src/HOL/Nat.thy
 author nipkow Thu Dec 11 08:52:50 2008 +0100 (2008-12-11) changeset 29106 25e28a4070f3 parent 28952 15a4b2cf8c34 child 29608 564ea783ace8 child 29667 53103fc8ffa3 permissions -rw-r--r--
Testfile for Stefan's code generator
     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, mod and dvd, see theory Divides).

     6 *)

     7

     8 header {* Natural numbers *}

     9

    10 theory Nat

    11 imports Inductive Ring_and_Field

    12 uses

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

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

    15   ("Tools/arith_data.ML")

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

    17   ("Tools/lin_arith.ML")

    18 begin

    19

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

    21

    22 typedecl ind

    23

    24 axiomatization

    25   Zero_Rep :: ind and

    26   Suc_Rep :: "ind => ind"

    27 where

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

    29   inj_Suc_Rep:          "inj Suc_Rep" and

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

    31

    32

    33 subsection {* Type nat *}

    34

    35 text {* Type definition *}

    36

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

    38 where

    39     Zero_RepI: "Nat Zero_Rep"

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

    41

    42 global

    43

    44 typedef (open Nat)

    45   nat = Nat

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

    47

    48 constdefs

    49   Suc ::   "nat => nat"

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

    51

    52 local

    53

    54 instantiation nat :: zero

    55 begin

    56

    57 definition Zero_nat_def [code del]:

    58   "0 = Abs_Nat Zero_Rep"

    59

    60 instance ..

    61

    62 end

    63

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

    65   apply (simp add: Zero_nat_def Suc_def Abs_Nat_inject [unfolded mem_def]

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

    67   done

    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 -- naming of variables differs *}

    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)" by simp

   200   show "1 * n = n" 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 text {* Difference distributes over multiplication *}

   284

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

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

   287

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

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

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

   291

   292

   293 subsubsection {* Multiplication *}

   294

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

   296   by (rule mult_assoc)

   297

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

   299   by (rule mult_commute)

   300

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

   302   by (rule right_distrib)

   303

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

   305   by (induct m) auto

   306

   307 lemmas nat_distrib =

   308   add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2

   309

   310 lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)"

   311   apply (induct m)

   312    apply simp

   313   apply (induct n)

   314    apply auto

   315   done

   316

   317 lemma one_eq_mult_iff [simp,noatp]: "(Suc 0 = m * n) = (m = 1 & n = 1)"

   318   apply (rule trans)

   319   apply (rule_tac [2] mult_eq_1_iff, fastsimp)

   320   done

   321

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

   323 proof -

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

   325   proof (induct n arbitrary: m)

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

   327   next

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

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

   330   qed

   331   then show ?thesis by auto

   332 qed

   333

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

   335   by (simp add: mult_commute)

   336

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

   338   by (subst mult_cancel1) simp

   339

   340

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

   342

   343 subsubsection {* Operation definition *}

   344

   345 instantiation nat :: linorder

   346 begin

   347

   348 primrec less_eq_nat where

   349   "(0\<Colon>nat) \<le> n \<longleftrightarrow> True"

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

   351

   352 declare less_eq_nat.simps [simp del]

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

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

   355

   356 definition less_nat where

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

   358

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

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

   361

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

   363   unfolding less_eq_Suc_le ..

   364

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

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

   367

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

   369   by (simp add: less_eq_Suc_le)

   370

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

   372   by simp

   373

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

   375   by (simp add: less_eq_Suc_le)

   376

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

   378   by (simp add: less_eq_Suc_le)

   379

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

   381   by (induct m arbitrary: n)

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

   383

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

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

   386

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

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

   389

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

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

   392

   393 instance

   394 proof

   395   fix n m :: nat

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

   397   proof (induct n arbitrary: m)

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

   399   next

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

   401   qed

   402 next

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

   404 next

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

   406   then show "n = m"

   407     by (induct n arbitrary: m)

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

   409 next

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

   411   then show "n \<le> q"

   412   proof (induct n arbitrary: m q)

   413     case 0 show ?case by simp

   414   next

   415     case (Suc n) then show ?case

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

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

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

   419   qed

   420 next

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

   422     by (induct n arbitrary: m)

