(* Title: HOL/Finite_Set.thy
ID: $Id$
Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
Additions by Jeremy Avigad in Feb 2004
*)
header {* Finite sets *}
theory Finite_Set
imports Divides Power Inductive
begin
subsection {* Definition and basic properties *}
consts Finites :: "'a set set"
syntax
finite :: "'a set => bool"
translations
"finite A" == "A : Finites"
inductive Finites
intros
emptyI [simp, intro!]: "{} : Finites"
insertI [simp, intro!]: "A : Finites ==> insert a A : Finites"
axclass finite \<subseteq> type
finite: "finite UNIV"
lemma ex_new_if_finite: -- "does not depend on def of finite at all"
assumes "\<not> finite (UNIV :: 'a set)" and "finite A"
shows "\<exists>a::'a. a \<notin> A"
proof -
from prems have "A \<noteq> UNIV" by blast
thus ?thesis by blast
qed
lemma finite_induct [case_names empty insert, induct set: Finites]:
"finite F ==>
P {} ==> (!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F"
-- {* Discharging @{text "x \<notin> F"} entails extra work. *}
proof -
assume "P {}" and
insert: "!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)"
assume "finite F"
thus "P F"
proof induct
show "P {}" .
fix x F assume F: "finite F" and P: "P F"
show "P (insert x F)"
proof cases
assume "x \<in> F"
hence "insert x F = F" by (rule insert_absorb)
with P show ?thesis by (simp only:)
next
assume "x \<notin> F"
from F this P show ?thesis by (rule insert)
qed
qed
qed
lemma finite_ne_induct[case_names singleton insert, consumes 2]:
assumes fin: "finite F" shows "F \<noteq> {} \<Longrightarrow>
\<lbrakk> \<And>x. P{x};
\<And>x F. \<lbrakk> finite F; F \<noteq> {}; x \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert x F) \<rbrakk>
\<Longrightarrow> P F"
using fin
proof induct
case empty thus ?case by simp
next
case (insert x F)
show ?case
proof cases
assume "F = {}" thus ?thesis using insert(4) by simp
next
assume "F \<noteq> {}" thus ?thesis using insert by blast
qed
qed
lemma finite_subset_induct [consumes 2, case_names empty insert]:
"finite F ==> F \<subseteq> A ==>
P {} ==> (!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)) ==>
P F"
proof -
assume "P {}" and insert:
"!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
assume "finite F"
thus "F \<subseteq> A ==> P F"
proof induct
show "P {}" .
fix x F assume "finite F" and "x \<notin> F"
and P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A"
show "P (insert x F)"
proof (rule insert)
from i show "x \<in> A" by blast
from i have "F \<subseteq> A" by blast
with P show "P F" .
qed
qed
qed
text{* Finite sets are the images of initial segments of natural numbers: *}
lemma finite_imp_nat_seg_image:
assumes fin: "finite A" shows "\<exists> (n::nat) f. A = f ` {i::nat. i<n}"
using fin
proof induct
case empty
show ?case
proof show "\<exists>f. {} = f ` {i::nat. i < 0}" by(simp add:image_def) qed
next
case (insert a A)
from insert.hyps obtain n f where "A = f ` {i::nat. i < n}" by blast
hence "insert a A = (%i. if i<n then f i else a) ` {i. i < n+1}"
by (auto simp add:image_def Ball_def)
thus ?case by blast
qed
lemma nat_seg_image_imp_finite:
"!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A"
proof (induct n)
case 0 thus ?case by simp
next
case (Suc n)
let ?B = "f ` {i. i < n}"
have finB: "finite ?B" by(rule Suc.hyps[OF refl])
show ?case
proof cases
assume "\<exists>k<n. f n = f k"
hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq)
thus ?thesis using finB by simp
next
assume "\<not>(\<exists> k<n. f n = f k)"
hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq)
thus ?thesis using finB by simp
qed
qed
lemma finite_conv_nat_seg_image:
"finite A = (\<exists> (n::nat) f. A = f ` {i::nat. i<n})"
by(blast intro: finite_imp_nat_seg_image nat_seg_image_imp_finite)
subsubsection{* Finiteness and set theoretic constructions *}
lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
-- {* The union of two finite sets is finite. *}
by (induct set: Finites) simp_all
lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
-- {* Every subset of a finite set is finite. *}
proof -
assume "finite B"
thus "!!A. A \<subseteq> B ==> finite A"
proof induct
case empty
thus ?case by simp
next
case (insert x F A)
have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" .
show "finite A"
proof cases
assume x: "x \<in> A"
with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff)
with r have "finite (A - {x})" .
hence "finite (insert x (A - {x}))" ..
also have "insert x (A - {x}) = A" by (rule insert_Diff)
finally show ?thesis .
next
show "A \<subseteq> F ==> ?thesis" .
assume "x \<notin> A"
with A show "A \<subseteq> F" by (simp add: subset_insert_iff)
qed
qed
qed
lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
-- {* The converse obviously fails. *}
by (blast intro: finite_subset)
lemma finite_insert [simp]: "finite (insert a A) = finite A"
apply (subst insert_is_Un)
apply (simp only: finite_Un, blast)
done
lemma finite_Union[simp, intro]:
"\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"
by (induct rule:finite_induct) simp_all
lemma finite_empty_induct:
"finite A ==>
P A ==> (!!a A. finite A ==> a:A ==> P A ==> P (A - {a})) ==> P {}"
proof -
assume "finite A"
and "P A" and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
have "P (A - A)"
proof -
fix c b :: "'a set"
presume c: "finite c" and b: "finite b"
and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})"
from c show "c \<subseteq> b ==> P (b - c)"
proof induct
case empty
from P1 show ?case by simp
next
case (insert x F)
have "P (b - F - {x})"
proof (rule P2)
from _ b show "finite (b - F)" by (rule finite_subset) blast
from insert show "x \<in> b - F" by simp
from insert show "P (b - F)" by simp
qed
also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric])
finally show ?case .
qed
next
show "A \<subseteq> A" ..
qed
thus "P {}" by simp
qed
lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)"
by (rule Diff_subset [THEN finite_subset])
lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"
apply (subst Diff_insert)
apply (case_tac "a : A - B")
apply (rule finite_insert [symmetric, THEN trans])
apply (subst insert_Diff, simp_all)
done
text {* Image and Inverse Image over Finite Sets *}
lemma finite_imageI[simp]: "finite F ==> finite (h ` F)"
-- {* The image of a finite set is finite. *}
by (induct set: Finites) simp_all
lemma finite_surj: "finite A ==> B <= f ` A ==> finite B"
apply (frule finite_imageI)
apply (erule finite_subset, assumption)
done
lemma finite_range_imageI:
"finite (range g) ==> finite (range (%x. f (g x)))"
apply (drule finite_imageI, simp)
done
lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A"
proof -
have aux: "!!A. finite (A - {}) = finite A" by simp
fix B :: "'a set"
assume "finite B"
thus "!!A. f`A = B ==> inj_on f A ==> finite A"
apply induct
apply simp
apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})")
apply clarify
apply (simp (no_asm_use) add: inj_on_def)
apply (blast dest!: aux [THEN iffD1], atomize)
apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl)
apply (frule subsetD [OF equalityD2 insertI1], clarify)
apply (rule_tac x = xa in bexI)
apply (simp_all add: inj_on_image_set_diff)
done
qed (rule refl)
lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}"
-- {* The inverse image of a singleton under an injective function
is included in a singleton. *}
apply (auto simp add: inj_on_def)
apply (blast intro: the_equality [symmetric])
done
lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)"
-- {* The inverse image of a finite set under an injective function
is finite. *}
apply (induct set: Finites, simp_all)
apply (subst vimage_insert)
apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton])
done
text {* The finite UNION of finite sets *}
lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)"
by (induct set: Finites) simp_all
text {*
Strengthen RHS to
@{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}?
We'd need to prove
@{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"}
by induction. *}
lemma finite_UN [simp]: "finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))"
by (blast intro: finite_UN_I finite_subset)
text {* Sigma of finite sets *}
lemma finite_SigmaI [simp]:
"finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)"
by (unfold Sigma_def) (blast intro!: finite_UN_I)
lemma finite_cartesian_product: "[| finite A; finite B |] ==>
finite (A <*> B)"
by (rule finite_SigmaI)
lemma finite_Prod_UNIV:
"finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)"
apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)")
apply (erule ssubst)
apply (erule finite_SigmaI, auto)
done
lemma finite_cartesian_productD1:
"[| finite (A <*> B); B \<noteq> {} |] ==> finite A"
apply (auto simp add: finite_conv_nat_seg_image)
apply (drule_tac x=n in spec)
apply (drule_tac x="fst o f" in spec)
apply (auto simp add: o_def)
prefer 2 apply (force dest!: equalityD2)
apply (drule equalityD1)
apply (rename_tac y x)
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)")
prefer 2 apply force
apply clarify
apply (rule_tac x=k in image_eqI, auto)
done
lemma finite_cartesian_productD2:
"[| finite (A <*> B); A \<noteq> {} |] ==> finite B"
apply (auto simp add: finite_conv_nat_seg_image)
apply (drule_tac x=n in spec)
apply (drule_tac x="snd o f" in spec)
apply (auto simp add: o_def)
prefer 2 apply (force dest!: equalityD2)
apply (drule equalityD1)
apply (rename_tac x y)
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)")
prefer 2 apply force
apply clarify
apply (rule_tac x=k in image_eqI, auto)
done
instance unit :: finite
proof
have "finite {()}" by simp
also have "{()} = UNIV" by auto
finally show "finite (UNIV :: unit set)" .
