(*  Title:      HOL/Library/Set_Algebras.thy

     Author:     Jeremy Avigad and Kevin Donnelly; Florian Haftmann, TUM

 *)

 header {* Algebraic operations on sets *}

 theory Set_Algebras

 imports Main

 begin

 text {*

   This library lifts operations like addition and muliplication to

   sets.  It was designed to support asymptotic calculations. See the

   comments at the top of theory @{text BigO}.

 *}

 instantiation set :: (plus) plus

 begin

 definition plus_set :: "'a::plus set \<Rightarrow> 'a set \<Rightarrow> 'a set" where

   set_plus_def: "A + B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a + b}"

 instance ..

 end

 instantiation set :: (times) times

 begin

 definition times_set :: "'a::times set \<Rightarrow> 'a set \<Rightarrow> 'a set" where

   set_times_def: "A * B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a * b}"

 instance ..

 end

 instantiation set :: (zero) zero

 begin

 definition

   set_zero[simp]: "0::('a::zero)set == {0}"

 instance ..

 end

 instantiation set :: (one) one

 begin

 definition

   set_one[simp]: "1::('a::one)set == {1}"

 instance ..

 end

 definition elt_set_plus :: "'a::plus \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "+o" 70) where

   "a +o B = {c. \<exists>b\<in>B. c = a + b}"

 definition elt_set_times :: "'a::times \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "*o" 80) where

   "a *o B = {c. \<exists>b\<in>B. c = a * b}"

 abbreviation (input) elt_set_eq :: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  (infix "=o" 50) where

   "x =o A \<equiv> x \<in> A"

 instance set :: (semigroup_add) semigroup_add

 by default (force simp add: set_plus_def add.assoc)

 instance set :: (ab_semigroup_add) ab_semigroup_add

 by default (force simp add: set_plus_def add.commute)

 instance set :: (monoid_add) monoid_add

 by default (simp_all add: set_plus_def)

 instance set :: (comm_monoid_add) comm_monoid_add

 by default (simp_all add: set_plus_def)

 instance set :: (semigroup_mult) semigroup_mult

 by default (force simp add: set_times_def mult.assoc)

 instance set :: (ab_semigroup_mult) ab_semigroup_mult

 by default (force simp add: set_times_def mult.commute)

 instance set :: (monoid_mult) monoid_mult

 by default (simp_all add: set_times_def)

 instance set :: (comm_monoid_mult) comm_monoid_mult

 by default (simp_all add: set_times_def)

 lemma set_plus_intro [intro]: "a : C ==> b : D ==> a + b : C + D"

   by (auto simp add: set_plus_def)

 lemma set_plus_elim:

   assumes "x \<in> A + B"

   obtains a b where "x = a + b" and "a \<in> A" and "b \<in> B"

   using assms unfolding set_plus_def by fast

 lemma set_plus_intro2 [intro]: "b : C ==> a + b : a +o C"

   by (auto simp add: elt_set_plus_def)

 lemma set_plus_rearrange: "((a::'a::comm_monoid_add) +o C) +

     (b +o D) = (a + b) +o (C + D)"

   apply (auto simp add: elt_set_plus_def set_plus_def add_ac)

    apply (rule_tac x = "ba + bb" in exI)

   apply (auto simp add: add_ac)

   apply (rule_tac x = "aa + a" in exI)

   apply (auto simp add: add_ac)

   done

 lemma set_plus_rearrange2: "(a::'a::semigroup_add) +o (b +o C) =

     (a + b) +o C"

   by (auto simp add: elt_set_plus_def add_assoc)

 lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) + C =

     a +o (B + C)"

   apply (auto simp add: elt_set_plus_def set_plus_def)

    apply (blast intro: add_ac)

   apply (rule_tac x = "a + aa" in exI)

   apply (rule conjI)

    apply (rule_tac x = "aa" in bexI)

     apply auto

   apply (rule_tac x = "ba" in bexI)

    apply (auto simp add: add_ac)

   done

 theorem set_plus_rearrange4: "C + ((a::'a::comm_monoid_add) +o D) =

     a +o (C + D)"

   apply (auto simp add: elt_set_plus_def set_plus_def add_ac)

    apply (rule_tac x = "aa + ba" in exI)

    apply (auto simp add: add_ac)

   done

 theorems set_plus_rearranges = set_plus_rearrange set_plus_rearrange2

   set_plus_rearrange3 set_plus_rearrange4

 lemma set_plus_mono [intro!]: "C <= D ==> a +o C <= a +o D"

   by (auto simp add: elt_set_plus_def)

 lemma set_plus_mono2 [intro]: "(C::('a::plus) set) <= D ==> E <= F ==>

     C + E <= D + F"

   by (auto simp add: set_plus_def)

 lemma set_plus_mono3 [intro]: "a : C ==> a +o D <= C + D"

   by (auto simp add: elt_set_plus_def set_plus_def)

