src/HOL/SetInterval.thy
author kleing
Sun Feb 19 13:21:32 2006 +0100 (2006-02-19)
changeset 19106 6e6b5b1fdc06
parent 19022 0e6ec4fd204c
child 19376 529b735edbf2
permissions -rw-r--r--
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
* added Complex/ex/ASeries_Complex (instantiation of the above for reals)
* added Complex/ex/HarmonicSeries (should really be in something like Complex/Library)

(these are contributions by Benjamin Porter, numbers 68 and 34 of
http://www.cs.ru.nl/~freek/100/)
     1 (*  Title:      HOL/SetInterval.thy
     2     ID:         $Id$
     3     Author:     Tobias Nipkow and Clemens Ballarin
     4                 Additions by Jeremy Avigad in March 2004
     5     Copyright   2000  TU Muenchen
     6 
     7 lessThan, greaterThan, atLeast, atMost and two-sided intervals
     8 *)
     9 
    10 header {* Set intervals *}
    11 
    12 theory SetInterval
    13 imports IntArith
    14 begin
    15 
    16 constdefs
    17   lessThan    :: "('a::ord) => 'a set"	("(1{..<_})")
    18   "{..<u} == {x. x<u}"
    19 
    20   atMost      :: "('a::ord) => 'a set"	("(1{.._})")
    21   "{..u} == {x. x<=u}"
    22 
    23   greaterThan :: "('a::ord) => 'a set"	("(1{_<..})")
    24   "{l<..} == {x. l<x}"
    25 
    26   atLeast     :: "('a::ord) => 'a set"	("(1{_..})")
    27   "{l..} == {x. l<=x}"
    28 
    29   greaterThanLessThan :: "['a::ord, 'a] => 'a set"  ("(1{_<..<_})")
    30   "{l<..<u} == {l<..} Int {..<u}"
    31 
    32   atLeastLessThan :: "['a::ord, 'a] => 'a set"      ("(1{_..<_})")
    33   "{l..<u} == {l..} Int {..<u}"
    34 
    35   greaterThanAtMost :: "['a::ord, 'a] => 'a set"    ("(1{_<.._})")
    36   "{l<..u} == {l<..} Int {..u}"
    37 
    38   atLeastAtMost :: "['a::ord, 'a] => 'a set"        ("(1{_.._})")
    39   "{l..u} == {l..} Int {..u}"
    40 
    41 (* Old syntax, will disappear! *)
    42 syntax
    43   "_lessThan"    :: "('a::ord) => 'a set"	("(1{.._'(})")
    44   "_greaterThan" :: "('a::ord) => 'a set"	("(1{')_..})")
    45   "_greaterThanLessThan" :: "['a::ord, 'a] => 'a set"  ("(1{')_.._'(})")
    46   "_atLeastLessThan" :: "['a::ord, 'a] => 'a set"      ("(1{_.._'(})")
    47   "_greaterThanAtMost" :: "['a::ord, 'a] => 'a set"    ("(1{')_.._})")
    48 translations
    49   "{..m(}" => "{..<m}"
    50   "{)m..}" => "{m<..}"
    51   "{)m..n(}" => "{m<..<n}"
    52   "{m..n(}" => "{m..<n}"
    53   "{)m..n}" => "{m<..n}"
    54 
    55 
    56 text{* A note of warning when using @{term"{..<n}"} on type @{typ
    57 nat}: it is equivalent to @{term"{0::nat..<n}"} but some lemmas involving
    58 @{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well. *}
    59 
    60 syntax
    61   "@UNION_le"   :: "nat => nat => 'b set => 'b set"       ("(3UN _<=_./ _)" 10)
    62   "@UNION_less" :: "nat => nat => 'b set => 'b set"       ("(3UN _<_./ _)" 10)
    63   "@INTER_le"   :: "nat => nat => 'b set => 'b set"       ("(3INT _<=_./ _)" 10)
    64   "@INTER_less" :: "nat => nat => 'b set => 'b set"       ("(3INT _<_./ _)" 10)
    65 
    66 syntax (input)
    67   "@UNION_le"   :: "nat => nat => 'b set => 'b set"       ("(3\<Union> _\<le>_./ _)" 10)
    68   "@UNION_less" :: "nat => nat => 'b set => 'b set"       ("(3\<Union> _<_./ _)" 10)
    69   "@INTER_le"   :: "nat => nat => 'b set => 'b set"       ("(3\<Inter> _\<le>_./ _)" 10)
    70   "@INTER_less" :: "nat => nat => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" 10)
    71 
    72 syntax (xsymbols)
    73   "@UNION_le"   :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Union>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)
    74   "@UNION_less" :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Union>(00\<^bsub>_ < _\<^esub>)/ _)" 10)
    75   "@INTER_le"   :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Inter>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)
