src/HOL/Library/Set_Algebras.thy
author haftmann
Sat, 05 Jul 2014 11:01:53 +0200
changeset 57514 bdc2c6b40bf2
parent 57512 cc97b347b301
child 58881 b9556a055632
permissions -rw-r--r--
prefer ac_simps collections over separate name bindings for add and mult
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
38622
86fc906dcd86 split and enriched theory SetsAndFunctions
haftmann
parents: 35267
diff changeset
     1
(*  Title:      HOL/Library/Set_Algebras.thy
86fc906dcd86 split and enriched theory SetsAndFunctions
haftmann
parents: 35267
diff changeset
     2
    Author:     Jeremy Avigad and Kevin Donnelly; Florian Haftmann, TUM
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
     3
*)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
     4
38622
86fc906dcd86 split and enriched theory SetsAndFunctions
haftmann
parents: 35267
diff changeset
     5
header {* Algebraic operations on sets *}
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
     6
38622
86fc906dcd86 split and enriched theory SetsAndFunctions
haftmann
parents: 35267
diff changeset
     7
theory Set_Algebras
30738
0842e906300c normalized imports
haftmann
parents: 29667
diff changeset
     8
imports Main
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
     9
begin
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
    10
19736
wenzelm
parents: 19656
diff changeset
    11
text {*
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
    12
  This library lifts operations like addition and multiplication to
38622
86fc906dcd86 split and enriched theory SetsAndFunctions
haftmann
parents: 35267
diff changeset
    13
  sets.  It was designed to support asymptotic calculations. See the
86fc906dcd86 split and enriched theory SetsAndFunctions
haftmann
parents: 35267
diff changeset
    14
  comments at the top of theory @{text BigO}.
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
    15
*}
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
    16
47443
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    17
instantiation set :: (plus) plus
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    18
begin
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    19
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    20
definition plus_set :: "'a::plus set \<Rightarrow> 'a set \<Rightarrow> 'a set" where
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    21
  set_plus_def: "A + B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a + b}"
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    22
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    23
instance ..
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    24
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    25
end
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    26
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    27
instantiation set :: (times) times
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    28
begin
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    29
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    30
definition times_set :: "'a::times set \<Rightarrow> 'a set \<Rightarrow> 'a set" where
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    31
  set_times_def: "A * B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a * b}"
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    32
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    33
instance ..
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    34
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    35
end
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    36
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    37
instantiation set :: (zero) zero
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    38
begin
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    39
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    40
definition
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
    41
  set_zero[simp]: "(0::'a::zero set) = {0}"
47443
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    42
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    43
instance ..
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    44
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    45
end
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
    46
47443
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    47
instantiation set :: (one) one
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    48
begin
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    49
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    50
definition
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
    51
  set_one[simp]: "(1::'a::one set) = {1}"
47443
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    52
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    53
instance ..
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    54
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    55
end
25594
43c718438f9f switched import from Main to PreList
haftmann
parents: 23477
diff changeset
    56
38622
86fc906dcd86 split and enriched theory SetsAndFunctions
haftmann
parents: 35267
diff changeset
    57
definition elt_set_plus :: "'a::plus \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "+o" 70) where
86fc906dcd86 split and enriched theory SetsAndFunctions
haftmann
parents: 35267
diff changeset
    58
  "a +o B = {c. \<exists>b\<in>B. c = a + b}"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
    59
38622
86fc906dcd86 split and enriched theory SetsAndFunctions
haftmann
parents: 35267
diff changeset
    60
definition elt_set_times :: "'a::times \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "*o" 80) where
86fc906dcd86 split and enriched theory SetsAndFunctions
haftmann
parents: 35267
diff changeset
    61
  "a *o B = {c. \<exists>b\<in>B. c = a * b}"
25594
43c718438f9f switched import from Main to PreList
haftmann
parents: 23477
diff changeset
    62
38622
86fc906dcd86 split and enriched theory SetsAndFunctions
haftmann
parents: 35267
diff changeset
    63
abbreviation (input) elt_set_eq :: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  (infix "=o" 50) where
86fc906dcd86 split and enriched theory SetsAndFunctions
haftmann
parents: 35267
diff changeset
    64
  "x =o A \<equiv> x \<in> A"
25594
43c718438f9f switched import from Main to PreList
