src/HOL/Library/Set_Algebras.thy
 author wenzelm Wed Jun 17 11:03:05 2015 +0200 (2015-06-17) changeset 60500 903bb1495239 parent 58881 b9556a055632 child 60679 ade12ef2773c permissions -rw-r--r--
isabelle update_cartouches;
```     1 (*  Title:      HOL/Library/Set_Algebras.thy
```
```     2     Author:     Jeremy Avigad and Kevin Donnelly; Florian Haftmann, TUM
```
```     3 *)
```
```     4
```
```     5 section \<open>Algebraic operations on sets\<close>
```
```     6
```
```     7 theory Set_Algebras
```
```     8 imports Main
```
```     9 begin
```
```    10
```
```    11 text \<open>
```
```    12   This library lifts operations like addition and multiplication to
```
```    13   sets.  It was designed to support asymptotic calculations. See the
```
```    14   comments at the top of theory @{text BigO}.
```
```    15 \<close>
```
```    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 \<in> C \<Longrightarrow> b \<in> D \<Longrightarrow> a + b \<in> 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 \<in> C \<Longrightarrow> a + b \<in> a +o C"
```
```    99   by (auto simp add: elt_set_plus_def)
```
```   100
```
```   101 lemma set_plus_rearrange:
```
```   102   "((a::'a::comm_monoid_add) +o C) + (b +o D) = (a + b) +o (C + D)"
```
```   103   apply (auto simp add: elt_set_plus_def set_plus_def ac_simps)
```
```   104    apply (rule_tac x = "ba + bb" in exI)
```
```   105   apply (auto simp add: ac_simps)
```
```   106   apply (rule_tac x = "aa + a" in exI)
```
```   107   apply (auto simp add: ac_simps)
```
```   108   done
```
```   109
```
```   110 lemma set_plus_rearrange2: "(a::'a::semigroup_add) +o (b +o C) = (a + b) +o C"
```
```   111   by (auto simp add: elt_set_plus_def add.assoc)
```
```   112
```
```   113 lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) + C = a +o (B + C)"
```
```   114   apply (auto simp add: elt_set_plus_def set_plus_def)
```
```   115    apply (blast intro: ac_simps)
```
```   116   apply (rule_tac x = "a + aa" in exI)
```
```   117   apply (rule conjI)
```
```   118    apply (rule_tac x = "aa" in bexI)
```
```   119     apply auto
```
```   120   apply (rule_tac x = "ba" in bexI)
```
```   121    apply (auto simp add: ac_simps)
```
```   122   done
```
```   123
```
```   124 theorem set_plus_rearrange4: "C + ((a::'a::comm_monoid_add) +o D) = a +o (C + D)"
```
```   125   apply (auto simp add: elt_set_plus_def set_plus_def ac_simps)
```
```   126    apply (rule_tac x = "aa + ba" in exI)
```
```   127    apply (auto simp add: ac_simps)
```
```   128   done
```
```   129
```
```   130 theorems set_plus_rearranges = set_plus_rearrange set_plus_rearrange2
```
```   131   set_plus_rearrange3 set_plus_rearrange4
```
```   132
```
```   133 lemma set_plus_mono [intro!]: "C \<subseteq> D \<Longrightarrow> a +o C \<subseteq> a +o D"
```
```   134   by (auto simp add: elt_set_plus_def)
```
```   135
```
```   136 lemma set_plus_mono2 [intro]: "(C::'a::plus set) \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C + E \<subseteq> D + F"
```
```   137   by (auto simp add: set_plus_def)
```
```   138
```
```   139 lemma set_plus_mono3 [intro]: "a \<in> C \<Longrightarrow> a +o D \<subseteq> C + D"
```
```   140   by (auto simp add: elt_set_plus_def set_plus_def)
```
```   141
```
```   142 lemma set_plus_mono4 [intro]: "(a::'a::comm_monoid_add) \<in> C \<Longrightarrow> a +o D \<subseteq> D + C"
```
```   143   by (auto simp add: elt_set_plus_def set_plus_def ac_simps)
```
```   144
```
```   145 lemma set_plus_mono5: "a \<in> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> a +o B \<subseteq> C + D"
```
```   146   apply (subgoal_tac "a +o B \<subseteq> 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 \<subseteq> D \<Longrightarrow> x \<in> a +o C \<Longrightarrow> x \<in> a +o D"
```
