src/HOL/Library/Extended_Nat.thy
 author wenzelm Fri Oct 09 20:26:03 2015 +0200 (2015-10-09) changeset 61378 3e04c9ca001a parent 61076 bdc1e2f0a86a child 61384 9f5145281888 permissions -rw-r--r--
discontinued specific HTML syntax;
1 (*  Title:      HOL/Library/Extended_Nat.thy
2     Author:     David von Oheimb, TU Muenchen;  Florian Haftmann, TU Muenchen
3     Contributions: David Trachtenherz, TU Muenchen
4 *)
6 section \<open>Extended natural numbers (i.e. with infinity)\<close>
8 theory Extended_Nat
9 imports Main Countable Order_Continuity
10 begin
12 class infinity =
13   fixes infinity :: "'a"
15 notation (xsymbols)
16   infinity  ("\<infinity>")
19 subsection \<open>Type definition\<close>
21 text \<open>
22   We extend the standard natural numbers by a special value indicating
23   infinity.
24 \<close>
26 typedef enat = "UNIV :: nat option set" ..
28 text \<open>TODO: introduce enat as coinductive datatype, enat is just @{const of_nat}\<close>
30 definition enat :: "nat \<Rightarrow> enat" where
31   "enat n = Abs_enat (Some n)"
33 instantiation enat :: infinity
34 begin
36 definition "\<infinity> = Abs_enat None"
37 instance ..
39 end
41 instance enat :: countable
42 proof
43   show "\<exists>to_nat::enat \<Rightarrow> nat. inj to_nat"
44     by (rule exI[of _ "to_nat \<circ> Rep_enat"]) (simp add: inj_on_def Rep_enat_inject)
45 qed
47 old_rep_datatype enat "\<infinity> :: enat"
48 proof -
49   fix P i assume "\<And>j. P (enat j)" "P \<infinity>"
50   then show "P i"
51   proof induct
52     case (Abs_enat y) then show ?case
53       by (cases y rule: option.exhaust)
54          (auto simp: enat_def infinity_enat_def)
55   qed
56 qed (auto simp add: enat_def infinity_enat_def Abs_enat_inject)
58 declare [[coercion "enat::nat\<Rightarrow>enat"]]
60 lemmas enat2_cases = enat.exhaust[case_product enat.exhaust]
61 lemmas enat3_cases = enat.exhaust[case_product enat.exhaust enat.exhaust]
63 lemma not_infinity_eq [iff]: "(x \<noteq> \<infinity>) = (\<exists>i. x = enat i)"
64   by (cases x) auto
66 lemma not_enat_eq [iff]: "(\<forall>y. x \<noteq> enat y) = (x = \<infinity>)"
67   by (cases x) auto
69 primrec the_enat :: "enat \<Rightarrow> nat"
70   where "the_enat (enat n) = n"
73 subsection \<open>Constructors and numbers\<close>
75 instantiation enat :: "{zero, one}"
76 begin
78 definition
79   "0 = enat 0"
81 definition
82   "1 = enat 1"
84 instance ..
