src/HOL/Word/Misc_Numeric.thy
 author huffman Wed Sep 07 09:02:58 2011 -0700 (2011-09-07) changeset 44821 a92f65e174cf parent 39910 10097e0a9dbd child 44939 5930d35c976d permissions -rw-r--r--
avoid using legacy theorem names
1 (*
2   Author:  Jeremy Dawson, NICTA
3 *)
5 header {* Useful Numerical Lemmas *}
7 theory Misc_Numeric
8 imports Main Parity
9 begin
11 lemma the_elemI: "y = {x} ==> the_elem y = x"
12   by simp
14 lemmas split_split = prod.split
15 lemmas split_split_asm = prod.split_asm
16 lemmas split_splits = split_split split_split_asm
18 lemmas funpow_0 = funpow.simps(1)
19 lemmas funpow_Suc = funpow.simps(2)
21 lemma nonemptyE: "S ~= {} ==> (!!x. x : S ==> R) ==> R" by auto
23 lemma gt_or_eq_0: "0 < y \<or> 0 = (y::nat)" by arith
25 declare iszero_0 [iff]
27 lemmas xtr1 = xtrans(1)
28 lemmas xtr2 = xtrans(2)
29 lemmas xtr3 = xtrans(3)
30 lemmas xtr4 = xtrans(4)
31 lemmas xtr5 = xtrans(5)
32 lemmas xtr6 = xtrans(6)
33 lemmas xtr7 = xtrans(7)
34 lemmas xtr8 = xtrans(8)
39 lemma sum_imp_diff: "j = k + i ==> j - i = (k :: nat)" by arith
41 lemma nobm1:
42   "0 < (number_of w :: nat) ==>
43    number_of w - (1 :: nat) = number_of (Int.pred w)"
44   apply (unfold nat_number_of_def One_nat_def nat_1 [symmetric] pred_def)
45   apply (simp add: number_of_eq nat_diff_distrib [symmetric])
46   done
48 lemma zless2: "0 < (2 :: int)" by arith
50 lemmas zless2p [simp] = zless2 [THEN zero_less_power]
51 lemmas zle2p [simp] = zless2p [THEN order_less_imp_le]
53 lemmas pos_mod_sign2 = zless2 [THEN pos_mod_sign [where b = "2::int"]]
54 lemmas pos_mod_bound2 = zless2 [THEN pos_mod_bound [where b = "2::int"]]
56 -- "the inverse(s) of @{text number_of}"
57 lemma nmod2: "n mod (2::int) = 0 | n mod 2 = 1" by arith
59 lemma emep1:
60   "even n ==> even d ==> 0 <= d ==> (n + 1) mod (d :: int) = (n mod d) + 1"
62   apply (safe dest!: even_equiv_def [THEN iffD1])
63   apply (subst pos_zmod_mult_2)
64    apply arith
66  done
68 lemmas eme1p = emep1 [simplified add_commute]
70 lemma le_diff_eq': "(a \<le> c - b) = (b + a \<le> (c::int))" by arith
72 lemma less_diff_eq': "(a < c - b) = (b + a < (c::int))" by arith
74 lemma diff_le_eq': "(a - b \<le> c) = (a \<le> b + (c::int))" by arith
76 lemma diff_less_eq': "(a - b < c) = (a < b + (c::int))" by arith
78 lemmas m1mod2k = zless2p [THEN zmod_minus1]
79 lemmas m1mod22k = mult_pos_pos [OF zless2 zless2p, THEN zmod_minus1]
80 lemmas p1mod22k' = zless2p [THEN order_less_imp_le, THEN pos_zmod_mult_2]
81 lemmas z1pmod2' = zero_le_one [THEN pos_zmod_mult_2, simplified]
82 lemmas z1pdiv2' = zero_le_one [THEN pos_zdiv_mult_2, simplified]
84 lemma p1mod22k:
85   "(2 * b + 1) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n) + (1::int)"
88 lemma z1pmod2:
89   "(2 * b + 1) mod 2 = (1::int)" by arith
91 lemma z1pdiv2:
92   "(2 * b + 1) div 2 = (b::int)" by arith
94 lemmas zdiv_le_dividend = xtr3 [OF div_by_1 [symmetric] zdiv_mono2,
95   simplified int_one_le_iff_zero_less, simplified, standard]
97 lemma axxbyy:
98   "a + m + m = b + n + n ==> (a = 0 | a = 1) ==> (b = 0 | b = 1) ==>
99    a = b & m = (n :: int)" by arith
101 lemma axxmod2:
102   "(1 + x + x) mod 2 = (1 :: int) & (0 + x + x) mod 2 = (0 :: int)" by arith
104 lemma axxdiv2:
105   "(1 + x + x) div 2 = (x :: int) & (0 + x + x) div 2 = (x :: int)"  by arith
107 lemmas iszero_minus = trans [THEN trans,
108   OF iszero_def neg_equal_0_iff_equal iszero_def [symmetric], standard]
111   standard]
