src/HOL/Nat.thy
 author blanchet Thu Sep 11 18:54:36 2014 +0200 (2014-09-11) changeset 58306 117ba6cbe414 parent 58189 9d714be4f028 child 58377 c6f93b8d2d8e permissions -rw-r--r--
renamed 'rep_datatype' to 'old_rep_datatype' (HOL)
     1 (*  Title:      HOL/Nat.thy

     2     Author:     Tobias Nipkow and Lawrence C Paulson and Markus Wenzel

     3

     4 Type "nat" is a linear order, and a datatype; arithmetic operators + -

     5 and * (for div and mod, see theory Divides).

     6 *)

     7

     8 header {* Natural numbers *}

     9

    10 theory Nat

    11 imports Inductive Typedef Fun Fields

    12 begin

    13

    14 ML_file "~~/src/Tools/rat.ML"

    15

    16 named_theorems arith "arith facts -- only ground formulas"

    17 ML_file "Tools/arith_data.ML"

    18 ML_file "~~/src/Provers/Arith/fast_lin_arith.ML"

    19

    20

    21 subsection {* Type @{text ind} *}

    22

    23 typedecl ind

    24

    25 axiomatization Zero_Rep :: ind and Suc_Rep :: "ind => ind" where

    26   -- {* the axiom of infinity in 2 parts *}

    27   Suc_Rep_inject:       "Suc_Rep x = Suc_Rep y ==> x = y" and

    28   Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"

    29

    30 subsection {* Type nat *}

    31

    32 text {* Type definition *}

    33

    34 inductive Nat :: "ind \<Rightarrow> bool" where

    35   Zero_RepI: "Nat Zero_Rep"

    36 | Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)"

    37

    38 typedef nat = "{n. Nat n}"

    39   morphisms Rep_Nat Abs_Nat

    40   using Nat.Zero_RepI by auto

    41

    42 lemma Nat_Rep_Nat:

    43   "Nat (Rep_Nat n)"

    44   using Rep_Nat by simp

    45

    46 lemma Nat_Abs_Nat_inverse:

    47   "Nat n \<Longrightarrow> Rep_Nat (Abs_Nat n) = n"

    48   using Abs_Nat_inverse by simp

    49

    50 lemma Nat_Abs_Nat_inject:

    51   "Nat n \<Longrightarrow> Nat m \<Longrightarrow> Abs_Nat n = Abs_Nat m \<longleftrightarrow> n = m"

    52   using Abs_Nat_inject by simp

    53

    54 instantiation nat :: zero

    55 begin

    56

    57 definition Zero_nat_def:

    58   "0 = Abs_Nat Zero_Rep"

    59

    60 instance ..

    61

    62 end

    63

    64 definition Suc :: "nat \<Rightarrow> nat" where

    65   "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"

    66

    67 lemma Suc_not_Zero: "Suc m \<noteq> 0"

    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)

    69

    70 lemma Zero_not_Suc: "0 \<noteq> Suc m"

    71   by (rule not_sym, rule Suc_not_Zero not_sym)

    72

    73 lemma Suc_Rep_inject': "Suc_Rep x = Suc_Rep y \<longleftrightarrow> x = y"

    74   by (rule iffI, rule Suc_Rep_inject) simp_all

    75

    76 lemma nat_induct0:

    77   fixes n

    78   assumes "P 0" and "\<And>n. P n \<Longrightarrow> P (Suc n)"

    79   shows "P n"

    80 using assms

    81 apply (unfold Zero_nat_def Suc_def)

    82 apply (rule Rep_Nat_inverse [THEN subst]) -- {* types force good instantiation *}

    83 apply (erule Nat_Rep_Nat [THEN Nat.induct])

    84 apply (iprover elim: Nat_Abs_Nat_inverse [THEN subst])

    85 done

    86

    87 free_constructors case_nat for

    88     "0 \<Colon> nat"

    89   | Suc pred

    90 where

    91   "pred (0 \<Colon> nat) = (0 \<Colon> nat)"

    92     apply atomize_elim

    93     apply (rename_tac n, induct_tac n rule: nat_induct0, auto)

    94    apply (simp add: Suc_def Nat_Abs_Nat_inject Nat_Rep_Nat Suc_RepI Suc_Rep_inject'

    95      Rep_Nat_inject)

    96   apply (simp only: Suc_not_Zero)

    97   done

    98

    99 -- {* Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}. *}

   100 setup {* Sign.mandatory_path "old" *}

   101

   102 old_rep_datatype "0 \<Colon> nat" Suc

   103   apply (erule nat_induct0, assumption)

   104  apply (rule nat.inject)

   105 apply (rule nat.distinct(1))

   106 done

   107

   108 setup {* Sign.parent_path *}

   109

   110 -- {* But erase the prefix for properties that are not generated by @{text free_constructors}. *}

   111 setup {* Sign.mandatory_path "nat" *}

   112

   113 declare

   114   old.nat.inject[iff del]

   115   old.nat.distinct(1)[simp del, induct_simp del]

   116

   117 lemmas induct = old.nat.induct

   118 lemmas inducts = old.nat.inducts

   119 lemmas rec = old.nat.rec

   120 lemmas simps = nat.inject nat.distinct nat.case nat.rec

   121

   122 setup {* Sign.parent_path *}

   123

   124 abbreviation rec_nat :: "'a \<Rightarrow> (nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a" where

   125   "rec_nat \<equiv> old.rec_nat"

   126

   127 declare nat.sel[code del]

   128

   129 hide_const (open) Nat.pred -- {* hide everything related to the selector *}

   130 hide_fact

   131   nat.case_eq_if

   132   nat.collapse

   133   nat.expand

   134   nat.sel

   135   nat.exhaust_sel

   136   nat.split_sel

   137   nat.split_sel_asm

   138

   139 lemma nat_exhaust [case_names 0 Suc, cases type: nat]:

   140   -- {* for backward compatibility -- names of variables differ *}

   141   "(y = 0 \<Longrightarrow> P) \<Longrightarrow> (\<And>nat. y = Suc nat \<Longrightarrow> P) \<Longrightarrow> P"

   142 by (rule old.nat.exhaust)

   143

   144 lemma nat_induct [case_names 0 Suc, induct type: nat]:

   145   -- {* for backward compatibility -- names of variables differ *}

   146   fixes n

   147   assumes "P 0" and "\<And>n. P n \<Longrightarrow> P (Suc n)"

   148   shows "P n"

   149 using assms by (rule nat.induct)

   150

   151 hide_fact

   152   nat_exhaust

   153   nat_induct0

   154

   155 text {* Injectiveness and distinctness lemmas *}

   156

   157 lemma inj_Suc[simp]: "inj_on Suc N"

   158   by (simp add: inj_on_def)

   159

   160 lemma Suc_neq_Zero: "Suc m = 0 \<Longrightarrow> R"

   161 by (rule notE, rule Suc_not_Zero)

   162

   163 lemma Zero_neq_Suc: "0 = Suc m \<Longrightarrow> R"

   164 by (rule Suc_neq_Zero, erule sym)

   165

   166 lemma Suc_inject: "Suc x = Suc y \<Longrightarrow> x = y"

   167 by (rule inj_Suc [THEN injD])

   168

   169 lemma n_not_Suc_n: "n \<noteq> Suc n"

   170 by (induct n) simp_all

   171

   172 lemma Suc_n_not_n: "Suc n \<noteq> n"

   173 by (rule not_sym, rule n_not_Suc_n)

   174

   175 text {* A special form of induction for reasoning

   176   about @{term "m < n"} and @{term "m - n"} *}

   177

   178 lemma diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==>

   179     (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n"

   180   apply (rule_tac x = m in spec)

   181   apply (induct n)

   182   prefer 2

   183   apply (rule allI)

   184   apply (induct_tac x, iprover+)

   185   done

   186

   187

   188 subsection {* Arithmetic operators *}

   189

   190 instantiation nat :: comm_monoid_diff

   191 begin

   192

   193 primrec plus_nat where

   194   add_0:      "0 + n = (n\<Colon>nat)"

   195 | add_Suc:  "Suc m + n = Suc (m + n)"

   196

   197 lemma add_0_right [simp]: "m + 0 = (m::nat)"

   198   by (induct m) simp_all

   199

   200 lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"

   201   by (induct m) simp_all

   202

   203 declare add_0 [code]

   204

   205 lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"

   206   by simp

   207

   208 primrec minus_nat where

   209   diff_0 [code]: "m - 0 = (m\<Colon>nat)"

   210 | diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"

   211

   212 declare diff_Suc [simp del]

   213

   214 lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"

   215   by (induct n) (simp_all add: diff_Suc)

   216

   217 lemma diff_Suc_Suc [simp, code]: "Suc m - Suc n = m - n"

   218   by (induct n) (simp_all add: diff_Suc)