haftmann
parents: 23477
diff changeset
    65
47443
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    66
instance set :: (semigroup_add) semigroup_add
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
    67
  by default (force simp add: set_plus_def add.assoc)
25594
43c718438f9f switched import from Main to PreList
haftmann
parents: 23477
diff changeset
    68
47443
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    69
instance set :: (ab_semigroup_add) ab_semigroup_add
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
    70
  by default (force simp add: set_plus_def add.commute)
25594
43c718438f9f switched import from Main to PreList
haftmann
parents: 23477
diff changeset
    71
47443
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    72
instance set :: (monoid_add) monoid_add
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
    73
  by default (simp_all add: set_plus_def)
25594
43c718438f9f switched import from Main to PreList
haftmann
parents: 23477
diff changeset
    74
47443
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    75
instance set :: (comm_monoid_add) comm_monoid_add
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
    76
  by default (simp_all add: set_plus_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
    77
47443
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    78
instance set :: (semigroup_mult) semigroup_mult
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
    79
  by default (force simp add: set_times_def mult.assoc)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
    80
47443
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    81
instance set :: (ab_semigroup_mult) ab_semigroup_mult
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
    82
  by default (force simp add: set_times_def mult.commute)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
    83
47443
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    84
instance set :: (monoid_mult) monoid_mult
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
    85
  by default (simp_all add: set_times_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
    86
47443
aeff49a3369b backported Set_Algebras to use type classes (basically reverting b3e8d5ec721d from 2008)
krauss
parents: 44890
diff changeset
    87
instance set :: (comm_monoid_mult) comm_monoid_mult
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
    88
  by default (simp_all add: set_times_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
    89
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
    90
lemma set_plus_intro [intro]: "a \<in> C \<Longrightarrow> b \<in> D \<Longrightarrow> a + b \<in> C + D"
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
    91
  by (auto simp add: set_plus_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
    92
53596
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
    93
lemma set_plus_elim:
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
    94
  assumes "x \<in> A + B"
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
    95
  obtains a b where "x = a + b" and "a \<in> A" and "b \<in> B"
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
    96
  using assms unfolding set_plus_def by fast
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
    97
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
    98
lemma set_plus_intro2 [intro]: "b \<in> C \<Longrightarrow> a + b \<in> a +o C"
19736
wenzelm
parents: 19656
diff changeset
    99
  by (auto simp add: elt_set_plus_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   100
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   101
lemma set_plus_rearrange:
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   102
  "((a::'a::comm_monoid_add) +o C) + (b +o D) = (a + b) +o (C + D)"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   103
  apply (auto simp add: elt_set_plus_def set_plus_def ac_simps)
19736
wenzelm
parents: 19656
diff changeset
   104
   apply (rule_tac x = "ba + bb" in exI)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   105
  apply (auto simp add: ac_simps)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   106
  apply (rule_tac x = "aa + a" in exI)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   107
  apply (auto simp add: ac_simps)
19736
wenzelm
parents: 19656
diff changeset
   108
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   109
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   110
lemma set_plus_rearrange2: "(a::'a::semigroup_add) +o (b +o C) = (a + b) +o C"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56899
diff changeset
   111
  by (auto simp add: elt_set_plus_def add.assoc)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   112
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   113
lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) + C = a +o (B + C)"
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   114
  apply (auto simp add: elt_set_plus_def set_plus_def)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   115
   apply (blast intro: ac_simps)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   116
  apply (rule_tac x = "a + aa" in exI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   117
  apply (rule conjI)
19736
wenzelm
parents: 19656
diff changeset
   118
   apply (rule_tac x = "aa" in bexI)
wenzelm
parents: 19656
diff changeset
   119
    apply auto
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   120
  apply (rule_tac x = "ba" in bexI)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   121
   apply (auto simp add: ac_simps)
19736
wenzelm
parents: 19656
diff changeset
   122
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   123
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   124
theorem set_plus_rearrange4: "C + ((a::'a::comm_monoid_add) +o D) = a +o (C + D)"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   125
  apply (auto simp add: elt_set_plus_def set_plus_def ac_simps)
19736
wenzelm
parents: 19656
diff changeset
   126
   apply (rule_tac x = "aa + ba" in exI)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   127
   apply (auto simp add: ac_simps)
19736
wenzelm
parents: 19656
diff changeset
   128
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   129
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   130
theorems set_plus_rearranges = set_plus_rearrange set_plus_rearrange2