86 end
88 definition eSuc :: "enat \<Rightarrow> enat" where
89   "eSuc i = (case i of enat n \<Rightarrow> enat (Suc n) | \<infinity> \<Rightarrow> \<infinity>)"
91 lemma enat_0 [code_post]: "enat 0 = 0"
94 lemma enat_1 [code_post]: "enat 1 = 1"
97 lemma enat_0_iff: "enat x = 0 \<longleftrightarrow> x = 0" "0 = enat x \<longleftrightarrow> x = 0"
98   by (auto simp add: zero_enat_def)
100 lemma enat_1_iff: "enat x = 1 \<longleftrightarrow> x = 1" "1 = enat x \<longleftrightarrow> x = 1"
101   by (auto simp add: one_enat_def)
103 lemma one_eSuc: "1 = eSuc 0"
104   by (simp add: zero_enat_def one_enat_def eSuc_def)
106 lemma infinity_ne_i0 [simp]: "(\<infinity>::enat) \<noteq> 0"
109 lemma i0_ne_infinity [simp]: "0 \<noteq> (\<infinity>::enat)"
112 lemma zero_one_enat_neq [simp]:
113   "\<not> 0 = (1::enat)"
114   "\<not> 1 = (0::enat)"
115   unfolding zero_enat_def one_enat_def by simp_all
117 lemma infinity_ne_i1 [simp]: "(\<infinity>::enat) \<noteq> 1"
120 lemma i1_ne_infinity [simp]: "1 \<noteq> (\<infinity>::enat)"
123 lemma eSuc_enat: "eSuc (enat n) = enat (Suc n)"
126 lemma eSuc_infinity [simp]: "eSuc \<infinity> = \<infinity>"
129 lemma eSuc_ne_0 [simp]: "eSuc n \<noteq> 0"
130   by (simp add: eSuc_def zero_enat_def split: enat.splits)
132 lemma zero_ne_eSuc [simp]: "0 \<noteq> eSuc n"
133   by (rule eSuc_ne_0 [symmetric])
135 lemma eSuc_inject [simp]: "eSuc m = eSuc n \<longleftrightarrow> m = n"
136   by (simp add: eSuc_def split: enat.splits)
138 lemma eSuc_enat_iff: "eSuc x = enat y \<longleftrightarrow> (\<exists>n. y = Suc n \<and> x = enat n)"
139   by (cases y) (auto simp: enat_0 eSuc_enat[symmetric])
141 lemma enat_eSuc_iff: "enat y = eSuc x \<longleftrightarrow> (\<exists>n. y = Suc n \<and> enat n = x)"
142   by (cases y) (auto simp: enat_0 eSuc_enat[symmetric])
147 begin
149 definition [nitpick_simp]:
150   "m + n = (case m of \<infinity> \<Rightarrow> \<infinity> | enat m \<Rightarrow> (case n of \<infinity> \<Rightarrow> \<infinity> | enat n \<Rightarrow> enat (m + n)))"
152 lemma plus_enat_simps [simp, code]:
153   fixes q :: enat
154   shows "enat m + enat n = enat (m + n)"
155     and "\<infinity> + q = \<infinity>"
156     and "q + \<infinity> = \<infinity>"
157   by (simp_all add: plus_enat_def split: enat.splits)
159 instance
160 proof
161   fix n m q :: enat
162   show "n + m + q = n + (m + q)"
163     by (cases n m q rule: enat3_cases) auto
164   show "n + m = m + n"
165     by (cases n m rule: enat2_cases) auto
166   show "0 + n = n"
167     by (cases n) (simp_all add: zero_enat_def)
168 qed
170 end
172 lemma eSuc_plus_1:
173   "eSuc n = n + 1"
174   by (cases n) (simp_all add: eSuc_enat one_enat_def)
176 lemma plus_1_eSuc:
177   "1 + q = eSuc q"
178   "q + 1 = eSuc q"
179   by (simp_all add: eSuc_plus_1 ac_simps)
181 lemma iadd_Suc: "eSuc m + n = eSuc (m + n)"
182   by (simp_all add: eSuc_plus_1 ac_simps)
184 lemma iadd_Suc_right: "m + eSuc n = eSuc (m + n)"
187 lemma iadd_is_0: "(m + n = (0::enat)) = (m = 0 \<and> n = 0)"
188   by (cases m, cases n, simp_all add: zero_enat_def)
190 subsection \<open>Multiplication\<close>
192 instantiation enat :: comm_semiring_1
193 begin
195 definition times_enat_def [nitpick_simp]:
196   "m * n = (case m of \<infinity> \<Rightarrow> if n = 0 then 0 else \<infinity> | enat m \<Rightarrow>
197     (case n of \<infinity> \<Rightarrow> if m = 0 then 0 else \<infinity> | enat n \<Rightarrow> enat (m * n)))"
199 lemma times_enat_simps [simp, code]:
200   "enat m * enat n = enat (m * n)"
201   "\<infinity> * \<infinity> = (\<infinity>::enat)"
202   "\<infinity> * enat n = (if n = 0 then 0 else \<infinity>)"
203   "enat m * \<infinity> = (if m = 0 then 0 else \<infinity>)"
204   unfolding times_enat_def zero_enat_def
205   by (simp_all split: enat.split)
207 instance
208 proof
209   fix a b c :: enat
210   show "(a * b) * c = a * (b * c)"
211     unfolding times_enat_def zero_enat_def
212     by (simp split: enat.split)
213   show "a * b = b * a"
214     unfolding times_enat_def zero_enat_def
215     by (simp split: enat.split)
216   show "1 * a = a"
217     unfolding times_enat_def zero_enat_def one_enat_def
218     by (simp split: enat.split)
219   show "(a + b) * c = a * c + b * c"
220     unfolding times_enat_def zero_enat_def
221     by (simp split: enat.split add: distrib_right)
222   show "0 * a = 0"
223     unfolding times_enat_def zero_enat_def
224     by (simp split: enat.split)
225   show "a * 0 = 0"
226     unfolding times_enat_def zero_enat_def
227     by (simp split: enat.split)
228   show "(0::enat) \<noteq> 1"
229     unfolding zero_enat_def one_enat_def
230     by simp
231 qed
233 end
235 lemma mult_eSuc: "eSuc m * n = n + m * n"
236   unfolding eSuc_plus_1 by (simp add: algebra_simps)
238 lemma mult_eSuc_right: "m * eSuc n = m + m * n"
239   unfolding eSuc_plus_1 by (simp add: algebra_simps)
241 lemma of_nat_eq_enat: "of_nat n = enat n"
242   apply (induct n)
244   apply (simp add: plus_1_eSuc eSuc_enat)
245   done
247 instance enat :: semiring_char_0
248 proof
249   have "inj enat" by (rule injI) simp
250   then show "inj (\<lambda>n. of_nat n :: enat)" by (simp add: of_nat_eq_enat)
251 qed
253 lemma imult_is_0 [simp]: "((m::enat) * n = 0) = (m = 0 \<or> n = 0)"
254   by (auto simp add: times_enat_def zero_enat_def split: enat.split)
256 lemma imult_is_infinity: "((a::enat) * b = \<infinity>) = (a = \<infinity> \<and> b \<noteq> 0 \<or> b = \<infinity> \<and> a \<noteq> 0)"
257   by (auto simp add: times_enat_def zero_enat_def split: enat.split)
260 subsection \<open>Numerals\<close>
262 lemma numeral_eq_enat:
263   "numeral k = enat (numeral k)"
264   using of_nat_eq_enat [of "numeral k"] by simp
266 lemma enat_numeral [code_abbrev]:
267   "enat (numeral k) = numeral k"
268   using numeral_eq_enat ..
270 lemma infinity_ne_numeral [simp]: "(\<infinity>::enat) \<noteq> numeral k"
273 lemma numeral_ne_infinity [simp]: "numeral k \<noteq> (\<infinity>::enat)"
276 lemma eSuc_numeral [simp]: "eSuc (numeral k) = numeral (k + Num.One)"
277   by (simp only: eSuc_plus_1 numeral_plus_one)
279 subsection \<open>Subtraction\<close>
281 instantiation enat :: minus
282 begin
284 definition diff_enat_def:
285 "a - b = (case a of (enat x) \<Rightarrow> (case b of (enat y) \<Rightarrow> enat (x - y) | \<infinity> \<Rightarrow> 0)
286           | \<infinity> \<Rightarrow> \<infinity>)"
288 instance ..
