src/HOL/Nat.thy
author blanchet
Thu Sep 11 18:54:36 2014 +0200 (2014-09-11)
changeset 58306 117ba6cbe414
parent 58189 9d714be4f028
child 58377 c6f93b8d2d8e
permissions -rw-r--r--
renamed 'rep_datatype' to 'old_rep_datatype' (HOL)
     1 (*  Title:      HOL/Nat.thy
     2     Author:     Tobias Nipkow and Lawrence C Paulson and Markus Wenzel
     3 
     4 Type "nat" is a linear order, and a datatype; arithmetic operators + -
     5 and * (for div and mod, see theory Divides).
     6 *)
     7 
     8 header {* Natural numbers *}
     9 
    10 theory Nat
    11 imports Inductive Typedef Fun Fields
    12 begin
    13 
    14 ML_file "~~/src/Tools/rat.ML"
    15 
    16 named_theorems arith "arith facts -- only ground formulas"
    17 ML_file "Tools/arith_data.ML"
    18 ML_file "~~/src/Provers/Arith/fast_lin_arith.ML"
    19 
    20 
    21 subsection {* Type @{text ind} *}
    22 
    23 typedecl ind
    24 
    25 axiomatization Zero_Rep :: ind and Suc_Rep :: "ind => ind" where
    26   -- {* the axiom of infinity in 2 parts *}
    27   Suc_Rep_inject:       "Suc_Rep x = Suc_Rep y ==> x = y" and
    28   Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"
    29 
    30 subsection {* Type nat *}
    31 
    32 text {* Type definition *}
    33 
    34 inductive Nat :: "ind \<Rightarrow> bool" where
    35   Zero_RepI: "Nat Zero_Rep"
    36 | Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)"
    37 
    38 typedef nat = "{n. Nat n}"
    39   morphisms Rep_Nat Abs_Nat
    40   using Nat.Zero_RepI by auto
    41 
    42 lemma Nat_Rep_Nat:
    43   "Nat (Rep_Nat n)"
    44   using Rep_Nat by simp
    45 
    46 lemma Nat_Abs_Nat_inverse:
    47   "Nat n \<Longrightarrow> Rep_Nat (Abs_Nat n) = n"
    48   using Abs_Nat_inverse by simp
    49 
    50 lemma Nat_Abs_Nat_inject:
    51   "Nat n \<Longrightarrow> Nat m \<Longrightarrow> Abs_Nat n = Abs_Nat m \<longleftrightarrow> n = m"
    52   using Abs_Nat_inject by simp
    53 
    54 instantiation nat :: zero
    55 begin
    56 
    57 definition Zero_nat_def:
    58   "0 = Abs_Nat Zero_Rep"
    59 
    60 instance ..
    61 
    62 end
    63 
    64 definition Suc :: "nat \<Rightarrow> nat" where
    65   "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"
    66 
    67 lemma Suc_not_Zero: "Suc m \<noteq> 0"
    68   by (simp add: Zero_nat_def Suc_def Suc_RepI Zero_RepI Nat_Abs_Nat_inject Suc_Rep_not_Zero_Rep Nat_Rep_Nat)
    69 
    70 lemma Zero_not_Suc: "0 \<noteq> Suc m"
    71   by (rule not_sym, rule Suc_not_Zero not_sym)
    72 
    73 lemma Suc_Rep_inject': "Suc_Rep x = Suc_Rep y \<longleftrightarrow> x = y"
    74   by (rule iffI, rule Suc_Rep_inject) simp_all
    75 
    76 lemma nat_induct0:
    77   fixes n
    78   assumes "P 0" and "\<And>n. P n \<Longrightarrow> P (Suc n)"
    79   shows "P n"
    80 using assms
    81 apply (unfold Zero_nat_def Suc_def)
    82 apply (rule Rep_Nat_inverse [THEN subst]) -- {* types force good instantiation *}
    83 apply (erule Nat_Rep_Nat [THEN Nat.induct])
    84 apply (iprover elim: Nat_Abs_Nat_inverse [THEN subst])
    85 done
    86 
    87 free_constructors case_nat for
    88     "0 \<Colon> nat"
    89   | Suc pred
    90 where
    91   "pred (0 \<Colon> nat) = (0 \<Colon> nat)"
    92     apply atomize_elim
    93     apply (rename_tac n, induct_tac n rule: nat_induct0, auto)
    94    apply (simp add: Suc_def Nat_Abs_Nat_inject Nat_Rep_Nat Suc_RepI Suc_Rep_inject'
    95      Rep_Nat_inject)
    96   apply (simp only: Suc_not_Zero)
    97   done
    98 
    99 -- {* Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}. *}
   100 setup {* Sign.mandatory_path "old" *}
   101 
   102 old_rep_datatype "0 \<Colon> nat" Suc
   103   apply (erule nat_induct0, assumption)
   104  apply (rule nat.inject)
   105 apply (rule nat.distinct(1))
   106 done
   107 
   108 setup {* Sign.parent_path *}
   109 
   110 -- {* But erase the prefix for properties that are not generated by @{text free_constructors}. *}
   111 setup {* Sign.mandatory_path "nat" *}
   112 
   113 declare
   114   old.nat.inject[iff del]
   115   old.nat.distinct(1)[simp del, induct_simp del]
   116 
   117 lemmas induct = old.nat.induct
   118 lemmas inducts = old.nat.inducts
   119 lemmas rec = old.nat.rec
   120 lemmas simps = nat.inject nat.distinct nat.case nat.rec
   121 
   122 setup {* Sign.parent_path *}
   123 
   124 abbreviation rec_nat :: "'a \<Rightarrow> (nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a" where
   125   "rec_nat \<equiv> old.rec_nat"
   126 
   127 declare nat.sel[code del]
   128 
   129 hide_const (open) Nat.pred -- {* hide everything related to the selector *}
   130 hide_fact
   131   nat.case_eq_if
   132   nat.collapse
   133   nat.expand
   134   nat.sel
   135   nat.exhaust_sel
   136   nat.split_sel
   137   nat.split_sel_asm
   138 
   139 lemma nat_exhaust [case_names 0 Suc, cases type: nat]:
   140   -- {* for backward compatibility -- names of variables differ *}
   141   "(y = 0 \<Longrightarrow> P) \<Longrightarrow> (\<And>nat. y = Suc nat \<Longrightarrow> P) \<Longrightarrow> P"
   142 by (rule old.nat.exhaust)
   143 
   144 lemma nat_induct [case_names 0 Suc, induct type: nat]:
   145   -- {* for backward compatibility -- names of variables differ *}
   146   fixes n
   147   assumes "P 0" and "\<And>n. P n \<Longrightarrow> P (Suc n)"
   148   shows "P n"
   149 using assms by (rule nat.induct)
   150 
   151 hide_fact
   152   nat_exhaust
   153   nat_induct0
   154 
   155 text {* Injectiveness and distinctness lemmas *}
   156 
   157 lemma inj_Suc[simp]: "inj_on Suc N"
   158   by (simp add: inj_on_def)
   159 
   160 lemma Suc_neq_Zero: "Suc m = 0 \<Longrightarrow> R"
   161 by (rule notE, rule Suc_not_Zero)
   162 
   163 lemma Zero_neq_Suc: "0 = Suc m \<Longrightarrow> R"
   164 by (rule Suc_neq_Zero, erule sym)
   165 
   166 lemma Suc_inject: "Suc x = Suc y \<Longrightarrow> x = y"
   167 by (rule inj_Suc [THEN injD])
   168 
   169 lemma n_not_Suc_n: "n \<noteq> Suc n"
   170 by (induct n) simp_all
   171 
   172 lemma Suc_n_not_n: "Suc n \<noteq> n"
   173 by (rule not_sym, rule n_not_Suc_n)
   174 
   175 text {* A special form of induction for reasoning
   176   about @{term "m < n"} and @{term "m - n"} *}
   177 
   178 lemma diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==>
   179     (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n"
   180   apply (rule_tac x = m in spec)
   181   apply (induct n)
   182   prefer 2
   183   apply (rule allI)
   184   apply (induct_tac x, iprover+)
   185   done
   186 
   187 
   188 subsection {* Arithmetic operators *}
   189 
   190 instantiation nat :: comm_monoid_diff
   191 begin
   192 
   193 primrec plus_nat where
   194   add_0:      "0 + n = (n\<Colon>nat)"
   195 | add_Suc:  "Suc m + n = Suc (m + n)"
   196 
   197 lemma add_0_right [simp]: "m + 0 = (m::nat)"
   198   by (induct m) simp_all
   199 
   200 lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
   201   by (induct m) simp_all
   202 
   203 declare add_0 [code]
   204 
   205 lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
   206   by simp
   207 
   208 primrec minus_nat where
   209   diff_0 [code]: "m - 0 = (m\<Colon>nat)"
   210 | diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
   211 
   212 declare diff_Suc [simp del]
   213 
   214 lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"
   215   by (induct n) (simp_all add: diff_Suc)
   216 
   217 lemma diff_Suc_Suc [simp, code]: "Suc m - Suc n = m - n"
   218   by (induct n) (simp_all add: diff_Suc)
   219 
   220 instance proof
   221   fix n m q :: nat
   222   show "(n + m) + q = n + (m + q)" by (induct n) simp_all
   223   show "n + m = m + n" by (induct n) simp_all
   224   show "0 + n = n" by simp
   225   show "n - 0 = n" by simp
   226   show "0 - n = 0" by simp
   227   show "(q + n) - (q + m) = n - m" by (induct q) simp_all
   228   show "n - m - q = n - (m + q)" by (induct q) (simp_all add: diff_Suc)
   229 qed
   230 
   231 end
   232 
   233 hide_fact (open) add_0 add_0_right diff_0
   234 
   235 instantiation nat :: comm_semiring_1_cancel
   236 begin
   237 
   238 definition
   239   One_nat_def [simp]: "1 = Suc 0"
   240 
   241 primrec times_nat where
   242   mult_0:     "0 * n = (0\<Colon>nat)"
   243 | mult_Suc: "Suc m * n = n + (m * n)"
   244 
   245 lemma mult_0_right [simp]: "(m::nat) * 0 = 0"
   246   by (induct m) simp_all
   247 
   248 lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
   249   by (induct m) (simp_all add: add.left_commute)
   250 
   251 lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"
   252   by (induct m) (simp_all add: add.assoc)
   253 
   254 instance proof
   255   fix n m q :: nat
   256   show "0 \<noteq> (1::nat)" unfolding One_nat_def by simp
   257   show "1 * n = n" unfolding One_nat_def by simp
   258   show "n * m = m * n" by (induct n) simp_all
   259   show "(n * m) * q = n * (m * q)" by (induct n) (simp_all add: add_mult_distrib)
   260   show "(n + m) * q = n * q + m * q" by (rule add_mult_distrib)
   261   assume "n + m = n + q" thus "m = q" by (induct n) simp_all
   262 qed
   263 
   264 end
   265 
   266 subsubsection {* Addition *}
   267 
