src/HOL/Nat.thy
 author haftmann Fri Jun 19 07:53:35 2015 +0200 (2015-06-19) changeset 60517 f16e4fb20652 parent 60427 b4b672f09270 child 60562 24af00b010cf permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
     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 section {* 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 ML {*

   156 val nat_basic_lfp_sugar =

   157   let

   158     val ctr_sugar = the (Ctr_Sugar.ctr_sugar_of_global @{theory} @{type_name nat});

   159     val recx = Logic.varify_types_global @{term rec_nat};

   160     val C = body_type (fastype_of recx);

   161   in

   162     {T = HOLogic.natT, fp_res_index = 0, C = C, fun_arg_Tsss = [[], [[HOLogic.natT, C]]],

   163      ctr_sugar = ctr_sugar, recx = recx, rec_thms = @{thms nat.rec}}

   164   end;

   165 *}

   166

   167 setup {*

   168 let

   169   fun basic_lfp_sugars_of _ [@{typ nat}] _ _ ctxt =

   170       ([], [0], [nat_basic_lfp_sugar], [], [], TrueI (*dummy*), [], false, ctxt)

   171     | basic_lfp_sugars_of bs arg_Ts callers callssss ctxt =

   172       BNF_LFP_Rec_Sugar.default_basic_lfp_sugars_of bs arg_Ts callers callssss ctxt;

   173 in

   174   BNF_LFP_Rec_Sugar.register_lfp_rec_extension

   175     {nested_simps = [], is_new_datatype = K (K true), basic_lfp_sugars_of = basic_lfp_sugars_of,

   176      rewrite_nested_rec_call = NONE}

   177 end

   178 *}

   179

   180 text {* Injectiveness and distinctness lemmas *}

   181

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

   183   by (simp add: inj_on_def)

   184

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

   186 by (rule notE, rule Suc_not_Zero)

   187

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

   189 by (rule Suc_neq_Zero, erule sym)

   190

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

   192 by (rule inj_Suc [THEN injD])

   193

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

   195 by (induct n) simp_all

   196

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

   198 by (rule not_sym, rule n_not_Suc_n)

   199

   200 text {* A special form of induction for reasoning

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

   202

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

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

   205   apply (rule_tac x = m in spec)

   206   apply (induct n)

   207   prefer 2

   208   apply (rule allI)

   209   apply (induct_tac x, iprover+)

   210   done

   211

   212

   213 subsection {* Arithmetic operators *}

   214

   215 instantiation nat :: comm_monoid_diff

   216 begin

   217

   218 primrec plus_nat where

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

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

   221

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

   223   by (induct m) simp_all

   224

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

   226   by (induct m) simp_all

   227

   228 declare add_0 [code]

   229

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

   231   by simp

   232

   233 primrec minus_nat where

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

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

   236

   237 declare diff_Suc [simp del]

   238

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

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

   241

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

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

   244

   245 instance proof

   246   fix n m q :: nat

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

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

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

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

   251   show "0 + n = n" by simp

   252   show "0 - n = 0" by simp

   253 qed

   254

   255 end

   256

   257 hide_fact (open) add_0 add_0_right diff_0

   258

   259 instantiation nat :: comm_semiring_1_cancel

   260 begin

   261

   262 definition

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

   264

   265 primrec times_nat where

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

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

   268

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

   270   by (induct m) simp_all

   271

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

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

   274

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

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

   277

   278 instance proof

   279   fix n m q :: nat

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

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

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

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

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

   285 qed

   286

   287 end

   288

   289 subsubsection {* Addition *}

   290

   291 lemma nat_add_left_cancel:

   292   fixes k m n :: nat

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

   294   by (fact add_left_cancel)

   295

   296 lemma nat_add_right_cancel:

   297   fixes k m n :: nat

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

   299   by (fact add_right_cancel)

   300

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

   302

   303 lemma add_is_0 [iff]:

   304   fixes m n :: nat

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

   306   by (cases m) simp_all

   307

   308 lemma add_is_1:

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

   310   by (cases m) simp_all

   311

   312 lemma one_is_add:

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

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

   315

   316 lemma add_eq_self_zero:

   317   fixes m n :: nat

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

   319   by (induct m) simp_all

   320

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

   322   apply (induct k)

   323    apply simp

   324   apply(drule comp_inj_on[OF _ inj_Suc])

   325   apply (simp add:o_def)

   326   done

   327

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

   329   unfolding One_nat_def by simp

   330

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

   332   unfolding One_nat_def by simp

   333

   334

   335 subsubsection {* Difference *}

   336

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

   338   by (fact diff_cancel)

   339

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

   341   by (fact diff_diff_add)

   342

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

   344   by (simp add: diff_diff_left)

   345

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

   347   by (fact diff_right_commute)

   348

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

   350   by (fact add_diff_cancel_left')

   351

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

   353   by (fact add_diff_cancel_right')

   354

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

   356   by (fact add_diff_cancel_left)

   357

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

   359   by (fact add_diff_cancel_right)

   360

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

   362   by (fact diff_add_zero)

   363

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

   365   unfolding One_nat_def by simp

   366

   367 text {* Difference distributes over multiplication *}

   368

   369 instance nat :: comm_semiring_1_diff_distrib

   370 proof

   371   fix k m n :: nat

   372   show "k * ((m::nat) - n) = (k * m) - (k * n)"

   373     by (induct m n rule: diff_induct) simp_all

   374 qed

   375

   376 lemma diff_mult_distrib:

   377   "((m::nat) - n) * k = (m * k) - (n * k)"

   378   by (fact left_diff_distrib')

   379

   380 lemma diff_mult_distrib2:

   381   "k * ((m::nat) - n) = (k * m) - (k * n)"

   382   by (fact right_diff_distrib')

   383

   384

   385 subsubsection {* Multiplication *}

   386

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

   388   by (fact distrib_left)

   389

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

   391   by (induct m) auto

   392

   393 lemmas nat_distrib =

   394   add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2

   395

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

   397   apply (induct m)

   398    apply simp

   399   apply (induct n)

   400    apply auto

   401   done

   402

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

   404   apply (rule trans)

   405   apply (rule_tac [2] mult_eq_1_iff, fastforce)

   406   done

   407

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

   409   unfolding One_nat_def by (rule mult_eq_1_iff)

   410

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

   412   unfolding One_nat_def by (rule one_eq_mult_iff)

   413

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

   415 proof -

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

   417   proof (induct n arbitrary: m)

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

   419   next

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

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

   422   qed

   423   then show ?thesis by auto

   424 qed

   425

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

   427   by (simp add: mult.commute)

   428

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

   430   by (subst mult_cancel1) simp

   431

   432

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

   434

   435 subsubsection {* Operation definition *}

   436

   437 instantiation nat :: linorder

   438 begin

   439

   440 primrec less_eq_nat where

   441   "(0\<Colon>nat) \<le> n \<longleftrightarrow> True"

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

   443

   444 declare less_eq_nat.simps [simp del]

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

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

   447

   448 definition less_nat where

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

   450

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

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

   453

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

   455   unfolding less_eq_Suc_le ..