   219

   220 instance proof

   221   fix n m q :: nat

   222   show "(n + m) + q = n + (m + q)" by (induct n) simp_all

   223   show "n + m = m + n" by (induct n) simp_all

   224   show "0 + n = n" by simp

   225   show "n - 0 = n" by simp

   226   show "0 - n = 0" by simp

   227   show "(q + n) - (q + m) = n - m" by (induct q) simp_all

   228   show "n - m - q = n - (m + q)" by (induct q) (simp_all add: diff_Suc)

   229 qed

   230

   231 end

   232

   233 hide_fact (open) add_0 add_0_right diff_0

   234

   235 instantiation nat :: comm_semiring_1_cancel

   236 begin

   237

   238 definition

   239   One_nat_def [simp]: "1 = Suc 0"

   240

   241 primrec times_nat where

   242   mult_0:     "0 * n = (0\<Colon>nat)"

   243 | mult_Suc: "Suc m * n = n + (m * n)"

   244

   245 lemma mult_0_right [simp]: "(m::nat) * 0 = 0"

   246   by (induct m) simp_all

   247

   248 lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"

   249   by (induct m) (simp_all add: add.left_commute)

   250

   251 lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"

   252   by (induct m) (simp_all add: add.assoc)

   253

   254 instance proof

   255   fix n m q :: nat

   256   show "0 \<noteq> (1::nat)" unfolding One_nat_def by simp

   257   show "1 * n = n" unfolding One_nat_def by simp

   258   show "n * m = m * n" by (induct n) simp_all

   259   show "(n * m) * q = n * (m * q)" by (induct n) (simp_all add: add_mult_distrib)

   260   show "(n + m) * q = n * q + m * q" by (rule add_mult_distrib)

   261   assume "n + m = n + q" thus "m = q" by (induct n) simp_all

   262 qed

   263

   264 end

   265

   266 subsubsection {* Addition *}

   267

   268 lemma nat_add_left_cancel:

   269   fixes k m n :: nat

   270   shows "k + m = k + n \<longleftrightarrow> m = n"

   271   by (fact add_left_cancel)

   272

   273 lemma nat_add_right_cancel:

   274   fixes k m n :: nat

   275   shows "m + k = n + k \<longleftrightarrow> m = n"

   276   by (fact add_right_cancel)

   277

   278 text {* Reasoning about @{text "m + 0 = 0"}, etc. *}

   279

   280 lemma add_is_0 [iff]:

   281   fixes m n :: nat

   282   shows "(m + n = 0) = (m = 0 & n = 0)"

   283   by (cases m) simp_all

   284

   285 lemma add_is_1:

   286   "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"

   287   by (cases m) simp_all

   288

   289 lemma one_is_add:

   290   "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"

   291   by (rule trans, rule eq_commute, rule add_is_1)

   292

   293 lemma add_eq_self_zero:

   294   fixes m n :: nat

   295   shows "m + n = m \<Longrightarrow> n = 0"

   296   by (induct m) simp_all

   297

   298 lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N"

   299   apply (induct k)

   300    apply simp

   301   apply(drule comp_inj_on[OF _ inj_Suc])

   302   apply (simp add:o_def)

   303   done

   304

   305 lemma Suc_eq_plus1: "Suc n = n + 1"

   306   unfolding One_nat_def by simp

   307

   308 lemma Suc_eq_plus1_left: "Suc n = 1 + n"

   309   unfolding One_nat_def by simp

   310

   311

   312 subsubsection {* Difference *}

   313

   314 lemma diff_self_eq_0 [simp]: "(m\<Colon>nat) - m = 0"

   315   by (fact diff_cancel)

   316

   317 lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"

   318   by (fact diff_diff_add)

   319

   320 lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"

   321   by (simp add: diff_diff_left)

   322

   323 lemma diff_commute: "(i::nat) - j - k = i - k - j"

   324   by (fact diff_right_commute)

   325

   326 lemma diff_add_inverse: "(n + m) - n = (m::nat)"

   327   by (fact add_diff_cancel_left')

   328

   329 lemma diff_add_inverse2: "(m + n) - n = (m::nat)"

   330   by (fact add_diff_cancel_right')

   331

   332 lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"

   333   by (fact comm_monoid_diff_class.add_diff_cancel_left)

   334

   335 lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"

   336   by (fact add_diff_cancel_right)

   337

   338 lemma diff_add_0: "n - (n + m) = (0::nat)"

   339   by (fact diff_add_zero)

   340

   341 lemma diff_Suc_1 [simp]: "Suc n - 1 = n"

   342   unfolding One_nat_def by simp

   343

   344 text {* Difference distributes over multiplication *}

   345

   346 lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"

   347 by (induct m n rule: diff_induct) (simp_all add: diff_cancel)

   348

   349 lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"

   350 by (simp add: diff_mult_distrib mult.commute [of k])

   351   -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}

   352

   353

   354 subsubsection {* Multiplication *}

   355

   356 lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"

   357   by (fact distrib_left)

   358

   359 lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"

   360   by (induct m) auto

   361

   362 lemmas nat_distrib =

   363   add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2

   364

   365 lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = Suc 0 & n = Suc 0)"

   366   apply (induct m)

   367    apply simp

   368   apply (induct n)

   369    apply auto

   370   done

   371

   372 lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = Suc 0 & n = Suc 0)"

   373   apply (rule trans)

   374   apply (rule_tac [2] mult_eq_1_iff, fastforce)

   375   done

   376

   377 lemma nat_mult_eq_1_iff [simp]: "m * n = (1::nat) \<longleftrightarrow> m = 1 \<and> n = 1"

   378   unfolding One_nat_def by (rule mult_eq_1_iff)

   379

   380 lemma nat_1_eq_mult_iff [simp]: "(1::nat) = m * n \<longleftrightarrow> m = 1 \<and> n = 1"

   381   unfolding One_nat_def by (rule one_eq_mult_iff)

   382

   383 lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"

   384 proof -

   385   have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"

   386   proof (induct n arbitrary: m)

   387     case 0 then show "m = 0" by simp

   388   next

   389     case (Suc n) then show "m = Suc n"

   390       by (cases m) (simp_all add: eq_commute [of "0"])

   391   qed

   392   then show ?thesis by auto

   393 qed

   394

   395 lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"

   396   by (simp add: mult.commute)

   397

   398 lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"

   399   by (subst mult_cancel1) simp

   400

   401

   402 subsection {* Orders on @{typ nat} *}

   403

   404 subsubsection {* Operation definition *}

   405

   406 instantiation nat :: linorder

   407 begin

   408

   409 primrec less_eq_nat where

   410   "(0\<Colon>nat) \<le> n \<longleftrightarrow> True"

   411 | "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"

   412

   413 declare less_eq_nat.simps [simp del]

   414 lemma le0 [iff]: "0 \<le> (n\<Colon>nat)" by (simp add: less_eq_nat.simps)

   415 lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by simp

   416

   417 definition less_nat where

   418   less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"

   419

   420 lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"

   421   by (simp add: less_eq_nat.simps(2))

   422

   423 lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"

   424   unfolding less_eq_Suc_le ..

   425

   426 lemma le_0_eq [iff]: "(n\<Colon>nat) \<le> 0 \<longleftrightarrow> n = 0"

   427   by (induct n) (simp_all add: less_eq_nat.simps(2))

   428

   429 lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)"

   430   by (simp add: less_eq_Suc_le)

   431

   432 lemma less_nat_zero_code [code]: "n < (0\<Colon>nat) \<longleftrightarrow> False"

   433   by simp

   434

   435 lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"

   436   by (simp add: less_eq_Suc_le)

   437

   438 lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"

   439   by (simp add: less_eq_Suc_le)

   440

   441 lemma Suc_less_eq2: "Suc n < m \<longleftrightarrow> (\<exists>m'. m = Suc m' \<and> n < m')"

   442   by (cases m) auto

   443

   444 lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"

   445   by (induct m arbitrary: n)

   446     (simp_all add: less_eq_nat.simps(2) split: nat.splits)

   447

   448 lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"

   449   by (cases n) (auto intro: le_SucI)

   450

   451 lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"

   452   by (simp add: less_eq_Suc_le) (erule Suc_leD)

   453

   454 lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"

   455   by (simp add: less_eq_Suc_le) (erule Suc_leD)

   456

   457 instance

   458 proof

   459   fix n m :: nat

   460   show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n"

   461   proof (induct n arbitrary: m)

   462     case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le)

   463   next

   464     case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le)

   465   qed

   466 next

   467   fix n :: nat show "n \<le> n" by (induct n) simp_all

   468 next

   469   fix n m :: nat assume "n \<le> m" and "m \<le> n"

   470   then show "n = m"

   471     by (induct n arbitrary: m)

   472       (simp_all add: less_eq_nat.simps(2) split: nat.splits)

   473 next

   474   fix n m q :: nat assume "n \<le> m" and "m \<le> q"

   475   then show "n \<le> q"

   476   proof (induct n arbitrary: m q)

