src/HOL/Nat.thy
author wenzelm
Wed Mar 04 19:53:18 2015 +0100 (2015-03-04)
changeset 59582 0fbed69ff081
parent 59000 6eb0725503fc
child 59815 cce82e360c2f
permissions -rw-r--r--
tuned signature -- prefer qualified names;
clasohm@923
     1
(*  Title:      HOL/Nat.thy
wenzelm@21243
     2
    Author:     Tobias Nipkow and Lawrence C Paulson and Markus Wenzel
clasohm@923
     3
wenzelm@9436
     4
Type "nat" is a linear order, and a datatype; arithmetic operators + -
haftmann@30496
     5
and * (for div and mod, see theory Divides).
clasohm@923
     6
*)
clasohm@923
     7
wenzelm@58889
     8
section {* Natural numbers *}
berghofe@13449
     9
nipkow@15131
    10
theory Nat
haftmann@35121
    11
imports Inductive Typedef Fun Fields
nipkow@15131
    12
begin
berghofe@13449
    13
wenzelm@48891
    14
ML_file "~~/src/Tools/rat.ML"
wenzelm@57952
    15
wenzelm@57952
    16
named_theorems arith "arith facts -- only ground formulas"
wenzelm@48891
    17
ML_file "Tools/arith_data.ML"
wenzelm@48891
    18
ML_file "~~/src/Provers/Arith/fast_lin_arith.ML"
wenzelm@48891
    19
wenzelm@48891
    20
berghofe@13449
    21
subsection {* Type @{text ind} *}
berghofe@13449
    22
berghofe@13449
    23
typedecl ind
berghofe@13449
    24
haftmann@44325
    25
axiomatization Zero_Rep :: ind and Suc_Rep :: "ind => ind" where
berghofe@13449
    26
  -- {* the axiom of infinity in 2 parts *}
krauss@34208
    27
  Suc_Rep_inject:       "Suc_Rep x = Suc_Rep y ==> x = y" and
paulson@14267
    28
  Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"
wenzelm@19573
    29
berghofe@13449
    30
subsection {* Type nat *}
berghofe@13449
    31
berghofe@13449
    32
text {* Type definition *}
berghofe@13449
    33
haftmann@44325
    34
inductive Nat :: "ind \<Rightarrow> bool" where
haftmann@44325
    35
  Zero_RepI: "Nat Zero_Rep"
haftmann@44325
    36
| Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)"
berghofe@13449
    37
wenzelm@49834
    38
typedef nat = "{n. Nat n}"
wenzelm@45696
    39
  morphisms Rep_Nat Abs_Nat
haftmann@44278
    40
  using Nat.Zero_RepI by auto
haftmann@44278
    41
haftmann@44278
    42
lemma Nat_Rep_Nat:
haftmann@44278
    43
  "Nat (Rep_Nat n)"
haftmann@44278
    44
  using Rep_Nat by simp
berghofe@13449
    45
haftmann@44278
    46
lemma Nat_Abs_Nat_inverse:
haftmann@44278
    47
  "Nat n \<Longrightarrow> Rep_Nat (Abs_Nat n) = n"
haftmann@44278
    48
  using Abs_Nat_inverse by simp
haftmann@44278
    49
haftmann@44278
    50
lemma Nat_Abs_Nat_inject:
haftmann@44278
    51
  "Nat n \<Longrightarrow> Nat m \<Longrightarrow> Abs_Nat n = Abs_Nat m \<longleftrightarrow> n = m"
haftmann@44278
    52
  using Abs_Nat_inject by simp
berghofe@13449
    53
haftmann@25510
    54
instantiation nat :: zero
haftmann@25510
    55
begin
haftmann@25510
    56
haftmann@37767
    57
definition Zero_nat_def:
haftmann@25510
    58
  "0 = Abs_Nat Zero_Rep"
haftmann@25510
    59
haftmann@25510
    60
instance ..
haftmann@25510
    61
haftmann@25510
    62
end
haftmann@24995
    63
haftmann@44278
    64
definition Suc :: "nat \<Rightarrow> nat" where
haftmann@44278
    65
  "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"
haftmann@44278
    66
haftmann@27104
    67
lemma Suc_not_Zero: "Suc m \<noteq> 0"
haftmann@44278
    68
  by (simp add: Zero_nat_def Suc_def Suc_RepI Zero_RepI Nat_Abs_Nat_inject Suc_Rep_not_Zero_Rep Nat_Rep_Nat)
berghofe@13449
    69
haftmann@27104
    70
lemma Zero_not_Suc: "0 \<noteq> Suc m"
berghofe@13449
    71
  by (rule not_sym, rule Suc_not_Zero not_sym)
berghofe@13449
    72
krauss@34208
    73
lemma Suc_Rep_inject': "Suc_Rep x = Suc_Rep y \<longleftrightarrow> x = y"
krauss@34208
    74
  by (rule iffI, rule Suc_Rep_inject) simp_all
krauss@34208
    75
blanchet@55417
    76
lemma nat_induct0:
blanchet@55417
    77
  fixes n
blanchet@55417
    78
  assumes "P 0" and "\<And>n. P n \<Longrightarrow> P (Suc n)"
blanchet@55417
    79
  shows "P n"
blanchet@55417
    80
using assms
blanchet@55417
    81
apply (unfold Zero_nat_def Suc_def)
blanchet@55417
    82
apply (rule Rep_Nat_inverse [THEN subst]) -- {* types force good instantiation *}
blanchet@55417
    83
apply (erule Nat_Rep_Nat [THEN Nat.induct])
blanchet@55417
    84
apply (iprover elim: Nat_Abs_Nat_inverse [THEN subst])
blanchet@55417
    85
done
blanchet@55417
    86
blanchet@55469
    87
free_constructors case_nat for
blanchet@57200
    88
    "0 \<Colon> nat"
blanchet@55469
    89
  | Suc pred
blanchet@57200
    90
where
blanchet@57200
    91
  "pred (0 \<Colon> nat) = (0 \<Colon> nat)"
blanchet@58189
    92
    apply atomize_elim
blanchet@58189
    93
    apply (rename_tac n, induct_tac n rule: nat_induct0, auto)
blanchet@58189
    94
   apply (simp add: Suc_def Nat_Abs_Nat_inject Nat_Rep_Nat Suc_RepI Suc_Rep_inject'
blanchet@58189
    95
     Rep_Nat_inject)
blanchet@58189
    96
  apply (simp only: Suc_not_Zero)
blanchet@58189
    97
  done
blanchet@55417
    98
blanchet@58306
    99
-- {* Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}. *}
blanchet@55417
   100
setup {* Sign.mandatory_path "old" *}
blanchet@55417
   101
blanchet@58306
   102
old_rep_datatype "0 \<Colon> nat" Suc
blanchet@55417
   103
  apply (erule nat_induct0, assumption)
blanchet@55417
   104
 apply (rule nat.inject)
blanchet@55417
   105
apply (rule nat.distinct(1))
blanchet@55417
   106
done
blanchet@55417
   107
blanchet@55417
   108
setup {* Sign.parent_path *}
blanchet@55417
   109
blanchet@55468
   110
-- {* But erase the prefix for properties that are not generated by @{text free_constructors}. *}
blanchet@55417
   111
setup {* Sign.mandatory_path "nat" *}
blanchet@55417
   112
blanchet@55417
   113
declare
blanchet@55417
   114
  old.nat.inject[iff del]
blanchet@55417
   115
  old.nat.distinct(1)[simp del, induct_simp del]
blanchet@55417
   116
blanchet@55417
   117
lemmas induct = old.nat.induct
blanchet@55417
   118
lemmas inducts = old.nat.inducts
blanchet@55642
   119
lemmas rec = old.nat.rec
blanchet@55642
   120
lemmas simps = nat.inject nat.distinct nat.case nat.rec
blanchet@55417
   121
blanchet@55417
   122
setup {* Sign.parent_path *}
blanchet@55417
   123
blanchet@55417
   124
abbreviation rec_nat :: "'a \<Rightarrow> (nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a" where
blanchet@55417
   125
  "rec_nat \<equiv> old.rec_nat"
blanchet@55417
   126
blanchet@55424
   127
declare nat.sel[code del]
blanchet@55424
   128
blanchet@55443
   129
hide_const (open) Nat.pred -- {* hide everything related to the selector *}
blanchet@55417
   130
hide_fact
blanchet@55417
   131
  nat.case_eq_if
blanchet@55417
   132
  nat.collapse
blanchet@55417
   133
  nat.expand
blanchet@55417
   134
  nat.sel
blanchet@57983
   135
  nat.exhaust_sel
blanchet@57983
   136
  nat.split_sel
blanchet@57983
   137
  nat.split_sel_asm
blanchet@55417
   138
blanchet@55417
   139
lemma nat_exhaust [case_names 0 Suc, cases type: nat]:
blanchet@55417
   140
  -- {* for backward compatibility -- names of variables differ *}
blanchet@55417
   141
  "(y = 0 \<Longrightarrow> P) \<Longrightarrow> (\<And>nat. y = Suc nat \<Longrightarrow> P) \<Longrightarrow> P"
blanchet@55423
   142
by (rule old.nat.exhaust)
berghofe@13449
   143
haftmann@27104
   144
lemma nat_induct [case_names 0 Suc, induct type: nat]:
haftmann@30686
   145
  -- {* for backward compatibility -- names of variables differ *}
haftmann@27104
   146
  fixes n
blanchet@55417
   147
  assumes "P 0" and "\<And>n. P n \<Longrightarrow> P (Suc n)"
haftmann@27104
   148
  shows "P n"
blanchet@55417
   149
using assms by (rule nat.induct)
berghofe@13449
   150
blanchet@55417
   151
hide_fact
blanchet@55417
   152
  nat_exhaust
blanchet@55417
   153
  nat_induct0
haftmann@24995
   154
blanchet@58389
   155
ML {*
blanchet@58389
   156
val nat_basic_lfp_sugar =
blanchet@58389
   157
  let
blanchet@58389
   158
    val ctr_sugar = the (Ctr_Sugar.ctr_sugar_of_global @{theory} @{type_name nat});
blanchet@58389
   159
    val recx = Logic.varify_types_global @{term rec_nat};
blanchet@58389
   160
    val C = body_type (fastype_of recx);
blanchet@58389
   161
  in
blanchet@58389
   162
    {T = HOLogic.natT, fp_res_index = 0, C = C, fun_arg_Tsss = [[], [[HOLogic.natT, C]]],
blanchet@58389
   163
     ctr_sugar = ctr_sugar, recx = recx, rec_thms = @{thms nat.rec}}
blanchet@58389
   164
  end;
blanchet@58389
   165
*}
blanchet@58389
   166
blanchet@58389
   167
setup {*
blanchet@58389
   168
let
blanchet@58389
   169
  fun basic_lfp_sugars_of _ [@{typ nat}] _ _ ctxt =
blanchet@58389
   170
      ([], [0], [nat_basic_lfp_sugar], [], [], TrueI (*dummy*), [], false, ctxt)
blanchet@58389
   171
    | basic_lfp_sugars_of bs arg_Ts callers callssss ctxt =
blanchet@58389
   172
      BNF_LFP_Rec_Sugar.default_basic_lfp_sugars_of bs arg_Ts callers callssss ctxt;
blanchet@58389
   173
in
blanchet@58389
   174
  BNF_LFP_Rec_Sugar.register_lfp_rec_extension
blanchet@58389
   175
    {nested_simps = [], is_new_datatype = K (K true), basic_lfp_sugars_of = basic_lfp_sugars_of,
blanchet@58389
   176
     rewrite_nested_rec_call = NONE}
blanchet@58389
   177
end
blanchet@58389
   178
*}
blanchet@58389
   179
haftmann@24995
   180
text {* Injectiveness and distinctness lemmas *}
haftmann@24995
   181
haftmann@27104
   182
lemma inj_Suc[simp]: "inj_on Suc N"
haftmann@27104
   183
  by (simp add: inj_on_def)
haftmann@27104
   184
haftmann@26072
   185
lemma Suc_neq_Zero: "Suc m = 0 \<Longrightarrow> R"
nipkow@25162
   186
by (rule notE, rule Suc_not_Zero)
haftmann@24995
   187
haftmann@26072
   188
lemma Zero_neq_Suc: "0 = Suc m \<Longrightarrow> R"
nipkow@25162
   189
by (rule Suc_neq_Zero, erule sym)
haftmann@24995
   190
haftmann@26072
   191
lemma Suc_inject: "Suc x = Suc y \<Longrightarrow> x = y"
nipkow@25162
   192
by (rule inj_Suc [THEN injD])
haftmann@24995
   193
paulson@14267
   194
lemma n_not_Suc_n: "n \<noteq> Suc n"
nipkow@25162
   195
by (induct n) simp_all
berghofe@13449
   196
haftmann@26072
   197
lemma Suc_n_not_n: "Suc n \<noteq> n"
nipkow@25162
   198
by (rule not_sym, rule n_not_Suc_n)
berghofe@13449
   199
berghofe@13449
   200
text {* A special form of induction for reasoning
berghofe@13449
   201
  about @{term "m < n"} and @{term "m - n"} *}
berghofe@13449
   202
haftmann@26072
   203
lemma diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==>
berghofe@13449
   204
    (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n"
paulson@14208
   205
  apply (rule_tac x = m in spec)
paulson@15251
   206
  apply (induct n)
berghofe@13449
   207
  prefer 2
berghofe@13449
   208
  apply (rule allI)
nipkow@17589
   209
  apply (induct_tac x, iprover+)
berghofe@13449
   210
  done
berghofe@13449
   211
haftmann@24995
   212
haftmann@24995
   213
subsection {* Arithmetic operators *}
haftmann@24995
   214
haftmann@49388
   215
instantiation nat :: comm_monoid_diff
haftmann@25571
   216
begin
haftmann@24995
   217
blanchet@55575
   218
primrec plus_nat where
haftmann@25571
   219
  add_0:      "0 + n = (n\<Colon>nat)"
haftmann@44325
   220
| add_Suc:  "Suc m + n = Suc (m + n)"
haftmann@24995
   221
haftmann@26072
   222
