Theory Disjoint_Sets

(*  Author:     Johannes Hölzl, TU München

section ‹Partitions and Disjoint Sets›

theory Disjoint_Sets
  imports FuncSet

lemma mono_imp_UN_eq_last: "mono A  (in. A i) = A n"
  unfolding mono_def by auto

subsection ‹Set of Disjoint Sets›

abbreviation disjoint :: "'a set set  bool" where "disjoint  pairwise disjnt"

lemma disjoint_def: "disjoint A  (aA. bA. a  b  a  b = {})"
  unfolding pairwise_def disjnt_def by auto

lemma disjointI:
  "(a b. a  A  b  A  a  b  a  b = {})  disjoint A"
  unfolding disjoint_def by auto

lemma disjointD:
  "disjoint A  a  A  b  A  a  b  a  b = {}"
  unfolding disjoint_def by auto

lemma disjoint_image: "inj_on f (A)  disjoint A  disjoint ((`) f ` A)"
  unfolding inj_on_def disjoint_def by blast

lemma assumes "disjoint (A  B)"
      shows   disjoint_unionD1: "disjoint A" and disjoint_unionD2: "disjoint B"
  using assms by (simp_all add: disjoint_def)
lemma disjoint_INT:
  assumes *: "i. i  I  disjoint (F i)"
  shows "disjoint {iI. X i | X. iI. X i  F i}"
proof (safe intro!: disjointI del: equalityI)
  fix A B :: "'a  'b set" assume "(iI. A i)  (iI. B i)"
  then obtain i where "A i  B i" "i  I"
    by auto
  moreover assume "iI. A i  F i" "iI. B i  F i"
  ultimately show "(iI. A i)  (iI. B i) = {}"
    using *[OF iI, THEN disjointD, of "A i" "B i"]
    by (auto simp flip: INT_Int_distrib)

lemma diff_Union_pairwise_disjoint:
  assumes "pairwise disjnt 𝒜" "  𝒜"
  shows "𝒜 -  = (𝒜 - )"
proof -
  have "False"
    if x: "x  A" "x  B" and AB: "A  𝒜" "A  " "B  " for x A B
  proof -
    have "A  B = {}"
      using assms disjointD AB by blast
  with x show ?thesis
    by blast
  then show ?thesis by auto

lemma Int_Union_pairwise_disjoint:
  assumes "pairwise disjnt (𝒜  )"
  shows "𝒜   = (𝒜  )"
proof -
  have "False"
    if x: "x  A" "x  B" and AB: "A  𝒜" "A  " "B  " for x A B
  proof -
    have "A  B = {}"
      using assms disjointD AB by blast
  with x show ?thesis
    by blast
  then show ?thesis by auto

lemma psubset_Union_pairwise_disjoint:
  assumes : "pairwise disjnt " and "𝒜   - {{}}"
  shows "𝒜  "
  unfolding psubset_eq
  show "𝒜  "
    using assms by blast
  have "𝒜  " "( - 𝒜  ( - {{}}))  {}"
    using assms by blast+
  then show "𝒜  "
    using diff_Union_pairwise_disjoint [OF ] by blast

subsubsection "Family of Disjoint Sets"

definition disjoint_family_on :: "('i  'a set)  'i set  bool" where
  "disjoint_family_on A S  (mS. nS. m  n  A m  A n = {})"

abbreviation "disjoint_family A  disjoint_family_on A UNIV"

lemma disjoint_family_elem_disjnt:
  assumes "infinite A" "finite C"
      and df: "disjoint_family_on B A"
  obtains x where "x  A" "disjnt C (B x)"
proof -
  have "False" if *: "x  A. y. y  C  y  B x"
  proof -
    obtain g where g: "x  A. g x  C  g x  B x"
      using * by metis
    with df have "inj_on g A"
      by (fastforce simp add: inj_on_def disjoint_family_on_def)
    then have "infinite (g ` A)"
      using infinite A finite_image_iff by blast
    then show False
      by (meson finite C finite_subset g image_subset_iff)
  then show ?thesis
    by (force simp: disjnt_iff intro: that)

lemma disjoint_family_onD:
  "disjoint_family_on A I  i  I  j  I  i  j  A i  A j = {}"
  by (auto simp: disjoint_family_on_def)

