src/HOL/SetInterval.thy
author nipkow
Mon Sep 01 19:16:40 2008 +0200 (2008-09-01)
changeset 28068 f6b2d1995171
parent 27656 d4f6e64ee7cc
child 28853 69eb69659bf3
permissions -rw-r--r--
cleaned up code generation for {.._} and {..<_}
moved lemmas into SetInterval where they belong
nipkow@8924
     1
(*  Title:      HOL/SetInterval.thy
nipkow@8924
     2
    ID:         $Id$
ballarin@13735
     3
    Author:     Tobias Nipkow and Clemens Ballarin
paulson@14485
     4
                Additions by Jeremy Avigad in March 2004
paulson@8957
     5
    Copyright   2000  TU Muenchen
nipkow@8924
     6
ballarin@13735
     7
lessThan, greaterThan, atLeast, atMost and two-sided intervals
nipkow@8924
     8
*)
nipkow@8924
     9
wenzelm@14577
    10
header {* Set intervals *}
wenzelm@14577
    11
nipkow@15131
    12
theory SetInterval
haftmann@25919
    13
imports Int
nipkow@15131
    14
begin
nipkow@8924
    15
nipkow@24691
    16
context ord
nipkow@24691
    17
begin
nipkow@24691
    18
definition
haftmann@25062
    19
  lessThan    :: "'a => 'a set"	("(1{..<_})") where
haftmann@25062
    20
  "{..<u} == {x. x < u}"
nipkow@24691
    21
nipkow@24691
    22
definition
haftmann@25062
    23
  atMost      :: "'a => 'a set"	("(1{.._})") where
haftmann@25062
    24
  "{..u} == {x. x \<le> u}"
nipkow@24691
    25
nipkow@24691
    26
definition
haftmann@25062
    27
  greaterThan :: "'a => 'a set"	("(1{_<..})") where
haftmann@25062
    28
  "{l<..} == {x. l<x}"
nipkow@24691
    29
nipkow@24691
    30
definition
haftmann@25062
    31
  atLeast     :: "'a => 'a set"	("(1{_..})") where
haftmann@25062
    32
  "{l..} == {x. l\<le>x}"
nipkow@24691
    33
nipkow@24691
    34
definition
haftmann@25062
    35
  greaterThanLessThan :: "'a => 'a => 'a set"  ("(1{_<..<_})") where
haftmann@25062
    36
  "{l<..<u} == {l<..} Int {..<u}"
nipkow@24691
    37
nipkow@24691
    38
definition
haftmann@25062
    39
  atLeastLessThan :: "'a => 'a => 'a set"      ("(1{_..<_})") where
haftmann@25062
    40
  "{l..<u} == {l..} Int {..<u}"
nipkow@24691
    41
nipkow@24691
    42
definition
haftmann@25062
    43
  greaterThanAtMost :: "'a => 'a => 'a set"    ("(1{_<.._})") where
haftmann@25062
    44
  "{l<..u} == {l<..} Int {..u}"
nipkow@24691
    45
nipkow@24691
    46
definition
haftmann@25062
    47
  atLeastAtMost :: "'a => 'a => 'a set"        ("(1{_.._})") where
haftmann@25062
    48
  "{l..u} == {l..} Int {..u}"
nipkow@24691
    49
nipkow@24691
    50
end
nipkow@8924
    51
ballarin@13735
    52
nipkow@15048
    53
text{* A note of warning when using @{term"{..<n}"} on type @{typ
nipkow@15048
    54
nat}: it is equivalent to @{term"{0::nat..<n}"} but some lemmas involving
nipkow@15052
    55
@{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well. *}
nipkow@15048
    56
kleing@14418
    57
syntax
kleing@14418
    58
  "@UNION_le"   :: "nat => nat => 'b set => 'b set"       ("(3UN _<=_./ _)" 10)
kleing@14418
    59
  "@UNION_less" :: "nat => nat => 'b set => 'b set"       ("(3UN _<_./ _)" 10)
kleing@14418
    60
  "@INTER_le"   :: "nat => nat => 'b set => 'b set"       ("(3INT _<=_./ _)" 10)
kleing@14418
    61
  "@INTER_less" :: "nat => nat => 'b set => 'b set"       ("(3INT _<_./ _)" 10)
kleing@14418
    62
kleing@14418
    63
syntax (input)
kleing@14418
    64
  "@UNION_le"   :: "nat => nat => 'b set => 'b set"       ("(3\<Union> _\<le>_./ _)" 10)
kleing@14418
    65
  "@UNION_less" :: "nat => nat => 'b set => 'b set"       ("(3\<Union> _<_./ _)" 10)
kleing@14418
    66
  "@INTER_le"   :: "nat => nat => 'b set => 'b set"       ("(3\<Inter> _\<le>_./ _)" 10)
kleing@14418
    67
  "@INTER_less" :: "nat => nat => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" 10)
kleing@14418
    68
kleing@14418
    69
syntax (xsymbols)
wenzelm@14846
    70
  "@UNION_le"   :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Union>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)
wenzelm@14846
    71
  "@UNION_less" :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Union>(00\<^bsub>_ < _\<^esub>)/ _)" 10)
wenzelm@14846
    72
  "@INTER_le"   :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Inter>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)
wenzelm@14846
    73
  "@INTER_less" :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Inter>(00\<^bsub>_ < _\<^esub>)/ _)" 10)
kleing@14418
    74
kleing@14418
    75
translations
kleing@14418
    76
  "UN i<=n. A"  == "UN i:{..n}. A"
nipkow@15045
    77
  "UN i<n. A"   == "UN i:{..<n}. A"
kleing@14418
    78
  "INT i<=n. A" == "INT i:{..n}. A"
nipkow@15045
    79
  "INT i<n. A"  == "INT i:{..<n}. A"
kleing@14418
    80
kleing@14418
    81
paulson@14485
    82
subsection {* Various equivalences *}
ballarin@13735
    83
haftmann@25062
    84
lemma (in ord) lessThan_iff [iff]: "(i: lessThan k) = (i<k)"
paulson@13850
    85
by (simp add: lessThan_def)
ballarin@13735
    86
paulson@15418