290 end
292 lemma idiff_enat_enat [simp, code]: "enat a - enat b = enat (a - b)"
295 lemma idiff_infinity [simp, code]: "\<infinity> - n = (\<infinity>::enat)"
298 lemma idiff_infinity_right [simp, code]: "enat a - \<infinity> = 0"
301 lemma idiff_0 [simp]: "(0::enat) - n = 0"
302   by (cases n, simp_all add: zero_enat_def)
304 lemmas idiff_enat_0 [simp] = idiff_0 [unfolded zero_enat_def]
306 lemma idiff_0_right [simp]: "(n::enat) - 0 = n"
307   by (cases n) (simp_all add: zero_enat_def)
309 lemmas idiff_enat_0_right [simp] = idiff_0_right [unfolded zero_enat_def]
311 lemma idiff_self [simp]: "n \<noteq> \<infinity> \<Longrightarrow> (n::enat) - n = 0"
312   by (auto simp: zero_enat_def)
314 lemma eSuc_minus_eSuc [simp]: "eSuc n - eSuc m = n - m"
315   by (simp add: eSuc_def split: enat.split)
317 lemma eSuc_minus_1 [simp]: "eSuc n - 1 = n"
318   by (simp add: one_enat_def eSuc_enat[symmetric] zero_enat_def[symmetric])
320 (*lemmas idiff_self_eq_0_enat = idiff_self_eq_0[unfolded zero_enat_def]*)
322 subsection \<open>Ordering\<close>
325 begin
327 definition [nitpick_simp]:
328   "m \<le> n = (case n of enat n1 \<Rightarrow> (case m of enat m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False)
329     | \<infinity> \<Rightarrow> True)"
331 definition [nitpick_simp]:
332   "m < n = (case m of enat m1 \<Rightarrow> (case n of enat n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True)
333     | \<infinity> \<Rightarrow> False)"
335 lemma enat_ord_simps [simp]:
336   "enat m \<le> enat n \<longleftrightarrow> m \<le> n"
337   "enat m < enat n \<longleftrightarrow> m < n"
338   "q \<le> (\<infinity>::enat)"
339   "q < (\<infinity>::enat) \<longleftrightarrow> q \<noteq> \<infinity>"
340   "(\<infinity>::enat) \<le> q \<longleftrightarrow> q = \<infinity>"
341   "(\<infinity>::enat) < q \<longleftrightarrow> False"
342   by (simp_all add: less_eq_enat_def less_enat_def split: enat.splits)
344 lemma numeral_le_enat_iff[simp]:
345   shows "numeral m \<le> enat n \<longleftrightarrow> numeral m \<le> n"
346 by (auto simp: numeral_eq_enat)
348 lemma numeral_less_enat_iff[simp]:
349   shows "numeral m < enat n \<longleftrightarrow> numeral m < n"
350 by (auto simp: numeral_eq_enat)
352 lemma enat_ord_code [code]:
353   "enat m \<le> enat n \<longleftrightarrow> m \<le> n"
354   "enat m < enat n \<longleftrightarrow> m < n"
355   "q \<le> (\<infinity>::enat) \<longleftrightarrow> True"
356   "enat m < \<infinity> \<longleftrightarrow> True"
357   "\<infinity> \<le> enat n \<longleftrightarrow> False"
358   "(\<infinity>::enat) < q \<longleftrightarrow> False"
359   by simp_all
361 instance
362   by standard (auto simp add: less_eq_enat_def less_enat_def plus_enat_def split: enat.splits)
364 end
366 instance enat :: ordered_comm_semiring
367 proof
368   fix a b c :: enat
369   assume "a \<le> b" and "0 \<le> c"
370   thus "c * a \<le> c * b"
371     unfolding times_enat_def less_eq_enat_def zero_enat_def
372     by (simp split: enat.splits)
373 qed
375 (* BH: These equations are already proven generally for any type in
376 class linordered_semidom. However, enat is not in that class because
377 it does not have the cancellation property. Would it be worthwhile to
378 a generalize linordered_semidom to a new class that includes enat? *)
380 lemma enat_ord_number [simp]:
381   "(numeral m :: enat) \<le> numeral n \<longleftrightarrow> (numeral m :: nat) \<le> numeral n"
382   "(numeral m :: enat) < numeral n \<longleftrightarrow> (numeral m :: nat) < numeral n"
385 lemma i0_lb [simp]: "(0::enat) \<le> n"
386   by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
388 lemma ile0_eq [simp]: "n \<le> (0::enat) \<longleftrightarrow> n = 0"
389   by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
391 lemma infinity_ileE [elim!]: "\<infinity> \<le> enat m \<Longrightarrow> R"
392   by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
394 lemma infinity_ilessE [elim!]: "\<infinity> < enat m \<Longrightarrow> R"
395   by simp
397 lemma not_iless0 [simp]: "\<not> n < (0::enat)"
398   by (simp add: zero_enat_def less_enat_def split: enat.splits)
400 lemma i0_less [simp]: "(0::enat) < n \<longleftrightarrow> n \<noteq> 0"
401   by (simp add: zero_enat_def less_enat_def split: enat.splits)
403 lemma eSuc_ile_mono [simp]: "eSuc n \<le> eSuc m \<longleftrightarrow> n \<le> m"
404   by (simp add: eSuc_def less_eq_enat_def split: enat.splits)
406 lemma eSuc_mono [simp]: "eSuc n < eSuc m \<longleftrightarrow> n < m"
407   by (simp add: eSuc_def less_enat_def split: enat.splits)
409 lemma ile_eSuc [simp]: "n \<le> eSuc n"
410   by (simp add: eSuc_def less_eq_enat_def split: enat.splits)
412 lemma not_eSuc_ilei0 [simp]: "\<not> eSuc n \<le> 0"
413   by (simp add: zero_enat_def eSuc_def less_eq_enat_def split: enat.splits)
415 lemma i0_iless_eSuc [simp]: "0 < eSuc n"
416   by (simp add: zero_enat_def eSuc_def less_enat_def split: enat.splits)
418 lemma iless_eSuc0[simp]: "(n < eSuc 0) = (n = 0)"
419   by (simp add: zero_enat_def eSuc_def less_enat_def split: enat.split)
421 lemma ileI1: "m < n \<Longrightarrow> eSuc m \<le> n"
422   by (simp add: eSuc_def less_eq_enat_def less_enat_def split: enat.splits)
424 lemma Suc_ile_eq: "enat (Suc m) \<le> n \<longleftrightarrow> enat m < n"
425   by (cases n) auto
427 lemma iless_Suc_eq [simp]: "enat m < eSuc n \<longleftrightarrow> enat m \<le> n"
428   by (auto simp add: eSuc_def less_enat_def split: enat.splits)
430 lemma imult_infinity: "(0::enat) < n \<Longrightarrow> \<infinity> * n = \<infinity>"
431   by (simp add: zero_enat_def less_enat_def split: enat.splits)
433 lemma imult_infinity_right: "(0::enat) < n \<Longrightarrow> n * \<infinity> = \<infinity>"
434   by (simp add: zero_enat_def less_enat_def split: enat.splits)
436 lemma enat_0_less_mult_iff: "(0 < (m::enat) * n) = (0 < m \<and> 0 < n)"
437   by (simp only: i0_less imult_is_0, simp)
439 lemma mono_eSuc: "mono eSuc"
443 lemma min_enat_simps [simp]:
444   "min (enat m) (enat n) = enat (min m n)"
445   "min q 0 = 0"
446   "min 0 q = 0"
447   "min q (\<infinity>::enat) = q"
448   "min (\<infinity>::enat) q = q"
449   by (auto simp add: min_def)
451 lemma max_enat_simps [simp]:
452   "max (enat m) (enat n) = enat (max m n)"
453   "max q 0 = q"
454   "max 0 q = q"
455   "max q \<infinity> = (\<infinity>::enat)"
456   "max \<infinity> q = (\<infinity>::enat)"
459 lemma enat_ile: "n \<le> enat m \<Longrightarrow> \<exists>k. n = enat k"
460   by (cases n) simp_all
462 lemma enat_iless: "n < enat m \<Longrightarrow> \<exists>k. n = enat k"
463   by (cases n) simp_all
465 lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. enat k < Y j"
466 apply (induct_tac k)
467  apply (simp (no_asm) only: enat_0)
