src/HOL/Library/Set_Algebras.thy
author Andreas Lochbihler
Fri Sep 20 10:09:16 2013 +0200 (2013-09-20)
changeset 53745 788730ab7da4
parent 53596 d29d63460d84
child 54230 b1d955791529
permissions -rw-r--r--
prefer Code.abort over code_abort
     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 instantiation set :: (zero) zero
    38 begin
    39 
    40 definition
    41   set_zero[simp]: "0::('a::zero)set == {0}"
    42 
    43 instance ..
    44 
    45 end
    46  
    47 instantiation set :: (one) one
    48 begin
    49 
    50 definition
    51   set_one[simp]: "1::('a::one)set == {1}"
    52 
    53 instance ..
    54 
    55 end
    56 
    57 definition elt_set_plus :: "'a::plus \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "+o" 70) where
    58   "a +o B = {c. \<exists>b\<in>B. c = a + b}"
    59 
    60 definition elt_set_times :: "'a::times \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "*o" 80) where
    61   "a *o B = {c. \<exists>b\<in>B. c = a * b}"
    62 
    63 abbreviation (input) elt_set_eq :: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  (infix "=o" 50) where
    64   "x =o A \<equiv> x \<in> A"
    65 
    66 instance set :: (semigroup_add) semigroup_add
    67 by default (force simp add: set_plus_def add.assoc)
    68 
    69 instance set :: (ab_semigroup_add) ab_semigroup_add
    70 by default (force simp add: set_plus_def add.commute)
    71 
    72 instance set :: (monoid_add) monoid_add
    73 by default (simp_all add: set_plus_def)
    74 
    75 instance set :: (comm_monoid_add) comm_monoid_add
    76 by default (simp_all add: set_plus_def)
    77 
    78 instance set :: (semigroup_mult) semigroup_mult
    79 by default (force simp add: set_times_def mult.assoc)
    80 
    81 instance set :: (ab_semigroup_mult) ab_semigroup_mult
    82 by default (force simp add: set_times_def mult.commute)
    83 
    84 instance set :: (monoid_mult) monoid_mult
    85 by default (simp_all add: set_times_def)
    86 
    87 instance set :: (comm_monoid_mult) comm_monoid_mult
    88 by default (simp_all add: set_times_def)
    89 
    90 lemma set_plus_intro [intro]: "a : C ==> b : D ==> a + b : C + D"
    91   by (auto simp add: set_plus_def)
    92 
    93 lemma set_plus_elim:
    94   assumes "x \<in> A + B"
    95   obtains a b where "x = a + b" and "a \<in> A" and "b \<in> B"
    96   using assms unfolding set_plus_def by fast
    97 
    98 lemma set_plus_intro2 [intro]: "b : C ==> a + b : a +o C"
    99   by (auto simp add: elt_set_plus_def)
   100 
   101 lemma set_plus_rearrange: "((a::'a::comm_monoid_add) +o C) +
   102     (b +o D) = (a + b) +o (C + D)"
   103   apply (auto simp add: elt_set_plus_def set_plus_def add_ac)
   104    apply (rule_tac x = "ba + bb" in exI)
   105   apply (auto simp add: add_ac)
   106   apply (rule_tac x = "aa + a" in exI)
   107   apply (auto simp add: add_ac)
   108   done
   109 
   110 lemma set_plus_rearrange2: "(a::'a::semigroup_add) +o (b +o C) =
   111     (a + b) +o C"
   112   by (auto simp add: elt_set_plus_def add_assoc)
   113 
   114 lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) + C =
   115     a +o (B + C)"
   116   apply (auto simp add: elt_set_plus_def set_plus_def)
   117    apply (blast intro: add_ac)
   118   apply (rule_tac x = "a + aa" in exI)
   119   apply (rule conjI)
   120    apply (rule_tac x = "aa" in bexI)
   121     apply auto
   122   apply (rule_tac x = "ba" in bexI)
   123    apply (auto simp add: add_ac)
   124   done
   125 
   126 theorem set_plus_rearrange4: "C + ((a::'a::comm_monoid_add) +o D) =
