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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "ground_2/notation/functions/uparrow_1.ma".
16 include "ground_2/notation/functions/downarrow_1.ma".
17 include "arithmetics/nat.ma".
18 include "ground_2/lib/relations.ma".
20 (* ARITHMETICAL PROPERTIES **************************************************)
22 interpretation "nat successor" 'UpArrow m = (S m).
24 interpretation "nat predecessor" 'DownArrow m = (pred m).
26 interpretation "nat min" 'and x y = (min x y).
28 interpretation "nat max" 'or x y = (max x y).
30 (* Iota equations ***********************************************************)
32 lemma pred_O: pred 0 = 0.
35 lemma pred_S: ∀m. pred (S m) = m.
38 lemma plus_S1: ∀x,y. ↑(x+y) = (↑x) + y.
41 lemma max_O1: ∀n. n = (0 ∨ n).
44 lemma max_O2: ∀n. n = (n ∨ 0).
47 lemma max_SS: ∀n1,n2. ↑(n1∨n2) = (↑n1 ∨ ↑n2).
48 #n1 #n2 elim (decidable_le n1 n2) #H normalize
49 [ >(le_to_leb_true … H) | >(not_le_to_leb_false … H) ] -H //
52 (* Equalities ***************************************************************)
54 lemma plus_SO: ∀n. n + 1 = ↑n.
57 lemma minus_plus_m_m_commutative: ∀n,m:nat. n = m + n - m.
60 lemma plus_to_minus_2: ∀m1,m2,n1,n2. n1 ≤ m1 → n2 ≤ m2 →
61 m1+n2 = m2+n1 → m1-n1 = m2-n2.
62 #m1 #m2 #n1 #n2 #H1 #H2 #H
63 @plus_to_minus >plus_minus_associative //
66 (* Note: uses minus_minus_comm, minus_plus_m_m, commutative_plus, plus_minus *)
67 lemma plus_minus_minus_be: ∀x,y,z. y ≤ z → z ≤ x → (x - z) + (z - y) = x - y.
68 #x #z #y #Hzy #Hyx >plus_minus // >commutative_plus >plus_minus //
71 lemma lt_succ_pred: ∀m,n. n < m → m = ↑↓m.
72 #m #n #Hm >S_pred /2 width=2 by ltn_to_ltO/
75 fact plus_minus_minus_be_aux: ∀i,x,y,z. y ≤ z → z ≤ x → i = z - y → x - z + i = x - y.
76 /2 width=1 by plus_minus_minus_be/ qed-.
78 lemma le_plus_minus: ∀m,n,p. p ≤ n → m + n - p = m + (n - p).
79 /2 by plus_minus/ qed-.
81 lemma le_plus_minus_comm: ∀n,m,p. p ≤ m → m + n - p = m - p + n.
82 /2 by plus_minus/ qed-.
84 lemma minus_minus_comm3: ∀n,x,y,z. n-x-y-z = n-y-z-x.
87 lemma idempotent_max: ∀n:nat. n = (n ∨ n).
88 #n normalize >le_to_leb_true //
91 lemma associative_max: associative … max.
93 @(leb_elim x y) normalize #Hxy
94 @(leb_elim y z) normalize #Hyz //
95 [1,2: >le_to_leb_true /2 width=3 by transitive_le/
96 | >not_le_to_leb_false /4 width=3 by lt_to_not_le, not_le_to_lt, transitive_lt/
97 >not_le_to_leb_false //
101 (* Properties ***************************************************************)
103 lemma eq_nat_dec: ∀n1,n2:nat. Decidable (n1 = n2).
104 #n1 elim n1 -n1 [| #n1 #IHn1 ] * [2,4: #n2 ]
105 [1,4: @or_intror #H destruct
106 | elim (IHn1 n2) -IHn1 /3 width=1 by or_intror, or_introl/
107 | /2 width=1 by or_introl/
111 lemma lt_or_eq_or_gt: ∀m,n. ∨∨ m < n | n = m | n < m.
112 #m #n elim (lt_or_ge m n) /2 width=1 by or3_intro0/
113 #H elim H -m /2 width=1 by or3_intro1/
114 #m #Hm * /3 width=1 by not_le_to_lt, le_S_S, or3_intro2/
117 lemma monotonic_le_minus_l2: ∀x1,x2,y,z. x1 ≤ x2 → x1 - y - z ≤ x2 - y - z.
118 /3 width=1 by monotonic_le_minus_l/ qed.
120 lemma minus_le_trans_sn: ∀x1,x2. x1 ≤ x2 → ∀x. x1-x ≤ x2.
121 /2 width=3 by transitive_le/ qed.
123 lemma le_plus_to_minus_l: ∀a,b,c. a + b ≤ c → b ≤ c-a.
124 /2 width=1 by le_plus_to_minus_r/
127 lemma le_plus_to_minus_comm: ∀n,m,p. n ≤ p+m → n-p ≤ m.
128 /2 width=1 by le_plus_to_minus/ qed-.