115 lemma zmod_uminus: "- ((a :: int) mod b) mod b = -a mod b"
116   by (simp add : zmod_zminus1_eq_if)
118 lemma zmod_zsub_distrib: "((a::int) - b) mod c = (a mod c - b mod c) mod c"
119   apply (unfold diff_int_def)
120   apply (rule trans [OF _ mod_add_eq [symmetric]])
122   done
124 lemma zmod_zsub_right_eq: "((a::int) - b) mod c = (a - b mod c) mod c"
125   apply (unfold diff_int_def)
126   apply (rule trans [OF _ mod_add_right_eq [symmetric]])
128   done
130 lemma zmod_zsub_left_eq: "((a::int) - b) mod c = (a mod c - b) mod c"
131   by (rule mod_add_left_eq [where b = "- b", simplified diff_int_def [symmetric]])
133 lemma zmod_zsub_self [simp]:
134   "((b :: int) - a) mod a = b mod a"
137 lemma zmod_zmult1_eq_rev:
138   "b * a mod c = b mod c * a mod (c::int)"
140   apply (subst zmod_zmult1_eq)
141   apply simp
142   done
144 lemmas rdmods [symmetric] = zmod_uminus [symmetric]
148 lemma mod_plus_right:
149   "((a + x) mod m = (b + x) mod m) = (a mod m = b mod (m :: nat))"
150   apply (induct x)
152   apply arith
153   done
155 lemma nat_minus_mod: "(n - n mod m) mod m = (0 :: nat)"
156   by (induct n) (simp_all add : mod_Suc)
158 lemmas nat_minus_mod_plus_right = trans [OF nat_minus_mod mod_0 [symmetric],
159   THEN mod_plus_right [THEN iffD2], standard, simplified]
161 lemmas push_mods' = mod_add_eq [standard]
162   mod_mult_eq [standard] zmod_zsub_distrib [standard]
163   zmod_uminus [symmetric, standard]
165 lemmas push_mods = push_mods' [THEN eq_reflection, standard]
166 lemmas pull_mods = push_mods [symmetric] rdmods [THEN eq_reflection, standard]
167 lemmas mod_simps =
168   mod_mult_self2_is_0 [THEN eq_reflection]
169   mod_mult_self1_is_0 [THEN eq_reflection]
170   mod_mod_trivial [THEN eq_reflection]
172 lemma nat_mod_eq:
173   "!!b. b < n ==> a mod n = b mod n ==> a mod n = (b :: nat)"
174   by (induct a) auto
176 lemmas nat_mod_eq' = refl [THEN [2] nat_mod_eq]
178 lemma nat_mod_lem:
179   "(0 :: nat) < n ==> b < n = (b mod n = b)"
180   apply safe
181    apply (erule nat_mod_eq')
182   apply (erule subst)
183   apply (erule mod_less_divisor)
184   done
187   "(x :: nat) < z ==> y < z ==>
188    (x + y) mod z = (if x + y < z then x + y else x + y - z)"
189   apply (rule nat_mod_eq)
190    apply auto
191   apply (rule trans)
192    apply (rule le_mod_geq)
193    apply simp
194   apply (rule nat_mod_eq')
195   apply arith
196   done
198 lemma mod_nat_sub:
199   "(x :: nat) < z ==> (x - y) mod z = x - y"
200   by (rule nat_mod_eq') arith
202 lemma int_mod_lem:
203   "(0 :: int) < n ==> (0 <= b & b < n) = (b mod n = b)"
204   apply safe
205     apply (erule (1) mod_pos_pos_trivial)
206    apply (erule_tac [!] subst)
207    apply auto
208   done
210 lemma int_mod_eq:
211   "(0 :: int) <= b ==> b < n ==> a mod n = b mod n ==> a mod n = b"
212   by clarsimp (rule mod_pos_pos_trivial)
214 lemmas int_mod_eq' = refl [THEN [3] int_mod_eq]
216 lemma int_mod_le: "0 <= a ==> 0 < (n :: int) ==> a mod n <= a"
217   apply (cases "a < n")
218    apply (auto dest: mod_pos_pos_trivial pos_mod_bound [where a=a])
219   done
221 lemma int_mod_le': "0 <= b - n ==> 0 < (n :: int) ==> b mod n <= b - n"
222   by (rule int_mod_le [where a = "b - n" and n = n, simplified])
224 lemma int_mod_ge: "a < n ==> 0 < (n :: int) ==> a <= a mod n"
225   apply (cases "0 <= a")
226    apply (drule (1) mod_pos_pos_trivial)
227    apply simp
228   apply (rule order_trans [OF _ pos_mod_sign])
229    apply simp
230   apply assumption
231   done
