--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/SetsAndFunctions.thy Mon Jul 25 18:54:49 2005 +0200
@@ -0,0 +1,376 @@
+(* Title: SetsAndFunctions.thy
+ 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 BigO.thy
+*}
+
+subsection {* Basic definitions *}
+
+instance set :: (plus)plus
+by intro_classes
+
+instance fun :: (type,plus)plus
+by intro_classes
+
+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
+by intro_classes
+
+instance fun :: (type,times)times
+by intro_classes
+
+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
+by intro_classes
+
+defs (overloaded)
+ func_minus: "- f == (%x. - f x)"
+ func_diff: "f - g == %x. f x - g x"
+
+instance fun :: (type,zero)zero
+by intro_classes
+
+instance set :: (zero)zero
+by(intro_classes)
+
+defs (overloaded)
+ func_zero: "0::(('a::type) => ('b::zero)) == %x. 0"
+ set_zero: "0::('a::zero)set == {0}"
+
+instance fun :: (type,one)one
+by intro_classes
+
+instance set :: (one)one
+by intro_classes
+
+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