src/HOL/Nat.thy
author paulson
Tue Oct 19 18:18:45 2004 +0200 (2004-10-19)
changeset 15251 bb6f072c8d10
parent 15140 322485b816ac
child 15281 bd4611956c7b
permissions -rw-r--r--
converted some induct_tac to induct
clasohm@923
     1
(*  Title:      HOL/Nat.thy
clasohm@923
     2
    ID:         $Id$
wenzelm@9436
     3
    Author:     Tobias Nipkow and Lawrence C Paulson
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
nipkow@15140
    12
imports Wellfounded_Recursion Ring_and_Field
nipkow@15131
    13
begin
berghofe@13449
    14
berghofe@13449
    15
subsection {* Type @{text ind} *}
berghofe@13449
    16
berghofe@13449
    17
typedecl ind
berghofe@13449
    18
berghofe@13449
    19
consts
berghofe@13449
    20
  Zero_Rep      :: ind
berghofe@13449
    21
  Suc_Rep       :: "ind => ind"
berghofe@13449
    22
berghofe@13449
    23
axioms
berghofe@13449
    24
  -- {* the axiom of infinity in 2 parts *}
berghofe@13449
    25
  inj_Suc_Rep:          "inj Suc_Rep"
paulson@14267
    26
  Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"
berghofe@13449
    27
berghofe@13449
    28
berghofe@13449
    29
subsection {* Type nat *}
berghofe@13449
    30
berghofe@13449
    31
text {* Type definition *}
berghofe@13449
    32
berghofe@13449
    33
consts
berghofe@13449
    34
  Nat :: "ind set"
berghofe@13449
    35
berghofe@13449
    36
inductive Nat
berghofe@13449
    37
intros
berghofe@13449
    38
  Zero_RepI: "Zero_Rep : Nat"
berghofe@13449
    39
  Suc_RepI: "i : Nat ==> Suc_Rep i : Nat"
berghofe@13449
    40
berghofe@13449
    41
global
berghofe@13449
    42
berghofe@13449
    43
typedef (open Nat)
paulson@14208
    44
  nat = Nat by (rule exI, rule Nat.Zero_RepI)
berghofe@13449
    45
wenzelm@14691
    46
instance nat :: "{ord, zero, one}" ..
berghofe@13449
    47
berghofe@13449
    48
berghofe@13449
    49
text {* Abstract constants and syntax *}
berghofe@13449
    50
berghofe@13449
    51
consts
berghofe@13449
    52
  Suc :: "nat => nat"
berghofe@13449
    53
  pred_nat :: "(nat * nat) set"
berghofe@13449
    54
berghofe@13449
    55
local
berghofe@13449
    56
berghofe@13449
    57
defs
berghofe@13449
    58
  Zero_nat_def: "0 == Abs_Nat Zero_Rep"
berghofe@13449
    59
  Suc_def: "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))"
berghofe@13449
    60
  One_nat_def [simp]: "1 == Suc 0"
berghofe@13449
    61
berghofe@13449
    62
  -- {* nat operations *}
berghofe@13449
    63
  pred_nat_def: "pred_nat == {(m, n). n = Suc m}"
berghofe@13449
    64
berghofe@13449
    65
  less_def: "m < n == (m, n) : trancl pred_nat"
berghofe@13449
    66
paulson@14267
    67
  le_def: "m \<le> (n::nat) == ~ (n < m)"
berghofe@13449
    68
berghofe@13449
    69
berghofe@13449
    70
text {* Induction *}
clasohm@923
    71
berghofe@13449
    72
theorem nat_induct: "P 0 ==> (!!n. P n ==> P (Suc n)) ==> P n"
berghofe@13449
    73
  apply (unfold Zero_nat_def Suc_def)
berghofe@13449
    74
  apply (rule Rep_Nat_inverse [THEN subst]) -- {* types force good instantiation *}
berghofe@13449
    75
  apply (erule Rep_Nat [THEN Nat.induct])
berghofe@13449
    76
  apply (rules elim: Abs_Nat_inverse [THEN subst])
berghofe@13449
    77
  done
berghofe@13449
    78
berghofe@13449
    79
berghofe@13449
    80
text {* Isomorphisms: @{text Abs_Nat} and @{text Rep_Nat} *}
berghofe@13449
    81
berghofe@13449
    82
lemma inj_Rep_Nat: "inj Rep_Nat"
paulson@13585
    83
  apply (rule inj_on_inverseI)
berghofe@13449
    84
  apply (rule Rep_Nat_inverse)
berghofe@13449
    85
  done
berghofe@13449
    86
berghofe@13449
    87
lemma inj_on_Abs_Nat: "inj_on Abs_Nat Nat"
berghofe@13449
    88
  apply (rule inj_on_inverseI)
berghofe@13449
    89
  apply (erule Abs_Nat_inverse)
berghofe@13449
    90
  done
berghofe@13449
    91
berghofe@13449
    92
text {* Distinctness of constructors *}
berghofe@13449
    93
paulson@14267
    94
lemma Suc_not_Zero [iff]: "Suc m \<noteq> 0"
berghofe@13449
    95
  apply (unfold Zero_nat_def Suc_def)
berghofe@13449
    96
  apply (rule inj_on_Abs_Nat [THEN inj_on_contraD])
berghofe@13449
    97
  apply (rule Suc_Rep_not_Zero_Rep)
berghofe@13449
    98
  apply (rule Rep_Nat Nat.Suc_RepI Nat.Zero_RepI)+
berghofe@13449
    99
  done
berghofe@13449
   100
paulson@14267
   101
lemma Zero_not_Suc [iff]: "0 \<noteq> Suc m"
berghofe@13449
   102
  by (rule not_sym, rule Suc_not_Zero not_sym)
berghofe@13449
   103
berghofe@13449
   104
lemma Suc_neq_Zero: "Suc m = 0 ==> R"
berghofe@13449
   105
  by (rule notE, rule Suc_not_Zero)
berghofe@13449
   106
berghofe@13449
   107
lemma Zero_neq_Suc: "0 = Suc m ==> R"
berghofe@13449
   108
  by (rule Suc_neq_Zero, erule sym)
berghofe@13449
   109
berghofe@13449
   110
text {* Injectiveness of @{term Suc} *}
berghofe@13449
   111
berghofe@13449
   112
lemma inj_Suc: "inj Suc"
berghofe@13449
   113
  apply (unfold Suc_def)
paulson@13585
   114
  apply (rule inj_onI)
berghofe@13449
   115
  apply (drule inj_on_Abs_Nat [THEN inj_onD])
berghofe@13449
   116
  apply (rule Rep_Nat Nat.Suc_RepI)+
berghofe@13449
   117
  apply (drule inj_Suc_Rep [THEN injD])
berghofe@13449
   118
  apply (erule inj_Rep_Nat [THEN injD])
berghofe@13449
   119
  done
berghofe@13449
   120
berghofe@13449
   121
lemma Suc_inject: "Suc x = Suc y ==> x = y"
berghofe@13449
   122
  by (rule inj_Suc [THEN injD])
berghofe@13449
   123
berghofe@13449
   124
lemma Suc_Suc_eq [iff]: "(Suc m = Suc n) = (m = n)"
berghofe@13449
   125
  apply (rule iffI)
berghofe@13449
   126
  apply (erule Suc_inject)
berghofe@13449
   127
  apply (erule arg_cong)
berghofe@13449
   128
  done
berghofe@13449
   129
paulson@14267
   130
lemma nat_not_singleton: "(\<forall>x. x = (0::nat)) = False"
berghofe@13449
   131
  by auto
berghofe@13449
   132
berghofe@13449
   133
text {* @{typ nat} is a datatype *}
wenzelm@9436
   134
berghofe@5188
   135
rep_datatype nat
berghofe@13449