lemma disjoint_family_subset: "disjoint_family A  (x. B x  A x)  disjoint_family B"
  by (force simp add: disjoint_family_on_def)

lemma disjoint_family_on_insert:
  "i  I  disjoint_family_on A (insert i I)  A i  (iI. A i) = {}  disjoint_family_on A I"
  by (fastforce simp: disjoint_family_on_def)

lemma disjoint_family_on_bisimulation:
  assumes "disjoint_family_on f S"
  and "n m. n  S  m  S  n  m  f n  f m = {}  g n  g m = {}"
  shows "disjoint_family_on g S"
  using assms unfolding disjoint_family_on_def by auto

lemma disjoint_family_on_mono:
  "A  B  disjoint_family_on f B  disjoint_family_on f A"
  unfolding disjoint_family_on_def by auto

lemma disjoint_family_Suc:
  "(n. A n  A (Suc n))  disjoint_family (λi. A (Suc i) - A i)"
  using lift_Suc_mono_le[of A]
  by (auto simp add: disjoint_family_on_def)
     (metis insert_absorb insert_subset le_SucE le_antisym not_le_imp_less less_imp_le)

lemma disjoint_family_on_disjoint_image:
  "disjoint_family_on A I  disjoint (A ` I)"
  unfolding disjoint_family_on_def disjoint_def by force
lemma disjoint_family_on_vimageI: "disjoint_family_on F I  disjoint_family_on (λi. f -` F i) I"
  by (auto simp: disjoint_family_on_def)

lemma disjoint_image_disjoint_family_on:
  assumes d: "disjoint (A ` I)" and i: "inj_on A I"
  shows "disjoint_family_on A I"
  unfolding disjoint_family_on_def
proof (intro ballI impI)
  fix n m assume nm: "m  I" "n  I" and "n  m"
  with i[THEN inj_onD, of n m] show "A n  A m = {}"
    by (intro disjointD[OF d]) auto

lemma disjoint_family_on_iff_disjoint_image:
  assumes "i. i  I  A i  {}"
  shows "disjoint_family_on A I  disjoint (A ` I)  inj_on A I"
  assume "disjoint_family_on A I"
  then show "disjoint (A ` I)  inj_on A I"
    by (metis (mono_tags, lifting) assms disjoint_family_onD disjoint_family_on_disjoint_image inf.idem inj_onI)
qed (use disjoint_image_disjoint_family_on in metis)

lemma card_UN_disjoint':
  assumes "disjoint_family_on A I" "i. i  I  finite (A i)" "finite I"
  shows "card (iI. A i) = (iI. card (A i))"
  using assms   by (simp add: card_UN_disjoint disjoint_family_on_def)

lemma disjoint_UN:
  assumes F: "i. i  I  disjoint (F i)" and *: "disjoint_family_on (λi. (F i)) I"
  shows "disjoint (iI. F i)"
proof (safe intro!: disjointI del: equalityI)
  fix A B i j assume "A  B" "A  F i" "i  I" "B  F j" "j  I"
  show "A  B = {}"
  proof cases
    assume "i = j" with F[of i] i  I A  F i B  F j A  B show "A  B = {}"
      by (auto dest: disjointD)
    assume "i  j"
    with * iI jI have "((F i))  ((F j)) = {}"
      by (rule disjoint_family_onD)
    with AF i iI BF j jI
    show "A  B = {}"
      by auto

lemma distinct_list_bind: 
  assumes "distinct xs" "x. x  set xs  distinct (f x)" 
          "disjoint_family_on (set  f) (set xs)"
  shows   "distinct (List.bind xs f)"
  using assms
  by (induction xs)
     (auto simp: disjoint_family_on_def distinct_map inj_on_def set_list_bind)