   456

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

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

   459

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

   461   by (simp add: less_eq_Suc_le)

   462

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

   464   by simp

   465

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

   467   by (simp add: less_eq_Suc_le)

   468

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

   470   by (simp add: less_eq_Suc_le)

   471

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

   473   by (cases m) auto

   474

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

   476   by (induct m arbitrary: n)

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

   478

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

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

   481

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

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

   484

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

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

   487

   488 instance

   489 proof

   490   fix n m :: nat

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

   492   proof (induct n arbitrary: m)

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

   494   next

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

   496   qed

   497 next

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

   499 next

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

   501   then show "n = m"

   502     by (induct n arbitrary: m)

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

   504 next

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

   506   then show "n \<le> q"

   507   proof (induct n arbitrary: m q)

   508     case 0 show ?case by simp

   509   next

   510     case (Suc n) then show ?case

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

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

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

   514   qed

   515 next

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

   517     by (induct n arbitrary: m)

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

   519 qed

   520

   521 end

   522

   523 instantiation nat :: order_bot

   524 begin

   525

   526 definition bot_nat :: nat where

   527   "bot_nat = 0"

   528

   529 instance proof

   530 qed (simp add: bot_nat_def)

   531

   532 end

   533

   534 instance nat :: no_top

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

   536

   537

   538 subsubsection {* Introduction properties *}

   539

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

   541   by (simp add: less_Suc_eq_le)

   542

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

   544   by (simp add: less_Suc_eq_le)

   545

   546

   547 subsubsection {* Elimination properties *}

   548

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

   550   by (rule order_less_irrefl)

   551

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

   553   by (rule not_sym) (rule less_imp_neq)

   554

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

   556   by (rule less_imp_neq)

   557

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

   559   by (rule notE, rule less_not_refl)

   560

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

   562   by (rule notE) (rule not_less0)

   563

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

   565   unfolding less_Suc_eq_le le_less ..

   566

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

   568   by (simp add: less_Suc_eq)

   569

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

   571   unfolding One_nat_def by (rule less_Suc0)

   572

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

   574   by simp

   575

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

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

   578   unfolding not_less less_Suc_eq_le by (rule antisym)

   579

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

   581   by (rule linorder_neq_iff)

   582

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

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

   585   shows "P n m"

   586   apply (rule less_linear [THEN disjE])

   587   apply (erule_tac [2] disjE)

   588   apply (erule lessCase)

   589   apply (erule sym [THEN eqCase])

   590   apply (erule major)

   591   done

   592

   593

   594 subsubsection {* Inductive (?) properties *}

   595

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

   597   unfolding less_eq_Suc_le [of m] le_less by simp

   598

   599 lemma lessE:

   600   assumes major: "i < k"

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

   602   shows P

   603 proof -

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

   605     unfolding less_eq_Suc_le by (induct k) simp_all

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

   607     by (clarsimp simp add: less_le)

   608   with p1 p2 show P by auto

   609 qed

   610

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

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

   613   apply (rule major [THEN lessE])

   614   apply (rule eq, blast)

   615   apply (rule less, blast)

   616   done

   617

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

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

   620   apply (rule major [THEN lessE])

   621   apply (erule lessI [THEN minor])

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

   623   done

   624

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

   626   by simp

   627

   628 lemma less_trans_Suc:

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

   630   apply (induct k, simp_all)

   631   apply (insert le)

   632   apply (simp add: less_Suc_eq)

   633   apply (blast dest: Suc_lessD)

   634   done

   635

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

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

   638   unfolding not_less less_Suc_eq_le ..

   639

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

   641   unfolding not_le Suc_le_eq ..

   642

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

   644

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

   646   unfolding less_Suc_eq_le .

   647

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

   649   unfolding not_le less_Suc_eq_le ..

   650

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

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

   653

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

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

   656

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

   658   unfolding Suc_le_eq .

   659

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

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

   662   unfolding Suc_le_eq .

   663

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

   665   unfolding less_eq_Suc_le by (rule Suc_leD)

   666

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

   668 lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq

   669

   670

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

   672

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

   674   unfolding le_less .

   675

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

   677   by (rule le_less)

   678

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

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

   681   by auto

   682

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

   684   by simp

   685

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

   687   by (rule order_trans)

   688

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

   690   by (rule antisym)

   691

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

   693   by (rule less_le)

   694

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

   696   unfolding less_le ..

   697

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

   699   by (rule linear)

   700

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

   702

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

   704   unfolding less_Suc_eq_le by auto

   705

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

   707   unfolding not_less by (rule le_less_Suc_eq)

   708

   709 lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq

   710

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

   712 by (cases n) simp_all

   713

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

   715 by (cases n) simp_all

   716

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

   718 by (cases n) simp_all

   719

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

   721 by (cases n) simp_all

   722

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

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

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

   726

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

   728 by (fast intro: not0_implies_Suc)

   729

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

   731 using neq0_conv by blast

   732

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

   734 by (induct m') simp_all

   735

   736 text {* Useful in certain inductive arguments *}

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

   738 by (cases m) simp_all

   739

   740

   741 subsubsection {* Monotonicity of Addition *}

   742

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

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

   745

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

   747 unfolding One_nat_def by (rule Suc_pred)

   748

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

   750 by (induct k) simp_all

   751

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

   753 by (induct k) simp_all

   754

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

   756 by(auto dest:gr0_implies_Suc)

   757

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

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

   760 by (induct k) simp_all

   761

   762 text {* strict, in both arguments *}

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

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

   765   apply (induct j, simp_all)

   766   done

   767

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

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

   770   apply (induct n)

   771   apply (simp_all add: order_le_less)

   772   apply (blast elim!: less_SucE

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

   774   done

   775

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

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

   778

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

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

   781 apply(auto simp: gr0_conv_Suc)

   782 apply (induct_tac m)

   783 apply (simp_all add: add_less_mono)

   784 done

   785

   786 text{*The naturals form an ordered @{text semidom}*}

   787 instance nat :: linordered_semidom

   788 proof

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

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

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

   792   show "\<And>m n :: nat. m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> m * n \<noteq> 0" by simp

   793 qed

   794

   795

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

   797

   798 lemma mono_Suc: "mono Suc"

   799 by (rule monoI) simp

   800

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

   802 by (rule min_absorb1) simp

   803

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

   805 by (rule min_absorb2) simp

   806

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

   808 by (simp add: mono_Suc min_of_mono)

   809

   810 lemma min_Suc1:

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

   812 by (simp split: nat.split)

   813

   814 lemma min_Suc2:

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

   816 by (simp split: nat.split)

   817

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

   819 by (rule max_absorb2) simp

   820

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

   822 by (rule max_absorb1) simp

   823

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

   825 by (simp add: mono_Suc max_of_mono)

