(* Title: HOL/Finite_Set.thy
Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
with contributions by Jeremy Avigad
*)
header {* Finite sets *}
theory Finite_Set
imports Nat Product_Type Power
begin
subsection {* Definition and basic properties *}
inductive finite :: "'a set => bool"
where
emptyI [simp, intro!]: "finite {}"
| insertI [simp, intro!]: "finite A ==> finite (insert a A)"
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 assms have "A \<noteq> UNIV" by blast
thus ?thesis by blast
qed
lemma finite_induct [case_names empty insert, induct set: finite]:
"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 {}" by fact
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 `P {x}` by simp
next
assume "F \<noteq> {}"
thus ?thesis using insert by blast
qed
qed
lemma finite_subset_induct [consumes 2, case_names empty insert]:
assumes "finite F" and "F \<subseteq> A"
and empty: "P {}"
and insert: "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)"
shows "P F"
proof -
from `finite F` and `F \<subseteq> A`
show ?thesis
proof induct
show "P {}" by fact
next
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" .
show "finite F" by fact
show "x \<notin> F" by fact
qed
qed
qed
text{* A finite choice principle. Does not need the SOME choice operator. *}
lemma finite_set_choice:
"finite A \<Longrightarrow> ALL x:A. (EX y. P x y) \<Longrightarrow> EX f. ALL x:A. P x (f x)"
proof (induct set: finite)
case empty thus ?case by simp
next
case (insert a A)
then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto
show ?case (is "EX f. ?P f")
proof
show "?P(%x. if x = a then b else f x)" using f ab by auto
qed
qed
text{* Finite sets are the images of initial segments of natural numbers: *}
lemma finite_imp_nat_seg_image_inj_on:
assumes fin: "finite A"
shows "\<exists> (n::nat) f. A = f ` {i. i<n} & inj_on f {i. i<n}"
using fin
proof induct
case empty
show ?case
proof show "\<exists>f. {} = f ` {i::nat. i < 0} & inj_on f {i. i<0}" by simp
qed
next
case (insert a A)
have notinA: "a \<notin> A" by fact
from insert.hyps obtain n f
where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast
hence "insert a A = f(n:=a) ` {i. i < Suc n}"
"inj_on (f(n:=a)) {i. i < Suc n}" using notinA
by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq)
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: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on)
lemma finite_Collect_less_nat[iff]: "finite{n::nat. n<k}"
by(fastsimp simp: finite_conv_nat_seg_image)
subsubsection{* Finiteness and set theoretic constructions *}
lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
by (induct set: finite) 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})" by fact+
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" using x by (rule insert_Diff)
finally show ?thesis .
next
show "A \<subseteq> F ==> ?thesis" by fact
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_Collect_disjI[simp]:
"finite{x. P x | Q x} = (finite{x. P x} & finite{x. Q x})"
by(simp add:Collect_disj_eq)
lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
-- {* The converse obviously fails. *}
by (blast intro: finite_subset)
lemma finite_Collect_conjI [simp, intro]:
"finite{x. P x} | finite{x. Q x} ==> finite{x. P x & Q x}"
-- {* The converse obviously fails. *}
by(simp add:Collect_conj_eq)
lemma finite_Collect_le_nat[iff]: "finite{n::nat. n<=k}"
by(simp add: le_eq_less_or_eq)
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_Inter[intro]: "EX A:M. finite(A) \<Longrightarrow> finite(Inter M)"
by (blast intro: Inter_lower finite_subset)
lemma finite_INT[intro]: "EX x:I. finite(A x) \<Longrightarrow> finite(INT x:I. A x)"
by (blast intro: INT_lower finite_subset)
lemma finite_empty_induct:
assumes "finite A"
and "P A"
and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})"
shows "P {}"
proof -
have "P (A - A)"
proof -
{
fix c b :: "'a set"
assume 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})"
have "c \<subseteq> b ==> P (b - c)"
using 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
}
then show ?thesis by this (simp_all add: assms)
qed
then show ?thesis by simp
qed
lemma finite_Diff [simp]: "finite A ==> finite (A - B)"
by (rule Diff_subset [THEN finite_subset])
lemma finite_Diff2 [simp]:
assumes "finite B" shows "finite (A - B) = finite A"
proof -
have "finite A \<longleftrightarrow> finite((A-B) Un (A Int B))" by(simp add: Un_Diff_Int)
also have "\<dots> \<longleftrightarrow> finite(A-B)" using `finite B` by(simp)
finally show ?thesis ..
qed
lemma finite_compl[simp]:
"finite(A::'a set) \<Longrightarrow> finite(-A) = finite(UNIV::'a set)"
by(simp add:Compl_eq_Diff_UNIV)
lemma finite_Collect_not[simp]:
"finite{x::'a. P x} \<Longrightarrow> finite{x. ~P x} = finite(UNIV::'a set)"
by(simp add:Collect_neg_eq)
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: finite) simp_all
lemma finite_image_set [simp]:
"finite {x. P x} \<Longrightarrow> finite { f x | x. P x }"
by (simp add: image_Collect [symmetric])
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 add: range_composition)
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: finite)
apply 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: finite) 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)
lemma finite_Collect_bex[simp]: "finite A \<Longrightarrow>
finite{x. EX y:A. Q x y} = (ALL y:A. finite{x. Q x y})"
apply(subgoal_tac "{x. EX y:A. Q x y} = UNION A (%y. {x. Q x y})")
apply auto
done
lemma finite_Collect_bounded_ex[simp]: "finite{y. P y} \<Longrightarrow>
finite{x. EX y. P y & Q x y} = (ALL y. P y \<longrightarrow> finite{x. Q x y})"
apply(subgoal_tac "{x. EX y. P y & Q x y} = UNION {y. P y} (%y. {x. Q x y})")
apply auto
done
lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)"
by (simp add: Plus_def)
lemma finite_PlusD:
fixes A :: "'a set" and B :: "'b set"
assumes fin: "finite (A <+> B)"
shows "finite A" "finite B"
proof -
have "Inl ` A \<subseteq> A <+> B" by auto
hence "finite (Inl ` A :: ('a + 'b) set)" using fin by(rule finite_subset)
thus "finite A" by(rule finite_imageD)(auto intro: inj_onI)
next
have "Inr ` B \<subseteq> A <+> B" by auto
hence "finite (Inr ` B :: ('a + 'b) set)" using fin by(rule finite_subset)
thus "finite B" by(rule finite_imageD)(auto intro: inj_onI)
qed
lemma finite_Plus_iff[simp]: "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B"
by(auto intro: finite_PlusD finite_Plus)
lemma finite_Plus_UNIV_iff[simp]:
"finite (UNIV :: ('a + 'b) set) =
(finite (UNIV :: 'a set) & finite (UNIV :: 'b set))"
by(subst UNIV_Plus_UNIV[symmetric])(rule finite_Plus_iff)
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
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_Collect_subsets[simp,intro]: "finite A \<Longrightarrow> finite{B. B \<subseteq> A}"
by(simp add: Pow_def[symmetric])
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"
by(blast intro: finite_subset[OF subset_Pow_Union])
lemma finite_subset_image:
assumes "finite B"
shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C"
using assms proof(induct)
case empty thus ?case by simp
next
case insert thus ?case
by (clarsimp simp del: image_insert simp add: image_insert[symmetric])
blast
qed
subsection {* Class @{text finite} *}
setup {* Sign.add_path "finite" *} -- {*FIXME: name tweaking*}
class finite =
assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)"
setup {* Sign.parent_path *}
hide const finite
context finite
begin
lemma finite [simp]: "finite (A \<Colon> 'a set)"
by (rule subset_UNIV finite_UNIV finite_subset)+
end
lemma UNIV_unit [noatp]:
"UNIV = {()}" by auto
instance unit :: finite
by default (simp add: UNIV_unit)
lemma UNIV_bool [noatp]:
"UNIV = {False, True}" by auto
instance bool :: finite
by default (simp add: UNIV_bool)
instance * :: (finite, finite) finite
by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
lemma inj_graph: "inj (%f. {(x, y). y = f x})"
by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq)
instance "fun" :: (finite, finite) finite
proof
show "finite (UNIV :: ('a => 'b) set)"
proof (rule finite_imageD)
let ?graph = "%f::'a => 'b. {(x, y). y = f x}"
have "range ?graph \<subseteq> Pow UNIV" by simp
moreover have "finite (Pow (UNIV :: ('a * 'b) set))"
by (simp only: finite_Pow_iff finite)
ultimately show "finite (range ?graph)"
by (rule finite_subset)
show "inj ?graph" by (rule inj_graph)
qed
qed
instance "+" :: (finite, finite) finite
by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
subsection {* A fold functional for finite sets *}
text {* The intended behaviour is
@{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
if @{text f} is ``left-commutative'':
*}
locale fun_left_comm =
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
assumes fun_left_comm: "f x (f y z) = f y (f x z)"
begin
text{* On a functional level it looks much nicer: *}
lemma fun_comp_comm: "f x \<circ> f y = f y \<circ> f x"
by (simp add: fun_left_comm expand_fun_eq)
end
inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where
emptyI [intro]: "fold_graph f z {} z" |
insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y
\<Longrightarrow> fold_graph f z (insert x A) (f x y)"
inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where
[code del]: "fold f z A = (THE y. fold_graph f z A y)"
text{*A tempting alternative for the definiens is
@{term "if finite A then THE y. fold_graph f z A y else e"}.
It allows the removal of finiteness assumptions from the theorems
@{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}.