lemma bij_betw_UNION_disjoint:
  assumes disj: "disjoint_family_on A' I"
  assumes bij: "i. i  I  bij_betw f (A i) (A' i)"
  shows   "bij_betw f (iI. A i) (iI. A' i)"
unfolding bij_betw_def
  from bij show eq: "f ` (A ` I) = (A' ` I)"
    by (auto simp: bij_betw_def image_UN)
  show "inj_on f ((A ` I))"
  proof (rule inj_onI, clarify)
    fix i j x y assume A: "i  I" "j  I" "x  A i" "y  A j" and B: "f x = f y"
    from A bij[of i] bij[of j] have "f x  A' i" "f y  A' j"
      by (auto simp: bij_betw_def)
    with B have "A' i  A' j  {}" by auto
    with disj A have "i = j" unfolding disjoint_family_on_def by blast
    with A B bij[of i] show "x = y" by (auto simp: bij_betw_def dest: inj_onD)

lemma disjoint_union: "disjoint C  disjoint B  C  B = {}  disjoint (C  B)"
  using disjoint_UN[of "{C, B}" "λx. x"] by (auto simp add: disjoint_family_on_def)

text ‹
  Sum/product of the union of a finite disjoint family
context comm_monoid_set

lemma UNION_disjoint_family:
  assumes "finite I" and "iI. finite (A i)"
    and "disjoint_family_on A I"
  shows "F g ((A ` I)) = F (λx. F g (A x)) I"
  using assms unfolding disjoint_family_on_def  by (rule UNION_disjoint)

lemma Union_disjoint_sets:
  assumes "AC. finite A" and "disjoint C"
  shows "F g (C) = (F  F) g C"
  using assms unfolding disjoint_def  by (rule Union_disjoint)


text ‹
  The union of an infinite disjoint family of non-empty sets is infinite.
lemma infinite_disjoint_family_imp_infinite_UNION:
  assumes "¬finite A" "x. x  A  f x  {}" "disjoint_family_on f A"
  shows   "¬finite ((f ` A))"
proof -
  define g where "g x = (SOME y. y  f x)" for x
  have g: "g x  f x" if "x  A" for x
    unfolding g_def by (rule someI_ex, insert assms(2) that) blast
  have inj_on_g: "inj_on g A"
  proof (rule inj_onI, rule ccontr)
    fix x y assume A: "x  A" "y  A" "g x = g y" "x  y"
    with g[of x] g[of y] have "g x  f x" "g x  f y" by auto
    with A x  y assms show False
      by (auto simp: disjoint_family_on_def inj_on_def)
  from g have "g ` A  (f ` A)" by blast
  moreover from inj_on_g ¬finite A have "¬finite (g ` A)"
    using finite_imageD by blast
  ultimately show ?thesis using finite_subset by blast

subsection ‹Construct Disjoint Sequences›

definition disjointed :: "(nat  'a set)  nat  'a set" where
  "disjointed A n = A n - (i{0..<n}. A i)"

lemma finite_UN_disjointed_eq: "(i{0..<n}. disjointed A i) = (i{0..<n}. A i)"
proof (induct n)
  case 0 show ?case by simp
  case (Suc n)
  thus ?case by (simp add: atLeastLessThanSuc disjointed_def)

lemma UN_disjointed_eq: "(i. disjointed A i) = (i. A i)"
  by (rule UN_finite2_eq [where k=0])
     (simp add: finite_UN_disjointed_eq)

lemma less_disjoint_disjointed: "m < n  disjointed A m  disjointed A n = {}"
  by (auto simp add: disjointed_def)

lemma disjoint_family_disjointed: "disjoint_family (disjointed A)"
  by (simp add: disjoint_family_on_def)
     (metis neq_iff Int_commute less_disjoint_disjointed)

lemma disjointed_subset: "disjointed A n  A n"
  by (auto simp add: disjointed_def)

lemma disjointed_0[simp]: "disjointed A 0 = A 0"
  by (simp add: disjointed_def)

lemma disjointed_mono: "mono A  disjointed A (Suc n) = A (Suc n) - A n"
  using mono_imp_UN_eq_last[of A] by (simp add: disjointed_def atLeastLessThanSuc_atLeastAtMost atLeast0AtMost)

subsection ‹Partitions›

text ‹
  Partitions termP of a set termA. We explicitly disallow empty sets.

