src/HOL/SetInterval.thy
author nipkow
Wed Aug 26 16:13:19 2009 +0200 (2009-08-26)
changeset 32408 a1a85b0a26f7
parent 32400 6c62363cf0d7
child 32436 10cd49e0c067
permissions -rw-r--r--
new interval lemma
CCS example for predicate compiler
nipkow@8924
     1
(*  Title:      HOL/SetInterval.thy
ballarin@13735
     2
    Author:     Tobias Nipkow and Clemens Ballarin
paulson@14485
     3
                Additions by Jeremy Avigad in March 2004
paulson@8957
     4
    Copyright   2000  TU Muenchen
nipkow@8924
     5
ballarin@13735
     6
lessThan, greaterThan, atLeast, atMost and two-sided intervals
nipkow@8924
     7
*)
nipkow@8924
     8
wenzelm@14577
     9
header {* Set intervals *}
wenzelm@14577
    10
nipkow@15131
    11
theory SetInterval
haftmann@25919
    12
imports Int
nipkow@15131
    13
begin
nipkow@8924
    14
nipkow@24691
    15
context ord
nipkow@24691
    16
begin
nipkow@24691
    17
definition
haftmann@25062
    18
  lessThan    :: "'a => 'a set"	("(1{..<_})") where
haftmann@25062
    19
  "{..<u} == {x. x < u}"
nipkow@24691
    20
nipkow@24691
    21
definition
haftmann@25062
    22
  atMost      :: "'a => 'a set"	("(1{.._})") where
haftmann@25062
    23
  "{..u} == {x. x \<le> u}"
nipkow@24691
    24
nipkow@24691
    25
definition
haftmann@25062
    26
  greaterThan :: "'a => 'a set"	("(1{_<..})") where
haftmann@25062
    27
  "{l<..} == {x. l<x}"
nipkow@24691
    28
nipkow@24691
    29
definition
haftmann@25062
    30
  atLeast     :: "'a => 'a set"	("(1{_..})") where
haftmann@25062
    31
  "{l..} == {x. l\<le>x}"
nipkow@24691
    32
nipkow@24691
    33
definition
haftmann@25062
    34
  greaterThanLessThan :: "'a => 'a => 'a set"  ("(1{_<..<_})") where
haftmann@25062
    35
  "{l<..<u} == {l<..} Int {..<u}"
nipkow@24691
    36
nipkow@24691
    37
definition
haftmann@25062
    38
  atLeastLessThan :: "'a => 'a => 'a set"      ("(1{_..<_})") where
haftmann@25062
    39
  "{l..<u} == {l..} Int {..<u}"
nipkow@24691
    40
nipkow@24691
    41
definition
haftmann@25062
    42
  greaterThanAtMost :: "'a => 'a => 'a set"    ("(1{_<.._})") where
haftmann@25062
    43
  "{l<..u} == {l<..} Int {..u}"
nipkow@24691
    44
nipkow@24691
    45
definition
haftmann@25062
    46
  atLeastAtMost :: "'a => 'a => 'a set"        ("(1{_.._})") where
haftmann@25062
    47
  "{l..u} == {l..} Int {..u}"
nipkow@24691
    48
nipkow@24691
    49
end
nipkow@8924
    50
ballarin@13735
    51
nipkow@15048
    52
text{* A note of warning when using @{term"{..<n}"} on type @{typ
nipkow@15048
    53
nat}: it is equivalent to @{term"{0::nat..<n}"} but some lemmas involving
nipkow@15052
    54
@{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well. *}
nipkow@15048
    55
kleing@14418
    56
syntax
nipkow@30384
    57
  "@UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3UN _<=_./ _)" 10)
nipkow@30384
    58
  "@UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3UN _<_./ _)" 10)
nipkow@30384
    59
  "@INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3INT _<=_./ _)" 10)
nipkow@30384
    60
  "@INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3INT _<_./ _)" 10)
kleing@14418
    61
nipkow@30372
    62
syntax (xsymbols)
nipkow@30384
    63
  "@UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _\<le>_./ _)" 10)
nipkow@30384
    64
  "@UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _<_./ _)" 10)
nipkow@30384
    65
  "@INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _\<le>_./ _)" 10)
nipkow@30384
    66
  "@INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" 10)
kleing@14418
    67
nipkow@30372
    68
syntax (latex output)
nipkow@30384
    69
  "@UNION_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ \<le> _)/ _)" 10)
nipkow@30384
    70
  "@UNION_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ < _)/ _)" 10)
nipkow@30384
    71
  "@INTER_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ \<le> _)/ _)" 10)
nipkow@30384
    72
  "@INTER_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ < _)/ _)" 10)
kleing@14418
    73
kleing@14418
    74
translations
kleing@14418
    75
  "UN i<=n. A"  == "UN i:{..n}. A"
nipkow@15045
    76
  "UN i<n. A"   == "UN i:{..<n}. A"
kleing@14418
    77
  "INT i<=n. A" == "INT i:{..n}. A"
nipkow@15045
    78
  "INT i<n. A"  == "INT i:{..<n}. A"
kleing@14418
    79
kleing@14418
    80
paulson@14485
    81
subsection {* Various equivalences *}
ballarin@13735
    82
haftmann@25062
    83
lemma (in ord) lessThan_iff [iff]: "(i: lessThan k) = (i<k)"
paulson@13850
    84
by (simp add: lessThan_def)
ballarin@13735
    85
paulson@15418
    86
lemma Compl_lessThan [simp]:
ballarin@13735
    87
    "!!k:: 'a::linorder. -lessThan k = atLeast k"
paulson@13850
    88
apply (auto simp add: lessThan_def atLeast_def)
ballarin@13735
    89
done
ballarin@13735
    90
paulson@13850
    91
lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"
paulson@13850
    92
by auto
ballarin@13735
    93
haftmann@25062
    94
lemma (in ord) greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"
paulson@13850
    95
by (simp add: greaterThan_def)
ballarin@13735
    96
paulson@15418
    97
lemma Compl_greaterThan [simp]:
ballarin@13735
    98
    "!!k:: 'a::linorder. -greaterThan k = atMost k"
haftmann@26072
    99
  by (auto simp add: greaterThan_def atMost_def)
ballarin@13735
   100
paulson@13850
   101
lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
paulson@13850
   102
apply (subst Compl_greaterThan [symmetric])
paulson@15418
   103
apply (rule double_complement)
ballarin@13735
   104
done
ballarin@13735
   105
haftmann@25062
   106
lemma (in ord) atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
paulson@13850
   107
by (simp add: atLeast_def)
ballarin@13735
   108
paulson@15418
   109
lemma Compl_atLeast [simp]:
ballarin@13735
   110
    "!!k:: 'a::linorder. -atLeast k = lessThan k"
haftmann@26072
   111
  by (auto simp add: lessThan_def atLeast_def)
ballarin@13735
   112
haftmann@25062
   113
lemma (in ord) atMost_iff [iff]: "(i: atMost k) = (i<=k)"
paulson@13850
   114
by (simp add: atMost_def)
ballarin@13735
   115
paulson@14485
   116
lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
paulson@14485
   117
by (blast intro: order_antisym)
paulson@13850
   118
paulson@13850
   119
paulson@14485
   120
subsection {* Logical Equivalences for Set Inclusion and Equality *}
paulson@13850
   121
paulson@13850
   122
lemma atLeast_subset_iff [iff]:
paulson@15418
   123
     "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::order))"
paulson@15418
   124
by (blast intro: order_trans)
paulson@13850
   125
paulson@13850
   126
lemma atLeast_eq_iff [iff]:
paulson@15418
   127
     "(atLeast x = atLeast y) = (x = (y::'a::linorder))"
paulson@13850
   128
by (blast intro: order_antisym order_trans)
paulson@13850
   129
paulson@13850
   130
lemma greaterThan_subset_iff [iff]:
paulson@15418
   131
     "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
