src/HOL/SetInterval.thy
author nipkow
Mon May 23 19:39:45 2005 +0200 (2005-05-23)
changeset 16052 880b0e786c1b
parent 16041 5a8736668ced
child 16102 c5f6726d9bb1
permissions -rw-r--r--
tuned setsum rewrites
     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 atLeastAtMost_singleton [simp]: "{a::'a::order..a} = {a}";
   200   by (auto simp add: atLeastAtMost_def atMost_def atLeast_def);
   201 
   202 subsection {* Intervals of natural numbers *}
   203 
   204 subsubsection {* The Constant @{term lessThan} *}
   205 
   206 lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
   207 by (simp add: lessThan_def)
   208 
   209 lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
   210 by (simp add: lessThan_def less_Suc_eq, blast)
   211 
   212 lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
   213 by (simp add: lessThan_def atMost_def less_Suc_eq_le)
   214 
   215 lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
   216 by blast
   217 
   218 subsubsection {* The Constant @{term greaterThan} *}
   219 
   220 lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
   221 apply (simp add: greaterThan_def)
   222 apply (blast dest: gr0_conv_Suc [THEN iffD1])
   223 done
   224 
   225 lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
   226 apply (simp add: greaterThan_def)
   227 apply (auto elim: linorder_neqE)
   228 done
   229 
   230 lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
   231 by blast
   232 
   233 subsubsection {* The Constant @{term atLeast} *}
   234 
   235 lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
   236 by (unfold atLeast_def UNIV_def, simp)
   237 
   238 lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
   239 apply (simp add: atLeast_def)
   240 apply (simp add: Suc_le_eq)
   241 apply (simp add: order_le_less, blast)
   242 done
   243 
   244 lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
   245   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
   246 
   247 lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
   248 by blast
   249 
   250 subsubsection {* The Constant @{term atMost} *}
   251 
   252 lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
   253 by (simp add: atMost_def)
   254 
   255 lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
   256 apply (simp add: atMost_def)
   257 apply (simp add: less_Suc_eq order_le_less, blast)
   258 done
   259 
   260 lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
   261 by blast
   262 
   263 subsubsection {* The Constant @{term atLeastLessThan} *}
   264 
   265 text{*But not a simprule because some concepts are better left in terms
   266   of @{term atLeastLessThan}*}
   267 lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
   268 by(simp add:lessThan_def atLeastLessThan_def)
   269 (*
   270 lemma atLeastLessThan0 [simp]: "{m..<0::nat} = {}"
   271 by (simp add: atLeastLessThan_def)
   272 *)
   273 subsubsection {* Intervals of nats with @{term Suc} *}
   274 
   275 text{*Not a simprule because the RHS is too messy.*}
   276 lemma atLeastLessThanSuc:
   277     "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
   278 by (auto simp add: atLeastLessThan_def)
   279 
   280 lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
   281 by (auto simp add: atLeastLessThan_def)
   282 (*
   283 lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
   284 by (induct k, simp_all add: atLeastLessThanSuc)
   285 
   286 lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
   287 by (auto simp add: atLeastLessThan_def)
   288 *)
   289 lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
   290   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
   291 
   292 lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
   293   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
   294     greaterThanAtMost_def)
   295 
   296 lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
   297   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
   298     greaterThanLessThan_def)
   299 
   300 lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
   301 by (auto simp add: atLeastAtMost_def)
   302 
   303 subsubsection {* Finiteness *}
   304 
   305 lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
   306   by (induct k) (simp_all add: lessThan_Suc)
   307 
   308 lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
   309   by (induct k) (simp_all add: atMost_Suc)
   310 
   311 lemma finite_greaterThanLessThan [iff]:
   312   fixes l :: nat shows "finite {l<..<u}"
   313 by (simp add: greaterThanLessThan_def)
   314 
   315 lemma finite_atLeastLessThan [iff]:
   316   fixes l :: nat shows "finite {l..<u}"
   317 by (simp add: atLeastLessThan_def)
   318 
   319 lemma finite_greaterThanAtMost [iff]:
   320   fixes l :: nat shows "finite {l<..u}"
   321 by (simp add: greaterThanAtMost_def)
   322 
   323 lemma finite_atLeastAtMost [iff]:
   324   fixes l :: nat shows "finite {l..u}"
   325 by (simp add: atLeastAtMost_def)
   326 
   327 lemma bounded_nat_set_is_finite:
   328     "(ALL i:N. i < (n::nat)) ==> finite N"
   329   -- {* A bounded set of natural numbers is finite. *}
   330   apply (rule finite_subset)
   331    apply (rule_tac [2] finite_lessThan, auto)
   332   done
   333 
   334 subsubsection {* Cardinality *}
   335 
   336 lemma card_lessThan [simp]: "card {..<u} = u"