lemma add_0_right [simp]: "m + 0 = (m::nat)"
haftmann@26072
   223
  by (induct m) simp_all
haftmann@26072
   224
haftmann@26072
   225
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
haftmann@26072
   226
  by (induct m) simp_all
haftmann@26072
   227
haftmann@28514
   228
declare add_0 [code]
haftmann@28514
   229
haftmann@26072
   230
lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
haftmann@26072
   231
  by simp
haftmann@26072
   232
blanchet@55575
   233
primrec minus_nat where
haftmann@39793
   234
  diff_0 [code]: "m - 0 = (m\<Colon>nat)"
haftmann@39793
   235
| diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
haftmann@24995
   236
haftmann@28514
   237
declare diff_Suc [simp del]
haftmann@26072
   238
haftmann@26072
   239
lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"
haftmann@26072
   240
  by (induct n) (simp_all add: diff_Suc)
haftmann@26072
   241
haftmann@26072
   242
lemma diff_Suc_Suc [simp, code]: "Suc m - Suc n = m - n"
haftmann@26072
   243
  by (induct n) (simp_all add: diff_Suc)
haftmann@26072
   244
haftmann@26072
   245
instance proof
haftmann@26072
   246
  fix n m q :: nat
haftmann@26072
   247
  show "(n + m) + q = n + (m + q)" by (induct n) simp_all
haftmann@26072
   248
  show "n + m = m + n" by (induct n) simp_all
haftmann@26072
   249
  show "0 + n = n" by simp
haftmann@49388
   250
  show "n - 0 = n" by simp
haftmann@49388
   251
  show "0 - n = 0" by simp
haftmann@49388
   252
  show "(q + n) - (q + m) = n - m" by (induct q) simp_all
haftmann@49388
   253
  show "n - m - q = n - (m + q)" by (induct q) (simp_all add: diff_Suc)
haftmann@26072
   254
qed
haftmann@26072
   255
haftmann@26072
   256
end
haftmann@26072
   257
wenzelm@36176
   258
hide_fact (open) add_0 add_0_right diff_0
haftmann@35047
   259
haftmann@26072
   260
instantiation nat :: comm_semiring_1_cancel
haftmann@26072
   261
begin
haftmann@26072
   262
haftmann@26072
   263
definition
huffman@47108
   264
  One_nat_def [simp]: "1 = Suc 0"
haftmann@26072
   265
blanchet@55575
   266
primrec times_nat where
haftmann@25571
   267
  mult_0:     "0 * n = (0\<Colon>nat)"
haftmann@44325
   268
| mult_Suc: "Suc m * n = n + (m * n)"
haftmann@25571
   269
haftmann@26072
   270
lemma mult_0_right [simp]: "(m::nat) * 0 = 0"
haftmann@26072
   271
  by (induct m) simp_all
haftmann@26072
   272
haftmann@26072
   273
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
haftmann@57512
   274
  by (induct m) (simp_all add: add.left_commute)
haftmann@26072
   275
haftmann@26072
   276
lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"
haftmann@57512
   277
  by (induct m) (simp_all add: add.assoc)
haftmann@26072
   278
haftmann@26072
   279
instance proof
haftmann@26072
   280
  fix n m q :: nat
huffman@30079
   281
  show "0 \<noteq> (1::nat)" unfolding One_nat_def by simp
huffman@30079
   282
  show "1 * n = n" unfolding One_nat_def by simp
haftmann@26072
   283
  show "n * m = m * n" by (induct n) simp_all
haftmann@26072
   284
  show "(n * m) * q = n * (m * q)" by (induct n) (simp_all add: add_mult_distrib)
haftmann@26072
   285
  show "(n + m) * q = n * q + m * q" by (rule add_mult_distrib)
haftmann@26072
   286
  assume "n + m = n + q" thus "m = q" by (induct n) simp_all
haftmann@26072
   287
qed
haftmann@25571
   288
haftmann@25571
   289
end
haftmann@24995
   290
haftmann@26072
   291
subsubsection {* Addition *}
haftmann@26072
   292
haftmann@57512
   293
lemma nat_add_left_cancel:
haftmann@57512
   294
  fixes k m n :: nat
haftmann@57512
   295
  shows "k + m = k + n \<longleftrightarrow> m = n"
haftmann@57512
   296
  by (fact add_left_cancel)
haftmann@26072
   297
haftmann@57512
   298
lemma nat_add_right_cancel:
haftmann@57512
   299
  fixes k m n :: nat
haftmann@57512
   300
  shows "m + k = n + k \<longleftrightarrow> m = n"
haftmann@57512
   301
  by (fact add_right_cancel)
haftmann@26072
   302
haftmann@26072
   303
text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
haftmann@26072
   304
haftmann@26072
   305
lemma add_is_0 [iff]:
haftmann@26072
   306
  fixes m n :: nat
haftmann@26072
   307
  shows "(m + n = 0) = (m = 0 & n = 0)"
haftmann@26072
   308
  by (cases m) simp_all
haftmann@26072
   309
haftmann@26072
   310
lemma add_is_1:
haftmann@26072
   311
  "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
haftmann@26072
   312
  by (cases m) simp_all
haftmann@26072
   313
haftmann@26072
   314
lemma one_is_add:
haftmann@26072
   315
  "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
haftmann@26072
   316
  by (rule trans, rule eq_commute, rule add_is_1)
haftmann@26072
   317
haftmann@26072
   318
lemma add_eq_self_zero:
haftmann@26072
   319
  fixes m n :: nat
haftmann@26072
   320
  shows "m + n = m \<Longrightarrow> n = 0"
haftmann@26072
   321
  by (induct m) simp_all
haftmann@26072
   322
haftmann@26072
   323
lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N"
haftmann@26072
   324
  apply (induct k)
haftmann@26072
   325
   apply simp
haftmann@26072
   326
  apply(drule comp_inj_on[OF _ inj_Suc])
haftmann@26072
   327
  apply (simp add:o_def)
haftmann@26072
   328
  done
haftmann@26072
   329
huffman@47208
   330
lemma Suc_eq_plus1: "Suc n = n + 1"
huffman@47208
   331
  unfolding One_nat_def by simp
huffman@47208
   332
huffman@47208
   333
lemma Suc_eq_plus1_left: "Suc n = 1 + n"
huffman@47208
   334
  unfolding One_nat_def by simp
huffman@47208
   335
haftmann@26072
   336
haftmann@26072
   337
subsubsection {* Difference *}
haftmann@26072
   338
haftmann@26072
   339
lemma diff_self_eq_0 [simp]: "(m\<Colon>nat) - m = 0"
haftmann@57512
   340
  by (fact diff_cancel)
haftmann@26072
   341
haftmann@26072
   342
lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
haftmann@57512
   343
  by (fact diff_diff_add)
haftmann@26072
   344
haftmann@26072
   345
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
haftmann@26072
   346
  by (simp add: diff_diff_left)
haftmann@26072
   347
haftmann@26072
   348
lemma diff_commute: "(i::nat) - j - k = i - k - j"
haftmann@57512
   349
  by (fact diff_right_commute)
haftmann@26072
   350
haftmann@26072
   351
lemma diff_add_inverse: "(n + m) - n = (m::nat)"
haftmann@57512
   352
  by (fact add_diff_cancel_left')
haftmann@26072
   353
haftmann@26072
   354
lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
haftmann@57512
   355
  by (fact add_diff_cancel_right')
haftmann@26072
   356
haftmann@26072
   357
lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
haftmann@57512
   358
  by (fact comm_monoid_diff_class.add_diff_cancel_left)
haftmann@26072
   359
haftmann@26072
   360
lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
haftmann@57512
   361
  by (fact add_diff_cancel_right)
haftmann@26072
   362
haftmann@26072
   363
lemma diff_add_0: "n - (n + m) = (0::nat)"
haftmann@57512
   364
  by (fact diff_add_zero)
haftmann@26072
   365
huffman@30093
   366
lemma diff_Suc_1 [simp]: "Suc n - 1 = n"
huffman@30093
   367
  unfolding One_nat_def by simp
huffman@30093
   368
haftmann@26072
   369
text {* Difference distributes over multiplication *}
haftmann@26072
   370
haftmann@26072
   371
lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"
haftmann@26072
   372
by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
haftmann@26072
   373
haftmann@26072
   374
lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
haftmann@57512
   375
by (simp add: diff_mult_distrib mult.commute [of k])
haftmann@26072
   376
  -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
haftmann@26072
   377
haftmann@26072
   378
haftmann@26072
   379
subsubsection {* Multiplication *}
haftmann@26072
   380
haftmann@26072
   381
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
haftmann@57512
   382
  by (fact distrib_left)
haftmann@26072
   383
haftmann@26072
   384
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
haftmann@26072
   385
  by (induct m) auto
haftmann@26072
   386
haftmann@26072
   387
lemmas nat_distrib =
haftmann@26072
   388
  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
haftmann@26072
   389
huffman@30079
   390
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = Suc 0 & n = Suc 0)"
haftmann@26072
   391
  apply (induct m)
haftmann@26072
   392
   apply simp
haftmann@26072
   393
  apply (induct n)
haftmann@26072
   394
   apply auto
haftmann@26072
   395
  done
haftmann@26072
   396
blanchet@54147
   397
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = Suc 0 & n = Suc 0)"
haftmann@26072
   398
  apply (rule trans)
nipkow@44890
   399
  apply (rule_tac [2] mult_eq_1_iff, fastforce)
haftmann@26072
   400
  done
haftmann@26072
   401
huffman@30079
   402
lemma nat_mult_eq_1_iff [simp]: "m * n = (1::nat) \<longleftrightarrow> m = 1 \<and> n = 1"
huffman@30079
   403
  unfolding One_nat_def by (rule mult_eq_1_iff)
huffman@30079
   404
huffman@30079
   405
lemma nat_1_eq_mult_iff [simp]: "(1::nat) = m * n \<longleftrightarrow> m = 1 \<and> n = 1"
huffman@30079
   406
  unfolding One_nat_def by (rule one_eq_mult_iff)
huffman@30079
   407
haftmann@26072
   408
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
haftmann@26072
   409
proof -
haftmann@26072
   410
  have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
haftmann@26072
   411
  proof (induct n arbitrary: m)
haftmann@26072
   412
    case 0 then show "m = 0" by simp
haftmann@26072
   413
  next
haftmann@26072
   414
    case (Suc n) then show "m = Suc n"
haftmann@26072
   415
      by (cases m) (simp_all add: eq_commute [of "0"])
haftmann@26072
   416
  qed
haftmann@26072
   417
  then show ?thesis by auto
haftmann@26072
   418
qed
haftmann@26072
   419
haftmann@26072
   420
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
haftmann@57512
   421
  by (simp add: mult.commute)
haftmann@26072
   422
haftmann@26072
   423
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
haftmann@26072
   424
  by (subst mult_cancel1) simp
haftmann@26072
   425
haftmann@24995
   426
haftmann@24995
   427
subsection {* Orders on @{typ nat} *}
haftmann@24995
   428
haftmann@26072
   429
subsubsection {* Operation definition *}
haftmann@24995
   430
haftmann@26072
   431
instantiation nat :: linorder
haftmann@25510
   432
begin
haftmann@25510
   433
blanchet@55575
   434
primrec less_eq_nat where
haftmann@26072
   435
  "(0\<Colon>nat) \<le> n \<longleftrightarrow> True"
haftmann@44325
   436
| "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
haftmann@26072
   437
haftmann@28514
   438
declare less_eq_nat.simps [simp del]
haftmann@26072
   439
lemma le0 [iff]: "0 \<le> (n\<Colon>nat)" by (simp add: less_eq_nat.simps)
haftmann@54223
   440
lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by simp
haftmann@26072
   441
haftmann@26072
   442
definition less_nat where
haftmann@28514
   443
  less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   444
haftmann@26072
   445
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
haftmann@26072
   446
  by (simp add: less_eq_nat.simps(2))
haftmann@26072
   447
haftmann@26072
   448
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
haftmann@26072
   449
  unfolding less_eq_Suc_le ..
