src/HOL/Library/Set_Algebras.thy
 author krauss Thu Apr 12 23:07:01 2012 +0200 (2012-04-12) changeset 47445 69e96e5500df parent 47444 d21c95af2df7 child 47446 ed0795caec95 permissions -rw-r--r--
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
```     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_intro2 [intro]: "b : C ==> a + b : a +o C"
```
```    94   by (auto simp add: elt_set_plus_def)
```
```    95
```
```    96 lemma set_plus_rearrange: "((a::'a::comm_monoid_add) +o C) +
```
```    97     (b +o D) = (a + b) +o (C + D)"
```
```    98   apply (auto simp add: elt_set_plus_def set_plus_def add_ac)
```
```    99    apply (rule_tac x = "ba + bb" in exI)
```
```   100   apply (auto simp add: add_ac)
```
```   101   apply (rule_tac x = "aa + a" in exI)
```
```   102   apply (auto simp add: add_ac)
```
```   103   done
```
```   104
```
```   105 lemma set_plus_rearrange2: "(a::'a::semigroup_add) +o (b +o C) =
```
```   106     (a + b) +o C"
```
```   107   by (auto simp add: elt_set_plus_def add_assoc)
```
```   108
```
```   109 lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) + C =
```
```   110     a +o (B + C)"
```
```   111   apply (auto simp add: elt_set_plus_def set_plus_def)
```
```   112    apply (blast intro: add_ac)
```
```   113   apply (rule_tac x = "a + aa" in exI)
```
```   114   apply (rule conjI)
```
```   115    apply (rule_tac x = "aa" in bexI)
```
```   116     apply auto
```
```   117   apply (rule_tac x = "ba" in bexI)
```
```   118    apply (auto simp add: add_ac)
```
```   119   done
```
```   120
```
```   121 theorem set_plus_rearrange4: "C + ((a::'a::comm_monoid_add) +o D) =
```
```   122     a +o (C + D)"
```
```   123   apply (auto simp add: elt_set_plus_def set_plus_def add_ac)
```
```   124    apply (rule_tac x = "aa + ba" in exI)
```
```   125    apply (auto simp add: add_ac)
```
```   126   done
```
```   127
```
```   128 theorems set_plus_rearranges = set_plus_rearrange set_plus_rearrange2
```
```   129   set_plus_rearrange3 set_plus_rearrange4
```
```   130
```
```   131 lemma set_plus_mono [intro!]: "C <= D ==> a +o C <= a +o D"
```
```   132   by (auto simp add: elt_set_plus_def)
```
```   133
```
```   134 lemma set_plus_mono2 [intro]: "(C::('a::plus) set) <= D ==> E <= F ==>
```
```   135     C + E <= D + F"
```
```   136   by (auto simp add: set_plus_def)
```
```   137
```
```   138 lemma set_plus_mono3 [intro]: "a : C ==> a +o D <= C + D"
```
```   139   by (auto simp add: elt_set_plus_def set_plus_def)
```
```   140
```
```   141 lemma set_plus_mono4 [intro]: "(a::'a::comm_monoid_add) : C ==>
```
```   142     a +o D <= D + C"
```
```   143   by (auto simp add: elt_set_plus_def set_plus_def add_ac)
```
```   144
```
```   145 lemma set_plus_mono5: "a:C ==> B <= D ==> a +o B <= C + D"
```
```   146   apply (subgoal_tac "a +o B <= a +o D")
```
```   147    apply (erule order_trans)
```
```   148    apply (erule set_plus_mono3)
```
```   149   apply (erule set_plus_mono)
```
```   150   done
```
```   151
```
```   152 lemma set_plus_mono_b: "C <= D ==> x : a +o C
```
```   153     ==> x : a +o D"
```
```   154   apply (frule set_plus_mono)
```
```   155   apply auto
```
```   156   done
```
```   157
```
```   158 lemma set_plus_mono2_b: "C <= D ==> E <= F ==> x : C + E ==>
```
```   159     x : D + F"
```
```   160   apply (frule set_plus_mono2)
```
```   161    prefer 2
```
```   162    apply force
```
```   163   apply assumption
```
```   164   done
```
```   165
```
```   166 lemma set_plus_mono3_b: "a : C ==> x : a +o D ==> x : C + D"
```
```   167   apply (frule set_plus_mono3)
```
```   168   apply auto
```
```   169   done
```
```   170
```
```   171 lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C ==>
```
```   172     x : a +o D ==> x : D + C"
```
```   173   apply (frule set_plus_mono4)
```
```   174   apply auto
```
```   175   done
```
```   176
```
```   177 lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C"
```
```   178   by (auto simp add: elt_set_plus_def)
```
```   179
```
```   180 lemma set_zero_plus2: "(0::'a::comm_monoid_add) : A ==> B <= A + B"
```
```   181   apply (auto simp add: set_plus_def)
```
```   182   apply (rule_tac x = 0 in bexI)
```
```   183    apply (rule_tac x = x in bexI)
```
```   184     apply (auto simp add: add_ac)
```
```   185   done
```