   826

   827 lemma max_Suc1:

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

   829 by (simp split: nat.split)

   830

   831 lemma max_Suc2:

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

   833 by (simp split: nat.split)

   834

   835 lemma nat_mult_min_left:

   836   fixes m n q :: nat

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

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

   839

   840 lemma nat_mult_min_right:

   841   fixes m n q :: nat

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

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

   844

   845 lemma nat_add_max_left:

   846   fixes m n q :: nat

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

   848   by (simp add: max_def)

   849

   850 lemma nat_add_max_right:

   851   fixes m n q :: nat

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

   853   by (simp add: max_def)

   854

   855 lemma nat_mult_max_left:

   856   fixes m n q :: nat

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

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

   859

   860 lemma nat_mult_max_right:

   861   fixes m n q :: nat

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

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

   864

   865

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

   867

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

   869

   870 instance nat :: wellorder proof

   871   fix P and n :: nat

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

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

   874   proof (induct n)

   875     case (0 n)

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

   877     thus ?case using 0 by auto

   878   next

   879     case (Suc m n)

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

   881     thus ?case

   882     proof

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

   884     next

   885       assume n: "n = Suc m"

   886       show "P n"

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

   888     qed

   889   qed

   890   then show "P n" by auto

   891 qed

   892

   893

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

   895 by (rule Least_equality[OF _ le0])

   896

   897 lemma Least_Suc:

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

   899   apply (cases n, auto)

   900   apply (frule LeastI)

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

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

   903   apply (erule_tac [2] Least_le)

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

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

   906   apply (blast intro: order_antisym)

   907   done

   908

   909 lemma Least_Suc2:

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

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

   912   apply simp

   913   done

   914

   915 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)"

   916   apply (cases n)

   917    apply blast

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

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

   920   done

   921

   922 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)"

   923   unfolding One_nat_def

   924   apply (cases n)

   925    apply blast

   926   apply (frule (1) ex_least_nat_le)

   927   apply (erule exE)

   928   apply (case_tac k)

   929    apply simp

   930   apply (rename_tac k1)

   931   apply (rule_tac x=k1 in exI)

   932   apply (auto simp add: less_eq_Suc_le)

   933   done

   934

   935 lemma nat_less_induct:

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

   937   using assms less_induct by blast

   938

   939 lemma measure_induct_rule [case_names less]:

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

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

   942   shows "P a"

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

   944

   945 text {* old style induction rules: *}

   946 lemma measure_induct:

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

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

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

   950

   951 lemma full_nat_induct:

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

   953   shows "P n"

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

   955

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

   957 lemma less_Suc_induct:

   958   assumes less:  "i < j"

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

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

   961   shows "P i j"

   962 proof -

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

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

   965   proof (induct k)

   966     case 0

   967     show ?case by (simp add: step)

   968   next

   969     case (Suc k)

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

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

   972     from trans[OF this lessI Suc step]

   973     show ?case by simp

   974   qed

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

   976 qed

   977

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

   979 Provided by Roelof Oosterhuis.

   980 $P(n)$ is true for all $n\in\mathbb{N}$ if

   981 \begin{itemize}

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

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

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

   985 \end{itemize} *}

   986

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

   988 lemma infinite_descent:

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

   990 by (induct n rule: less_induct) auto

   991

   992 lemma infinite_descent0[case_names 0 smaller]:

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

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

   995

   996 text {*

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

   998 $P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and

   999 \begin{itemize}

  1000 \item case 0'': given $V(x)=0$ prove $P(x)$,

  1001 \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 $~\neg P(y)$.

  1002 \end{itemize}

  1003 NB: the proof also shows how to use the previous lemma. *}

  1004

  1005 corollary infinite_descent0_measure [case_names 0 smaller]:

  1006   assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"

  1007     and   A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"

  1008   shows "P x"

  1009 proof -

  1010   obtain n where "n = V x" by auto

  1011   moreover have "\<And>x. V x = n \<Longrightarrow> P x"

  1012   proof (induct n rule: infinite_descent0)

  1013     case 0 -- "i.e. $V(x) = 0$"

  1014     with A0 show "P x" by auto

  1015   next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"

  1016     case (smaller n)

  1017     then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto

  1018     with A1 obtain y where "V y < V x \<and> \<not> P y" by auto

  1019     with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto

  1020     then show ?case by auto

  1021   qed

  1022   ultimately show "P x" by auto

  1023 qed

  1024

  1025 text{* Again, without explicit base case: *}

  1026 lemma infinite_descent_measure:

  1027 assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"

  1028 proof -

  1029   from assms obtain n where "n = V x" by auto

  1030   moreover have "!!x. V x = n \<Longrightarrow> P x"

  1031   proof (induct n rule: infinite_descent, auto)

  1032     fix x assume "\<not> P x"

  1033     with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto

  1034   qed

  1035   ultimately show "P x" by auto

  1036 qed

  1037

  1038 text {* A [clumsy] way of lifting @{text "<"}

  1039   monotonicity to @{text "\<le>"} monotonicity *}

  1040 lemma less_mono_imp_le_mono:

  1041   "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"

  1042 by (simp add: order_le_less) (blast)

  1043

  1044

  1045 text {* non-strict, in 1st argument *}

  1046 lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"

  1047 by (rule add_right_mono)

  1048

  1049 text {* non-strict, in both arguments *}

  1050 lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"

  1051 by (rule add_mono)

  1052

  1053 lemma le_add2: "n \<le> ((m + n)::nat)"

  1054 by (insert add_right_mono [of 0 m n], simp)

  1055

  1056 lemma le_add1: "n \<le> ((n + m)::nat)"

  1057 by (simp add: add.commute, rule le_add2)

  1058

  1059 lemma less_add_Suc1: "i < Suc (i + m)"

  1060 by (rule le_less_trans, rule le_add1, rule lessI)

  1061

  1062 lemma less_add_Suc2: "i < Suc (m + i)"

  1063 by (rule le_less_trans, rule le_add2, rule lessI)

  1064

  1065 lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"

  1066 by (iprover intro!: less_add_Suc1 less_imp_Suc_add)

  1067

  1068 lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"

  1069 by (rule le_trans, assumption, rule le_add1)

  1070

  1071 lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"

  1072 by (rule le_trans, assumption, rule le_add2)

  1073

  1074 lemma trans_less_add1: "(i::nat) < j ==> i < j + m"

  1075 by (rule less_le_trans, assumption, rule le_add1)

  1076

  1077 lemma trans_less_add2: "(i::nat) < j ==> i < m + j"

  1078 by (rule less_le_trans, assumption, rule le_add2)

  1079

  1080 lemma add_lessD1: "i + j < (k::nat) ==> i < k"

  1081 apply (rule le_less_trans [of _ "i+j"])

  1082 apply (simp_all add: le_add1)

  1083 done

  1084

  1085 lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"

  1086 apply (rule notI)