    76   "@INTER_less" :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Inter>(00\<^bsub>_ < _\<^esub>)/ _)" 10)
    77 
    78 translations
    79   "UN i<=n. A"  == "UN i:{..n}. A"
    80   "UN i<n. A"   == "UN i:{..<n}. A"
    81   "INT i<=n. A" == "INT i:{..n}. A"
    82   "INT i<n. A"  == "INT i:{..<n}. A"
    83 
    84 
    85 subsection {* Various equivalences *}
    86 
    87 lemma lessThan_iff [iff]: "(i: lessThan k) = (i<k)"
    88 by (simp add: lessThan_def)
    89 
    90 lemma Compl_lessThan [simp]:
    91     "!!k:: 'a::linorder. -lessThan k = atLeast k"
    92 apply (auto simp add: lessThan_def atLeast_def)
    93 done
    94 
    95 lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"
    96 by auto
    97 
    98 lemma greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"
    99 by (simp add: greaterThan_def)
   100 
   101 lemma Compl_greaterThan [simp]:
   102     "!!k:: 'a::linorder. -greaterThan k = atMost k"
   103 apply (simp add: greaterThan_def atMost_def le_def, auto)
   104 done
   105 
   106 lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
   107 apply (subst Compl_greaterThan [symmetric])
   108 apply (rule double_complement)
   109 done
   110 
   111 lemma atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
   112 by (simp add: atLeast_def)
   113 
   114 lemma Compl_atLeast [simp]:
   115     "!!k:: 'a::linorder. -atLeast k = lessThan k"
   116 apply (simp add: lessThan_def atLeast_def le_def, auto)
   117 done
   118 
   119 lemma atMost_iff [iff]: "(i: atMost k) = (i<=k)"
   120 by (simp add: atMost_def)
   121 
   122 lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
   123 by (blast intro: order_antisym)
   124 
   125 
   126 subsection {* Logical Equivalences for Set Inclusion and Equality *}
   127 
   128 lemma atLeast_subset_iff [iff]:
   129      "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::order))"
   130 by (blast intro: order_trans)
   131 
   132 lemma atLeast_eq_iff [iff]:
   133      "(atLeast x = atLeast y) = (x = (y::'a::linorder))"
   134 by (blast intro: order_antisym order_trans)
   135 
   136 lemma greaterThan_subset_iff [iff]:
   137      "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
   138 apply (auto simp add: greaterThan_def)
   139  apply (subst linorder_not_less [symmetric], blast)
   140 done
   141 
   142 lemma greaterThan_eq_iff [iff]:
   143      "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
   144 apply (rule iffI)
   145  apply (erule equalityE)
   146  apply (simp_all add: greaterThan_subset_iff)
   147 done
   148 
   149 lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::order))"
   150 by (blast intro: order_trans)
   151 
   152 lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))"
   153 by (blast intro: order_antisym order_trans)
   154 
   155 lemma lessThan_subset_iff [iff]:
   156      "(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))"
   157 apply (auto simp add: lessThan_def)
   158  apply (subst linorder_not_less [symmetric], blast)
   159 done
   160 
   161 lemma lessThan_eq_iff [iff]:
   162      "(lessThan x = lessThan y) = (x = (y::'a::linorder))"
   163 apply (rule iffI)
   164  apply (erule equalityE)
   165  apply (simp_all add: lessThan_subset_iff)
   166 done
   167 
   168 
   169 subsection {*Two-sided intervals*}
   170 
   171 lemma greaterThanLessThan_iff [simp]:
   172   "(i : {l<..<u}) = (l < i & i < u)"
   173 by (simp add: greaterThanLessThan_def)
   174 
   175 lemma atLeastLessThan_iff [simp]:
   176   "(i : {l..<u}) = (l <= i & i < u)"
   177 by (simp add: atLeastLessThan_def)
   178 
   179 lemma greaterThanAtMost_iff [simp]:
   180   "(i : {l<..u}) = (l < i & i <= u)"
   181 by (simp add: greaterThanAtMost_def)
   182 
   183 lemma atLeastAtMost_iff [simp]:
   184   "(i : {l..u}) = (l <= i & i <= u)"
   185 by (simp add: atLeastAtMost_def)
   186 
   187 text {* The above four lemmas could be declared as iffs.
