src/HOL/SetInterval.thy
 author nipkow Wed Aug 26 16:13:19 2009 +0200 (2009-08-26) changeset 32408 a1a85b0a26f7 parent 32400 6c62363cf0d7 child 32436 10cd49e0c067 permissions -rw-r--r--
new interval lemma
CCS example for predicate compiler
1 (*  Title:      HOL/SetInterval.thy
2     Author:     Tobias Nipkow and Clemens Ballarin
6 lessThan, greaterThan, atLeast, atMost and two-sided intervals
7 *)
9 header {* Set intervals *}
11 theory SetInterval
12 imports Int
13 begin
15 context ord
16 begin
17 definition
18   lessThan    :: "'a => 'a set"	("(1{..<_})") where
19   "{..<u} == {x. x < u}"
21 definition
22   atMost      :: "'a => 'a set"	("(1{.._})") where
23   "{..u} == {x. x \<le> u}"
25 definition
26   greaterThan :: "'a => 'a set"	("(1{_<..})") where
27   "{l<..} == {x. l<x}"
29 definition
30   atLeast     :: "'a => 'a set"	("(1{_..})") where
31   "{l..} == {x. l\<le>x}"
33 definition
34   greaterThanLessThan :: "'a => 'a => 'a set"  ("(1{_<..<_})") where
35   "{l<..<u} == {l<..} Int {..<u}"
37 definition
38   atLeastLessThan :: "'a => 'a => 'a set"      ("(1{_..<_})") where
39   "{l..<u} == {l..} Int {..<u}"
41 definition
42   greaterThanAtMost :: "'a => 'a => 'a set"    ("(1{_<.._})") where
43   "{l<..u} == {l<..} Int {..u}"
45 definition
46   atLeastAtMost :: "'a => 'a => 'a set"        ("(1{_.._})") where
47   "{l..u} == {l..} Int {..u}"
49 end
52 text{* A note of warning when using @{term"{..<n}"} on type @{typ
53 nat}: it is equivalent to @{term"{0::nat..<n}"} but some lemmas involving
54 @{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well. *}
56 syntax
57   "@UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3UN _<=_./ _)" 10)
58   "@UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3UN _<_./ _)" 10)
59   "@INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3INT _<=_./ _)" 10)
60   "@INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3INT _<_./ _)" 10)
62 syntax (xsymbols)
63   "@UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _\<le>_./ _)" 10)
64   "@UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _<_./ _)" 10)
65   "@INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _\<le>_./ _)" 10)
66   "@INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" 10)
68 syntax (latex output)
69   "@UNION_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ \<le> _)/ _)" 10)
70   "@UNION_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ < _)/ _)" 10)
71   "@INTER_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ \<le> _)/ _)" 10)
72   "@INTER_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ < _)/ _)" 10)
74 translations
75   "UN i<=n. A"  == "UN i:{..n}. A"
76   "UN i<n. A"   == "UN i:{..<n}. A"
77   "INT i<=n. A" == "INT i:{..n}. A"
78   "INT i<n. A"  == "INT i:{..<n}. A"
81 subsection {* Various equivalences *}
83 lemma (in ord) lessThan_iff [iff]: "(i: lessThan k) = (i<k)"
86 lemma Compl_lessThan [simp]:
87     "!!k:: 'a::linorder. -lessThan k = atLeast k"
88 apply (auto simp add: lessThan_def atLeast_def)
89 done
91 lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"
92 by auto
94 lemma (in ord) greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"
97 lemma Compl_greaterThan [simp]:
98     "!!k:: 'a::linorder. -greaterThan k = atMost k"
99   by (auto simp add: greaterThan_def atMost_def)
101 lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
102 apply (subst Compl_greaterThan [symmetric])
103 apply (rule double_complement)
104 done
106 lemma (in ord) atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
109 lemma Compl_atLeast [simp]:
110     "!!k:: 'a::linorder. -atLeast k = lessThan k"
111   by (auto simp add: lessThan_def atLeast_def)
113 lemma (in ord) atMost_iff [iff]: "(i: atMost k) = (i<=k)"
116 lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
117 by (blast intro: order_antisym)
120 subsection {* Logical Equivalences for Set Inclusion and Equality *}
122 lemma atLeast_subset_iff [iff]:
123      "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::order))"
124 by (blast intro: order_trans)
126 lemma atLeast_eq_iff [iff]:
127      "(atLeast x = atLeast y) = (x = (y::'a::linorder))"
128 by (blast intro: order_antisym order_trans)
130 lemma greaterThan_subset_iff [iff]:
131      "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
132 apply (auto simp add: greaterThan_def)
133  apply (subst linorder_not_less [symmetric], blast)
134 done
136 lemma greaterThan_eq_iff [iff]:
137      "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
138 apply (rule iffI)
139  apply (erule equalityE)
140  apply simp_all
141 done
143 lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::order))"
144 by (blast intro: order_trans)
146 lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))"
147 by (blast intro: order_antisym order_trans)
149 lemma lessThan_subset_iff [iff]:
150      "(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))"
151 apply (auto simp add: lessThan_def)
152  apply (subst linorder_not_less [symmetric], blast)
153 done
155 lemma lessThan_eq_iff [iff]:
156      "(lessThan x = lessThan y) = (x = (y::'a::linorder))"
157 apply (rule iffI)
158  apply (erule equalityE)
159  apply simp_all
160 done
163 subsection {*Two-sided intervals*}
165 context ord
166 begin
168 lemma greaterThanLessThan_iff [simp,noatp]:
169   "(i : {l<..<u}) = (l < i & i < u)"
172 lemma atLeastLessThan_iff [simp,noatp]:
173   "(i : {l..<u}) = (l <= i & i < u)"
176 lemma greaterThanAtMost_iff [simp,noatp]:
177   "(i : {l<..u}) = (l < i & i <= u)"
180 lemma atLeastAtMost_iff [simp,noatp]:
181   "(i : {l..u}) = (l <= i & i <= u)"
184 text {* The above four lemmas could be declared as iffs.
185   If we do so, a call to blast in Hyperreal/Star.ML, lemma @{text STAR_Int}
186   seems to take forever (more than one hour). *}
187 end
189 subsubsection{* Emptyness, singletons, subset *}
191 context order
192 begin
194 lemma atLeastatMost_empty[simp]:
195   "b < a \<Longrightarrow> {a..b} = {}"
196 by(auto simp: atLeastAtMost_def atLeast_def atMost_def)
198 lemma atLeastatMost_empty_iff[simp]:
199   "{a..b} = {} \<longleftrightarrow> (~ a <= b)"
200 by auto (blast intro: order_trans)
202 lemma atLeastatMost_empty_iff2[simp]:
203   "{} = {a..b} \<longleftrightarrow> (~ a <= b)"
204 by auto (blast intro: order_trans)
206 lemma atLeastLessThan_empty[simp]:
207   "b <= a \<Longrightarrow> {a..<b} = {}"
208 by(auto simp: atLeastLessThan_def)
210 lemma atLeastLessThan_empty_iff[simp]:
211   "{a..<b} = {} \<longleftrightarrow> (~ a < b)"
212 by auto (blast intro: le_less_trans)
214 lemma atLeastLessThan_empty_iff2[simp]:
215   "{} = {a..<b} \<longleftrightarrow> (~ a < b)"
216 by auto (blast intro: le_less_trans)
218 lemma greaterThanAtMost_empty[simp]: "l \<le> k ==> {k<..l} = {}"
219 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)
221 lemma greaterThanAtMost_empty_iff[simp]: "{k<..l} = {} \<longleftrightarrow> ~ k < l"
222 by auto (blast intro: less_le_trans)
224 lemma greaterThanAtMost_empty_iff2[simp]: "{} = {k<..l} \<longleftrightarrow> ~ k < l"
225 by auto (blast intro: less_le_trans)
227 lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..<l} = {}"
228 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)
230 lemma atLeastAtMost_singleton [simp]: "{a..a} = {a}"
231 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)
233 lemma atLeastatMost_subset_iff[simp]:
234   "{a..b} <= {c..d} \<longleftrightarrow> (~ a <= b) | c <= a & b <= d"
235 unfolding atLeastAtMost_def atLeast_def atMost_def
236 by (blast intro: order_trans)
238 lemma atLeastatMost_psubset_iff:
239   "{a..b} < {c..d} \<longleftrightarrow>
240    ((~ a <= b) | c <= a & b <= d & (c < a | b < d))  &  c <= d"
241 by(simp add: psubset_eq expand_set_eq less_le_not_le)(blast intro: order_trans)
243 end
245 lemma (in linorder) atLeastLessThan_subset_iff:
246   "{a..<b} <= {c..<d} \<Longrightarrow> b <= a | c<=a & b<=d"
247 apply (auto simp:subset_eq Ball_def)
248 apply(frule_tac x=a in spec)
249 apply(erule_tac x=d in allE)
251 done
253 subsection {* Intervals of natural numbers *}
255 subsubsection {* The Constant @{term lessThan} *}
257 lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
260 lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
261 by (simp add: lessThan_def less_Suc_eq, blast)
263 lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
264 by (simp add: lessThan_def atMost_def less_Suc_eq_le)
266 lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
267 by blast
269 subsubsection {* The Constant @{term greaterThan} *}
271 lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
273 apply (blast dest: gr0_conv_Suc [THEN iffD1])
274 done
276 lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
278 apply (auto elim: linorder_neqE)
279 done
281 lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
282 by blast
284 subsubsection {* The Constant @{term atLeast} *}