   337   by (induct u, simp_all add: lessThan_Suc)
   338 
   339 lemma card_atMost [simp]: "card {..u} = Suc u"
   340   by (simp add: lessThan_Suc_atMost [THEN sym])
   341 
   342 lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
   343   apply (subgoal_tac "card {l..<u} = card {..<u-l}")
   344   apply (erule ssubst, rule card_lessThan)
   345   apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
   346   apply (erule subst)
   347   apply (rule card_image)
   348   apply (simp add: inj_on_def)
   349   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
   350   apply arith
   351   apply (rule_tac x = "x - l" in exI)
   352   apply arith
   353   done
   354 
   355 lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
   356   by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
   357 
   358 lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
   359   by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
   360 
   361 lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
   362   by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
   363 
   364 subsection {* Intervals of integers *}
   365 
   366 lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
   367   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
   368 
   369 lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
   370   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
   371 
   372 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
   373     "{l+1..<u} = {l<..<u::int}"
   374   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
   375 
   376 subsubsection {* Finiteness *}
   377 
   378 lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
   379     {(0::int)..<u} = int ` {..<nat u}"
   380   apply (unfold image_def lessThan_def)
   381   apply auto
   382   apply (rule_tac x = "nat x" in exI)
   383   apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
   384   done
   385 
   386 lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
   387   apply (case_tac "0 \<le> u")
   388   apply (subst image_atLeastZeroLessThan_int, assumption)
   389   apply (rule finite_imageI)
   390   apply auto
   391   done
   392 
   393 lemma image_atLeastLessThan_int_shift:
   394     "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
   395   apply (auto simp add: image_def)
   396   apply (rule_tac x = "x - l" in bexI)
   397   apply auto
   398   done
   399 
   400 lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
   401   apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
   402   apply (erule subst)
   403   apply (rule finite_imageI)
   404   apply (rule finite_atLeastZeroLessThan_int)
   405   apply (rule image_atLeastLessThan_int_shift)
   406   done
   407 
   408 lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
   409   by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
   410 
   411 lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
   412   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
   413 
   414 lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
   415   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
   416 
   417 subsubsection {* Cardinality *}
   418 
   419 lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
   420   apply (case_tac "0 \<le> u")
   421   apply (subst image_atLeastZeroLessThan_int, assumption)
   422   apply (subst card_image)
   423   apply (auto simp add: inj_on_def)
   424   done
   425 
   426 lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
   427   apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
   428   apply (erule ssubst, rule card_atLeastZeroLessThan_int)
   429   apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
   430   apply (erule subst)
   431   apply (rule card_image)
   432   apply (simp add: inj_on_def)
   433   apply (rule image_atLeastLessThan_int_shift)
   434   done
   435 
   436 lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
   437   apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
   438   apply (auto simp add: compare_rls)
   439   done
   440 
   441 lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
   442   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
   443 
   444 lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
   445   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
   446 
   447 
   448 subsection {*Lemmas useful with the summation operator setsum*}
   449 
   450 text {* For examples, see Algebra/poly/UnivPoly.thy *}
   451 
   452 subsubsection {* Disjoint Unions *}
   453 
   454 text {* Singletons and open intervals *}
   455 
   456 lemma ivl_disj_un_singleton:
   457   "{l::'a::linorder} Un {l<..} = {l..}"
   458   "{..<u} Un {u::'a::linorder} = {..u}"
   459   "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
   460   "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
   461   "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
   462   "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
   463 by auto
   464 
   465 text {* One- and two-sided intervals *}
   466 
   467 lemma ivl_disj_un_one:
   468   "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
   469   "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
   470   "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
   471   "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
   472   "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