paulson@15418
   132
apply (auto simp add: greaterThan_def)
paulson@15418
   133
 apply (subst linorder_not_less [symmetric], blast)
paulson@13850
   134
done
paulson@13850
   135
paulson@13850
   136
lemma greaterThan_eq_iff [iff]:
paulson@15418
   137
     "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
paulson@15418
   138
apply (rule iffI)
paulson@15418
   139
 apply (erule equalityE)
haftmann@29709
   140
 apply simp_all
paulson@13850
   141
done
paulson@13850
   142
paulson@15418
   143
lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::order))"
paulson@13850
   144
by (blast intro: order_trans)
paulson@13850
   145
paulson@15418
   146
lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))"
paulson@13850
   147
by (blast intro: order_antisym order_trans)
paulson@13850
   148
paulson@13850
   149
lemma lessThan_subset_iff [iff]:
paulson@15418
   150
     "(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))"
paulson@15418
   151
apply (auto simp add: lessThan_def)
paulson@15418
   152
 apply (subst linorder_not_less [symmetric], blast)
paulson@13850
   153
done
paulson@13850
   154
paulson@13850
   155
lemma lessThan_eq_iff [iff]:
paulson@15418
   156
     "(lessThan x = lessThan y) = (x = (y::'a::linorder))"
paulson@15418
   157
apply (rule iffI)
paulson@15418
   158
 apply (erule equalityE)
haftmann@29709
   159
 apply simp_all
ballarin@13735
   160
done
ballarin@13735
   161
ballarin@13735
   162
paulson@13850
   163
subsection {*Two-sided intervals*}
ballarin@13735
   164
nipkow@24691
   165
context ord
nipkow@24691
   166
begin
nipkow@24691
   167
paulson@24286
   168
lemma greaterThanLessThan_iff [simp,noatp]:
haftmann@25062
   169
  "(i : {l<..<u}) = (l < i & i < u)"
ballarin@13735
   170
by (simp add: greaterThanLessThan_def)
ballarin@13735
   171
paulson@24286
   172
lemma atLeastLessThan_iff [simp,noatp]:
haftmann@25062
   173
  "(i : {l..<u}) = (l <= i & i < u)"
ballarin@13735
   174
by (simp add: atLeastLessThan_def)
ballarin@13735
   175
paulson@24286
   176
lemma greaterThanAtMost_iff [simp,noatp]:
haftmann@25062
   177
  "(i : {l<..u}) = (l < i & i <= u)"
ballarin@13735
   178
by (simp add: greaterThanAtMost_def)
ballarin@13735
   179
paulson@24286
   180
lemma atLeastAtMost_iff [simp,noatp]:
haftmann@25062
   181
  "(i : {l..u}) = (l <= i & i <= u)"
ballarin@13735
   182
by (simp add: atLeastAtMost_def)
ballarin@13735
   183
wenzelm@14577
   184
text {* The above four lemmas could be declared as iffs.
wenzelm@14577
   185
  If we do so, a call to blast in Hyperreal/Star.ML, lemma @{text STAR_Int}
wenzelm@14577
   186
  seems to take forever (more than one hour). *}
nipkow@24691
   187
end
ballarin@13735
   188
nipkow@32400
   189
subsubsection{* Emptyness, singletons, subset *}
nipkow@15554
   190
nipkow@24691
   191
context order
nipkow@24691
   192
begin
nipkow@15554
   193
nipkow@32400
   194
lemma atLeastatMost_empty[simp]:
nipkow@32400
   195
  "b < a \<Longrightarrow> {a..b} = {}"
nipkow@32400
   196
by(auto simp: atLeastAtMost_def atLeast_def atMost_def)
nipkow@32400
   197
nipkow@32400
   198
lemma atLeastatMost_empty_iff[simp]:
nipkow@32400
   199
  "{a..b} = {} \<longleftrightarrow> (~ a <= b)"
nipkow@32400
   200
by auto (blast intro: order_trans)
nipkow@32400
   201
nipkow@32400
   202
lemma atLeastatMost_empty_iff2[simp]:
nipkow@32400
   203
  "{} = {a..b} \<longleftrightarrow> (~ a <= b)"
nipkow@32400
   204
by auto (blast intro: order_trans)
nipkow@32400
   205
nipkow@32400
   206
lemma atLeastLessThan_empty[simp]:
nipkow@32400
   207
  "b <= a \<Longrightarrow> {a..<b} = {}"
nipkow@32400
   208
by(auto simp: atLeastLessThan_def)
nipkow@24691
   209
nipkow@32400
   210
lemma atLeastLessThan_empty_iff[simp]:
nipkow@32400
   211
  "{a..<b} = {} \<longleftrightarrow> (~ a < b)"
nipkow@32400
   212
by auto (blast intro: le_less_trans)
nipkow@32400
   213
nipkow@32400
   214
lemma atLeastLessThan_empty_iff2[simp]:
nipkow@32400
   215
  "{} = {a..<b} \<longleftrightarrow> (~ a < b)"
nipkow@32400
   216
by auto (blast intro: le_less_trans)
nipkow@15554
   217
nipkow@32400
   218
lemma greaterThanAtMost_empty[simp]: "l \<le> k ==> {k<..l} = {}"
nipkow@17719
   219
by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)
nipkow@17719
   220
nipkow@32400
   221
lemma greaterThanAtMost_empty_iff[simp]: "{k<..l} = {} \<longleftrightarrow> ~ k < l"
nipkow@32400
   222
by auto (blast intro: less_le_trans)
nipkow@32400
   223
nipkow@32400
   224
lemma greaterThanAtMost_empty_iff2[simp]: "{} = {k<..l} \<longleftrightarrow> ~ k < l"
nipkow@32400
   225
by auto (blast intro: less_le_trans)
nipkow@32400
   226
haftmann@29709
   227
lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..<l} = {}"
nipkow@17719
   228
by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)
nipkow@17719
   229
haftmann@25062
   230
lemma atLeastAtMost_singleton [simp]: "{a..a} = {a}"
nipkow@24691
   231
by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)
nipkow@24691
   232
nipkow@32400
   233
lemma atLeastatMost_subset_iff[simp]:
nipkow@32400
   234
  "{a..b} <= {c..d} \<longleftrightarrow> (~ a <= b) | c <= a & b <= d"
nipkow@32400
   235
unfolding atLeastAtMost_def atLeast_def atMost_def
nipkow@32400
   236
by (blast intro: order_trans)
nipkow@32400
   237
nipkow@32400
   238
lemma atLeastatMost_psubset_iff:
nipkow@32400
   239
  "{a..b} < {c..d} \<longleftrightarrow>
nipkow@32400
   240
   ((~ a <= b) | c <= a & b <= d & (c < a | b < d))  &  c <= d"
nipkow@32400
   241
by(simp add: psubset_eq expand_set_eq less_le_not_le)(blast intro: order_trans)
nipkow@32400
   242
nipkow@24691
   243
end
paulson@14485
   244
nipkow@32408
   245
lemma (in linorder) atLeastLessThan_subset_iff:
nipkow@32408
   246
  "{a..<b} <= {c..<d} \<Longrightarrow> b <= a | c<=a & b<=d"
nipkow@32408
   247
apply (auto simp:subset_eq Ball_def)
nipkow@32408
   248
apply(frule_tac x=a in spec)
nipkow@32408
   249
apply(erule_tac x=d in allE)
nipkow@32408
   250
apply (simp add: less_imp_le)
nipkow@32408
   251
done
nipkow@32408
   252
paulson@14485
   253
subsection {* Intervals of natural numbers *}
paulson@14485
   254
paulson@15047
   255
subsubsection {* The Constant @{term lessThan} *}
paulson@15047
   256
paulson@14485
   257
lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
paulson@14485
   258
by (simp add: lessThan_def)
paulson@14485
   259
paulson@14485
   260
lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
paulson@14485
   261
by (simp add: lessThan_def less_Suc_eq, blast)
paulson@14485
   262
paulson@14485
   263
lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
paulson@14485
   264
by (simp add: lessThan_def atMost_def less_Suc_eq_le)
paulson@14485
   265
paulson@14485
   266
lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
paulson@14485
   267
by blast
paulson@14485
   268
paulson@15047
   269
subsubsection {* The Constant @{term greaterThan} *}
paulson@15047
