src/HOL/Nat.thy
author wenzelm
Fri Mar 28 19:43:54 2008 +0100 (2008-03-28)
changeset 26462 dac4e2bce00d
parent 26335 961bbcc9d85b
child 26748 4d51ddd6aa5c
permissions -rw-r--r--
avoid rebinding of existing facts;
clasohm@923
     1
(*  Title:      HOL/Nat.thy
clasohm@923
     2
    ID:         $Id$
wenzelm@21243
     3
    Author:     Tobias Nipkow and Lawrence C Paulson and Markus Wenzel
clasohm@923
     4
wenzelm@9436
     5
Type "nat" is a linear order, and a datatype; arithmetic operators + -
wenzelm@9436
     6
and * (for div, mod and dvd, see theory Divides).
clasohm@923
     7
*)
clasohm@923
     8
berghofe@13449
     9
header {* Natural numbers *}
berghofe@13449
    10
nipkow@15131
    11
theory Nat
haftmann@26072
    12
imports Inductive Ring_and_Field
haftmann@23263
    13
uses
haftmann@23263
    14
  "~~/src/Tools/rat.ML"
haftmann@23263
    15
  "~~/src/Provers/Arith/cancel_sums.ML"
haftmann@23263
    16
  ("arith_data.ML")
wenzelm@24091
    17
  "~~/src/Provers/Arith/fast_lin_arith.ML"
wenzelm@24091
    18
  ("Tools/lin_arith.ML")
nipkow@15131
    19
begin
berghofe@13449
    20
berghofe@13449
    21
subsection {* Type @{text ind} *}
berghofe@13449
    22
berghofe@13449
    23
typedecl ind
berghofe@13449
    24
wenzelm@19573
    25
axiomatization
wenzelm@19573
    26
  Zero_Rep :: ind and
wenzelm@19573
    27
  Suc_Rep :: "ind => ind"
wenzelm@19573
    28
where
berghofe@13449
    29
  -- {* the axiom of infinity in 2 parts *}
wenzelm@19573
    30
  inj_Suc_Rep:          "inj Suc_Rep" and
paulson@14267
    31
  Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"
wenzelm@19573
    32
berghofe@13449
    33
berghofe@13449
    34
subsection {* Type nat *}
berghofe@13449
    35
berghofe@13449
    36
text {* Type definition *}
berghofe@13449
    37
haftmann@26072
    38
inductive Nat :: "ind \<Rightarrow> bool"
berghofe@22262
    39
where
haftmann@26072
    40
    Zero_RepI: "Nat Zero_Rep"
haftmann@26072
    41
  | Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)"
berghofe@13449
    42
berghofe@13449
    43
global
berghofe@13449
    44
berghofe@13449
    45
typedef (open Nat)
haftmann@26072
    46
  nat = "Collect Nat"
haftmann@26072
    47
  by (rule exI, rule CollectI, rule Nat.Zero_RepI)
berghofe@13449
    48
haftmann@26072
    49
constdefs
berghofe@13449
    50
  Suc :: "nat => nat"
haftmann@26072
    51
  Suc_def:      "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))"
berghofe@13449
    52
berghofe@13449
    53
local
berghofe@13449
    54
haftmann@25510
    55
instantiation nat :: zero
haftmann@25510
    56
begin
haftmann@25510
    57
haftmann@25510
    58
definition Zero_nat_def [code func del]:
haftmann@25510
    59
  "0 = Abs_Nat Zero_Rep"
haftmann@25510
    60
haftmann@25510
    61
instance ..
haftmann@25510
    62
haftmann@25510
    63
end
haftmann@24995
    64
haftmann@26072
    65
lemma nat_induct: "P 0 ==> (!!n. P n ==> P (Suc n)) ==> P n"
berghofe@13449
    66
  apply (unfold Zero_nat_def Suc_def)
berghofe@13449
    67
  apply (rule Rep_Nat_inverse [THEN subst]) -- {* types force good instantiation *}
haftmann@26072
    68
  apply (erule Rep_Nat [THEN CollectD, THEN Nat.induct])
haftmann@26072
    69
  apply (iprover elim: Abs_Nat_inverse [OF CollectI, THEN subst])
berghofe@13449
    70
  done
berghofe@13449
    71
paulson@14267
    72
lemma Suc_not_Zero [iff]: "Suc m \<noteq> 0"
haftmann@26072
    73
  by (simp add: Zero_nat_def Suc_def
haftmann@26072
    74
    Abs_Nat_inject Rep_Nat [THEN CollectD] Suc_RepI Zero_RepI
haftmann@26072
    75
      Suc_Rep_not_Zero_Rep)
berghofe@13449
    76
paulson@14267
    77
lemma Zero_not_Suc [iff]: "0 \<noteq> Suc m"
berghofe@13449
    78
  by (rule not_sym, rule Suc_not_Zero not_sym)
berghofe@13449
    79
nipkow@16733
    80
lemma inj_Suc[simp]: "inj_on Suc N"
haftmann@26072
    81
  by (simp add: Suc_def inj_on_def Abs_Nat_inject Rep_Nat [THEN CollectD] Suc_RepI
wenzelm@22718
    82
                inj_Suc_Rep [THEN inj_eq] Rep_Nat_inject)
berghofe@13449
    83
haftmann@26072
    84
lemma Suc_Suc_eq [iff]: "Suc m = Suc n \<longleftrightarrow> m = n"
paulson@15413
    85
  by (rule inj_Suc [THEN inj_eq])
berghofe@13449
    86
berghofe@5188
    87
rep_datatype nat
berghofe@13449
    88
  distinct  Suc_not_Zero Zero_not_Suc
berghofe@13449
    89
  inject    Suc_Suc_eq
haftmann@21411
    90
  induction nat_induct
haftmann@21411
    91
haftmann@21411
    92
declare nat.induct [case_names 0 Suc, induct type: nat]
haftmann@21411
    93
declare nat.exhaust [case_names 0 Suc, cases type: nat]
berghofe@13449
    94
wenzelm@21672
    95
lemmas nat_rec_0 = nat.recs(1)
wenzelm@21672
    96
  and nat_rec_Suc = nat.recs(2)
wenzelm@21672
    97
wenzelm@21672
    98
lemmas nat_case_0 = nat.cases(1)
wenzelm@21672
    99
  and nat_case_Suc = nat.cases(2)
wenzelm@21672
   100
haftmann@24995
   101
haftmann@24995
   102
text {* Injectiveness and distinctness lemmas *}
haftmann@24995
   103
haftmann@26072
   104
lemma Suc_neq_Zero: "Suc m = 0 \<Longrightarrow> R"
nipkow@25162
   105
by (rule notE, rule Suc_not_Zero)
haftmann@24995
   106
haftmann@26072
   107
lemma Zero_neq_Suc: "0 = Suc m \<Longrightarrow> R"
nipkow@25162
   108
by (rule Suc_neq_Zero, erule sym)
haftmann@24995
   109
haftmann@26072
   110
lemma Suc_inject: "Suc x = Suc y \<Longrightarrow> x = y"
nipkow@25162
   111
by (rule inj_Suc [THEN injD])
haftmann@24995
   112
paulson@14267
   113
lemma n_not_Suc_n: "n \<noteq> Suc n"
nipkow@25162
   114
by (induct n) simp_all
berghofe@13449
   115
haftmann@26072
   116
lemma Suc_n_not_n: "Suc n \<noteq> n"
nipkow@25162
   117
by (rule not_sym, rule n_not_Suc_n)
berghofe@13449
   118
berghofe@13449
   119
text {* A special form of induction for reasoning
berghofe@13449
   120
  about @{term "m < n"} and @{term "m - n"} *}
berghofe@13449
   121
haftmann@26072
   122
lemma diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==>
berghofe@13449
   123
    (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n"
paulson@14208
   124
  apply (rule_tac x = m in spec)
paulson@15251
   125
  apply (induct n)
berghofe@13449
   126
  prefer 2
berghofe@13449
   127
  apply (rule allI)
nipkow@17589
   128
  apply (induct_tac x, iprover+)
berghofe@13449
   129
  done
berghofe@13449
   130
haftmann@24995
   131
haftmann@24995
   132
subsection {* Arithmetic operators *}
haftmann@24995
   133
haftmann@26072
   134
instantiation nat :: "{minus, comm_monoid_add}"
haftmann@25571
   135
begin
haftmann@24995
   136
haftmann@25571
   137
primrec plus_nat
haftmann@25571
   138
where
haftmann@25571
   139
  add_0:      "0 + n = (n\<Colon>nat)"
haftmann@25571
   140
  | add_Suc:  "Suc m + n = Suc (m + n)"
haftmann@24995
   141
haftmann@26072
   142
lemma add_0_right [simp]: "m + 0 = (m::nat)"
haftmann@26072
   143
  by (induct m) simp_all
haftmann@26072
   144
haftmann@26072
   145
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
haftmann@26072
   146
  by (induct m) simp_all
haftmann@26072
   147
haftmann@26072
   148
lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
haftmann@26072
   149
  by simp
haftmann@26072
   150
haftmann@25571
   151
primrec minus_nat
haftmann@25571
   152
where
haftmann@25571
   153
  diff_0:     "m - 0 = (m\<Colon>nat)"
haftmann@25571
   154
  | diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
haftmann@24995
   155
haftmann@26072
   156
declare diff_Suc [simp del, code del]
haftmann@26072
   157