haftmann@26072
   450
haftmann@26072
   451
lemma le_0_eq [iff]: "(n\<Colon>nat) \<le> 0 \<longleftrightarrow> n = 0"
haftmann@26072
   452
  by (induct n) (simp_all add: less_eq_nat.simps(2))
haftmann@26072
   453
haftmann@26072
   454
lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)"
haftmann@26072
   455
  by (simp add: less_eq_Suc_le)
haftmann@26072
   456
haftmann@26072
   457
lemma less_nat_zero_code [code]: "n < (0\<Colon>nat) \<longleftrightarrow> False"
haftmann@26072
   458
  by simp
haftmann@26072
   459
haftmann@26072
   460
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
haftmann@26072
   461
  by (simp add: less_eq_Suc_le)
haftmann@26072
   462
haftmann@26072
   463
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
haftmann@26072
   464
  by (simp add: less_eq_Suc_le)
haftmann@26072
   465
hoelzl@56194
   466
lemma Suc_less_eq2: "Suc n < m \<longleftrightarrow> (\<exists>m'. m = Suc m' \<and> n < m')"
hoelzl@56194
   467
  by (cases m) auto
hoelzl@56194
   468
haftmann@26072
   469
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
haftmann@26072
   470
  by (induct m arbitrary: n)
haftmann@26072
   471
    (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   472
haftmann@26072
   473
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
haftmann@26072
   474
  by (cases n) (auto intro: le_SucI)
haftmann@26072
   475
haftmann@26072
   476
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
haftmann@26072
   477
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@24995
   478
haftmann@26072
   479
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
haftmann@26072
   480
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@25510
   481
wenzelm@26315
   482
instance
wenzelm@26315
   483
proof
haftmann@26072
   484
  fix n m :: nat
haftmann@27679
   485
  show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n" 
haftmann@26072
   486
  proof (induct n arbitrary: m)
haftmann@27679
   487
    case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   488
  next
haftmann@27679
   489
    case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   490
  qed
haftmann@26072
   491
next
haftmann@26072
   492
  fix n :: nat show "n \<le> n" by (induct n) simp_all
haftmann@26072
   493
next
haftmann@26072
   494
  fix n m :: nat assume "n \<le> m" and "m \<le> n"
haftmann@26072
   495
  then show "n = m"
haftmann@26072
   496
    by (induct n arbitrary: m)
haftmann@26072
   497
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   498
next
haftmann@26072
   499
  fix n m q :: nat assume "n \<le> m" and "m \<le> q"
haftmann@26072
   500
  then show "n \<le> q"
haftmann@26072
   501
  proof (induct n arbitrary: m q)
haftmann@26072
   502
    case 0 show ?case by simp
haftmann@26072
   503
  next
haftmann@26072
   504
    case (Suc n) then show ?case
haftmann@26072
   505
      by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   506
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   507
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   508
  qed
haftmann@26072
   509
next
haftmann@26072
   510
  fix n m :: nat show "n \<le> m \<or> m \<le> n"
haftmann@26072
   511
    by (induct n arbitrary: m)
haftmann@26072
   512
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   513
qed
haftmann@25510
   514
haftmann@25510
   515
end
berghofe@13449
   516
haftmann@52729
   517
instantiation nat :: order_bot
haftmann@29652
   518
begin
haftmann@29652
   519
haftmann@29652
   520
definition bot_nat :: nat where
haftmann@29652
   521
  "bot_nat = 0"
haftmann@29652
   522
haftmann@29652
   523
instance proof
haftmann@29652
   524
qed (simp add: bot_nat_def)
haftmann@29652
   525
haftmann@29652
   526
end
haftmann@29652
   527
hoelzl@51329
   528
instance nat :: no_top
haftmann@52289
   529
  by default (auto intro: less_Suc_eq_le [THEN iffD2])
haftmann@52289
   530
hoelzl@51329
   531
haftmann@26072
   532
subsubsection {* Introduction properties *}
berghofe@13449
   533
haftmann@26072
   534
lemma lessI [iff]: "n < Suc n"
haftmann@26072
   535
  by (simp add: less_Suc_eq_le)
berghofe@13449
   536
haftmann@26072
   537
lemma zero_less_Suc [iff]: "0 < Suc n"
haftmann@26072
   538
  by (simp add: less_Suc_eq_le)
berghofe@13449
   539
berghofe@13449
   540
berghofe@13449
   541
subsubsection {* Elimination properties *}
berghofe@13449
   542
berghofe@13449
   543
lemma less_not_refl: "~ n < (n::nat)"
haftmann@26072
   544
  by (rule order_less_irrefl)
berghofe@13449
   545
wenzelm@26335
   546
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)"
wenzelm@26335
   547
  by (rule not_sym) (rule less_imp_neq) 
berghofe@13449
   548
paulson@14267
   549
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
haftmann@26072
   550
  by (rule less_imp_neq)
berghofe@13449
   551
wenzelm@26335
   552
lemma less_irrefl_nat: "(n::nat) < n ==> R"
wenzelm@26335
   553
  by (rule notE, rule less_not_refl)
berghofe@13449
   554
berghofe@13449
   555
lemma less_zeroE: "(n::nat) < 0 ==> R"
haftmann@26072
   556
  by (rule notE) (rule not_less0)
berghofe@13449
   557
berghofe@13449
   558
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
haftmann@26072
   559
  unfolding less_Suc_eq_le le_less ..
berghofe@13449
   560
huffman@30079
   561
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
haftmann@26072
   562
  by (simp add: less_Suc_eq)
berghofe@13449
   563
blanchet@54147
   564
lemma less_one [iff]: "(n < (1::nat)) = (n = 0)"
huffman@30079
   565
  unfolding One_nat_def by (rule less_Suc0)
berghofe@13449
   566
berghofe@13449
   567
lemma Suc_mono: "m < n ==> Suc m < Suc n"
haftmann@26072
   568
  by simp
berghofe@13449
   569
nipkow@14302
   570
text {* "Less than" is antisymmetric, sort of *}
nipkow@14302
   571
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
haftmann@26072
   572
  unfolding not_less less_Suc_eq_le by (rule antisym)
nipkow@14302
   573
paulson@14267
   574
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
haftmann@26072
   575
  by (rule linorder_neq_iff)
berghofe@13449
   576
berghofe@13449
   577
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
berghofe@13449
   578
  and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
berghofe@13449
   579
  shows "P n m"
berghofe@13449
   580
  apply (rule less_linear [THEN disjE])
berghofe@13449
   581
  apply (erule_tac [2] disjE)
berghofe@13449
   582
  apply (erule lessCase)
berghofe@13449
   583
  apply (erule sym [THEN eqCase])
berghofe@13449
   584
  apply (erule major)
berghofe@13449
   585
  done
berghofe@13449
   586
berghofe@13449
   587
berghofe@13449
   588
subsubsection {* Inductive (?) properties *}
berghofe@13449
   589
paulson@14267
   590
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
haftmann@26072
   591
  unfolding less_eq_Suc_le [of m] le_less by simp 
berghofe@13449
   592
haftmann@26072
   593
lemma lessE:
haftmann@26072
   594
  assumes major: "i < k"
haftmann@26072
   595
  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
haftmann@26072
   596
  shows P
haftmann@26072
   597
proof -
haftmann@26072
   598
  from major have "\<exists>j. i \<le> j \<and> k = Suc j"
haftmann@26072
   599
    unfolding less_eq_Suc_le by (induct k) simp_all
haftmann@26072
   600
  then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
haftmann@26072
   601
    by (clarsimp simp add: less_le)
haftmann@26072
   602
  with p1 p2 show P by auto
haftmann@26072
   603
qed
haftmann@26072
   604
haftmann@26072
   605
lemma less_SucE: assumes major: "m < Suc n"
haftmann@26072
   606
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
haftmann@26072
   607
  apply (rule major [THEN lessE])
haftmann@26072
   608
  apply (rule eq, blast)
haftmann@26072
   609
  apply (rule less, blast)
berghofe@13449
   610
  done
berghofe@13449
   611
berghofe@13449
   612
lemma Suc_lessE: assumes major: "Suc i < k"
berghofe@13449
   613
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
berghofe@13449
   614
  apply (rule major [THEN lessE])
berghofe@13449
   615
  apply (erule lessI [THEN minor])
paulson@14208
   616
  apply (erule Suc_lessD [THEN minor], assumption)
berghofe@13449
   617
  done
berghofe@13449
   618
berghofe@13449
   619
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
haftmann@26072
   620
  by simp
berghofe@13449
   621
berghofe@13449
   622
lemma less_trans_Suc:
berghofe@13449
   623
  assumes le: "i < j" shows "j < k ==> Suc i < k"
paulson@14208
   624
  apply (induct k, simp_all)
berghofe@13449
   625
  apply (insert le)
berghofe@13449
   626
  apply (simp add: less_Suc_eq)
berghofe@13449
   627
  apply (blast dest: Suc_lessD)
berghofe@13449
   628
  done
berghofe@13449
   629
berghofe@13449
   630
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
haftmann@26072
   631
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
haftmann@26072
   632
  unfolding not_less less_Suc_eq_le ..
berghofe@13449
   633
haftmann@26072
   634
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   635
  unfolding not_le Suc_le_eq ..
wenzelm@21243
   636
haftmann@24995
   637
text {* Properties of "less than or equal" *}
berghofe@13449
   638
paulson@14267
   639
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
haftmann@26072
   640
  unfolding less_Suc_eq_le .
berghofe@13449
   641
paulson@14267
   642
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
haftmann@26072
   643
  unfolding not_le less_Suc_eq_le ..
berghofe@13449
   644
paulson@14267
   645
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
haftmann@26072
   646
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
berghofe@13449
   647
paulson@14267
   648
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
haftmann@26072
   649
  by (drule le_Suc_eq [THEN iffD1], iprover+)
berghofe@13449
   650
paulson@14267
   651
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
haftmann@26072
   652
  unfolding Suc_le_eq .
berghofe@13449
   653
berghofe@13449
   654
text {* Stronger version of @{text Suc_leD} *}
paulson@14267
   655
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
haftmann@26072
   656
  unfolding Suc_le_eq .
berghofe@13449
   657
wenzelm@26315
   658
lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)"
haftmann@26072
   659
  unfolding less_eq_Suc_le by (rule Suc_leD)
berghofe@13449
   660
paulson@14267
   661
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
wenzelm@26315
   662
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
berghofe@13449
   663
berghofe@13449
   664
paulson@14267
   665
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
berghofe@13449
   666
paulson@14267
   667
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
haftmann@26072
   668
  unfolding le_less .
berghofe@13449
   669
paulson@14267
   670
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
haftmann@26072
   671
  by (rule le_less)
berghofe@13449
   672
wenzelm@22718
   673
text {* Useful with @{text blast}. *}
paulson@14267
   674
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
haftmann@26072
   675
  by auto
berghofe@13449
   676
paulson@14267
   677
lemma le_refl: "n \<le> (n::nat)"
haftmann@26072
   678
  by simp
berghofe@13449
   679
paulson@14267
   680
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
haftmann@26072
   681
  by (rule order_trans)
berghofe@13449
   682
nipkow@33657
   683
lemma le_antisym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
haftmann@26072
   684
  by (rule antisym)
berghofe@13449
   685
paulson@14267
   686
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
haftmann@26072
   687
  by (rule less_le)
berghofe@13449
   688
paulson@14267
   689
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
haftmann@26072
   690
  unfolding less_le ..
berghofe@13449
   691
haftmann@26072
   692
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
haftmann@26072
   693
  by (rule linear)
paulson@14341
   694
wenzelm@22718
   695
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
nipkow@15921
   696
haftmann@26072
   697
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
haftmann@26072
   698
  unfolding less_Suc_eq_le by auto
berghofe@13449
   699
haftmann@26072
   700
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
haftmann@26072
   701
  unfolding not_less by (rule le_less_Suc_eq)
berghofe@13449
   702
berghofe@13449
   703
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
berghofe@13449
   704
paulson@14267
   705
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   706
by (cases n) simp_all
nipkow@25162
   707
nipkow@25162
   708
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   709
by (cases n) simp_all
berghofe@13449
   710
wenzelm@22718
   711
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
nipkow@25162
   712
by (cases n) simp_all
berghofe@13449
   713
nipkow@25162
   714
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
nipkow@25162
   715
by (cases n) simp_all
nipkow@25140
   716
berghofe@13449
   717
text {* This theorem is useful with @{text blast} *}
berghofe@13449
   718
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
nipkow@25162
   719
by (rule neq0_conv[THEN iffD1], iprover)
berghofe@13449
   720
paulson@14267
   721
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
nipkow@25162
   722
by (fast intro: not0_implies_Suc)
berghofe@13449
   723
blanchet@54147
   724
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"
nipkow@25134
   725
using neq0_conv by blast
berghofe@13449
   726
paulson@14267
   727
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
nipkow@25162
   728
by (induct m') simp_all
berghofe@13449
   729
berghofe@13449
   730
text {* Useful in certain inductive arguments *}
paulson@14267
   731
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
nipkow@25162
   732
by (cases m) simp_all
berghofe@13449
   733
berghofe@13449
   734