   268 lemma nat_add_left_cancel:
   269   fixes k m n :: nat
   270   shows "k + m = k + n \<longleftrightarrow> m = n"
   271   by (fact add_left_cancel)
   272 
   273 lemma nat_add_right_cancel:
   274   fixes k m n :: nat
   275   shows "m + k = n + k \<longleftrightarrow> m = n"
   276   by (fact add_right_cancel)
   277 
   278 text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
   279 
   280 lemma add_is_0 [iff]:
   281   fixes m n :: nat
   282   shows "(m + n = 0) = (m = 0 & n = 0)"
   283   by (cases m) simp_all
   284 
   285 lemma add_is_1:
   286   "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
   287   by (cases m) simp_all
   288 
   289 lemma one_is_add:
   290   "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
   291   by (rule trans, rule eq_commute, rule add_is_1)
   292 
   293 lemma add_eq_self_zero:
   294   fixes m n :: nat
   295   shows "m + n = m \<Longrightarrow> n = 0"
   296   by (induct m) simp_all
   297 
   298 lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N"
   299   apply (induct k)
   300    apply simp
   301   apply(drule comp_inj_on[OF _ inj_Suc])
   302   apply (simp add:o_def)
   303   done
   304 
   305 lemma Suc_eq_plus1: "Suc n = n + 1"
   306   unfolding One_nat_def by simp
   307 
   308 lemma Suc_eq_plus1_left: "Suc n = 1 + n"
   309   unfolding One_nat_def by simp
   310 
   311 
   312 subsubsection {* Difference *}
   313 
   314 lemma diff_self_eq_0 [simp]: "(m\<Colon>nat) - m = 0"
   315   by (fact diff_cancel)
   316 
   317 lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
   318   by (fact diff_diff_add)
   319 
   320 lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
   321   by (simp add: diff_diff_left)
   322 
   323 lemma diff_commute: "(i::nat) - j - k = i - k - j"
   324   by (fact diff_right_commute)
   325 
   326 lemma diff_add_inverse: "(n + m) - n = (m::nat)"
   327   by (fact add_diff_cancel_left')
   328 
   329 lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
   330   by (fact add_diff_cancel_right')
   331 
   332 lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
   333   by (fact comm_monoid_diff_class.add_diff_cancel_left)
   334 
   335 lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
   336   by (fact add_diff_cancel_right)
   337 
   338 lemma diff_add_0: "n - (n + m) = (0::nat)"
   339   by (fact diff_add_zero)
   340 
   341 lemma diff_Suc_1 [simp]: "Suc n - 1 = n"
   342   unfolding One_nat_def by simp
   343 
   344 text {* Difference distributes over multiplication *}
   345 
   346 lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"
   347 by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
   348 
   349 lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
   350 by (simp add: diff_mult_distrib mult.commute [of k])
   351   -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
   352 
   353 
   354 subsubsection {* Multiplication *}
   355 
   356 lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
   357   by (fact distrib_left)
   358 
   359 lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
   360   by (induct m) auto
   361 
   362 lemmas nat_distrib =
   363   add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
   364 
   365 lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = Suc 0 & n = Suc 0)"
   366   apply (induct m)
   367    apply simp
   368   apply (induct n)
   369    apply auto
   370   done
   371 
   372 lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = Suc 0 & n = Suc 0)"
   373   apply (rule trans)
   374   apply (rule_tac [2] mult_eq_1_iff, fastforce)
   375   done
   376 
   377 lemma nat_mult_eq_1_iff [simp]: "m * n = (1::nat) \<longleftrightarrow> m = 1 \<and> n = 1"
   378   unfolding One_nat_def by (rule mult_eq_1_iff)
   379 
   380 lemma nat_1_eq_mult_iff [simp]: "(1::nat) = m * n \<longleftrightarrow> m = 1 \<and> n = 1"
   381   unfolding One_nat_def by (rule one_eq_mult_iff)
   382 
   383 lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
   384 proof -
   385   have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
   386   proof (induct n arbitrary: m)
   387     case 0 then show "m = 0" by simp
   388   next
   389     case (Suc n) then show "m = Suc n"
   390       by (cases m) (simp_all add: eq_commute [of "0"])
   391   qed
   392   then show ?thesis by auto
   393 qed
   394 
   395 lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
   396   by (simp add: mult.commute)
   397 
   398 lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
   399   by (subst mult_cancel1) simp
   400 
   401 
   402 subsection {* Orders on @{typ nat} *}
   403 
   404 subsubsection {* Operation definition *}
   405 
   406 instantiation nat :: linorder
   407 begin
   408 
   409 primrec less_eq_nat where
   410   "(0\<Colon>nat) \<le> n \<longleftrightarrow> True"
   411 | "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
   412 
   413 declare less_eq_nat.simps [simp del]
   414 lemma le0 [iff]: "0 \<le> (n\<Colon>nat)" by (simp add: less_eq_nat.simps)
   415 lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by simp
   416 
   417 definition less_nat where
   418   less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"
   419 
   420 lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
   421   by (simp add: less_eq_nat.simps(2))
   422 
   423 lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
   424   unfolding less_eq_Suc_le ..
   425 
   426 lemma le_0_eq [iff]: "(n\<Colon>nat) \<le> 0 \<longleftrightarrow> n = 0"
   427   by (induct n) (simp_all add: less_eq_nat.simps(2))
   428 
   429 lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)"
   430   by (simp add: less_eq_Suc_le)
   431 
   432 lemma less_nat_zero_code [code]: "n < (0\<Colon>nat) \<longleftrightarrow> False"
   433   by simp
   434 
   435 lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
   436   by (simp add: less_eq_Suc_le)
   437 
   438 lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
   439   by (simp add: less_eq_Suc_le)
   440 
   441 lemma Suc_less_eq2: "Suc n < m \<longleftrightarrow> (\<exists>m'. m = Suc m' \<and> n < m')"
   442   by (cases m) auto
   443 
   444 lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
   445   by (induct m arbitrary: n)
   446     (simp_all add: less_eq_nat.simps(2) split: nat.splits)
   447 
   448 lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
   449   by (cases n) (auto intro: le_SucI)
   450 
   451 lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
   452   by (simp add: less_eq_Suc_le) (erule Suc_leD)
   453 
   454 lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
   455   by (simp add: less_eq_Suc_le) (erule Suc_leD)
   456 
   457 instance
   458 proof
   459   fix n m :: nat
   460   show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n" 
   461   proof (induct n arbitrary: m)
   462     case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
   463   next
   464     case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
   465   qed
   466 next
   467   fix n :: nat show "n \<le> n" by (induct n) simp_all
   468 next
   469   fix n m :: nat assume "n \<le> m" and "m \<le> n"
   470   then show "n = m"
   471     by (induct n arbitrary: m)
   472       (simp_all add: less_eq_nat.simps(2) split: nat.splits)
   473 next
   474   fix n m q :: nat assume "n \<le> m" and "m \<le> q"
   475   then show "n \<le> q"
   476   proof (induct n arbitrary: m q)
   477     case 0 show ?case by simp
   478   next
   479     case (Suc n) then show ?case
   480       by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
   481         simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
   482         simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
   483   qed
   484 next
   485   fix n m :: nat show "n \<le> m \<or> m \<le> n"
   486     by (induct n arbitrary: m)
   487       (simp_all add: less_eq_nat.simps(2) split: nat.splits)
   488 qed
   489 
   490 end
   491 
   492 instantiation nat :: order_bot
   493 begin
   494 
   495 definition bot_nat :: nat where
   496   "bot_nat = 0"
   497 
   498 instance proof
   499 qed (simp add: bot_nat_def)
   500 
   501 end
   502 
   503 instance nat :: no_top
   504   by default (auto intro: less_Suc_eq_le [THEN iffD2])
   505 
   506 
   507 subsubsection {* Introduction properties *}
   508 
   509 lemma lessI [iff]: "n < Suc n"
   510   by (simp add: less_Suc_eq_le)
   511 
   512 lemma zero_less_Suc [iff]: "0 < Suc n"
   513   by (simp add: less_Suc_eq_le)
   514 
   515 
   516 subsubsection {* Elimination properties *}
   517 
   518 lemma less_not_refl: "~ n < (n::nat)"
   519   by (rule order_less_irrefl)
   520 
   521 lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)"
   522   by (rule not_sym) (rule less_imp_neq) 
   523 
   524 lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
   525   by (rule less_imp_neq)
   526 
   527 lemma less_irrefl_nat: "(n::nat) < n ==> R"
   528   by (rule notE, rule less_not_refl)
   529 
   530 lemma less_zeroE: "(n::nat) < 0 ==> R"
   531   by (rule notE) (rule not_less0)
   532 
   533 lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
   534   unfolding less_Suc_eq_le le_less ..
