src/HOL/Nat.thy
 author haftmann Fri Nov 27 08:41:10 2009 +0100 (2009-11-27) changeset 33963 977b94b64905 parent 33657 a4179bf442d1 child 34208 a7acd6c68d9b permissions -rw-r--r--
renamed former datatype.ML to datatype_data.ML; datatype.ML provides uniform view on datatype.ML and datatype_rep_proofs.ML
     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 Product_Type Ring_and_Field

    12 uses

    13   "~~/src/Tools/rat.ML"

    14   "~~/src/Provers/Arith/cancel_sums.ML"

    15   "Tools/arith_data.ML"

    16   ("Tools/nat_arith.ML")

    17   "~~/src/Provers/Arith/fast_lin_arith.ML"

    18   ("Tools/lin_arith.ML")

    19 begin

    20

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

    22

    23 typedecl ind

    24

    25 axiomatization

    26   Zero_Rep :: ind and

    27   Suc_Rep :: "ind => ind"

    28 where

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

    30   inj_Suc_Rep:          "inj Suc_Rep" and

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

    32

    33

    34 subsection {* Type nat *}

    35

    36 text {* Type definition *}

    37

    38 inductive Nat :: "ind \<Rightarrow> bool"

    39 where

    40     Zero_RepI: "Nat Zero_Rep"

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

    42

    43 global

    44

    45 typedef (open Nat)

    46   nat = Nat

    47   by (rule exI, unfold mem_def, rule Nat.Zero_RepI)

    48

    49 constdefs

    50   Suc ::   "nat => nat"

    51   Suc_def: "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))"

    52

    53 local

    54

    55 instantiation nat :: zero

    56 begin

    57

    58 definition Zero_nat_def [code del]:

    59   "0 = Abs_Nat Zero_Rep"

    60

    61 instance ..

    62

    63 end

    64

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

    66   by (simp add: Zero_nat_def Suc_def Abs_Nat_inject [unfolded mem_def]

    67     Rep_Nat [unfolded mem_def] Suc_RepI Zero_RepI Suc_Rep_not_Zero_Rep [unfolded mem_def])

    68

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

    70   by (rule not_sym, rule Suc_not_Zero not_sym)

    71

    72 rep_datatype "0 \<Colon> nat" Suc

    73   apply (unfold Zero_nat_def Suc_def)

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

    75      apply (erule Rep_Nat [unfolded mem_def, THEN Nat.induct])

    76      apply (iprover elim: Abs_Nat_inverse [unfolded mem_def, THEN subst])

    77     apply (simp_all add: Abs_Nat_inject [unfolded mem_def] Rep_Nat [unfolded mem_def]

    78       Suc_RepI Zero_RepI Suc_Rep_not_Zero_Rep [unfolded mem_def]

    79       Suc_Rep_not_Zero_Rep [unfolded mem_def, symmetric]

    80       inj_Suc_Rep [THEN inj_eq] Rep_Nat_inject)

    81   done

    82

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

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

    85   fixes n

    86   assumes "P 0"

    87     and "\<And>n. P n \<Longrightarrow> P (Suc n)"

    88   shows "P n"

    89   using assms by (rule nat.induct)

    90

    91 declare nat.exhaust [case_names 0 Suc, cases type: nat]

    92

    93 lemmas nat_rec_0 = nat.recs(1)

    94   and nat_rec_Suc = nat.recs(2)

    95

    96 lemmas nat_case_0 = nat.cases(1)

    97   and nat_case_Suc = nat.cases(2)

    98

    99

   100 text {* Injectiveness and distinctness lemmas *}

   101

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

   103   by (simp add: inj_on_def)

   104

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

   106 by (rule notE, rule Suc_not_Zero)

   107

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

   109 by (rule Suc_neq_Zero, erule sym)

   110

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

   112 by (rule inj_Suc [THEN injD])

   113

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

   115 by (induct n) simp_all

   116

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

   118 by (rule not_sym, rule n_not_Suc_n)

   119

   120 text {* A special form of induction for reasoning

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

   122

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

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

   125   apply (rule_tac x = m in spec)

   126   apply (induct n)

   127   prefer 2

   128   apply (rule allI)

   129   apply (induct_tac x, iprover+)

   130   done

   131

   132

   133 subsection {* Arithmetic operators *}

   134

   135 instantiation nat :: "{minus, comm_monoid_add}"

   136 begin

   137

   138 primrec plus_nat

   139 where

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

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

   142

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

   144   by (induct m) simp_all

   145

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

   147   by (induct m) simp_all

   148

   149 declare add_0 [code]

   150

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

   152   by simp

   153

   154 primrec minus_nat

   155 where

   156   diff_0:     "m - 0 = (m\<Colon>nat)"

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

   158

   159 declare diff_Suc [simp del]

   160 declare diff_0 [code]

   161

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

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

   164

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

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

   167

   168 instance proof

   169   fix n m q :: nat

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

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

   172   show "0 + n = n" by simp

   173 qed

   174

   175 end

   176

   177 instantiation nat :: comm_semiring_1_cancel

   178 begin

   179

   180 definition

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

   182

   183 primrec times_nat

   184 where

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

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

   187

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

   189   by (induct m) simp_all

   190

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

   192   by (induct m) (simp_all add: add_left_commute)

   193

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

   195   by (induct m) (simp_all add: add_assoc)

   196

   197 instance proof

   198   fix n m q :: nat

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

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

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

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

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

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

   205 qed

   206

   207 end

   208

   209 subsubsection {* Addition *}

   210

   211 lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)"