   270
paulson@14485
   271
lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
paulson@14485
   272
apply (simp add: greaterThan_def)
paulson@14485
   273
apply (blast dest: gr0_conv_Suc [THEN iffD1])
paulson@14485
   274
done
paulson@14485
   275
paulson@14485
   276
lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
paulson@14485
   277
apply (simp add: greaterThan_def)
paulson@14485
   278
apply (auto elim: linorder_neqE)
paulson@14485
   279
done
paulson@14485
   280
paulson@14485
   281
lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
paulson@14485
   282
by blast
paulson@14485
   283
paulson@15047
   284
subsubsection {* The Constant @{term atLeast} *}
paulson@15047
   285
paulson@14485
   286
lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
paulson@14485
   287
by (unfold atLeast_def UNIV_def, simp)
paulson@14485
   288
paulson@14485
   289
lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
paulson@14485
   290
apply (simp add: atLeast_def)
paulson@14485
   291
apply (simp add: Suc_le_eq)
paulson@14485
   292
apply (simp add: order_le_less, blast)
paulson@14485
   293
done
paulson@14485
   294
paulson@14485
   295
lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
paulson@14485
   296
  by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
paulson@14485
   297
paulson@14485
   298
lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
paulson@14485
   299
by blast
paulson@14485
   300
paulson@15047
   301
subsubsection {* The Constant @{term atMost} *}
paulson@15047
   302
paulson@14485
   303
lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
paulson@14485
   304
by (simp add: atMost_def)
paulson@14485
   305
paulson@14485
   306
lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
paulson@14485
   307
apply (simp add: atMost_def)
paulson@14485
   308
apply (simp add: less_Suc_eq order_le_less, blast)
paulson@14485
   309
done
paulson@14485
   310
paulson@14485
   311
lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
paulson@14485
   312
by blast
paulson@14485
   313
paulson@15047
   314
subsubsection {* The Constant @{term atLeastLessThan} *}
paulson@15047
   315
nipkow@28068
   316
text{*The orientation of the following 2 rules is tricky. The lhs is
nipkow@24449
   317
defined in terms of the rhs.  Hence the chosen orientation makes sense
nipkow@24449
   318
in this theory --- the reverse orientation complicates proofs (eg
nipkow@24449
   319
nontermination). But outside, when the definition of the lhs is rarely
nipkow@24449
   320
used, the opposite orientation seems preferable because it reduces a
nipkow@24449
   321
specific concept to a more general one. *}
nipkow@28068
   322
paulson@15047
   323
lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
nipkow@15042
   324
by(simp add:lessThan_def atLeastLessThan_def)
nipkow@24449
   325
nipkow@28068
   326
lemma atLeast0AtMost: "{0..n::nat} = {..n}"
nipkow@28068
   327
by(simp add:atMost_def atLeastAtMost_def)
nipkow@28068
   328
haftmann@31998
   329
declare atLeast0LessThan[symmetric, code_unfold]
haftmann@31998
   330
        atLeast0AtMost[symmetric, code_unfold]
nipkow@24449
   331
nipkow@24449
   332
lemma atLeastLessThan0: "{m..<0::nat} = {}"
paulson@15047
   333
by (simp add: atLeastLessThan_def)
nipkow@24449
   334
paulson@15047
   335
subsubsection {* Intervals of nats with @{term Suc} *}
paulson@15047
   336
paulson@15047
   337
text{*Not a simprule because the RHS is too messy.*}
paulson@15047
   338
lemma atLeastLessThanSuc:
paulson@15047
   339
    "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
paulson@15418
   340
by (auto simp add: atLeastLessThan_def)
paulson@15047
   341
paulson@15418
   342
lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
paulson@15047
   343
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   344
(*
paulson@15047
   345
lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
paulson@15047
   346
by (induct k, simp_all add: atLeastLessThanSuc)
paulson@15047
   347
paulson@15047
   348
lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
paulson@15047
   349
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   350
*)
nipkow@15045
   351
lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
paulson@14485
   352
  by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   353
paulson@15418
   354
lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
paulson@15418
   355
  by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
paulson@14485
   356
    greaterThanAtMost_def)
paulson@14485
   357
paulson@15418
   358
lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
paulson@15418
   359
  by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
paulson@14485
   360
    greaterThanLessThan_def)
paulson@14485
   361
nipkow@15554
   362
lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
nipkow@15554
   363
by (auto simp add: atLeastAtMost_def)
nipkow@15554
   364
nipkow@16733
   365
subsubsection {* Image *}
nipkow@16733
   366
nipkow@16733
   367
lemma image_add_atLeastAtMost:
nipkow@16733
   368
  "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
nipkow@16733
   369
proof
nipkow@16733
   370
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   371
next
nipkow@16733
   372
  show "?B \<subseteq> ?A"
nipkow@16733
   373
  proof
nipkow@16733
   374
    fix n assume a: "n : ?B"
webertj@20217
   375
    hence "n - k : {i..j}" by auto
nipkow@16733
   376
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   377
    ultimately show "n : ?A" by blast
nipkow@16733
   378
  qed
nipkow@16733
   379
qed
nipkow@16733
   380
nipkow@16733
   381
lemma image_add_atLeastLessThan:
nipkow@16733
   382
  "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
nipkow@16733
   383
proof
nipkow@16733
   384
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   385
next
nipkow@16733
   386
  show "?B \<subseteq> ?A"
nipkow@16733
   387
  proof
nipkow@16733
   388
    fix n assume a: "n : ?B"
webertj@20217
   389
    hence "n - k : {i..<j}" by auto
nipkow@16733
   390
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   391
    ultimately show "n : ?A" by blast
nipkow@16733
   392
  qed
nipkow@16733
   393
qed
nipkow@16733
   394
nipkow@16733
   395
corollary image_Suc_atLeastAtMost[simp]:
nipkow@16733
   396
  "Suc ` {i..j} = {Suc i..Suc j}"
huffman@30079
   397
using image_add_atLeastAtMost[where k="Suc 0"] by simp
nipkow@16733
   398
nipkow@16733
   399
corollary image_Suc_atLeastLessThan[simp]:
nipkow@16733
   400
  "Suc ` {i..<j} = {Suc i..<Suc j}"
huffman@30079
   401
using image_add_atLeastLessThan[where k="Suc 0"] by simp
nipkow@16733
   402
nipkow@16733
   403
lemma image_add_int_atLeastLessThan:
nipkow@16733
   404
    "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
nipkow@16733
   405
  apply (auto simp add: image_def)
nipkow@16733
   406
  apply (rule_tac x = "x - l" in bexI)
nipkow@16733
   407
  apply auto
nipkow@16733
   408
