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--
added lemmas
kleing@24333
     1
(* 
kleing@24333
     2
  Author:  Jeremy Dawson, NICTA
huffman@24350
     3
*) 
kleing@24333
     4
huffman@24350
     5
header {* Useful Numerical Lemmas *}
kleing@24333
     6
haftmann@25592
     7
theory Num_Lemmas
haftmann@25592
     8
imports Main Parity
haftmann@25592
     9
begin
kleing@24333
    10
haftmann@26560
    11
lemma contentsI: "y = {x} ==> contents y = x" 
haftmann@26560
    12
  unfolding contents_def by auto -- {* FIXME move *}
huffman@26086
    13
huffman@24465
    14
lemmas split_split = prod.split [unfolded prod_case_split] 
huffman@24465
    15
lemmas split_split_asm = prod.split_asm [unfolded prod_case_split]
huffman@24465
    16
lemmas "split.splits" = split_split split_split_asm 
huffman@24465
    17
huffman@24465
    18
lemmas funpow_0 = funpow.simps(1)
kleing@24333
    19
lemmas funpow_Suc = funpow.simps(2)
huffman@24465
    20
chaieb@27570
    21
lemma nonemptyE: "S ~= {} ==> (!!x. x : S ==> R) ==> R" by auto
kleing@24333
    22
chaieb@27570
    23
lemma gt_or_eq_0: "0 < y \<or> 0 = (y::nat)" by arith 
kleing@24333
    24
huffman@24465
    25
declare iszero_0 [iff]
huffman@24465
    26
kleing@24333
    27
lemmas xtr1 = xtrans(1)
kleing@24333
    28
lemmas xtr2 = xtrans(2)
kleing@24333
    29
lemmas xtr3 = xtrans(3)
kleing@24333
    30
lemmas xtr4 = xtrans(4)
kleing@24333
    31
lemmas xtr5 = xtrans(5)
kleing@24333
    32
lemmas xtr6 = xtrans(6)
kleing@24333
    33
lemmas xtr7 = xtrans(7)
kleing@24333
    34
lemmas xtr8 = xtrans(8)
kleing@24333
    35
huffman@24465
    36
lemmas nat_simps = diff_add_inverse2 diff_add_inverse
huffman@24465
    37
lemmas nat_iffs = le_add1 le_add2
huffman@24465
    38
chaieb@27570
    39
lemma sum_imp_diff: "j = k + i ==> j - i = (k :: nat)" by arith
huffman@24465
    40
kleing@24333
    41
lemma nobm1:
kleing@24333
    42
  "0 < (number_of w :: nat) ==> 
chaieb@27570
    43
   number_of w - (1 :: nat) = number_of (Int.pred w)" 
kleing@24333
    44
  apply (unfold nat_number_of_def One_nat_def nat_1 [symmetric] pred_def)
kleing@24333
    45
  apply (simp add: number_of_eq nat_diff_distrib [symmetric])
kleing@24333
    46
  done
huffman@24465
    47
huffman@24465
    48
lemma of_int_power:
huffman@24465
    49
  "of_int (a ^ n) = (of_int a ^ n :: 'a :: {recpower, comm_ring_1})" 
huffman@24465
    50
  by (induct n) (auto simp add: power_Suc)
kleing@24333
    51
chaieb@27570
    52
lemma zless2: "0 < (2 :: int)" by arith
kleing@24333
    53
huffman@24465
    54
lemmas zless2p [simp] = zless2 [THEN zero_less_power]
huffman@24465
    55
lemmas zle2p [simp] = zless2p [THEN order_less_imp_le]
huffman@24465
    56
huffman@24465
    57
lemmas pos_mod_sign2 = zless2 [THEN pos_mod_sign [where b = "2::int"]]
huffman@24465
    58
lemmas pos_mod_bound2 = zless2 [THEN pos_mod_bound [where b = "2::int"]]
