src/HOL/SetInterval.thy
 author hoelzl Tue Jan 18 21:37:23 2011 +0100 (2011-01-18) changeset 41654 32fe42892983 parent 40703 d1fc454d6735 child 42891 e2f473671937 permissions -rw-r--r--
Gauge measure removed
     1 (*  Title:      HOL/SetInterval.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

     9 header {* Set intervals *}

    10

    11 theory SetInterval

    12 imports Int Nat_Transfer

    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 _<=_./ _)" [0, 0, 10] 10)

    58   "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3UN _<_./ _)" [0, 0, 10] 10)

    59   "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3INT _<=_./ _)" [0, 0, 10] 10)

    60   "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3INT _<_./ _)" [0, 0, 10] 10)

    61

    62 syntax (xsymbols)

    63   "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _\<le>_./ _)" [0, 0, 10] 10)

    64   "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _<_./ _)" [0, 0, 10] 10)

    65   "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _\<le>_./ _)" [0, 0, 10] 10)

    66   "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" [0, 0, 10] 10)

    67

    68 syntax (latex output)

    69   "_UNION_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ \<le> _)/ _)" [0, 0, 10] 10)

    70   "_UNION_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ < _)/ _)" [0, 0, 10] 10)

    71   "_INTER_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ \<le> _)/ _)" [0, 0, 10] 10)

    72   "_INTER_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ < _)/ _)" [0, 0, 10] 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 lemma lessThan_strict_subset_iff:

   163   fixes m n :: "'a::linorder"

   164   shows "{..<m} < {..<n} \<longleftrightarrow> m < n"

   165   by (metis leD lessThan_subset_iff linorder_linear not_less_iff_gr_or_eq psubset_eq)

   166

   167 subsection {*Two-sided intervals*}

   168

   169 context ord

   170 begin

   171

   172 lemma greaterThanLessThan_iff [simp,no_atp]:

   173   "(i : {l<..<u}) = (l < i & i < u)"

   174 by (simp add: greaterThanLessThan_def)

   175

   176 lemma atLeastLessThan_iff [simp,no_atp]:

   177   "(i : {l..<u}) = (l <= i & i < u)"

   178 by (simp add: atLeastLessThan_def)

   179

   180 lemma greaterThanAtMost_iff [simp,no_atp]:

   181   "(i : {l<..u}) = (l < i & i <= u)"

   182 by (simp add: greaterThanAtMost_def)

   183

   184 lemma atLeastAtMost_iff [simp,no_atp]:

   185   "(i : {l..u}) = (l <= i & i <= u)"

   186 by (simp add: atLeastAtMost_def)

   187

   188 text {* The above four lemmas could be declared as iffs. Unfortunately this

   189 breaks many proofs. Since it only helps blast, it is better to leave well

   190 alone *}

   191

   192 end

   193

   194 subsubsection{* Emptyness, singletons, subset *}

   195

   196 context order

   197 begin

   198

   199 lemma atLeastatMost_empty[simp]:

   200   "b < a \<Longrightarrow> {a..b} = {}"

   201 by(auto simp: atLeastAtMost_def atLeast_def atMost_def)

   202

   203 lemma atLeastatMost_empty_iff[simp]:

   204   "{a..b} = {} \<longleftrightarrow> (~ a <= b)"

   205 by auto (blast intro: order_trans)

   206

   207 lemma atLeastatMost_empty_iff2[simp]:

   208   "{} = {a..b} \<longleftrightarrow> (~ a <= b)"

   209 by auto (blast intro: order_trans)

   210

   211 lemma atLeastLessThan_empty[simp]:

   212   "b <= a \<Longrightarrow> {a..<b} = {}"

   213 by(auto simp: atLeastLessThan_def)

   214

   215 lemma atLeastLessThan_empty_iff[simp]:

   216   "{a..<b} = {} \<longleftrightarrow> (~ a < b)"

   217 by auto (blast intro: le_less_trans)

   218

   219 lemma atLeastLessThan_empty_iff2[simp]:

   220   "{} = {a..<b} \<longleftrightarrow> (~ a < b)"

   221 by auto (blast intro: le_less_trans)

   222

   223 lemma greaterThanAtMost_empty[simp]: "l \<le> k ==> {k<..l} = {}"

   224 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)

   225

   226 lemma greaterThanAtMost_empty_iff[simp]: "{k<..l} = {} \<longleftrightarrow> ~ k < l"

   227 by auto (blast intro: less_le_trans)

   228

   229 lemma greaterThanAtMost_empty_iff2[simp]: "{} = {k<..l} \<longleftrightarrow> ~ k < l"

   230 by auto (blast intro: less_le_trans)

   231

   232 lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..<l} = {}"

   233 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)

   234

   235 lemma atLeastAtMost_singleton [simp]: "{a..a} = {a}"

   236 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)

   237

   238 lemma atLeastAtMost_singleton': "a = b \<Longrightarrow> {a .. b} = {a}" by simp

   239

   240 lemma atLeastatMost_subset_iff[simp]:

   241   "{a..b} <= {c..d} \<longleftrightarrow> (~ a <= b) | c <= a & b <= d"

   242 unfolding atLeastAtMost_def atLeast_def atMost_def

   243 by (blast intro: order_trans)

   244

   245 lemma atLeastatMost_psubset_iff:

   246   "{a..b} < {c..d} \<longleftrightarrow>

   247    ((~ a <= b) | c <= a & b <= d & (c < a | b < d))  &  c <= d"

   248 by(simp add: psubset_eq set_eq_iff less_le_not_le)(blast intro: order_trans)

   249

   250 lemma atLeastAtMost_singleton_iff[simp]:

   251   "{a .. b} = {c} \<longleftrightarrow> a = b \<and> b = c"