   477     case 0 show ?case by simp

   478   next

   479     case (Suc n) then show ?case

   480       by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,

   481         simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,

   482         simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)

   483   qed

   484 next

   485   fix n m :: nat show "n \<le> m \<or> m \<le> n"

   486     by (induct n arbitrary: m)

   487       (simp_all add: less_eq_nat.simps(2) split: nat.splits)

   488 qed

   489

   490 end

   491

   492 instantiation nat :: order_bot

   493 begin

   494

   495 definition bot_nat :: nat where

   496   "bot_nat = 0"

   497

   498 instance proof

   499 qed (simp add: bot_nat_def)

   500

   501 end

   502

   503 instance nat :: no_top

   504   by default (auto intro: less_Suc_eq_le [THEN iffD2])

   505

   506

   507 subsubsection {* Introduction properties *}

   508

   509 lemma lessI [iff]: "n < Suc n"

   510   by (simp add: less_Suc_eq_le)

   511

   512 lemma zero_less_Suc [iff]: "0 < Suc n"

   513   by (simp add: less_Suc_eq_le)

   514

   515

   516 subsubsection {* Elimination properties *}

   517

   518 lemma less_not_refl: "~ n < (n::nat)"

   519   by (rule order_less_irrefl)

   520

   521 lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)"

   522   by (rule not_sym) (rule less_imp_neq)

   523

   524 lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"

   525   by (rule less_imp_neq)

   526

   527 lemma less_irrefl_nat: "(n::nat) < n ==> R"

   528   by (rule notE, rule less_not_refl)

   529

   530 lemma less_zeroE: "(n::nat) < 0 ==> R"

   531   by (rule notE) (rule not_less0)

   532

   533 lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"

   534   unfolding less_Suc_eq_le le_less ..

   535

   536 lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"

   537   by (simp add: less_Suc_eq)

   538

   539 lemma less_one [iff]: "(n < (1::nat)) = (n = 0)"

   540   unfolding One_nat_def by (rule less_Suc0)

   541

   542 lemma Suc_mono: "m < n ==> Suc m < Suc n"

   543   by simp

   544

   545 text {* "Less than" is antisymmetric, sort of *}

   546 lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"

   547   unfolding not_less less_Suc_eq_le by (rule antisym)

   548

   549 lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"

   550   by (rule linorder_neq_iff)

   551

   552 lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"

   553   and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"

   554   shows "P n m"

   555   apply (rule less_linear [THEN disjE])

   556   apply (erule_tac [2] disjE)

   557   apply (erule lessCase)

   558   apply (erule sym [THEN eqCase])

   559   apply (erule major)

   560   done

   561

   562

   563 subsubsection {* Inductive (?) properties *}

   564

   565 lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"

   566   unfolding less_eq_Suc_le [of m] le_less by simp

   567

   568 lemma lessE:

   569   assumes major: "i < k"

   570   and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"

   571   shows P

   572 proof -

   573   from major have "\<exists>j. i \<le> j \<and> k = Suc j"

   574     unfolding less_eq_Suc_le by (induct k) simp_all

   575   then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"

   576     by (clarsimp simp add: less_le)

   577   with p1 p2 show P by auto

   578 qed

   579

   580 lemma less_SucE: assumes major: "m < Suc n"

   581   and less: "m < n ==> P" and eq: "m = n ==> P" shows P

   582   apply (rule major [THEN lessE])

   583   apply (rule eq, blast)

   584   apply (rule less, blast)

   585   done

   586

   587 lemma Suc_lessE: assumes major: "Suc i < k"

   588   and minor: "!!j. i < j ==> k = Suc j ==> P" shows P

   589   apply (rule major [THEN lessE])

   590   apply (erule lessI [THEN minor])

   591   apply (erule Suc_lessD [THEN minor], assumption)

   592   done

   593

   594 lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"

   595   by simp

   596

   597 lemma less_trans_Suc:

   598   assumes le: "i < j" shows "j < k ==> Suc i < k"

   599   apply (induct k, simp_all)

   600   apply (insert le)

   601   apply (simp add: less_Suc_eq)

   602   apply (blast dest: Suc_lessD)

   603   done

   604

   605 text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}

   606 lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"

   607   unfolding not_less less_Suc_eq_le ..

   608

   609 lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"

   610   unfolding not_le Suc_le_eq ..

   611

   612 text {* Properties of "less than or equal" *}

   613

   614 lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"

   615   unfolding less_Suc_eq_le .

   616

   617 lemma Suc_n_not_le_n: "~ Suc n \<le> n"

   618   unfolding not_le less_Suc_eq_le ..

   619

   620 lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"

   621   by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)

   622

   623 lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"

   624   by (drule le_Suc_eq [THEN iffD1], iprover+)

   625

   626 lemma Suc_leI: "m < n ==> Suc(m) \<le> n"

   627   unfolding Suc_le_eq .

   628

   629 text {* Stronger version of @{text Suc_leD} *}

   630 lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"

   631   unfolding Suc_le_eq .

   632

   633 lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)"

   634   unfolding less_eq_Suc_le by (rule Suc_leD)

   635

   636 text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}

   637 lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq

   638

   639

   640 text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}

   641

   642 lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"

   643   unfolding le_less .

   644

   645 lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"

   646   by (rule le_less)

   647

   648 text {* Useful with @{text blast}. *}

   649 lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"

   650   by auto

   651

   652 lemma le_refl: "n \<le> (n::nat)"

   653   by simp

   654

   655 lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"

   656   by (rule order_trans)

   657

   658 lemma le_antisym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"

   659   by (rule antisym)

   660

   661 lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"

   662   by (rule less_le)

   663

   664 lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"

   665   unfolding less_le ..

   666

   667 lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"

   668   by (rule linear)

   669

   670 lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]

   671

   672 lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"

   673   unfolding less_Suc_eq_le by auto

   674

   675 lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"

   676   unfolding not_less by (rule le_less_Suc_eq)

   677

   678 lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq

   679

   680 lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"

   681 by (cases n) simp_all

   682

   683 lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"

   684 by (cases n) simp_all

   685

   686 lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"

   687 by (cases n) simp_all

   688

   689 lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"

   690 by (cases n) simp_all

   691

   692 text {* This theorem is useful with @{text blast} *}

   693 lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"

   694 by (rule neq0_conv[THEN iffD1], iprover)

   695

   696 lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"

   697 by (fast intro: not0_implies_Suc)

   698

   699 lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"

   700 using neq0_conv by blast

   701

   702 lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"

   703 by (induct m') simp_all

   704

   705 text {* Useful in certain inductive arguments *}

   706 lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"

   707 by (cases m) simp_all

   708

   709

   710 subsubsection {* Monotonicity of Addition *}

   711

   712 lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"

   713 by (simp add: diff_Suc split: nat.split)

   714

   715 lemma Suc_diff_1 [simp]: "0 < n ==> Suc (n - 1) = n"

   716 unfolding One_nat_def by (rule Suc_pred)

   717

   718 lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"

   719 by (induct k) simp_all

   720

   721 lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"

   722 by (induct k) simp_all

   723

   724 lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)"

   725 by(auto dest:gr0_implies_Suc)

   726

   727 text {* strict, in 1st argument *}

   728 lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"

   729 by (induct k) simp_all

   730

   731 text {* strict, in both arguments *}

   732 lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"

   733   apply (rule add_less_mono1 [THEN less_trans], assumption+)

   734   apply (induct j, simp_all)

   735   done

   736

   737 text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}

   738 lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"

   739   apply (induct n)

   740   apply (simp_all add: order_le_less)

   741   apply (blast elim!: less_SucE

   742                intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])

   743   done

   744

   745 lemma le_Suc_ex: "(k::nat) \<le> l \<Longrightarrow> (\<exists>n. l = k + n)"

   746   by (auto simp: less_Suc_eq_le[symmetric] dest: less_imp_Suc_add)

   747

   748 text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}

   749 lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"

   750 apply(auto simp: gr0_conv_Suc)

   751 apply (induct_tac m)

   752 apply (simp_all add: add_less_mono)

   753 done

   754

   755 text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}

   756 instance nat :: linordered_semidom

   757 proof

   758   show "0 < (1::nat)" by simp

   759   show "\<And>m n q :: nat. m \<le> n \<Longrightarrow> q + m \<le> q + n" by simp

   760   show "\<And>m n q :: nat. m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n" by (simp add: mult_less_mono2)

   761 qed

   762

   763 instance nat :: no_zero_divisors

   764 proof

   765   fix a::nat and b::nat show "a ~= 0 \<Longrightarrow> b ~= 0 \<Longrightarrow> a * b ~= 0" by auto

   766 qed

   767

   768

   769 subsubsection {* @{term min} and @{term max} *}

   770

   771 lemma mono_Suc: "mono Suc"

   772 by (rule monoI) simp

   773

   774 lemma min_0L [simp]: "min 0 n = (0::nat)"

