src/HOL/Set_Interval.thy
author wenzelm
Fri Mar 07 22:30:58 2014 +0100 (2014-03-07)
changeset 55990 41c6b99c5fb7
parent 55719 cdddd073bff8
child 56193 c726ecfb22b6
permissions -rw-r--r--
more antiquotations;
     1 (*  Title:      HOL/Set_Interval.thy
     2     Author:     Tobias Nipkow
     3     Author:     Clemens Ballarin
     4     Author:     Jeremy Avigad
     5 
     6 lessThan, greaterThan, atLeast, atMost and two-sided intervals
     7 
     8 Modern convention: Ixy stands for an interval where x and y
     9 describe the lower and upper bound and x,y : {c,o,i}
    10 where c = closed, o = open, i = infinite.
    11 Examples: Ico = {_ ..< _} and Ici = {_ ..}
    12 *)
    13 
    14 header {* Set intervals *}
    15 
    16 theory Set_Interval
    17 imports Lattices_Big Nat_Transfer
    18 begin
    19 
    20 context ord
    21 begin
    22 
    23 definition
    24   lessThan    :: "'a => 'a set" ("(1{..<_})") where
    25   "{..<u} == {x. x < u}"
    26 
    27 definition
    28   atMost      :: "'a => 'a set" ("(1{.._})") where
    29   "{..u} == {x. x \<le> u}"
    30 
    31 definition
    32   greaterThan :: "'a => 'a set" ("(1{_<..})") where
    33   "{l<..} == {x. l<x}"
    34 
    35 definition
    36   atLeast     :: "'a => 'a set" ("(1{_..})") where
    37   "{l..} == {x. l\<le>x}"
    38 
    39 definition
    40   greaterThanLessThan :: "'a => 'a => 'a set"  ("(1{_<..<_})") where
    41   "{l<..<u} == {l<..} Int {..<u}"
    42 
    43 definition
    44   atLeastLessThan :: "'a => 'a => 'a set"      ("(1{_..<_})") where
    45   "{l..<u} == {l..} Int {..<u}"
    46 
    47 definition
    48   greaterThanAtMost :: "'a => 'a => 'a set"    ("(1{_<.._})") where
    49   "{l<..u} == {l<..} Int {..u}"
    50 
    51 definition
    52   atLeastAtMost :: "'a => 'a => 'a set"        ("(1{_.._})") where
    53   "{l..u} == {l..} Int {..u}"
    54 
    55 end
    56 
    57 
    58 text{* A note of warning when using @{term"{..<n}"} on type @{typ
    59 nat}: it is equivalent to @{term"{0::nat..<n}"} but some lemmas involving
    60 @{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well. *}
    61 
    62 syntax
    63   "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3UN _<=_./ _)" [0, 0, 10] 10)
    64   "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3UN _<_./ _)" [0, 0, 10] 10)
    65   "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3INT _<=_./ _)" [0, 0, 10] 10)
    66   "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3INT _<_./ _)" [0, 0, 10] 10)
    67 
    68 syntax (xsymbols)
    69   "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _\<le>_./ _)" [0, 0, 10] 10)
    70   "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _<_./ _)" [0, 0, 10] 10)
    71   "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _\<le>_./ _)" [0, 0, 10] 10)
    72   "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" [0, 0, 10] 10)
    73 
    74 syntax (latex output)
    75   "_UNION_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ \<le> _)/ _)" [0, 0, 10] 10)
    76   "_UNION_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ < _)/ _)" [0, 0, 10] 10)
    77   "_INTER_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ \<le> _)/ _)" [0, 0, 10] 10)
    78   "_INTER_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ < _)/ _)" [0, 0, 10] 10)
    79 
    80 translations
    81   "UN i<=n. A"  == "UN i:{..n}. A"
    82   "UN i<n. A"   == "UN i:{..<n}. A"
    83   "INT i<=n. A" == "INT i:{..n}. A"
    84   "INT i<n. A"  == "INT i:{..<n}. A"
    85 
    86 
    87 subsection {* Various equivalences *}
    88 
    89 lemma (in ord) lessThan_iff [iff]: "(i: lessThan k) = (i<k)"
    90 by (simp add: lessThan_def)
    91 
    92 lemma Compl_lessThan [simp]:
    93     "!!k:: 'a::linorder. -lessThan k = atLeast k"
    94 apply (auto simp add: lessThan_def atLeast_def)
    95 done
    96 
    97 lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"
    98 by auto
    99 
   100 lemma (in ord) greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"
   101 by (simp add: greaterThan_def)
   102 
   103 lemma Compl_greaterThan [simp]:
   104     "!!k:: 'a::linorder. -greaterThan k = atMost k"
   105   by (auto simp add: greaterThan_def atMost_def)
   106 
   107 lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
   108 apply (subst Compl_greaterThan [symmetric])
   109 apply (rule double_complement)
   110 done
   111 
   112 lemma (in ord) atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
   113 by (simp add: atLeast_def)
   114 
   115 lemma Compl_atLeast [simp]:
   116     "!!k:: 'a::linorder. -atLeast k = lessThan k"
   117   by (auto simp add: lessThan_def atLeast_def)
   118 
   119 lemma (in ord) 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 lemma (in linorder) lessThan_Int_lessThan: "{ a <..} \<inter> { b <..} = { max a b <..}"
   126   by auto
   127 
   128 lemma (in linorder) greaterThan_Int_greaterThan: "{..< a} \<inter> {..< b} = {..< min a b}"
   129   by auto
   130 
   131 subsection {* Logical Equivalences for Set Inclusion and Equality *}
   132 
   133 lemma atLeast_subset_iff [iff]:
   134      "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::order))"
   135 by (blast intro: order_trans)
   136 
   137 lemma atLeast_eq_iff [iff]:
   138      "(atLeast x = atLeast y) = (x = (y::'a::linorder))"
   139 by (blast intro: order_antisym order_trans)
   140 
   141 lemma greaterThan_subset_iff [iff]:
   142      "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
   143 apply (auto simp add: greaterThan_def)
   144  apply (subst linorder_not_less [symmetric], blast)
   145 done
   146 
   147 lemma greaterThan_eq_iff [iff]:
   148      "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
   149 apply (rule iffI)
   150  apply (erule equalityE)
   151  apply simp_all
   152 done
   153 
   154 lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::order))"
   155 by (blast intro: order_trans)
   156 
   157 lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))"
   158 by (blast intro: order_antisym order_trans)
   159 
   160 lemma lessThan_subset_iff [iff]:
   161      "(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))"
   162 apply (auto simp add: lessThan_def)
   163  apply (subst linorder_not_less [symmetric], blast)
   164 done
   165 
   166 lemma lessThan_eq_iff [iff]:
   167      "(lessThan x = lessThan y) = (x = (y::'a::linorder))"
   168 apply (rule iffI)
   169  apply (erule equalityE)
   170  apply simp_all
   171 done
   172 
   173 lemma lessThan_strict_subset_iff:
   174   fixes m n :: "'a::linorder"
   175   shows "{..<m} < {..<n} \<longleftrightarrow> m < n"
   176   by (metis leD lessThan_subset_iff linorder_linear not_less_iff_gr_or_eq psubset_eq)
   177 
   178 subsection {*Two-sided intervals*}
   179 
   180 context ord
   181 begin
   182 
   183 lemma greaterThanLessThan_iff [simp]:
   184   "(i : {l<..<u}) = (l < i & i < u)"
   185 by (simp add: greaterThanLessThan_def)
   186 
   187 lemma atLeastLessThan_iff [simp]:
   188   "(i : {l..<u}) = (l <= i & i < u)"
   189 by (simp add: atLeastLessThan_def)
   190 
   191 lemma greaterThanAtMost_iff [simp]:
   192   "(i : {l<..u}) = (l < i & i <= u)"
   193 by (simp add: greaterThanAtMost_def)
   194 
   195 lemma atLeastAtMost_iff [simp]:
   196   "(i : {l..u}) = (l <= i & i <= u)"
   197 by (simp add: atLeastAtMost_def)
   198 
   199 text {* The above four lemmas could be declared as iffs. Unfortunately this
   200 breaks many proofs. Since it only helps blast, it is better to leave them
   201 alone. *}
   202 
   203 lemma greaterThanLessThan_eq: "{ a <..< b} = { a <..} \<inter> {..< b }"
   204   by auto
   205 
   206 end
   207 
   208 subsubsection{* Emptyness, singletons, subset *}
   209 
   210 context order
   211 begin
   212 
   213 lemma atLeastatMost_empty[simp]:
   214   "b < a \<Longrightarrow> {a..b} = {}"
   215 by(auto simp: atLeastAtMost_def atLeast_def atMost_def)
   216 
   217 lemma atLeastatMost_empty_iff[simp]:
   218   "{a..b} = {} \<longleftrightarrow> (~ a <= b)"
   219 by auto (blast intro: order_trans)
   220 
   221 lemma atLeastatMost_empty_iff2[simp]:
   222   "{} = {a..b} \<longleftrightarrow> (~ a <= b)"
   223 by auto (blast intro: order_trans)
   224 
   225 lemma atLeastLessThan_empty[simp]:
   226   "b <= a \<Longrightarrow> {a..<b} = {}"
   227 by(auto simp: atLeastLessThan_def)
   228 
   229 lemma atLeastLessThan_empty_iff[simp]:
   230   "{a..<b} = {} \<longleftrightarrow> (~ a < b)"
   231 by auto (blast intro: le_less_trans)
   232 
   233 lemma atLeastLessThan_empty_iff2[simp]:
   234   "{} = {a..<b} \<longleftrightarrow> (~ a < b)"
   235 by auto (blast intro: le_less_trans)
   236 
   237 lemma greaterThanAtMost_empty[simp]: "l \<le> k ==> {k<..l} = {}"
   238 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)
   239 
   240 lemma greaterThanAtMost_empty_iff[simp]: "{k<..l} = {} \<longleftrightarrow> ~ k < l"
   241 by auto (blast intro: less_le_trans)
   242 
   243 lemma greaterThanAtMost_empty_iff2[simp]: "{} = {k<..l} \<longleftrightarrow> ~ k < l"
   244 by auto (blast intro: less_le_trans)
   245 
   246 lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..<l} = {}"
   247 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)
   248 
   249 lemma atLeastAtMost_singleton [simp]: "{a..a} = {a}"
   250 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)
   251 
   252 lemma atLeastAtMost_singleton': "a = b \<Longrightarrow> {a .. b} = {a}" by simp
   253 
   254 lemma atLeastatMost_subset_iff[simp]:
   255   "{a..b} <= {c..d} \<longleftrightarrow> (~ a <= b) | c <= a & b <= d"
   256 unfolding atLeastAtMost_def atLeast_def atMost_def
   257 by (blast intro: order_trans)
   258 
   259 lemma atLeastatMost_psubset_iff:
   260   "{a..b} < {c..d} \<longleftrightarrow>
   261    ((~ a <= b) | c <= a & b <= d & (c < a | b < d))  &  c <= d"
   262 by(simp add: psubset_eq set_eq_iff less_le_not_le)(blast intro: order_trans)
   263 
   264 lemma Icc_eq_Icc[simp]:
   265   "{l..h} = {l'..h'} = (l=l' \<and> h=h' \<or> \<not> l\<le>h \<and> \<not> l'\<le>h')"
   266 by(simp add: order_class.eq_iff)(auto intro: order_trans)
   267 
   268 lemma atLeastAtMost_singleton_iff[simp]:
   269   "{a .. b} = {c} \<longleftrightarrow> a = b \<and> b = c"
   270 proof
   271   assume "{a..b} = {c}"
   272   hence *: "\<not> (\<not> a \<le> b)" unfolding atLeastatMost_empty_iff[symmetric] by simp
   273   with `{a..b} = {c}` have "c \<le> a \<and> b \<le> c" by auto
   274   with * show "a = b \<and> b = c" by auto
   275 qed simp
   276 
   277 lemma Icc_subset_Ici_iff[simp]:
   278   "{l..h} \<subseteq> {l'..} = (~ l\<le>h \<or> l\<ge>l')"
   279 by(auto simp: subset_eq intro: order_trans)
   280 
   281 lemma Icc_subset_Iic_iff[simp]:
   282   "{l..h} \<subseteq> {..h'} = (~ l\<le>h \<or> h\<le>h')"
   283 by(auto simp: subset_eq intro: order_trans)
   284 
   285 lemma not_Ici_eq_empty[simp]: "{l..} \<noteq> {}"
   286 by(auto simp: set_eq_iff)
   287 
   288 lemma not_Iic_eq_empty[simp]: "{..h} \<noteq> {}"
   289 by(auto simp: set_eq_iff)
   290 
   291 lemmas not_empty_eq_Ici_eq_empty[simp] = not_Ici_eq_empty[symmetric]
   292 lemmas not_empty_eq_Iic_eq_empty[simp] = not_Iic_eq_empty[symmetric]
   293 
   294 end
   295 
   296 context no_top
   297 begin
   298 
   299 (* also holds for no_bot but no_top should suffice *)
   300 lemma not_UNIV_le_Icc[simp]: "\<not> UNIV \<subseteq> {l..h}"
   301 using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)
   302 
   303 lemma not_UNIV_le_Iic[simp]: "\<not> UNIV \<subseteq> {..h}"
   304 using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)
   305 
   306 lemma not_Ici_le_Icc[simp]: "\<not> {l..} \<subseteq> {l'..h'}"
   307 using gt_ex[of h']
   308 by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
   309 
   310 lemma not_Ici_le_Iic[simp]: "\<not> {l..} \<subseteq> {..h'}"
   311 using gt_ex[of h']
   312 by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
   313 
   314 end
   315 
   316 context no_bot
   317 begin
   318 
   319 lemma not_UNIV_le_Ici[simp]: "\<not> UNIV \<subseteq> {l..}"
   320 using lt_ex[of l] by(auto simp: subset_eq less_le_not_le)
   321 
   322 lemma not_Iic_le_Icc[simp]: "\<not> {..h} \<subseteq> {l'..h'}"
   323 using lt_ex[of l']
   324 by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
   325 
   326 lemma not_Iic_le_Ici[simp]: "\<not> {..h} \<subseteq> {l'..}"
   327 using lt_ex[of l']
   328 by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
   329 
   330 end
   331 
   332 
   333 context no_top
   334 begin
   335 
   336 (* also holds for no_bot but no_top should suffice *)
   337 lemma not_UNIV_eq_Icc[simp]: "\<not> UNIV = {l'..h'}"
   338 using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)
   339 
   340 lemmas not_Icc_eq_UNIV[simp] = not_UNIV_eq_Icc[symmetric]
   341 
   342 lemma not_UNIV_eq_Iic[simp]: "\<not> UNIV = {..h'}"
   343 using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)
   344 
   345 lemmas not_Iic_eq_UNIV[simp] = not_UNIV_eq_Iic[symmetric]
   346 
   347 lemma not_Icc_eq_Ici[simp]: "\<not> {l..h} = {l'..}"
   348 unfolding atLeastAtMost_def using not_Ici_le_Iic[of l'] by blast
   349 
   350 lemmas not_Ici_eq_Icc[simp] = not_Icc_eq_Ici[symmetric]
   351 
   352 (* also holds for no_bot but no_top should suffice *)
   353 lemma not_Iic_eq_Ici[simp]: "\<not> {..h} = {l'..}"
   354 using not_Ici_le_Iic[of l' h] by blast
   355 
   356 lemmas not_Ici_eq_Iic[simp] = not_Iic_eq_Ici[symmetric]
   357 
   358 end
   359 
   360 context no_bot
   361 begin
   362 
   363 lemma not_UNIV_eq_Ici[simp]: "\<not> UNIV = {l'..}"
   364 using lt_ex[of l'] by(auto simp: set_eq_iff  less_le_not_le)
   365 
   366 lemmas not_Ici_eq_UNIV[simp] = not_UNIV_eq_Ici[symmetric]
   367 
   368 lemma not_Icc_eq_Iic[simp]: "\<not> {l..h} = {..h'}"
   369 unfolding atLeastAtMost_def using not_Iic_le_Ici[of h'] by blast
   370 
   371 lemmas not_Iic_eq_Icc[simp] = not_Icc_eq_Iic[symmetric]
   372 
   373 end
   374 
   375 
   376 context dense_linorder
   377 begin
   378 
   379 lemma greaterThanLessThan_empty_iff[simp]:
   380   "{ a <..< b } = {} \<longleftrightarrow> b \<le> a"
   381   using dense[of a b] by (cases "a < b") auto
   382 
   383 lemma greaterThanLessThan_empty_iff2[simp]:
   384   "{} = { a <..< b } \<longleftrightarrow> b \<le> a"
   385   using dense[of a b] by (cases "a < b") auto
   386 
   387 lemma atLeastLessThan_subseteq_atLeastAtMost_iff:
   388   "{a ..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
   389   using dense[of "max a d" "b"]
   390   by (force simp: subset_eq Ball_def not_less[symmetric])
   391 
   392 lemma greaterThanAtMost_subseteq_atLeastAtMost_iff:
   393   "{a <.. b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
   394   using dense[of "a" "min c b"]
   395   by (force simp: subset_eq Ball_def not_less[symmetric])
   396 
   397 lemma greaterThanLessThan_subseteq_atLeastAtMost_iff:
   398   "{a <..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
   399   using dense[of "a" "min c b"] dense[of "max a d" "b"]
   400   by (force simp: subset_eq Ball_def not_less[symmetric])
   401 
   402 lemma atLeastAtMost_subseteq_atLeastLessThan_iff:
   403   "{a .. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a \<le> b \<longrightarrow> c \<le> a \<and> b < d)"
   404   using dense[of "max a d" "b"]
   405   by (force simp: subset_eq Ball_def not_less[symmetric])
   406 
   407 lemma greaterThanAtMost_subseteq_atLeastLessThan_iff:
   408   "{a <.. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b < d)"
   409   using dense[of "a" "min c b"]
   410   by (force simp: subset_eq Ball_def not_less[symmetric])
   411 
   412 lemma greaterThanLessThan_subseteq_atLeastLessThan_iff:
   413   "{a <..< b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
   414   using dense[of "a" "min c b"] dense[of "max a d" "b"]
   415   by (force simp: subset_eq Ball_def not_less[symmetric])
   416 
   417 end
   418 
   419 context no_top
   420 begin
   421 
   422 lemma greaterThan_non_empty[simp]: "{x <..} \<noteq> {}"
   423   using gt_ex[of x] by auto
   424 
   425 end
   426 
   427 context no_bot
   428 begin
   429 
   430 lemma lessThan_non_empty[simp]: "{..< x} \<noteq> {}"
   431   using lt_ex[of x] by auto
