(*  Title:      HOL/Library/SetsAndFunctions.thy

    ID:         $Id$

    Author:     Jeremy Avigad and Kevin Donnelly

*)

header {* Operations on sets and functions *}

theory SetsAndFunctions

imports PreList

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 *}

instantiation set :: (plus) plus

begin

definition

  set_plus: "A + B == {c. EX a:A. EX b:B. c = a + b}"

instance ..

end

instantiation "fun" :: (type, plus) plus

begin

definition

  func_plus: "f + g == (%x. f x + g x)"

instance ..

end

instantiation set :: (times) times

begin

definition

  set_times:"A * B == {c. EX a:A. EX b:B. c = a * b}"

instance ..

end

instantiation "fun" :: (type, times) times

begin

definition

  func_times: "f * g == (%x. f x * g x)"

instance ..

end

instantiation "fun" :: (type, minus) minus

begin

definition

  func_minus: "- f == (%x. - f x)"

definition

  func_diff: "f - g == %x. f x - g x"

instance ..

end

instantiation set :: (zero) zero

begin

definition

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

instance ..

end

instantiation "fun" :: (type, zero) zero

begin

definition

  func_zero: "0::(('a::type) => ('b::zero)) == %x. 0"

instance ..

end

instantiation set :: (one) one

begin

definition

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

instance ..

end

instantiation "fun" :: (type, one) one

begin

definition

  func_one: "1::(('a::type) => ('b::one)) == %x. 1"

instance ..

end

definition

  elt_set_plus :: "'a::plus => 'a set => 'a set"  (infixl "+o" 70) where

  "a +o B = {c. EX b:B. c = a + b}"

definition

  elt_set_times :: "'a::times => 'a set => 'a set"  (infixl "*o" 80) where

  "a *o B = {c. EX b:B. c = a * b}"

abbreviation (input)

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

  "x =o A == x : A"

instance "fun" :: (type,semigroup_add)semigroup_add

  by default (auto simp add: func_plus add_assoc)

instance "fun" :: (type,comm_monoid_add)comm_monoid_add

  by default (auto simp add: func_zero func_plus add_ac)

instance "fun" :: (type,ab_group_add)ab_group_add

  apply default

   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 default

  apply (auto simp add: func_times mult_assoc)

  done

instance "fun" :: (type,comm_monoid_mult)comm_monoid_mult

  apply default

   apply (auto simp add: func_one func_times mult_ac)

  done

instance "fun" :: (type,comm_ring_1)comm_ring_1

  apply default

   apply (auto simp add: func_plus func_times func_minus func_diff ext

     func_one func_zero ring_simps)

  apply (drule fun_cong)

  apply simp

  done

instance set :: (semigroup_add)semigroup_add

  apply default

  apply (unfold set_plus)

  apply (force simp add: add_assoc)

  done

instance set :: (semigroup_mult)semigroup_mult

  apply default

  apply (unfold set_times)

  apply (force simp add: mult_assoc)

  done

instance set :: (comm_monoid_add)comm_monoid_add

  apply default

   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 default

   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_distribs)

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_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 intro!: subsetI simp add:

    elt_set_plus_def elt_set_times_def set_times

    set_plus 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)

end