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

   424 qed

   425

   426 end

   427

   428 subsubsection {* Introduction properties *}

   429

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

   431   by (simp add: less_Suc_eq_le)

   432

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

   434   by (simp add: less_Suc_eq_le)

   435

   436

   437 subsubsection {* Elimination properties *}

   438

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

   440   by (rule order_less_irrefl)

   441

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

   443   by (rule not_sym) (rule less_imp_neq)

   444

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

   446   by (rule less_imp_neq)

   447

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

   449   by (rule notE, rule less_not_refl)

   450

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

   452   by (rule notE) (rule not_less0)

   453

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

   455   unfolding less_Suc_eq_le le_less ..

   456

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

   458   by (simp add: less_Suc_eq)

   459

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

   461   by (simp add: less_Suc_eq)

   462

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

   464   by simp

   465

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

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

   468   unfolding not_less less_Suc_eq_le by (rule antisym)

   469

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

   471   by (rule linorder_neq_iff)

   472

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

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

   475   shows "P n m"

   476   apply (rule less_linear [THEN disjE])

   477   apply (erule_tac [2] disjE)

   478   apply (erule lessCase)

   479   apply (erule sym [THEN eqCase])

   480   apply (erule major)

   481   done

   482

   483

   484 subsubsection {* Inductive (?) properties *}

   485

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

   487   unfolding less_eq_Suc_le [of m] le_less by simp

   488

   489 lemma lessE:

   490   assumes major: "i < k"

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

   492   shows P

   493 proof -

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

   495     unfolding less_eq_Suc_le by (induct k) simp_all

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

   497     by (clarsimp simp add: less_le)

   498   with p1 p2 show P by auto

   499 qed

   500

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

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

   503   apply (rule major [THEN lessE])

   504   apply (rule eq, blast)

   505   apply (rule less, blast)

   506   done

   507

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

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

   510   apply (rule major [THEN lessE])

   511   apply (erule lessI [THEN minor])

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

   513   done

   514

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

   516   by simp

   517

   518 lemma less_trans_Suc:

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

   520   apply (induct k, simp_all)

   521   apply (insert le)

   522   apply (simp add: less_Suc_eq)

   523   apply (blast dest: Suc_lessD)

   524   done

   525

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

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

   528   unfolding not_less less_Suc_eq_le ..

   529

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

   531   unfolding not_le Suc_le_eq ..

   532

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

   534

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

   536   unfolding less_Suc_eq_le .

   537

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

   539   unfolding not_le less_Suc_eq_le ..

   540

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

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

   543

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

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

   546

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

   548   unfolding Suc_le_eq .

   549

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

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

   552   unfolding Suc_le_eq .

   553

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

   555   unfolding less_eq_Suc_le by (rule Suc_leD)

   556

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

   558 lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq

   559

   560

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

   562

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

   564   unfolding le_less .

   565

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

   567   by (rule le_less)

   568

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

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

   571   by auto

   572

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

   574   by simp

   575

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

   577   by (rule order_trans)

   578

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

   580   by (rule antisym)

   581

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

   583   by (rule less_le)

   584

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

   586   unfolding less_le ..

   587

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

   589   by (rule linear)

   590

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

   592

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

   594   unfolding less_Suc_eq_le by auto

   595

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

   597   unfolding not_less by (rule le_less_Suc_eq)

   598

   599 lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq

   600

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

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

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

   604 by simp

   605

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

   607 by simp

   608

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

   610 by (cases n) simp_all

   611

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

   613 by (cases n) simp_all

   614

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

   616 by (cases n) simp_all

   617

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

   619 by (cases n) simp_all

   620

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

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

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

   624

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

   626 by (fast intro: not0_implies_Suc)

   627

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

   629 using neq0_conv by blast

   630

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

   632 by (induct m') simp_all

   633

   634 text {* Useful in certain inductive arguments *}

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

   636 by (cases m) simp_all

   637

   638

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

   640

   641 lemma mono_Suc: "mono Suc"