huffman@24465
    59
huffman@24465
    60
-- "the inverse(s) of @{text number_of}"
chaieb@27570
    61
lemma nmod2: "n mod (2::int) = 0 | n mod 2 = 1" by arith
kleing@24333
    62
kleing@24333
    63
lemma emep1:
kleing@24333
    64
  "even n ==> even d ==> 0 <= d ==> (n + 1) mod (d :: int) = (n mod d) + 1"
kleing@24333
    65
  apply (simp add: add_commute)
kleing@24333
    66
  apply (safe dest!: even_equiv_def [THEN iffD1])
kleing@24333
    67
  apply (subst pos_zmod_mult_2)
kleing@24333
    68
   apply arith
kleing@24333
    69
  apply (simp add: zmod_zmult_zmult1)
kleing@24333
    70
 done
kleing@24333
    71
kleing@24333
    72
lemmas eme1p = emep1 [simplified add_commute]
kleing@24333
    73
chaieb@27570
    74
lemma le_diff_eq': "(a \<le> c - b) = (b + a \<le> (c::int))" by arith
huffman@24465
    75
chaieb@27570
    76
lemma less_diff_eq': "(a < c - b) = (b + a < (c::int))" by arith
huffman@24465
    77
chaieb@27570
    78
lemma diff_le_eq': "(a - b \<le> c) = (a \<le> b + (c::int))" by arith
huffman@24465
    79
chaieb@27570
    80
lemma diff_less_eq': "(a - b < c) = (a < b + (c::int))" by arith
kleing@24333
    81
kleing@24333
    82
lemmas m1mod2k = zless2p [THEN zmod_minus1]
huffman@24465
    83
lemmas m1mod22k = mult_pos_pos [OF zless2 zless2p, THEN zmod_minus1]
kleing@24333
    84
lemmas p1mod22k' = zless2p [THEN order_less_imp_le, THEN pos_zmod_mult_2]
huffman@24465
    85
lemmas z1pmod2' = zero_le_one [THEN pos_zmod_mult_2, simplified]
huffman@24465
    86
lemmas z1pdiv2' = zero_le_one [THEN pos_zdiv_mult_2, simplified]
kleing@24333
    87
kleing@24333
    88
lemma p1mod22k:
kleing@24333
    89
  "(2 * b + 1) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n) + (1::int)"
kleing@24333
    90
  by (simp add: p1mod22k' add_commute)
huffman@24465
    91
huffman@24465
    92
lemma z1pmod2:
chaieb@27570
    93
  "(2 * b + 1) mod 2 = (1::int)" by arith
huffman@24465
    94
  
huffman@24465
    95
lemma z1pdiv2:
chaieb@27570
    96
  "(2 * b + 1) div 2 = (b::int)" by arith
kleing@24333
    97
nipkow@30031
    98
lemmas zdiv_le_dividend = xtr3 [OF div_by_1 [symmetric] zdiv_mono2,
kleing@24333
    99
  simplified int_one_le_iff_zero_less, simplified, standard]
huffman@24465
   100
  
huffman@24465
   101
lemma axxbyy:
huffman@24465
   102
  "a + m + m = b + n + n ==> (a = 0 | a = 1) ==> (b = 0 | b = 1) ==>  
chaieb@27570
   103
   a = b & m = (n :: int)" by arith
huffman@24465
   104
huffman@24465
   105
lemma axxmod2:
chaieb@27570
   106
  "(1 + x + x) mod 2 = (1 :: int) & (0 + x + x) mod 2 = (0 :: int)" by arith
huffman@24465
   107
huffman@24465
   108
lemma axxdiv2:
chaieb@27570
   109
  "(1 + x + x) div 2 = (x :: int) & (0 + x + x) div 2 = (x :: int)"  by arith
huffman@24465
   110
huffman@24465
   111
lemmas iszero_minus = trans [THEN trans,
huffman@24465
   112