    87
lemma Compl_lessThan [simp]:
ballarin@13735
    88
    "!!k:: 'a::linorder. -lessThan k = atLeast k"
paulson@13850
    89
apply (auto simp add: lessThan_def atLeast_def)
ballarin@13735
    90
done
ballarin@13735
    91
paulson@13850
    92
lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"
paulson@13850
    93
by auto
ballarin@13735
    94
haftmann@25062
    95
lemma (in ord) greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"
paulson@13850
    96
by (simp add: greaterThan_def)
ballarin@13735
    97
paulson@15418
    98
lemma Compl_greaterThan [simp]:
ballarin@13735
    99
    "!!k:: 'a::linorder. -greaterThan k = atMost k"
haftmann@26072
   100
  by (auto simp add: greaterThan_def atMost_def)
ballarin@13735
   101
paulson@13850
   102
lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
paulson@13850
   103
apply (subst Compl_greaterThan [symmetric])
paulson@15418
   104
apply (rule double_complement)
ballarin@13735
   105
done
ballarin@13735
   106
haftmann@25062
   107
lemma (in ord) atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
paulson@13850
   108
by (simp add: atLeast_def)
ballarin@13735
   109
paulson@15418
   110
lemma Compl_atLeast [simp]:
ballarin@13735
   111
    "!!k:: 'a::linorder. -atLeast k = lessThan k"
haftmann@26072
   112
  by (auto simp add: lessThan_def atLeast_def)
ballarin@13735
   113
haftmann@25062
   114
lemma (in ord) atMost_iff [iff]: "(i: atMost k) = (i<=k)"
paulson@13850
   115
by (simp add: atMost_def)
ballarin@13735
   116
paulson@14485
   117
lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
paulson@14485
   118
by (blast intro: order_antisym)
paulson@13850
   119
paulson@13850
   120
paulson@14485
   121
subsection {* Logical Equivalences for Set Inclusion and Equality *}
paulson@13850
   122
paulson@13850
   123
lemma atLeast_subset_iff [iff]:
paulson@15418
   124
     "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::order))"
paulson@15418
   125
by (blast intro: order_trans)
paulson@13850
   126
paulson@13850
   127
lemma atLeast_eq_iff [iff]:
paulson@15418
   128
     "(atLeast x = atLeast y) = (x = (y::'a::linorder))"
paulson@13850
   129
by (blast intro: order_antisym order_trans)
paulson@13850
   130
paulson@13850
   131
lemma greaterThan_subset_iff [iff]:
paulson@15418
   132
     "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
paulson@15418
   133
apply (auto simp add: greaterThan_def)
paulson@15418
   134
 apply (subst linorder_not_less [symmetric], blast)
paulson@13850
   135
done
paulson@13850
   136
paulson@13850
   137
lemma greaterThan_eq_iff [iff]:
paulson@15418
   138
     "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
paulson@15418
   139
apply (rule iffI)
paulson@15418
   140
 apply (erule equalityE)
paulson@15418
   141
 apply (simp_all add: greaterThan_subset_iff)
paulson@13850
   142
done
paulson@13850
   143
paulson@15418
   144
lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::order))"
paulson@13850
   145
by (blast intro: order_trans)
paulson@13850
   146
paulson@15418
   147
lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))"
paulson@13850
   148
by (blast intro: order_antisym order_trans)
paulson@13850
   149
paulson@13850
   150
lemma lessThan_subset_iff [iff]:
paulson@15418
   151
     "(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))"
paulson@15418
   152
apply (auto simp add: lessThan_def)
paulson@15418
   153
 apply (subst linorder_not_less [symmetric], blast)
paulson@13850
   154
done
paulson@13850
   155
paulson@13850
   156
lemma lessThan_eq_iff [iff]:
paulson@15418
   157
     "(lessThan x = lessThan y) = (x = (y::'a::linorder))"
paulson@15418
   158
apply (rule iffI)
paulson@15418
   159
 apply (erule equalityE)
paulson@15418
   160
 apply (simp_all add: lessThan_subset_iff)
ballarin@13735
   161
done
ballarin@13735
   162
ballarin@13735
   163
paulson@13850
   164
subsection {*Two-sided intervals*}
ballarin@13735
   165
nipkow@24691
   166
context ord
nipkow@24691
   167
begin
nipkow@24691
   168
paulson@24286
   169
lemma greaterThanLessThan_iff [simp,noatp]:
haftmann@25062
   170
  "(i : {l<..<u}) = (l < i & i < u)"
ballarin@13735
   171
by (simp add: greaterThanLessThan_def)
ballarin@13735
   172
paulson@24286
   173
lemma atLeastLessThan_iff [simp,noatp]:
haftmann@25062
   174
  "(i : {l..<u}) = (l <= i & i < u)"
ballarin@13735
   175
by (simp add: atLeastLessThan_def)
ballarin@13735
   176
paulson@24286
   177
lemma greaterThanAtMost_iff [simp,noatp]:
haftmann@25062
   178
  "(i : {l<..u}) = (l < i & i <= u)"
ballarin@13735
   179
by (simp add: greaterThanAtMost_def)
ballarin@13735
   180
paulson@24286
   181
lemma atLeastAtMost_iff [simp,noatp]:
haftmann@25062
   182
  "(i : {l..u}) = (l <= i & i <= u)"
ballarin@13735
   183
by (simp add: atLeastAtMost_def)
ballarin@13735
   184
wenzelm@14577
   185
text {* The above four lemmas could be declared as iffs.
