src/HOL/ex/Set_Algebras.thy

+(*  Title:      HOL/ex/Set_Algebras.thy
+    Author:     Jeremy Avigad and Kevin Donnelly; Florian Haftmann, TUM
+*)
+
+header {* Algebraic operations on sets *}
+
+theory Set_Algebras
+imports Main Interpretation_with_Defs
+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}.
+*}
+
+definition set_plus :: "'a::plus set \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "\<oplus>" 65) where
+  "A \<oplus> B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a + b}"
+
+definition set_times :: "'a::times set \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "\<otimes>" 70) where
+  "A \<otimes> B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a * b}"
+
+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"
+
+interpretation set_add!: semigroup "set_plus :: 'a::semigroup_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" proof
+qed (force simp add: set_plus_def add.assoc)
+
+interpretation set_add!: abel_semigroup "set_plus :: 'a::ab_semigroup_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" proof
+qed (force simp add: set_plus_def add.commute)
+
+interpretation set_add!: monoid "set_plus :: 'a::monoid_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}" proof
+qed (simp_all add: set_plus_def)
+
+interpretation set_add!: comm_monoid "set_plus :: 'a::comm_monoid_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}" proof
+qed (simp add: set_plus_def)
+
+interpretation set_add!: monoid_add "set_plus :: 'a::monoid_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}"
+  defines listsum_set is set_add.listsum
+proof
+qed (simp_all add: set_add.assoc)
+
+interpretation set_add!: comm_monoid_add "set_plus :: 'a::comm_monoid_add set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}"
+  defines setsum_set is set_add.setsum
+  where "monoid_add.listsum set_plus {0::'a} = listsum_set"
+proof -
+  show "class.comm_monoid_add (set_plus :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set) {0}" proof
+  qed (simp_all add: set_add.commute)
+  then interpret set_add!: comm_monoid_add "set_plus :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{0}" .
+  show "monoid_add.listsum set_plus {0::'a} = listsum_set"
+    by (simp only: listsum_set_def)
+qed
+
+interpretation set_mult!: semigroup "set_times :: 'a::semigroup_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set" proof
+qed (force simp add: set_times_def mult.assoc)
+
+interpretation set_mult!: abel_semigroup "set_times :: 'a::ab_semigroup_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set" proof
+qed (force simp add: set_times_def mult.commute)
+
+interpretation set_mult!: monoid "set_times :: 'a::monoid_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{1}" proof
+qed (simp_all add: set_times_def)
+
+interpretation set_mult!: comm_monoid "set_times :: 'a::comm_monoid_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{1}" proof
+qed (simp add: set_times_def)
+
+interpretation set_mult!: monoid_mult "{1}" "set_times :: 'a::monoid_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set"
+  defines power_set is set_mult.power
+proof
+qed (simp_all add: set_mult.assoc)
+
+interpretation set_mult!: comm_monoid_mult "set_times :: 'a::comm_monoid_mult set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{1}"
+  defines setprod_set is set_mult.setprod
+  where "power.power {1} set_times = power_set"
+proof -
+  show "class.comm_monoid_mult (set_times :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set) {1}" proof
+  qed (simp_all add: set_mult.commute)
+  then interpret set_mult!: comm_monoid_mult "set_times :: 'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" "{1}" .
+  show "power.power {1} set_times = power_set"
+    by (simp add: power_set_def)
+qed
+
+lemma set_plus_intro [intro]: "a : C ==> b : D ==> a + b : C \<oplus> D"
+  by (auto simp add: set_plus_def)
+
+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) \<oplus>
+    (b +o D) = (a + b) +o (C \<oplus> 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) \<oplus> C =
+    a +o (B \<oplus> 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 \<oplus> ((a::'a::comm_monoid_add) +o D) =
+    a +o (C \<oplus> D)"
+  apply (auto intro!: subsetI 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 \<oplus> E <= D \<oplus> F"
+  by (auto simp add: set_plus_def)
+
+lemma set_plus_mono3 [intro]: "a : C ==> a +o D <= C \<oplus> 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 \<oplus> 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 \<oplus> 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 \<oplus> E ==>
+    x : D \<oplus> 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 \<oplus> 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 \<oplus> 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 \<oplus> B"
+  apply (auto intro!: subsetI 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 \<otimes> D"
+  by (auto simp add: set_times_def)
+
+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) \<otimes>
+    (b *o D) = (a * b) *o (C \<otimes> 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) \<otimes> C =
+    a *o (B \<otimes> 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 \<otimes> ((a::'a::comm_monoid_mult) *o D) =
+    a *o (C \<otimes> D)"
+  apply (auto intro!: subsetI 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 \<otimes> E <= D \<otimes> F"
+  by (auto simp add: set_times_def)
+
+lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C \<otimes> 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 \<otimes> 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 \<otimes> 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 \<otimes> E ==>
+    x : D \<otimes> 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 \<otimes> 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 \<otimes> 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 \<oplus> C) =
+    (a *o B) \<oplus> (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) \<otimes> D <=
+    a *o D \<oplus> C \<otimes> D"
+  apply (auto intro!: subsetI 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:
+  fixes S T :: "'n::semigroup_add set" shows "S \<oplus> T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
+  unfolding set_plus_def by (fastsimp simp: image_iff)
+
+lemma set_setsum_alt:
+  assumes fin: "finite I"
+  shows "setsum_set 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_set S (insert x F) = S x \<oplus> ?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 \<oplus> B)" "P {0}"
+    and f: "\<And>A B. P A  \<Longrightarrow> P B \<Longrightarrow> f (A \<oplus> B) = f A \<oplus> f B" "f {0} = {0}"
+  assumes all: "\<And>i. i \<in> I \<Longrightarrow> P (S i)"
+  shows "f (setsum_set S I) = setsum_set (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_set 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) \<oplus> f(B) = f(A \<oplus> B)" "f {0} = {0}"
+  shows "f (setsum_set S I) = setsum_set (f \<circ> S) I"
+  using setsum_set_cond_linear[of "\<lambda>x. True" f I S] assms by auto
+
+end