130 lemma le_inv_S1: ∀m,n. ↑m ≤ n → ∃∃p. m ≤ p & ↑p = n.
132 [ #H lapply (le_n_O_to_eq … H) -H
134 | /3 width=3 by monotonic_pred, ex2_intro/
138 (* Note: this might interfere with nat.ma *)
139 lemma monotonic_lt_pred: ∀m,n. m < n → 0 < m → pred m < pred n.
140 #m #n #Hmn #Hm whd >(S_pred … Hm)
141 @le_S_S_to_le >S_pred /2 width=3 by transitive_lt/
144 lemma lt_S_S: ∀x,y. x < y → ↑x < ↑y.
145 /2 width=1 by le_S_S/ qed.
147 lemma lt_S: ∀n,m. n < m → n < ↑m.
148 /2 width=1 by le_S/ qed.
150 lemma max_S1_le_S: ∀n1,n2,n. (n1 ∨ n2) ≤ n → (↑n1 ∨ n2) ≤ ↑n.
151 /4 width=2 by to_max, le_maxr, le_S_S, le_S/ qed-.
153 lemma max_S2_le_S: ∀n1,n2,n. (n1 ∨ n2) ≤ n → (n1 ∨ ↑n2) ≤ ↑n.
154 /2 width=1 by max_S1_le_S/ qed-.
156 (* Inversion & forward lemmas ***********************************************)
158 lemma lt_refl_false: ∀n. n < n → ⊥.
159 #n #H elim (lt_to_not_eq … H) -H /2 width=1 by/
162 lemma lt_zero_false: ∀n. n < 0 → ⊥.
163 #n #H elim (lt_to_not_le … H) -H /2 width=1 by/
166 lemma lt_le_false: ∀x,y. x < y → y ≤ x → ⊥.
167 /3 width=4 by lt_refl_false, lt_to_le_to_lt/ qed-.
169 lemma succ_inv_refl_sn: ∀x. ↑x = x → ⊥.
170 #x #H @(lt_le_false x (↑x)) //
173 lemma le_plus_xSy_O_false: ∀x,y. x + S y ≤ 0 → ⊥.
174 #x #y #H lapply (le_n_O_to_eq … H) -H <plus_n_Sm #H destruct
177 lemma le_plus_xySz_x_false: ∀y,z,x. x + y + S z ≤ x → ⊥.
178 #y #z #x elim x -x /3 width=1 by le_S_S_to_le/
179 #H elim (le_plus_xSy_O_false … H)
182 lemma plus_xySz_x_false: ∀z,x,y. x + y + S z = x → ⊥.
183 /2 width=4 by le_plus_xySz_x_false/ qed-.
185 lemma plus_xSy_x_false: ∀y,x. x + S y = x → ⊥.
186 /2 width=4 by plus_xySz_x_false/ qed-.
188 lemma pred_inv_fix_sn: ∀x. ↓x = x → 0 = x.
190 elim (succ_inv_refl_sn x) //
193 lemma discr_plus_xy_y: ∀x,y. x + y = y → x = 0.
196 lemma discr_plus_x_xy: ∀x,y. x = x + y → y = 0.
197 /2 width=2 by le_plus_minus_comm/ qed-.
199 lemma plus2_inv_le_sn: ∀m1,m2,n1,n2. m1 + n1 = m2 + n2 → m1 ≤ m2 → n2 ≤ n1.
200 #m1 #m2 #n1 #n2 #H #Hm
201 lapply (monotonic_le_plus_l n1 … Hm) -Hm >H -H
202 /2 width=2 by le_plus_to_le/
205 lemma lt_S_S_to_lt: ∀x,y. ↑x < ↑y → x < y.
206 /2 width=1 by le_S_S_to_le/ qed-.
208 (* Note this should go in nat.ma *)
209 lemma discr_x_minus_xy: ∀x,y. x = x - y → x = 0 ∨ y = 0.
210 #x @(nat_ind_plus … x) -x /2 width=1 by or_introl/
211 #x #_ #y @(nat_ind_plus … y) -y /2 width=1 by or_intror/
212 #y #_ >minus_plus_plus_l
213 #H lapply (discr_plus_xy_minus_xz … H) -H
217 lemma lt_inv_O1: ∀n. 0 < n → ∃m. ↑m = n.
218 * /2 width=2 by ex_intro/
219 #H cases (lt_le_false … H) -H //
222 lemma lt_inv_S1: ∀m,n. ↑m < n → ∃∃p. m < p & ↑p = n.
223 #m * /3 width=3 by lt_S_S_to_lt, ex2_intro/
224 #H cases (lt_le_false … H) -H //
227 lemma lt_inv_gen: ∀y,x. x < y → ∃∃z. x ≤ z & ↑z = y.
228 * /3 width=3 by le_S_S_to_le, ex2_intro/
229 #x #H elim (lt_le_false … H) -H //
232 lemma plus_inv_O3: ∀x,y. x + y = 0 → x = 0 ∧ y = 0.