   212   by (rule add_assoc)

   213

   214 lemma nat_add_commute: "m + n = n + (m::nat)"

   215   by (rule add_commute)

   216

   217 lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)"

   218   by (rule add_left_commute)

   219

   220 lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))"

   221   by (rule add_left_cancel)

   222

   223 lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))"

   224   by (rule add_right_cancel)

   225

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

   227

   228 lemma add_is_0 [iff]:

   229   fixes m n :: nat

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

   231   by (cases m) simp_all

   232

   233 lemma add_is_1:

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

   235   by (cases m) simp_all

   236

   237 lemma one_is_add:

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

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

   240

   241 lemma add_eq_self_zero:

   242   fixes m n :: nat

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

   244   by (induct m) simp_all

   245

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

   247   apply (induct k)

   248    apply simp

   249   apply(drule comp_inj_on[OF _ inj_Suc])

   250   apply (simp add:o_def)

   251   done

   252

   253

   254 subsubsection {* Difference *}

   255

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

   257   by (induct m) simp_all

   258

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

   260   by (induct i j rule: diff_induct) simp_all

   261

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

   263   by (simp add: diff_diff_left)

   264

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

   266   by (simp add: diff_diff_left add_commute)

   267

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

   269   by (induct n) simp_all

   270

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

   272   by (simp add: diff_add_inverse add_commute [of m n])

   273

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

   275   by (induct k) simp_all

   276

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

   278   by (simp add: diff_cancel add_commute)

   279

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

   281   by (induct n) simp_all

   282

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

   284   unfolding One_nat_def by simp

   285

   286 text {* Difference distributes over multiplication *}

   287

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

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

   290

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

   292 by (simp add: diff_mult_distrib mult_commute [of k])

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

   294

   295

   296 subsubsection {* Multiplication *}

   297

   298 lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)"

   299   by (rule mult_assoc)

   300

   301 lemma nat_mult_commute: "m * n = n * (m::nat)"

   302   by (rule mult_commute)

   303

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

   305   by (rule right_distrib)

   306

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

   308   by (induct m) auto

   309

   310 lemmas nat_distrib =

   311   add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2

   312

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

   314   apply (induct m)

   315    apply simp

   316   apply (induct n)

   317    apply auto

   318   done

   319

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

   321   apply (rule trans)

   322   apply (rule_tac [2] mult_eq_1_iff, fastsimp)

   323   done

   324

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

   326   unfolding One_nat_def by (rule mult_eq_1_iff)

   327

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

   329   unfolding One_nat_def by (rule one_eq_mult_iff)

   330

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

   332 proof -

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

   334   proof (induct n arbitrary: m)

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

   336   next

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

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

   339   qed

   340   then show ?thesis by auto

   341 qed

   342

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

   344   by (simp add: mult_commute)

   345

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

   347   by (subst mult_cancel1) simp

   348

   349

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

   351

   352 subsubsection {* Operation definition *}

   353

   354 instantiation nat :: linorder

   355 begin

   356

   357 primrec less_eq_nat where

   358   "(0\<Colon>nat) \<le> n \<longleftrightarrow> True"

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

   360

   361 declare less_eq_nat.simps [simp del]

   362 lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by (simp add: less_eq_nat.simps)

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

   364

   365 definition less_nat where

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

   367

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

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

   370

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

   372   unfolding less_eq_Suc_le ..

   373

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

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

   376

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

   378   by (simp add: less_eq_Suc_le)

   379

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

   381   by simp

   382

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

   384   by (simp add: less_eq_Suc_le)

   385

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

   387   by (simp add: less_eq_Suc_le)

   388

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

   390   by (induct m arbitrary: n)

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

   392

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

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

   395

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

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

   398

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

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

   401

   402 instance

   403 proof

   404   fix n m :: nat

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

   406   proof (induct n arbitrary: m)

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

   408   next

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

   410   qed

   411 next

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

   413 next

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

   415   then show "n = m"

   416     by (induct n arbitrary: m)

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

   418 next

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

   420   then show "n \<le> q"

   421   proof (induct n arbitrary: m q)

   422     case 0 show ?case by simp

   423   next

   424     case (Suc n) then show ?case

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

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

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

   428   qed

   429 next

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

   431     by (induct n arbitrary: m)

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

   433 qed

   434

   435 end

   436

   437 instantiation nat :: bot

   438 begin

   439

   440 definition bot_nat :: nat where

   441   "bot_nat = 0"

   442

   443 instance proof

   444 qed (simp add: bot_nat_def)

   445

   446 end

   447

   448 subsubsection {* Introduction properties *}

   449

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

   451   by (simp add: less_Suc_eq_le)

   452

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

   454   by (simp add: less_Suc_eq_le)

   455

   456

   457 subsubsection {* Elimination properties *}

   458

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

   460   by (rule order_less_irrefl)

   461

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

   463   by (rule not_sym) (rule less_imp_neq)

   464

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

   466   by (rule less_imp_neq)

   467

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

   469   by (rule notE, rule less_not_refl)

   470

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

   472   by (rule notE) (rule not_less0)

   473

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

   475   unfolding less_Suc_eq_le le_less ..

