src/HOL/SetInterval.thy
author nipkow
Mon Mar 09 12:24:01 2009 +0100 (2009-03-09)
changeset 30384 2f24531b2d3e
parent 30372 96d508968153
child 31017 2c227493ea56
permissions -rw-r--r--
fixed typing of UN/INT syntax
     1 (*  Title:      HOL/SetInterval.thy
     2     Author:     Tobias Nipkow and Clemens Ballarin
     3                 Additions by Jeremy Avigad in March 2004
     4     Copyright   2000  TU Muenchen
     5 
     6 lessThan, greaterThan, atLeast, atMost and two-sided intervals
     7 *)
     8 
     9 header {* Set intervals *}
    10 
    11 theory SetInterval
    12 imports Int
    13 begin
    14 
    15 context ord
    16 begin
    17 definition
    18   lessThan    :: "'a => 'a set"	("(1{..<_})") where
    19   "{..<u} == {x. x < u}"
    20 
    21 definition
    22   atMost      :: "'a => 'a set"	("(1{.._})") where
    23   "{..u} == {x. x \<le> u}"
    24 
    25 definition
    26   greaterThan :: "'a => 'a set"	("(1{_<..})") where
    27   "{l<..} == {x. l<x}"
    28 
    29 definition
    30   atLeast     :: "'a => 'a set"	("(1{_..})") where
    31   "{l..} == {x. l\<le>x}"
    32 
    33 definition
    34   greaterThanLessThan :: "'a => 'a => 'a set"  ("(1{_<..<_})") where
    35   "{l<..<u} == {l<..} Int {..<u}"
    36 
    37 definition
    38   atLeastLessThan :: "'a => 'a => 'a set"      ("(1{_..<_})") where
    39   "{l..<u} == {l..} Int {..<u}"
    40 
    41 definition
    42   greaterThanAtMost :: "'a => 'a => 'a set"    ("(1{_<.._})") where
    43   "{l<..u} == {l<..} Int {..u}"
    44 
    45 definition
    46   atLeastAtMost :: "'a => 'a => 'a set"        ("(1{_.._})") where
    47   "{l..u} == {l..} Int {..u}"
    48 
    49 end
    50 
    51 
    52 text{* A note of warning when using @{term"{..<n}"} on type @{typ
    53 nat}: it is equivalent to @{term"{0::nat..<n}"} but some lemmas involving
    54 @{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well. *}
    55 
    56 syntax
    57   "@UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3UN _<=_./ _)" 10)
    58   "@UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3UN _<_./ _)" 10)
    59   "@INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3INT _<=_./ _)" 10)
    60   "@INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3INT _<_./ _)" 10)
    61 
    62 syntax (xsymbols)
    63   "@UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _\<le>_./ _)" 10)
    64   "@UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _<_./ _)" 10)
    65   "@INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _\<le>_./ _)" 10)
    66   "@INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" 10)
    67 
    68 syntax (latex output)
    69   "@UNION_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ \<le> _)/ _)" 10)
    70   "@UNION_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ < _)/ _)" 10)
    71   "@INTER_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ \<le> _)/ _)" 10)
    72   "@INTER_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ < _)/ _)" 10)
    73 
    74 translations
    75   "UN i<=n. A"  == "UN i:{..n}. A"
    76   "UN i<n. A"   == "UN i:{..<n}. A"
    77   "INT i<=n. A" == "INT i:{..n}. A"
    78   "INT i<n. A"  == "INT i:{..<n}. A"
    79 
    80 
    81 subsection {* Various equivalences *}
    82 
    83 lemma (in ord) lessThan_iff [iff]: "(i: lessThan k) = (i<k)"
    84 by (simp add: lessThan_def)
    85 
    86 lemma Compl_lessThan [simp]:
    87     "!!k:: 'a::linorder. -lessThan k = atLeast k"
    88 apply (auto simp add: lessThan_def atLeast_def)
    89 done
    90 
    91 lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"
    92 by auto
    93 
    94 lemma (in ord) greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"
    95 by (simp add: greaterThan_def)
    96 
    97 lemma Compl_greaterThan [simp]:
    98     "!!k:: 'a::linorder. -greaterThan k = atMost k"
    99   by (auto simp add: greaterThan_def atMost_def)
   100 
   101 lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
   102 apply (subst Compl_greaterThan [symmetric])
   103 apply (rule double_complement)
   104 done
   105 
   106 lemma (in ord) atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
   107 by (simp add: atLeast_def)
   108 
   109 lemma Compl_atLeast [simp]:
   110     "!!k:: 'a::linorder. -atLeast k = lessThan k"
   111   by (auto simp add: lessThan_def atLeast_def)
   112 
   113 lemma (in ord) atMost_iff [iff]: "(i: atMost k) = (i<=k)"
   114 by (simp add: atMost_def)
   115 
   116 lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
   117 by (blast intro: order_antisym)
   118 
   119 
   120 subsection {* Logical Equivalences for Set Inclusion and Equality *}
   121 
   122 lemma atLeast_subset_iff [iff]:
   123      "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::order))"
   124 by (blast intro: order_trans)
   125 
   126 lemma atLeast_eq_iff [iff]:
   127      "(atLeast x = atLeast y) = (x = (y::'a::linorder))"
   128 by (blast intro: order_antisym order_trans)
   129 
   130 lemma greaterThan_subset_iff [iff]:
   131      "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
   132 apply (auto simp add: greaterThan_def)
   133  apply (subst linorder_not_less [symmetric], blast)
   134 done
   135 
   136 lemma greaterThan_eq_iff [iff]:
   137      "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
   138 apply (rule iffI)
   139  apply (erule equalityE)
   140  apply simp_all
   141 done
   142 
   143 lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::order))"
   144 by (blast intro: order_trans)
   145 
   146 lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))"
   147 by (blast intro: order_antisym order_trans)
   148 
   149 lemma lessThan_subset_iff [iff]:
   150      "(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))"
   151 apply (auto simp add: lessThan_def)
   152  apply (subst linorder_not_less [symmetric], blast)
   153 done
   154 
   155 lemma lessThan_eq_iff [iff]:
   156      "(lessThan x = lessThan y) = (x = (y::'a::linorder))"
   157 apply (rule iffI)
   158  apply (erule equalityE)
   159  apply simp_all
   160 done
   161 
   162 
   163 subsection {*Two-sided intervals*}
   164 
   165 context ord
   166 begin
   167 
   168 lemma greaterThanLessThan_iff [simp,noatp]:
   169   "(i : {l<..<u}) = (l < i & i < u)"
   170 by (simp add: greaterThanLessThan_def)
   171 
   172 lemma atLeastLessThan_iff [simp,noatp]:
   173   "(i : {l..<u}) = (l <= i & i < u)"
   174 by (simp add: atLeastLessThan_def)
   175 
   176 lemma greaterThanAtMost_iff [simp,noatp]:
   177   "(i : {l<..u}) = (l < i & i <= u)"
   178 by (simp add: greaterThanAtMost_def)
   179 
   180 lemma atLeastAtMost_iff [simp,noatp]:
   181   "(i : {l..u}) = (l <= i & i <= u)"
   182 by (simp add: atLeastAtMost_def)
   183 
   184 text {* The above four lemmas could be declared as iffs.