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   131
  set_plus_rearrange3 set_plus_rearrange4
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   132
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   133
lemma set_plus_mono [intro!]: "C \<subseteq> D \<Longrightarrow> a +o C \<subseteq> a +o D"
19736
wenzelm
parents: 19656
diff changeset
   134
  by (auto simp add: elt_set_plus_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   135
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   136
lemma set_plus_mono2 [intro]: "(C::'a::plus set) \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C + E \<subseteq> D + F"
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   137
  by (auto simp add: set_plus_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   138
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   139
lemma set_plus_mono3 [intro]: "a \<in> C \<Longrightarrow> a +o D \<subseteq> C + D"
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   140
  by (auto simp add: elt_set_plus_def set_plus_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   141
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   142
lemma set_plus_mono4 [intro]: "(a::'a::comm_monoid_add) \<in> C \<Longrightarrow> a +o D \<subseteq> D + C"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   143
  by (auto simp add: elt_set_plus_def set_plus_def ac_simps)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   144
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   145
lemma set_plus_mono5: "a \<in> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> a +o B \<subseteq> C + D"
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   146
  apply (subgoal_tac "a +o B \<subseteq> a +o D")
19736
wenzelm
parents: 19656
diff changeset
   147
   apply (erule order_trans)
wenzelm
parents: 19656
diff changeset
   148
   apply (erule set_plus_mono3)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   149
  apply (erule set_plus_mono)
19736
wenzelm
parents: 19656
diff changeset
   150
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   151
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   152
lemma set_plus_mono_b: "C \<subseteq> D \<Longrightarrow> x \<in> a +o C \<Longrightarrow> x \<in> a +o D"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   153
  apply (frule set_plus_mono)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   154
  apply auto
19736
wenzelm
parents: 19656
diff changeset
   155
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   156
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   157
lemma set_plus_mono2_b: "C \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> x \<in> C + E \<Longrightarrow> x \<in> D + F"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   158
  apply (frule set_plus_mono2)
19736
wenzelm
parents: 19656
diff changeset
   159
   prefer 2
wenzelm
parents: 19656
diff changeset
   160
   apply force
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   161
  apply assumption
19736
wenzelm
parents: 19656
diff changeset
   162
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   163
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   164
lemma set_plus_mono3_b: "a \<in> C \<Longrightarrow> x \<in> a +o D \<Longrightarrow> x \<in> C + D"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   165
  apply (frule set_plus_mono3)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   166
  apply auto
19736
wenzelm
parents: 19656
diff changeset
   167
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   168
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   169
lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C \<Longrightarrow> x \<in> a +o D \<Longrightarrow> x \<in> D + C"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   170
  apply (frule set_plus_mono4)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   171
  apply auto
19736
wenzelm
parents: 19656
diff changeset
   172
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   173
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   174
lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C"
19736
wenzelm
parents: 19656
diff changeset
   175
  by (auto simp add: elt_set_plus_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   176
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   177
lemma set_zero_plus2: "(0::'a::comm_monoid_add) \<in> A \<Longrightarrow> B \<subseteq> A + B"
44142
8e27e0177518 avoid warnings about duplicate rules
huffman
parents: 40887
diff changeset
   178
  apply (auto simp add: set_plus_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   179
  apply (rule_tac x = 0 in bexI)
19736
wenzelm
parents: 19656
diff changeset
   180
   apply (rule_tac x = x in bexI)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   181
    apply (auto simp add: ac_simps)
19736
wenzelm
parents: 19656
diff changeset
   182
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   183
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   184
lemma set_plus_imp_minus: "(a::'a::ab_group_add) : b +o C \<Longrightarrow> (a - b) \<in> C"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   185
  by (auto simp add: elt_set_plus_def ac_simps)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   186
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   187
lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C \<Longrightarrow> a \<in> b +o C"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   188
  apply (auto simp add: elt_set_plus_def ac_simps)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   189
  apply (subgoal_tac "a = (a + - b) + b")
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53596
diff changeset
   190
   apply (rule bexI, assumption)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   191
  apply (auto simp add: ac_simps)
19736
wenzelm
parents: 19656
diff changeset
   192
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   193
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   194
lemma set_minus_plus: "(a::'a::ab_group_add) - b \<in> C \<longleftrightarrow> a \<in> b +o C"
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   195
  by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   196
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   197
lemma set_times_intro [intro]: "a \<in> C \<Longrightarrow> b \<in> D \<Longrightarrow> a * b \<in> C * D"