  1087 apply (drule add_lessD1)

  1088 apply (erule less_irrefl [THEN notE])

  1089 done

  1090

  1091 lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"

  1092 by (simp add: add.commute)

  1093

  1094 lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"

  1095 apply (rule order_trans [of _ "m+k"])

  1096 apply (simp_all add: le_add1)

  1097 done

  1098

  1099 lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"

  1100 apply (simp add: add.commute)

  1101 apply (erule add_leD1)

  1102 done

  1103

  1104 lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"

  1105 by (blast dest: add_leD1 add_leD2)

  1106

  1107 text {* needs @{text "!!k"} for @{text ac_simps} to work *}

  1108 lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"

  1109 by (force simp del: add_Suc_right

  1110     simp add: less_iff_Suc_add add_Suc_right [symmetric] ac_simps)

  1111

  1112

  1113 subsubsection {* More results about difference *}

  1114

  1115 text {* Addition is the inverse of subtraction:

  1116   if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}

  1117 lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"

  1118 by (induct m n rule: diff_induct) simp_all

  1119

  1120 lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"

  1121 by (simp add: add_diff_inverse linorder_not_less)

  1122

  1123 lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"

  1124 by (simp add: add.commute)

  1125

  1126 lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"

  1127 by (induct m n rule: diff_induct) simp_all

  1128

  1129 lemma diff_less_Suc: "m - n < Suc m"

  1130 apply (induct m n rule: diff_induct)

  1131 apply (erule_tac [3] less_SucE)

  1132 apply (simp_all add: less_Suc_eq)

  1133 done

  1134

  1135 lemma diff_le_self [simp]: "m - n \<le> (m::nat)"

  1136 by (induct m n rule: diff_induct) (simp_all add: le_SucI)

  1137

  1138 lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"

  1139   by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])

  1140

  1141 instance nat :: ordered_cancel_comm_monoid_diff

  1142 proof

  1143   show "\<And>m n :: nat. m \<le> n \<longleftrightarrow> (\<exists>q. n = m + q)" by (fact le_iff_add)

  1144 qed

  1145

  1146 lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"

  1147 by (rule le_less_trans, rule diff_le_self)

  1148

  1149 lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"

  1150 by (cases n) (auto simp add: le_simps)

  1151

  1152 lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"

  1153 by (induct j k rule: diff_induct) simp_all

  1154

  1155 lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"

  1156 by (simp add: add.commute diff_add_assoc)

  1157

  1158 lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"

  1159 by (auto simp add: diff_add_inverse2)

  1160

  1161 lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"

  1162 by (induct m n rule: diff_induct) simp_all

  1163

  1164 lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"

  1165 by (rule iffD2, rule diff_is_0_eq)

  1166

  1167 lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"

  1168 by (induct m n rule: diff_induct) simp_all

  1169

  1170 lemma less_imp_add_positive:

  1171   assumes "i < j"

  1172   shows "\<exists>k::nat. 0 < k & i + k = j"

  1173 proof

  1174   from assms show "0 < j - i & i + (j - i) = j"

  1175     by (simp add: order_less_imp_le)

  1176 qed

  1177

  1178 text {* a nice rewrite for bounded subtraction *}

  1179 lemma nat_minus_add_max:

  1180   fixes n m :: nat

  1181   shows "n - m + m = max n m"

  1182     by (simp add: max_def not_le order_less_imp_le)

  1183

  1184 lemma nat_diff_split:

  1185   "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"

  1186     -- {* elimination of @{text -} on @{text nat} *}

  1187 by (cases "a < b")

  1188   (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse

  1189     not_less le_less dest!: add_eq_self_zero add_eq_self_zero[OF sym])

  1190

  1191 lemma nat_diff_split_asm:

  1192   "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"

  1193     -- {* elimination of @{text -} on @{text nat} in assumptions *}

  1194 by (auto split: nat_diff_split)

  1195

  1196 lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"

  1197   by simp

  1198

  1199 lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"

  1200   unfolding One_nat_def by (cases m) simp_all

  1201

  1202 lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"

  1203   unfolding One_nat_def by (cases m) simp_all

  1204

  1205 lemma Suc_diff_eq_diff_pred: "0 < n ==> Suc m - n = m - (n - 1)"

  1206   unfolding One_nat_def by (cases n) simp_all

  1207

  1208 lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"

  1209   unfolding One_nat_def by (cases m) simp_all

  1210

  1211 lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"

  1212   by (fact Let_def)

  1213

  1214

  1215 subsubsection {* Monotonicity of multiplication *}

  1216

  1217 lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"

  1218 by (simp add: mult_right_mono)

  1219

  1220 lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"

  1221 by (simp add: mult_left_mono)

  1222

  1223 text {* @{text "\<le>"} monotonicity, BOTH arguments *}

  1224 lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"

  1225 by (simp add: mult_mono)

  1226

  1227 lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"

  1228 by (simp add: mult_strict_right_mono)

  1229

  1230 text{*Differs from the standard @{text zero_less_mult_iff} in that

  1231       there are no negative numbers.*}

  1232 lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"

  1233   apply (induct m)

  1234    apply simp

  1235   apply (case_tac n)

  1236    apply simp_all

  1237   done

  1238

  1239 lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)"

  1240   apply (induct m)

  1241    apply simp

  1242   apply (case_tac n)

  1243    apply simp_all

  1244   done

  1245

  1246 lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"

  1247   apply (safe intro!: mult_less_mono1)

  1248   apply (cases k, auto)

  1249   apply (simp del: le_0_eq add: linorder_not_le [symmetric])

  1250   apply (blast intro: mult_le_mono1)

  1251   done

  1252

  1253 lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"

  1254 by (simp add: mult.commute [of k])

  1255

  1256 lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"

  1257 by (simp add: linorder_not_less [symmetric], auto)

  1258

  1259 lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"

  1260 by (simp add: linorder_not_less [symmetric], auto)

  1261

  1262 lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"

  1263 by (subst mult_less_cancel1) simp

  1264

  1265 lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"

  1266 by (subst mult_le_cancel1) simp

  1267

  1268 lemma le_square: "m \<le> m * (m::nat)"

  1269   by (cases m) (auto intro: le_add1)

  1270

  1271 lemma le_cube: "(m::nat) \<le> m * (m * m)"

  1272   by (cases m) (auto intro: le_add1)

  1273

  1274 text {* Lemma for @{text gcd} *}

  1275 lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"

  1276   apply (drule sym)

  1277   apply (rule disjCI)

  1278   apply (rule nat_less_cases, erule_tac [2] _)

  1279    apply (drule_tac [2] mult_less_mono2)

  1280     apply (auto)

  1281   done

  1282

  1283 lemma mono_times_nat:

  1284   fixes n :: nat

  1285   assumes "n > 0"

  1286   shows "mono (times n)"

  1287 proof

  1288   fix m q :: nat

  1289   assume "m \<le> q"

  1290   with assms show "n * m \<le> n * q" by simp

  1291 qed

  1292

  1293 text {* the lattice order on @{typ nat} *}

  1294

  1295 instantiation nat :: distrib_lattice

  1296 begin

  1297

  1298 definition

  1299   "(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min"

  1300

  1301 definition

  1302   "(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max"

  1303

  1304 instance by intro_classes

  1305   (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def

  1306     intro: order_less_imp_le antisym elim!: order_trans order_less_trans)

  1307

  1308 end

  1309

  1310

  1311 subsection {* Natural operation of natural numbers on functions *}

  1312

  1313 text {*

  1314   We use the same logical constant for the power operations on

  1315   functions and relations, in order to share the same syntax.