haftmann@26072
   158
lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"
haftmann@26072
   159
  by (induct n) (simp_all add: diff_Suc)
haftmann@26072
   160
haftmann@26072
   161
lemma diff_Suc_Suc [simp, code]: "Suc m - Suc n = m - n"
haftmann@26072
   162
  by (induct n) (simp_all add: diff_Suc)
haftmann@26072
   163
haftmann@26072
   164
instance proof
haftmann@26072
   165
  fix n m q :: nat
haftmann@26072
   166
  show "(n + m) + q = n + (m + q)" by (induct n) simp_all
haftmann@26072
   167
  show "n + m = m + n" by (induct n) simp_all
haftmann@26072
   168
  show "0 + n = n" by simp
haftmann@26072
   169
qed
haftmann@26072
   170
haftmann@26072
   171
end
haftmann@26072
   172
haftmann@26072
   173
instantiation nat :: comm_semiring_1_cancel
haftmann@26072
   174
begin
haftmann@26072
   175
haftmann@26072
   176
definition
haftmann@26072
   177
  One_nat_def [simp]: "1 = Suc 0"
haftmann@26072
   178
haftmann@25571
   179
primrec times_nat
haftmann@25571
   180
where
haftmann@25571
   181
  mult_0:     "0 * n = (0\<Colon>nat)"
haftmann@25571
   182
  | mult_Suc: "Suc m * n = n + (m * n)"
haftmann@25571
   183
haftmann@26072
   184
lemma mult_0_right [simp]: "(m::nat) * 0 = 0"
haftmann@26072
   185
  by (induct m) simp_all
haftmann@26072
   186
haftmann@26072
   187
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
haftmann@26072
   188
  by (induct m) (simp_all add: add_left_commute)
haftmann@26072
   189
haftmann@26072
   190
lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"
haftmann@26072
   191
  by (induct m) (simp_all add: add_assoc)
haftmann@26072
   192
haftmann@26072
   193
instance proof
haftmann@26072
   194
  fix n m q :: nat
haftmann@26072
   195
  show "0 \<noteq> (1::nat)" by simp
haftmann@26072
   196
  show "1 * n = n" by simp
haftmann@26072
   197
  show "n * m = m * n" by (induct n) simp_all
haftmann@26072
   198
  show "(n * m) * q = n * (m * q)" by (induct n) (simp_all add: add_mult_distrib)
haftmann@26072
   199
  show "(n + m) * q = n * q + m * q" by (rule add_mult_distrib)
haftmann@26072
   200
  assume "n + m = n + q" thus "m = q" by (induct n) simp_all
haftmann@26072
   201
qed
haftmann@25571
   202
haftmann@25571
   203
end
haftmann@24995
   204
haftmann@26072
   205
subsubsection {* Addition *}
haftmann@26072
   206
haftmann@26072
   207
lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)"
haftmann@26072
   208
  by (rule add_assoc)
haftmann@26072
   209
haftmann@26072
   210
lemma nat_add_commute: "m + n = n + (m::nat)"
haftmann@26072
   211
  by (rule add_commute)
haftmann@26072
   212
haftmann@26072
   213
lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)"
haftmann@26072
   214
  by (rule add_left_commute)
haftmann@26072
   215
haftmann@26072
   216
lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))"
haftmann@26072
   217
  by (rule add_left_cancel)
haftmann@26072
   218
haftmann@26072
   219
lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))"
haftmann@26072
   220
  by (rule add_right_cancel)
haftmann@26072
   221
haftmann@26072
   222
text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
haftmann@26072
   223
haftmann@26072
   224
lemma add_is_0 [iff]:
haftmann@26072
   225
  fixes m n :: nat
haftmann@26072
   226
  shows "(m + n = 0) = (m = 0 & n = 0)"
haftmann@26072
   227
  by (cases m) simp_all
haftmann@26072
   228
haftmann@26072
   229
lemma add_is_1:
haftmann@26072
   230
  "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
haftmann@26072
   231
  by (cases m) simp_all
haftmann@26072
   232
haftmann@26072
   233
lemma one_is_add:
haftmann@26072
   234
  "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
haftmann@26072
   235
  by (rule trans, rule eq_commute, rule add_is_1)
haftmann@26072
   236
haftmann@26072
   237
lemma add_eq_self_zero:
haftmann@26072
   238
  fixes m n :: nat
haftmann@26072
   239
  shows "m + n = m \<Longrightarrow> n = 0"
haftmann@26072
   240
  by (induct m) simp_all
haftmann@26072
   241
haftmann@26072
   242
lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N"
haftmann@26072
   243
  apply (induct k)
haftmann@26072
   244
   apply simp
haftmann@26072
   245
  apply(drule comp_inj_on[OF _ inj_Suc])
haftmann@26072
   246
  apply (simp add:o_def)
haftmann@26072
   247
  done
haftmann@26072
   248
haftmann@26072
   249
haftmann@26072
   250
subsubsection {* Difference *}
haftmann@26072
   251
haftmann@26072
   252
lemma diff_self_eq_0 [simp]: "(m\<Colon>nat) - m = 0"
haftmann@26072
   253
  by (induct m) simp_all
haftmann@26072
   254
haftmann@26072
   255
lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
haftmann@26072
   256
  by (induct i j rule: diff_induct) simp_all
haftmann@26072
   257
haftmann@26072
   258
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
haftmann@26072
   259
  by (simp add: diff_diff_left)
haftmann@26072
   260
haftmann@26072
   261
lemma diff_commute: "(i::nat) - j - k = i - k - j"
haftmann@26072
   262
  by (simp add: diff_diff_left add_commute)
haftmann@26072
   263
haftmann@26072
   264
lemma diff_add_inverse: "(n + m) - n = (m::nat)"
haftmann@26072
   265
  by (induct n) simp_all
haftmann@26072
   266
haftmann@26072
   267
lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
haftmann@26072
   268
  by (simp add: diff_add_inverse add_commute [of m n])
haftmann@26072
   269
haftmann@26072
   270
lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
haftmann@26072
   271
  by (induct k) simp_all
haftmann@26072
   272
haftmann@26072
   273
lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
haftmann@26072
   274
  by (simp add: diff_cancel add_commute)
haftmann@26072
   275
haftmann@26072
   276
lemma diff_add_0: "n - (n + m) = (0::nat)"
haftmann@26072
   277
  by (induct n) simp_all
haftmann@26072
   278
haftmann@26072
   279
text {* Difference distributes over multiplication *}
haftmann@26072
   280
haftmann@26072
   281
lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"
haftmann@26072
   282
by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
haftmann@26072
   283
haftmann@26072
   284
lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
haftmann@26072
   285
by (simp add: diff_mult_distrib mult_commute [of k])
haftmann@26072
   286
  -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
haftmann@26072
   287
haftmann@26072
   288
haftmann@26072
   289
subsubsection {* Multiplication *}
haftmann@26072
   290
haftmann@26072
   291
lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)"
haftmann@26072
   292
  by (rule mult_assoc)
haftmann@26072
   293
haftmann@26072
   294
lemma nat_mult_commute: "m * n = n * (m::nat)"
haftmann@26072
   295
  by (rule mult_commute)
haftmann@26072
   296
haftmann@26072
   297
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
haftmann@26072
   298
  by (rule right_distrib)
haftmann@26072
   299
haftmann@26072
   300
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
haftmann@26072
   301
  by (induct m) auto
haftmann@26072
   302
haftmann@26072
   303
lemmas nat_distrib =
haftmann@26072
   304
  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
haftmann@26072
   305
haftmann@26072
   306
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)"
haftmann@26072
   307
  apply (induct m)
haftmann@26072
   308
   apply simp
haftmann@26072
   309
  apply (induct n)
haftmann@26072
   310
   apply auto
haftmann@26072
   311
  done
haftmann@26072
   312
haftmann@26072
   313
