src/HOL/Nat.thy
 author haftmann Wed Sep 07 23:07:16 2011 +0200 (2011-09-07) changeset 44817 b63e445c8f6d parent 44325 84696670feb1 child 44848 f4d0b060c7ca permissions -rw-r--r--
lemmas about +, *, min, max on nat
1 (*  Title:      HOL/Nat.thy
2     Author:     Tobias Nipkow and Lawrence C Paulson and Markus Wenzel
4 Type "nat" is a linear order, and a datatype; arithmetic operators + -
5 and * (for div and mod, see theory Divides).
6 *)
8 header {* Natural numbers *}
10 theory Nat
11 imports Inductive Typedef Fun Fields
12 uses
13   "~~/src/Tools/rat.ML"
14   "~~/src/Provers/Arith/cancel_sums.ML"
15   "Tools/arith_data.ML"
16   ("Tools/nat_arith.ML")
17   "~~/src/Provers/Arith/fast_lin_arith.ML"
18   ("Tools/lin_arith.ML")
19 begin
21 subsection {* Type @{text ind} *}
23 typedecl ind
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"
30 subsection {* Type nat *}
32 text {* Type definition *}
34 inductive Nat :: "ind \<Rightarrow> bool" where
35   Zero_RepI: "Nat Zero_Rep"
36 | Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)"
38 typedef (open Nat) nat = "{n. Nat n}"
39   using Nat.Zero_RepI by auto
41 lemma Nat_Rep_Nat:
42   "Nat (Rep_Nat n)"
43   using Rep_Nat by simp
45 lemma Nat_Abs_Nat_inverse:
46   "Nat n \<Longrightarrow> Rep_Nat (Abs_Nat n) = n"
47   using Abs_Nat_inverse by simp
49 lemma Nat_Abs_Nat_inject:
50   "Nat n \<Longrightarrow> Nat m \<Longrightarrow> Abs_Nat n = Abs_Nat m \<longleftrightarrow> n = m"
51   using Abs_Nat_inject by simp
53 instantiation nat :: zero
54 begin
56 definition Zero_nat_def:
57   "0 = Abs_Nat Zero_Rep"
59 instance ..
61 end
63 definition Suc :: "nat \<Rightarrow> nat" where
64   "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"
66 lemma Suc_not_Zero: "Suc m \<noteq> 0"
67   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 lemma Zero_not_Suc: "0 \<noteq> Suc m"
70   by (rule not_sym, rule Suc_not_Zero not_sym)
72 lemma Suc_Rep_inject': "Suc_Rep x = Suc_Rep y \<longleftrightarrow> x = y"
73   by (rule iffI, rule Suc_Rep_inject) simp_all
75 rep_datatype "0 \<Colon> nat" Suc
76   apply (unfold Zero_nat_def Suc_def)
77   apply (rule Rep_Nat_inverse [THEN subst]) -- {* types force good instantiation *}
78    apply (erule Nat_Rep_Nat [THEN Nat.induct])
79    apply (iprover elim: Nat_Abs_Nat_inverse [THEN subst])
80     apply (simp_all add: Nat_Abs_Nat_inject Nat_Rep_Nat
81       Suc_RepI Zero_RepI Suc_Rep_not_Zero_Rep
82       Suc_Rep_not_Zero_Rep [symmetric]
83       Suc_Rep_inject' Rep_Nat_inject)
84   done
86 lemma nat_induct [case_names 0 Suc, induct type: nat]:
87   -- {* for backward compatibility -- names of variables differ *}
88   fixes n
89   assumes "P 0"
90     and "\<And>n. P n \<Longrightarrow> P (Suc n)"
91   shows "P n"
92   using assms by (rule nat.induct)
94 declare nat.exhaust [case_names 0 Suc, cases type: nat]
96 lemmas nat_rec_0 = nat.recs(1)
97   and nat_rec_Suc = nat.recs(2)
99 lemmas nat_case_0 = nat.cases(1)
100   and nat_case_Suc = nat.cases(2)
103 text {* Injectiveness and distinctness lemmas *}
105 lemma inj_Suc[simp]: "inj_on Suc N"
106   by (simp add: inj_on_def)
108 lemma Suc_neq_Zero: "Suc m = 0 \<Longrightarrow> R"
109 by (rule notE, rule Suc_not_Zero)
111 lemma Zero_neq_Suc: "0 = Suc m \<Longrightarrow> R"
112 by (rule Suc_neq_Zero, erule sym)
114 lemma Suc_inject: "Suc x = Suc y \<Longrightarrow> x = y"
115 by (rule inj_Suc [THEN injD])
117 lemma n_not_Suc_n: "n \<noteq> Suc n"
118 by (induct n) simp_all
120 lemma Suc_n_not_n: "Suc n \<noteq> n"
121 by (rule not_sym, rule n_not_Suc_n)
123 text {* A special form of induction for reasoning
124   about @{term "m < n"} and @{term "m - n"} *}
126 lemma diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==>
127     (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n"
128   apply (rule_tac x = m in spec)
129   apply (induct n)
130   prefer 2
131   apply (rule allI)
132   apply (induct_tac x, iprover+)
133   done
136 subsection {* Arithmetic operators *}
138 instantiation nat :: "{minus, comm_monoid_add}"
139 begin
141 primrec plus_nat where
142   add_0:      "0 + n = (n\<Colon>nat)"
143 | add_Suc:  "Suc m + n = Suc (m + n)"
145 lemma add_0_right [simp]: "m + 0 = (m::nat)"
146   by (induct m) simp_all
148 lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
149   by (induct m) simp_all
151 declare add_0 [code]
153 lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
154   by simp
156 primrec minus_nat where
157   diff_0 [code]: "m - 0 = (m\<Colon>nat)"
158 | diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
160 declare diff_Suc [simp del]
162 lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"
163   by (induct n) (simp_all add: diff_Suc)
165 lemma diff_Suc_Suc [simp, code]: "Suc m - Suc n = m - n"
166   by (induct n) (simp_all add: diff_Suc)
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
175 end
179 instantiation nat :: comm_semiring_1_cancel
180 begin
182 definition
183   One_nat_def [simp]: "1 = Suc 0"
185 primrec times_nat where
186   mult_0:     "0 * n = (0\<Colon>nat)"
187 | mult_Suc: "Suc m * n = n + (m * n)"
189 lemma mult_0_right [simp]: "(m::nat) * 0 = 0"
190   by (induct m) simp_all
192 lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
193   by (induct m) (simp_all add: add_left_commute)