   252 proof

   253   assume "{a..b} = {c}"

   254   hence "\<not> (\<not> a \<le> b)" unfolding atLeastatMost_empty_iff[symmetric] by simp

   255   moreover with {a..b} = {c} have "c \<le> a \<and> b \<le> c" by auto

   256   ultimately show "a = b \<and> b = c" by auto

   257 qed simp

   258

   259 end

   260

   261 lemma (in linorder) atLeastLessThan_subset_iff:

   262   "{a..<b} <= {c..<d} \<Longrightarrow> b <= a | c<=a & b<=d"

   263 apply (auto simp:subset_eq Ball_def)

   264 apply(frule_tac x=a in spec)

   265 apply(erule_tac x=d in allE)

   266 apply (simp add: less_imp_le)

   267 done

   268

   269 lemma atLeastLessThan_inj:

   270   fixes a b c d :: "'a::linorder"

   271   assumes eq: "{a ..< b} = {c ..< d}" and "a < b" "c < d"

   272   shows "a = c" "b = d"

   273 using assms by (metis atLeastLessThan_subset_iff eq less_le_not_le linorder_antisym_conv2 subset_refl)+

   274

   275 lemma atLeastLessThan_eq_iff:

   276   fixes a b c d :: "'a::linorder"

   277   assumes "a < b" "c < d"

   278   shows "{a ..< b} = {c ..< d} \<longleftrightarrow> a = c \<and> b = d"

   279   using atLeastLessThan_inj assms by auto

   280

   281 subsubsection {* Intersection *}

   282

   283 context linorder

   284 begin

   285

   286 lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"

   287 by auto

   288

   289 lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"

   290 by auto

   291

   292 lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"

   293 by auto

   294

   295 lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"

   296 by auto

   297

   298 lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"

   299 by auto

   300

   301 lemma Int_atLeastLessThan[simp]: "{a..<b} Int {c..<d} = {max a c ..< min b d}"

   302 by auto

   303

   304 lemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"

   305 by auto

   306

   307 lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"

   308 by auto

   309

   310 end

   311

   312

   313 subsection {* Intervals of natural numbers *}

   314

   315 subsubsection {* The Constant @{term lessThan} *}

   316

   317 lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"

   318 by (simp add: lessThan_def)

   319

   320 lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"

   321 by (simp add: lessThan_def less_Suc_eq, blast)

   322

   323 text {* The following proof is convinient in induction proofs where

   324 new elements get indices at the beginning. So it is used to transform

   325 @{term "{..<Suc n}"} to @{term "0::nat"} and @{term "{..< n}"}. *}

   326

   327 lemma lessThan_Suc_eq_insert_0: "{..<Suc n} = insert 0 (Suc  {..<n})"

   328 proof safe

   329   fix x assume "x < Suc n" "x \<notin> Suc  {..<n}"

   330   then have "x \<noteq> Suc (x - 1)" by auto

   331   with x < Suc n show "x = 0" by auto

   332 qed

   333

   334 lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"

   335 by (simp add: lessThan_def atMost_def less_Suc_eq_le)

   336

   337 lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"

   338 by blast

   339

   340 subsubsection {* The Constant @{term greaterThan} *}

   341

   342 lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"

   343 apply (simp add: greaterThan_def)

   344 apply (blast dest: gr0_conv_Suc [THEN iffD1])

   345 done

   346

   347 lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"

   348 apply (simp add: greaterThan_def)

   349 apply (auto elim: linorder_neqE)

   350 done

   351

   352 lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"

   353 by blast

   354

   355 subsubsection {* The Constant @{term atLeast} *}

   356

   357 lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"

   358 by (unfold atLeast_def UNIV_def, simp)

   359

   360 lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"

   361 apply (simp add: atLeast_def)

   362 apply (simp add: Suc_le_eq)

   363 apply (simp add: order_le_less, blast)

   364 done

   365

   366 lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"

   367   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)

   368

   369 lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"

   370 by blast

   371

   372 subsubsection {* The Constant @{term atMost} *}

   373

   374 lemma atMost_0 [simp]: "atMost (0::nat) = {0}"

   375 by (simp add: atMost_def)

   376

   377 lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"

   378 apply (simp add: atMost_def)

   379 apply (simp add: less_Suc_eq order_le_less, blast)

   380 done

   381

   382 lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"

   383 by blast

   384

   385 subsubsection {* The Constant @{term atLeastLessThan} *}

   386

   387 text{*The orientation of the following 2 rules is tricky. The lhs is

   388 defined in terms of the rhs.  Hence the chosen orientation makes sense

   389 in this theory --- the reverse orientation complicates proofs (eg

   390 nontermination). But outside, when the definition of the lhs is rarely

   391 used, the opposite orientation seems preferable because it reduces a

   392 specific concept to a more general one. *}

   393

   394 lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"

   395 by(simp add:lessThan_def atLeastLessThan_def)

   396

   397 lemma atLeast0AtMost: "{0..n::nat} = {..n}"

   398 by(simp add:atMost_def atLeastAtMost_def)

   399

   400 declare atLeast0LessThan[symmetric, code_unfold]

   401         atLeast0AtMost[symmetric, code_unfold]

   402

   403 lemma atLeastLessThan0: "{m..<0::nat} = {}"

   404 by (simp add: atLeastLessThan_def)

   405

   406 subsubsection {* Intervals of nats with @{term Suc} *}

   407

   408 text{*Not a simprule because the RHS is too messy.*}

   409 lemma atLeastLessThanSuc:

   410     "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"

   411 by (auto simp add: atLeastLessThan_def)

   412

   413 lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"

   414 by (auto simp add: atLeastLessThan_def)

   415 (*

   416 lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"

   417 by (induct k, simp_all add: atLeastLessThanSuc)