lemma one_eq_mult_iff [simp,noatp]: "(Suc 0 = m * n) = (m = 1 & n = 1)"
haftmann@26072
   314
  apply (rule trans)
haftmann@26072
   315
  apply (rule_tac [2] mult_eq_1_iff, fastsimp)
haftmann@26072
   316
  done
haftmann@26072
   317
haftmann@26072
   318
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
haftmann@26072
   319
proof -
haftmann@26072
   320
  have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
haftmann@26072
   321
  proof (induct n arbitrary: m)
haftmann@26072
   322
    case 0 then show "m = 0" by simp
haftmann@26072
   323
  next
haftmann@26072
   324
    case (Suc n) then show "m = Suc n"
haftmann@26072
   325
      by (cases m) (simp_all add: eq_commute [of "0"])
haftmann@26072
   326
  qed
haftmann@26072
   327
  then show ?thesis by auto
haftmann@26072
   328
qed
haftmann@26072
   329
haftmann@26072
   330
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
haftmann@26072
   331
  by (simp add: mult_commute)
haftmann@26072
   332
haftmann@26072
   333
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
haftmann@26072
   334
  by (subst mult_cancel1) simp
haftmann@26072
   335
haftmann@24995
   336
haftmann@24995
   337
subsection {* Orders on @{typ nat} *}
haftmann@24995
   338
haftmann@26072
   339
subsubsection {* Operation definition *}
haftmann@24995
   340
haftmann@26072
   341
instantiation nat :: linorder
haftmann@25510
   342
begin
haftmann@25510
   343
haftmann@26072
   344
primrec less_eq_nat where
haftmann@26072
   345
  "(0\<Colon>nat) \<le> n \<longleftrightarrow> True"
haftmann@26072
   346
  | "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
haftmann@26072
   347
haftmann@26072
   348
declare less_eq_nat.simps [simp del, code del]
haftmann@26072
   349
lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by (simp add: less_eq_nat.simps)
haftmann@26072
   350
lemma le0 [iff]: "0 \<le> (n\<Colon>nat)" by (simp add: less_eq_nat.simps)
haftmann@26072
   351
haftmann@26072
   352
definition less_nat where
haftmann@26072
   353
  less_eq_Suc_le [code func del]: "n < m \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   354
haftmann@26072
   355
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
haftmann@26072
   356
  by (simp add: less_eq_nat.simps(2))
haftmann@26072
   357
haftmann@26072
   358
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
haftmann@26072
   359
  unfolding less_eq_Suc_le ..
haftmann@26072
   360
haftmann@26072
   361
lemma le_0_eq [iff]: "(n\<Colon>nat) \<le> 0 \<longleftrightarrow> n = 0"
haftmann@26072
   362
  by (induct n) (simp_all add: less_eq_nat.simps(2))
haftmann@26072
   363
haftmann@26072
   364
lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)"
haftmann@26072
   365
  by (simp add: less_eq_Suc_le)
haftmann@26072
   366
haftmann@26072
   367
lemma less_nat_zero_code [code]: "n < (0\<Colon>nat) \<longleftrightarrow> False"
haftmann@26072
   368
  by simp
haftmann@26072
   369
haftmann@26072
   370
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
haftmann@26072
   371
  by (simp add: less_eq_Suc_le)
haftmann@26072
   372
haftmann@26072
   373
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
haftmann@26072
   374
  by (simp add: less_eq_Suc_le)
haftmann@26072
   375
haftmann@26072
   376
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
haftmann@26072
   377
  by (induct m arbitrary: n)
haftmann@26072
   378
    (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   379
haftmann@26072
   380
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
haftmann@26072
   381
  by (cases n) (auto intro: le_SucI)
haftmann@26072
   382
haftmann@26072
   383
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
haftmann@26072
   384
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@24995
   385
haftmann@26072
   386
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
haftmann@26072
   387
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@25510
   388
wenzelm@26315
   389
instance
wenzelm@26315
   390
proof
haftmann@26072
   391
  fix n m :: nat
haftmann@26072
   392
  have less_imp_le: "n < m \<Longrightarrow> n \<le> m"
haftmann@26072
   393
    unfolding less_eq_Suc_le by (erule Suc_leD)
haftmann@26072
   394
  have irrefl: "\<not> m < m" by (induct m) auto
haftmann@26072
   395
  have strict: "n \<le> m \<Longrightarrow> n \<noteq> m \<Longrightarrow> n < m"
haftmann@26072
   396
  proof (induct n arbitrary: m)
haftmann@26072
   397
    case 0 then show ?case
haftmann@26072
   398
      by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   399
  next
haftmann@26072
   400
    case (Suc n) then show ?case
haftmann@26072
   401
      by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   402
  qed
haftmann@26072
   403
  show "n < m \<longleftrightarrow> n \<le> m \<and> n \<noteq> m"
haftmann@26072
   404
    by (auto simp add: irrefl intro: less_imp_le strict)
haftmann@26072
   405
next
haftmann@26072
   406
  fix n :: nat show "n \<le> n" by (induct n) simp_all
haftmann@26072
   407
next
haftmann@26072
   408
  fix n m :: nat assume "n \<le> m" and "m \<le> n"
haftmann@26072
   409
  then show "n = m"
haftmann@26072
   410
    by (induct n arbitrary: m)
haftmann@26072
   411
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   412
next
haftmann@26072
   413
  fix n m q :: nat assume "n \<le> m" and "m \<le> q"
haftmann@26072
   414
  then show "n \<le> q"
haftmann@26072
   415
  proof (induct n arbitrary: m q)
haftmann@26072
   416
    case 0 show ?case by simp
haftmann@26072
   417
  next
haftmann@26072
   418
    case (Suc n) then show ?case
haftmann@26072
   419
      by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   420
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   421
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   422
  qed
haftmann@26072
   423
next
haftmann@26072
   424
  fix n m :: nat show "n \<le> m \<or> m \<le> n"
haftmann@26072
   425
    by (induct n arbitrary: m)
haftmann@26072
   426
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   427
qed
haftmann@25510
   428
haftmann@25510
   429
end
berghofe@13449
   430
haftmann@26072
   431
subsubsection {* Introduction properties *}
berghofe@13449
   432
haftmann@26072
   433
lemma lessI [iff]: "n < Suc n"
haftmann@26072
   434
  by (simp add: less_Suc_eq_le)
berghofe@13449
   435
haftmann@26072
   436
lemma zero_less_Suc [iff]: "0 < Suc n"
haftmann@26072
   437
  by (simp add: less_Suc_eq_le)
berghofe@13449
   438
berghofe@13449
   439
berghofe@13449
   440
subsubsection {* Elimination properties *}
berghofe@13449
   441
berghofe@13449
   442
lemma less_not_refl: "~ n < (n::nat)"
haftmann@26072
   443
  by (rule order_less_irrefl)
berghofe@13449
   444
wenzelm@26335
   445
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)"
wenzelm@26335
   446
  by (rule not_sym) (rule less_imp_neq) 
berghofe@13449
   447
paulson@14267
   448
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
haftmann@26072
   449
  by (rule less_imp_neq)
berghofe@13449
   450
wenzelm@26335
   451
lemma less_irrefl_nat: "(n::nat) < n ==> R"
wenzelm@26335
   452
  by (rule notE, rule less_not_refl)
berghofe@13449
   453
berghofe@13449
   454
lemma less_zeroE: "(n::nat) < 0 ==> R"
haftmann@26072
   455
  by (rule notE) (rule not_less0)
berghofe@13449
   456
berghofe@13449
   457
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
haftmann@26072
   458
  unfolding less_Suc_eq_le le_less ..