   473   "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
   474   "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
   475   "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
   476 by auto
   477 
   478 text {* Two- and two-sided intervals *}
   479 
   480 lemma ivl_disj_un_two:
   481   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
   482   "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
   483   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
   484   "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
   485   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
   486   "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
   487   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
   488   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
   489 by auto
   490 
   491 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
   492 
   493 subsubsection {* Disjoint Intersections *}
   494 
   495 text {* Singletons and open intervals *}
   496 
   497 lemma ivl_disj_int_singleton:
   498   "{l::'a::order} Int {l<..} = {}"
   499   "{..<u} Int {u} = {}"
   500   "{l} Int {l<..<u} = {}"
   501   "{l<..<u} Int {u} = {}"
   502   "{l} Int {l<..u} = {}"
   503   "{l..<u} Int {u} = {}"
   504   by simp+
   505 
   506 text {* One- and two-sided intervals *}
   507 
   508 lemma ivl_disj_int_one:
   509   "{..l::'a::order} Int {l<..<u} = {}"
   510   "{..<l} Int {l..<u} = {}"
   511   "{..l} Int {l<..u} = {}"
   512   "{..<l} Int {l..u} = {}"
   513   "{l<..u} Int {u<..} = {}"
   514   "{l<..<u} Int {u..} = {}"
   515   "{l..u} Int {u<..} = {}"
   516   "{l..<u} Int {u..} = {}"
   517   by auto
   518 
   519 text {* Two- and two-sided intervals *}
   520 
   521 lemma ivl_disj_int_two:
   522   "{l::'a::order<..<m} Int {m..<u} = {}"
   523   "{l<..m} Int {m<..<u} = {}"
   524   "{l..<m} Int {m..<u} = {}"
   525   "{l..m} Int {m<..<u} = {}"
   526   "{l<..<m} Int {m..u} = {}"
   527   "{l<..m} Int {m<..u} = {}"
   528   "{l..<m} Int {m..u} = {}"
   529   "{l..m} Int {m<..u} = {}"
   530   by auto
   531 
   532 lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
   533 
   534 subsubsection {* Some Differences *}
   535 
   536 lemma ivl_diff[simp]:
   537  "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
   538 by(auto)
   539 
   540 
   541 subsubsection {* Some Subset Conditions *}
   542 
   543 lemma ivl_subset[simp]:
   544  "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
   545 apply(auto simp:linorder_not_le)
   546 apply(rule ccontr)
   547 apply(insert linorder_le_less_linear[of i n])
   548 apply(clarsimp simp:linorder_not_le)
   549 apply(fastsimp)
   550 done
   551 
   552 
   553 subsection {* Summation indexed over intervals *}
   554 
   555 syntax
   556   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
   557   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
   558   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
   559   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
   560 syntax (xsymbols)
   561   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
   562   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
   563   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
   564   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
   565 syntax (HTML output)
   566   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
   567   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
   568   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
   569   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
   570 syntax (latex_sum output)
   571   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   572  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
   573   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   574  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
   575   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   576  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
   577   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   578  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
   579 
   580 translations
   581   "\<Sum>x=a..b. t" == "setsum (%x. t) {a..b}"
   582   "\<Sum>x=a..<b. t" == "setsum (%x. t) {a..<b}"
   583   "\<Sum>i\<le>n. t" == "setsum (\<lambda>i. t) {..n}"
   584   "\<Sum>i<n. t" == "setsum (\<lambda>i. t) {..<n}"
   585 
   586 text{* The above introduces some pretty alternative syntaxes for
   587 summation over intervals:
   588 \begin{center}
   589 \begin{tabular}{lll}
   590 Old & New & \LaTeX\\
   591 @{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\
   592 @{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\
   593 @{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
   594 @{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
   595 \end{tabular}
   596 \end{center}
   597 The left column shows the term before introduction of the new syntax,
   598 the middle column shows the new (default) syntax, and the right column
   599 shows a special syntax. The latter is only meaningful for latex output
   600 and has to be activated explicitly by setting the print mode to
   601 \texttt{latex\_sum} (e.g.\ via \texttt{mode=latex\_sum} in
   602 antiquotations). It is not the default \LaTeX\ output because it only
   603 works well with italic-style formulae, not tt-style.