```   186
```
```   187 lemma set_plus_imp_minus: "(a::'a::ab_group_add) : b +o C ==> (a - b) : C"
```
```   188   by (auto simp add: elt_set_plus_def add_ac diff_minus)
```
```   189
```
```   190 lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C ==> a : b +o C"
```
```   191   apply (auto simp add: elt_set_plus_def add_ac diff_minus)
```
```   192   apply (subgoal_tac "a = (a + - b) + b")
```
```   193    apply (rule bexI, assumption, assumption)
```
```   194   apply (auto simp add: add_ac)
```
```   195   done
```
```   196
```
```   197 lemma set_minus_plus: "((a::'a::ab_group_add) - b : C) = (a : b +o C)"
```
```   198   by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus,
```
```   199     assumption)
```
```   200
```
```   201 lemma set_times_intro [intro]: "a : C ==> b : D ==> a * b : C * D"
```
```   202   by (auto simp add: set_times_def)
```
```   203
```
```   204 lemma set_times_intro2 [intro!]: "b : C ==> a * b : a *o C"
```
```   205   by (auto simp add: elt_set_times_def)
```
```   206
```
```   207 lemma set_times_rearrange: "((a::'a::comm_monoid_mult) *o C) *
```
```   208     (b *o D) = (a * b) *o (C * D)"
```
```   209   apply (auto simp add: elt_set_times_def set_times_def)
```
```   210    apply (rule_tac x = "ba * bb" in exI)
```
```   211    apply (auto simp add: mult_ac)
```
```   212   apply (rule_tac x = "aa * a" in exI)
```
```   213   apply (auto simp add: mult_ac)
```
```   214   done
```
```   215
```
```   216 lemma set_times_rearrange2: "(a::'a::semigroup_mult) *o (b *o C) =
```
```   217     (a * b) *o C"
```
```   218   by (auto simp add: elt_set_times_def mult_assoc)
```
```   219
```
```   220 lemma set_times_rearrange3: "((a::'a::semigroup_mult) *o B) * C =
```
```   221     a *o (B * C)"
```
```   222   apply (auto simp add: elt_set_times_def set_times_def)
```
```   223    apply (blast intro: mult_ac)
```
```   224   apply (rule_tac x = "a * aa" in exI)
```
```   225   apply (rule conjI)
```
```   226    apply (rule_tac x = "aa" in bexI)
```
```   227     apply auto
```
```   228   apply (rule_tac x = "ba" in bexI)
```
```   229    apply (auto simp add: mult_ac)
```
```   230   done
```
```   231
```
```   232 theorem set_times_rearrange4: "C * ((a::'a::comm_monoid_mult) *o D) =
```
```   233     a *o (C * D)"
```
```   234   apply (auto simp add: elt_set_times_def set_times_def
```
```   235     mult_ac)
```
```   236    apply (rule_tac x = "aa * ba" in exI)
```
```   237    apply (auto simp add: mult_ac)
```
```   238   done
```
```   239
```
```   240 theorems set_times_rearranges = set_times_rearrange set_times_rearrange2
```
```   241   set_times_rearrange3 set_times_rearrange4
```
```   242
```
```   243 lemma set_times_mono [intro]: "C <= D ==> a *o C <= a *o D"
```
```   244   by (auto simp add: elt_set_times_def)
```
```   245
```
```   246 lemma set_times_mono2 [intro]: "(C::('a::times) set) <= D ==> E <= F ==>
```
```   247     C * E <= D * F"
```
```   248   by (auto simp add: set_times_def)
```
```   249
```
```   250 lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C * D"
```
```   251   by (auto simp add: elt_set_times_def set_times_def)
```
```   252
```
```   253 lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C ==>
```
```   254     a *o D <= D * C"
```
```   255   by (auto simp add: elt_set_times_def set_times_def mult_ac)
```
```   256
```
```   257 lemma set_times_mono5: "a:C ==> B <= D ==> a *o B <= C * D"
```
```   258   apply (subgoal_tac "a *o B <= a *o D")
```
```   259    apply (erule order_trans)
```
```   260    apply (erule set_times_mono3)
```
```   261   apply (erule set_times_mono)
```
```   262   done
```
```   263
```
```   264 lemma set_times_mono_b: "C <= D ==> x : a *o C
```
```   265     ==> x : a *o D"
```
```   266   apply (frule set_times_mono)
```
```   267   apply auto
```
```   268   done
```
```   269
```
```   270 lemma set_times_mono2_b: "C <= D ==> E <= F ==> x : C * E ==>
```
```   271     x : D * F"
```
```   272   apply (frule set_times_mono2)
```
```   273    prefer 2
```
```   274    apply force
```
```   275   apply assumption
```
```   276   done
```
```   277
```
```   278 lemma set_times_mono3_b: "a : C ==> x : a *o D ==> x : C * D"
```
```   279   apply (frule set_times_mono3)
```
```   280   apply auto
```
```   281   done
```