haftmann@26072
   735
subsubsection {* Monotonicity of Addition *}
berghofe@13449
   736
haftmann@26072
   737
lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"
haftmann@26072
   738
by (simp add: diff_Suc split: nat.split)
berghofe@13449
   739
huffman@30128
   740
lemma Suc_diff_1 [simp]: "0 < n ==> Suc (n - 1) = n"
huffman@30128
   741
unfolding One_nat_def by (rule Suc_pred)
huffman@30128
   742
paulson@14331
   743
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
nipkow@25162
   744
by (induct k) simp_all
berghofe@13449
   745
paulson@14331
   746
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
nipkow@25162
   747
by (induct k) simp_all
berghofe@13449
   748
nipkow@25162
   749
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)"
nipkow@25162
   750
by(auto dest:gr0_implies_Suc)
berghofe@13449
   751
paulson@14341
   752
text {* strict, in 1st argument *}
paulson@14341
   753
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
nipkow@25162
   754
by (induct k) simp_all
paulson@14341
   755
paulson@14341
   756
text {* strict, in both arguments *}
paulson@14341
   757
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
paulson@14341
   758
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
paulson@15251
   759
  apply (induct j, simp_all)
paulson@14341
   760
  done
paulson@14341
   761
paulson@14341
   762
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
paulson@14341
   763
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
paulson@14341
   764
  apply (induct n)
paulson@14341
   765
  apply (simp_all add: order_le_less)
wenzelm@22718
   766
  apply (blast elim!: less_SucE
haftmann@35047
   767
               intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
paulson@14341
   768
  done
paulson@14341
   769
hoelzl@56194
   770
lemma le_Suc_ex: "(k::nat) \<le> l \<Longrightarrow> (\<exists>n. l = k + n)"
hoelzl@56194
   771
  by (auto simp: less_Suc_eq_le[symmetric] dest: less_imp_Suc_add)
hoelzl@56194
   772
paulson@14341
   773
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
nipkow@25134
   774
lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"
nipkow@25134
   775
apply(auto simp: gr0_conv_Suc)
nipkow@25134
   776
apply (induct_tac m)
nipkow@25134
   777
apply (simp_all add: add_less_mono)
nipkow@25134
   778
done
paulson@14341
   779
nipkow@14740
   780
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
haftmann@35028
   781
instance nat :: linordered_semidom
paulson@14341
   782
proof
paulson@14348
   783
  show "0 < (1::nat)" by simp
haftmann@52289
   784
  show "\<And>m n q :: nat. m \<le> n \<Longrightarrow> q + m \<le> q + n" by simp
haftmann@52289
   785
  show "\<And>m n q :: nat. m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n" by (simp add: mult_less_mono2)
paulson@14267
   786
qed
paulson@14267
   787
haftmann@58952
   788
instance nat :: semiring_no_zero_divisors
nipkow@30056
   789
proof
haftmann@58952
   790
  fix m n :: nat
haftmann@58952
   791
  show "m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> m * n \<noteq> 0" by simp
nipkow@30056
   792
qed
nipkow@30056
   793
haftmann@44817
   794
haftmann@44817
   795
subsubsection {* @{term min} and @{term max} *}
haftmann@44817
   796
haftmann@44817
   797
lemma mono_Suc: "mono Suc"
haftmann@44817
   798
by (rule monoI) simp
haftmann@44817
   799
haftmann@44817
   800
lemma min_0L [simp]: "min 0 n = (0::nat)"
noschinl@45931
   801
by (rule min_absorb1) simp
haftmann@44817
   802
haftmann@44817
   803
lemma min_0R [simp]: "min n 0 = (0::nat)"
noschinl@45931
   804
by (rule min_absorb2) simp
haftmann@44817
   805
haftmann@44817
   806
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
haftmann@44817
   807
by (simp add: mono_Suc min_of_mono)
haftmann@44817
   808
haftmann@44817
   809
lemma min_Suc1:
haftmann@44817
   810
   "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
haftmann@44817
   811
by (simp split: nat.split)
haftmann@44817
   812
haftmann@44817
   813
lemma min_Suc2:
haftmann@44817
   814
   "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
haftmann@44817
   815
by (simp split: nat.split)
haftmann@44817
   816
haftmann@44817
   817
lemma max_0L [simp]: "max 0 n = (n::nat)"
noschinl@45931
   818
by (rule max_absorb2) simp
haftmann@44817
   819
haftmann@44817
   820
lemma max_0R [simp]: "max n 0 = (n::nat)"
noschinl@45931
   821
by (rule max_absorb1) simp
haftmann@44817
   822
haftmann@44817
   823
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
haftmann@44817
   824
by (simp add: mono_Suc max_of_mono)
haftmann@44817
   825
haftmann@44817
   826
lemma max_Suc1:
haftmann@44817
   827
   "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
haftmann@44817
   828
by (simp split: nat.split)
haftmann@44817
   829
haftmann@44817
   830
lemma max_Suc2:
haftmann@44817
   831
   "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
haftmann@44817
   832
by (simp split: nat.split)
paulson@14267
   833
haftmann@44817
   834
lemma nat_mult_min_left:
haftmann@44817
   835
  fixes m n q :: nat
haftmann@44817
   836
  shows "min m n * q = min (m * q) (n * q)"
haftmann@44817
   837
  by (simp add: min_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
haftmann@44817
   838
haftmann@44817
   839
lemma nat_mult_min_right:
haftmann@44817
   840
  fixes m n q :: nat
haftmann@44817
   841
  shows "m * min n q = min (m * n) (m * q)"
haftmann@44817
   842
  by (simp add: min_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
haftmann@44817
   843
haftmann@44817
   844
lemma nat_add_max_left:
haftmann@44817
   845
  fixes m n q :: nat
haftmann@44817
   846
  shows "max m n + q = max (m + q) (n + q)"
haftmann@44817
   847
  by (simp add: max_def)
haftmann@44817
   848
haftmann@44817
   849
lemma nat_add_max_right:
haftmann@44817
   850
  fixes m n q :: nat
haftmann@44817
   851
  shows "m + max n q = max (m + n) (m + q)"
haftmann@44817
   852
  by (simp add: max_def)
haftmann@44817
   853
haftmann@44817
   854
lemma nat_mult_max_left:
haftmann@44817
   855
  fixes m n q :: nat
haftmann@44817
   856
  shows "max m n * q = max (m * q) (n * q)"
haftmann@44817
   857
  by (simp add: max_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
haftmann@44817
   858
haftmann@44817
   859
lemma nat_mult_max_right:
haftmann@44817
   860
  fixes m n q :: nat
haftmann@44817
   861
  shows "m * max n q = max (m * n) (m * q)"
haftmann@44817
   862
  by (simp add: max_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
paulson@14267
   863
paulson@14267
   864
krauss@26748
   865
subsubsection {* Additional theorems about @{term "op \<le>"} *}
krauss@26748
   866
krauss@26748
   867
text {* Complete induction, aka course-of-values induction *}
krauss@26748
   868
haftmann@27823
   869
instance nat :: wellorder proof
haftmann@27823
   870
  fix P and n :: nat
haftmann@27823
   871
  assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n"
haftmann@27823
   872
  have "\<And>q. q \<le> n \<Longrightarrow> P q"
haftmann@27823
   873
  proof (induct n)
haftmann@27823
   874
    case (0 n)
krauss@26748
   875
    have "P 0" by (rule step) auto
krauss@26748
   876
    thus ?case using 0 by auto
krauss@26748
   877
  next
haftmann@27823
   878
    case (Suc m n)
haftmann@27823
   879
    then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)
krauss@26748
   880
    thus ?case
krauss@26748
   881
    proof
haftmann@27823
   882
      assume "n \<le> m" thus "P n" by (rule Suc(1))
krauss@26748
   883
    next
haftmann@27823
   884
      assume n: "n = Suc m"
haftmann@27823
   885
      show "P n"
haftmann@27823
   886
        by (rule step) (rule Suc(1), simp add: n le_simps)
krauss@26748
   887
    qed
krauss@26748
   888
  qed
haftmann@27823
   889
  then show "P n" by auto
krauss@26748
   890
qed
krauss@26748
   891
nipkow@57015
   892
nipkow@57015
   893
lemma Least_eq_0[simp]: "P(0::nat) \<Longrightarrow> Least P = 0"
nipkow@57015
   894
by (rule Least_equality[OF _ le0])
nipkow@57015
   895
haftmann@27823
   896
lemma Least_Suc:
haftmann@27823
   897
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
wenzelm@47988
   898
  apply (cases n, auto)
haftmann@27823
   899
  apply (frule LeastI)
haftmann@27823
   900
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
haftmann@27823
   901
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
haftmann@27823
   902
  apply (erule_tac [2] Least_le)
wenzelm@47988
   903
  apply (cases "LEAST x. P x", auto)
haftmann@27823
   904
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
haftmann@27823
   905
  apply (blast intro: order_antisym)
haftmann@27823
   906
  done
haftmann@27823
   907
haftmann@27823
   908
lemma Least_Suc2:
haftmann@27823
   909
   "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
haftmann@27823
   910
  apply (erule (1) Least_Suc [THEN ssubst])
haftmann@27823
   911
  apply simp
haftmann@27823
   912
  done
haftmann@27823
   913
haftmann@27823
   914
lemma ex_least_nat_le: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k\<le>n. (\<forall>i<k. \<not>P i) & P(k)"
haftmann@27823
   915
  apply (cases n)
haftmann@27823
   916
   apply blast
haftmann@27823
   917
  apply (rule_tac x="LEAST k. P(k)" in exI)
haftmann@27823
   918
  apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
haftmann@27823
   919
  done
haftmann@27823
   920
haftmann@27823
   921
lemma ex_least_nat_less: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k<n. (\<forall>i\<le>k. \<not>P i) & P(k+1)"
huffman@30079
   922
  unfolding One_nat_def
haftmann@27823
   923
  apply (cases n)
haftmann@27823
   924
   apply blast
haftmann@27823
   925
  apply (frule (1) ex_least_nat_le)
haftmann@27823
   926
  apply (erule exE)
haftmann@27823
   927
  apply (case_tac k)
haftmann@27823
   928
   apply simp
haftmann@27823
   929
  apply (rename_tac k1)
haftmann@27823
   930
  apply (rule_tac x=k1 in exI)
haftmann@27823
   931
  apply (auto simp add: less_eq_Suc_le)
haftmann@27823
   932
  done
haftmann@27823
   933
krauss@26748
   934
lemma nat_less_induct:
krauss@26748
   935
  assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
krauss@26748
   936
  using assms less_induct by blast
krauss@26748
   937
krauss@26748
   938
lemma measure_induct_rule [case_names less]:
krauss@26748
   939
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   940
  assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
krauss@26748
   941
  shows "P a"
krauss@26748
   942
by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step)
krauss@26748
   943
krauss@26748
   944
text {* old style induction rules: *}
krauss@26748
   945
lemma measure_induct:
krauss@26748
   946
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   947
  shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
krauss@26748
   948
  by (rule measure_induct_rule [of f P a]) iprover
krauss@26748
   949
krauss@26748
   950
lemma full_nat_induct:
krauss@26748
   951
  assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"
krauss@26748
   952
  shows "P n"
krauss@26748
   953
  by (rule less_induct) (auto intro: step simp:le_simps)
paulson@14267
   954
paulson@19870
   955
text{*An induction rule for estabilishing binary relations*}
wenzelm@22718
   956
lemma less_Suc_induct:
paulson@19870
   957
  assumes less:  "i < j"
paulson@19870
   958
     and  step:  "!!i. P i (Suc i)"
krauss@31714
   959
     and  trans: "!!i j k. i < j ==> j < k ==>  P i j ==> P j k ==> P i k"
paulson@19870
   960
  shows "P i j"
paulson@19870
   961
proof -
krauss@31714
   962
  from less obtain k where j: "j = Suc (i + k)" by (auto dest: less_imp_Suc_add)
wenzelm@22718
   963
  have "P i (Suc (i + k))"
paulson@19870
   964
  proof (induct k)
wenzelm@22718
   965
    case 0
wenzelm@22718
   966
    show ?case by (simp add: step)
paulson@19870
   967
  next
paulson@19870
   968
    case (Suc k)
krauss@31714
   969
    have "0 + i < Suc k + i" by (rule add_less_mono1) simp
haftmann@57512
   970
    hence "i < Suc (i + k)" by (simp add: add.commute)
krauss@31714
   971
    from trans[OF this lessI Suc step]
krauss@31714
   972
    show ?case by simp
paulson@19870
   973
  qed
wenzelm@22718
   974
  thus "P i j" by (simp add: j)
paulson@19870
   975
qed
paulson@19870
   976
krauss@26748
   977
text {* The method of infinite descent, frequently used in number theory.
krauss@26748
   978
Provided by Roelof Oosterhuis.
krauss@26748
   979
$P(n)$ is true for all $n\in\mathbb{N}$ if
krauss@26748
   980
\begin{itemize}
krauss@26748
   981
  \item case ``0'': given $n=0$ prove $P(n)$,
krauss@26748
   982
  \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
krauss@26748
   983
        a smaller integer $m$ such that $\neg P(m)$.
krauss@26748
   984
\end{itemize} *}
krauss@26748
   985
krauss@26748
   986
text{* A compact version without explicit base case: *}
krauss@26748
   987
lemma infinite_descent:
krauss@26748
   988
  "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow>  \<exists>m<n. \<not>  P m \<rbrakk> \<Longrightarrow>  P n"
wenzelm@47988
   989
by (induct n rule: less_induct) auto
krauss@26748
   990
krauss@26748
   991
lemma infinite_descent0[case_names 0 smaller]: 
krauss@26748
   992
  "\<lbrakk> P 0; !!n. n>0 \<Longrightarrow> \<not> P n \<Longrightarrow> (\<exists>m::nat. m < n \<and> \<not>P m) \<rbrakk> \<Longrightarrow> P n"
krauss@26748
   993
by (rule infinite_descent) (case_tac "n>0", auto)
krauss@26748
   994
krauss@26748
   995
text {*
krauss@26748
   996
Infinite descent using a mapping to $\mathbb{N}$:
krauss@26748
   997
$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
krauss@26748
   998
\begin{itemize}
krauss@26748
   999
\item case ``0'': given $V(x)=0$ prove $P(x)$,
krauss@26748
  1000
\item case ``smaller'': given $V(x)>0$ and $\neg P(x)$ prove there exists a $y \in D$ such that $V(y)<V(x)$ and $~\neg P(y)$.