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   198
  by (auto simp add: set_times_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   199
53596
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
   200
lemma set_times_elim:
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
   201
  assumes "x \<in> A * B"
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
   202
  obtains a b where "x = a * b" and "a \<in> A" and "b \<in> B"
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
   203
  using assms unfolding set_times_def by fast
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
   204
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   205
lemma set_times_intro2 [intro!]: "b \<in> C \<Longrightarrow> a * b \<in> a *o C"
19736
wenzelm
parents: 19656
diff changeset
   206
  by (auto simp add: elt_set_times_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   207
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   208
lemma set_times_rearrange:
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   209
  "((a::'a::comm_monoid_mult) *o C) * (b *o D) = (a * b) *o (C * D)"
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   210
  apply (auto simp add: elt_set_times_def set_times_def)
19736
wenzelm
parents: 19656
diff changeset
   211
   apply (rule_tac x = "ba * bb" in exI)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   212
   apply (auto simp add: ac_simps)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   213
  apply (rule_tac x = "aa * a" in exI)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   214
  apply (auto simp add: ac_simps)
19736
wenzelm
parents: 19656
diff changeset
   215
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   216
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   217
lemma set_times_rearrange2:
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   218
  "(a::'a::semigroup_mult) *o (b *o C) = (a * b) *o C"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 56899
diff changeset
   219
  by (auto simp add: elt_set_times_def mult.assoc)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   220
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   221
lemma set_times_rearrange3:
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   222
  "((a::'a::semigroup_mult) *o B) * C = a *o (B * C)"
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   223
  apply (auto simp add: elt_set_times_def set_times_def)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   224
   apply (blast intro: ac_simps)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   225
  apply (rule_tac x = "a * aa" in exI)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   226
  apply (rule conjI)
19736
wenzelm
parents: 19656
diff changeset
   227
   apply (rule_tac x = "aa" in bexI)
wenzelm
parents: 19656
diff changeset
   228
    apply auto
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   229
  apply (rule_tac x = "ba" in bexI)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   230
   apply (auto simp add: ac_simps)
19736
wenzelm
parents: 19656
diff changeset
   231
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   232
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   233
theorem set_times_rearrange4:
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   234
  "C * ((a::'a::comm_monoid_mult) *o D) = a *o (C * D)"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   235
  apply (auto simp add: elt_set_times_def set_times_def ac_simps)
19736
wenzelm
parents: 19656
diff changeset
   236
   apply (rule_tac x = "aa * ba" in exI)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   237
   apply (auto simp add: ac_simps)
19736
wenzelm
parents: 19656
diff changeset
   238
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   239
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   240
theorems set_times_rearranges = set_times_rearrange set_times_rearrange2
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   241
  set_times_rearrange3 set_times_rearrange4
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   242
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   243
lemma set_times_mono [intro]: "C \<subseteq> D \<Longrightarrow> a *o C \<subseteq> a *o D"
19736
wenzelm
parents: 19656
diff changeset
   244
  by (auto simp add: elt_set_times_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   245
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   246
lemma set_times_mono2 [intro]: "(C::'a::times set) \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C * E \<subseteq> D * F"
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   247
  by (auto simp add: set_times_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   248
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   249
lemma set_times_mono3 [intro]: "a \<in> C \<Longrightarrow> a *o D \<subseteq> C * D"
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   250
  by (auto simp add: elt_set_times_def set_times_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   251
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   252
lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C \<Longrightarrow> a *o D \<subseteq> D * C"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   253
  by (auto simp add: elt_set_times_def set_times_def ac_simps)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   254
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   255
lemma set_times_mono5: "a \<in> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> a *o B \<subseteq> C * D"
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   256
  apply (subgoal_tac "a *o B \<subseteq> a *o D")
19736
wenzelm
parents: 19656
diff changeset
   257
   apply (erule order_trans)
wenzelm
parents: 19656
diff changeset
   258
   apply (erule set_times_mono3)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   259
  apply (erule set_times_mono)
19736
wenzelm
parents: 19656
diff changeset
   260
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   261
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   262
lemma set_times_mono_b: "C \<subseteq> D \<Longrightarrow> x \<in> a *o C \<Longrightarrow> x \<in> a *o D"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   263
  apply (frule set_times_mono)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   264
  apply auto