 lemma set_plus_mono4 [intro]: "(a::'a::comm_monoid_add) : C ==>

     a +o D <= D + C"

   by (auto simp add: elt_set_plus_def set_plus_def add_ac)

 lemma set_plus_mono5: "a:C ==> B <= D ==> a +o B <= C + D"

   apply (subgoal_tac "a +o B <= a +o D")

    apply (erule order_trans)

    apply (erule set_plus_mono3)

   apply (erule set_plus_mono)

   done

 lemma set_plus_mono_b: "C <= D ==> x : a +o C

     ==> x : a +o D"

   apply (frule set_plus_mono)

   apply auto

   done

 lemma set_plus_mono2_b: "C <= D ==> E <= F ==> x : C + E ==>

     x : D + F"

   apply (frule set_plus_mono2)

    prefer 2

    apply force

   apply assumption

   done

 lemma set_plus_mono3_b: "a : C ==> x : a +o D ==> x : C + D"

   apply (frule set_plus_mono3)

   apply auto

   done

 lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C ==>

     x : a +o D ==> x : D + C"

   apply (frule set_plus_mono4)

   apply auto

   done

 lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C"

   by (auto simp add: elt_set_plus_def)

 lemma set_zero_plus2: "(0::'a::comm_monoid_add) : A ==> B <= A + B"

   apply (auto simp add: set_plus_def)

   apply (rule_tac x = 0 in bexI)

    apply (rule_tac x = x in bexI)

     apply (auto simp add: add_ac)

   done

 lemma set_plus_imp_minus: "(a::'a::ab_group_add) : b +o C ==> (a - b) : C"

   by (auto simp add: elt_set_plus_def add_ac diff_minus)

 lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C ==> a : b +o C"

   apply (auto simp add: elt_set_plus_def add_ac diff_minus)

   apply (subgoal_tac "a = (a + - b) + b")

    apply (rule bexI, assumption, assumption)

   apply (auto simp add: add_ac)

   done

 lemma set_minus_plus: "((a::'a::ab_group_add) - b : C) = (a : b +o C)"

   by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus,

     assumption)

 lemma set_times_intro [intro]: "a : C ==> b : D ==> a * b : C * D"

   by (auto simp add: set_times_def)

 lemma set_times_elim:

   assumes "x \<in> A * B"

   obtains a b where "x = a * b" and "a \<in> A" and "b \<in> B"

   using assms unfolding set_times_def by fast

 lemma set_times_intro2 [intro!]: "b : C ==> a * b : a *o C"

   by (auto simp add: elt_set_times_def)

 lemma set_times_rearrange: "((a::'a::comm_monoid_mult) *o C) *

     (b *o D) = (a * b) *o (C * D)"

   apply (auto simp add: elt_set_times_def set_times_def)

    apply (rule_tac x = "ba * bb" in exI)

    apply (auto simp add: mult_ac)

   apply (rule_tac x = "aa * a" in exI)

   apply (auto simp add: mult_ac)

   done

 lemma set_times_rearrange2: "(a::'a::semigroup_mult) *o (b *o C) =

     (a * b) *o C"

   by (auto simp add: elt_set_times_def mult_assoc)

 lemma set_times_rearrange3: "((a::'a::semigroup_mult) *o B) * C =

     a *o (B * C)"

   apply (auto simp add: elt_set_times_def set_times_def)

    apply (blast intro: mult_ac)

   apply (rule_tac x = "a * aa" in exI)

   apply (rule conjI)

    apply (rule_tac x = "aa" in bexI)

     apply auto

   apply (rule_tac x = "ba" in bexI)

    apply (auto simp add: mult_ac)

   done

 theorem set_times_rearrange4: "C * ((a::'a::comm_monoid_mult) *o D) =

     a *o (C * D)"

   apply (auto simp add: elt_set_times_def set_times_def

     mult_ac)

    apply (rule_tac x = "aa * ba" in exI)

    apply (auto simp add: mult_ac)

   done

 theorems set_times_rearranges = set_times_rearrange set_times_rearrange2

   set_times_rearrange3 set_times_rearrange4

 lemma set_times_mono [intro]: "C <= D ==> a *o C <= a *o D"

   by (auto simp add: elt_set_times_def)

 lemma set_times_mono2 [intro]: "(C::('a::times) set) <= D ==> E <= F ==>

     C * E <= D * F"

   by (auto simp add: set_times_def)

 lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C * D"

   by (auto simp add: elt_set_times_def set_times_def)

 lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C ==>

     a *o D <= D * C"

   by (auto simp add: elt_set_times_def set_times_def mult_ac)

 lemma set_times_mono5: "a:C ==> B <= D ==> a *o B <= C * D"

   apply (subgoal_tac "a *o B <= a *o D")

    apply (erule order_trans)

    apply (erule set_times_mono3)

   apply (erule set_times_mono)