195 lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"
196   by (induct m) (simp_all add: add_assoc)
198 instance proof
199   fix n m q :: nat
200   show "0 \<noteq> (1::nat)" unfolding One_nat_def by simp
201   show "1 * n = n" unfolding One_nat_def by simp
202   show "n * m = m * n" by (induct n) simp_all
203   show "(n * m) * q = n * (m * q)" by (induct n) (simp_all add: add_mult_distrib)
204   show "(n + m) * q = n * q + m * q" by (rule add_mult_distrib)
205   assume "n + m = n + q" thus "m = q" by (induct n) simp_all
206 qed
208 end
210 subsubsection {* Addition *}
212 lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)"
213   by (rule add_assoc)
215 lemma nat_add_commute: "m + n = n + (m::nat)"
216   by (rule add_commute)
218 lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)"
219   by (rule add_left_commute)
221 lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))"
222   by (rule add_left_cancel)
224 lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))"
225   by (rule add_right_cancel)
227 text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
229 lemma add_is_0 [iff]:
230   fixes m n :: nat
231   shows "(m + n = 0) = (m = 0 & n = 0)"
232   by (cases m) simp_all
235   "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
236   by (cases m) simp_all
239   "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
240   by (rule trans, rule eq_commute, rule add_is_1)
243   fixes m n :: nat
244   shows "m + n = m \<Longrightarrow> n = 0"
245   by (induct m) simp_all
247 lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N"
248   apply (induct k)
249    apply simp
250   apply(drule comp_inj_on[OF _ inj_Suc])
251   apply (simp add:o_def)
252   done
255 subsubsection {* Difference *}
257 lemma diff_self_eq_0 [simp]: "(m\<Colon>nat) - m = 0"
258   by (induct m) simp_all
260 lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
261   by (induct i j rule: diff_induct) simp_all
263 lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
264   by (simp add: diff_diff_left)
266 lemma diff_commute: "(i::nat) - j - k = i - k - j"
269 lemma diff_add_inverse: "(n + m) - n = (m::nat)"
270   by (induct n) simp_all
272 lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
275 lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
276   by (induct k) simp_all
278 lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
281 lemma diff_add_0: "n - (n + m) = (0::nat)"
282   by (induct n) simp_all
284 lemma diff_Suc_1 [simp]: "Suc n - 1 = n"
285   unfolding One_nat_def by simp
287 text {* Difference distributes over multiplication *}
289 lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"
290 by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
292 lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
293 by (simp add: diff_mult_distrib mult_commute [of k])
294   -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
297 subsubsection {* Multiplication *}
299 lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)"
300   by (rule mult_assoc)
302 lemma nat_mult_commute: "m * n = n * (m::nat)"
303   by (rule mult_commute)
305 lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
306   by (rule right_distrib)
308 lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
309   by (induct m) auto
311 lemmas nat_distrib =
314 lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = Suc 0 & n = Suc 0)"
315   apply (induct m)
316    apply simp
317   apply (induct n)
318    apply auto
319   done
321 lemma one_eq_mult_iff [simp,no_atp]: "(Suc 0 = m * n) = (m = Suc 0 & n = Suc 0)"
322   apply (rule trans)
323   apply (rule_tac [2] mult_eq_1_iff, fastsimp)
324   done
326 lemma nat_mult_eq_1_iff [simp]: "m * n = (1::nat) \<longleftrightarrow> m = 1 \<and> n = 1"
327   unfolding One_nat_def by (rule mult_eq_1_iff)
329 lemma nat_1_eq_mult_iff [simp]: "(1::nat) = m * n \<longleftrightarrow> m = 1 \<and> n = 1"
330   unfolding One_nat_def by (rule one_eq_mult_iff)
332 lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
333 proof -
334   have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
335   proof (induct n arbitrary: m)
336     case 0 then show "m = 0" by simp
337   next
338     case (Suc n) then show "m = Suc n"
339       by (cases m) (simp_all add: eq_commute [of "0"])
340   qed
341   then show ?thesis by auto
342 qed
344 lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
345   by (simp add: mult_commute)
347 lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
348   by (subst mult_cancel1) simp
351 subsection {* Orders on @{typ nat} *}
353 subsubsection {* Operation definition *}
355 instantiation nat :: linorder
356 begin
358 primrec less_eq_nat where
359   "(0\<Colon>nat) \<le> n \<longleftrightarrow> True"
360 | "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
362 declare less_eq_nat.simps [simp del]
363 lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by (simp add: less_eq_nat.simps)
364 lemma le0 [iff]: "0 \<le> (n\<Colon>nat)" by (simp add: less_eq_nat.simps)
366 definition less_nat where
367   less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"
369 lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
370   by (simp add: less_eq_nat.simps(2))
372 lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
373   unfolding less_eq_Suc_le ..
