src/HOL/Word/Num_Lemmas.thy
 author nipkow Fri Mar 06 17:38:47 2009 +0100 (2009-03-06) changeset 30313 b2441b0c8d38 parent 30242 aea5d7fa7ef5 child 30445 757ba2bb2b39 permissions -rw-r--r--
1 (*
2   Author:  Jeremy Dawson, NICTA
3 *)
5 header {* Useful Numerical Lemmas *}
7 theory Num_Lemmas
8 imports Main Parity
9 begin
11 lemma contentsI: "y = {x} ==> contents y = x"
12   unfolding contents_def by auto -- {* FIXME move *}
14 lemmas split_split = prod.split [unfolded prod_case_split]
15 lemmas split_split_asm = prod.split_asm [unfolded prod_case_split]
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 of_int_power:
49   "of_int (a ^ n) = (of_int a ^ n :: 'a :: {recpower, comm_ring_1})"
50   by (induct n) (auto simp add: power_Suc)
52 lemma zless2: "0 < (2 :: int)" by arith
54 lemmas zless2p [simp] = zless2 [THEN zero_less_power]
55 lemmas zle2p [simp] = zless2p [THEN order_less_imp_le]
57 lemmas pos_mod_sign2 = zless2 [THEN pos_mod_sign [where b = "2::int"]]
58 lemmas pos_mod_bound2 = zless2 [THEN pos_mod_bound [where b = "2::int"]]
60 -- "the inverse(s) of @{text number_of}"
61 lemma nmod2: "n mod (2::int) = 0 | n mod 2 = 1" by arith
63 lemma emep1:
64   "even n ==> even d ==> 0 <= d ==> (n + 1) mod (d :: int) = (n mod d) + 1"
66   apply (safe dest!: even_equiv_def [THEN iffD1])
67   apply (subst pos_zmod_mult_2)
68    apply arith
69   apply (simp add: zmod_zmult_zmult1)
70  done
72 lemmas eme1p = emep1 [simplified add_commute]
74 lemma le_diff_eq': "(a \<le> c - b) = (b + a \<le> (c::int))" by arith
76 lemma less_diff_eq': "(a < c - b) = (b + a < (c::int))" by arith
78 lemma diff_le_eq': "(a - b \<le> c) = (a \<le> b + (c::int))" by arith
80 lemma diff_less_eq': "(a - b < c) = (a < b + (c::int))" by arith
82 lemmas m1mod2k = zless2p [THEN zmod_minus1]
83 lemmas m1mod22k = mult_pos_pos [OF zless2 zless2p, THEN zmod_minus1]
84 lemmas p1mod22k' = zless2p [THEN order_less_imp_le, THEN pos_zmod_mult_2]
85 lemmas z1pmod2' = zero_le_one [THEN pos_zmod_mult_2, simplified]
86 lemmas z1pdiv2' = zero_le_one [THEN pos_zdiv_mult_2, simplified]
88 lemma p1mod22k:
89   "(2 * b + 1) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n) + (1::int)"
92 lemma z1pmod2:
93   "(2 * b + 1) mod 2 = (1::int)" by arith
95 lemma z1pdiv2:
96   "(2 * b + 1) div 2 = (b::int)" by arith
98 lemmas zdiv_le_dividend = xtr3 [OF div_by_1 [symmetric] zdiv_mono2,
99   simplified int_one_le_iff_zero_less, simplified, standard]
101 lemma axxbyy:
102   "a + m + m = b + n + n ==> (a = 0 | a = 1) ==> (b = 0 | b = 1) ==>
103    a = b & m = (n :: int)" by arith
105 lemma axxmod2:
106   "(1 + x + x) mod 2 = (1 :: int) & (0 + x + x) mod 2 = (0 :: int)" by arith
108 lemma axxdiv2:
109   "(1 + x + x) div 2 = (x :: int) & (0 + x + x) div 2 = (x :: int)"  by arith
111 lemmas iszero_minus = trans [THEN trans,
112   OF iszero_def neg_equal_0_iff_equal iszero_def [symmetric], standard]
115   standard]
117 lemmas add_diff_cancel2 = add_commute [THEN diff_eq_eq [THEN iffD2], standard]
119 lemma zmod_uminus: "- ((a :: int) mod b) mod b = -a mod b"
120   by (simp add : zmod_zminus1_eq_if)
122 lemma zmod_zsub_distrib: "((a::int) - b) mod c = (a mod c - b mod c) mod c"
123   apply (unfold diff_int_def)
124   apply (rule trans [OF _ mod_add_eq [symmetric]])
125   apply (simp add: zmod_uminus mod_add_eq [symmetric])
126   done
128 lemma zmod_zsub_right_eq: "((a::int) - b) mod c = (a - b mod c) mod c"
129   apply (unfold diff_int_def)
130   apply (rule trans [OF _ mod_add_right_eq [symmetric]])