  done
nipkow@16733
   409
nipkow@16733
   410
paulson@14485
   411
subsubsection {* Finiteness *}
paulson@14485
   412
nipkow@15045
   413
lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
paulson@14485
   414
  by (induct k) (simp_all add: lessThan_Suc)
paulson@14485
   415
paulson@14485
   416
lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
paulson@14485
   417
  by (induct k) (simp_all add: atMost_Suc)
paulson@14485
   418
paulson@14485
   419
lemma finite_greaterThanLessThan [iff]:
nipkow@15045
   420
  fixes l :: nat shows "finite {l<..<u}"
paulson@14485
   421
by (simp add: greaterThanLessThan_def)
paulson@14485
   422
paulson@14485
   423
lemma finite_atLeastLessThan [iff]:
nipkow@15045
   424
  fixes l :: nat shows "finite {l..<u}"
paulson@14485
   425
by (simp add: atLeastLessThan_def)
paulson@14485
   426
paulson@14485
   427
lemma finite_greaterThanAtMost [iff]:
nipkow@15045
   428
  fixes l :: nat shows "finite {l<..u}"
paulson@14485
   429
by (simp add: greaterThanAtMost_def)
paulson@14485
   430
paulson@14485
   431
lemma finite_atLeastAtMost [iff]:
paulson@14485
   432
  fixes l :: nat shows "finite {l..u}"
paulson@14485
   433
by (simp add: atLeastAtMost_def)
paulson@14485
   434
nipkow@28068
   435
text {* A bounded set of natural numbers is finite. *}
paulson@14485
   436
lemma bounded_nat_set_is_finite:
nipkow@24853
   437
  "(ALL i:N. i < (n::nat)) ==> finite N"
nipkow@28068
   438
apply (rule finite_subset)
nipkow@28068
   439
 apply (rule_tac [2] finite_lessThan, auto)
nipkow@28068
   440
done
nipkow@28068
   441
nipkow@31044
   442
text {* A set of natural numbers is finite iff it is bounded. *}
nipkow@31044
   443
lemma finite_nat_set_iff_bounded:
nipkow@31044
   444
  "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")
nipkow@31044
   445
proof
nipkow@31044
   446
  assume f:?F  show ?B
nipkow@31044
   447
    using Max_ge[OF `?F`, simplified less_Suc_eq_le[symmetric]] by blast
nipkow@31044
   448
next
nipkow@31044
   449
  assume ?B show ?F using `?B` by(blast intro:bounded_nat_set_is_finite)
nipkow@31044
   450
qed
nipkow@31044
   451
nipkow@31044
   452
lemma finite_nat_set_iff_bounded_le:
nipkow@31044
   453
  "finite(N::nat set) = (EX m. ALL n:N. n<=m)"
nipkow@31044
   454
apply(simp add:finite_nat_set_iff_bounded)
nipkow@31044
   455
apply(blast dest:less_imp_le_nat le_imp_less_Suc)
nipkow@31044
   456
done
nipkow@31044
   457
nipkow@28068
   458
lemma finite_less_ub:
nipkow@28068
   459
     "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
nipkow@28068
   460
by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
paulson@14485
   461
nipkow@24853
   462
text{* Any subset of an interval of natural numbers the size of the
nipkow@24853
   463
subset is exactly that interval. *}
nipkow@24853
   464
nipkow@24853
   465
lemma subset_card_intvl_is_intvl:
nipkow@24853
   466
  "A <= {k..<k+card A} \<Longrightarrow> A = {k..<k+card A}" (is "PROP ?P")
nipkow@24853
   467
proof cases
nipkow@24853
   468
  assume "finite A"
nipkow@24853
   469
  thus "PROP ?P"
nipkow@32006
   470
  proof(induct A rule:finite_linorder_max_induct)
nipkow@24853
   471
    case empty thus ?case by auto
nipkow@24853
   472
  next
nipkow@24853
   473
    case (insert A b)
nipkow@24853
   474
    moreover hence "b ~: A" by auto
nipkow@24853
   475
    moreover have "A <= {k..<k+card A}" and "b = k+card A"
nipkow@24853
   476
      using `b ~: A` insert by fastsimp+
nipkow@24853
   477
    ultimately show ?case by auto
nipkow@24853
   478
  qed
nipkow@24853
   479
next
nipkow@24853
   480
  assume "~finite A" thus "PROP ?P" by simp
nipkow@24853
   481
qed
nipkow@24853
   482
nipkow@24853
   483
paulson@14485
   484
subsubsection {* Cardinality *}
paulson@14485
   485
nipkow@15045
   486
lemma card_lessThan [simp]: "card {..<u} = u"
paulson@15251
   487
  by (induct u, simp_all add: lessThan_Suc)
paulson@14485
   488
paulson@14485
   489
lemma card_atMost [simp]: "card {..u} = Suc u"
paulson@14485
   490
  by (simp add: lessThan_Suc_atMost [THEN sym])
paulson@14485
   491
nipkow@15045
   492
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
nipkow@15045
   493
  apply (subgoal_tac "card {l..<u} = card {..<u-l}")
paulson@14485
   494
  apply (erule ssubst, rule card_lessThan)
nipkow@15045
   495
  apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
paulson@14485
   496
  apply (erule subst)
paulson@14485
   497
  apply (rule card_image)
paulson@14485
   498
  apply (simp add: inj_on_def)
paulson@14485
   499
  apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
paulson@14485
   500
  apply (rule_tac x = "x - l" in exI)
paulson@14485
   501
  apply arith
paulson@14485
   502
  done
paulson@14485
   503
paulson@15418
   504
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
paulson@14485
   505
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
paulson@14485
   506
paulson@15418
   507
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
paulson@14485
   508
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
paulson@14485
   509
nipkow@15045
   510
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
paulson@14485
   511
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
paulson@14485
   512
nipkow@26105
   513
lemma ex_bij_betw_nat_finite:
nipkow@26105
   514
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
nipkow@26105
   515
apply(drule finite_imp_nat_seg_image_inj_on)
nipkow@26105
   516
apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
nipkow@26105
   517
done
nipkow@26105
   518
nipkow@26105
   519
lemma ex_bij_betw_finite_nat:
nipkow@26105
   520
  "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
nipkow@26105
   521
by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
nipkow@26105
   522
nipkow@31438
   523
lemma finite_same_card_bij:
nipkow@31438
   524
  "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"
nipkow@31438
   525
apply(drule ex_bij_betw_finite_nat)
nipkow@31438
   526
apply(drule ex_bij_betw_nat_finite)
nipkow@31438
   527
apply(auto intro!:bij_betw_trans)
nipkow@31438
   528
done
nipkow@31438
   529
nipkow@31438
   530
lemma ex_bij_betw_nat_finite_1:
nipkow@31438
   531
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
nipkow@31438
   532
by (rule finite_same_card_bij) auto
nipkow@31438
   533
nipkow@26105
   534
paulson@14485
   535
subsection {* Intervals of integers *}
paulson@14485
   536
nipkow@15045
   537
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
paulson@14485
   538
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   539
paulson@15418
   540
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
paulson@14485
   541
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
paulson@14485
   542
paulson@15418
   543
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
paulson@15418
   544
    "{l+1..<u} = {l<..<u::int}"
paulson@14485
   545
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