qed
instance * :: (finite, finite) finite
proof
show "finite (UNIV :: ('a \<times> 'b) set)"
proof (rule finite_Prod_UNIV)
show "finite (UNIV :: 'a set)" by (rule finite)
show "finite (UNIV :: 'b set)" by (rule finite)
qed
qed
text {* The powerset of a finite set *}
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A"
proof
assume "finite (Pow A)"
with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast
thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp
next
assume "finite A"
thus "finite (Pow A)"
by induct (simp_all add: finite_UnI finite_imageI Pow_insert)
qed
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
by(blast intro: finite_subset[OF subset_Pow_Union])
lemma finite_converse [iff]: "finite (r^-1) = finite r"
apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
apply simp
apply (rule iffI)
apply (erule finite_imageD [unfolded inj_on_def])
apply (simp split add: split_split)
apply (erule finite_imageI)
apply (simp add: converse_def image_def, auto)
apply (rule bexI)
prefer 2 apply assumption
apply simp
done
text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi
Ehmety) *}
lemma finite_Field: "finite r ==> finite (Field r)"
-- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
apply (induct set: Finites)
apply (auto simp add: Field_def Domain_insert Range_insert)
done
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
apply clarify
apply (erule trancl_induct)
apply (auto simp add: Field_def)
done
lemma finite_trancl: "finite (r^+) = finite r"
apply auto
prefer 2
apply (rule trancl_subset_Field2 [THEN finite_subset])
apply (rule finite_SigmaI)
prefer 3
apply (blast intro: r_into_trancl' finite_subset)
apply (auto simp add: finite_Field)
done
subsection {* A fold functional for finite sets *}
text {* The intended behaviour is
@{text "fold f g z {x\<^isub>1, ..., x\<^isub>n} = f (g x\<^isub>1) (\<dots> (f (g x\<^isub>n) z)\<dots>)"}
if @{text f} is associative-commutative. For an application of @{text fold}
se the definitions of sums and products over finite sets.
*}
consts
foldSet :: "('a => 'a => 'a) => ('b => 'a) => 'a => ('b set \<times> 'a) set"
inductive "foldSet f g z"
intros
emptyI [intro]: "({}, z) : foldSet f g z"
insertI [intro]: "\<lbrakk> x \<notin> A; (A, y) : foldSet f g z \<rbrakk>
\<Longrightarrow> (insert x A, f (g x) y) : foldSet f g z"
inductive_cases empty_foldSetE [elim!]: "({}, x) : foldSet f g z"
constdefs
fold :: "('a => 'a => 'a) => ('b => 'a) => 'a => 'b set => 'a"
"fold f g z A == THE x. (A, x) : foldSet f g z"
lemma Diff1_foldSet:
"(A - {x}, y) : foldSet f g z ==> x: A ==> (A, f (g x) y) : foldSet f g z"
by (erule insert_Diff [THEN subst], rule foldSet.intros, auto)
lemma foldSet_imp_finite: "(A, x) : foldSet f g z ==> finite A"
by (induct set: foldSet) auto
lemma finite_imp_foldSet: "finite A ==> EX x. (A, x) : foldSet f g z"
by (induct set: Finites) auto
subsubsection {* Commutative monoids *}
locale ACf =
fixes f :: "'a => 'a => 'a" (infixl "\<cdot>" 70)
assumes commute: "x \<cdot> y = y \<cdot> x"
and assoc: "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
locale ACe = ACf +
fixes e :: 'a
assumes ident [simp]: "x \<cdot> e = x"
locale ACIf = ACf +
assumes idem: "x \<cdot> x = x"
lemma (in ACf) left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
proof -
have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp only: commute)
also have "... = y \<cdot> (z \<cdot> x)" by (simp only: assoc)
also have "z \<cdot> x = x \<cdot> z" by (simp only: commute)
finally show ?thesis .
qed
lemmas (in ACf) AC = assoc commute left_commute
lemma (in ACe) left_ident [simp]: "e \<cdot> x = x"
proof -
have "x \<cdot> e = x" by (rule ident)
thus ?thesis by (subst commute)
qed
lemma (in ACIf) idem2: "x \<cdot> (x \<cdot> y) = x \<cdot> y"
proof -
have "x \<cdot> (x \<cdot> y) = (x \<cdot> x) \<cdot> y" by(simp add:assoc)
also have "\<dots> = x \<cdot> y" by(simp add:idem)
finally show ?thesis .
qed
lemmas (in ACIf) ACI = AC idem idem2
text{* Instantiation of locales: *}
lemma ACf_add: "ACf (op + :: 'a::comm_monoid_add \<Rightarrow> 'a \<Rightarrow> 'a)"
by(fastsimp intro: ACf.intro add_assoc add_commute)
lemma ACe_add: "ACe (op +) (0::'a::comm_monoid_add)"
by(fastsimp intro: ACe.intro ACe_axioms.intro ACf_add)
lemma ACf_mult: "ACf (op * :: 'a::comm_monoid_mult \<Rightarrow> 'a \<Rightarrow> 'a)"
by(fast intro: ACf.intro mult_assoc ab_semigroup_mult.mult_commute)
lemma ACe_mult: "ACe (op *) (1::'a::comm_monoid_mult)"
by(fastsimp intro: ACe.intro ACe_axioms.intro ACf_mult)
subsubsection{*From @{term foldSet} to @{term fold}*}
(* only used in the next lemma, but in there twice *)
lemma card_lemma: assumes A1: "A = insert b B" and notinB: "b \<notin> B" and
card: "A = h`{i. i<Suc n}" and new: "\<not>(EX k<n. h n = h k)"
shows "EX h. B = h`{i. i<n}" (is "EX h. ?P h")
proof
let ?h = "%i. if h i = b then h n else h i"
show "B = ?h`{i. i<n}" (is "_ = ?r")
proof
show "B \<subseteq> ?r"
proof
fix u assume "u \<in> B"
hence uinA: "u \<in> A" and unotb: "u \<noteq> b" using A1 notinB by blast+
then obtain i\<^isub>u where below: "i\<^isub>u < Suc n" and [simp]: "u = h i\<^isub>u"
using card by(auto simp:image_def)
show "u \<in> ?r"
proof cases
assume "i\<^isub>u < n"
thus ?thesis using unotb by(fastsimp)
next
assume "\<not> i\<^isub>u < n"
with below have [simp]: "i\<^isub>u = n" by arith
obtain i\<^isub>k where i\<^isub>k: "i\<^isub>k < Suc n" and [simp]: "b = h i\<^isub>k"
using A1 card by blast
have "i\<^isub>k < n"
proof (rule ccontr)
assume "\<not> i\<^isub>k < n"
hence "i\<^isub>k = n" using i\<^isub>k by arith
thus False using unotb by simp
qed
thus ?thesis by(auto simp add:image_def)
qed
qed
next
show "?r \<subseteq> B"
proof
fix u assume "u \<in> ?r"
then obtain i\<^isub>u where below: "i\<^isub>u < n" and
or: "b = h i\<^isub>u \<and> u = h n \<or> h i\<^isub>u \<noteq> b \<and> h i\<^isub>u = u"
by(auto simp:image_def)
from or show "u \<in> B"
proof
assume [simp]: "b = h i\<^isub>u \<and> u = h n"
have "u \<in> A" using card by auto
moreover have "u \<noteq> b" using new below by auto
ultimately show "u \<in> B" using A1 by blast
next
assume "h i\<^isub>u \<noteq> b \<and> h i\<^isub>u = u"
moreover hence "u \<in> A" using card below by auto
ultimately show "u \<in> B" using A1 by blast
qed
qed
qed
qed
lemma (in ACf) foldSet_determ_aux:
"!!A x x' h. \<lbrakk> A = h`{i::nat. i<n}; (A,x) : foldSet f g z; (A,x') : foldSet f g z \<rbrakk>
\<Longrightarrow> x' = x"
proof (induct n)
case 0 thus ?case by auto
next
case (Suc n)
have IH: "!!A x x' h. \<lbrakk>A = h`{i::nat. i<n}; (A,x) \<in> foldSet f g z; (A,x') \<in> foldSet f g z\<rbrakk>
\<Longrightarrow> x' = x" and card: "A = h`{i. i<Suc n}"
and Afoldx: "(A, x) \<in> foldSet f g z" and Afoldy: "(A,x') \<in> foldSet f g z" .
show ?case
proof cases
assume "EX k<n. h n = h k"
--{*@{term h} is not injective, so the cardinality has not increased*}
hence card': "A = h ` {i. i < n}"
using card by (auto simp:image_def less_Suc_eq)
show ?thesis by(rule IH[OF card' Afoldx Afoldy])
next
assume new: "\<not>(EX k<n. h n = h k)"
show ?thesis
proof (rule foldSet.cases[OF Afoldx])
assume "(A, x) = ({}, z)" --{*fold of a singleton set*}
thus "x' = x" using Afoldy by (auto)
next
fix B b y
assume eq1: "(A, x) = (insert b B, g b \<cdot> y)"
and y: "(B,y) \<in> foldSet f g z" and notinB: "b \<notin> B"
hence A1: "A = insert b B" and x: "x = g b \<cdot> y" by auto
show ?thesis
proof (rule foldSet.cases[OF Afoldy])
assume "(A,x') = ({}, z)"
thus ?thesis using A1 by auto
next
fix C c r
assume eq2: "(A,x') = (insert c C, g c \<cdot> r)"
and r: "(C,r) \<in> foldSet f g z" and notinC: "c \<notin> C"
hence A2: "A = insert c C" and x': "x' = g c \<cdot> r" by auto
obtain hB where lessB: "B = hB ` {i. i<n}"
using card_lemma[OF A1 notinB card new] by auto
obtain hC where lessC: "C = hC ` {i. i<n}"
using card_lemma[OF A2 notinC card new] by auto
show ?thesis
proof cases
assume "b = c"
then moreover have "B = C" using A1 A2 notinB notinC by auto
ultimately show ?thesis using IH[OF lessB] y r x x' by auto
next
assume diff: "b \<noteq> c"
let ?D = "B - {c}"
have B: "B = insert c ?D" and C: "C = insert b ?D"
using A1 A2 notinB notinC diff by(blast elim!:equalityE)+
have "finite A" by(rule foldSet_imp_finite[OF Afoldx])
with A1 have "finite ?D" by simp
then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g z"
using finite_imp_foldSet by rules
moreover have cinB: "c \<in> B" using B by(auto)
ultimately have "(B,g c \<cdot> d) \<in> foldSet f g z"
by(rule Diff1_foldSet)
hence "g c \<cdot> d = y" by(rule IH[OF lessB y])
moreover have "g b \<cdot> d = r"
proof (rule IH[OF lessC r])
show "(C,g b \<cdot> d) \<in> foldSet f g z" using C notinB Dfoldd
by fastsimp
qed
ultimately show ?thesis using x x' by(auto simp:AC)
qed
qed
qed
qed
qed
(* The same proof, but using card
lemma (in ACf) foldSet_determ_aux:
"!!A x x'. \<lbrakk> card A < n; (A,x) : foldSet f g e; (A,x') : foldSet f g e \<rbrakk>
\<Longrightarrow> x' = x"
proof (induct n)
case 0 thus ?case by simp
next
case (Suc n)
have IH: "!!A x x'. \<lbrakk>card A < n; (A,x) \<in> foldSet f g e; (A,x') \<in> foldSet f g e\<rbrakk>
\<Longrightarrow> x' = x" and card: "card A < Suc n"
and Afoldx: "(A, x) \<in> foldSet f g e" and Afoldy: "(A,x') \<in> foldSet f g e" .