   418

   419 lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"

   420 by (auto simp add: atLeastLessThan_def)

   421 *)

   422 lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"

   423   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)

   424

   425 lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"

   426   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def

   427     greaterThanAtMost_def)

   428

   429 lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"

   430   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def

   431     greaterThanLessThan_def)

   432

   433 lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"

   434 by (auto simp add: atLeastAtMost_def)

   435

   436 lemma atLeastLessThan_add_Un: "i \<le> j \<Longrightarrow> {i..<j+k} = {i..<j} \<union> {j..<j+k::nat}"

   437   apply (induct k)

   438   apply (simp_all add: atLeastLessThanSuc)

   439   done

   440

   441 subsubsection {* Image *}

   442

   443 lemma image_add_atLeastAtMost:

   444   "(%n::nat. n+k)  {i..j} = {i+k..j+k}" (is "?A = ?B")

   445 proof

   446   show "?A \<subseteq> ?B" by auto

   447 next

   448   show "?B \<subseteq> ?A"

   449   proof

   450     fix n assume a: "n : ?B"

   451     hence "n - k : {i..j}" by auto

   452     moreover have "n = (n - k) + k" using a by auto

   453     ultimately show "n : ?A" by blast

   454   qed

   455 qed

   456

   457 lemma image_add_atLeastLessThan:

   458   "(%n::nat. n+k)  {i..<j} = {i+k..<j+k}" (is "?A = ?B")

   459 proof

   460   show "?A \<subseteq> ?B" by auto

   461 next

   462   show "?B \<subseteq> ?A"

   463   proof

   464     fix n assume a: "n : ?B"

   465     hence "n - k : {i..<j}" by auto

   466     moreover have "n = (n - k) + k" using a by auto

   467     ultimately show "n : ?A" by blast

   468   qed

   469 qed

   470

   471 corollary image_Suc_atLeastAtMost[simp]:

   472   "Suc  {i..j} = {Suc i..Suc j}"

   473 using image_add_atLeastAtMost[where k="Suc 0"] by simp

   474

   475 corollary image_Suc_atLeastLessThan[simp]:

   476   "Suc  {i..<j} = {Suc i..<Suc j}"

   477 using image_add_atLeastLessThan[where k="Suc 0"] by simp

   478

   479 lemma image_add_int_atLeastLessThan:

   480     "(%x. x + (l::int))  {0..<u-l} = {l..<u}"

   481   apply (auto simp add: image_def)

   482   apply (rule_tac x = "x - l" in bexI)

   483   apply auto

   484   done

   485

   486 lemma image_minus_const_atLeastLessThan_nat:

   487   fixes c :: nat

   488   shows "(\<lambda>i. i - c)  {x ..< y} =

   489       (if c < y then {x - c ..< y - c} else if x < y then {0} else {})"

   490     (is "_ = ?right")

   491 proof safe

   492   fix a assume a: "a \<in> ?right"

   493   show "a \<in> (\<lambda>i. i - c)  {x ..< y}"

   494   proof cases

   495     assume "c < y" with a show ?thesis

   496       by (auto intro!: image_eqI[of _ _ "a + c"])

   497   next

   498     assume "\<not> c < y" with a show ?thesis

   499       by (auto intro!: image_eqI[of _ _ x] split: split_if_asm)

   500   qed

   501 qed auto

   502

   503 context ordered_ab_group_add

   504 begin

   505

   506 lemma

   507   fixes x :: 'a

   508   shows image_uminus_greaterThan[simp]: "uminus  {x<..} = {..<-x}"

   509   and image_uminus_atLeast[simp]: "uminus  {x..} = {..-x}"

   510 proof safe

   511   fix y assume "y < -x"

   512   hence *:  "x < -y" using neg_less_iff_less[of "-y" x] by simp

   513   have "- (-y) \<in> uminus  {x<..}"

   514     by (rule imageI) (simp add: *)

   515   thus "y \<in> uminus  {x<..}" by simp

   516 next

   517   fix y assume "y \<le> -x"

   518   have "- (-y) \<in> uminus  {x..}"

   519     by (rule imageI) (insert y \<le> -x[THEN le_imp_neg_le], simp)

   520   thus "y \<in> uminus  {x..}" by simp

   521 qed simp_all

   522

   523 lemma

   524   fixes x :: 'a

   525   shows image_uminus_lessThan[simp]: "uminus  {..<x} = {-x<..}"

   526   and image_uminus_atMost[simp]: "uminus  {..x} = {-x..}"

   527 proof -

   528   have "uminus  {..<x} = uminus  uminus  {-x<..}"

   529     and "uminus  {..x} = uminus  uminus  {-x..}" by simp_all

   530   thus "uminus  {..<x} = {-x<..}" and "uminus  {..x} = {-x..}"

   531     by (simp_all add: image_image

   532         del: image_uminus_greaterThan image_uminus_atLeast)

   533 qed

   534

   535 lemma

   536   fixes x :: 'a

   537   shows image_uminus_atLeastAtMost[simp]: "uminus  {x..y} = {-y..-x}"

   538   and image_uminus_greaterThanAtMost[simp]: "uminus  {x<..y} = {-y..<-x}"

   539   and image_uminus_atLeastLessThan[simp]: "uminus  {x..<y} = {-y<..-x}"

   540   and image_uminus_greaterThanLessThan[simp]: "uminus  {x<..<y} = {-y<..<-x}"

   541   by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def

   542       greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)

   543 end

   544

   545 subsubsection {* Finiteness *}

   546

   547 lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"

   548   by (induct k) (simp_all add: lessThan_Suc)

   549

   550 lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"

   551   by (induct k) (simp_all add: atMost_Suc)