233 lemma int_mod_ge': "b < 0 ==> 0 < (n :: int) ==> b + n <= b mod n"
234   by (rule int_mod_ge [where a = "b + n" and n = n, simplified])
237   "(x :: int) < z ==> y < z ==> 0 <= y ==> 0 <= x ==> 0 <= z ==>
238    (x + y) mod z = (if x + y < z then x + y else x + y - z)"
239   by (auto intro: int_mod_eq)
241 lemma mod_sub_if_z:
242   "(x :: int) < z ==> y < z ==> 0 <= y ==> 0 <= x ==> 0 <= z ==>
243    (x - y) mod z = (if y <= x then x - y else x - y + z)"
244   by (auto intro: int_mod_eq)
246 lemmas zmde = zmod_zdiv_equality [THEN diff_eq_eq [THEN iffD2], symmetric]
247 lemmas mcl = mult_cancel_left [THEN iffD1, THEN make_pos_rule]
249 (* already have this for naturals, div_mult_self1/2, but not for ints *)
250 lemma zdiv_mult_self: "m ~= (0 :: int) ==> (a + m * n) div m = a div m + n"
251   apply (rule mcl)
252    prefer 2
253    apply (erule asm_rl)
254   apply (simp add: zmde ring_distribs)
255   done
257 (** Rep_Integ **)
258 lemma eqne: "equiv A r ==> X : A // r ==> X ~= {}"
259   unfolding equiv_def refl_on_def quotient_def Image_def by auto
261 lemmas Rep_Integ_ne = Integ.Rep_Integ
262   [THEN equiv_intrel [THEN eqne, simplified Integ_def [symmetric]], standard]
264 lemmas riq = Integ.Rep_Integ [simplified Integ_def]
265 lemmas intrel_refl = refl [THEN equiv_intrel_iff [THEN iffD1], standard]
266 lemmas Rep_Integ_equiv = quotient_eq_iff
267   [OF equiv_intrel riq riq, simplified Integ.Rep_Integ_inject, standard]
268 lemmas Rep_Integ_same =
269   Rep_Integ_equiv [THEN intrel_refl [THEN rev_iffD2], standard]
271 lemma RI_int: "(a, 0) : Rep_Integ (int a)"
272   unfolding int_def by auto
274 lemmas RI_intrel [simp] = UNIV_I [THEN quotientI,
275   THEN Integ.Abs_Integ_inverse [simplified Integ_def], standard]
277 lemma RI_minus: "(a, b) : Rep_Integ x ==> (b, a) : Rep_Integ (- x)"
278   apply (rule_tac z=x in eq_Abs_Integ)
279   apply (clarsimp simp: minus)
280   done
283   "(a, b) : Rep_Integ x ==> (c, d) : Rep_Integ y ==>
284    (a + c, b + d) : Rep_Integ (x + y)"
285   apply (rule_tac z=x in eq_Abs_Integ)
286   apply (rule_tac z=y in eq_Abs_Integ)
288   done
290 lemma mem_same: "a : S ==> a = b ==> b : S"
291   by fast
293 (* two alternative proofs of this *)
294 lemma RI_eq_diff': "(a, b) : Rep_Integ (int a - int b)"
295   apply (unfold diff_minus)
296   apply (rule mem_same)
297    apply (rule RI_minus RI_add RI_int)+
298   apply simp
299   done
301 lemma RI_eq_diff: "((a, b) : Rep_Integ x) = (int a - int b = x)"
302   apply safe
303    apply (rule Rep_Integ_same)
304     prefer 2
305     apply (erule asm_rl)
306    apply (rule RI_eq_diff')+
307   done
309 lemma mod_power_lem:
310   "a > 1 ==> a ^ n mod a ^ m = (if m <= n then 0 else (a :: int) ^ n)"
311   apply clarsimp
312   apply safe
313    apply (simp add: dvd_eq_mod_eq_0 [symmetric])
314    apply (drule le_iff_add [THEN iffD1])
316   apply (rule mod_pos_pos_trivial)
317    apply (simp)
318   apply (rule power_strict_increasing)
319    apply auto
320   done
322 lemma min_pm [simp]: "min a b + (a - b) = (a :: nat)" by arith
324 lemmas min_pm1 [simp] = trans [OF add_commute min_pm]
326 lemma rev_min_pm [simp]: "min b a + (a - b) = (a::nat)" by arith
328 lemmas rev_min_pm1 [simp] = trans [OF add_commute rev_min_pm]
330 lemma pl_pl_rels:
331   "a + b = c + d ==>
332    a >= c & b <= d | a <= c & b >= (d :: nat)" by arith
334 lemmas pl_pl_rels' = add_commute [THEN [2] trans, THEN pl_pl_rels]
336 lemma minus_eq: "(m - k = m) = (k = 0 | m = (0 :: nat))"  by arith
