# Theory SetsAndFunctions

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theory SetsAndFunctions
imports Main
begin

    (*  Title:      HOL/Library/SetsAndFunctions.thy
        ID:         $Id: SetsAndFunctions.thy,v 1.3 2005/08/28 14:04:45 wenzelm Exp $
        Author:     Jeremy Avigad and Kevin Donnelly
    *)

    header {* Operations on sets and functions *}

    theory SetsAndFunctions
    imports Main
    begin

    text {*
    This library lifts operations like addition and muliplication to sets and
    functions of appropriate types. It was designed to support asymptotic
    calculations. See the comments at the top of theory @{text BigO}.
    *}

    subsection {* Basic definitions *}

    instance set :: (plus) plus ..
    instance fun :: (type, plus) plus ..

    defs (overloaded)
      func_plus: "f + g == (%x. f x + g x)"
      set_plus: "A + B == {c. EX a:A. EX b:B. c = a + b}"

    instance set :: (times) times ..
    instance fun :: (type, times) times ..

    defs (overloaded)
      func_times: "f * g == (%x. f x * g x)"
      set_times:"A * B == {c. EX a:A. EX b:B. c = a * b}"

    instance fun :: (type, minus) minus ..

    defs (overloaded)
      func_minus: "- f == (%x. - f x)"
      func_diff: "f - g == %x. f x - g x"

    instance fun :: (type, zero) zero ..
    instance set :: (zero) zero ..

    defs (overloaded)
      func_zero: "0::(('a::type) => ('b::zero)) == %x. 0"
      set_zero: "0::('a::zero)set == {0}"

    instance fun :: (type, one) one ..
    instance set :: (one) one ..

    defs (overloaded)
      func_one: "1::(('a::type) => ('b::one)) == %x. 1"
      set_one: "1::('a::one)set == {1}"

    constdefs
      elt_set_plus :: "'a::plus => 'a set => 'a set"    (infixl "+o" 70)
      "a +o B == {c. EX b:B. c = a + b}"

      elt_set_times :: "'a::times => 'a set => 'a set"  (infixl "*o" 80)
      "a *o B == {c. EX b:B. c = a * b}"

    syntax
      "elt_set_eq" :: "'a => 'a set => bool"      (infix "=o" 50)

    translations
      "x =o A" => "x : A"

    instance fun :: (type,semigroup_add)semigroup_add
      apply intro_classes
      apply (auto simp add: func_plus add_assoc)
    done

    instance fun :: (type,comm_monoid_add)comm_monoid_add
      apply intro_classes
      apply (auto simp add: func_zero func_plus add_ac)
    done

    instance fun :: (type,ab_group_add)ab_group_add
      apply intro_classes
      apply (simp add: func_minus func_plus func_zero)
      apply (simp add: func_minus func_plus func_diff diff_minus)
    done

    instance fun :: (type,semigroup_mult)semigroup_mult
      apply intro_classes
      apply (auto simp add: func_times mult_assoc)
    done

    instance fun :: (type,comm_monoid_mult)comm_monoid_mult
      apply intro_classes
      apply (auto simp add: func_one func_times mult_ac)
    done

    instance fun :: (type,comm_ring_1)comm_ring_1
      apply intro_classes
      apply (auto simp add: func_plus func_times func_minus func_diff ext
        func_one func_zero ring_eq_simps)
      apply (drule fun_cong)
      apply simp
    done

    instance set :: (semigroup_add)semigroup_add
      apply intro_classes
      apply (unfold set_plus)
      apply (force simp add: add_assoc)
    done

    instance set :: (semigroup_mult)semigroup_mult
      apply intro_classes
      apply (unfold set_times)
      apply (force simp add: mult_assoc)
    done

    instance set :: (comm_monoid_add)comm_monoid_add
      apply intro_classes
      apply (unfold set_plus)
      apply (force simp add: add_ac)
      apply (unfold set_zero)
      apply force
    done

    instance set :: (comm_monoid_mult)comm_monoid_mult
      apply intro_classes
      apply (unfold set_times)
      apply (force simp add: mult_ac)
      apply (unfold set_one)
      apply force
    done

    subsection {* Basic properties *}

    lemma set_plus_intro [intro]: "a : C ==> b : D ==> a + b : C + D"
    by (auto simp add: set_plus)

    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 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)
      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 intro!: subsetI simp add: elt_set_plus_def set_plus 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)

    lemma set_plus_mono3 [intro]: "a : C ==> a +o D <= C + D"
    by (auto simp add: elt_set_plus_def set_plus)

    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 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 intro!: subsetI simp add: set_plus)
      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)

    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)
      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)
      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 intro!: subsetI simp add: elt_set_times_def set_times
        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)

    lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C * D"
    by (auto simp add: elt_set_times_def set_times)

