src/HOL/SetInterval.thy
author paulson
Tue Oct 19 18:18:45 2004 +0200 (2004-10-19)
changeset 15251 bb6f072c8d10
parent 15140 322485b816ac
child 15402 97204f3b4705
permissions -rw-r--r--
converted some induct_tac to induct
     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 add: greaterThan_subset_iff order_antisym, simp) 
   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 add: lessThan_subset_iff order_antisym, simp) 
   166 done
   167 
   168 
   169 subsection {*Two-sided intervals*}
   170 
   171 text {* @{text greaterThanLessThan} *}
   172 
   173 lemma greaterThanLessThan_iff [simp]:
   174   "(i : {l<..<u}) = (l < i & i < u)"
   175 by (simp add: greaterThanLessThan_def)
   176 
   177 text {* @{text atLeastLessThan} *}
   178 
   179 lemma atLeastLessThan_iff [simp]:
   180   "(i : {l..<u}) = (l <= i & i < u)"
   181 by (simp add: atLeastLessThan_def)
   182 
   183 text {* @{text greaterThanAtMost} *}
   184 
   185 lemma greaterThanAtMost_iff [simp]:
   186   "(i : {l<..u}) = (l < i & i <= u)"
   187 by (simp add: greaterThanAtMost_def)
   188 
   189 text {* @{text atLeastAtMost} *}
   190 
   191 lemma atLeastAtMost_iff [simp]:
   192   "(i : {l..u}) = (l <= i & i <= u)"
   193 by (simp add: atLeastAtMost_def)
   194 
   195 text {* The above four lemmas could be declared as iffs.
   196   If we do so, a call to blast in Hyperreal/Star.ML, lemma @{text STAR_Int}
   197   seems to take forever (more than one hour). *}
   198 
   199 
   200 subsection {* Intervals of natural numbers *}
   201 
   202 subsubsection {* The Constant @{term lessThan} *}
   203 
   204 lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
   205 by (simp add: lessThan_def)
   206 
   207 lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
   208 by (simp add: lessThan_def less_Suc_eq, blast)
   209 
   210 lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
   211 by (simp add: lessThan_def atMost_def less_Suc_eq_le)
   212 
   213 lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
   214 by blast
   215 
   216 subsubsection {* The Constant @{term greaterThan} *}
   217 
   218 lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
   219 apply (simp add: greaterThan_def)
   220 apply (blast dest: gr0_conv_Suc [THEN iffD1])
   221 done
   222 
   223 lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
   224 apply (simp add: greaterThan_def)
   225 apply (auto elim: linorder_neqE)
   226 done
   227 
   228 lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
   229 by blast
   230 
   231 subsubsection {* The Constant @{term atLeast} *}
   232 
   233 lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
   234 by (unfold atLeast_def UNIV_def, simp)
   235 
   236 lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
   237 apply (simp add: atLeast_def)
   238 apply (simp add: Suc_le_eq)
   239 apply (simp add: order_le_less, blast)
   240 done
   241 
   242 lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
   243   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
   244 
   245 lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
   246 by blast
   247 
   248 subsubsection {* The Constant @{term atMost} *}
   249 
   250 lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
   251 by (simp add: atMost_def)
   252 
   253 lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
   254 apply (simp add: atMost_def)
   255 apply (simp add: less_Suc_eq order_le_less, blast)
   256 done
   257 
   258 lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
   259 by blast
   260 
   261 subsubsection {* The Constant @{term atLeastLessThan} *}
   262 
   263 text{*But not a simprule because some concepts are better left in terms
   264   of @{term atLeastLessThan}*}
   265 lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
   266 by(simp add:lessThan_def atLeastLessThan_def)
   267 
   268 lemma atLeastLessThan0 [simp]: "{m..<0::nat} = {}"
   269 by (simp add: atLeastLessThan_def)
   270 
   271 lemma atLeastLessThan_self [simp]: "{n::'a::order..<n} = {}"
   272 by (auto simp add: atLeastLessThan_def)
   273 
   274 lemma atLeastLessThan_empty: "n \<le> m ==> {m..<n::'a::order} = {}"
   275 by (auto simp add: atLeastLessThan_def)
   276 
   277 subsubsection {* Intervals of nats with @{term Suc} *}
   278 
   279 text{*Not a simprule because the RHS is too messy.*}
   280 lemma atLeastLessThanSuc:
   281     "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
   282 by (auto simp add: atLeastLessThan_def) 
   283 
   284 lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}" 
   285 by (auto simp add: atLeastLessThan_def)
   286 
   287 lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
   288 by (induct k, simp_all add: atLeastLessThanSuc)
   289 
   290 lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
   291 by (auto simp add: atLeastLessThan_def)
   292 
   293 lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
   294   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
   295 
   296 lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"  
   297   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def 
   298     greaterThanAtMost_def)
   299 
   300 lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"  
   301   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def 
   302     greaterThanLessThan_def)
   303 
   304 subsubsection {* Finiteness *}
   305 
   306 lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
   307   by (induct k) (simp_all add: lessThan_Suc)
   308 
   309 lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