   188   If we do so, a call to blast in Hyperreal/Star.ML, lemma @{text STAR_Int}
   189   seems to take forever (more than one hour). *}
   190 
   191 subsubsection{* Emptyness and singletons *}
   192 
   193 lemma atLeastAtMost_empty [simp]: "n < m ==> {m::'a::order..n} = {}";
   194   by (auto simp add: atLeastAtMost_def atMost_def atLeast_def);
   195 
   196 lemma atLeastLessThan_empty[simp]: "n \<le> m ==> {m..<n::'a::order} = {}"
   197 by (auto simp add: atLeastLessThan_def)
   198 
   199 lemma greaterThanAtMost_empty[simp]:"l \<le> k ==> {k<..(l::'a::order)} = {}"
   200 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)
   201 
   202 lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..(l::'a::order)} = {}"
   203 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)
   204 
   205 lemma atLeastAtMost_singleton [simp]: "{a::'a::order..a} = {a}";
   206 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def);
   207 
   208 subsection {* Intervals of natural numbers *}
   209 
   210 subsubsection {* The Constant @{term lessThan} *}
   211 
   212 lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
   213 by (simp add: lessThan_def)
   214 
   215 lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
   216 by (simp add: lessThan_def less_Suc_eq, blast)
   217 
   218 lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
   219 by (simp add: lessThan_def atMost_def less_Suc_eq_le)
   220 
   221 lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
   222 by blast
   223 
   224 subsubsection {* The Constant @{term greaterThan} *}
   225 
   226 lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
   227 apply (simp add: greaterThan_def)
   228 apply (blast dest: gr0_conv_Suc [THEN iffD1])
   229 done
   230 
   231 lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
   232 apply (simp add: greaterThan_def)
   233 apply (auto elim: linorder_neqE)
   234 done
   235 
   236 lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
   237 by blast
   238 
   239 subsubsection {* The Constant @{term atLeast} *}
   240 
   241 lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
   242 by (unfold atLeast_def UNIV_def, simp)
   243 
   244 lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
   245 apply (simp add: atLeast_def)
   246 apply (simp add: Suc_le_eq)
   247 apply (simp add: order_le_less, blast)
   248 done
   249 
   250 lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
   251   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
   252 
   253 lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
   254 by blast
   255 
   256 subsubsection {* The Constant @{term atMost} *}
   257 
   258 lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
   259 by (simp add: atMost_def)
   260 
   261 lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
   262 apply (simp add: atMost_def)
   263 apply (simp add: less_Suc_eq order_le_less, blast)
   264 done
   265 
   266 lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
   267 by blast
   268 
   269 subsubsection {* The Constant @{term atLeastLessThan} *}
   270 
   271 text{*But not a simprule because some concepts are better left in terms
   272   of @{term atLeastLessThan}*}
   273 lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
   274 by(simp add:lessThan_def atLeastLessThan_def)
   275 (*
   276 lemma atLeastLessThan0 [simp]: "{m..<0::nat} = {}"
   277 by (simp add: atLeastLessThan_def)
   278 *)
   279 subsubsection {* Intervals of nats with @{term Suc} *}
   280 
   281 text{*Not a simprule because the RHS is too messy.*}
   282 lemma atLeastLessThanSuc:
   283     "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
   284 by (auto simp add: atLeastLessThan_def)
   285 
   286 lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
   287 by (auto simp add: atLeastLessThan_def)
   288 (*
   289 lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
   290 by (induct k, simp_all add: atLeastLessThanSuc)
   291 
   292 lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
   293 by (auto simp add: atLeastLessThan_def)
   294 *)
   295 lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
   296   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
   297 
   298 lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
   299   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
   300     greaterThanAtMost_def)
   301 
   302 lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