wenzelm@14577
   186
  If we do so, a call to blast in Hyperreal/Star.ML, lemma @{text STAR_Int}
wenzelm@14577
   187
  seems to take forever (more than one hour). *}
nipkow@24691
   188
end
ballarin@13735
   189
nipkow@15554
   190
subsubsection{* Emptyness and singletons *}
nipkow@15554
   191
nipkow@24691
   192
context order
nipkow@24691
   193
begin
nipkow@15554
   194
haftmann@25062
   195
lemma atLeastAtMost_empty [simp]: "n < m ==> {m..n} = {}";
nipkow@24691
   196
by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)
nipkow@24691
   197
haftmann@25062
   198
lemma atLeastLessThan_empty[simp]: "n \<le> m ==> {m..<n} = {}"
nipkow@15554
   199
by (auto simp add: atLeastLessThan_def)
nipkow@15554
   200
haftmann@25062
   201
lemma greaterThanAtMost_empty[simp]:"l \<le> k ==> {k<..l} = {}"
nipkow@17719
   202
by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)
nipkow@17719
   203
haftmann@25062
   204
lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..l} = {}"
nipkow@17719
   205
by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)
nipkow@17719
   206
haftmann@25062
   207
lemma atLeastAtMost_singleton [simp]: "{a..a} = {a}"
nipkow@24691
   208
by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)
nipkow@24691
   209
nipkow@24691
   210
end
paulson@14485
   211
paulson@14485
   212
subsection {* Intervals of natural numbers *}
paulson@14485
   213
paulson@15047
   214
subsubsection {* The Constant @{term lessThan} *}
paulson@15047
   215
paulson@14485
   216
lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
paulson@14485
   217
by (simp add: lessThan_def)
paulson@14485
   218
paulson@14485
   219
lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
paulson@14485
   220
by (simp add: lessThan_def less_Suc_eq, blast)
paulson@14485
   221
paulson@14485
   222
lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
paulson@14485
   223
by (simp add: lessThan_def atMost_def less_Suc_eq_le)
paulson@14485
   224
paulson@14485
   225
lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
paulson@14485
   226
by blast
paulson@14485
   227
paulson@15047
   228
subsubsection {* The Constant @{term greaterThan} *}
paulson@15047
   229
paulson@14485
   230
lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
paulson@14485
   231
apply (simp add: greaterThan_def)
paulson@14485
   232
apply (blast dest: gr0_conv_Suc [THEN iffD1])
paulson@14485
   233
done
paulson@14485
   234
paulson@14485
   235
lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
paulson@14485
   236
apply (simp add: greaterThan_def)
paulson@14485
   237
apply (auto elim: linorder_neqE)
paulson@14485
   238
done
paulson@14485
   239
paulson@14485
   240
lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
paulson@14485
   241
by blast
paulson@14485
   242
paulson@15047
   243
subsubsection {* The Constant @{term atLeast} *}
paulson@15047
   244
paulson@14485
   245
lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
paulson@14485
   246
by (unfold atLeast_def UNIV_def, simp)
paulson@14485
   247
paulson@14485
   248
lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
paulson@14485
   249
apply (simp add: atLeast_def)
paulson@14485
   250
apply (simp add: Suc_le_eq)
paulson@14485
   251
apply (simp add: order_le_less, blast)
paulson@14485
   252
done
paulson@14485
   253
paulson@14485
   254
lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
paulson@14485
   255
  by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
paulson@14485
   256
paulson@14485
   257
lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
paulson@14485
   258
by blast
paulson@14485
   259
paulson@15047
   260
subsubsection {* The Constant @{term atMost} *}
paulson@15047
   261
paulson@14485
   262
lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
paulson@14485
   263
by (simp add: atMost_def)
paulson@14485
   264
paulson@14485
   265
lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
paulson@14485
   266
apply (simp add: atMost_def)
paulson@14485
   267
apply (simp add: less_Suc_eq order_le_less, blast)
paulson@14485
   268
done
paulson@14485
   269
paulson@14485
   270
lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
paulson@14485
   271
by blast
paulson@14485
   272
paulson@15047
   273
subsubsection {* The Constant @{term atLeastLessThan} *}
paulson@15047
   274
nipkow@28068
   275
text{*The orientation of the following 2 rules is tricky. The lhs is
nipkow@24449
   276
defined in terms of the rhs.  Hence the chosen orientation makes sense
nipkow@24449
   277
in this theory --- the reverse orientation complicates proofs (eg
nipkow@24449
   278
nontermination). But outside, when the definition of the lhs is rarely
nipkow@24449
   279
used, the opposite orientation seems preferable because it reduces a
nipkow@24449
   280
specific concept to a more general one. *}
nipkow@28068
   281
paulson@15047
   282
lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
nipkow@15042
   283
by(simp add:lessThan_def atLeastLessThan_def)
nipkow@24449
   284
nipkow@28068
   285
lemma atLeast0AtMost: "{0..n::nat} = {..n}"
nipkow@28068
   286
by(simp add:atMost_def atLeastAtMost_def)
nipkow@28068
   287
nipkow@24449
   288
declare atLeast0LessThan[symmetric, code unfold]
nipkow@28068
   289
        atLeast0AtMost[symmetric, code unfold]
nipkow@24449
   290
nipkow@24449
   291
lemma atLeastLessThan0: "{m..<0::nat} = {}"
paulson@15047
   292
by (simp add: atLeastLessThan_def)
nipkow@24449
   293
paulson@15047
   294
subsubsection {* Intervals of nats with @{term Suc} *}
paulson@15047
   295
paulson@15047
   296
text{*Not a simprule because the RHS is too messy.*}
paulson@15047
   297
lemma atLeastLessThanSuc:
paulson@15047
   298
    "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
paulson@15418
   299
by (auto simp add: atLeastLessThan_def)
paulson@15047
   300
paulson@15418
   301
lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
paulson@15047
   302
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   303
(*
paulson@15047
   304
lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
paulson@15047
   305
by (induct k, simp_all add: atLeastLessThanSuc)
paulson@15047
   306
paulson@15047
   307
lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
