Theory HOL.Complete_Lattices

(*  Title:      HOL/Complete_Lattices.thy
    Author:     Tobias Nipkow
    Author:     Lawrence C Paulson
    Author:     Markus Wenzel
    Author:     Florian Haftmann
    Author:     Viorel Preoteasa (Complete Distributive Lattices)     
*)

section ‹Complete lattices›

theory Complete_Lattices
  imports Fun
begin

subsection ‹Syntactic infimum and supremum operations›

class Inf =
  fixes Inf :: "'a set  'a"  (" _" [900] 900)

class Sup =
  fixes Sup :: "'a set  'a"  (" _" [900] 900)

syntax
  "_INF1"     :: "pttrns  'b  'b"           ("(3INF _./ _)" [0, 10] 10)
  "_INF"      :: "pttrn  'a set  'b  'b"  ("(3INF __./ _)" [0, 0, 10] 10)
  "_SUP1"     :: "pttrns  'b  'b"           ("(3SUP _./ _)" [0, 10] 10)
  "_SUP"      :: "pttrn  'a set  'b  'b"  ("(3SUP __./ _)" [0, 0, 10] 10)

syntax
  "_INF1"     :: "pttrns  'b  'b"           ("(3_./ _)" [0, 10] 10)
  "_INF"      :: "pttrn  'a set  'b  'b"  ("(3__./ _)" [0, 0, 10] 10)
  "_SUP1"     :: "pttrns  'b  'b"           ("(3_./ _)" [0, 10] 10)
  "_SUP"      :: "pttrn  'a set  'b  'b"  ("(3__./ _)" [0, 0, 10] 10)

translations
  "x y. f"    "x. y. f"
  "x. f"      "(CONST range (λx. f))"
  "xA. f"    "CONST Inf ((λx. f) ` A)"
  "x y. f"    "x. y. f"
  "x. f"      "(CONST range (λx. f))"
  "xA. f"    "CONST Sup ((λx. f) `  A)"

context Inf
begin

lemma INF_image: " (g ` f ` A) =  ((g  f) ` A)"
  by (simp add: image_comp)

lemma INF_identity_eq [simp]: "(xA. x) = A"
  by simp

lemma INF_id_eq [simp]: "(id ` A) = A"
  by simp

lemma INF_cong: "A = B  (x. x  B  C x = D x)  (C ` A) = (D ` B)"
  by (simp add: image_def)

lemma INF_cong_simp:
  "A = B  (x. x  B =simp=> C x = D x)  (C ` A) = (D ` B)"
  unfolding simp_implies_def by (fact INF_cong)

end

context Sup
begin

lemma SUP_image: " (g ` f ` A) =  ((g  f) ` A)"
by(fact Inf.INF_image)

lemma SUP_identity_eq [simp]: "(xA. x) = A"
by(fact Inf.INF_identity_eq)

lemma SUP_id_eq [simp]: "(id ` A) = A"
by(fact Inf.INF_id_eq)

lemma SUP_cong: "A = B  (x. x  B  C x = D x)  (C ` A) = (D ` B)"
by (fact Inf.INF_cong)

lemma SUP_cong_simp:
  "A = B  (x. x  B =simp=> C x = D x)  (C ` A) = (D ` B)"
by (fact Inf.INF_cong_simp)

end


subsection ‹Abstract complete lattices›

text ‹A complete lattice always has a bottom and a top,
so we include them into the following type class,
along with assumptions that define bottom and top
in terms of infimum and supremum.›

class complete_lattice = lattice + Inf + Sup + bot + top +
  assumes Inf_lower: "x  A  A  x"
    and Inf_greatest: "(x. x  A  z  x)  z  A"
    and Sup_upper: "x  A  x  A"
    and Sup_least: "(x. x  A  x  z)  A  z"
    and Inf_empty [simp]: "{} = "
    and Sup_empty [simp]: "{} = "
begin

subclass bounded_lattice
proof
  fix a
  show "  a"
    by (auto intro: Sup_least simp only: Sup_empty [symmetric])
  show "a  "
    by (auto intro: Inf_greatest simp only: Inf_empty [symmetric])
qed

lemma dual_complete_lattice: "class.complete_lattice Sup Inf sup (≥) (>) inf  "
  by (auto intro!: class.complete_lattice.intro dual_lattice)
    (unfold_locales, (fact Inf_empty Sup_empty Sup_upper Sup_least Inf_lower Inf_greatest)+)

end

context complete_lattice
begin

lemma Sup_eqI:
  "(y. y  A  y  x)  (y. (z. z  A  z  y)  x  y)  A = x"
  by (blast intro: order.antisym Sup_least Sup_upper)

lemma Inf_eqI:
  "(i. i  A  x  i)  (y. (i. i  A  y  i)  y  x)  A = x"
  by (blast intro: order.antisym Inf_greatest Inf_lower)