   552

   553 lemma finite_greaterThanLessThan [iff]:

   554   fixes l :: nat shows "finite {l<..<u}"

   555 by (simp add: greaterThanLessThan_def)

   556

   557 lemma finite_atLeastLessThan [iff]:

   558   fixes l :: nat shows "finite {l..<u}"

   559 by (simp add: atLeastLessThan_def)

   560

   561 lemma finite_greaterThanAtMost [iff]:

   562   fixes l :: nat shows "finite {l<..u}"

   563 by (simp add: greaterThanAtMost_def)

   564

   565 lemma finite_atLeastAtMost [iff]:

   566   fixes l :: nat shows "finite {l..u}"

   567 by (simp add: atLeastAtMost_def)

   568

   569 text {* A bounded set of natural numbers is finite. *}

   570 lemma bounded_nat_set_is_finite:

   571   "(ALL i:N. i < (n::nat)) ==> finite N"

   572 apply (rule finite_subset)

   573  apply (rule_tac [2] finite_lessThan, auto)

   574 done

   575

   576 text {* A set of natural numbers is finite iff it is bounded. *}

   577 lemma finite_nat_set_iff_bounded:

   578   "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")

   579 proof

   580   assume f:?F  show ?B

   581     using Max_ge[OF ?F, simplified less_Suc_eq_le[symmetric]] by blast

   582 next

   583   assume ?B show ?F using ?B by(blast intro:bounded_nat_set_is_finite)

   584 qed

   585

   586 lemma finite_nat_set_iff_bounded_le:

   587   "finite(N::nat set) = (EX m. ALL n:N. n<=m)"

   588 apply(simp add:finite_nat_set_iff_bounded)

   589 apply(blast dest:less_imp_le_nat le_imp_less_Suc)

   590 done

   591

   592 lemma finite_less_ub:

   593      "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"

   594 by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)

   595

   596 text{* Any subset of an interval of natural numbers the size of the

   597 subset is exactly that interval. *}

   598

   599 lemma subset_card_intvl_is_intvl:

   600   "A <= {k..<k+card A} \<Longrightarrow> A = {k..<k+card A}" (is "PROP ?P")

   601 proof cases

   602   assume "finite A"

   603   thus "PROP ?P"

   604   proof(induct A rule:finite_linorder_max_induct)

   605     case empty thus ?case by auto

   606   next

   607     case (insert b A)

   608     moreover hence "b ~: A" by auto

   609     moreover have "A <= {k..<k+card A}" and "b = k+card A"

   610       using b ~: A insert by fastsimp+

   611     ultimately show ?case by auto

   612   qed

   613 next

   614   assume "~finite A" thus "PROP ?P" by simp

   615 qed

   616

   617

   618 subsubsection {* Proving Inclusions and Equalities between Unions *}

   619

   620 lemma UN_le_eq_Un0:

   621   "(\<Union>i\<le>n::nat. M i) = (\<Union>i\<in>{1..n}. M i) \<union> M 0" (is "?A = ?B")

   622 proof

   623   show "?A <= ?B"

   624   proof

   625     fix x assume "x : ?A"

   626     then obtain i where i: "i\<le>n" "x : M i" by auto

   627     show "x : ?B"

   628     proof(cases i)

   629       case 0 with i show ?thesis by simp

   630     next

   631       case (Suc j) with i show ?thesis by auto

   632     qed

   633   qed

   634 next

   635   show "?B <= ?A" by auto

   636 qed

   637

   638 lemma UN_le_add_shift:

   639   "(\<Union>i\<le>n::nat. M(i+k)) = (\<Union>i\<in>{k..n+k}. M i)" (is "?A = ?B")

   640 proof

   641   show "?A <= ?B" by fastsimp

   642 next

   643   show "?B <= ?A"

   644   proof

   645     fix x assume "x : ?B"

   646     then obtain i where i: "i : {k..n+k}" "x : M(i)" by auto

   647     hence "i-k\<le>n & x : M((i-k)+k)" by auto

   648     thus "x : ?A" by blast

   649   qed

   650 qed

   651

   652 lemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"

   653   by (auto simp add: atLeast0LessThan)

   654

   655 lemma UN_finite_subset: "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> C) \<Longrightarrow> (\<Union>n. A n) \<subseteq> C"

   656   by (subst UN_UN_finite_eq [symmetric]) blast

   657

   658 lemma UN_finite2_subset:

   659      "(!!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)"

   660   apply (rule UN_finite_subset)

   661   apply (subst UN_UN_finite_eq [symmetric, of B])

   662   apply blast

   663   done

   664

   665 lemma UN_finite2_eq:

   666   "(!!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)"

   667   apply (rule subset_antisym)

   668    apply (rule UN_finite2_subset, blast)

   669  apply (rule UN_finite2_subset [where k=k])

   670  apply (force simp add: atLeastLessThan_add_Un [of 0])

   671  done

   672

   673

   674 subsubsection {* Cardinality *}

   675

   676 lemma card_lessThan [simp]: "card {..<u} = u"

   677   by (induct u, simp_all add: lessThan_Suc)

   678

   679 lemma card_atMost [simp]: "card {..u} = Suc u"

   680   by (simp add: lessThan_Suc_atMost [THEN sym])

   681

   682 lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"

   683   apply (subgoal_tac "card {l..<u} = card {..<u-l}")

   684   apply (erule ssubst, rule card_lessThan)

   685   apply (subgoal_tac "(%x. x + l)  {..<u-l} = {l..<u}")

   686   apply (erule subst)

   687   apply (rule card_image)

   688   apply (simp add: inj_on_def)

   689   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)

   690   apply (rule_tac x = "x - l" in exI)

   691   apply arith

   692   done

   693

   694 lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"

   695   by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)

   696

   697 lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"