The proofs become ugly. It is not worth the effort. (???) *}
lemma Diff1_fold_graph:
"fold_graph f z (A - {x}) y \<Longrightarrow> x \<in> A \<Longrightarrow> fold_graph f z A (f x y)"
by (erule insert_Diff [THEN subst], rule fold_graph.intros, auto)
lemma fold_graph_imp_finite: "fold_graph f z A x \<Longrightarrow> finite A"
by (induct set: fold_graph) auto
lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x"
by (induct set: finite) auto
subsubsection{*From @{const fold_graph} to @{term fold}*}
lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})"
by (auto simp add: less_Suc_eq)
lemma insert_image_inj_on_eq:
"[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A;
inj_on h {i. i < Suc m}|]
==> A = h ` {i. i < m}"
apply (auto simp add: image_less_Suc inj_on_def)
apply (blast intro: less_trans)
done
lemma insert_inj_onE:
assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A"
and inj_on: "inj_on h {i::nat. i<n}"
shows "\<exists>hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n"
proof (cases n)
case 0 thus ?thesis using aA by auto
next
case (Suc m)
have nSuc: "n = Suc m" by fact
have mlessn: "m<n" by (simp add: nSuc)
from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE)
let ?hm = "Fun.swap k m h"
have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn
by (simp add: inj_on_swap_iff inj_on)
show ?thesis
proof (intro exI conjI)
show "inj_on ?hm {i. i < m}" using inj_hm
by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on)
show "m<n" by (rule mlessn)
show "A = ?hm ` {i. i < m}"
proof (rule insert_image_inj_on_eq)
show "inj_on (Fun.swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp
show "?hm m \<notin> A" by (simp add: swap_def hkeq anot)
show "insert (?hm m) A = ?hm ` {i. i < Suc m}"
using aA hkeq nSuc klessn
by (auto simp add: swap_def image_less_Suc fun_upd_image
less_Suc_eq inj_on_image_set_diff [OF inj_on])
qed
qed
qed
context fun_left_comm
begin
lemma fold_graph_determ_aux:
"A = h`{i::nat. i<n} \<Longrightarrow> inj_on h {i. i<n}
\<Longrightarrow> fold_graph f z A x \<Longrightarrow> fold_graph f z A x'
\<Longrightarrow> x' = x"
proof (induct n arbitrary: A x x' h rule: less_induct)
case (less n)
have IH: "\<And>m h A x x'. m < n \<Longrightarrow> A = h ` {i. i<m}
\<Longrightarrow> inj_on h {i. i<m} \<Longrightarrow> fold_graph f z A x
\<Longrightarrow> fold_graph f z A x' \<Longrightarrow> x' = x" by fact
have Afoldx: "fold_graph f z A x" and Afoldx': "fold_graph f z A x'"
and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" by fact+
show ?case
proof (rule fold_graph.cases [OF Afoldx])
assume "A = {}" and "x = z"
with Afoldx' show "x' = x" by auto
next
fix B b u
assume AbB: "A = insert b B" and x: "x = f b u"
and notinB: "b \<notin> B" and Bu: "fold_graph f z B u"
show "x'=x"
proof (rule fold_graph.cases [OF Afoldx'])
assume "A = {}" and "x' = z"
with AbB show "x' = x" by blast
next
fix C c v
assume AcC: "A = insert c C" and x': "x' = f c v"
and notinC: "c \<notin> C" and Cv: "fold_graph f z C v"
from A AbB have Beq: "insert b B = h`{i. i<n}" by simp
from insert_inj_onE [OF Beq notinB injh]
obtain hB mB where inj_onB: "inj_on hB {i. i < mB}"
and Beq: "B = hB ` {i. i < mB}" and lessB: "mB < n" by auto
from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp
from insert_inj_onE [OF Ceq notinC injh]
obtain hC mC where inj_onC: "inj_on hC {i. i < mC}"
and Ceq: "C = hC ` {i. i < mC}" and lessC: "mC < n" by auto
show "x'=x"
proof cases
assume "b=c"
then moreover have "B = C" using AbB AcC notinB notinC by auto
ultimately show ?thesis using Bu Cv x x' IH [OF lessC Ceq inj_onC]
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 AbB AcC notinB notinC diff by(blast elim!:equalityE)+
have "finite A" by(rule fold_graph_imp_finite [OF Afoldx])
with AbB have "finite ?D" by simp
then obtain d where Dfoldd: "fold_graph f z ?D d"
using finite_imp_fold_graph by iprover
moreover have cinB: "c \<in> B" using B by auto
ultimately have "fold_graph f z B (f c d)" by(rule Diff1_fold_graph)
hence "f c d = u" by (rule IH [OF lessB Beq inj_onB Bu])
moreover have "f b d = v"
proof (rule IH[OF lessC Ceq inj_onC Cv])
show "fold_graph f z C (f b d)" using C notinB Dfoldd by fastsimp
qed
ultimately show ?thesis
using fun_left_comm [of c b] x x' by (auto simp add: o_def)
qed
qed
qed
qed
lemma fold_graph_determ:
"fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x"
apply (frule fold_graph_imp_finite [THEN finite_imp_nat_seg_image_inj_on])
apply (blast intro: fold_graph_determ_aux [rule_format])
done
lemma fold_equality:
"fold_graph f z A y \<Longrightarrow> fold f z A = y"
by (unfold fold_def) (blast intro: fold_graph_determ)
text{* The base case for @{text fold}: *}
lemma (in -) fold_empty [simp]: "fold f z {} = z"
by (unfold fold_def) blast
text{* The various recursion equations for @{const fold}: *}
lemma fold_insert_aux: "x \<notin> A
\<Longrightarrow> fold_graph f z (insert x A) v \<longleftrightarrow>
(\<exists>y. fold_graph f z A y \<and> v = f x y)"
apply auto
apply (rule_tac A1 = A and f1 = f in finite_imp_fold_graph [THEN exE])
apply (fastsimp dest: fold_graph_imp_finite)
apply (blast intro: fold_graph_determ)
done
lemma fold_insert [simp]:
"finite A ==> x \<notin> A ==> fold f z (insert x A) = f x (fold f z A)"
apply (simp add: fold_def fold_insert_aux)
apply (rule the_equality)
apply (auto intro: finite_imp_fold_graph
cong add: conj_cong simp add: fold_def[symmetric] fold_equality)
done
lemma fold_fun_comm:
"finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A"
proof (induct rule: finite_induct)
case empty then show ?case by simp
next
case (insert y A) then show ?case
by (simp add: fun_left_comm[of x])
qed
lemma fold_insert2:
"finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
by (simp add: fold_insert fold_fun_comm)
lemma fold_rec:
assumes "finite A" and "x \<in> A"
shows "fold f z A = f x (fold f z (A - {x}))"
proof -
have A: "A = insert x (A - {x})" using `x \<in> A` by blast
then have "fold f z A = fold f z (insert x (A - {x}))" by simp
also have "\<dots> = f x (fold f z (A - {x}))"
by (rule fold_insert) (simp add: `finite A`)+
finally show ?thesis .
qed
lemma fold_insert_remove:
assumes "finite A"
shows "fold f z (insert x A) = f x (fold f z (A - {x}))"
proof -
from `finite A` have "finite (insert x A)" by auto
moreover have "x \<in> insert x A" by auto
ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))"
by (rule fold_rec)
then show ?thesis by simp
qed
end
text{* A simplified version for idempotent functions: *}
locale fun_left_comm_idem = fun_left_comm +
assumes fun_left_idem: "f x (f x z) = f x z"
begin
text{* The nice version: *}
lemma fun_comp_idem : "f x o f x = f x"
by (simp add: fun_left_idem expand_fun_eq)
lemma fold_insert_idem:
assumes fin: "finite A"
shows "fold f z (insert x A) = f x (fold f z A)"
proof cases
assume "x \<in> A"
then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert)
then show ?thesis using assms by (simp add:fun_left_idem)
next
assume "x \<notin> A" then show ?thesis using assms by simp
qed
declare fold_insert[simp del] fold_insert_idem[simp]
lemma fold_insert_idem2:
"finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A"
by(simp add:fold_fun_comm)
end
context ab_semigroup_idem_mult
begin
lemma fun_left_comm_idem: "fun_left_comm_idem(op *)"
apply unfold_locales
apply (simp add: mult_ac)
apply (simp add: mult_idem mult_assoc[symmetric])
done
end
context lower_semilattice
begin
lemma ab_semigroup_idem_mult_inf: "ab_semigroup_idem_mult inf"
proof qed (rule inf_assoc inf_commute inf_idem)+
lemma fold_inf_insert[simp]: "finite A \<Longrightarrow> fold inf b (insert a A) = inf a (fold inf b A)"
by(rule fun_left_comm_idem.fold_insert_idem[OF ab_semigroup_idem_mult.fun_left_comm_idem[OF ab_semigroup_idem_mult_inf]])
lemma inf_le_fold_inf: "finite A \<Longrightarrow> ALL a:A. b \<le> a \<Longrightarrow> inf b c \<le> fold inf c A"
by (induct pred: finite) (auto intro: le_infI1)
lemma fold_inf_le_inf: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> fold inf b A \<le> inf a b"
proof(induct arbitrary: a pred:finite)
case empty thus ?case by simp
next
case (insert x A)
show ?case
proof cases
assume "A = {}" thus ?thesis using insert by simp
next
assume "A \<noteq> {}" thus ?thesis using insert by (auto intro: le_infI2)
qed
qed
end
context upper_semilattice
begin
lemma ab_semigroup_idem_mult_sup: "ab_semigroup_idem_mult sup"
by (rule lower_semilattice.ab_semigroup_idem_mult_inf)(rule dual_semilattice)
lemma fold_sup_insert[simp]: "finite A \<Longrightarrow> fold sup b (insert a A) = sup a (fold sup b A)"
by(rule lower_semilattice.fold_inf_insert)(rule dual_semilattice)
lemma fold_sup_le_sup: "finite A \<Longrightarrow> ALL a:A. a \<le> b \<Longrightarrow> fold sup c A \<le> sup b c"
by(rule lower_semilattice.inf_le_fold_inf)(rule dual_semilattice)
lemma sup_le_fold_sup: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a b \<le> fold sup b A"
by(rule lower_semilattice.fold_inf_le_inf)(rule dual_semilattice)
end
subsubsection{* The derived combinator @{text fold_image} *}
definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
where "fold_image f g = fold (%x y. f (g x) y)"
lemma fold_image_empty[simp]: "fold_image f g z {} = z"
by(simp add:fold_image_def)
context ab_semigroup_mult
begin
lemma fold_image_insert[simp]:
assumes "finite A" and "a \<notin> A"
shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)"
proof -
interpret I: fun_left_comm "%x y. (g x) * y"
by unfold_locales (simp add: mult_ac)
show ?thesis using assms by(simp add:fold_image_def)
qed
(*
lemma fold_commute:
"finite A ==> (!!z. x * (fold times g z A) = fold times g (x * z) A)"
apply (induct set: finite)
apply simp
apply (simp add: mult_left_commute [of x])
done
lemma fold_nest_Un_Int:
"finite A ==> finite B
==> fold times g (fold times g z B) A = fold times g (fold times g z (A Int B)) (A Un B)"
apply (induct set: finite)
apply simp
apply (simp add: fold_commute Int_insert_left insert_absorb)
done
lemma fold_nest_Un_disjoint:
"finite A ==> finite B ==> A Int B = {}
==> fold times g z (A Un B) = fold times g (fold times g z B) A"
by (simp add: fold_nest_Un_Int)
*)
lemma fold_image_reindex:
assumes fin: "finite A"
shows "inj_on h A \<Longrightarrow> fold_image times g z (h`A) = fold_image times (g\<circ>h) z A"
using fin by induct auto
(*
text{*
Fusion theorem, as described in Graham Hutton's paper,
A Tutorial on the Universality and Expressiveness of Fold,
JFP 9:4 (355-372), 1999.
*}
lemma fold_fusion:
assumes "ab_semigroup_mult g"
assumes fin: "finite A"
and hyp: "\<And>x y. h (g x y) = times x (h y)"
shows "h (fold g j w A) = fold times j (h w) A"
proof -
class_interpret ab_semigroup_mult [g] by fact
show ?thesis using fin hyp by (induct set: finite) simp_all
qed
*)
lemma fold_image_cong:
"finite A \<Longrightarrow>
(!!x. x:A ==> g x = h x) ==> fold_image times g z A = fold_image times h z A"
apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times 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
end
context comm_monoid_mult
begin
lemma fold_image_Un_Int:
"finite A ==> finite B ==>
fold_image times g 1 A * fold_image times g 1 B =
fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)"
by (induct set: finite)
(auto simp add: mult_ac insert_absorb Int_insert_left)
corollary fold_Un_disjoint:
"finite A ==> finite B ==> A Int B = {} ==>
fold_image times g 1 (A Un B) =
fold_image times g 1 A * fold_image times g 1 B"
by (simp add: fold_image_Un_Int)
lemma fold_image_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_image times g 1 (UNION I A) =
fold_image times (%i. fold_image times g 1 (A i)) 1 I"
apply (induct set: finite, 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 fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==>
fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A =
fold_image times (split g) 1 (SIGMA x:A. B x)"
apply (subst Sigma_def)
apply (subst fold_image_UN_disjoint, assumption, simp)
apply blast
apply (erule fold_image_cong)
apply (subst fold_image_UN_disjoint, simp, simp)
apply blast
apply simp
done
lemma fold_image_distrib: "finite A \<Longrightarrow>
fold_image times (%x. g x * h x) 1 A =
fold_image times g 1 A * fold_image times h 1 A"
by (erule finite_induct) (simp_all add: mult_ac)
lemma fold_image_related:
assumes Re: "R e e"
and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)"
and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)"
using fS by (rule finite_subset_induct) (insert assms, auto)
lemma fold_image_eq_general:
assumes fS: "finite S"
and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y"
and f12: "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x"
shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'"
proof-
from h f12 have hS: "h ` S = S'" by auto
{fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
from f12 h H have "x = y" by auto }
hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp
also have "\<dots> = fold_image (op *) (f2 o h) e S"
using fold_image_reindex[OF fS hinj, of f2 e] .
also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e]
by blast
finally show ?thesis ..