   535 
   536 lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
   537   by (simp add: less_Suc_eq)
   538 
   539 lemma less_one [iff]: "(n < (1::nat)) = (n = 0)"
   540   unfolding One_nat_def by (rule less_Suc0)
   541 
   542 lemma Suc_mono: "m < n ==> Suc m < Suc n"
   543   by simp
   544 
   545 text {* "Less than" is antisymmetric, sort of *}
   546 lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
   547   unfolding not_less less_Suc_eq_le by (rule antisym)
   548 
   549 lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
   550   by (rule linorder_neq_iff)
   551 
   552 lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
   553   and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
   554   shows "P n m"
   555   apply (rule less_linear [THEN disjE])
   556   apply (erule_tac [2] disjE)
   557   apply (erule lessCase)
   558   apply (erule sym [THEN eqCase])
   559   apply (erule major)
   560   done
   561 
   562 
   563 subsubsection {* Inductive (?) properties *}
   564 
   565 lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
   566   unfolding less_eq_Suc_le [of m] le_less by simp 
   567 
   568 lemma lessE:
   569   assumes major: "i < k"
   570   and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
   571   shows P
   572 proof -
   573   from major have "\<exists>j. i \<le> j \<and> k = Suc j"
   574     unfolding less_eq_Suc_le by (induct k) simp_all
   575   then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
   576     by (clarsimp simp add: less_le)
   577   with p1 p2 show P by auto
   578 qed
   579 
   580 lemma less_SucE: assumes major: "m < Suc n"
   581   and less: "m < n ==> P" and eq: "m = n ==> P" shows P
   582   apply (rule major [THEN lessE])
   583   apply (rule eq, blast)
   584   apply (rule less, blast)
   585   done
   586 
   587 lemma Suc_lessE: assumes major: "Suc i < k"
   588   and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
   589   apply (rule major [THEN lessE])
   590   apply (erule lessI [THEN minor])
   591   apply (erule Suc_lessD [THEN minor], assumption)
   592   done
   593 
   594 lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
   595   by simp
   596 
   597 lemma less_trans_Suc:
   598   assumes le: "i < j" shows "j < k ==> Suc i < k"
   599   apply (induct k, simp_all)
   600   apply (insert le)
   601   apply (simp add: less_Suc_eq)
   602   apply (blast dest: Suc_lessD)
   603   done
   604 
   605 text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
   606 lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
   607   unfolding not_less less_Suc_eq_le ..
   608 
   609 lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
   610   unfolding not_le Suc_le_eq ..
   611 
   612 text {* Properties of "less than or equal" *}
   613 
   614 lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
   615   unfolding less_Suc_eq_le .
   616 
   617 lemma Suc_n_not_le_n: "~ Suc n \<le> n"
   618   unfolding not_le less_Suc_eq_le ..
   619 
   620 lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
   621   by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
   622 
   623 lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
   624   by (drule le_Suc_eq [THEN iffD1], iprover+)
   625 
   626 lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
   627   unfolding Suc_le_eq .
   628 
   629 text {* Stronger version of @{text Suc_leD} *}
   630 lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
   631   unfolding Suc_le_eq .
   632 
   633 lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)"
   634   unfolding less_eq_Suc_le by (rule Suc_leD)
   635 
   636 text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
   637 lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
   638 
   639 
   640 text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
   641 
   642 lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
   643   unfolding le_less .
   644 
   645 lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
   646   by (rule le_less)
   647 
   648 text {* Useful with @{text blast}. *}
   649 lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
   650   by auto
   651 
   652 lemma le_refl: "n \<le> (n::nat)"
   653   by simp
   654 
   655 lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
   656   by (rule order_trans)
   657 
   658 lemma le_antisym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
   659   by (rule antisym)
   660 
   661 lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
   662   by (rule less_le)
   663 
   664 lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
   665   unfolding less_le ..
   666 
   667 lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
   668   by (rule linear)
   669 
   670 lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
   671 
   672 lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
   673   unfolding less_Suc_eq_le by auto
   674 
   675 lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
   676   unfolding not_less by (rule le_less_Suc_eq)
   677 
   678 lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
   679 
   680 lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
   681 by (cases n) simp_all
   682 
   683 lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"
   684 by (cases n) simp_all
   685 
   686 lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
   687 by (cases n) simp_all
   688 
   689 lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
   690 by (cases n) simp_all
   691 
   692 text {* This theorem is useful with @{text blast} *}
   693 lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
   694 by (rule neq0_conv[THEN iffD1], iprover)
   695 
   696 lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
   697 by (fast intro: not0_implies_Suc)
   698 
   699 lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"
   700 using neq0_conv by blast
   701 
   702 lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
   703 by (induct m') simp_all
   704 
   705 text {* Useful in certain inductive arguments *}
   706 lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
   707 by (cases m) simp_all
   708 
   709 
   710 subsubsection {* Monotonicity of Addition *}
   711 
   712 lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"
   713 by (simp add: diff_Suc split: nat.split)
   714 
   715 lemma Suc_diff_1 [simp]: "0 < n ==> Suc (n - 1) = n"
   716 unfolding One_nat_def by (rule Suc_pred)
   717 
   718 lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
   719 by (induct k) simp_all
   720 
   721 lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
   722 by (induct k) simp_all
   723 
   724 lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)"
   725 by(auto dest:gr0_implies_Suc)
   726 
   727 text {* strict, in 1st argument *}
   728 lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
   729 by (induct k) simp_all
   730 
   731 text {* strict, in both arguments *}
   732 lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
   733   apply (rule add_less_mono1 [THEN less_trans], assumption+)
   734   apply (induct j, simp_all)
   735   done
   736 
   737 text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
   738 lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
   739   apply (induct n)
   740   apply (simp_all add: order_le_less)
   741   apply (blast elim!: less_SucE
   742                intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
   743   done
   744 
   745 lemma le_Suc_ex: "(k::nat) \<le> l \<Longrightarrow> (\<exists>n. l = k + n)"
   746   by (auto simp: less_Suc_eq_le[symmetric] dest: less_imp_Suc_add)
   747 
   748 text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
   749 lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"
   750 apply(auto simp: gr0_conv_Suc)
   751 apply (induct_tac m)
   752 apply (simp_all add: add_less_mono)
   753 done
   754 
   755 text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
   756 instance nat :: linordered_semidom
   757 proof
   758   show "0 < (1::nat)" by simp
   759   show "\<And>m n q :: nat. m \<le> n \<Longrightarrow> q + m \<le> q + n" by simp