paulson@15047
   308
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   309
*)
nipkow@15045
   310
lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
paulson@14485
   311
  by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   312
paulson@15418
   313
lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
paulson@15418
   314
  by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
paulson@14485
   315
    greaterThanAtMost_def)
paulson@14485
   316
paulson@15418
   317
lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
paulson@15418
   318
  by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
paulson@14485
   319
    greaterThanLessThan_def)
paulson@14485
   320
nipkow@15554
   321
lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
nipkow@15554
   322
by (auto simp add: atLeastAtMost_def)
nipkow@15554
   323
nipkow@16733
   324
subsubsection {* Image *}
nipkow@16733
   325
nipkow@16733
   326
lemma image_add_atLeastAtMost:
nipkow@16733
   327
  "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
nipkow@16733
   328
proof
nipkow@16733
   329
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   330
next
nipkow@16733
   331
  show "?B \<subseteq> ?A"
nipkow@16733
   332
  proof
nipkow@16733
   333
    fix n assume a: "n : ?B"
webertj@20217
   334
    hence "n - k : {i..j}" by auto
nipkow@16733
   335
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   336
    ultimately show "n : ?A" by blast
nipkow@16733
   337
  qed
nipkow@16733
   338
qed
nipkow@16733
   339
nipkow@16733
   340
lemma image_add_atLeastLessThan:
nipkow@16733
   341
  "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
nipkow@16733
   342
proof
nipkow@16733
   343
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   344
next
nipkow@16733
   345
  show "?B \<subseteq> ?A"
nipkow@16733
   346
  proof
nipkow@16733
   347
    fix n assume a: "n : ?B"
webertj@20217
   348
    hence "n - k : {i..<j}" by auto
nipkow@16733
   349
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   350
    ultimately show "n : ?A" by blast
nipkow@16733
   351
  qed
nipkow@16733
   352
qed
nipkow@16733
   353
nipkow@16733
   354
corollary image_Suc_atLeastAtMost[simp]:
nipkow@16733
   355
  "Suc ` {i..j} = {Suc i..Suc j}"
nipkow@16733
   356
using image_add_atLeastAtMost[where k=1] by simp
nipkow@16733
   357
nipkow@16733
   358
corollary image_Suc_atLeastLessThan[simp]:
nipkow@16733
   359
  "Suc ` {i..<j} = {Suc i..<Suc j}"
nipkow@16733
   360
using image_add_atLeastLessThan[where k=1] by simp
nipkow@16733
   361
nipkow@16733
   362
lemma image_add_int_atLeastLessThan:
nipkow@16733
   363
    "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
nipkow@16733
   364
  apply (auto simp add: image_def)
nipkow@16733
   365
  apply (rule_tac x = "x - l" in bexI)
nipkow@16733
   366
  apply auto
nipkow@16733
   367
  done
nipkow@16733
   368
nipkow@16733
   369
paulson@14485
   370
subsubsection {* Finiteness *}
paulson@14485
   371
nipkow@15045
   372
lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
paulson@14485
   373
  by (induct k) (simp_all add: lessThan_Suc)
paulson@14485
   374
paulson@14485
   375
lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
paulson@14485
   376
  by (induct k) (simp_all add: atMost_Suc)
paulson@14485
   377
paulson@14485
   378
lemma finite_greaterThanLessThan [iff]:
nipkow@15045
   379
  fixes l :: nat shows "finite {l<..<u}"
paulson@14485
   380
by (simp add: greaterThanLessThan_def)
paulson@14485
   381
paulson@14485
   382
lemma finite_atLeastLessThan [iff]:
nipkow@15045
   383
  fixes l :: nat shows "finite {l..<u}"
paulson@14485
   384
by (simp add: atLeastLessThan_def)
paulson@14485
   385
paulson@14485
   386
lemma finite_greaterThanAtMost [iff]:
nipkow@15045
   387
  fixes l :: nat shows "finite {l<..u}"
paulson@14485
   388
by (simp add: greaterThanAtMost_def)
paulson@14485
   389
paulson@14485
   390
lemma finite_atLeastAtMost [iff]:
paulson@14485
   391
  fixes l :: nat shows "finite {l..u}"
paulson@14485
   392
by (simp add: atLeastAtMost_def)
paulson@14485
   393
nipkow@28068
   394
text {* A bounded set of natural numbers is finite. *}
paulson@14485
   395
lemma bounded_nat_set_is_finite:
nipkow@24853
   396
  "(ALL i:N. i < (n::nat)) ==> finite N"
nipkow@28068
   397
apply (rule finite_subset)
nipkow@28068
   398
 apply (rule_tac [2] finite_lessThan, auto)
nipkow@28068
   399
done
nipkow@28068
   400
nipkow@28068
   401
lemma finite_less_ub:
nipkow@28068
   402
     "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
nipkow@28068
   403
by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
paulson@14485
   404
nipkow@24853
   405
text{* Any subset of an interval of natural numbers the size of the
nipkow@24853
   406
subset is exactly that interval. *}
nipkow@24853
   407
nipkow@24853
   408
lemma subset_card_intvl_is_intvl:
nipkow@24853
   409
  "A <= {k..<k+card A} \<Longrightarrow> A = {k..<k+card A}" (is "PROP ?P")
nipkow@24853
   410
proof cases
nipkow@24853
   411
  assume "finite A"
nipkow@24853
   412
  thus "PROP ?P"
nipkow@24853
   413
  proof(induct A rule:finite_linorder_induct)
nipkow@24853
   414
    case empty thus ?case by auto
nipkow@24853
   415
  next
nipkow@24853
   416
    case (insert A b)
nipkow@24853
   417
    moreover hence "b ~: A" by auto
nipkow@24853
   418
    moreover have "A <= {k..<k+card A}" and "b = k+card A"
nipkow@24853
   419
      using `b ~: A` insert by fastsimp+
nipkow@24853
   420
    ultimately show ?case by auto
nipkow@24853
   421
  qed
nipkow@24853
   422
next
nipkow@24853
   423
  assume "~finite A" thus "PROP ?P" by simp
nipkow@24853
   424
qed
nipkow@24853
   425
nipkow@24853
   426
paulson@14485
   427
subsubsection {* Cardinality *}
paulson@14485
   428
nipkow@15045
   429
lemma card_lessThan [simp]: "card {..<u} = u"
paulson@15251
   430
  by (induct u, simp_all add: lessThan_Suc)
paulson@14485
   431
paulson@14485
   432
lemma card_atMost [simp]: "card {..u} = Suc u"
paulson@14485
   433
  by (simp add: lessThan_Suc_atMost [THEN sym])
paulson@14485
   434
nipkow@15045
   435
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
nipkow@15045
   436
  apply (subgoal_tac "card {l..<u} = card {..<u-l}")
paulson@14485
   437
  apply (erule ssubst, rule card_lessThan)