   642 by (rule monoI) simp

   643

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

   645 by (rule min_leastL) simp

   646

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

   648 by (rule min_leastR) simp

   649

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

   651 by (simp add: mono_Suc min_of_mono)

   652

   653 lemma min_Suc1:

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

   655 by (simp split: nat.split)

   656

   657 lemma min_Suc2:

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

   659 by (simp split: nat.split)

   660

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

   662 by (rule max_leastL) simp

   663

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

   665 by (rule max_leastR) simp

   666

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

   668 by (simp add: mono_Suc max_of_mono)

   669

   670 lemma max_Suc1:

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

   672 by (simp split: nat.split)

   673

   674 lemma max_Suc2:

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

   676 by (simp split: nat.split)

   677

   678

   679 subsubsection {* Monotonicity of Addition *}

   680

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

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

   683

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

   685 by (induct k) simp_all

   686

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

   688 by (induct k) simp_all

   689

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

   691 by(auto dest:gr0_implies_Suc)

   692

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

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

   695 by (induct k) simp_all

   696

   697 text {* strict, in both arguments *}

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

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

   700   apply (induct j, simp_all)

   701   done

   702

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

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

   705   apply (induct n)

   706   apply (simp_all add: order_le_less)

   707   apply (blast elim!: less_SucE

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

   709   done

   710

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

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

   713 apply(auto simp: gr0_conv_Suc)

   714 apply (induct_tac m)

   715 apply (simp_all add: add_less_mono)

   716 done

   717

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

   719 instance nat :: ordered_semidom

   720 proof

   721   fix i j k :: nat

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

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

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

   725 qed

   726

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

   728 by simp

   729

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

   731 by simp

   732

   733

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

   735

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

   737

   738 instance nat :: wellorder proof

   739   fix P and n :: nat

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

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

   742   proof (induct n)

   743     case (0 n)

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

   745     thus ?case using 0 by auto

   746   next

   747     case (Suc m n)

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

   749     thus ?case

   750     proof

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

   752     next

   753       assume n: "n = Suc m"

   754       show "P n"

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

   756     qed

   757   qed

   758   then show "P n" by auto

   759 qed

   760

   761 lemma Least_Suc:

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

   763   apply (case_tac "n", auto)

   764   apply (frule LeastI)

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

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

   767   apply (erule_tac [2] Least_le)

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

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

   770   apply (blast intro: order_antisym)

   771   done

   772

   773 lemma Least_Suc2:

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

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

   776   apply simp

   777   done

   778

   779 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)"

   780   apply (cases n)

   781    apply blast

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

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

   784   done

   785

   786 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)"

   787   apply (cases n)

   788    apply blast

   789   apply (frule (1) ex_least_nat_le)

   790   apply (erule exE)

   791   apply (case_tac k)

   792    apply simp

   793   apply (rename_tac k1)

   794   apply (rule_tac x=k1 in exI)

   795   apply (auto simp add: less_eq_Suc_le)

   796   done

   797

   798 lemma nat_less_induct:

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

   800   using assms less_induct by blast

   801

   802 lemma measure_induct_rule [case_names less]:

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

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

   805   shows "P a"

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

   807

   808 text {* old style induction rules: *}

   809 lemma measure_induct:

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

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

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

   813

   814 lemma full_nat_induct:

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

   816   shows "P n"

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

   818

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

   820 lemma less_Suc_induct:

   821   assumes less:  "i < j"

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

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

   824   shows "P i j"

   825 proof -

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

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

   828   proof (induct k)

   829     case 0

   830     show ?case by (simp add: step)

   831   next

   832     case (Suc k)

   833     thus ?case by (auto intro: assms)

   834   qed

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

   836 qed

   837

   838 lemma nat_induct2: "[|P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))|] ==> P n"

   839   apply (rule nat_less_induct)

   840   apply (case_tac n)

   841   apply (case_tac [2] nat)

   842   apply (blast intro: less_trans)+

   843   done

   844

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

   846 Provided by Roelof Oosterhuis.

   847 $P(n)$ is true for all $n\in\mathbb{N}$ if

   848 \begin{itemize}

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

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

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

   852 \end{itemize} *}

   853

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

   855 lemma infinite_descent:

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

   857 by (induct n rule: less_induct, auto)

   858

   859 lemma infinite_descent0[case_names 0 smaller]:

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

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

   862

   863 text {*

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

   865 $P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and

   866 \begin{itemize}

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

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

   869 \end{itemize}

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

   871

   872 corollary infinite_descent0_measure [case_names 0 smaller]:

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

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

   875   shows "P x"

   876 proof -

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

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

   879   proof (induct n rule: infinite_descent0)

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

   881     with A0 show "P x" by auto

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

   883     case (smaller n)

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

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

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

   887     then show ?case by auto

   888   qed

   889   ultimately show "P x" by auto

   890 qed

   891

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

   893 lemma infinite_descent_measure:

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

   895 proof -

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

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

   898   proof (induct n rule: infinite_descent, auto)

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

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

   901   qed

   902   ultimately show "P x" by auto

   903 qed

   904

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

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

   907 lemma less_mono_imp_le_mono:

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

   909 by (simp add: order_le_less) (blast)

   910

   911

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

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

   914 by (rule add_right_mono)

   915

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

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

   918 by (rule add_mono)

   919

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

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

   922

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

   924 by (simp add: add_commute, rule le_add2)

   925

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

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

   928

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

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

   931

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

   933 by (iprover intro!: less_add_Suc1 less_imp_Suc_add)

   934

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

   936 by (rule le_trans, assumption, rule le_add1)

   937

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

   939 by (rule le_trans, assumption, rule le_add2)

   940

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

   942 by (rule less_le_trans, assumption, rule le_add1)

   943

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

   945 by (rule less_le_trans, assumption, rule le_add2)

   946

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

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

   949 apply (simp_all add: le_add1)

   950 done

   951

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

   953 apply (rule notI)

   954 apply (drule add_lessD1)

   955 apply (erule less_irrefl [THEN notE])

   956 done

   957

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

   959 by (simp add: add_commute)

   960

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

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

   963 apply (simp_all add: le_add1)

   964 done

   965

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

   967 apply (simp add: add_commute)

   968 apply (erule add_leD1)

   969 done

   970

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

   972 by (blast dest: add_leD1 add_leD2)

   973

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

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

   976 by (force simp del: add_Suc_right

   977     simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)

   978

   979

   980 subsubsection {* More results about difference *}

   981

   982 text {* Addition is the inverse of subtraction:

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

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

   985 by (induct m n rule: diff_induct) simp_all

   986

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

   988 by (simp add: add_diff_inverse linorder_not_less)

   989

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

   991 by (simp add: add_commute)

   992

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

   994 by (induct m n rule: diff_induct) simp_all

   995

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

   997 apply (induct m n rule: diff_induct)

   998 apply (erule_tac [3] less_SucE)

   999 apply (simp_all add: less_Suc_eq)

  1000 done

  1001

  1002 lemma diff_le_self [simp]: "m - n \<le> (m::nat)"

  1003 by (induct m n rule: diff_induct) (simp_all add: le_SucI)

  1004

  1005 lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"

  1006   by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])

  1007

  1008 lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"

  1009 by (rule le_less_trans, rule diff_le_self)

  1010

  1011 lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"

  1012 by (cases n) (auto simp add: le_simps)

  1013

  1014 lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"

  1015 by (induct j k rule: diff_induct) simp_all

  1016

  1017 lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"

  1018 by (simp add: add_commute diff_add_assoc)

  1019

  1020 lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"

  1021 by (auto simp add: diff_add_inverse2)

  1022

  1023 lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"

  1024 by (induct m n rule: diff_induct) simp_all

  1025

  1026 lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"

  1027 by (rule iffD2, rule diff_is_0_eq)

  1028

  1029 lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"

  1030 by (induct m n rule: diff_induct) simp_all

  1031

  1032 lemma less_imp_add_positive:

  1033   assumes "i < j"

  1034   shows "\<exists>k::nat. 0 < k & i + k = j"

  1035 proof

  1036   from assms show "0 < j - i & i + (j - i) = j"

  1037     by (simp add: order_less_imp_le)