krauss@26748
  1001
\end{itemize}
krauss@26748
  1002
NB: the proof also shows how to use the previous lemma. *}
krauss@26748
  1003
krauss@26748
  1004
corollary infinite_descent0_measure [case_names 0 smaller]:
krauss@26748
  1005
  assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
krauss@26748
  1006
    and   A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
krauss@26748
  1007
  shows "P x"
krauss@26748
  1008
proof -
krauss@26748
  1009
  obtain n where "n = V x" by auto
krauss@26748
  1010
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
krauss@26748
  1011
  proof (induct n rule: infinite_descent0)
krauss@26748
  1012
    case 0 -- "i.e. $V(x) = 0$"
krauss@26748
  1013
    with A0 show "P x" by auto
krauss@26748
  1014
  next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
krauss@26748
  1015
    case (smaller n)
krauss@26748
  1016
    then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
krauss@26748
  1017
    with A1 obtain y where "V y < V x \<and> \<not> P y" by auto
krauss@26748
  1018
    with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
krauss@26748
  1019
    then show ?case by auto
krauss@26748
  1020
  qed
krauss@26748
  1021
  ultimately show "P x" by auto
krauss@26748
  1022
qed
krauss@26748
  1023
krauss@26748
  1024
text{* Again, without explicit base case: *}
krauss@26748
  1025
lemma infinite_descent_measure:
krauss@26748
  1026
assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
krauss@26748
  1027
proof -
krauss@26748
  1028
  from assms obtain n where "n = V x" by auto
krauss@26748
  1029
  moreover have "!!x. V x = n \<Longrightarrow> P x"
krauss@26748
  1030
  proof (induct n rule: infinite_descent, auto)
krauss@26748
  1031
    fix x assume "\<not> P x"
krauss@26748
  1032
    with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
krauss@26748
  1033
  qed
krauss@26748
  1034
  ultimately show "P x" by auto
krauss@26748
  1035
qed
krauss@26748
  1036
paulson@14267
  1037
text {* A [clumsy] way of lifting @{text "<"}
paulson@14267
  1038
  monotonicity to @{text "\<le>"} monotonicity *}
paulson@14267
  1039
lemma less_mono_imp_le_mono:
nipkow@24438
  1040
  "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
nipkow@24438
  1041
by (simp add: order_le_less) (blast)
nipkow@24438
  1042
paulson@14267
  1043
paulson@14267
  1044
text {* non-strict, in 1st argument *}
paulson@14267
  1045
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
nipkow@24438
  1046
by (rule add_right_mono)
paulson@14267
  1047
paulson@14267
  1048
text {* non-strict, in both arguments *}
paulson@14267
  1049
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
nipkow@24438
  1050
by (rule add_mono)
paulson@14267
  1051
paulson@14267
  1052
lemma le_add2: "n \<le> ((m + n)::nat)"
nipkow@24438
  1053
by (insert add_right_mono [of 0 m n], simp)
berghofe@13449
  1054
paulson@14267
  1055
lemma le_add1: "n \<le> ((n + m)::nat)"
haftmann@57512
  1056
by (simp add: add.commute, rule le_add2)
berghofe@13449
  1057
berghofe@13449
  1058
lemma less_add_Suc1: "i < Suc (i + m)"
nipkow@24438
  1059
by (rule le_less_trans, rule le_add1, rule lessI)
berghofe@13449
  1060
berghofe@13449
  1061
lemma less_add_Suc2: "i < Suc (m + i)"
nipkow@24438
  1062
by (rule le_less_trans, rule le_add2, rule lessI)
berghofe@13449
  1063
paulson@14267
  1064
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
nipkow@24438
  1065
by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
berghofe@13449
  1066
paulson@14267
  1067
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
nipkow@24438
  1068
by (rule le_trans, assumption, rule le_add1)
berghofe@13449
  1069
paulson@14267
  1070
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
nipkow@24438
  1071
by (rule le_trans, assumption, rule le_add2)
berghofe@13449
  1072
berghofe@13449
  1073
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
nipkow@24438
  1074
by (rule less_le_trans, assumption, rule le_add1)
berghofe@13449
  1075
berghofe@13449
  1076
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
nipkow@24438
  1077
by (rule less_le_trans, assumption, rule le_add2)
berghofe@13449
  1078
berghofe@13449
  1079
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
nipkow@24438
  1080
apply (rule le_less_trans [of _ "i+j"])
nipkow@24438
  1081
apply (simp_all add: le_add1)
nipkow@24438
  1082
done
berghofe@13449
  1083
berghofe@13449
  1084
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
nipkow@24438
  1085
apply (rule notI)
wenzelm@26335
  1086
apply (drule add_lessD1)
wenzelm@26335
  1087
apply (erule less_irrefl [THEN notE])
nipkow@24438
  1088
done
berghofe@13449
  1089
berghofe@13449
  1090
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
haftmann@57512
  1091
by (simp add: add.commute)
berghofe@13449
  1092
paulson@14267
  1093
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
nipkow@24438
  1094
apply (rule order_trans [of _ "m+k"])
nipkow@24438
  1095
apply (simp_all add: le_add1)
nipkow@24438
  1096
done
berghofe@13449
  1097
paulson@14267
  1098
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
haftmann@57512
  1099
apply (simp add: add.commute)
nipkow@24438
  1100
apply (erule add_leD1)
nipkow@24438
  1101
done
berghofe@13449
  1102
paulson@14267
  1103
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
nipkow@24438
  1104
by (blast dest: add_leD1 add_leD2)
berghofe@13449
  1105
haftmann@57514
  1106
text {* needs @{text "!!k"} for @{text ac_simps} to work *}
berghofe@13449
  1107
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
nipkow@24438
  1108
by (force simp del: add_Suc_right
haftmann@57514
  1109
    simp add: less_iff_Suc_add add_Suc_right [symmetric] ac_simps)
berghofe@13449
  1110
berghofe@13449
  1111
haftmann@26072
  1112
subsubsection {* More results about difference *}
berghofe@13449
  1113
berghofe@13449
  1114
text {* Addition is the inverse of subtraction:
paulson@14267
  1115
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
berghofe@13449
  1116
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
nipkow@24438
  1117
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1118
paulson@14267
  1119
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
nipkow@24438
  1120
by (simp add: add_diff_inverse linorder_not_less)
berghofe@13449
  1121
paulson@14267
  1122
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
haftmann@57512
  1123
by (simp add: add.commute)
berghofe@13449
  1124
paulson@14267
  1125
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
nipkow@24438
  1126
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1127
berghofe@13449
  1128
lemma diff_less_Suc: "m - n < Suc m"
nipkow@24438
  1129
apply (induct m n rule: diff_induct)
nipkow@24438
  1130
apply (erule_tac [3] less_SucE)
nipkow@24438
  1131
apply (simp_all add: less_Suc_eq)
nipkow@24438
  1132
done
berghofe@13449
  1133
paulson@14267
  1134
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
nipkow@24438
  1135
by (induct m n rule: diff_induct) (simp_all add: le_SucI)
berghofe@13449
  1136
haftmann@26072
  1137
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
haftmann@26072
  1138
  by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])
haftmann@26072
  1139
haftmann@52289
  1140
instance nat :: ordered_cancel_comm_monoid_diff
haftmann@52289
  1141
proof
haftmann@52289
  1142
  show "\<And>m n :: nat. m \<le> n \<longleftrightarrow> (\<exists>q. n = m + q)" by (fact le_iff_add)
haftmann@52289
  1143
qed
haftmann@52289
  1144
berghofe@13449
  1145
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
nipkow@24438
  1146
by (rule le_less_trans, rule diff_le_self)
berghofe@13449
  1147
berghofe@13449
  1148
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
nipkow@24438
  1149
by (cases n) (auto simp add: le_simps)
berghofe@13449
  1150
paulson@14267
  1151
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
nipkow@24438
  1152
by (induct j k rule: diff_induct) simp_all
berghofe@13449
  1153
paulson@14267
  1154
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
haftmann@57512
  1155
by (simp add: add.commute diff_add_assoc)
berghofe@13449
  1156
paulson@14267
  1157
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
nipkow@24438
  1158
by (auto simp add: diff_add_inverse2)
berghofe@13449
  1159
paulson@14267
  1160
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
nipkow@24438
  1161
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1162
paulson@14267
  1163
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
nipkow@24438
  1164
by (rule iffD2, rule diff_is_0_eq)
berghofe@13449
  1165
berghofe@13449
  1166
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
nipkow@24438
  1167
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1168
wenzelm@22718
  1169
lemma less_imp_add_positive:
wenzelm@22718
  1170
  assumes "i < j"
wenzelm@22718
  1171
  shows "\<exists>k::nat. 0 < k & i + k = j"
wenzelm@22718
  1172
proof
wenzelm@22718
  1173
  from assms show "0 < j - i & i + (j - i) = j"
huffman@23476
  1174
    by (simp add: order_less_imp_le)
wenzelm@22718
  1175
qed
wenzelm@9436
  1176
haftmann@26072
  1177
text {* a nice rewrite for bounded subtraction *}
haftmann@26072
  1178
lemma nat_minus_add_max:
haftmann@26072
  1179
  fixes n m :: nat
haftmann@26072
  1180
  shows "n - m + m = max n m"
haftmann@26072
  1181
    by (simp add: max_def not_le order_less_imp_le)
berghofe@13449
  1182
haftmann@26072
  1183
lemma nat_diff_split:
haftmann@26072
  1184
  "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
haftmann@26072
  1185
    -- {* elimination of @{text -} on @{text nat} *}
haftmann@26072
  1186
by (cases "a < b")
haftmann@26072
  1187
  (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse
thomas@57492
  1188
    not_less le_less dest!: add_eq_self_zero add_eq_self_zero[OF sym])
berghofe@13449
  1189
haftmann@26072
  1190
lemma nat_diff_split_asm:
haftmann@26072
  1191
  "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
haftmann@26072
  1192
    -- {* elimination of @{text -} on @{text nat} in assumptions *}
haftmann@26072
  1193
by (auto split: nat_diff_split)
berghofe@13449
  1194
huffman@47255
  1195
lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
huffman@47255
  1196
  by simp
huffman@47255
  1197
huffman@47255
  1198
lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
huffman@47255
  1199
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
  1200
huffman@47255
  1201
lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
huffman@47255
  1202
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
  1203
huffman@47255
  1204
lemma Suc_diff_eq_diff_pred: "0 < n ==> Suc m - n = m - (n - 1)"
huffman@47255
  1205
  unfolding One_nat_def by (cases n) simp_all
huffman@47255
  1206
huffman@47255
  1207
lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
huffman@47255
  1208
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
  1209
huffman@47255
  1210
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
huffman@47255
  1211
  by (fact Let_def)
huffman@47255
  1212
berghofe@13449
  1213
blanchet@58377
  1214
subsubsection {* Monotonicity of multiplication *}
berghofe@13449
  1215
paulson@14267
  1216
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
nipkow@24438
  1217
by (simp add: mult_right_mono)
berghofe@13449
  1218
paulson@14267
  1219
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
nipkow@24438
  1220
by (simp add: mult_left_mono)
berghofe@13449
  1221
paulson@14267
  1222
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
paulson@14267
  1223
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
nipkow@24438
  1224
by (simp add: mult_mono)
berghofe@13449
  1225
berghofe@13449
  1226
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
nipkow@24438
  1227
by (simp add: mult_strict_right_mono)
berghofe@13449
  1228
paulson@14266
  1229
text{*Differs from the standard @{text zero_less_mult_iff} in that
paulson@14266
  1230
      there are no negative numbers.*}
paulson@14266
  1231
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
berghofe@13449
  1232
  apply (induct m)
wenzelm@22718
  1233
   apply simp
wenzelm@22718
  1234
  apply (case_tac n)
wenzelm@22718
  1235
   apply simp_all
berghofe@13449
  1236
  done
berghofe@13449
  1237
huffman@30079
  1238
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)"
berghofe@13449
  1239
  apply (induct m)
wenzelm@22718
  1240
   apply simp
wenzelm@22718
  1241
  apply (case_tac n)
wenzelm@22718
  1242
   apply simp_all
berghofe@13449
  1243
  done
berghofe@13449
  1244
paulson@14341
  1245
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
berghofe@13449
  1246
  apply (safe intro!: mult_less_mono1)
wenzelm@47988
  1247
  apply (cases k, auto)
berghofe@13449
  1248
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
berghofe@13449
  1249
  apply (blast intro: mult_le_mono1)
berghofe@13449
  1250
  done
berghofe@13449
  1251
berghofe@13449
  1252
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
haftmann@57512
  1253
by (simp add: mult.commute [of k])
berghofe@13449
  1254
paulson@14267
  1255
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
nipkow@24438
  1256
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1257
paulson@14267
  1258
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
nipkow@24438
  1259
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1260
berghofe@13449
  1261
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
nipkow@24438
  1262
by (subst mult_less_cancel1) simp
berghofe@13449
  1263
paulson@14267
  1264
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
nipkow@24438
  1265
by (subst mult_le_cancel1) simp
berghofe@13449
  1266
haftmann@26072
  1267
lemma le_square: "m \<le> m * (m::nat)"
haftmann@26072
  1268
  by (cases m) (auto intro: le_add1)
haftmann@26072
  1269
haftmann@26072
  1270
lemma le_cube: "(m::nat) \<le> m * (m * m)"
haftmann@26072
  1271
  by (cases m) (auto intro: le_add1)
berghofe@13449
  1272
berghofe@13449
  1273
text {* Lemma for @{text gcd} *}
huffman@30128
  1274
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
berghofe@13449
  1275
  apply (drule sym)
berghofe@13449
  1276
  apply (rule disjCI)
berghofe@13449
  1277
  apply (rule nat_less_cases, erule_tac [2] _)
paulson@25157
  1278
   apply (drule_tac [2] mult_less_mono2)
nipkow@25162
  1279
    apply (auto)
berghofe@13449
  1280
  done
wenzelm@9436
  1281
haftmann@51263
  1282
lemma mono_times_nat:
haftmann@51263
  1283
  fixes n :: nat
haftmann@51263
  1284
  assumes "n > 0"
haftmann@51263
  1285
  shows "mono (times n)"
haftmann@51263
  1286
proof
haftmann@51263
  1287
  fix m q :: nat
haftmann@51263
  1288
  assume "m \<le> q"
haftmann@51263
  1289
  with assms show "n * m \<le> n * q" by simp
haftmann@51263
  1290
qed
haftmann@51263
  1291
haftmann@26072
  1292
text {* the lattice order on @{typ nat} *}
haftmann@24995
  1293
haftmann@26072
  1294
instantiation nat :: distrib_lattice
haftmann@26072
  1295
begin
haftmann@24995
  1296
haftmann@26072
  1297
definition
haftmann@26072
  1298
  "(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min"
haftmann@24995
  1299
haftmann@26072
  1300
definition
haftmann@26072
  1301
  "(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max"
haftmann@24995
  1302
haftmann@26072
  1303
instance by intro_classes
haftmann@26072
  1304
  (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
haftmann@26072
  1305
    intro: order_less_imp_le antisym elim!: order_trans order_less_trans)
haftmann@24995
  1306
haftmann@26072
  1307
end
haftmann@24995
  1308
haftmann@24995
  1309
haftmann@30954
  1310
subsection {* Natural operation of natural numbers on functions *}
haftmann@30954
  1311
haftmann@30971
  1312
text {*
haftmann@30971
  1313
  We use the same logical constant for the power operations on
haftmann@30971
  1314
  functions and relations, in order to share the same syntax.