from card have "card A < n \<or> card A = n" by arith
thus ?case
proof
assume less: "card A < n"
show ?thesis by(rule IH[OF less Afoldx Afoldy])
next
assume cardA: "card A = n"
show ?thesis
proof (rule foldSet.cases[OF Afoldx])
assume "(A, x) = ({}, e)"
thus "x' = x" using Afoldy by (auto)
next
fix B b y
assume eq1: "(A, x) = (insert b B, g b \<cdot> y)"
and y: "(B,y) \<in> foldSet f g e" and notinB: "b \<notin> B"
hence A1: "A = insert b B" and x: "x = g b \<cdot> y" by auto
show ?thesis
proof (rule foldSet.cases[OF Afoldy])
assume "(A,x') = ({}, e)"
thus ?thesis using A1 by auto
next
fix C c z
assume eq2: "(A,x') = (insert c C, g c \<cdot> z)"
and z: "(C,z) \<in> foldSet f g e" and notinC: "c \<notin> C"
hence A2: "A = insert c C" and x': "x' = g c \<cdot> z" by auto
have finA: "finite A" by(rule foldSet_imp_finite[OF Afoldx])
with cardA A1 notinB have less: "card B < n" by simp
show ?thesis
proof cases
assume "b = c"
then moreover have "B = C" using A1 A2 notinB notinC by auto
ultimately show ?thesis using IH[OF less] y z x x' by auto
next
assume diff: "b \<noteq> c"
let ?D = "B - {c}"
have B: "B = insert c ?D" and C: "C = insert b ?D"
using A1 A2 notinB notinC diff by(blast elim!:equalityE)+
have "finite ?D" using finA A1 by simp
then obtain d where Dfoldd: "(?D,d) \<in> foldSet f g e"
using finite_imp_foldSet by rules
moreover have cinB: "c \<in> B" using B by(auto)
ultimately have "(B,g c \<cdot> d) \<in> foldSet f g e"
by(rule Diff1_foldSet)
hence "g c \<cdot> d = y" by(rule IH[OF less y])
moreover have "g b \<cdot> d = z"
proof (rule IH[OF _ z])
show "card C < n" using C cardA A1 notinB finA cinB
by(auto simp:card_Diff1_less)
next
show "(C,g b \<cdot> d) \<in> foldSet f g e" using C notinB Dfoldd
by fastsimp
qed
ultimately show ?thesis using x x' by(auto simp:AC)
qed
qed
qed
qed
qed
*)
lemma (in ACf) foldSet_determ:
"(A, x) : foldSet f g z ==> (A, y) : foldSet f g z ==> y = x"
apply(frule foldSet_imp_finite)
apply(simp add:finite_conv_nat_seg_image)
apply(blast intro: foldSet_determ_aux [rule_format])
done
lemma (in ACf) fold_equality: "(A, y) : foldSet f g z ==> fold f g z A = y"
by (unfold fold_def) (blast intro: foldSet_determ)
text{* The base case for @{text fold}: *}
lemma fold_empty [simp]: "fold f g z {} = z"
by (unfold fold_def) blast
lemma (in ACf) fold_insert_aux: "x \<notin> A ==>
((insert x A, v) : foldSet f g z) =
(EX y. (A, y) : foldSet f g z & v = f (g x) y)"
apply auto
apply (rule_tac A1 = A and f1 = f in finite_imp_foldSet [THEN exE])
apply (fastsimp dest: foldSet_imp_finite)
apply (blast intro: foldSet_determ)
done
text{* The recursion equation for @{text fold}: *}
lemma (in ACf) fold_insert[simp]:
"finite A ==> x \<notin> A ==> fold f g z (insert x A) = f (g x) (fold f g z A)"
apply (unfold fold_def)
apply (simp add: fold_insert_aux)
apply (rule the_equality)
apply (auto intro: finite_imp_foldSet
cong add: conj_cong simp add: fold_def [symmetric] fold_equality)
done
declare
empty_foldSetE [rule del] foldSet.intros [rule del]
-- {* Delete rules to do with @{text foldSet} relation. *}
text{* A simplified version for idempotent functions: *}
lemma (in ACIf) fold_insert2:
assumes finA: "finite A"
shows "fold (op \<cdot>) g z (insert a A) = g a \<cdot> fold f g z A"
proof cases
assume "a \<in> A"
then obtain B where A: "A = insert a B" and disj: "a \<notin> B"
by(blast dest: mk_disjoint_insert)
show ?thesis
proof -
from finA A have finB: "finite B" by(blast intro: finite_subset)
have "fold f g z (insert a A) = fold f g z (insert a B)" using A by simp
also have "\<dots> = (g a) \<cdot> (fold f g z B)"
using finB disj by(simp)
also have "\<dots> = g a \<cdot> fold f g z A"
using A finB disj by(simp add:idem assoc[symmetric])
finally show ?thesis .
qed
next
assume "a \<notin> A"
with finA show ?thesis by simp
qed
lemma (in ACIf) foldI_conv_id:
"finite A \<Longrightarrow> fold f g z A = fold f id z (g ` A)"
by(erule finite_induct)(simp_all add: fold_insert2 del: fold_insert)
subsubsection{*Lemmas about @{text fold}*}
lemma (in ACf) fold_commute:
"finite A ==> (!!z. f x (fold f g z A) = fold f g (f x z) A)"
apply (induct set: Finites, simp)
apply (simp add: left_commute [of x])
done
lemma (in ACf) fold_nest_Un_Int:
"finite A ==> finite B
==> fold f g (fold f g z B) A = fold f g (fold f g z (A Int B)) (A Un B)"
apply (induct set: Finites, simp)
apply (simp add: fold_commute Int_insert_left insert_absorb)
done
lemma (in ACf) fold_nest_Un_disjoint:
"finite A ==> finite B ==> A Int B = {}
==> fold f g z (A Un B) = fold f g (fold f g z B) A"
by (simp add: fold_nest_Un_Int)
lemma (in ACf) fold_reindex:
assumes fin: "finite A"
shows "inj_on h A \<Longrightarrow> fold f g z (h ` A) = fold f (g \<circ> h) z A"
using fin apply (induct)
apply simp
apply simp
done
lemma (in ACe) fold_Un_Int:
"finite A ==> finite B ==>
fold f g e A \<cdot> fold f g e B =
fold f g e (A Un B) \<cdot> fold f g e (A Int B)"
apply (induct set: Finites, simp)
apply (simp add: AC insert_absorb Int_insert_left)
done
corollary (in ACe) fold_Un_disjoint:
"finite A ==> finite B ==> A Int B = {} ==>
fold f g e (A Un B) = fold f g e A \<cdot> fold f g e B"
by (simp add: fold_Un_Int)
lemma (in ACe) fold_UN_disjoint:
"\<lbrakk> finite I; ALL i:I. finite (A i);
ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk>
\<Longrightarrow> fold f g e (UNION I A) =
fold f (%i. fold f g e (A i)) e I"
apply (induct set: Finites, simp, atomize)
apply (subgoal_tac "ALL i:F. x \<noteq> i")
prefer 2 apply blast
apply (subgoal_tac "A x Int UNION F A = {}")
prefer 2 apply blast
apply (simp add: fold_Un_disjoint)
done
lemma (in ACf) fold_cong:
"finite A \<Longrightarrow> (!!x. x:A ==> g x = h x) ==> fold f g z A = fold f h z A"
apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold f g z C = fold f h z C")
apply simp
apply (erule finite_induct, simp)
apply (simp add: subset_insert_iff, clarify)
apply (subgoal_tac "finite C")
prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl])
apply (subgoal_tac "C = insert x (C - {x})")
prefer 2 apply blast
apply (erule ssubst)
apply (drule spec)
apply (erule (1) notE impE)
apply (simp add: Ball_def del: insert_Diff_single)
done
lemma (in ACe) fold_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
fold f (%x. fold f (g x) e (B x)) e A =
fold f (split g) e (SIGMA x:A. B x)"
apply (subst Sigma_def)
apply (subst fold_UN_disjoint)
apply assumption
apply simp
apply blast
apply (erule fold_cong)
apply (subst fold_UN_disjoint)
apply simp
apply simp
apply blast
apply (simp)
done
lemma (in ACe) fold_distrib: "finite A \<Longrightarrow>
fold f (%x. f (g x) (h x)) e A = f (fold f g e A) (fold f h e A)"
apply (erule finite_induct)
apply simp
apply (simp add:AC)
done
subsection {* Generalized summation over a set *}
constdefs
setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
"setsum f A == if finite A then fold (op +) f 0 A else 0"
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
written @{text"\<Sum>x\<in>A. e"}. *}
syntax
"_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add" ("(3SUM _:_. _)" [0, 51, 10] 10)
syntax (xsymbols)
"_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add" ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
syntax (HTML output)
"_setsum" :: "idt => 'a set => 'b => 'b::comm_monoid_add" ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
translations -- {* Beware of argument permutation! *}
"SUM i:A. b" == "setsum (%i. b) A"
"\<Sum>i\<in>A. b" == "setsum (%i. b) A"
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter
@{text"\<Sum>x|P. e"}. *}