   604 
   605 Note that for uniformity on @{typ nat} it is better to use
   606 @{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
   607 not provide all lemmas available for @{term"{m..<n}"} also in the
   608 special form for @{term"{..<n}"}. *}
   609 
   610 (* FIXME change the simplifier's treatment of congruence rules?? *)
   611 
   612 text{* This congruence rule should be used for sums over intervals as
   613 the standard theorem @{text[source]setsum_cong} does not work well
   614 with the simplifier who adds the unsimplified premise @{term"x:B"} to
   615 the context. *}
   616 
   617 lemma setsum_ivl_cong:
   618  "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
   619  setsum f {a..<b} = setsum g {c..<d}"
   620 by(rule setsum_cong, simp_all)
   621 
   622 (* FIXME why are the following simp rules but the corresponding eqns
   623 on intervals are not? *)
   624 
   625 lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
   626 by (simp add:atMost_Suc add_ac)
   627 
   628 lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
   629 by (simp add:lessThan_Suc add_ac)
   630 
   631 lemma setsum_cl_ivl_Suc[simp]:
   632   "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
   633 by (auto simp:add_ac atLeastAtMostSuc_conv)
   634 
   635 lemma setsum_op_ivl_Suc[simp]:
   636   "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
   637 by (auto simp:add_ac atLeastLessThanSuc)
   638 (*
   639 lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
   640     (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
   641 by (auto simp:add_ac atLeastAtMostSuc_conv)
   642 *)
   643 lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
   644   setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
   645 by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
   646 
   647 lemma setsum_diff_nat_ivl:
   648 fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
   649 shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
   650   setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
   651 using setsum_add_nat_ivl [of m n p f,symmetric]
   652 apply (simp add: add_ac)
   653 done
   654 
   655 lemma setsum_shift_bounds_nat_ivl:
   656   "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
   657 by (induct "n", auto simp:atLeastLessThanSuc)
   658 
   659 
   660 ML
   661 {*
   662 val Compl_atLeast = thm "Compl_atLeast";
   663 val Compl_atMost = thm "Compl_atMost";
   664 val Compl_greaterThan = thm "Compl_greaterThan";
   665 val Compl_lessThan = thm "Compl_lessThan";
   666 val INT_greaterThan_UNIV = thm "INT_greaterThan_UNIV";
   667 val UN_atLeast_UNIV = thm "UN_atLeast_UNIV";
   668 val UN_atMost_UNIV = thm "UN_atMost_UNIV";
   669 val UN_lessThan_UNIV = thm "UN_lessThan_UNIV";
   670 val atLeastAtMost_def = thm "atLeastAtMost_def";
   671 val atLeastAtMost_iff = thm "atLeastAtMost_iff";
   672 val atLeastLessThan_def  = thm "atLeastLessThan_def";
   673 val atLeastLessThan_iff = thm "atLeastLessThan_iff";
   674 val atLeast_0 = thm "atLeast_0";
   675 val atLeast_Suc = thm "atLeast_Suc";
   676 val atLeast_def      = thm "atLeast_def";
   677 val atLeast_iff = thm "atLeast_iff";
   678 val atMost_0 = thm "atMost_0";
   679 val atMost_Int_atLeast = thm "atMost_Int_atLeast";
   680 val atMost_Suc = thm "atMost_Suc";
   681 val atMost_def       = thm "atMost_def";
   682 val atMost_iff = thm "atMost_iff";
   683 val greaterThanAtMost_def  = thm "greaterThanAtMost_def";
   684 val greaterThanAtMost_iff = thm "greaterThanAtMost_iff";
   685 val greaterThanLessThan_def  = thm "greaterThanLessThan_def";
   686 val greaterThanLessThan_iff = thm "greaterThanLessThan_iff";
   687 val greaterThan_0 = thm "greaterThan_0";
   688 val greaterThan_Suc = thm "greaterThan_Suc";
   689 val greaterThan_def  = thm "greaterThan_def";
   690 val greaterThan_iff = thm "greaterThan_iff";
   691 val ivl_disj_int = thms "ivl_disj_int";
   692 val ivl_disj_int_one = thms "ivl_disj_int_one";
   693 val ivl_disj_int_singleton = thms "ivl_disj_int_singleton";
   694 val ivl_disj_int_two = thms "ivl_disj_int_two";
   695 val ivl_disj_un = thms "ivl_disj_un";
   696 val ivl_disj_un_one = thms "ivl_disj_un_one";
   697 val ivl_disj_un_singleton = thms "ivl_disj_un_singleton";
   698 val ivl_disj_un_two = thms "ivl_disj_un_two";
   699 val lessThan_0 = thm "lessThan_0";
   700 val lessThan_Suc = thm "lessThan_Suc";
   701 val lessThan_Suc_atMost = thm "lessThan_Suc_atMost";
   702 val lessThan_def     = thm "lessThan_def";
   703 val lessThan_iff = thm "lessThan_iff";
   704 val single_Diff_lessThan = thm "single_Diff_lessThan";
   705 
   706 val bounded_nat_set_is_finite = thm "bounded_nat_set_is_finite";
   707 val finite_atMost = thm "finite_atMost";
   708 val finite_lessThan = thm "finite_lessThan";
   709 *}
   710 
   711 end