   185   If we do so, a call to blast in Hyperreal/Star.ML, lemma @{text STAR_Int}
   186   seems to take forever (more than one hour). *}
   187 end
   188 
   189 subsubsection{* Emptyness and singletons *}
   190 
   191 context order
   192 begin
   193 
   194 lemma atLeastAtMost_empty [simp]: "n < m ==> {m..n} = {}";
   195 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)
   196 
   197 lemma atLeastLessThan_empty[simp]: "n \<le> m ==> {m..<n} = {}"
   198 by (auto simp add: atLeastLessThan_def)
   199 
   200 lemma greaterThanAtMost_empty[simp]:"l \<le> k ==> {k<..l} = {}"
   201 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)
   202 
   203 lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..<l} = {}"
   204 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)
   205 
   206 lemma atLeastAtMost_singleton [simp]: "{a..a} = {a}"
   207 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)
   208 
   209 end
   210 
   211 subsection {* Intervals of natural numbers *}
   212 
   213 subsubsection {* The Constant @{term lessThan} *}
   214 
   215 lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
   216 by (simp add: lessThan_def)
   217 
   218 lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
   219 by (simp add: lessThan_def less_Suc_eq, blast)
   220 
   221 lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
   222 by (simp add: lessThan_def atMost_def less_Suc_eq_le)
   223 
   224 lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
   225 by blast
   226 
   227 subsubsection {* The Constant @{term greaterThan} *}
   228 
   229 lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
   230 apply (simp add: greaterThan_def)
   231 apply (blast dest: gr0_conv_Suc [THEN iffD1])
   232 done
   233 
   234 lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
   235 apply (simp add: greaterThan_def)
   236 apply (auto elim: linorder_neqE)
   237 done
   238 
   239 lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
   240 by blast
   241 
   242 subsubsection {* The Constant @{term atLeast} *}
   243 
   244 lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
   245 by (unfold atLeast_def UNIV_def, simp)
   246 
   247 lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
   248 apply (simp add: atLeast_def)
   249 apply (simp add: Suc_le_eq)
   250 apply (simp add: order_le_less, blast)
   251 done
   252 
   253 lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
   254   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
   255 
   256 lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
   257 by blast
   258 
   259 subsubsection {* The Constant @{term atMost} *}
   260 
   261 lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
   262 by (simp add: atMost_def)
   263 
   264 lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
   265 apply (simp add: atMost_def)
   266 apply (simp add: less_Suc_eq order_le_less, blast)
   267 done
   268 
   269 lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
   270 by blast
   271 
   272 subsubsection {* The Constant @{term atLeastLessThan} *}
   273 
   274 text{*The orientation of the following 2 rules is tricky. The lhs is
   275 defined in terms of the rhs.  Hence the chosen orientation makes sense
   276 in this theory --- the reverse orientation complicates proofs (eg
   277 nontermination). But outside, when the definition of the lhs is rarely
   278 used, the opposite orientation seems preferable because it reduces a
   279 specific concept to a more general one. *}
   280 
   281 lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
   282 by(simp add:lessThan_def atLeastLessThan_def)
   283 
   284 lemma atLeast0AtMost: "{0..n::nat} = {..n}"
   285 by(simp add:atMost_def atLeastAtMost_def)
   286 
   287 declare atLeast0LessThan[symmetric, code unfold]
   288         atLeast0AtMost[symmetric, code unfold]
   289 
   290 lemma atLeastLessThan0: "{m..<0::nat} = {}"
   291 by (simp add: atLeastLessThan_def)
   292 
   293 subsubsection {* Intervals of nats with @{term Suc} *}
   294 
   295 text{*Not a simprule because the RHS is too messy.*}
   296 lemma atLeastLessThanSuc:
   297     "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
   298 by (auto simp add: atLeastLessThan_def)
   299 
   300 lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
   301 by (auto simp add: atLeastLessThan_def)
   302 (*
   303 lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
   304 by (induct k, simp_all add: atLeastLessThanSuc)
   305 
   306 lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
   307 by (auto simp add: atLeastLessThan_def)
   308 *)
   309 lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
   310   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
   311 
   312 lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
   313   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
   314     greaterThanAtMost_def)
   315 
   316 lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
   317   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
   318     greaterThanLessThan_def)
   319 
   320 lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
   321 by (auto simp add: atLeastAtMost_def)
   322 
   323 subsubsection {* Image *}
   324 
   325 lemma image_add_atLeastAtMost:
   326   "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