  OF iszero_def neg_equal_0_iff_equal iszero_def [symmetric], standard]
kleing@24333
   113
kleing@24333
   114
lemmas zadd_diff_inverse = trans [OF diff_add_cancel [symmetric] add_commute,
kleing@24333
   115
  standard]
kleing@24333
   116
kleing@24333
   117
lemmas add_diff_cancel2 = add_commute [THEN diff_eq_eq [THEN iffD2], standard]
kleing@24333
   118
kleing@24333
   119
lemma zmod_uminus: "- ((a :: int) mod b) mod b = -a mod b"
kleing@24333
   120
  by (simp add : zmod_zminus1_eq_if)
huffman@24465
   121
huffman@24465
   122
lemma zmod_zsub_distrib: "((a::int) - b) mod c = (a mod c - b mod c) mod c"
huffman@24465
   123
  apply (unfold diff_int_def)
nipkow@29948
   124
  apply (rule trans [OF _ mod_add_eq [symmetric]])
nipkow@29948
   125
  apply (simp add: zmod_uminus mod_add_eq [symmetric])
huffman@24465
   126
  done
kleing@24333
   127
kleing@24333
   128
lemma zmod_zsub_right_eq: "((a::int) - b) mod c = (a - b mod c) mod c"
kleing@24333
   129
  apply (unfold diff_int_def)
nipkow@30034
   130
  apply (rule trans [OF _ mod_add_right_eq [symmetric]])
nipkow@30034
   131
  apply (simp add : zmod_uminus mod_add_right_eq [symmetric])
kleing@24333
   132
  done
kleing@24333
   133
wenzelm@25349
   134
lemma zmod_zsub_left_eq: "((a::int) - b) mod c = (a mod c - b) mod c"
nipkow@30034
   135
  by (rule mod_add_left_eq [where b = "- b", simplified diff_int_def [symmetric]])
wenzelm@25349
   136
kleing@24333
   137
lemma zmod_zsub_self [simp]: 
kleing@24333
   138
  "((b :: int) - a) mod a = b mod a"
kleing@24333
   139
  by (simp add: zmod_zsub_right_eq)
kleing@24333
   140
kleing@24333
   141
lemma zmod_zmult1_eq_rev:
kleing@24333
   142
  "b * a mod c = b mod c * a mod (c::int)"
kleing@24333
   143
  apply (simp add: mult_commute)
kleing@24333
   144
  apply (subst zmod_zmult1_eq)
kleing@24333
   145
  apply simp
kleing@24333
   146
  done
kleing@24333
   147
kleing@24333
   148
lemmas rdmods [symmetric] = zmod_uminus [symmetric]
nipkow@30034
   149
  zmod_zsub_left_eq zmod_zsub_right_eq mod_add_left_eq
nipkow@30034
   150
  mod_add_right_eq zmod_zmult1_eq zmod_zmult1_eq_rev
kleing@24333
   151
kleing@24333
   152
lemma mod_plus_right:
kleing@24333
   153
  "((a + x) mod m = (b + x) mod m) = (a mod m = b mod (m :: nat))"
kleing@24333
   154
  apply (induct x)
kleing@24333
   155
   apply (simp_all add: mod_Suc)
kleing@24333
   156
  apply arith
kleing@24333
   157
  done
kleing@24333
   158
huffman@24465
   159
lemma nat_minus_mod: "(n - n mod m) mod m = (0 :: nat)"
huffman@24465
   160
  by (induct n) (simp_all add : mod_Suc)
huffman@24465
   161
huffman@24465
   162
lemmas nat_minus_mod_plus_right = trans [OF nat_minus_mod mod_0 [symmetric],
huffman@24465
   163
  THEN mod_plus_right [THEN iffD2], standard, simplified]
huffman@24465
   164
nipkow@29948
   165
lemmas push_mods' = mod_add_eq [standard]
nipkow@29948
   166