   432 
   433 end
   434 
   435 lemma (in linorder) atLeastLessThan_subset_iff:
   436   "{a..<b} <= {c..<d} \<Longrightarrow> b <= a | c<=a & b<=d"
   437 apply (auto simp:subset_eq Ball_def)
   438 apply(frule_tac x=a in spec)
   439 apply(erule_tac x=d in allE)
   440 apply (simp add: less_imp_le)
   441 done
   442 
   443 lemma atLeastLessThan_inj:
   444   fixes a b c d :: "'a::linorder"
   445   assumes eq: "{a ..< b} = {c ..< d}" and "a < b" "c < d"
   446   shows "a = c" "b = d"
   447 using assms by (metis atLeastLessThan_subset_iff eq less_le_not_le linorder_antisym_conv2 subset_refl)+
   448 
   449 lemma atLeastLessThan_eq_iff:
   450   fixes a b c d :: "'a::linorder"
   451   assumes "a < b" "c < d"
   452   shows "{a ..< b} = {c ..< d} \<longleftrightarrow> a = c \<and> b = d"
   453   using atLeastLessThan_inj assms by auto
   454 
   455 lemma (in order_bot) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot"
   456 by (auto simp: set_eq_iff intro: le_bot)
   457 
   458 lemma (in order_top) atMost_eq_UNIV_iff: "{..x} = UNIV \<longleftrightarrow> x = top"
   459 by (auto simp: set_eq_iff intro: top_le)
   460 
   461 lemma (in bounded_lattice) atLeastAtMost_eq_UNIV_iff:
   462   "{x..y} = UNIV \<longleftrightarrow> (x = bot \<and> y = top)"
   463 by (auto simp: set_eq_iff intro: top_le le_bot)
   464 
   465 
   466 subsubsection {* Intersection *}
   467 
   468 context linorder
   469 begin
   470 
   471 lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"
   472 by auto
   473 
   474 lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"
   475 by auto
   476 
   477 lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"
   478 by auto
   479 
   480 lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"
   481 by auto
   482 
   483 lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"
   484 by auto
   485 
   486 lemma Int_atLeastLessThan[simp]: "{a..<b} Int {c..<d} = {max a c ..< min b d}"
   487 by auto
   488 
   489 lemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"
   490 by auto
   491 
   492 lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"
   493 by auto
   494 
   495 lemma Int_atMost[simp]: "{..a} \<inter> {..b} = {.. min a b}"
   496   by (auto simp: min_def)
   497 
   498 end
   499 
   500 context complete_lattice
   501 begin
   502 
   503 lemma
   504   shows Sup_atLeast[simp]: "Sup {x ..} = top"
   505     and Sup_greaterThanAtLeast[simp]: "x < top \<Longrightarrow> Sup {x <..} = top"
   506     and Sup_atMost[simp]: "Sup {.. y} = y"
   507     and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y"
   508     and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y"
   509   by (auto intro!: Sup_eqI)
   510 
   511 lemma
   512   shows Inf_atMost[simp]: "Inf {.. x} = bot"
   513     and Inf_atMostLessThan[simp]: "top < x \<Longrightarrow> Inf {..< x} = bot"
   514     and Inf_atLeast[simp]: "Inf {x ..} = x"
   515     and Inf_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Inf { x .. y} = x"
   516     and Inf_atLeastLessThan[simp]: "x < y \<Longrightarrow> Inf { x ..< y} = x"
   517   by (auto intro!: Inf_eqI)
   518 
   519 end
   520 
   521 lemma
   522   fixes x y :: "'a :: {complete_lattice, dense_linorder}"
   523   shows Sup_lessThan[simp]: "Sup {..< y} = y"
   524     and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y"
   525     and Sup_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Sup { x <..< y} = y"
   526     and Inf_greaterThan[simp]: "Inf {x <..} = x"
   527     and Inf_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Inf { x <.. y} = x"
   528     and Inf_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Inf { x <..< y} = x"
   529   by (auto intro!: Inf_eqI Sup_eqI intro: dense_le dense_le_bounded dense_ge dense_ge_bounded)
   530 
   531 subsection {* Intervals of natural numbers *}
   532 
   533 subsubsection {* The Constant @{term lessThan} *}
   534 
   535 lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
   536 by (simp add: lessThan_def)
   537 
   538 lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
   539 by (simp add: lessThan_def less_Suc_eq, blast)
   540 
   541 text {* The following proof is convenient in induction proofs where
   542 new elements get indices at the beginning. So it is used to transform
   543 @{term "{..<Suc n}"} to @{term "0::nat"} and @{term "{..< n}"}. *}
   544 
   545 lemma lessThan_Suc_eq_insert_0: "{..<Suc n} = insert 0 (Suc ` {..<n})"
   546 proof safe
   547   fix x assume "x < Suc n" "x \<notin> Suc ` {..<n}"
   548   then have "x \<noteq> Suc (x - 1)" by auto
   549   with `x < Suc n` show "x = 0" by auto
   550 qed
   551 
   552 lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
   553 by (simp add: lessThan_def atMost_def less_Suc_eq_le)
   554 
   555 lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
   556 by blast
   557 
   558 subsubsection {* The Constant @{term greaterThan} *}
   559 
   560 lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
   561 apply (simp add: greaterThan_def)
   562 apply (blast dest: gr0_conv_Suc [THEN iffD1])
   563 done
   564 
   565 lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
   566 apply (simp add: greaterThan_def)
   567 apply (auto elim: linorder_neqE)
   568 done
   569 
   570 lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
   571 by blast
   572 
   573 subsubsection {* The Constant @{term atLeast} *}
   574 
   575 lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
   576 by (unfold atLeast_def UNIV_def, simp)
   577 
   578 lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
   579 apply (simp add: atLeast_def)
   580 apply (simp add: Suc_le_eq)
   581 apply (simp add: order_le_less, blast)
   582 done
   583 
   584 lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
   585   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
   586 
   587 lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
   588 by blast
   589 
   590 subsubsection {* The Constant @{term atMost} *}
   591 
   592 lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
   593 by (simp add: atMost_def)
   594 
   595 lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
   596 apply (simp add: atMost_def)
   597 apply (simp add: less_Suc_eq order_le_less, blast)
   598 done
   599 
   600 lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
   601 by blast
   602 
   603 subsubsection {* The Constant @{term atLeastLessThan} *}
   604 
   605 text{*The orientation of the following 2 rules is tricky. The lhs is
   606 defined in terms of the rhs.  Hence the chosen orientation makes sense
   607 in this theory --- the reverse orientation complicates proofs (eg
   608 nontermination). But outside, when the definition of the lhs is rarely
   609 used, the opposite orientation seems preferable because it reduces a
   610 specific concept to a more general one. *}
   611 
   612 lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
   613 by(simp add:lessThan_def atLeastLessThan_def)
   614 
   615 lemma atLeast0AtMost: "{0..n::nat} = {..n}"
   616 by(simp add:atMost_def atLeastAtMost_def)
   617 
   618 declare atLeast0LessThan[symmetric, code_unfold]
   619         atLeast0AtMost[symmetric, code_unfold]
   620 
   621 lemma atLeastLessThan0: "{m..<0::nat} = {}"
   622 by (simp add: atLeastLessThan_def)
   623 
   624 subsubsection {* Intervals of nats with @{term Suc} *}
   625 
   626 text{*Not a simprule because the RHS is too messy.*}
   627 lemma atLeastLessThanSuc:
   628     "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
   629 by (auto simp add: atLeastLessThan_def)
   630 
   631 lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
   632 by (auto simp add: atLeastLessThan_def)
   633 (*
   634 lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
   635 by (induct k, simp_all add: atLeastLessThanSuc)
   636 
   637 lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
   638 by (auto simp add: atLeastLessThan_def)
   639 *)
   640 lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
   641   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
   642 