```   153   apply (frule set_plus_mono)
```
```   154   apply auto
```
```   155   done
```
```   156
```
```   157 lemma set_plus_mono2_b: "C \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> x \<in> C + E \<Longrightarrow> x \<in> D + F"
```
```   158   apply (frule set_plus_mono2)
```
```   159    prefer 2
```
```   160    apply force
```
```   161   apply assumption
```
```   162   done
```
```   163
```
```   164 lemma set_plus_mono3_b: "a \<in> C \<Longrightarrow> x \<in> a +o D \<Longrightarrow> x \<in> C + D"
```
```   165   apply (frule set_plus_mono3)
```
```   166   apply auto
```
```   167   done
```
```   168
```
```   169 lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C \<Longrightarrow> x \<in> a +o D \<Longrightarrow> x \<in> D + C"
```
```   170   apply (frule set_plus_mono4)
```
```   171   apply auto
```
```   172   done
```
```   173
```
```   174 lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C"
```
```   175   by (auto simp add: elt_set_plus_def)
```
```   176
```
```   177 lemma set_zero_plus2: "(0::'a::comm_monoid_add) \<in> A \<Longrightarrow> B \<subseteq> A + B"
```
```   178   apply (auto simp add: set_plus_def)
```
```   179   apply (rule_tac x = 0 in bexI)
```
```   180    apply (rule_tac x = x in bexI)
```
```   181     apply (auto simp add: ac_simps)
```
```   182   done
```
```   183
```
```   184 lemma set_plus_imp_minus: "(a::'a::ab_group_add) : b +o C \<Longrightarrow> (a - b) \<in> C"
```
```   185   by (auto simp add: elt_set_plus_def ac_simps)
```
```   186
```
```   187 lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C \<Longrightarrow> a \<in> b +o C"
```
```   188   apply (auto simp add: elt_set_plus_def ac_simps)
```
```   189   apply (subgoal_tac "a = (a + - b) + b")
```
```   190    apply (rule bexI, assumption)
```
```   191   apply (auto simp add: ac_simps)
```
```   192   done
```
```   193
```
```   194 lemma set_minus_plus: "(a::'a::ab_group_add) - b \<in> C \<longleftrightarrow> a \<in> b +o C"
```
```   195   by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus)
```
```   196
```
```   197 lemma set_times_intro [intro]: "a \<in> C \<Longrightarrow> b \<in> D \<Longrightarrow> a * b \<in> C * D"
```
```   198   by (auto simp add: set_times_def)
```
```   199
```
```   200 lemma set_times_elim:
```
```   201   assumes "x \<in> A * B"
```
```   202   obtains a b where "x = a * b" and "a \<in> A" and "b \<in> B"
```
```   203   using assms unfolding set_times_def by fast
```
```   204
```
```   205 lemma set_times_intro2 [intro!]: "b \<in> C \<Longrightarrow> a * b \<in> a *o C"
```
```   206   by (auto simp add: elt_set_times_def)
```
```   207
```
```   208 lemma set_times_rearrange:
```
```   209   "((a::'a::comm_monoid_mult) *o C) * (b *o D) = (a * b) *o (C * D)"
```
```   210   apply (auto simp add: elt_set_times_def set_times_def)
```
```   211    apply (rule_tac x = "ba * bb" in exI)
```
```   212    apply (auto simp add: ac_simps)
```
```   213   apply (rule_tac x = "aa * a" in exI)
```
```   214   apply (auto simp add: ac_simps)
```
```   215   done
```
```   216
```
```   217 lemma set_times_rearrange2:
```
```   218   "(a::'a::semigroup_mult) *o (b *o C) = (a * b) *o C"
```
```   219   by (auto simp add: elt_set_times_def mult.assoc)
```
```   220
```
```   221 lemma set_times_rearrange3:
```
```   222   "((a::'a::semigroup_mult) *o B) * C = a *o (B * C)"
```
```   223   apply (auto simp add: elt_set_times_def set_times_def)
```
```   224    apply (blast intro: ac_simps)
```
```   225   apply (rule_tac x = "a * aa" in exI)
```
```   226   apply (rule conjI)
```
```   227    apply (rule_tac x = "aa" in bexI)
```
```   228     apply auto
```
```   229   apply (rule_tac x = "ba" in bexI)
```
```   230    apply (auto simp add: ac_simps)
```
```   231   done
```