qed
lemma fold_image_eq_general_inverses:
assumes fS: "finite S"
and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"
shows "fold_image (op *) f e S = fold_image (op *) g e T"
(* metis solves it, but not yet available here *)
apply (rule fold_image_eq_general[OF fS, of T h g f e])
apply (rule ballI)
apply (frule kh)
apply (rule ex1I[])
apply blast
apply clarsimp
apply (drule hk) apply simp
apply (rule sym)
apply (erule conjunct1[OF conjunct2[OF hk]])
apply (rule ballI)
apply (drule hk)
apply blast
done
end
subsection {* Generalized summation over a set *}
interpretation comm_monoid_add: comm_monoid_mult "0::'a::comm_monoid_add" "op +"
proof qed (auto intro: add_assoc add_commute)
definition setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add"
where "setsum f A == if finite A then fold_image (op +) f 0 A else 0"
abbreviation
Setsum ("\<Sum>_" [1000] 999) where
"\<Sum>A == setsum (%x. x) A"
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is
written @{text"\<Sum>x\<in>A. e"}. *}
syntax
"_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3SUM _:_. _)" [0, 51, 10] 10)
syntax (xsymbols)
"_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
syntax (HTML output)
"_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10)
translations -- {* Beware of argument permutation! *}
"SUM i:A. b" == "CONST setsum (%i. b) A"
"\<Sum>i\<in>A. b" == "CONST 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" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10)
syntax (xsymbols)
"_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
syntax (HTML output)
"_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10)
translations
"SUM x|P. t" => "CONST setsum (%x. t) {x. P}"
"\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}"
print_translation {*
let
fun 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)
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 comm_monoid_add.fold_image_reindex 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_reindex_nonzero:
assumes fS: "finite S"
and nz: "\<And> x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0"
shows "setsum h (f ` S) = setsum (h o f) S"
using nz
proof(induct rule: finite_induct[OF fS])
case 1 thus ?case by simp
next
case (2 x F)
{assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
then obtain y where y: "y \<in> F" "f x = f y" by auto
from "2.hyps" y have xy: "x \<noteq> y" by auto
from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
also have "\<dots> = setsum (h o f) (insert x F)"
unfolding setsum_insert[OF `finite F` `x\<notin>F`]
using h0
apply simp
apply (rule "2.hyps"(3))
apply (rule_tac y="y" in "2.prems")
apply simp_all
done
finally have ?case .}
moreover
{assume fxF: "f x \<notin> f ` F"
have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)"
using fxF "2.hyps" by simp
also have "\<dots> = setsum (h o f) (insert x F)"
unfolding setsum_insert[OF `finite F` `x\<notin>F`]
apply simp
apply (rule cong[OF refl[of "op + (h (f x))"]])
apply (rule "2.hyps"(3))
apply (rule_tac y="y" in "2.prems")
apply simp_all
done
finally have ?case .}
ultimately show ?case by blast
qed
lemma setsum_cong:
"A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B"
by(fastsimp simp: setsum_def intro: comm_monoid_add.fold_image_cong)
lemma strong_setsum_cong[cong]:
"A = B ==> (!!x. x:B =simp=> f x = g x)
==> setsum (%x. f x) A = setsum (%x. g x) B"
by(fastsimp simp: simp_implies_def setsum_def intro: comm_monoid_add.fold_image_cong)
lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A";
by (rule setsum_cong[OF refl], auto);
lemma setsum_reindex_cong:
"[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|]
==> setsum h B = setsum g A"
by (simp add: setsum_reindex cong: setsum_cong)
lemma setsum_0[simp]: "setsum (%i. 0) A = 0"
apply (clarsimp simp: setsum_def)
apply (erule finite_induct, auto)
done
lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0"
by(simp add:setsum_cong)
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 comm_monoid_add.fold_image_Un_Int [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)
lemma setsum_mono_zero_left:
assumes fT: "finite T" and ST: "S \<subseteq> T"
and z: "\<forall>i \<in> T - S. f i = 0"
shows "setsum f S = setsum f T"
proof-
have eq: "T = S \<union> (T - S)" using ST by blast
have d: "S \<inter> (T - S) = {}" using ST by blast
from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset)
show ?thesis
by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z])
qed
lemma setsum_mono_zero_right:
"finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. f i = 0 \<Longrightarrow> setsum f T = setsum f S"
by(blast intro!: setsum_mono_zero_left[symmetric])
lemma setsum_mono_zero_cong_left:
assumes fT: "finite T" and ST: "S \<subseteq> T"
and z: "\<forall>i \<in> T - S. g i = 0"
and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
shows "setsum f S = setsum g T"
proof-
have eq: "T = S \<union> (T - S)" using ST by blast
have d: "S \<inter> (T - S) = {}" using ST by blast
from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset)
show ?thesis
using fg by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z])
qed
lemma setsum_mono_zero_cong_right:
assumes fT: "finite T" and ST: "S \<subseteq> T"
and z: "\<forall>i \<in> T - S. f i = 0"
and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
shows "setsum f T = setsum g S"
using setsum_mono_zero_cong_left[OF fT ST z] fg[symmetric] by auto
lemma setsum_delta:
assumes fS: "finite S"
shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
proof-
let ?f = "(\<lambda>k. if k=a then b k else 0)"
{assume a: "a \<notin> S"
hence "\<forall> k\<in> S. ?f k = 0" by simp
hence ?thesis using a by simp}
moreover
{assume a: "a \<in> S"
let ?A = "S - {a}"
let ?B = "{a}"
have eq: "S = ?A \<union> ?B" using a by blast
have dj: "?A \<inter> ?B = {}" by simp
from fS have fAB: "finite ?A" "finite ?B" by auto
have "setsum ?f S = setsum ?f ?A + setsum ?f ?B"
using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
by simp
then have ?thesis using a by simp}
ultimately show ?thesis by blast
qed
lemma setsum_delta':
assumes fS: "finite S" shows
"setsum (\<lambda>k. if a = k then b k else 0) S =
(if a\<in> S then b a else 0)"
using setsum_delta[OF fS, of a b, symmetric]
by (auto intro: setsum_cong)
lemma setsum_restrict_set:
assumes fA: "finite A"
shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A"
proof-
from fA have fab: "finite (A \<inter> B)" by auto
have aba: "A \<inter> B \<subseteq> A" by blast
let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0"
from setsum_mono_zero_left[OF fA aba, of ?g]
show ?thesis by simp
qed
lemma setsum_cases:
assumes fA: "finite A"
shows "setsum (\<lambda>x. if x \<in> B then f x else g x) A =
setsum f (A \<inter> B) + setsum g (A \<inter> - B)"
proof-
have a: "A = A \<inter> B \<union> A \<inter> -B" "(A \<inter> B) \<inter> (A \<inter> -B) = {}"
by blast+
from fA
have f: "finite (A \<inter> B)" "finite (A \<inter> -B)" by auto
let ?g = "\<lambda>x. if x \<in> B then f x else g x"
from setsum_Un_disjoint[OF f a(2), of ?g] a(1)
show ?thesis by simp
qed
(*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 comm_monoid_add.fold_image_UN_disjoint 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>(x,y)\<in>(SIGMA x:A. B x). f x y)"
by(simp add:setsum_def comm_monoid_add.fold_image_Sigma 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>(x,y) \<in> A <*> B. f x y)"
apply (cases "finite A")
apply (cases "finite B")
apply (simp add: setsum_Sigma)
apply (cases "A={}", simp)
apply (simp)
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 comm_monoid_add.fold_image_distrib)
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: finite) auto
lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
(setsum f A = Suc 0) = (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
apply(erule finite_induct)
apply (auto simp add:add_is_1)
done
lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
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: algebra_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: algebra_simps)
lemma (in comm_monoid_mult) fold_image_1: "finite S \<Longrightarrow> (\<forall>x\<in>S. f x = 1) \<Longrightarrow> fold_image op * f 1 S = 1"
apply (induct set: finite)
apply simp by (auto simp add: fold_image_insert)
lemma (in comm_monoid_mult) fold_image_Un_one:
assumes fS: "finite S" and fT: "finite T"
and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
shows "fold_image (op *) f 1 (S \<union> T) = fold_image (op *) f 1 S * fold_image (op *) f 1 T"
proof-
have "fold_image op * f 1 (S \<inter> T) = 1"
apply (rule fold_image_1)
using fS fT I0 by auto
with fold_image_Un_Int[OF fS fT] show ?thesis by simp
qed
lemma setsum_eq_general_reverses:
assumes fS: "finite S" and fT: "finite T"
and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"
shows "setsum f S = setsum g T"
apply (simp add: setsum_def fS fT)
apply (rule comm_monoid_add.fold_image_eq_general_inverses[OF fS])
apply (erule kh)
apply (erule hk)
done
lemma setsum_Un_zero:
assumes fS: "finite S" and fT: "finite T"
and I0: "\<forall>x \<in> S\<inter>T. f x = 0"
shows "setsum f (S \<union> T) = setsum f S + setsum f T"
using fS fT
apply (simp add: setsum_def)
apply (rule comm_monoid_add.fold_image_Un_one)
using I0 by auto
lemma setsum_UNION_zero:
assumes fS: "finite S" and fSS: "\<forall>T \<in> S. finite T"
and f0: "\<And>T1 T2 x. T1\<in>S \<Longrightarrow> T2\<in>S \<Longrightarrow> T1 \<noteq> T2 \<Longrightarrow> x \<in> T1 \<Longrightarrow> x \<in> T2 \<Longrightarrow> f x = 0"
shows "setsum f (\<Union>S) = setsum (\<lambda>T. setsum f T) S"
using fSS f0
proof(induct rule: finite_induct[OF fS])
case 1 thus ?case by simp
next
case (2 T F)
then have fTF: "finite T" "\<forall>T\<in>F. finite T" "finite F" and TF: "T \<notin> F"
and H: "setsum f (\<Union> F) = setsum (setsum f) F" by (auto simp add: finite_insert)
from fTF have fUF: "finite (\<Union>F)" by (auto intro: finite_Union)
from "2.prems" TF fTF
show ?case
by (auto simp add: H[symmetric] intro: setsum_Un_zero[OF fTF(1) fUF, of f])
qed
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)
lemma setsum_diff1'[rule_format]:
"finite A \<Longrightarrow> a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x)"
apply (erule finite_induct[where F=A and P="% A. (a \<in> A \<longrightarrow> (\<Sum> x \<in> A. f x) = f a + (\<Sum> x \<in> (A - {a}). f x))"])
apply (auto simp add: insert_Diff_if add_ac)
done
lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
unfolding setsum_diff1'[OF assms] by auto
(* By Jeremy Siek: *)
lemma setsum_diff_nat:
assumes "finite B" and "B \<subseteq> A"
shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
using assms
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 fastsimp
qed
next
case False
thus ?thesis
by (simp add: setsum_def)
qed
lemma setsum_strict_mono:
fixes f :: "'a \<Rightarrow> 'b::{pordered_cancel_ab_semigroup_add,comm_monoid_add}"
assumes "finite A" "A \<noteq> {}"
and "!!x. x:A \<Longrightarrow> f x < g x"
shows "setsum f A < setsum g A"
using prems
proof (induct rule: finite_ne_induct)
case singleton thus ?case by simp
next
case insert thus ?case by (auto simp: add_strict_mono)
qed
lemma setsum_negf:
"setsum (%x. - (f x)::'a::ab_group_add) A = - setsum f A"
proof (cases "finite A")
case True thus ?thesis by (induct set: finite) auto
next
case False thus ?thesis by (simp add: setsum_def)
qed
lemma setsum_subtractf:
"setsum (%x. ((f x)::'a::ab_group_add) - g x) A =
setsum f A - setsum g A"
proof (cases "finite A")
case True thus ?thesis by (simp add: diff_minus setsum_addf setsum_negf)
next
case False thus ?thesis by (simp add: setsum_def)
qed
lemma setsum_nonneg:
assumes nn: "\<forall>x\<in>A. (0::'a::{pordered_ab_semigroup_add,comm_monoid_add}) \<le> f x"
shows "0 \<le> setsum f A"
proof (cases "finite A")
case True thus ?thesis using nn
proof induct
case empty then show ?case by simp
next
case (insert x F)
then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
with insert show ?case by simp
qed
next
case False thus ?thesis by (simp add: setsum_def)
qed
lemma setsum_nonpos:
assumes np: "\<forall>x\<in>A. f x \<le> (0::'a::{pordered_ab_semigroup_add,comm_monoid_add})"
shows "setsum f A \<le> 0"
proof (cases "finite A")
case True thus ?thesis using np
proof induct
case empty then show ?case by simp
next
case (insert x F)
then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
with insert show ?case by simp
qed
next
case False thus ?thesis by (simp add: setsum_def)
qed
lemma setsum_mono2:
fixes f :: "'a \<Rightarrow> 'b :: {pordered_ab_semigroup_add_imp_le,comm_monoid_add}"
assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
shows "setsum f A \<le> setsum f B"
proof -
have "setsum f A \<le> setsum f A + setsum f (B-A)"
by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
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_mono3: "finite B ==> A <= B ==>
ALL x: B - A.