   760   show "\<And>m n q :: nat. m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n" by (simp add: mult_less_mono2)
   761 qed
   762 
   763 instance nat :: no_zero_divisors
   764 proof
   765   fix a::nat and b::nat show "a ~= 0 \<Longrightarrow> b ~= 0 \<Longrightarrow> a * b ~= 0" by auto
   766 qed
   767 
   768 
   769 subsubsection {* @{term min} and @{term max} *}
   770 
   771 lemma mono_Suc: "mono Suc"
   772 by (rule monoI) simp
   773 
   774 lemma min_0L [simp]: "min 0 n = (0::nat)"
   775 by (rule min_absorb1) simp
   776 
   777 lemma min_0R [simp]: "min n 0 = (0::nat)"
   778 by (rule min_absorb2) simp
   779 
   780 lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
   781 by (simp add: mono_Suc min_of_mono)
   782 
   783 lemma min_Suc1:
   784    "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
   785 by (simp split: nat.split)
   786 
   787 lemma min_Suc2:
   788    "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
   789 by (simp split: nat.split)
   790 
   791 lemma max_0L [simp]: "max 0 n = (n::nat)"
   792 by (rule max_absorb2) simp
   793 
   794 lemma max_0R [simp]: "max n 0 = (n::nat)"
   795 by (rule max_absorb1) simp
   796 
   797 lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
   798 by (simp add: mono_Suc max_of_mono)
   799 
   800 lemma max_Suc1:
   801    "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
   802 by (simp split: nat.split)
   803 
   804 lemma max_Suc2:
   805    "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
   806 by (simp split: nat.split)
   807 
   808 lemma nat_mult_min_left:
   809   fixes m n q :: nat
   810   shows "min m n * q = min (m * q) (n * q)"
   811   by (simp add: min_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
   812 
   813 lemma nat_mult_min_right:
   814   fixes m n q :: nat
   815   shows "m * min n q = min (m * n) (m * q)"
   816   by (simp add: min_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
   817 
   818 lemma nat_add_max_left:
   819   fixes m n q :: nat
   820   shows "max m n + q = max (m + q) (n + q)"
   821   by (simp add: max_def)
   822 
   823 lemma nat_add_max_right:
   824   fixes m n q :: nat
   825   shows "m + max n q = max (m + n) (m + q)"
   826   by (simp add: max_def)
   827 
   828 lemma nat_mult_max_left:
   829   fixes m n q :: nat
   830   shows "max m n * q = max (m * q) (n * q)"
   831   by (simp add: max_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
   832 
   833 lemma nat_mult_max_right:
   834   fixes m n q :: nat
   835   shows "m * max n q = max (m * n) (m * q)"
   836   by (simp add: max_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
   837 
   838 
   839 subsubsection {* Additional theorems about @{term "op \<le>"} *}
   840 
   841 text {* Complete induction, aka course-of-values induction *}
   842 
   843 instance nat :: wellorder proof
   844   fix P and n :: nat
   845   assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n"
   846   have "\<And>q. q \<le> n \<Longrightarrow> P q"
   847   proof (induct n)
   848     case (0 n)
   849     have "P 0" by (rule step) auto
   850     thus ?case using 0 by auto
   851   next
   852     case (Suc m n)
   853     then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)
   854     thus ?case
   855     proof
   856       assume "n \<le> m" thus "P n" by (rule Suc(1))
   857     next
   858       assume n: "n = Suc m"
   859       show "P n"
   860         by (rule step) (rule Suc(1), simp add: n le_simps)
   861     qed
   862   qed
   863   then show "P n" by auto
   864 qed
   865 
   866 
   867 lemma Least_eq_0[simp]: "P(0::nat) \<Longrightarrow> Least P = 0"
   868 by (rule Least_equality[OF _ le0])
   869 
   870 lemma Least_Suc:
   871      "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
   872   apply (cases n, auto)
   873   apply (frule LeastI)
   874   apply (drule_tac P = "%x. P (Suc x) " in LeastI)
   875   apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
   876   apply (erule_tac [2] Least_le)
   877   apply (cases "LEAST x. P x", auto)
   878   apply (drule_tac P = "%x. P (Suc x) " in Least_le)
   879   apply (blast intro: order_antisym)
   880   done
   881 
   882 lemma Least_Suc2:
   883    "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
   884   apply (erule (1) Least_Suc [THEN ssubst])
   885   apply simp
   886   done
   887 
   888 lemma ex_least_nat_le: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k\<le>n. (\<forall>i<k. \<not>P i) & P(k)"
   889   apply (cases n)
   890    apply blast
   891   apply (rule_tac x="LEAST k. P(k)" in exI)
   892   apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
   893   done
   894 
   895 lemma ex_least_nat_less: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k<n. (\<forall>i\<le>k. \<not>P i) & P(k+1)"
   896   unfolding One_nat_def
   897   apply (cases n)
   898    apply blast
   899   apply (frule (1) ex_least_nat_le)
   900   apply (erule exE)
   901   apply (case_tac k)
   902    apply simp
   903   apply (rename_tac k1)
   904   apply (rule_tac x=k1 in exI)
   905   apply (auto simp add: less_eq_Suc_le)
   906   done
   907 
   908 lemma nat_less_induct:
   909   assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
   910   using assms less_induct by blast
   911 
   912 lemma measure_induct_rule [case_names less]:
   913   fixes f :: "'a \<Rightarrow> nat"
   914   assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
   915   shows "P a"
   916 by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step)
   917 
   918 text {* old style induction rules: *}
   919 lemma measure_induct:
   920   fixes f :: "'a \<Rightarrow> nat"
   921   shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
   922   by (rule measure_induct_rule [of f P a]) iprover
   923 
   924 lemma full_nat_induct:
   925   assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"
   926   shows "P n"
   927   by (rule less_induct) (auto intro: step simp:le_simps)
   928 
   929 text{*An induction rule for estabilishing binary relations*}
   930 lemma less_Suc_induct:
   931   assumes less:  "i < j"
   932      and  step:  "!!i. P i (Suc i)"
   933      and  trans: "!!i j k. i < j ==> j < k ==>  P i j ==> P j k ==> P i k"
   934   shows "P i j"
   935 proof -
   936   from less obtain k where j: "j = Suc (i + k)" by (auto dest: less_imp_Suc_add)
   937   have "P i (Suc (i + k))"
   938   proof (induct k)
   939     case 0
   940     show ?case by (simp add: step)
   941   next
   942     case (Suc k)
   943     have "0 + i < Suc k + i" by (rule add_less_mono1) simp
   944     hence "i < Suc (i + k)" by (simp add: add.commute)
   945     from trans[OF this lessI Suc step]
   946     show ?case by simp
   947   qed
   948   thus "P i j" by (simp add: j)
   949 qed
   950 
   951 text {* The method of infinite descent, frequently used in number theory.
   952 Provided by Roelof Oosterhuis.
   953 $P(n)$ is true for all $n\in\mathbb{N}$ if
   954 \begin{itemize}
   955   \item case ``0'': given $n=0$ prove $P(n)$,
   956   \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
   957         a smaller integer $m$ such that $\neg P(m)$.
   958 \end{itemize} *}
   959 
   960 text{* A compact version without explicit base case: *}
   961 lemma infinite_descent:
   962   "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow>  \<exists>m<n. \<not>  P m \<rbrakk> \<Longrightarrow>  P n"
   963 by (induct n rule: less_induct) auto
   964 
   965 lemma infinite_descent0[case_names 0 smaller]: 
   966   "\<lbrakk> P 0; !!n. n>0 \<Longrightarrow> \<not> P n \<Longrightarrow> (\<exists>m::nat. m < n \<and> \<not>P m) \<rbrakk> \<Longrightarrow> P n"
   967 by (rule infinite_descent) (case_tac "n>0", auto)
   968 
   969 text {*
   970 Infinite descent using a mapping to $\mathbb{N}$:
   971 $P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
   972 \begin{itemize}
   973 \item case ``0'': given $V(x)=0$ prove $P(x)$,
   974 \item case ``smaller'': given $V(x)>0$ and $\neg P(x)$ prove there exists a $y \in D$ such that $V(y)<V(x)$ and $~\neg P(y)$.