   476

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

   478   by (simp add: less_Suc_eq)

   479

   480 lemma less_one [iff, noatp]: "(n < (1::nat)) = (n = 0)"

   481   unfolding One_nat_def by (rule less_Suc0)

   482

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

   484   by simp

   485

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

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

   488   unfolding not_less less_Suc_eq_le by (rule antisym)

   489

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

   491   by (rule linorder_neq_iff)

   492

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

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

   495   shows "P n m"

   496   apply (rule less_linear [THEN disjE])

   497   apply (erule_tac [2] disjE)

   498   apply (erule lessCase)

   499   apply (erule sym [THEN eqCase])

   500   apply (erule major)

   501   done

   502

   503

   504 subsubsection {* Inductive (?) properties *}

   505

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

   507   unfolding less_eq_Suc_le [of m] le_less by simp

   508

   509 lemma lessE:

   510   assumes major: "i < k"

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

   512   shows P

   513 proof -

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

   515     unfolding less_eq_Suc_le by (induct k) simp_all

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

   517     by (clarsimp simp add: less_le)

   518   with p1 p2 show P by auto

   519 qed

   520

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

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

   523   apply (rule major [THEN lessE])

   524   apply (rule eq, blast)

   525   apply (rule less, blast)

   526   done

   527

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

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

   530   apply (rule major [THEN lessE])

   531   apply (erule lessI [THEN minor])

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

   533   done

   534

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

   536   by simp

   537

   538 lemma less_trans_Suc:

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

   540   apply (induct k, simp_all)

   541   apply (insert le)

   542   apply (simp add: less_Suc_eq)

   543   apply (blast dest: Suc_lessD)

   544   done

   545

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

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

   548   unfolding not_less less_Suc_eq_le ..

   549

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

   551   unfolding not_le Suc_le_eq ..

   552

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

   554

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

   556   unfolding less_Suc_eq_le .

   557

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

   559   unfolding not_le less_Suc_eq_le ..

   560

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

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

   563

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

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

   566

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

   568   unfolding Suc_le_eq .

   569

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

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

   572   unfolding Suc_le_eq .

   573

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

   575   unfolding less_eq_Suc_le by (rule Suc_leD)

   576

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

   578 lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq

   579

   580

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

   582

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

   584   unfolding le_less .

   585

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

   587   by (rule le_less)

   588

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

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

   591   by auto

   592

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

   594   by simp

   595

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

   597   by (rule order_trans)

   598

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

   600   by (rule antisym)

   601

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

   603   by (rule less_le)

   604

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

   606   unfolding less_le ..

   607

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

   609   by (rule linear)

   610

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

   612

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

   614   unfolding less_Suc_eq_le by auto

   615

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

   617   unfolding not_less by (rule le_less_Suc_eq)

   618

   619 lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq

   620

   621 text {* These two rules ease the use of primitive recursion.

   622 NOTE USE OF @{text "=="} *}

   623 lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"

   624 by simp

   625

   626 lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"

   627 by simp

   628

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

   630 by (cases n) simp_all

   631

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

   633 by (cases n) simp_all

   634

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

   636 by (cases n) simp_all

   637

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

   639 by (cases n) simp_all

   640

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

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

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

   644

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

   646 by (fast intro: not0_implies_Suc)

   647

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

   649 using neq0_conv by blast

   650

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

   652 by (induct m') simp_all

   653

   654 text {* Useful in certain inductive arguments *}

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

   656 by (cases m) simp_all

   657

   658

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

   660

   661 lemma mono_Suc: "mono Suc"

   662 by (rule monoI) simp

   663

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

   665 by (rule min_leastL) simp

   666

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

   668 by (rule min_leastR) simp

   669

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

   671 by (simp add: mono_Suc min_of_mono)

   672

   673 lemma min_Suc1:

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

   675 by (simp split: nat.split)

   676

   677 lemma min_Suc2:

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

   679 by (simp split: nat.split)

   680

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

   682 by (rule max_leastL) simp

   683

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

   685 by (rule max_leastR) simp

   686

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

   688 by (simp add: mono_Suc max_of_mono)

   689

   690 lemma max_Suc1:

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

   692 by (simp split: nat.split)

   693

   694 lemma max_Suc2:

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

   696 by (simp split: nat.split)

   697

   698

   699 subsubsection {* Monotonicity of Addition *}

   700

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

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

   703

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

   705 unfolding One_nat_def by (rule Suc_pred)

   706

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

   708 by (induct k) simp_all

   709

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

   711 by (induct k) simp_all

   712

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

   714 by(auto dest:gr0_implies_Suc)

   715

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

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

   718 by (induct k) simp_all

   719

   720 text {* strict, in both arguments *}

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

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

   723   apply (induct j, simp_all)

   724   done

   725

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

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

   728   apply (induct n)

   729   apply (simp_all add: order_le_less)

   730   apply (blast elim!: less_SucE

   731                intro!: add_0_right [symmetric] add_Suc_right [symmetric])

   732   done

   733

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

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

   736 apply(auto simp: gr0_conv_Suc)

   737 apply (induct_tac m)

   738 apply (simp_all add: add_less_mono)

   739 done

   740

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

   742 instance nat :: ordered_semidom

   743 proof

   744   fix i j k :: nat

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

   746   show "i \<le> j ==> k + i \<le> k + j" by simp

   747   show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2)

   748 qed

   749

   750 instance nat :: no_zero_divisors

   751 proof

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

   753 qed

   754

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

   756 by simp

   757

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

   759 by simp

   760

   761

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

   763

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

   765

   766 instance nat :: wellorder proof

   767   fix P and n :: nat

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

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

   770   proof (induct n)

   771     case (0 n)

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

   773     thus ?case using 0 by auto

   774   next

   775     case (Suc m n)

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

   777     thus ?case

   778     proof

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

   780     next

   781       assume n: "n = Suc m"