   327 proof
   328   show "?A \<subseteq> ?B" by auto
   329 next
   330   show "?B \<subseteq> ?A"
   331   proof
   332     fix n assume a: "n : ?B"
   333     hence "n - k : {i..j}" by auto
   334     moreover have "n = (n - k) + k" using a by auto
   335     ultimately show "n : ?A" by blast
   336   qed
   337 qed
   338 
   339 lemma image_add_atLeastLessThan:
   340   "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
   341 proof
   342   show "?A \<subseteq> ?B" by auto
   343 next
   344   show "?B \<subseteq> ?A"
   345   proof
   346     fix n assume a: "n : ?B"
   347     hence "n - k : {i..<j}" by auto
   348     moreover have "n = (n - k) + k" using a by auto
   349     ultimately show "n : ?A" by blast
   350   qed
   351 qed
   352 
   353 corollary image_Suc_atLeastAtMost[simp]:
   354   "Suc ` {i..j} = {Suc i..Suc j}"
   355 using image_add_atLeastAtMost[where k="Suc 0"] by simp
   356 
   357 corollary image_Suc_atLeastLessThan[simp]:
   358   "Suc ` {i..<j} = {Suc i..<Suc j}"
   359 using image_add_atLeastLessThan[where k="Suc 0"] by simp
   360 
   361 lemma image_add_int_atLeastLessThan:
   362     "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
   363   apply (auto simp add: image_def)
   364   apply (rule_tac x = "x - l" in bexI)
   365   apply auto
   366   done
   367 
   368 
   369 subsubsection {* Finiteness *}
   370 
   371 lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
   372   by (induct k) (simp_all add: lessThan_Suc)
   373 
   374 lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
   375   by (induct k) (simp_all add: atMost_Suc)
   376 
   377 lemma finite_greaterThanLessThan [iff]:
   378   fixes l :: nat shows "finite {l<..<u}"
   379 by (simp add: greaterThanLessThan_def)
   380 
   381 lemma finite_atLeastLessThan [iff]:
   382   fixes l :: nat shows "finite {l..<u}"
   383 by (simp add: atLeastLessThan_def)
   384 
   385 lemma finite_greaterThanAtMost [iff]:
   386   fixes l :: nat shows "finite {l<..u}"
   387 by (simp add: greaterThanAtMost_def)
   388 
   389 lemma finite_atLeastAtMost [iff]:
   390   fixes l :: nat shows "finite {l..u}"
   391 by (simp add: atLeastAtMost_def)
   392 
   393 text {* A bounded set of natural numbers is finite. *}
   394 lemma bounded_nat_set_is_finite:
   395   "(ALL i:N. i < (n::nat)) ==> finite N"
   396 apply (rule finite_subset)
   397  apply (rule_tac [2] finite_lessThan, auto)
   398 done
   399 
   400 lemma finite_less_ub:
   401      "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
   402 by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
   403 
   404 text{* Any subset of an interval of natural numbers the size of the
   405 subset is exactly that interval. *}
   406 
   407 lemma subset_card_intvl_is_intvl:
   408   "A <= {k..<k+card A} \<Longrightarrow> A = {k..<k+card A}" (is "PROP ?P")
   409 proof cases
   410   assume "finite A"
   411   thus "PROP ?P"
   412   proof(induct A rule:finite_linorder_induct)
   413     case empty thus ?case by auto
   414   next
   415     case (insert A b)
   416     moreover hence "b ~: A" by auto
   417     moreover have "A <= {k..<k+card A}" and "b = k+card A"
   418       using `b ~: A` insert by fastsimp+
   419     ultimately show ?case by auto
   420   qed
   421 next
   422   assume "~finite A" thus "PROP ?P" by simp
   423 qed
   424 
   425 
   426 subsubsection {* Cardinality *}
   427 
   428 lemma card_lessThan [simp]: "card {..<u} = u"
   429   by (induct u, simp_all add: lessThan_Suc)
   430 
   431 lemma card_atMost [simp]: "card {..u} = Suc u"
   432   by (simp add: lessThan_Suc_atMost [THEN sym])
   433 
   434 lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
   435   apply (subgoal_tac "card {l..<u} = card {..<u-l}")
   436   apply (erule ssubst, rule card_lessThan)
   437   apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
   438   apply (erule subst)
   439   apply (rule card_image)
   440   apply (simp add: inj_on_def)
   441   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
   442   apply (rule_tac x = "x - l" in exI)
   443   apply arith
   444   done
   445 
   446 lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
   447   by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
   448 
   449 lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
   450   by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
   451 
   452 lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
   453   by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
   454 
   455 
   456 lemma ex_bij_betw_nat_finite:
   457   "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
   458 apply(drule finite_imp_nat_seg_image_inj_on)
   459 apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
   460 done
   461 
   462 lemma ex_bij_betw_finite_nat:
   463   "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
   464 by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
   465 
   466 
   467 subsection {* Intervals of integers *}
   468 
   469 lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
   470   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
   471 
   472 lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
   473   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
   474 
   475 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
   476     "{l+1..<u} = {l<..<u::int}"