  mod_mult_eq [standard] zmod_zsub_distrib [standard]
huffman@24465
   167
  zmod_uminus [symmetric, standard]
huffman@24465
   168
huffman@24465
   169
lemmas push_mods = push_mods' [THEN eq_reflection, standard]
huffman@24465
   170
lemmas pull_mods = push_mods [symmetric] rdmods [THEN eq_reflection, standard]
huffman@24465
   171
lemmas mod_simps = 
nipkow@30034
   172
  mod_mult_self2_is_0 [THEN eq_reflection]
nipkow@30034
   173
  mod_mult_self1_is_0 [THEN eq_reflection]
huffman@24465
   174
  mod_mod_trivial [THEN eq_reflection]
huffman@24465
   175
kleing@24333
   176
lemma nat_mod_eq:
kleing@24333
   177
  "!!b. b < n ==> a mod n = b mod n ==> a mod n = (b :: nat)" 
kleing@24333
   178
  by (induct a) auto
kleing@24333
   179
kleing@24333
   180
lemmas nat_mod_eq' = refl [THEN [2] nat_mod_eq]
kleing@24333
   181
kleing@24333
   182
lemma nat_mod_lem: 
kleing@24333
   183
  "(0 :: nat) < n ==> b < n = (b mod n = b)"
kleing@24333
   184
  apply safe
kleing@24333
   185
   apply (erule nat_mod_eq')
kleing@24333
   186
  apply (erule subst)
kleing@24333
   187
  apply (erule mod_less_divisor)
kleing@24333
   188
  done
kleing@24333
   189
kleing@24333
   190
lemma mod_nat_add: 
kleing@24333
   191
  "(x :: nat) < z ==> y < z ==> 
kleing@24333
   192
   (x + y) mod z = (if x + y < z then x + y else x + y - z)"
kleing@24333
   193
  apply (rule nat_mod_eq)
kleing@24333
   194
   apply auto
kleing@24333
   195
  apply (rule trans)
kleing@24333
   196
   apply (rule le_mod_geq)
kleing@24333
   197
   apply simp
kleing@24333
   198
  apply (rule nat_mod_eq')
kleing@24333
   199
  apply arith
kleing@24333
   200
  done
huffman@24465
   201
huffman@24465
   202
lemma mod_nat_sub: 
huffman@24465
   203
  "(x :: nat) < z ==> (x - y) mod z = x - y"
huffman@24465
   204
  by (rule nat_mod_eq') arith
kleing@24333
   205
kleing@24333
   206
lemma int_mod_lem: 
kleing@24333
   207
  "(0 :: int) < n ==> (0 <= b & b < n) = (b mod n = b)"
kleing@24333
   208
  apply safe
kleing@24333
   209
    apply (erule (1) mod_pos_pos_trivial)
kleing@24333
   210
   apply (erule_tac [!] subst)
kleing@24333
   211
   apply auto
kleing@24333
   212
  done
kleing@24333
   213
kleing@24333
   214
lemma int_mod_eq:
kleing@24333
   215
  "(0 :: int) <= b ==> b < n ==> a mod n = b mod n ==> a mod n = b"
kleing@24333
   216
  by clarsimp (rule mod_pos_pos_trivial)
kleing@24333
   217
kleing@24333
   218
lemmas int_mod_eq' = refl [THEN [3] int_mod_eq]
kleing@24333
   219
kleing@24333
   220
lemma int_mod_le: "0 <= a ==> 0 < (n :: int) ==> a mod n <= a"
kleing@24333
   221
  apply (cases "a < n")
kleing@24333
   222
   apply (auto dest: mod_pos_pos_trivial pos_mod_bound [where a=a])
kleing@24333
   223
  done
kleing@24333
   224
wenzelm@25349
   225
lemma int_mod_le': "0 <= b - n ==> 0 < (n :: int) ==> b mod n <= b - n"
wenzelm@25349
   226
  by (rule int_mod_le [where a = "b - n" and n = n, simplified])