paulson@14485
   546
paulson@14485
   547
subsubsection {* Finiteness *}
paulson@14485
   548
paulson@15418
   549
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
nipkow@15045
   550
    {(0::int)..<u} = int ` {..<nat u}"
paulson@14485
   551
  apply (unfold image_def lessThan_def)
paulson@14485
   552
  apply auto
paulson@14485
   553
  apply (rule_tac x = "nat x" in exI)
paulson@14485
   554
  apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
paulson@14485
   555
  done
paulson@14485
   556
nipkow@15045
   557
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
paulson@14485
   558
  apply (case_tac "0 \<le> u")
paulson@14485
   559
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   560
  apply (rule finite_imageI)
paulson@14485
   561
  apply auto
paulson@14485
   562
  done
paulson@14485
   563
nipkow@15045
   564
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
nipkow@15045
   565
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   566
  apply (erule subst)
paulson@14485
   567
  apply (rule finite_imageI)
paulson@14485
   568
  apply (rule finite_atLeastZeroLessThan_int)
nipkow@16733
   569
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
   570
  done
paulson@14485
   571
paulson@15418
   572
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
paulson@14485
   573
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
paulson@14485
   574
paulson@15418
   575
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
paulson@14485
   576
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   577
paulson@15418
   578
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
paulson@14485
   579
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   580
nipkow@24853
   581
paulson@14485
   582
subsubsection {* Cardinality *}
paulson@14485
   583
nipkow@15045
   584
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
paulson@14485
   585
  apply (case_tac "0 \<le> u")
paulson@14485
   586
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   587
  apply (subst card_image)
paulson@14485
   588
  apply (auto simp add: inj_on_def)
paulson@14485
   589
  done
paulson@14485
   590
nipkow@15045
   591
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
nipkow@15045
   592
  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
paulson@14485
   593
  apply (erule ssubst, rule card_atLeastZeroLessThan_int)
nipkow@15045
   594
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   595
  apply (erule subst)
paulson@14485
   596
  apply (rule card_image)
paulson@14485
   597
  apply (simp add: inj_on_def)
nipkow@16733
   598
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
   599
  done
paulson@14485
   600
paulson@14485
   601
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
nipkow@29667
   602
apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
nipkow@29667
   603
apply (auto simp add: algebra_simps)
nipkow@29667
   604
done
paulson@14485
   605
paulson@15418
   606
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
nipkow@29667
   607
by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   608
nipkow@15045
   609
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
nipkow@29667
   610
by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   611
bulwahn@27656
   612
lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
bulwahn@27656
   613
proof -
bulwahn@27656
   614
  have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
bulwahn@27656
   615
  with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
bulwahn@27656
   616
qed
bulwahn@27656
   617
bulwahn@27656
   618
lemma card_less:
bulwahn@27656
   619
assumes zero_in_M: "0 \<in> M"
bulwahn@27656
   620
shows "card {k \<in> M. k < Suc i} \<noteq> 0"
bulwahn@27656
   621
proof -
bulwahn@27656
   622
  from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
bulwahn@27656
   623
  with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
bulwahn@27656
   624
qed
bulwahn@27656
   625
bulwahn@27656
   626
lemma card_less_Suc2: "0 \<notin> M \<Longrightarrow> card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"
huffman@30079
   627
apply (rule card_bij_eq [of "Suc" _ _ "\<lambda>x. x - Suc 0"])
bulwahn@27656
   628
apply simp
bulwahn@27656
   629
apply fastsimp
bulwahn@27656
   630
apply auto
bulwahn@27656
   631
apply (rule inj_on_diff_nat)
bulwahn@27656
   632
apply auto
bulwahn@27656
   633
apply (case_tac x)
bulwahn@27656
   634
apply auto
bulwahn@27656
   635
apply (case_tac xa)
bulwahn@27656
   636
apply auto
bulwahn@27656
   637
apply (case_tac xa)
bulwahn@27656
   638
apply auto
bulwahn@27656
   639
done
bulwahn@27656
   640
bulwahn@27656
   641
lemma card_less_Suc:
bulwahn@27656
   642
  assumes zero_in_M: "0 \<in> M"
bulwahn@27656
   643
    shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
bulwahn@27656
   644
proof -
bulwahn@27656
   645
  from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp
bulwahn@27656
   646
  hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"
bulwahn@27656
   647
    by (auto simp only: insert_Diff)
bulwahn@27656
   648
  have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto
bulwahn@27656
   649
  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
   650
    apply (subst card_insert)
bulwahn@27656
   651
    apply simp_all
bulwahn@27656
   652
    apply (subst b)
bulwahn@27656
   653
    apply (subst card_less_Suc2[symmetric])
bulwahn@27656
   654
    apply simp_all
bulwahn@27656
   655
    done
bulwahn@27656
   656
  with c show ?thesis by simp
bulwahn@27656
   657
qed
bulwahn@27656
   658
paulson@14485
   659
paulson@13850
   660
subsection {*Lemmas useful with the summation operator setsum*}
paulson@13850
   661
ballarin@16102
   662
text {* For examples, see Algebra/poly/UnivPoly2.thy *}
ballarin@13735
   663
wenzelm@14577
   664
subsubsection {* Disjoint Unions *}
ballarin@13735
   665
wenzelm@14577
   666
text {* Singletons and open intervals *}
ballarin@13735
   667
ballarin@13735
   668
lemma ivl_disj_un_singleton:
nipkow@15045
   669
  "{l::'a::linorder} Un {l<..} = {l..}"
nipkow@15045
   670
  "{..<u} Un {u::'a::linorder} = {..u}"
nipkow@15045
   671
  "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
nipkow@15045
   672
  "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
nipkow@15045
   673
  "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
nipkow@15045
   674
  "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
ballarin@14398
   675
by auto
ballarin@13735
   676
wenzelm@14577
   677
text {* One- and two-sided intervals *}
ballarin@13735
   678
ballarin@13735
   679
lemma ivl_disj_un_one:
nipkow@15045
   680
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
nipkow@15045
   681
  "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
nipkow@15045
   682
  "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
nipkow@15045
   683
  "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