   477   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
   478 
   479 subsubsection {* Finiteness *}
   480 
   481 lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
   482     {(0::int)..<u} = int ` {..<nat u}"
   483   apply (unfold image_def lessThan_def)
   484   apply auto
   485   apply (rule_tac x = "nat x" in exI)
   486   apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
   487   done
   488 
   489 lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
   490   apply (case_tac "0 \<le> u")
   491   apply (subst image_atLeastZeroLessThan_int, assumption)
   492   apply (rule finite_imageI)
   493   apply auto
   494   done
   495 
   496 lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
   497   apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
   498   apply (erule subst)
   499   apply (rule finite_imageI)
   500   apply (rule finite_atLeastZeroLessThan_int)
   501   apply (rule image_add_int_atLeastLessThan)
   502   done
   503 
   504 lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
   505   by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
   506 
   507 lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
   508   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
   509 
   510 lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
   511   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
   512 
   513 
   514 subsubsection {* Cardinality *}
   515 
   516 lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
   517   apply (case_tac "0 \<le> u")
   518   apply (subst image_atLeastZeroLessThan_int, assumption)
   519   apply (subst card_image)
   520   apply (auto simp add: inj_on_def)
   521   done
   522 
   523 lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
   524   apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
   525   apply (erule ssubst, rule card_atLeastZeroLessThan_int)
   526   apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
   527   apply (erule subst)
   528   apply (rule card_image)
   529   apply (simp add: inj_on_def)
   530   apply (rule image_add_int_atLeastLessThan)
   531   done
   532 
   533 lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
   534 apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
   535 apply (auto simp add: algebra_simps)
   536 done
   537 
   538 lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
   539 by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
   540 
   541 lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
   542 by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
   543 
   544 lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
   545 proof -
   546   have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
   547   with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
   548 qed
   549 
   550 lemma card_less:
   551 assumes zero_in_M: "0 \<in> M"
   552 shows "card {k \<in> M. k < Suc i} \<noteq> 0"
   553 proof -
   554   from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
   555   with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
   556 qed
   557 
   558 lemma card_less_Suc2: "0 \<notin> M \<Longrightarrow> card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"
   559 apply (rule card_bij_eq [of "Suc" _ _ "\<lambda>x. x - Suc 0"])
   560 apply simp
   561 apply fastsimp
   562 apply auto
   563 apply (rule inj_on_diff_nat)
   564 apply auto
   565 apply (case_tac x)
   566 apply auto
   567 apply (case_tac xa)
   568 apply auto
   569 apply (case_tac xa)
   570 apply auto
   571 done
   572 
   573 lemma card_less_Suc:
   574   assumes zero_in_M: "0 \<in> M"
   575     shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
   576 proof -
   577   from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp
   578   hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"
   579     by (auto simp only: insert_Diff)
   580   have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto
   581   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}))"
   582     apply (subst card_insert)
   583     apply simp_all
   584     apply (subst b)
   585     apply (subst card_less_Suc2[symmetric])
   586     apply simp_all
   587     done
   588   with c show ?thesis by simp
   589 qed
   590 
   591 
   592 subsection {*Lemmas useful with the summation operator setsum*}
   593 
   594 text {* For examples, see Algebra/poly/UnivPoly2.thy *}
   595 
   596 subsubsection {* Disjoint Unions *}
   597 
   598 text {* Singletons and open intervals *}
   599 
   600 lemma ivl_disj_un_singleton:
   601   "{l::'a::linorder} Un {l<..} = {l..}"
   602   "{..<u} Un {u::'a::linorder} = {..u}"
   603   "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
   604   "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
   605   "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
   606   "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
   607 by auto
   608 
   609 text {* One- and two-sided intervals *}
   610 
   611 lemma ivl_disj_un_one:
   612   "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
   613   "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
   614   "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
   615   "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
   616   "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