   975 \end{itemize}
   976 NB: the proof also shows how to use the previous lemma. *}
   977 
   978 corollary infinite_descent0_measure [case_names 0 smaller]:
   979   assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
   980     and   A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
   981   shows "P x"
   982 proof -
   983   obtain n where "n = V x" by auto
   984   moreover have "\<And>x. V x = n \<Longrightarrow> P x"
   985   proof (induct n rule: infinite_descent0)
   986     case 0 -- "i.e. $V(x) = 0$"
   987     with A0 show "P x" by auto
   988   next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
   989     case (smaller n)
   990     then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
   991     with A1 obtain y where "V y < V x \<and> \<not> P y" by auto
   992     with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
   993     then show ?case by auto
   994   qed
   995   ultimately show "P x" by auto
   996 qed
   997 
   998 text{* Again, without explicit base case: *}
   999 lemma infinite_descent_measure:
  1000 assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
  1001 proof -
  1002   from assms obtain n where "n = V x" by auto
  1003   moreover have "!!x. V x = n \<Longrightarrow> P x"
  1004   proof (induct n rule: infinite_descent, auto)
  1005     fix x assume "\<not> P x"
  1006     with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
  1007   qed
  1008   ultimately show "P x" by auto
  1009 qed
  1010 
  1011 text {* A [clumsy] way of lifting @{text "<"}
  1012   monotonicity to @{text "\<le>"} monotonicity *}
  1013 lemma less_mono_imp_le_mono:
  1014   "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
  1015 by (simp add: order_le_less) (blast)
  1016 
  1017 
  1018 text {* non-strict, in 1st argument *}
  1019 lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
  1020 by (rule add_right_mono)
  1021 
  1022 text {* non-strict, in both arguments *}
  1023 lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
  1024 by (rule add_mono)
  1025 
  1026 lemma le_add2: "n \<le> ((m + n)::nat)"
  1027 by (insert add_right_mono [of 0 m n], simp)
  1028 
  1029 lemma le_add1: "n \<le> ((n + m)::nat)"
  1030 by (simp add: add.commute, rule le_add2)
  1031 
  1032 lemma less_add_Suc1: "i < Suc (i + m)"
  1033 by (rule le_less_trans, rule le_add1, rule lessI)
  1034 
  1035 lemma less_add_Suc2: "i < Suc (m + i)"
  1036 by (rule le_less_trans, rule le_add2, rule lessI)
  1037 
  1038 lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
  1039 by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
  1040 
  1041 lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
  1042 by (rule le_trans, assumption, rule le_add1)
  1043 
  1044 lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
  1045 by (rule le_trans, assumption, rule le_add2)
  1046 
  1047 lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
  1048 by (rule less_le_trans, assumption, rule le_add1)
  1049 
  1050 lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
  1051 by (rule less_le_trans, assumption, rule le_add2)
  1052 
  1053 lemma add_lessD1: "i + j < (k::nat) ==> i < k"
  1054 apply (rule le_less_trans [of _ "i+j"])
  1055 apply (simp_all add: le_add1)
  1056 done
  1057 
  1058 lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
  1059 apply (rule notI)
  1060 apply (drule add_lessD1)
  1061 apply (erule less_irrefl [THEN notE])
  1062 done
  1063 
  1064 lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
  1065 by (simp add: add.commute)
  1066 
  1067 lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
  1068 apply (rule order_trans [of _ "m+k"])
  1069 apply (simp_all add: le_add1)
  1070 done
  1071 
  1072 lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
  1073 apply (simp add: add.commute)
  1074 apply (erule add_leD1)
  1075 done
  1076 
  1077 lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
  1078 by (blast dest: add_leD1 add_leD2)
  1079 
  1080 text {* needs @{text "!!k"} for @{text ac_simps} to work *}
  1081 lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
  1082 by (force simp del: add_Suc_right
  1083     simp add: less_iff_Suc_add add_Suc_right [symmetric] ac_simps)
  1084 
  1085 
  1086 subsubsection {* More results about difference *}
  1087 
  1088 text {* Addition is the inverse of subtraction:
  1089   if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
  1090 lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
  1091 by (induct m n rule: diff_induct) simp_all
  1092 
  1093 lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
  1094 by (simp add: add_diff_inverse linorder_not_less)
  1095 
  1096 lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
  1097 by (simp add: add.commute)
  1098 
  1099 lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
  1100 by (induct m n rule: diff_induct) simp_all
  1101 
  1102 lemma diff_less_Suc: "m - n < Suc m"
  1103 apply (induct m n rule: diff_induct)
  1104 apply (erule_tac [3] less_SucE)
  1105 apply (simp_all add: less_Suc_eq)
  1106 done
  1107 
  1108 lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
  1109 by (induct m n rule: diff_induct) (simp_all add: le_SucI)
  1110 
  1111 lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
  1112   by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])
  1113 
  1114 instance nat :: ordered_cancel_comm_monoid_diff
  1115 proof
  1116   show "\<And>m n :: nat. m \<le> n \<longleftrightarrow> (\<exists>q. n = m + q)" by (fact le_iff_add)
  1117 qed
  1118 
  1119 lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
  1120 by (rule le_less_trans, rule diff_le_self)
  1121 
  1122 lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
  1123 by (cases n) (auto simp add: le_simps)
  1124 
  1125 lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
  1126 by (induct j k rule: diff_induct) simp_all
  1127 
  1128 lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
  1129 by (simp add: add.commute diff_add_assoc)
  1130 
  1131 lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
  1132 by (auto simp add: diff_add_inverse2)
  1133 
  1134 lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
  1135 by (induct m n rule: diff_induct) simp_all
  1136 
  1137 lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
  1138 by (rule iffD2, rule diff_is_0_eq)
  1139 
  1140 lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
  1141 by (induct m n rule: diff_induct) simp_all
  1142 
  1143 lemma less_imp_add_positive:
  1144   assumes "i < j"
  1145   shows "\<exists>k::nat. 0 < k & i + k = j"
  1146 proof
  1147   from assms show "0 < j - i & i + (j - i) = j"
  1148     by (simp add: order_less_imp_le)
  1149 qed
  1150 
  1151 text {* a nice rewrite for bounded subtraction *}
  1152 lemma nat_minus_add_max:
  1153   fixes n m :: nat
  1154   shows "n - m + m = max n m"
  1155     by (simp add: max_def not_le order_less_imp_le)
  1156 
  1157 lemma nat_diff_split:
  1158   "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
  1159     -- {* elimination of @{text -} on @{text nat} *}
  1160 by (cases "a < b")
  1161   (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse
  1162     not_less le_less dest!: add_eq_self_zero add_eq_self_zero[OF sym])
  1163 
  1164 lemma nat_diff_split_asm:
  1165   "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
  1166     -- {* elimination of @{text -} on @{text nat} in assumptions *}
  1167 by (auto split: nat_diff_split)
  1168 
  1169 lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
  1170   by simp
  1171 
  1172 lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
  1173   unfolding One_nat_def by (cases m) simp_all
  1174 
  1175 lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
  1176   unfolding One_nat_def by (cases m) simp_all
  1177 
  1178 lemma Suc_diff_eq_diff_pred: "0 < n ==> Suc m - n = m - (n - 1)"
  1179   unfolding One_nat_def by (cases n) simp_all
  1180 
  1181 lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
  1182   unfolding One_nat_def by (cases m) simp_all
  1183 
  1184 lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
  1185   by (fact Let_def)
  1186 
  1187 
  1188 subsubsection {* Monotonicity of Multiplication *}
  1189 
  1190 lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
  1191 by (simp add: mult_right_mono)
  1192 
  1193 lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
  1194 by (simp add: mult_left_mono)
  1195 
  1196 text {* @{text "\<le>"} monotonicity, BOTH arguments *}
  1197 lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
  1198 by (simp add: mult_mono)
  1199 
  1200 lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
  1201 by (simp add: mult_strict_right_mono)
  1202 
  1203 text{*Differs from the standard @{text zero_less_mult_iff} in that
  1204       there are no negative numbers.*}
  1205 lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
  1206   apply (induct m)
  1207    apply simp
  1208   apply (case_tac n)
  1209    apply simp_all
  1210   done
  1211 
  1212 lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)"
  1213   apply (induct m)
  1214    apply simp
  1215   apply (case_tac n)
  1216    apply simp_all
  1217   done
  1218 
  1219 lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
  1220   apply (safe intro!: mult_less_mono1)
  1221   apply (cases k, auto)
  1222   apply (simp del: le_0_eq add: linorder_not_le [symmetric])
  1223   apply (blast intro: mult_le_mono1)
  1224   done
  1225 
  1226 lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
  1227 by (simp add: mult.commute [of k])
  1228 
  1229 lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
  1230 by (simp add: linorder_not_less [symmetric], auto)
  1231 
  1232 lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
  1233 by (simp add: linorder_not_less [symmetric], auto)
  1234 
  1235 lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
  1236 by (subst mult_less_cancel1) simp
  1237 
  1238 lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
  1239 by (subst mult_le_cancel1) simp
  1240 
  1241 lemma le_square: "m \<le> m * (m::nat)"
  1242   by (cases m) (auto intro: le_add1)
  1243 
  1244 lemma le_cube: "(m::nat) \<le> m * (m * m)"
  1245   by (cases m) (auto intro: le_add1)
  1246 
  1247 text {* Lemma for @{text gcd} *}
  1248 lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
  1249   apply (drule sym)
  1250   apply (rule disjCI)
  1251   apply (rule nat_less_cases, erule_tac [2] _)
  1252    apply (drule_tac [2] mult_less_mono2)
  1253     apply (auto)
  1254   done
  1255 
  1256 lemma mono_times_nat:
  1257   fixes n :: nat
  1258   assumes "n > 0"
  1259   shows "mono (times n)"
  1260 proof
  1261   fix m q :: nat
  1262   assume "m \<le> q"
  1263   with assms show "n * m \<le> n * q" by simp
  1264 qed
  1265 
  1266 text {* the lattice order on @{typ nat} *}
  1267 
  1268 instantiation nat :: distrib_lattice
  1269 begin
  1270 
  1271 definition
  1272   "(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min"
  1273 
  1274 definition
  1275   "(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max"
  1276 
  1277 instance by intro_classes
  1278   (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
  1279     intro: order_less_imp_le antisym elim!: order_trans order_less_trans)
  1280 
  1281 end
  1282 
  1283 
  1284 subsection {* Natural operation of natural numbers on functions *}
  1285 
  1286 text {*
  1287   We use the same logical constant for the power operations on
  1288   functions and relations, in order to share the same syntax.
