src/HOL/Library/Set_Algebras.thy
author krauss
Thu Apr 12 22:55:11 2012 +0200 (2012-04-12)
changeset 47444 d21c95af2df7
parent 47443 aeff49a3369b
child 47445 69e96e5500df
permissions -rw-r--r--
removed "setsum_set", now subsumed by generic setsum
     1 (*  Title:      HOL/Library/Set_Algebras.thy
     2     Author:     Jeremy Avigad and Kevin Donnelly; Florian Haftmann, TUM
     3 *)
     4 
     5 header {* Algebraic operations on sets *}
     6 
     7 theory Set_Algebras
     8 imports Main
     9 begin
    10 
    11 text {*
    12   This library lifts operations like addition and muliplication to
    13   sets.  It was designed to support asymptotic calculations. See the
    14   comments at the top of theory @{text BigO}.
    15 *}
    16 
    17 instantiation set :: (plus) plus
    18 begin
    19 
    20 definition plus_set :: "'a::plus set \<Rightarrow> 'a set \<Rightarrow> 'a set" where
    21   set_plus_def: "A + B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a + b}"
    22 
    23 instance ..
    24 
    25 end
    26 
    27 instantiation set :: (times) times
    28 begin
    29 
    30 definition times_set :: "'a::times set \<Rightarrow> 'a set \<Rightarrow> 'a set" where
    31   set_times_def: "A * B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a * b}"
    32 
    33 instance ..
    34 
    35 end
    36 
    37 
    38 text {* Legacy syntax: *}
    39 
    40 abbreviation (input) set_plus :: "'a::plus set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<oplus>" 65) where
    41   "A \<oplus> B \<equiv> A + B"
    42 abbreviation (input) set_times :: "'a::times set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<otimes>" 70) where
    43   "A \<otimes> B \<equiv> A * B"
    44 
    45 instantiation set :: (zero) zero
    46 begin
    47 
    48 definition
    49   set_zero[simp]: "0::('a::zero)set == {0}"
    50 
    51 instance ..
    52 
    53 end
    54  
    55 instantiation set :: (one) one
    56 begin
    57 
    58 definition
    59   set_one[simp]: "1::('a::one)set == {1}"
    60 
    61 instance ..
    62 
    63 end
    64 
    65 definition elt_set_plus :: "'a::plus \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "+o" 70) where
    66   "a +o B = {c. \<exists>b\<in>B. c = a + b}"
    67 
    68 definition elt_set_times :: "'a::times \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "*o" 80) where
    69   "a *o B = {c. \<exists>b\<in>B. c = a * b}"
    70 
    71 abbreviation (input) elt_set_eq :: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  (infix "=o" 50) where
    72   "x =o A \<equiv> x \<in> A"
    73 
    74 instance set :: (semigroup_add) semigroup_add
    75 by default (force simp add: set_plus_def add.assoc)
    76 
    77 instance set :: (ab_semigroup_add) ab_semigroup_add
    78 by default (force simp add: set_plus_def add.commute)
    79 
    80 instance set :: (monoid_add) monoid_add
    81 by default (simp_all add: set_plus_def)
    82 
    83 instance set :: (comm_monoid_add) comm_monoid_add
    84 by default (simp_all add: set_plus_def)
    85 
    86 instance set :: (semigroup_mult) semigroup_mult
    87 by default (force simp add: set_times_def mult.assoc)
    88 
    89 instance set :: (ab_semigroup_mult) ab_semigroup_mult
    90 by default (force simp add: set_times_def mult.commute)
    91 
    92 instance set :: (monoid_mult) monoid_mult
    93 by default (simp_all add: set_times_def)
    94 
    95 instance set :: (comm_monoid_mult) comm_monoid_mult
    96 by default (simp_all add: set_times_def)
    97 
    98 lemma set_plus_intro [intro]: "a : C ==> b : D ==> a + b : C \<oplus> D"
    99   by (auto simp add: set_plus_def)
   100 