kleing@24333
   227
kleing@24333
   228
lemma int_mod_ge: "a < n ==> 0 < (n :: int) ==> a <= a mod n"
kleing@24333
   229
  apply (cases "0 <= a")
kleing@24333
   230
   apply (drule (1) mod_pos_pos_trivial)
kleing@24333
   231
   apply simp
kleing@24333
   232
  apply (rule order_trans [OF _ pos_mod_sign])
kleing@24333
   233
   apply simp
kleing@24333
   234
  apply assumption
kleing@24333
   235
  done
kleing@24333
   236
wenzelm@25349
   237
lemma int_mod_ge': "b < 0 ==> 0 < (n :: int) ==> b + n <= b mod n"
wenzelm@25349
   238
  by (rule int_mod_ge [where a = "b + n" and n = n, simplified])
kleing@24333
   239
kleing@24333
   240
lemma mod_add_if_z:
kleing@24333
   241
  "(x :: int) < z ==> y < z ==> 0 <= y ==> 0 <= x ==> 0 <= z ==> 
kleing@24333
   242
   (x + y) mod z = (if x + y < z then x + y else x + y - z)"
kleing@24333
   243
  by (auto intro: int_mod_eq)
kleing@24333
   244
kleing@24333
   245
lemma mod_sub_if_z:
kleing@24333
   246
  "(x :: int) < z ==> y < z ==> 0 <= y ==> 0 <= x ==> 0 <= z ==> 
kleing@24333
   247
   (x - y) mod z = (if y <= x then x - y else x - y + z)"
kleing@24333
   248
  by (auto intro: int_mod_eq)
huffman@24465
   249
huffman@24465
   250
lemmas zmde = zmod_zdiv_equality [THEN diff_eq_eq [THEN iffD2], symmetric]
huffman@24465
   251
lemmas mcl = mult_cancel_left [THEN iffD1, THEN make_pos_rule]
huffman@24465
   252
huffman@24465
   253
(* already have this for naturals, div_mult_self1/2, but not for ints *)
huffman@24465
   254
lemma zdiv_mult_self: "m ~= (0 :: int) ==> (a + m * n) div m = a div m + n"
huffman@24465
   255
  apply (rule mcl)
huffman@24465
   256
   prefer 2
huffman@24465
   257
   apply (erule asm_rl)
huffman@24465
   258
  apply (simp add: zmde ring_distribs)
huffman@24465
   259
  done
huffman@24465
   260
huffman@24465
   261
(** Rep_Integ **)
huffman@24465
   262
lemma eqne: "equiv A r ==> X : A // r ==> X ~= {}"
nipkow@30198
   263
  unfolding equiv_def refl_on_def quotient_def Image_def by auto
huffman@24465
   264
huffman@24465
   265
lemmas Rep_Integ_ne = Integ.Rep_Integ 
huffman@24465
   266
  [THEN equiv_intrel [THEN eqne, simplified Integ_def [symmetric]], standard]
huffman@24465
   267
huffman@24465
   268
lemmas riq = Integ.Rep_Integ [simplified Integ_def]
huffman@24465
   269
lemmas intrel_refl = refl [THEN equiv_intrel_iff [THEN iffD1], standard]
huffman@24465
   270
lemmas Rep_Integ_equiv = quotient_eq_iff
huffman@24465
   271
  [OF equiv_intrel riq riq, simplified Integ.Rep_Integ_inject, standard]
huffman@24465
   272
lemmas Rep_Integ_same = 
huffman@24465
   273
  Rep_Integ_equiv [THEN intrel_refl [THEN rev_iffD2], standard]
huffman@24465
   274
huffman@24465
   275
lemma RI_int: "(a, 0) : Rep_Integ (int a)"
huffman@24465
   276
  unfolding int_def by auto
huffman@24465
   277
huffman@24465
   278
lemmas RI_intrel [simp] = UNIV_I [THEN quotientI,