berghofe@13449
   459
haftmann@26072
   460
lemma less_one [iff, noatp]: "(n < (1::nat)) = (n = 0)"
haftmann@26072
   461
  by (simp add: less_Suc_eq)
berghofe@13449
   462
berghofe@13449
   463
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
haftmann@26072
   464
  by (simp add: less_Suc_eq)
berghofe@13449
   465
berghofe@13449
   466
lemma Suc_mono: "m < n ==> Suc m < Suc n"
haftmann@26072
   467
  by simp
berghofe@13449
   468
nipkow@14302
   469
text {* "Less than" is antisymmetric, sort of *}
nipkow@14302
   470
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
haftmann@26072
   471
  unfolding not_less less_Suc_eq_le by (rule antisym)
nipkow@14302
   472
paulson@14267
   473
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
haftmann@26072
   474
  by (rule linorder_neq_iff)
berghofe@13449
   475
berghofe@13449
   476
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
berghofe@13449
   477
  and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
berghofe@13449
   478
  shows "P n m"
berghofe@13449
   479
  apply (rule less_linear [THEN disjE])
berghofe@13449
   480
  apply (erule_tac [2] disjE)
berghofe@13449
   481
  apply (erule lessCase)
berghofe@13449
   482
  apply (erule sym [THEN eqCase])
berghofe@13449
   483
  apply (erule major)
berghofe@13449
   484
  done
berghofe@13449
   485
berghofe@13449
   486
berghofe@13449
   487
subsubsection {* Inductive (?) properties *}
berghofe@13449
   488
paulson@14267
   489
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
haftmann@26072
   490
  unfolding less_eq_Suc_le [of m] le_less by simp 
berghofe@13449
   491
haftmann@26072
   492
lemma lessE:
haftmann@26072
   493
  assumes major: "i < k"
haftmann@26072
   494
  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
haftmann@26072
   495
  shows P
haftmann@26072
   496
proof -
haftmann@26072
   497
  from major have "\<exists>j. i \<le> j \<and> k = Suc j"
haftmann@26072
   498
    unfolding less_eq_Suc_le by (induct k) simp_all
haftmann@26072
   499
  then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
haftmann@26072
   500
    by (clarsimp simp add: less_le)
haftmann@26072
   501
  with p1 p2 show P by auto
haftmann@26072
   502
qed
haftmann@26072
   503
haftmann@26072
   504
lemma less_SucE: assumes major: "m < Suc n"
haftmann@26072
   505
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
haftmann@26072
   506
  apply (rule major [THEN lessE])
haftmann@26072
   507
  apply (rule eq, blast)
haftmann@26072
   508
  apply (rule less, blast)
berghofe@13449
   509
  done
berghofe@13449
   510
berghofe@13449
   511
lemma Suc_lessE: assumes major: "Suc i < k"
berghofe@13449
   512
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
berghofe@13449
   513
  apply (rule major [THEN lessE])
berghofe@13449
   514
  apply (erule lessI [THEN minor])
paulson@14208
   515
  apply (erule Suc_lessD [THEN minor], assumption)
berghofe@13449
   516
  done
berghofe@13449
   517
berghofe@13449
   518
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
haftmann@26072
   519
  by simp
berghofe@13449
   520
berghofe@13449
   521
lemma less_trans_Suc:
berghofe@13449
   522
  assumes le: "i < j" shows "j < k ==> Suc i < k"
paulson@14208
   523
  apply (induct k, simp_all)
berghofe@13449
   524
  apply (insert le)
berghofe@13449
   525
  apply (simp add: less_Suc_eq)
berghofe@13449
   526
  apply (blast dest: Suc_lessD)
berghofe@13449
   527
  done
berghofe@13449
   528
berghofe@13449
   529
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
haftmann@26072
   530
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
haftmann@26072
   531
  unfolding not_less less_Suc_eq_le ..
berghofe@13449
   532
haftmann@26072
   533
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   534
  unfolding not_le Suc_le_eq ..
wenzelm@21243
   535
haftmann@24995
   536
text {* Properties of "less than or equal" *}
berghofe@13449
   537
paulson@14267
   538
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
haftmann@26072
   539
  unfolding less_Suc_eq_le .
berghofe@13449
   540
paulson@14267
   541
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
haftmann@26072
   542
  unfolding not_le less_Suc_eq_le ..
berghofe@13449
   543
paulson@14267
   544
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
haftmann@26072
   545
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
berghofe@13449
   546
paulson@14267
   547
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
haftmann@26072
   548
  by (drule le_Suc_eq [THEN iffD1], iprover+)
berghofe@13449
   549
paulson@14267
   550
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
haftmann@26072
   551
  unfolding Suc_le_eq .
berghofe@13449
   552
berghofe@13449
   553
text {* Stronger version of @{text Suc_leD} *}
paulson@14267
   554
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
haftmann@26072
   555
  unfolding Suc_le_eq .
berghofe@13449
   556
wenzelm@26315
   557
lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)"
haftmann@26072
   558
  unfolding less_eq_Suc_le by (rule Suc_leD)
berghofe@13449
   559
paulson@14267
   560
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
wenzelm@26315
   561
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
berghofe@13449
   562
berghofe@13449
   563
paulson@14267
   564
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
berghofe@13449
   565
paulson@14267
   566
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
haftmann@26072
   567
  unfolding le_less .
berghofe@13449
   568
paulson@14267
   569
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
haftmann@26072
   570
  by (rule le_less)
berghofe@13449
   571
wenzelm@22718
   572
text {* Useful with @{text blast}. *}
paulson@14267
   573
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
haftmann@26072
   574
  by auto
berghofe@13449
   575
paulson@14267
   576
lemma le_refl: "n \<le> (n::nat)"
haftmann@26072
   577
  by simp
berghofe@13449
   578
paulson@14267
   579
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
haftmann@26072
   580
  by (rule order_trans)
berghofe@13449
   581
paulson@14267
   582
lemma le_anti_sym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
haftmann@26072
   583
  by (rule antisym)
berghofe@13449
   584
paulson@14267
   585
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
haftmann@26072
   586
  by (rule less_le)
berghofe@13449
   587
paulson@14267
   588
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
haftmann@26072
   589
  unfolding less_le ..
berghofe@13449
   590
haftmann@26072
   591
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
haftmann@26072
   592
  by (rule linear)
paulson@14341
   593
wenzelm@22718
   594
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
nipkow@15921
   595
haftmann@26072
   596
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
haftmann@26072
   597
  unfolding less_Suc_eq_le by auto
berghofe@13449
   598
haftmann@26072
   599
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
haftmann@26072
   600
  unfolding not_less by (rule le_less_Suc_eq)
berghofe@13449
   601
berghofe@13449
   602
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
berghofe@13449
   603
wenzelm@22718
   604
text {* These two rules ease the use of primitive recursion.