   303   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
   304     greaterThanLessThan_def)
   305 
   306 lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
   307 by (auto simp add: atLeastAtMost_def)
   308 
   309 subsubsection {* Image *}
   310 
   311 lemma image_add_atLeastAtMost:
   312   "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
   313 proof
   314   show "?A \<subseteq> ?B" by auto
   315 next
   316   show "?B \<subseteq> ?A"
   317   proof
   318     fix n assume a: "n : ?B"
   319     hence "n - k : {i..j}" by auto arith+
   320     moreover have "n = (n - k) + k" using a by auto
   321     ultimately show "n : ?A" by blast
   322   qed
   323 qed
   324 
   325 lemma image_add_atLeastLessThan:
   326   "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
   327 proof
   328   show "?A \<subseteq> ?B" by auto
   329 next
   330   show "?B \<subseteq> ?A"
   331   proof
   332     fix n assume a: "n : ?B"
   333     hence "n - k : {i..<j}" by auto arith+
   334     moreover have "n = (n - k) + k" using a by auto
   335     ultimately show "n : ?A" by blast
   336   qed
   337 qed
   338 
   339 corollary image_Suc_atLeastAtMost[simp]:
   340   "Suc ` {i..j} = {Suc i..Suc j}"
   341 using image_add_atLeastAtMost[where k=1] by simp
   342 
   343 corollary image_Suc_atLeastLessThan[simp]:
   344   "Suc ` {i..<j} = {Suc i..<Suc j}"
   345 using image_add_atLeastLessThan[where k=1] by simp
   346 
   347 lemma image_add_int_atLeastLessThan:
   348     "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
   349   apply (auto simp add: image_def)
   350   apply (rule_tac x = "x - l" in bexI)
   351   apply auto
   352   done
   353 
   354 
   355 subsubsection {* Finiteness *}
   356 
   357 lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
   358   by (induct k) (simp_all add: lessThan_Suc)
   359 
   360 lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
   361   by (induct k) (simp_all add: atMost_Suc)
   362 
   363 lemma finite_greaterThanLessThan [iff]:
   364   fixes l :: nat shows "finite {l<..<u}"
   365 by (simp add: greaterThanLessThan_def)
   366 
   367 lemma finite_atLeastLessThan [iff]:
   368   fixes l :: nat shows "finite {l..<u}"
   369 by (simp add: atLeastLessThan_def)
   370 
   371 lemma finite_greaterThanAtMost [iff]:
   372   fixes l :: nat shows "finite {l<..u}"
   373 by (simp add: greaterThanAtMost_def)
   374 
   375 lemma finite_atLeastAtMost [iff]:
   376   fixes l :: nat shows "finite {l..u}"
   377 by (simp add: atLeastAtMost_def)
   378 
   379 lemma bounded_nat_set_is_finite:
   380     "(ALL i:N. i < (n::nat)) ==> finite N"
   381   -- {* A bounded set of natural numbers is finite. *}
   382   apply (rule finite_subset)
   383    apply (rule_tac [2] finite_lessThan, auto)
   384   done
   385 
   386 subsubsection {* Cardinality *}
   387 
   388 lemma card_lessThan [simp]: "card {..<u} = u"
   389   by (induct u, simp_all add: lessThan_Suc)
   390 
   391 lemma card_atMost [simp]: "card {..u} = Suc u"
   392   by (simp add: lessThan_Suc_atMost [THEN sym])
   393 
   394 lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
   395   apply (subgoal_tac "card {l..<u} = card {..<u-l}")
   396   apply (erule ssubst, rule card_lessThan)
   397   apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
   398   apply (erule subst)
   399   apply (rule card_image)
   400   apply (simp add: inj_on_def)
   401   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
   402   apply arith
   403   apply (rule_tac x = "x - l" in exI)
   404   apply arith
   405   done
   406 
   407 lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
   408   by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
   409 
   410 lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
   411   by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
   412 
   413 lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
   414   by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
   415 
   416 subsection {* Intervals of integers *}
   417 
   418 lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
   419   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
   420 
   421 lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
   422   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
   423 
   424 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
   425     "{l+1..<u} = {l<..<u::int}"
   426   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