haftmann@30971
  1315
*}
haftmann@30971
  1316
haftmann@45965
  1317
consts compow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@30971
  1318
haftmann@45965
  1319
abbreviation compower :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^^" 80) where
haftmann@30971
  1320
  "f ^^ n \<equiv> compow n f"
haftmann@30971
  1321
haftmann@30971
  1322
notation (latex output)
haftmann@30971
  1323
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30971
  1324
haftmann@30971
  1325
notation (HTML output)
haftmann@30971
  1326
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30971
  1327
haftmann@30971
  1328
text {* @{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f} *}
haftmann@30971
  1329
haftmann@30971
  1330
overloading
haftmann@30971
  1331
  funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
haftmann@30971
  1332
begin
haftmann@30954
  1333
blanchet@55575
  1334
primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@44325
  1335
  "funpow 0 f = id"
haftmann@44325
  1336
| "funpow (Suc n) f = f o funpow n f"
haftmann@30954
  1337
haftmann@30971
  1338
end
haftmann@30971
  1339
haftmann@49723
  1340
lemma funpow_Suc_right:
haftmann@49723
  1341
  "f ^^ Suc n = f ^^ n \<circ> f"
haftmann@49723
  1342
proof (induct n)
haftmann@49723
  1343
  case 0 then show ?case by simp
haftmann@49723
  1344
next
haftmann@49723
  1345
  fix n
haftmann@49723
  1346
  assume "f ^^ Suc n = f ^^ n \<circ> f"
haftmann@49723
  1347
  then show "f ^^ Suc (Suc n) = f ^^ Suc n \<circ> f"
haftmann@49723
  1348
    by (simp add: o_assoc)
haftmann@49723
  1349
qed
haftmann@49723
  1350
haftmann@49723
  1351
lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right
haftmann@49723
  1352
haftmann@30971
  1353
text {* for code generation *}
haftmann@30971
  1354
haftmann@30971
  1355
definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@46028
  1356
  funpow_code_def [code_abbrev]: "funpow = compow"
haftmann@30954
  1357
haftmann@30971
  1358
lemma [code]:
haftmann@37430
  1359
  "funpow (Suc n) f = f o funpow n f"
haftmann@30971
  1360
  "funpow 0 f = id"
haftmann@37430
  1361
  by (simp_all add: funpow_code_def)
haftmann@30971
  1362
wenzelm@36176
  1363
hide_const (open) funpow
haftmann@30954
  1364
haftmann@30954
  1365
lemma funpow_add:
haftmann@30971
  1366
  "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
haftmann@30954
  1367
  by (induct m) simp_all
haftmann@30954
  1368
haftmann@37430
  1369
lemma funpow_mult:
haftmann@37430
  1370
  fixes f :: "'a \<Rightarrow> 'a"
haftmann@37430
  1371
  shows "(f ^^ m) ^^ n = f ^^ (m * n)"
haftmann@37430
  1372
  by (induct n) (simp_all add: funpow_add)
haftmann@37430
  1373
haftmann@30954
  1374
lemma funpow_swap1:
haftmann@30971
  1375
  "f ((f ^^ n) x) = (f ^^ n) (f x)"
haftmann@30954
  1376
proof -
haftmann@30971
  1377
  have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp
haftmann@30971
  1378
  also have "\<dots>  = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)
haftmann@30971
  1379
  also have "\<dots> = (f ^^ n) (f x)" by simp
haftmann@30954
  1380
  finally show ?thesis .
haftmann@30954
  1381
qed
haftmann@30954
  1382
haftmann@38621
  1383
lemma comp_funpow:
haftmann@38621
  1384
  fixes f :: "'a \<Rightarrow> 'a"
haftmann@38621
  1385
  shows "comp f ^^ n = comp (f ^^ n)"
haftmann@38621
  1386
  by (induct n) simp_all
haftmann@30954
  1387
hoelzl@54496
  1388
lemma Suc_funpow[simp]: "Suc ^^ n = (op + n)"
hoelzl@54496
  1389
  by (induct n) simp_all
hoelzl@54496
  1390
hoelzl@54496
  1391
lemma id_funpow[simp]: "id ^^ n = id"
hoelzl@54496
  1392
  by (induct n) simp_all
haftmann@38621
  1393
hoelzl@59000
  1394
lemma funpow_mono:
hoelzl@59000
  1395
  fixes f :: "'a \<Rightarrow> ('a::lattice)"
hoelzl@59000
  1396
  shows "mono f \<Longrightarrow> A \<le> B \<Longrightarrow> (f ^^ n) A \<le> (f ^^ n) B"
hoelzl@59000
  1397
  by (induct n arbitrary: A B)
hoelzl@59000
  1398
     (auto simp del: funpow.simps(2) simp add: funpow_Suc_right mono_def)
hoelzl@59000
  1399
nipkow@45833
  1400
subsection {* Kleene iteration *}
nipkow@45833
  1401
haftmann@52729
  1402
lemma Kleene_iter_lpfp:
haftmann@52729
  1403
assumes "mono f" and "f p \<le> p" shows "(f^^k) (bot::'a::order_bot) \<le> p"
nipkow@45833
  1404
proof(induction k)
nipkow@45833
  1405
  case 0 show ?case by simp
nipkow@45833
  1406
next
nipkow@45833
  1407
  case Suc
nipkow@45833
  1408
  from monoD[OF assms(1) Suc] assms(2)
nipkow@45833
  1409
  show ?case by simp
nipkow@45833
  1410
qed
nipkow@45833
  1411
nipkow@45833
  1412
lemma lfp_Kleene_iter: assumes "mono f" and "(f^^Suc k) bot = (f^^k) bot"
nipkow@45833
  1413
shows "lfp f = (f^^k) bot"
nipkow@45833
  1414
proof(rule antisym)
nipkow@45833
  1415
  show "lfp f \<le> (f^^k) bot"
nipkow@45833
  1416
  proof(rule lfp_lowerbound)
nipkow@45833
  1417
    show "f ((f^^k) bot) \<le> (f^^k) bot" using assms(2) by simp
nipkow@45833
  1418
  qed
nipkow@45833
  1419
next
nipkow@45833
  1420
  show "(f^^k) bot \<le> lfp f"
nipkow@45833
  1421
    using Kleene_iter_lpfp[OF assms(1)] lfp_unfold[OF assms(1)] by simp
nipkow@45833
  1422
qed
nipkow@45833
  1423
nipkow@45833
  1424
blanchet@58377
  1425
subsection {* Embedding of the naturals into any @{text semiring_1}: @{term of_nat} *}
haftmann@24196
  1426
haftmann@24196
  1427
context semiring_1
haftmann@24196
  1428
begin
haftmann@24196
  1429
haftmann@38621
  1430
definition of_nat :: "nat \<Rightarrow> 'a" where
haftmann@38621
  1431
  "of_nat n = (plus 1 ^^ n) 0"
haftmann@38621
  1432
haftmann@38621
  1433
lemma of_nat_simps [simp]:
haftmann@38621
  1434
  shows of_nat_0: "of_nat 0 = 0"
haftmann@38621
  1435
    and of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
haftmann@38621
  1436
  by (simp_all add: of_nat_def)
haftmann@25193
  1437
haftmann@25193
  1438
lemma of_nat_1 [simp]: "of_nat 1 = 1"
haftmann@38621
  1439
  by (simp add: of_nat_def)
haftmann@25193
  1440
haftmann@25193
  1441
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
haftmann@57514
  1442
  by (induct m) (simp_all add: ac_simps)
haftmann@25193
  1443
haftmann@25193
  1444
lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"
haftmann@57514
  1445
  by (induct m) (simp_all add: ac_simps distrib_right)
haftmann@25193
  1446
blanchet@55575
  1447
primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@28514
  1448
  "of_nat_aux inc 0 i = i"
haftmann@44325
  1449
| "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- {* tail recursive *}
haftmann@25928
  1450
haftmann@30966
  1451
lemma of_nat_code:
haftmann@28514
  1452
  "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"
haftmann@28514
  1453
proof (induct n)
haftmann@28514
  1454
  case 0 then show ?case by simp
haftmann@28514
  1455
next
haftmann@28514
  1456
  case (Suc n)
haftmann@28514
  1457
  have "\<And>i. of_nat_aux (\<lambda>i. i + 1) n (i + 1) = of_nat_aux (\<lambda>i. i + 1) n i + 1"
haftmann@28514
  1458
    by (induct n) simp_all
haftmann@28514
  1459
  from this [of 0] have "of_nat_aux (\<lambda>i. i + 1) n 1 = of_nat_aux (\<lambda>i. i + 1) n 0 + 1"
haftmann@28514
  1460
    by simp
haftmann@57512
  1461
  with Suc show ?case by (simp add: add.commute)
haftmann@28514
  1462
qed
haftmann@30966
  1463
haftmann@24196
  1464
end
haftmann@24196
  1465
bulwahn@45231
  1466
declare of_nat_code [code]
haftmann@30966
  1467
haftmann@26072
  1468
text{*Class for unital semirings with characteristic zero.
haftmann@26072
  1469
 Includes non-ordered rings like the complex numbers.*}
haftmann@26072
  1470
haftmann@26072
  1471
class semiring_char_0 = semiring_1 +
haftmann@38621
  1472
  assumes inj_of_nat: "inj of_nat"
haftmann@26072
  1473
begin
haftmann@26072
  1474
haftmann@38621
  1475
lemma of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"
haftmann@38621
  1476
  by (auto intro: inj_of_nat injD)
haftmann@38621
  1477
haftmann@26072
  1478
text{*Special cases where either operand is zero*}
haftmann@26072
  1479
blanchet@54147
  1480
lemma of_nat_0_eq_iff [simp]: "0 = of_nat n \<longleftrightarrow> 0 = n"
haftmann@38621
  1481
  by (fact of_nat_eq_iff [of 0 n, unfolded of_nat_0])
haftmann@26072
  1482
blanchet@54147
  1483
lemma of_nat_eq_0_iff [simp]: "of_nat m = 0 \<longleftrightarrow> m = 0"
haftmann@38621
  1484
  by (fact of_nat_eq_iff [of m 0, unfolded of_nat_0])
haftmann@26072
  1485
haftmann@26072
  1486
end
haftmann@26072
  1487
haftmann@35028
  1488
context linordered_semidom
haftmann@25193
  1489
begin
haftmann@25193
  1490
huffman@47489
  1491
lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"
huffman@47489
  1492
  by (induct n) simp_all
haftmann@25193
  1493
huffman@47489
  1494
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"
huffman@47489
  1495
  by (simp add: not_less)
haftmann@25193
  1496
haftmann@25193
  1497
lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
huffman@47489
  1498
  by (induct m n rule: diff_induct, simp_all add: add_pos_nonneg)
haftmann@25193
  1499
haftmann@26072
  1500
lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
haftmann@26072
  1501
  by (simp add: not_less [symmetric] linorder_not_less [symmetric])
haftmann@25193
  1502
huffman@47489
  1503
lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"
huffman@47489
  1504
  by simp
huffman@47489
  1505
huffman@47489
  1506
lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
huffman@47489
  1507
  by simp
huffman@47489
  1508
haftmann@35028
  1509
text{*Every @{text linordered_semidom} has characteristic zero.*}
haftmann@25193
  1510
haftmann@38621
  1511
subclass semiring_char_0 proof
haftmann@38621
  1512
qed (auto intro!: injI simp add: eq_iff)
haftmann@25193
  1513
haftmann@25193
  1514
text{*Special cases where either operand is zero*}
haftmann@25193
  1515
blanchet@54147
  1516
lemma of_nat_le_0_iff [simp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"
haftmann@25193
  1517
  by (rule of_nat_le_iff [of _ 0, simplified])
haftmann@25193
  1518
haftmann@26072
  1519
lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"
haftmann@26072
  1520
  by (rule of_nat_less_iff [of 0, simplified])
haftmann@26072
  1521
haftmann@26072
  1522
end
haftmann@26072
  1523
haftmann@26072
  1524
context ring_1
haftmann@26072
  1525
begin
haftmann@26072
  1526
haftmann@26072
  1527
lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"
nipkow@29667
  1528
by (simp add: algebra_simps of_nat_add [symmetric])
haftmann@26072
  1529
haftmann@26072
  1530
end
haftmann@26072
  1531
haftmann@35028
  1532
context linordered_idom
haftmann@26072
  1533
begin
haftmann@26072
  1534
haftmann@26072
  1535
lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"
haftmann@26072
  1536
  unfolding abs_if by auto
haftmann@26072
  1537
haftmann@25193
  1538
end
haftmann@25193
  1539
haftmann@25193
  1540
lemma of_nat_id [simp]: "of_nat n = n"
huffman@35216
  1541
  by (induct n) simp_all
haftmann@25193
  1542
haftmann@25193
  1543
lemma of_nat_eq_id [simp]: "of_nat = id"
nipkow@39302
  1544
  by (auto simp add: fun_eq_iff)
haftmann@25193
  1545
haftmann@25193
  1546
blanchet@58377
  1547
subsection {* The set of natural numbers *}
haftmann@25193
  1548
haftmann@26072
  1549
context semiring_1
haftmann@25193
  1550
begin
haftmann@25193
  1551
haftmann@37767
  1552
definition Nats  :: "'a set" where
haftmann@37767
  1553
  "Nats = range of_nat"
haftmann@26072
  1554
haftmann@26072
  1555
notation (xsymbols)
haftmann@26072
  1556
  Nats  ("\<nat>")
haftmann@25193
  1557
haftmann@26072
  1558
lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"
haftmann@26072
  1559
  by (simp add: Nats_def)
haftmann@26072
  1560
haftmann@26072
  1561
lemma Nats_0 [simp]: "0 \<in> \<nat>"
haftmann@26072
  1562
apply (simp add: Nats_def)
haftmann@26072
  1563
apply (rule range_eqI)
haftmann@26072
  1564
apply (rule of_nat_0 [symmetric])
haftmann@26072
  1565
done
haftmann@25193
  1566
haftmann@26072
  1567
lemma Nats_1 [simp]: "1 \<in> \<nat>"
haftmann@26072
  1568
apply (simp add: Nats_def)
haftmann@26072
  1569
apply (rule range_eqI)
haftmann@26072
  1570
apply (rule of_nat_1 [symmetric])
haftmann@26072
  1571
done
haftmann@25193
  1572
haftmann@26072
  1573
lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"
haftmann@26072
  1574
apply (auto simp add: Nats_def)
haftmann@26072
  1575
apply (rule range_eqI)
haftmann@26072
  1576
apply (rule of_nat_add [symmetric])
haftmann@26072
  1577
done
haftmann@26072
  1578
haftmann@26072
  1579
lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"
haftmann@26072
  1580
apply (auto simp add: Nats_def)
haftmann@26072
  1581
apply (rule range_eqI)
haftmann@26072
  1582
apply (rule of_nat_mult [symmetric])
haftmann@26072
  1583
done
haftmann@25193
  1584
huffman@35633
  1585
lemma Nats_cases [cases set: Nats]:
huffman@35633
  1586
  assumes "x \<in> \<nat>"
huffman@35633
  1587
  obtains (of_nat) n where "x = of_nat n"
huffman@35633
  1588
  unfolding Nats_def
huffman@35633
  1589
proof -
huffman@35633
  1590
  from `x \<in> \<nat>` have "x \<in> range of_nat" unfolding Nats_def .
