1 (* Title: HOL/SetInterval.thy
3 Author: Tobias Nipkow and Clemens Ballarin
4 Additions by Jeremy Avigad in March 2004
5 Copyright 2000 TU Muenchen
7 lessThan, greaterThan, atLeast, atMost and two-sided intervals
10 header {* Set intervals *}
17 lessThan :: "('a::ord) => 'a set" ("(1{..<_})")
20 atMost :: "('a::ord) => 'a set" ("(1{.._})")
23 greaterThan :: "('a::ord) => 'a set" ("(1{_<..})")
26 atLeast :: "('a::ord) => 'a set" ("(1{_..})")
29 greaterThanLessThan :: "['a::ord, 'a] => 'a set" ("(1{_<..<_})")
30 "{l<..<u} == {l<..} Int {..<u}"
32 atLeastLessThan :: "['a::ord, 'a] => 'a set" ("(1{_..<_})")
33 "{l..<u} == {l..} Int {..<u}"
35 greaterThanAtMost :: "['a::ord, 'a] => 'a set" ("(1{_<.._})")
36 "{l<..u} == {l<..} Int {..u}"
38 atLeastAtMost :: "['a::ord, 'a] => 'a set" ("(1{_.._})")
39 "{l..u} == {l..} Int {..u}"
41 text{* A note of warning when using @{term"{..<n}"} on type @{typ
42 nat}: it is equivalent to @{term"{0::nat..<n}"} but some lemmas involving
43 @{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well. *}
46 "@UNION_le" :: "nat => nat => 'b set => 'b set" ("(3UN _<=_./ _)" 10)
47 "@UNION_less" :: "nat => nat => 'b set => 'b set" ("(3UN _<_./ _)" 10)
48 "@INTER_le" :: "nat => nat => 'b set => 'b set" ("(3INT _<=_./ _)" 10)
49 "@INTER_less" :: "nat => nat => 'b set => 'b set" ("(3INT _<_./ _)" 10)
52 "@UNION_le" :: "nat => nat => 'b set => 'b set" ("(3\<Union> _\<le>_./ _)" 10)
53 "@UNION_less" :: "nat => nat => 'b set => 'b set" ("(3\<Union> _<_./ _)" 10)
54 "@INTER_le" :: "nat => nat => 'b set => 'b set" ("(3\<Inter> _\<le>_./ _)" 10)
55 "@INTER_less" :: "nat => nat => 'b set => 'b set" ("(3\<Inter> _<_./ _)" 10)
58 "@UNION_le" :: "nat \<Rightarrow> nat => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)
59 "@UNION_less" :: "nat \<Rightarrow> nat => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_ < _\<^esub>)/ _)" 10)
60 "@INTER_le" :: "nat \<Rightarrow> nat => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)
61 "@INTER_less" :: "nat \<Rightarrow> nat => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_ < _\<^esub>)/ _)" 10)
64 "UN i<=n. A" == "UN i:{..n}. A"
65 "UN i<n. A" == "UN i:{..<n}. A"
66 "INT i<=n. A" == "INT i:{..n}. A"
67 "INT i<n. A" == "INT i:{..<n}. A"
70 subsection {* Various equivalences *}
72 lemma lessThan_iff [iff]: "(i: lessThan k) = (i<k)"
73 by (simp add: lessThan_def)
75 lemma Compl_lessThan [simp]:
76 "!!k:: 'a::linorder. -lessThan k = atLeast k"
77 apply (auto simp add: lessThan_def atLeast_def)
80 lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"
83 lemma greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"
84 by (simp add: greaterThan_def)
86 lemma Compl_greaterThan [simp]:
87 "!!k:: 'a::linorder. -greaterThan k = atMost k"
88 apply (simp add: greaterThan_def atMost_def le_def, auto)
91 lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
92 apply (subst Compl_greaterThan [symmetric])
93 apply (rule double_complement)
96 lemma atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
97 by (simp add: atLeast_def)
99 lemma Compl_atLeast [simp]:
100 "!!k:: 'a::linorder. -atLeast k = lessThan k"
101 apply (simp add: lessThan_def atLeast_def le_def, auto)
104 lemma atMost_iff [iff]: "(i: atMost k) = (i<=k)"
105 by (simp add: atMost_def)
107 lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
108 by (blast intro: order_antisym)
111 subsection {* Logical Equivalences for Set Inclusion and Equality *}
113 lemma atLeast_subset_iff [iff]:
114 "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::order))"
115 by (blast intro: order_trans)
117 lemma atLeast_eq_iff [iff]:
118 "(atLeast x = atLeast y) = (x = (y::'a::linorder))"
119 by (blast intro: order_antisym order_trans)
121 lemma greaterThan_subset_iff [iff]:
122 "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
123 apply (auto simp add: greaterThan_def)
124 apply (subst linorder_not_less [symmetric], blast)
127 lemma greaterThan_eq_iff [iff]:
128 "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
130 apply (erule equalityE)
131 apply (simp_all add: greaterThan_subset_iff)
134 lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::order))"
135 by (blast intro: order_trans)
137 lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))"
138 by (blast intro: order_antisym order_trans)
140 lemma lessThan_subset_iff [iff]:
141 "(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))"
142 apply (auto simp add: lessThan_def)
143 apply (subst linorder_not_less [symmetric], blast)
146 lemma lessThan_eq_iff [iff]:
147 "(lessThan x = lessThan y) = (x = (y::'a::linorder))"
149 apply (erule equalityE)
150 apply (simp_all add: lessThan_subset_iff)
154 subsection {*Two-sided intervals*}
156 lemma greaterThanLessThan_iff [simp,noatp]:
157 "(i : {l<..<u}) = (l < i & i < u)"
158 by (simp add: greaterThanLessThan_def)
160 lemma atLeastLessThan_iff [simp,noatp]:
161 "(i : {l..<u}) = (l <= i & i < u)"
162 by (simp add: atLeastLessThan_def)
164 lemma greaterThanAtMost_iff [simp,noatp]:
165 "(i : {l<..u}) = (l < i & i <= u)"