   101 lemma set_plus_intro2 [intro]: "b : C ==> a + b : a +o C"
   102   by (auto simp add: elt_set_plus_def)
   103 
   104 lemma set_plus_rearrange: "((a::'a::comm_monoid_add) +o C) \<oplus>
   105     (b +o D) = (a + b) +o (C \<oplus> D)"
   106   apply (auto simp add: elt_set_plus_def set_plus_def add_ac)
   107    apply (rule_tac x = "ba + bb" in exI)
   108   apply (auto simp add: add_ac)
   109   apply (rule_tac x = "aa + a" in exI)
   110   apply (auto simp add: add_ac)
   111   done
   112 
   113 lemma set_plus_rearrange2: "(a::'a::semigroup_add) +o (b +o C) =
   114     (a + b) +o C"
   115   by (auto simp add: elt_set_plus_def add_assoc)
   116 
   117 lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) \<oplus> C =
   118     a +o (B \<oplus> C)"
   119   apply (auto simp add: elt_set_plus_def set_plus_def)
   120    apply (blast intro: add_ac)
   121   apply (rule_tac x = "a + aa" in exI)
   122   apply (rule conjI)
   123    apply (rule_tac x = "aa" in bexI)
   124     apply auto
   125   apply (rule_tac x = "ba" in bexI)
   126    apply (auto simp add: add_ac)
   127   done
   128 
   129 theorem set_plus_rearrange4: "C \<oplus> ((a::'a::comm_monoid_add) +o D) =
   130     a +o (C \<oplus> D)"
   131   apply (auto simp add: elt_set_plus_def set_plus_def add_ac)
   132    apply (rule_tac x = "aa + ba" in exI)
   133    apply (auto simp add: add_ac)
   134   done
   135 
   136 theorems set_plus_rearranges = set_plus_rearrange set_plus_rearrange2
   137   set_plus_rearrange3 set_plus_rearrange4
   138 
   139 lemma set_plus_mono [intro!]: "C <= D ==> a +o C <= a +o D"
   140   by (auto simp add: elt_set_plus_def)
   141 
   142 lemma set_plus_mono2 [intro]: "(C::('a::plus) set) <= D ==> E <= F ==>
   143     C \<oplus> E <= D \<oplus> F"
   144   by (auto simp add: set_plus_def)
   145 
   146 lemma set_plus_mono3 [intro]: "a : C ==> a +o D <= C \<oplus> D"
   147   by (auto simp add: elt_set_plus_def set_plus_def)
   148 
   149 lemma set_plus_mono4 [intro]: "(a::'a::comm_monoid_add) : C ==>
   150     a +o D <= D \<oplus> C"
   151   by (auto simp add: elt_set_plus_def set_plus_def add_ac)
   152 
   153 lemma set_plus_mono5: "a:C ==> B <= D ==> a +o B <= C \<oplus> D"
   154   apply (subgoal_tac "a +o B <= a +o D")
   155    apply (erule order_trans)
   156    apply (erule set_plus_mono3)
   157   apply (erule set_plus_mono)
   158   done
   159 
   160 lemma set_plus_mono_b: "C <= D ==> x : a +o C
   161     ==> x : a +o D"
   162   apply (frule set_plus_mono)
   163   apply auto
   164   done
   165 
   166 lemma set_plus_mono2_b: "C <= D ==> E <= F ==> x : C \<oplus> E ==>
   167     x : D \<oplus> F"
   168   apply (frule set_plus_mono2)
   169    prefer 2
   170    apply force
   171   apply assumption
   172   done
   173 
   174 lemma set_plus_mono3_b: "a : C ==> x : a +o D ==> x : C \<oplus> D"
   175   apply (frule set_plus_mono3)
   176   apply auto
   177   done
   178 
   179 lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C ==>
   180     x : a +o D ==> x : D \<oplus> C"
   181   apply (frule set_plus_mono4)
   182   apply auto
   183   done
   184 
   185 lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C"
   186   by (auto simp add: elt_set_plus_def)
   187 
   188 lemma set_zero_plus2: "(0::'a::comm_monoid_add) : A ==> B <= A \<oplus> B"
   189   apply (auto simp add: set_plus_def)
   190   apply (rule_tac x = 0 in bexI)
   191    apply (rule_tac x = x in bexI)
   192     apply (auto simp add: add_ac)
   193   done
   194 
   195 lemma set_plus_imp_minus: "(a::'a::ab_group_add) : b +o C ==> (a - b) : C"
   196   by (auto simp add: elt_set_plus_def add_ac diff_minus)
   197 