375 lemma le_0_eq [iff]: "(n\<Colon>nat) \<le> 0 \<longleftrightarrow> n = 0"
376   by (induct n) (simp_all add: less_eq_nat.simps(2))
378 lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)"
379   by (simp add: less_eq_Suc_le)
381 lemma less_nat_zero_code [code]: "n < (0\<Colon>nat) \<longleftrightarrow> False"
382   by simp
384 lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
385   by (simp add: less_eq_Suc_le)
387 lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
388   by (simp add: less_eq_Suc_le)
390 lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
391   by (induct m arbitrary: n)
392     (simp_all add: less_eq_nat.simps(2) split: nat.splits)
394 lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
395   by (cases n) (auto intro: le_SucI)
397 lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
398   by (simp add: less_eq_Suc_le) (erule Suc_leD)
400 lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
401   by (simp add: less_eq_Suc_le) (erule Suc_leD)
403 instance
404 proof
405   fix n m :: nat
406   show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n"
407   proof (induct n arbitrary: m)
408     case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
409   next
410     case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
411   qed
412 next
413   fix n :: nat show "n \<le> n" by (induct n) simp_all
414 next
415   fix n m :: nat assume "n \<le> m" and "m \<le> n"
416   then show "n = m"
417     by (induct n arbitrary: m)
418       (simp_all add: less_eq_nat.simps(2) split: nat.splits)
419 next
420   fix n m q :: nat assume "n \<le> m" and "m \<le> q"
421   then show "n \<le> q"
422   proof (induct n arbitrary: m q)
423     case 0 show ?case by simp
424   next
425     case (Suc n) then show ?case
426       by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
427         simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
428         simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
429   qed
430 next
431   fix n m :: nat show "n \<le> m \<or> m \<le> n"
432     by (induct n arbitrary: m)
433       (simp_all add: less_eq_nat.simps(2) split: nat.splits)
434 qed
436 end
438 instantiation nat :: bot
439 begin
441 definition bot_nat :: nat where
442   "bot_nat = 0"
444 instance proof
445 qed (simp add: bot_nat_def)
447 end
449 subsubsection {* Introduction properties *}
451 lemma lessI [iff]: "n < Suc n"
452   by (simp add: less_Suc_eq_le)
454 lemma zero_less_Suc [iff]: "0 < Suc n"
455   by (simp add: less_Suc_eq_le)
458 subsubsection {* Elimination properties *}
460 lemma less_not_refl: "~ n < (n::nat)"
461   by (rule order_less_irrefl)
463 lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)"
464   by (rule not_sym) (rule less_imp_neq)
466 lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
467   by (rule less_imp_neq)
469 lemma less_irrefl_nat: "(n::nat) < n ==> R"
470   by (rule notE, rule less_not_refl)
472 lemma less_zeroE: "(n::nat) < 0 ==> R"
473   by (rule notE) (rule not_less0)
475 lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
476   unfolding less_Suc_eq_le le_less ..
478 lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
479   by (simp add: less_Suc_eq)
481 lemma less_one [iff, no_atp]: "(n < (1::nat)) = (n = 0)"
482   unfolding One_nat_def by (rule less_Suc0)
484 lemma Suc_mono: "m < n ==> Suc m < Suc n"
485   by simp
487 text {* "Less than" is antisymmetric, sort of *}
488 lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
489   unfolding not_less less_Suc_eq_le by (rule antisym)
491 lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
492   by (rule linorder_neq_iff)
494 lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
495   and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
496   shows "P n m"
497   apply (rule less_linear [THEN disjE])
498   apply (erule_tac [2] disjE)
499   apply (erule lessCase)
500   apply (erule sym [THEN eqCase])
501   apply (erule major)
502   done
505 subsubsection {* Inductive (?) properties *}
507 lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
508   unfolding less_eq_Suc_le [of m] le_less by simp
510 lemma lessE:
511   assumes major: "i < k"
512   and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
513   shows P
514 proof -
515   from major have "\<exists>j. i \<le> j \<and> k = Suc j"
516     unfolding less_eq_Suc_le by (induct k) simp_all
517   then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
518     by (clarsimp simp add: less_le)
519   with p1 p2 show P by auto
520 qed
522 lemma less_SucE: assumes major: "m < Suc n"
523   and less: "m < n ==> P" and eq: "m = n ==> P" shows P
524   apply (rule major [THEN lessE])
525   apply (rule eq, blast)
526   apply (rule less, blast)
527   done
529 lemma Suc_lessE: assumes major: "Suc i < k"
530   and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
531   apply (rule major [THEN lessE])
532   apply (erule lessI [THEN minor])
533   apply (erule Suc_lessD [THEN minor], assumption)
534   done
536 lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
537   by simp
539 lemma less_trans_Suc:
540   assumes le: "i < j" shows "j < k ==> Suc i < k"
541   apply (induct k, simp_all)
542   apply (insert le)
543   apply (simp add: less_Suc_eq)
544   apply (blast dest: Suc_lessD)
545   done
547 text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
548 lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
549   unfolding not_less less_Suc_eq_le ..
551 lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
552   unfolding not_le Suc_le_eq ..
554 text {* Properties of "less than or equal" *}
556 lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
557   unfolding less_Suc_eq_le .
559 lemma Suc_n_not_le_n: "~ Suc n \<le> n"
560   unfolding not_le less_Suc_eq_le ..
562 lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
563   by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
565 lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
566   by (drule le_Suc_eq [THEN iffD1], iprover+)
568 lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
569   unfolding Suc_le_eq .
571 text {* Stronger version of @{text Suc_leD} *}
572 lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
573   unfolding Suc_le_eq .
575 lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)"
576   unfolding less_eq_Suc_le by (rule Suc_leD)
578 text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
579 lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
582 text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
584 lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
585   unfolding le_less .
587 lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
588   by (rule le_less)
590 text {* Useful with @{text blast}. *}
591 lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
592   by auto
594 lemma le_refl: "n \<le> (n::nat)"
595   by simp
597 lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
598   by (rule order_trans)
600 lemma le_antisym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
601   by (rule antisym)
603 lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
604   by (rule less_le)
606 lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
607   unfolding less_le ..
609 lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
610   by (rule linear)
612 lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
614 lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
615   unfolding less_Suc_eq_le by auto
617 lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
618   unfolding not_less by (rule le_less_Suc_eq)
620 lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
622 text {* These two rules ease the use of primitive recursion.