```   232
```
```   233 theorem set_times_rearrange4:
```
```   234   "C * ((a::'a::comm_monoid_mult) *o D) = a *o (C * D)"
```
```   235   apply (auto simp add: elt_set_times_def set_times_def ac_simps)
```
```   236    apply (rule_tac x = "aa * ba" in exI)
```
```   237    apply (auto simp add: ac_simps)
```
```   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 \<subseteq> D \<Longrightarrow> a *o C \<subseteq> a *o D"
```
```   244   by (auto simp add: elt_set_times_def)
```
```   245
```
```   246 lemma set_times_mono2 [intro]: "(C::'a::times set) \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C * E \<subseteq> D * F"
```
```   247   by (auto simp add: set_times_def)
```
```   248
```
```   249 lemma set_times_mono3 [intro]: "a \<in> C \<Longrightarrow> a *o D \<subseteq> C * D"
```
```   250   by (auto simp add: elt_set_times_def set_times_def)
```
```   251
```
```   252 lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C \<Longrightarrow> a *o D \<subseteq> D * C"
```
```   253   by (auto simp add: elt_set_times_def set_times_def ac_simps)
```
```   254
```
```   255 lemma set_times_mono5: "a \<in> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> a *o B \<subseteq> C * D"
```
```   256   apply (subgoal_tac "a *o B \<subseteq> a *o D")
```
```   257    apply (erule order_trans)
```
```   258    apply (erule set_times_mono3)
```
```   259   apply (erule set_times_mono)
```
```   260   done
```
```   261
```
```   262 lemma set_times_mono_b: "C \<subseteq> D \<Longrightarrow> x \<in> a *o C \<Longrightarrow> x \<in> a *o D"
```
```   263   apply (frule set_times_mono)
```
```   264   apply auto
```
```   265   done
```
```   266
```
```   267 lemma set_times_mono2_b: "C \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> x \<in> C * E \<Longrightarrow> x \<in> D * F"
```
```   268   apply (frule set_times_mono2)
```
```   269    prefer 2
```
```   270    apply force
```
```   271   apply assumption
```
```   272   done
```
```   273
```
```   274 lemma set_times_mono3_b: "a \<in> C \<Longrightarrow> x \<in> a *o D \<Longrightarrow> x \<in> C * D"
```
```   275   apply (frule set_times_mono3)
```
```   276   apply auto
```
```   277   done
```
```   278
```
```   279 lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) \<in> C \<Longrightarrow> x \<in> a *o D \<Longrightarrow> x \<in> D * C"
```
```   280   apply (frule set_times_mono4)
```
```   281   apply auto
```
```   282   done
```
```   283
```
```   284 lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C"
```
```   285   by (auto simp add: elt_set_times_def)
```
```   286
```
```   287 lemma set_times_plus_distrib:
```
```   288   "(a::'a::semiring) *o (b +o C) = (a * b) +o (a *o C)"
```
```   289   by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs)
```
```   290
```
```   291 lemma set_times_plus_distrib2:
```
```   292   "(a::'a::semiring) *o (B + C) = (a *o B) + (a *o C)"
```
```   293   apply (auto simp add: set_plus_def elt_set_times_def ring_distribs)
```
```   294    apply blast
```
```   295   apply (rule_tac x = "b + bb" in exI)
```
```   296   apply (auto simp add: ring_distribs)
```
```   297   done
```
```   298
```
```   299 lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) * D \<subseteq> a *o D + C * D"
```
```   300   apply (auto simp add:
```
```   301     elt_set_plus_def elt_set_times_def set_times_def
```
```   302     set_plus_def ring_distribs)
```
```   303   apply auto
```
```   304   done
```
```   305
```
```   306 theorems set_times_plus_distribs =
```
```   307   set_times_plus_distrib
```
```   308   set_times_plus_distrib2
```
```   309
```
```   310 lemma set_neg_intro: "(a::'a::ring_1) \<in> (- 1) *o C \<Longrightarrow> - a \<in> C"
```
```   311   by (auto simp add: elt_set_times_def)
```
```   312
```