definition partition_on :: "'a set  'a set set  bool"
  "partition_on A P  P = A  disjoint P  {}  P"

lemma partition_onI:
  "P = A  (p q. p  P  q  P  p  q  disjnt p q)  {}  P  partition_on A P"
  by (auto simp: partition_on_def pairwise_def)

lemma partition_onD1: "partition_on A P  A = P"
  by (auto simp: partition_on_def)

lemma partition_onD2: "partition_on A P  disjoint P"
  by (auto simp: partition_on_def)

lemma partition_onD3: "partition_on A P  {}  P"
  by (auto simp: partition_on_def)

subsection ‹Constructions of partitions›

lemma partition_on_empty: "partition_on {} P  P = {}"
  unfolding partition_on_def by fastforce

lemma partition_on_space: "A  {}  partition_on A {A}"
  by (auto simp: partition_on_def disjoint_def)

lemma partition_on_singletons: "partition_on A ((λx. {x}) ` A)"
  by (auto simp: partition_on_def disjoint_def)

lemma partition_on_transform:
  assumes P: "partition_on A P"
  assumes F_UN: "(F ` P) = F (P)" and F_disjnt: "p q. p  P  q  P  disjnt p q  disjnt (F p) (F q)"
  shows "partition_on (F A) (F ` P - {{}})"
proof -
  have "(F ` P - {{}}) = F A"
    unfolding P[THEN partition_onD1] F_UN[symmetric] by auto
  with P show ?thesis
    by (auto simp add: partition_on_def pairwise_def intro!: F_disjnt)

lemma partition_on_restrict: "partition_on A P  partition_on (B  A) ((∩) B ` P - {{}})"
  by (intro partition_on_transform) (auto simp: disjnt_def)

lemma partition_on_vimage: "partition_on A P  partition_on (f -` A) ((-`) f ` P - {{}})"
  by (intro partition_on_transform) (auto simp: disjnt_def)

lemma partition_on_inj_image:
  assumes P: "partition_on A P" and f: "inj_on f A"
  shows "partition_on (f ` A) ((`) f ` P - {{}})"
proof (rule partition_on_transform[OF P])
  show "p  P  q  P  disjnt p q  disjnt (f ` p) (f ` q)" for p q
    using f[THEN inj_onD] P[THEN partition_onD1] by (auto simp: disjnt_def)
qed auto

lemma partition_on_insert:
  assumes "disjnt p (P)"
  shows "partition_on A (insert p P)  partition_on (A-p) P  p  A  p  {}"
  using assms
  by (auto simp: partition_on_def disjnt_iff pairwise_insert)

subsection ‹Finiteness of partitions›

lemma finitely_many_partition_on:
  assumes "finite A"
  shows "finite {P. partition_on A P}"
proof (rule finite_subset)
  show "{P. partition_on A P}  Pow (Pow A)"
    unfolding partition_on_def by auto
  show "finite (Pow (Pow A))"
    using assms by simp

lemma finite_elements: "finite A  partition_on A P  finite P"
  using partition_onD1[of A P] by (simp add: finite_UnionD)

lemma product_partition:
  assumes "partition_on A P" and "p. p  P  finite p" 
  shows "card A = (pP. card p)"
  using assms unfolding partition_on_def  by (meson card_Union_disjoint)

subsection ‹Equivalence of partitions and equivalence classes›

lemma partition_on_quotient:
  assumes r: "equiv A r"
  shows "partition_on A (A // r)"
proof (rule partition_onI)
  from r have "refl_on A r"
    by (auto elim: equivE)
  then show "(A // r) = A" "{}  A // r"
    by (auto simp: refl_on_def quotient_def)

  fix p q assume "p  A // r" "q  A // r" "p  q"
  then obtain x y where "x  A" "y  A" "p = r `` {x}" "q = r `` {y}"
    by (auto simp: quotient_def)
  with r equiv_class_eq_iff[OF r, of x y] p  q show "disjnt p q"
    by (auto simp: disjnt_equiv_class)

lemma equiv_partition_on:
  assumes P: "partition_on A P"
  shows "equiv A {(x, y). p  P. x  p  y  p}"
proof (rule equivI)
  have "A = P"
    using P by (auto simp: partition_on_def)

  have "{(x, y). p  P. x  p  y  p}  A × A"
    unfolding A = P by blast
  then show "refl_on A {(x, y). pP. x  p  y  p}"
    unfolding refl_on_def A = P by auto
  show "trans {(x, y). pP. x  p  y  p}"
    using P by (auto simp only: trans_def disjoint_def partition_on_def)
  show "sym {(x, y). pP. x  p  y  p}"
    by (auto simp only: sym_def)