   643 lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
   644   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
   645     greaterThanAtMost_def)
   646 
   647 lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
   648   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
   649     greaterThanLessThan_def)
   650 
   651 lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
   652 by (auto simp add: atLeastAtMost_def)
   653 
   654 lemma atLeastAtMost_insertL: "m \<le> n \<Longrightarrow> insert m {Suc m..n} = {m ..n}"
   655 by auto
   656 
   657 text {* The analogous result is useful on @{typ int}: *}
   658 (* here, because we don't have an own int section *)
   659 lemma atLeastAtMostPlus1_int_conv:
   660   "m <= 1+n \<Longrightarrow> {m..1+n} = insert (1+n) {m..n::int}"
   661   by (auto intro: set_eqI)
   662 
   663 lemma atLeastLessThan_add_Un: "i \<le> j \<Longrightarrow> {i..<j+k} = {i..<j} \<union> {j..<j+k::nat}"
   664   apply (induct k) 
   665   apply (simp_all add: atLeastLessThanSuc)   
   666   done
   667 
   668 subsubsection {* Image *}
   669 
   670 lemma image_add_atLeastAtMost:
   671   "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
   672 proof
   673   show "?A \<subseteq> ?B" by auto
   674 next
   675   show "?B \<subseteq> ?A"
   676   proof
   677     fix n assume a: "n : ?B"
   678     hence "n - k : {i..j}" by auto
   679     moreover have "n = (n - k) + k" using a by auto
   680     ultimately show "n : ?A" by blast
   681   qed
   682 qed
   683 
   684 lemma image_add_atLeastLessThan:
   685   "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
   686 proof
   687   show "?A \<subseteq> ?B" by auto
   688 next
   689   show "?B \<subseteq> ?A"
   690   proof
   691     fix n assume a: "n : ?B"
   692     hence "n - k : {i..<j}" by auto
   693     moreover have "n = (n - k) + k" using a by auto
   694     ultimately show "n : ?A" by blast
   695   qed
   696 qed
   697 
   698 corollary image_Suc_atLeastAtMost[simp]:
   699   "Suc ` {i..j} = {Suc i..Suc j}"
   700 using image_add_atLeastAtMost[where k="Suc 0"] by simp
   701 
   702 corollary image_Suc_atLeastLessThan[simp]:
   703   "Suc ` {i..<j} = {Suc i..<Suc j}"
   704 using image_add_atLeastLessThan[where k="Suc 0"] by simp
   705 
   706 lemma image_add_int_atLeastLessThan:
   707     "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
   708   apply (auto simp add: image_def)
   709   apply (rule_tac x = "x - l" in bexI)
   710   apply auto
   711   done
   712 
   713 lemma image_minus_const_atLeastLessThan_nat:
   714   fixes c :: nat
   715   shows "(\<lambda>i. i - c) ` {x ..< y} =
   716       (if c < y then {x - c ..< y - c} else if x < y then {0} else {})"
   717     (is "_ = ?right")
   718 proof safe
   719   fix a assume a: "a \<in> ?right"
   720   show "a \<in> (\<lambda>i. i - c) ` {x ..< y}"
   721   proof cases
   722     assume "c < y" with a show ?thesis
   723       by (auto intro!: image_eqI[of _ _ "a + c"])
   724   next
   725     assume "\<not> c < y" with a show ?thesis
   726       by (auto intro!: image_eqI[of _ _ x] split: split_if_asm)
   727   qed
   728 qed auto
   729 
   730 lemma image_int_atLeastLessThan: "int ` {a..<b} = {int a..<int b}"
   731   by (auto intro!: image_eqI [where x = "nat x" for x])
   732 
   733 context ordered_ab_group_add
   734 begin
   735 
   736 lemma
   737   fixes x :: 'a
   738   shows image_uminus_greaterThan[simp]: "uminus ` {x<..} = {..<-x}"
   739   and image_uminus_atLeast[simp]: "uminus ` {x..} = {..-x}"
   740 proof safe
   741   fix y assume "y < -x"
   742   hence *:  "x < -y" using neg_less_iff_less[of "-y" x] by simp
   743   have "- (-y) \<in> uminus ` {x<..}"
   744     by (rule imageI) (simp add: *)
   745   thus "y \<in> uminus ` {x<..}" by simp
   746 next
   747   fix y assume "y \<le> -x"
   748   have "- (-y) \<in> uminus ` {x..}"
   749     by (rule imageI) (insert `y \<le> -x`[THEN le_imp_neg_le], simp)
   750   thus "y \<in> uminus ` {x..}" by simp
   751 qed simp_all
   752 
   753 lemma
   754   fixes x :: 'a
   755   shows image_uminus_lessThan[simp]: "uminus ` {..<x} = {-x<..}"
   756   and image_uminus_atMost[simp]: "uminus ` {..x} = {-x..}"
   757 proof -
   758   have "uminus ` {..<x} = uminus ` uminus ` {-x<..}"
   759     and "uminus ` {..x} = uminus ` uminus ` {-x..}" by simp_all
   760   thus "uminus ` {..<x} = {-x<..}" and "uminus ` {..x} = {-x..}"
   761     by (simp_all add: image_image
   762         del: image_uminus_greaterThan image_uminus_atLeast)
   763 qed
   764 
   765 lemma
   766   fixes x :: 'a
   767   shows image_uminus_atLeastAtMost[simp]: "uminus ` {x..y} = {-y..-x}"
   768   and image_uminus_greaterThanAtMost[simp]: "uminus ` {x<..y} = {-y..<-x}"
   769   and image_uminus_atLeastLessThan[simp]: "uminus ` {x..<y} = {-y<..-x}"
   770   and image_uminus_greaterThanLessThan[simp]: "uminus ` {x<..<y} = {-y<..<-x}"
   771   by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def
   772       greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)
   773 end
   774 
   775 subsubsection {* Finiteness *}
   776 
   777 lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
   778   by (induct k) (simp_all add: lessThan_Suc)
   779 
   780 lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
   781   by (induct k) (simp_all add: atMost_Suc)
   782 
   783 lemma finite_greaterThanLessThan [iff]:
   784   fixes l :: nat shows "finite {l<..<u}"
   785 by (simp add: greaterThanLessThan_def)
   786 
   787 lemma finite_atLeastLessThan [iff]:
   788   fixes l :: nat shows "finite {l..<u}"
   789 by (simp add: atLeastLessThan_def)
   790 
   791 lemma finite_greaterThanAtMost [iff]:
   792   fixes l :: nat shows "finite {l<..u}"
   793 by (simp add: greaterThanAtMost_def)
   794 
   795 lemma finite_atLeastAtMost [iff]:
   796   fixes l :: nat shows "finite {l..u}"
   797 by (simp add: atLeastAtMost_def)
   798 
   799 text {* A bounded set of natural numbers is finite. *}
   800 lemma bounded_nat_set_is_finite:
   801   "(ALL i:N. i < (n::nat)) ==> finite N"
   802 apply (rule finite_subset)
   803  apply (rule_tac [2] finite_lessThan, auto)
   804 done
   805 
   806 text {* A set of natural numbers is finite iff it is bounded. *}
   807 lemma finite_nat_set_iff_bounded:
   808   "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")
   809 proof
   810   assume f:?F  show ?B
   811     using Max_ge[OF `?F`, simplified less_Suc_eq_le[symmetric]] by blast
   812 next
   813   assume ?B show ?F using `?B` by(blast intro:bounded_nat_set_is_finite)
   814 qed
   815 
   816 lemma finite_nat_set_iff_bounded_le:
   817   "finite(N::nat set) = (EX m. ALL n:N. n<=m)"
   818 apply(simp add:finite_nat_set_iff_bounded)
   819 apply(blast dest:less_imp_le_nat le_imp_less_Suc)
   820 done
   821 
   822 lemma finite_less_ub:
   823      "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
   824 by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
   825 
   826 text{* Any subset of an interval of natural numbers the size of the
   827 subset is exactly that interval. *}
   828 
   829 lemma subset_card_intvl_is_intvl:
   830   assumes "A \<subseteq> {k..<k + card A}"
   831   shows "A = {k..<k + card A}"
   832 proof (cases "finite A")
   833   case True
   834   from this and assms show ?thesis
   835   proof (induct A rule: finite_linorder_max_induct)
   836     case empty thus ?case by auto
   837   next
   838     case (insert b A)
   839     hence *: "b \<notin> A" by auto
   840     with insert have "A <= {k..<k + card A}" and "b = k + card A"
   841       by fastforce+
   842     with insert * show ?case by auto
   843   qed
   844 next
   845   case False
   846   with assms show ?thesis by simp
   847 qed
   848 
   849 
   850 subsubsection {* Proving Inclusions and Equalities between Unions *}
   851 
   852 lemma UN_le_eq_Un0:
   853   "(\<Union>i\<le>n::nat. M i) = (\<Union>i\<in>{1..n}. M i) \<union> M 0" (is "?A = ?B")
   854 proof
   855   show "?A <= ?B"
   856   proof
   857     fix x assume "x : ?A"
   858     then obtain i where i: "i\<le>n" "x : M i" by auto
   859     show "x : ?B"
   860     proof(cases i)