huffman@35633
  1591
  then obtain n where "x = of_nat n" ..
huffman@35633
  1592
  then show thesis ..
huffman@35633
  1593
qed
huffman@35633
  1594
huffman@35633
  1595
lemma Nats_induct [case_names of_nat, induct set: Nats]:
huffman@35633
  1596
  "x \<in> \<nat> \<Longrightarrow> (\<And>n. P (of_nat n)) \<Longrightarrow> P x"
huffman@35633
  1597
  by (rule Nats_cases) auto
huffman@35633
  1598
haftmann@25193
  1599
end
haftmann@25193
  1600
haftmann@25193
  1601
blanchet@58377
  1602
subsection {* Further arithmetic facts concerning the natural numbers *}
wenzelm@21243
  1603
haftmann@22845
  1604
lemma subst_equals:
haftmann@22845
  1605
  assumes 1: "t = s" and 2: "u = t"
haftmann@22845
  1606
  shows "u = s"
haftmann@22845
  1607
  using 2 1 by (rule trans)
haftmann@22845
  1608
wenzelm@48891
  1609
ML_file "Tools/nat_arith.ML"
huffman@48559
  1610
huffman@48559
  1611
simproc_setup nateq_cancel_sums
huffman@48559
  1612
  ("(l::nat) + m = n" | "(l::nat) = m + n" | "Suc m = n" | "m = Suc n") =
wenzelm@54742
  1613
  {* fn phi => try o Nat_Arith.cancel_eq_conv *}
huffman@48559
  1614
huffman@48559
  1615
simproc_setup natless_cancel_sums
huffman@48559
  1616
  ("(l::nat) + m < n" | "(l::nat) < m + n" | "Suc m < n" | "m < Suc n") =
wenzelm@54742
  1617
  {* fn phi => try o Nat_Arith.cancel_less_conv *}
huffman@48559
  1618
huffman@48559
  1619
simproc_setup natle_cancel_sums
huffman@48559
  1620
  ("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" | "Suc m \<le> n" | "m \<le> Suc n") =
wenzelm@54742
  1621
  {* fn phi => try o Nat_Arith.cancel_le_conv *}
huffman@48559
  1622
huffman@48559
  1623
simproc_setup natdiff_cancel_sums
huffman@48559
  1624
  ("(l::nat) + m - n" | "(l::nat) - (m + n)" | "Suc m - n" | "m - Suc n") =
wenzelm@54742
  1625
  {* fn phi => try o Nat_Arith.cancel_diff_conv *}
wenzelm@24091
  1626
wenzelm@48891
  1627
ML_file "Tools/lin_arith.ML"
haftmann@31100
  1628
setup {* Lin_Arith.global_setup *}
haftmann@30686
  1629
declaration {* K Lin_Arith.setup *}
wenzelm@24091
  1630
wenzelm@43595
  1631
simproc_setup fast_arith_nat ("(m::nat) < n" | "(m::nat) <= n" | "(m::nat) = n") =
wenzelm@59582
  1632
  {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (Thm.term_of ct) *}
wenzelm@43595
  1633
(* Because of this simproc, the arithmetic solver is really only
wenzelm@43595
  1634
useful to detect inconsistencies among the premises for subgoals which are
wenzelm@43595
  1635
*not* themselves (in)equalities, because the latter activate
wenzelm@43595
  1636
fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
wenzelm@43595
  1637
solver all the time rather than add the additional check. *)
wenzelm@43595
  1638
wenzelm@43595
  1639
wenzelm@21243
  1640
lemmas [arith_split] = nat_diff_split split_min split_max
wenzelm@21243
  1641
nipkow@27625
  1642
context order
nipkow@27625
  1643
begin
nipkow@27625
  1644
nipkow@27625
  1645
lemma lift_Suc_mono_le:
haftmann@53986
  1646
  assumes mono: "\<And>n. f n \<le> f (Suc n)" and "n \<le> n'"
krauss@27627
  1647
  shows "f n \<le> f n'"
krauss@27627
  1648
proof (cases "n < n'")
krauss@27627
  1649
  case True
haftmann@53986
  1650
  then show ?thesis
haftmann@53986
  1651
    by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
haftmann@53986
  1652
qed (insert `n \<le> n'`, auto) -- {* trivial for @{prop "n = n'"} *}
nipkow@27625
  1653
hoelzl@56020
  1654
lemma lift_Suc_antimono_le:
hoelzl@56020
  1655
  assumes mono: "\<And>n. f n \<ge> f (Suc n)" and "n \<le> n'"
hoelzl@56020
  1656
  shows "f n \<ge> f n'"
hoelzl@56020
  1657
proof (cases "n < n'")
hoelzl@56020
  1658
  case True
hoelzl@56020
  1659
  then show ?thesis
hoelzl@56020
  1660
    by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
hoelzl@56020
  1661
qed (insert `n \<le> n'`, auto) -- {* trivial for @{prop "n = n'"} *}
hoelzl@56020
  1662
nipkow@27625
  1663
lemma lift_Suc_mono_less:
haftmann@53986
  1664
  assumes mono: "\<And>n. f n < f (Suc n)" and "n < n'"
krauss@27627
  1665
  shows "f n < f n'"
krauss@27627
  1666
using `n < n'`
haftmann@53986
  1667
by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
nipkow@27625
  1668
nipkow@27789
  1669
lemma lift_Suc_mono_less_iff:
haftmann@53986
  1670
  "(\<And>n. f n < f (Suc n)) \<Longrightarrow> f n < f m \<longleftrightarrow> n < m"
haftmann@53986
  1671
  by (blast intro: less_asym' lift_Suc_mono_less [of f]
haftmann@53986
  1672
    dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq [THEN iffD1])
nipkow@27789
  1673
nipkow@27625
  1674
end
nipkow@27625
  1675
haftmann@53986
  1676
lemma mono_iff_le_Suc:
haftmann@53986
  1677
  "mono f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
haftmann@37387
  1678
  unfolding mono_def by (auto intro: lift_Suc_mono_le [of f])
nipkow@27625
  1679
hoelzl@56020
  1680
lemma antimono_iff_le_Suc:
hoelzl@56020
  1681
  "antimono f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
hoelzl@56020
  1682
  unfolding antimono_def by (auto intro: lift_Suc_antimono_le [of f])
hoelzl@56020
  1683
nipkow@27789
  1684
lemma mono_nat_linear_lb:
haftmann@53986
  1685
  fixes f :: "nat \<Rightarrow> nat"
haftmann@53986
  1686
  assumes "\<And>m n. m < n \<Longrightarrow> f m < f n"
haftmann@53986
  1687
  shows "f m + k \<le> f (m + k)"
haftmann@53986
  1688
proof (induct k)
haftmann@53986
  1689
  case 0 then show ?case by simp
haftmann@53986
  1690
next
haftmann@53986
  1691
  case (Suc k)
haftmann@53986
  1692
  then have "Suc (f m + k) \<le> Suc (f (m + k))" by simp
haftmann@53986
  1693
  also from assms [of "m + k" "Suc (m + k)"] have "Suc (f (m + k)) \<le> f (Suc (m + k))"
haftmann@53986
  1694
    by (simp add: Suc_le_eq)
haftmann@53986
  1695
  finally show ?case by simp
haftmann@53986
  1696
qed
nipkow@27789
  1697
nipkow@27789
  1698
wenzelm@21243
  1699
text{*Subtraction laws, mostly by Clemens Ballarin*}
wenzelm@21243
  1700
wenzelm@21243
  1701
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
nipkow@24438
  1702
by arith
wenzelm@21243
  1703
wenzelm@21243
  1704
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
nipkow@24438
  1705
by arith
wenzelm@21243
  1706
haftmann@51173
  1707
lemma less_diff_conv2:
haftmann@51173
  1708
  fixes j k i :: nat
haftmann@51173
  1709
  assumes "k \<le> j"
haftmann@51173
  1710
  shows "j - k < i \<longleftrightarrow> j < i + k"
haftmann@51173
  1711
  using assms by arith
haftmann@51173
  1712
wenzelm@21243
  1713
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
nipkow@24438
  1714
by arith
wenzelm@21243
  1715
wenzelm@21243
  1716
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
haftmann@57512
  1717
  by (fact le_diff_conv2) -- {* FIXME delete *}
wenzelm@21243
  1718
wenzelm@21243
  1719
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
nipkow@24438
  1720
by arith
wenzelm@21243
  1721
wenzelm@21243
  1722
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
haftmann@57512
  1723
  by (fact le_add_diff) -- {* FIXME delete *}
wenzelm@21243
  1724
wenzelm@21243
  1725
(*Replaces the previous diff_less and le_diff_less, which had the stronger
wenzelm@21243
  1726
  second premise n\<le>m*)
wenzelm@21243
  1727
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
nipkow@24438
  1728
by arith
wenzelm@21243
  1729
haftmann@26072
  1730
text {* Simplification of relational expressions involving subtraction *}
wenzelm@21243
  1731
wenzelm@21243
  1732
lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
nipkow@24438
  1733
by (simp split add: nat_diff_split)
wenzelm@21243
  1734
wenzelm@36176
  1735
hide_fact (open) diff_diff_eq
haftmann@35064
  1736
wenzelm@21243
  1737
lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
nipkow@24438
  1738
by (auto split add: nat_diff_split)
wenzelm@21243
  1739
wenzelm@21243
  1740
lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
nipkow@24438
  1741
by (auto split add: nat_diff_split)
wenzelm@21243
  1742
wenzelm@21243
  1743
lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
nipkow@24438
  1744
by (auto split add: nat_diff_split)
wenzelm@21243
  1745
wenzelm@21243
  1746
text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
wenzelm@21243
  1747
wenzelm@21243
  1748
(* Monotonicity of subtraction in first argument *)
wenzelm@21243
  1749
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
nipkow@24438
  1750
by (simp split add: nat_diff_split)
wenzelm@21243
  1751
wenzelm@21243
  1752
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
nipkow@24438
  1753
by (simp split add: nat_diff_split)
wenzelm@21243
  1754
wenzelm@21243
  1755
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
nipkow@24438
  1756
by (simp split add: nat_diff_split)
wenzelm@21243
  1757
wenzelm@21243
  1758
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
nipkow@24438
  1759
by (simp split add: nat_diff_split)
wenzelm@21243
  1760
bulwahn@26143
  1761
lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"
nipkow@32437
  1762
by auto
bulwahn@26143
  1763
bulwahn@26143
  1764
lemma inj_on_diff_nat: 
bulwahn@26143
  1765
  assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"
bulwahn@26143
  1766
  shows "inj_on (\<lambda>n. n - k) N"
bulwahn@26143
  1767
proof (rule inj_onI)
bulwahn@26143
  1768
  fix x y
bulwahn@26143
  1769
  assume a: "x \<in> N" "y \<in> N" "x - k = y - k"
bulwahn@26143
  1770
  with k_le_n have "x - k + k = y - k + k" by auto
bulwahn@26143
  1771
  with a k_le_n show "x = y" by auto
bulwahn@26143
  1772
qed
bulwahn@26143
  1773
haftmann@26072
  1774
text{*Rewriting to pull differences out*}
haftmann@26072
  1775
haftmann@26072
  1776
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
haftmann@26072
  1777
by arith
haftmann@26072
  1778
haftmann@26072
  1779
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
haftmann@26072
  1780
by arith
haftmann@26072
  1781
haftmann@26072
  1782
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
haftmann@26072
  1783
by arith
haftmann@26072
  1784
noschinl@45933
  1785
lemma Suc_diff_Suc: "n < m \<Longrightarrow> Suc (m - Suc n) = m - n"
noschinl@45933
  1786
by simp
noschinl@45933
  1787
bulwahn@46350
  1788
(*The others are
bulwahn@46350
  1789
      i - j - k = i - (j + k),
bulwahn@46350
  1790
      k \<le> j ==> j - k + i = j + i - k,
bulwahn@46350
  1791
      k \<le> j ==> i + (j - k) = i + j - k *)
bulwahn@46350
  1792
lemmas add_diff_assoc = diff_add_assoc [symmetric]
bulwahn@46350
  1793
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
bulwahn@46350
  1794
declare diff_diff_left [simp]  add_diff_assoc [simp] add_diff_assoc2[simp]
bulwahn@46350
  1795
bulwahn@46350
  1796
text{*At present we prove no analogue of @{text not_less_Least} or @{text
bulwahn@46350
  1797
Least_Suc}, since there appears to be no need.*}
bulwahn@46350
  1798
wenzelm@21243
  1799
text{*Lemmas for ex/Factorization*}
wenzelm@21243
  1800
wenzelm@21243
  1801
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
nipkow@24438
  1802
by (cases m) auto
wenzelm@21243
  1803
wenzelm@21243
  1804
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
nipkow@24438
  1805
by (cases m) auto
wenzelm@21243
  1806
wenzelm@21243
  1807
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
nipkow@24438
  1808
by (cases m) auto
wenzelm@21243
  1809
krauss@23001
  1810
text {* Specialized induction principles that work "backwards": *}
krauss@23001
  1811
krauss@23001
  1812
lemma inc_induct[consumes 1, case_names base step]:
hoelzl@54411
  1813
  assumes less: "i \<le> j"
krauss@23001
  1814
  assumes base: "P j"
hoelzl@54411
  1815