   617   "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
   618   "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
   619   "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
   620 by auto
   621 
   622 text {* Two- and two-sided intervals *}
   623 
   624 lemma ivl_disj_un_two:
   625   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
   626   "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
   627   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
   628   "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
   629   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
   630   "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
   631   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
   632   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
   633 by auto
   634 
   635 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
   636 
   637 subsubsection {* Disjoint Intersections *}
   638 
   639 text {* Singletons and open intervals *}
   640 
   641 lemma ivl_disj_int_singleton:
   642   "{l::'a::order} Int {l<..} = {}"
   643   "{..<u} Int {u} = {}"
   644   "{l} Int {l<..<u} = {}"
   645   "{l<..<u} Int {u} = {}"
   646   "{l} Int {l<..u} = {}"
   647   "{l..<u} Int {u} = {}"
   648   by simp+
   649 
   650 text {* One- and two-sided intervals *}
   651 
   652 lemma ivl_disj_int_one:
   653   "{..l::'a::order} Int {l<..<u} = {}"
   654   "{..<l} Int {l..<u} = {}"
   655   "{..l} Int {l<..u} = {}"
   656   "{..<l} Int {l..u} = {}"
   657   "{l<..u} Int {u<..} = {}"
   658   "{l<..<u} Int {u..} = {}"
   659   "{l..u} Int {u<..} = {}"
   660   "{l..<u} Int {u..} = {}"
   661   by auto
   662 
   663 text {* Two- and two-sided intervals *}
   664 
   665 lemma ivl_disj_int_two:
   666   "{l::'a::order<..<m} Int {m..<u} = {}"
   667   "{l<..m} Int {m<..<u} = {}"
   668   "{l..<m} Int {m..<u} = {}"
   669   "{l..m} Int {m<..<u} = {}"
   670   "{l<..<m} Int {m..u} = {}"
   671   "{l<..m} Int {m<..u} = {}"
   672   "{l..<m} Int {m..u} = {}"
   673   "{l..m} Int {m<..u} = {}"
   674   by auto
   675 
   676 lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
   677 
   678 subsubsection {* Some Differences *}
   679 
   680 lemma ivl_diff[simp]:
   681  "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
   682 by(auto)
   683 
   684 
   685 subsubsection {* Some Subset Conditions *}
   686 
   687 lemma ivl_subset [simp,noatp]:
   688  "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
   689 apply(auto simp:linorder_not_le)
   690 apply(rule ccontr)
   691 apply(insert linorder_le_less_linear[of i n])
   692 apply(clarsimp simp:linorder_not_le)
   693 apply(fastsimp)
   694 done
   695 
   696 
   697 subsection {* Summation indexed over intervals *}
   698 
   699 syntax
   700   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
   701   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
   702   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
   703   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
   704 syntax (xsymbols)
   705   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
   706   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
   707   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
   708   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
   709 syntax (HTML output)
   710   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
   711   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
   712   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
   713   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
   714 syntax (latex_sum output)
   715   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   716  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
   717   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   718  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
   719   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   720  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
   721   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   722  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
   723 
   724 translations
   725   "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"
   726   "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"
   727   "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"
   728   "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"
   729 
   730 text{* The above introduces some pretty alternative syntaxes for
   731 summation over intervals:
   732 \begin{center}
   733 \begin{tabular}{lll}
   734 Old & New & \LaTeX\\
   735 @{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\
   736 @{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\
   737 @{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
   738 @{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
   739 \end{tabular}
   740 \end{center}
   741 The left column shows the term before introduction of the new syntax,
   742 the middle column shows the new (default) syntax, and the right column
   743 shows a special syntax. The latter is only meaningful for latex output
   744 and has to be activated explicitly by setting the print mode to
   745 @{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
   746 antiquotations). It is not the default \LaTeX\ output because it only
   747 works well with italic-style formulae, not tt-style.