paulson@14341
   605
NOTE USE OF @{text "=="} *}
berghofe@13449
   606
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"
nipkow@25162
   607
by simp
berghofe@13449
   608
berghofe@13449
   609
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"
nipkow@25162
   610
by simp
berghofe@13449
   611
paulson@14267
   612
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   613
by (cases n) simp_all
nipkow@25162
   614
nipkow@25162
   615
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   616
by (cases n) simp_all
berghofe@13449
   617
wenzelm@22718
   618
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
nipkow@25162
   619
by (cases n) simp_all
berghofe@13449
   620
nipkow@25162
   621
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
nipkow@25162
   622
by (cases n) simp_all
nipkow@25140
   623
berghofe@13449
   624
text {* This theorem is useful with @{text blast} *}
berghofe@13449
   625
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
nipkow@25162
   626
by (rule neq0_conv[THEN iffD1], iprover)
berghofe@13449
   627
paulson@14267
   628
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
nipkow@25162
   629
by (fast intro: not0_implies_Suc)
berghofe@13449
   630
paulson@24286
   631
lemma not_gr0 [iff,noatp]: "!!n::nat. (~ (0 < n)) = (n = 0)"
nipkow@25134
   632
using neq0_conv by blast
berghofe@13449
   633
paulson@14267
   634
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
nipkow@25162
   635
by (induct m') simp_all
berghofe@13449
   636
berghofe@13449
   637
text {* Useful in certain inductive arguments *}
paulson@14267
   638
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
nipkow@25162
   639
by (cases m) simp_all
berghofe@13449
   640
berghofe@13449
   641
haftmann@26072
   642
subsubsection {* @{term min} and @{term max} *}
berghofe@13449
   643
haftmann@25076
   644
lemma mono_Suc: "mono Suc"
nipkow@25162
   645
by (rule monoI) simp
haftmann@25076
   646
berghofe@13449
   647
lemma min_0L [simp]: "min 0 n = (0::nat)"
nipkow@25162
   648
by (rule min_leastL) simp
berghofe@13449
   649
berghofe@13449
   650
lemma min_0R [simp]: "min n 0 = (0::nat)"
nipkow@25162
   651
by (rule min_leastR) simp
berghofe@13449
   652
berghofe@13449
   653
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
nipkow@25162
   654
by (simp add: mono_Suc min_of_mono)
berghofe@13449
   655
paulson@22191
   656
lemma min_Suc1:
paulson@22191
   657
   "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
nipkow@25162
   658
by (simp split: nat.split)
paulson@22191
   659
paulson@22191
   660
lemma min_Suc2:
paulson@22191
   661
   "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
nipkow@25162
   662
by (simp split: nat.split)
paulson@22191
   663
berghofe@13449
   664
lemma max_0L [simp]: "max 0 n = (n::nat)"
nipkow@25162
   665
by (rule max_leastL) simp
berghofe@13449
   666
berghofe@13449
   667
lemma max_0R [simp]: "max n 0 = (n::nat)"
nipkow@25162
   668
by (rule max_leastR) simp
berghofe@13449
   669
berghofe@13449
   670
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
nipkow@25162
   671
by (simp add: mono_Suc max_of_mono)
berghofe@13449
   672
paulson@22191
   673
lemma max_Suc1:
paulson@22191
   674
   "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
nipkow@25162
   675
by (simp split: nat.split)
paulson@22191
   676
paulson@22191
   677
lemma max_Suc2:
paulson@22191
   678
   "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
nipkow@25162
   679
by (simp split: nat.split)
paulson@22191
   680
berghofe@13449
   681
haftmann@26072
   682
subsubsection {* Monotonicity of Addition *}
berghofe@13449
   683
haftmann@26072
   684
lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"
haftmann@26072
   685
by (simp add: diff_Suc split: nat.split)
berghofe@13449
   686
paulson@14331
   687
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
nipkow@25162
   688
by (induct k) simp_all
berghofe@13449
   689
paulson@14331
   690
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
nipkow@25162
   691
by (induct k) simp_all
berghofe@13449
   692
nipkow@25162
   693
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)"
nipkow@25162
   694
by(auto dest:gr0_implies_Suc)
berghofe@13449
   695
paulson@14341
   696
text {* strict, in 1st argument *}
paulson@14341
   697
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
nipkow@25162
   698
by (induct k) simp_all
paulson@14341
   699
paulson@14341
   700
text {* strict, in both arguments *}
paulson@14341
   701
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
paulson@14341
   702
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
paulson@15251
   703
  apply (induct j, simp_all)
paulson@14341
   704
  done
paulson@14341
   705
paulson@14341
   706
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
paulson@14341
   707
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
paulson@14341
   708
  apply (induct n)
paulson@14341
   709
  apply (simp_all add: order_le_less)
wenzelm@22718
   710
  apply (blast elim!: less_SucE
paulson@14341
   711
               intro!: add_0_right [symmetric] add_Suc_right [symmetric])
paulson@14341
   712
  done
paulson@14341
   713
paulson@14341
   714
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
nipkow@25134
   715
lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"
nipkow@25134
   716
apply(auto simp: gr0_conv_Suc)
nipkow@25134
   717
apply (induct_tac m)
nipkow@25134
   718
apply (simp_all add: add_less_mono)
nipkow@25134
   719
done
paulson@14341
   720
nipkow@14740
   721
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
obua@14738
   722
instance nat :: ordered_semidom
paulson@14341
   723
proof
paulson@14341
   724
  fix i j k :: nat
paulson@14348
   725
  show "0 < (1::nat)" by simp
paulson@14267
   726
  show "i \<le> j ==> k + i \<le> k + j" by simp
paulson@14267
   727
  show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2)
paulson@14267
   728
qed
paulson@14267
   729
paulson@14267
   730
lemma nat_mult_1: "(1::nat) * n = n"
nipkow@25162
   731
by simp
paulson@14267
   732
paulson@14267
   733
lemma nat_mult_1_right: "n * (1::nat) = n"
nipkow@25162
   734
by simp
paulson@14267
   735
paulson@14267
   736
haftmann@26072
   737
subsubsection {* Additional theorems about "less than" *}
paulson@14267
   738
paulson@19870
   739
text{*An induction rule for estabilishing binary relations*}
wenzelm@22718
   740
lemma less_Suc_induct:
paulson@19870
   741
  assumes less:  "i < j"
paulson@19870
   742
     and  step:  "!!i. P i (Suc i)"
paulson@19870
   743
     and  trans: "!!i j k. P i j ==> P j k ==> P i k"
paulson@19870
   744
  shows "P i j"
paulson@19870
   745
proof -
wenzelm@22718
   746
  from less obtain k where j: "j = Suc(i+k)" by (auto dest: less_imp_Suc_add)
wenzelm@22718
   747
  have "P i (Suc (i + k))"
paulson@19870
   748
  proof (induct k)
wenzelm@22718
   749
    case 0
wenzelm@22718
   750
    show ?case by (simp add: step)
paulson@19870
   751
  next
paulson@19870
   752
    case (Suc k)
wenzelm@22718
   753
    thus ?case by (auto intro: assms)
paulson@19870
   754
  qed
wenzelm@22718
   755
  thus "P i j" by (simp add: j)
paulson@19870
   756
qed
paulson@19870
   757
paulson@14267
   758
text {* A [clumsy] way of lifting @{text "<"}
paulson@14267
   759
  monotonicity to @{text "\<le>"} monotonicity *}
paulson@14267
   760
lemma less_mono_imp_le_mono:
nipkow@24438
   761
  "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
nipkow@24438
   762
by (simp add: order_le_less) (blast)
nipkow@24438
   763
paulson@14267
   764
paulson@14267
   765
text {* non-strict, in 1st argument *}
paulson@14267
   766
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
nipkow@24438
   767
by (rule add_right_mono)
paulson@14267
   768
paulson@14267
   769
text {* non-strict, in both arguments *}
paulson@14267
   770
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
nipkow@24438
   771
by (rule add_mono)
paulson@14267
   772
paulson@14267
   773
lemma le_add2: "n \<le> ((m + n)::nat)"
nipkow@24438
   774
by (insert add_right_mono [of 0 m n], simp)
berghofe@13449
   775
paulson@14267
   776
lemma le_add1: "n \<le> ((n + m)::nat)"
nipkow@24438
   777
by (simp add: add_commute, rule le_add2)
berghofe@13449
   778
berghofe@13449
   779
lemma less_add_Suc1: "i < Suc (i + m)"
nipkow@24438
   780
by (rule le_less_trans, rule le_add1, rule lessI)
berghofe@13449
   781
berghofe@13449
   782
lemma less_add_Suc2: "i < Suc (m + i)"
nipkow@24438
   783
by (rule le_less_trans, rule le_add2, rule lessI)
berghofe@13449
   784
paulson@14267
   785
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
nipkow@24438
   786
by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
berghofe@13449
   787
paulson@14267
   788
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
nipkow@24438
   789
by (rule le_trans, assumption, rule le_add1)
berghofe@13449
   790
paulson@14267
   791
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
nipkow@24438
   792
by (rule le_trans, assumption, rule le_add2)
berghofe@13449
   793
berghofe@13449
   794
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
nipkow@24438
   795
by (rule less_le_trans, assumption, rule le_add1)
berghofe@13449
   796
berghofe@13449
   797
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
nipkow@24438
   798
by (rule less_le_trans, assumption, rule le_add2)
berghofe@13449
   799
berghofe@13449
   800
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
nipkow@24438
   801
apply (rule le_less_trans [of _ "i+j"])
nipkow@24438
   802
apply (simp_all add: le_add1)
nipkow@24438
   803
done
berghofe@13449
   804
berghofe@13449
   805
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
nipkow@24438
   806
apply (rule notI)
wenzelm@26335
   807
apply (drule add_lessD1)
wenzelm@26335
   808
apply (erule less_irrefl [THEN notE])
nipkow@24438
   809
done
berghofe@13449
   810
berghofe@13449
   811
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
nipkow@24438
   812
by (simp add: add_commute not_add_less1)
berghofe@13449
   813
paulson@14267
   814
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
nipkow@24438
   815
apply (rule order_trans [of _ "m+k"])
nipkow@24438
   816
apply (simp_all add: le_add1)
nipkow@24438
   817
done
berghofe@13449
   818
paulson@14267
   819
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
nipkow@24438
   820
apply (simp add: add_commute)
nipkow@24438
   821
apply (erule add_leD1)
nipkow@24438
   822
done
berghofe@13449
   823
paulson@14267
   824
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