468  apply (fast intro: le_less_trans [OF i0_lb])
469 apply (erule exE)
470 apply (drule spec)
471 apply (erule exE)
472 apply (drule ileI1)
473 apply (rule eSuc_enat [THEN subst])
474 apply (rule exI)
475 apply (erule (1) le_less_trans)
476 done
478 lemma eSuc_max: "eSuc (max x y) = max (eSuc x) (eSuc y)"
479   by (simp add: eSuc_def split: enat.split)
481 lemma eSuc_Max:
482   assumes "finite A" "A \<noteq> {}"
483   shows "eSuc (Max A) = Max (eSuc ` A)"
484 using assms proof induction
485   case (insert x A)
486   thus ?case by(cases "A = {}")(simp_all add: eSuc_max)
487 qed simp
489 instantiation enat :: "{order_bot, order_top}"
490 begin
492 definition bot_enat :: enat where "bot_enat = 0"
493 definition top_enat :: enat where "top_enat = \<infinity>"
495 instance
496   by standard (simp_all add: bot_enat_def top_enat_def)
498 end
500 lemma finite_enat_bounded:
501   assumes le_fin: "\<And>y. y \<in> A \<Longrightarrow> y \<le> enat n"
502   shows "finite A"
503 proof (rule finite_subset)
504   show "finite (enat ` {..n})" by blast
505   have "A \<subseteq> {..enat n}" using le_fin by fastforce
506   also have "\<dots> \<subseteq> enat ` {..n}"
507     apply (rule subsetI)
508     subgoal for x by (cases x) auto
509     done
510   finally show "A \<subseteq> enat ` {..n}" .
511 qed
514 subsection \<open>Cancellation simprocs\<close>
516 lemma enat_add_left_cancel: "a + b = a + c \<longleftrightarrow> a = (\<infinity>::enat) \<or> b = c"
517   unfolding plus_enat_def by (simp split: enat.split)
519 lemma enat_add_left_cancel_le: "a + b \<le> a + c \<longleftrightarrow> a = (\<infinity>::enat) \<or> b \<le> c"
520   unfolding plus_enat_def by (simp split: enat.split)
522 lemma enat_add_left_cancel_less: "a + b < a + c \<longleftrightarrow> a \<noteq> (\<infinity>::enat) \<and> b < c"
523   unfolding plus_enat_def by (simp split: enat.split)
525 ML \<open>
526 structure Cancel_Enat_Common =
527 struct
528   (* copied from src/HOL/Tools/nat_numeral_simprocs.ML *)
529   fun find_first_t _    _ []         = raise TERM("find_first_t", [])
530     | find_first_t past u (t::terms) =
531           if u aconv t then (rev past @ terms)
532           else find_first_t (t::past) u terms
534   fun dest_summing (Const (@{const_name Groups.plus}, _) \$ t \$ u, ts) =
535         dest_summing (t, dest_summing (u, ts))
536     | dest_summing (t, ts) = t :: ts
538   val mk_sum = Arith_Data.long_mk_sum
539   fun dest_sum t = dest_summing (t, [])
540   val find_first = find_first_t []
541   val trans_tac = Numeral_Simprocs.trans_tac
542   val norm_ss =
543     simpset_of (put_simpset HOL_basic_ss @{context}
545   fun norm_tac ctxt = ALLGOALS (simp_tac (put_simpset norm_ss ctxt))
546   fun simplify_meta_eq ctxt cancel_th th =
547     Arith_Data.simplify_meta_eq [] ctxt
548       ([th, cancel_th] MRS trans)
549   fun mk_eq (a, b) = HOLogic.mk_Trueprop (HOLogic.mk_eq (a, b))
550 end
552 structure Eq_Enat_Cancel = ExtractCommonTermFun
553 (open Cancel_Enat_Common
554   val mk_bal = HOLogic.mk_eq
555   val dest_bal = HOLogic.dest_bin @{const_name HOL.eq} @{typ enat}
556   fun simp_conv _ _ = SOME @{thm enat_add_left_cancel}
557 )
559 structure Le_Enat_Cancel = ExtractCommonTermFun
560 (open Cancel_Enat_Common
561   val mk_bal = HOLogic.mk_binrel @{const_name Orderings.less_eq}