   done

 lemma set_times_mono_b: "C <= D ==> x : a *o C

     ==> x : a *o D"

   apply (frule set_times_mono)

   apply auto

   done

 lemma set_times_mono2_b: "C <= D ==> E <= F ==> x : C * E ==>

     x : D * F"

   apply (frule set_times_mono2)

    prefer 2

    apply force

   apply assumption

   done

 lemma set_times_mono3_b: "a : C ==> x : a *o D ==> x : C * D"

   apply (frule set_times_mono3)

   apply auto

   done

 lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) : C ==>

     x : a *o D ==> x : D * C"

   apply (frule set_times_mono4)

   apply auto

   done

 lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C"

   by (auto simp add: elt_set_times_def)

 lemma set_times_plus_distrib: "(a::'a::semiring) *o (b +o C)=

     (a * b) +o (a *o C)"

   by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs)

 lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B + C) =

     (a *o B) + (a *o C)"

   apply (auto simp add: set_plus_def elt_set_times_def ring_distribs)

    apply blast

   apply (rule_tac x = "b + bb" in exI)

   apply (auto simp add: ring_distribs)

   done

 lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) * D <=

     a *o D + C * D"

   apply (auto simp add:

     elt_set_plus_def elt_set_times_def set_times_def

     set_plus_def ring_distribs)

   apply auto

   done

 theorems set_times_plus_distribs =

   set_times_plus_distrib

   set_times_plus_distrib2

 lemma set_neg_intro: "(a::'a::ring_1) : (- 1) *o C ==>

     - a : C"

   by (auto simp add: elt_set_times_def)

 lemma set_neg_intro2: "(a::'a::ring_1) : C ==>

     - a : (- 1) *o C"

   by (auto simp add: elt_set_times_def)

 lemma set_plus_image: "S + T = (\<lambda>(x, y). x + y) ` (S \<times> T)"

   unfolding set_plus_def by (fastforce simp: image_iff)

 lemma set_times_image: "S * T = (\<lambda>(x, y). x * y) ` (S \<times> T)"

   unfolding set_times_def by (fastforce simp: image_iff)

 lemma finite_set_plus:

   assumes "finite s" and "finite t" shows "finite (s + t)"

   using assms unfolding set_plus_image by simp

 lemma finite_set_times:

   assumes "finite s" and "finite t" shows "finite (s * t)"

   using assms unfolding set_times_image by simp

 lemma set_setsum_alt:

   assumes fin: "finite I"

   shows "setsum S I = {setsum s I |s. \<forall>i\<in>I. s i \<in> S i}"

     (is "_ = ?setsum I")

 using fin proof induct

   case (insert x F)

   have "setsum S (insert x F) = S x + ?setsum F"

     using insert.hyps by auto

   also have "...= {s x + setsum s F |s. \<forall> i\<in>insert x F. s i \<in> S i}"

     unfolding set_plus_def

   proof safe

     fix y s assume "y \<in> S x" "\<forall>i\<in>F. s i \<in> S i"

     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)"

       using insert.hyps

       by (intro exI[of _ "\<lambda>i. if i \<in> F then s i else y"]) (auto simp add: set_plus_def)

   qed auto

   finally show ?case

     using insert.hyps by auto

 qed auto

 lemma setsum_set_cond_linear:

   fixes f :: "('a::comm_monoid_add) set \<Rightarrow> ('b::comm_monoid_add) set"

   assumes [intro!]: "\<And>A B. P A  \<Longrightarrow> P B  \<Longrightarrow> P (A + B)" "P {0}"

     and f: "\<And>A B. P A  \<Longrightarrow> P B \<Longrightarrow> f (A + B) = f A + f B" "f {0} = {0}"

   assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)"

   shows "f (setsum S I) = setsum (f \<circ> S) I"

 proof cases

   assume "finite I" from this all show ?thesis

   proof induct

     case (insert x F)

     from `finite F` `\<And>i. i \<in> insert x F \<Longrightarrow> P (S i)` have "P (setsum S F)"

       by induct auto

     with insert show ?case

       by (simp, subst f) auto

   qed (auto intro!: f)

 qed (auto intro!: f)

 lemma setsum_set_linear:

   fixes f :: "('a::comm_monoid_add) set => ('b::comm_monoid_add) set"

   assumes "\<And>A B. f(A) + f(B) = f(A + B)" "f {0} = {0}"

   shows "f (setsum S I) = setsum (f \<circ> S) I"

   using setsum_set_cond_linear[of "\<lambda>x. True" f I S] assms by auto

 lemma set_times_Un_distrib:

   "A * (B \<union> C) = A * B \<union> A * C"

   "(A \<union> B) * C = A * C \<union> B * C"

 by (auto simp: set_times_def)

 lemma set_times_UNION_distrib:

   "A * UNION I M = UNION I (%i. A * M i)"

   "UNION I M * A = UNION I (%i. M i * A)"

 by (auto simp: set_times_def)

 end