286 lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
287 by (unfold atLeast_def UNIV_def, simp)
289 lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
292 apply (simp add: order_le_less, blast)
293 done
295 lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
296   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
298 lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
299 by blast
301 subsubsection {* The Constant @{term atMost} *}
303 lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
306 lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
308 apply (simp add: less_Suc_eq order_le_less, blast)
309 done
311 lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
312 by blast
314 subsubsection {* The Constant @{term atLeastLessThan} *}
316 text{*The orientation of the following 2 rules is tricky. The lhs is
317 defined in terms of the rhs.  Hence the chosen orientation makes sense
318 in this theory --- the reverse orientation complicates proofs (eg
319 nontermination). But outside, when the definition of the lhs is rarely
320 used, the opposite orientation seems preferable because it reduces a
321 specific concept to a more general one. *}
323 lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
326 lemma atLeast0AtMost: "{0..n::nat} = {..n}"
329 declare atLeast0LessThan[symmetric, code_unfold]
330         atLeast0AtMost[symmetric, code_unfold]
332 lemma atLeastLessThan0: "{m..<0::nat} = {}"
335 subsubsection {* Intervals of nats with @{term Suc} *}
337 text{*Not a simprule because the RHS is too messy.*}
338 lemma atLeastLessThanSuc:
339     "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
340 by (auto simp add: atLeastLessThan_def)
342 lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
343 by (auto simp add: atLeastLessThan_def)
344 (*
345 lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
346 by (induct k, simp_all add: atLeastLessThanSuc)
348 lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
349 by (auto simp add: atLeastLessThan_def)
350 *)
351 lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
352   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
354 lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
355   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
356     greaterThanAtMost_def)
358 lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
359   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
360     greaterThanLessThan_def)
362 lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
363 by (auto simp add: atLeastAtMost_def)
365 subsubsection {* Image *}
368   "(%n::nat. n+k)  {i..j} = {i+k..j+k}" (is "?A = ?B")
369 proof
370   show "?A \<subseteq> ?B" by auto
371 next
372   show "?B \<subseteq> ?A"
373   proof
374     fix n assume a: "n : ?B"
375     hence "n - k : {i..j}" by auto
376     moreover have "n = (n - k) + k" using a by auto
377     ultimately show "n : ?A" by blast
378   qed
379 qed
382   "(%n::nat. n+k)  {i..<j} = {i+k..<j+k}" (is "?A = ?B")
383 proof
384   show "?A \<subseteq> ?B" by auto
385 next
386   show "?B \<subseteq> ?A"
387   proof
388     fix n assume a: "n : ?B"
389     hence "n - k : {i..<j}" by auto
390     moreover have "n = (n - k) + k" using a by auto
391     ultimately show "n : ?A" by blast
392   qed
393 qed
395 corollary image_Suc_atLeastAtMost[simp]:
396   "Suc  {i..j} = {Suc i..Suc j}"
397 using image_add_atLeastAtMost[where k="Suc 0"] by simp
399 corollary image_Suc_atLeastLessThan[simp]:
400   "Suc  {i..<j} = {Suc i..<Suc j}"
401 using image_add_atLeastLessThan[where k="Suc 0"] by simp
404     "(%x. x + (l::int))  {0..<u-l} = {l..<u}"
405   apply (auto simp add: image_def)
406   apply (rule_tac x = "x - l" in bexI)
407   apply auto
408   done
411 subsubsection {* Finiteness *}
413 lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
414   by (induct k) (simp_all add: lessThan_Suc)
416 lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
417   by (induct k) (simp_all add: atMost_Suc)
419 lemma finite_greaterThanLessThan [iff]:
420   fixes l :: nat shows "finite {l<..<u}"
423 lemma finite_atLeastLessThan [iff]:
424   fixes l :: nat shows "finite {l..<u}"
427 lemma finite_greaterThanAtMost [iff]:
428   fixes l :: nat shows "finite {l<..u}"
431 lemma finite_atLeastAtMost [iff]:
432   fixes l :: nat shows "finite {l..u}"
435 text {* A bounded set of natural numbers is finite. *}
436 lemma bounded_nat_set_is_finite:
437   "(ALL i:N. i < (n::nat)) ==> finite N"
438 apply (rule finite_subset)
439  apply (rule_tac [2] finite_lessThan, auto)
440 done
442 text {* A set of natural numbers is finite iff it is bounded. *}
443 lemma finite_nat_set_iff_bounded:
444   "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")
445 proof
446   assume f:?F  show ?B
447     using Max_ge[OF ?F, simplified less_Suc_eq_le[symmetric]] by blast
448 next
449   assume ?B show ?F using ?B by(blast intro:bounded_nat_set_is_finite)
450 qed
452 lemma finite_nat_set_iff_bounded_le:
453   "finite(N::nat set) = (EX m. ALL n:N. n<=m)"
455 apply(blast dest:less_imp_le_nat le_imp_less_Suc)
456 done
458 lemma finite_less_ub:
459      "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
460 by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
462 text{* Any subset of an interval of natural numbers the size of the
463 subset is exactly that interval. *}
465 lemma subset_card_intvl_is_intvl:
466   "A <= {k..<k+card A} \<Longrightarrow> A = {k..<k+card A}" (is "PROP ?P")
467 proof cases
468   assume "finite A"
469   thus "PROP ?P"
470   proof(induct A rule:finite_linorder_max_induct)
471     case empty thus ?case by auto
472   next
473     case (insert A b)
474     moreover hence "b ~: A" by auto
475     moreover have "A <= {k..<k+card A}" and "b = k+card A"
476       using b ~: A insert by fastsimp+
477     ultimately show ?case by auto
478   qed
479 next
480   assume "~finite A" thus "PROP ?P" by simp
481 qed
484 subsubsection {* Cardinality *}
486 lemma card_lessThan [simp]: "card {..<u} = u"
487   by (induct u, simp_all add: lessThan_Suc)
489 lemma card_atMost [simp]: "card {..u} = Suc u"
490   by (simp add: lessThan_Suc_atMost [THEN sym])
492 lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
493   apply (subgoal_tac "card {l..<u} = card {..<u-l}")
494   apply (erule ssubst, rule card_lessThan)
495   apply (subgoal_tac "(%x. x + l)  {..<u-l} = {l..<u}")
496   apply (erule subst)
497   apply (rule card_image)
499   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
500   apply (rule_tac x = "x - l" in exI)
501   apply arith
502   done
504 lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
505   by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
507 lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
508   by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
510 lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
511   by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
513 lemma ex_bij_betw_nat_finite:
514   "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
515 apply(drule finite_imp_nat_seg_image_inj_on)
516 apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
517 done
519 lemma ex_bij_betw_finite_nat:
520   "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
521 by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
523 lemma finite_same_card_bij:
524   "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"
525 apply(drule ex_bij_betw_finite_nat)
526 apply(drule ex_bij_betw_nat_finite)
527 apply(auto intro!:bij_betw_trans)
528 done
530 lemma ex_bij_betw_nat_finite_1:
531   "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
532 by (rule finite_same_card_bij) auto
535 subsection {* Intervals of integers *}
537 lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
538   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
540 lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
541   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
543 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
544     "{l+1..<u} = {l<..<u::int}"
545   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
547 subsubsection {* Finiteness *}
549 lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
550     {(0::int)..<u} = int  {..<nat u}"
551   apply (unfold image_def lessThan_def)
552   apply auto
553   apply (rule_tac x = "nat x" in exI)
554   apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
555   done
557 lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
558   apply (case_tac "0 \<le> u")
559   apply (subst image_atLeastZeroLessThan_int, assumption)
560   apply (rule finite_imageI)
561   apply auto
562   done
564 lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
565   apply (subgoal_tac "(%x. x + l)  {0..<u-l} = {l..<u}")
566   apply (erule subst)
567   apply (rule finite_imageI)
568   apply (rule finite_atLeastZeroLessThan_int)