166 by (simp add: greaterThanAtMost_def)
168 lemma atLeastAtMost_iff [simp,noatp]:
169 "(i : {l..u}) = (l <= i & i <= u)"
170 by (simp add: atLeastAtMost_def)
172 text {* The above four lemmas could be declared as iffs.
173 If we do so, a call to blast in Hyperreal/Star.ML, lemma @{text STAR_Int}
174 seems to take forever (more than one hour). *}
176 subsubsection{* Emptyness and singletons *}
178 lemma atLeastAtMost_empty [simp]: "n < m ==> {m::'a::order..n} = {}";
179 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def);
181 lemma atLeastLessThan_empty[simp]: "n \<le> m ==> {m..<n::'a::order} = {}"
182 by (auto simp add: atLeastLessThan_def)
184 lemma greaterThanAtMost_empty[simp]:"l \<le> k ==> {k<..(l::'a::order)} = {}"
185 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)
187 lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..(l::'a::order)} = {}"
188 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)
190 lemma atLeastAtMost_singleton [simp]: "{a::'a::order..a} = {a}";
191 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def);
193 subsection {* Intervals of natural numbers *}
195 subsubsection {* The Constant @{term lessThan} *}
197 lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
198 by (simp add: lessThan_def)
200 lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
201 by (simp add: lessThan_def less_Suc_eq, blast)
203 lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
204 by (simp add: lessThan_def atMost_def less_Suc_eq_le)
206 lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
209 subsubsection {* The Constant @{term greaterThan} *}
211 lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
212 apply (simp add: greaterThan_def)
213 apply (blast dest: gr0_conv_Suc [THEN iffD1])
216 lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
217 apply (simp add: greaterThan_def)
218 apply (auto elim: linorder_neqE)
221 lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
224 subsubsection {* The Constant @{term atLeast} *}
226 lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
227 by (unfold atLeast_def UNIV_def, simp)
229 lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
230 apply (simp add: atLeast_def)
231 apply (simp add: Suc_le_eq)
232 apply (simp add: order_le_less, blast)
235 lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
236 by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
238 lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
241 subsubsection {* The Constant @{term atMost} *}
243 lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
244 by (simp add: atMost_def)
246 lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
247 apply (simp add: atMost_def)
248 apply (simp add: less_Suc_eq order_le_less, blast)
251 lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
254 subsubsection {* The Constant @{term atLeastLessThan} *}
256 text{*But not a simprule because some concepts are better left in terms
257 of @{term atLeastLessThan}*}
258 lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
259 by(simp add:lessThan_def atLeastLessThan_def)
261 lemma atLeastLessThan0 [simp]: "{m..<0::nat} = {}"
262 by (simp add: atLeastLessThan_def)
264 subsubsection {* Intervals of nats with @{term Suc} *}
266 text{*Not a simprule because the RHS is too messy.*}
267 lemma atLeastLessThanSuc:
268 "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
269 by (auto simp add: atLeastLessThan_def)
271 lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
272 by (auto simp add: atLeastLessThan_def)
274 lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
275 by (induct k, simp_all add: atLeastLessThanSuc)
277 lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
278 by (auto simp add: atLeastLessThan_def)
280 lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
281 by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
283 lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
284 by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
285 greaterThanAtMost_def)
287 lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
288 by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
289 greaterThanLessThan_def)
291 lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
292 by (auto simp add: atLeastAtMost_def)
294 subsubsection {* Image *}
296 lemma image_add_atLeastAtMost:
297 "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
299 show "?A \<subseteq> ?B" by auto
301 show "?B \<subseteq> ?A"
303 fix n assume a: "n : ?B"
304 hence "n - k : {i..j}" by auto
305 moreover have "n = (n - k) + k" using a by auto
306 ultimately show "n : ?A" by blast
310 lemma image_add_atLeastLessThan:
311 "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
313 show "?A \<subseteq> ?B" by auto
315 show "?B \<subseteq> ?A"
317 fix n assume a: "n : ?B"
318 hence "n - k : {i..<j}" by auto
319 moreover have "n = (n - k) + k" using a by auto
320 ultimately show "n : ?A" by blast
324 corollary image_Suc_atLeastAtMost[simp]:
325 "Suc ` {i..j} = {Suc i..Suc j}"
326 using image_add_atLeastAtMost[where k=1] by simp
328 corollary image_Suc_atLeastLessThan[simp]:
329 "Suc ` {i..<j} = {Suc i..<Suc j}"
330 using image_add_atLeastLessThan[where k=1] by simp
332 lemma image_add_int_atLeastLessThan:
333 "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
334 apply (auto simp add: image_def)
335 apply (rule_tac x = "x - l" in bexI)
340 subsubsection {* Finiteness *}
342 lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
343 by (induct k) (simp_all add: lessThan_Suc)
345 lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
346 by (induct k) (simp_all add: atMost_Suc)
348 lemma finite_greaterThanLessThan [iff]:
349 fixes l :: nat shows "finite {l<..<u}"
350 by (simp add: greaterThanLessThan_def)
352 lemma finite_atLeastLessThan [iff]:
353 fixes l :: nat shows "finite {l..<u}"
354 by (simp add: atLeastLessThan_def)
356 lemma finite_greaterThanAtMost [iff]:
357 fixes l :: nat shows "finite {l<..u}"
358 by (simp add: greaterThanAtMost_def)
360 lemma finite_atLeastAtMost [iff]:
361 fixes l :: nat shows "finite {l..u}"
362 by (simp add: atLeastAtMost_def)
364 lemma bounded_nat_set_is_finite:
365 "(ALL i:N. i < (n::nat)) ==> finite N"
366 -- {* A bounded set of natural numbers is finite. *}
367 apply (rule finite_subset)
368 apply (rule_tac [2] finite_lessThan, auto)
371 subsubsection {* Cardinality *}
373 lemma card_lessThan [simp]: "card {..<u} = u"
374 by (induct u, simp_all add: lessThan_Suc)
376 lemma card_atMost [simp]: "card {..u} = Suc u"
377 by (simp add: lessThan_Suc_atMost [THEN sym])
379 lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
380 apply (subgoal_tac "card {l..<u} = card {..<u-l}")
381 apply (erule ssubst, rule card_lessThan)
382 apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
384 apply (rule card_image)
385 apply (simp add: inj_on_def)
386 apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
387 apply (rule_tac x = "x - l" in exI)
391 lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
392 by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
394 lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
395 by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
397 lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
398 by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
400 subsection {* Intervals of integers *}
402 lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
403 by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
405 lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
406 by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
408 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
409 "{l+1..<u} = {l<..<u::int}"
410 by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
412 subsubsection {* Finiteness *}
414 lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
415 {(0::int)..<u} = int ` {..<nat u}"
416 apply (unfold image_def lessThan_def)
418 apply (rule_tac x = "nat x" in exI)
419 apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
422 lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
423 apply (case_tac "0 \<le> u")
424 apply (subst image_atLeastZeroLessThan_int, assumption)
425 apply (rule finite_imageI)
429 lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
430 apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
432 apply (rule finite_imageI)
433 apply (rule finite_atLeastZeroLessThan_int)
434 apply (rule image_add_int_atLeastLessThan)
437 lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
438 by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
440 lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
441 by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
443 lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
444 by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
446 subsubsection {* Cardinality *}
448 lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
449 apply (case_tac "0 \<le> u")
450 apply (subst image_atLeastZeroLessThan_int, assumption)
451 apply (subst card_image)
452 apply (auto simp add: inj_on_def)
455 lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
456 apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
457 apply (erule ssubst, rule card_atLeastZeroLessThan_int)
458 apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
460 apply (rule card_image)
461 apply (simp add: inj_on_def)
462 apply (rule image_add_int_atLeastLessThan)
465 lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
466 apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
467 apply (auto simp add: compare_rls)
470 lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
471 by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
473 lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
474 by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
477 subsection {*Lemmas useful with the summation operator setsum*}
479 text {* For examples, see Algebra/poly/UnivPoly2.thy *}
481 subsubsection {* Disjoint Unions *}
483 text {* Singletons and open intervals *}
485 lemma ivl_disj_un_singleton:
486 "{l::'a::linorder} Un {l<..} = {l..}"
487 "{..<u} Un {u::'a::linorder} = {..u}"