   427 
   428 subsubsection {* Finiteness *}
   429 
   430 lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
   431     {(0::int)..<u} = int ` {..<nat u}"
   432   apply (unfold image_def lessThan_def)
   433   apply auto
   434   apply (rule_tac x = "nat x" in exI)
   435   apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
   436   done
   437 
   438 lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
   439   apply (case_tac "0 \<le> u")
   440   apply (subst image_atLeastZeroLessThan_int, assumption)
   441   apply (rule finite_imageI)
   442   apply auto
   443   done
   444 
   445 lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
   446   apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
   447   apply (erule subst)
   448   apply (rule finite_imageI)
   449   apply (rule finite_atLeastZeroLessThan_int)
   450   apply (rule image_add_int_atLeastLessThan)
   451   done
   452 
   453 lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
   454   by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
   455 
   456 lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
   457   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
   458 
   459 lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
   460   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
   461 
   462 subsubsection {* Cardinality *}
   463 
   464 lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
   465   apply (case_tac "0 \<le> u")
   466   apply (subst image_atLeastZeroLessThan_int, assumption)
   467   apply (subst card_image)
   468   apply (auto simp add: inj_on_def)
   469   done
   470 
   471 lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
   472   apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
   473   apply (erule ssubst, rule card_atLeastZeroLessThan_int)
   474   apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
   475   apply (erule subst)
   476   apply (rule card_image)
   477   apply (simp add: inj_on_def)
   478   apply (rule image_add_int_atLeastLessThan)
   479   done
   480 
   481 lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
   482   apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
   483   apply (auto simp add: compare_rls)
   484   done
   485 
   486 lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
   487   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
   488 
   489 lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
   490   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
   491 
   492 
   493 subsection {*Lemmas useful with the summation operator setsum*}
   494 
   495 text {* For examples, see Algebra/poly/UnivPoly2.thy *}
   496 
   497 subsubsection {* Disjoint Unions *}
   498 
   499 text {* Singletons and open intervals *}
   500 
   501 lemma ivl_disj_un_singleton:
   502   "{l::'a::linorder} Un {l<..} = {l..}"
   503   "{..<u} Un {u::'a::linorder} = {..u}"
   504   "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
   505   "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
   506   "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
   507   "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
   508 by auto
   509 
   510 text {* One- and two-sided intervals *}
   511 
   512 lemma ivl_disj_un_one:
   513   "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
   514   "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
   515   "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
   516   "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
   517   "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
   518   "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
   519   "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
   520   "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
   521 by auto
   522 
   523 text {* Two- and two-sided intervals *}
   524 
   525 lemma ivl_disj_un_two:
   526   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
   527   "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
   528   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
   529   "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
   530   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
   531   "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
   532   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
   533   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
   534 by auto
   535 
   536 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