```   313 lemma set_neg_intro2: "(a::'a::ring_1) \<in> C \<Longrightarrow> - a \<in> (- 1) *o C"
```
```   314   by (auto simp add: elt_set_times_def)
```
```   315
```
```   316 lemma set_plus_image: "S + T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
```
```   317   unfolding set_plus_def by (fastforce simp: image_iff)
```
```   318
```
```   319 lemma set_times_image: "S * T = (\<lambda>(x, y). x * y) ` (S \<times> T)"
```
```   320   unfolding set_times_def by (fastforce simp: image_iff)
```
```   321
```
```   322 lemma finite_set_plus: "finite s \<Longrightarrow> finite t \<Longrightarrow> finite (s + t)"
```
```   323   unfolding set_plus_image by simp
```
```   324
```
```   325 lemma finite_set_times: "finite s \<Longrightarrow> finite t \<Longrightarrow> finite (s * t)"
```
```   326   unfolding set_times_image by simp
```
```   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
```
```   333 proof induct
```
```   334   case empty
```
```   335   then show ?case by simp
```
```   336 next
```
```   337   case (insert x F)
```
```   338   have "setsum S (insert x F) = S x + ?setsum F"
```
```   339     using insert.hyps by auto
```
```   340   also have "\<dots> = {s x + setsum s F |s. \<forall> i\<in>insert x F. s i \<in> S i}"
```
```   341     unfolding set_plus_def
```
```   342   proof safe
```
```   343     fix y s
```
```   344     assume "y \<in> S x" "\<forall>i\<in>F. s i \<in> S i"
```
```   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)"
```
```   346       using insert.hyps
```
```   347       by (intro exI[of _ "\<lambda>i. if i \<in> F then s i else y"]) (auto simp add: set_plus_def)
```
```   348   qed auto
```
```   349   finally show ?case
```
```   350     using insert.hyps by auto
```
```   351 qed
```
```   352
```
```   353 lemma setsum_set_cond_linear:
```
```   354   fixes f :: "'a::comm_monoid_add set \<Rightarrow> 'b::comm_monoid_add set"
```
```   355   assumes [intro!]: "\<And>A B. P A  \<Longrightarrow> P B  \<Longrightarrow> P (A + B)" "P {0}"
```
```   356     and f: "\<And>A B. P A  \<Longrightarrow> P B \<Longrightarrow> f (A + B) = f A + f B" "f {0} = {0}"
```
```   357   assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)"
```
```   358   shows "f (setsum S I) = setsum (f \<circ> S) I"
```
```   359 proof (cases "finite I")
```
```   360   case True
```
```   361   from this all show ?thesis
```
```   362   proof induct
```
```   363     case empty
```
```   364     then show ?case by (auto intro!: f)
```
```   365   next
```
```   366     case (insert x F)
```
```   367     from \<open>finite F\<close> \<open>\<And>i. i \<in> insert x F \<Longrightarrow> P (S i)\<close> have "P (setsum S F)"
```
```   368       by induct auto
```
```   369     with insert show ?case
```
```   370       by (simp, subst f) auto
```
```   371   qed
```
```   372 next
```
```   373   case False
```
```   374   then show ?thesis by (auto intro!: f)
```
```   375 qed
```
```   376
```
```   377 lemma setsum_set_linear:
```
```   378   fixes f :: "'a::comm_monoid_add set \<Rightarrow> 'b::comm_monoid_add set"
```
```   379   assumes "\<And>A B. f(A) + f(B) = f(A + B)" "f {0} = {0}"
```
```   380   shows "f (setsum S I) = setsum (f \<circ> S) I"
```
```   381   using setsum_set_cond_linear[of "\<lambda>x. True" f I S] assms by auto
```
```   382
```
```   383 lemma set_times_Un_distrib:
```
```   384   "A * (B \<union> C) = A * B \<union> A * C"
```
```   385   "(A \<union> B) * C = A * C \<union> B * C"
```
```   386   by (auto simp: set_times_def)
```
```   387
```
```   388 lemma set_times_UNION_distrib:
```
```   389   "A * UNION I M = (\<Union>i\<in>I. A * M i)"
```
```   390   "UNION I M * A = (\<Union>i\<in>I. M i * A)"
```
```   391   by (auto simp: set_times_def)
```
```   392
```
```   393 end
```