  1038 qed

  1039

  1040 text {* a nice rewrite for bounded subtraction *}

  1041 lemma nat_minus_add_max:

  1042   fixes n m :: nat

  1043   shows "n - m + m = max n m"

  1044     by (simp add: max_def not_le order_less_imp_le)

  1045

  1046 lemma nat_diff_split:

  1047   "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"

  1048     -- {* elimination of @{text -} on @{text nat} *}

  1049 by (cases "a < b")

  1050   (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse

  1051     not_less le_less dest!: sym [of a] sym [of b] add_eq_self_zero)

  1052

  1053 lemma nat_diff_split_asm:

  1054   "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"

  1055     -- {* elimination of @{text -} on @{text nat} in assumptions *}

  1056 by (auto split: nat_diff_split)

  1057

  1058

  1059 subsubsection {* Monotonicity of Multiplication *}

  1060

  1061 lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"

  1062 by (simp add: mult_right_mono)

  1063

  1064 lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"

  1065 by (simp add: mult_left_mono)

  1066

  1067 text {* @{text "\<le>"} monotonicity, BOTH arguments *}

  1068 lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"

  1069 by (simp add: mult_mono)

  1070

  1071 lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"

  1072 by (simp add: mult_strict_right_mono)

  1073

  1074 text{*Differs from the standard @{text zero_less_mult_iff} in that

  1075       there are no negative numbers.*}

  1076 lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"

  1077   apply (induct m)

  1078    apply simp

  1079   apply (case_tac n)

  1080    apply simp_all

  1081   done

  1082

  1083 lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)"

  1084   apply (induct m)

  1085    apply simp

  1086   apply (case_tac n)

  1087    apply simp_all

  1088   done

  1089

  1090 lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"

  1091   apply (safe intro!: mult_less_mono1)

  1092   apply (case_tac k, auto)

  1093   apply (simp del: le_0_eq add: linorder_not_le [symmetric])

  1094   apply (blast intro: mult_le_mono1)

  1095   done

  1096

  1097 lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"

  1098 by (simp add: mult_commute [of k])

  1099

  1100 lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"

  1101 by (simp add: linorder_not_less [symmetric], auto)

  1102

  1103 lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"

  1104 by (simp add: linorder_not_less [symmetric], auto)

  1105

  1106 lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"

  1107 by (subst mult_less_cancel1) simp

  1108

  1109 lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"

  1110 by (subst mult_le_cancel1) simp

  1111

  1112 lemma le_square: "m \<le> m * (m::nat)"

  1113   by (cases m) (auto intro: le_add1)

  1114

  1115 lemma le_cube: "(m::nat) \<le> m * (m * m)"

  1116   by (cases m) (auto intro: le_add1)

  1117

  1118 text {* Lemma for @{text gcd} *}

  1119 lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"

  1120   apply (drule sym)

  1121   apply (rule disjCI)

  1122   apply (rule nat_less_cases, erule_tac [2] _)

  1123    apply (drule_tac [2] mult_less_mono2)

  1124     apply (auto)

  1125   done

  1126

  1127 text {* the lattice order on @{typ nat} *}

  1128

  1129 instantiation nat :: distrib_lattice

  1130 begin

  1131

  1132 definition

  1133   "(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min"

  1134

  1135 definition

  1136   "(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max"

  1137

  1138 instance by intro_classes

  1139   (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def

  1140     intro: order_less_imp_le antisym elim!: order_trans order_less_trans)

  1141

  1142 end

  1143

  1144

  1145 subsection {* Embedding of the Naturals into any

  1146   @{text semiring_1}: @{term of_nat} *}

  1147

  1148 context semiring_1

  1149 begin

  1150

  1151 primrec

  1152   of_nat :: "nat \<Rightarrow> 'a"

  1153 where

  1154   of_nat_0:     "of_nat 0 = 0"

  1155   | of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"

  1156

  1157 lemma of_nat_1 [simp]: "of_nat 1 = 1"

  1158   by simp

  1159

  1160 lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"

  1161   by (induct m) (simp_all add: add_ac)

  1162

  1163 lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"