   782       show "P n"

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

   784     qed

   785   qed

   786   then show "P n" by auto

   787 qed

   788

   789 lemma Least_Suc:

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

   791   apply (case_tac "n", auto)

   792   apply (frule LeastI)

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

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

   795   apply (erule_tac [2] Least_le)

   796   apply (case_tac "LEAST x. P x", auto)

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

   798   apply (blast intro: order_antisym)

   799   done

   800

   801 lemma Least_Suc2:

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

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

   804   apply simp

   805   done

   806

   807 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)"

   808   apply (cases n)

   809    apply blast

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

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

   812   done

   813

   814 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)"

   815   unfolding One_nat_def

   816   apply (cases n)

   817    apply blast

   818   apply (frule (1) ex_least_nat_le)

   819   apply (erule exE)

   820   apply (case_tac k)

   821    apply simp

   822   apply (rename_tac k1)

   823   apply (rule_tac x=k1 in exI)

   824   apply (auto simp add: less_eq_Suc_le)

   825   done

   826

   827 lemma nat_less_induct:

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

   829   using assms less_induct by blast

   830

   831 lemma measure_induct_rule [case_names less]:

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

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

   834   shows "P a"

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

   836

   837 text {* old style induction rules: *}

   838 lemma measure_induct:

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

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

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

   842

   843 lemma full_nat_induct:

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

   845   shows "P n"

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

   847

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

   849 lemma less_Suc_induct:

   850   assumes less:  "i < j"

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

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

   853   shows "P i j"

   854 proof -

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

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

   857   proof (induct k)

   858     case 0

   859     show ?case by (simp add: step)

   860   next

   861     case (Suc k)

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

   863     hence "i < Suc (i + k)" by (simp add: add_commute)

   864     from trans[OF this lessI Suc step]

   865     show ?case by simp

   866   qed

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

   868 qed

   869

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

   871 Provided by Roelof Oosterhuis.

   872 $P(n)$ is true for all $n\in\mathbb{N}$ if

   873 \begin{itemize}

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

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

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

   877 \end{itemize} *}

   878

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

   880 lemma infinite_descent:

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

   882 by (induct n rule: less_induct, auto)

   883

   884 lemma infinite_descent0[case_names 0 smaller]:

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

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

   887

   888 text {*

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

   890 $P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and

   891 \begin{itemize}

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

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

   894 \end{itemize}

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

   896

   897 corollary infinite_descent0_measure [case_names 0 smaller]:

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

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

   900   shows "P x"

   901 proof -

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

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

   904   proof (induct n rule: infinite_descent0)

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

   906     with A0 show "P x" by auto

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

   908     case (smaller n)

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

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

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

   912     then show ?case by auto

   913   qed

   914   ultimately show "P x" by auto

   915 qed

   916

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

   918 lemma infinite_descent_measure:

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

   920 proof -

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

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

   923   proof (induct n rule: infinite_descent, auto)

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

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

   926   qed

   927   ultimately show "P x" by auto

   928 qed

   929

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

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

   932 lemma less_mono_imp_le_mono:

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

   934 by (simp add: order_le_less) (blast)

   935

   936

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

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

   939 by (rule add_right_mono)

   940

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

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

   943 by (rule add_mono)

   944

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

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

   947

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

   949 by (simp add: add_commute, rule le_add2)

   950

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

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

   953

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

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

   956

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

   958 by (iprover intro!: less_add_Suc1 less_imp_Suc_add)

   959

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

   961 by (rule le_trans, assumption, rule le_add1)

   962

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

   964 by (rule le_trans, assumption, rule le_add2)

   965

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

   967 by (rule less_le_trans, assumption, rule le_add1)

   968

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

   970 by (rule less_le_trans, assumption, rule le_add2)

   971

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

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

   974 apply (simp_all add: le_add1)

   975 done

   976

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

   978 apply (rule notI)

   979 apply (drule add_lessD1)

   980 apply (erule less_irrefl [THEN notE])

   981 done

   982

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

   984 by (simp add: add_commute)

   985

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

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

   988 apply (simp_all add: le_add1)

   989 done

   990

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

   992 apply (simp add: add_commute)

   993 apply (erule add_leD1)

   994 done

   995

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

   997 by (blast dest: add_leD1 add_leD2)

   998

   999 text {* needs @{text "!!k"} for @{text add_ac} to work *}

  1000 lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"

  1001 by (force simp del: add_Suc_right

  1002     simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)

  1003

  1004

  1005 subsubsection {* More results about difference *}

  1006

  1007 text {* Addition is the inverse of subtraction:

  1008   if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}

  1009 lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"

  1010 by (induct m n rule: diff_induct) simp_all

  1011

  1012 lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"

  1013 by (simp add: add_diff_inverse linorder_not_less)

  1014

  1015 lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"

  1016 by (simp add: add_commute)

  1017

  1018 lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"

  1019 by (induct m n rule: diff_induct) simp_all

  1020

  1021 lemma diff_less_Suc: "m - n < Suc m"

  1022 apply (induct m n rule: diff_induct)

  1023 apply (erule_tac [3] less_SucE)

  1024 apply (simp_all add: less_Suc_eq)

  1025 done

  1026

  1027 lemma diff_le_self [simp]: "m - n \<le> (m::nat)"

  1028 by (induct m n rule: diff_induct) (simp_all add: le_SucI)

  1029

  1030 lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"

  1031   by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])

  1032

  1033 lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"