syntax
"_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
syntax (xsymbols)
"_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
syntax (HTML output)
"_qsetsum" :: "idt \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
translations
"SUM x|P. t" => "setsum (%x. t) {x. P}"
"\<Sum>x|P. t" => "setsum (%x. t) {x. P}"
text{* Finally we abbreviate @{term"\<Sum>x\<in>A. x"} by @{text"\<Sum>A"}. *}
syntax
"_Setsum" :: "'a set => 'a::comm_monoid_mult" ("\<Sum>_" [1000] 999)
parse_translation {*
let
fun Setsum_tr [A] = Syntax.const "setsum" $ Abs ("", dummyT, Bound 0) $ A
in [("_Setsum", Setsum_tr)] end;
*}
print_translation {*
let
fun setsum_tr' [Abs(_,_,Bound 0), A] = Syntax.const "_Setsum" $ A
| setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] =
if x<>y then raise Match
else let val x' = Syntax.mark_bound x
val t' = subst_bound(x',t)
val P' = subst_bound(x',P)
in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end
in
[("setsum", setsum_tr')]
end
*}
lemma setsum_empty [simp]: "setsum f {} = 0"
by (simp add: setsum_def)
lemma setsum_insert [simp]:
"finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F"
by (simp add: setsum_def ACf.fold_insert [OF ACf_add])
lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0"
by (simp add: setsum_def)
lemma setsum_reindex:
"inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B"
by(auto simp add: setsum_def ACf.fold_reindex[OF ACf_add] dest!:finite_imageD)
lemma setsum_reindex_id:
"inj_on f B ==> setsum f B = setsum id (f ` B)"
by (auto simp add: setsum_reindex)
lemma setsum_cong:
"A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
by(fastsimp simp: setsum_def intro: ACf.fold_cong[OF ACf_add])
lemma setsum_reindex_cong:
"[|inj_on f A; B = f ` A; !!a. g a = h (f a)|]
==> setsum h B = setsum g A"
by (simp add: setsum_reindex cong: setsum_cong)
lemma setsum_0: "setsum (%i. 0) A = 0"
apply (clarsimp simp: setsum_def)
apply (erule finite_induct, auto simp:ACf.fold_insert [OF ACf_add])
done
lemma setsum_0': "ALL a:F. f a = 0 ==> setsum f F = 0"
apply (subgoal_tac "setsum f F = setsum (%x. 0) F")
apply (erule ssubst, rule setsum_0)
apply (rule setsum_cong, auto)
done
lemma setsum_Un_Int: "finite A ==> finite B ==>
setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"
-- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
by(simp add: setsum_def ACe.fold_Un_Int[OF ACe_add,symmetric])
lemma setsum_Un_disjoint: "finite A ==> finite B
==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B"
by (subst setsum_Un_Int [symmetric], auto)
(*But we can't get rid of finite I. If infinite, although the rhs is 0,
the lhs need not be, since UNION I A could still be finite.*)
lemma setsum_UN_disjoint:
"finite I ==> (ALL i:I. finite (A i)) ==>
(ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))"
by(simp add: setsum_def ACe.fold_UN_disjoint[OF ACe_add] cong: setsum_cong)
text{*No need to assume that @{term C} is finite. If infinite, the rhs is
directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*}
lemma setsum_Union_disjoint:
"[| (ALL A:C. finite A);
(ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
==> setsum f (Union C) = setsum (setsum f) C"
apply (cases "finite C")
prefer 2 apply (force dest: finite_UnionD simp add: setsum_def)
apply (frule setsum_UN_disjoint [of C id f])
apply (unfold Union_def id_def, assumption+)
done
(*But we can't get rid of finite A. If infinite, although the lhs is 0,
the rhs need not be, since SIGMA A B could still be finite.*)
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
(\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) =
(\<Sum>z\<in>(SIGMA x:A. B x). f (fst z) (snd z))"
by(simp add:setsum_def ACe.fold_Sigma[OF ACe_add] split_def cong:setsum_cong)
text{*Here we can eliminate the finiteness assumptions, by cases.*}
lemma setsum_cartesian_product:
"(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>z\<in>A <*> B. f (fst z) (snd z))"
apply (cases "finite A")
apply (cases "finite B")
apply (simp add: setsum_Sigma)
apply (cases "A={}", simp)
apply (simp add: setsum_0)
apply (auto simp add: setsum_def
dest: finite_cartesian_productD1 finite_cartesian_productD2)
done
lemma setsum_addf: "setsum (%x. f x + g x) A = (setsum f A + setsum g A)"
by(simp add:setsum_def ACe.fold_distrib[OF ACe_add])
subsubsection {* Properties in more restricted classes of structures *}
lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
apply (case_tac "finite A")
prefer 2 apply (simp add: setsum_def)
apply (erule rev_mp)
apply (erule finite_induct, auto)
done
lemma setsum_eq_0_iff [simp]:
"finite F ==> (setsum f F = 0) = (ALL a:F. f a = (0::nat))"
by (induct set: Finites) auto
lemma setsum_Un_nat: "finite A ==> finite B ==>
(setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
-- {* For the natural numbers, we have subtraction. *}
by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
lemma setsum_Un: "finite A ==> finite B ==>
(setsum f (A Un B) :: 'a :: ab_group_add) =
setsum f A + setsum f B - setsum f (A Int B)"
by (subst setsum_Un_Int [symmetric], auto simp add: ring_eq_simps)
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
(if a:A then setsum f A - f a else setsum f A)"
apply (case_tac "finite A")
prefer 2 apply (simp add: setsum_def)
apply (erule finite_induct)
apply (auto simp add: insert_Diff_if)
apply (drule_tac a = a in mk_disjoint_insert, auto)
done
lemma setsum_diff1: "finite A \<Longrightarrow>
(setsum f (A - {a}) :: ('a::ab_group_add)) =
(if a:A then setsum f A - f a else setsum f A)"
by (erule finite_induct) (auto simp add: insert_Diff_if)
(* By Jeremy Siek: *)
lemma setsum_diff_nat:
assumes finB: "finite B"
shows "B \<subseteq> A \<Longrightarrow> (setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
using finB
proof (induct)
show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
next
fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
and xFinA: "insert x F \<subseteq> A"
and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
by (simp add: setsum_diff1_nat)
from xFinA have "F \<subseteq> A" by simp
with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
by simp
from xnotinF have "A - insert x F = (A - F) - {x}" by auto
with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
by simp
from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
by simp
thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
qed
lemma setsum_diff:
assumes le: "finite A" "B \<subseteq> A"
shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
proof -
from le have finiteB: "finite B" using finite_subset by auto
show ?thesis using finiteB le
proof (induct)
case empty
thus ?case by auto
next
case (insert x F)
thus ?case using le finiteB
by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
qed
qed
lemma setsum_mono:
assumes le: "\<And>i. i\<in>K \<Longrightarrow> f (i::'a) \<le> ((g i)::('b::{comm_monoid_add, pordered_ab_semigroup_add}))"
shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
proof (cases "finite K")
case True
thus ?thesis using le
proof (induct)
case empty
thus ?case by simp
next
case insert
thus ?case using add_mono
by force
qed
next
case False
thus ?thesis
by (simp add: setsum_def)
qed
lemma setsum_mono2_nat:
assumes fin: "finite B" and sub: "A \<subseteq> B"
shows "setsum f A \<le> (setsum f B :: nat)"
proof -
have "setsum f A \<le> setsum f A + setsum f (B-A)" by arith
also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
by (simp add:setsum_Un_disjoint del:Un_Diff_cancel)
also have "A \<union> (B-A) = B" using sub by blast
finally show ?thesis .