570   done
572 lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
573   by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
575 lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
576   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
578 lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
579   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
582 subsubsection {* Cardinality *}
584 lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
585   apply (case_tac "0 \<le> u")
586   apply (subst image_atLeastZeroLessThan_int, assumption)
587   apply (subst card_image)
588   apply (auto simp add: inj_on_def)
589   done
591 lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
592   apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
593   apply (erule ssubst, rule card_atLeastZeroLessThan_int)
594   apply (subgoal_tac "(%x. x + l)  {0..<u-l} = {l..<u}")
595   apply (erule subst)
596   apply (rule card_image)
599   done
601 lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
602 apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
603 apply (auto simp add: algebra_simps)
604 done
606 lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
607 by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
609 lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
610 by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
612 lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
613 proof -
614   have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
615   with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
616 qed
618 lemma card_less:
619 assumes zero_in_M: "0 \<in> M"
620 shows "card {k \<in> M. k < Suc i} \<noteq> 0"
621 proof -
622   from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
623   with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
624 qed
626 lemma card_less_Suc2: "0 \<notin> M \<Longrightarrow> card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"
627 apply (rule card_bij_eq [of "Suc" _ _ "\<lambda>x. x - Suc 0"])
628 apply simp
629 apply fastsimp
630 apply auto
631 apply (rule inj_on_diff_nat)
632 apply auto
633 apply (case_tac x)
634 apply auto
635 apply (case_tac xa)
636 apply auto
637 apply (case_tac xa)
638 apply auto
639 done
641 lemma card_less_Suc:
642   assumes zero_in_M: "0 \<in> M"
643     shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
644 proof -
645   from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp
646   hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"
647     by (auto simp only: insert_Diff)
648   have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto
649   from finite_M_bounded_by_nat[of "\<lambda>x. x \<in> M" "Suc i"] have "Suc (card {k. Suc k \<in> M \<and> k < i}) = card (insert 0 ({k \<in> M. k < Suc i} - {0}))"
650     apply (subst card_insert)
651     apply simp_all
652     apply (subst b)
653     apply (subst card_less_Suc2[symmetric])
654     apply simp_all
655     done
656   with c show ?thesis by simp
657 qed
660 subsection {*Lemmas useful with the summation operator setsum*}
662 text {* For examples, see Algebra/poly/UnivPoly2.thy *}
664 subsubsection {* Disjoint Unions *}
666 text {* Singletons and open intervals *}
668 lemma ivl_disj_un_singleton:
669   "{l::'a::linorder} Un {l<..} = {l..}"
670   "{..<u} Un {u::'a::linorder} = {..u}"
671   "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
672   "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
673   "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
674   "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
675 by auto
677 text {* One- and two-sided intervals *}
679 lemma ivl_disj_un_one:
680   "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
681   "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
682   "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
683   "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
684   "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
685   "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
686   "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
687   "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
688 by auto
690 text {* Two- and two-sided intervals *}
692 lemma ivl_disj_un_two:
693   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
694   "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
695   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
696   "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