0 <= ((f x)::'a::{comm_monoid_add,pordered_ab_semigroup_add}) ==>
setsum f A <= setsum f B"
apply (subgoal_tac "setsum f B = setsum f A + setsum f (B - A)")
apply (erule ssubst)
apply (subgoal_tac "setsum f A + 0 <= setsum f A + setsum f (B - A)")
apply simp
apply (rule add_left_mono)
apply (erule setsum_nonneg)
apply (subst setsum_Un_disjoint [THEN sym])
apply (erule finite_subset, assumption)
apply (rule finite_subset)
prefer 2
apply assumption
apply (auto simp add: sup_absorb2)
done
lemma setsum_right_distrib:
fixes f :: "'a => ('b::semiring_0)"
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_left_distrib:
"setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
proof (cases "finite A")
case True
then show ?thesis
proof induct
case empty thus ?case by simp
next
case (insert x A) thus ?case by (simp add: left_distrib)
qed
next
case False thus ?thesis by (simp add: setsum_def)
qed
lemma setsum_divide_distrib:
"setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
proof (cases "finite A")
case True
then show ?thesis
proof induct
case empty thus ?case by simp
next
case (insert x A) thus ?case by (simp add: add_divide_distrib)
qed
next
case False thus ?thesis by (simp add: setsum_def)
qed
lemma setsum_abs[iff]:
fixes f :: "'a => ('b::pordered_ab_group_add_abs)"
shows "abs (setsum f A) \<le> setsum (%i. abs(f i)) 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 (auto intro: abs_triangle_ineq order_trans)
qed
next
case False thus ?thesis by (simp add: setsum_def)
qed
lemma setsum_abs_ge_zero[iff]:
fixes f :: "'a => ('b::pordered_ab_group_add_abs)"
shows "0 \<le> setsum (%i. abs(f i)) 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 (auto simp: add_nonneg_nonneg)
qed
next
case False thus ?thesis by (simp add: setsum_def)
qed
lemma abs_setsum_abs[simp]:
fixes f :: "'a => ('b::pordered_ab_group_add_abs)"
shows "abs (\<Sum>a\<in>A. abs(f a)) = (\<Sum>a\<in>A. abs(f a))"
proof (cases "finite A")
case True
thus ?thesis
proof induct
case empty thus ?case by simp
next
case (insert a A)
hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>" using insert by simp
also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
by (simp del: abs_of_nonneg)
also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
finally show ?case .
qed
next
case False thus ?thesis by (simp add: setsum_def)
qed
lemma setsum_Plus:
fixes A :: "'a set" and B :: "'b set"
assumes fin: "finite A" "finite B"
shows "setsum f (A <+> B) = setsum (f \<circ> Inl) A + setsum (f \<circ> Inr) B"
proof -
have "A <+> B = Inl ` A \<union> Inr ` B" by auto
moreover from fin have "finite (Inl ` A :: ('a + 'b) set)" "finite (Inr ` B :: ('a + 'b) set)"
by(auto intro: finite_imageI)
moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('a + 'b) set)" by auto
moreover have "inj_on (Inl :: 'a \<Rightarrow> 'a + 'b) A" "inj_on (Inr :: 'b \<Rightarrow> 'a + 'b) B" by(auto intro: inj_onI)
ultimately show ?thesis using fin by(simp add: setsum_Un_disjoint setsum_reindex)
qed
text {* Commuting outer and inner summation *}
lemma swap_inj_on:
"inj_on (%(i, j). (j, i)) (A \<times> B)"
by (unfold inj_on_def) fast
lemma swap_product:
"(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
by (simp add: split_def image_def) blast
lemma setsum_commute:
"(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
proof (simp add: setsum_cartesian_product)
have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
(\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
(is "?s = _")
apply (simp add: setsum_reindex [where f = "%(i, j). (j, i)"] swap_inj_on)
apply (simp add: split_def)
done
also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
(is "_ = ?t")
apply (simp add: swap_product)
done
finally show "?s = ?t" .
qed
lemma setsum_product:
fixes f :: "'a => ('b::semiring_0)"
shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum_commute)
subsection {* Generalized product over a set *}
definition setprod :: "('a => 'b) => 'a set => 'b::comm_monoid_mult"
where "setprod f A == if finite A then fold_image (op *) f 1 A else 1"
abbreviation
Setprod ("\<Prod>_" [1000] 999) where
"\<Prod>A == setprod (%x. x) A"
syntax
"_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(3PROD _:_. _)" [0, 51, 10] 10)
syntax (xsymbols)
"_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
syntax (HTML output)
"_setprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult" ("(3\<Prod>_\<in>_. _)" [0, 51, 10] 10)
translations -- {* Beware of argument permutation! *}
"PROD i:A. b" == "CONST setprod (%i. b) A"
"\<Prod>i\<in>A. b" == "CONST setprod (%i. b) A"
text{* Instead of @{term"\<Prod>x\<in>{x. P}. e"} we introduce the shorter
@{text"\<Prod>x|P. e"}. *}
syntax
"_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3PROD _ |/ _./ _)" [0,0,10] 10)
syntax (xsymbols)
"_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
syntax (HTML output)
"_qsetprod" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Prod>_ | (_)./ _)" [0,0,10] 10)
translations
"PROD x|P. t" => "CONST setprod (%x. t) {x. P}"
"\<Prod>x|P. t" => "CONST setprod (%x. t) {x. P}"
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)
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 fold_image_reindex 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: fold_image_cong)
lemma strong_setprod_cong[cong]:
"A = B ==> (!!x. x:B =simp=> f x = g x) ==> setprod f A = setprod g B"
by(fastsimp simp: simp_implies_def setprod_def intro: fold_image_cong)
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 strong_setprod_reindex_cong: assumes i: "inj_on f A"
and B: "B = f ` A" and eq: "\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x"
shows "setprod h B = setprod g A"
proof-
have "setprod h B = setprod (h o f) A"
by (simp add: B setprod_reindex[OF i, of h])
then show ?thesis apply simp
apply (rule setprod_cong)
apply simp
by (simp add: eq)
qed
lemma setprod_Un_one:
assumes fS: "finite S" and fT: "finite T"
and I0: "\<forall>x \<in> S\<inter>T. f x = 1"
shows "setprod f (S \<union> T) = setprod f S * setprod f T"
using fS fT
apply (simp add: setprod_def)
apply (rule fold_image_Un_one)
using I0 by auto
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 fold_image_Un_Int[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_mono_one_left:
assumes fT: "finite T" and ST: "S \<subseteq> T"
and z: "\<forall>i \<in> T - S. f i = 1"
shows "setprod f S = setprod f T"
proof-
have eq: "T = S \<union> (T - S)" using ST by blast
have d: "S \<inter> (T - S) = {}" using ST by blast
from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset)
show ?thesis
by (simp add: setprod_Un_disjoint[OF f d, unfolded eq[symmetric]] setprod_1'[OF z])
qed
lemmas setprod_mono_one_right = setprod_mono_one_left [THEN sym]
lemma setprod_delta:
assumes fS: "finite S"
shows "setprod (\<lambda>k. if k=a then b k else 1) S = (if a \<in> S then b a else 1)"
proof-
let ?f = "(\<lambda>k. if k=a then b k else 1)"
{assume a: "a \<notin> S"
hence "\<forall> k\<in> S. ?f k = 1" by simp
hence ?thesis using a by (simp add: setprod_1 cong add: setprod_cong) }
moreover
{assume a: "a \<in> S"
let ?A = "S - {a}"
let ?B = "{a}"
have eq: "S = ?A \<union> ?B" using a by blast
have dj: "?A \<inter> ?B = {}" by simp
from fS have fAB: "finite ?A" "finite ?B" by auto
have fA1: "setprod ?f ?A = 1" apply (rule setprod_1') by auto
have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
by simp
then have ?thesis using a by (simp add: fA1 cong add: setprod_cong cong del: if_weak_cong)}
ultimately show ?thesis by blast
qed
lemma setprod_delta':
assumes fS: "finite S" shows
"setprod (\<lambda>k. if a = k then b k else 1) S =
(if a\<in> S then b a else 1)"
using setprod_delta[OF fS, of a b, symmetric]
by (auto intro: setprod_cong)
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 fold_image_UN_disjoint 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\<in>A. (\<Prod>y\<in> B x. f x y)) =
(\<Prod>(x,y)\<in>(SIGMA x:A. B x). f x y)"
by(simp add:setprod_def fold_image_Sigma split_def cong:setprod_cong)
text{*Here we can eliminate the finiteness assumptions, by cases.*}
lemma setprod_cartesian_product:
"(\<Prod>x\<in>A. (\<Prod>y\<in> B. f x y)) = (\<Prod>(x,y)\<in>(A <*> B). f x y)"
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 fold_image_distrib)
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: finite) auto
lemma setprod_zero:
"finite A ==> EX x: A. f x = (0::'a::comm_semiring_1) ==> setprod f A = 0"
apply (induct set: finite, force, clarsimp)
apply (erule disjE, auto)
done
lemma setprod_nonneg [rule_format]:
"(ALL x: A. (0::'a::ordered_semidom) \<le> f x) --> 0 \<le> setprod f A"
by (cases "finite A", induct set: finite, simp_all add: mult_nonneg_nonneg)
lemma setprod_pos [rule_format]: "(ALL x: A. (0::'a::ordered_semidom) < f x)
--> 0 < setprod f A"
by (cases "finite A", induct set: finite, simp_all add: mult_pos_pos)
lemma setprod_zero_iff[simp]: "finite A ==>
(setprod f A = (0::'a::{comm_semiring_1,no_zero_divisors})) =
(EX x: A. f x = 0)"
by (erule finite_induct, auto simp:no_zero_divisors)
lemma setprod_pos_nat:
"finite S ==> (ALL x : S. f x > (0::nat)) ==> setprod f S > 0"
using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
lemma setprod_pos_nat_iff[simp]:
"finite S ==> (setprod f S > 0) = (ALL x : S. f x > (0::nat))"
using setprod_zero_iff by(simp del:neq0_conv add:neq0_conv[symmetric])
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)"
by (subst setprod_Un_Int [symmetric], auto)
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)"
by (erule finite_induct) (auto simp add: insert_Diff_if)
lemma setprod_inversef:
fixes f :: "'b \<Rightarrow> 'a::{field,division_by_zero}"
shows "finite A ==> setprod (inverse \<circ> f) A = inverse (setprod f A)"
by (erule finite_induct) auto
lemma setprod_dividef:
fixes f :: "'b \<Rightarrow> 'a::{field,division_by_zero}"
shows "finite A
==> 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
lemma setprod_dvd_setprod [rule_format]:
"(ALL x : A. f x dvd g x) \<longrightarrow> setprod f A dvd setprod g A"