   537 
   538 subsubsection {* Disjoint Intersections *}
   539 
   540 text {* Singletons and open intervals *}
   541 
   542 lemma ivl_disj_int_singleton:
   543   "{l::'a::order} Int {l<..} = {}"
   544   "{..<u} Int {u} = {}"
   545   "{l} Int {l<..<u} = {}"
   546   "{l<..<u} Int {u} = {}"
   547   "{l} Int {l<..u} = {}"
   548   "{l..<u} Int {u} = {}"
   549   by simp+
   550 
   551 text {* One- and two-sided intervals *}
   552 
   553 lemma ivl_disj_int_one:
   554   "{..l::'a::order} Int {l<..<u} = {}"
   555   "{..<l} Int {l..<u} = {}"
   556   "{..l} Int {l<..u} = {}"
   557   "{..<l} Int {l..u} = {}"
   558   "{l<..u} Int {u<..} = {}"
   559   "{l<..<u} Int {u..} = {}"
   560   "{l..u} Int {u<..} = {}"
   561   "{l..<u} Int {u..} = {}"
   562   by auto
   563 
   564 text {* Two- and two-sided intervals *}
   565 
   566 lemma ivl_disj_int_two:
   567   "{l::'a::order<..<m} Int {m..<u} = {}"
   568   "{l<..m} Int {m<..<u} = {}"
   569   "{l..<m} Int {m..<u} = {}"
   570   "{l..m} Int {m<..<u} = {}"
   571   "{l<..<m} Int {m..u} = {}"
   572   "{l<..m} Int {m<..u} = {}"
   573   "{l..<m} Int {m..u} = {}"
   574   "{l..m} Int {m<..u} = {}"
   575   by auto
   576 
   577 lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
   578 
   579 subsubsection {* Some Differences *}
   580 
   581 lemma ivl_diff[simp]:
   582  "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
   583 by(auto)
   584 
   585 
   586 subsubsection {* Some Subset Conditions *}
   587 
   588 lemma ivl_subset[simp]:
   589  "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
   590 apply(auto simp:linorder_not_le)
   591 apply(rule ccontr)
   592 apply(insert linorder_le_less_linear[of i n])
   593 apply(clarsimp simp:linorder_not_le)
   594 apply(fastsimp)
   595 done
   596 
   597 
   598 subsection {* Summation indexed over intervals *}
   599 
   600 syntax
   601   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
   602   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
   603   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
   604   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
   605 syntax (xsymbols)
   606   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
   607   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
   608   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
   609   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
   610 syntax (HTML output)
   611   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
   612   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
   613   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
   614   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
   615 syntax (latex_sum output)
   616   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   617  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
   618   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   619  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
   620   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   621  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
   622   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   623  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
   624 
   625 translations
   626   "\<Sum>x=a..b. t" == "setsum (%x. t) {a..b}"
   627   "\<Sum>x=a..<b. t" == "setsum (%x. t) {a..<b}"
   628   "\<Sum>i\<le>n. t" == "setsum (\<lambda>i. t) {..n}"
   629   "\<Sum>i<n. t" == "setsum (\<lambda>i. t) {..<n}"
   630 
   631 text{* The above introduces some pretty alternative syntaxes for
   632 summation over intervals:
   633 \begin{center}
   634 \begin{tabular}{lll}
   635 Old & New & \LaTeX\\
   636 @{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\
   637 @{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\
   638 @{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
   639 @{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
   640 \end{tabular}
   641 \end{center}
   642 The left column shows the term before introduction of the new syntax,
   643 the middle column shows the new (default) syntax, and the right column
   644 shows a special syntax. The latter is only meaningful for latex output
   645 and has to be activated explicitly by setting the print mode to
   646 \texttt{latex\_sum} (e.g.\ via \texttt{mode=latex\_sum} in
   647 antiquotations). It is not the default \LaTeX\ output because it only
   648 works well with italic-style formulae, not tt-style.