  1316 *}

  1317

  1318 consts compow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"

  1319

  1320 abbreviation compower :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^^" 80) where

  1321   "f ^^ n \<equiv> compow n f"

  1322

  1323 notation (latex output)

  1324   compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)

  1325

  1326 notation (HTML output)

  1327   compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)

  1328

  1329 text {* @{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f} *}

  1330

  1331 overloading

  1332   funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"

  1333 begin

  1334

  1335 primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where

  1336   "funpow 0 f = id"

  1337 | "funpow (Suc n) f = f o funpow n f"

  1338

  1339 end

  1340

  1341 lemma funpow_Suc_right:

  1342   "f ^^ Suc n = f ^^ n \<circ> f"

  1343 proof (induct n)

  1344   case 0 then show ?case by simp

  1345 next

  1346   fix n

  1347   assume "f ^^ Suc n = f ^^ n \<circ> f"

  1348   then show "f ^^ Suc (Suc n) = f ^^ Suc n \<circ> f"

  1349     by (simp add: o_assoc)

  1350 qed

  1351

  1352 lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right

  1353

  1354 text {* for code generation *}

  1355

  1356 definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where

  1357   funpow_code_def [code_abbrev]: "funpow = compow"

  1358

  1359 lemma [code]:

  1360   "funpow (Suc n) f = f o funpow n f"

  1361   "funpow 0 f = id"

  1362   by (simp_all add: funpow_code_def)

  1363

  1364 hide_const (open) funpow

  1365

  1366 lemma funpow_add:

  1367   "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"

  1368   by (induct m) simp_all

  1369

  1370 lemma funpow_mult:

  1371   fixes f :: "'a \<Rightarrow> 'a"

  1372   shows "(f ^^ m) ^^ n = f ^^ (m * n)"

  1373   by (induct n) (simp_all add: funpow_add)

  1374

  1375 lemma funpow_swap1:

  1376   "f ((f ^^ n) x) = (f ^^ n) (f x)"

  1377 proof -

  1378   have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp

  1379   also have "\<dots>  = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)

  1380   also have "\<dots> = (f ^^ n) (f x)" by simp

  1381   finally show ?thesis .

  1382 qed

  1383

  1384 lemma comp_funpow:

  1385   fixes f :: "'a \<Rightarrow> 'a"

  1386   shows "comp f ^^ n = comp (f ^^ n)"

  1387   by (induct n) simp_all

  1388

  1389 lemma Suc_funpow[simp]: "Suc ^^ n = (op + n)"

  1390   by (induct n) simp_all

  1391

  1392 lemma id_funpow[simp]: "id ^^ n = id"

  1393   by (induct n) simp_all

  1394

  1395 lemma funpow_mono:

  1396   fixes f :: "'a \<Rightarrow> ('a::lattice)"

  1397   shows "mono f \<Longrightarrow> A \<le> B \<Longrightarrow> (f ^^ n) A \<le> (f ^^ n) B"

  1398   by (induct n arbitrary: A B)

  1399      (auto simp del: funpow.simps(2) simp add: funpow_Suc_right mono_def)

  1400

  1401 subsection {* Kleene iteration *}

  1402

  1403 lemma Kleene_iter_lpfp:

  1404 assumes "mono f" and "f p \<le> p" shows "(f^^k) (bot::'a::order_bot) \<le> p"

  1405 proof(induction k)

  1406   case 0 show ?case by simp

  1407 next

  1408   case Suc

  1409   from monoD[OF assms(1) Suc] assms(2)

  1410   show ?case by simp

  1411 qed

  1412

  1413 lemma lfp_Kleene_iter: assumes "mono f" and "(f^^Suc k) bot = (f^^k) bot"

  1414 shows "lfp f = (f^^k) bot"

  1415 proof(rule antisym)

  1416   show "lfp f \<le> (f^^k) bot"

  1417   proof(rule lfp_lowerbound)

  1418     show "f ((f^^k) bot) \<le> (f^^k) bot" using assms(2) by simp

  1419   qed

  1420 next

  1421   show "(f^^k) bot \<le> lfp f"

  1422     using Kleene_iter_lpfp[OF assms(1)] lfp_unfold[OF assms(1)] by simp

  1423 qed

  1424

  1425

  1426 subsection {* Embedding of the naturals into any @{text semiring_1}: @{term of_nat} *}

  1427

  1428 context semiring_1

  1429 begin

  1430

  1431 definition of_nat :: "nat \<Rightarrow> 'a" where

  1432   "of_nat n = (plus 1 ^^ n) 0"

  1433

  1434 lemma of_nat_simps [simp]:

  1435   shows of_nat_0: "of_nat 0 = 0"

  1436     and of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"

  1437   by (simp_all add: of_nat_def)

  1438

  1439 lemma of_nat_1 [simp]: "of_nat 1 = 1"

  1440   by (simp add: of_nat_def)

  1441

  1442 lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"

  1443   by (induct m) (simp_all add: ac_simps)

  1444

  1445 lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"

  1446   by (induct m) (simp_all add: ac_simps distrib_right)

  1447

  1448 primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where

  1449   "of_nat_aux inc 0 i = i"

  1450 | "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- {* tail recursive *}

  1451

  1452 lemma of_nat_code:

  1453   "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"

  1454 proof (induct n)

  1455   case 0 then show ?case by simp

  1456 next

  1457   case (Suc n)

  1458   have "\<And>i. of_nat_aux (\<lambda>i. i + 1) n (i + 1) = of_nat_aux (\<lambda>i. i + 1) n i + 1"

  1459     by (induct n) simp_all

  1460   from this [of 0] have "of_nat_aux (\<lambda>i. i + 1) n 1 = of_nat_aux (\<lambda>i. i + 1) n 0 + 1"

  1461     by simp

  1462   with Suc show ?case by (simp add: add.commute)

  1463 qed

  1464

  1465 end

  1466

  1467 declare of_nat_code [code]

  1468

  1469 text{*Class for unital semirings with characteristic zero.