697   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
698   "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
699   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
700   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
701 by auto
703 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
705 subsubsection {* Disjoint Intersections *}
707 text {* Singletons and open intervals *}
709 lemma ivl_disj_int_singleton:
710   "{l::'a::order} Int {l<..} = {}"
711   "{..<u} Int {u} = {}"
712   "{l} Int {l<..<u} = {}"
713   "{l<..<u} Int {u} = {}"
714   "{l} Int {l<..u} = {}"
715   "{l..<u} Int {u} = {}"
716   by simp+
718 text {* One- and two-sided intervals *}
720 lemma ivl_disj_int_one:
721   "{..l::'a::order} Int {l<..<u} = {}"
722   "{..<l} Int {l..<u} = {}"
723   "{..l} Int {l<..u} = {}"
724   "{..<l} Int {l..u} = {}"
725   "{l<..u} Int {u<..} = {}"
726   "{l<..<u} Int {u..} = {}"
727   "{l..u} Int {u<..} = {}"
728   "{l..<u} Int {u..} = {}"
729   by auto
731 text {* Two- and two-sided intervals *}
733 lemma ivl_disj_int_two:
734   "{l::'a::order<..<m} Int {m..<u} = {}"
735   "{l<..m} Int {m<..<u} = {}"
736   "{l..<m} Int {m..<u} = {}"
737   "{l..m} Int {m<..<u} = {}"
738   "{l<..<m} Int {m..u} = {}"
739   "{l<..m} Int {m<..u} = {}"
740   "{l..<m} Int {m..u} = {}"
741   "{l..m} Int {m<..u} = {}"
742   by auto
744 lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
746 subsubsection {* Some Differences *}
748 lemma ivl_diff[simp]:
749  "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
750 by(auto)
753 subsubsection {* Some Subset Conditions *}
755 lemma ivl_subset [simp,noatp]:
756  "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
757 apply(auto simp:linorder_not_le)
758 apply(rule ccontr)
759 apply(insert linorder_le_less_linear[of i n])
760 apply(clarsimp simp:linorder_not_le)
761 apply(fastsimp)
762 done
765 subsection {* Summation indexed over intervals *}
767 syntax
768   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
769   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
770   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
771   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
772 syntax (xsymbols)
773   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
774   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
775   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
776   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
777 syntax (HTML output)
778   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
779   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
780   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
781   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
782 syntax (latex_sum output)
783   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
784  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
785   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
786  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
787   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
788  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
789   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
790  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
792 translations
793   "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"
794   "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"
795   "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"
796   "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"
798 text{* The above introduces some pretty alternative syntaxes for
799 summation over intervals:
800 \begin{center}
801 \begin{tabular}{lll}
802 Old & New & \LaTeX\\
803 @{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\
804 @{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\
805 @{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
806 @{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
807 \end{tabular}
808 \end{center}
809 The left column shows the term before introduction of the new syntax,
810 the middle column shows the new (default) syntax, and the right column
811 shows a special syntax. The latter is only meaningful for latex output
812 and has to be activated explicitly by setting the print mode to
813 @{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
814 antiquotations). It is not the default \LaTeX\ output because it only
815 works well with italic-style formulae, not tt-style.