qed
lemma setsum_negf: "finite A ==> setsum (%x. - (f x)::'a::ab_group_add) A =
- setsum f A"
by (induct set: Finites, auto)
lemma setsum_subtractf: "finite A ==> setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
setsum f A - setsum g A"
by (simp add: diff_minus setsum_addf setsum_negf)
lemma setsum_nonneg: "[| finite A;
\<forall>x \<in> A. (0::'a::{pordered_ab_semigroup_add, comm_monoid_add}) \<le> f x |] ==>
0 \<le> setsum f A";
apply (induct set: Finites, auto)
apply (subgoal_tac "0 + 0 \<le> f x + setsum f F", simp)
apply (blast intro: add_mono)
done
lemma setsum_nonpos: "[| finite A;
\<forall>x \<in> A. f x \<le> (0::'a::{pordered_ab_semigroup_add, comm_monoid_add}) |] ==>
setsum f A \<le> 0";
apply (induct set: Finites, auto)
apply (subgoal_tac "f x + setsum f F \<le> 0 + 0", simp)
apply (blast intro: add_mono)
done
lemma setsum_mult:
fixes f :: "'a => ('b::semiring_0_cancel)"
shows "r * setsum f A = setsum (%n. r * f n) A"
proof (cases "finite A")
case True
thus ?thesis
proof (induct)
case empty thus ?case by simp
next
case (insert x A) thus ?case by (simp add: right_distrib)
qed
next
case False thus ?thesis by (simp add: setsum_def)
qed
lemma setsum_abs:
fixes f :: "'a => ('b::lordered_ab_group_abs)"
assumes fin: "finite A"
shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) A"
using fin
proof (induct)
case empty thus ?case by simp
next
case (insert x A)
thus ?case by (auto intro: abs_triangle_ineq order_trans)
qed
lemma setsum_abs_ge_zero:
fixes f :: "'a => ('b::lordered_ab_group_abs)"
assumes fin: "finite A"
shows "0 \<le> setsum (%i. abs(f i)) A"
using fin
proof (induct)
case empty thus ?case by simp
next
case (insert x A) thus ?case by (auto intro: order_trans)
qed
subsection {* Generalized product over a set *}
constdefs
setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult"
"setprod f A == if finite A then fold (op *) f 1 A else 1"
syntax
"_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult" ("(3\<Prod>_:_. _)" [0, 51, 10] 10)
syntax (xsymbols)
"_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult" ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
syntax (HTML output)
"_setprod" :: "idt => 'a set => 'b => 'b::comm_monoid_mult" ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
translations
"\<Prod>i:A. b" == "setprod (%i. b) A" -- {* Beware of argument permutation! *}
syntax
"_Setprod" :: "'a set => 'a::comm_monoid_mult" ("\<Prod>_" [1000] 999)
parse_translation {*
let
fun Setprod_tr [A] = Syntax.const "setprod" $ Abs ("", dummyT, Bound 0) $ A
in [("_Setprod", Setprod_tr)] end;
*}
print_translation {*
let fun setprod_tr' [Abs(x,Tx,t), A] =
if t = Bound 0 then Syntax.const "_Setprod" $ A else raise Match
in
[("setprod", setprod_tr')]
end
*}
lemma setprod_empty [simp]: "setprod f {} = 1"
by (auto simp add: setprod_def)
lemma setprod_insert [simp]: "[| finite A; a \<notin> A |] ==>
setprod f (insert a A) = f a * setprod f A"
by (simp add: setprod_def ACf.fold_insert [OF ACf_mult])
lemma setprod_infinite [simp]: "~ finite A ==> setprod f A = 1"
by (simp add: setprod_def)
lemma setprod_reindex:
"inj_on f B ==> setprod h (f ` B) = setprod (h \<circ> f) B"
by(auto simp: setprod_def ACf.fold_reindex[OF ACf_mult] dest!:finite_imageD)
lemma setprod_reindex_id: "inj_on f B ==> setprod f B = setprod id (f ` B)"
by (auto simp add: setprod_reindex)
lemma setprod_cong:
"A = B ==> (!!x. x:B ==> f x = g x) ==> setprod f A = setprod g B"
by(fastsimp simp: setprod_def intro: ACf.fold_cong[OF ACf_mult])
lemma setprod_reindex_cong: "inj_on f A ==>
B = f ` A ==> g = h \<circ> f ==> setprod h B = setprod g A"
by (frule setprod_reindex, simp)
lemma setprod_1: "setprod (%i. 1) A = 1"
apply (case_tac "finite A")
apply (erule finite_induct, auto simp add: mult_ac)
done
lemma setprod_1': "ALL a:F. f a = 1 ==> setprod f F = 1"
apply (subgoal_tac "setprod f F = setprod (%x. 1) F")
apply (erule ssubst, rule setprod_1)
apply (rule setprod_cong, auto)
done
lemma setprod_Un_Int: "finite A ==> finite B
==> setprod g (A Un B) * setprod g (A Int B) = setprod g A * setprod g B"
by(simp add: setprod_def ACe.fold_Un_Int[OF ACe_mult,symmetric])
lemma setprod_Un_disjoint: "finite A ==> finite B
==> A Int B = {} ==> setprod g (A Un B) = setprod g A * setprod g B"
by (subst setprod_Un_Int [symmetric], auto)
lemma setprod_UN_disjoint:
"finite I ==> (ALL i:I. finite (A i)) ==>
(ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
setprod f (UNION I A) = setprod (%i. setprod f (A i)) I"
by(simp add: setprod_def ACe.fold_UN_disjoint[OF ACe_mult] cong: setprod_cong)
lemma setprod_Union_disjoint:
"[| (ALL A:C. finite A);
(ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |]
==> setprod f (Union C) = setprod (setprod f) C"
apply (cases "finite C")
prefer 2 apply (force dest: finite_UnionD simp add: setprod_def)
apply (frule setprod_UN_disjoint [of C id f])
apply (unfold Union_def id_def, assumption+)
done
lemma setprod_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
(\<Prod>x:A. (\<Prod>y: B x. f x y)) =
(\<Prod>z:(SIGMA x:A. B x). f (fst z) (snd z))"
by(simp add:setprod_def ACe.fold_Sigma[OF ACe_mult] split_def cong:setprod_cong)
text{*Here we can eliminate the finiteness assumptions, by cases.*}
lemma setprod_cartesian_product:
"(\<Prod>x:A. (\<Prod>y: B. f x y)) = (\<Prod>z:(A <*> B). f (fst z) (snd z))"
apply (cases "finite A")
apply (cases "finite B")
apply (simp add: setprod_Sigma)
apply (cases "A={}", simp)
apply (simp add: setprod_1)
apply (auto simp add: setprod_def
dest: finite_cartesian_productD1 finite_cartesian_productD2)
done
lemma setprod_timesf:
"setprod (%x. f x * g x) A = (setprod f A * setprod g A)"
by(simp add:setprod_def ACe.fold_distrib[OF ACe_mult])
subsubsection {* Properties in more restricted classes of structures *}
lemma setprod_eq_1_iff [simp]:
"finite F ==> (setprod f F = 1) = (ALL a:F. f a = (1::nat))"
by (induct set: Finites) auto
lemma setprod_zero:
"finite A ==> EX x: A. f x = (0::'a::comm_semiring_1_cancel) ==> setprod f A = 0"
apply (induct set: Finites, force, clarsimp)
apply (erule disjE, auto)
done
lemma setprod_nonneg [rule_format]:
"(ALL x: A. (0::'a::ordered_idom) \<le> f x) --> 0 \<le> setprod f A"
apply (case_tac "finite A")
apply (induct set: Finites, force, clarsimp)
apply (subgoal_tac "0 * 0 \<le> f x * setprod f F", force)
apply (rule mult_mono, assumption+)
apply (auto simp add: setprod_def)
done
lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_idom) < f x)
--> 0 < setprod f A"
apply (case_tac "finite A")
apply (induct set: Finites, force, clarsimp)
apply (subgoal_tac "0 * 0 < f x * setprod f F", force)
apply (rule mult_strict_mono, assumption+)
apply (auto simp add: setprod_def)
done
lemma setprod_nonzero [rule_format]:
"(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
finite A ==> (ALL x: A. f x \<noteq> (0::'a)) --> setprod f A \<noteq> 0"
apply (erule finite_induct, auto)
done
lemma setprod_zero_eq:
"(ALL x y. (x::'a::comm_semiring_1_cancel) * y = 0 --> x = 0 | y = 0) ==>
finite A ==> (setprod f A = (0::'a)) = (EX x: A. f x = 0)"
apply (insert setprod_zero [of A f] setprod_nonzero [of A f], blast)
done
lemma setprod_nonzero_field:
"finite A ==> (ALL x: A. f x \<noteq> (0::'a::field)) ==> setprod f A \<noteq> 0"
apply (rule setprod_nonzero, auto)
done
lemma setprod_zero_eq_field:
"finite A ==> (setprod f A = (0::'a::field)) = (EX x: A. f x = 0)"
apply (rule setprod_zero_eq, auto)
done
lemma setprod_Un: "finite A ==> finite B ==> (ALL x: A Int B. f x \<noteq> 0) ==>
(setprod f (A Un B) :: 'a ::{field})
= setprod f A * setprod f B / setprod f (A Int B)"
apply (subst setprod_Un_Int [symmetric], auto)
apply (subgoal_tac "finite (A Int B)")
apply (frule setprod_nonzero_field [of "A Int B" f], assumption)
apply (subst times_divide_eq_right [THEN sym], auto simp add: divide_self)
done
lemma setprod_diff1: "finite A ==> f a \<noteq> 0 ==>
(setprod f (A - {a}) :: 'a :: {field}) =
(if a:A then setprod f A / f a else setprod f A)"
apply (erule finite_induct)
apply (auto simp add: insert_Diff_if)
apply (subgoal_tac "f a * setprod f F / f a = setprod f F * f a / f a")
apply (erule ssubst)
apply (subst times_divide_eq_right [THEN sym])
apply (auto simp add: mult_ac times_divide_eq_right divide_self)
done
lemma setprod_inversef: "finite A ==>
ALL x: A. f x \<noteq> (0::'a::{field,division_by_zero}) ==>
setprod (inverse \<circ> f) A = inverse (setprod f A)"
apply (erule finite_induct)
apply (simp, simp)
done
lemma setprod_dividef:
"[|finite A;
\<forall>x \<in> A. g x \<noteq> (0::'a::{field,division_by_zero})|]
==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
apply (subgoal_tac
"setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
apply (erule ssubst)
apply (subst divide_inverse)
apply (subst setprod_timesf)
apply (subst setprod_inversef, assumption+, rule refl)
apply (rule setprod_cong, rule refl)
apply (subst divide_inverse, auto)
done
subsection {* Finite cardinality *}
text {* This definition, although traditional, is ugly to work with:
@{text "card A == LEAST n. EX f. A = {f i | i. i < n}"}.