   775 by (rule min_absorb1) simp

   776

   777 lemma min_0R [simp]: "min n 0 = (0::nat)"

   778 by (rule min_absorb2) simp

   779

   780 lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"

   781 by (simp add: mono_Suc min_of_mono)

   782

   783 lemma min_Suc1:

   784    "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"

   785 by (simp split: nat.split)

   786

   787 lemma min_Suc2:

   788    "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"

   789 by (simp split: nat.split)

   790

   791 lemma max_0L [simp]: "max 0 n = (n::nat)"

   792 by (rule max_absorb2) simp

   793

   794 lemma max_0R [simp]: "max n 0 = (n::nat)"

   795 by (rule max_absorb1) simp

   796

   797 lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"

   798 by (simp add: mono_Suc max_of_mono)

   799

   800 lemma max_Suc1:

   801    "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"

   802 by (simp split: nat.split)

   803

   804 lemma max_Suc2:

   805    "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"

   806 by (simp split: nat.split)

   807

   808 lemma nat_mult_min_left:

   809   fixes m n q :: nat

   810   shows "min m n * q = min (m * q) (n * q)"

   811   by (simp add: min_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)

   812

   813 lemma nat_mult_min_right:

   814   fixes m n q :: nat

   815   shows "m * min n q = min (m * n) (m * q)"

   816   by (simp add: min_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)

   817

   818 lemma nat_add_max_left:

   819   fixes m n q :: nat

   820   shows "max m n + q = max (m + q) (n + q)"

   821   by (simp add: max_def)

   822

   823 lemma nat_add_max_right:

   824   fixes m n q :: nat

   825   shows "m + max n q = max (m + n) (m + q)"

   826   by (simp add: max_def)

   827

   828 lemma nat_mult_max_left:

   829   fixes m n q :: nat

   830   shows "max m n * q = max (m * q) (n * q)"

   831   by (simp add: max_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)

   832

   833 lemma nat_mult_max_right:

   834   fixes m n q :: nat

   835   shows "m * max n q = max (m * n) (m * q)"

   836   by (simp add: max_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)

   837

   838

   839 subsubsection {* Additional theorems about @{term "op \<le>"} *}

   840

   841 text {* Complete induction, aka course-of-values induction *}

   842

   843 instance nat :: wellorder proof

   844   fix P and n :: nat

   845   assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n"

   846   have "\<And>q. q \<le> n \<Longrightarrow> P q"

   847   proof (induct n)

   848     case (0 n)

   849     have "P 0" by (rule step) auto

   850     thus ?case using 0 by auto

   851   next

   852     case (Suc m n)

   853     then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)

   854     thus ?case

   855     proof

   856       assume "n \<le> m" thus "P n" by (rule Suc(1))

   857     next

   858       assume n: "n = Suc m"

   859       show "P n"

   860         by (rule step) (rule Suc(1), simp add: n le_simps)

   861     qed

   862   qed

   863   then show "P n" by auto

   864 qed

   865

   866

   867 lemma Least_eq_0[simp]: "P(0::nat) \<Longrightarrow> Least P = 0"

   868 by (rule Least_equality[OF _ le0])

   869

   870 lemma Least_Suc:

   871      "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"

   872   apply (cases n, auto)

   873   apply (frule LeastI)

   874   apply (drule_tac P = "%x. P (Suc x) " in LeastI)

   875   apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")

   876   apply (erule_tac [2] Least_le)

   877   apply (cases "LEAST x. P x", auto)

   878   apply (drule_tac P = "%x. P (Suc x) " in Least_le)

   879   apply (blast intro: order_antisym)

   880   done

   881

   882 lemma Least_Suc2:

   883    "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"

   884   apply (erule (1) Least_Suc [THEN ssubst])

   885   apply simp

   886   done

   887

   888 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)"

   889   apply (cases n)

   890    apply blast

   891   apply (rule_tac x="LEAST k. P(k)" in exI)

   892   apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)

   893   done

   894

   895 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)"

   896   unfolding One_nat_def

   897   apply (cases n)

   898    apply blast

   899   apply (frule (1) ex_least_nat_le)

   900   apply (erule exE)

   901   apply (case_tac k)

   902    apply simp

   903   apply (rename_tac k1)

   904   apply (rule_tac x=k1 in exI)

   905   apply (auto simp add: less_eq_Suc_le)

   906   done

   907

   908 lemma nat_less_induct:

   909   assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"

   910   using assms less_induct by blast

   911

   912 lemma measure_induct_rule [case_names less]:

   913   fixes f :: "'a \<Rightarrow> nat"

   914   assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"

   915   shows "P a"

   916 by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step)

   917

   918 text {* old style induction rules: *}

   919 lemma measure_induct:

   920   fixes f :: "'a \<Rightarrow> nat"

   921   shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"

   922   by (rule measure_induct_rule [of f P a]) iprover

   923

   924 lemma full_nat_induct:

   925   assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"

   926   shows "P n"

   927   by (rule less_induct) (auto intro: step simp:le_simps)

   928

   929 text{*An induction rule for estabilishing binary relations*}

   930 lemma less_Suc_induct:

   931   assumes less:  "i < j"

   932      and  step:  "!!i. P i (Suc i)"

   933      and  trans: "!!i j k. i < j ==> j < k ==>  P i j ==> P j k ==> P i k"

   934   shows "P i j"

   935 proof -

   936   from less obtain k where j: "j = Suc (i + k)" by (auto dest: less_imp_Suc_add)

   937   have "P i (Suc (i + k))"

   938   proof (induct k)

   939     case 0

   940     show ?case by (simp add: step)

   941   next

   942     case (Suc k)

   943     have "0 + i < Suc k + i" by (rule add_less_mono1) simp

   944     hence "i < Suc (i + k)" by (simp add: add.commute)

   945     from trans[OF this lessI Suc step]

   946     show ?case by simp

   947   qed

   948   thus "P i j" by (simp add: j)

   949 qed

   950

   951 text {* The method of infinite descent, frequently used in number theory.

   952 Provided by Roelof Oosterhuis.

   953 $P(n)$ is true for all $n\in\mathbb{N}$ if

   954 \begin{itemize}

   955   \item case 0'': given $n=0$ prove $P(n)$,

   956   \item case smaller'': given $n>0$ and $\neg P(n)$ prove there exists

   957         a smaller integer $m$ such that $\neg P(m)$.

   958 \end{itemize} *}

   959

   960 text{* A compact version without explicit base case: *}

   961 lemma infinite_descent:

   962   "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow>  \<exists>m<n. \<not>  P m \<rbrakk> \<Longrightarrow>  P n"

   963 by (induct n rule: less_induct) auto

   964

   965 lemma infinite_descent0[case_names 0 smaller]:

   966   "\<lbrakk> P 0; !!n. n>0 \<Longrightarrow> \<not> P n \<Longrightarrow> (\<exists>m::nat. m < n \<and> \<not>P m) \<rbrakk> \<Longrightarrow> P n"

   967 by (rule infinite_descent) (case_tac "n>0", auto)

   968

   969 text {*

   970 Infinite descent using a mapping to $\mathbb{N}$:

   971 $P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and

   972 \begin{itemize}

   973 \item case 0'': given $V(x)=0$ prove $P(x)$,

   974 \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 $~\neg P(y)$.

   975 \end{itemize}

   976 NB: the proof also shows how to use the previous lemma. *}

   977

   978 corollary infinite_descent0_measure [case_names 0 smaller]:

   979   assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"

   980     and   A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"

   981   shows "P x"

   982 proof -

   983   obtain n where "n = V x" by auto

   984   moreover have "\<And>x. V x = n \<Longrightarrow> P x"

   985   proof (induct n rule: infinite_descent0)

   986     case 0 -- "i.e. $V(x) = 0$"

   987     with A0 show "P x" by auto

   988   next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"

   989     case (smaller n)

   990     then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto

   991     with A1 obtain y where "V y < V x \<and> \<not> P y" by auto

   992     with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto

   993     then show ?case by auto

   994   qed

   995   ultimately show "P x" by auto

   996 qed

   997

   998 text{* Again, without explicit base case: *}

   999 lemma infinite_descent_measure:

  1000 assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"

  1001 proof -

  1002   from assms obtain n where "n = V x" by auto

  1003   moreover have "!!x. V x = n \<Longrightarrow> P x"

  1004   proof (induct n rule: infinite_descent, auto)

  1005     fix x assume "\<not> P x"

  1006     with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto

  1007   qed

  1008   ultimately show "P x" by auto

  1009 qed

  1010

  1011 text {* A [clumsy] way of lifting @{text "<"}

  1012   monotonicity to @{text "\<le>"} monotonicity *}

  1013 lemma less_mono_imp_le_mono:

  1014   "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"

  1015 by (simp add: order_le_less) (blast)

  1016

  1017

  1018 text {* non-strict, in 1st argument *}

  1019 lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"

  1020 by (rule add_right_mono)