817 Note that for uniformity on @{typ nat} it is better to use
818 @{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
819 not provide all lemmas available for @{term"{m..<n}"} also in the
820 special form for @{term"{..<n}"}. *}
822 text{* This congruence rule should be used for sums over intervals as
823 the standard theorem @{text[source]setsum_cong} does not work well
824 with the simplifier who adds the unsimplified premise @{term"x:B"} to
825 the context. *}
827 lemma setsum_ivl_cong:
828  "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
829  setsum f {a..<b} = setsum g {c..<d}"
830 by(rule setsum_cong, simp_all)
832 (* FIXME why are the following simp rules but the corresponding eqns
833 on intervals are not? *)
835 lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
838 lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
841 lemma setsum_cl_ivl_Suc[simp]:
842   "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
845 lemma setsum_op_ivl_Suc[simp]:
846   "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
848 (*
849 lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
850     (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
852 *)
855   fixes n :: nat
856   assumes mn: "m <= n"
857   shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
858 proof -
859   from mn
860   have "{m..n} = {m} \<union> {m<..n}"
861     by (auto intro: ivl_disj_un_singleton)
862   hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
864   also have "\<dots> = ?rhs" by simp
865   finally show ?thesis .
866 qed
869   "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"
873   "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"
874 apply(insert setsum_head_Suc[of m "n - Suc 0" f])
875 apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)
876 done
878 lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"
879   shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
880 proof-
881   have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using m \<le> n+1 by auto
882   thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint
883     atLeastSucAtMost_greaterThanAtMost)
884 qed
886 lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
887   setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
888 by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
890 lemma setsum_diff_nat_ivl:
891 fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
892 shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
893   setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
894 using setsum_add_nat_ivl [of m n p f,symmetric]
896 done
898 lemma setsum_natinterval_difff:
899   fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
900   shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
901           (if m <= n then f m - f(n + 1) else 0)"
902 by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)
904 lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]
906 lemma setsum_setsum_restrict:
907   "finite S \<Longrightarrow> finite T \<Longrightarrow> setsum (\<lambda>x. setsum (\<lambda>y. f x y) {y. y\<in> T \<and> R x y}) S = setsum (\<lambda>y. setsum (\<lambda>x. f x y) {x. x \<in> S \<and> R x y}) T"
908   by (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)
909      (rule setsum_commute)
911 lemma setsum_image_gen: assumes fS: "finite S"
912   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f  S)"
913 proof-
914   { fix x assume "x \<in> S" then have "{y. y\<in> fS \<and> f x = y} = {f x}" by auto }
915   hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> fS \<and> f x = y}) S"
916     by simp
917   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
918     by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])
919   finally show ?thesis .
920 qed
922 lemma setsum_multicount_gen:
923   assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
924   shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")
925 proof-
926   have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto
927   also have "\<dots> = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]
928     using assms(3) by auto
929   finally show ?thesis .