   861       case 0 with i show ?thesis by simp
   862     next
   863       case (Suc j) with i show ?thesis by auto
   864     qed
   865   qed
   866 next
   867   show "?B <= ?A" by auto
   868 qed
   869 
   870 lemma UN_le_add_shift:
   871   "(\<Union>i\<le>n::nat. M(i+k)) = (\<Union>i\<in>{k..n+k}. M i)" (is "?A = ?B")
   872 proof
   873   show "?A <= ?B" by fastforce
   874 next
   875   show "?B <= ?A"
   876   proof
   877     fix x assume "x : ?B"
   878     then obtain i where i: "i : {k..n+k}" "x : M(i)" by auto
   879     hence "i-k\<le>n & x : M((i-k)+k)" by auto
   880     thus "x : ?A" by blast
   881   qed
   882 qed
   883 
   884 lemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"
   885   by (auto simp add: atLeast0LessThan) 
   886 
   887 lemma UN_finite_subset: "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> C) \<Longrightarrow> (\<Union>n. A n) \<subseteq> C"
   888   by (subst UN_UN_finite_eq [symmetric]) blast
   889 
   890 lemma UN_finite2_subset: 
   891      "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> (\<Union>i\<in>{0..<n+k}. B i)) \<Longrightarrow> (\<Union>n. A n) \<subseteq> (\<Union>n. B n)"
   892   apply (rule UN_finite_subset)
   893   apply (subst UN_UN_finite_eq [symmetric, of B]) 
   894   apply blast
   895   done
   896 
   897 lemma UN_finite2_eq:
   898   "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) = (\<Union>i\<in>{0..<n+k}. B i)) \<Longrightarrow> (\<Union>n. A n) = (\<Union>n. B n)"
   899   apply (rule subset_antisym)
   900    apply (rule UN_finite2_subset, blast)
   901  apply (rule UN_finite2_subset [where k=k])
   902  apply (force simp add: atLeastLessThan_add_Un [of 0])
   903  done
   904 
   905 
   906 subsubsection {* Cardinality *}
   907 
   908 lemma card_lessThan [simp]: "card {..<u} = u"
   909   by (induct u, simp_all add: lessThan_Suc)
   910 
   911 lemma card_atMost [simp]: "card {..u} = Suc u"
   912   by (simp add: lessThan_Suc_atMost [THEN sym])
   913 
   914 lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
   915   apply (subgoal_tac "card {l..<u} = card {..<u-l}")
   916   apply (erule ssubst, rule card_lessThan)
   917   apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
   918   apply (erule subst)
   919   apply (rule card_image)
   920   apply (simp add: inj_on_def)
   921   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
   922   apply (rule_tac x = "x - l" in exI)
   923   apply arith
   924   done
   925 
   926 lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
   927   by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
   928 
   929 lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
   930   by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
   931 
   932 lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
   933   by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
   934 
   935 lemma ex_bij_betw_nat_finite:
   936   "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
   937 apply(drule finite_imp_nat_seg_image_inj_on)
   938 apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
   939 done
   940 
   941 lemma ex_bij_betw_finite_nat:
   942   "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
   943 by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
   944 
   945 lemma finite_same_card_bij:
   946   "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"
   947 apply(drule ex_bij_betw_finite_nat)
   948 apply(drule ex_bij_betw_nat_finite)
   949 apply(auto intro!:bij_betw_trans)
   950 done
   951 
   952 lemma ex_bij_betw_nat_finite_1:
   953   "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
   954 by (rule finite_same_card_bij) auto
   955 
   956 lemma bij_betw_iff_card:
   957   assumes FIN: "finite A" and FIN': "finite B"
   958   shows BIJ: "(\<exists>f. bij_betw f A B) \<longleftrightarrow> (card A = card B)"
   959 using assms
   960 proof(auto simp add: bij_betw_same_card)
   961   assume *: "card A = card B"
   962   obtain f where "bij_betw f A {0 ..< card A}"
   963   using FIN ex_bij_betw_finite_nat by blast
   964   moreover obtain g where "bij_betw g {0 ..< card B} B"
   965   using FIN' ex_bij_betw_nat_finite by blast
   966   ultimately have "bij_betw (g o f) A B"
   967   using * by (auto simp add: bij_betw_trans)
   968   thus "(\<exists>f. bij_betw f A B)" by blast
   969 qed
   970 
   971 lemma inj_on_iff_card_le:
   972   assumes FIN: "finite A" and FIN': "finite B"
   973   shows "(\<exists>f. inj_on f A \<and> f ` A \<le> B) = (card A \<le> card B)"
   974 proof (safe intro!: card_inj_on_le)
   975   assume *: "card A \<le> card B"
   976   obtain f where 1: "inj_on f A" and 2: "f ` A = {0 ..< card A}"
   977   using FIN ex_bij_betw_finite_nat unfolding bij_betw_def by force
   978   moreover obtain g where "inj_on g {0 ..< card B}" and 3: "g ` {0 ..< card B} = B"
   979   using FIN' ex_bij_betw_nat_finite unfolding bij_betw_def by force
   980   ultimately have "inj_on g (f ` A)" using subset_inj_on[of g _ "f ` A"] * by force
   981   hence "inj_on (g o f) A" using 1 comp_inj_on by blast
   982   moreover
   983   {have "{0 ..< card A} \<le> {0 ..< card B}" using * by force
   984    with 2 have "f ` A  \<le> {0 ..< card B}" by blast
   985    hence "(g o f) ` A \<le> B" unfolding comp_def using 3 by force
   986   }
   987   ultimately show "(\<exists>f. inj_on f A \<and> f ` A \<le> B)" by blast
   988 qed (insert assms, auto)
   989 
   990 subsection {* Intervals of integers *}
   991 
   992 lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
   993   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
   994 
   995 lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
   996   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
   997 
   998 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
   999     "{l+1..<u} = {l<..<u::int}"
  1000   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
  1001 
  1002 subsubsection {* Finiteness *}
  1003 
  1004 lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
  1005     {(0::int)..<u} = int ` {..<nat u}"
  1006   apply (unfold image_def lessThan_def)
  1007   apply auto
  1008   apply (rule_tac x = "nat x" in exI)
  1009   apply (auto simp add: zless_nat_eq_int_zless [THEN sym])
  1010   done
  1011 
  1012 lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
  1013   apply (cases "0 \<le> u")
  1014   apply (subst image_atLeastZeroLessThan_int, assumption)
  1015   apply (rule finite_imageI)
  1016   apply auto
  1017   done
  1018 
  1019 lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
  1020   apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
  1021   apply (erule subst)
  1022   apply (rule finite_imageI)
  1023   apply (rule finite_atLeastZeroLessThan_int)
  1024   apply (rule image_add_int_atLeastLessThan)
  1025   done
  1026 
  1027 lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
  1028   by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
  1029 
  1030 lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
  1031   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
  1032 
  1033 lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
  1034   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
  1035 
  1036 
  1037 subsubsection {* Cardinality *}
  1038 
  1039 lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
  1040   apply (cases "0 \<le> u")
  1041   apply (subst image_atLeastZeroLessThan_int, assumption)
  1042   apply (subst card_image)
  1043   apply (auto simp add: inj_on_def)
  1044   done
  1045 
  1046 lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
  1047   apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
  1048   apply (erule ssubst, rule card_atLeastZeroLessThan_int)
  1049   apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
  1050   apply (erule subst)
  1051   apply (rule card_image)
  1052   apply (simp add: inj_on_def)
  1053   apply (rule image_add_int_atLeastLessThan)
  1054   done
  1055 
  1056 lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
  1057 apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
  1058 apply (auto simp add: algebra_simps)
  1059 done
  1060 
