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theory SetsAndFunctions(* Title: HOL/Library/SetsAndFunctions.thy ID: $Id: SetsAndFunctions.thy,v 1.13 2008/05/07 08:59:24 berghofe Exp $ Author: Jeremy Avigad and Kevin Donnelly *) header {* Operations on sets and functions *} theory SetsAndFunctions imports ATP_Linkup 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 *} definition set_plus :: "('a::plus) set => 'a set => 'a set" (infixl "⊕" 65) where "A ⊕ B == {c. EX a:A. EX b:B. c = a + b}" instantiation "fun" :: (type, plus) plus begin definition func_plus: "f + g == (%x. f x + g x)" instance .. end definition set_times :: "('a::times) set => 'a set => 'a set" (infixl "⊗" 70) where "A ⊗ B == {c. EX a:A. EX b:B. c = a * b}" instantiation "fun" :: (type, times) times begin definition func_times: "f * g == (%x. f x * g x)" instance .. end instantiation "fun" :: (type, zero) zero begin definition func_zero: "0::(('a::type) => ('b::zero)) == %x. 0" 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: fun_Compl_def func_plus func_zero) apply (simp add: fun_Compl_def func_plus fun_diff_def 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 fun_Compl_def fun_diff_def ext func_one func_zero ring_simps) apply (drule fun_cong) apply simp done interpretation set_semigroup_add: semigroup_add ["op ⊕ :: ('a::semigroup_add) set => 'a set => 'a set"] apply default apply (unfold set_plus_def) apply (force simp add: add_assoc) done interpretation set_semigroup_mult: semigroup_mult ["op ⊗ :: ('a::semigroup_mult) set => 'a set => 'a set"] apply default apply (unfold set_times_def) apply (force simp add: mult_assoc) done interpretation set_comm_monoid_add: comm_monoid_add ["{0}" "op ⊕ :: ('a::comm_monoid_add) set => 'a set => 'a set"] apply default apply (unfold set_plus_def) apply (force simp add: add_ac) apply force done interpretation set_comm_monoid_mult: comm_monoid_mult ["{1}" "op ⊗ :: ('a::comm_monoid_mult) set => 'a set => 'a set"] apply default apply (unfold set_times_def) apply (force simp add: mult_ac) 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_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) ⊕ (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 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 ⊕ 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 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 ⊗ 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) ⊗ (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 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 ⊗ 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 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) end
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)
theorem 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)
theorem 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
theorem 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