623 NOTE USE OF @{text "=="} *}
624 lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"
625 by simp
627 lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"
628 by simp
630 lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
631 by (cases n) simp_all
633 lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"
634 by (cases n) simp_all
636 lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
637 by (cases n) simp_all
639 lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
640 by (cases n) simp_all
642 text {* This theorem is useful with @{text blast} *}
643 lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
644 by (rule neq0_conv[THEN iffD1], iprover)
646 lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
647 by (fast intro: not0_implies_Suc)
649 lemma not_gr0 [iff,no_atp]: "!!n::nat. (~ (0 < n)) = (n = 0)"
650 using neq0_conv by blast
652 lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
653 by (induct m') simp_all
655 text {* Useful in certain inductive arguments *}
656 lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
657 by (cases m) simp_all
660 subsubsection {* Monotonicity of Addition *}
662 lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"
663 by (simp add: diff_Suc split: nat.split)
665 lemma Suc_diff_1 [simp]: "0 < n ==> Suc (n - 1) = n"
666 unfolding One_nat_def by (rule Suc_pred)
668 lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
669 by (induct k) simp_all
671 lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
672 by (induct k) simp_all
674 lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)"
675 by(auto dest:gr0_implies_Suc)
677 text {* strict, in 1st argument *}
678 lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
679 by (induct k) simp_all
681 text {* strict, in both arguments *}
682 lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
683   apply (rule add_less_mono1 [THEN less_trans], assumption+)
684   apply (induct j, simp_all)
685   done
687 text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
688 lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
689   apply (induct n)
690   apply (simp_all add: order_le_less)
691   apply (blast elim!: less_SucE
693   done
695 text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
696 lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"
697 apply(auto simp: gr0_conv_Suc)
698 apply (induct_tac m)
700 done
702 text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
703 instance nat :: linordered_semidom
704 proof
705   fix i j k :: nat
706   show "0 < (1::nat)" by simp
707   show "i \<le> j ==> k + i \<le> k + j" by simp
708   show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2)
709 qed
711 instance nat :: no_zero_divisors
712 proof
713   fix a::nat and b::nat show "a ~= 0 \<Longrightarrow> b ~= 0 \<Longrightarrow> a * b ~= 0" by auto
714 qed
717 subsubsection {* @{term min} and @{term max} *}
719 lemma mono_Suc: "mono Suc"
720 by (rule monoI) simp
722 lemma min_0L [simp]: "min 0 n = (0::nat)"
723 by (rule min_leastL) simp
725 lemma min_0R [simp]: "min n 0 = (0::nat)"
726 by (rule min_leastR) simp
728 lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
729 by (simp add: mono_Suc min_of_mono)
731 lemma min_Suc1:
732    "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
733 by (simp split: nat.split)
735 lemma min_Suc2:
736    "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
737 by (simp split: nat.split)
739 lemma max_0L [simp]: "max 0 n = (n::nat)"
740 by (rule max_leastL) simp
742 lemma max_0R [simp]: "max n 0 = (n::nat)"
743 by (rule max_leastR) simp
745 lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
746 by (simp add: mono_Suc max_of_mono)
748 lemma max_Suc1:
749    "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
750 by (simp split: nat.split)
752 lemma max_Suc2:
753    "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
754 by (simp split: nat.split)
757   fixes m n q :: nat
758   shows "min m n + q = min (m + q) (n + q)"
759   by (simp add: min_def)
762   fixes m n q :: nat
763   shows "m + min n q = min (m + n) (m + q)"
764   by (simp add: min_def)
766 lemma nat_mult_min_left:
767   fixes m n q :: nat
768   shows "min m n * q = min (m * q) (n * q)"
769   by (simp add: min_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
771 lemma nat_mult_min_right:
772   fixes m n q :: nat
773   shows "m * min n q = min (m * n) (m * q)"
774   by (simp add: min_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
777   fixes m n q :: nat
778   shows "max m n + q = max (m + q) (n + q)"
779   by (simp add: max_def)
782   fixes m n q :: nat
783   shows "m + max n q = max (m + n) (m + q)"
784   by (simp add: max_def)
786 lemma nat_mult_max_left:
787   fixes m n q :: nat
788   shows "max m n * q = max (m * q) (n * q)"
789   by (simp add: max_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
791 lemma nat_mult_max_right:
792   fixes m n q :: nat
793   shows "m * max n q = max (m * n) (m * q)"
794   by (simp add: max_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
797 subsubsection {* Additional theorems about @{term "op \<le>"} *}
799 text {* Complete induction, aka course-of-values induction *}
801 instance nat :: wellorder proof
802   fix P and n :: nat
803   assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n"
804   have "\<And>q. q \<le> n \<Longrightarrow> P q"
805   proof (induct n)
806     case (0 n)
807     have "P 0" by (rule step) auto
808     thus ?case using 0 by auto
809   next
810     case (Suc m n)
811     then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)
812     thus ?case
813     proof
814       assume "n \<le> m" thus "P n" by (rule Suc(1))
815     next
816       assume n: "n = Suc m"
817       show "P n"
818         by (rule step) (rule Suc(1), simp add: n le_simps)
819     qed
820   qed
821   then show "P n" by auto
822 qed
824 lemma Least_Suc:
825      "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
826   apply (case_tac "n", auto)
827   apply (frule LeastI)
828   apply (drule_tac P = "%x. P (Suc x) " in LeastI)
829   apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
830   apply (erule_tac [2] Least_le)
831   apply (case_tac "LEAST x. P x", auto)
832   apply (drule_tac P = "%x. P (Suc x) " in Least_le)
833   apply (blast intro: order_antisym)
834   done
836 lemma Least_Suc2:
837    "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
838   apply (erule (1) Least_Suc [THEN ssubst])
839   apply simp
840   done
842 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)"
843   apply (cases n)
844    apply blast
845   apply (rule_tac x="LEAST k. P(k)" in exI)
846   apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
847   done
849 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)"
850   unfolding One_nat_def
851   apply (cases n)
852    apply blast
853   apply (frule (1) ex_least_nat_le)
854   apply (erule exE)
855   apply (case_tac k)
856    apply simp
857   apply (rename_tac k1)
858   apply (rule_tac x=k1 in exI)
859   apply (auto simp add: less_eq_Suc_le)
860   done
862 lemma nat_less_induct:
863   assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
864   using assms less_induct by blast
866 lemma measure_induct_rule [case_names less]:
867   fixes f :: "'a \<Rightarrow> nat"
868   assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
869   shows "P a"
870 by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step)
872 text {* old style induction rules: *}
873 lemma measure_induct:
874   fixes f :: "'a \<Rightarrow> nat"
875   shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
876   by (rule measure_induct_rule [of f P a]) iprover
878 lemma full_nat_induct:
879   assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"
880   shows "P n"
881   by (rule less_induct) (auto intro: step simp:le_simps)
883 text{*An induction rule for estabilishing binary relations*}
884 lemma less_Suc_induct:
885   assumes less:  "i < j"
886      and  step:  "!!i. P i (Suc i)"
887      and  trans: "!!i j k. i < j ==> j < k ==>  P i j ==> P j k ==> P i k"
888   shows "P i j"
889 proof -
890   from less obtain k where j: "j = Suc (i + k)" by (auto dest: less_imp_Suc_add)
891   have "P i (Suc (i + k))"
892   proof (induct k)
893     case 0
894     show ?case by (simp add: step)
895   next
896     case (Suc k)
897     have "0 + i < Suc k + i" by (rule add_less_mono1) simp
898     hence "i < Suc (i + k)" by (simp add: add_commute)
899     from trans[OF this lessI Suc step]
900     show ?case by simp
901   qed
902   thus "P i j" by (simp add: j)
903 qed
905 text {* The method of infinite descent, frequently used in number theory.