  1289 *}
  1290 
  1291 consts compow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
  1292 
  1293 abbreviation compower :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^^" 80) where
  1294   "f ^^ n \<equiv> compow n f"
  1295 
  1296 notation (latex output)
  1297   compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
  1298 
  1299 notation (HTML output)
  1300   compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
  1301 
  1302 text {* @{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f} *}
  1303 
  1304 overloading
  1305   funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
  1306 begin
  1307 
  1308 primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
  1309   "funpow 0 f = id"
  1310 | "funpow (Suc n) f = f o funpow n f"
  1311 
  1312 end
  1313 
  1314 lemma funpow_Suc_right:
  1315   "f ^^ Suc n = f ^^ n \<circ> f"
  1316 proof (induct n)
  1317   case 0 then show ?case by simp
  1318 next
  1319   fix n
  1320   assume "f ^^ Suc n = f ^^ n \<circ> f"
  1321   then show "f ^^ Suc (Suc n) = f ^^ Suc n \<circ> f"
  1322     by (simp add: o_assoc)
  1323 qed
  1324 
  1325 lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right
  1326 
  1327 text {* for code generation *}
  1328 
  1329 definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
  1330   funpow_code_def [code_abbrev]: "funpow = compow"
  1331 
  1332 lemma [code]:
  1333   "funpow (Suc n) f = f o funpow n f"
  1334   "funpow 0 f = id"
  1335   by (simp_all add: funpow_code_def)
  1336 
  1337 hide_const (open) funpow
  1338 
  1339 lemma funpow_add:
  1340   "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
  1341   by (induct m) simp_all
  1342 
  1343 lemma funpow_mult:
  1344   fixes f :: "'a \<Rightarrow> 'a"
  1345   shows "(f ^^ m) ^^ n = f ^^ (m * n)"
  1346   by (induct n) (simp_all add: funpow_add)
  1347 
  1348 lemma funpow_swap1:
  1349   "f ((f ^^ n) x) = (f ^^ n) (f x)"
  1350 proof -
  1351   have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp
  1352   also have "\<dots>  = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)
  1353   also have "\<dots> = (f ^^ n) (f x)" by simp
  1354   finally show ?thesis .
  1355 qed
  1356 
  1357 lemma comp_funpow:
  1358   fixes f :: "'a \<Rightarrow> 'a"
  1359   shows "comp f ^^ n = comp (f ^^ n)"
  1360   by (induct n) simp_all
  1361 
  1362 lemma Suc_funpow[simp]: "Suc ^^ n = (op + n)"
  1363   by (induct n) simp_all
  1364 
  1365 lemma id_funpow[simp]: "id ^^ n = id"
  1366   by (induct n) simp_all
  1367 
  1368 subsection {* Kleene iteration *}
  1369 
  1370 lemma Kleene_iter_lpfp:
  1371 assumes "mono f" and "f p \<le> p" shows "(f^^k) (bot::'a::order_bot) \<le> p"
  1372 proof(induction k)
  1373   case 0 show ?case by simp
  1374 next
  1375   case Suc
  1376   from monoD[OF assms(1) Suc] assms(2)
  1377   show ?case by simp
  1378 qed
  1379 
  1380 lemma lfp_Kleene_iter: assumes "mono f" and "(f^^Suc k) bot = (f^^k) bot"
  1381 shows "lfp f = (f^^k) bot"
  1382 proof(rule antisym)
  1383   show "lfp f \<le> (f^^k) bot"
  1384   proof(rule lfp_lowerbound)
  1385     show "f ((f^^k) bot) \<le> (f^^k) bot" using assms(2) by simp
  1386   qed
  1387 next
  1388   show "(f^^k) bot \<le> lfp f"
  1389     using Kleene_iter_lpfp[OF assms(1)] lfp_unfold[OF assms(1)] by simp
  1390 qed
  1391 
  1392 
  1393 subsection {* Embedding of the Naturals into any @{text semiring_1}: @{term of_nat} *}
  1394 
  1395 context semiring_1
  1396 begin
  1397 
  1398 definition of_nat :: "nat \<Rightarrow> 'a" where
  1399   "of_nat n = (plus 1 ^^ n) 0"
  1400 
  1401 lemma of_nat_simps [simp]:
  1402   shows of_nat_0: "of_nat 0 = 0"
  1403     and of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
  1404   by (simp_all add: of_nat_def)
  1405 
  1406 lemma of_nat_1 [simp]: "of_nat 1 = 1"
  1407   by (simp add: of_nat_def)
  1408 
  1409 lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
  1410   by (induct m) (simp_all add: ac_simps)
  1411 
  1412 lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"
  1413   by (induct m) (simp_all add: ac_simps distrib_right)
  1414 
  1415 primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where
  1416   "of_nat_aux inc 0 i = i"
  1417 | "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- {* tail recursive *}
  1418 
  1419 lemma of_nat_code:
  1420   "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"
  1421 proof (induct n)
  1422   case 0 then show ?case by simp
  1423 next
  1424   case (Suc n)
  1425   have "\<And>i. of_nat_aux (\<lambda>i. i + 1) n (i + 1) = of_nat_aux (\<lambda>i. i + 1) n i + 1"
  1426     by (induct n) simp_all
  1427   from this [of 0] have "of_nat_aux (\<lambda>i. i + 1) n 1 = of_nat_aux (\<lambda>i. i + 1) n 0 + 1"
  1428     by simp
  1429   with Suc show ?case by (simp add: add.commute)
  1430 qed
  1431 
  1432 end
  1433 
  1434 declare of_nat_code [code]
  1435 
  1436 text{*Class for unital semirings with characteristic zero.
  1437  Includes non-ordered rings like the complex numbers.*}
  1438 
  1439 class semiring_char_0 = semiring_1 +
  1440   assumes inj_of_nat: "inj of_nat"
  1441 begin
  1442 
  1443 lemma of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"
  1444   by (auto intro: inj_of_nat injD)
  1445 
  1446 text{*Special cases where either operand is zero*}
  1447 
  1448 lemma of_nat_0_eq_iff [simp]: "0 = of_nat n \<longleftrightarrow> 0 = n"
  1449   by (fact of_nat_eq_iff [of 0 n, unfolded of_nat_0])
  1450 
  1451 lemma of_nat_eq_0_iff [simp]: "of_nat m = 0 \<longleftrightarrow> m = 0"
  1452   by (fact of_nat_eq_iff [of m 0, unfolded of_nat_0])
  1453 
  1454 end
  1455 
  1456 context linordered_semidom
  1457 begin
  1458 
  1459 lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"
  1460   by (induct n) simp_all
  1461 
  1462 lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"
  1463   by (simp add: not_less)
  1464 
  1465 lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
  1466   by (induct m n rule: diff_induct, simp_all add: add_pos_nonneg)
  1467 
  1468 lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
  1469   by (simp add: not_less [symmetric] linorder_not_less [symmetric])
  1470 
  1471 lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"
  1472   by simp
  1473 
  1474 lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
  1475   by simp
  1476 
  1477 text{*Every @{text linordered_semidom} has characteristic zero.*}
  1478 
  1479 subclass semiring_char_0 proof
  1480 qed (auto intro!: injI simp add: eq_iff)
  1481 
  1482 text{*Special cases where either operand is zero*}
  1483 
  1484 lemma of_nat_le_0_iff [simp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"
  1485   by (rule of_nat_le_iff [of _ 0, simplified])
  1486 
  1487 lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"
  1488   by (rule of_nat_less_iff [of 0, simplified])
  1489 
  1490 end
  1491 
  1492 context ring_1
  1493 begin
  1494 
  1495 lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"
  1496 by (simp add: algebra_simps of_nat_add [symmetric])
  1497 
  1498 end
  1499 
  1500 context linordered_idom
  1501 begin
  1502 
  1503 lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"
  1504   unfolding abs_if by auto
  1505 
  1506 end
  1507 
  1508 lemma of_nat_id [simp]: "of_nat n = n"
  1509   by (induct n) simp_all
  1510 
  1511 lemma of_nat_eq_id [simp]: "of_nat = id"
  1512   by (auto simp add: fun_eq_iff)
  1513 
  1514 
  1515 subsection {* The Set of Natural Numbers *}
  1516 
  1517 context semiring_1
  1518 begin
  1519 
  1520 definition Nats  :: "'a set" where
  1521   "Nats = range of_nat"
  1522 
  1523 notation (xsymbols)
  1524   Nats  ("\<nat>")
  1525 
  1526 lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"
  1527   by (simp add: Nats_def)
  1528 
  1529 lemma Nats_0 [simp]: "0 \<in> \<nat>"
  1530 apply (simp add: Nats_def)
  1531 apply (rule range_eqI)
  1532 apply (rule of_nat_0 [symmetric])
  1533 done
  1534 
  1535 lemma Nats_1 [simp]: "1 \<in> \<nat>"
  1536 apply (simp add: Nats_def)
  1537 apply (rule range_eqI)
  1538 apply (rule of_nat_1 [symmetric])
  1539 done
  1540 
  1541 lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"
  1542 apply (auto simp add: Nats_def)
  1543 apply (rule range_eqI)
  1544 apply (rule of_nat_add [symmetric])
  1545 done
  1546 
  1547 lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"
  1548 apply (auto simp add: Nats_def)
  1549 apply (rule range_eqI)
  1550 apply (rule of_nat_mult [symmetric])
  1551 done
  1552 
  1553 lemma Nats_cases [cases set: Nats]:
  1554   assumes "x \<in> \<nat>"
  1555   obtains (of_nat) n where "x = of_nat n"
  1556   unfolding Nats_def
  1557 proof -
  1558   from `x \<in> \<nat>` have "x \<in> range of_nat" unfolding Nats_def .