488 "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
489 "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
490 "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
491 "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
494 text {* One- and two-sided intervals *}
496 lemma ivl_disj_un_one:
497 "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
498 "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
499 "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
500 "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
501 "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
502 "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
503 "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
504 "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
507 text {* Two- and two-sided intervals *}
509 lemma ivl_disj_un_two:
510 "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
511 "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
512 "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
513 "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
514 "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
515 "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
516 "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
517 "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
520 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
522 subsubsection {* Disjoint Intersections *}
524 text {* Singletons and open intervals *}
526 lemma ivl_disj_int_singleton:
527 "{l::'a::order} Int {l<..} = {}"
528 "{..<u} Int {u} = {}"
529 "{l} Int {l<..<u} = {}"
530 "{l<..<u} Int {u} = {}"
531 "{l} Int {l<..u} = {}"
532 "{l..<u} Int {u} = {}"
535 text {* One- and two-sided intervals *}
537 lemma ivl_disj_int_one:
538 "{..l::'a::order} Int {l<..<u} = {}"
539 "{..<l} Int {l..<u} = {}"
540 "{..l} Int {l<..u} = {}"
541 "{..<l} Int {l..u} = {}"
542 "{l<..u} Int {u<..} = {}"
543 "{l<..<u} Int {u..} = {}"
544 "{l..u} Int {u<..} = {}"
545 "{l..<u} Int {u..} = {}"
548 text {* Two- and two-sided intervals *}
550 lemma ivl_disj_int_two:
551 "{l::'a::order<..<m} Int {m..<u} = {}"
552 "{l<..m} Int {m<..<u} = {}"
553 "{l..<m} Int {m..<u} = {}"
554 "{l..m} Int {m<..<u} = {}"
555 "{l<..<m} Int {m..u} = {}"
556 "{l<..m} Int {m<..u} = {}"
557 "{l..<m} Int {m..u} = {}"
558 "{l..m} Int {m<..u} = {}"
561 lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
563 subsubsection {* Some Differences *}
565 lemma ivl_diff[simp]:
566 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
570 subsubsection {* Some Subset Conditions *}
572 lemma ivl_subset [simp,noatp]:
573 "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
574 apply(auto simp:linorder_not_le)
576 apply(insert linorder_le_less_linear[of i n])
577 apply(clarsimp simp:linorder_not_le)
582 subsection {* Summation indexed over intervals *}
585 "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
586 "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
587 "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
588 "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
590 "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
591 "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
592 "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
593 "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
595 "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
596 "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
597 "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
598 "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
599 syntax (latex_sum output)
600 "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
601 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
602 "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
603 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
604 "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
605 ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
606 "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
607 ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
610 "\<Sum>x=a..b. t" == "setsum (%x. t) {a..b}"
611 "\<Sum>x=a..<b. t" == "setsum (%x. t) {a..<b}"
612 "\<Sum>i\<le>n. t" == "setsum (\<lambda>i. t) {..n}"
613 "\<Sum>i<n. t" == "setsum (\<lambda>i. t) {..<n}"
615 text{* The above introduces some pretty alternative syntaxes for
616 summation over intervals:
620 @{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\
621 @{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\
622 @{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
623 @{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
626 The left column shows the term before introduction of the new syntax,
627 the middle column shows the new (default) syntax, and the right column
628 shows a special syntax. The latter is only meaningful for latex output
629 and has to be activated explicitly by setting the print mode to
630 @{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
631 antiquotations). It is not the default \LaTeX\ output because it only
632 works well with italic-style formulae, not tt-style.