   748 
   749 Note that for uniformity on @{typ nat} it is better to use
   750 @{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
   751 not provide all lemmas available for @{term"{m..<n}"} also in the
   752 special form for @{term"{..<n}"}. *}
   753 
   754 text{* This congruence rule should be used for sums over intervals as
   755 the standard theorem @{text[source]setsum_cong} does not work well
   756 with the simplifier who adds the unsimplified premise @{term"x:B"} to
   757 the context. *}
   758 
   759 lemma setsum_ivl_cong:
   760  "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
   761  setsum f {a..<b} = setsum g {c..<d}"
   762 by(rule setsum_cong, simp_all)
   763 
   764 (* FIXME why are the following simp rules but the corresponding eqns
   765 on intervals are not? *)
   766 
   767 lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
   768 by (simp add:atMost_Suc add_ac)
   769 
   770 lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
   771 by (simp add:lessThan_Suc add_ac)
   772 
   773 lemma setsum_cl_ivl_Suc[simp]:
   774   "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
   775 by (auto simp:add_ac atLeastAtMostSuc_conv)
   776 
   777 lemma setsum_op_ivl_Suc[simp]:
   778   "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
   779 by (auto simp:add_ac atLeastLessThanSuc)
   780 (*
   781 lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
   782     (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
   783 by (auto simp:add_ac atLeastAtMostSuc_conv)
   784 *)
   785 
   786 lemma setsum_head:
   787   fixes n :: nat
   788   assumes mn: "m <= n" 
   789   shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
   790 proof -
   791   from mn
   792   have "{m..n} = {m} \<union> {m<..n}"
   793     by (auto intro: ivl_disj_un_singleton)
   794   hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
   795     by (simp add: atLeast0LessThan)
   796   also have "\<dots> = ?rhs" by simp
   797   finally show ?thesis .
   798 qed
   799 
   800 lemma setsum_head_Suc:
   801   "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"
   802 by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)
   803 
   804 lemma setsum_head_upt_Suc:
   805   "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"
   806 apply(insert setsum_head_Suc[of m "n - Suc 0" f])
   807 apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)
   808 done
   809 
   810 
   811 lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
   812   setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
   813 by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
   814 
   815 lemma setsum_diff_nat_ivl:
   816 fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
   817 shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
   818   setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
   819 using setsum_add_nat_ivl [of m n p f,symmetric]
   820 apply (simp add: add_ac)
   821 done
   822 
   823 
   824 subsection{* Shifting bounds *}
   825 
   826 lemma setsum_shift_bounds_nat_ivl:
   827   "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
   828 by (induct "n", auto simp:atLeastLessThanSuc)
   829 
   830 lemma setsum_shift_bounds_cl_nat_ivl:
   831   "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
   832 apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
   833 apply (simp add:image_add_atLeastAtMost o_def)
   834 done
   835 
   836 corollary setsum_shift_bounds_cl_Suc_ivl:
   837   "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
   838 by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
   839 
   840 corollary setsum_shift_bounds_Suc_ivl:
   841   "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
   842 by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])
   843 
   844 lemma setsum_shift_lb_Suc0_0:
   845   "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"
   846 by(simp add:setsum_head_Suc)
   847 
   848 lemma setsum_shift_lb_Suc0_0_upt:
   849   "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"
   850 apply(cases k)apply simp
   851 apply(simp add:setsum_head_upt_Suc)
   852 done
   853 
   854 subsection {* The formula for geometric sums *}
   855 
   856 lemma geometric_sum:
   857   "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) =
   858   (x ^ n - 1) / (x - 1::'a::{field, recpower})"
   859 by (induct "n") (simp_all add:field_simps power_Suc)
   860 
   861 subsection {* The formula for arithmetic sums *}
   862 
   863 lemma gauss_sum:
   864   "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =
   865    of_nat n*((of_nat n)+1)"
   866 proof (induct n)
   867   case 0
   868   show ?case by simp
   869 next
   870   case (Suc n)
   871   then show ?case by (simp add: algebra_simps)