233 /2 width=1 by plus_le_0/ qed-.
235 lemma plus_inv_S3_sn: ∀x1,x2,x3. x1+x2 = ↑x3 →
236 ∨∨ ∧∧ x1 = 0 & x2 = ↑x3
237 | ∃∃y1. x1 = ↑y1 & y1 + x2 = x3.
238 * /3 width=1 by or_introl, conj/
239 #x1 #x2 #x3 <plus_S1 #H destruct
240 /3 width=3 by ex2_intro, or_intror/
243 lemma plus_inv_S3_dx: ∀x2,x1,x3. x1+x2 = ↑x3 →
244 ∨∨ ∧∧ x2 = 0 & x1 = ↑x3
245 | ∃∃y2. x2 = ↑y2 & x1 + y2 = x3.
246 * /3 width=1 by or_introl, conj/
247 #x2 #x1 #x3 <plus_n_Sm #H destruct
248 /3 width=3 by ex2_intro, or_intror/
251 lemma max_inv_O3: ∀x,y. (x ∨ y) = 0 → 0 = x ∧ 0 = y.
252 /4 width=2 by le_maxr, le_maxl, le_n_O_to_eq, conj/
255 lemma zero_eq_plus: ∀x,y. 0 = x + y → 0 = x ∧ 0 = y.
256 * /2 width=1 by conj/ #x #y normalize #H destruct
259 lemma nat_split: ∀x. x = 0 ∨ ∃y. ↑y = x.
260 * /3 width=2 by ex_intro, or_introl, or_intror/
263 lemma lt_elim: ∀R:relation nat.
265 (∀n1,n2. R n1 n2 → R (↑n1) (↑n2)) →
266 ∀n2,n1. n1 < n2 → R n1 n2.
267 #R #IH1 #IH2 #n2 elim n2 -n2
268 [ #n1 #H elim (lt_le_false … H) -H //
269 | #n2 #IH * /4 width=1 by lt_S_S_to_lt/
273 lemma le_elim: ∀R:relation nat.
275 (∀n1,n2. R n1 n2 → R (↑n1) (↑n2)) →
276 ∀n1,n2. n1 ≤ n2 → R n1 n2.
277 #R #IH1 #IH2 #n1 #n2 @(nat_elim2 … n1 n2) -n1 -n2
278 /4 width=1 by monotonic_pred/ -IH1 -IH2
279 #n1 #H elim (lt_le_false … H) -H //
282 (* Iterators ****************************************************************)
284 (* Note: see also: lib/arithemetics/bigops.ma *)
285 rec definition iter (n:nat) (B:Type[0]) (op: B → B) (nil: B) ≝
288 | S k ⇒ op (iter k B op nil)
291 interpretation "iterated function" 'exp op n = (iter n ? op).
293 lemma iter_O: ∀B:Type[0]. ∀f:B→B.∀b. f^0 b = b.
296 lemma iter_S: ∀B:Type[0]. ∀f:B→B.∀b,l. f^(S l) b = f (f^l b).
299 lemma iter_n_Sm: ∀B:Type[0]. ∀f:B→B. ∀b,l. f^l (f b) = f (f^l b).
300 #B #f #b #l elim l -l normalize //
303 lemma iter_plus: ∀B:Type[0]. ∀f:B→B. ∀b,l1,l2. f^(l1+l2) b = f^l1 (f^l2 b).
304 #B #f #b #l1 elim l1 -l1 normalize //
307 (* Trichotomy operator ******************************************************)
309 (* Note: this is "if eqb n1 n2 then a2 else if leb n1 n2 then a1 else a3" *)
310 rec definition tri (A:Type[0]) n1 n2 a1 a2 a3 on n1 : A ≝
312 [ O ⇒ match n2 with [ O ⇒ a2 | S n2 ⇒ a1 ]
313 | S n1 ⇒ match n2 with [ O ⇒ a3 | S n2 ⇒ tri A n1 n2 a1 a2 a3 ]
316 lemma tri_lt: ∀A,a1,a2,a3,n2,n1. n1 < n2 → tri A n1 n2 a1 a2 a3 = a1.
317 #A #a1 #a2 #a3 #n2 elim n2 -n2
318 [ #n1 #H elim (lt_zero_false … H)
319 | #n2 #IH #n1 elim n1 -n1 /3 width=1 by monotonic_lt_pred/
323 lemma tri_eq: ∀A,a1,a2,a3,n. tri A n n a1 a2 a3 = a2.
324 #A #a1 #a2 #a3 #n elim n -n normalize //
327 lemma tri_gt: ∀A,a1,a2,a3,n1,n2. n2 < n1 → tri A n1 n2 a1 a2 a3 = a3.
328 #A #a1 #a2 #a3 #n1 elim n1 -n1
329 [ #n2 #H elim (lt_zero_false … H)
330 | #n1 #IH #n2 elim n2 -n2 /3 width=1 by monotonic_lt_pred/