   310   by (induct k) (simp_all add: atMost_Suc)
   311 
   312 lemma finite_greaterThanLessThan [iff]:
   313   fixes l :: nat shows "finite {l<..<u}"
   314 by (simp add: greaterThanLessThan_def)
   315 
   316 lemma finite_atLeastLessThan [iff]:
   317   fixes l :: nat shows "finite {l..<u}"
   318 by (simp add: atLeastLessThan_def)
   319 
   320 lemma finite_greaterThanAtMost [iff]:
   321   fixes l :: nat shows "finite {l<..u}"
   322 by (simp add: greaterThanAtMost_def)
   323 
   324 lemma finite_atLeastAtMost [iff]:
   325   fixes l :: nat shows "finite {l..u}"
   326 by (simp add: atLeastAtMost_def)
   327 
   328 lemma bounded_nat_set_is_finite:
   329     "(ALL i:N. i < (n::nat)) ==> finite N"
   330   -- {* A bounded set of natural numbers is finite. *}
   331   apply (rule finite_subset)
   332    apply (rule_tac [2] finite_lessThan, auto)
   333   done
   334 
   335 subsubsection {* Cardinality *}
   336 
   337 lemma card_lessThan [simp]: "card {..<u} = u"
   338   by (induct u, simp_all add: lessThan_Suc)
   339 
   340 lemma card_atMost [simp]: "card {..u} = Suc u"
   341   by (simp add: lessThan_Suc_atMost [THEN sym])
   342 
   343 lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
   344   apply (subgoal_tac "card {l..<u} = card {..<u-l}")
   345   apply (erule ssubst, rule card_lessThan)
   346   apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
   347   apply (erule subst)
   348   apply (rule card_image)
   349   apply (rule finite_lessThan)
   350   apply (simp add: inj_on_def)
   351   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
   352   apply arith
   353   apply (rule_tac x = "x - l" in exI)
   354   apply arith
   355   done
   356 
   357 lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"  
   358   by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
   359 
   360 lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l" 
   361   by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
   362 
   363 lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
   364   by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
   365 
   366 subsection {* Intervals of integers *}
   367 
   368 lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
   369   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
   370 
   371 lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"  
   372   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
   373 
   374 lemma atLeastPlusOneLessThan_greaterThanLessThan_int: 
   375     "{l+1..<u} = {l<..<u::int}"  
   376   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
   377 
   378 subsubsection {* Finiteness *}
   379 
   380 lemma image_atLeastZeroLessThan_int: "0 \<le> u ==> 
   381     {(0::int)..<u} = int ` {..<nat u}"
   382   apply (unfold image_def lessThan_def)
   383   apply auto
   384   apply (rule_tac x = "nat x" in exI)
   385   apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
   386   done
   387 
   388 lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
   389   apply (case_tac "0 \<le> u")
   390   apply (subst image_atLeastZeroLessThan_int, assumption)
   391   apply (rule finite_imageI)
   392   apply auto
   393   apply (subgoal_tac "{0..<u} = {}")
   394   apply auto
   395   done
   396 
   397 lemma image_atLeastLessThan_int_shift: 
   398     "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
   399   apply (auto simp add: image_def atLeastLessThan_iff)
   400   apply (rule_tac x = "x - l" in bexI)
   401   apply auto
   402   done
   403 
   404 lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
   405   apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
   406   apply (erule subst)
   407   apply (rule finite_imageI)
   408   apply (rule finite_atLeastZeroLessThan_int)
   409   apply (rule image_atLeastLessThan_int_shift)
   410   done
   411 
   412 lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}" 
   413   by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
   414 
   415 lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}" 
   416   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
   417 
   418 lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}" 
   419   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
   420 
   421 subsubsection {* Cardinality *}
   422 
   423 lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
   424   apply (case_tac "0 \<le> u")
   425   apply (subst image_atLeastZeroLessThan_int, assumption)
   426   apply (subst card_image)
   427   apply (auto simp add: inj_on_def)
   428   done
   429 
   430 lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
   431   apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
   432   apply (erule ssubst, rule card_atLeastZeroLessThan_int)
   433   apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
   434   apply (erule subst)
   435   apply (rule card_image)
   436   apply (rule finite_atLeastZeroLessThan_int)
   437   apply (simp add: inj_on_def)
   438   apply (rule image_atLeastLessThan_int_shift)
   439   done
   440 
   441 lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
   442   apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
   443   apply (auto simp add: compare_rls)
   444   done
   445 
   446 lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)" 
   447   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
   448 
   449 lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
   450   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
   451 
   452 
   453 subsection {*Lemmas useful with the summation operator setsum*}
   454 