nipkow@15045
   438
  apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
paulson@14485
   439
  apply (erule subst)
paulson@14485
   440
  apply (rule card_image)
paulson@14485
   441
  apply (simp add: inj_on_def)
paulson@14485
   442
  apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
paulson@14485
   443
  apply (rule_tac x = "x - l" in exI)
paulson@14485
   444
  apply arith
paulson@14485
   445
  done
paulson@14485
   446
paulson@15418
   447
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
paulson@14485
   448
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
paulson@14485
   449
paulson@15418
   450
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
paulson@14485
   451
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
paulson@14485
   452
nipkow@15045
   453
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
paulson@14485
   454
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
paulson@14485
   455
nipkow@26105
   456
nipkow@26105
   457
lemma ex_bij_betw_nat_finite:
nipkow@26105
   458
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
nipkow@26105
   459
apply(drule finite_imp_nat_seg_image_inj_on)
nipkow@26105
   460
apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
nipkow@26105
   461
done
nipkow@26105
   462
nipkow@26105
   463
lemma ex_bij_betw_finite_nat:
nipkow@26105
   464
  "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
nipkow@26105
   465
by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
nipkow@26105
   466
nipkow@26105
   467
paulson@14485
   468
subsection {* Intervals of integers *}
paulson@14485
   469
nipkow@15045
   470
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
paulson@14485
   471
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   472
paulson@15418
   473
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
paulson@14485
   474
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
paulson@14485
   475
paulson@15418
   476
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
paulson@15418
   477
    "{l+1..<u} = {l<..<u::int}"
paulson@14485
   478
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
paulson@14485
   479
paulson@14485
   480
subsubsection {* Finiteness *}
paulson@14485
   481
paulson@15418
   482
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
nipkow@15045
   483
    {(0::int)..<u} = int ` {..<nat u}"
paulson@14485
   484
  apply (unfold image_def lessThan_def)
paulson@14485
   485
  apply auto
paulson@14485
   486
  apply (rule_tac x = "nat x" in exI)
paulson@14485
   487
  apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
paulson@14485
   488
  done
paulson@14485
   489
nipkow@15045
   490
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
paulson@14485
   491
  apply (case_tac "0 \<le> u")
paulson@14485
   492
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   493
  apply (rule finite_imageI)
paulson@14485
   494
  apply auto
paulson@14485
   495
  done
paulson@14485
   496
nipkow@15045
   497
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
nipkow@15045
   498
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   499
  apply (erule subst)
paulson@14485
   500
  apply (rule finite_imageI)
paulson@14485
   501
  apply (rule finite_atLeastZeroLessThan_int)
nipkow@16733
   502
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
   503
  done
paulson@14485
   504
paulson@15418
   505
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
paulson@14485
   506
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
paulson@14485
   507
paulson@15418
   508
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
paulson@14485
   509
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   510
paulson@15418
   511
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
paulson@14485
   512
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   513
nipkow@24853
   514
paulson@14485
   515
subsubsection {* Cardinality *}
paulson@14485
   516
nipkow@15045
   517
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
paulson@14485
   518
  apply (case_tac "0 \<le> u")
paulson@14485
   519
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   520
  apply (subst card_image)
paulson@14485
   521
  apply (auto simp add: inj_on_def)
paulson@14485
   522
  done
paulson@14485
   523
nipkow@15045
   524
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
nipkow@15045
   525
  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
paulson@14485
   526
  apply (erule ssubst, rule card_atLeastZeroLessThan_int)
nipkow@15045
   527
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   528
  apply (erule subst)
paulson@14485
   529
  apply (rule card_image)
paulson@14485
   530
  apply (simp add: inj_on_def)
nipkow@16733
   531
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
   532
  done
paulson@14485
   533
paulson@14485
   534
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
paulson@14485
   535
  apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
paulson@14485
   536
  apply (auto simp add: compare_rls)
paulson@14485
   537
  done
paulson@14485
   538
paulson@15418
   539
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
paulson@14485
   540
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   541
nipkow@15045
   542
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
paulson@14485
   543
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   544
bulwahn@27656
   545
lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
bulwahn@27656
   546
proof -
bulwahn@27656
   547
  have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
bulwahn@27656
   548
  with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
bulwahn@27656
   549
qed
bulwahn@27656
   550
bulwahn@27656
   551
lemma card_less:
bulwahn@27656
   552
assumes zero_in_M: "0 \<in> M"
bulwahn@27656
   553
shows "card {k \<in> M. k < Suc i} \<noteq> 0"
bulwahn@27656
   554
proof -
bulwahn@27656
   555
  from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
bulwahn@27656
   556
  with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
bulwahn@27656
   557
qed
bulwahn@27656
   558
bulwahn@27656
   559
lemma card_less_Suc2: "0 \<notin> M \<Longrightarrow> card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"