   136
  distinct  Suc_not_Zero Zero_not_Suc
berghofe@13449
   137
  inject    Suc_Suc_eq
berghofe@13449
   138
  induction nat_induct
berghofe@13449
   139
paulson@14267
   140
lemma n_not_Suc_n: "n \<noteq> Suc n"
berghofe@13449
   141
  by (induct n) simp_all
berghofe@13449
   142
paulson@14267
   143
lemma Suc_n_not_n: "Suc t \<noteq> t"
berghofe@13449
   144
  by (rule not_sym, rule n_not_Suc_n)
berghofe@13449
   145
berghofe@13449
   146
text {* A special form of induction for reasoning
berghofe@13449
   147
  about @{term "m < n"} and @{term "m - n"} *}
berghofe@13449
   148
berghofe@13449
   149
theorem diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==>
berghofe@13449
   150
    (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n"
paulson@14208
   151
  apply (rule_tac x = m in spec)
paulson@15251
   152
  apply (induct n)
berghofe@13449
   153
  prefer 2
berghofe@13449
   154
  apply (rule allI)
paulson@14208
   155
  apply (induct_tac x, rules+)
berghofe@13449
   156
  done
berghofe@13449
   157
berghofe@13449
   158
subsection {* Basic properties of "less than" *}
berghofe@13449
   159
berghofe@13449
   160
lemma wf_pred_nat: "wf pred_nat"
paulson@14208
   161
  apply (unfold wf_def pred_nat_def, clarify)
paulson@14208
   162
  apply (induct_tac x, blast+)
berghofe@13449
   163
  done
berghofe@13449
   164
berghofe@13449
   165
lemma wf_less: "wf {(x, y::nat). x < y}"
berghofe@13449
   166
  apply (unfold less_def)
paulson@14208
   167
  apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_subset], blast)
berghofe@13449
   168
  done
berghofe@13449
   169
berghofe@13449
   170
lemma less_eq: "((m, n) : pred_nat^+) = (m < n)"
berghofe@13449
   171
  apply (unfold less_def)
berghofe@13449
   172
  apply (rule refl)
berghofe@13449
   173
  done
berghofe@13449
   174
berghofe@13449
   175
subsubsection {* Introduction properties *}
berghofe@13449
   176
berghofe@13449
   177
lemma less_trans: "i < j ==> j < k ==> i < (k::nat)"
berghofe@13449
   178
  apply (unfold less_def)
paulson@14208
   179
  apply (rule trans_trancl [THEN transD], assumption+)
berghofe@13449
   180
  done
berghofe@13449
   181
berghofe@13449
   182
lemma lessI [iff]: "n < Suc n"
berghofe@13449
   183
  apply (unfold less_def pred_nat_def)
berghofe@13449
   184
  apply (simp add: r_into_trancl)
berghofe@13449
   185
  done
berghofe@13449
   186
berghofe@13449
   187
lemma less_SucI: "i < j ==> i < Suc j"
paulson@14208
   188
  apply (rule less_trans, assumption)
berghofe@13449
   189
  apply (rule lessI)
berghofe@13449
   190
  done
berghofe@13449
   191
berghofe@13449
   192
lemma zero_less_Suc [iff]: "0 < Suc n"
berghofe@13449
   193
  apply (induct n)
berghofe@13449
   194
  apply (rule lessI)
berghofe@13449
   195
  apply (erule less_trans)
berghofe@13449
   196
  apply (rule lessI)
berghofe@13449
   197
  done
berghofe@13449
   198
berghofe@13449
   199
subsubsection {* Elimination properties *}
berghofe@13449
   200
berghofe@13449
   201
lemma less_not_sym: "n < m ==> ~ m < (n::nat)"
berghofe@13449
   202
  apply (unfold less_def)
berghofe@13449
   203
  apply (blast intro: wf_pred_nat wf_trancl [THEN wf_asym])
berghofe@13449
   204
  done
berghofe@13449
   205
berghofe@13449
   206
lemma less_asym:
berghofe@13449
   207
  assumes h1: "(n::nat) < m" and h2: "~ P ==> m < n" shows P
berghofe@13449
   208
  apply (rule contrapos_np)
berghofe@13449
   209
  apply (rule less_not_sym)
berghofe@13449
   210
  apply (rule h1)
berghofe@13449
   211
  apply (erule h2)
berghofe@13449
   212
  done
berghofe@13449
   213
berghofe@13449
   214
lemma less_not_refl: "~ n < (n::nat)"
berghofe@13449
   215
  apply (unfold less_def)
berghofe@13449
   216
  apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_not_refl])
berghofe@13449
   217
  done
berghofe@13449
   218
berghofe@13449
   219
lemma less_irrefl [elim!]: "(n::nat) < n ==> R"
berghofe@13449
   220
  by (rule notE, rule less_not_refl)
berghofe@13449
   221
paulson@14267
   222
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)" by blast
berghofe@13449
   223
paulson@14267
   224
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
berghofe@13449
   225
  by (rule not_sym, rule less_not_refl2)
berghofe@13449
   226
berghofe@13449
   227
lemma lessE:
berghofe@13449
   228
  assumes major: "i < k"
berghofe@13449
   229
  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
berghofe@13449
   230
  shows P
paulson@14208
   231
  apply (rule major [unfolded less_def pred_nat_def, THEN tranclE], simp_all)
berghofe@13449
   232
  apply (erule p1)
berghofe@13449
   233
  apply (rule p2)
paulson@14208
   234
  apply (simp add: less_def pred_nat_def, assumption)
berghofe@13449
   235
  done
berghofe@13449
   236
berghofe@13449
   237
lemma not_less0 [iff]: "~ n < (0::nat)"
berghofe@13449
   238
  by (blast elim: lessE)
berghofe@13449
   239
berghofe@13449
   240
lemma less_zeroE: "(n::nat) < 0 ==> R"
berghofe@13449
   241
  by (rule notE, rule not_less0)
berghofe@13449
   242
berghofe@13449
   243
lemma less_SucE: assumes major: "m < Suc n"
berghofe@13449
   244
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
berghofe@13449
   245
  apply (rule major [THEN lessE])
paulson@14208
   246
  apply (rule eq, blast)
paulson@14208
   247
  apply (rule less, blast)
berghofe@13449
   248
  done
berghofe@13449
   249
berghofe@13449
   250
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
berghofe@13449
   251
  by (blast elim!: less_SucE intro: less_trans)
berghofe@13449
   252
berghofe@13449
   253
lemma less_one [iff]: "(n < (1::nat)) = (n = 0)"
berghofe@13449
   254
  by (simp add: less_Suc_eq)
berghofe@13449
   255
berghofe@13449
   256
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
berghofe@13449
   257
  by (simp add: less_Suc_eq)
berghofe@13449
   258
berghofe@13449
   259
lemma Suc_mono: "m < n ==> Suc m < Suc n"
