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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_le.ma".
17 (* NATURAL NUMBERS WITH INFINITY ********************************************)
19 (* strict order relation *)
20 inductive ylt: relation ynat ≝
21 | ylt_inj: ∀m,n. m < n → ylt m n
22 | ylt_Y : ∀m:nat. ylt m (∞)
25 interpretation "ynat 'less than'" 'lt x y = (ylt x y).
27 (* Basic forward lemmas *****************************************************)
29 lemma ylt_fwd_gen: ∀x,y. x < y → ∃m. x = yinj m.
30 #x #y * -x -y /2 width=2 by ex_intro/
33 lemma ylt_fwd_lt_O1: ∀x,y:ynat. x < y → 0 < y.
34 #x #y #H elim H -x -y /3 width=2 by ylt_inj, ltn_to_ltO/
37 (* Basic inversion lemmas ***************************************************)
39 fact ylt_inv_inj2_aux: ∀x,y. x < y → ∀n. y = yinj n →
40 ∃∃m. m < n & x = yinj m.
42 [ #x #y #Hxy #n #Hy elim (le_inv_S1 … Hxy) -Hxy
43 #m #Hm #H destruct /3 width=3 by le_S_S, ex2_intro/
48 lemma ylt_inv_inj2: ∀x,n. x < yinj n →
49 ∃∃m. m < n & x = yinj m.
50 /2 width=3 by ylt_inv_inj2_aux/ qed-.
52 lemma ylt_inv_inj: ∀m,n. yinj m < yinj n → m < n.
53 #m #n #H elim (ylt_inv_inj2 … H) -H
57 lemma ylt_inv_Y1: ∀n. ∞ < n → ⊥.
58 #n #H elim (ylt_fwd_gen … H) -H
62 lemma ylt_inv_Y2: ∀x:ynat. x < ∞ → ∃n. x = yinj n.
63 * /2 width=2 by ex_intro/
64 #H elim (ylt_inv_Y1 … H)
67 lemma ylt_inv_O1: ∀n:ynat. 0 < n → ⫯⫰n = n.
68 * // #n #H lapply (ylt_inv_inj … H) -H normalize
69 /3 width=1 by S_pred, eq_f/
72 (* Inversion lemmas on successor ********************************************)
74 fact ylt_inv_succ1_aux: ∀x,y:ynat. x < y → ∀m. x = ⫯m → m < ⫰y ∧ ⫯⫰y = y.
76 [ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H
77 #n #H1 #H2 destruct elim (le_inv_S1 … Hxy) -Hxy
78 #m #Hnm #H destruct /3 width=1 by ylt_inj, conj/
79 | #x #y #H elim (ysucc_inv_inj_sn … H) -H
80 #m #H #_ destruct /2 width=1 by ylt_Y, conj/
84 lemma ylt_inv_succ1: ∀m,y:ynat. ⫯m < y → m < ⫰y ∧ ⫯⫰y = y.
85 /2 width=3 by ylt_inv_succ1_aux/ qed-.
87 lemma ylt_inv_succ: ∀m,n. ⫯m < ⫯n → m < n.
88 #m #n #H elim (ylt_inv_succ1 … H) -H //
91 (* Forward lemmas on successor **********************************************)
93 fact ylt_fwd_succ2_aux: ∀x,y. x < y → ∀n. y = ⫯n → x ≤ n.
95 [ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H
96 #n #H1 #H2 destruct /3 width=1 by yle_inj, le_S_S_to_le/
97 | #x #n #H lapply (ysucc_inv_Y_sn … H) -H //
101 lemma ylt_fwd_succ2: ∀m,n. m < ⫯n → m ≤ n.
102 /2 width=3 by ylt_fwd_succ2_aux/ qed-.
104 (* inversion and forward lemmas on order ************************************)
106 lemma ylt_fwd_le_succ1: ∀m,n. m < n → ⫯m ≤ n.
107 #m #n * -m -n /2 width=1 by yle_inj/
110 lemma ylt_fwd_le_pred2: ∀x,y:ynat. x < y → x ≤ ⫰y.
111 #x #y #H elim H -x -y /3 width=1 by yle_inj, monotonic_pred/
114 lemma ylt_fwd_le: ∀m:ynat. ∀n:ynat. m < n → m ≤ n.
115 #m #n * -m -n /3 width=1 by lt_to_le, yle_inj/
118 lemma ylt_yle_false: ∀m:ynat. ∀n:ynat. m < n → n ≤ m → ⊥.
120 [ #m #n #Hmn #H lapply (yle_inv_inj … H) -H
121 #H elim (lt_refl_false n) /2 width=3 by le_to_lt_to_lt/
122 | #m #H lapply (yle_inv_Y1 … H) -H
127 lemma ylt_inv_le: ∀x,y. x < y → x < ∞ ∧ ⫯x ≤ y.
128 #x #y #H elim H -x -y /3 width=1 by yle_inj, conj/
131 (* Basic properties *********************************************************)
133 lemma ylt_O1: ∀x:ynat. ⫯⫰x = x → 0 < x.
134 * // * /2 width=1 by ylt_inj/ normalize
138 (* Properties on predecessor ************************************************)
140 lemma ylt_pred: ∀m,n:ynat. m < n → 0 < m → ⫰m < ⫰n.
