(*  Title:      HOL/Finite_Set.thy

    ID:         $Id$

    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel

                with contributions by Jeremy Avigad

*)

header {* Finite sets *}

theory Finite_Set

imports Datatype Divides Transitive_Closure

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{* 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)

subsubsection{* Finiteness and set theoretic constructions *}

lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"

  -- {* The union of two finite sets is finite. *}

  by (induct set: 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_Collect_subset[simp]: "finite A \<Longrightarrow> finite{x \<in> A. P x}"

using finite_subset[of "{x \<in> A. P x}" "A"] by blast

lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"

  by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)

lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"

  -- {* The converse obviously fails. *}

  by (blast intro: finite_subset)

lemma finite_insert [simp]: "finite (insert a A) = finite A"

  apply (subst insert_is_Un)

  apply (simp only: finite_Un, blast)

  done

lemma finite_Union[simp, intro]:

 "\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)"

by (induct rule:finite_induct) simp_all

lemma finite_empty_induct:

  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 B ==> finite (B - Ba)"

  by (rule Diff_subset [THEN finite_subset])

lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"

  apply (subst Diff_insert)

  apply (case_tac "a : A - B")

   apply (rule finite_insert [symmetric, THEN trans])

   apply (subst insert_Diff, simp_all)

  done

lemma finite_Diff_singleton [simp]: "finite (A - {a}) = finite A"

  by simp

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_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_Plus: "[| finite A; finite B |] ==> finite (A <+> B)"

by (simp add: Plus_def)

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_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A"

by(blast intro: finite_subset[OF subset_Pow_Union])

lemma finite_converse [iff]: "finite (r^-1) = finite r"

  apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")

   apply simp

   apply (rule iffI)

    apply (erule finite_imageD [unfolded inj_on_def])

    apply (simp split add: split_split)

   apply (erule finite_imageI)

  apply (simp add: converse_def image_def, auto)

  apply (rule bexI)

   prefer 2 apply assumption

  apply simp

  done

text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi

Ehmety) *}

lemma finite_Field: "finite r ==> finite (Field r)"

  -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}

  apply (induct set: finite)

   apply (auto simp add: Field_def Domain_insert Range_insert)

  done

lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"

  apply clarify

  apply (erule trancl_induct)

   apply (auto simp add: Field_def)

  done

lemma finite_trancl: "finite (r^+) = finite r"

  apply auto

   prefer 2

   apply (rule trancl_subset_Field2 [THEN finite_subset])

   apply (rule finite_SigmaI)

    prefer 3

    apply (blast intro: r_into_trancl' finite_subset)

   apply (auto simp add: finite_Field)

  done

subsection {* Class @{text finite}  *}

setup {* Sign.add_path "finite" *} -- {*FIXME: name tweaking*}

class finite = itself +

  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)

instance option :: (finite) finite

  by default (simp add: insert_None_conv_UNIV [symmetric])

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

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 I.fold_insert)

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 apply induct

 apply simp

apply simp

done

(*

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 -

  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)

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_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)

(*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])