   455 text {* For examples, see Algebra/poly/UnivPoly.thy *}
   456 
   457 subsubsection {* Disjoint Unions *}
   458 
   459 text {* Singletons and open intervals *}
   460 
   461 lemma ivl_disj_un_singleton:
   462   "{l::'a::linorder} Un {l<..} = {l..}"
   463   "{..<u} Un {u::'a::linorder} = {..u}"
   464   "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
   465   "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
   466   "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
   467   "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
   468 by auto
   469 
   470 text {* One- and two-sided intervals *}
   471 
   472 lemma ivl_disj_un_one:
   473   "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
   474   "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
   475   "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
   476   "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
   477   "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
   478   "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
   479   "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
   480   "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
   481 by auto
   482 
   483 text {* Two- and two-sided intervals *}
   484 
   485 lemma ivl_disj_un_two:
   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   "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
   490   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
   491   "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
   492   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
   493   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
   494 by auto
   495 
   496 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
   497 
   498 subsubsection {* Disjoint Intersections *}
   499 
   500 text {* Singletons and open intervals *}
   501 
   502 lemma ivl_disj_int_singleton:
   503   "{l::'a::order} Int {l<..} = {}"
   504   "{..<u} Int {u} = {}"
   505   "{l} Int {l<..<u} = {}"
   506   "{l<..<u} Int {u} = {}"
   507   "{l} Int {l<..u} = {}"
   508   "{l..<u} Int {u} = {}"
   509   by simp+
   510 
   511 text {* One- and two-sided intervals *}
   512 
   513 lemma ivl_disj_int_one:
   514   "{..l::'a::order} Int {l<..<u} = {}"
   515   "{..<l} Int {l..<u} = {}"
   516   "{..l} Int {l<..u} = {}"
   517   "{..<l} Int {l..u} = {}"
   518   "{l<..u} Int {u<..} = {}"
   519   "{l<..<u} Int {u..} = {}"
   520   "{l..u} Int {u<..} = {}"
   521   "{l..<u} Int {u..} = {}"
   522   by auto
   523 
   524 text {* Two- and two-sided intervals *}
   525 
   526 lemma ivl_disj_int_two:
   527   "{l::'a::order<..<m} Int {m..<u} = {}"
   528   "{l<..m} Int {m<..<u} = {}"
   529   "{l..<m} Int {m..<u} = {}"
   530   "{l..m} Int {m<..<u} = {}"
   531   "{l<..<m} Int {m..u} = {}"
   532   "{l<..m} Int {m<..u} = {}"
   533   "{l..<m} Int {m..u} = {}"
   534   "{l..m} Int {m<..u} = {}"
   535   by auto
   536 
   537 lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
   538 
   539 
   540 subsection {* Summation indexed over intervals *}
   541 
   542 syntax
   543   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
   544   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
   545   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
   546 syntax (xsymbols)
   547   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
   548   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
   549   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
   550 syntax (HTML output)
   551   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
   552   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
   553   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
   554 syntax (latex_sum output)
   555   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   556  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
   557   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   558  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
   559   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   560  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
   561 
   562 translations
   563   "\<Sum>x=a..b. t" == "setsum (%x. t) {a..b}"
   564   "\<Sum>x=a..<b. t" == "setsum (%x. t) {a..<b}"
   565   "\<Sum>i<n. t" == "setsum (\<lambda>i. t) {..<n}"
   566 
   567 text{* The above introduces some pretty alternative syntaxes for
   568 summation over intervals:
   569 \begin{center}
   570 \begin{tabular}{lll}
   571 Old & New & \LaTeX\\
   572 @{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\
   573 @{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\
   574 @{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
   575 \end{tabular}
   576 \end{center}
   577 The left column shows the term before introduction of the new syntax,
   578 the middle column shows the new (default) syntax, and the right column
   579 shows a special syntax. The latter is only meaningful for latex output
   580 and has to be activated explicitly by setting the print mode to
   581 \texttt{latex\_sum} (e.g.\ via \texttt{mode=latex\_sum} in
   582 antiquotations). It is not the default \LaTeX\ output because it only
   583 works well with italic-style formulae, not tt-style.
   584 
   585 Note that for uniformity on @{typ nat} it is better to use
   586 @{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
   587 not provide all lemmas available for @{term"{m..<n}"} also in the
   588 special form for @{term"{..<n}"}. *}
   589 
   590 
   591 lemma Summation_Suc[simp]: "(\<Sum>i < Suc n. b i) = b n + (\<Sum>i < n. b i)"
   592 by (simp add:lessThan_Suc)
   593 
   594 end