   872 qed
   873 
   874 theorem arith_series_general:
   875   "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
   876   of_nat n * (a + (a + of_nat(n - 1)*d))"
   877 proof cases
   878   assume ngt1: "n > 1"
   879   let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
   880   have
   881     "(\<Sum>i\<in>{..<n}. a+?I i*d) =
   882      ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
   883     by (rule setsum_addf)
   884   also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
   885   also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
   886     unfolding One_nat_def
   887     by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)
   888   also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"
   889     by (simp add: left_distrib right_distrib)
   890   also from ngt1 have "{1..<n} = {1..n - 1}"
   891     by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
   892   also from ngt1
   893   have "(1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..n - 1}. ?I i) = ((1+1)*?n*a + d*?I (n - 1)*?I n)"
   894     by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)
   895        (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
   896   finally show ?thesis by (simp add: algebra_simps)
   897 next
   898   assume "\<not>(n > 1)"
   899   hence "n = 1 \<or> n = 0" by auto
   900   thus ?thesis by (auto simp: algebra_simps)
   901 qed
   902 
   903 lemma arith_series_nat:
   904   "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
   905 proof -
   906   have
   907     "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
   908     of_nat(n) * (a + (a + of_nat(n - 1)*d))"
   909     by (rule arith_series_general)
   910   thus ?thesis
   911     unfolding One_nat_def by (auto simp add: of_nat_id)
   912 qed
   913 
   914 lemma arith_series_int:
   915   "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
   916   of_nat n * (a + (a + of_nat(n - 1)*d))"
   917 proof -
   918   have
   919     "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
   920     of_nat(n) * (a + (a + of_nat(n - 1)*d))"
   921     by (rule arith_series_general)
   922   thus ?thesis by simp
   923 qed
   924 
   925 lemma sum_diff_distrib:
   926   fixes P::"nat\<Rightarrow>nat"
   927   shows
   928   "\<forall>x. Q x \<le> P x  \<Longrightarrow>
   929   (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
   930 proof (induct n)
   931   case 0 show ?case by simp
   932 next
   933   case (Suc n)
   934 
   935   let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
   936   let ?rhs = "\<Sum>x<n. P x - Q x"
   937 
   938   from Suc have "?lhs = ?rhs" by simp
   939   moreover
   940   from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
   941   moreover
   942   from Suc have
   943     "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
   944     by (subst diff_diff_left[symmetric],
   945         subst diff_add_assoc2)
   946        (auto simp: diff_add_assoc2 intro: setsum_mono)
   947   ultimately
   948   show ?case by simp
   949 qed
   950 
   951 subsection {* Products indexed over intervals *}
   952 
   953 syntax
   954   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)
   955   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)
   956   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)
   957   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)
   958 syntax (xsymbols)
   959   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
   960   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
   961   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
   962   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
   963 syntax (HTML output)
   964   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
   965   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
   966   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
   967   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
   968 syntax (latex_prod output)
   969   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   970  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
   971   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   972  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
   973   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   974  ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
   975   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
   976  ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
   977 
   978 translations
   979   "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"
   980   "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"
   981   "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"
   982   "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"
   983 
   984 end