nipkow@15045
   684
  "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
nipkow@15045
   685
  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
nipkow@15045
   686
  "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
nipkow@15045
   687
  "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
ballarin@14398
   688
by auto
ballarin@13735
   689
wenzelm@14577
   690
text {* Two- and two-sided intervals *}
ballarin@13735
   691
ballarin@13735
   692
lemma ivl_disj_un_two:
nipkow@15045
   693
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
nipkow@15045
   694
  "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
nipkow@15045
   695
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
nipkow@15045
   696
  "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
nipkow@15045
   697
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
nipkow@15045
   698
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
nipkow@15045
   699
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
nipkow@15045
   700
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
ballarin@14398
   701
by auto
ballarin@13735
   702
ballarin@13735
   703
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
ballarin@13735
   704
wenzelm@14577
   705
subsubsection {* Disjoint Intersections *}
ballarin@13735
   706
wenzelm@14577
   707
text {* Singletons and open intervals *}
ballarin@13735
   708
ballarin@13735
   709
lemma ivl_disj_int_singleton:
nipkow@15045
   710
  "{l::'a::order} Int {l<..} = {}"
nipkow@15045
   711
  "{..<u} Int {u} = {}"
nipkow@15045
   712
  "{l} Int {l<..<u} = {}"
nipkow@15045
   713
  "{l<..<u} Int {u} = {}"
nipkow@15045
   714
  "{l} Int {l<..u} = {}"
nipkow@15045
   715
  "{l..<u} Int {u} = {}"
ballarin@13735
   716
  by simp+
ballarin@13735
   717
wenzelm@14577
   718
text {* One- and two-sided intervals *}
ballarin@13735
   719
ballarin@13735
   720
lemma ivl_disj_int_one:
nipkow@15045
   721
  "{..l::'a::order} Int {l<..<u} = {}"
nipkow@15045
   722
  "{..<l} Int {l..<u} = {}"
nipkow@15045
   723
  "{..l} Int {l<..u} = {}"
nipkow@15045
   724
  "{..<l} Int {l..u} = {}"
nipkow@15045
   725
  "{l<..u} Int {u<..} = {}"
nipkow@15045
   726
  "{l<..<u} Int {u..} = {}"
nipkow@15045
   727
  "{l..u} Int {u<..} = {}"
nipkow@15045
   728
  "{l..<u} Int {u..} = {}"
ballarin@14398
   729
  by auto
ballarin@13735
   730
wenzelm@14577
   731
text {* Two- and two-sided intervals *}
ballarin@13735
   732
ballarin@13735
   733
lemma ivl_disj_int_two:
nipkow@15045
   734
  "{l::'a::order<..<m} Int {m..<u} = {}"
nipkow@15045
   735
  "{l<..m} Int {m<..<u} = {}"
nipkow@15045
   736
  "{l..<m} Int {m..<u} = {}"
nipkow@15045
   737
  "{l..m} Int {m<..<u} = {}"
nipkow@15045
   738
  "{l<..<m} Int {m..u} = {}"
nipkow@15045
   739
  "{l<..m} Int {m<..u} = {}"
nipkow@15045
   740
  "{l..<m} Int {m..u} = {}"
nipkow@15045
   741
  "{l..m} Int {m<..u} = {}"
ballarin@14398
   742
  by auto
ballarin@13735
   743
ballarin@13735
   744
lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
ballarin@13735
   745
nipkow@15542
   746
subsubsection {* Some Differences *}
nipkow@15542
   747
nipkow@15542
   748
lemma ivl_diff[simp]:
nipkow@15542
   749
 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
nipkow@15542
   750
by(auto)
nipkow@15542
   751
nipkow@15542
   752
nipkow@15542
   753
subsubsection {* Some Subset Conditions *}
nipkow@15542
   754
paulson@24286
   755
lemma ivl_subset [simp,noatp]:
nipkow@15542
   756
 "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
nipkow@15542
   757
apply(auto simp:linorder_not_le)
nipkow@15542
   758
apply(rule ccontr)
nipkow@15542
   759
apply(insert linorder_le_less_linear[of i n])
nipkow@15542
   760
apply(clarsimp simp:linorder_not_le)
nipkow@15542
   761
apply(fastsimp)
nipkow@15542
   762
done
nipkow@15542
   763
nipkow@15041
   764
nipkow@15042
   765
subsection {* Summation indexed over intervals *}
nipkow@15042
   766
nipkow@15042
   767
syntax
nipkow@15042
   768
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   769
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
   770
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
nipkow@16052
   771
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
nipkow@15042
   772
syntax (xsymbols)
nipkow@15042
   773
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   774
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
   775
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
   776
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15042
   777
syntax (HTML output)
nipkow@15042
   778
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   779
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
   780
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
   781
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15056
   782
syntax (latex_sum output)
nipkow@15052
   783
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
   784
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@15052
   785
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
   786
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@16052
   787
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
   788
 ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15052
   789
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
   790
 ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15041
   791
nipkow@15048
   792
translations
nipkow@28853
   793
  "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"
nipkow@28853
   794
  "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"
nipkow@28853
   795
  "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"
nipkow@28853
   796
  "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"
nipkow@15041
   797
nipkow@15052
   798
text{* The above introduces some pretty alternative syntaxes for
nipkow@15056
   799
summation over intervals:
nipkow@15052
   800
\begin{center}
nipkow@15052
   801
\begin{tabular}{lll}
nipkow@15056
   802
Old & New & \LaTeX\\
nipkow@15056
   803
@{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
   804
@{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
   805
@{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
nipkow@15056
   806
@{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
nipkow@15052
   807
\end{tabular}
nipkow@15052
   808
\end{center}
nipkow@15056
   809
The left column shows the term before introduction of the new syntax,
nipkow@15056
   810
the middle column shows the new (default) syntax, and the right column
nipkow@15056
   811
shows a special syntax. The latter is only meaningful for latex output
nipkow@15056
   812
and has to be activated explicitly by setting the print mode to
wenzelm@21502
   813
@{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
nipkow@15056
   814
antiquotations). It is not the default \LaTeX\ output because it only
nipkow@15056
   815
works well with italic-style formulae, not tt-style.