  1034 by (rule le_less_trans, rule diff_le_self)

  1035

  1036 lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"

  1037 by (cases n) (auto simp add: le_simps)

  1038

  1039 lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"

  1040 by (induct j k rule: diff_induct) simp_all

  1041

  1042 lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"

  1043 by (simp add: add_commute diff_add_assoc)

  1044

  1045 lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"

  1046 by (auto simp add: diff_add_inverse2)

  1047

  1048 lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"

  1049 by (induct m n rule: diff_induct) simp_all

  1050

  1051 lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"

  1052 by (rule iffD2, rule diff_is_0_eq)

  1053

  1054 lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"

  1055 by (induct m n rule: diff_induct) simp_all

  1056

  1057 lemma less_imp_add_positive:

  1058   assumes "i < j"

  1059   shows "\<exists>k::nat. 0 < k & i + k = j"

  1060 proof

  1061   from assms show "0 < j - i & i + (j - i) = j"

  1062     by (simp add: order_less_imp_le)

  1063 qed

  1064

  1065 text {* a nice rewrite for bounded subtraction *}

  1066 lemma nat_minus_add_max:

  1067   fixes n m :: nat

  1068   shows "n - m + m = max n m"

  1069     by (simp add: max_def not_le order_less_imp_le)

  1070

  1071 lemma nat_diff_split:

  1072   "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"

  1073     -- {* elimination of @{text -} on @{text nat} *}

  1074 by (cases "a < b")

  1075   (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse

  1076     not_less le_less dest!: sym [of a] sym [of b] add_eq_self_zero)

  1077

  1078 lemma nat_diff_split_asm:

  1079   "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"

  1080     -- {* elimination of @{text -} on @{text nat} in assumptions *}

  1081 by (auto split: nat_diff_split)

  1082

  1083

  1084 subsubsection {* Monotonicity of Multiplication *}

  1085

  1086 lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"

  1087 by (simp add: mult_right_mono)

  1088

  1089 lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"

  1090 by (simp add: mult_left_mono)

  1091

  1092 text {* @{text "\<le>"} monotonicity, BOTH arguments *}

  1093 lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"

  1094 by (simp add: mult_mono)

  1095

  1096 lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"

  1097 by (simp add: mult_strict_right_mono)

  1098

  1099 text{*Differs from the standard @{text zero_less_mult_iff} in that

  1100       there are no negative numbers.*}

  1101 lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"

  1102   apply (induct m)

  1103    apply simp

  1104   apply (case_tac n)

  1105    apply simp_all

  1106   done

  1107

  1108 lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)"

  1109   apply (induct m)

  1110    apply simp

  1111   apply (case_tac n)

  1112    apply simp_all

  1113   done

  1114

  1115 lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"

  1116   apply (safe intro!: mult_less_mono1)

  1117   apply (case_tac k, auto)

  1118   apply (simp del: le_0_eq add: linorder_not_le [symmetric])

  1119   apply (blast intro: mult_le_mono1)

  1120   done

  1121

  1122 lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"

  1123 by (simp add: mult_commute [of k])

  1124

  1125 lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"

  1126 by (simp add: linorder_not_less [symmetric], auto)

  1127

  1128 lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"

  1129 by (simp add: linorder_not_less [symmetric], auto)

  1130

  1131 lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"

  1132 by (subst mult_less_cancel1) simp

  1133

  1134 lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"

  1135 by (subst mult_le_cancel1) simp

  1136

  1137 lemma le_square: "m \<le> m * (m::nat)"

  1138   by (cases m) (auto intro: le_add1)

  1139

  1140 lemma le_cube: "(m::nat) \<le> m * (m * m)"

  1141   by (cases m) (auto intro: le_add1)

  1142

  1143 text {* Lemma for @{text gcd} *}

  1144 lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"

  1145   apply (drule sym)

  1146   apply (rule disjCI)

  1147   apply (rule nat_less_cases, erule_tac [2] _)

  1148    apply (drule_tac [2] mult_less_mono2)

  1149     apply (auto)

  1150   done

  1151

  1152 text {* the lattice order on @{typ nat} *}

  1153

  1154 instantiation nat :: distrib_lattice

  1155 begin

  1156

  1157 definition

  1158   "(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min"

  1159

  1160 definition

  1161   "(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max"

  1162

  1163 instance by intro_classes

  1164   (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def

  1165     intro: order_less_imp_le antisym elim!: order_trans order_less_trans)

  1166

  1167 end

  1168

  1169

  1170 subsection {* Natural operation of natural numbers on functions *}

  1171

  1172 text {*

  1173   We use the same logical constant for the power operations on

  1174   functions and relations, in order to share the same syntax.

  1175 *}

  1176

  1177 consts compow :: "nat \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"

  1178

  1179 abbreviation compower :: "('a \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'b" (infixr "^^" 80) where

  1180   "f ^^ n \<equiv> compow n f"

  1181

  1182 notation (latex output)

  1183   compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)

  1184

  1185 notation (HTML output)

  1186   compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)

  1187

  1188 text {* @{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f} *}

  1189

  1190 overloading

  1191   funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"

  1192 begin

  1193

  1194 primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where

  1195     "funpow 0 f = id"

  1196   | "funpow (Suc n) f = f o funpow n f"

  1197

  1198 end

  1199

  1200 text {* for code generation *}

  1201

  1202 definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where

  1203   funpow_code_def [code_post]: "funpow = compow"

  1204

  1205 lemmas [code_unfold] = funpow_code_def [symmetric]

  1206

  1207 lemma [code]:

  1208   "funpow 0 f = id"

  1209   "funpow (Suc n) f = f o funpow n f"

  1210   unfolding funpow_code_def by simp_all

  1211

  1212 hide (open) const funpow

  1213

  1214 lemma funpow_add:

  1215   "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"

  1216   by (induct m) simp_all

  1217

  1218 lemma funpow_swap1:

  1219   "f ((f ^^ n) x) = (f ^^ n) (f x)"

  1220 proof -

  1221   have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp

  1222   also have "\<dots>  = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)

  1223   also have "\<dots> = (f ^^ n) (f x)" by simp

  1224   finally show ?thesis .