lemma SUP_eqI:
  "(i. i  A  f i  x)  (y. (i. i  A  f i  y)  x  y)  (iA. f i) = x"
  using Sup_eqI [of "f ` A" x] by auto

lemma INF_eqI:
  "(i. i  A  x  f i)  (y. (i. i  A  f i  y)  x  y)  (iA. f i) = x"
  using Inf_eqI [of "f ` A" x] by auto

lemma INF_lower: "i  A  (iA. f i)  f i"
  using Inf_lower [of _ "f ` A"] by simp

lemma INF_greatest: "(i. i  A  u  f i)  u  (iA. f i)"
  using Inf_greatest [of "f ` A"] by auto

lemma SUP_upper: "i  A  f i  (iA. f i)"
  using Sup_upper [of _ "f ` A"] by simp

lemma SUP_least: "(i. i  A  f i  u)  (iA. f i)  u"
  using Sup_least [of "f ` A"] by auto

lemma Inf_lower2: "u  A  u  v  A  v"
  using Inf_lower [of u A] by auto

lemma INF_lower2: "i  A  f i  u  (iA. f i)  u"
  using INF_lower [of i A f] by auto

lemma Sup_upper2: "u  A  v  u  v  A"
  using Sup_upper [of u A] by auto

lemma SUP_upper2: "i  A  u  f i  u  (iA. f i)"
  using SUP_upper [of i A f] by auto

lemma le_Inf_iff: "b  A  (aA. b  a)"
  by (auto intro: Inf_greatest dest: Inf_lower)

lemma le_INF_iff: "u  (iA. f i)  (iA. u  f i)"
  using le_Inf_iff [of _ "f ` A"] by simp

lemma Sup_le_iff: "A  b  (aA. a  b)"
  by (auto intro: Sup_least dest: Sup_upper)

lemma SUP_le_iff: "(iA. f i)  u  (iA. f i  u)"
  using Sup_le_iff [of "f ` A"] by simp

lemma Inf_insert [simp]: "(insert a A) = a  A"
  by (auto intro: le_infI le_infI1 le_infI2 order.antisym Inf_greatest Inf_lower)

lemma INF_insert: "(xinsert a A. f x) = f a  (f ` A)"
  by simp

lemma Sup_insert [simp]: "(insert a A) = a  A"
  by (auto intro: le_supI le_supI1 le_supI2 order.antisym Sup_least Sup_upper)

lemma SUP_insert: "(xinsert a A. f x) = f a  (f ` A)"
  by simp

lemma INF_empty: "(x{}. f x) = "
  by simp

lemma SUP_empty: "(x{}. f x) = "
  by simp

lemma Inf_UNIV [simp]: "UNIV = "
  by (auto intro!: order.antisym Inf_lower)

lemma Sup_UNIV [simp]: "UNIV = "
  by (auto intro!: order.antisym Sup_upper)

lemma Inf_eq_Sup: "A = {b. a  A. b  a}"
  by (auto intro: order.antisym Inf_lower Inf_greatest Sup_upper Sup_least)

lemma Sup_eq_Inf:  "A = {b. a  A. a  b}"
  by (auto intro: order.antisym Inf_lower Inf_greatest Sup_upper Sup_least)

lemma Inf_superset_mono: "B  A  A  B"
  by (auto intro: Inf_greatest Inf_lower)

lemma Sup_subset_mono: "A  B  A  B"
  by (auto intro: Sup_least Sup_upper)

lemma Inf_mono:
  assumes "b. b  B  aA. a  b"
  shows "A  B"
proof (rule Inf_greatest)
  fix b assume "b  B"
  with assms obtain a where "a  A" and "a  b" by blast
  from a  A have "A  a" by (rule Inf_lower)
  with a  b show "A  b" by auto
qed

lemma INF_mono: "(m. m  B  nA. f n  g m)  (nA. f n)  (nB. g n)"
  using Inf_mono [of "g ` B" "f ` A"] by auto

lemma INF_mono': "(x. f x  g x)  (xA. f x)  (xA. g x)"
  by (rule INF_mono) auto

lemma Sup_mono:
  assumes "a. a  A  bB. a  b"
  shows "A  B"
proof (rule Sup_least)
  fix a assume "a  A"
  with assms obtain b where "b  B" and "a  b" by blast
  from b  B have "b  B" by (rule Sup_upper)
  with a  b show "a  B" by auto
qed

lemma SUP_mono: "(n. n  A  mB. f n  g m)  (nA. f n)  (nB. g n)"
  using Sup_mono [of "f ` A" "g ` B"] by auto

