓y = ↓(x+y).
+/2 width=1 by yplus_pred1/ qed-.
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