  1164   by (induct m) (simp_all add: add_ac left_distrib)

  1165

  1166 primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where

  1167   "of_nat_aux inc 0 i = i"

  1168   | "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- {* tail recursive *}

  1169

  1170 lemma of_nat_code [code, code unfold, code inline del]:

  1171   "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"

  1172 proof (induct n)

  1173   case 0 then show ?case by simp

  1174 next

  1175   case (Suc n)

  1176   have "\<And>i. of_nat_aux (\<lambda>i. i + 1) n (i + 1) = of_nat_aux (\<lambda>i. i + 1) n i + 1"

  1177     by (induct n) simp_all

  1178   from this [of 0] have "of_nat_aux (\<lambda>i. i + 1) n 1 = of_nat_aux (\<lambda>i. i + 1) n 0 + 1"

  1179     by simp

  1180   with Suc show ?case by (simp add: add_commute)

  1181 qed

  1182

  1183 end

  1184

  1185 text{*Class for unital semirings with characteristic zero.

  1186  Includes non-ordered rings like the complex numbers.*}

  1187

  1188 class semiring_char_0 = semiring_1 +

  1189   assumes of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"

  1190 begin

  1191

  1192 text{*Special cases where either operand is zero*}

  1193

  1194 lemma of_nat_0_eq_iff [simp, noatp]: "0 = of_nat n \<longleftrightarrow> 0 = n"

  1195   by (rule of_nat_eq_iff [of 0, simplified])

  1196

  1197 lemma of_nat_eq_0_iff [simp, noatp]: "of_nat m = 0 \<longleftrightarrow> m = 0"

  1198   by (rule of_nat_eq_iff [of _ 0, simplified])

  1199

  1200 lemma inj_of_nat: "inj of_nat"

  1201   by (simp add: inj_on_def)

  1202

  1203 end

  1204

  1205 context ordered_semidom

  1206 begin

  1207

  1208 lemma zero_le_imp_of_nat: "0 \<le> of_nat m"

  1209   apply (induct m, simp_all)

  1210   apply (erule order_trans)

  1211   apply (rule ord_le_eq_trans [OF _ add_commute])

  1212   apply (rule less_add_one [THEN less_imp_le])

  1213   done

  1214

  1215 lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"

  1216   apply (induct m n rule: diff_induct, simp_all)

  1217   apply (insert add_less_le_mono [OF zero_less_one zero_le_imp_of_nat], force)

  1218   done

  1219

  1220 lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"

  1221   apply (induct m n rule: diff_induct, simp_all)

  1222   apply (insert zero_le_imp_of_nat)

  1223   apply (force simp add: not_less [symmetric])

  1224   done

  1225

  1226 lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"

  1227   by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)

  1228

  1229 lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"

  1230   by (simp add: not_less [symmetric] linorder_not_less [symmetric])

  1231

  1232 text{*Every @{text ordered_semidom} has characteristic zero.*}

  1233

  1234 subclass semiring_char_0

  1235   proof qed (simp add: eq_iff order_eq_iff)

  1236

  1237 text{*Special cases where either operand is zero*}

  1238

  1239 lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"

  1240   by (rule of_nat_le_iff [of 0, simplified])

  1241

  1242 lemma of_nat_le_0_iff [simp, noatp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"

  1243   by (rule of_nat_le_iff [of _ 0, simplified])

  1244

  1245 lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"

  1246   by (rule of_nat_less_iff [of 0, simplified])

  1247

  1248 lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"

  1249   by (rule of_nat_less_iff [of _ 0, simplified])

  1250

  1251 end

  1252

  1253 context ring_1

  1254 begin

  1255

  1256 lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"

  1257   by (simp add: compare_rls of_nat_add [symmetric])

  1258

  1259 end

  1260

  1261 context ordered_idom

  1262 begin

  1263

  1264 lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"

  1265   unfolding abs_if by auto

  1266

  1267 end

  1268

  1269 lemma of_nat_id [simp]: "of_nat n = n"

  1270   by (induct n) auto

  1271

  1272 lemma of_nat_eq_id [simp]: "of_nat = id"

  1273   by (auto simp add: expand_fun_eq)