berghofe@13449
   260
  by (induct n) (fast elim: less_trans lessE)+
berghofe@13449
   261
berghofe@13449
   262
text {* "Less than" is a linear ordering *}
berghofe@13449
   263
lemma less_linear: "m < n | m = n | n < (m::nat)"
paulson@15251
   264
  apply (induct m)
paulson@15251
   265
  apply (induct n)
berghofe@13449
   266
  apply (rule refl [THEN disjI1, THEN disjI2])
berghofe@13449
   267
  apply (rule zero_less_Suc [THEN disjI1])
berghofe@13449
   268
  apply (blast intro: Suc_mono less_SucI elim: lessE)
berghofe@13449
   269
  done
berghofe@13449
   270
nipkow@14302
   271
text {* "Less than" is antisymmetric, sort of *}
nipkow@14302
   272
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
nipkow@14302
   273
apply(simp only:less_Suc_eq)
nipkow@14302
   274
apply blast
nipkow@14302
   275
done
nipkow@14302
   276
paulson@14267
   277
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
berghofe@13449
   278
  using less_linear by blast
berghofe@13449
   279
berghofe@13449
   280
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
berghofe@13449
   281
  and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
berghofe@13449
   282
  shows "P n m"
berghofe@13449
   283
  apply (rule less_linear [THEN disjE])
berghofe@13449
   284
  apply (erule_tac [2] disjE)
berghofe@13449
   285
  apply (erule lessCase)
berghofe@13449
   286
  apply (erule sym [THEN eqCase])
berghofe@13449
   287
  apply (erule major)
berghofe@13449
   288
  done
berghofe@13449
   289
berghofe@13449
   290
berghofe@13449
   291
subsubsection {* Inductive (?) properties *}
berghofe@13449
   292
paulson@14267
   293
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
berghofe@13449
   294
  apply (simp add: nat_neq_iff)
berghofe@13449
   295
  apply (blast elim!: less_irrefl less_SucE elim: less_asym)
berghofe@13449
   296
  done
berghofe@13449
   297
berghofe@13449
   298
lemma Suc_lessD: "Suc m < n ==> m < n"
berghofe@13449
   299
  apply (induct n)
berghofe@13449
   300
  apply (fast intro!: lessI [THEN less_SucI] elim: less_trans lessE)+
berghofe@13449
   301
  done
berghofe@13449
   302
berghofe@13449
   303
lemma Suc_lessE: assumes major: "Suc i < k"
berghofe@13449
   304
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
berghofe@13449
   305
  apply (rule major [THEN lessE])
berghofe@13449
   306
  apply (erule lessI [THEN minor])
paulson@14208
   307
  apply (erule Suc_lessD [THEN minor], assumption)
berghofe@13449
   308
  done
berghofe@13449
   309
berghofe@13449
   310
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
berghofe@13449
   311
  by (blast elim: lessE dest: Suc_lessD)
wenzelm@4104
   312
berghofe@13449
   313
lemma Suc_less_eq [iff]: "(Suc m < Suc n) = (m < n)"
berghofe@13449
   314
  apply (rule iffI)
berghofe@13449
   315
  apply (erule Suc_less_SucD)
berghofe@13449
   316
  apply (erule Suc_mono)
berghofe@13449
   317
  done
berghofe@13449
   318
berghofe@13449
   319
lemma less_trans_Suc:
berghofe@13449
   320
  assumes le: "i < j" shows "j < k ==> Suc i < k"
paulson@14208
   321
  apply (induct k, simp_all)
berghofe@13449
   322
  apply (insert le)
berghofe@13449
   323
  apply (simp add: less_Suc_eq)
berghofe@13449
   324
  apply (blast dest: Suc_lessD)
berghofe@13449
   325
  done
berghofe@13449
   326
berghofe@13449
   327
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
berghofe@13449
   328
lemma not_less_eq: "(~ m < n) = (n < Suc m)"
paulson@14208
   329
by (rule_tac m = m and n = n in diff_induct, simp_all)
berghofe@13449
   330
berghofe@13449
   331
text {* Complete induction, aka course-of-values induction *}
berghofe@13449
   332
lemma nat_less_induct:
paulson@14267
   333
  assumes prem: "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
berghofe@13449
   334
  apply (rule_tac a=n in wf_induct)
berghofe@13449
   335
  apply (rule wf_pred_nat [THEN wf_trancl])
berghofe@13449
   336
  apply (rule prem)
paulson@14208
   337
  apply (unfold less_def, assumption)
berghofe@13449
   338
  done
berghofe@13449
   339
paulson@14131
   340
lemmas less_induct = nat_less_induct [rule_format, case_names less]
paulson@14131
   341
paulson@14131
   342
subsection {* Properties of "less than or equal" *}
berghofe@13449
   343
berghofe@13449
   344
text {* Was @{text le_eq_less_Suc}, but this orientation is more useful *}
paulson@14267
   345
lemma less_Suc_eq_le: "(m < Suc n) = (m \<le> n)"
berghofe@13449
   346
  by (unfold le_def, rule not_less_eq [symmetric])
berghofe@13449
   347
paulson@14267
   348
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
berghofe@13449
   349
  by (rule less_Suc_eq_le [THEN iffD2])
berghofe@13449
   350
paulson@14267
   351
lemma le0 [iff]: "(0::nat) \<le> n"
berghofe@13449
   352
  by (unfold le_def, rule not_less0)
berghofe@13449
   353
paulson@14267
   354
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
berghofe@13449
   355
  by (simp add: le_def)
berghofe@13449
   356
paulson@14267
   357
lemma le_0_eq [iff]: "((i::nat) \<le> 0) = (i = 0)"
berghofe@13449
   358
  by (induct i) (simp_all add: le_def)
berghofe@13449
   359
paulson@14267
   360
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
berghofe@13449
   361
  by (simp del: less_Suc_eq_le add: less_Suc_eq_le [symmetric] less_Suc_eq)
berghofe@13449
   362
paulson@14267
   363
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
berghofe@13449
   364
  by (drule le_Suc_eq [THEN iffD1], rules+)
berghofe@13449
   365
paulson@14267
   366
lemma leI: "~ n < m ==> m \<le> (n::nat)" by (simp add: le_def)
berghofe@13449
   367
paulson@14267
   368
lemma leD: "m \<le> n ==> ~ n < (m::nat)"
berghofe@13449
   369
  by (simp add: le_def)
berghofe@13449
   370
berghofe@13449
   371
lemmas leE = leD [elim_format]
berghofe@13449
   372
paulson@14267
   373
lemma not_less_iff_le: "(~ n < m) = (m \<le> (n::nat))"
berghofe@13449
   374