   649 
   650 Note that for uniformity on @{typ nat} it is better to use
   651 @{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
   652 not provide all lemmas available for @{term"{m..<n}"} also in the
   653 special form for @{term"{..<n}"}. *}
   654 
   655 text{* This congruence rule should be used for sums over intervals as
   656 the standard theorem @{text[source]setsum_cong} does not work well
   657 with the simplifier who adds the unsimplified premise @{term"x:B"} to
   658 the context. *}
   659 
   660 lemma setsum_ivl_cong:
   661  "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
   662  setsum f {a..<b} = setsum g {c..<d}"
   663 by(rule setsum_cong, simp_all)
   664 
   665 (* FIXME why are the following simp rules but the corresponding eqns
   666 on intervals are not? *)
   667 
   668 lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
   669 by (simp add:atMost_Suc add_ac)
   670 
   671 lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
   672 by (simp add:lessThan_Suc add_ac)
   673 
   674 lemma setsum_cl_ivl_Suc[simp]:
   675   "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
   676 by (auto simp:add_ac atLeastAtMostSuc_conv)
   677 
   678 lemma setsum_op_ivl_Suc[simp]:
   679   "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
   680 by (auto simp:add_ac atLeastLessThanSuc)
   681 (*
   682 lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
   683     (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
   684 by (auto simp:add_ac atLeastAtMostSuc_conv)
   685 *)
   686 lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
   687   setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
   688 by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
   689 
   690 lemma setsum_diff_nat_ivl:
   691 fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
   692 shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
   693   setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
   694 using setsum_add_nat_ivl [of m n p f,symmetric]
   695 apply (simp add: add_ac)
   696 done
   697 
   698 subsection{* Shifting bounds *}
   699 
   700 lemma setsum_shift_bounds_nat_ivl:
   701   "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
   702 by (induct "n", auto simp:atLeastLessThanSuc)
   703 
   704 lemma setsum_shift_bounds_cl_nat_ivl:
   705   "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
   706 apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
   707 apply (simp add:image_add_atLeastAtMost o_def)
   708 done
   709 
   710 corollary setsum_shift_bounds_cl_Suc_ivl:
   711   "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
   712 by (simp add:setsum_shift_bounds_cl_nat_ivl[where k=1,simplified])
   713 
   714 corollary setsum_shift_bounds_Suc_ivl:
   715   "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
   716 by (simp add:setsum_shift_bounds_nat_ivl[where k=1,simplified])
   717 
   718 lemma setsum_head:
   719   fixes n :: nat
   720   assumes mn: "m <= n" 
   721   shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
   722 proof -
   723   from mn
   724   have "{m..n} = {m} \<union> {m<..n}"
   725     by (auto intro: ivl_disj_un_singleton)
   726   hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
   727     by (simp add: atLeast0LessThan)
   728   also have "\<dots> = ?rhs" by simp
   729   finally show ?thesis .
   730 qed
   731 
   732 lemma setsum_head_upt:
   733   fixes m::nat
   734   assumes m: "0 < m"
   735   shows "(\<Sum>x<m. P x) = P 0 + (\<Sum>x\<in>{1..<m}. P x)"
   736 proof -
   737   have "(\<Sum>x<m. P x) = (\<Sum>x\<in>{0..<m}. P x)" 
   738     by (simp add: atLeast0LessThan)
   739   also 
   740   from m 
   741   have "\<dots> = (\<Sum>x\<in>{0..m - 1}. P x)"
   742     by (cases m) (auto simp add: atLeastLessThanSuc_atLeastAtMost)
   743   also
   744   have "\<dots> = P 0 + (\<Sum>x\<in>{0<..m - 1}. P x)"
   745     by (simp add: setsum_head)
   746   also 
   747   from m 
   748   have "{0<..m - 1} = {1..<m}" 
   749     by (cases m) (auto simp add: atLeastLessThanSuc_atLeastAtMost)
   750   finally show ?thesis .