  1274

  1275

  1276 subsection {* The Set of Natural Numbers *}

  1277

  1278 context semiring_1

  1279 begin

  1280

  1281 definition

  1282   Nats  :: "'a set" where

  1283   [code del]: "Nats = range of_nat"

  1284

  1285 notation (xsymbols)

  1286   Nats  ("\<nat>")

  1287

  1288 lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"

  1289   by (simp add: Nats_def)

  1290

  1291 lemma Nats_0 [simp]: "0 \<in> \<nat>"

  1292 apply (simp add: Nats_def)

  1293 apply (rule range_eqI)

  1294 apply (rule of_nat_0 [symmetric])

  1295 done

  1296

  1297 lemma Nats_1 [simp]: "1 \<in> \<nat>"

  1298 apply (simp add: Nats_def)

  1299 apply (rule range_eqI)

  1300 apply (rule of_nat_1 [symmetric])

  1301 done

  1302

  1303 lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"

  1304 apply (auto simp add: Nats_def)

  1305 apply (rule range_eqI)

  1306 apply (rule of_nat_add [symmetric])

  1307 done

  1308

  1309 lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"

  1310 apply (auto simp add: Nats_def)

  1311 apply (rule range_eqI)

  1312 apply (rule of_nat_mult [symmetric])

  1313 done

  1314

  1315 end

  1316

  1317

  1318 subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}

  1319

  1320 lemma subst_equals:

  1321   assumes 1: "t = s" and 2: "u = t"

  1322   shows "u = s"

  1323   using 2 1 by (rule trans)

  1324

  1325 use "Tools/arith_data.ML"

  1326 declaration {* K ArithData.setup *}

  1327

  1328 use "Tools/lin_arith.ML"

  1329 declaration {* K LinArith.setup *}

  1330

  1331 lemmas [arith_split] = nat_diff_split split_min split_max

  1332

  1333

  1334 context order

  1335 begin

  1336

  1337 lemma lift_Suc_mono_le:

  1338   assumes mono: "!!n. f n \<le> f(Suc n)" and "n\<le>n'"

  1339   shows "f n \<le> f n'"

  1340 proof (cases "n < n'")

  1341   case True

  1342   thus ?thesis

  1343     by (induct n n' rule: less_Suc_induct[consumes 1]) (auto intro: mono)

  1344 qed (insert n \<le> n', auto) -- {*trivial for @{prop "n = n'"} *}

  1345

  1346 lemma lift_Suc_mono_less:

  1347   assumes mono: "!!n. f n < f(Suc n)" and "n < n'"

  1348   shows "f n < f n'"

  1349 using n < n'

  1350 by (induct n n' rule: less_Suc_induct[consumes 1]) (auto intro: mono)

  1351

  1352 lemma lift_Suc_mono_less_iff:

  1353   "(!!n. f n < f(Suc n)) \<Longrightarrow> f(n) < f(m) \<longleftrightarrow> n<m"

  1354 by(blast intro: less_asym' lift_Suc_mono_less[of f]

  1355          dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq[THEN iffD1])

  1356

  1357 end

  1358

  1359

  1360 lemma mono_nat_linear_lb:

  1361   "(!!m n::nat. m<n \<Longrightarrow> f m < f n) \<Longrightarrow> f(m)+k \<le> f(m+k)"

  1362 apply(induct_tac k)

  1363  apply simp

  1364 apply(erule_tac x="m+n" in meta_allE)

  1365 apply(erule_tac x="m+n+1" in meta_allE)

  1366 apply simp

  1367 done

  1368

  1369

  1370 text{*Subtraction laws, mostly by Clemens Ballarin*}

  1371

  1372 lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"

  1373 by arith

  1374

  1375 lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"

  1376 by arith

  1377

  1378 lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"

  1379 by arith

  1380

  1381 lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"

  1382 by arith

  1383

  1384 lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"

  1385 by arith

  1386

  1387 lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"

  1388 by arith

  1389

  1390 (*Replaces the previous diff_less and le_diff_less, which had the stronger

  1391   second premise n\<le>m*)

  1392 lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"