But now that we have @{text setsum} things are easy:
*}
constdefs
card :: "'a set => nat"
"card A == setsum (%x. 1::nat) A"
lemma card_empty [simp]: "card {} = 0"
by (simp add: card_def)
lemma card_infinite [simp]: "~ finite A ==> card A = 0"
by (simp add: card_def)
lemma card_eq_setsum: "card A = setsum (%x. 1) A"
by (simp add: card_def)
lemma card_insert_disjoint [simp]:
"finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
by(simp add: card_def ACf.fold_insert[OF ACf_add])
lemma card_insert_if:
"finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
by (simp add: insert_absorb)
lemma card_0_eq [simp]: "finite A ==> (card A = 0) = (A = {})"
apply auto
apply (drule_tac a = x in mk_disjoint_insert, clarify)
apply (auto)
done
lemma card_eq_0_iff: "(card A = 0) = (A = {} | ~ finite A)"
by auto
lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
apply(simp del:insert_Diff_single)
done
lemma card_Diff_singleton:
"finite A ==> x: A ==> card (A - {x}) = card A - 1"
by (simp add: card_Suc_Diff1 [symmetric])
lemma card_Diff_singleton_if:
"finite A ==> card (A-{x}) = (if x : A then card A - 1 else card A)"
by (simp add: card_Diff_singleton)
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
by (simp add: card_insert_if card_Suc_Diff1)
lemma card_insert_le: "finite A ==> card A <= card (insert x A)"
by (simp add: card_insert_if)
lemma card_mono: "\<lbrakk> finite B; A \<subseteq> B \<rbrakk> \<Longrightarrow> card A \<le> card B"
by (simp add: card_def setsum_mono2_nat)
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
apply (induct set: Finites, simp, clarify)
apply (subgoal_tac "finite A & A - {x} <= F")
prefer 2 apply (blast intro: finite_subset, atomize)
apply (drule_tac x = "A - {x}" in spec)
apply (simp add: card_Diff_singleton_if split add: split_if_asm)
apply (case_tac "card A", auto)
done
lemma psubset_card_mono: "finite B ==> A < B ==> card A < card B"
apply (simp add: psubset_def linorder_not_le [symmetric])
apply (blast dest: card_seteq)
done
lemma card_Un_Int: "finite A ==> finite B
==> card A + card B = card (A Un B) + card (A Int B)"
by(simp add:card_def setsum_Un_Int)
lemma card_Un_disjoint: "finite A ==> finite B
==> A Int B = {} ==> card (A Un B) = card A + card B"
by (simp add: card_Un_Int)
lemma card_Diff_subset:
"finite B ==> B <= A ==> card (A - B) = card A - card B"
by(simp add:card_def setsum_diff_nat)
lemma card_Diff1_less: "finite A ==> x: A ==> card (A - {x}) < card A"
apply (rule Suc_less_SucD)
apply (simp add: card_Suc_Diff1)
done
lemma card_Diff2_less:
"finite A ==> x: A ==> y: A ==> card (A - {x} - {y}) < card A"
apply (case_tac "x = y")
apply (simp add: card_Diff1_less)
apply (rule less_trans)
prefer 2 apply (auto intro!: card_Diff1_less)
done
lemma card_Diff1_le: "finite A ==> card (A - {x}) <= card A"
apply (case_tac "x : A")
apply (simp_all add: card_Diff1_less less_imp_le)
done
lemma card_psubset: "finite B ==> A \<subseteq> B ==> card A < card B ==> A < B"
by (erule psubsetI, blast)
lemma insert_partition:
"\<lbrakk> x \<notin> F; \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<rbrakk>
\<Longrightarrow> x \<inter> \<Union> F = {}"
by auto
(* main cardinality theorem *)
lemma card_partition [rule_format]:
"finite C ==>
finite (\<Union> C) -->
(\<forall>c\<in>C. card c = k) -->
(\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = {}) -->
k * card(C) = card (\<Union> C)"
apply (erule finite_induct, simp)
apply (simp add: card_insert_disjoint card_Un_disjoint insert_partition
finite_subset [of _ "\<Union> (insert x F)"])
done
lemma setsum_constant_nat: "(\<Sum>x\<in>A. y) = (card A) * y"
-- {* Generalized to any @{text comm_semiring_1_cancel} in
@{text IntDef} as @{text setsum_constant}. *}
apply (cases "finite A")
apply (erule finite_induct, auto)
done
lemma setprod_constant: "finite A ==> (\<Prod>x: A. (y::'a::recpower)) = y^(card A)"
apply (erule finite_induct)
apply (auto simp add: power_Suc)
done
subsubsection {* Cardinality of unions *}
lemma card_UN_disjoint:
"finite I ==> (ALL i:I. finite (A i)) ==>
(ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
apply (simp add: card_def)
apply (subgoal_tac
"setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I")
apply (simp add: setsum_UN_disjoint)
apply (simp add: setsum_constant_nat cong: setsum_cong)
done
lemma card_Union_disjoint:
"finite C ==> (ALL A:C. finite A) ==>
(ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) ==>
card (Union C) = setsum card C"
apply (frule card_UN_disjoint [of C id])
apply (unfold Union_def id_def, assumption+)
done
subsubsection {* Cardinality of image *}
text{*The image of a finite set can be expressed using @{term fold}.*}
lemma image_eq_fold: "finite A ==> f ` A = fold (op Un) (%x. {f x}) {} A"
apply (erule finite_induct, simp)
apply (subst ACf.fold_insert)
apply (auto simp add: ACf_def)
done
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
apply (induct set: Finites, simp)
apply (simp add: le_SucI finite_imageI card_insert_if)
done
lemma card_image: "inj_on f A ==> card (f ` A) = card A"
by(simp add:card_def setsum_reindex o_def)
lemma endo_inj_surj: "finite A ==> f ` A \<subseteq> A ==> inj_on f A ==> f ` A = A"
by (simp add: card_seteq card_image)
lemma eq_card_imp_inj_on:
"[| finite A; card(f ` A) = card A |] ==> inj_on f A"
apply(induct rule:finite_induct)
apply simp
apply(frule card_image_le[where f = f])
apply(simp add:card_insert_if split:if_splits)
done
lemma inj_on_iff_eq_card:
"finite A ==> inj_on f A = (card(f ` A) = card A)"
by(blast intro: card_image eq_card_imp_inj_on)
lemma card_inj_on_le:
"[|inj_on f A; f ` A \<subseteq> B; finite B |] ==> card A \<le> card B"
apply (subgoal_tac "finite A")
apply (force intro: card_mono simp add: card_image [symmetric])
apply (blast intro: finite_imageD dest: finite_subset)
done
lemma card_bij_eq:
"[|inj_on f A; f ` A \<subseteq> B; inj_on g B; g ` B \<subseteq> A;
finite A; finite B |] ==> card A = card B"
by (auto intro: le_anti_sym card_inj_on_le)
subsubsection {* Cardinality of products *}
(*
lemma SigmaI_insert: "y \<notin> A ==>
(SIGMA x:(insert y A). B x) = (({y} <*> (B y)) \<union> (SIGMA x: A. B x))"
by auto
*)
lemma card_SigmaI [simp]:
"\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
\<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
by(simp add:card_def setsum_Sigma)
lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
apply (cases "finite A")
apply (cases "finite B")
apply (simp add: setsum_constant_nat)
apply (auto simp add: card_eq_0_iff
dest: finite_cartesian_productD1 finite_cartesian_productD2)
done
lemma card_cartesian_product_singleton: "card({x} <*> A) = card(A)"
by (simp add: card_cartesian_product)
subsubsection {* Cardinality of the Powerset *}
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A" (* FIXME numeral 2 (!?) *)
apply (induct set: Finites)
apply (simp_all add: Pow_insert)
apply (subst card_Un_disjoint, blast)
apply (blast intro: finite_imageI, blast)
apply (subgoal_tac "inj_on (insert x) (Pow F)")
apply (simp add: card_image Pow_insert)
apply (unfold inj_on_def)
apply (blast elim!: equalityE)
done
text {* Relates to equivalence classes. Based on a theorem of
F. Kammüller's. *}
lemma dvd_partition:
"finite (Union C) ==>
ALL c : C. k dvd card c ==>
(ALL c1: C. ALL c2: C. c1 \<noteq> c2 --> c1 Int c2 = {}) ==>
k dvd card (Union C)"
apply(frule finite_UnionD)
apply(rotate_tac -1)
apply (induct set: Finites, simp_all, clarify)
apply (subst card_Un_disjoint)
apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
done
subsubsection {* Theorems about @{text "choose"} *}
text {*
\medskip Basic theorem about @{text "choose"}. By Florian
Kamm\"uller, tidied by LCP.
*}
lemma card_s_0_eq_empty:
"finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"
apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
apply (simp cong add: rev_conj_cong)
done
lemma choose_deconstruct: "finite M ==> x \<notin> M
==> {s. s <= insert x M & card(s) = Suc k}
= {s. s <= M & card(s) = Suc k} Un
{s. EX t. t <= M & card(t) = k & s = insert x t}"
apply safe
apply (auto intro: finite_subset [THEN card_insert_disjoint])
apply (drule_tac x = "xa - {x}" in spec)
apply (subgoal_tac "x \<notin> xa", auto)
apply (erule rev_mp, subst card_Diff_singleton)
apply (auto intro: finite_subset)
done
text{*There are as many subsets of @{term A} having cardinality @{term k}
as there are sets obtained from the former by inserting a fixed element
@{term x} into each.*}
lemma constr_bij:
"[|finite A; x \<notin> A|] ==>
card {B. EX C. C <= A & card(C) = k & B = insert x C} =
card {B. B <= A & card(B) = k}"
apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq)
apply (auto elim!: equalityE simp add: inj_on_def)
apply (subst Diff_insert0, auto)
txt {* finiteness of the two sets *}
apply (rule_tac [2] B = "Pow (A)" in finite_subset)
apply (rule_tac B = "Pow (insert x A)" in finite_subset)
apply fast+
done
text {*
Main theorem: combinatorial statement about number of subsets of a set.