  1061 lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
  1062 by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
  1063 
  1064 lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
  1065 by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
  1066 
  1067 lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
  1068 proof -
  1069   have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
  1070   with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
  1071 qed
  1072 
  1073 lemma card_less:
  1074 assumes zero_in_M: "0 \<in> M"
  1075 shows "card {k \<in> M. k < Suc i} \<noteq> 0"
  1076 proof -
  1077   from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
  1078   with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
  1079 qed
  1080 
  1081 lemma card_less_Suc2: "0 \<notin> M \<Longrightarrow> card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"
  1082 apply (rule card_bij_eq [of Suc _ _ "\<lambda>x. x - Suc 0"])
  1083 apply simp
  1084 apply fastforce
  1085 apply auto
  1086 apply (rule inj_on_diff_nat)
  1087 apply auto
  1088 apply (case_tac x)
  1089 apply auto
  1090 apply (case_tac xa)
  1091 apply auto
  1092 apply (case_tac xa)
  1093 apply auto
  1094 done
  1095 
  1096 lemma card_less_Suc:
  1097   assumes zero_in_M: "0 \<in> M"
  1098     shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
  1099 proof -
  1100   from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp
  1101   hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"
  1102     by (auto simp only: insert_Diff)
  1103   have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto
  1104   from finite_M_bounded_by_nat[of "\<lambda>x. x \<in> M" "Suc i"] have "Suc (card {k. Suc k \<in> M \<and> k < i}) = card (insert 0 ({k \<in> M. k < Suc i} - {0}))"
  1105     apply (subst card_insert)
  1106     apply simp_all
  1107     apply (subst b)
  1108     apply (subst card_less_Suc2[symmetric])
  1109     apply simp_all
  1110     done
  1111   with c show ?thesis by simp
  1112 qed
  1113 
  1114 
  1115 subsection {*Lemmas useful with the summation operator setsum*}
  1116 
  1117 text {* For examples, see Algebra/poly/UnivPoly2.thy *}
  1118 
  1119 subsubsection {* Disjoint Unions *}
  1120 
  1121 text {* Singletons and open intervals *}
  1122 
  1123 lemma ivl_disj_un_singleton:
  1124   "{l::'a::linorder} Un {l<..} = {l..}"
  1125   "{..<u} Un {u::'a::linorder} = {..u}"
  1126   "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
  1127   "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
  1128   "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
  1129   "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
  1130 by auto
  1131 
  1132 text {* One- and two-sided intervals *}
  1133 
  1134 lemma ivl_disj_un_one:
  1135   "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
  1136   "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
  1137   "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
  1138   "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
  1139   "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
  1140   "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
  1141   "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
  1142   "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
  1143 by auto
  1144 
  1145 text {* Two- and two-sided intervals *}
  1146 
  1147 lemma ivl_disj_un_two:
  1148   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
  1149   "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
  1150   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
  1151   "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
  1152   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
  1153   "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
  1154   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
  1155   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
  1156 by auto
  1157 
  1158 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
  1159 
  1160 subsubsection {* Disjoint Intersections *}
  1161 
  1162 text {* One- and two-sided intervals *}
  1163 
  1164 lemma ivl_disj_int_one:
  1165   "{..l::'a::order} Int {l<..<u} = {}"
  1166   "{..<l} Int {l..<u} = {}"
  1167   "{..l} Int {l<..u} = {}"
  1168   "{..<l} Int {l..u} = {}"
  1169   "{l<..u} Int {u<..} = {}"
  1170   "{l<..<u} Int {u..} = {}"
  1171   "{l..u} Int {u<..} = {}"
  1172   "{l..<u} Int {u..} = {}"
  1173   by auto
  1174 
  1175 text {* Two- and two-sided intervals *}
  1176 
  1177 lemma ivl_disj_int_two:
  1178   "{l::'a::order<..<m} Int {m..<u} = {}"
  1179   "{l<..m} Int {m<..<u} = {}"
  1180   "{l..<m} Int {m..<u} = {}"
  1181   "{l..m} Int {m<..<u} = {}"
  1182   "{l<..<m} Int {m..u} = {}"
  1183   "{l<..m} Int {m<..u} = {}"
  1184   "{l..<m} Int {m..u} = {}"
  1185   "{l..m} Int {m<..u} = {}"
  1186   by auto
  1187 
  1188 lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two
  1189 
  1190 subsubsection {* Some Differences *}
  1191 
  1192 lemma ivl_diff[simp]:
  1193  "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
  1194 by(auto)
  1195 
  1196 
  1197 subsubsection {* Some Subset Conditions *}
  1198 
  1199 lemma ivl_subset [simp]:
  1200  "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
  1201 apply(auto simp:linorder_not_le)
  1202 apply(rule ccontr)
  1203 apply(insert linorder_le_less_linear[of i n])
  1204 apply(clarsimp simp:linorder_not_le)
  1205 apply(fastforce)
  1206 done
  1207 
  1208 
  1209 subsection {* Summation indexed over intervals *}
  1210 
  1211 syntax
  1212   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
  1213   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
  1214   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
  1215   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
  1216 syntax (xsymbols)
  1217   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
  1218   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
  1219   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
  1220   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
  1221 syntax (HTML output)
  1222   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
  1223   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
  1224   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
  1225   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
  1226 syntax (latex_sum output)
  1227   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  1228  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
  1229   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  1230  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
  1231   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  1232  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
  1233   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  1234  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
  1235 
  1236 translations
  1237   "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"
  1238   "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"
  1239   "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"
  1240   "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"
  1241 
  1242 text{* The above introduces some pretty alternative syntaxes for
  1243 summation over intervals:
  1244 \begin{center}
  1245 \begin{tabular}{lll}
  1246 Old & New & \LaTeX\\
  1247 @{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\
  1248 @{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\
  1249 @{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
  1250 @{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
  1251 \end{tabular}
  1252 \end{center}
  1253 The left column shows the term before introduction of the new syntax,
  1254 the middle column shows the new (default) syntax, and the right column
  1255 shows a special syntax. The latter is only meaningful for latex output
  1256 and has to be activated explicitly by setting the print mode to
  1257 @{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
  1258 antiquotations). It is not the default \LaTeX\ output because it only
  1259 works well with italic-style formulae, not tt-style.