906 Provided by Roelof Oosterhuis.
907 $P(n)$ is true for all $n\in\mathbb{N}$ if
908 \begin{itemize}
909   \item case 0'': given $n=0$ prove $P(n)$,
910   \item case smaller'': given $n>0$ and $\neg P(n)$ prove there exists
911         a smaller integer $m$ such that $\neg P(m)$.
912 \end{itemize} *}
914 text{* A compact version without explicit base case: *}
915 lemma infinite_descent:
916   "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow>  \<exists>m<n. \<not>  P m \<rbrakk> \<Longrightarrow>  P n"
917 by (induct n rule: less_induct, auto)
919 lemma infinite_descent0[case_names 0 smaller]:
920   "\<lbrakk> P 0; !!n. n>0 \<Longrightarrow> \<not> P n \<Longrightarrow> (\<exists>m::nat. m < n \<and> \<not>P m) \<rbrakk> \<Longrightarrow> P n"
921 by (rule infinite_descent) (case_tac "n>0", auto)
923 text {*
924 Infinite descent using a mapping to $\mathbb{N}$:
925 $P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
926 \begin{itemize}
927 \item case 0'': given $V(x)=0$ prove $P(x)$,
928 \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 $~\neg P(y)$.
929 \end{itemize}
930 NB: the proof also shows how to use the previous lemma. *}
932 corollary infinite_descent0_measure [case_names 0 smaller]:
933   assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
934     and   A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
935   shows "P x"
936 proof -
937   obtain n where "n = V x" by auto
938   moreover have "\<And>x. V x = n \<Longrightarrow> P x"
939   proof (induct n rule: infinite_descent0)
940     case 0 -- "i.e. $V(x) = 0$"
941     with A0 show "P x" by auto
942   next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
943     case (smaller n)
944     then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
945     with A1 obtain y where "V y < V x \<and> \<not> P y" by auto
946     with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
947     then show ?case by auto
948   qed
949   ultimately show "P x" by auto
950 qed
952 text{* Again, without explicit base case: *}
953 lemma infinite_descent_measure:
954 assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
955 proof -
956   from assms obtain n where "n = V x" by auto
957   moreover have "!!x. V x = n \<Longrightarrow> P x"
958   proof (induct n rule: infinite_descent, auto)
959     fix x assume "\<not> P x"
960     with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
961   qed
962   ultimately show "P x" by auto
963 qed
965 text {* A [clumsy] way of lifting @{text "<"}
966   monotonicity to @{text "\<le>"} monotonicity *}
967 lemma less_mono_imp_le_mono:
968   "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
969 by (simp add: order_le_less) (blast)
972 text {* non-strict, in 1st argument *}
973 lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
974 by (rule add_right_mono)
976 text {* non-strict, in both arguments *}
977 lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
978 by (rule add_mono)
980 lemma le_add2: "n \<le> ((m + n)::nat)"
981 by (insert add_right_mono [of 0 m n], simp)
983 lemma le_add1: "n \<le> ((n + m)::nat)"
986 lemma less_add_Suc1: "i < Suc (i + m)"
987 by (rule le_less_trans, rule le_add1, rule lessI)
989 lemma less_add_Suc2: "i < Suc (m + i)"
990 by (rule le_less_trans, rule le_add2, rule lessI)
992 lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
995 lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
996 by (rule le_trans, assumption, rule le_add1)
998 lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
999 by (rule le_trans, assumption, rule le_add2)
1001 lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
1002 by (rule less_le_trans, assumption, rule le_add1)
1004 lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
1005 by (rule less_le_trans, assumption, rule le_add2)
1007 lemma add_lessD1: "i + j < (k::nat) ==> i < k"
1008 apply (rule le_less_trans [of _ "i+j"])
1010 done
1012 lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
1013 apply (rule notI)
1014 apply (drule add_lessD1)
1015 apply (erule less_irrefl [THEN notE])
1016 done
1018 lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
1021 lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
1022 apply (rule order_trans [of _ "m+k"])
1024 done
1026 lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
1028 apply (erule add_leD1)
1029 done
1031 lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
1034 text {* needs @{text "!!k"} for @{text add_ac} to work *}
1035 lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
1036 by (force simp del: add_Suc_right
1040 subsubsection {* More results about difference *}
1042 text {* Addition is the inverse of subtraction:
1043   if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
1044 lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
1045 by (induct m n rule: diff_induct) simp_all
1047 lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
1050 lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
1053 lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
1054 by (induct m n rule: diff_induct) simp_all
1056 lemma diff_less_Suc: "m - n < Suc m"
1057 apply (induct m n rule: diff_induct)
1058 apply (erule_tac [3] less_SucE)
1059 apply (simp_all add: less_Suc_eq)
1060 done
1062 lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
1063 by (induct m n rule: diff_induct) (simp_all add: le_SucI)
1065 lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
1066   by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])
1068 lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
1069 by (rule le_less_trans, rule diff_le_self)
1071 lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
1072 by (cases n) (auto simp add: le_simps)
1074 lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
1075 by (induct j k rule: diff_induct) simp_all
1077 lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
1080 lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
1083 lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
1084 by (induct m n rule: diff_induct) simp_all
1086 lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
1087 by (rule iffD2, rule diff_is_0_eq)
1089 lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
1090 by (induct m n rule: diff_induct) simp_all
1093   assumes "i < j"
1094   shows "\<exists>k::nat. 0 < k & i + k = j"
1095 proof
1096   from assms show "0 < j - i & i + (j - i) = j"
1097     by (simp add: order_less_imp_le)
1098 qed
1100 text {* a nice rewrite for bounded subtraction *}
1102   fixes n m :: nat
1103   shows "n - m + m = max n m"
1104     by (simp add: max_def not_le order_less_imp_le)
1106 lemma nat_diff_split:
1107   "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
1108     -- {* elimination of @{text -} on @{text nat} *}
1109 by (cases "a < b")
1110   (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse
1111     not_less le_less dest!: sym [of a] sym [of b] add_eq_self_zero)
1113 lemma nat_diff_split_asm:
1114   "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
1115     -- {* elimination of @{text -} on @{text nat} in assumptions *}
1116 by (auto split: nat_diff_split)
1119 subsubsection {* Monotonicity of Multiplication *}
1121 lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
1122 by (simp add: mult_right_mono)
1124 lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
1125 by (simp add: mult_left_mono)
1127 text {* @{text "\<le>"} monotonicity, BOTH arguments *}
1128 lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
1129 by (simp add: mult_mono)
1131 lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
1132 by (simp add: mult_strict_right_mono)
1134 text{*Differs from the standard @{text zero_less_mult_iff} in that
1135       there are no negative numbers.*}
1136 lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
1137   apply (induct m)
1138    apply simp
1139   apply (case_tac n)
1140    apply simp_all
1141   done
1143 lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)"
1144   apply (induct m)
1145    apply simp
1146   apply (case_tac n)
1147    apply simp_all
1148   done
1150 lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
1151   apply (safe intro!: mult_less_mono1)
1152   apply (case_tac k, auto)
1153   apply (simp del: le_0_eq add: linorder_not_le [symmetric])
1154   apply (blast intro: mult_le_mono1)
1155   done
1157 lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
1158 by (simp add: mult_commute [of k])
1160 lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
1161 by (simp add: linorder_not_less [symmetric], auto)
1163 lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
1164 by (simp add: linorder_not_less [symmetric], auto)
1166 lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
1167 by (subst mult_less_cancel1) simp
1169 lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
1170 by (subst mult_le_cancel1) simp
1172 lemma le_square: "m \<le> m * (m::nat)"
1173   by (cases m) (auto intro: le_add1)
1175 lemma le_cube: "(m::nat) \<le> m * (m * m)"
1176   by (cases m) (auto intro: le_add1)
1178 text {* Lemma for @{text gcd} *}
1179 lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
1180   apply (drule sym)
1181   apply (rule disjCI)
1182   apply (rule nat_less_cases, erule_tac [2] _)
1183    apply (drule_tac [2] mult_less_mono2)
1184     apply (auto)
1185   done
1187 text {* the lattice order on @{typ nat} *}
1189 instantiation nat :: distrib_lattice
1190 begin
1192 definition
1193   "(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min"
1195 definition
1196   "(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max"
1198 instance by intro_classes
1199   (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
1200     intro: order_less_imp_le antisym elim!: order_trans order_less_trans)
1202 end
1205 subsection {* Natural operation of natural numbers on functions *}
1207 text {*
1208   We use the same logical constant for the power operations on
1209   functions and relations, in order to share the same syntax.
1210 *}
1212 consts compow :: "nat \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
1214 abbreviation compower :: "('a \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'b" (infixr "^^" 80) where
1215   "f ^^ n \<equiv> compow n f"
1217 notation (latex output)
1218   compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
1220 notation (HTML output)
1221   compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
1223 text {* @{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f} *}
1226   funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
1227 begin
1229 primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
1230   "funpow 0 f = id"
1231 | "funpow (Suc n) f = f o funpow n f"
1233 end
1235 text {* for code generation *}
1237 definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
1238   funpow_code_def [code_post]: "funpow = compow"
1240 lemmas [code_unfold] = funpow_code_def [symmetric]
1242 lemma [code]:
1243   "funpow (Suc n) f = f o funpow n f"
1244   "funpow 0 f = id"
1245   by (simp_all add: funpow_code_def)
1247 hide_const (open) funpow
1250   "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
1251   by (induct m) simp_all
1253 lemma funpow_mult:
1254   fixes f :: "'a \<Rightarrow> 'a"
1255   shows "(f ^^ m) ^^ n = f ^^ (m * n)"
1256   by (induct n) (simp_all add: funpow_add)
1258 lemma funpow_swap1:
1259   "f ((f ^^ n) x) = (f ^^ n) (f x)"
1260 proof -
1261   have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp
1262   also have "\<dots>  = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)
1263   also have "\<dots> = (f ^^ n) (f x)" by simp
1264   finally show ?thesis .
1265 qed
1267 lemma comp_funpow:
1268   fixes f :: "'a \<Rightarrow> 'a"
1269   shows "comp f ^^ n = comp (f ^^ n)"
1270   by (induct n) simp_all
1273 subsection {* Embedding of the Naturals into any @{text semiring_1}: @{term of_nat} *}
1275 context semiring_1
1276 begin
1278 definition of_nat :: "nat \<Rightarrow> 'a" where
1279   "of_nat n = (plus 1 ^^ n) 0"
1281 lemma of_nat_simps [simp]:
1282   shows of_nat_0: "of_nat 0 = 0"
1283     and of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
1284   by (simp_all add: of_nat_def)
1286 lemma of_nat_1 [simp]: "of_nat 1 = 1"
1287   by (simp add: of_nat_def)
1289 lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
1290   by (induct m) (simp_all add: add_ac)
1292 lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"
1293   by (induct m) (simp_all add: add_ac left_distrib)
1295 primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where
1296   "of_nat_aux inc 0 i = i"
1297 | "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- {* tail recursive *}
1299 lemma of_nat_code:
1300   "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"
1301 proof (induct n)
1302   case 0 then show ?case by simp
1303 next
1304   case (Suc n)
1305   have "\<And>i. of_nat_aux (\<lambda>i. i + 1) n (i + 1) = of_nat_aux (\<lambda>i. i + 1) n i + 1"
1306     by (induct n) simp_all
1307   from this [of 0] have "of_nat_aux (\<lambda>i. i + 1) n 1 = of_nat_aux (\<lambda>i. i + 1) n 0 + 1"
1308     by simp
1309   with Suc show ?case by (simp add: add_commute)
1310 qed
1312 end
1314 declare of_nat_code [code, code_unfold, code_inline del]
1316 text{*Class for unital semirings with characteristic zero.