nipkow@24438
   825
by (blast dest: add_leD1 add_leD2)
berghofe@13449
   826
berghofe@13449
   827
text {* needs @{text "!!k"} for @{text add_ac} to work *}
berghofe@13449
   828
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
nipkow@24438
   829
by (force simp del: add_Suc_right
berghofe@13449
   830
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
berghofe@13449
   831
berghofe@13449
   832
haftmann@26072
   833
subsubsection {* More results about difference *}
berghofe@13449
   834
berghofe@13449
   835
text {* Addition is the inverse of subtraction:
paulson@14267
   836
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
berghofe@13449
   837
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
nipkow@24438
   838
by (induct m n rule: diff_induct) simp_all
berghofe@13449
   839
paulson@14267
   840
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
nipkow@24438
   841
by (simp add: add_diff_inverse linorder_not_less)
berghofe@13449
   842
paulson@14267
   843
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
nipkow@24438
   844
by (simp add: le_add_diff_inverse add_commute)
berghofe@13449
   845
paulson@14267
   846
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
nipkow@24438
   847
by (induct m n rule: diff_induct) simp_all
berghofe@13449
   848
berghofe@13449
   849
lemma diff_less_Suc: "m - n < Suc m"
nipkow@24438
   850
apply (induct m n rule: diff_induct)
nipkow@24438
   851
apply (erule_tac [3] less_SucE)
nipkow@24438
   852
apply (simp_all add: less_Suc_eq)
nipkow@24438
   853
done
berghofe@13449
   854
paulson@14267
   855
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
nipkow@24438
   856
by (induct m n rule: diff_induct) (simp_all add: le_SucI)
berghofe@13449
   857
haftmann@26072
   858
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
haftmann@26072
   859
  by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])
haftmann@26072
   860
berghofe@13449
   861
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
nipkow@24438
   862
by (rule le_less_trans, rule diff_le_self)
berghofe@13449
   863
berghofe@13449
   864
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
nipkow@24438
   865
by (cases n) (auto simp add: le_simps)
berghofe@13449
   866
paulson@14267
   867
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
nipkow@24438
   868
by (induct j k rule: diff_induct) simp_all
berghofe@13449
   869
paulson@14267
   870
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
nipkow@24438
   871
by (simp add: add_commute diff_add_assoc)
berghofe@13449
   872
paulson@14267
   873
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
nipkow@24438
   874
by (auto simp add: diff_add_inverse2)
berghofe@13449
   875
paulson@14267
   876
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
nipkow@24438
   877
by (induct m n rule: diff_induct) simp_all
berghofe@13449
   878
paulson@14267
   879
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
nipkow@24438
   880
by (rule iffD2, rule diff_is_0_eq)
berghofe@13449
   881
berghofe@13449
   882
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
nipkow@24438
   883
by (induct m n rule: diff_induct) simp_all
berghofe@13449
   884
wenzelm@22718
   885
lemma less_imp_add_positive:
wenzelm@22718
   886
  assumes "i < j"
wenzelm@22718
   887
  shows "\<exists>k::nat. 0 < k & i + k = j"
wenzelm@22718
   888
proof
wenzelm@22718
   889
  from assms show "0 < j - i & i + (j - i) = j"
huffman@23476
   890
    by (simp add: order_less_imp_le)
wenzelm@22718
   891
qed
wenzelm@9436
   892
haftmann@26072
   893
text {* a nice rewrite for bounded subtraction *}
haftmann@26072
   894
lemma nat_minus_add_max:
haftmann@26072
   895
  fixes n m :: nat
haftmann@26072
   896
  shows "n - m + m = max n m"
haftmann@26072
   897
    by (simp add: max_def not_le order_less_imp_le)
berghofe@13449
   898
haftmann@26072
   899
lemma nat_diff_split:
haftmann@26072
   900
  "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
haftmann@26072
   901
    -- {* elimination of @{text -} on @{text nat} *}
haftmann@26072
   902
by (cases "a < b")
haftmann@26072
   903
  (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse
haftmann@26072
   904
    not_less le_less dest!: sym [of a] sym [of b] add_eq_self_zero)
berghofe@13449
   905
haftmann@26072
   906
lemma nat_diff_split_asm:
haftmann@26072
   907
  "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
haftmann@26072
   908
    -- {* elimination of @{text -} on @{text nat} in assumptions *}
haftmann@26072
   909
by (auto split: nat_diff_split)
berghofe@13449
   910
berghofe@13449
   911
haftmann@26072
   912
subsubsection {* Monotonicity of Multiplication *}
berghofe@13449
   913
paulson@14267
   914
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
nipkow@24438
   915
by (simp add: mult_right_mono)
berghofe@13449
   916
paulson@14267
   917
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
nipkow@24438
   918
by (simp add: mult_left_mono)
berghofe@13449
   919
paulson@14267
   920
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
paulson@14267
   921
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
nipkow@24438
   922
by (simp add: mult_mono)
berghofe@13449
   923
berghofe@13449
   924
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
nipkow@24438
   925
by (simp add: mult_strict_right_mono)
berghofe@13449
   926
paulson@14266
   927
text{*Differs from the standard @{text zero_less_mult_iff} in that
paulson@14266
   928
      there are no negative numbers.*}
paulson@14266
   929
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
berghofe@13449
   930
  apply (induct m)
wenzelm@22718
   931
   apply simp
wenzelm@22718
   932
  apply (case_tac n)
wenzelm@22718
   933
   apply simp_all
berghofe@13449
   934
  done
berghofe@13449
   935
paulson@14267
   936
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)"
berghofe@13449
   937
  apply (induct m)
wenzelm@22718
   938
   apply simp
wenzelm@22718
   939
  apply (case_tac n)
wenzelm@22718
   940
   apply simp_all
berghofe@13449
   941
  done
berghofe@13449
   942
paulson@14341
   943
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
berghofe@13449
   944
  apply (safe intro!: mult_less_mono1)
paulson@14208
   945
  apply (case_tac k, auto)
berghofe@13449
   946
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
berghofe@13449
   947
  apply (blast intro: mult_le_mono1)
berghofe@13449
   948
  done
berghofe@13449
   949
berghofe@13449
   950
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
nipkow@24438
   951
by (simp add: mult_commute [of k])
berghofe@13449
   952
paulson@14267
   953
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
nipkow@24438
   954
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
   955
paulson@14267
   956
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
nipkow@24438
   957
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
   958
berghofe@13449
   959
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
nipkow@24438
   960
by (subst mult_less_cancel1) simp
berghofe@13449
   961
paulson@14267
   962
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
nipkow@24438
   963
by (subst mult_le_cancel1) simp
berghofe@13449
   964
haftmann@26072
   965
lemma le_square: "m \<le> m * (m::nat)"
haftmann@26072
   966
  by (cases m) (auto intro: le_add1)
haftmann@26072
   967
haftmann@26072
   968
lemma le_cube: "(m::nat) \<le> m * (m * m)"
haftmann@26072
   969
  by (cases m) (auto intro: le_add1)
berghofe@13449
   970
berghofe@13449
   971
text {* Lemma for @{text gcd} *}
berghofe@13449
   972
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
berghofe@13449
   973
  apply (drule sym)
berghofe@13449
   974
  apply (rule disjCI)
berghofe@13449
   975
  apply (rule nat_less_cases, erule_tac [2] _)
paulson@25157
   976
   apply (drule_tac [2] mult_less_mono2)
nipkow@25162
   977
    apply (auto)
berghofe@13449
   978
  done
wenzelm@9436
   979
haftmann@26072
   980
text {* the lattice order on @{typ nat} *}
haftmann@24995
   981
haftmann@26072
   982
instantiation nat :: distrib_lattice
haftmann@26072
   983
begin
haftmann@24995
   984
haftmann@26072
   985
definition
haftmann@26072
   986
  "(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min"
haftmann@24995
   987
haftmann@26072
   988
definition
haftmann@26072
   989
  "(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max"
haftmann@24995
   990
haftmann@26072
   991
instance by intro_classes
haftmann@26072
   992
  (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
haftmann@26072
   993
    intro: order_less_imp_le antisym elim!: order_trans order_less_trans)
haftmann@24995
   994
haftmann@26072
   995
end
haftmann@24995
   996
haftmann@24995
   997
haftmann@25193
   998
subsection {* Embedding of the Naturals into any
haftmann@25193
   999
  @{text semiring_1}: @{term of_nat} *}
haftmann@24196
  1000
haftmann@24196
  1001
context semiring_1
haftmann@24196
  1002
begin
haftmann@24196
  1003
haftmann@25559
  1004
primrec
haftmann@25559
  1005
  of_nat :: "nat \<Rightarrow> 'a"
haftmann@25559
  1006
where
haftmann@25559
  1007
  of_nat_0:     "of_nat 0 = 0"
haftmann@25559
  1008
  | of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
haftmann@25193
  1009
haftmann@25193
  1010
lemma of_nat_1 [simp]: "of_nat 1 = 1"
haftmann@25193
  1011
  by simp
haftmann@25193
  1012
haftmann@25193
  1013
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
haftmann@25193
  1014
  by (induct m) (simp_all add: add_ac)
haftmann@25193
  1015
haftmann@25193
  1016
lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"
haftmann@25193
  1017
  by (induct m) (simp_all add: add_ac left_distrib)
haftmann@25193
  1018
haftmann@25928
  1019
definition
haftmann@25928
  1020
  of_nat_aux :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@25928
  1021
where
haftmann@25928
  1022
  [code func del]: "of_nat_aux n i = of_nat n + i"
haftmann@25928
  1023
haftmann@25928
  1024
lemma of_nat_aux_code [code]:
haftmann@25928
  1025
  "of_nat_aux 0 i = i"
haftmann@25928
  1026
  "of_nat_aux (Suc n) i = of_nat_aux n (i + 1)" -- {* tail recursive *}
haftmann@25928
  1027
  by (simp_all add: of_nat_aux_def add_ac)
haftmann@25928
  1028
haftmann@25928
  1029
lemma of_nat_code [code]:
haftmann@25928
  1030
  "of_nat n = of_nat_aux n 0"
haftmann@25928
  1031
  by (simp add: of_nat_aux_def)
haftmann@25928
  1032
haftmann@24196
  1033
end
haftmann@24196
  1034
haftmann@26072
  1035
text{*Class for unital semirings with characteristic zero.