  1559   then obtain n where "x = of_nat n" ..
  1560   then show thesis ..
  1561 qed
  1562 
  1563 lemma Nats_induct [case_names of_nat, induct set: Nats]:
  1564   "x \<in> \<nat> \<Longrightarrow> (\<And>n. P (of_nat n)) \<Longrightarrow> P x"
  1565   by (rule Nats_cases) auto
  1566 
  1567 end
  1568 
  1569 
  1570 subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}
  1571 
  1572 lemma subst_equals:
  1573   assumes 1: "t = s" and 2: "u = t"
  1574   shows "u = s"
  1575   using 2 1 by (rule trans)
  1576 
  1577 setup Arith_Data.setup
  1578 
  1579 ML_file "Tools/nat_arith.ML"
  1580 
  1581 simproc_setup nateq_cancel_sums
  1582   ("(l::nat) + m = n" | "(l::nat) = m + n" | "Suc m = n" | "m = Suc n") =
  1583   {* fn phi => try o Nat_Arith.cancel_eq_conv *}
  1584 
  1585 simproc_setup natless_cancel_sums
  1586   ("(l::nat) + m < n" | "(l::nat) < m + n" | "Suc m < n" | "m < Suc n") =
  1587   {* fn phi => try o Nat_Arith.cancel_less_conv *}
  1588 
  1589 simproc_setup natle_cancel_sums
  1590   ("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" | "Suc m \<le> n" | "m \<le> Suc n") =
  1591   {* fn phi => try o Nat_Arith.cancel_le_conv *}
  1592 
  1593 simproc_setup natdiff_cancel_sums
  1594   ("(l::nat) + m - n" | "(l::nat) - (m + n)" | "Suc m - n" | "m - Suc n") =
  1595   {* fn phi => try o Nat_Arith.cancel_diff_conv *}
  1596 
  1597 ML_file "Tools/lin_arith.ML"
  1598 setup {* Lin_Arith.global_setup *}
  1599 declaration {* K Lin_Arith.setup *}
  1600 
  1601 simproc_setup fast_arith_nat ("(m::nat) < n" | "(m::nat) <= n" | "(m::nat) = n") =
  1602   {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}
  1603 (* Because of this simproc, the arithmetic solver is really only
  1604 useful to detect inconsistencies among the premises for subgoals which are
  1605 *not* themselves (in)equalities, because the latter activate
  1606 fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
  1607 solver all the time rather than add the additional check. *)
  1608 
  1609 
  1610 lemmas [arith_split] = nat_diff_split split_min split_max
  1611 
  1612 context order
  1613 begin
  1614 
  1615 lemma lift_Suc_mono_le:
  1616   assumes mono: "\<And>n. f n \<le> f (Suc n)" and "n \<le> n'"
  1617   shows "f n \<le> f n'"
  1618 proof (cases "n < n'")
  1619   case True
  1620   then show ?thesis
  1621     by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
  1622 qed (insert `n \<le> n'`, auto) -- {* trivial for @{prop "n = n'"} *}
  1623 
  1624 lemma lift_Suc_antimono_le:
  1625   assumes mono: "\<And>n. f n \<ge> f (Suc n)" and "n \<le> n'"
  1626   shows "f n \<ge> f n'"
  1627 proof (cases "n < n'")
  1628   case True
  1629   then show ?thesis
  1630     by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
  1631 qed (insert `n \<le> n'`, auto) -- {* trivial for @{prop "n = n'"} *}
  1632 
  1633 lemma lift_Suc_mono_less:
  1634   assumes mono: "\<And>n. f n < f (Suc n)" and "n < n'"
  1635   shows "f n < f n'"
  1636 using `n < n'`
  1637 by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
  1638 
  1639 lemma lift_Suc_mono_less_iff:
  1640   "(\<And>n. f n < f (Suc n)) \<Longrightarrow> f n < f m \<longleftrightarrow> n < m"
  1641   by (blast intro: less_asym' lift_Suc_mono_less [of f]
  1642     dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq [THEN iffD1])
  1643 
  1644 end
  1645 
  1646 lemma mono_iff_le_Suc:
  1647   "mono f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
  1648   unfolding mono_def by (auto intro: lift_Suc_mono_le [of f])
  1649 
  1650 lemma antimono_iff_le_Suc:
  1651   "antimono f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
  1652   unfolding antimono_def by (auto intro: lift_Suc_antimono_le [of f])
  1653 
  1654 lemma mono_nat_linear_lb:
  1655   fixes f :: "nat \<Rightarrow> nat"
  1656   assumes "\<And>m n. m < n \<Longrightarrow> f m < f n"
  1657   shows "f m + k \<le> f (m + k)"
  1658 proof (induct k)
  1659   case 0 then show ?case by simp
  1660 next
  1661   case (Suc k)
  1662   then have "Suc (f m + k) \<le> Suc (f (m + k))" by simp
  1663   also from assms [of "m + k" "Suc (m + k)"] have "Suc (f (m + k)) \<le> f (Suc (m + k))"
  1664     by (simp add: Suc_le_eq)
  1665   finally show ?case by simp
  1666 qed
  1667 
  1668 
  1669 text{*Subtraction laws, mostly by Clemens Ballarin*}
  1670 
  1671 lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
  1672 by arith
  1673 
  1674 lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
  1675 by arith
  1676 
  1677 lemma less_diff_conv2:
  1678   fixes j k i :: nat
  1679   assumes "k \<le> j"
  1680   shows "j - k < i \<longleftrightarrow> j < i + k"
  1681   using assms by arith
  1682 
  1683 lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
  1684 by arith
  1685 
  1686 lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
  1687   by (fact le_diff_conv2) -- {* FIXME delete *}
  1688 
  1689 lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
  1690 by arith
  1691 
  1692 lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
  1693   by (fact le_add_diff) -- {* FIXME delete *}
  1694 
  1695 (*Replaces the previous diff_less and le_diff_less, which had the stronger
  1696   second premise n\<le>m*)
  1697 lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
  1698 by arith
  1699 
  1700 text {* Simplification of relational expressions involving subtraction *}
  1701 
  1702 lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
  1703 by (simp split add: nat_diff_split)
  1704 
  1705 hide_fact (open) diff_diff_eq
  1706 
  1707 lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
  1708 by (auto split add: nat_diff_split)
  1709 
  1710 lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
  1711 by (auto split add: nat_diff_split)
  1712 
  1713 lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
  1714 by (auto split add: nat_diff_split)
  1715 
  1716 text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
  1717 
  1718 (* Monotonicity of subtraction in first argument *)
  1719 lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
  1720 by (simp split add: nat_diff_split)
  1721 
  1722 lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
  1723 by (simp split add: nat_diff_split)
  1724 
  1725 lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
  1726 by (simp split add: nat_diff_split)
  1727 
  1728 lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
  1729 by (simp split add: nat_diff_split)
  1730 
  1731 lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"
  1732 by auto
  1733 
  1734 lemma inj_on_diff_nat: 
  1735   assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"
  1736   shows "inj_on (\<lambda>n. n - k) N"
  1737 proof (rule inj_onI)
  1738   fix x y
  1739   assume a: "x \<in> N" "y \<in> N" "x - k = y - k"
  1740   with k_le_n have "x - k + k = y - k + k" by auto
  1741   with a k_le_n show "x = y" by auto
  1742 qed
  1743 
  1744 text{*Rewriting to pull differences out*}
  1745 
  1746 lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
  1747 by arith
  1748 
  1749 lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
  1750 by arith
  1751 
  1752 lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
  1753 by arith
  1754 
  1755 lemma Suc_diff_Suc: "n < m \<Longrightarrow> Suc (m - Suc n) = m - n"
  1756 by simp
  1757 
  1758 (*The others are
  1759       i - j - k = i - (j + k),
  1760       k \<le> j ==> j - k + i = j + i - k,
  1761       k \<le> j ==> i + (j - k) = i + j - k *)
  1762 lemmas add_diff_assoc = diff_add_assoc [symmetric]
  1763 lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
  1764 declare diff_diff_left [simp]  add_diff_assoc [simp] add_diff_assoc2[simp]
  1765 
  1766 text{*At present we prove no analogue of @{text not_less_Least} or @{text
  1767 Least_Suc}, since there appears to be no need.*}
  1768 
  1769 text{*Lemmas for ex/Factorization*}
  1770 
  1771 lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
  1772 by (cases m) auto
  1773 