   751 qed
   752 
   753 subsection {* The formula for geometric sums *}
   754 
   755 lemma geometric_sum:
   756   "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) =
   757   (x ^ n - 1) / (x - 1::'a::{field, recpower, division_by_zero})"
   758   apply (induct "n", auto)
   759   apply (rule_tac c = "x - 1" in field_mult_cancel_right_lemma)
   760   apply (auto simp add: mult_assoc left_distrib)
   761   apply (simp add: right_distrib diff_minus mult_commute power_Suc)
   762   done
   763 
   764 
   765 
   766 lemma sum_diff_distrib:
   767   fixes P::"nat\<Rightarrow>nat"
   768   shows
   769   "\<forall>x. Q x \<le> P x  \<Longrightarrow>
   770   (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
   771 proof (induct n)
   772   case 0 show ?case by simp
   773 next
   774   case (Suc n)
   775 
   776   let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
   777   let ?rhs = "\<Sum>x<n. P x - Q x"
   778 
   779   from Suc have "?lhs = ?rhs" by simp
   780   moreover
   781   from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
   782   moreover
   783   from Suc have
   784     "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
   785     by (subst diff_diff_left[symmetric],
   786         subst diff_add_assoc2)
   787        (auto simp: diff_add_assoc2 intro: setsum_mono)
   788   ultimately
   789   show ?case by simp
   790 qed
   791 
   792 
   793 ML
   794 {*
   795 val Compl_atLeast = thm "Compl_atLeast";
   796 val Compl_atMost = thm "Compl_atMost";
   797 val Compl_greaterThan = thm "Compl_greaterThan";
   798 val Compl_lessThan = thm "Compl_lessThan";
   799 val INT_greaterThan_UNIV = thm "INT_greaterThan_UNIV";
   800 val UN_atLeast_UNIV = thm "UN_atLeast_UNIV";
   801 val UN_atMost_UNIV = thm "UN_atMost_UNIV";
   802 val UN_lessThan_UNIV = thm "UN_lessThan_UNIV";
   803 val atLeastAtMost_def = thm "atLeastAtMost_def";
   804 val atLeastAtMost_iff = thm "atLeastAtMost_iff";
   805 val atLeastLessThan_def  = thm "atLeastLessThan_def";
   806 val atLeastLessThan_iff = thm "atLeastLessThan_iff";
   807 val atLeast_0 = thm "atLeast_0";
   808 val atLeast_Suc = thm "atLeast_Suc";
   809 val atLeast_def      = thm "atLeast_def";
   810 val atLeast_iff = thm "atLeast_iff";
   811 val atMost_0 = thm "atMost_0";
   812 val atMost_Int_atLeast = thm "atMost_Int_atLeast";
   813 val atMost_Suc = thm "atMost_Suc";
   814 val atMost_def       = thm "atMost_def";
   815 val atMost_iff = thm "atMost_iff";
   816 val greaterThanAtMost_def  = thm "greaterThanAtMost_def";
   817 val greaterThanAtMost_iff = thm "greaterThanAtMost_iff";
   818 val greaterThanLessThan_def  = thm "greaterThanLessThan_def";
   819 val greaterThanLessThan_iff = thm "greaterThanLessThan_iff";
   820 val greaterThan_0 = thm "greaterThan_0";
   821 val greaterThan_Suc = thm "greaterThan_Suc";
   822 val greaterThan_def  = thm "greaterThan_def";
   823 val greaterThan_iff = thm "greaterThan_iff";
   824 val ivl_disj_int = thms "ivl_disj_int";
   825 val ivl_disj_int_one = thms "ivl_disj_int_one";
   826 val ivl_disj_int_singleton = thms "ivl_disj_int_singleton";
   827 val ivl_disj_int_two = thms "ivl_disj_int_two";
   828 val ivl_disj_un = thms "ivl_disj_un";
   829 val ivl_disj_un_one = thms "ivl_disj_un_one";
   830 val ivl_disj_un_singleton = thms "ivl_disj_un_singleton";
   831 val ivl_disj_un_two = thms "ivl_disj_un_two";
   832 val lessThan_0 = thm "lessThan_0";
   833 val lessThan_Suc = thm "lessThan_Suc";
   834 val lessThan_Suc_atMost = thm "lessThan_Suc_atMost";
   835 val lessThan_def     = thm "lessThan_def";
   836 val lessThan_iff = thm "lessThan_iff";
   837 val single_Diff_lessThan = thm "single_Diff_lessThan";
   838 
   839 val bounded_nat_set_is_finite = thm "bounded_nat_set_is_finite";
   840 val finite_atMost = thm "finite_atMost";
   841 val finite_lessThan = thm "finite_lessThan";
   842 *}
   843 
   844 end