  by (blast intro: leI elim: leE)
berghofe@13449
   375
paulson@14267
   376
lemma not_leE: "~ m \<le> n ==> n<(m::nat)"
berghofe@13449
   377
  by (simp add: le_def)
berghofe@13449
   378
paulson@14267
   379
lemma not_le_iff_less: "(~ n \<le> m) = (m < (n::nat))"
berghofe@13449
   380
  by (simp add: le_def)
berghofe@13449
   381
paulson@14267
   382
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
berghofe@13449
   383
  apply (simp add: le_def less_Suc_eq)
berghofe@13449
   384
  apply (blast elim!: less_irrefl less_asym)
berghofe@13449
   385
  done -- {* formerly called lessD *}
berghofe@13449
   386
paulson@14267
   387
lemma Suc_leD: "Suc(m) \<le> n ==> m \<le> n"
berghofe@13449
   388
  by (simp add: le_def less_Suc_eq)
berghofe@13449
   389
berghofe@13449
   390
text {* Stronger version of @{text Suc_leD} *}
paulson@14267
   391
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
berghofe@13449
   392
  apply (simp add: le_def less_Suc_eq)
berghofe@13449
   393
  using less_linear
berghofe@13449
   394
  apply blast
berghofe@13449
   395
  done
berghofe@13449
   396
paulson@14267
   397
lemma Suc_le_eq: "(Suc m \<le> n) = (m < n)"
berghofe@13449
   398
  by (blast intro: Suc_leI Suc_le_lessD)
berghofe@13449
   399
paulson@14267
   400
lemma le_SucI: "m \<le> n ==> m \<le> Suc n"
berghofe@13449
   401
  by (unfold le_def) (blast dest: Suc_lessD)
berghofe@13449
   402
paulson@14267
   403
lemma less_imp_le: "m < n ==> m \<le> (n::nat)"
berghofe@13449
   404
  by (unfold le_def) (blast elim: less_asym)
berghofe@13449
   405
paulson@14267
   406
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
berghofe@13449
   407
lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq
berghofe@13449
   408
berghofe@13449
   409
paulson@14267
   410
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
berghofe@13449
   411
paulson@14267
   412
lemma le_imp_less_or_eq: "m \<le> n ==> m < n | m = (n::nat)"
berghofe@13449
   413
  apply (unfold le_def)
berghofe@13449
   414
  using less_linear
berghofe@13449
   415
  apply (blast elim: less_irrefl less_asym)
berghofe@13449
   416
  done
berghofe@13449
   417
paulson@14267
   418
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
berghofe@13449
   419
  apply (unfold le_def)
berghofe@13449
   420
  using less_linear
berghofe@13449
   421
  apply (blast elim!: less_irrefl elim: less_asym)
berghofe@13449
   422
  done
berghofe@13449
   423
paulson@14267
   424
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
berghofe@13449
   425
  by (rules intro: less_or_eq_imp_le le_imp_less_or_eq)
berghofe@13449
   426
berghofe@13449
   427
text {* Useful with @{text Blast}. *}
paulson@14267
   428
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
berghofe@13449
   429
  by (rule less_or_eq_imp_le, rule disjI2)
berghofe@13449
   430
paulson@14267
   431
lemma le_refl: "n \<le> (n::nat)"
berghofe@13449
   432
  by (simp add: le_eq_less_or_eq)
berghofe@13449
   433
paulson@14267
   434
lemma le_less_trans: "[| i \<le> j; j < k |] ==> i < (k::nat)"
berghofe@13449
   435
  by (blast dest!: le_imp_less_or_eq intro: less_trans)
berghofe@13449
   436
paulson@14267
   437
lemma less_le_trans: "[| i < j; j \<le> k |] ==> i < (k::nat)"
berghofe@13449
   438
  by (blast dest!: le_imp_less_or_eq intro: less_trans)
berghofe@13449
   439
paulson@14267
   440
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
berghofe@13449
   441
  by (blast dest!: le_imp_less_or_eq intro: less_or_eq_imp_le less_trans)
berghofe@13449
   442
paulson@14267
   443
lemma le_anti_sym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
berghofe@13449
   444
  by (blast dest!: le_imp_less_or_eq elim!: less_irrefl elim: less_asym)
berghofe@13449
   445
paulson@14267
   446
lemma Suc_le_mono [iff]: "(Suc n \<le> Suc m) = (n \<le> m)"
berghofe@13449
   447
  by (simp add: le_simps)
berghofe@13449
   448
berghofe@13449
   449
text {* Axiom @{text order_less_le} of class @{text order}: *}
paulson@14267
   450
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
berghofe@13449
   451
  by (simp add: le_def nat_neq_iff) (blast elim!: less_asym)
berghofe@13449
   452
paulson@14267
   453
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
berghofe@13449
   454
  by (rule iffD2, rule nat_less_le, rule conjI)
berghofe@13449
   455
berghofe@13449
   456
text {* Axiom @{text linorder_linear} of class @{text linorder}: *}
paulson@14267
   457
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
berghofe@13449
   458
  apply (simp add: le_eq_less_or_eq)
berghofe@13449
   459
  using less_linear
berghofe@13449
   460
  apply blast
berghofe@13449
   461
  done
berghofe@13449
   462
paulson@14341
   463
text {* Type {@typ nat} is a wellfounded linear order *}
paulson@14341
   464
wenzelm@14691
   465
instance nat :: "{order, linorder, wellorder}"
wenzelm@14691
   466
  by intro_classes
wenzelm@14691
   467
    (assumption |
wenzelm@14691
   468
      rule le_refl le_trans le_anti_sym nat_less_le nat_le_linear wf_less)+
paulson@14341
   469
berghofe@13449
   470
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
berghofe@13449
   471
  by (blast elim!: less_SucE)
berghofe@13449
   472
berghofe@13449
   473
text {*
berghofe@13449
   474
  Rewrite @{term "n < Suc m"} to @{term "n = m"}
paulson@14267
   475
  if @{term "~ n < m"} or @{term "m \<le> n"} hold.
berghofe@13449
   476
  Not suitable as default simprules because they often lead to looping
berghofe@13449
   477
*}
paulson@14267
   478
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
berghofe@13449
   479
  by (rule not_less_less_Suc_eq, rule leD)
berghofe@13449
   480
berghofe@13449
   481
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
berghofe@13449
   482
berghofe@13449
   483
berghofe@13449
   484
text {*
berghofe@13449
   485
  Re-orientation of the equations @{text "0 = x"} and @{text "1 = x"}. 