haftmann@26072
  1036
 Includes non-ordered rings like the complex numbers.*}
haftmann@26072
  1037
haftmann@26072
  1038
class semiring_char_0 = semiring_1 +
haftmann@26072
  1039
  assumes of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"
haftmann@26072
  1040
begin
haftmann@26072
  1041
haftmann@26072
  1042
text{*Special cases where either operand is zero*}
haftmann@26072
  1043
haftmann@26072
  1044
lemma of_nat_0_eq_iff [simp, noatp]: "0 = of_nat n \<longleftrightarrow> 0 = n"
haftmann@26072
  1045
  by (rule of_nat_eq_iff [of 0, simplified])
haftmann@26072
  1046
haftmann@26072
  1047
lemma of_nat_eq_0_iff [simp, noatp]: "of_nat m = 0 \<longleftrightarrow> m = 0"
haftmann@26072
  1048
  by (rule of_nat_eq_iff [of _ 0, simplified])
haftmann@26072
  1049
haftmann@26072
  1050
lemma inj_of_nat: "inj of_nat"
haftmann@26072
  1051
  by (simp add: inj_on_def)
haftmann@26072
  1052
haftmann@26072
  1053
end
haftmann@26072
  1054
haftmann@25193
  1055
context ordered_semidom
haftmann@25193
  1056
begin
haftmann@25193
  1057
haftmann@25193
  1058
lemma zero_le_imp_of_nat: "0 \<le> of_nat m"
haftmann@25193
  1059
  apply (induct m, simp_all)
haftmann@25193
  1060
  apply (erule order_trans)
haftmann@25193
  1061
  apply (rule ord_le_eq_trans [OF _ add_commute])
haftmann@25193
  1062
  apply (rule less_add_one [THEN less_imp_le])
haftmann@25193
  1063
  done
haftmann@25193
  1064
haftmann@25193
  1065
lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"
haftmann@25193
  1066
  apply (induct m n rule: diff_induct, simp_all)
haftmann@25193
  1067
  apply (insert add_less_le_mono [OF zero_less_one zero_le_imp_of_nat], force)
haftmann@25193
  1068
  done
haftmann@25193
  1069
haftmann@25193
  1070
lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
haftmann@25193
  1071
  apply (induct m n rule: diff_induct, simp_all)
haftmann@25193
  1072
  apply (insert zero_le_imp_of_nat)
haftmann@25193
  1073
  apply (force simp add: not_less [symmetric])
haftmann@25193
  1074
  done
haftmann@25193
  1075
haftmann@25193
  1076
lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
haftmann@25193
  1077
  by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
haftmann@25193
  1078
haftmann@26072
  1079
lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
haftmann@26072
  1080
  by (simp add: not_less [symmetric] linorder_not_less [symmetric])
haftmann@25193
  1081
haftmann@26072
  1082
text{*Every @{text ordered_semidom} has characteristic zero.*}
haftmann@25193
  1083
haftmann@26072
  1084
subclass semiring_char_0
haftmann@26072
  1085
  by unfold_locales (simp add: eq_iff order_eq_iff)
haftmann@25193
  1086
haftmann@25193
  1087
text{*Special cases where either operand is zero*}
haftmann@25193
  1088
haftmann@25193
  1089
lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"
haftmann@25193
  1090
  by (rule of_nat_le_iff [of 0, simplified])
haftmann@25193
  1091
haftmann@25193
  1092
lemma of_nat_le_0_iff [simp, noatp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"
haftmann@25193
  1093
  by (rule of_nat_le_iff [of _ 0, simplified])
haftmann@25193
  1094
haftmann@26072
  1095
lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"
haftmann@26072
  1096
  by (rule of_nat_less_iff [of 0, simplified])
haftmann@26072
  1097
haftmann@26072
  1098
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"
haftmann@26072
  1099
  by (rule of_nat_less_iff [of _ 0, simplified])
haftmann@26072
  1100
haftmann@26072
  1101
end
haftmann@26072
  1102
haftmann@26072
  1103
context ring_1
haftmann@26072
  1104
begin
haftmann@26072
  1105
haftmann@26072
  1106
lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"
haftmann@26072
  1107
  by (simp add: compare_rls of_nat_add [symmetric])
haftmann@26072
  1108
haftmann@26072
  1109
end
haftmann@26072
  1110
haftmann@26072
  1111
context ordered_idom
haftmann@26072
  1112
begin
haftmann@26072
  1113
haftmann@26072
  1114
lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"
haftmann@26072
  1115
  unfolding abs_if by auto
haftmann@26072
  1116
haftmann@25193
  1117
end
haftmann@25193
  1118
haftmann@25193
  1119
lemma of_nat_id [simp]: "of_nat n = n"
haftmann@25193
  1120
  by (induct n) auto
haftmann@25193
  1121
haftmann@25193
  1122
lemma of_nat_eq_id [simp]: "of_nat = id"
haftmann@25193
  1123
  by (auto simp add: expand_fun_eq)
haftmann@25193
  1124
haftmann@25193
  1125
haftmann@26149
  1126
subsection {* The Set of Natural Numbers *}
haftmann@25193
  1127
haftmann@26072
  1128
context semiring_1
haftmann@25193
  1129
begin
haftmann@25193
  1130
haftmann@26072
  1131
definition
haftmann@26072
  1132
  Nats  :: "'a set" where
haftmann@26072
  1133
  "Nats = range of_nat"
haftmann@26072
  1134
haftmann@26072
  1135
notation (xsymbols)
haftmann@26072
  1136
  Nats  ("\<nat>")
haftmann@25193
  1137
haftmann@26072
  1138
lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"
haftmann@26072
  1139
  by (simp add: Nats_def)
haftmann@26072
  1140
haftmann@26072
  1141
lemma Nats_0 [simp]: "0 \<in> \<nat>"
haftmann@26072
  1142
apply (simp add: Nats_def)
haftmann@26072
  1143
apply (rule range_eqI)
haftmann@26072
  1144
apply (rule of_nat_0 [symmetric])
haftmann@26072
  1145
done
haftmann@25193
  1146
haftmann@26072
  1147
lemma Nats_1 [simp]: "1 \<in> \<nat>"
haftmann@26072
  1148
apply (simp add: Nats_def)
haftmann@26072
  1149
apply (rule range_eqI)
haftmann@26072
  1150
apply (rule of_nat_1 [symmetric])
haftmann@26072
  1151
done
haftmann@25193
  1152
haftmann@26072
  1153
lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"
haftmann@26072
  1154
apply (auto simp add: Nats_def)
haftmann@26072
  1155
apply (rule range_eqI)
haftmann@26072
  1156
apply (rule of_nat_add [symmetric])
haftmann@26072
  1157
done
haftmann@26072
  1158
haftmann@26072
  1159
lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"
haftmann@26072
  1160
apply (auto simp add: Nats_def)
haftmann@26072
  1161
apply (rule range_eqI)
haftmann@26072
  1162
apply (rule of_nat_mult [symmetric])
haftmann@26072
  1163
done
haftmann@25193
  1164
haftmann@25193
  1165
end
haftmann@25193
  1166
haftmann@25193
  1167
wenzelm@21243
  1168
subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}
wenzelm@21243
  1169
haftmann@22845
  1170
lemma subst_equals:
haftmann@22845
  1171
  assumes 1: "t = s" and 2: "u = t"
haftmann@22845
  1172
  shows "u = s"
haftmann@22845
  1173
  using 2 1 by (rule trans)
haftmann@22845
  1174
wenzelm@21243
  1175
use "arith_data.ML"
haftmann@26101
  1176
declaration {* K ArithData.setup *}
wenzelm@24091
  1177
wenzelm@24091
  1178
use "Tools/lin_arith.ML"
wenzelm@24091
  1179
declaration {* K LinArith.setup *}
wenzelm@24091
  1180
wenzelm@21243
  1181
lemmas [arith_split] = nat_diff_split split_min split_max
wenzelm@21243
  1182
wenzelm@21243
  1183
text{*Subtraction laws, mostly by Clemens Ballarin*}
wenzelm@21243
  1184
wenzelm@21243
  1185
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
nipkow@24438
  1186
by arith
wenzelm@21243