  1774 lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
  1775 by (cases m) auto
  1776 
  1777 lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
  1778 by (cases m) auto
  1779 
  1780 text {* Specialized induction principles that work "backwards": *}
  1781 
  1782 lemma inc_induct[consumes 1, case_names base step]:
  1783   assumes less: "i \<le> j"
  1784   assumes base: "P j"
  1785   assumes step: "\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P (Suc n) \<Longrightarrow> P n"
  1786   shows "P i"
  1787   using less step
  1788 proof (induct d\<equiv>"j - i" arbitrary: i)
  1789   case (0 i)
  1790   hence "i = j" by simp
  1791   with base show ?case by simp
  1792 next
  1793   case (Suc d n)
  1794   hence "n \<le> n" "n < j" "P (Suc n)"
  1795     by simp_all
  1796   then show "P n" by fact
  1797 qed
  1798 
  1799 lemma strict_inc_induct[consumes 1, case_names base step]:
  1800   assumes less: "i < j"
  1801   assumes base: "!!i. j = Suc i ==> P i"
  1802   assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
  1803   shows "P i"
  1804   using less
  1805 proof (induct d=="j - i - 1" arbitrary: i)
  1806   case (0 i)
  1807   with `i < j` have "j = Suc i" by simp
  1808   with base show ?case by simp
  1809 next
  1810   case (Suc d i)
  1811   hence "i < j" "P (Suc i)"
  1812     by simp_all
  1813   thus "P i" by (rule step)
  1814 qed
  1815 
  1816 lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
  1817   using inc_induct[of "k - i" k P, simplified] by blast
  1818 
  1819 lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
  1820   using inc_induct[of 0 k P] by blast
  1821 
  1822 text {* Further induction rule similar to @{thm inc_induct} *}
  1823 
  1824 lemma dec_induct[consumes 1, case_names base step]:
  1825   "i \<le> j \<Longrightarrow> P i \<Longrightarrow> (\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n \<Longrightarrow> P (Suc n)) \<Longrightarrow> P j"
  1826   by (induct j arbitrary: i) (auto simp: le_Suc_eq)
  1827  
  1828 subsection {* The divides relation on @{typ nat} *}
  1829 
  1830 lemma dvd_1_left [iff]: "Suc 0 dvd k"
  1831 unfolding dvd_def by simp
  1832 
  1833 lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
  1834 by (simp add: dvd_def)
  1835 
  1836 lemma nat_dvd_1_iff_1 [simp]: "m dvd (1::nat) \<longleftrightarrow> m = 1"
  1837 by (simp add: dvd_def)
  1838 
  1839 lemma dvd_antisym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
  1840   unfolding dvd_def
  1841   by (force dest: mult_eq_self_implies_10 simp add: mult.assoc)
  1842 
  1843 text {* @{term "op dvd"} is a partial order *}
  1844 
  1845 interpretation dvd: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n"
  1846   proof qed (auto intro: dvd_refl dvd_trans dvd_antisym)
  1847 
  1848 lemma dvd_diff_nat[simp]: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
  1849 unfolding dvd_def
  1850 by (blast intro: diff_mult_distrib2 [symmetric])
  1851 
  1852 lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
  1853   apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
  1854   apply (blast intro: dvd_add)
  1855   done
  1856 
  1857 lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
  1858 by (drule_tac m = m in dvd_diff_nat, auto)
  1859 
  1860 lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
  1861   apply (rule iffI)
  1862    apply (erule_tac [2] dvd_add)
  1863    apply (rule_tac [2] dvd_refl)
  1864   apply (subgoal_tac "n = (n+k) -k")
  1865    prefer 2 apply simp
  1866   apply (erule ssubst)
  1867   apply (erule dvd_diff_nat)
  1868   apply (rule dvd_refl)
  1869   done
  1870 
  1871 lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
  1872   unfolding dvd_def
  1873   apply (erule exE)
  1874   apply (simp add: ac_simps)
  1875   done
  1876 
  1877 lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
  1878   apply auto
  1879    apply (subgoal_tac "m*n dvd m*1")
  1880    apply (drule dvd_mult_cancel, auto)
  1881   done
  1882 
  1883 lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
  1884   apply (subst mult.commute)
  1885   apply (erule dvd_mult_cancel1)
  1886   done
  1887 
  1888 lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
  1889 by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
  1890 
  1891 lemma nat_dvd_not_less:
  1892   fixes m n :: nat
  1893   shows "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"
  1894 by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
  1895 
  1896 lemma dvd_plusE:
  1897   fixes m n q :: nat
  1898   assumes "m dvd n + q" "m dvd n"
  1899   obtains "m dvd q"
  1900 proof (cases "m = 0")
  1901   case True with assms that show thesis by simp
  1902 next
  1903   case False then have "m > 0" by simp
  1904   from assms obtain r s where "n = m * r" and "n + q = m * s" by (blast elim: dvdE)
  1905   then have *: "m * r + q = m * s" by simp
  1906   show thesis proof (cases "r \<le> s")
  1907     case False then have "s < r" by (simp add: not_le)
  1908     with * have "m * r + q - m * s = m * s - m * s" by simp
  1909     then have "m * r + q - m * s = 0" by simp
  1910     with `m > 0` `s < r` have "m * r - m * s + q = 0" by (unfold less_le_not_le) auto
  1911     then have "m * (r - s) + q = 0" by auto
  1912     then have "m * (r - s) = 0" by simp
  1913     then have "m = 0 \<or> r - s = 0" by simp
  1914     with `s < r` have "m = 0" by (simp add: less_le_not_le)
  1915     with `m > 0` show thesis by auto
  1916   next
  1917     case True with * have "m * r + q - m * r = m * s - m * r" by simp
  1918     with `m > 0` `r \<le> s` have "m * r - m * r + q = m * s - m * r" by simp
  1919     then have "q = m * (s - r)" by (simp add: diff_mult_distrib2)
  1920     with assms that show thesis by (auto intro: dvdI)
  1921   qed
  1922 qed
  1923 
  1924 lemma dvd_plus_eq_right:
  1925   fixes m n q :: nat
  1926   assumes "m dvd n"
  1927   shows "m dvd n + q \<longleftrightarrow> m dvd q"
  1928   using assms by (auto elim: dvd_plusE)
  1929 
  1930 lemma dvd_plus_eq_left:
  1931   fixes m n q :: nat
  1932   assumes "m dvd q"
  1933   shows "m dvd n + q \<longleftrightarrow> m dvd n"
  1934   using assms by (simp add: dvd_plus_eq_right add.commute [of n])
  1935 
  1936 lemma less_eq_dvd_minus:
  1937   fixes m n :: nat
  1938   assumes "m \<le> n"
  1939   shows "m dvd n \<longleftrightarrow> m dvd n - m"
  1940 proof -
  1941   from assms have "n = m + (n - m)" by simp
  1942   then obtain q where "n = m + q" ..
  1943   then show ?thesis by (simp add: dvd_reduce add.commute [of m])
  1944 qed
  1945 
  1946 lemma dvd_minus_self:
  1947   fixes m n :: nat
  1948   shows "m dvd n - m \<longleftrightarrow> n < m \<or> m dvd n"
  1949   by (cases "n < m") (auto elim!: dvdE simp add: not_less le_imp_diff_is_add)
  1950 
  1951 lemma dvd_minus_add:
  1952   fixes m n q r :: nat
  1953   assumes "q \<le> n" "q \<le> r * m"
  1954   shows "m dvd n - q \<longleftrightarrow> m dvd n + (r * m - q)"
  1955 proof -
  1956   have "m dvd n - q \<longleftrightarrow> m dvd r * m + (n - q)"
  1957     by (auto elim: dvd_plusE)
  1958   also from assms have "\<dots> \<longleftrightarrow> m dvd r * m + n - q" by simp
  1959   also from assms have "\<dots> \<longleftrightarrow> m dvd (r * m - q) + n" by simp
  1960   also have "\<dots> \<longleftrightarrow> m dvd n + (r * m - q)" by (simp add: add.commute)
  1961   finally show ?thesis .
  1962 qed
  1963 
  1964 
  1965 subsection {* aliases *}
  1966 
  1967 lemma nat_mult_1: "(1::nat) * n = n"
  1968   by (rule mult_1_left)
  1969  
  1970 lemma nat_mult_1_right: "n * (1::nat) = n"
  1971   by (rule mult_1_right)
  1972 
  1973 
  1974 subsection {* size of a datatype value *}
  1975 
  1976 class size =
  1977   fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded} *}
  1978 
  1979 
  1980 subsection {* code module namespace *}
  1981 
  1982 code_identifier
  1983   code_module Nat \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
  1984 
  1985 hide_const (open) of_nat_aux
  1986 
  1987 end