berghofe@13449
   486
  No longer added as simprules (they loop) 
berghofe@13449
   487
  but via @{text reorient_simproc} in Bin
berghofe@13449
   488
*}
berghofe@13449
   489
berghofe@13449
   490
text {* Polymorphic, not just for @{typ nat} *}
berghofe@13449
   491
lemma zero_reorient: "(0 = x) = (x = 0)"
berghofe@13449
   492
  by auto
berghofe@13449
   493
berghofe@13449
   494
lemma one_reorient: "(1 = x) = (x = 1)"
berghofe@13449
   495
  by auto
berghofe@13449
   496
berghofe@13449
   497
subsection {* Arithmetic operators *}
oheimb@1660
   498
wenzelm@12338
   499
axclass power < type
wenzelm@10435
   500
paulson@3370
   501
consts
berghofe@13449
   502
  power :: "('a::power) => nat => 'a"            (infixr "^" 80)
paulson@3370
   503
wenzelm@9436
   504
berghofe@13449
   505
text {* arithmetic operators @{text "+ -"} and @{text "*"} *}
berghofe@13449
   506
wenzelm@14691
   507
instance nat :: "{plus, minus, times, power}" ..
wenzelm@9436
   508
berghofe@13449
   509
text {* size of a datatype value; overloaded *}
berghofe@13449
   510
consts size :: "'a => nat"
wenzelm@9436
   511
berghofe@13449
   512
primrec
berghofe@13449
   513
  add_0:    "0 + n = n"
berghofe@13449
   514
  add_Suc:  "Suc m + n = Suc (m + n)"
berghofe@13449
   515
berghofe@13449
   516
primrec
berghofe@13449
   517
  diff_0:   "m - 0 = m"
berghofe@13449
   518
  diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
wenzelm@9436
   519
wenzelm@9436
   520
primrec
berghofe@13449
   521
  mult_0:   "0 * n = 0"
berghofe@13449
   522
  mult_Suc: "Suc m * n = n + (m * n)"
berghofe@13449
   523
paulson@14341
   524
text {* These two rules ease the use of primitive recursion. 
paulson@14341
   525
NOTE USE OF @{text "=="} *}
berghofe@13449
   526
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"
berghofe@13449
   527
  by simp
berghofe@13449
   528
berghofe@13449
   529
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"
berghofe@13449
   530
  by simp
berghofe@13449
   531
paulson@14267
   532
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
berghofe@13449
   533
  by (case_tac n) simp_all
berghofe@13449
   534
paulson@14267
   535
lemma gr_implies_not0: "!!n::nat. m<n ==> n \<noteq> 0"
berghofe@13449
   536
  by (case_tac n) simp_all
berghofe@13449
   537
paulson@14267
   538
lemma neq0_conv [iff]: "!!n::nat. (n \<noteq> 0) = (0 < n)"
berghofe@13449
   539
  by (case_tac n) simp_all
berghofe@13449
   540
berghofe@13449
   541
text {* This theorem is useful with @{text blast} *}
berghofe@13449
   542
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
berghofe@13449
   543
  by (rule iffD1, rule neq0_conv, rules)
berghofe@13449
   544
paulson@14267
   545
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
berghofe@13449
   546
  by (fast intro: not0_implies_Suc)
berghofe@13449
   547
berghofe@13449
   548
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"
berghofe@13449
   549
  apply (rule iffI)
paulson@14208
   550
  apply (rule ccontr, simp_all)
berghofe@13449
   551
  done
berghofe@13449
   552
paulson@14267
   553
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
berghofe@13449
   554
  by (induct m') simp_all
berghofe@13449
   555
berghofe@13449
   556
text {* Useful in certain inductive arguments *}
paulson@14267
   557
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
berghofe@13449
   558
  by (case_tac m) simp_all
berghofe@13449
   559
paulson@14341
   560
lemma nat_induct2: "[|P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))|] ==> P n"
berghofe@13449
   561
  apply (rule nat_less_induct)
berghofe@13449
   562
  apply (case_tac n)
berghofe@13449
   563
  apply (case_tac [2] nat)
berghofe@13449
   564
  apply (blast intro: less_trans)+
berghofe@13449
   565
  done
berghofe@13449
   566
berghofe@13449
   567
subsection {* @{text LEAST} theorems for type @{typ nat} by specialization *}
berghofe@13449
   568
berghofe@13449
   569
lemmas LeastI = wellorder_LeastI
berghofe@13449
   570
lemmas Least_le = wellorder_Least_le
berghofe@13449
   571
lemmas not_less_Least = wellorder_not_less_Least
berghofe@13449
   572
paulson@14267
   573
lemma Least_Suc:
paulson@14267
   574
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
paulson@14208
   575
  apply (case_tac "n", auto)
berghofe@13449
   576
  apply (frule LeastI)
berghofe@13449
   577
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
paulson@14267
   578
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
berghofe@13449
   579
  apply (erule_tac [2] Least_le)
paulson@14208
   580
  apply (case_tac "LEAST x. P x", auto)
berghofe@13449
   581
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
berghofe@13449
   582
  apply (blast intro: order_antisym)
berghofe@13449
   583
  done
berghofe@13449
   584
paulson@14267
   585
lemma Least_Suc2:
paulson@14267
   586
     "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
paulson@14267
   587
  by (erule (1) Least_Suc [THEN ssubst], simp)
berghofe@13449
   588
berghofe@13449
   589
paulson@14265
   590
berghofe@13449
   591
subsection {* @{term min} and @{term max} *}
berghofe@13449
   592
berghofe@13449
   593
lemma min_0L [simp]: "min 0 n = (0::nat)"
berghofe@13449
   594
  by (rule min_leastL) simp
berghofe@13449
   595
berghofe@13449
   596
lemma min_0R [simp]: "min n 0 = (0::nat)"
berghofe@13449
   597
  by (rule min_leastR) simp
berghofe@13449
   598
berghofe@13449
   599
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
berghofe@13449
   600
  by (simp add: min_of_mono)
berghofe@13449
   601
berghofe@13449
   602
lemma max_0L [simp]: "max 0 n = (n::nat)"
berghofe@13449
   603
  by (rule max_leastL) simp
berghofe@13449
   604
berghofe@13449
   605
lemma max_0R [simp]: "max n 0 = (n::nat)"
berghofe@13449
   606
  by (rule max_leastR) simp
berghofe@13449
   607
berghofe@13449
   608
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
berghofe@13449
   609
  by (simp add: max_of_mono)
berghofe@13449
   610
berghofe@13449
   611
berghofe@13449
   612
subsection {* Basic rewrite rules for the arithmetic operators *}
berghofe@13449
   613
berghofe@13449
   614
text {* Difference *}
berghofe@13449
   615
berghofe@14193
   616
lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"
paulson@15251
   617
  by (induct n) simp_all
berghofe@13449
   618
berghofe@14193
   619
lemma diff_Suc_Suc [simp, code]: "Suc(m) - Suc(n) = m - n"
paulson@15251
   620
  by (induct n) simp_all
berghofe@13449
   621
berghofe@13449
   622
berghofe@13449
   623
text {*
berghofe@13449
   624
  Could be (and is, below) generalized in various ways
berghofe@13449
   625
  However, none of the generalizations are currently in the simpset,
berghofe@13449
   626
  and I dread to think what happens if I put them in
berghofe@13449
   627
*}
berghofe@13449
   628
lemma Suc_pred [simp]: "0 < n ==> Suc (n - Suc 0) = n"
berghofe@13449
   629
  by (simp split add: nat.split)
berghofe@13449
   630
berghofe@14193
   631
declare diff_Suc [simp del, code del]
berghofe@13449
   632
berghofe@13449
   633
berghofe@13449
   634
subsection {* Addition *}
berghofe@13449
   635
berghofe@13449