131   apply (simp add : zmod_uminus mod_add_right_eq [symmetric])
132   done
134 lemma zmod_zsub_left_eq: "((a::int) - b) mod c = (a mod c - b) mod c"
135   by (rule mod_add_left_eq [where b = "- b", simplified diff_int_def [symmetric]])
137 lemma zmod_zsub_self [simp]:
138   "((b :: int) - a) mod a = b mod a"
139   by (simp add: zmod_zsub_right_eq)
141 lemma zmod_zmult1_eq_rev:
142   "b * a mod c = b mod c * a mod (c::int)"
143   apply (simp add: mult_commute)
144   apply (subst zmod_zmult1_eq)
145   apply simp
146   done
148 lemmas rdmods [symmetric] = zmod_uminus [symmetric]
149   zmod_zsub_left_eq zmod_zsub_right_eq mod_add_left_eq
150   mod_add_right_eq zmod_zmult1_eq zmod_zmult1_eq_rev
152 lemma mod_plus_right:
153   "((a + x) mod m = (b + x) mod m) = (a mod m = b mod (m :: nat))"
154   apply (induct x)
155    apply (simp_all add: mod_Suc)
156   apply arith
157   done
159 lemma nat_minus_mod: "(n - n mod m) mod m = (0 :: nat)"
160   by (induct n) (simp_all add : mod_Suc)
162 lemmas nat_minus_mod_plus_right = trans [OF nat_minus_mod mod_0 [symmetric],
163   THEN mod_plus_right [THEN iffD2], standard, simplified]
165 lemmas push_mods' = mod_add_eq [standard]
166   mod_mult_eq [standard] zmod_zsub_distrib [standard]
167   zmod_uminus [symmetric, standard]
169 lemmas push_mods = push_mods' [THEN eq_reflection, standard]
170 lemmas pull_mods = push_mods [symmetric] rdmods [THEN eq_reflection, standard]
171 lemmas mod_simps =
172   mod_mult_self2_is_0 [THEN eq_reflection]
173   mod_mult_self1_is_0 [THEN eq_reflection]
174   mod_mod_trivial [THEN eq_reflection]
176 lemma nat_mod_eq:
177   "!!b. b < n ==> a mod n = b mod n ==> a mod n = (b :: nat)"
178   by (induct a) auto
180 lemmas nat_mod_eq' = refl [THEN [2] nat_mod_eq]
182 lemma nat_mod_lem:
183   "(0 :: nat) < n ==> b < n = (b mod n = b)"
184   apply safe
185    apply (erule nat_mod_eq')
186   apply (erule subst)
187   apply (erule mod_less_divisor)
188   done
191   "(x :: nat) < z ==> y < z ==>
192    (x + y) mod z = (if x + y < z then x + y else x + y - z)"
193   apply (rule nat_mod_eq)
194    apply auto
195   apply (rule trans)
196    apply (rule le_mod_geq)
197    apply simp
198   apply (rule nat_mod_eq')
199   apply arith
200   done
202 lemma mod_nat_sub:
203   "(x :: nat) < z ==> (x - y) mod z = x - y"
204   by (rule nat_mod_eq') arith
206 lemma int_mod_lem:
207   "(0 :: int) < n ==> (0 <= b & b < n) = (b mod n = b)"
208   apply safe
209     apply (erule (1) mod_pos_pos_trivial)
210    apply (erule_tac [!] subst)
211    apply auto
212   done
214 lemma int_mod_eq:
215   "(0 :: int) <= b ==> b < n ==> a mod n = b mod n ==> a mod n = b"
216   by clarsimp (rule mod_pos_pos_trivial)
218 lemmas int_mod_eq' = refl [THEN [3] int_mod_eq]
220 lemma int_mod_le: "0 <= a ==> 0 < (n :: int) ==> a mod n <= a"
221   apply (cases "a < n")
222    apply (auto dest: mod_pos_pos_trivial pos_mod_bound [where a=a])
223   done
225 lemma int_mod_le': "0 <= b - n ==> 0 < (n :: int) ==> b mod n <= b - n"
226   by (rule int_mod_le [where a = "b - n" and n = n, simplified])
228 lemma int_mod_ge: "a < n ==> 0 < (n :: int) ==> a <= a mod n"
229   apply (cases "0 <= a")
230    apply (drule (1) mod_pos_pos_trivial)
231    apply simp
232   apply (rule order_trans [OF _ pos_mod_sign])
233    apply simp
234   apply assumption
235   done
237 lemma int_mod_ge': "b < 0 ==> 0 < (n :: int) ==> b + n <= b mod n"
238   by (rule int_mod_ge [where a = "b + n" and n = n, simplified])
241   "(x :: int) < z ==> y < z ==> 0 <= y ==> 0 <= x ==> 0 <= z ==>