562   val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} @{typ enat}
563   fun simp_conv _ _ = SOME @{thm enat_add_left_cancel_le}
564 )
566 structure Less_Enat_Cancel = ExtractCommonTermFun
567 (open Cancel_Enat_Common
568   val mk_bal = HOLogic.mk_binrel @{const_name Orderings.less}
569   val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} @{typ enat}
570   fun simp_conv _ _ = SOME @{thm enat_add_left_cancel_less}
571 )
572 \<close>
574 simproc_setup enat_eq_cancel
575   ("(l::enat) + m = n" | "(l::enat) = m + n") =
576   \<open>fn phi => fn ctxt => fn ct => Eq_Enat_Cancel.proc ctxt (Thm.term_of ct)\<close>
578 simproc_setup enat_le_cancel
579   ("(l::enat) + m \<le> n" | "(l::enat) \<le> m + n") =
580   \<open>fn phi => fn ctxt => fn ct => Le_Enat_Cancel.proc ctxt (Thm.term_of ct)\<close>
582 simproc_setup enat_less_cancel
583   ("(l::enat) + m < n" | "(l::enat) < m + n") =
584   \<open>fn phi => fn ctxt => fn ct => Less_Enat_Cancel.proc ctxt (Thm.term_of ct)\<close>
586 text \<open>TODO: add regression tests for these simprocs\<close>
588 text \<open>TODO: add simprocs for combining and cancelling numerals\<close>
590 subsection \<open>Well-ordering\<close>
592 lemma less_enatE:
593   "[| n < enat m; !!k. n = enat k ==> k < m ==> P |] ==> P"
594 by (induct n) auto
596 lemma less_infinityE:
597   "[| n < \<infinity>; !!k. n = enat k ==> P |] ==> P"
598 by (induct n) auto
600 lemma enat_less_induct:
601   assumes prem: "!!n. \<forall>m::enat. m < n --> P m ==> P n" shows "P n"
602 proof -
603   have P_enat: "!!k. P (enat k)"
604     apply (rule nat_less_induct)
605     apply (rule prem, clarify)
606     apply (erule less_enatE, simp)
607     done
608   show ?thesis
609   proof (induct n)
610     fix nat
611     show "P (enat nat)" by (rule P_enat)
612   next
613     show "P \<infinity>"
614       apply (rule prem, clarify)
615       apply (erule less_infinityE)
617       done
618   qed
619 qed
621 instance enat :: wellorder
622 proof
623   fix P and n
624   assume hyp: "(\<And>n::enat. (\<And>m::enat. m < n \<Longrightarrow> P m) \<Longrightarrow> P n)"
625   show "P n" by (blast intro: enat_less_induct hyp)
626 qed
628 subsection \<open>Complete Lattice\<close>
630 instantiation enat :: complete_lattice
631 begin
633 definition inf_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where
634   "inf_enat = min"
636 definition sup_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where
637   "sup_enat = max"
639 definition Inf_enat :: "enat set \<Rightarrow> enat" where
640   "Inf_enat A = (if A = {} then \<infinity> else (LEAST x. x \<in> A))"
642 definition Sup_enat :: "enat set \<Rightarrow> enat" where
643   "Sup_enat A = (if A = {} then 0 else if finite A then Max A else \<infinity>)"
644 instance
645 proof
646   fix x :: "enat" and A :: "enat set"
647   { assume "x \<in> A" then show "Inf A \<le> x"
648       unfolding Inf_enat_def by (auto intro: Least_le) }
649   { assume "\<And>y. y \<in> A \<Longrightarrow> x \<le> y" then show "x \<le> Inf A"
650       unfolding Inf_enat_def
651       by (cases "A = {}") (auto intro: LeastI2_ex) }
652   { assume "x \<in> A" then show "x \<le> Sup A"
653       unfolding Sup_enat_def by (cases "finite A") auto }
654   { assume "\<And>y. y \<in> A \<Longrightarrow> y \<le> x" then show "Sup A \<le> x"
655       unfolding Sup_enat_def using finite_enat_bounded by auto }