bulwahn@27656
   560
apply (rule card_bij_eq [of "Suc" _ _ "\<lambda>x. x - 1"])
bulwahn@27656
   561
apply simp
bulwahn@27656
   562
apply fastsimp
bulwahn@27656
   563
apply auto
bulwahn@27656
   564
apply (rule inj_on_diff_nat)
bulwahn@27656
   565
apply auto
bulwahn@27656
   566
apply (case_tac x)
bulwahn@27656
   567
apply auto
bulwahn@27656
   568
apply (case_tac xa)
bulwahn@27656
   569
apply auto
bulwahn@27656
   570
apply (case_tac xa)
bulwahn@27656
   571
apply auto
bulwahn@27656
   572
apply (auto simp add: finite_M_bounded_by_nat)
bulwahn@27656
   573
done
bulwahn@27656
   574
bulwahn@27656
   575
lemma card_less_Suc:
bulwahn@27656
   576
  assumes zero_in_M: "0 \<in> M"
bulwahn@27656
   577
    shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
bulwahn@27656
   578
proof -
bulwahn@27656
   579
  from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp
bulwahn@27656
   580
  hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"
bulwahn@27656
   581
    by (auto simp only: insert_Diff)
bulwahn@27656
   582
  have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto
bulwahn@27656
   583
  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}))"
bulwahn@27656
   584
    apply (subst card_insert)
bulwahn@27656
   585
    apply simp_all
bulwahn@27656
   586
    apply (subst b)
bulwahn@27656
   587
    apply (subst card_less_Suc2[symmetric])
bulwahn@27656
   588
    apply simp_all
bulwahn@27656
   589
    done
bulwahn@27656
   590
  with c show ?thesis by simp
bulwahn@27656
   591
qed
bulwahn@27656
   592
paulson@14485
   593
paulson@13850
   594
subsection {*Lemmas useful with the summation operator setsum*}
paulson@13850
   595
ballarin@16102
   596
text {* For examples, see Algebra/poly/UnivPoly2.thy *}
ballarin@13735
   597
wenzelm@14577
   598
subsubsection {* Disjoint Unions *}
ballarin@13735
   599
wenzelm@14577
   600
text {* Singletons and open intervals *}
ballarin@13735
   601
ballarin@13735
   602
lemma ivl_disj_un_singleton:
nipkow@15045
   603
  "{l::'a::linorder} Un {l<..} = {l..}"
nipkow@15045
   604
  "{..<u} Un {u::'a::linorder} = {..u}"
nipkow@15045
   605
  "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
nipkow@15045
   606
  "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
nipkow@15045
   607
  "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
nipkow@15045
   608
  "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
ballarin@14398
   609
by auto
ballarin@13735
   610
wenzelm@14577
   611
text {* One- and two-sided intervals *}
ballarin@13735
   612
ballarin@13735
   613
lemma ivl_disj_un_one:
nipkow@15045
   614
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
nipkow@15045
   615
  "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
nipkow@15045
   616
  "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
nipkow@15045
   617
  "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
nipkow@15045
   618
  "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
nipkow@15045
   619
  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
nipkow@15045
   620
  "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
nipkow@15045
   621
  "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
ballarin@14398
   622
by auto
ballarin@13735
   623
wenzelm@14577
   624
text {* Two- and two-sided intervals *}
ballarin@13735
   625
ballarin@13735
   626
lemma ivl_disj_un_two:
nipkow@15045
   627
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
nipkow@15045
   628
  "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
nipkow@15045
   629
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
nipkow@15045
   630
  "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
nipkow@15045
   631
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
nipkow@15045
   632
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
nipkow@15045
   633
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
nipkow@15045
   634
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
ballarin@14398
   635
by auto
ballarin@13735
   636
ballarin@13735
   637
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
ballarin@13735
   638
wenzelm@14577
   639
subsubsection {* Disjoint Intersections *}
ballarin@13735
   640
wenzelm@14577
   641
text {* Singletons and open intervals *}
ballarin@13735
   642
ballarin@13735
   643
lemma ivl_disj_int_singleton:
nipkow@15045
   644
  "{l::'a::order} Int {l<..} = {}"
nipkow@15045
   645
  "{..<u} Int {u} = {}"
nipkow@15045
   646
  "{l} Int {l<..<u} = {}"
nipkow@15045
   647
  "{l<..<u} Int {u} = {}"
nipkow@15045
   648
  "{l} Int {l<..u} = {}"
nipkow@15045
   649
  "{l..<u} Int {u} = {}"
ballarin@13735
   650
  by simp+
ballarin@13735
   651
wenzelm@14577
   652
text {* One- and two-sided intervals *}
ballarin@13735
   653
ballarin@13735
   654
lemma ivl_disj_int_one:
nipkow@15045
   655
  "{..l::'a::order} Int {l<..<u} = {}"
nipkow@15045
   656
  "{..<l} Int {l..<u} = {}"
nipkow@15045
   657
  "{..l} Int {l<..u} = {}"
nipkow@15045
   658
  "{..<l} Int {l..u} = {}"
nipkow@15045
   659
  "{l<..u} Int {u<..} = {}"
nipkow@15045
   660
  "{l<..<u} Int {u..} = {}"
nipkow@15045
   661
  "{l..u} Int {u<..} = {}"
nipkow@15045
   662
  "{l..<u} Int {u..} = {}"
ballarin@14398
   663
  by auto
ballarin@13735
   664
wenzelm@14577
   665
text {* Two- and two-sided intervals *}
ballarin@13735
   666
ballarin@13735
   667
lemma ivl_disj_int_two:
nipkow@15045
   668
  "{l::'a::order<..<m} Int {m..<u} = {}"
nipkow@15045
   669
  "{l<..m} Int {m<..<u} = {}"
nipkow@15045
   670
  "{l..<m} Int {m..<u} = {}"
nipkow@15045
   671
  "{l..m} Int {m<..<u} = {}"
nipkow@15045
   672
  "{l<..<m} Int {m..u} = {}"
nipkow@15045
   673
  "{l<..m} Int {m<..u} = {}"
nipkow@15045
   674
  "{l..<m} Int {m..u} = {}"
nipkow@15045
   675
  "{l..m} Int {m<..u} = {}"
ballarin@14398
   676
  by auto
ballarin@13735
   677
ballarin@13735
   678
lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
ballarin@13735
   679
nipkow@15542
   680
subsubsection {* Some Differences *}
nipkow@15542
   681
nipkow@15542
   682
lemma ivl_diff[simp]:
nipkow@15542
   683
 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
nipkow@15542
   684
by(auto)
nipkow@15542
   685
nipkow@15542
   686
nipkow@15542
   687