huffman@24465
   279
  THEN Integ.Abs_Integ_inverse [simplified Integ_def], standard]
huffman@24465
   280
huffman@24465
   281
lemma RI_minus: "(a, b) : Rep_Integ x ==> (b, a) : Rep_Integ (- x)"
huffman@24465
   282
  apply (rule_tac z=x in eq_Abs_Integ)
huffman@24465
   283
  apply (clarsimp simp: minus)
huffman@24465
   284
  done
kleing@24333
   285
huffman@24465
   286
lemma RI_add: 
huffman@24465
   287
  "(a, b) : Rep_Integ x ==> (c, d) : Rep_Integ y ==> 
huffman@24465
   288
   (a + c, b + d) : Rep_Integ (x + y)"
huffman@24465
   289
  apply (rule_tac z=x in eq_Abs_Integ)
huffman@24465
   290
  apply (rule_tac z=y in eq_Abs_Integ) 
huffman@24465
   291
  apply (clarsimp simp: add)
huffman@24465
   292
  done
huffman@24465
   293
huffman@24465
   294
lemma mem_same: "a : S ==> a = b ==> b : S"
huffman@24465
   295
  by fast
huffman@24465
   296
huffman@24465
   297
(* two alternative proofs of this *)
huffman@24465
   298
lemma RI_eq_diff': "(a, b) : Rep_Integ (int a - int b)"
huffman@24465
   299
  apply (unfold diff_def)
huffman@24465
   300
  apply (rule mem_same)
huffman@24465
   301
   apply (rule RI_minus RI_add RI_int)+
huffman@24465
   302
  apply simp
huffman@24465
   303
  done
huffman@24465
   304
huffman@24465
   305
lemma RI_eq_diff: "((a, b) : Rep_Integ x) = (int a - int b = x)"
huffman@24465
   306
  apply safe
huffman@24465
   307
   apply (rule Rep_Integ_same)
huffman@24465
   308
    prefer 2
huffman@24465
   309
    apply (erule asm_rl)
huffman@24465
   310
   apply (rule RI_eq_diff')+
huffman@24465
   311
  done
huffman@24465
   312
huffman@24465
   313
lemma mod_power_lem:
huffman@24465
   314
  "a > 1 ==> a ^ n mod a ^ m = (if m <= n then 0 else (a :: int) ^ n)"
huffman@24465
   315
  apply clarsimp
huffman@24465
   316
  apply safe
nipkow@30042
   317
   apply (simp add: dvd_eq_mod_eq_0 [symmetric])
huffman@24465
   318
   apply (drule le_iff_add [THEN iffD1])
huffman@24465
   319
   apply (force simp: zpower_zadd_distrib)
huffman@24465
   320
  apply (rule mod_pos_pos_trivial)
nipkow@25875
   321
   apply (simp)
huffman@24465
   322
  apply (rule power_strict_increasing)
huffman@24465
   323
   apply auto
huffman@24465
   324
  done
kleing@24333
   325
chaieb@27570
   326
lemma min_pm [simp]: "min a b + (a - b) = (a :: nat)" by arith
kleing@24333
   327
  
kleing@24333
   328
lemmas min_pm1 [simp] = trans [OF add_commute min_pm]
kleing@24333
   329
chaieb@27570
   330
lemma rev_min_pm [simp]: "min b a + (a - b) = (a::nat)" by arith
kleing@24333
   331
kleing@24333
   332
lemmas rev_min_pm1 [simp] = trans [OF add_commute rev_min_pm]
kleing@24333
   333
huffman@24465
   334
lemma pl_pl_rels: 
huffman@24465
   335
  "a + b = c + d ==> 
chaieb@27570
   336
   a >= c & b <= d | a <= c & b >= (d :: nat)" by arith
huffman@24465
   337
huffman@24465
   338