  1260 
  1261 Note that for uniformity on @{typ nat} it is better to use
  1262 @{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
  1263 not provide all lemmas available for @{term"{m..<n}"} also in the
  1264 special form for @{term"{..<n}"}. *}
  1265 
  1266 text{* This congruence rule should be used for sums over intervals as
  1267 the standard theorem @{text[source]setsum_cong} does not work well
  1268 with the simplifier who adds the unsimplified premise @{term"x:B"} to
  1269 the context. *}
  1270 
  1271 lemma setsum_ivl_cong:
  1272  "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
  1273  setsum f {a..<b} = setsum g {c..<d}"
  1274 by(rule setsum_cong, simp_all)
  1275 
  1276 (* FIXME why are the following simp rules but the corresponding eqns
  1277 on intervals are not? *)
  1278 
  1279 lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
  1280 by (simp add:atMost_Suc add_ac)
  1281 
  1282 lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
  1283 by (simp add:lessThan_Suc add_ac)
  1284 
  1285 lemma setsum_cl_ivl_Suc[simp]:
  1286   "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
  1287 by (auto simp:add_ac atLeastAtMostSuc_conv)
  1288 
  1289 lemma setsum_op_ivl_Suc[simp]:
  1290   "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
  1291 by (auto simp:add_ac atLeastLessThanSuc)
  1292 (*
  1293 lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
  1294     (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
  1295 by (auto simp:add_ac atLeastAtMostSuc_conv)
  1296 *)
  1297 
  1298 lemma setsum_head:
  1299   fixes n :: nat
  1300   assumes mn: "m <= n" 
  1301   shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
  1302 proof -
  1303   from mn
  1304   have "{m..n} = {m} \<union> {m<..n}"
  1305     by (auto intro: ivl_disj_un_singleton)
  1306   hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
  1307     by (simp add: atLeast0LessThan)
  1308   also have "\<dots> = ?rhs" by simp
  1309   finally show ?thesis .
  1310 qed
  1311 
  1312 lemma setsum_head_Suc:
  1313   "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"
  1314 by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)
  1315 
  1316 lemma setsum_head_upt_Suc:
  1317   "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"
  1318 apply(insert setsum_head_Suc[of m "n - Suc 0" f])
  1319 apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)
  1320 done
  1321 
  1322 lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"
  1323   shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
  1324 proof-
  1325   have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using `m \<le> n+1` by auto
  1326   thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint
  1327     atLeastSucAtMost_greaterThanAtMost)
  1328 qed
  1329 
  1330 lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
  1331   setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
  1332 by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
  1333 
  1334 lemma setsum_diff_nat_ivl:
  1335 fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
  1336 shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
  1337   setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
  1338 using setsum_add_nat_ivl [of m n p f,symmetric]
  1339 apply (simp add: add_ac)
  1340 done
  1341 
  1342 lemma setsum_natinterval_difff:
  1343   fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
  1344   shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
  1345           (if m <= n then f m - f(n + 1) else 0)"
  1346 by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)
  1347 
  1348 lemma setsum_restrict_set':
  1349   "finite A \<Longrightarrow> setsum f {x \<in> A. x \<in> B} = (\<Sum>x\<in>A. if x \<in> B then f x else 0)"
  1350   by (simp add: setsum_restrict_set [symmetric] Int_def)
  1351 
  1352 lemma setsum_restrict_set'':
  1353   "finite A \<Longrightarrow> setsum f {x \<in> A. P x} = (\<Sum>x\<in>A. if P x  then f x else 0)"
  1354   by (simp add: setsum_restrict_set' [of A f "{x. P x}", simplified mem_Collect_eq])
  1355 
  1356 lemma setsum_setsum_restrict:
  1357   "finite S \<Longrightarrow> finite T \<Longrightarrow>
  1358     setsum (\<lambda>x. setsum (\<lambda>y. f x y) {y. y \<in> T \<and> R x y}) S = setsum (\<lambda>y. setsum (\<lambda>x. f x y) {x. x \<in> S \<and> R x y}) T"
  1359   by (simp add: setsum_restrict_set'') (rule setsum_commute)
  1360 
  1361 lemma setsum_image_gen: assumes fS: "finite S"
  1362   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
  1363 proof-
  1364   { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }
  1365   hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
  1366     by simp
  1367   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
  1368     by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])
  1369   finally show ?thesis .
  1370 qed
  1371 
  1372 lemma setsum_le_included:
  1373   fixes f :: "'a \<Rightarrow> 'b::ordered_comm_monoid_add"
  1374   assumes "finite s" "finite t"
  1375   and "\<forall>y\<in>t. 0 \<le> g y" "(\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<le> g y)"
  1376   shows "setsum f s \<le> setsum g t"
  1377 proof -
  1378   have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"
  1379   proof (rule setsum_mono)
  1380     fix y assume "y \<in> s"
  1381     with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto
  1382     with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")
  1383       using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]
  1384       by (auto intro!: setsum_mono2)
  1385   qed
  1386   also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"
  1387     using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)
  1388   also have "... \<le> setsum g t"
  1389     using assms by (auto simp: setsum_image_gen[symmetric])
  1390   finally show ?thesis .
  1391 qed
  1392 
  1393 lemma setsum_multicount_gen:
  1394   assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
  1395   shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")
  1396 proof-
  1397   have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto
  1398   also have "\<dots> = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]
  1399     using assms(3) by auto
  1400   finally show ?thesis .
  1401 qed
  1402 
  1403 lemma setsum_multicount:
  1404   assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
  1405   shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
  1406 proof-
  1407   have "?l = setsum (\<lambda>i. k) T" by(rule setsum_multicount_gen)(auto simp:assms)
  1408   also have "\<dots> = ?r" by(simp add: mult_commute)
  1409   finally show ?thesis by auto
  1410 qed
  1411 
  1412 
  1413 subsection{* Shifting bounds *}
  1414 
  1415 lemma setsum_shift_bounds_nat_ivl:
  1416   "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
  1417 by (induct "n", auto simp:atLeastLessThanSuc)
  1418 
  1419 lemma setsum_shift_bounds_cl_nat_ivl:
  1420   "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
  1421 apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
  1422 apply (simp add:image_add_atLeastAtMost o_def)
  1423 done
  1424 
  1425 corollary setsum_shift_bounds_cl_Suc_ivl:
  1426   "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
  1427 by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
  1428 
  1429 corollary setsum_shift_bounds_Suc_ivl:
  1430   "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
  1431 by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])
  1432 
  1433 lemma setsum_shift_lb_Suc0_0:
  1434   "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"
  1435 by(simp add:setsum_head_Suc)
  1436 
  1437 lemma setsum_shift_lb_Suc0_0_upt:
  1438   "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"
  1439 apply(cases k)apply simp
  1440 apply(simp add:setsum_head_upt_Suc)
  1441 done
  1442 
  1443 lemma setsum_atMost_Suc_shift:
  1444   fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"
  1445   shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
  1446 proof (induct n)
  1447   case 0 show ?case by simp
  1448 next
  1449   case (Suc n) note IH = this
  1450   have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"
  1451     by (rule setsum_atMost_Suc)
  1452   also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
  1453     by (rule IH)
  1454   also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =
  1455              f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"
  1456     by (rule add_assoc)
  1457   also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"
  1458     by (rule setsum_atMost_Suc [symmetric])
  1459   finally show ?case .