242    (x + y) mod z = (if x + y < z then x + y else x + y - z)"
243   by (auto intro: int_mod_eq)
245 lemma mod_sub_if_z:
246   "(x :: int) < z ==> y < z ==> 0 <= y ==> 0 <= x ==> 0 <= z ==>
247    (x - y) mod z = (if y <= x then x - y else x - y + z)"
248   by (auto intro: int_mod_eq)
250 lemmas zmde = zmod_zdiv_equality [THEN diff_eq_eq [THEN iffD2], symmetric]
251 lemmas mcl = mult_cancel_left [THEN iffD1, THEN make_pos_rule]
253 (* already have this for naturals, div_mult_self1/2, but not for ints *)
254 lemma zdiv_mult_self: "m ~= (0 :: int) ==> (a + m * n) div m = a div m + n"
255   apply (rule mcl)
256    prefer 2
257    apply (erule asm_rl)
258   apply (simp add: zmde ring_distribs)
259   done
261 (** Rep_Integ **)
262 lemma eqne: "equiv A r ==> X : A // r ==> X ~= {}"
263   unfolding equiv_def refl_on_def quotient_def Image_def by auto
265 lemmas Rep_Integ_ne = Integ.Rep_Integ
266   [THEN equiv_intrel [THEN eqne, simplified Integ_def [symmetric]], standard]
268 lemmas riq = Integ.Rep_Integ [simplified Integ_def]
269 lemmas intrel_refl = refl [THEN equiv_intrel_iff [THEN iffD1], standard]
270 lemmas Rep_Integ_equiv = quotient_eq_iff
271   [OF equiv_intrel riq riq, simplified Integ.Rep_Integ_inject, standard]
272 lemmas Rep_Integ_same =
273   Rep_Integ_equiv [THEN intrel_refl [THEN rev_iffD2], standard]
275 lemma RI_int: "(a, 0) : Rep_Integ (int a)"
276   unfolding int_def by auto
278 lemmas RI_intrel [simp] = UNIV_I [THEN quotientI,
279   THEN Integ.Abs_Integ_inverse [simplified Integ_def], standard]
281 lemma RI_minus: "(a, b) : Rep_Integ x ==> (b, a) : Rep_Integ (- x)"
282   apply (rule_tac z=x in eq_Abs_Integ)
283   apply (clarsimp simp: minus)
284   done
287   "(a, b) : Rep_Integ x ==> (c, d) : Rep_Integ y ==>
288    (a + c, b + d) : Rep_Integ (x + y)"
289   apply (rule_tac z=x in eq_Abs_Integ)
290   apply (rule_tac z=y in eq_Abs_Integ)
291   apply (clarsimp simp: add)
292   done
294 lemma mem_same: "a : S ==> a = b ==> b : S"
295   by fast
297 (* two alternative proofs of this *)
298 lemma RI_eq_diff': "(a, b) : Rep_Integ (int a - int b)"
299   apply (unfold diff_def)
300   apply (rule mem_same)
301    apply (rule RI_minus RI_add RI_int)+
302   apply simp
303   done
305 lemma RI_eq_diff: "((a, b) : Rep_Integ x) = (int a - int b = x)"
306   apply safe
307    apply (rule Rep_Integ_same)
308     prefer 2
309     apply (erule asm_rl)
310    apply (rule RI_eq_diff')+
311   done
313 lemma mod_power_lem:
314   "a > 1 ==> a ^ n mod a ^ m = (if m <= n then 0 else (a :: int) ^ n)"
315   apply clarsimp
316   apply safe
317    apply (simp add: dvd_eq_mod_eq_0 [symmetric])
318    apply (drule le_iff_add [THEN iffD1])
319    apply (force simp: zpower_zadd_distrib)
320   apply (rule mod_pos_pos_trivial)
321    apply (simp)
322   apply (rule power_strict_increasing)
323    apply auto
324   done
326 lemma min_pm [simp]: "min a b + (a - b) = (a :: nat)" by arith
328 lemmas min_pm1 [simp] = trans [OF add_commute min_pm]
330 lemma rev_min_pm [simp]: "min b a + (a - b) = (a::nat)" by arith
332 lemmas rev_min_pm1 [simp] = trans [OF add_commute rev_min_pm]
334 lemma pl_pl_rels:
335   "a + b = c + d ==>
336    a >= c & b <= d | a <= c & b >= (d :: nat)" by arith
338 lemmas pl_pl_rels' = add_commute [THEN [2] trans, THEN pl_pl_rels]
340 lemma minus_eq: "(m - k = m) = (k = 0 | m = (0 :: nat))"  by arith
342 lemma pl_pl_mm: "(a :: nat) + b = c + d ==> a - c = d - b"  by arith
344 lemmas pl_pl_mm' = add_commute [THEN [2] trans, THEN pl_pl_mm]