19736
wenzelm
parents: 19656
diff changeset
   265
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   266
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   267
lemma set_times_mono2_b: "C \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> x \<in> C * E \<Longrightarrow> x \<in> D * F"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   268
  apply (frule set_times_mono2)
19736
wenzelm
parents: 19656
diff changeset
   269
   prefer 2
wenzelm
parents: 19656
diff changeset
   270
   apply force
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   271
  apply assumption
19736
wenzelm
parents: 19656
diff changeset
   272
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   273
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   274
lemma set_times_mono3_b: "a \<in> C \<Longrightarrow> x \<in> a *o D \<Longrightarrow> x \<in> C * D"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   275
  apply (frule set_times_mono3)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   276
  apply auto
19736
wenzelm
parents: 19656
diff changeset
   277
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   278
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   279
lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) \<in> C \<Longrightarrow> x \<in> a *o D \<Longrightarrow> x \<in> D * C"
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   280
  apply (frule set_times_mono4)
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   281
  apply auto
19736
wenzelm
parents: 19656
diff changeset
   282
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   283
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   284
lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C"
19736
wenzelm
parents: 19656
diff changeset
   285
  by (auto simp add: elt_set_times_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   286
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   287
lemma set_times_plus_distrib:
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   288
  "(a::'a::semiring) *o (b +o C) = (a * b) +o (a *o C)"
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 21404
diff changeset
   289
  by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   290
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   291
lemma set_times_plus_distrib2:
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   292
  "(a::'a::semiring) *o (B + C) = (a *o B) + (a *o C)"
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   293
  apply (auto simp add: set_plus_def elt_set_times_def ring_distribs)
19736
wenzelm
parents: 19656
diff changeset
   294
   apply blast
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   295
  apply (rule_tac x = "b + bb" in exI)
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 21404
diff changeset
   296
  apply (auto simp add: ring_distribs)
19736
wenzelm
parents: 19656
diff changeset
   297
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   298
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   299
lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) * D \<subseteq> a *o D + C * D"
44142
8e27e0177518 avoid warnings about duplicate rules
huffman
parents: 40887
diff changeset
   300
  apply (auto simp add:
26814
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   301
    elt_set_plus_def elt_set_times_def set_times_def
b3e8d5ec721d Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents: 25764
diff changeset
   302
    set_plus_def ring_distribs)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   303
  apply auto
19736
wenzelm
parents: 19656
diff changeset
   304
  done
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   305
19380
b808efaa5828 tuned syntax/abbreviations;
wenzelm
parents: 17161
diff changeset
   306
theorems set_times_plus_distribs =
b808efaa5828 tuned syntax/abbreviations;
wenzelm
parents: 17161
diff changeset
   307
  set_times_plus_distrib
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   308
  set_times_plus_distrib2
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   309
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   310
lemma set_neg_intro: "(a::'a::ring_1) \<in> (- 1) *o C \<Longrightarrow> - a \<in> C"
19736
wenzelm
parents: 19656
diff changeset
   311
  by (auto simp add: elt_set_times_def)
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   312
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   313
lemma set_neg_intro2: "(a::'a::ring_1) \<in> C \<Longrightarrow> - a \<in> (- 1) *o C"
19736
wenzelm
parents: 19656
diff changeset
   314
  by (auto simp add: elt_set_times_def)
wenzelm
parents: 19656
diff changeset
   315
53596
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
   316
lemma set_plus_image: "S + T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44142
diff changeset
   317
  unfolding set_plus_def by (fastforce simp: image_iff)
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   318
53596
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
   319
lemma set_times_image: "S * T = (\<lambda>(x, y). x * y) ` (S \<times> T)"
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
   320
  unfolding set_times_def by (fastforce simp: image_iff)
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
   321
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   322
lemma finite_set_plus: "finite s \<Longrightarrow> finite t \<Longrightarrow> finite (s + t)"
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   323
  unfolding set_plus_image by simp
53596
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
   324
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   325
lemma finite_set_times: "finite s \<Longrightarrow> finite t \<Longrightarrow> finite (s * t)"
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   326
  unfolding set_times_image by simp
53596
d29d63460d84 new lemmas
huffman
parents: 47446
diff changeset
   327
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   328
lemma set_setsum_alt:
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   329
  assumes fin: "finite I"
47444
d21c95af2df7 removed "setsum_set", now subsumed by generic setsum
krauss
parents: 47443
diff changeset
   330
  shows "setsum S I = {setsum s I |s. \<forall>i\<in>I. s i \<in> S i}"
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   331
    (is "_ = ?setsum I")
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   332