  1470  Includes non-ordered rings like the complex numbers.*}

  1471

  1472 class semiring_char_0 = semiring_1 +

  1473   assumes inj_of_nat: "inj of_nat"

  1474 begin

  1475

  1476 lemma of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"

  1477   by (auto intro: inj_of_nat injD)

  1478

  1479 text{*Special cases where either operand is zero*}

  1480

  1481 lemma of_nat_0_eq_iff [simp]: "0 = of_nat n \<longleftrightarrow> 0 = n"

  1482   by (fact of_nat_eq_iff [of 0 n, unfolded of_nat_0])

  1483

  1484 lemma of_nat_eq_0_iff [simp]: "of_nat m = 0 \<longleftrightarrow> m = 0"

  1485   by (fact of_nat_eq_iff [of m 0, unfolded of_nat_0])

  1486

  1487 lemma of_nat_neq_0 [simp]:

  1488   "of_nat (Suc n) \<noteq> 0"

  1489   unfolding of_nat_eq_0_iff by simp

  1490

  1491 lemma of_nat_0_neq [simp]:

  1492   "0 \<noteq> of_nat (Suc n)"

  1493   unfolding of_nat_0_eq_iff by simp

  1494

  1495 end

  1496

  1497 context linordered_semidom

  1498 begin

  1499

  1500 lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"

  1501   by (induct n) simp_all

  1502

  1503 lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"

  1504   by (simp add: not_less)

  1505

  1506 lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"

  1507   by (induct m n rule: diff_induct, simp_all add: add_pos_nonneg)

  1508

  1509 lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"

  1510   by (simp add: not_less [symmetric] linorder_not_less [symmetric])

  1511

  1512 lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"

  1513   by simp

  1514

  1515 lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"

  1516   by simp

  1517

  1518 text{*Every @{text linordered_semidom} has characteristic zero.*}

  1519

  1520 subclass semiring_char_0 proof

  1521 qed (auto intro!: injI simp add: eq_iff)

  1522

  1523 text{*Special cases where either operand is zero*}

  1524

  1525 lemma of_nat_le_0_iff [simp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"

  1526   by (rule of_nat_le_iff [of _ 0, simplified])

  1527

  1528 lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"

  1529   by (rule of_nat_less_iff [of 0, simplified])

  1530

  1531 end

  1532

  1533 context ring_1

  1534 begin

  1535

  1536 lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"

  1537 by (simp add: algebra_simps of_nat_add [symmetric])

  1538

  1539 end

  1540

  1541 context linordered_idom

  1542 begin

  1543

  1544 lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"

  1545   unfolding abs_if by auto

  1546

  1547 end

  1548

  1549 lemma of_nat_id [simp]: "of_nat n = n"

  1550   by (induct n) simp_all

  1551

  1552 lemma of_nat_eq_id [simp]: "of_nat = id"

  1553   by (auto simp add: fun_eq_iff)

  1554

  1555

  1556 subsection {* The set of natural numbers *}

  1557

  1558 context semiring_1

  1559 begin

  1560

  1561 definition Nats  :: "'a set" where

  1562   "Nats = range of_nat"

  1563

  1564 notation (xsymbols)

  1565   Nats  ("\<nat>")

  1566

  1567 lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"

  1568   by (simp add: Nats_def)

  1569

  1570 lemma Nats_0 [simp]: "0 \<in> \<nat>"

  1571 apply (simp add: Nats_def)

  1572 apply (rule range_eqI)

  1573 apply (rule of_nat_0 [symmetric])

  1574 done

  1575

  1576 lemma Nats_1 [simp]: "1 \<in> \<nat>"

  1577 apply (simp add: Nats_def)

  1578 apply (rule range_eqI)

  1579 apply (rule of_nat_1 [symmetric])

  1580 done

  1581

  1582 lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"

  1583 apply (auto simp add: Nats_def)

  1584 apply (rule range_eqI)

  1585 apply (rule of_nat_add [symmetric])

  1586 done

  1587

  1588 lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"

  1589 apply (auto simp add: Nats_def)

  1590 apply (rule range_eqI)

  1591 apply (rule of_nat_mult [symmetric])

  1592 done

  1593

  1594 lemma Nats_cases [cases set: Nats]:

  1595   assumes "x \<in> \<nat>"

  1596   obtains (of_nat) n where "x = of_nat n"

  1597   unfolding Nats_def

  1598 proof -

  1599   from x \<in> \<nat> have "x \<in> range of_nat" unfolding Nats_def .

  1600   then obtain n where "x = of_nat n" ..

  1601   then show thesis ..

  1602 qed

  1603

  1604 lemma Nats_induct [case_names of_nat, induct set: Nats]:

  1605   "x \<in> \<nat> \<Longrightarrow> (\<And>n. P (of_nat n)) \<Longrightarrow> P x"

  1606   by (rule Nats_cases) auto

  1607

  1608 end

  1609

  1610

  1611 subsection {* Further arithmetic facts concerning the natural numbers *}

  1612

  1613 lemma subst_equals:

  1614   assumes 1: "t = s" and 2: "u = t"

  1615   shows "u = s"

  1616   using 2 1 by (rule trans)

  1617

  1618 ML_file "Tools/nat_arith.ML"

  1619

  1620 simproc_setup nateq_cancel_sums

  1621   ("(l::nat) + m = n" | "(l::nat) = m + n" | "Suc m = n" | "m = Suc n") =

  1622   {* fn phi => try o Nat_Arith.cancel_eq_conv *}

  1623

  1624 simproc_setup natless_cancel_sums

  1625   ("(l::nat) + m < n" | "(l::nat) < m + n" | "Suc m < n" | "m < Suc n") =

  1626   {* fn phi => try o Nat_Arith.cancel_less_conv *}

  1627

  1628 simproc_setup natle_cancel_sums

  1629   ("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" | "Suc m \<le> n" | "m \<le> Suc n") =

  1630   {* fn phi => try o Nat_Arith.cancel_le_conv *}

  1631

  1632 simproc_setup natdiff_cancel_sums

  1633   ("(l::nat) + m - n" | "(l::nat) - (m + n)" | "Suc m - n" | "m - Suc n") =

  1634   {* fn phi => try o Nat_Arith.cancel_diff_conv *}

  1635

  1636 ML_file "Tools/lin_arith.ML"

  1637 setup {* Lin_Arith.global_setup *}

  1638 declaration {* K Lin_Arith.setup *}

  1639

  1640 simproc_setup fast_arith_nat ("(m::nat) < n" | "(m::nat) <= n" | "(m::nat) = n") =

  1641   {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (Thm.term_of ct) *}

  1642 (* Because of this simproc, the arithmetic solver is really only

  1643 useful to detect inconsistencies among the premises for subgoals which are

  1644 *not* themselves (in)equalities, because the latter activate

  1645 fast_nat_arith_simproc anyway. However, it seems cheaper to activate the

  1646 solver all the time rather than add the additional check. *)

  1647

  1648

  1649 lemmas [arith_split] = nat_diff_split split_min split_max

  1650

  1651 context order

  1652 begin

  1653

  1654 lemma lift_Suc_mono_le:

  1655   assumes mono: "\<And>n. f n \<le> f (Suc n)" and "n \<le> n'"

  1656   shows "f n \<le> f n'"

  1657 proof (cases "n < n'")

  1658   case True

  1659   then show ?thesis

  1660     by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)

  1661 qed (insert n \<le> n', auto) -- {* trivial for @{prop "n = n'"} *}

  1662

  1663 lemma lift_Suc_antimono_le:

  1664   assumes mono: "\<And>n. f n \<ge> f (Suc n)" and "n \<le> n'"

  1665   shows "f n \<ge> f n'"

  1666 proof (cases "n < n'")

  1667   case True

  1668   then show ?thesis

  1669     by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)

  1670 qed (insert n \<le> n', auto) -- {* trivial for @{prop "n = n'"} *}

  1671

  1672 lemma lift_Suc_mono_less:

  1673   assumes mono: "\<And>n. f n < f (Suc n)" and "n < n'"

  1674   shows "f n < f n'"

  1675 using n < n'

  1676 by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)

  1677

  1678 lemma lift_Suc_mono_less_iff:

  1679   "(\<And>n. f n < f (Suc n)) \<Longrightarrow> f n < f m \<longleftrightarrow> n < m"

  1680   by (blast intro: less_asym' lift_Suc_mono_less [of f]

  1681     dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq [THEN iffD1])

  1682

  1683 end

  1684

  1685 lemma mono_iff_le_Suc:

  1686   "mono f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"

  1687   unfolding mono_def by (auto intro: lift_Suc_mono_le [of f])

  1688

  1689 lemma antimono_iff_le_Suc:

  1690   "antimono f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"

  1691   unfolding antimono_def by (auto intro: lift_Suc_antimono_le [of f])

  1692

  1693 lemma mono_nat_linear_lb:

  1694   fixes f :: "nat \<Rightarrow> nat"

  1695   assumes "\<And>m n. m < n \<Longrightarrow> f m < f n"

  1696   shows "f m + k \<le> f (m + k)"

  1697 proof (induct k)

  1698   case 0 then show ?case by simp

  1699 next

  1700   case (Suc k)

  1701   then have "Suc (f m + k) \<le> Suc (f (m + k))" by simp

  1702   also from assms [of "m + k" "Suc (m + k)"] have "Suc (f (m + k)) \<le> f (Suc (m + k))"

  1703     by (simp add: Suc_le_eq)

  1704   finally show ?case by simp

  1705 qed

  1706

  1707

  1708 text{*Subtraction laws, mostly by Clemens Ballarin*}

  1709

  1710 lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"

  1711 by arith

  1712

  1713 lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"

  1714 by arith

  1715

  1716 lemma less_diff_conv2:

  1717   fixes j k i :: nat

  1718   assumes "k \<le> j"

  1719   shows "j - k < i \<longleftrightarrow> j < i + k"

  1720   using assms by arith

  1721

  1722 lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"

  1723 by arith

  1724

  1725 lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"

  1726   by (fact le_diff_conv2) -- {* FIXME delete *}

  1727

  1728 lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"

  1729 by arith

  1730

  1731 lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"

  1732   by (fact le_add_diff) -- {* FIXME delete *}

  1733

  1734 (*Replaces the previous diff_less and le_diff_less, which had the stronger

  1735   second premise n\<le>m*)

  1736 lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"

  1737 by arith

  1738

  1739 text {* Simplification of relational expressions involving subtraction *}

  1740

  1741 lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"

  1742 by (simp split add: nat_diff_split)

  1743

  1744 hide_fact (open) diff_diff_eq

  1745

  1746 lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"

  1747 by (auto split add: nat_diff_split)

  1748

  1749 lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"

  1750 by (auto split add: nat_diff_split)

  1751

  1752 lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"

  1753 by (auto split add: nat_diff_split)

  1754

  1755 text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}

  1756

  1757 (* Monotonicity of subtraction in first argument *)

  1758 lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"

  1759 by (simp split add: nat_diff_split)

  1760

  1761 lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"

  1762 by (simp split add: nat_diff_split)

  1763

  1764 lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"

  1765 by (simp split add: nat_diff_split)

  1766

  1767 lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"

  1768 by (simp split add: nat_diff_split)

  1769

  1770 lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"

  1771 by auto

  1772

  1773 lemma inj_on_diff_nat:

  1774   assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"

  1775   shows "inj_on (\<lambda>n. n - k) N"

  1776 proof (rule inj_onI)

  1777   fix x y

  1778   assume a: "x \<in> N" "y \<in> N" "x - k = y - k"

  1779   with k_le_n have "x - k + k = y - k + k" by auto

  1780   with a k_le_n show "x = y" by auto

  1781 qed

  1782

  1783 text{*Rewriting to pull differences out*}

  1784

  1785 lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"

  1786 by arith

  1787

  1788 lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"

  1789 by arith

  1790

  1791 lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"

  1792 by arith

  1793

  1794 lemma Suc_diff_Suc: "n < m \<Longrightarrow> Suc (m - Suc n) = m - n"

  1795 by simp

  1796

  1797 (*The others are

  1798       i - j - k = i - (j + k),

  1799       k \<le> j ==> j - k + i = j + i - k,

  1800       k \<le> j ==> i + (j - k) = i + j - k *)

  1801 lemmas add_diff_assoc = diff_add_assoc [symmetric]

  1802 lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]

  1803 declare diff_diff_left [simp]  add_diff_assoc [simp] add_diff_assoc2[simp]

  1804

  1805 text{*At present we prove no analogue of @{text not_less_Least} or @{text

  1806 Least_Suc}, since there appears to be no need.*}

  1807

  1808 text{*Lemmas for ex/Factorization*}

  1809

  1810 lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"

  1811 by (cases m) auto

  1812

  1813 lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"

  1814 by (cases m) auto

  1815

  1816 lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"

  1817 by (cases m) auto

  1818

  1819 text {* Specialized induction principles that work "backwards": *}

  1820

  1821 lemma inc_induct[consumes 1, case_names base step]:

  1822   assumes less: "i \<le> j"

  1823   assumes base: "P j"

  1824   assumes step: "\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P (Suc n) \<Longrightarrow> P n"

  1825   shows "P i"

  1826   using less step

  1827 proof (induct d\<equiv>"j - i" arbitrary: i)

  1828   case (0 i)

  1829   hence "i = j" by simp

  1830   with base show ?case by simp

  1831 next

  1832   case (Suc d n)

  1833   hence "n \<le> n" "n < j" "P (Suc n)"

  1834     by simp_all

  1835   then show "P n" by fact

  1836 qed

  1837

  1838 lemma strict_inc_induct[consumes 1, case_names base step]:

  1839   assumes less: "i < j"

  1840   assumes base: "!!i. j = Suc i ==> P i"

  1841   assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"

  1842   shows "P i"