lemmas pl_pl_rels' = add_commute [THEN [2] trans, THEN pl_pl_rels]
huffman@24465
   339
chaieb@27570
   340
lemma minus_eq: "(m - k = m) = (k = 0 | m = (0 :: nat))"  by arith
huffman@24465
   341
chaieb@27570
   342
lemma pl_pl_mm: "(a :: nat) + b = c + d ==> a - c = d - b"  by arith
huffman@24465
   343
huffman@24465
   344
lemmas pl_pl_mm' = add_commute [THEN [2] trans, THEN pl_pl_mm]
huffman@24465
   345
 
chaieb@27570
   346
lemma min_minus [simp] : "min m (m - k) = (m - k :: nat)" by arith
kleing@24333
   347
  
kleing@24333
   348
lemmas min_minus' [simp] = trans [OF min_max.inf_commute min_minus]
kleing@24333
   349
huffman@24465
   350
lemma nat_no_eq_iff: 
huffman@24465
   351
  "(number_of b :: int) >= 0 ==> (number_of c :: int) >= 0 ==> 
chaieb@27570
   352
   (number_of b = (number_of c :: nat)) = (b = c)" 
chaieb@27570
   353
  apply (unfold nat_number_of_def) 
huffman@24465
   354
  apply safe
huffman@24465
   355
  apply (drule (2) eq_nat_nat_iff [THEN iffD1])
huffman@24465
   356
  apply (simp add: number_of_eq)
huffman@24465
   357
  done
huffman@24465
   358
kleing@24333
   359
lemmas dme = box_equals [OF div_mod_equality add_0_right add_0_right]
kleing@24333
   360
lemmas dtle = xtr3 [OF dme [symmetric] le_add1]
kleing@24333
   361
lemmas th2 = order_trans [OF order_refl [THEN [2] mult_le_mono] dtle]
kleing@24333
   362
kleing@24333
   363
lemma td_gal: 
kleing@24333
   364
  "0 < c ==> (a >= b * c) = (a div c >= (b :: nat))"
kleing@24333
   365
  apply safe
kleing@24333
   366
   apply (erule (1) xtr4 [OF div_le_mono div_mult_self_is_m])
kleing@24333
   367
  apply (erule th2)
kleing@24333
   368
  done
kleing@24333
   369
  
haftmann@26072
   370
lemmas td_gal_lt = td_gal [simplified not_less [symmetric], simplified]
kleing@24333
   371
kleing@24333
   372
lemma div_mult_le: "(a :: nat) div b * b <= a"
kleing@24333
   373
  apply (cases b)
kleing@24333
   374
   prefer 2
kleing@24333
   375
   apply (rule order_refl [THEN th2])
kleing@24333
   376
  apply auto
kleing@24333
   377
  done
kleing@24333
   378
kleing@24333
   379
lemmas sdl = split_div_lemma [THEN iffD1, symmetric]
kleing@24333
   380
kleing@24333
   381
lemma given_quot: "f > (0 :: nat) ==> (f * l + (f - 1)) div f = l"
kleing@24333
   382
  by (rule sdl, assumption) (simp (no_asm))
kleing@24333
   383
kleing@24333
   384
lemma given_quot_alt: "f > (0 :: nat) ==> (l * f + f - Suc 0) div f = l"
kleing@24333
   385
  apply (frule given_quot)
kleing@24333
   386
  apply (rule trans)
kleing@24333
   387
   prefer 2
kleing@24333
   388
   apply (erule asm_rl)
kleing@24333
   389
  apply (rule_tac f="%n. n div f" in arg_cong)
kleing@24333
   390
  apply (simp add : mult_ac)
kleing@24333
   391
  done
kleing@24333
   392
    
huffman@24465
   393
lemma diff_mod_le: "(a::nat) < d ==> b dvd d ==> a - a mod b <= d - b"
huffman@24465
   394
  apply (unfold dvd_def)