634 Note that for uniformity on @{typ nat} it is better to use
635 @{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
636 not provide all lemmas available for @{term"{m..<n}"} also in the
637 special form for @{term"{..<n}"}. *}
639 text{* This congruence rule should be used for sums over intervals as
640 the standard theorem @{text[source]setsum_cong} does not work well
641 with the simplifier who adds the unsimplified premise @{term"x:B"} to
644 lemma setsum_ivl_cong:
645 "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
646 setsum f {a..<b} = setsum g {c..<d}"
647 by(rule setsum_cong, simp_all)
649 (* FIXME why are the following simp rules but the corresponding eqns
650 on intervals are not? *)
652 lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
653 by (simp add:atMost_Suc add_ac)
655 lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
656 by (simp add:lessThan_Suc add_ac)
658 lemma setsum_cl_ivl_Suc[simp]:
659 "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
660 by (auto simp:add_ac atLeastAtMostSuc_conv)
662 lemma setsum_op_ivl_Suc[simp]:
663 "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
664 by (auto simp:add_ac atLeastLessThanSuc)
666 lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
667 (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
668 by (auto simp:add_ac atLeastAtMostSuc_conv)
670 lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
671 setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
672 by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
674 lemma setsum_diff_nat_ivl:
675 fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
676 shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
677 setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
678 using setsum_add_nat_ivl [of m n p f,symmetric]
679 apply (simp add: add_ac)
682 subsection{* Shifting bounds *}
684 lemma setsum_shift_bounds_nat_ivl:
685 "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
686 by (induct "n", auto simp:atLeastLessThanSuc)
688 lemma setsum_shift_bounds_cl_nat_ivl:
689 "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
690 apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
691 apply (simp add:image_add_atLeastAtMost o_def)
694 corollary setsum_shift_bounds_cl_Suc_ivl:
695 "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
696 by (simp add:setsum_shift_bounds_cl_nat_ivl[where k=1,simplified])
698 corollary setsum_shift_bounds_Suc_ivl:
699 "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
700 by (simp add:setsum_shift_bounds_nat_ivl[where k=1,simplified])
705 shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
708 have "{m..n} = {m} \<union> {m<..n}"
709 by (auto intro: ivl_disj_un_singleton)
710 hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
711 by (simp add: atLeast0LessThan)
712 also have "\<dots> = ?rhs" by simp
713 finally show ?thesis .
716 lemma setsum_head_upt:
719 shows "(\<Sum>x<m. P x) = P 0 + (\<Sum>x\<in>{1..<m}. P x)"
721 have "(\<Sum>x<m. P x) = (\<Sum>x\<in>{0..<m}. P x)"
722 by (simp add: atLeast0LessThan)
725 have "\<dots> = (\<Sum>x\<in>{0..m - 1}. P x)"
726 by (cases m) (auto simp add: atLeastLessThanSuc_atLeastAtMost)
728 have "\<dots> = P 0 + (\<Sum>x\<in>{0<..m - 1}. P x)"
729 by (simp add: setsum_head)
732 have "{0<..m - 1} = {1..<m}"
733 by (cases m) (auto simp add: atLeastLessThanSuc_atLeastAtMost)
734 finally show ?thesis .