338 lemma pl_pl_mm: "(a :: nat) + b = c + d ==> a - c = d - b"  by arith
340 lemmas pl_pl_mm' = add_commute [THEN [2] trans, THEN pl_pl_mm]
342 lemma min_minus [simp] : "min m (m - k) = (m - k :: nat)" by arith
344 lemmas min_minus' [simp] = trans [OF min_max.inf_commute min_minus]
346 lemma nat_no_eq_iff:
347   "(number_of b :: int) >= 0 ==> (number_of c :: int) >= 0 ==>
348    (number_of b = (number_of c :: nat)) = (b = c)"
349   apply (unfold nat_number_of_def)
350   apply safe
351   apply (drule (2) eq_nat_nat_iff [THEN iffD1])
353   done
356 lemmas dtle = xtr3 [OF dme [symmetric] le_add1]
357 lemmas th2 = order_trans [OF order_refl [THEN [2] mult_le_mono] dtle]
359 lemma td_gal:
360   "0 < c ==> (a >= b * c) = (a div c >= (b :: nat))"
361   apply safe
362    apply (erule (1) xtr4 [OF div_le_mono div_mult_self_is_m])
363   apply (erule th2)
364   done
366 lemmas td_gal_lt = td_gal [simplified not_less [symmetric], simplified]
368 lemma div_mult_le: "(a :: nat) div b * b <= a"
369   apply (cases b)
370    prefer 2
371    apply (rule order_refl [THEN th2])
372   apply auto
373   done
375 lemmas sdl = split_div_lemma [THEN iffD1, symmetric]
377 lemma given_quot: "f > (0 :: nat) ==> (f * l + (f - 1)) div f = l"
378   by (rule sdl, assumption) (simp (no_asm))
380 lemma given_quot_alt: "f > (0 :: nat) ==> (l * f + f - Suc 0) div f = l"
381   apply (frule given_quot)
382   apply (rule trans)
383    prefer 2
384    apply (erule asm_rl)
385   apply (rule_tac f="%n. n div f" in arg_cong)
386   apply (simp add : mult_ac)
387   done
389 lemma diff_mod_le: "(a::nat) < d ==> b dvd d ==> a - a mod b <= d - b"
390   apply (unfold dvd_def)
391   apply clarify
392   apply (case_tac k)
393    apply clarsimp
394   apply clarify
395   apply (cases "b > 0")
396    apply (drule mult_commute [THEN xtr1])
397    apply (frule (1) td_gal_lt [THEN iffD1])
398    apply (clarsimp simp: le_simps)
399    apply (rule mult_div_cancel [THEN [2] xtr4])
400    apply (rule mult_mono)
401       apply auto
402   done
404 lemma less_le_mult':
405   "w * c < b * c ==> 0 \<le> c ==> (w + 1) * c \<le> b * (c::int)"
406   apply (rule mult_right_mono)
408    apply (erule (1) mult_right_less_imp_less)
409   apply assumption
410   done
412 lemmas less_le_mult = less_le_mult' [simplified left_distrib, simplified]
414 lemmas less_le_mult_minus = iffD2 [OF le_diff_eq less_le_mult,
415   simplified left_diff_distrib, standard]
417 lemma lrlem':
418   assumes d: "(i::nat) \<le> j \<or> m < j'"
419   assumes R1: "i * k \<le> j * k \<Longrightarrow> R"
420   assumes R2: "Suc m * k' \<le> j' * k' \<Longrightarrow> R"
421   shows "R" using d
422   apply safe
423    apply (rule R1, erule mult_le_mono1)
424   apply (rule R2, erule Suc_le_eq [THEN iffD2 [THEN mult_le_mono1]])
425   done
427 lemma lrlem: "(0::nat) < sc ==>
428     (sc - n + (n + lb * n) <= m * n) = (sc + lb * n <= m * n)"
429   apply safe
430    apply arith
431   apply (case_tac "sc >= n")
432    apply arith
433   apply (insert linorder_le_less_linear [of m lb])
434   apply (erule_tac k=n and k'=n in lrlem')
435    apply arith
436   apply simp
437   done
439 lemma gen_minus: "0 < n ==> f n = f (Suc (n - 1))"
440   by auto
442 lemma mpl_lem: "j <= (i :: nat) ==> k < j ==> i - j + k < i" by arith
444 lemma nonneg_mod_div:
445   "0 <= a ==> 0 <= b ==> 0 <= (a mod b :: int) & 0 <= a div b"
446   apply (cases "b = 0", clarsimp)
447   apply (auto intro: pos_imp_zdiv_nonneg_iff [THEN iffD2])
448   done
450 end