  1021

  1022 text {* non-strict, in both arguments *}

  1023 lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"

  1024 by (rule add_mono)

  1025

  1026 lemma le_add2: "n \<le> ((m + n)::nat)"

  1027 by (insert add_right_mono [of 0 m n], simp)

  1028

  1029 lemma le_add1: "n \<le> ((n + m)::nat)"

  1030 by (simp add: add.commute, rule le_add2)

  1031

  1032 lemma less_add_Suc1: "i < Suc (i + m)"

  1033 by (rule le_less_trans, rule le_add1, rule lessI)

  1034

  1035 lemma less_add_Suc2: "i < Suc (m + i)"

  1036 by (rule le_less_trans, rule le_add2, rule lessI)

  1037

  1038 lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"

  1039 by (iprover intro!: less_add_Suc1 less_imp_Suc_add)

  1040

  1041 lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"

  1042 by (rule le_trans, assumption, rule le_add1)

  1043

  1044 lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"

  1045 by (rule le_trans, assumption, rule le_add2)

  1046

  1047 lemma trans_less_add1: "(i::nat) < j ==> i < j + m"

  1048 by (rule less_le_trans, assumption, rule le_add1)

  1049

  1050 lemma trans_less_add2: "(i::nat) < j ==> i < m + j"

  1051 by (rule less_le_trans, assumption, rule le_add2)

  1052

  1053 lemma add_lessD1: "i + j < (k::nat) ==> i < k"

  1054 apply (rule le_less_trans [of _ "i+j"])

  1055 apply (simp_all add: le_add1)

  1056 done

  1057

  1058 lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"

  1059 apply (rule notI)

  1060 apply (drule add_lessD1)

  1061 apply (erule less_irrefl [THEN notE])

  1062 done

  1063

  1064 lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"

  1065 by (simp add: add.commute)

  1066

  1067 lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"

  1068 apply (rule order_trans [of _ "m+k"])

  1069 apply (simp_all add: le_add1)

  1070 done

  1071

  1072 lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"

  1073 apply (simp add: add.commute)

  1074 apply (erule add_leD1)

  1075 done

  1076

  1077 lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"

  1078 by (blast dest: add_leD1 add_leD2)

  1079

  1080 text {* needs @{text "!!k"} for @{text ac_simps} to work *}

  1081 lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"

  1082 by (force simp del: add_Suc_right

  1083     simp add: less_iff_Suc_add add_Suc_right [symmetric] ac_simps)

  1084

  1085

  1086 subsubsection {* More results about difference *}

  1087

  1088 text {* Addition is the inverse of subtraction:

  1089   if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}

  1090 lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"

  1091 by (induct m n rule: diff_induct) simp_all

  1092

  1093 lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"

  1094 by (simp add: add_diff_inverse linorder_not_less)

  1095

  1096 lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"

  1097 by (simp add: add.commute)

  1098

  1099 lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"

  1100 by (induct m n rule: diff_induct) simp_all

  1101

  1102 lemma diff_less_Suc: "m - n < Suc m"

  1103 apply (induct m n rule: diff_induct)

  1104 apply (erule_tac [3] less_SucE)

  1105 apply (simp_all add: less_Suc_eq)

  1106 done

  1107

  1108 lemma diff_le_self [simp]: "m - n \<le> (m::nat)"

  1109 by (induct m n rule: diff_induct) (simp_all add: le_SucI)

  1110

  1111 lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"

  1112   by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])

  1113

  1114 instance nat :: ordered_cancel_comm_monoid_diff

  1115 proof

  1116   show "\<And>m n :: nat. m \<le> n \<longleftrightarrow> (\<exists>q. n = m + q)" by (fact le_iff_add)

  1117 qed

  1118

  1119 lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"

  1120 by (rule le_less_trans, rule diff_le_self)

  1121

  1122 lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"

  1123 by (cases n) (auto simp add: le_simps)

  1124

  1125 lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"

  1126 by (induct j k rule: diff_induct) simp_all

  1127

  1128 lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"

  1129 by (simp add: add.commute diff_add_assoc)

  1130

  1131 lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"

  1132 by (auto simp add: diff_add_inverse2)

  1133

  1134 lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"

  1135 by (induct m n rule: diff_induct) simp_all

  1136

  1137 lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"

  1138 by (rule iffD2, rule diff_is_0_eq)

  1139

  1140 lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"

  1141 by (induct m n rule: diff_induct) simp_all

  1142

  1143 lemma less_imp_add_positive:

  1144   assumes "i < j"

  1145   shows "\<exists>k::nat. 0 < k & i + k = j"

  1146 proof

  1147   from assms show "0 < j - i & i + (j - i) = j"

  1148     by (simp add: order_less_imp_le)

  1149 qed

  1150

  1151 text {* a nice rewrite for bounded subtraction *}

  1152 lemma nat_minus_add_max:

  1153   fixes n m :: nat

  1154   shows "n - m + m = max n m"

  1155     by (simp add: max_def not_le order_less_imp_le)

  1156

  1157 lemma nat_diff_split:

  1158   "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"

  1159     -- {* elimination of @{text -} on @{text nat} *}

  1160 by (cases "a < b")

  1161   (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse

  1162     not_less le_less dest!: add_eq_self_zero add_eq_self_zero[OF sym])

  1163

  1164 lemma nat_diff_split_asm:

  1165   "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"

  1166     -- {* elimination of @{text -} on @{text nat} in assumptions *}

  1167 by (auto split: nat_diff_split)

  1168

  1169 lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"

  1170   by simp

  1171

  1172 lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"

  1173   unfolding One_nat_def by (cases m) simp_all

  1174

  1175 lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"

  1176   unfolding One_nat_def by (cases m) simp_all

  1177

  1178 lemma Suc_diff_eq_diff_pred: "0 < n ==> Suc m - n = m - (n - 1)"

  1179   unfolding One_nat_def by (cases n) simp_all

  1180

  1181 lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"

  1182   unfolding One_nat_def by (cases m) simp_all

  1183

  1184 lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"

  1185   by (fact Let_def)

  1186

  1187

  1188 subsubsection {* Monotonicity of Multiplication *}

  1189

  1190 lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"

  1191 by (simp add: mult_right_mono)

  1192

  1193 lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"

  1194 by (simp add: mult_left_mono)

  1195

  1196 text {* @{text "\<le>"} monotonicity, BOTH arguments *}

  1197 lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"

  1198 by (simp add: mult_mono)

  1199

  1200 lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"

  1201 by (simp add: mult_strict_right_mono)

  1202

  1203 text{*Differs from the standard @{text zero_less_mult_iff} in that

  1204       there are no negative numbers.*}

  1205 lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"

  1206   apply (induct m)

  1207    apply simp

  1208   apply (case_tac n)

  1209    apply simp_all

  1210   done

  1211

  1212 lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)"

  1213   apply (induct m)

  1214    apply simp

  1215   apply (case_tac n)

  1216    apply simp_all

  1217   done

  1218

  1219 lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"

  1220   apply (safe intro!: mult_less_mono1)

  1221   apply (cases k, auto)

  1222   apply (simp del: le_0_eq add: linorder_not_le [symmetric])

  1223   apply (blast intro: mult_le_mono1)

  1224   done

  1225

  1226 lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"

  1227 by (simp add: mult.commute [of k])

  1228

  1229 lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"

  1230 by (simp add: linorder_not_less [symmetric], auto)

  1231

  1232 lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"

  1233 by (simp add: linorder_not_less [symmetric], auto)

  1234

  1235 lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"

  1236 by (subst mult_less_cancel1) simp

  1237

  1238 lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"

  1239 by (subst mult_le_cancel1) simp

  1240

  1241 lemma le_square: "m \<le> m * (m::nat)"

  1242   by (cases m) (auto intro: le_add1)

  1243

  1244 lemma le_cube: "(m::nat) \<le> m * (m * m)"

  1245   by (cases m) (auto intro: le_add1)

  1246

  1247 text {* Lemma for @{text gcd} *}

  1248 lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"

  1249   apply (drule sym)

  1250   apply (rule disjCI)

  1251   apply (rule nat_less_cases, erule_tac [2] _)

  1252    apply (drule_tac [2] mult_less_mono2)

  1253     apply (auto)

  1254   done

  1255

  1256 lemma mono_times_nat:

  1257   fixes n :: nat

  1258   assumes "n > 0"

  1259   shows "mono (times n)"

  1260 proof

  1261   fix m q :: nat

  1262   assume "m \<le> q"

  1263   with assms show "n * m \<le> n * q" by simp

  1264 qed

  1265

  1266 text {* the lattice order on @{typ nat} *}

  1267

  1268 instantiation nat :: distrib_lattice

  1269 begin

  1270

  1271 definition

  1272   "(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min"

  1273

  1274 definition

  1275   "(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max"

  1276

  1277 instance by intro_classes

  1278   (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def

  1279     intro: order_less_imp_le antisym elim!: order_trans order_less_trans)

  1280

  1281 end

  1282

  1283

  1284 subsection {* Natural operation of natural numbers on functions *}

  1285

  1286 text {*

  1287   We use the same logical constant for the power operations on

  1288   functions and relations, in order to share the same syntax.