   698   by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)

   699

   700 lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"

   701   by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)

   702

   703 lemma ex_bij_betw_nat_finite:

   704   "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"

   705 apply(drule finite_imp_nat_seg_image_inj_on)

   706 apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)

   707 done

   708

   709 lemma ex_bij_betw_finite_nat:

   710   "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"

   711 by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)

   712

   713 lemma finite_same_card_bij:

   714   "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"

   715 apply(drule ex_bij_betw_finite_nat)

   716 apply(drule ex_bij_betw_nat_finite)

   717 apply(auto intro!:bij_betw_trans)

   718 done

   719

   720 lemma ex_bij_betw_nat_finite_1:

   721   "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"

   722 by (rule finite_same_card_bij) auto

   723

   724 lemma bij_betw_iff_card:

   725   assumes FIN: "finite A" and FIN': "finite B"

   726   shows BIJ: "(\<exists>f. bij_betw f A B) \<longleftrightarrow> (card A = card B)"

   727 using assms

   728 proof(auto simp add: bij_betw_same_card)

   729   assume *: "card A = card B"

   730   obtain f where "bij_betw f A {0 ..< card A}"

   731   using FIN ex_bij_betw_finite_nat by blast

   732   moreover obtain g where "bij_betw g {0 ..< card B} B"

   733   using FIN' ex_bij_betw_nat_finite by blast

   734   ultimately have "bij_betw (g o f) A B"

   735   using * by (auto simp add: bij_betw_trans)

   736   thus "(\<exists>f. bij_betw f A B)" by blast

   737 qed

   738

   739 lemma inj_on_iff_card_le:

   740   assumes FIN: "finite A" and FIN': "finite B"

   741   shows "(\<exists>f. inj_on f A \<and> f  A \<le> B) = (card A \<le> card B)"

   742 proof (safe intro!: card_inj_on_le)

   743   assume *: "card A \<le> card B"

   744   obtain f where 1: "inj_on f A" and 2: "f  A = {0 ..< card A}"

   745   using FIN ex_bij_betw_finite_nat unfolding bij_betw_def by force

   746   moreover obtain g where "inj_on g {0 ..< card B}" and 3: "g  {0 ..< card B} = B"

   747   using FIN' ex_bij_betw_nat_finite unfolding bij_betw_def by force

   748   ultimately have "inj_on g (f  A)" using subset_inj_on[of g _ "f  A"] * by force

   749   hence "inj_on (g o f) A" using 1 comp_inj_on by blast

   750   moreover

   751   {have "{0 ..< card A} \<le> {0 ..< card B}" using * by force

   752    with 2 have "f  A  \<le> {0 ..< card B}" by blast

   753    hence "(g o f)  A \<le> B" unfolding comp_def using 3 by force

   754   }

   755   ultimately show "(\<exists>f. inj_on f A \<and> f  A \<le> B)" by blast

   756 qed (insert assms, auto)

   757

   758 subsection {* Intervals of integers *}

   759

   760 lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"

   761   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)

   762

   763 lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"

   764   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)

   765

   766 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:

   767     "{l+1..<u} = {l<..<u::int}"

   768   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)

   769

   770 subsubsection {* Finiteness *}

   771

   772 lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>

   773     {(0::int)..<u} = int  {..<nat u}"

   774   apply (unfold image_def lessThan_def)

   775   apply auto

   776   apply (rule_tac x = "nat x" in exI)

   777   apply (auto simp add: zless_nat_eq_int_zless [THEN sym])

   778   done

   779

   780 lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"

   781   apply (case_tac "0 \<le> u")

   782   apply (subst image_atLeastZeroLessThan_int, assumption)

   783   apply (rule finite_imageI)

   784   apply auto

   785   done

   786

   787 lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"

   788   apply (subgoal_tac "(%x. x + l)  {0..<u-l} = {l..<u}")

   789   apply (erule subst)

   790   apply (rule finite_imageI)

   791   apply (rule finite_atLeastZeroLessThan_int)

   792   apply (rule image_add_int_atLeastLessThan)

   793   done

   794

   795 lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"

   796   by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)

   797

   798 lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"

   799   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)

   800

   801 lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"

   802   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)

   803

   804

   805 subsubsection {* Cardinality *}

   806

   807 lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"

   808   apply (case_tac "0 \<le> u")

   809   apply (subst image_atLeastZeroLessThan_int, assumption)

   810   apply (subst card_image)

   811   apply (auto simp add: inj_on_def)

   812   done

   813

   814 lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"

   815   apply (subgoal_tac "card {l..<u} = card {0..<u-l}")

   816   apply (erule ssubst, rule card_atLeastZeroLessThan_int)

   817   apply (subgoal_tac "(%x. x + l)  {0..<u-l} = {l..<u}")

   818   apply (erule subst)

   819   apply (rule card_image)

   820   apply (simp add: inj_on_def)

   821   apply (rule image_add_int_atLeastLessThan)

   822   done

   823

   824 lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"

   825 apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])

   826 apply (auto simp add: algebra_simps)

   827 done

   828

   829 lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"

   830 by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)

   831

   832 lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"

   833 by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)

   834

   835 lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"

   836 proof -

   837   have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto

   838   with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)

   839 qed

   840

   841 lemma card_less:

   842 assumes zero_in_M: "0 \<in> M"

   843 shows "card {k \<in> M. k < Suc i} \<noteq> 0"

   844 proof -

   845   from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto

   846   with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)

   847 qed

   848

   849 lemma card_less_Suc2: "0 \<notin> M \<Longrightarrow> card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"

   850 apply (rule card_bij_eq [of Suc _ _ "\<lambda>x. x - Suc 0"])

   851 apply simp

   852 apply fastsimp

   853 apply auto

   854 apply (rule inj_on_diff_nat)