   198 lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C ==> a : b +o C"
   199   apply (auto simp add: elt_set_plus_def add_ac diff_minus)
   200   apply (subgoal_tac "a = (a + - b) + b")
   201    apply (rule bexI, assumption, assumption)
   202   apply (auto simp add: add_ac)
   203   done
   204 
   205 lemma set_minus_plus: "((a::'a::ab_group_add) - b : C) = (a : b +o C)"
   206   by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus,
   207     assumption)
   208 
   209 lemma set_times_intro [intro]: "a : C ==> b : D ==> a * b : C \<otimes> D"
   210   by (auto simp add: set_times_def)
   211 
   212 lemma set_times_intro2 [intro!]: "b : C ==> a * b : a *o C"
   213   by (auto simp add: elt_set_times_def)
   214 
   215 lemma set_times_rearrange: "((a::'a::comm_monoid_mult) *o C) \<otimes>
   216     (b *o D) = (a * b) *o (C \<otimes> D)"
   217   apply (auto simp add: elt_set_times_def set_times_def)
   218    apply (rule_tac x = "ba * bb" in exI)
   219    apply (auto simp add: mult_ac)
   220   apply (rule_tac x = "aa * a" in exI)
   221   apply (auto simp add: mult_ac)
   222   done
   223 
   224 lemma set_times_rearrange2: "(a::'a::semigroup_mult) *o (b *o C) =
   225     (a * b) *o C"
   226   by (auto simp add: elt_set_times_def mult_assoc)
   227 
   228 lemma set_times_rearrange3: "((a::'a::semigroup_mult) *o B) \<otimes> C =
   229     a *o (B \<otimes> C)"
   230   apply (auto simp add: elt_set_times_def set_times_def)
   231    apply (blast intro: mult_ac)
   232   apply (rule_tac x = "a * aa" in exI)
   233   apply (rule conjI)
   234    apply (rule_tac x = "aa" in bexI)
   235     apply auto
   236   apply (rule_tac x = "ba" in bexI)
   237    apply (auto simp add: mult_ac)
   238   done
   239 
   240 theorem set_times_rearrange4: "C \<otimes> ((a::'a::comm_monoid_mult) *o D) =
   241     a *o (C \<otimes> D)"
   242   apply (auto simp add: elt_set_times_def set_times_def
   243     mult_ac)
   244    apply (rule_tac x = "aa * ba" in exI)
   245    apply (auto simp add: mult_ac)
   246   done
   247 
   248 theorems set_times_rearranges = set_times_rearrange set_times_rearrange2
   249   set_times_rearrange3 set_times_rearrange4
   250 
   251 lemma set_times_mono [intro]: "C <= D ==> a *o C <= a *o D"
   252   by (auto simp add: elt_set_times_def)
   253 
   254 lemma set_times_mono2 [intro]: "(C::('a::times) set) <= D ==> E <= F ==>
   255     C \<otimes> E <= D \<otimes> F"
   256   by (auto simp add: set_times_def)
   257 
   258 lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C \<otimes> D"
   259   by (auto simp add: elt_set_times_def set_times_def)
   260 
   261 lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C ==>
   262     a *o D <= D \<otimes> C"
   263   by (auto simp add: elt_set_times_def set_times_def mult_ac)
   264 
   265 lemma set_times_mono5: "a:C ==> B <= D ==> a *o B <= C \<otimes> D"
   266   apply (subgoal_tac "a *o B <= a *o D")
   267    apply (erule order_trans)
   268    apply (erule set_times_mono3)
   269   apply (erule set_times_mono)
   270   done
   271 
   272 lemma set_times_mono_b: "C <= D ==> x : a *o C
   273     ==> x : a *o D"
   274   apply (frule set_times_mono)
   275   apply auto
   276   done
   277 
   278 lemma set_times_mono2_b: "C <= D ==> E <= F ==> x : C \<otimes> E ==>
   279     x : D \<otimes> F"
   280   apply (frule set_times_mono2)
   281    prefer 2
   282    apply force
   283   apply assumption
   284   done
   285 
   286 lemma set_times_mono3_b: "a : C ==> x : a *o D ==> x : C \<otimes> D"
   287   apply (frule set_times_mono3)
   288   apply auto
   289   done
   290 
   291 lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) : C ==>
   292     x : a *o D ==> x : D \<otimes> C"