  1187
wenzelm@21243
  1188
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
nipkow@24438
  1189
by arith
wenzelm@21243
  1190
wenzelm@21243
  1191
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
nipkow@24438
  1192
by arith
wenzelm@21243
  1193
wenzelm@21243
  1194
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
nipkow@24438
  1195
by arith
wenzelm@21243
  1196
wenzelm@21243
  1197
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
nipkow@24438
  1198
by arith
wenzelm@21243
  1199
wenzelm@21243
  1200
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
nipkow@24438
  1201
by arith
wenzelm@21243
  1202
wenzelm@21243
  1203
(*Replaces the previous diff_less and le_diff_less, which had the stronger
wenzelm@21243
  1204
  second premise n\<le>m*)
wenzelm@21243
  1205
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
nipkow@24438
  1206
by arith
wenzelm@21243
  1207
haftmann@26072
  1208
text {* Simplification of relational expressions involving subtraction *}
wenzelm@21243
  1209
wenzelm@21243
  1210
lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
nipkow@24438
  1211
by (simp split add: nat_diff_split)
wenzelm@21243
  1212
wenzelm@21243
  1213
lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
nipkow@24438
  1214
by (auto split add: nat_diff_split)
wenzelm@21243
  1215
wenzelm@21243
  1216
lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
nipkow@24438
  1217
by (auto split add: nat_diff_split)
wenzelm@21243
  1218
wenzelm@21243
  1219
lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
nipkow@24438
  1220
by (auto split add: nat_diff_split)
wenzelm@21243
  1221
wenzelm@21243
  1222
text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
wenzelm@21243
  1223
wenzelm@21243
  1224
(* Monotonicity of subtraction in first argument *)
wenzelm@21243
  1225
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
nipkow@24438
  1226
by (simp split add: nat_diff_split)
wenzelm@21243
  1227
wenzelm@21243
  1228
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
nipkow@24438
  1229
by (simp split add: nat_diff_split)
wenzelm@21243
  1230
wenzelm@21243
  1231
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
nipkow@24438
  1232
by (simp split add: nat_diff_split)
wenzelm@21243
  1233
wenzelm@21243
  1234
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
nipkow@24438
  1235
by (simp split add: nat_diff_split)
wenzelm@21243
  1236
bulwahn@26143
  1237
lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"
bulwahn@26143
  1238
unfolding min_def by auto
bulwahn@26143
  1239
bulwahn@26143
  1240
lemma inj_on_diff_nat: 
bulwahn@26143
  1241
  assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"
bulwahn@26143
  1242
  shows "inj_on (\<lambda>n. n - k) N"
bulwahn@26143
  1243
proof (rule inj_onI)
bulwahn@26143
  1244
  fix x y
bulwahn@26143
  1245
  assume a: "x \<in> N" "y \<in> N" "x - k = y - k"
bulwahn@26143
  1246
  with k_le_n have "x - k + k = y - k + k" by auto
bulwahn@26143
  1247
  with a k_le_n show "x = y" by auto
bulwahn@26143
  1248
qed
bulwahn@26143
  1249
haftmann@26072
  1250
text{*Rewriting to pull differences out*}
haftmann@26072
  1251
haftmann@26072
  1252
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
haftmann@26072
  1253
by arith
haftmann@26072
  1254
haftmann@26072
  1255
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
haftmann@26072
  1256
by arith
haftmann@26072
  1257
haftmann@26072
  1258
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
haftmann@26072
  1259
by arith
haftmann@26072
  1260
wenzelm@21243
  1261
text{*Lemmas for ex/Factorization*}
wenzelm@21243
  1262
wenzelm@21243
  1263
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
nipkow@24438
  1264
by (cases m) auto
wenzelm@21243
  1265
wenzelm@21243
  1266
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
nipkow@24438
  1267
by (cases m) auto
wenzelm@21243
  1268
wenzelm@21243
  1269
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
nipkow@24438
  1270
by (cases m) auto
wenzelm@21243
  1271
krauss@23001
  1272
text {* Specialized induction principles that work "backwards": *}
krauss@23001
  1273
krauss@23001
  1274
lemma inc_induct[consumes 1, case_names base step]:
krauss@23001
  1275
  assumes less: "i <= j"
krauss@23001
  1276
  assumes base: "P j"
krauss@23001
  1277
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1278
  shows "P i"
krauss@23001
  1279
  using less
krauss@23001
  1280
proof (induct d=="j - i" arbitrary: i)
krauss@23001
  1281
  case (0 i)
krauss@23001
  1282
  hence "i = j" by simp
krauss@23001
  1283
  with base show ?case by simp
krauss@23001
  1284
next
krauss@23001
  1285
  case (Suc d i)
krauss@23001
  1286
  hence "i < j" "P (Suc i)"
krauss@23001
  1287
    by simp_all
krauss@23001
  1288
  thus "P i" by (rule step)
krauss@23001
  1289
qed
krauss@23001
  1290
krauss@23001
  1291
lemma strict_inc_induct[consumes 1, case_names base step]:
krauss@23001
  1292
  assumes less: "i < j"
krauss@23001
  1293
  assumes base: "!!i. j = Suc i ==> P i"
krauss@23001
  1294
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1295
  shows "P i"
krauss@23001
  1296
  using less
krauss@23001
  1297
proof (induct d=="j - i - 1" arbitrary: i)
krauss@23001
  1298
  case (0 i)
krauss@23001
  1299
  with `i < j` have "j = Suc i" by simp
krauss@23001
  1300
  with base show ?case by simp
krauss@23001
  1301
next
krauss@23001
  1302
  case (Suc d i)
krauss@23001
  1303
  hence "i < j" "P (Suc i)"
krauss@23001
  1304
    by simp_all
krauss@23001
  1305
  thus "P i" by (rule step)
krauss@23001
  1306
qed
krauss@23001
  1307
krauss@23001
  1308
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
krauss@23001
  1309
  using inc_induct[of "k - i" k P, simplified] by blast
krauss@23001
  1310
krauss@23001
  1311
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
krauss@23001
  1312
  using inc_induct[of 0 k P] by blast
wenzelm@21243
  1313
haftmann@26072
  1314
lemma nat_not_singleton: "(\<forall>x. x = (0::nat)) = False"
haftmann@26072
  1315
  by auto
wenzelm@21243
  1316
wenzelm@21243
  1317
(*The others are
wenzelm@21243
  1318
      i - j - k = i - (j + k),
wenzelm@21243
  1319
      k \<le> j ==> j - k + i = j + i - k,
wenzelm@21243
  1320
      k \<le> j ==> i + (j - k) = i + j - k *)
wenzelm@21243
  1321
lemmas add_diff_assoc = diff_add_assoc [symmetric]
wenzelm@21243
  1322
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
haftmann@26072
  1323
declare diff_diff_left [simp]  add_diff_assoc [simp] add_diff_assoc2[simp]
wenzelm@21243
  1324
wenzelm@21243
  1325
text{*At present we prove no analogue of @{text not_less_Least} or @{text
wenzelm@21243
  1326
Least_Suc}, since there appears to be no need.*}
wenzelm@21243
  1327
haftmann@26072
  1328
subsection {* size of a datatype value *}
haftmann@25193
  1329
haftmann@26072
  1330
class size = type +
haftmann@26072
  1331
  fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded_Recursion} *}
haftmann@23852
  1332
haftmann@25193
  1333
end