   855 apply auto

   856 apply (case_tac x)

   857 apply auto

   858 apply (case_tac xa)

   859 apply auto

   860 apply (case_tac xa)

   861 apply auto

   862 done

   863

   864 lemma card_less_Suc:

   865   assumes zero_in_M: "0 \<in> M"

   866     shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"

   867 proof -

   868   from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp

   869   hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"

   870     by (auto simp only: insert_Diff)

   871   have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto

   872   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}))"

   873     apply (subst card_insert)

   874     apply simp_all

   875     apply (subst b)

   876     apply (subst card_less_Suc2[symmetric])

   877     apply simp_all

   878     done

   879   with c show ?thesis by simp

   880 qed

   881

   882

   883 subsection {*Lemmas useful with the summation operator setsum*}

   884

   885 text {* For examples, see Algebra/poly/UnivPoly2.thy *}

   886

   887 subsubsection {* Disjoint Unions *}

   888

   889 text {* Singletons and open intervals *}

   890

   891 lemma ivl_disj_un_singleton:

   892   "{l::'a::linorder} Un {l<..} = {l..}"

   893   "{..<u} Un {u::'a::linorder} = {..u}"

   894   "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"

   895   "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"

   896   "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"

   897   "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"

   898 by auto

   899

   900 text {* One- and two-sided intervals *}

   901

   902 lemma ivl_disj_un_one:

   903   "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"

   904   "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"

   905   "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"

   906   "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"

   907   "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"

   908   "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"

   909   "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"

   910   "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"

   911 by auto

   912

   913 text {* Two- and two-sided intervals *}

   914

   915 lemma ivl_disj_un_two:

   916   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"

   917   "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"

   918   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"

   919   "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"

   920   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"

   921   "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"

   922   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"

   923   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"

   924 by auto

   925

   926 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two

   927

   928 subsubsection {* Disjoint Intersections *}

   929

   930 text {* One- and two-sided intervals *}

   931

   932 lemma ivl_disj_int_one:

   933   "{..l::'a::order} Int {l<..<u} = {}"

   934   "{..<l} Int {l..<u} = {}"

   935   "{..l} Int {l<..u} = {}"

   936   "{..<l} Int {l..u} = {}"

   937   "{l<..u} Int {u<..} = {}"

   938   "{l<..<u} Int {u..} = {}"

   939   "{l..u} Int {u<..} = {}"

   940   "{l..<u} Int {u..} = {}"

   941   by auto

   942

   943 text {* Two- and two-sided intervals *}

   944

   945 lemma ivl_disj_int_two:

   946   "{l::'a::order<..<m} Int {m..<u} = {}"

   947   "{l<..m} Int {m<..<u} = {}"

   948   "{l..<m} Int {m..<u} = {}"

   949   "{l..m} Int {m<..<u} = {}"

   950   "{l<..<m} Int {m..u} = {}"

   951   "{l<..m} Int {m<..u} = {}"

   952   "{l..<m} Int {m..u} = {}"

   953   "{l..m} Int {m<..u} = {}"

   954   by auto

   955

   956 lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two

   957

   958 subsubsection {* Some Differences *}

   959

   960 lemma ivl_diff[simp]:

   961  "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"

   962 by(auto)

   963

   964

   965 subsubsection {* Some Subset Conditions *}

   966

   967 lemma ivl_subset [simp,no_atp]:

   968  "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"

   969 apply(auto simp:linorder_not_le)

   970 apply(rule ccontr)

   971 apply(insert linorder_le_less_linear[of i n])

   972 apply(clarsimp simp:linorder_not_le)

   973 apply(fastsimp)

   974 done

   975

   976

   977 subsection {* Summation indexed over intervals *}

   978

   979 syntax

   980   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)

   981   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)

   982   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)

   983   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)

   984 syntax (xsymbols)

   985   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)

   986   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)

   987   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)

   988   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)

   989 syntax (HTML output)

   990   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)

   991   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)

   992   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)

   993   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)

   994 syntax (latex_sum output)

   995   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

   996  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

   997   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

   998  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

   999   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1000  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

  1001   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1002  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

  1003

  1004 translations

  1005   "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"

  1006   "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"

  1007   "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"

  1008   "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"

  1009

  1010 text{* The above introduces some pretty alternative syntaxes for

  1011 summation over intervals:

  1012 \begin{center}

  1013 \begin{tabular}{lll}

  1014 Old & New & \LaTeX\\

  1015 @{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\

  1016 @{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\

  1017 @{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\

  1018 @{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}

  1019 \end{tabular}

  1020 \end{center}

  1021 The left column shows the term before introduction of the new syntax,

  1022 the middle column shows the new (default) syntax, and the right column

  1023 shows a special syntax. The latter is only meaningful for latex output

  1024 and has to be activated explicitly by setting the print mode to

  1025 @{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in

  1026 antiquotations). It is not the default \LaTeX\ output because it only

  1027 works well with italic-style formulae, not tt-style.