   293   apply (frule set_times_mono4)
   294   apply auto
   295   done
   296 
   297 lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C"
   298   by (auto simp add: elt_set_times_def)
   299 
   300 lemma set_times_plus_distrib: "(a::'a::semiring) *o (b +o C)=
   301     (a * b) +o (a *o C)"
   302   by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs)
   303 
   304 lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B \<oplus> C) =
   305     (a *o B) \<oplus> (a *o C)"
   306   apply (auto simp add: set_plus_def elt_set_times_def ring_distribs)
   307    apply blast
   308   apply (rule_tac x = "b + bb" in exI)
   309   apply (auto simp add: ring_distribs)
   310   done
   311 
   312 lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) \<otimes> D <=
   313     a *o D \<oplus> C \<otimes> D"
   314   apply (auto simp add:
   315     elt_set_plus_def elt_set_times_def set_times_def
   316     set_plus_def ring_distribs)
   317   apply auto
   318   done
   319 
   320 theorems set_times_plus_distribs =
   321   set_times_plus_distrib
   322   set_times_plus_distrib2
   323 
   324 lemma set_neg_intro: "(a::'a::ring_1) : (- 1) *o C ==>
   325     - a : C"
   326   by (auto simp add: elt_set_times_def)
   327 
   328 lemma set_neg_intro2: "(a::'a::ring_1) : C ==>
   329     - a : (- 1) *o C"
   330   by (auto simp add: elt_set_times_def)
   331 
   332 lemma set_plus_image:
   333   fixes S T :: "'n::semigroup_add set" shows "S \<oplus> T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
   334   unfolding set_plus_def by (fastforce simp: image_iff)
   335 
   336 lemma set_setsum_alt:
   337   assumes fin: "finite I"
   338   shows "setsum S I = {setsum s I |s. \<forall>i\<in>I. s i \<in> S i}"
   339     (is "_ = ?setsum I")
   340 using fin proof induct
   341   case (insert x F)
   342   have "setsum S (insert x F) = S x \<oplus> ?setsum F"
   343     using insert.hyps by auto
   344   also have "...= {s x + setsum s F |s. \<forall> i\<in>insert x F. s i \<in> S i}"
   345     unfolding set_plus_def
   346   proof safe
   347     fix y s assume "y \<in> S x" "\<forall>i\<in>F. s i \<in> S i"
   348     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)"
   349       using insert.hyps
   350       by (intro exI[of _ "\<lambda>i. if i \<in> F then s i else y"]) (auto simp add: set_plus_def)
   351   qed auto
   352   finally show ?case
   353     using insert.hyps by auto
   354 qed auto
   355 
   356 lemma setsum_set_cond_linear:
   357   fixes f :: "('a::comm_monoid_add) set \<Rightarrow> ('b::comm_monoid_add) set"
   358   assumes [intro!]: "\<And>A B. P A  \<Longrightarrow> P B  \<Longrightarrow> P (A \<oplus> B)" "P {0}"
   359     and f: "\<And>A B. P A  \<Longrightarrow> P B \<Longrightarrow> f (A \<oplus> B) = f A \<oplus> f B" "f {0} = {0}"
   360   assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)"
   361   shows "f (setsum S I) = setsum (f \<circ> S) I"
   362 proof cases
   363   assume "finite I" from this all show ?thesis
   364   proof induct
   365     case (insert x F)
   366     from `finite F` `\<And>i. i \<in> insert x F \<Longrightarrow> P (S i)` have "P (setsum S F)"
   367       by induct auto
   368     with insert show ?case
   369       by (simp, subst f) auto
   370   qed (auto intro!: f)
   371 qed (auto intro!: f)
   372 
   373 lemma setsum_set_linear:
   374   fixes f :: "('a::comm_monoid_add) set => ('b::comm_monoid_add) set"
   375   assumes "\<And>A B. f(A) \<oplus> f(B) = f(A \<oplus> B)" "f {0} = {0}"
   376   shows "f (setsum S I) = setsum (f \<circ> S) I"
   377   using setsum_set_cond_linear[of "\<lambda>x. True" f I S] assms by auto
   378 
   379 end