   127     a +o (C + D)"
   128   apply (auto simp add: elt_set_plus_def set_plus_def add_ac)
   129    apply (rule_tac x = "aa + ba" in exI)
   130    apply (auto simp add: add_ac)
   131   done
   132 
   133 theorems set_plus_rearranges = set_plus_rearrange set_plus_rearrange2
   134   set_plus_rearrange3 set_plus_rearrange4
   135 
   136 lemma set_plus_mono [intro!]: "C <= D ==> a +o C <= a +o D"
   137   by (auto simp add: elt_set_plus_def)
   138 
   139 lemma set_plus_mono2 [intro]: "(C::('a::plus) set) <= D ==> E <= F ==>
   140     C + E <= D + F"
   141   by (auto simp add: set_plus_def)
   142 
   143 lemma set_plus_mono3 [intro]: "a : C ==> a +o D <= C + D"
   144   by (auto simp add: elt_set_plus_def set_plus_def)
   145 
   146 lemma set_plus_mono4 [intro]: "(a::'a::comm_monoid_add) : C ==>
   147     a +o D <= D + C"
   148   by (auto simp add: elt_set_plus_def set_plus_def add_ac)
   149 
   150 lemma set_plus_mono5: "a:C ==> B <= D ==> a +o B <= C + D"
   151   apply (subgoal_tac "a +o B <= a +o D")
   152    apply (erule order_trans)
   153    apply (erule set_plus_mono3)
   154   apply (erule set_plus_mono)
   155   done
   156 
   157 lemma set_plus_mono_b: "C <= D ==> x : a +o C
   158     ==> x : a +o D"
   159   apply (frule set_plus_mono)
   160   apply auto
   161   done
   162 
   163 lemma set_plus_mono2_b: "C <= D ==> E <= F ==> x : C + E ==>
   164     x : D + F"
   165   apply (frule set_plus_mono2)
   166    prefer 2
   167    apply force
   168   apply assumption
   169   done
   170 
   171 lemma set_plus_mono3_b: "a : C ==> x : a +o D ==> x : C + D"
   172   apply (frule set_plus_mono3)
   173   apply auto
   174   done
   175 
   176 lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C ==>
   177     x : a +o D ==> x : D + C"
   178   apply (frule set_plus_mono4)
   179   apply auto
   180   done
   181 
   182 lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C"
   183   by (auto simp add: elt_set_plus_def)
   184 
   185 lemma set_zero_plus2: "(0::'a::comm_monoid_add) : A ==> B <= A + B"
   186   apply (auto simp add: set_plus_def)
   187   apply (rule_tac x = 0 in bexI)
   188    apply (rule_tac x = x in bexI)
   189     apply (auto simp add: add_ac)
   190   done
   191 
   192 lemma set_plus_imp_minus: "(a::'a::ab_group_add) : b +o C ==> (a - b) : C"
   193   by (auto simp add: elt_set_plus_def add_ac diff_minus)
   194 
   195 lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C ==> a : b +o C"
   196   apply (auto simp add: elt_set_plus_def add_ac diff_minus)
   197   apply (subgoal_tac "a = (a + - b) + b")
   198    apply (rule bexI, assumption, assumption)
   199   apply (auto simp add: add_ac)
   200   done
   201 
   202 lemma set_minus_plus: "((a::'a::ab_group_add) - b : C) = (a : b +o C)"
   203   by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus,
   204     assumption)
   205 
   206 lemma set_times_intro [intro]: "a : C ==> b : D ==> a * b : C * D"
   207   by (auto simp add: set_times_def)
   208 
   209 lemma set_times_elim:
   210   assumes "x \<in> A * B"
   211   obtains a b where "x = a * b" and "a \<in> A" and "b \<in> B"
   212   using assms unfolding set_times_def by fast
   213 
   214 lemma set_times_intro2 [intro!]: "b : C ==> a * b : a *o C"
   215   by (auto simp add: elt_set_times_def)
   216 
   217 lemma set_times_rearrange: "((a::'a::comm_monoid_mult) *o C) *
   218     (b *o D) = (a * b) *o (C * D)"
   219   apply (auto simp add: elt_set_times_def set_times_def)