nipkow@15052
   816
nipkow@15052
   817
Note that for uniformity on @{typ nat} it is better to use
nipkow@15052
   818
@{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
nipkow@15052
   819
not provide all lemmas available for @{term"{m..<n}"} also in the
nipkow@15052
   820
special form for @{term"{..<n}"}. *}
nipkow@15052
   821
nipkow@15542
   822
text{* This congruence rule should be used for sums over intervals as
nipkow@15542
   823
the standard theorem @{text[source]setsum_cong} does not work well
nipkow@15542
   824
with the simplifier who adds the unsimplified premise @{term"x:B"} to
nipkow@15542
   825
the context. *}
nipkow@15542
   826
nipkow@15542
   827
lemma setsum_ivl_cong:
nipkow@15542
   828
 "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
nipkow@15542
   829
 setsum f {a..<b} = setsum g {c..<d}"
nipkow@15542
   830
by(rule setsum_cong, simp_all)
nipkow@15041
   831
nipkow@16041
   832
(* FIXME why are the following simp rules but the corresponding eqns
nipkow@16041
   833
on intervals are not? *)
nipkow@16041
   834
nipkow@16052
   835
lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
nipkow@16052
   836
by (simp add:atMost_Suc add_ac)
nipkow@16052
   837
nipkow@16041
   838
lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
nipkow@16041
   839
by (simp add:lessThan_Suc add_ac)
nipkow@15041
   840
nipkow@15911
   841
lemma setsum_cl_ivl_Suc[simp]:
nipkow@15561
   842
  "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
nipkow@15561
   843
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@15561
   844
nipkow@15911
   845
lemma setsum_op_ivl_Suc[simp]:
nipkow@15561
   846
  "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
nipkow@15561
   847
by (auto simp:add_ac atLeastLessThanSuc)
nipkow@16041
   848
(*
nipkow@15561
   849
lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
nipkow@15561
   850
    (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
nipkow@15561
   851
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@16041
   852
*)
nipkow@28068
   853
nipkow@28068
   854
lemma setsum_head:
nipkow@28068
   855
  fixes n :: nat
nipkow@28068
   856
  assumes mn: "m <= n" 
nipkow@28068
   857
  shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
nipkow@28068
   858
proof -
nipkow@28068
   859
  from mn
nipkow@28068
   860
  have "{m..n} = {m} \<union> {m<..n}"
nipkow@28068
   861
    by (auto intro: ivl_disj_un_singleton)
nipkow@28068
   862
  hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
nipkow@28068
   863
    by (simp add: atLeast0LessThan)
nipkow@28068
   864
  also have "\<dots> = ?rhs" by simp
nipkow@28068
   865
  finally show ?thesis .
nipkow@28068
   866
qed
nipkow@28068
   867
nipkow@28068
   868
lemma setsum_head_Suc:
nipkow@28068
   869
  "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"
nipkow@28068
   870
by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)
nipkow@28068
   871
nipkow@28068
   872
lemma setsum_head_upt_Suc:
nipkow@28068
   873
  "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"
huffman@30079
   874
apply(insert setsum_head_Suc[of m "n - Suc 0" f])
nipkow@29667
   875
apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)
nipkow@28068
   876
done
nipkow@28068
   877
nipkow@31501
   878
lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"
nipkow@31501
   879
  shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
nipkow@31501
   880
proof-
nipkow@31501
   881
  have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using `m \<le> n+1` by auto
nipkow@31501
   882
  thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint
nipkow@31501
   883
    atLeastSucAtMost_greaterThanAtMost)
nipkow@31501
   884
qed
nipkow@28068
   885
nipkow@15539
   886
lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
   887
  setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
nipkow@15539
   888
by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
nipkow@15539
   889
nipkow@15539
   890
lemma setsum_diff_nat_ivl:
nipkow@15539
   891
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
nipkow@15539
   892
shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
   893
  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
nipkow@15539
   894
using setsum_add_nat_ivl [of m n p f,symmetric]
nipkow@15539
   895
apply (simp add: add_ac)
nipkow@15539
   896
done
nipkow@15539
   897
nipkow@31505
   898
lemma setsum_natinterval_difff:
nipkow@31505
   899
  fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
nipkow@31505
   900
  shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
nipkow@31505
   901
          (if m <= n then f m - f(n + 1) else 0)"
nipkow@31505
   902
by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)
nipkow@31505
   903
nipkow@31509
   904
lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]
nipkow@31509
   905
nipkow@31509
   906
lemma setsum_setsum_restrict:
nipkow@31509
   907
  "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"
nipkow@31509
   908
  by (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)
nipkow@31509
   909
     (rule setsum_commute)
nipkow@31509
   910
nipkow@31509
   911
lemma setsum_image_gen: assumes fS: "finite S"
nipkow@31509
   912
  shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
nipkow@31509
   913
proof-
nipkow@31509
   914
  { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }
nipkow@31509
   915
  hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
nipkow@31509
   916
    by simp
nipkow@31509
   917
  also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
nipkow@31509
   918
    by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])
nipkow@31509
   919
  finally show ?thesis .
nipkow@31509
   920
qed
nipkow@31509
   921
nipkow@31509
   922
lemma setsum_multicount_gen:
nipkow@31509
   923
  assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
nipkow@31509
   924
  shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")
nipkow@31509
   925
proof-
nipkow@31509
   926
  have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto
nipkow@31509
   927
  also have "\<dots> = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]
nipkow@31509
   928
    using assms(3) by auto
nipkow@31509
   929
  finally show ?thesis .