930 qed
932 lemma setsum_multicount:
933   assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
934   shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
935 proof-
936   have "?l = setsum (\<lambda>i. k) T" by(rule setsum_multicount_gen)(auto simp:assms)
937   also have "\<dots> = ?r" by(simp add: setsum_constant mult_commute)
938   finally show ?thesis by auto
939 qed
942 subsection{* Shifting bounds *}
944 lemma setsum_shift_bounds_nat_ivl:
945   "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
946 by (induct "n", auto simp:atLeastLessThanSuc)
948 lemma setsum_shift_bounds_cl_nat_ivl:
949   "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
950 apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
952 done
954 corollary setsum_shift_bounds_cl_Suc_ivl:
955   "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
956 by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
958 corollary setsum_shift_bounds_Suc_ivl:
959   "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
960 by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])
962 lemma setsum_shift_lb_Suc0_0:
963   "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"
966 lemma setsum_shift_lb_Suc0_0_upt:
967   "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"
968 apply(cases k)apply simp
970 done
972 subsection {* The formula for geometric sums *}
974 lemma geometric_sum:
975   "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) =
976   (x ^ n - 1) / (x - 1::'a::{field})"
977 by (induct "n") (simp_all add:field_simps power_Suc)
979 subsection {* The formula for arithmetic sums *}
981 lemma gauss_sum:
982   "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =
983    of_nat n*((of_nat n)+1)"
984 proof (induct n)
985   case 0
986   show ?case by simp
987 next
988   case (Suc n)
989   then show ?case by (simp add: algebra_simps)
990 qed
992 theorem arith_series_general:
993   "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
994   of_nat n * (a + (a + of_nat(n - 1)*d))"
995 proof cases
996   assume ngt1: "n > 1"
997   let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
998   have
999     "(\<Sum>i\<in>{..<n}. a+?I i*d) =
1000      ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
1002   also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
1003   also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
1004     unfolding One_nat_def
1005     by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)
1006   also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"
1007     by (simp add: left_distrib right_distrib)
1008   also from ngt1 have "{1..<n} = {1..n - 1}"
1009     by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
1010   also from ngt1
1011   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)"
1012     by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)
1014   finally show ?thesis by (simp add: algebra_simps)
1015 next
1016   assume "\<not>(n > 1)"
1017   hence "n = 1 \<or> n = 0" by auto
1018   thus ?thesis by (auto simp: algebra_simps)
1019 qed
1021 lemma arith_series_nat:
1022   "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
1023 proof -
1024   have
1025     "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
1026     of_nat(n) * (a + (a + of_nat(n - 1)*d))"
1027     by (rule arith_series_general)
1028   thus ?thesis
1029     unfolding One_nat_def by (auto simp add: of_nat_id)
1030 qed
1032 lemma arith_series_int:
1033   "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
1034   of_nat n * (a + (a + of_nat(n - 1)*d))"
1035 proof -
1036   have
1037     "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
1038     of_nat(n) * (a + (a + of_nat(n - 1)*d))"
1039     by (rule arith_series_general)
1040   thus ?thesis by simp
1041 qed
1043 lemma sum_diff_distrib:
1044   fixes P::"nat\<Rightarrow>nat"
1045   shows
1046   "\<forall>x. Q x \<le> P x  \<Longrightarrow>
1047   (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
1048 proof (induct n)
1049   case 0 show ?case by simp
1050 next
1051   case (Suc n)
1053   let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
1054   let ?rhs = "\<Sum>x<n. P x - Q x"
1056   from Suc have "?lhs = ?rhs" by simp
1057   moreover
1058   from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
1059   moreover
1060   from Suc have
1061     "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
1062     by (subst diff_diff_left[symmetric],
1064        (auto simp: diff_add_assoc2 intro: setsum_mono)
1065   ultimately
1066   show ?case by simp
1067 qed
1069 subsection {* Products indexed over intervals *}
1071 syntax
1072   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)
1073   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)
1074   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)
1075   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)
1076 syntax (xsymbols)
1077   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
1078   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
1079   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
1080   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
1081 syntax (HTML output)
1082   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
1083   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
1084   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
1085   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
1086 syntax (latex_prod output)
1087   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
1088  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
1089   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
1090  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
1091   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
1092  ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
1093   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
1094  ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
1096 translations
1097   "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"
1098   "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"
1099   "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"
1100   "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"
1102 end