  1289 *}

  1290

  1291 consts compow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"

  1292

  1293 abbreviation compower :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^^" 80) where

  1294   "f ^^ n \<equiv> compow n f"

  1295

  1296 notation (latex output)

  1297   compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)

  1298

  1299 notation (HTML output)

  1300   compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)

  1301

  1302 text {* @{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f} *}

  1303

  1304 overloading

  1305   funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"

  1306 begin

  1307

  1308 primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where

  1309   "funpow 0 f = id"

  1310 | "funpow (Suc n) f = f o funpow n f"

  1311

  1312 end

  1313

  1314 lemma funpow_Suc_right:

  1315   "f ^^ Suc n = f ^^ n \<circ> f"

  1316 proof (induct n)

  1317   case 0 then show ?case by simp

  1318 next

  1319   fix n

  1320   assume "f ^^ Suc n = f ^^ n \<circ> f"

  1321   then show "f ^^ Suc (Suc n) = f ^^ Suc n \<circ> f"

  1322     by (simp add: o_assoc)

  1323 qed

  1324

  1325 lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right

  1326

  1327 text {* for code generation *}

  1328

  1329 definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where

  1330   funpow_code_def [code_abbrev]: "funpow = compow"

  1331

  1332 lemma [code]:

  1333   "funpow (Suc n) f = f o funpow n f"

  1334   "funpow 0 f = id"

  1335   by (simp_all add: funpow_code_def)

  1336

  1337 hide_const (open) funpow

  1338

  1339 lemma funpow_add:

  1340   "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"

  1341   by (induct m) simp_all

  1342

  1343 lemma funpow_mult:

  1344   fixes f :: "'a \<Rightarrow> 'a"

  1345   shows "(f ^^ m) ^^ n = f ^^ (m * n)"

  1346   by (induct n) (simp_all add: funpow_add)

  1347

  1348 lemma funpow_swap1:

  1349   "f ((f ^^ n) x) = (f ^^ n) (f x)"

  1350 proof -

  1351   have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp

  1352   also have "\<dots>  = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)

  1353   also have "\<dots> = (f ^^ n) (f x)" by simp

  1354   finally show ?thesis .

  1355 qed

  1356

  1357 lemma comp_funpow:

  1358   fixes f :: "'a \<Rightarrow> 'a"

  1359   shows "comp f ^^ n = comp (f ^^ n)"

  1360   by (induct n) simp_all

  1361

  1362 lemma Suc_funpow[simp]: "Suc ^^ n = (op + n)"

  1363   by (induct n) simp_all

  1364

  1365 lemma id_funpow[simp]: "id ^^ n = id"

  1366   by (induct n) simp_all

  1367

  1368 subsection {* Kleene iteration *}

  1369

  1370 lemma Kleene_iter_lpfp:

  1371 assumes "mono f" and "f p \<le> p" shows "(f^^k) (bot::'a::order_bot) \<le> p"

  1372 proof(induction k)

  1373   case 0 show ?case by simp

  1374 next

  1375   case Suc

  1376   from monoD[OF assms(1) Suc] assms(2)

  1377   show ?case by simp

  1378 qed

  1379

  1380 lemma lfp_Kleene_iter: assumes "mono f" and "(f^^Suc k) bot = (f^^k) bot"

  1381 shows "lfp f = (f^^k) bot"

  1382 proof(rule antisym)

  1383   show "lfp f \<le> (f^^k) bot"

  1384   proof(rule lfp_lowerbound)

  1385     show "f ((f^^k) bot) \<le> (f^^k) bot" using assms(2) by simp

  1386   qed

  1387 next

  1388   show "(f^^k) bot \<le> lfp f"

  1389     using Kleene_iter_lpfp[OF assms(1)] lfp_unfold[OF assms(1)] by simp

  1390 qed

  1391

  1392

  1393 subsection {* Embedding of the Naturals into any @{text semiring_1}: @{term of_nat} *}

  1394

  1395 context semiring_1

  1396 begin

  1397

  1398 definition of_nat :: "nat \<Rightarrow> 'a" where

  1399   "of_nat n = (plus 1 ^^ n) 0"

  1400

  1401 lemma of_nat_simps [simp]:

  1402   shows of_nat_0: "of_nat 0 = 0"

  1403     and of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"

  1404   by (simp_all add: of_nat_def)

  1405

  1406 lemma of_nat_1 [simp]: "of_nat 1 = 1"

  1407   by (simp add: of_nat_def)

  1408

  1409 lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"

  1410   by (induct m) (simp_all add: ac_simps)

  1411

  1412 lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"

  1413   by (induct m) (simp_all add: ac_simps distrib_right)

  1414

  1415 primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where

  1416   "of_nat_aux inc 0 i = i"

  1417 | "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- {* tail recursive *}

  1418

  1419 lemma of_nat_code:

  1420   "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"

  1421 proof (induct n)

  1422   case 0 then show ?case by simp

  1423 next

  1424   case (Suc n)

  1425   have "\<And>i. of_nat_aux (\<lambda>i. i + 1) n (i + 1) = of_nat_aux (\<lambda>i. i + 1) n i + 1"

  1426     by (induct n) simp_all

  1427   from this [of 0] have "of_nat_aux (\<lambda>i. i + 1) n 1 = of_nat_aux (\<lambda>i. i + 1) n 0 + 1"

  1428     by simp

  1429   with Suc show ?case by (simp add: add.commute)

  1430 qed

  1431

  1432 end

  1433

  1434 declare of_nat_code [code]

  1435

  1436 text{*Class for unital semirings with characteristic zero.

  1437  Includes non-ordered rings like the complex numbers.*}

  1438

  1439 class semiring_char_0 = semiring_1 +

  1440   assumes inj_of_nat: "inj of_nat"

  1441 begin

  1442

  1443 lemma of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"

  1444   by (auto intro: inj_of_nat injD)

  1445

  1446 text{*Special cases where either operand is zero*}

  1447

  1448 lemma of_nat_0_eq_iff [simp]: "0 = of_nat n \<longleftrightarrow> 0 = n"

  1449   by (fact of_nat_eq_iff [of 0 n, unfolded of_nat_0])

  1450

  1451 lemma of_nat_eq_0_iff [simp]: "of_nat m = 0 \<longleftrightarrow> m = 0"

  1452   by (fact of_nat_eq_iff [of m 0, unfolded of_nat_0])

  1453

  1454 end

  1455

  1456 context linordered_semidom

  1457 begin

  1458

  1459 lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"

  1460   by (induct n) simp_all

  1461

  1462 lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"

  1463   by (simp add: not_less)

  1464

  1465 lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"

  1466   by (induct m n rule: diff_induct, simp_all add: add_pos_nonneg)

  1467

  1468 lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"

  1469   by (simp add: not_less [symmetric] linorder_not_less [symmetric])

  1470

  1471 lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"

  1472   by simp

  1473

  1474 lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"

  1475   by simp

  1476

  1477 text{*Every @{text linordered_semidom} has characteristic zero.*}

  1478

  1479 subclass semiring_char_0 proof

  1480 qed (auto intro!: injI simp add: eq_iff)

  1481

  1482 text{*Special cases where either operand is zero*}

  1483

  1484 lemma of_nat_le_0_iff [simp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"

  1485   by (rule of_nat_le_iff [of _ 0, simplified])

  1486

  1487 lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"

  1488   by (rule of_nat_less_iff [of 0, simplified])

  1489

  1490 end

  1491

  1492 context ring_1

  1493 begin

  1494

  1495 lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"

  1496 by (simp add: algebra_simps of_nat_add [symmetric])

  1497

  1498 end

  1499

  1500 context linordered_idom

  1501 begin

  1502

  1503 lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"

  1504   unfolding abs_if by auto

  1505

  1506 end

  1507

  1508 lemma of_nat_id [simp]: "of_nat n = n"

  1509   by (induct n) simp_all

  1510

  1511 lemma of_nat_eq_id [simp]: "of_nat = id"

  1512   by (auto simp add: fun_eq_iff)

  1513

  1514

  1515 subsection {* The Set of Natural Numbers *}

  1516

  1517 context semiring_1

  1518 begin

  1519

  1520 definition Nats  :: "'a set" where

  1521   "Nats = range of_nat"

  1522

  1523 notation (xsymbols)

  1524   Nats  ("\<nat>")

  1525

  1526 lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"

  1527   by (simp add: Nats_def)

  1528

  1529 lemma Nats_0 [simp]: "0 \<in> \<nat>"

  1530 apply (simp add: Nats_def)

  1531 apply (rule range_eqI)

  1532 apply (rule of_nat_0 [symmetric])

  1533 done

  1534

  1535 lemma Nats_1 [simp]: "1 \<in> \<nat>"

  1536 apply (simp add: Nats_def)

  1537 apply (rule range_eqI)

  1538 apply (rule of_nat_1 [symmetric])

  1539 done

  1540

  1541 lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"

  1542 apply (auto simp add: Nats_def)

  1543 apply (rule range_eqI)

  1544 apply (rule of_nat_add [symmetric])

  1545 done

  1546

  1547 lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"

  1548 apply (auto simp add: Nats_def)

  1549 apply (rule range_eqI)

  1550 apply (rule of_nat_mult [symmetric])

  1551 done

  1552

  1553 lemma Nats_cases [cases set: Nats]:

  1554   assumes "x \<in> \<nat>"

  1555   obtains (of_nat) n where "x = of_nat n"

  1556   unfolding Nats_def

  1557 proof -

  1558   from x \<in> \<nat> have "x \<in> range of_nat" unfolding Nats_def .

