(* Inversion lemmas on successor ********************************************)
-fact yle_inv_succ1_aux: ∀x,y. x ≤ y → ∀m. x = ⫯m → m ≤ ⫰y ∧ y = ⫯⫰y.
+fact yle_inv_succ1_aux: ∀x,y. x ≤ y → ∀m. x = ⫯m → m ≤ ⫰y ∧ ⫯⫰y = y.
#x #y * -x -y
[ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H
#n #H1 #H2 destruct elim (le_inv_S1 … Hxy) -Hxy
]
qed-.
-lemma yle_inv_succ1: ∀m,y. ⫯m ≤ y → m ≤ ⫰y ∧ y = ⫯⫰y.
+lemma yle_inv_succ1: ∀m,y. ⫯m ≤ y → m ≤ ⫰y ∧ ⫯⫰y = y.
/2 width=3 by yle_inv_succ1_aux/ qed-.
lemma yle_inv_succ: ∀m,n. ⫯m ≤ ⫯n → m ≤ n.
lemma yle_refl_pred_sn: ∀x. ⫰x ≤ x.
/2 width=1 by yle_refl, yle_pred_sn/ qed.
+lemma yle_pred: ∀m,n. m ≤ n → ⫰m ≤ ⫰n.
+#m #n * -m -n /3 width=1 by yle_inj, monotonic_pred/
+qed.
+
(* Properties on successor **************************************************)
lemma yle_succ: ∀m,n. m ≤ n → ⫯m ≤ ⫯n.
interpretation "ynat 'less than'" 'lt x y = (ylt x y).
+(* Basic forward lemmas *****************************************************)
+
+lemma ylt_inv_gen: ∀x,y. x < y → ∃m. x = yinj m.
+#x #y * -x -y /2 width=2 by ex_intro/
+qed-.
+
(* Basic inversion lemmas ***************************************************)
fact ylt_inv_inj2_aux: ∀x,y. x < y → ∀n. y = yinj n →
#x #Hx #H destruct //
qed-.
-fact ylt_inv_Y2_aux: ∀x,y. x < y → y = ∞ → ∃m. x = yinj m.
-#x #y * -x -y /2 width=2 by ex_intro/
+lemma ylt_inv_Y1: ∀n. ∞ < n → ⊥.
+#n #H elim (ylt_inv_gen … H) -H
+#y #H destruct
qed-.
-lemma ylt_inv_Y2: ∀x. x < ∞ → ∃m. x = yinj m.
-/2 width=3 by ylt_inv_Y2_aux/ qed-.
-
lemma ylt_inv_O1: ∀n. 0 < n → ⫯⫰n = n.
* // #n #H lapply (ylt_inv_inj … H) -H normalize
/3 width=1 by S_pred, eq_f/
(* Inversion lemmas on successor ********************************************)
-fact ylt_inv_succ1_aux: ∀x,y. x < y → ∀m. x = ⫯m → m < ⫰y ∧ y = ⫯⫰y.
+fact ylt_inv_succ1_aux: ∀x,y. x < y → ∀m. x = ⫯m → m < ⫰y ∧ ⫯⫰y = y.
#x #y * -x -y
[ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H
#n #H1 #H2 destruct elim (le_inv_S1 … Hxy) -Hxy
]
qed-.
-lemma ylt_inv_succ1: ∀m,y. ⫯m < y → m < ⫰y ∧ y = ⫯⫰y.
+lemma ylt_inv_succ1: ∀m,y. ⫯m < y → m < ⫰y ∧ ⫯⫰y = y.
/2 width=3 by ylt_inv_succ1_aux/ qed-.
lemma ylt_inv_succ: ∀m,n. ⫯m < ⫯n → m < n.
]
qed-.
+(* Basic properties *********************************************************)
+
+lemma ylt_O: ∀x. ⫯⫰(yinj x) = yinj x → 0 < x.
+* /2 width=1 by/ normalize
+#H destruct
+qed.
+
(* Properties on successor **************************************************)
lemma ylt_O_succ: ∀n. 0 < ⫯n.
* /2 width=1 by ylt_inj/
qed.
-(* Properties on yle ********************************************************)
+lemma ylt_succ: ∀m,n. m < n → ⫯m < ⫯n.
+#m #n #H elim H -m -n /3 width=1 by ylt_inj, le_S_S/
+qed.
+
+(* Properties on order ******************************************************)
-lemma yle_to_ylt_or_eq: ∀m:ynat. ∀n:ynat. m ≤ n → m < n ∨ m = n.
+lemma yle_split_eq: ∀m:ynat. ∀n:ynat. m ≤ n → m < n ∨ m = n.
#m #n * -m -n
[ #m #n #Hmn elim (le_to_or_lt_eq … Hmn) -Hmn
/3 width=1 by or_introl, ylt_inj/
]
qed-.
+lemma ylt_split: ∀m,n:ynat. m < n ∨ n ≤ m..
+#m #n elim (yle_split m n) /2 width=1 by or_intror/
+#H elim (yle_split_eq … H) -H /2 width=1 by or_introl, or_intror/
+qed-.
+
lemma ylt_yle_trans: ∀x:ynat. ∀y:ynat. ∀z:ynat. y ≤ z → x < y → x < z.
#x #y #z * -y -z
[ #y #z #Hyz #H elim (ylt_inv_inj2 … H) -H
#x #y * -x -y
[ #x #y #Hxy >yminus_inj
/3 width=4 by yle_inj, monotonic_le_minus_l/
-| * // #m >yminus_Y_inj //
+| * //
]
qed-.
#n #IHn >IHn //
qed.
+lemma yminus_O1: ∀x:ynat. 0 - 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.
lemma yminus_succ: ∀n,m. ⫯m - ⫯n = m - n.
* // #n * [2: >yminus_Y_inj // ]
#m >yminus_inj //
-qed.
+qed.
+
+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-.
(* Properties on order ******************************************************)
| >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 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.
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, yle_trans/ qed.
+
(* Forward lemmas on order **************************************************)
lemma yle_fwd_plus_sn2: ∀x,y,z. x + y ≤ z → y ≤ z.
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: ∀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: ∀x,y:ynat.∀z:nat. z + x ≤ z + y → x ≤ y.
+/2 width=2 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/
+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-.
+
+(* 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-.
+
+(* Properties on strict order ***********************************************)
+
+lemma monotonic_ylt_plus_dx: ∀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: ∀x,y. x < y → ∀z:nat. yinj z + x < yinj z + y.
+/2 width=1 by monotonic_ylt_plus_dx/ 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.
+
+(* Forward lemmas on minus **************************************************)
+
+lemma yle_plus_to_minus_inj2: ∀x,z:ynat. ∀y:nat. x + y ≤ z → x ≤ z - y.
+/2 width=1 by monotonic_yle_minus_dx/ qed-.
+
+lemma yle_plus_to_minus_inj1: ∀x,z:ynat. ∀y:nat. y + x ≤ z → x ≤ z - y.
+/2 width=1 by yle_plus_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_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-.
+
+(* 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_plus_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-.
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