apply (cases "finite A")
apply (induct set: finite)
apply (auto simp add: dvd_def)
apply (rule_tac x = "k * ka" in exI)
apply (simp add: algebra_simps)
done
lemma setprod_dvd_setprod_subset:
"finite B \<Longrightarrow> A <= B \<Longrightarrow> setprod f A dvd setprod f B"
apply (subgoal_tac "setprod f B = setprod f A * setprod f (B - A)")
apply (unfold dvd_def, blast)
apply (subst setprod_Un_disjoint [symmetric])
apply (auto elim: finite_subset intro: setprod_cong)
done
lemma setprod_dvd_setprod_subset2:
"finite B \<Longrightarrow> A <= B \<Longrightarrow> ALL x : A. (f x::'a::comm_semiring_1) dvd g x \<Longrightarrow>
setprod f A dvd setprod g B"
apply (rule dvd_trans)
apply (rule setprod_dvd_setprod, erule (1) bspec)
apply (erule (1) setprod_dvd_setprod_subset)
done
lemma dvd_setprod: "finite A \<Longrightarrow> i:A \<Longrightarrow>
(f i ::'a::comm_semiring_1) dvd setprod f A"
by (induct set: finite) (auto intro: dvd_mult)
lemma dvd_setsum [rule_format]: "(ALL i : A. d dvd f i) \<longrightarrow>
(d::'a::comm_semiring_1) dvd (SUM x : A. f x)"
apply (cases "finite A")
apply (induct set: finite)
apply 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:
*}
definition card :: "'a set \<Rightarrow> nat" where
"card A = setsum (\<lambda>x. 1) A"
lemmas card_eq_setsum = card_def
lemma card_empty [simp]: "card {} = 0"
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)
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_infinite [simp]: "~ finite A ==> card A = 0"
by (simp add: card_def)
lemma card_ge_0_finite:
"card A > 0 \<Longrightarrow> finite A"
by (rule ccontr) simp
lemma card_0_eq [simp,noatp]: "finite A ==> (card A = 0) = (A = {})"
apply auto
apply (drule_tac a = x in mk_disjoint_insert, clarify, auto)
done
lemma finite_UNIV_card_ge_0:
"finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
by (rule ccontr) simp
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_Diff_insert[simp]:
assumes "finite A" and "a:A" and "a ~: B"
shows "card(A - insert a B) = card(A - B) - 1"
proof -
have "A - insert a B = (A - B) - {a}" using assms by blast
then show ?thesis using assms by(simp add:card_Diff_singleton)
qed
lemma card_insert: "finite A ==> card (insert x A) = Suc (card (A - {x}))"
by (simp add: card_insert_if card_Suc_Diff1 del:card_Diff_insert)
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)
lemma card_seteq: "finite B ==> (!!A. A <= B ==> card B <= card A ==> A = B)"
apply (induct set: finite, 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_eq 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 del:card_Diff_insert)
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 del:card_Diff_insert)
apply (rule less_trans)
prefer 2 apply (auto intro!: card_Diff1_less simp del:card_Diff_insert)
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
lemma finite_psubset_induct[consumes 1, case_names psubset]:
assumes "finite A" and "!!A. finite A \<Longrightarrow> (!!B. finite B \<Longrightarrow> B \<subset> A \<Longrightarrow> P(B)) \<Longrightarrow> P(A)" shows "P A"
using assms(1)
proof (induct A rule: measure_induct_rule[where f=card])
case (less A)
show ?case
proof(rule assms(2)[OF less(2)])
fix B assume "finite B" "B \<subset> A"
show "P B" by(rule less(1)[OF psubset_card_mono[OF less(2) `B \<subset> A`] `finite B`])
qed
qed
text{* 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 card_eq_UNIV_imp_eq_UNIV:
assumes fin: "finite (UNIV :: 'a set)"
and card: "card A = card (UNIV :: 'a set)"
shows "A = (UNIV :: 'a set)"
proof
show "A \<subseteq> UNIV" by simp
show "UNIV \<subseteq> A"
proof
fix x
show "x \<in> A"
proof (rule ccontr)
assume "x \<notin> A"
then have "A \<subset> UNIV" by auto
with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
with card show False by simp
qed
qed
qed
text{*The form of a finite set of given cardinality*}
lemma card_eq_SucD:
assumes "card A = Suc k"
shows "\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={})"
proof -
have fin: "finite A" using assms by (auto intro: ccontr)
moreover have "card A \<noteq> 0" using assms by auto
ultimately obtain b where b: "b \<in> A" by auto
show ?thesis
proof (intro exI conjI)
show "A = insert b (A-{b})" using b by blast
show "b \<notin> A - {b}" by blast
show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
using assms b fin by(fastsimp dest:mk_disjoint_insert)+
qed
qed
lemma card_Suc_eq:
"(card A = Suc k) =
(\<exists>b B. A = insert b B & b \<notin> B & card B = k & (k=0 \<longrightarrow> B={}))"
apply(rule iffI)
apply(erule card_eq_SucD)
apply(auto)
apply(subst card_insert)
apply(auto intro:ccontr)
done
lemma finite_fun_UNIVD2:
assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
shows "finite (UNIV :: 'b set)"
proof -
from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
by(rule finite_imageI)
moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
by(rule UNIV_eq_I) auto
ultimately show "finite (UNIV :: 'b set)" by simp
qed
lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y"
apply (cases "finite A")
apply (erule finite_induct)
apply (auto simp add: algebra_simps)
done
lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{comm_monoid_mult})) = y^(card A)"
apply (erule finite_induct)
apply (auto simp add: power_Suc)
done
lemma setprod_gen_delta:
assumes fS: "finite S"
shows "setprod (\<lambda>k. if k=a then b k else c) S = (if a \<in> S then (b a ::'a::{comm_monoid_mult}) * c^ (card S - 1) else c^ card S)"
proof-
let ?f = "(\<lambda>k. if k=a then b k else c)"
{assume a: "a \<notin> S"
hence "\<forall> k\<in> S. ?f k = c" by simp
hence ?thesis using a setprod_constant[OF fS, of c] by (simp add: setprod_1 cong add: setprod_cong) }
moreover
{assume a: "a \<in> S"
let ?A = "S - {a}"
let ?B = "{a}"
have eq: "S = ?A \<union> ?B" using a by blast
have dj: "?A \<inter> ?B = {}" by simp
from fS have fAB: "finite ?A" "finite ?B" by auto
have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
apply (rule setprod_cong) by auto
have cA: "card ?A = card S - 1" using fS a by auto
have fA1: "setprod ?f ?A = c ^ card ?A" unfolding fA0 apply (rule setprod_constant) using fS by auto
have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
using setprod_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
by simp
then have ?thesis using a cA
by (simp add: fA1 ring_simps cong add: setprod_cong cong del: if_weak_cong)}
ultimately show ?thesis by blast
qed
lemma setsum_bounded:
assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, pordered_ab_semigroup_add})"
shows "setsum f A \<le> of_nat(card A) * K"
proof (cases "finite A")
case True
thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
next
case False thus ?thesis by (simp add: setsum_def)
qed
lemma card_UNIV_unit: "card (UNIV :: unit set) = 1"
unfolding UNIV_unit by simp
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 del: setsum_constant)
apply (subgoal_tac
"setsum (%i. card (A i)) I = setsum (%i. (setsum (%x. 1) (A i))) I")
apply (simp add: setsum_UN_disjoint del: setsum_constant)
apply (simp 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_image}.*}
lemma image_eq_fold_image:
"finite A ==> f ` A = fold_image (op Un) (%x. {f x}) {} A"
proof (induct rule: finite_induct)
case empty then show ?case by simp
next
interpret ab_semigroup_mult "op Un"
proof qed auto
case insert
then show ?case by simp
qed
lemma card_image_le: "finite A ==> card (f ` A) <= card A"
apply (induct set: finite)
apply 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 del:setsum_constant)
lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
by(auto simp: card_image bij_betw_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 del:setsum_constant)
lemma card_cartesian_product: "card (A <*> B) = card(A) * card(B)"
apply (cases "finite A")
apply (cases "finite B")
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 sums *}
lemma card_Plus:
assumes "finite A" and "finite B"
shows "card (A <+> B) = card A + card B"
proof -
have "Inl`A \<inter> Inr`B = {}" by fast
with assms show ?thesis
unfolding Plus_def
by (simp add: card_Un_disjoint card_image)
qed
lemma card_Plus_conv_if:
"card (A <+> B) = (if finite A \<and> finite B then card(A) + card(B) else 0)"
by(auto simp: card_def setsum_Plus simp del: setsum_constant)
subsubsection {* Cardinality of the Powerset *}
lemma card_Pow: "finite A ==> card (Pow A) = Suc (Suc 0) ^ card A" (* FIXME numeral 2 (!?) *)
apply (induct set: finite)
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. *}
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: finite, simp_all, clarify)
apply (subst card_Un_disjoint)
apply (auto simp add: dvd_add disjoint_eq_subset_Compl)
done
subsubsection {* Relating injectivity and surjectivity *}
lemma finite_surj_inj: "finite(A) \<Longrightarrow> A <= f`A \<Longrightarrow> inj_on f A"
apply(rule eq_card_imp_inj_on, assumption)
apply(frule finite_imageI)
apply(drule (1) card_seteq)
apply(erule card_image_le)
apply simp
done
lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
by (blast intro: finite_surj_inj subset_UNIV dest:surj_range)
lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
by(fastsimp simp:surj_def dest!: endo_inj_surj)
corollary infinite_UNIV_nat[iff]: "~finite(UNIV::nat set)"
proof
assume "finite(UNIV::nat set)"
with finite_UNIV_inj_surj[of Suc]
show False by simp (blast dest: Suc_neq_Zero surjD)
qed
(* Often leads to bogus ATP proofs because of reduced type information, hence noatp *)
lemma infinite_UNIV_char_0[noatp]:
"\<not> finite (UNIV::'a::semiring_char_0 set)"
proof
assume "finite (UNIV::'a set)"
with subset_UNIV have "finite (range of_nat::'a set)"
by (rule finite_subset)
moreover have "inj (of_nat::nat \<Rightarrow> 'a)"
by (simp add: inj_on_def)
ultimately have "finite (UNIV::nat set)"
by (rule finite_imageD)
then show "False"
by (simp add: infinite_UNIV_nat)
qed
subsection{* A fold functional for non-empty sets *}
text{* Does not require start value. *}
inductive
fold1Set :: "('a => 'a => 'a) => 'a set => 'a => bool"
for f :: "'a => 'a => 'a"