   636
lemma add_0_right [simp]: "m + 0 = (m::nat)"
berghofe@13449
   637
  by (induct m) simp_all
berghofe@13449
   638
berghofe@13449
   639
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
berghofe@13449
   640
  by (induct m) simp_all
berghofe@13449
   641
berghofe@14193
   642
lemma [code]: "Suc m + n = m + Suc n" by simp
berghofe@14193
   643
berghofe@13449
   644
berghofe@13449
   645
text {* Associative law for addition *}
paulson@14267
   646
lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)"
berghofe@13449
   647
  by (induct m) simp_all
berghofe@13449
   648
berghofe@13449
   649
text {* Commutative law for addition *}
paulson@14267
   650
lemma nat_add_commute: "m + n = n + (m::nat)"
berghofe@13449
   651
  by (induct m) simp_all
berghofe@13449
   652
paulson@14267
   653
lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)"
berghofe@13449
   654
  apply (rule mk_left_commute [of "op +"])
paulson@14267
   655
  apply (rule nat_add_assoc)
paulson@14267
   656
  apply (rule nat_add_commute)
berghofe@13449
   657
  done
berghofe@13449
   658
paulson@14331
   659
lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))"
berghofe@13449
   660
  by (induct k) simp_all
berghofe@13449
   661
paulson@14331
   662
lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))"
berghofe@13449
   663
  by (induct k) simp_all
berghofe@13449
   664
paulson@14331
   665
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
berghofe@13449
   666
  by (induct k) simp_all
berghofe@13449
   667
paulson@14331
   668
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
berghofe@13449
   669
  by (induct k) simp_all
berghofe@13449
   670
berghofe@13449
   671
text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
berghofe@13449
   672
berghofe@13449
   673
lemma add_is_0 [iff]: "!!m::nat. (m + n = 0) = (m = 0 & n = 0)"
berghofe@13449
   674
  by (case_tac m) simp_all
berghofe@13449
   675
berghofe@13449
   676
lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
berghofe@13449
   677
  by (case_tac m) simp_all
berghofe@13449
   678
berghofe@13449
   679
lemma one_is_add: "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
berghofe@13449
   680
  by (rule trans, rule eq_commute, rule add_is_1)
berghofe@13449
   681
berghofe@13449
   682
lemma add_gr_0 [iff]: "!!m::nat. (0 < m + n) = (0 < m | 0 < n)"
berghofe@13449
   683
  by (simp del: neq0_conv add: neq0_conv [symmetric])
berghofe@13449
   684
berghofe@13449
   685
lemma add_eq_self_zero: "!!m::nat. m + n = m ==> n = 0"
berghofe@13449
   686
  apply (drule add_0_right [THEN ssubst])
paulson@14267
   687
  apply (simp add: nat_add_assoc del: add_0_right)
berghofe@13449
   688
  done
berghofe@13449
   689
paulson@14267
   690
paulson@14267
   691
subsection {* Multiplication *}
paulson@14267
   692
paulson@14267
   693
text {* right annihilation in product *}
paulson@14267
   694
lemma mult_0_right [simp]: "(m::nat) * 0 = 0"
paulson@14267
   695
  by (induct m) simp_all
paulson@14267
   696
paulson@14267
   697
text {* right successor law for multiplication *}
paulson@14267
   698
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
paulson@14267
   699
  by (induct m) (simp_all add: nat_add_left_commute)
paulson@14267
   700
paulson@14267
   701
text {* Commutative law for multiplication *}
paulson@14267
   702
lemma nat_mult_commute: "m * n = n * (m::nat)"
paulson@14267
   703
  by (induct m) simp_all
paulson@14267
   704
paulson@14267
   705
text {* addition distributes over multiplication *}
paulson@14267
   706
lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"
paulson@14267
   707
  by (induct m) (simp_all add: nat_add_assoc nat_add_left_commute)
paulson@14267
   708
paulson@14267
   709
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
paulson@14267
   710
  by (induct m) (simp_all add: nat_add_assoc)
paulson@14267
   711
paulson@14267
   712
text {* Associative law for multiplication *}
paulson@14267
   713
lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)"
paulson@14267
   714
  by (induct m) (simp_all add: add_mult_distrib)
paulson@14267
   715
paulson@14267
   716
nipkow@14740
   717
text{*The naturals form a @{text comm_semiring_1_cancel}*}
obua@14738
   718
instance nat :: comm_semiring_1_cancel
paulson@14267
   719
proof
paulson@14267
   720
  fix i j k :: nat
paulson@14267
   721
  show "(i + j) + k = i + (j + k)" by (rule nat_add_assoc)
paulson@14267
   722
  show "i + j = j + i" by (rule nat_add_commute)
paulson@14267
   723
  show "0 + i = i" by simp
paulson@14267
   724
  show "(i * j) * k = i * (j * k)" by (rule nat_mult_assoc)
paulson@14267
   725
  show "i * j = j * i" by (rule nat_mult_commute)
paulson@14267
   726
  show "1 * i = i" by simp
paulson@14267
   727
  show "(i + j) * k = i * k + j * k" by (simp add: add_mult_distrib)
paulson@14267
   728
  show "0 \<noteq> (1::nat)" by simp
paulson@14341
   729
  assume "k+i = k+j" thus "i=j" by simp
paulson@14341
   730
qed
paulson@14341
   731
paulson@14341
   732
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
paulson@15251
   733
  apply (induct m)
paulson@14341
   734
  apply (induct_tac [2] n, simp_all)
paulson@14341
   735
  done
paulson@14341
   736
paulson@14341
   737
subsection {* Monotonicity of Addition *}
paulson@14341
   738
paulson@14341
   739
text {* strict, in 1st argument *}
paulson@14341
   740
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
paulson@14341
   741
  by (induct k) simp_all
paulson@14341
   742
paulson@14341
   743
text {* strict, in both arguments *}
paulson@14341
   744
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
paulson@14341
   745
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
paulson@15251
   746
  apply (induct j, simp_all)
paulson@14341
   747
  done
paulson@14341
   748
paulson@14341
   749
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
paulson@14341
   750
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
paulson@14341
   751
  apply (induct n)
paulson@14341
   752
  apply (simp_all add: order_le_less)
paulson@14341
   753
  apply (blast elim!: less_SucE 
paulson@14341
   754
               intro!: add_0_right [symmetric] add_Suc_right [symmetric])
paulson@14341
   755
  done
paulson@14341
   756
paulson@14341
   757
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
paulson@14341
   758
lemma mult_less_mono2: "(i::nat) < j ==> 0 < k ==> k * i < k * j"
paulson@14341
   759
  apply (erule_tac m1 = 0 in less_imp_Suc_add [THEN exE], simp)
paulson@14341
   760
  apply (induct_tac x) 
paulson@14341
   761
  apply (simp_all add: add_less_mono)
paulson@14341
   762
  done
paulson@14341
   763
paulson@14341
   764
nipkow@14740
   765
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
obua@14738
   766
instance nat :: ordered_semidom
paulson@14341
   767
proof
paulson@14341
   768
  fix i j k :: nat
paulson@14348
   769
  show "0 < (1::nat)" by simp
paulson@14267
   770
  show "i \<le> j ==> k + i \<le> k + j" by simp
paulson@14267
   771
  show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2)
paulson@14267
   772
qed
paulson@14267
   773
paulson@14267