  1559   then obtain n where "x = of_nat n" ..

  1560   then show thesis ..

  1561 qed

  1562

  1563 lemma Nats_induct [case_names of_nat, induct set: Nats]:

  1564   "x \<in> \<nat> \<Longrightarrow> (\<And>n. P (of_nat n)) \<Longrightarrow> P x"

  1565   by (rule Nats_cases) auto

  1566

  1567 end

  1568

  1569

  1570 subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}

  1571

  1572 lemma subst_equals:

  1573   assumes 1: "t = s" and 2: "u = t"

  1574   shows "u = s"

  1575   using 2 1 by (rule trans)

  1576

  1577 setup Arith_Data.setup

  1578

  1579 ML_file "Tools/nat_arith.ML"

  1580

  1581 simproc_setup nateq_cancel_sums

  1582   ("(l::nat) + m = n" | "(l::nat) = m + n" | "Suc m = n" | "m = Suc n") =

  1583   {* fn phi => try o Nat_Arith.cancel_eq_conv *}

  1584

  1585 simproc_setup natless_cancel_sums

  1586   ("(l::nat) + m < n" | "(l::nat) < m + n" | "Suc m < n" | "m < Suc n") =

  1587   {* fn phi => try o Nat_Arith.cancel_less_conv *}

  1588

  1589 simproc_setup natle_cancel_sums

  1590   ("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" | "Suc m \<le> n" | "m \<le> Suc n") =

  1591   {* fn phi => try o Nat_Arith.cancel_le_conv *}

  1592

  1593 simproc_setup natdiff_cancel_sums

  1594   ("(l::nat) + m - n" | "(l::nat) - (m + n)" | "Suc m - n" | "m - Suc n") =

  1595   {* fn phi => try o Nat_Arith.cancel_diff_conv *}

  1596

  1597 ML_file "Tools/lin_arith.ML"

  1598 setup {* Lin_Arith.global_setup *}

  1599 declaration {* K Lin_Arith.setup *}

  1600

  1601 simproc_setup fast_arith_nat ("(m::nat) < n" | "(m::nat) <= n" | "(m::nat) = n") =

  1602   {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}

  1603 (* Because of this simproc, the arithmetic solver is really only

  1604 useful to detect inconsistencies among the premises for subgoals which are

  1605 *not* themselves (in)equalities, because the latter activate

  1606 fast_nat_arith_simproc anyway. However, it seems cheaper to activate the

  1607 solver all the time rather than add the additional check. *)

  1608

  1609

  1610 lemmas [arith_split] = nat_diff_split split_min split_max

  1611

  1612 context order

  1613 begin

  1614

  1615 lemma lift_Suc_mono_le:

  1616   assumes mono: "\<And>n. f n \<le> f (Suc n)" and "n \<le> n'"

  1617   shows "f n \<le> f n'"

  1618 proof (cases "n < n'")

  1619   case True

  1620   then show ?thesis

  1621     by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)

  1622 qed (insert n \<le> n', auto) -- {* trivial for @{prop "n = n'"} *}

  1623

  1624 lemma lift_Suc_antimono_le:

  1625   assumes mono: "\<And>n. f n \<ge> f (Suc n)" and "n \<le> n'"

  1626   shows "f n \<ge> f n'"

  1627 proof (cases "n < n'")

  1628   case True

  1629   then show ?thesis

  1630     by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)

  1631 qed (insert n \<le> n', auto) -- {* trivial for @{prop "n = n'"} *}

  1632

  1633 lemma lift_Suc_mono_less:

  1634   assumes mono: "\<And>n. f n < f (Suc n)" and "n < n'"

  1635   shows "f n < f n'"

  1636 using n < n'

  1637 by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)

  1638

  1639 lemma lift_Suc_mono_less_iff:

  1640   "(\<And>n. f n < f (Suc n)) \<Longrightarrow> f n < f m \<longleftrightarrow> n < m"

  1641   by (blast intro: less_asym' lift_Suc_mono_less [of f]

  1642     dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq [THEN iffD1])

  1643

  1644 end

  1645

  1646 lemma mono_iff_le_Suc:

  1647   "mono f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"

  1648   unfolding mono_def by (auto intro: lift_Suc_mono_le [of f])

  1649

  1650 lemma antimono_iff_le_Suc:

  1651   "antimono f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"

  1652   unfolding antimono_def by (auto intro: lift_Suc_antimono_le [of f])

  1653

  1654 lemma mono_nat_linear_lb:

  1655   fixes f :: "nat \<Rightarrow> nat"

  1656   assumes "\<And>m n. m < n \<Longrightarrow> f m < f n"

  1657   shows "f m + k \<le> f (m + k)"

  1658 proof (induct k)

  1659   case 0 then show ?case by simp

  1660 next

  1661   case (Suc k)

  1662   then have "Suc (f m + k) \<le> Suc (f (m + k))" by simp

  1663   also from assms [of "m + k" "Suc (m + k)"] have "Suc (f (m + k)) \<le> f (Suc (m + k))"

  1664     by (simp add: Suc_le_eq)

  1665   finally show ?case by simp

  1666 qed

  1667

  1668

  1669 text{*Subtraction laws, mostly by Clemens Ballarin*}

  1670

  1671 lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"

  1672 by arith

  1673

  1674 lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"

  1675 by arith

  1676

  1677 lemma less_diff_conv2:

  1678   fixes j k i :: nat

  1679   assumes "k \<le> j"

  1680   shows "j - k < i \<longleftrightarrow> j < i + k"

  1681   using assms by arith

  1682

  1683 lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"

  1684 by arith

  1685

  1686 lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"

  1687   by (fact le_diff_conv2) -- {* FIXME delete *}

  1688

  1689 lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"

  1690 by arith

  1691

  1692 lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"

  1693   by (fact le_add_diff) -- {* FIXME delete *}

  1694

  1695 (*Replaces the previous diff_less and le_diff_less, which had the stronger

  1696   second premise n\<le>m*)

  1697 lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"

  1698 by arith

  1699

  1700 text {* Simplification of relational expressions involving subtraction *}

  1701

  1702 lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"

  1703 by (simp split add: nat_diff_split)

  1704

  1705 hide_fact (open) diff_diff_eq

  1706

  1707 lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"

  1708 by (auto split add: nat_diff_split)

  1709

  1710 lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"

  1711 by (auto split add: nat_diff_split)

  1712

  1713 lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"

  1714 by (auto split add: nat_diff_split)

  1715

  1716 text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}

  1717

  1718 (* Monotonicity of subtraction in first argument *)

  1719 lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"

  1720 by (simp split add: nat_diff_split)

  1721

  1722 lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"

  1723 by (simp split add: nat_diff_split)

  1724

  1725 lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"

  1726 by (simp split add: nat_diff_split)

  1727

  1728 lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"

  1729 by (simp split add: nat_diff_split)

  1730

  1731 lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"

  1732 by auto

  1733

  1734 lemma inj_on_diff_nat:

  1735   assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"

  1736   shows "inj_on (\<lambda>n. n - k) N"

  1737 proof (rule inj_onI)

  1738   fix x y

  1739   assume a: "x \<in> N" "y \<in> N" "x - k = y - k"

  1740   with k_le_n have "x - k + k = y - k + k" by auto

  1741   with a k_le_n show "x = y" by auto

  1742 qed

  1743

  1744 text{*Rewriting to pull differences out*}

  1745

  1746 lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"

  1747 by arith

  1748

  1749 lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"

  1750 by arith

  1751

  1752 lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"

  1753 by arith

  1754

  1755 lemma Suc_diff_Suc: "n < m \<Longrightarrow> Suc (m - Suc n) = m - n"

  1756 by simp

  1757

  1758 (*The others are

  1759       i - j - k = i - (j + k),

  1760       k \<le> j ==> j - k + i = j + i - k,

  1761       k \<le> j ==> i + (j - k) = i + j - k *)

  1762 lemmas add_diff_assoc = diff_add_assoc [symmetric]