  1028

  1029 Note that for uniformity on @{typ nat} it is better to use

  1030 @{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may

  1031 not provide all lemmas available for @{term"{m..<n}"} also in the

  1032 special form for @{term"{..<n}"}. *}

  1033

  1034 text{* This congruence rule should be used for sums over intervals as

  1035 the standard theorem @{text[source]setsum_cong} does not work well

  1036 with the simplifier who adds the unsimplified premise @{term"x:B"} to

  1037 the context. *}

  1038

  1039 lemma setsum_ivl_cong:

  1040  "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>

  1041  setsum f {a..<b} = setsum g {c..<d}"

  1042 by(rule setsum_cong, simp_all)

  1043

  1044 (* FIXME why are the following simp rules but the corresponding eqns

  1045 on intervals are not? *)

  1046

  1047 lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"

  1048 by (simp add:atMost_Suc add_ac)

  1049

  1050 lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"

  1051 by (simp add:lessThan_Suc add_ac)

  1052

  1053 lemma setsum_cl_ivl_Suc[simp]:

  1054   "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"

  1055 by (auto simp:add_ac atLeastAtMostSuc_conv)

  1056

  1057 lemma setsum_op_ivl_Suc[simp]:

  1058   "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"

  1059 by (auto simp:add_ac atLeastLessThanSuc)

  1060 (*

  1061 lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>

  1062     (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"

  1063 by (auto simp:add_ac atLeastAtMostSuc_conv)

  1064 *)

  1065

  1066 lemma setsum_head:

  1067   fixes n :: nat

  1068   assumes mn: "m <= n"

  1069   shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")

  1070 proof -

  1071   from mn

  1072   have "{m..n} = {m} \<union> {m<..n}"

  1073     by (auto intro: ivl_disj_un_singleton)

  1074   hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"

  1075     by (simp add: atLeast0LessThan)

  1076   also have "\<dots> = ?rhs" by simp

  1077   finally show ?thesis .

  1078 qed

  1079

  1080 lemma setsum_head_Suc:

  1081   "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"

  1082 by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)

  1083

  1084 lemma setsum_head_upt_Suc:

  1085   "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"

  1086 apply(insert setsum_head_Suc[of m "n - Suc 0" f])

  1087 apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)

  1088 done

  1089

  1090 lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"

  1091   shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"

  1092 proof-

  1093   have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using m \<le> n+1 by auto

  1094   thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint

  1095     atLeastSucAtMost_greaterThanAtMost)

  1096 qed

  1097

  1098 lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>

  1099   setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"

  1100 by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)

  1101

  1102 lemma setsum_diff_nat_ivl:

  1103 fixes f :: "nat \<Rightarrow> 'a::ab_group_add"

  1104 shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>

  1105   setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"

  1106 using setsum_add_nat_ivl [of m n p f,symmetric]

  1107 apply (simp add: add_ac)

  1108 done

  1109

  1110 lemma setsum_natinterval_difff:

  1111   fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"

  1112   shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =

  1113           (if m <= n then f m - f(n + 1) else 0)"

  1114 by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)

  1115

  1116 lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]

  1117

  1118 lemma setsum_setsum_restrict:

  1119   "finite S \<Longrightarrow> finite T \<Longrightarrow> 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"

  1120   by (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)

  1121      (rule setsum_commute)

  1122

  1123 lemma setsum_image_gen: assumes fS: "finite S"

  1124   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f  S)"

  1125 proof-

  1126   { fix x assume "x \<in> S" then have "{y. y\<in> fS \<and> f x = y} = {f x}" by auto }

  1127   hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> fS \<and> f x = y}) S"

  1128     by simp

  1129   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f  S)"

  1130     by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])

  1131   finally show ?thesis .

  1132 qed

  1133

  1134 lemma setsum_le_included:

  1135   fixes f :: "'a \<Rightarrow> 'b::ordered_comm_monoid_add"

  1136   assumes "finite s" "finite t"

  1137   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)"

  1138   shows "setsum f s \<le> setsum g t"

  1139 proof -

  1140   have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"

  1141   proof (rule setsum_mono)

  1142     fix y assume "y \<in> s"

  1143     with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto

  1144     with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")

  1145       using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]

  1146       by (auto intro!: setsum_mono2)

  1147   qed

  1148   also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i  t)"

  1149     using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)

  1150   also have "... \<le> setsum g t"

  1151     using assms by (auto simp: setsum_image_gen[symmetric])

  1152   finally show ?thesis .

  1153 qed

  1154

  1155 lemma setsum_multicount_gen:

  1156   assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"

  1157   shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")

  1158 proof-

  1159   have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto

  1160   also have "\<dots> = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]

  1161     using assms(3) by auto

  1162   finally show ?thesis .

  1163 qed

  1164

  1165 lemma setsum_multicount:

  1166   assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"

  1167   shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")

  1168 proof-

  1169   have "?l = setsum (\<lambda>i. k) T" by(rule setsum_multicount_gen)(auto simp:assms)

  1170   also have "\<dots> = ?r" by(simp add: mult_commute)

  1171   finally show ?thesis by auto

  1172 qed

  1173

  1174

  1175 subsection{* Shifting bounds *}

  1176

  1177 lemma setsum_shift_bounds_nat_ivl:

  1178   "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"

  1179 by (induct "n", auto simp:atLeastLessThanSuc)

  1180

  1181 lemma setsum_shift_bounds_cl_nat_ivl:

  1182   "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"

  1183 apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])

  1184 apply (simp add:image_add_atLeastAtMost o_def)

  1185 done

  1186

  1187 corollary setsum_shift_bounds_cl_Suc_ivl:

  1188   "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"

  1189 by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])

  1190

  1191 corollary setsum_shift_bounds_Suc_ivl:

  1192   "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"

  1193 by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])

  1194

  1195 lemma setsum_shift_lb_Suc0_0:

  1196   "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"

  1197 by(simp add:setsum_head_Suc)

  1198

  1199 lemma setsum_shift_lb_Suc0_0_upt:

  1200   "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"

  1201 apply(cases k)apply simp

  1202 apply(simp add:setsum_head_upt_Suc)

  1203 done

  1204

  1205 subsection {* The formula for geometric sums *}

  1206

  1207 lemma geometric_sum:

  1208   assumes "x \<noteq> 1"

  1209   shows "(\<Sum>i=0..<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"

  1210 proof -

  1211   from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all

  1212   moreover have "(\<Sum>i=0..<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"