subsubsection {* Some Subset Conditions *}
nipkow@15542
   688
paulson@24286
   689
lemma ivl_subset [simp,noatp]:
nipkow@15542
   690
 "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
nipkow@15542
   691
apply(auto simp:linorder_not_le)
nipkow@15542
   692
apply(rule ccontr)
nipkow@15542
   693
apply(insert linorder_le_less_linear[of i n])
nipkow@15542
   694
apply(clarsimp simp:linorder_not_le)
nipkow@15542
   695
apply(fastsimp)
nipkow@15542
   696
done
nipkow@15542
   697
nipkow@15041
   698
nipkow@15042
   699
subsection {* Summation indexed over intervals *}
nipkow@15042
   700
nipkow@15042
   701
syntax
nipkow@15042
   702
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   703
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
   704
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
nipkow@16052
   705
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
nipkow@15042
   706
syntax (xsymbols)
nipkow@15042
   707
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   708
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
   709
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
   710
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15042
   711
syntax (HTML output)
nipkow@15042
   712
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   713
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
   714
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
   715
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15056
   716
syntax (latex_sum output)
nipkow@15052
   717
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
   718
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@15052
   719
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
   720
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@16052
   721
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
   722
 ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15052
   723
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
   724
 ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15041
   725
nipkow@15048
   726
translations
nipkow@15048
   727
  "\<Sum>x=a..b. t" == "setsum (%x. t) {a..b}"
nipkow@15048
   728
  "\<Sum>x=a..<b. t" == "setsum (%x. t) {a..<b}"
nipkow@16052
   729
  "\<Sum>i\<le>n. t" == "setsum (\<lambda>i. t) {..n}"
nipkow@15048
   730
  "\<Sum>i<n. t" == "setsum (\<lambda>i. t) {..<n}"
nipkow@15041
   731
nipkow@15052
   732
text{* The above introduces some pretty alternative syntaxes for
nipkow@15056
   733
summation over intervals:
nipkow@15052
   734
\begin{center}
nipkow@15052
   735
\begin{tabular}{lll}
nipkow@15056
   736
Old & New & \LaTeX\\
nipkow@15056
   737
@{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\
nipkow@15056
   738
@{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\
nipkow@16052
   739
@{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
nipkow@15056
   740
@{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
nipkow@15052
   741
\end{tabular}
nipkow@15052
   742
\end{center}
nipkow@15056
   743
The left column shows the term before introduction of the new syntax,
nipkow@15056
   744
the middle column shows the new (default) syntax, and the right column
nipkow@15056
   745
shows a special syntax. The latter is only meaningful for latex output
nipkow@15056
   746
and has to be activated explicitly by setting the print mode to
wenzelm@21502
   747
@{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
nipkow@15056
   748
antiquotations). It is not the default \LaTeX\ output because it only
nipkow@15056
   749
works well with italic-style formulae, not tt-style.
nipkow@15052
   750
nipkow@15052
   751
Note that for uniformity on @{typ nat} it is better to use
nipkow@15052
   752
@{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
nipkow@15052
   753
not provide all lemmas available for @{term"{m..<n}"} also in the
nipkow@15052
   754
special form for @{term"{..<n}"}. *}
nipkow@15052
   755
nipkow@15542
   756
text{* This congruence rule should be used for sums over intervals as
nipkow@15542
   757
the standard theorem @{text[source]setsum_cong} does not work well
nipkow@15542
   758
with the simplifier who adds the unsimplified premise @{term"x:B"} to
nipkow@15542
   759
the context. *}
nipkow@15542
   760
nipkow@15542
   761
lemma setsum_ivl_cong:
nipkow@15542
   762
 "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
nipkow@15542
   763
 setsum f {a..<b} = setsum g {c..<d}"
nipkow@15542
   764
by(rule setsum_cong, simp_all)
nipkow@15041
   765
nipkow@16041
   766
(* FIXME why are the following simp rules but the corresponding eqns
nipkow@16041
   767
on intervals are not? *)
nipkow@16041
   768
nipkow@16052
   769
lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
nipkow@16052
   770
by (simp add:atMost_Suc add_ac)
nipkow@16052
   771
nipkow@16041
   772
lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
nipkow@16041
   773
by (simp add:lessThan_Suc add_ac)
nipkow@15041
   774
nipkow@15911
   775
lemma setsum_cl_ivl_Suc[simp]:
nipkow@15561
   776
  "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
nipkow@15561
   777
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@15561
   778
nipkow@15911
   779
lemma setsum_op_ivl_Suc[simp]:
nipkow@15561
   780
  "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
nipkow@15561
   781
by (auto simp:add_ac atLeastLessThanSuc)
nipkow@16041
   782
(*
nipkow@15561
   783
lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
nipkow@15561
   784
    (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
nipkow@15561
   785
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@16041
   786
*)
nipkow@28068
   787
nipkow@28068
   788
lemma setsum_head:
nipkow@28068
   789
  fixes n :: nat
nipkow@28068
   790
  assumes mn: "m <= n" 
nipkow@28068
   791
  shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
nipkow@28068
   792
proof -
nipkow@28068
   793
  from mn
nipkow@28068
   794
  have "{m..n} = {m} \<union> {m<..n}"
nipkow@28068
   795
    by (auto intro: ivl_disj_un_singleton)
nipkow@28068
   796
  hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
nipkow@28068
   797
    by (simp add: atLeast0LessThan)
nipkow@28068
   798
  also have "\<dots> = ?rhs" by simp
nipkow@28068
   799
  finally show ?thesis .