  1225 qed

  1226

  1227

  1228 subsection {* Embedding of the Naturals into any

  1229   @{text semiring_1}: @{term of_nat} *}

  1230

  1231 context semiring_1

  1232 begin

  1233

  1234 primrec

  1235   of_nat :: "nat \<Rightarrow> 'a"

  1236 where

  1237   of_nat_0:     "of_nat 0 = 0"

  1238   | of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"

  1239

  1240 lemma of_nat_1 [simp]: "of_nat 1 = 1"

  1241   unfolding One_nat_def by simp

  1242

  1243 lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"

  1244   by (induct m) (simp_all add: add_ac)

  1245

  1246 lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"

  1247   by (induct m) (simp_all add: add_ac left_distrib)

  1248

  1249 primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where

  1250   "of_nat_aux inc 0 i = i"

  1251   | "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- {* tail recursive *}

  1252

  1253 lemma of_nat_code:

  1254   "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"

  1255 proof (induct n)

  1256   case 0 then show ?case by simp

  1257 next

  1258   case (Suc n)

  1259   have "\<And>i. of_nat_aux (\<lambda>i. i + 1) n (i + 1) = of_nat_aux (\<lambda>i. i + 1) n i + 1"

  1260     by (induct n) simp_all

  1261   from this [of 0] have "of_nat_aux (\<lambda>i. i + 1) n 1 = of_nat_aux (\<lambda>i. i + 1) n 0 + 1"

  1262     by simp

  1263   with Suc show ?case by (simp add: add_commute)

  1264 qed

  1265

  1266 end

  1267

  1268 declare of_nat_code [code, code_unfold, code_inline del]

  1269

  1270 text{*Class for unital semirings with characteristic zero.

  1271  Includes non-ordered rings like the complex numbers.*}

  1272

  1273 class semiring_char_0 = semiring_1 +

  1274   assumes of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"

  1275 begin

  1276

  1277 text{*Special cases where either operand is zero*}

  1278

  1279 lemma of_nat_0_eq_iff [simp, noatp]: "0 = of_nat n \<longleftrightarrow> 0 = n"

  1280   by (rule of_nat_eq_iff [of 0 n, unfolded of_nat_0])

  1281

  1282 lemma of_nat_eq_0_iff [simp, noatp]: "of_nat m = 0 \<longleftrightarrow> m = 0"

  1283   by (rule of_nat_eq_iff [of m 0, unfolded of_nat_0])

  1284

  1285 lemma inj_of_nat: "inj of_nat"

  1286   by (simp add: inj_on_def)

  1287

  1288 end

  1289

  1290 context ordered_semidom

  1291 begin

  1292

  1293 lemma zero_le_imp_of_nat: "0 \<le> of_nat m"

  1294   apply (induct m, simp_all)

  1295   apply (erule order_trans)

  1296   apply (rule ord_le_eq_trans [OF _ add_commute])

  1297   apply (rule less_add_one [THEN less_imp_le])

  1298   done

  1299

  1300 lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"

  1301   apply (induct m n rule: diff_induct, simp_all)

  1302   apply (insert add_less_le_mono [OF zero_less_one zero_le_imp_of_nat], force)

  1303   done

  1304

  1305 lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"

  1306   apply (induct m n rule: diff_induct, simp_all)

  1307   apply (insert zero_le_imp_of_nat)

  1308   apply (force simp add: not_less [symmetric])

  1309   done

  1310

  1311 lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"

  1312   by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)

  1313

  1314 lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"

  1315   by (simp add: not_less [symmetric] linorder_not_less [symmetric])

  1316

  1317 text{*Every @{text ordered_semidom} has characteristic zero.*}

  1318

  1319 subclass semiring_char_0

  1320   proof qed (simp add: eq_iff order_eq_iff)

  1321

  1322 text{*Special cases where either operand is zero*}

  1323

  1324 lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"

  1325   by (rule of_nat_le_iff [of 0, simplified])

  1326

  1327 lemma of_nat_le_0_iff [simp, noatp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"

  1328   by (rule of_nat_le_iff [of _ 0, simplified])

  1329

  1330 lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"

  1331   by (rule of_nat_less_iff [of 0, simplified])

  1332

  1333 lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"

  1334   by (rule of_nat_less_iff [of _ 0, simplified])

  1335

  1336 end

  1337

  1338 context ring_1

  1339 begin

  1340

  1341 lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"

  1342 by (simp add: algebra_simps of_nat_add [symmetric])

  1343

  1344 end

  1345

  1346 context ordered_idom

  1347 begin

  1348

  1349 lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"

  1350   unfolding abs_if by auto

  1351

  1352 end

  1353

  1354 lemma of_nat_id [simp]: "of_nat n = n"

  1355   by (induct n) (auto simp add: One_nat_def)

  1356

  1357 lemma of_nat_eq_id [simp]: "of_nat = id"