where
fold1Set_insertI [intro]:
"\<lbrakk> fold_graph f a A x; a \<notin> A \<rbrakk> \<Longrightarrow> fold1Set f (insert a A) x"
constdefs
fold1 :: "('a => 'a => 'a) => 'a set => 'a"
"fold1 f A == THE x. fold1Set f A x"
lemma fold1Set_nonempty:
"fold1Set f A x \<Longrightarrow> A \<noteq> {}"
by(erule fold1Set.cases, simp_all)
inductive_cases empty_fold1SetE [elim!]: "fold1Set f {} x"
inductive_cases insert_fold1SetE [elim!]: "fold1Set f (insert a X) x"
lemma fold1Set_sing [iff]: "(fold1Set f {a} b) = (a = b)"
by (blast intro: fold_graph.intros elim: fold_graph.cases)
lemma fold1_singleton [simp]: "fold1 f {a} = a"
by (unfold fold1_def) blast
lemma finite_nonempty_imp_fold1Set:
"\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> EX x. fold1Set f A x"
apply (induct A rule: finite_induct)
apply (auto dest: finite_imp_fold_graph [of _ f])
done
text{*First, some lemmas about @{const fold_graph}.*}
context ab_semigroup_mult
begin
lemma fun_left_comm: "fun_left_comm(op *)"
by unfold_locales (simp add: mult_ac)
lemma fold_graph_insert_swap:
assumes fold: "fold_graph times (b::'a) A y" and "b \<notin> A"
shows "fold_graph times z (insert b A) (z * y)"
proof -
interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
from assms show ?thesis
proof (induct rule: fold_graph.induct)
case emptyI thus ?case by (force simp add: fold_insert_aux mult_commute)
next
case (insertI x A y)
have "fold_graph times z (insert x (insert b A)) (x * (z * y))"
using insertI by force --{*how does @{term id} get unfolded?*}
thus ?case by (simp add: insert_commute mult_ac)
qed
qed
lemma fold_graph_permute_diff:
assumes fold: "fold_graph times b A x"
shows "!!a. \<lbrakk>a \<in> A; b \<notin> A\<rbrakk> \<Longrightarrow> fold_graph times a (insert b (A-{a})) x"
using fold
proof (induct rule: fold_graph.induct)
case emptyI thus ?case by simp
next
case (insertI x A y)
have "a = x \<or> a \<in> A" using insertI by simp
thus ?case
proof
assume "a = x"
with insertI show ?thesis
by (simp add: id_def [symmetric], blast intro: fold_graph_insert_swap)
next
assume ainA: "a \<in> A"
hence "fold_graph times a (insert x (insert b (A - {a}))) (x * y)"
using insertI by force
moreover
have "insert x (insert b (A - {a})) = insert b (insert x A - {a})"
using ainA insertI by blast
ultimately show ?thesis by simp
qed
qed
lemma fold1_eq_fold:
assumes "finite A" "a \<notin> A" shows "fold1 times (insert a A) = fold times a A"
proof -
interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
from assms show ?thesis
apply (simp add: fold1_def fold_def)
apply (rule the_equality)
apply (best intro: fold_graph_determ theI dest: finite_imp_fold_graph [of _ times])
apply (rule sym, clarify)
apply (case_tac "Aa=A")
apply (best intro: the_equality fold_graph_determ)
apply (subgoal_tac "fold_graph times a A x")
apply (best intro: the_equality fold_graph_determ)
apply (subgoal_tac "insert aa (Aa - {a}) = A")
prefer 2 apply (blast elim: equalityE)
apply (auto dest: fold_graph_permute_diff [where a=a])
done
qed
lemma nonempty_iff: "(A \<noteq> {}) = (\<exists>x B. A = insert x B & x \<notin> B)"
apply safe
apply simp
apply (drule_tac x=x in spec)
apply (drule_tac x="A-{x}" in spec, auto)
done
lemma fold1_insert:
assumes nonempty: "A \<noteq> {}" and A: "finite A" "x \<notin> A"
shows "fold1 times (insert x A) = x * fold1 times A"
proof -
interpret fun_left_comm "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a" by (rule fun_left_comm)
from nonempty obtain a A' where "A = insert a A' & a ~: A'"
by (auto simp add: nonempty_iff)
with A show ?thesis
by (simp add: insert_commute [of x] fold1_eq_fold eq_commute)
qed
end
context ab_semigroup_idem_mult
begin
lemma fold1_insert_idem [simp]:
assumes nonempty: "A \<noteq> {}" and A: "finite A"
shows "fold1 times (insert x A) = x * fold1 times A"
proof -
interpret fun_left_comm_idem "op *::'a \<Rightarrow> 'a \<Rightarrow> 'a"
by (rule fun_left_comm_idem)
from nonempty obtain a A' where A': "A = insert a A' & a ~: A'"
by (auto simp add: nonempty_iff)
show ?thesis
proof cases
assume "a = x"
thus ?thesis
proof cases
assume "A' = {}"
with prems show ?thesis by (simp add: mult_idem)
next
assume "A' \<noteq> {}"
with prems show ?thesis
by (simp add: fold1_insert mult_assoc [symmetric] mult_idem)
qed
next
assume "a \<noteq> x"
with prems show ?thesis
by (simp add: insert_commute fold1_eq_fold fold_insert_idem)
qed
qed
lemma hom_fold1_commute:
assumes hom: "!!x y. h (x * y) = h x * h y"
and N: "finite N" "N \<noteq> {}" shows "h (fold1 times N) = fold1 times (h ` N)"
using N proof (induct rule: finite_ne_induct)
case singleton thus ?case by simp
next
case (insert n N)
then have "h (fold1 times (insert n N)) = h (n * fold1 times N)" by simp
also have "\<dots> = h n * h (fold1 times N)" by(rule hom)
also have "h (fold1 times N) = fold1 times (h ` N)" by(rule insert)
also have "times (h n) \<dots> = fold1 times (insert (h n) (h ` N))"
using insert by(simp)
also have "insert (h n) (h ` N) = h ` insert n N" by simp
finally show ?case .
qed
lemma fold1_eq_fold_idem:
assumes "finite A"
shows "fold1 times (insert a A) = fold times a A"
proof (cases "a \<in> A")
case False
with assms show ?thesis by (simp add: fold1_eq_fold)
next
interpret fun_left_comm_idem times by (fact fun_left_comm_idem)
case True then obtain b B
where A: "A = insert a B" and "a \<notin> B" by (rule set_insert)
with assms have "finite B" by auto
then have "fold times a (insert a B) = fold times (a * a) B"
using `a \<notin> B` by (rule fold_insert2)
then show ?thesis
using `a \<notin> B` `finite B` by (simp add: fold1_eq_fold A)
qed
end
text{* Now the recursion rules for definitions: *}
lemma fold1_singleton_def: "g = fold1 f \<Longrightarrow> g {a} = a"
by(simp add:fold1_singleton)
lemma (in ab_semigroup_mult) fold1_insert_def:
"\<lbrakk> g = fold1 times; finite A; x \<notin> A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
by (simp add:fold1_insert)
lemma (in ab_semigroup_idem_mult) fold1_insert_idem_def:
"\<lbrakk> g = fold1 times; finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> g (insert x A) = x * g A"
by simp
subsubsection{* Determinacy for @{term fold1Set} *}
(*Not actually used!!*)
(*
context ab_semigroup_mult
begin
lemma fold_graph_permute:
"[|fold_graph times id b (insert a A) x; a \<notin> A; b \<notin> A|]
==> fold_graph times id a (insert b A) x"
apply (cases "a=b")
apply (auto dest: fold_graph_permute_diff)
done
lemma fold1Set_determ:
"fold1Set times A x ==> fold1Set times A y ==> y = x"
proof (clarify elim!: fold1Set.cases)
fix A x B y a b
assume Ax: "fold_graph times id a A x"
assume By: "fold_graph times id b B y"
assume anotA: "a \<notin> A"
assume bnotB: "b \<notin> B"
assume eq: "insert a A = insert b B"
show "y=x"
proof cases
assume same: "a=b"
hence "A=B" using anotA bnotB eq by (blast elim!: equalityE)
thus ?thesis using Ax By same by (blast intro: fold_graph_determ)
next
assume diff: "a\<noteq>b"
let ?D = "B - {a}"
have B: "B = insert a ?D" and A: "A = insert b ?D"
and aB: "a \<in> B" and bA: "b \<in> A"
using eq anotA bnotB diff by (blast elim!:equalityE)+
with aB bnotB By
have "fold_graph times id a (insert b ?D) y"
by (auto intro: fold_graph_permute simp add: insert_absorb)
moreover
have "fold_graph times id a (insert b ?D) x"
by (simp add: A [symmetric] Ax)
ultimately show ?thesis by (blast intro: fold_graph_determ)
qed
qed
lemma fold1Set_equality: "fold1Set times A y ==> fold1 times A = y"
by (unfold fold1_def) (blast intro: fold1Set_determ)
end
*)
declare
empty_fold_graphE [rule del] fold_graph.intros [rule del]
empty_fold1SetE [rule del] insert_fold1SetE [rule del]
-- {* No more proofs involve these relations. *}
subsubsection {* Lemmas about @{text fold1} *}
context ab_semigroup_mult
begin
lemma fold1_Un:
assumes A: "finite A" "A \<noteq> {}"
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow> A Int B = {} \<Longrightarrow>
fold1 times (A Un B) = fold1 times A * fold1 times B"
using A by (induct rule: finite_ne_induct)
(simp_all add: fold1_insert mult_assoc)
lemma fold1_in:
assumes A: "finite (A)" "A \<noteq> {}" and elem: "\<And>x y. x * y \<in> {x,y}"
shows "fold1 times 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
end
lemma (in ab_semigroup_idem_mult) fold1_Un2:
assumes A: "finite A" "A \<noteq> {}"
shows "finite B \<Longrightarrow> B \<noteq> {} \<Longrightarrow>
fold1 times (A Un B) = fold1 times A * fold1 times B"
using A
proof(induct rule:finite_ne_induct)
case singleton thus ?case by simp
next
case insert thus ?case by (simp add: mult_assoc)
qed
subsubsection {* Fold1 in lattices with @{const inf} and @{const sup} *}
text{*
As an application of @{text fold1} we define infimum
and supremum in (not necessarily complete!) lattices
over (non-empty) sets by means of @{text fold1}.
*}
context lower_semilattice
begin
lemma below_fold1_iff:
assumes "finite A" "A \<noteq> {}"
shows "x \<le> fold1 inf A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
proof -
interpret ab_semigroup_idem_mult inf
by (rule ab_semigroup_idem_mult_inf)
show ?thesis using assms by (induct rule: finite_ne_induct) simp_all
qed
lemma fold1_belowI:
assumes "finite A"
and "a \<in> A"
shows "fold1 inf A \<le> a"
proof -
from assms have "A \<noteq> {}" by auto
from `finite A` `A \<noteq> {}` `a \<in> A` show ?thesis
proof (induct rule: finite_ne_induct)
case singleton thus ?case by simp
next
interpret ab_semigroup_idem_mult inf
by (rule ab_semigroup_idem_mult_inf)
case (insert x F)
from insert(5) have "a = x \<or> a \<in> F" by simp
thus ?case
proof
assume "a = x" thus ?thesis using insert
by (simp add: mult_ac)
next
assume "a \<in> F"
hence bel: "fold1 inf F \<le> a" by (rule insert)
have "inf (fold1 inf (insert x F)) a = inf x (inf (fold1 inf F) a)"
using insert by (simp add: mult_ac)
also have "inf (fold1 inf F) a = fold1 inf F"
using bel by (auto intro: antisym)
also have "inf x \<dots> = fold1 inf (insert x F)"
using insert by (simp add: mult_ac)
finally have aux: "inf (fold1 inf (insert x F)) a = fold1 inf (insert x F)" .