  1393 by arith

  1394

  1395 text {* Simplification of relational expressions involving subtraction *}

  1396

  1397 lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"

  1398 by (simp split add: nat_diff_split)

  1399

  1400 lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"

  1401 by (auto split add: nat_diff_split)

  1402

  1403 lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"

  1404 by (auto split add: nat_diff_split)

  1405

  1406 lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"

  1407 by (auto split add: nat_diff_split)

  1408

  1409 text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}

  1410

  1411 (* Monotonicity of subtraction in first argument *)

  1412 lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"

  1413 by (simp split add: nat_diff_split)

  1414

  1415 lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"

  1416 by (simp split add: nat_diff_split)

  1417

  1418 lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"

  1419 by (simp split add: nat_diff_split)

  1420

  1421 lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"

  1422 by (simp split add: nat_diff_split)

  1423

  1424 lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"

  1425 unfolding min_def by auto

  1426

  1427 lemma inj_on_diff_nat:

  1428   assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"

  1429   shows "inj_on (\<lambda>n. n - k) N"

  1430 proof (rule inj_onI)

  1431   fix x y

  1432   assume a: "x \<in> N" "y \<in> N" "x - k = y - k"

  1433   with k_le_n have "x - k + k = y - k + k" by auto

  1434   with a k_le_n show "x = y" by auto

  1435 qed

  1436

  1437 text{*Rewriting to pull differences out*}

  1438

  1439 lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"

  1440 by arith

  1441

  1442 lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"

  1443 by arith

  1444

  1445 lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"

  1446 by arith

  1447

  1448 text{*Lemmas for ex/Factorization*}

  1449

  1450 lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"

  1451 by (cases m) auto

  1452

  1453 lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"

  1454 by (cases m) auto

  1455

  1456 lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"

  1457 by (cases m) auto

  1458

  1459 text {* Specialized induction principles that work "backwards": *}

  1460

  1461 lemma inc_induct[consumes 1, case_names base step]:

  1462   assumes less: "i <= j"

  1463   assumes base: "P j"

  1464   assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"

  1465   shows "P i"

  1466   using less

  1467 proof (induct d=="j - i" arbitrary: i)

  1468   case (0 i)

  1469   hence "i = j" by simp

  1470   with base show ?case by simp

  1471 next

  1472   case (Suc d i)

  1473   hence "i < j" "P (Suc i)"

  1474     by simp_all

  1475   thus "P i" by (rule step)

  1476 qed

  1477

  1478 lemma strict_inc_induct[consumes 1, case_names base step]:

  1479   assumes less: "i < j"

  1480   assumes base: "!!i. j = Suc i ==> P i"

  1481   assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"

  1482   shows "P i"

  1483   using less

  1484 proof (induct d=="j - i - 1" arbitrary: i)

  1485   case (0 i)

  1486   with i < j have "j = Suc i" by simp

  1487   with base show ?case by simp

  1488 next

  1489   case (Suc d i)

  1490   hence "i < j" "P (Suc i)"

  1491     by simp_all

  1492   thus "P i" by (rule step)

  1493 qed

  1494

  1495 lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"

  1496   using inc_induct[of "k - i" k P, simplified] by blast

  1497

  1498 lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"

  1499   using inc_induct[of 0 k P] by blast

  1500

  1501 lemma nat_not_singleton: "(\<forall>x. x = (0::nat)) = False"

  1502   by auto

  1503

  1504 (*The others are

  1505       i - j - k = i - (j + k),

  1506       k \<le> j ==> j - k + i = j + i - k,

  1507       k \<le> j ==> i + (j - k) = i + j - k *)

  1508 lemmas add_diff_assoc = diff_add_assoc [symmetric]

  1509 lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]

  1510 declare diff_diff_left [simp]  add_diff_assoc [simp] add_diff_assoc2[simp]

  1511

  1512 text{*At present we prove no analogue of @{text not_less_Least} or @{text

  1513 Least_Suc}, since there appears to be no need.*}

  1514

  1515

  1516 subsection {* size of a datatype value *}

  1517

  1518 class size = type +

  1519   fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded} *}

  1520

  1521 end