  1358   by (auto simp add: expand_fun_eq)

  1359

  1360

  1361 subsection {* The Set of Natural Numbers *}

  1362

  1363 context semiring_1

  1364 begin

  1365

  1366 definition

  1367   Nats  :: "'a set" where

  1368   [code del]: "Nats = range of_nat"

  1369

  1370 notation (xsymbols)

  1371   Nats  ("\<nat>")

  1372

  1373 lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"

  1374   by (simp add: Nats_def)

  1375

  1376 lemma Nats_0 [simp]: "0 \<in> \<nat>"

  1377 apply (simp add: Nats_def)

  1378 apply (rule range_eqI)

  1379 apply (rule of_nat_0 [symmetric])

  1380 done

  1381

  1382 lemma Nats_1 [simp]: "1 \<in> \<nat>"

  1383 apply (simp add: Nats_def)

  1384 apply (rule range_eqI)

  1385 apply (rule of_nat_1 [symmetric])

  1386 done

  1387

  1388 lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"

  1389 apply (auto simp add: Nats_def)

  1390 apply (rule range_eqI)

  1391 apply (rule of_nat_add [symmetric])

  1392 done

  1393

  1394 lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"

  1395 apply (auto simp add: Nats_def)

  1396 apply (rule range_eqI)

  1397 apply (rule of_nat_mult [symmetric])

  1398 done

  1399

  1400 end

  1401

  1402

  1403 subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}

  1404

  1405 lemma subst_equals:

  1406   assumes 1: "t = s" and 2: "u = t"

  1407   shows "u = s"

  1408   using 2 1 by (rule trans)

  1409

  1410 setup Arith_Data.setup

  1411

  1412 use "Tools/nat_arith.ML"

  1413 declaration {* K Nat_Arith.setup *}

  1414

  1415 use "Tools/lin_arith.ML"

  1416 setup {* Lin_Arith.global_setup *}

  1417 declaration {* K Lin_Arith.setup *}

  1418

  1419 lemmas [arith_split] = nat_diff_split split_min split_max

  1420

  1421 context order

  1422 begin

  1423

  1424 lemma lift_Suc_mono_le:

  1425   assumes mono: "!!n. f n \<le> f(Suc n)" and "n\<le>n'"

  1426   shows "f n \<le> f n'"

  1427 proof (cases "n < n'")

  1428   case True

  1429   thus ?thesis

  1430     by (induct n n' rule: less_Suc_induct[consumes 1]) (auto intro: mono)

  1431 qed (insert n \<le> n', auto) -- {*trivial for @{prop "n = n'"} *}

  1432

  1433 lemma lift_Suc_mono_less:

  1434   assumes mono: "!!n. f n < f(Suc n)" and "n < n'"

  1435   shows "f n < f n'"

  1436 using n < n'

  1437 by (induct n n' rule: less_Suc_induct[consumes 1]) (auto intro: mono)

  1438

  1439 lemma lift_Suc_mono_less_iff:

  1440   "(!!n. f n < f(Suc n)) \<Longrightarrow> f(n) < f(m) \<longleftrightarrow> n<m"

  1441 by(blast intro: less_asym' lift_Suc_mono_less[of f]

  1442          dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq[THEN iffD1])

  1443

  1444 end

  1445

  1446 lemma mono_iff_le_Suc: "mono f = (\<forall>n. f n \<le> f (Suc n))"

  1447 unfolding mono_def

  1448 by (auto intro:lift_Suc_mono_le[of f])

  1449

  1450 lemma mono_nat_linear_lb:

  1451   "(!!m n::nat. m<n \<Longrightarrow> f m < f n) \<Longrightarrow> f(m)+k \<le> f(m+k)"

  1452 apply(induct_tac k)

  1453  apply simp

  1454 apply(erule_tac x="m+n" in meta_allE)

  1455 apply(erule_tac x="Suc(m+n)" in meta_allE)

  1456 apply simp

  1457 done

  1458

  1459

  1460 text{*Subtraction laws, mostly by Clemens Ballarin*}

  1461

  1462 lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"

  1463 by arith

  1464

  1465 lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"

  1466 by arith

  1467

  1468 lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"

  1469 by arith

  1470

  1471 lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"

  1472 by arith

  1473

  1474 lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"

  1475 by arith

  1476

  1477 lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"

  1478 by arith

  1479

  1480 (*Replaces the previous diff_less and le_diff_less, which had the stronger

  1481   second premise n\<le>m*)

  1482 lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"

  1483 by arith

  1484

  1485 text {* Simplification of relational expressions involving subtraction *}

  1486

  1487 lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"

  1488 by (simp split add: nat_diff_split)

  1489

  1490 lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"

  1491 by (auto split add: nat_diff_split)

  1492

  1493 lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"

  1494 by (auto split add: nat_diff_split)

  1495

  1496 lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"

  1497 by (auto split add: nat_diff_split)

  1498

  1499 text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}

  1500

  1501 (* Monotonicity of subtraction in first argument *)

  1502 lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"

  1503 by (simp split add: nat_diff_split)

  1504

  1505 lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"

  1506 by (simp split add: nat_diff_split)

  1507

  1508 lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"

  1509 by (simp split add: nat_diff_split)

  1510

  1511 lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"

  1512 by (simp split add: nat_diff_split)

  1513

  1514 lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"

  1515 by auto

  1516

  1517 lemma inj_on_diff_nat:

  1518   assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"

  1519   shows "inj_on (\<lambda>n. n - k) N"