   774
lemma nat_mult_1: "(1::nat) * n = n"
paulson@14267
   775
  by simp
paulson@14267
   776
paulson@14267
   777
lemma nat_mult_1_right: "n * (1::nat) = n"
paulson@14267
   778
  by simp
paulson@14267
   779
paulson@14267
   780
paulson@14267
   781
subsection {* Additional theorems about "less than" *}
paulson@14267
   782
paulson@14267
   783
text {* A [clumsy] way of lifting @{text "<"}
paulson@14267
   784
  monotonicity to @{text "\<le>"} monotonicity *}
paulson@14267
   785
lemma less_mono_imp_le_mono:
paulson@14267
   786
  assumes lt_mono: "!!i j::nat. i < j ==> f i < f j"
paulson@14267
   787
  and le: "i \<le> j" shows "f i \<le> ((f j)::nat)" using le
paulson@14267
   788
  apply (simp add: order_le_less)
paulson@14267
   789
  apply (blast intro!: lt_mono)
paulson@14267
   790
  done
paulson@14267
   791
paulson@14267
   792
text {* non-strict, in 1st argument *}
paulson@14267
   793
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
paulson@14267
   794
  by (rule add_right_mono)
paulson@14267
   795
paulson@14267
   796
text {* non-strict, in both arguments *}
paulson@14267
   797
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
paulson@14267
   798
  by (rule add_mono)
paulson@14267
   799
paulson@14267
   800
lemma le_add2: "n \<le> ((m + n)::nat)"
paulson@14341
   801
  by (insert add_right_mono [of 0 m n], simp) 
berghofe@13449
   802
paulson@14267
   803
lemma le_add1: "n \<le> ((n + m)::nat)"
paulson@14341
   804
  by (simp add: add_commute, rule le_add2)
berghofe@13449
   805
berghofe@13449
   806
lemma less_add_Suc1: "i < Suc (i + m)"
berghofe@13449
   807
  by (rule le_less_trans, rule le_add1, rule lessI)
berghofe@13449
   808
berghofe@13449
   809
lemma less_add_Suc2: "i < Suc (m + i)"
berghofe@13449
   810
  by (rule le_less_trans, rule le_add2, rule lessI)
berghofe@13449
   811
paulson@14267
   812
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
berghofe@13449
   813
  by (rules intro!: less_add_Suc1 less_imp_Suc_add)
berghofe@13449
   814
paulson@14267
   815
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
berghofe@13449
   816
  by (rule le_trans, assumption, rule le_add1)
berghofe@13449
   817
paulson@14267
   818
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
berghofe@13449
   819
  by (rule le_trans, assumption, rule le_add2)
berghofe@13449
   820
berghofe@13449
   821
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
berghofe@13449
   822
  by (rule less_le_trans, assumption, rule le_add1)
berghofe@13449
   823
berghofe@13449
   824
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
berghofe@13449
   825
  by (rule less_le_trans, assumption, rule le_add2)
berghofe@13449
   826
berghofe@13449
   827
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
paulson@14341
   828
  apply (rule le_less_trans [of _ "i+j"]) 
paulson@14341
   829
  apply (simp_all add: le_add1)
berghofe@13449
   830
  done
berghofe@13449
   831
berghofe@13449
   832
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
berghofe@13449
   833
  apply (rule notI)
berghofe@13449
   834
  apply (erule add_lessD1 [THEN less_irrefl])
berghofe@13449
   835
  done
berghofe@13449
   836
berghofe@13449
   837
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
berghofe@13449
   838
  by (simp add: add_commute not_add_less1)
berghofe@13449
   839
paulson@14267
   840
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
paulson@14341
   841
  apply (rule order_trans [of _ "m+k"]) 
paulson@14341
   842
  apply (simp_all add: le_add1)
paulson@14341
   843
  done
berghofe@13449
   844
paulson@14267
   845
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
berghofe@13449
   846
  apply (simp add: add_commute)
berghofe@13449
   847
  apply (erule add_leD1)
berghofe@13449
   848
  done
berghofe@13449
   849
paulson@14267
   850
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
berghofe@13449
   851
  by (blast dest: add_leD1 add_leD2)
berghofe@13449
   852
berghofe@13449
   853
text {* needs @{text "!!k"} for @{text add_ac} to work *}
berghofe@13449
   854
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
berghofe@13449
   855
  by (force simp del: add_Suc_right
berghofe@13449
   856
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
berghofe@13449
   857
berghofe@13449
   858
berghofe@13449
   859
berghofe@13449
   860
subsection {* Difference *}
berghofe@13449
   861
berghofe@13449
   862
lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0"
berghofe@13449
   863
  by (induct m) simp_all
berghofe@13449
   864
berghofe@13449
   865
text {* Addition is the inverse of subtraction:
paulson@14267
   866
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
berghofe@13449
   867
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
berghofe@13449
   868
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   869
paulson@14267
   870
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
berghofe@13449
   871
  by (simp add: add_diff_inverse not_less_iff_le)
berghofe@13449
   872
paulson@14267
   873
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
berghofe@13449
   874
  by (simp add: le_add_diff_inverse add_commute)
berghofe@13449
   875
berghofe@13449
   876
berghofe@13449
   877
subsection {* More results about difference *}
berghofe@13449
   878
paulson@14267
   879
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
berghofe@13449
   880
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   881
berghofe@13449
   882
lemma diff_less_Suc: "m - n < Suc m"
berghofe@13449
   883
  apply (induct m n rule: diff_induct)
berghofe@13449
   884
  apply (erule_tac [3] less_SucE)
berghofe@13449
   885
  apply (simp_all add: less_Suc_eq)
berghofe@13449
   886
  done
berghofe@13449
   887
paulson@14267
   888
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
berghofe@13449
   889
  by (induct m n rule: diff_induct) (simp_all add: le_SucI)
berghofe@13449
   890
berghofe@13449
   891
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
berghofe@13449
   892
  by (rule le_less_trans, rule diff_le_self)
berghofe@13449
   893
berghofe@13449
   894
lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
berghofe@13449
   895
  by (induct i j rule: diff_induct) simp_all
berghofe@13449
   896
berghofe@13449
   897
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
berghofe@13449
   898
  by (simp add: diff_diff_left)
berghofe@13449
   899
berghofe@13449
   900
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
paulson@14208
   901
  apply (case_tac "n", safe)
berghofe@13449
   902
  apply (simp add: le_simps)
berghofe@13449
   903
  done
berghofe@13449
   904
berghofe@13449
   905
text {* This and the next few suggested by Florian Kammueller *}
berghofe@13449
   906
lemma diff_commute: "(i::nat) - j - k = i - k - j"
berghofe@13449
   907
  by (simp add: diff_diff_left add_commute)
berghofe@13449
   908
paulson@14267
   909
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
berghofe@13449
   910
  by (induct j k rule: diff_induct) simp_all
berghofe@13449
   911
paulson@14267
   912
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