*}
lemma n_sub_lemma:
"!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
apply (induct k)
apply (simp add: card_s_0_eq_empty, atomize)
apply (rotate_tac -1, erule finite_induct)
apply (simp_all (no_asm_simp) cong add: conj_cong
add: card_s_0_eq_empty choose_deconstruct)
apply (subst card_Un_disjoint)
prefer 4 apply (force simp add: constr_bij)
prefer 3 apply force
prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
finite_subset [of _ "Pow (insert x F)", standard])
apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
done
theorem n_subsets:
"finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
by (simp add: n_sub_lemma)
subsection{* A fold functional for non-empty sets *}
text{* Does not require start value. *}
consts
foldSet1 :: "('a => 'a => 'a) => ('a set \<times> 'a) set"
inductive "foldSet1 f"
intros
foldSet1_singletonI [intro]: "({a}, a) : foldSet1 f"
foldSet1_insertI [intro]:
"\<lbrakk> (A, x) : foldSet1 f; a \<notin> A; A \<noteq> {} \<rbrakk>
\<Longrightarrow> (insert a A, f a x) : foldSet1 f"
constdefs
fold1 :: "('a => 'a => 'a) => 'a set => 'a"
"fold1 f A == THE x. (A, x) : foldSet1 f"
lemma foldSet1_nonempty:
"(A, x) : foldSet1 f \<Longrightarrow> A \<noteq> {}"
by(erule foldSet1.cases, simp_all)
inductive_cases empty_foldSet1E [elim!]: "({}, x) : foldSet1 f"
lemma foldSet1_sing[iff]: "(({a},b) : foldSet1 f) = (a = b)"
apply(rule iffI)
prefer 2 apply fast
apply (erule foldSet1.cases)
apply blast
apply (erule foldSet1.cases)
apply blast
apply blast
done
lemma Diff1_foldSet1:
"(A - {x}, y) : foldSet1 f ==> x: A ==> (A, f x y) : foldSet1 f"
by (erule insert_Diff [THEN subst], rule foldSet1.intros,
auto dest!:foldSet1_nonempty)
lemma foldSet1_imp_finite: "(A, x) : foldSet1 f ==> finite A"
by (induct set: foldSet1) auto
lemma finite_nonempty_imp_foldSet1:
"\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. (A, x) : foldSet1 f"
by (induct set: Finites) auto
lemma (in ACf) foldSet1_determ_aux:
"!!A x y. \<lbrakk> card A < n; (A, x) : foldSet1 f; (A, y) : foldSet1 f \<rbrakk> \<Longrightarrow> y = x"
proof (induct n)
case 0 thus ?case by simp
next
case (Suc n)
have IH: "!!A x y. \<lbrakk>card A < n; (A, x) \<in> foldSet1 f; (A, y) \<in> foldSet1 f\<rbrakk>
\<Longrightarrow> y = x" and card: "card A < Suc n"
and Afoldx: "(A, x) \<in> foldSet1 f" and Afoldy: "(A, y) \<in> foldSet1 f" .
from card have "card A < n \<or> card A = n" by arith
thus ?case
proof
assume less: "card A < n"
show ?thesis by(rule IH[OF less Afoldx Afoldy])
next
assume cardA: "card A = n"
show ?thesis
proof (rule foldSet1.cases[OF Afoldx])
fix a assume "(A, x) = ({a}, a)"
thus "y = x" using Afoldy by (simp add:foldSet1_sing)
next
fix Ax ax x'
assume eq1: "(A, x) = (insert ax Ax, ax \<cdot> x')"
and x': "(Ax, x') \<in> foldSet1 f" and notinx: "ax \<notin> Ax"
and Axnon: "Ax \<noteq> {}"
hence A1: "A = insert ax Ax" and x: "x = ax \<cdot> x'" by auto
show ?thesis
proof (rule foldSet1.cases[OF Afoldy])
fix ay assume "(A, y) = ({ay}, ay)"
thus ?thesis using eq1 x' Axnon notinx
by (fastsimp simp:foldSet1_sing)
next
fix Ay ay y'
assume eq2: "(A, y) = (insert ay Ay, ay \<cdot> y')"
and y': "(Ay, y') \<in> foldSet1 f" and notiny: "ay \<notin> Ay"
and Aynon: "Ay \<noteq> {}"
hence A2: "A = insert ay Ay" and y: "y = ay \<cdot> y'" by auto
have finA: "finite A" by(rule foldSet1_imp_finite[OF Afoldx])
with cardA A1 notinx have less: "card Ax < n" by simp
show ?thesis
proof cases
assume "ax = ay"
then moreover have "Ax = Ay" using A1 A2 notinx notiny by auto
ultimately show ?thesis using IH[OF less x'] y' eq1 eq2 by auto
next
assume diff: "ax \<noteq> ay"
let ?B = "Ax - {ay}"
have Ax: "Ax = insert ay ?B" and Ay: "Ay = insert ax ?B"
using A1 A2 notinx notiny diff by(blast elim!:equalityE)+
show ?thesis
proof cases
assume "?B = {}"
with Ax Ay show ?thesis using x' y' x y by(simp add:commute)
next
assume Bnon: "?B \<noteq> {}"
moreover have "finite ?B" using finA A1 by simp
ultimately obtain b where Bfoldb: "(?B,b) \<in> foldSet1 f"
using finite_nonempty_imp_foldSet1 by(blast)
moreover have ayinAx: "ay \<in> Ax" using Ax by(auto)
ultimately have "(Ax,ay\<cdot>b) \<in> foldSet1 f" by(rule Diff1_foldSet1)
hence "ay\<cdot>b = x'" by(rule IH[OF less x'])
moreover have "ax\<cdot>b = y'"
proof (rule IH[OF _ y'])
show "card Ay < n" using Ay cardA A1 notinx finA ayinAx
by(auto simp:card_Diff1_less)
next
show "(Ay,ax\<cdot>b) \<in> foldSet1 f" using Ay notinx Bfoldb Bnon
by fastsimp
qed
ultimately show ?thesis using x y by(auto simp:AC)
qed
qed
qed
qed
qed
qed
lemma (in ACf) foldSet1_determ:
"(A, x) : foldSet1 f ==> (A, y) : foldSet1 f ==> y = x"
by (blast intro: foldSet1_determ_aux [rule_format])
lemma (in ACf) foldSet1_equality: "(A, y) : foldSet1 f ==> fold1 f A = y"
by (unfold fold1_def) (blast intro: foldSet1_determ)
lemma fold1_singleton[simp]: "fold1 f {a} = a"
by (unfold fold1_def) blast
lemma (in ACf) foldSet1_insert_aux: "x \<notin> A ==> A \<noteq> {} \<Longrightarrow>
((insert x A, v) : foldSet1 f) =
(EX y. (A, y) : foldSet1 f & v = f x y)"
apply auto
apply (rule_tac A1 = A and f1 = f in finite_nonempty_imp_foldSet1 [THEN exE])
apply (fastsimp dest: foldSet1_imp_finite)
apply blast
apply (blast intro: foldSet1_determ)
done
lemma (in ACf) fold1_insert:
"finite A ==> x \<notin> A ==> A \<noteq> {} \<Longrightarrow> fold1 f (insert x A) = f x (fold1 f A)"
apply (unfold fold1_def)
apply (simp add: foldSet1_insert_aux)
apply (rule the_equality)
apply (auto intro: finite_nonempty_imp_foldSet1
cong add: conj_cong simp add: fold1_def [symmetric] foldSet1_equality)
done
lemma (in ACIf) fold1_insert2[simp]:
assumes finA: "finite A" and nonA: "A \<noteq> {}"
shows "fold1 f (insert a A) = f a (fold1 f A)"
proof cases
assume "a \<in> A"
then obtain B where A: "A = insert a B" and disj: "a \<notin> B"
by(blast dest: mk_disjoint_insert)
show ?thesis
proof cases
assume "B = {}"
thus ?thesis using A by(simp add:idem fold1_singleton)
next
assume nonB: "B \<noteq> {}"
from finA A have finB: "finite B" by(blast intro: finite_subset)
have "fold1 f (insert a A) = fold1 f (insert a B)" using A by simp
also have "\<dots> = f a (fold1 f B)"
using finB nonB disj by(simp add: fold1_insert)
also have "\<dots> = f a (fold1 f A)"
using A finB nonB disj by(simp add:idem fold1_insert assoc[symmetric])
finally show ?thesis .
qed
next
assume "a \<notin> A"
with finA nonA show ?thesis by(simp add:fold1_insert)
qed
text{* Now the recursion rules for definitions: *}
lemma fold1_singleton_def: "g \<equiv> fold1 f \<Longrightarrow> g {a} = a"
by(simp add:fold1_singleton)
lemma (in ACf) fold1_insert_def:
"\<lbrakk> g \<equiv> fold1 f; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
by(simp add:fold1_insert)
lemma (in ACIf) fold1_insert2_def:
"\<lbrakk> g \<equiv> fold1 f; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g(insert x A) = x \<cdot> (g A)"
by(simp add:fold1_insert2)
subsubsection{* Semi-Lattices *}
locale ACIfSL = ACIf +
fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<preceq>" 50)
assumes below_def: "x \<preceq> y = (x\<cdot>y = x)"
locale ACIfSLlin = ACIfSL +
assumes lin: "x\<cdot>y \<in> {x,y}"
lemma (in ACIfSL) below_refl[simp]: "x \<preceq> x"
by(simp add: below_def idem)
lemma (in ACIfSL) below_f_conv[simp]: "x \<preceq> y \<cdot> z = (x \<preceq> y \<and> x \<preceq> z)"
proof
assume "x \<preceq> y \<cdot> z"
hence xyzx: "x \<cdot> (y \<cdot> z) = x" by(simp add: below_def)
have "x \<cdot> y = x"
proof -
have "x \<cdot> y = (x \<cdot> (y \<cdot> z)) \<cdot> y" by(rule subst[OF xyzx], rule refl)
also have "\<dots> = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
also have "\<dots> = x" by(rule xyzx)
finally show ?thesis .