    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 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_distrib)

    lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B + C) =
      (a *o B) + (a *o C)"
      apply (auto simp add: set_plus elt_set_times_def ring_distrib)
      apply blast
      apply (rule_tac x = "b + bb" in exI)
      apply (auto simp add: ring_distrib)
    done

    lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) * D <=
        a *o D + C * D"
      apply (auto intro!: subsetI simp add:
        elt_set_plus_def elt_set_times_def set_times
        set_plus ring_distrib)
      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)

    end

### Basic definitions

### Basic properties

lemma set_plus_intro:

      [| a ∈ C; b ∈ D |] ==> a + b ∈ C + D

lemma set_plus_intro2:

      b ∈ C ==> a + b ∈ a +o C

lemma set_plus_rearrange:

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

lemma set_plus_rearrange2:

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

lemma set_plus_rearrange3:

      a +o B + C = a +o (B + C)

theorem set_plus_rearrange4:

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

theorems set_plus_rearranges:

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

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

      a +o B + C = a +o (B + C)

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

theorems set_plus_rearranges:

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

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

      a +o B + C = a +o (B + C)

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

lemma set_plus_mono:

      C ⊆ D ==> a +o C ⊆ a +o D

lemma set_plus_mono2:

      [| C ⊆ D; E ⊆ F |] ==> C + E ⊆ D + F

lemma set_plus_mono3:

      a ∈ C ==> a +o D ⊆ C + D

lemma set_plus_mono4:

      a ∈ C ==> a +o D ⊆ D + C

lemma set_plus_mono5:

      [| a ∈ C; B ⊆ D |] ==> a +o B ⊆ C + D

lemma set_plus_mono_b:

      [| C ⊆ D; x ∈ a +o C |] ==> x ∈ a +o D

lemma set_plus_mono2_b:

      [| C ⊆ D; E ⊆ F; x ∈ C + E |] ==> x ∈ D + F

lemma set_plus_mono3_b:

      [| a ∈ C; x ∈ a +o D |] ==> x ∈ C + D

lemma set_plus_mono4_b:

      [| a ∈ C; x ∈ a +o D |] ==> x ∈ D + C

lemma set_zero_plus:

      (0::'a) +o C = C

lemma set_zero_plus2:

      (0::'a) ∈ A ==> B ⊆ A + B

lemma set_plus_imp_minus:

      a ∈ b +o C ==> a - b ∈ C

lemma set_minus_imp_plus:

      a - b ∈ C ==> a ∈ b +o C

lemma set_minus_plus:

      (a - b ∈ C) = (a ∈ b +o C)

lemma set_times_intro:

      [| a ∈ C; b ∈ D |] ==> a * b ∈ C * D

lemma set_times_intro2:

      b ∈ C ==> a * b ∈ a *o C

lemma set_times_rearrange:

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

lemma set_times_rearrange2:

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

lemma set_times_rearrange3:

      a *o B * C = a *o (B * C)

theorem set_times_rearrange4:

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

theorems set_times_rearranges:

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

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

      a *o B * C = a *o (B * C)

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

theorems set_times_rearranges:

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

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

      a *o B * C = a *o (B * C)

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

lemma set_times_mono:

      C ⊆ D ==> a *o C ⊆ a *o D

lemma set_times_mono2:

      [| C ⊆ D; E ⊆ F |] ==> C * E ⊆ D * F

lemma set_times_mono3:

      a ∈ C ==> a *o D ⊆ C * D

lemma set_times_mono4:

      a ∈ C ==> a *o D ⊆ D * C

lemma set_times_mono5:

      [| a ∈ C; B ⊆ D |] ==> a *o B ⊆ C * D

lemma set_times_mono_b:

      [| C ⊆ D; x ∈ a *o C |] ==> x ∈ a *o D

lemma set_times_mono2_b:

      [| C ⊆ D; E ⊆ F; x ∈ C * E |] ==> x ∈ D * F

lemma set_times_mono3_b:

      [| a ∈ C; x ∈ a *o D |] ==> x ∈ C * D

lemma set_times_mono4_b:

      [| a ∈ C; x ∈ a *o D |] ==> x ∈ D * C

lemma set_one_times:

      (1::'a) *o C = C

lemma set_times_plus_distrib:

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

lemma set_times_plus_distrib2:

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

lemma set_times_plus_distrib3:

      a +o C * D ⊆ a *o D + C * D

theorems set_times_plus_distribs:

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

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

theorems set_times_plus_distribs:

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

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

lemma set_neg_intro:

      a ∈ - (1::'a) *o C ==> - a ∈ C

lemma set_neg_intro2:

      a ∈ C ==> - a ∈ - (1::'a) *o C