   220    apply (rule_tac x = "ba * bb" in exI)
   221    apply (auto simp add: mult_ac)
   222   apply (rule_tac x = "aa * a" in exI)
   223   apply (auto simp add: mult_ac)
   224   done
   225 
   226 lemma set_times_rearrange2: "(a::'a::semigroup_mult) *o (b *o C) =
   227     (a * b) *o C"
   228   by (auto simp add: elt_set_times_def mult_assoc)
   229 
   230 lemma set_times_rearrange3: "((a::'a::semigroup_mult) *o B) * C =
   231     a *o (B * C)"
   232   apply (auto simp add: elt_set_times_def set_times_def)
   233    apply (blast intro: mult_ac)
   234   apply (rule_tac x = "a * aa" in exI)
   235   apply (rule conjI)
   236    apply (rule_tac x = "aa" in bexI)
   237     apply auto
   238   apply (rule_tac x = "ba" in bexI)
   239    apply (auto simp add: mult_ac)
   240   done
   241 
   242 theorem set_times_rearrange4: "C * ((a::'a::comm_monoid_mult) *o D) =
   243     a *o (C * D)"
   244   apply (auto simp add: elt_set_times_def set_times_def
   245     mult_ac)
   246    apply (rule_tac x = "aa * ba" in exI)
   247    apply (auto simp add: mult_ac)
   248   done
   249 
   250 theorems set_times_rearranges = set_times_rearrange set_times_rearrange2
   251   set_times_rearrange3 set_times_rearrange4
   252 
   253 lemma set_times_mono [intro]: "C <= D ==> a *o C <= a *o D"
   254   by (auto simp add: elt_set_times_def)
   255 
   256 lemma set_times_mono2 [intro]: "(C::('a::times) set) <= D ==> E <= F ==>
   257     C * E <= D * F"
   258   by (auto simp add: set_times_def)
   259 
   260 lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C * D"
   261   by (auto simp add: elt_set_times_def set_times_def)
   262 
   263 lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C ==>
   264     a *o D <= D * C"
   265   by (auto simp add: elt_set_times_def set_times_def mult_ac)
   266 
   267 lemma set_times_mono5: "a:C ==> B <= D ==> a *o B <= C * D"
   268   apply (subgoal_tac "a *o B <= a *o D")
   269    apply (erule order_trans)
   270    apply (erule set_times_mono3)
   271   apply (erule set_times_mono)
   272   done
   273 
   274 lemma set_times_mono_b: "C <= D ==> x : a *o C
   275     ==> x : a *o D"
   276   apply (frule set_times_mono)
   277   apply auto
   278   done
   279 
   280 lemma set_times_mono2_b: "C <= D ==> E <= F ==> x : C * E ==>
   281     x : D * F"
   282   apply (frule set_times_mono2)
   283    prefer 2
   284    apply force
   285   apply assumption
   286   done
   287 
   288 lemma set_times_mono3_b: "a : C ==> x : a *o D ==> x : C * D"
   289   apply (frule set_times_mono3)
   290   apply auto
   291   done
   292 
   293 lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) : C ==>
   294     x : a *o D ==> x : D * C"
   295   apply (frule set_times_mono4)
   296   apply auto
   297   done
   298 
   299 lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C"
   300   by (auto simp add: elt_set_times_def)
   301 
   302 lemma set_times_plus_distrib: "(a::'a::semiring) *o (b +o C)=
   303     (a * b) +o (a *o C)"
   304   by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs)
   305 
   306 lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B + C) =
   307     (a *o B) + (a *o C)"
   308   apply (auto simp add: set_plus_def elt_set_times_def ring_distribs)
   309    apply blast
   310   apply (rule_tac x = "b + bb" in exI)
   311   apply (auto simp add: ring_distribs)
   312   done
   313 
   314 lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) * D <=
   315     a *o D + C * D"
   316   apply (auto simp add:
   317     elt_set_plus_def elt_set_times_def set_times_def
   318     set_plus_def ring_distribs)
   319   apply auto
   320   done
   321 
   322 theorems set_times_plus_distribs =
   323   set_times_plus_distrib
   324   set_times_plus_distrib2
   325 
   326 lemma set_neg_intro: "(a::'a::ring_1) : (- 1) *o C ==>
   327     - a : C"
   328   by (auto simp add: elt_set_times_def)
   329 
   330 lemma set_neg_intro2: "(a::'a::ring_1) : C ==>
   331     - a : (- 1) *o C"
   332   by (auto simp add: elt_set_times_def)
   333 
   334 lemma set_plus_image: "S + T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
   335   unfolding set_plus_def by (fastforce simp: image_iff)
   336 
   337 lemma set_times_image: "S * T = (\<lambda>(x, y). x * y) ` (S \<times> T)"
   338   unfolding set_times_def by (fastforce simp: image_iff)
   339 
   340 lemma finite_set_plus:
   341   assumes "finite s" and "finite t" shows "finite (s + t)"
   342   using assms unfolding set_plus_image by simp
   343 
   344 lemma finite_set_times:
   345   assumes "finite s" and "finite t" shows "finite (s * t)"
   346   using assms unfolding set_times_image by simp
   347 
   348 lemma set_setsum_alt:
   349   assumes fin: "finite I"
   350   shows "setsum S I = {setsum s I |s. \<forall>i\<in>I. s i \<in> S i}"
   351     (is "_ = ?setsum I")
   352 using fin proof induct
   353   case (insert x F)
   354   have "setsum S (insert x F) = S x + ?setsum F"
   355     using insert.hyps by auto
   356   also have "...= {s x + setsum s F |s. \<forall> i\<in>insert x F. s i \<in> S i}"
   357     unfolding set_plus_def
   358   proof safe
   359     fix y s assume "y \<in> S x" "\<forall>i\<in>F. s i \<in> S i"
   360     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)"
   361       using insert.hyps
   362       by (intro exI[of _ "\<lambda>i. if i \<in> F then s i else y"]) (auto simp add: set_plus_def)
   363   qed auto
   364   finally show ?case
   365     using insert.hyps by auto
   366 qed auto
   367 
   368 lemma setsum_set_cond_linear:
   369   fixes f :: "('a::comm_monoid_add) set \<Rightarrow> ('b::comm_monoid_add) set"
   370   assumes [intro!]: "\<And>A B. P A  \<Longrightarrow> P B  \<Longrightarrow> P (A + B)" "P {0}"
   371     and f: "\<And>A B. P A  \<Longrightarrow> P B \<Longrightarrow> f (A + B) = f A + f B" "f {0} = {0}"
   372   assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)"
   373   shows "f (setsum S I) = setsum (f \<circ> S) I"
   374 proof cases
   375   assume "finite I" from this all show ?thesis
   376   proof induct
   377     case (insert x F)
   378     from `finite F` `\<And>i. i \<in> insert x F \<Longrightarrow> P (S i)` have "P (setsum S F)"
   379       by induct auto
   380     with insert show ?case
   381       by (simp, subst f) auto
   382   qed (auto intro!: f)
   383 qed (auto intro!: f)
   384 
   385 lemma setsum_set_linear:
   386   fixes f :: "('a::comm_monoid_add) set => ('b::comm_monoid_add) set"
   387   assumes "\<And>A B. f(A) + f(B) = f(A + B)" "f {0} = {0}"
   388   shows "f (setsum S I) = setsum (f \<circ> S) I"
   389   using setsum_set_cond_linear[of "\<lambda>x. True" f I S] assms by auto
   390 
   391 lemma set_times_Un_distrib:
   392   "A * (B \<union> C) = A * B \<union> A * C"
   393   "(A \<union> B) * C = A * C \<union> B * C"
   394 by (auto simp: set_times_def)
   395 
   396 lemma set_times_UNION_distrib:
   397   "A * UNION I M = UNION I (%i. A * M i)"
   398   "UNION I M * A = UNION I (%i. M i * A)"
   399 by (auto simp: set_times_def)
   400 
   401 end