lemma partition_on_eq_quotient:
  assumes P: "partition_on A P"
  shows "A // {(x, y). p  P. x  p  y  p} = P"
  unfolding quotient_def
proof safe
  fix x assume "x  A"
  then obtain p where "p  P" "x  p" "q. q  P  x  q  p = q"
    using P by (auto simp: partition_on_def disjoint_def)
  then have "{y. pP. x  p  y  p} = p"
    by (safe intro!: bexI[of _ p]) simp
  then show "{(x, y). pP. x  p  y  p} `` {x}  P"
    by (simp add: p  P)
  fix p assume "p  P"
  then have "p  {}"
    using P by (auto simp: partition_on_def)
  then obtain x where "x  p"
    by auto
  then have "x  A" "q. q  P  x  q  p = q"
    using P p  P by (auto simp: partition_on_def disjoint_def)
  with pP x  p have "{y. pP. x  p  y  p} = p"
    by (safe intro!: bexI[of _ p]) simp
  then show "p  (xA. {{(x, y). pP. x  p  y  p} `` {x}})"
    by (auto intro: x  A)

lemma partition_on_alt: "partition_on A P  (r. equiv A r  P = A // r)"
  by (auto simp: partition_on_eq_quotient intro!: partition_on_quotient intro: equiv_partition_on)

subsection ‹Refinement of partitions›

definition refines :: "'a set  'a set set  'a set set  bool"
  where "refines A P Q 
        partition_on A P  partition_on A Q  (XP. YQ. X  Y)" 

lemma refines_refl: "partition_on A P  refines A P P"
  using refines_def by blast

lemma refines_asym1:
  assumes "refines A P Q" "refines A Q P"
  shows "P  Q"
  fix X
  assume "X  P"
  then obtain Y X' where "Y  Q" "X  Y" "X'  P" "Y  X'"
    by (meson assms refines_def)
  then have "X' = X"
    using assms(2) unfolding partition_on_def refines_def
    by (metis X  P X  Y disjnt_self_iff_empty disjnt_subset1 pairwiseD)
  then show "X  Q"
    using X  Y Y  Q Y  X' by force

lemma refines_asym: "refines A P Q; refines A Q P  P=Q"
  by (meson antisym_conv refines_asym1)

lemma refines_trans: "refines A P Q; refines A Q R  refines A P R"
  by (meson order.trans refines_def)

lemma refines_obtains_subset:
  assumes "refines A P Q" "q  Q"
  shows "partition_on q {p  P. p  q}"
proof -
  have "p  q  disjnt p q" if "p  P" for p
    using that assms unfolding refines_def partition_on_def disjoint_def
    by (metis disjnt_def disjnt_subset1)
  with assms have "q  Union {p  P. p  q}"
    using assms 
   by (clarsimp simp: refines_def disjnt_iff partition_on_def) (metis Union_iff)
  with assms have "q = Union {p  P. p  q}"
    by auto
  then show ?thesis
    using assms by (auto simp: refines_def disjoint_def partition_on_def)

subsection ‹The coarsest common refinement of a set of partitions›

definition common_refinement :: "'a set set set  'a set set"
  where "common_refinement 𝒫  (f  (ΠE P𝒫. P). { (f ` 𝒫)}) - {{}}" 

text ‹With non-extensional function space›
lemma common_refinement: "common_refinement 𝒫 = (f  (Π P𝒫. P). { (f ` 𝒫)}) - {{}}" 
  (is "?lhs = ?rhs")
  show "?rhs  ?lhs"
    apply (clarsimp simp add: common_refinement_def PiE_def Ball_def)
    by (metis restrict_Pi_cancel image_restrict_eq restrict_extensional)
qed (auto simp add: common_refinement_def PiE_def)

lemma common_refinement_exists: "X  common_refinement 𝒫; P  𝒫  RP. X  R"
  by (auto simp add: common_refinement)