  using fin
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   333
proof induct
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   334
  case empty
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   335
  then show ?case by simp
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   336
next
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   337
  case (insert x F)
47445
69e96e5500df Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents: 47444
diff changeset
   338
  have "setsum S (insert x F) = S x + ?setsum F"
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   339
    using insert.hyps by auto
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   340
  also have "\<dots> = {s x + setsum s F |s. \<forall> i\<in>insert x F. s i \<in> S i}"
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   341
    unfolding set_plus_def
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   342
  proof safe
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   343
    fix y s
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   344
    assume "y \<in> S x" "\<forall>i\<in>F. s i \<in> S i"
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   345
    then show "\<exists>s'. y + setsum s F = s' x + setsum s' F \<and> (\<forall>i\<in>insert x F. s' i \<in> S i)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   346
      using insert.hyps
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   347
      by (intro exI[of _ "\<lambda>i. if i \<in> F then s i else y"]) (auto simp add: set_plus_def)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   348
  qed auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   349
  finally show ?case
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   350
    using insert.hyps by auto
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   351
qed
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   352
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   353
lemma setsum_set_cond_linear:
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   354
  fixes f :: "'a::comm_monoid_add set \<Rightarrow> 'b::comm_monoid_add set"
47445
69e96e5500df Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents: 47444
diff changeset
   355
  assumes [intro!]: "\<And>A B. P A  \<Longrightarrow> P B  \<Longrightarrow> P (A + B)" "P {0}"
69e96e5500df Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents: 47444
diff changeset
   356
    and f: "\<And>A B. P A  \<Longrightarrow> P B \<Longrightarrow> f (A + B) = f A + f B" "f {0} = {0}"
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   357
  assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)"
47444
d21c95af2df7 removed "setsum_set", now subsumed by generic setsum
krauss
parents: 47443
diff changeset
   358
  shows "f (setsum S I) = setsum (f \<circ> S) I"
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   359
proof (cases "finite I")
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   360
  case True
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   361
  from this all show ?thesis
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   362
  proof induct
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   363
    case empty
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   364
    then show ?case by (auto intro!: f)
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   365
  next
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   366
    case (insert x F)
47444
d21c95af2df7 removed "setsum_set", now subsumed by generic setsum
krauss
parents: 47443
diff changeset
   367
    from `finite F` `\<And>i. i \<in> insert x F \<Longrightarrow> P (S i)` have "P (setsum S F)"
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   368
      by induct auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   369
    with insert show ?case
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   370
      by (simp, subst f) auto
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   371
  qed
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   372
next
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   373
  case False
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   374
  then show ?thesis by (auto intro!: f)
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   375
qed
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   376
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   377
lemma setsum_set_linear:
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   378
  fixes f :: "'a::comm_monoid_add set \<Rightarrow> 'b::comm_monoid_add set"
47445
69e96e5500df Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents: 47444
diff changeset
   379
  assumes "\<And>A B. f(A) + f(B) = f(A + B)" "f {0} = {0}"
47444
d21c95af2df7 removed "setsum_set", now subsumed by generic setsum
krauss
parents: 47443
diff changeset
   380
  shows "f (setsum S I) = setsum (f \<circ> S) I"
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   381
  using setsum_set_cond_linear[of "\<lambda>x. True" f I S] assms by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 39302
diff changeset
   382
47446
ed0795caec95 distributivity of * over Un and UNION
krauss
parents: 47445
diff changeset
   383
lemma set_times_Un_distrib:
ed0795caec95 distributivity of * over Un and UNION
krauss
parents: 47445
diff changeset
   384
  "A * (B \<union> C) = A * B \<union> A * C"
ed0795caec95 distributivity of * over Un and UNION
krauss
parents: 47445
diff changeset
   385
  "(A \<union> B) * C = A * C \<union> B * C"
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   386
  by (auto simp: set_times_def)
47446
ed0795caec95 distributivity of * over Un and UNION
krauss
parents: 47445
diff changeset
   387
ed0795caec95 distributivity of * over Un and UNION
krauss
parents: 47445
diff changeset
   388
lemma set_times_UNION_distrib:
56899
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   389
  "A * UNION I M = (\<Union>i\<in>I. A * M i)"
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   390
  "UNION I M * A = (\<Union>i\<in>I. M i * A)"
9b9f4abaaa7e more symbols;
wenzelm
parents: 54230
diff changeset
   391
  by (auto simp: set_times_def)
47446
ed0795caec95 distributivity of * over Un and UNION
krauss
parents: 47445
diff changeset
   392
16908
d374530bfaaa Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff changeset
   393
end