nipkow@28068
   800
qed
nipkow@28068
   801
nipkow@28068
   802
lemma setsum_head_Suc:
nipkow@28068
   803
  "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"
nipkow@28068
   804
by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)
nipkow@28068
   805
nipkow@28068
   806
lemma setsum_head_upt_Suc:
nipkow@28068
   807
  "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"
nipkow@28068
   808
apply(insert setsum_head_Suc[of m "n - 1" f])
nipkow@28068
   809
apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] ring_simps)
nipkow@28068
   810
done
nipkow@28068
   811
nipkow@28068
   812
nipkow@15539
   813
lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
   814
  setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
nipkow@15539
   815
by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
nipkow@15539
   816
nipkow@15539
   817
lemma setsum_diff_nat_ivl:
nipkow@15539
   818
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
nipkow@15539
   819
shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
   820
  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
nipkow@15539
   821
using setsum_add_nat_ivl [of m n p f,symmetric]
nipkow@15539
   822
apply (simp add: add_ac)
nipkow@15539
   823
done
nipkow@15539
   824
nipkow@28068
   825
nipkow@16733
   826
subsection{* Shifting bounds *}
nipkow@16733
   827
nipkow@15539
   828
lemma setsum_shift_bounds_nat_ivl:
nipkow@15539
   829
  "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
nipkow@15539
   830
by (induct "n", auto simp:atLeastLessThanSuc)
nipkow@15539
   831
nipkow@16733
   832
lemma setsum_shift_bounds_cl_nat_ivl:
nipkow@16733
   833
  "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
nipkow@16733
   834
apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
nipkow@16733
   835
apply (simp add:image_add_atLeastAtMost o_def)
nipkow@16733
   836
done
nipkow@16733
   837
nipkow@16733
   838
corollary setsum_shift_bounds_cl_Suc_ivl:
nipkow@16733
   839
  "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
nipkow@16733
   840
by (simp add:setsum_shift_bounds_cl_nat_ivl[where k=1,simplified])
nipkow@16733
   841
nipkow@16733
   842
corollary setsum_shift_bounds_Suc_ivl:
nipkow@16733
   843
  "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
nipkow@16733
   844
by (simp add:setsum_shift_bounds_nat_ivl[where k=1,simplified])
nipkow@16733
   845
nipkow@28068
   846
lemma setsum_shift_lb_Suc0_0:
nipkow@28068
   847
  "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"
nipkow@28068
   848
by(simp add:setsum_head_Suc)
kleing@19106
   849
nipkow@28068
   850
lemma setsum_shift_lb_Suc0_0_upt:
nipkow@28068
   851
  "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"
nipkow@28068
   852
apply(cases k)apply simp
nipkow@28068
   853
apply(simp add:setsum_head_upt_Suc)
nipkow@28068
   854
done
kleing@19022
   855
ballarin@17149
   856
subsection {* The formula for geometric sums *}
ballarin@17149
   857
ballarin@17149
   858
lemma geometric_sum:
ballarin@17149
   859
  "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) =
huffman@22713
   860
  (x ^ n - 1) / (x - 1::'a::{field, recpower})"
nipkow@23496
   861
by (induct "n") (simp_all add:field_simps power_Suc)
ballarin@17149
   862
kleing@19469
   863
subsection {* The formula for arithmetic sums *}
kleing@19469
   864
kleing@19469
   865
lemma gauss_sum:
huffman@23277
   866
  "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =
kleing@19469
   867
   of_nat n*((of_nat n)+1)"
kleing@19469
   868
proof (induct n)
kleing@19469
   869
  case 0
kleing@19469
   870
  show ?case by simp
kleing@19469
   871
next
kleing@19469
   872
  case (Suc n)
nipkow@23477
   873
  then show ?case by (simp add: ring_simps)
kleing@19469
   874
qed
kleing@19469
   875
kleing@19469
   876
theorem arith_series_general:
huffman@23277
   877
  "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
   878
  of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469
   879
proof cases
kleing@19469
   880
  assume ngt1: "n > 1"
kleing@19469
   881
  let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
kleing@19469
   882
  have
kleing@19469
   883
    "(\<Sum>i\<in>{..<n}. a+?I i*d) =
kleing@19469
   884
     ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
kleing@19469
   885
    by (rule setsum_addf)
kleing@19469
   886
  also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
kleing@19469
   887
  also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
nipkow@28068
   888
    by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)
kleing@19469
   889
  also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"
kleing@19469
   890
    by (simp add: left_distrib right_distrib)
kleing@19469
   891
  also from ngt1 have "{1..<n} = {1..n - 1}"
nipkow@28068
   892
    by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
nipkow@28068
   893
  also from ngt1
kleing@19469
   894
  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)"
kleing@19469
   895
    by (simp only: mult_ac gauss_sum [of "n - 1"])
huffman@23431
   896
       (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
kleing@19469
   897
  finally show ?thesis by (simp add: mult_ac add_ac right_distrib)
kleing@19469
   898
next
kleing@19469
   899
  assume "\<not>(n > 1)"
kleing@19469
   900
  hence "n = 1 \<or> n = 0" by auto
kleing@19469
   901
  thus ?thesis by (auto simp: mult_ac right_distrib)
kleing@19469
   902
qed
kleing@19469
   903
kleing@19469
   904
lemma arith_series_nat:
kleing@19469
   905
  "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
kleing@19469
   906
proof -
kleing@19469
   907
  have
kleing@19469
   908
    "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
kleing@19469
   909
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469
   910
    by (rule arith_series_general)
kleing@19469
   911
  thus ?thesis by (auto simp add: of_nat_id)
kleing@19469
   912
qed
kleing@19469
   913
kleing@19469
   914
lemma arith_series_int:
kleing@19469
   915
  "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
   916
  of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469
   917
proof -
kleing@19469
   918
  have
kleing@19469
   919
    "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
   920
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469
   921
    by (rule arith_series_general)
kleing@19469
   922
  thus ?thesis by simp
kleing@19469
   923
qed
paulson@15418
   924
kleing@19022
   925
lemma sum_diff_distrib:
kleing@19022
   926
  fixes P::"nat\<Rightarrow>nat"
kleing@19022
   927
  shows
kleing@19022
   928
  "\<forall>x. Q x \<le> P x  \<Longrightarrow>
kleing@19022
   929
  (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
kleing@19022
   930
proof (induct n)
kleing@19022
   931
  case 0 show ?case by simp
kleing@19022
   932
next
kleing@19022
   933
  case (Suc n)
kleing@19022
   934
kleing@19022
   935
  let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
kleing@19022
   936
  let ?rhs = "\<Sum>x<n. P x - Q x"
kleing@19022
   937
kleing@19022
   938
  from Suc have "?lhs = ?rhs" by simp
kleing@19022
   939
  moreover
kleing@19022
   940
  from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
kleing@19022
   941
  moreover
kleing@19022
   942
  from Suc have
kleing@19022
   943
    "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
kleing@19022
   944
    by (subst diff_diff_left[symmetric],
kleing@19022
   945
        subst diff_add_assoc2)
kleing@19022
   946
       (auto simp: diff_add_assoc2 intro: setsum_mono)
kleing@19022
   947
  ultimately
kleing@19022
   948
  show ?case by simp
kleing@19022
   949
qed
kleing@19022
   950
nipkow@8924
   951
end