  1213   proof (induct n)

  1214     case 0 then show ?case by simp

  1215   next

  1216     case (Suc n)

  1217     moreover with y \<noteq> 0 have "(1 + y) ^ n = (y * inverse y) * (1 + y) ^ n" by simp

  1218     ultimately show ?case by (simp add: field_simps divide_inverse)

  1219   qed

  1220   ultimately show ?thesis by simp

  1221 qed

  1222

  1223

  1224 subsection {* The formula for arithmetic sums *}

  1225

  1226 lemma gauss_sum:

  1227   "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =

  1228    of_nat n*((of_nat n)+1)"

  1229 proof (induct n)

  1230   case 0

  1231   show ?case by simp

  1232 next

  1233   case (Suc n)

  1234   then show ?case by (simp add: algebra_simps)

  1235 qed

  1236

  1237 theorem arith_series_general:

  1238   "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

  1239   of_nat n * (a + (a + of_nat(n - 1)*d))"

  1240 proof cases

  1241   assume ngt1: "n > 1"

  1242   let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"

  1243   have

  1244     "(\<Sum>i\<in>{..<n}. a+?I i*d) =

  1245      ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"

  1246     by (rule setsum_addf)

  1247   also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp

  1248   also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"

  1249     unfolding One_nat_def

  1250     by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)

  1251   also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"

  1252     by (simp add: left_distrib right_distrib)

  1253   also from ngt1 have "{1..<n} = {1..n - 1}"

  1254     by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)

  1255   also from ngt1

  1256   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)"

  1257     by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)

  1258        (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])

  1259   finally show ?thesis by (simp add: algebra_simps)

  1260 next

  1261   assume "\<not>(n > 1)"

  1262   hence "n = 1 \<or> n = 0" by auto

  1263   thus ?thesis by (auto simp: algebra_simps)

  1264 qed

  1265

  1266 lemma arith_series_nat:

  1267   "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"

  1268 proof -

  1269   have

  1270     "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =

  1271     of_nat(n) * (a + (a + of_nat(n - 1)*d))"

  1272     by (rule arith_series_general)

  1273   thus ?thesis

  1274     unfolding One_nat_def by auto

  1275 qed

  1276

  1277 lemma arith_series_int:

  1278   "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

  1279   of_nat n * (a + (a + of_nat(n - 1)*d))"

  1280 proof -

  1281   have

  1282     "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

  1283     of_nat(n) * (a + (a + of_nat(n - 1)*d))"

  1284     by (rule arith_series_general)

  1285   thus ?thesis by simp

  1286 qed

  1287

  1288 lemma sum_diff_distrib:

  1289   fixes P::"nat\<Rightarrow>nat"

  1290   shows

  1291   "\<forall>x. Q x \<le> P x  \<Longrightarrow>

  1292   (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"

  1293 proof (induct n)

  1294   case 0 show ?case by simp

  1295 next

  1296   case (Suc n)

  1297

  1298   let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"

  1299   let ?rhs = "\<Sum>x<n. P x - Q x"

  1300

  1301   from Suc have "?lhs = ?rhs" by simp

  1302   moreover

  1303   from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp

  1304   moreover

  1305   from Suc have

  1306     "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"

  1307     by (subst diff_diff_left[symmetric],

  1308         subst diff_add_assoc2)

  1309        (auto simp: diff_add_assoc2 intro: setsum_mono)

  1310   ultimately

  1311   show ?case by simp

  1312 qed

  1313

  1314 subsection {* Products indexed over intervals *}

  1315

  1316 syntax

  1317   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)

  1318   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)

  1319   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)

  1320   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)

  1321 syntax (xsymbols)

  1322   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)

  1323   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)

  1324   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)

  1325   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)

  1326 syntax (HTML output)

  1327   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)

  1328   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)

  1329   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)

  1330   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)

  1331 syntax (latex_prod output)

  1332   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1333  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

  1334   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1335  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

  1336   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1337  ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

  1338   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1339  ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

  1340

  1341 translations

  1342   "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"

  1343   "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"

  1344   "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"

  1345   "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"

  1346

  1347 subsection {* Transfer setup *}

  1348

  1349 lemma transfer_nat_int_set_functions:

  1350     "{..n} = nat  {0..int n}"

  1351     "{m..n} = nat  {int m..int n}"  (* need all variants of these! *)

  1352   apply (auto simp add: image_def)

  1353   apply (rule_tac x = "int x" in bexI)

  1354   apply auto

  1355   apply (rule_tac x = "int x" in bexI)

  1356   apply auto

  1357   done

  1358

  1359 lemma transfer_nat_int_set_function_closures:

  1360     "x >= 0 \<Longrightarrow> nat_set {x..y}"

  1361   by (simp add: nat_set_def)

  1362

  1363 declare transfer_morphism_nat_int[transfer add

  1364   return: transfer_nat_int_set_functions

  1365     transfer_nat_int_set_function_closures

  1366 ]

  1367

  1368 lemma transfer_int_nat_set_functions:

  1369     "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int  {nat m..nat n}"

  1370   by (simp only: is_nat_def transfer_nat_int_set_functions

  1371     transfer_nat_int_set_function_closures

  1372     transfer_nat_int_set_return_embed nat_0_le

  1373     cong: transfer_nat_int_set_cong)

  1374

  1375 lemma transfer_int_nat_set_function_closures:

  1376     "is_nat x \<Longrightarrow> nat_set {x..y}"

  1377   by (simp only: transfer_nat_int_set_function_closures is_nat_def)

  1378

  1379 declare transfer_morphism_int_nat[transfer add

  1380   return: transfer_int_nat_set_functions

  1381     transfer_int_nat_set_function_closures

  1382 ]

  1383

  1384 end
`