moreover have "inf (fold1 inf (insert x F)) a \<le> a" by simp
ultimately show ?thesis by simp
qed
qed
qed
end
context lattice
begin
definition
Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^bsub>fin\<^esub>_" [900] 900)
where
"Inf_fin = fold1 inf"
definition
Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^bsub>fin\<^esub>_" [900] 900)
where
"Sup_fin = fold1 sup"
lemma Inf_le_Sup [simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>\<^bsub>fin\<^esub>A \<le> \<Squnion>\<^bsub>fin\<^esub>A"
apply(unfold Sup_fin_def Inf_fin_def)
apply(subgoal_tac "EX a. a:A")
prefer 2 apply blast
apply(erule exE)
apply(rule order_trans)
apply(erule (1) fold1_belowI)
apply(erule (1) lower_semilattice.fold1_belowI [OF dual_semilattice])
done
lemma sup_Inf_absorb [simp]:
"finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a (\<Sqinter>\<^bsub>fin\<^esub>A) = a"
apply(subst sup_commute)
apply(simp add: Inf_fin_def sup_absorb2 fold1_belowI)
done
lemma inf_Sup_absorb [simp]:
"finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^bsub>fin\<^esub>A) = a"
by (simp add: Sup_fin_def inf_absorb1
lower_semilattice.fold1_belowI [OF dual_semilattice])
end
context distrib_lattice
begin
lemma sup_Inf1_distrib:
assumes "finite A"
and "A \<noteq> {}"
shows "sup x (\<Sqinter>\<^bsub>fin\<^esub>A) = \<Sqinter>\<^bsub>fin\<^esub>{sup x a|a. a \<in> A}"
proof -
interpret ab_semigroup_idem_mult inf
by (rule ab_semigroup_idem_mult_inf)
from assms show ?thesis
by (simp add: Inf_fin_def image_def
hom_fold1_commute [where h="sup x", OF sup_inf_distrib1])
(rule arg_cong [where f="fold1 inf"], blast)
qed
lemma sup_Inf2_distrib:
assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
shows "sup (\<Sqinter>\<^bsub>fin\<^esub>A) (\<Sqinter>\<^bsub>fin\<^esub>B) = \<Sqinter>\<^bsub>fin\<^esub>{sup a b|a b. a \<in> A \<and> b \<in> B}"
using A proof (induct rule: finite_ne_induct)
case singleton thus ?case
by (simp add: sup_Inf1_distrib [OF B] fold1_singleton_def [OF Inf_fin_def])
next
interpret ab_semigroup_idem_mult inf
by (rule ab_semigroup_idem_mult_inf)
case (insert x A)
have finB: "finite {sup x b |b. b \<in> B}"
by(rule finite_surj[where f = "sup x", OF B(1)], auto)
have finAB: "finite {sup a b |a b. a \<in> A \<and> b \<in> B}"
proof -
have "{sup a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {sup a b})"
by blast
thus ?thesis by(simp add: insert(1) B(1))
qed
have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
have "sup (\<Sqinter>\<^bsub>fin\<^esub>(insert x A)) (\<Sqinter>\<^bsub>fin\<^esub>B) = sup (inf x (\<Sqinter>\<^bsub>fin\<^esub>A)) (\<Sqinter>\<^bsub>fin\<^esub>B)"
using insert by (simp add: fold1_insert_idem_def [OF Inf_fin_def])
also have "\<dots> = inf (sup x (\<Sqinter>\<^bsub>fin\<^esub>B)) (sup (\<Sqinter>\<^bsub>fin\<^esub>A) (\<Sqinter>\<^bsub>fin\<^esub>B))" by(rule sup_inf_distrib2)
also have "\<dots> = inf (\<Sqinter>\<^bsub>fin\<^esub>{sup x b|b. b \<in> B}) (\<Sqinter>\<^bsub>fin\<^esub>{sup a b|a b. a \<in> A \<and> b \<in> B})"
using insert by(simp add:sup_Inf1_distrib[OF B])
also have "\<dots> = \<Sqinter>\<^bsub>fin\<^esub>({sup x b |b. b \<in> B} \<union> {sup a b |a b. a \<in> A \<and> b \<in> B})"
(is "_ = \<Sqinter>\<^bsub>fin\<^esub>?M")
using B insert
by (simp add: Inf_fin_def fold1_Un2 [OF finB _ finAB ne])
also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
by blast
finally show ?case .
qed
lemma inf_Sup1_distrib:
assumes "finite A" and "A \<noteq> {}"
shows "inf x (\<Squnion>\<^bsub>fin\<^esub>A) = \<Squnion>\<^bsub>fin\<^esub>{inf x a|a. a \<in> A}"
proof -
interpret ab_semigroup_idem_mult sup
by (rule ab_semigroup_idem_mult_sup)
from assms show ?thesis
by (simp add: Sup_fin_def image_def hom_fold1_commute [where h="inf x", OF inf_sup_distrib1])
(rule arg_cong [where f="fold1 sup"], blast)
qed
lemma inf_Sup2_distrib:
assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
shows "inf (\<Squnion>\<^bsub>fin\<^esub>A) (\<Squnion>\<^bsub>fin\<^esub>B) = \<Squnion>\<^bsub>fin\<^esub>{inf a b|a b. a \<in> A \<and> b \<in> B}"
using A proof (induct rule: finite_ne_induct)
case singleton thus ?case
by(simp add: inf_Sup1_distrib [OF B] fold1_singleton_def [OF Sup_fin_def])
next
case (insert x A)
have finB: "finite {inf x b |b. b \<in> B}"
by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)
have finAB: "finite {inf a b |a b. a \<in> A \<and> b \<in> B}"
proof -
have "{inf a b |a b. a \<in> A \<and> b \<in> B} = (UN a:A. UN b:B. {inf a b})"
by blast
thus ?thesis by(simp add: insert(1) B(1))
qed
have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
interpret ab_semigroup_idem_mult sup
by (rule ab_semigroup_idem_mult_sup)
have "inf (\<Squnion>\<^bsub>fin\<^esub>(insert x A)) (\<Squnion>\<^bsub>fin\<^esub>B) = inf (sup x (\<Squnion>\<^bsub>fin\<^esub>A)) (\<Squnion>\<^bsub>fin\<^esub>B)"
using insert by (simp add: fold1_insert_idem_def [OF Sup_fin_def])
also have "\<dots> = sup (inf x (\<Squnion>\<^bsub>fin\<^esub>B)) (inf (\<Squnion>\<^bsub>fin\<^esub>A) (\<Squnion>\<^bsub>fin\<^esub>B))" by(rule inf_sup_distrib2)
also have "\<dots> = sup (\<Squnion>\<^bsub>fin\<^esub>{inf x b|b. b \<in> B}) (\<Squnion>\<^bsub>fin\<^esub>{inf a b|a b. a \<in> A \<and> b \<in> B})"
using insert by(simp add:inf_Sup1_distrib[OF B])
also have "\<dots> = \<Squnion>\<^bsub>fin\<^esub>({inf x b |b. b \<in> B} \<union> {inf a b |a b. a \<in> A \<and> b \<in> B})"
(is "_ = \<Squnion>\<^bsub>fin\<^esub>?M")
using B insert
by (simp add: Sup_fin_def fold1_Un2 [OF finB _ finAB ne])
also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
by blast
finally show ?case .
qed
end
context complete_lattice
begin
text {*
Coincidence on finite sets in complete lattices:
*}
lemma Inf_fin_Inf:
assumes "finite A" and "A \<noteq> {}"
shows "\<Sqinter>\<^bsub>fin\<^esub>A = Inf A"
proof -
interpret ab_semigroup_idem_mult inf
by (rule ab_semigroup_idem_mult_inf)
from assms show ?thesis
unfolding Inf_fin_def by (induct A set: finite)
(simp_all add: Inf_insert_simp)
qed
lemma Sup_fin_Sup:
assumes "finite A" and "A \<noteq> {}"
shows "\<Squnion>\<^bsub>fin\<^esub>A = Sup A"
proof -
interpret ab_semigroup_idem_mult sup
by (rule ab_semigroup_idem_mult_sup)
from assms show ?thesis
unfolding Sup_fin_def by (induct A set: finite)
(simp_all add: Sup_insert_simp)
qed
end
subsubsection {* Fold1 in linear orders with @{const min} and @{const max} *}
text{*
As an application of @{text fold1} we define minimum
and maximum in (not necessarily complete!) linear orders
over (non-empty) sets by means of @{text fold1}.
*}
context linorder
begin
lemma ab_semigroup_idem_mult_min:
"ab_semigroup_idem_mult min"
proof qed (auto simp add: min_def)
lemma ab_semigroup_idem_mult_max:
"ab_semigroup_idem_mult max"
proof qed (auto simp add: max_def)
lemma max_lattice:
"lower_semilattice (op \<ge>) (op >) max"
by (fact min_max.dual_semilattice)
lemma dual_max:
"ord.max (op \<ge>) = min"
by (auto simp add: ord.max_def_raw min_def expand_fun_eq)
lemma dual_min:
"ord.min (op \<ge>) = max"
by (auto simp add: ord.min_def_raw max_def expand_fun_eq)
lemma strict_below_fold1_iff:
assumes "finite A" and "A \<noteq> {}"
shows "x < fold1 min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
proof -
interpret ab_semigroup_idem_mult min
by (rule ab_semigroup_idem_mult_min)
from assms show ?thesis
by (induct rule: finite_ne_induct)
(simp_all add: fold1_insert)
qed
lemma fold1_below_iff:
assumes "finite A" and "A \<noteq> {}"
shows "fold1 min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
proof -
interpret ab_semigroup_idem_mult min
by (rule ab_semigroup_idem_mult_min)
from assms show ?thesis
by (induct rule: finite_ne_induct)
(simp_all add: fold1_insert min_le_iff_disj)
qed
lemma fold1_strict_below_iff:
assumes "finite A" and "A \<noteq> {}"
shows "fold1 min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
proof -
interpret ab_semigroup_idem_mult min
by (rule ab_semigroup_idem_mult_min)
from assms show ?thesis
by (induct rule: finite_ne_induct)
(simp_all add: fold1_insert min_less_iff_disj)
qed
lemma fold1_antimono:
assumes "A \<noteq> {}" and "A \<subseteq> B" and "finite B"
shows "fold1 min B \<le> fold1 min A"
proof cases
assume "A = B" thus ?thesis by simp
next
interpret ab_semigroup_idem_mult min
by (rule ab_semigroup_idem_mult_min)
assume "A \<noteq> B"
have B: "B = A \<union> (B-A)" using `A \<subseteq> B` by blast
have "fold1 min B = fold1 min (A \<union> (B-A))" by(subst B)(rule refl)
also have "\<dots> = min (fold1 min A) (fold1 min (B-A))"
proof -
have "finite A" by(rule finite_subset[OF `A \<subseteq> B` `finite B`])
moreover have "finite(B-A)" by(rule finite_Diff[OF `finite B`]) (* by(blast intro:finite_Diff prems) fails *)
moreover have "(B-A) \<noteq> {}" using prems by blast
moreover have "A Int (B-A) = {}" using prems by blast
ultimately show ?thesis using `A \<noteq> {}` by (rule_tac fold1_Un)
qed
also have "\<dots> \<le> fold1 min A" by (simp add: min_le_iff_disj)
finally show ?thesis .