  1460 qed
  1461 
  1462 lemma setsum_last_plus: "n \<noteq> 0 \<Longrightarrow> (\<Sum>i = 0..n. f i) = f n + (\<Sum>i = 0..n - Suc 0. f i)"
  1463   using atLeastAtMostSuc_conv [of 0 "n - 1"]
  1464   by auto
  1465 
  1466 lemma nested_setsum_swap:
  1467      "(\<Sum>i = 0..n. (\<Sum>j = 0..<i. a i j)) = (\<Sum>j = 0..<n. \<Sum>i = Suc j..n. a i j)"
  1468   by (induction n) (auto simp: setsum_addf)
  1469 
  1470 
  1471 subsection {* The formula for geometric sums *}
  1472 
  1473 lemma geometric_sum:
  1474   assumes "x \<noteq> 1"
  1475   shows "(\<Sum>i=0..<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"
  1476 proof -
  1477   from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all
  1478   moreover have "(\<Sum>i=0..<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"
  1479   proof (induct n)
  1480     case 0 then show ?case by simp
  1481   next
  1482     case (Suc n)
  1483     moreover from Suc `y \<noteq> 0` have "(1 + y) ^ n = (y * inverse y) * (1 + y) ^ n" by simp 
  1484     ultimately show ?case by (simp add: field_simps divide_inverse)
  1485   qed
  1486   ultimately show ?thesis by simp
  1487 qed
  1488 
  1489 
  1490 subsection {* The formula for arithmetic sums *}
  1491 
  1492 lemma gauss_sum:
  1493   "(2::'a::comm_semiring_1)*(\<Sum>i\<in>{1..n}. of_nat i) =
  1494    of_nat n*((of_nat n)+1)"
  1495 proof (induct n)
  1496   case 0
  1497   show ?case by simp
  1498 next
  1499   case (Suc n)
  1500   then show ?case
  1501     by (simp add: algebra_simps add: one_add_one [symmetric] del: one_add_one)
  1502       (* FIXME: make numeral cancellation simprocs work for semirings *)
  1503 qed
  1504 
  1505 theorem arith_series_general:
  1506   "(2::'a::comm_semiring_1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
  1507   of_nat n * (a + (a + of_nat(n - 1)*d))"
  1508 proof cases
  1509   assume ngt1: "n > 1"
  1510   let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
  1511   have
  1512     "(\<Sum>i\<in>{..<n}. a+?I i*d) =
  1513      ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
  1514     by (rule setsum_addf)
  1515   also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
  1516   also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
  1517     unfolding One_nat_def
  1518     by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)
  1519   also have "2*\<dots> = 2*?n*a + d*2*(\<Sum>i\<in>{1..<n}. ?I i)"
  1520     by (simp add: algebra_simps)
  1521   also from ngt1 have "{1..<n} = {1..n - 1}"
  1522     by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
  1523   also from ngt1
  1524   have "2*?n*a + d*2*(\<Sum>i\<in>{1..n - 1}. ?I i) = (2*?n*a + d*?I (n - 1)*?I n)"
  1525     by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)
  1526        (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
  1527   finally show ?thesis
  1528     unfolding mult_2 by (simp add: algebra_simps)
  1529 next
  1530   assume "\<not>(n > 1)"
  1531   hence "n = 1 \<or> n = 0" by auto
  1532   thus ?thesis by (auto simp: mult_2)
  1533 qed
  1534 
  1535 lemma arith_series_nat:
  1536   "(2::nat) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
  1537 proof -
  1538   have
  1539     "2 * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
  1540     of_nat(n) * (a + (a + of_nat(n - 1)*d))"
  1541     by (rule arith_series_general)
  1542   thus ?thesis
  1543     unfolding One_nat_def by auto
  1544 qed
  1545 
  1546 lemma arith_series_int:
  1547   "2 * (\<Sum>i\<in>{..<n}. a + int i * d) = int n * (a + (a + int(n - 1)*d))"
  1548   by (fact arith_series_general) (* FIXME: duplicate *)
  1549 
  1550 lemma sum_diff_distrib:
  1551   fixes P::"nat\<Rightarrow>nat"
  1552   shows
  1553   "\<forall>x. Q x \<le> P x  \<Longrightarrow>
  1554   (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
  1555 proof (induct n)
  1556   case 0 show ?case by simp
  1557 next
  1558   case (Suc n)
  1559 
  1560   let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
  1561   let ?rhs = "\<Sum>x<n. P x - Q x"
  1562 
  1563   from Suc have "?lhs = ?rhs" by simp
  1564   moreover
  1565   from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
  1566   moreover
  1567   from Suc have
  1568     "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
  1569     by (subst diff_diff_left[symmetric],
  1570         subst diff_add_assoc2)
  1571        (auto simp: diff_add_assoc2 intro: setsum_mono)
  1572   ultimately
  1573   show ?case by simp
  1574 qed
  1575 
  1576 lemma nat_diff_setsum_reindex:
  1577   fixes x :: "'a::{comm_ring,monoid_mult}"
  1578   shows "(\<Sum>i=0..<n. f (n - Suc i)) = (\<Sum>i=0..<n. f i)"
  1579 apply (subst setsum_reindex_cong [of "%i. n - Suc i" "{0..< n}"])
  1580 apply (auto simp: inj_on_def)
  1581 apply (rule_tac x="n - Suc x" in image_eqI, auto)
  1582 done
  1583 
  1584 subsection {* Products indexed over intervals *}
  1585 
  1586 syntax
  1587   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)
  1588   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)
  1589   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)
  1590   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)
  1591 syntax (xsymbols)
  1592   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
  1593   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
  1594   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
  1595   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
  1596 syntax (HTML output)
  1597   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
  1598   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
  1599   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
  1600   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
  1601 syntax (latex_prod output)
  1602   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  1603  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
  1604   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  1605  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
  1606   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  1607  ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
  1608   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
  1609  ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
  1610 
  1611 translations
  1612   "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"
  1613   "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"
  1614   "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"
  1615   "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"
  1616 
  1617 subsection {* Transfer setup *}
  1618 
  1619 lemma transfer_nat_int_set_functions:
  1620     "{..n} = nat ` {0..int n}"
  1621     "{m..n} = nat ` {int m..int n}"  (* need all variants of these! *)
  1622   apply (auto simp add: image_def)
  1623   apply (rule_tac x = "int x" in bexI)
  1624   apply auto
  1625   apply (rule_tac x = "int x" in bexI)
  1626   apply auto
  1627   done
  1628 
  1629 lemma transfer_nat_int_set_function_closures:
  1630     "x >= 0 \<Longrightarrow> nat_set {x..y}"
  1631   by (simp add: nat_set_def)
  1632 
  1633 declare transfer_morphism_nat_int[transfer add
  1634   return: transfer_nat_int_set_functions
  1635     transfer_nat_int_set_function_closures
  1636 ]
  1637 
  1638 lemma transfer_int_nat_set_functions:
  1639     "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int ` {nat m..nat n}"
  1640   by (simp only: is_nat_def transfer_nat_int_set_functions
  1641     transfer_nat_int_set_function_closures
  1642     transfer_nat_int_set_return_embed nat_0_le
  1643     cong: transfer_nat_int_set_cong)
  1644 
  1645 lemma transfer_int_nat_set_function_closures:
  1646     "is_nat x \<Longrightarrow> nat_set {x..y}"
  1647   by (simp only: transfer_nat_int_set_function_closures is_nat_def)
  1648 
  1649 declare transfer_morphism_int_nat[transfer add
  1650   return: transfer_int_nat_set_functions
  1651     transfer_int_nat_set_function_closures
  1652 ]
  1653 
  1654 lemma setprod_int_plus_eq: "setprod int {i..i+j} =  \<Prod>{int i..int (i+j)}"
  1655   by (induct j) (auto simp add: atLeastAtMostSuc_conv atLeastAtMostPlus1_int_conv)
  1656 
  1657 lemma setprod_int_eq: "setprod int {i..j} =  \<Prod>{int i..int j}"
  1658 proof (cases "i \<le> j")
  1659   case True
  1660   then show ?thesis
  1661     by (metis Nat.le_iff_add setprod_int_plus_eq)
  1662 next
  1663   case False
  1664   then show ?thesis
  1665     by auto
  1666 qed
  1667 
  1668 lemma setprod_power_distrib:
  1669   fixes f :: "'a \<Rightarrow> 'b::comm_semiring_1"
  1670   shows "setprod f A ^ n = setprod (\<lambda>x. (f x)^n) A"
  1671 proof (cases "finite A") 
  1672   case True then show ?thesis 
  1673     by (induct A rule: finite_induct) (auto simp add: power_mult_distrib)
  1674 next
  1675   case False then show ?thesis 
  1676     by (metis setprod_infinite power_one)
  1677 qed
  1678 
  1679 end