  1763 lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]

  1764 declare diff_diff_left [simp]  add_diff_assoc [simp] add_diff_assoc2[simp]

  1765

  1766 text{*At present we prove no analogue of @{text not_less_Least} or @{text

  1767 Least_Suc}, since there appears to be no need.*}

  1768

  1769 text{*Lemmas for ex/Factorization*}

  1770

  1771 lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"

  1772 by (cases m) auto

  1773

  1774 lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"

  1775 by (cases m) auto

  1776

  1777 lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"

  1778 by (cases m) auto

  1779

  1780 text {* Specialized induction principles that work "backwards": *}

  1781

  1782 lemma inc_induct[consumes 1, case_names base step]:

  1783   assumes less: "i \<le> j"

  1784   assumes base: "P j"

  1785   assumes step: "\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P (Suc n) \<Longrightarrow> P n"

  1786   shows "P i"

  1787   using less step

  1788 proof (induct d\<equiv>"j - i" arbitrary: i)

  1789   case (0 i)

  1790   hence "i = j" by simp

  1791   with base show ?case by simp

  1792 next

  1793   case (Suc d n)

  1794   hence "n \<le> n" "n < j" "P (Suc n)"

  1795     by simp_all

  1796   then show "P n" by fact

  1797 qed

  1798

  1799 lemma strict_inc_induct[consumes 1, case_names base step]:

  1800   assumes less: "i < j"

  1801   assumes base: "!!i. j = Suc i ==> P i"

  1802   assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"

  1803   shows "P i"

  1804   using less

  1805 proof (induct d=="j - i - 1" arbitrary: i)

  1806   case (0 i)

  1807   with i < j have "j = Suc i" by simp

  1808   with base show ?case by simp

  1809 next

  1810   case (Suc d i)

  1811   hence "i < j" "P (Suc i)"

  1812     by simp_all

  1813   thus "P i" by (rule step)

  1814 qed

  1815

  1816 lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"

  1817   using inc_induct[of "k - i" k P, simplified] by blast

  1818

  1819 lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"

  1820   using inc_induct[of 0 k P] by blast

  1821

  1822 text {* Further induction rule similar to @{thm inc_induct} *}

  1823

  1824 lemma dec_induct[consumes 1, case_names base step]:

  1825   "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"

  1826   by (induct j arbitrary: i) (auto simp: le_Suc_eq)

  1827

  1828 subsection {* The divides relation on @{typ nat} *}

  1829

  1830 lemma dvd_1_left [iff]: "Suc 0 dvd k"

  1831 unfolding dvd_def by simp

  1832

  1833 lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"

  1834 by (simp add: dvd_def)

  1835

  1836 lemma nat_dvd_1_iff_1 [simp]: "m dvd (1::nat) \<longleftrightarrow> m = 1"

  1837 by (simp add: dvd_def)

  1838

  1839 lemma dvd_antisym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"

  1840   unfolding dvd_def

  1841   by (force dest: mult_eq_self_implies_10 simp add: mult.assoc)

  1842

  1843 text {* @{term "op dvd"} is a partial order *}

  1844

  1845 interpretation dvd: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n"

  1846   proof qed (auto intro: dvd_refl dvd_trans dvd_antisym)

  1847

  1848 lemma dvd_diff_nat[simp]: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"

  1849 unfolding dvd_def

  1850 by (blast intro: diff_mult_distrib2 [symmetric])

  1851

  1852 lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"

  1853   apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])

  1854   apply (blast intro: dvd_add)

  1855   done

  1856

  1857 lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"

  1858 by (drule_tac m = m in dvd_diff_nat, auto)

  1859

  1860 lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"

  1861   apply (rule iffI)

  1862    apply (erule_tac [2] dvd_add)

  1863    apply (rule_tac [2] dvd_refl)

  1864   apply (subgoal_tac "n = (n+k) -k")

  1865    prefer 2 apply simp

  1866   apply (erule ssubst)

  1867   apply (erule dvd_diff_nat)

  1868   apply (rule dvd_refl)

  1869   done

  1870

  1871 lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"

  1872   unfolding dvd_def

  1873   apply (erule exE)

  1874   apply (simp add: ac_simps)

  1875   done

  1876

  1877 lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"

  1878   apply auto

  1879    apply (subgoal_tac "m*n dvd m*1")

  1880    apply (drule dvd_mult_cancel, auto)

  1881   done

  1882

  1883 lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"

  1884   apply (subst mult.commute)

  1885   apply (erule dvd_mult_cancel1)

  1886   done

  1887

  1888 lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"

  1889 by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)

  1890

  1891 lemma nat_dvd_not_less:

  1892   fixes m n :: nat

  1893   shows "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"

  1894 by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)

  1895

  1896 lemma dvd_plusE:

  1897   fixes m n q :: nat

  1898   assumes "m dvd n + q" "m dvd n"

  1899   obtains "m dvd q"

  1900 proof (cases "m = 0")

  1901   case True with assms that show thesis by simp

  1902 next

  1903   case False then have "m > 0" by simp

  1904   from assms obtain r s where "n = m * r" and "n + q = m * s" by (blast elim: dvdE)

  1905   then have *: "m * r + q = m * s" by simp

  1906   show thesis proof (cases "r \<le> s")

  1907     case False then have "s < r" by (simp add: not_le)

  1908     with * have "m * r + q - m * s = m * s - m * s" by simp

  1909     then have "m * r + q - m * s = 0" by simp

  1910     with m > 0 s < r have "m * r - m * s + q = 0" by (unfold less_le_not_le) auto

  1911     then have "m * (r - s) + q = 0" by auto

  1912     then have "m * (r - s) = 0" by simp

  1913     then have "m = 0 \<or> r - s = 0" by simp

  1914     with s < r have "m = 0" by (simp add: less_le_not_le)

  1915     with m > 0 show thesis by auto

  1916   next

  1917     case True with * have "m * r + q - m * r = m * s - m * r" by simp

  1918     with m > 0 r \<le> s have "m * r - m * r + q = m * s - m * r" by simp

  1919     then have "q = m * (s - r)" by (simp add: diff_mult_distrib2)

  1920     with assms that show thesis by (auto intro: dvdI)

  1921   qed

  1922 qed

  1923

  1924 lemma dvd_plus_eq_right:

  1925   fixes m n q :: nat

  1926   assumes "m dvd n"

  1927   shows "m dvd n + q \<longleftrightarrow> m dvd q"

  1928   using assms by (auto elim: dvd_plusE)

  1929

  1930 lemma dvd_plus_eq_left:

  1931   fixes m n q :: nat

  1932   assumes "m dvd q"

  1933   shows "m dvd n + q \<longleftrightarrow> m dvd n"

  1934   using assms by (simp add: dvd_plus_eq_right add.commute [of n])

  1935

  1936 lemma less_eq_dvd_minus:

  1937   fixes m n :: nat

  1938   assumes "m \<le> n"

  1939   shows "m dvd n \<longleftrightarrow> m dvd n - m"

  1940 proof -

  1941   from assms have "n = m + (n - m)" by simp

  1942   then obtain q where "n = m + q" ..

  1943   then show ?thesis by (simp add: dvd_reduce add.commute [of m])

  1944 qed

  1945

  1946 lemma dvd_minus_self:

  1947   fixes m n :: nat

  1948   shows "m dvd n - m \<longleftrightarrow> n < m \<or> m dvd n"

  1949   by (cases "n < m") (auto elim!: dvdE simp add: not_less le_imp_diff_is_add)

  1950

  1951 lemma dvd_minus_add:

  1952   fixes m n q r :: nat

  1953   assumes "q \<le> n" "q \<le> r * m"

  1954   shows "m dvd n - q \<longleftrightarrow> m dvd n + (r * m - q)"

  1955 proof -

  1956   have "m dvd n - q \<longleftrightarrow> m dvd r * m + (n - q)"

  1957     by (auto elim: dvd_plusE)

  1958   also from assms have "\<dots> \<longleftrightarrow> m dvd r * m + n - q" by simp

  1959   also from assms have "\<dots> \<longleftrightarrow> m dvd (r * m - q) + n" by simp

  1960   also have "\<dots> \<longleftrightarrow> m dvd n + (r * m - q)" by (simp add: add.commute)

  1961   finally show ?thesis .

  1962 qed

  1963

  1964

  1965 subsection {* aliases *}

  1966

  1967 lemma nat_mult_1: "(1::nat) * n = n"

  1968   by (rule mult_1_left)

  1969

  1970 lemma nat_mult_1_right: "n * (1::nat) = n"

  1971   by (rule mult_1_right)

  1972

  1973

  1974 subsection {* size of a datatype value *}

  1975

  1976 class size =

  1977   fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded} *}

  1978

  1979

  1980 subsection {* code module namespace *}

  1981

  1982 code_identifier

  1983   code_module Nat \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith

  1984

  1985 hide_const (open) of_nat_aux

  1986

  1987 end