  1520 proof (rule inj_onI)

  1521   fix x y

  1522   assume a: "x \<in> N" "y \<in> N" "x - k = y - k"

  1523   with k_le_n have "x - k + k = y - k + k" by auto

  1524   with a k_le_n show "x = y" by auto

  1525 qed

  1526

  1527 text{*Rewriting to pull differences out*}

  1528

  1529 lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"

  1530 by arith

  1531

  1532 lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"

  1533 by arith

  1534

  1535 lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"

  1536 by arith

  1537

  1538 text{*Lemmas for ex/Factorization*}

  1539

  1540 lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"

  1541 by (cases m) auto

  1542

  1543 lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"

  1544 by (cases m) auto

  1545

  1546 lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"

  1547 by (cases m) auto

  1548

  1549 text {* Specialized induction principles that work "backwards": *}

  1550

  1551 lemma inc_induct[consumes 1, case_names base step]:

  1552   assumes less: "i <= j"

  1553   assumes base: "P j"

  1554   assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"

  1555   shows "P i"

  1556   using less

  1557 proof (induct d=="j - i" arbitrary: i)

  1558   case (0 i)

  1559   hence "i = j" by simp

  1560   with base show ?case by simp

  1561 next

  1562   case (Suc d i)

  1563   hence "i < j" "P (Suc i)"

  1564     by simp_all

  1565   thus "P i" by (rule step)

  1566 qed

  1567

  1568 lemma strict_inc_induct[consumes 1, case_names base step]:

  1569   assumes less: "i < j"

  1570   assumes base: "!!i. j = Suc i ==> P i"

  1571   assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"

  1572   shows "P i"

  1573   using less

  1574 proof (induct d=="j - i - 1" arbitrary: i)

  1575   case (0 i)

  1576   with i < j have "j = Suc i" by simp

  1577   with base show ?case by simp

  1578 next

  1579   case (Suc d i)

  1580   hence "i < j" "P (Suc i)"

  1581     by simp_all

  1582   thus "P i" by (rule step)

  1583 qed

  1584

  1585 lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"

  1586   using inc_induct[of "k - i" k P, simplified] by blast

  1587

  1588 lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"

  1589   using inc_induct[of 0 k P] by blast

  1590

  1591 (*The others are

  1592       i - j - k = i - (j + k),

  1593       k \<le> j ==> j - k + i = j + i - k,

  1594       k \<le> j ==> i + (j - k) = i + j - k *)

  1595 lemmas add_diff_assoc = diff_add_assoc [symmetric]

  1596 lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]

  1597 declare diff_diff_left [simp]  add_diff_assoc [simp] add_diff_assoc2[simp]

  1598

  1599 text{*At present we prove no analogue of @{text not_less_Least} or @{text

  1600 Least_Suc}, since there appears to be no need.*}

  1601

  1602

  1603 subsection {* The divides relation on @{typ nat} *}

  1604

  1605 lemma dvd_1_left [iff]: "Suc 0 dvd k"

  1606 unfolding dvd_def by simp

  1607

  1608 lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"

  1609 by (simp add: dvd_def)

  1610

  1611 lemma nat_dvd_1_iff_1 [simp]: "m dvd (1::nat) \<longleftrightarrow> m = 1"

  1612 by (simp add: dvd_def)

  1613

  1614 lemma dvd_antisym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"

  1615   unfolding dvd_def

  1616   by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)

  1617

  1618 text {* @{term "op dvd"} is a partial order *}

  1619

  1620 interpretation dvd: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n"

  1621   proof qed (auto intro: dvd_refl dvd_trans dvd_antisym)

  1622

  1623 lemma dvd_diff_nat[simp]: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"

  1624 unfolding dvd_def

  1625 by (blast intro: diff_mult_distrib2 [symmetric])

  1626

  1627 lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"

  1628   apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])

  1629   apply (blast intro: dvd_add)

  1630   done

  1631

  1632 lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"

  1633 by (drule_tac m = m in dvd_diff_nat, auto)

  1634

  1635 lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"

  1636   apply (rule iffI)

  1637    apply (erule_tac [2] dvd_add)

  1638    apply (rule_tac [2] dvd_refl)

  1639   apply (subgoal_tac "n = (n+k) -k")

  1640    prefer 2 apply simp

  1641   apply (erule ssubst)

  1642   apply (erule dvd_diff_nat)

  1643   apply (rule dvd_refl)

  1644   done

  1645

  1646 lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"

  1647   unfolding dvd_def

  1648   apply (erule exE)

  1649   apply (simp add: mult_ac)

  1650   done

  1651

  1652 lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"

  1653   apply auto

  1654    apply (subgoal_tac "m*n dvd m*1")

  1655    apply (drule dvd_mult_cancel, auto)

  1656   done

  1657

  1658 lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"

  1659   apply (subst mult_commute)

  1660   apply (erule dvd_mult_cancel1)

  1661   done

  1662

  1663 lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"

  1664 by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)

  1665

  1666 lemma nat_dvd_not_less:

  1667   fixes m n :: nat

  1668   shows "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"

  1669 by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)

  1670

  1671

  1672 subsection {* size of a datatype value *}

  1673

  1674 class size =

  1675   fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded} *}

  1676

  1677

  1678 subsection {* code module namespace *}

  1679

  1680 code_modulename SML

  1681   Nat Arith

  1682

  1683 code_modulename OCaml

  1684   Nat Arith

  1685

  1686 code_modulename Haskell

  1687   Nat Arith

  1688

  1689 end