qed
moreover have "x \<cdot> z = x"
proof -
have "x \<cdot> z = (x \<cdot> (y \<cdot> z)) \<cdot> z" by(rule subst[OF xyzx], rule refl)
also have "\<dots> = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
also have "\<dots> = x" by(rule xyzx)
finally show ?thesis .
qed
ultimately show "x \<preceq> y \<and> x \<preceq> z" by(simp add: below_def)
next
assume a: "x \<preceq> y \<and> x \<preceq> z"
hence y: "x \<cdot> y = x" and z: "x \<cdot> z = x" by(simp_all add: below_def)
have "x \<cdot> (y \<cdot> z) = (x \<cdot> y) \<cdot> z" by(simp add:assoc)
also have "x \<cdot> y = x" using a by(simp_all add: below_def)
also have "x \<cdot> z = x" using a by(simp_all add: below_def)
finally show "x \<preceq> y \<cdot> z" by(simp_all add: below_def)
qed
lemma (in ACIfSLlin) above_f_conv:
"x \<cdot> y \<preceq> z = (x \<preceq> z \<or> y \<preceq> z)"
proof
assume a: "x \<cdot> y \<preceq> z"
have "x \<cdot> y = x \<or> x \<cdot> y = y" using lin[of x y] by simp
thus "x \<preceq> z \<or> y \<preceq> z"
proof
assume "x \<cdot> y = x" hence "x \<preceq> z" by(rule subst)(rule a) thus ?thesis ..
next
assume "x \<cdot> y = y" hence "y \<preceq> z" by(rule subst)(rule a) thus ?thesis ..
qed
next
assume "x \<preceq> z \<or> y \<preceq> z"
thus "x \<cdot> y \<preceq> z"
proof
assume a: "x \<preceq> z"
have "(x \<cdot> y) \<cdot> z = (x \<cdot> z) \<cdot> y" by(simp add:ACI)
also have "x \<cdot> z = x" using a by(simp add:below_def)
finally show "x \<cdot> y \<preceq> z" by(simp add:below_def)
next
assume a: "y \<preceq> z"
have "(x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)" by(simp add:ACI)
also have "y \<cdot> z = y" using a by(simp add:below_def)
finally show "x \<cdot> y \<preceq> z" by(simp add:below_def)
qed
qed
subsubsection{* Lemmas about {@text fold1} *}
lemma (in ACf) fold1_Un:
assumes A: "finite A" "A \<noteq> {}"
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
fold1 f (A Un B) = f (fold1 f A) (fold1 f B)"
using A
proof(induct rule:finite_ne_induct)
case singleton thus ?case by(simp add:fold1_insert)
next
case insert thus ?case by (simp add:fold1_insert assoc)
qed
lemma (in ACIf) fold1_Un2:
assumes A: "finite A" "A \<noteq> {}"
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
fold1 f (A Un B) = f (fold1 f A) (fold1 f B)"
using A
proof(induct rule:finite_ne_induct)
case singleton thus ?case by(simp add:fold1_insert2)
next
case insert thus ?case by (simp add:fold1_insert2 assoc)
qed
lemma (in ACf) fold1_in:
assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x\<cdot>y \<in> {x,y}"
shows "fold1 f A \<in> A"
using A
proof (induct rule:finite_ne_induct)
case singleton thus ?case by(simp)
next
case insert thus ?case using elem by (force simp add:fold1_insert)
qed
lemma (in ACIfSL) below_fold1_iff:
assumes A: "finite A" "A \<noteq> {}"
shows "x \<preceq> fold1 f A = (\<forall>a\<in>A. x \<preceq> a)"
using A
by(induct rule:finite_ne_induct) simp_all
lemma (in ACIfSL) fold1_belowI:
assumes A: "finite A" "A \<noteq> {}"
shows "a \<in> A \<Longrightarrow> fold1 f A \<preceq> a"
using A
proof (induct rule:finite_ne_induct)
case singleton thus ?case by simp
next
case (insert x F)
from insert(4) have "a = x \<or> a \<in> F" by simp
thus ?case
proof
assume "a = x" thus ?thesis using insert by(simp add:below_def ACI)
next
assume "a \<in> F"
hence bel: "fold1 op \<cdot> F \<preceq> a" by(rule insert)
have "fold1 op \<cdot> (insert x F) \<cdot> a = x \<cdot> (fold1 op \<cdot> F \<cdot> a)"
using insert by(simp add:below_def ACI)
also have "fold1 op \<cdot> F \<cdot> a = fold1 op \<cdot> F"
using bel by(simp add:below_def ACI)
also have "x \<cdot> \<dots> = fold1 op \<cdot> (insert x F)"
using insert by(simp add:below_def ACI)
finally show ?thesis by(simp add:below_def)
qed
qed
lemma (in ACIfSLlin) fold1_below_iff:
assumes A: "finite A" "A \<noteq> {}"
shows "fold1 f A \<preceq> x = (\<exists>a\<in>A. a \<preceq> x)"
using A
by(induct rule:finite_ne_induct)(simp_all add:above_f_conv)
subsection{*Min and Max*}
text{* As an application of @{text fold1} we define the minimal and
maximal element of a (non-empty) set over a linear order. *}
constdefs
Min :: "('a::linorder)set => 'a"
"Min == fold1 min"
Max :: "('a::linorder)set => 'a"
"Max == fold1 max"
text{* Before we can do anything, we need to show that @{text min} and
@{text max} are ACI and the ordering is linear: *}
lemma ACf_min: "ACf(min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
apply(rule ACf.intro)
apply(auto simp:min_def)
done
lemma ACIf_min: "ACIf(min:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
apply(rule ACIf.intro[OF ACf_min])
apply(rule ACIf_axioms.intro)
apply(auto simp:min_def)
done
lemma ACf_max: "ACf(max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
apply(rule ACf.intro)
apply(auto simp:max_def)
done
lemma ACIf_max: "ACIf(max:: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
apply(rule ACIf.intro[OF ACf_max])
apply(rule ACIf_axioms.intro)
apply(auto simp:max_def)
done
lemma ACIfSL_min: "ACIfSL(min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a) (op \<le>)"
apply(rule ACIfSL.intro)
apply(rule ACf_min)
apply(rule ACIf.axioms[OF ACIf_min])
apply(rule ACIfSL_axioms.intro)
apply(auto simp:min_def)
done
lemma ACIfSLlin_min: "ACIfSLlin(min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a) (op \<le>)"
apply(rule ACIfSLlin.intro)
apply(rule ACf_min)
apply(rule ACIf.axioms[OF ACIf_min])
apply(rule ACIfSL.axioms[OF ACIfSL_min])
apply(rule ACIfSLlin_axioms.intro)
apply(auto simp:min_def)
done
lemma ACIfSL_max: "ACIfSL(max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a) (%x y. y\<le>x)"
apply(rule ACIfSL.intro)
apply(rule ACf_max)
apply(rule ACIf.axioms[OF ACIf_max])
apply(rule ACIfSL_axioms.intro)
apply(auto simp:max_def)
done
lemma ACIfSLlin_max: "ACIfSLlin(max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a) (%x y. y\<le>x)"
apply(rule ACIfSLlin.intro)
apply(rule ACf_max)
apply(rule ACIf.axioms[OF ACIf_max])
apply(rule ACIfSL.axioms[OF ACIfSL_max])
apply(rule ACIfSLlin_axioms.intro)
apply(auto simp:max_def)
done
text{* Now we instantiate the recursion equations and declare them
simplification rules: *}
declare
fold1_singleton_def[OF Min_def, simp]
ACIf.fold1_insert2_def[OF ACIf_min Min_def, simp]
fold1_singleton_def[OF Max_def, simp]
ACIf.fold1_insert2_def[OF ACIf_max Max_def, simp]
text{* Now we instantiate some @{text fold1} properties: *}
lemma Min_in [simp]:
shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Min A \<in> A"
using ACf.fold1_in[OF ACf_min]
by(fastsimp simp: Min_def min_def)
lemma Max_in [simp]:
shows "finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> Max A \<in> A"
using ACf.fold1_in[OF ACf_max]
by(fastsimp simp: Max_def max_def)
lemma Min_le [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> Min A \<le> x"
by(simp add: Min_def ACIfSL.fold1_belowI[OF ACIfSL_min])
lemma Max_ge [simp]: "\<lbrakk> finite A; A \<noteq> {}; x \<in> A \<rbrakk> \<Longrightarrow> x \<le> Max A"
by(simp add: Max_def ACIfSL.fold1_belowI[OF ACIfSL_max])
lemma Min_ge_iff[simp]:
"\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (x \<le> Min A) = (\<forall>a\<in>A. x \<le> a)"
by(simp add: Min_def ACIfSL.below_fold1_iff[OF ACIfSL_min])
lemma Max_le_iff[simp]:
"\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (Max A \<le> x) = (\<forall>a\<in>A. a \<le> x)"
by(simp add: Max_def ACIfSL.below_fold1_iff[OF ACIfSL_max])
lemma Min_le_iff:
"\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (Min A \<le> x) = (\<exists>a\<in>A. a \<le> x)"
by(simp add: Min_def ACIfSLlin.fold1_below_iff[OF ACIfSLlin_min])
lemma Max_ge_iff:
"\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> (x \<le> Max A) = (\<exists>a\<in>A. x \<le> a)"
by(simp add: Max_def ACIfSLlin.fold1_below_iff[OF ACIfSLlin_max])
end