1317  Includes non-ordered rings like the complex numbers.*}
1319 class semiring_char_0 = semiring_1 +
1320   assumes inj_of_nat: "inj of_nat"
1321 begin
1323 lemma of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"
1324   by (auto intro: inj_of_nat injD)
1326 text{*Special cases where either operand is zero*}
1328 lemma of_nat_0_eq_iff [simp, no_atp]: "0 = of_nat n \<longleftrightarrow> 0 = n"
1329   by (fact of_nat_eq_iff [of 0 n, unfolded of_nat_0])
1331 lemma of_nat_eq_0_iff [simp, no_atp]: "of_nat m = 0 \<longleftrightarrow> m = 0"
1332   by (fact of_nat_eq_iff [of m 0, unfolded of_nat_0])
1334 end
1336 context linordered_semidom
1337 begin
1339 lemma zero_le_imp_of_nat: "0 \<le> of_nat m"
1340   by (induct m) simp_all
1342 lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"
1343   apply (induct m n rule: diff_induct, simp_all)
1344   apply (rule add_pos_nonneg [OF zero_less_one zero_le_imp_of_nat])
1345   done
1347 lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
1348   apply (induct m n rule: diff_induct, simp_all)
1349   apply (insert zero_le_imp_of_nat)
1350   apply (force simp add: not_less [symmetric])
1351   done
1353 lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
1354   by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
1356 lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
1357   by (simp add: not_less [symmetric] linorder_not_less [symmetric])
1359 text{*Every @{text linordered_semidom} has characteristic zero.*}
1361 subclass semiring_char_0 proof
1362 qed (auto intro!: injI simp add: eq_iff)
1364 text{*Special cases where either operand is zero*}
1366 lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"
1367   by (rule of_nat_le_iff [of 0, simplified])
1369 lemma of_nat_le_0_iff [simp, no_atp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"
1370   by (rule of_nat_le_iff [of _ 0, simplified])
1372 lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"
1373   by (rule of_nat_less_iff [of 0, simplified])
1375 lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"
1376   by (rule of_nat_less_iff [of _ 0, simplified])
1378 end
1380 context ring_1
1381 begin
1383 lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"
1384 by (simp add: algebra_simps of_nat_add [symmetric])
1386 end
1388 context linordered_idom
1389 begin
1391 lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"
1392   unfolding abs_if by auto
1394 end
1396 lemma of_nat_id [simp]: "of_nat n = n"
1397   by (induct n) simp_all
1399 lemma of_nat_eq_id [simp]: "of_nat = id"
1400   by (auto simp add: fun_eq_iff)
1403 subsection {* The Set of Natural Numbers *}
1405 context semiring_1
1406 begin
1408 definition Nats  :: "'a set" where
1409   "Nats = range of_nat"
1411 notation (xsymbols)
1412   Nats  ("\<nat>")
1414 lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"
1415   by (simp add: Nats_def)
1417 lemma Nats_0 [simp]: "0 \<in> \<nat>"
1418 apply (simp add: Nats_def)
1419 apply (rule range_eqI)
1420 apply (rule of_nat_0 [symmetric])
1421 done
1423 lemma Nats_1 [simp]: "1 \<in> \<nat>"
1424 apply (simp add: Nats_def)
1425 apply (rule range_eqI)
1426 apply (rule of_nat_1 [symmetric])
1427 done
1429 lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"
1430 apply (auto simp add: Nats_def)
1431 apply (rule range_eqI)
1432 apply (rule of_nat_add [symmetric])
1433 done
1435 lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"
1436 apply (auto simp add: Nats_def)
1437 apply (rule range_eqI)
1438 apply (rule of_nat_mult [symmetric])
1439 done
1441 lemma Nats_cases [cases set: Nats]:
1442   assumes "x \<in> \<nat>"
1443   obtains (of_nat) n where "x = of_nat n"
1444   unfolding Nats_def
1445 proof -
1446   from x \<in> \<nat> have "x \<in> range of_nat" unfolding Nats_def .
1447   then obtain n where "x = of_nat n" ..
1448   then show thesis ..
1449 qed
1451 lemma Nats_induct [case_names of_nat, induct set: Nats]:
1452   "x \<in> \<nat> \<Longrightarrow> (\<And>n. P (of_nat n)) \<Longrightarrow> P x"
1453   by (rule Nats_cases) auto
1455 end
1458 subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}
1460 lemma subst_equals:
1461   assumes 1: "t = s" and 2: "u = t"
1462   shows "u = s"
1463   using 2 1 by (rule trans)
1465 setup Arith_Data.setup
1467 use "Tools/nat_arith.ML"
1468 declaration {* K Nat_Arith.setup *}
1470 use "Tools/lin_arith.ML"
1471 setup {* Lin_Arith.global_setup *}
1472 declaration {* K Lin_Arith.setup *}
1474 simproc_setup fast_arith_nat ("(m::nat) < n" | "(m::nat) <= n" | "(m::nat) = n") =
1475   {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}
1476 (* Because of this simproc, the arithmetic solver is really only
1477 useful to detect inconsistencies among the premises for subgoals which are
1478 *not* themselves (in)equalities, because the latter activate
1479 fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
1480 solver all the time rather than add the additional check. *)
1483 lemmas [arith_split] = nat_diff_split split_min split_max
1485 context order
1486 begin
1488 lemma lift_Suc_mono_le:
1489   assumes mono: "!!n. f n \<le> f(Suc n)" and "n\<le>n'"
1490   shows "f n \<le> f n'"
1491 proof (cases "n < n'")
1492   case True
1493   thus ?thesis
1494     by (induct n n' rule: less_Suc_induct[consumes 1]) (auto intro: mono)
1495 qed (insert n \<le> n', auto) -- {*trivial for @{prop "n = n'"} *}
1497 lemma lift_Suc_mono_less:
1498   assumes mono: "!!n. f n < f(Suc n)" and "n < n'"
1499   shows "f n < f n'"
1500 using n < n'
1501 by (induct n n' rule: less_Suc_induct[consumes 1]) (auto intro: mono)
1503 lemma lift_Suc_mono_less_iff:
1504   "(!!n. f n < f(Suc n)) \<Longrightarrow> f(n) < f(m) \<longleftrightarrow> n<m"
1505 by(blast intro: less_asym' lift_Suc_mono_less[of f]
1506          dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq[THEN iffD1])
1508 end
1510 lemma mono_iff_le_Suc: "mono f = (\<forall>n. f n \<le> f (Suc n))"
1511   unfolding mono_def by (auto intro: lift_Suc_mono_le [of f])
1513 lemma mono_nat_linear_lb:
1514   "(!!m n::nat. m<n \<Longrightarrow> f m < f n) \<Longrightarrow> f(m)+k \<le> f(m+k)"
1515 apply(induct_tac k)
1516  apply simp
1517 apply(erule_tac x="m+n" in meta_allE)
1518 apply(erule_tac x="Suc(m+n)" in meta_allE)
1519 apply simp
1520 done
1523 text{*Subtraction laws, mostly by Clemens Ballarin*}
1525 lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
1526 by arith
1528 lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
1529 by arith
1531 lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
1532 by arith
1534 lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
1535 by arith