737 subsection {* The formula for geometric sums *}
740 "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) =
741 (x ^ n - 1) / (x - 1::'a::{field, recpower})"
742 by (induct "n") (simp_all add:field_simps power_Suc)
744 subsection {* The formula for arithmetic sums *}
747 "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =
748 of_nat n*((of_nat n)+1)"
754 then show ?case by (simp add: ring_simps)
757 theorem arith_series_general:
758 "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
759 of_nat n * (a + (a + of_nat(n - 1)*d))"
762 let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
764 "(\<Sum>i\<in>{..<n}. a+?I i*d) =
765 ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
766 by (rule setsum_addf)
767 also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
768 also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
769 by (simp add: setsum_right_distrib setsum_head_upt mult_ac)
770 also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"
771 by (simp add: left_distrib right_distrib)
772 also from ngt1 have "{1..<n} = {1..n - 1}"
773 by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
775 have "(1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..n - 1}. ?I i) = ((1+1)*?n*a + d*?I (n - 1)*?I n)"
776 by (simp only: mult_ac gauss_sum [of "n - 1"])
777 (simp add: mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
778 finally show ?thesis by (simp add: mult_ac add_ac right_distrib)
780 assume "\<not>(n > 1)"
781 hence "n = 1 \<or> n = 0" by auto
782 thus ?thesis by (auto simp: mult_ac right_distrib)
785 lemma arith_series_nat:
786 "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
789 "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
790 of_nat(n) * (a + (a + of_nat(n - 1)*d))"
791 by (rule arith_series_general)
792 thus ?thesis by (auto simp add: of_nat_id)
795 lemma arith_series_int:
796 "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
797 of_nat n * (a + (a + of_nat(n - 1)*d))"
800 "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
801 of_nat(n) * (a + (a + of_nat(n - 1)*d))"
802 by (rule arith_series_general)
806 lemma sum_diff_distrib:
807 fixes P::"nat\<Rightarrow>nat"
809 "\<forall>x. Q x \<le> P x \<Longrightarrow>
810 (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
812 case 0 show ?case by simp
816 let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
817 let ?rhs = "\<Sum>x<n. P x - Q x"
819 from Suc have "?lhs = ?rhs" by simp
821 from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
824 "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
825 by (subst diff_diff_left[symmetric],
826 subst diff_add_assoc2)
827 (auto simp: diff_add_assoc2 intro: setsum_mono)
835 val Compl_atLeast = thm "Compl_atLeast";
836 val Compl_atMost = thm "Compl_atMost";
837 val Compl_greaterThan = thm "Compl_greaterThan";
838 val Compl_lessThan = thm "Compl_lessThan";
839 val INT_greaterThan_UNIV = thm "INT_greaterThan_UNIV";
840 val UN_atLeast_UNIV = thm "UN_atLeast_UNIV";
841 val UN_atMost_UNIV = thm "UN_atMost_UNIV";
842 val UN_lessThan_UNIV = thm "UN_lessThan_UNIV";
843 val atLeastAtMost_def = thm "atLeastAtMost_def";
844 val atLeastAtMost_iff = thm "atLeastAtMost_iff";
845 val atLeastLessThan_def = thm "atLeastLessThan_def";
846 val atLeastLessThan_iff = thm "atLeastLessThan_iff";
847 val atLeast_0 = thm "atLeast_0";
848 val atLeast_Suc = thm "atLeast_Suc";
849 val atLeast_def = thm "atLeast_def";
850 val atLeast_iff = thm "atLeast_iff";
851 val atMost_0 = thm "atMost_0";
852 val atMost_Int_atLeast = thm "atMost_Int_atLeast";
853 val atMost_Suc = thm "atMost_Suc";
854 val atMost_def = thm "atMost_def";
855 val atMost_iff = thm "atMost_iff";
856 val greaterThanAtMost_def = thm "greaterThanAtMost_def";
857 val greaterThanAtMost_iff = thm "greaterThanAtMost_iff";
858 val greaterThanLessThan_def = thm "greaterThanLessThan_def";
859 val greaterThanLessThan_iff = thm "greaterThanLessThan_iff";
860 val greaterThan_0 = thm "greaterThan_0";
861 val greaterThan_Suc = thm "greaterThan_Suc";
862 val greaterThan_def = thm "greaterThan_def";
863 val greaterThan_iff = thm "greaterThan_iff";
864 val ivl_disj_int = thms "ivl_disj_int";
865 val ivl_disj_int_one = thms "ivl_disj_int_one";
866 val ivl_disj_int_singleton = thms "ivl_disj_int_singleton";
867 val ivl_disj_int_two = thms "ivl_disj_int_two";
868 val ivl_disj_un = thms "ivl_disj_un";
869 val ivl_disj_un_one = thms "ivl_disj_un_one";
870 val ivl_disj_un_singleton = thms "ivl_disj_un_singleton";
871 val ivl_disj_un_two = thms "ivl_disj_un_two";
872 val lessThan_0 = thm "lessThan_0";
873 val lessThan_Suc = thm "lessThan_Suc";
874 val lessThan_Suc_atMost = thm "lessThan_Suc_atMost";
875 val lessThan_def = thm "lessThan_def";
876 val lessThan_iff = thm "lessThan_iff";
877 val single_Diff_lessThan = thm "single_Diff_lessThan";
879 val bounded_nat_set_is_finite = thm "bounded_nat_set_is_finite";
880 val finite_atMost = thm "finite_atMost";
881 val finite_lessThan = thm "finite_lessThan";