berghofe@13449
   913
  by (simp add: add_commute diff_add_assoc)
berghofe@13449
   914
berghofe@13449
   915
lemma diff_add_inverse: "(n + m) - n = (m::nat)"
berghofe@13449
   916
  by (induct n) simp_all
berghofe@13449
   917
berghofe@13449
   918
lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
berghofe@13449
   919
  by (simp add: diff_add_assoc)
berghofe@13449
   920
paulson@14267
   921
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
berghofe@13449
   922
  apply safe
berghofe@13449
   923
  apply (simp_all add: diff_add_inverse2)
berghofe@13449
   924
  done
berghofe@13449
   925
paulson@14267
   926
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
berghofe@13449
   927
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   928
paulson@14267
   929
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
berghofe@13449
   930
  by (rule iffD2, rule diff_is_0_eq)
berghofe@13449
   931
berghofe@13449
   932
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
berghofe@13449
   933
  by (induct m n rule: diff_induct) simp_all
berghofe@13449
   934
paulson@14267
   935
lemma less_imp_add_positive: "i < j  ==> \<exists>k::nat. 0 < k & i + k = j"
berghofe@13449
   936
  apply (rule_tac x = "j - i" in exI)
berghofe@13449
   937
  apply (simp (no_asm_simp) add: add_diff_inverse less_not_sym)
berghofe@13449
   938
  done
wenzelm@9436
   939
berghofe@13449
   940
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
berghofe@13449
   941
  apply (induct k i rule: diff_induct)
berghofe@13449
   942
  apply (simp_all (no_asm))
berghofe@13449
   943
  apply rules
berghofe@13449
   944
  done
berghofe@13449
   945
berghofe@13449
   946
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
berghofe@13449
   947
  apply (rule diff_self_eq_0 [THEN subst])
paulson@14208
   948
  apply (rule zero_induct_lemma, rules+)
berghofe@13449
   949
  done
berghofe@13449
   950
berghofe@13449
   951
lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
berghofe@13449
   952
  by (induct k) simp_all
berghofe@13449
   953
berghofe@13449
   954
lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
berghofe@13449
   955
  by (simp add: diff_cancel add_commute)
berghofe@13449
   956
berghofe@13449
   957
lemma diff_add_0: "n - (n + m) = (0::nat)"
berghofe@13449
   958
  by (induct n) simp_all
berghofe@13449
   959
berghofe@13449
   960
berghofe@13449
   961
text {* Difference distributes over multiplication *}
berghofe@13449
   962
berghofe@13449
   963
lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"
berghofe@13449
   964
  by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
berghofe@13449
   965
berghofe@13449
   966
lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
berghofe@13449
   967
  by (simp add: diff_mult_distrib mult_commute [of k])
berghofe@13449
   968
  -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
berghofe@13449
   969
berghofe@13449
   970
lemmas nat_distrib =
berghofe@13449
   971
  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
berghofe@13449
   972
berghofe@13449
   973
berghofe@13449
   974
subsection {* Monotonicity of Multiplication *}
berghofe@13449
   975
paulson@14267
   976
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
paulson@14341
   977
  by (simp add: mult_right_mono) 
berghofe@13449
   978
paulson@14267
   979
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
paulson@14341
   980
  by (simp add: mult_left_mono) 
berghofe@13449
   981
paulson@14267
   982
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
paulson@14267
   983
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
paulson@14341
   984
  by (simp add: mult_mono) 
berghofe@13449
   985
berghofe@13449
   986
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
paulson@14341
   987
  by (simp add: mult_strict_right_mono) 
berghofe@13449
   988
paulson@14266
   989
text{*Differs from the standard @{text zero_less_mult_iff} in that
paulson@14266
   990
      there are no negative numbers.*}
paulson@14266
   991
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
berghofe@13449
   992
  apply (induct m)
paulson@14208
   993
  apply (case_tac [2] n, simp_all)
berghofe@13449
   994
  done
berghofe@13449
   995
paulson@14267
   996
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)"
berghofe@13449
   997
  apply (induct m)
paulson@14208
   998
  apply (case_tac [2] n, simp_all)
berghofe@13449
   999
  done
berghofe@13449
  1000
berghofe@13449
  1001
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)"
paulson@15251
  1002
  apply (induct m, simp)
paulson@15251
  1003
  apply (induct n, simp, fastsimp)
berghofe@13449
  1004
  done
berghofe@13449
  1005
berghofe@13449
  1006
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = 1 & n = 1)"
berghofe@13449
  1007
  apply (rule trans)
paulson@14208
  1008
  apply (rule_tac [2] mult_eq_1_iff, fastsimp)
berghofe@13449
  1009
  done
berghofe@13449
  1010
paulson@14341
  1011
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
berghofe@13449
  1012
  apply (safe intro!: mult_less_mono1)
paulson@14208
  1013
  apply (case_tac k, auto)
berghofe@13449
  1014
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
berghofe@13449
  1015
  apply (blast intro: mult_le_mono1)
berghofe@13449
  1016
  done
berghofe@13449
  1017
berghofe@13449
  1018
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
paulson@14341
  1019
  by (simp add: mult_commute [of k])
berghofe@13449
  1020
paulson@14267
  1021
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
paulson@14208
  1022
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1023
paulson@14267
  1024
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
paulson@14208
  1025
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1026
paulson@14341
  1027
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
paulson@14208
  1028
  apply (cut_tac less_linear, safe, auto)
berghofe@13449
  1029
  apply (drule mult_less_mono1, assumption, simp)+
berghofe@13449
  1030
  done
berghofe@13449
  1031
berghofe@13449
  1032
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
paulson@14341
  1033
  by (simp add: mult_commute [of k])
berghofe@13449
  1034
berghofe@13449
  1035
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
berghofe@13449
  1036
  by (subst mult_less_cancel1) simp
berghofe@13449
  1037
paulson@14267
  1038
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
berghofe@13449
  1039
  by (subst mult_le_cancel1) simp
berghofe@13449
  1040
berghofe@13449
  1041
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
berghofe@13449
  1042
  by (subst mult_cancel1) simp
berghofe@13449
  1043
berghofe@13449
  1044
text {* Lemma for @{text gcd} *}
berghofe@13449
  1045
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
berghofe@13449
  1046
  apply (drule sym)
berghofe@13449
  1047
  apply (rule disjCI)
berghofe@13449
  1048
  apply (rule nat_less_cases, erule_tac [2] _)
berghofe@13449
  1049
  apply (fastsimp elim!: less_SucE)
berghofe@13449
  1050
  apply (fastsimp dest: mult_less_mono2)
berghofe@13449
  1051
  done
wenzelm@9436
  1052
clasohm@923
  1053
end