1537 lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
1538 by arith
1540 lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
1541 by arith
1543 (*Replaces the previous diff_less and le_diff_less, which had the stronger
1544   second premise n\<le>m*)
1545 lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
1546 by arith
1548 text {* Simplification of relational expressions involving subtraction *}
1550 lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
1551 by (simp split add: nat_diff_split)
1553 hide_fact (open) diff_diff_eq
1555 lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
1556 by (auto split add: nat_diff_split)
1558 lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
1559 by (auto split add: nat_diff_split)
1561 lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
1562 by (auto split add: nat_diff_split)
1564 text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
1566 (* Monotonicity of subtraction in first argument *)
1567 lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
1568 by (simp split add: nat_diff_split)
1570 lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
1571 by (simp split add: nat_diff_split)
1573 lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
1574 by (simp split add: nat_diff_split)
1576 lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
1577 by (simp split add: nat_diff_split)
1579 lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"
1580 by auto
1582 lemma inj_on_diff_nat:
1583   assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"
1584   shows "inj_on (\<lambda>n. n - k) N"
1585 proof (rule inj_onI)
1586   fix x y
1587   assume a: "x \<in> N" "y \<in> N" "x - k = y - k"
1588   with k_le_n have "x - k + k = y - k + k" by auto
1589   with a k_le_n show "x = y" by auto
1590 qed
1592 text{*Rewriting to pull differences out*}
1594 lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
1595 by arith
1597 lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
1598 by arith
1600 lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
1601 by arith
1603 text{*Lemmas for ex/Factorization*}
1605 lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
1606 by (cases m) auto
1608 lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
1609 by (cases m) auto
1611 lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
1612 by (cases m) auto
1614 text {* Specialized induction principles that work "backwards": *}
1616 lemma inc_induct[consumes 1, case_names base step]:
1617   assumes less: "i <= j"
1618   assumes base: "P j"
1619   assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
1620   shows "P i"
1621   using less
1622 proof (induct d=="j - i" arbitrary: i)
1623   case (0 i)
1624   hence "i = j" by simp
1625   with base show ?case by simp
1626 next
1627   case (Suc d i)
1628   hence "i < j" "P (Suc i)"
1629     by simp_all
1630   thus "P i" by (rule step)
1631 qed
1633 lemma strict_inc_induct[consumes 1, case_names base step]:
1634   assumes less: "i < j"
1635   assumes base: "!!i. j = Suc i ==> P i"
1636   assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
1637   shows "P i"
1638   using less
1639 proof (induct d=="j - i - 1" arbitrary: i)
1640   case (0 i)
1641   with i < j have "j = Suc i" by simp
1642   with base show ?case by simp
1643 next
1644   case (Suc d i)
1645   hence "i < j" "P (Suc i)"
1646     by simp_all
1647   thus "P i" by (rule step)
1648 qed
1650 lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
1651   using inc_induct[of "k - i" k P, simplified] by blast
1653 lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
1654   using inc_induct[of 0 k P] by blast
1656 (*The others are
1657       i - j - k = i - (j + k),
1658       k \<le> j ==> j - k + i = j + i - k,
1659       k \<le> j ==> i + (j - k) = i + j - k *)
1662 declare diff_diff_left [simp]  add_diff_assoc [simp] add_diff_assoc2[simp]
1664 text{*At present we prove no analogue of @{text not_less_Least} or @{text
1665 Least_Suc}, since there appears to be no need.*}
1668 subsection {* The divides relation on @{typ nat} *}
1670 lemma dvd_1_left [iff]: "Suc 0 dvd k"
1671 unfolding dvd_def by simp
1673 lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
1674 by (simp add: dvd_def)
1676 lemma nat_dvd_1_iff_1 [simp]: "m dvd (1::nat) \<longleftrightarrow> m = 1"
1677 by (simp add: dvd_def)
1679 lemma dvd_antisym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
1680   unfolding dvd_def
1681   by (force dest: mult_eq_self_implies_10 simp add: mult_assoc)
1683 text {* @{term "op dvd"} is a partial order *}
1685 interpretation dvd: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n"
1686   proof qed (auto intro: dvd_refl dvd_trans dvd_antisym)
1688 lemma dvd_diff_nat[simp]: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
1689 unfolding dvd_def
1690 by (blast intro: diff_mult_distrib2 [symmetric])
1692 lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
1693   apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
1694   apply (blast intro: dvd_add)
1695   done
1697 lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
1698 by (drule_tac m = m in dvd_diff_nat, auto)
1700 lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
1701   apply (rule iffI)
1702    apply (erule_tac [2] dvd_add)
1703    apply (rule_tac [2] dvd_refl)
1704   apply (subgoal_tac "n = (n+k) -k")
1705    prefer 2 apply simp
1706   apply (erule ssubst)
1707   apply (erule dvd_diff_nat)
1708   apply (rule dvd_refl)
1709   done
1711 lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
1712   unfolding dvd_def
1713   apply (erule exE)
1714   apply (simp add: mult_ac)
1715   done
1717 lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
1718   apply auto
1719    apply (subgoal_tac "m*n dvd m*1")
1720    apply (drule dvd_mult_cancel, auto)
1721   done
1723 lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
1724   apply (subst mult_commute)
1725   apply (erule dvd_mult_cancel1)
1726   done
1728 lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
1729 by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
1731 lemma nat_dvd_not_less:
1732   fixes m n :: nat
1733   shows "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"
1734 by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
1737 subsection {* aliasses *}
1739 lemma nat_mult_1: "(1::nat) * n = n"
1740   by simp
1742 lemma nat_mult_1_right: "n * (1::nat) = n"
1743   by simp
1746 subsection {* size of a datatype value *}
1748 class size =
1749   fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded} *}
1752 subsection {* code module namespace *}
1754 code_modulename SML
1755   Nat Arith
1757 code_modulename OCaml
1758   Nat Arith