  1843   using less

  1844 proof (induct d=="j - i - 1" arbitrary: i)

  1845   case (0 i)

  1846   with i < j have "j = Suc i" by simp

  1847   with base show ?case by simp

  1848 next

  1849   case (Suc d i)

  1850   hence "i < j" "P (Suc i)"

  1851     by simp_all

  1852   thus "P i" by (rule step)

  1853 qed

  1854

  1855 lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"

  1856   using inc_induct[of "k - i" k P, simplified] by blast

  1857

  1858 lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"

  1859   using inc_induct[of 0 k P] by blast

  1860

  1861 text {* Further induction rule similar to @{thm inc_induct} *}

  1862

  1863 lemma dec_induct[consumes 1, case_names base step]:

  1864   "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"

  1865   by (induct j arbitrary: i) (auto simp: le_Suc_eq)

  1866

  1867 subsection \<open> Monotonicity of funpow \<close>

  1868

  1869 lemma funpow_increasing:

  1870   fixes f :: "'a \<Rightarrow> ('a::{lattice, order_top})"

  1871   shows "m \<le> n \<Longrightarrow> mono f \<Longrightarrow> (f ^^ n) \<top> \<le> (f ^^ m) \<top>"

  1872   by (induct rule: inc_induct)

  1873      (auto simp del: funpow.simps(2) simp add: funpow_Suc_right

  1874            intro: order_trans[OF _ funpow_mono])

  1875

  1876 lemma funpow_decreasing:

  1877   fixes f :: "'a \<Rightarrow> ('a::{lattice, order_bot})"

  1878   shows "m \<le> n \<Longrightarrow> mono f \<Longrightarrow> (f ^^ m) \<bottom> \<le> (f ^^ n) \<bottom>"

  1879   by (induct rule: dec_induct)

  1880      (auto simp del: funpow.simps(2) simp add: funpow_Suc_right

  1881            intro: order_trans[OF _ funpow_mono])

  1882

  1883 lemma mono_funpow:

  1884   fixes Q :: "'a::{lattice, order_bot} \<Rightarrow> 'a"

  1885   shows "mono Q \<Longrightarrow> mono (\<lambda>i. (Q ^^ i) \<bottom>)"

  1886   by (auto intro!: funpow_decreasing simp: mono_def)

  1887

  1888 lemma antimono_funpow:

  1889   fixes Q :: "'a::{lattice, order_top} \<Rightarrow> 'a"

  1890   shows "mono Q \<Longrightarrow> antimono (\<lambda>i. (Q ^^ i) \<top>)"

  1891   by (auto intro!: funpow_increasing simp: antimono_def)

  1892

  1893 subsection {* The divides relation on @{typ nat} *}

  1894

  1895 lemma dvd_1_left [iff]: "Suc 0 dvd k"

  1896 unfolding dvd_def by simp

  1897

  1898 lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"

  1899 by (simp add: dvd_def)

  1900

  1901 lemma nat_dvd_1_iff_1 [simp]: "m dvd (1::nat) \<longleftrightarrow> m = 1"

  1902 by (simp add: dvd_def)

  1903

  1904 lemma dvd_antisym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"

  1905   unfolding dvd_def

  1906   by (force dest: mult_eq_self_implies_10 simp add: mult.assoc)

  1907

  1908 text {* @{term "op dvd"} is a partial order *}

  1909

  1910 interpretation dvd: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n"

  1911   proof qed (auto intro: dvd_refl dvd_trans dvd_antisym)

  1912

  1913 lemma dvd_diff_nat[simp]: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"

  1914 unfolding dvd_def

  1915 by (blast intro: diff_mult_distrib2 [symmetric])

  1916

  1917 lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"

  1918   apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])

  1919   apply (blast intro: dvd_add)

  1920   done

  1921

  1922 lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"

  1923 by (drule_tac m = m in dvd_diff_nat, auto)

  1924

  1925 lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"

  1926   unfolding dvd_def

  1927   apply (erule exE)

  1928   apply (simp add: ac_simps)

  1929   done

  1930

  1931 lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"

  1932   apply auto

  1933    apply (subgoal_tac "m*n dvd m*1")

  1934    apply (drule dvd_mult_cancel, auto)

  1935   done

  1936

  1937 lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"

  1938   apply (subst mult.commute)

  1939   apply (erule dvd_mult_cancel1)

  1940   done

  1941

  1942 lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"

  1943 by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)

  1944

  1945 lemma nat_dvd_not_less:

  1946   fixes m n :: nat

  1947   shows "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"

  1948 by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)

  1949

  1950 lemma less_eq_dvd_minus:

  1951   fixes m n :: nat

  1952   assumes "m \<le> n"

  1953   shows "m dvd n \<longleftrightarrow> m dvd n - m"

  1954 proof -

  1955   from assms have "n = m + (n - m)" by simp

  1956   then obtain q where "n = m + q" ..

  1957   then show ?thesis by (simp add: add.commute [of m])

  1958 qed

  1959

  1960 lemma dvd_minus_self:

  1961   fixes m n :: nat

  1962   shows "m dvd n - m \<longleftrightarrow> n < m \<or> m dvd n"

  1963   by (cases "n < m") (auto elim!: dvdE simp add: not_less le_imp_diff_is_add)

  1964

  1965 lemma dvd_minus_add:

  1966   fixes m n q r :: nat

  1967   assumes "q \<le> n" "q \<le> r * m"

  1968   shows "m dvd n - q \<longleftrightarrow> m dvd n + (r * m - q)"

  1969 proof -

  1970   have "m dvd n - q \<longleftrightarrow> m dvd r * m + (n - q)"

  1971     using dvd_add_times_triv_left_iff [of m r] by simp

  1972   also from assms have "\<dots> \<longleftrightarrow> m dvd r * m + n - q" by simp

  1973   also from assms have "\<dots> \<longleftrightarrow> m dvd (r * m - q) + n" by simp

  1974   also have "\<dots> \<longleftrightarrow> m dvd n + (r * m - q)" by (simp add: add.commute)

  1975   finally show ?thesis .

  1976 qed

  1977

  1978

  1979 subsection {* Aliases *}

  1980

  1981 lemma nat_mult_1: "(1::nat) * n = n"

  1982   by (fact mult_1_left)

  1983

  1984 lemma nat_mult_1_right: "n * (1::nat) = n"

  1985   by (fact mult_1_right)

  1986

  1987

  1988 subsection {* Size of a datatype value *}

  1989

  1990 class size =

  1991   fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded} *}

  1992

  1993 instantiation nat :: size

  1994 begin

  1995

  1996 definition size_nat where

  1997   [simp, code]: "size (n \<Colon> nat) = n"

  1998

  1999 instance ..

  2000

  2001 end

  2002

  2003

  2004 subsection {* Code module namespace *}

  2005

  2006 code_identifier

  2007   code_module Nat \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith

  2008

  2009 hide_const (open) of_nat_aux

  2010

  2011 end