huffman@24465
   395
  apply clarify
huffman@24465
   396
  apply (case_tac k)
huffman@24465
   397
   apply clarsimp
huffman@24465
   398
  apply clarify
huffman@24465
   399
  apply (cases "b > 0")
huffman@24465
   400
   apply (drule mult_commute [THEN xtr1])
huffman@24465
   401
   apply (frule (1) td_gal_lt [THEN iffD1])
huffman@24465
   402
   apply (clarsimp simp: le_simps)
huffman@24465
   403
   apply (rule mult_div_cancel [THEN [2] xtr4])
huffman@24465
   404
   apply (rule mult_mono)
huffman@24465
   405
      apply auto
huffman@24465
   406
  done
huffman@24465
   407
kleing@24333
   408
lemma less_le_mult':
kleing@24333
   409
  "w * c < b * c ==> 0 \<le> c ==> (w + 1) * c \<le> b * (c::int)"
kleing@24333
   410
  apply (rule mult_right_mono)
kleing@24333
   411
   apply (rule zless_imp_add1_zle)
kleing@24333
   412
   apply (erule (1) mult_right_less_imp_less)
kleing@24333
   413
  apply assumption
kleing@24333
   414
  done
kleing@24333
   415
kleing@24333
   416
lemmas less_le_mult = less_le_mult' [simplified left_distrib, simplified]
huffman@24465
   417
huffman@24465
   418
lemmas less_le_mult_minus = iffD2 [OF le_diff_eq less_le_mult, 
huffman@24465
   419
  simplified left_diff_distrib, standard]
kleing@24333
   420
kleing@24333
   421
lemma lrlem':
kleing@24333
   422
  assumes d: "(i::nat) \<le> j \<or> m < j'"
kleing@24333
   423
  assumes R1: "i * k \<le> j * k \<Longrightarrow> R"
kleing@24333
   424
  assumes R2: "Suc m * k' \<le> j' * k' \<Longrightarrow> R"
kleing@24333
   425
  shows "R" using d
kleing@24333
   426
  apply safe
kleing@24333
   427
   apply (rule R1, erule mult_le_mono1)
kleing@24333
   428
  apply (rule R2, erule Suc_le_eq [THEN iffD2 [THEN mult_le_mono1]])
kleing@24333
   429
  done
kleing@24333
   430
kleing@24333
   431
lemma lrlem: "(0::nat) < sc ==>
kleing@24333
   432
    (sc - n + (n + lb * n) <= m * n) = (sc + lb * n <= m * n)"
kleing@24333
   433
  apply safe
kleing@24333
   434
   apply arith
kleing@24333
   435
  apply (case_tac "sc >= n")
kleing@24333
   436
   apply arith
kleing@24333
   437
  apply (insert linorder_le_less_linear [of m lb])
kleing@24333
   438
  apply (erule_tac k=n and k'=n in lrlem')
kleing@24333
   439
   apply arith
kleing@24333
   440
  apply simp
kleing@24333
   441
  done
kleing@24333
   442
kleing@24333
   443
lemma gen_minus: "0 < n ==> f n = f (Suc (n - 1))"
kleing@24333
   444
  by auto
kleing@24333
   445
chaieb@27570
   446
lemma mpl_lem: "j <= (i :: nat) ==> k < j ==> i - j + k < i" by arith
kleing@24333
   447
huffman@24465
   448
lemma nonneg_mod_div:
huffman@24465
   449
  "0 <= a ==> 0 <= b ==> 0 <= (a mod b :: int) & 0 <= a div b"
huffman@24465
   450
  apply (cases "b = 0", clarsimp)
huffman@24465
   451
  apply (auto intro: pos_imp_zdiv_nonneg_iff [THEN iffD2])
huffman@24465
   452
  done
huffman@24399
   453
kleing@24333
   454
end