qed
definition
Min :: "'a set \<Rightarrow> 'a"
where
"Min = fold1 min"
definition
Max :: "'a set \<Rightarrow> 'a"
where
"Max = fold1 max"
lemmas Min_singleton [simp] = fold1_singleton_def [OF Min_def]
lemmas Max_singleton [simp] = fold1_singleton_def [OF Max_def]
lemma Min_insert [simp]:
assumes "finite A" and "A \<noteq> {}"
shows "Min (insert x A) = min x (Min A)"
proof -
interpret ab_semigroup_idem_mult min
by (rule ab_semigroup_idem_mult_min)
from assms show ?thesis by (rule fold1_insert_idem_def [OF Min_def])
qed
lemma Max_insert [simp]:
assumes "finite A" and "A \<noteq> {}"
shows "Max (insert x A) = max x (Max A)"
proof -
interpret ab_semigroup_idem_mult max
by (rule ab_semigroup_idem_mult_max)
from assms show ?thesis by (rule fold1_insert_idem_def [OF Max_def])
qed
lemma Min_in [simp]:
assumes "finite A" and "A \<noteq> {}"
shows "Min A \<in> A"
proof -
interpret ab_semigroup_idem_mult min
by (rule ab_semigroup_idem_mult_min)
from assms fold1_in show ?thesis by (fastsimp simp: Min_def min_def)
qed
lemma Max_in [simp]:
assumes "finite A" and "A \<noteq> {}"
shows "Max A \<in> A"
proof -
interpret ab_semigroup_idem_mult max
by (rule ab_semigroup_idem_mult_max)
from assms fold1_in [of A] show ?thesis by (fastsimp simp: Max_def max_def)
qed
lemma Min_Un:
assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"
shows "Min (A \<union> B) = min (Min A) (Min B)"
proof -
interpret ab_semigroup_idem_mult min
by (rule ab_semigroup_idem_mult_min)
from assms show ?thesis
by (simp add: Min_def fold1_Un2)
qed
lemma Max_Un:
assumes "finite A" and "A \<noteq> {}" and "finite B" and "B \<noteq> {}"
shows "Max (A \<union> B) = max (Max A) (Max B)"
proof -
interpret ab_semigroup_idem_mult max
by (rule ab_semigroup_idem_mult_max)
from assms show ?thesis
by (simp add: Max_def fold1_Un2)
qed
lemma hom_Min_commute:
assumes "\<And>x y. h (min x y) = min (h x) (h y)"
and "finite N" and "N \<noteq> {}"
shows "h (Min N) = Min (h ` N)"
proof -
interpret ab_semigroup_idem_mult min
by (rule ab_semigroup_idem_mult_min)
from assms show ?thesis
by (simp add: Min_def hom_fold1_commute)
qed
lemma hom_Max_commute:
assumes "\<And>x y. h (max x y) = max (h x) (h y)"
and "finite N" and "N \<noteq> {}"
shows "h (Max N) = Max (h ` N)"
proof -
interpret ab_semigroup_idem_mult max
by (rule ab_semigroup_idem_mult_max)
from assms show ?thesis
by (simp add: Max_def hom_fold1_commute [of h])
qed
lemma Min_le [simp]:
assumes "finite A" and "x \<in> A"
shows "Min A \<le> x"
using assms by (simp add: Min_def min_max.fold1_belowI)
lemma Max_ge [simp]:
assumes "finite A" and "x \<in> A"
shows "x \<le> Max A"
proof -
interpret lower_semilattice "op \<ge>" "op >" max
by (rule max_lattice)
from assms show ?thesis by (simp add: Max_def fold1_belowI)
qed
lemma Min_ge_iff [simp, noatp]:
assumes "finite A" and "A \<noteq> {}"
shows "x \<le> Min A \<longleftrightarrow> (\<forall>a\<in>A. x \<le> a)"
using assms by (simp add: Min_def min_max.below_fold1_iff)
lemma Max_le_iff [simp, noatp]:
assumes "finite A" and "A \<noteq> {}"
shows "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
proof -
interpret lower_semilattice "op \<ge>" "op >" max
by (rule max_lattice)
from assms show ?thesis by (simp add: Max_def below_fold1_iff)
qed
lemma Min_gr_iff [simp, noatp]:
assumes "finite A" and "A \<noteq> {}"
shows "x < Min A \<longleftrightarrow> (\<forall>a\<in>A. x < a)"
using assms by (simp add: Min_def strict_below_fold1_iff)
lemma Max_less_iff [simp, noatp]:
assumes "finite A" and "A \<noteq> {}"
shows "Max A < x \<longleftrightarrow> (\<forall>a\<in>A. a < x)"
proof -
interpret dual: linorder "op \<ge>" "op >"
by (rule dual_linorder)
from assms show ?thesis
by (simp add: Max_def dual.strict_below_fold1_iff [folded dual.dual_max])
qed
lemma Min_le_iff [noatp]:
assumes "finite A" and "A \<noteq> {}"
shows "Min A \<le> x \<longleftrightarrow> (\<exists>a\<in>A. a \<le> x)"
using assms by (simp add: Min_def fold1_below_iff)
lemma Max_ge_iff [noatp]:
assumes "finite A" and "A \<noteq> {}"
shows "x \<le> Max A \<longleftrightarrow> (\<exists>a\<in>A. x \<le> a)"
proof -
interpret dual: linorder "op \<ge>" "op >"
by (rule dual_linorder)
from assms show ?thesis
by (simp add: Max_def dual.fold1_below_iff [folded dual.dual_max])
qed
lemma Min_less_iff [noatp]:
assumes "finite A" and "A \<noteq> {}"
shows "Min A < x \<longleftrightarrow> (\<exists>a\<in>A. a < x)"
using assms by (simp add: Min_def fold1_strict_below_iff)
lemma Max_gr_iff [noatp]:
assumes "finite A" and "A \<noteq> {}"
shows "x < Max A \<longleftrightarrow> (\<exists>a\<in>A. x < a)"
proof -
interpret dual: linorder "op \<ge>" "op >"
by (rule dual_linorder)
from assms show ?thesis
by (simp add: Max_def dual.fold1_strict_below_iff [folded dual.dual_max])
qed
lemma Min_eqI:
assumes "finite A"
assumes "\<And>y. y \<in> A \<Longrightarrow> y \<ge> x"
and "x \<in> A"
shows "Min A = x"
proof (rule antisym)
from `x \<in> A` have "A \<noteq> {}" by auto
with assms show "Min A \<ge> x" by simp
next
from assms show "x \<ge> Min A" by simp
qed
lemma Max_eqI:
assumes "finite A"
assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
and "x \<in> A"
shows "Max A = x"
proof (rule antisym)
from `x \<in> A` have "A \<noteq> {}" by auto
with assms show "Max A \<le> x" by simp
next
from assms show "x \<le> Max A" by simp
qed
lemma Min_antimono:
assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
shows "Min N \<le> Min M"
using assms by (simp add: Min_def fold1_antimono)
lemma Max_mono:
assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
shows "Max M \<le> Max N"
proof -
interpret dual: linorder "op \<ge>" "op >"
by (rule dual_linorder)
from assms show ?thesis
by (simp add: Max_def dual.fold1_antimono [folded dual.dual_max])
qed
lemma finite_linorder_max_induct[consumes 1, case_names empty insert]:
"finite A \<Longrightarrow> P {} \<Longrightarrow>
(!!A b. finite A \<Longrightarrow> ALL a:A. a < b \<Longrightarrow> P A \<Longrightarrow> P(insert b A))
\<Longrightarrow> P A"
proof (induct rule: finite_psubset_induct)
fix A :: "'a set"
assume IH: "!! B. finite B \<Longrightarrow> B < A \<Longrightarrow> P {} \<Longrightarrow>
(!!A b. finite A \<Longrightarrow> (\<forall>a\<in>A. a<b) \<Longrightarrow> P A \<Longrightarrow> P (insert b A))
\<Longrightarrow> P B"
and "finite A" and "P {}"
and step: "!!A b. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)"
show "P A"
proof (cases "A = {}")
assume "A = {}" thus "P A" using `P {}` by simp
next
let ?B = "A - {Max A}" let ?A = "insert (Max A) ?B"
assume "A \<noteq> {}"
with `finite A` have "Max A : A" by auto
hence A: "?A = A" using insert_Diff_single insert_absorb by auto
moreover have "finite ?B" using `finite A` by simp
ultimately have "P ?B" using `P {}` step IH by blast
moreover have "\<forall>a\<in>?B. a < Max A" using Max_ge [OF `finite A`] by fastsimp
ultimately show "P A" using A insert_Diff_single step[OF `finite ?B`] by fastsimp
qed
qed
lemma finite_linorder_min_induct[consumes 1, case_names empty insert]:
"\<lbrakk>finite A; P {}; \<And>A b. \<lbrakk>finite A; \<forall>a\<in>A. b < a; P A\<rbrakk> \<Longrightarrow> P (insert b A)\<rbrakk> \<Longrightarrow> P A"
by(rule linorder.finite_linorder_max_induct[OF dual_linorder])
end
context ordered_ab_semigroup_add
begin
lemma add_Min_commute:
fixes k
assumes "finite N" and "N \<noteq> {}"
shows "k + Min N = Min {k + m | m. m \<in> N}"
proof -
have "\<And>x y. k + min x y = min (k + x) (k + y)"
by (simp add: min_def not_le)
(blast intro: antisym less_imp_le add_left_mono)
with assms show ?thesis
using hom_Min_commute [of "plus k" N]
by simp (blast intro: arg_cong [where f = Min])
qed
lemma add_Max_commute:
fixes k
assumes "finite N" and "N \<noteq> {}"
shows "k + Max N = Max {k + m | m. m \<in> N}"
proof -
have "\<And>x y. k + max x y = max (k + x) (k + y)"
by (simp add: max_def not_le)
(blast intro: antisym less_imp_le add_left_mono)
with assms show ?thesis
using hom_Max_commute [of "plus k" N]
by simp (blast intro: arg_cong [where f = Max])
qed
end
subsection {* Expressing set operations via @{const fold} *}
lemma (in fun_left_comm) fun_left_comm_apply:
"fun_left_comm (\<lambda>x. f (g x))"
proof
qed (simp_all add: fun_left_comm)
lemma (in fun_left_comm_idem) fun_left_comm_idem_apply:
"fun_left_comm_idem (\<lambda>x. f (g x))"
by (rule fun_left_comm_idem.intro, rule fun_left_comm_apply, unfold_locales)
(simp_all add: fun_left_idem)
lemma fun_left_comm_idem_insert:
"fun_left_comm_idem insert"
proof
qed auto
lemma fun_left_comm_idem_remove:
"fun_left_comm_idem (\<lambda>x A. A - {x})"
proof
qed auto
lemma fun_left_comm_idem_inter:
"fun_left_comm_idem op \<inter>"
proof
qed auto
lemma fun_left_comm_idem_union:
"fun_left_comm_idem op \<union>"
proof
qed auto
lemma union_fold_insert:
assumes "finite A"
shows "A \<union> B = fold insert B A"
proof -
interpret fun_left_comm_idem insert by (fact fun_left_comm_idem_insert)
from `finite A` show ?thesis by (induct A arbitrary: B) simp_all
qed
lemma minus_fold_remove:
assumes "finite A"
shows "B - A = fold (\<lambda>x A. A - {x}) B A"
proof -
interpret fun_left_comm_idem "\<lambda>x A. A - {x}" by (fact fun_left_comm_idem_remove)
from `finite A` show ?thesis by (induct A arbitrary: B) auto
qed
lemma inter_Inter_fold_inter:
assumes "finite A"
shows "B \<inter> Inter A = fold (op \<inter>) B A"
proof -
interpret fun_left_comm_idem "op \<inter>" by (fact fun_left_comm_idem_inter)
from `finite A` show ?thesis by (induct A arbitrary: B)
(simp_all add: fold_fun_comm Int_commute)
qed
lemma union_Union_fold_union:
assumes "finite A"
shows "B \<union> Union A = fold (op \<union>) B A"
proof -
interpret fun_left_comm_idem "op \<union>" by (fact fun_left_comm_idem_union)
from `finite A` show ?thesis by (induct A arbitrary: B)
(simp_all add: fold_fun_comm Un_commute)
qed
lemma Inter_fold_inter:
assumes "finite A"
shows "Inter A = fold (op \<inter>) UNIV A"
using assms inter_Inter_fold_inter [of A UNIV] by simp
lemma Union_fold_union:
assumes "finite A"
shows "Union A = fold (op \<union>) {} A"
using assms union_Union_fold_union [of A "{}"] by simp
lemma inter_INTER_fold_inter:
assumes "finite A"
shows "B \<inter> INTER A f = fold (\<lambda>A. op \<inter> (f A)) B A" (is "?inter = ?fold")
proof (rule sym)
interpret fun_left_comm_idem "op \<inter>" by (fact fun_left_comm_idem_inter)
interpret fun_left_comm_idem "\<lambda>A. op \<inter> (f A)" by (fact fun_left_comm_idem_apply)
from `finite A` show "?fold = ?inter" by (induct A arbitrary: B) auto
qed
lemma union_UNION_fold_union:
assumes "finite A"
shows "B \<union> UNION A f = fold (\<lambda>A. op \<union> (f A)) B A" (is "?union = ?fold")
proof (rule sym)
interpret fun_left_comm_idem "op \<union>" by (fact fun_left_comm_idem_union)
interpret fun_left_comm_idem "\<lambda>A. op \<union> (f A)" by (fact fun_left_comm_idem_apply)
from `finite A` show "?fold = ?union" by (induct A arbitrary: B) auto
qed
lemma INTER_fold_inter:
assumes "finite A"
shows "INTER A f = fold (\<lambda>A. op \<inter> (f A)) UNIV A"
using assms inter_INTER_fold_inter [of A UNIV] by simp
lemma UNION_fold_union:
assumes "finite A"
shows "UNION A f = fold (\<lambda>A. op \<union> (f A)) {} A"
using assms union_UNION_fold_union [of A "{}"] by simp
end