142 /4 width=1 by ylt_inv_inj, ylt_inj, monotonic_lt_pred/
145 (* Properties on successor **************************************************)
147 lemma ylt_O_succ: ∀n. 0 < ⫯n.
148 * /2 width=1 by ylt_inj/
151 lemma ylt_succ: ∀m,n. m < n → ⫯m < ⫯n.
152 #m #n #H elim H -m -n /3 width=1 by ylt_inj, le_S_S/
155 lemma ylt_succ_Y: ∀x. x < ∞ → ⫯x < ∞.
156 * /2 width=1 by/ qed.
158 lemma yle_succ1_inj: ∀x. ∀y:ynat. ⫯yinj x ≤ y → x < y.
159 #x * /3 width=1 by yle_inv_inj, ylt_inj/
162 lemma ylt_succ2_refl: ∀x,y:ynat. x < y → x < ⫯x.
163 #x #y #H elim (ylt_fwd_gen … H) -y /2 width=1 by ylt_inj/
166 (* Properties on order ******************************************************)
168 lemma yle_split_eq: ∀m,n:ynat. m ≤ n → m < n ∨ m = n.
170 [ #m #n #Hmn elim (le_to_or_lt_eq … Hmn) -Hmn
171 /3 width=1 by or_introl, ylt_inj/
172 | * /2 width=1 by or_introl, ylt_Y/
176 lemma ylt_split: ∀m,n:ynat. m < n ∨ n ≤ m.
177 #m #n elim (yle_split m n) /2 width=1 by or_intror/
178 #H elim (yle_split_eq … H) -H /2 width=1 by or_introl, or_intror/
181 lemma ylt_split_eq: ∀m,n:ynat. ∨∨ m < n | n = m | n < m.
182 #m #n elim (ylt_split m n) /2 width=1 by or3_intro0/
183 #H elim (yle_split_eq … H) -H /2 width=1 by or3_intro1, or3_intro2/
186 lemma ylt_yle_trans: ∀x:ynat. ∀y:ynat. ∀z:ynat. y ≤ z → x < y → x < z.
188 [ #y #z #Hyz #H elim (ylt_inv_inj2 … H) -H
189 #m #Hm #H destruct /3 width=3 by ylt_inj, lt_to_le_to_lt/
194 lemma yle_ylt_trans: ∀x:ynat. ∀y:ynat. ∀z:ynat. y < z → x ≤ y → x < z.
196 [ #y #z #Hyz #H elim (yle_inv_inj2 … H) -H
197 #m #Hm #H destruct /3 width=3 by ylt_inj, le_to_lt_to_lt/
198 | #y #H elim (yle_inv_inj2 … H) -H //
202 lemma yle_inv_succ1_lt: ∀x,y:ynat. ⫯x ≤ y → 0 < y ∧ x ≤ ⫰y.
203 #x #y #H elim (yle_inv_succ1 … H) -H /3 width=1 by ylt_O1, conj/
206 lemma yle_lt: ∀x,y. x < ∞ → ⫯x ≤ y → x < y.
207 #x * // #y #H elim (ylt_inv_Y2 … H) -H #n #H destruct
208 /3 width=1 by ylt_inj, yle_inv_inj/
211 (* Main properties **********************************************************)
213 theorem ylt_trans: Transitive … ylt.
216 #z #H lapply (ylt_inv_inj … H) -H
217 /3 width=3 by transitive_lt, ylt_inj/ (**) (* full auto too slow *)
218 | #x #z #H elim (ylt_yle_false … H) //
222 (* Elimination principles ***************************************************)
224 fact ynat_ind_lt_le_aux: ∀R:predicate ynat.
225 (∀y. (∀x. x < y → R x) → R y) →
226 ∀y:nat. ∀x. x ≤ y → R x.
228 [ #x #H >(yle_inv_O2 … H) -x
229 @IH -IH #x #H elim (ylt_yle_false … H) -H //
230 | /5 width=3 by ylt_yle_trans, ylt_fwd_succ2/
234 fact ynat_ind_lt_aux: ∀R:predicate ynat.
235 (∀y. (∀x. x < y → R x) → R y) →
237 /4 width=2 by ynat_ind_lt_le_aux/ qed-.
239 lemma ynat_ind_lt: ∀R:predicate ynat.
240 (∀y. (∀x. x < y → R x) → R y) →
242 #R #IH * /4 width=1 by ynat_ind_lt_aux/
243 @IH #x #H elim (ylt_inv_Y2 … H) -H
244 #n #H destruct /4 width=1 by ynat_ind_lt_aux/
247 fact ynat_f_ind_aux: ∀A. ∀f:A→ynat. ∀R:predicate A.
248 (∀x. (∀a. f a < x → R a) → ∀a. f a = x → R a) →
250 #A #f #R #IH #x @(ynat_ind_lt … x) -x
254 lemma ynat_f_ind: ∀A. ∀f:A→ynat. ∀R:predicate A.
255 (∀x. (∀a. f a < x → R a) → ∀a. f a = x → R a) → ∀a. R a.
257 @(ynat_f_ind_aux … IH) -IH [2: // | skip ]