346 lemma min_minus [simp] : "min m (m - k) = (m - k :: nat)" by arith
348 lemmas min_minus' [simp] = trans [OF min_max.inf_commute min_minus]
350 lemma nat_no_eq_iff:
351   "(number_of b :: int) >= 0 ==> (number_of c :: int) >= 0 ==>
352    (number_of b = (number_of c :: nat)) = (b = c)"
353   apply (unfold nat_number_of_def)
354   apply safe
355   apply (drule (2) eq_nat_nat_iff [THEN iffD1])
356   apply (simp add: number_of_eq)
357   done
359 lemmas dme = box_equals [OF div_mod_equality add_0_right add_0_right]
360 lemmas dtle = xtr3 [OF dme [symmetric] le_add1]
361 lemmas th2 = order_trans [OF order_refl [THEN [2] mult_le_mono] dtle]
363 lemma td_gal:
364   "0 < c ==> (a >= b * c) = (a div c >= (b :: nat))"
365   apply safe
366    apply (erule (1) xtr4 [OF div_le_mono div_mult_self_is_m])
367   apply (erule th2)
368   done
370 lemmas td_gal_lt = td_gal [simplified not_less [symmetric], simplified]
372 lemma div_mult_le: "(a :: nat) div b * b <= a"
373   apply (cases b)
374    prefer 2
375    apply (rule order_refl [THEN th2])
376   apply auto
377   done
379 lemmas sdl = split_div_lemma [THEN iffD1, symmetric]
381 lemma given_quot: "f > (0 :: nat) ==> (f * l + (f - 1)) div f = l"
382   by (rule sdl, assumption) (simp (no_asm))
384 lemma given_quot_alt: "f > (0 :: nat) ==> (l * f + f - Suc 0) div f = l"
385   apply (frule given_quot)
386   apply (rule trans)
387    prefer 2
388    apply (erule asm_rl)
389   apply (rule_tac f="%n. n div f" in arg_cong)
390   apply (simp add : mult_ac)
391   done
393 lemma diff_mod_le: "(a::nat) < d ==> b dvd d ==> a - a mod b <= d - b"
394   apply (unfold dvd_def)
395   apply clarify
396   apply (case_tac k)
397    apply clarsimp
398   apply clarify
399   apply (cases "b > 0")
400    apply (drule mult_commute [THEN xtr1])
401    apply (frule (1) td_gal_lt [THEN iffD1])
402    apply (clarsimp simp: le_simps)
403    apply (rule mult_div_cancel [THEN [2] xtr4])
404    apply (rule mult_mono)
405       apply auto
406   done
408 lemma less_le_mult':
409   "w * c < b * c ==> 0 \<le> c ==> (w + 1) * c \<le> b * (c::int)"
410   apply (rule mult_right_mono)
411    apply (rule zless_imp_add1_zle)
412    apply (erule (1) mult_right_less_imp_less)
413   apply assumption
414   done
416 lemmas less_le_mult = less_le_mult' [simplified left_distrib, simplified]
418 lemmas less_le_mult_minus = iffD2 [OF le_diff_eq less_le_mult,
419   simplified left_diff_distrib, standard]
421 lemma lrlem':
422   assumes d: "(i::nat) \<le> j \<or> m < j'"
423   assumes R1: "i * k \<le> j * k \<Longrightarrow> R"
424   assumes R2: "Suc m * k' \<le> j' * k' \<Longrightarrow> R"
425   shows "R" using d
426   apply safe
427    apply (rule R1, erule mult_le_mono1)
428   apply (rule R2, erule Suc_le_eq [THEN iffD2 [THEN mult_le_mono1]])
429   done
431 lemma lrlem: "(0::nat) < sc ==>
432     (sc - n + (n + lb * n) <= m * n) = (sc + lb * n <= m * n)"
433   apply safe
434    apply arith
435   apply (case_tac "sc >= n")
436    apply arith
437   apply (insert linorder_le_less_linear [of m lb])
438   apply (erule_tac k=n and k'=n in lrlem')
439    apply arith
440   apply simp
441   done
443 lemma gen_minus: "0 < n ==> f n = f (Suc (n - 1))"
444   by auto
446 lemma mpl_lem: "j <= (i :: nat) ==> k < j ==> i - j + k < i" by arith
448 lemma nonneg_mod_div:
449   "0 <= a ==> 0 <= b ==> 0 <= (a mod b :: int) & 0 <= a div b"
450   apply (cases "b = 0", clarsimp)
451   apply (auto intro: pos_imp_zdiv_nonneg_iff [THEN iffD2])
452   done
454 end