```   282
```
```   283 lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) : C ==>
```
```   284     x : a *o D ==> x : D * C"
```
```   285   apply (frule set_times_mono4)
```
```   286   apply auto
```
```   287   done
```
```   288
```
```   289 lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C"
```
```   290   by (auto simp add: elt_set_times_def)
```
```   291
```
```   292 lemma set_times_plus_distrib: "(a::'a::semiring) *o (b +o C)=
```
```   293     (a * b) +o (a *o C)"
```
```   294   by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs)
```
```   295
```
```   296 lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B + C) =
```
```   297     (a *o B) + (a *o C)"
```
```   298   apply (auto simp add: set_plus_def elt_set_times_def ring_distribs)
```
```   299    apply blast
```
```   300   apply (rule_tac x = "b + bb" in exI)
```
```   301   apply (auto simp add: ring_distribs)
```
```   302   done
```
```   303
```
```   304 lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) * D <=
```
```   305     a *o D + C * D"
```
```   306   apply (auto simp add:
```
```   307     elt_set_plus_def elt_set_times_def set_times_def
```
```   308     set_plus_def ring_distribs)
```
```   309   apply auto
```
```   310   done
```
```   311
```
```   312 theorems set_times_plus_distribs =
```
```   313   set_times_plus_distrib
```
```   314   set_times_plus_distrib2
```
```   315
```
```   316 lemma set_neg_intro: "(a::'a::ring_1) : (- 1) *o C ==>
```
```   317     - a : C"
```
```   318   by (auto simp add: elt_set_times_def)
```
```   319
```
```   320 lemma set_neg_intro2: "(a::'a::ring_1) : C ==>
```
```   321     - a : (- 1) *o C"
```
```   322   by (auto simp add: elt_set_times_def)
```
```   323
```
```   324 lemma set_plus_image:
```
```   325   fixes S T :: "'n::semigroup_add set" shows "S + T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
```
```   326   unfolding set_plus_def by (fastforce simp: image_iff)
```
```   327
```
```   328 lemma set_setsum_alt:
```
```   329   assumes fin: "finite I"
```
```   330   shows "setsum S I = {setsum s I |s. \<forall>i\<in>I. s i \<in> S i}"
```
```   331     (is "_ = ?setsum I")
```
```   332 using fin proof induct
```
```   333   case (insert x F)
```
```   334   have "setsum S (insert x F) = S x + ?setsum F"
```
```   335     using insert.hyps by auto
```
```   336   also have "...= {s x + setsum s F |s. \<forall> i\<in>insert x F. s i \<in> S i}"
```
```   337     unfolding set_plus_def
```
```   338   proof safe
```
```   339     fix y s assume "y \<in> S x" "\<forall>i\<in>F. s i \<in> S i"
```
```   340     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)"
```
```   341       using insert.hyps
```
```   342       by (intro exI[of _ "\<lambda>i. if i \<in> F then s i else y"]) (auto simp add: set_plus_def)
```
```   343   qed auto
```
```   344   finally show ?case
```
```   345     using insert.hyps by auto
```
```   346 qed auto
```
```   347
```
```   348 lemma setsum_set_cond_linear:
```
```   349   fixes f :: "('a::comm_monoid_add) set \<Rightarrow> ('b::comm_monoid_add) set"
```
```   350   assumes [intro!]: "\<And>A B. P A  \<Longrightarrow> P B  \<Longrightarrow> P (A + B)" "P {0}"
```
```   351     and f: "\<And>A B. P A  \<Longrightarrow> P B \<Longrightarrow> f (A + B) = f A + f B" "f {0} = {0}"
```
```   352   assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)"
```
```   353   shows "f (setsum S I) = setsum (f \<circ> S) I"
```
```   354 proof cases
```
```   355   assume "finite I" from this all show ?thesis
```
```   356   proof induct
```
```   357     case (insert x F)
```
```   358     from `finite F` `\<And>i. i \<in> insert x F \<Longrightarrow> P (S i)` have "P (setsum S F)"
```
```   359       by induct auto
```
```   360     with insert show ?case
```
```   361       by (simp, subst f) auto
```
```   362   qed (auto intro!: f)
```
```   363 qed (auto intro!: f)
```
```   364
```
```   365 lemma setsum_set_linear:
```
```   366   fixes f :: "('a::comm_monoid_add) set => ('b::comm_monoid_add) set"
```
```   367   assumes "\<And>A B. f(A) + f(B) = f(A + B)" "f {0} = {0}"
```
```   368   shows "f (setsum S I) = setsum (f \<circ> S) I"
```
```   369   using setsum_set_cond_linear[of "\<lambda>x. True" f I S] assms by auto
```
```   370
```
```   371 end
```