lemma Union_common_refinement: " (common_refinement 𝒫) = ( P𝒫. P)"
  show "( P𝒫. P)   (common_refinement 𝒫)"
  proof (clarsimp simp: common_refinement)
    fix x
    assume "P𝒫. XP. x  X"
    then obtain F where F: "P. P𝒫  F P  P  x  F P"
      by metis
    then have "x   (F ` 𝒫)"
      by force
    with F show "X(xΠ P𝒫. P. { (x ` 𝒫)}) - {{}}. x  X"
      by (auto simp add: Pi_iff Bex_def)
qed (auto simp: common_refinement_def)

lemma partition_on_common_refinement:
  assumes A: "P. P  𝒫  partition_on A P" and "𝒫  {}"
  shows "partition_on A (common_refinement 𝒫)"
proof (rule partition_onI)
  show " (common_refinement 𝒫) = A"
    using assms by (simp add: partition_on_def Union_common_refinement)
  fix P Q
  assume "P  common_refinement 𝒫" and "Q  common_refinement 𝒫" and "P  Q"
  then obtain f g where f: "f  (ΠE P𝒫. P)" and P: "P =  (f ` 𝒫)" and "P  {}"
                  and   g: "g  (ΠE P𝒫. P)" and Q: "Q =  (g ` 𝒫)" and "Q  {}"
    by (auto simp add: common_refinement_def)
  have "f=g" if "x  P" "x  Q" for x
  proof (rule extensionalityI [of _ 𝒫])
    fix R
    assume "R  𝒫"
    with that P Q f g A [unfolded partition_on_def, OF R  𝒫]
    show "f R = g R"
      by (metis INT_E Int_iff PiE_iff disjointD emptyE)
  qed (use PiE_iff f g in auto)
  then show "disjnt P Q"
    by (metis P Q P  Q disjnt_iff) 
qed (simp add: common_refinement_def)

lemma refines_common_refinement:
  assumes "P. P  𝒫  partition_on A P" "P  𝒫"
  shows "refines A (common_refinement 𝒫) P"
  unfolding refines_def
proof (intro conjI strip)
  fix X
  assume "X  common_refinement 𝒫"
  with assms show "YP. X  Y"
    by (auto simp: common_refinement_def)
qed (use assms partition_on_common_refinement in auto)

text ‹The common refinement is itself refined by any other›
lemma common_refinement_coarsest:
  assumes "P. P  𝒫  partition_on A P" "partition_on A R" "P. P  𝒫  refines A R P" "𝒫  {}"
  shows "refines A R (common_refinement 𝒫)"
  unfolding refines_def
proof (intro conjI ballI partition_on_common_refinement)
  fix X
  assume "X  R"
  have "p  P. X  p" if "P  𝒫" for P
    by (meson X  R assms(3) refines_def that)
  then obtain F where f: "P. P  𝒫  F P  P  X  F P"
    by metis
  with partition_on A R X  R 𝒫  {}
  have " (F ` 𝒫)  common_refinement 𝒫"
    apply (simp add: partition_on_def common_refinement Pi_iff Bex_def)
    by (metis (no_types, lifting) cINF_greatest subset_empty)
  with f show "Ycommon_refinement 𝒫. X  Y"
    by (metis 𝒫  {} cINF_greatest)
qed (use assms in auto)

lemma finite_common_refinement:
  assumes "finite 𝒫" "P. P  𝒫  finite P"
  shows "finite (common_refinement 𝒫)"
proof -
  have "finite (ΠE P𝒫. P)"
    by (simp add: assms finite_PiE)
  then show ?thesis
    by (auto simp: common_refinement_def)

lemma card_common_refinement:
  assumes "finite 𝒫" "P. P  𝒫  finite P"
  shows "card (common_refinement 𝒫)  (P  𝒫. card P)"
proof -
  have "card (common_refinement 𝒫)  card (f  (ΠE P𝒫. P). { (f ` 𝒫)})"
    unfolding common_refinement_def by (meson card_Diff1_le)
  also have "  (f(ΠE P𝒫. P). card{ (f ` 𝒫)})"
    by (metis assms finite_PiE card_UN_le)
  also have " = card(ΠE P𝒫. P)"
    by simp
  also have " = (P  𝒫. card P)"
    by (simp add: assms(1) card_PiE dual_order.eq_iff)
  finally show ?thesis .