  assumes step: "\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P (Suc n) \<Longrightarrow> P n"
krauss@23001
  1816
  shows "P i"
hoelzl@54411
  1817
  using less step
hoelzl@54411
  1818
proof (induct d\<equiv>"j - i" arbitrary: i)
krauss@23001
  1819
  case (0 i)
krauss@23001
  1820
  hence "i = j" by simp
krauss@23001
  1821
  with base show ?case by simp
krauss@23001
  1822
next
hoelzl@54411
  1823
  case (Suc d n)
hoelzl@54411
  1824
  hence "n \<le> n" "n < j" "P (Suc n)"
krauss@23001
  1825
    by simp_all
hoelzl@54411
  1826
  then show "P n" by fact
krauss@23001
  1827
qed
krauss@23001
  1828
krauss@23001
  1829
lemma strict_inc_induct[consumes 1, case_names base step]:
krauss@23001
  1830
  assumes less: "i < j"
krauss@23001
  1831
  assumes base: "!!i. j = Suc i ==> P i"
krauss@23001
  1832
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1833
  shows "P i"
krauss@23001
  1834
  using less
krauss@23001
  1835
proof (induct d=="j - i - 1" arbitrary: i)
krauss@23001
  1836
  case (0 i)
krauss@23001
  1837
  with `i < j` have "j = Suc i" by simp
krauss@23001
  1838
  with base show ?case by simp
krauss@23001
  1839
next
krauss@23001
  1840
  case (Suc d i)
krauss@23001
  1841
  hence "i < j" "P (Suc i)"
krauss@23001
  1842
    by simp_all
krauss@23001
  1843
  thus "P i" by (rule step)
krauss@23001
  1844
qed
krauss@23001
  1845
krauss@23001
  1846
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
krauss@23001
  1847
  using inc_induct[of "k - i" k P, simplified] by blast
krauss@23001
  1848
krauss@23001
  1849
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
krauss@23001
  1850
  using inc_induct[of 0 k P] by blast
wenzelm@21243
  1851
bulwahn@46351
  1852
text {* Further induction rule similar to @{thm inc_induct} *}
nipkow@27625
  1853
bulwahn@46351
  1854
lemma dec_induct[consumes 1, case_names base step]:
hoelzl@54411
  1855
  "i \<le> j \<Longrightarrow> P i \<Longrightarrow> (\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n \<Longrightarrow> P (Suc n)) \<Longrightarrow> P j"
bulwahn@46351
  1856
  by (induct j arbitrary: i) (auto simp: le_Suc_eq)
hoelzl@59000
  1857
hoelzl@59000
  1858
subsection \<open> Monotonicity of funpow \<close>
hoelzl@59000
  1859
hoelzl@59000
  1860
lemma funpow_increasing:
hoelzl@59000
  1861
  fixes f :: "'a \<Rightarrow> ('a::{lattice, order_top})"
hoelzl@59000
  1862
  shows "m \<le> n \<Longrightarrow> mono f \<Longrightarrow> (f ^^ n) \<top> \<le> (f ^^ m) \<top>"
hoelzl@59000
  1863
  by (induct rule: inc_induct)
hoelzl@59000
  1864
     (auto simp del: funpow.simps(2) simp add: funpow_Suc_right
hoelzl@59000
  1865
           intro: order_trans[OF _ funpow_mono])
hoelzl@59000
  1866
hoelzl@59000
  1867
lemma funpow_decreasing:
hoelzl@59000
  1868
  fixes f :: "'a \<Rightarrow> ('a::{lattice, order_bot})"
hoelzl@59000
  1869
  shows "m \<le> n \<Longrightarrow> mono f \<Longrightarrow> (f ^^ m) \<bottom> \<le> (f ^^ n) \<bottom>"
hoelzl@59000
  1870
  by (induct rule: dec_induct)
hoelzl@59000
  1871
     (auto simp del: funpow.simps(2) simp add: funpow_Suc_right
hoelzl@59000
  1872
           intro: order_trans[OF _ funpow_mono])
hoelzl@59000
  1873
hoelzl@59000
  1874
lemma mono_funpow:
hoelzl@59000
  1875
  fixes Q :: "('i \<Rightarrow> 'a::{lattice, order_bot}) \<Rightarrow> ('i \<Rightarrow> 'a)"
hoelzl@59000
  1876
  shows "mono Q \<Longrightarrow> mono (\<lambda>i. (Q ^^ i) \<bottom>)"
hoelzl@59000
  1877
  by (auto intro!: funpow_decreasing simp: mono_def)
blanchet@58377
  1878
haftmann@33274
  1879
subsection {* The divides relation on @{typ nat} *}
haftmann@33274
  1880
haftmann@58647
  1881
instance nat :: semiring_dvd
haftmann@58647
  1882
proof
haftmann@58647
  1883
  fix m n q :: nat
haftmann@58647
  1884
  show "m dvd q * m + n \<longleftrightarrow> m dvd n" (is "?P \<longleftrightarrow> ?Q")
haftmann@58647
  1885
  proof
haftmann@58647
  1886
    assume ?Q then show ?P by simp
haftmann@58647
  1887
  next
haftmann@58647
  1888
    assume ?P then obtain d where *: "q * m + n = m * d" ..
haftmann@58647
  1889
    show ?Q
haftmann@58647
  1890
    proof (cases "n = 0")
haftmann@58647
  1891
      case True then show ?Q by simp
haftmann@58647
  1892
    next
haftmann@58647
  1893
      case False
haftmann@58647
  1894
      with * have "q * m < m * d"
haftmann@58647
  1895
        using less_add_eq_less [of 0 n "q * m" "m * d"] by simp
haftmann@58647
  1896
      then have "q \<le> d" by (auto intro: ccontr simp add: mult.commute [of m])
haftmann@58647
  1897
      with * [symmetric] have "n = m * (d - q)"
haftmann@58647
  1898
        by (simp add: diff_mult_distrib2 mult.commute [of m])
haftmann@58647
  1899
      then show ?Q ..
haftmann@58647
  1900
    qed
haftmann@58647
  1901
  qed
haftmann@58647
  1902
  assume "m dvd n + q" and "m dvd n"
haftmann@58647
  1903
  show "m dvd q"
haftmann@58647
  1904
  proof -
haftmann@58647
  1905
    from `m dvd n` obtain d where "n = m * d" ..
haftmann@58647
  1906
    moreover from `m dvd n + q` obtain e where "n + q = m * e" ..
haftmann@58647
  1907
    ultimately have *: "m * d + q = m * e" by simp
haftmann@58647
  1908
    show "m dvd q"
haftmann@58647
  1909
    proof (cases "q = 0")
haftmann@58647
  1910
      case True then show ?thesis by simp
haftmann@58647
  1911
    next
haftmann@58647
  1912
      case False
haftmann@58647
  1913
      with * have "m * d < m * e"
haftmann@58647
  1914
        using less_add_eq_less [of 0 q "m * d" "m * e"] by simp
haftmann@58647
  1915
      then have "d \<le> e" by (auto intro: ccontr)
haftmann@58647
  1916
      with * have "q = m * (e - d)"
haftmann@58647
  1917
        by (simp add: diff_mult_distrib2)
haftmann@58647
  1918
      then show ?thesis ..
haftmann@58647
  1919
    qed
haftmann@58647
  1920
  qed
haftmann@58647
  1921
qed
haftmann@58647
  1922
haftmann@33274
  1923
lemma dvd_1_left [iff]: "Suc 0 dvd k"
haftmann@33274
  1924
unfolding dvd_def by simp
haftmann@33274
  1925
haftmann@33274
  1926
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
haftmann@33274
  1927
by (simp add: dvd_def)
haftmann@33274
  1928
haftmann@33274
  1929
lemma nat_dvd_1_iff_1 [simp]: "m dvd (1::nat) \<longleftrightarrow> m = 1"
haftmann@33274
  1930
by (simp add: dvd_def)
haftmann@33274
  1931
nipkow@33657
  1932
lemma dvd_antisym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
haftmann@33274
  1933
  unfolding dvd_def
haftmann@57512
  1934
  by (force dest: mult_eq_self_implies_10 simp add: mult.assoc)
haftmann@33274
  1935
haftmann@33274
  1936
text {* @{term "op dvd"} is a partial order *}
haftmann@33274
  1937
haftmann@33274
  1938
interpretation dvd: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n"
nipkow@33657
  1939
  proof qed (auto intro: dvd_refl dvd_trans dvd_antisym)
haftmann@33274
  1940
haftmann@33274
  1941
lemma dvd_diff_nat[simp]: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
haftmann@33274
  1942
unfolding dvd_def
haftmann@33274
  1943
by (blast intro: diff_mult_distrib2 [symmetric])
haftmann@33274
  1944
haftmann@33274
  1945
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
haftmann@33274
  1946
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
haftmann@33274
  1947
  apply (blast intro: dvd_add)
haftmann@33274
  1948
  done
haftmann@33274
  1949
haftmann@33274
  1950
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
haftmann@33274
  1951
by (drule_tac m = m in dvd_diff_nat, auto)
haftmann@33274
  1952
haftmann@33274
  1953
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
haftmann@33274
  1954
  unfolding dvd_def
haftmann@33274
  1955
  apply (erule exE)
haftmann@57514
  1956
  apply (simp add: ac_simps)
haftmann@33274
  1957
  done
haftmann@33274
  1958
haftmann@33274
  1959
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
haftmann@33274
  1960
  apply auto
haftmann@33274
  1961
   apply (subgoal_tac "m*n dvd m*1")
haftmann@33274
  1962
   apply (drule dvd_mult_cancel, auto)
haftmann@33274
  1963
  done
haftmann@33274
  1964
haftmann@33274
  1965
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
haftmann@57512
  1966
  apply (subst mult.commute)
haftmann@33274
  1967
  apply (erule dvd_mult_cancel1)
haftmann@33274
  1968
  done
haftmann@33274
  1969
haftmann@33274
  1970
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
haftmann@33274
  1971
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
haftmann@33274
  1972
haftmann@33274
  1973
lemma nat_dvd_not_less:
haftmann@33274
  1974
  fixes m n :: nat
haftmann@33274
  1975
  shows "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"
haftmann@33274
  1976
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
haftmann@33274
  1977
haftmann@54222
  1978
lemma less_eq_dvd_minus:
haftmann@51173
  1979
  fixes m n :: nat
haftmann@54222
  1980
  assumes "m \<le> n"
haftmann@54222
  1981
  shows "m dvd n \<longleftrightarrow> m dvd n - m"
haftmann@51173
  1982
proof -
haftmann@54222
  1983
  from assms have "n = m + (n - m)" by simp
haftmann@51173
  1984
  then obtain q where "n = m + q" ..
haftmann@58647
  1985
  then show ?thesis by (simp add: add.commute [of m])
haftmann@51173
  1986
qed
haftmann@51173
  1987
haftmann@51173
  1988
lemma dvd_minus_self:
haftmann@51173
  1989
  fixes m n :: nat
haftmann@51173
  1990
  shows "m dvd n - m \<longleftrightarrow> n < m \<or> m dvd n"
haftmann@51173
  1991
  by (cases "n < m") (auto elim!: dvdE simp add: not_less le_imp_diff_is_add)
haftmann@51173
  1992
haftmann@51173
  1993
lemma dvd_minus_add:
haftmann@51173
  1994
  fixes m n q r :: nat
haftmann@51173
  1995
  assumes "q \<le> n" "q \<le> r * m"
haftmann@51173
  1996
  shows "m dvd n - q \<longleftrightarrow> m dvd n + (r * m - q)"
haftmann@51173
  1997
proof -
haftmann@51173
  1998
  have "m dvd n - q \<longleftrightarrow> m dvd r * m + (n - q)"
haftmann@58649
  1999
    using dvd_add_times_triv_left_iff [of m r] by simp
wenzelm@53374
  2000
  also from assms have "\<dots> \<longleftrightarrow> m dvd r * m + n - q" by simp
wenzelm@53374
  2001
  also from assms have "\<dots> \<longleftrightarrow> m dvd (r * m - q) + n" by simp
haftmann@57512
  2002
  also have "\<dots> \<longleftrightarrow> m dvd n + (r * m - q)" by (simp add: add.commute)
haftmann@51173
  2003
  finally show ?thesis .
haftmann@51173
  2004
qed
haftmann@51173
  2005
haftmann@33274
  2006
blanchet@58377
  2007
subsection {* Aliases *}
haftmann@44817
  2008
haftmann@44817
  2009
lemma nat_mult_1: "(1::nat) * n = n"
haftmann@58647
  2010
  by (fact mult_1_left)
haftmann@44817
  2011
 
haftmann@44817
  2012
lemma nat_mult_1_right: "n * (1::nat) = n"
haftmann@58647
  2013
  by (fact mult_1_right)
haftmann@58647
  2014
haftmann@44817
  2015
blanchet@58377
  2016
subsection {* Size of a datatype value *}
haftmann@25193
  2017
haftmann@29608
  2018
class size =
krauss@26748
  2019
  fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded} *}
haftmann@23852
  2020
blanchet@58377
  2021
instantiation nat :: size
blanchet@58377
  2022
begin
blanchet@58377
  2023
blanchet@58377
  2024
definition size_nat where
blanchet@58377
  2025
  [simp, code]: "size (n \<Colon> nat) = n"
blanchet@58377
  2026
blanchet@58377
  2027
instance ..
blanchet@58377
  2028
blanchet@58377
  2029
end
blanchet@58377
  2030
blanchet@58377
  2031
blanchet@58377
  2032
subsection {* Code module namespace *}
haftmann@33364
  2033
haftmann@52435
  2034
code_identifier
haftmann@52435
  2035
  code_module Nat \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  2036
huffman@47108
  2037
hide_const (open) of_nat_aux
huffman@47108
  2038
haftmann@25193
  2039
end