nipkow@31509
   930
qed
nipkow@31509
   931
nipkow@31509
   932
lemma setsum_multicount:
nipkow@31509
   933
  assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
nipkow@31509
   934
  shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
nipkow@31509
   935
proof-
nipkow@31509
   936
  have "?l = setsum (\<lambda>i. k) T" by(rule setsum_multicount_gen)(auto simp:assms)
nipkow@31509
   937
  also have "\<dots> = ?r" by(simp add: setsum_constant mult_commute)
nipkow@31509
   938
  finally show ?thesis by auto
nipkow@31509
   939
qed
nipkow@31509
   940
nipkow@28068
   941
nipkow@16733
   942
subsection{* Shifting bounds *}
nipkow@16733
   943
nipkow@15539
   944
lemma setsum_shift_bounds_nat_ivl:
nipkow@15539
   945
  "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
nipkow@15539
   946
by (induct "n", auto simp:atLeastLessThanSuc)
nipkow@15539
   947
nipkow@16733
   948
lemma setsum_shift_bounds_cl_nat_ivl:
nipkow@16733
   949
  "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
nipkow@16733
   950
apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
nipkow@16733
   951
apply (simp add:image_add_atLeastAtMost o_def)
nipkow@16733
   952
done
nipkow@16733
   953
nipkow@16733
   954
corollary setsum_shift_bounds_cl_Suc_ivl:
nipkow@16733
   955
  "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
huffman@30079
   956
by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
   957
nipkow@16733
   958
corollary setsum_shift_bounds_Suc_ivl:
nipkow@16733
   959
  "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
huffman@30079
   960
by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
   961
nipkow@28068
   962
lemma setsum_shift_lb_Suc0_0:
nipkow@28068
   963
  "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"
nipkow@28068
   964
by(simp add:setsum_head_Suc)
kleing@19106
   965
nipkow@28068
   966
lemma setsum_shift_lb_Suc0_0_upt:
nipkow@28068
   967
  "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"
nipkow@28068
   968
apply(cases k)apply simp
nipkow@28068
   969
apply(simp add:setsum_head_upt_Suc)
nipkow@28068
   970
done
kleing@19022
   971
ballarin@17149
   972
subsection {* The formula for geometric sums *}
ballarin@17149
   973
ballarin@17149
   974
lemma geometric_sum:
ballarin@17149
   975
  "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) =
haftmann@31017
   976
  (x ^ n - 1) / (x - 1::'a::{field})"
nipkow@23496
   977
by (induct "n") (simp_all add:field_simps power_Suc)
ballarin@17149
   978
kleing@19469
   979
subsection {* The formula for arithmetic sums *}
kleing@19469
   980
kleing@19469
   981
lemma gauss_sum:
huffman@23277
   982
  "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =
kleing@19469
   983
   of_nat n*((of_nat n)+1)"
kleing@19469
   984
proof (induct n)
kleing@19469
   985
  case 0
kleing@19469
   986
  show ?case by simp
kleing@19469
   987
next
kleing@19469
   988
  case (Suc n)
nipkow@29667
   989
  then show ?case by (simp add: algebra_simps)
kleing@19469
   990
qed
kleing@19469
   991
kleing@19469
   992
theorem arith_series_general:
huffman@23277
   993
  "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
   994
  of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469
   995
proof cases
kleing@19469
   996
  assume ngt1: "n > 1"
kleing@19469
   997
  let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
kleing@19469
   998
  have
kleing@19469
   999
    "(\<Sum>i\<in>{..<n}. a+?I i*d) =
kleing@19469
  1000
     ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
kleing@19469
  1001
    by (rule setsum_addf)
kleing@19469
  1002
  also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
kleing@19469
  1003
  also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
huffman@30079
  1004
    unfolding One_nat_def
nipkow@28068
  1005
    by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)
kleing@19469
  1006
  also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"
kleing@19469
  1007
    by (simp add: left_distrib right_distrib)
kleing@19469
  1008
  also from ngt1 have "{1..<n} = {1..n - 1}"
nipkow@28068
  1009
    by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
nipkow@28068
  1010
  also from ngt1
kleing@19469
  1011
  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)"
huffman@30079
  1012
    by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)
huffman@23431
  1013
       (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
nipkow@29667
  1014
  finally show ?thesis by (simp add: algebra_simps)
kleing@19469
  1015
next
kleing@19469
  1016
  assume "\<not>(n > 1)"
kleing@19469
  1017
  hence "n = 1 \<or> n = 0" by auto
nipkow@29667
  1018
  thus ?thesis by (auto simp: algebra_simps)
kleing@19469
  1019
qed
kleing@19469
  1020
kleing@19469
  1021
lemma arith_series_nat:
kleing@19469
  1022
  "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
kleing@19469
  1023
proof -
kleing@19469
  1024
  have
kleing@19469
  1025
    "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
kleing@19469
  1026
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1027
    by (rule arith_series_general)
huffman@30079
  1028
  thus ?thesis
huffman@30079
  1029
    unfolding One_nat_def by (auto simp add: of_nat_id)
kleing@19469
  1030
qed
kleing@19469
  1031
kleing@19469
  1032
lemma arith_series_int:
kleing@19469
  1033
  "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
  1034
  of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1035
proof -
kleing@19469
  1036
  have
kleing@19469
  1037
    "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
  1038
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1039
    by (rule arith_series_general)
kleing@19469
  1040
  thus ?thesis by simp
kleing@19469
  1041
qed
paulson@15418
  1042
kleing@19022
  1043
lemma sum_diff_distrib:
kleing@19022
  1044
  fixes P::"nat\<Rightarrow>nat"
kleing@19022
  1045
  shows
kleing@19022
  1046
  "\<forall>x. Q x \<le> P x  \<Longrightarrow>
kleing@19022
  1047
  (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
kleing@19022
  1048
proof (induct n)
kleing@19022
  1049
  case 0 show ?case by simp
kleing@19022
  1050
next
kleing@19022
  1051
  case (Suc n)
kleing@19022
  1052
kleing@19022
  1053
  let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
kleing@19022
  1054
  let ?rhs = "\<Sum>x<n. P x - Q x"
kleing@19022
  1055
kleing@19022
  1056
  from Suc have "?lhs = ?rhs" by simp
kleing@19022
  1057
  moreover
kleing@19022
  1058
  from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
kleing@19022
  1059
  moreover
kleing@19022
  1060
  from Suc have
kleing@19022
  1061
    "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
kleing@19022
  1062
    by (subst diff_diff_left[symmetric],
kleing@19022
  1063
        subst diff_add_assoc2)
kleing@19022
  1064
       (auto simp: diff_add_assoc2 intro: setsum_mono)
kleing@19022
  1065
  ultimately
kleing@19022
  1066
  show ?case by simp
kleing@19022
  1067
qed
kleing@19022
  1068
paulson@29960
  1069
subsection {* Products indexed over intervals *}
paulson@29960
  1070
paulson@29960
  1071
syntax
paulson@29960
  1072
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1073
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1074
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)
paulson@29960
  1075
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)
paulson@29960
  1076
syntax (xsymbols)
paulson@29960
  1077
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1078
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1079
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
  1080
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
  1081
syntax (HTML output)
paulson@29960
  1082
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1083
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1084
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
  1085
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
  1086
syntax (latex_prod output)
paulson@29960
  1087
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1088
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
  1089
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1090
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
  1091
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1092
 ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1093
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1094
 ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1095
paulson@29960
  1096
translations
paulson@29960
  1097
  "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"
paulson@29960
  1098
  "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"
paulson@29960
  1099
  "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"
paulson@29960
  1100
  "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"
paulson@29960
  1101
nipkow@8924
  1102
end