lemma SUP_mono': "(x. f x  g x)  (xA. f x)  (xA. g x)"
  by (rule SUP_mono) auto

lemma INF_superset_mono: "B  A  (x. x  B  f x  g x)  (xA. f x)  (xB. g x)"
  ― ‹The last inclusion is POSITIVE!›
  by (blast intro: INF_mono dest: subsetD)

lemma SUP_subset_mono: "A  B  (x. x  A  f x  g x)  (xA. f x)  (xB. g x)"
  by (blast intro: SUP_mono dest: subsetD)

lemma Inf_less_eq:
  assumes "v. v  A  v  u"
    and "A  {}"
  shows "A  u"
proof -
  from A  {} obtain v where "v  A" by blast
  moreover from v  A assms(1) have "v  u" by blast
  ultimately show ?thesis by (rule Inf_lower2)
qed

lemma less_eq_Sup:
  assumes "v. v  A  u  v"
    and "A  {}"
  shows "u  A"
proof -
  from A  {} obtain v where "v  A" by blast
  moreover from v  A assms(1) have "u  v" by blast
  ultimately show ?thesis by (rule Sup_upper2)
qed

lemma INF_eq:
  assumes "i. i  A  jB. f i  g j"
    and "j. j  B  iA. g j  f i"
  shows "(f ` A) = (g ` B)"
  by (intro order.antisym INF_greatest) (blast intro: INF_lower2 dest: assms)+

lemma SUP_eq:
  assumes "i. i  A  jB. f i  g j"
    and "j. j  B  iA. g j  f i"
  shows "(f ` A) = (g ` B)"
  by (intro order.antisym SUP_least) (blast intro: SUP_upper2 dest: assms)+

lemma less_eq_Inf_inter: "A  B  (A  B)"
  by (auto intro: Inf_greatest Inf_lower)

lemma Sup_inter_less_eq: "(A  B)  A  B "
  by (auto intro: Sup_least Sup_upper)

lemma Inf_union_distrib: "(A  B) = A  B"
  by (rule order.antisym) (auto intro: Inf_greatest Inf_lower le_infI1 le_infI2)

lemma INF_union: "(i  A  B. M i) = (i  A. M i)  (iB. M i)"
  by (auto intro!: order.antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)

lemma Sup_union_distrib: "(A  B) = A  B"
  by (rule order.antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)

lemma SUP_union: "(i  A  B. M i) = (i  A. M i)  (iB. M i)"
  by (auto intro!: order.antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)

lemma INF_inf_distrib: "(aA. f a)  (aA. g a) = (aA. f a  g a)"
  by (rule order.antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)

lemma SUP_sup_distrib: "(aA. f a)  (aA. g a) = (aA. f a  g a)"
  (is "?L = ?R")
proof (rule order.antisym)
  show "?L  ?R"
    by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
  show "?R  ?L"
    by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper)
qed

lemma Inf_top_conv [simp]:
  "A =   (xA. x = )"
  " = A  (xA. x = )"
proof -
  show "A =   (xA. x = )"
  proof
    assume "xA. x = "
    then have "A = {}  A = {}" by auto
    then show "A = " by auto
  next
    assume "A = "
    show "xA. x = "
    proof (rule ccontr)
      assume "¬ (xA. x = )"
      then obtain x where "x  A" and "x  " by blast
      then obtain B where "A = insert x B" by blast
      with A =  x   show False by simp
    qed
  qed
  then show " = A  (xA. x = )" by auto
qed

lemma INF_top_conv [simp]:
  "(xA. B x) =   (xA. B x = )"
  " = (xA. B x)  (xA. B x = )"
  using Inf_top_conv [of "B ` A"] by simp_all

lemma Sup_bot_conv [simp]:
  "A =   (xA. x = )"
  " = A  (xA. x = )"
  using dual_complete_lattice
  by (rule complete_lattice.Inf_top_conv)+

lemma SUP_bot_conv [simp]:
  "(xA. B x) =   (xA. B x = )"
  " = (xA. B x)  (xA. B x = )"
  using Sup_bot_conv [of "B ` A"] by simp_all

lemma INF_constant: "(yA. c) = (if A = {} then  else c)"
  by (auto intro: order.antisym INF_lower INF_greatest)

lemma SUP_constant: "(yA. c) = (if A = {} then  else c)"
  by (auto intro: order.antisym SUP_upper SUP_least)

lemma INF_const [simp]: "A  {}  (iA. f) = f"
  by (simp add: INF_constant)

lemma SUP_const [simp]: "A  {}  (iA. f) = f"
  by (simp add: SUP_constant)

lemma INF_top [simp]: "(xA. ) = "
  by (cases "A = {}") simp_all

lemma SUP_bot [simp]: "(xA. ) = "
  by (cases "A = {}") simp_all

lemma INF_commute: "(iA. jB. f i j) = (jB. iA. f i j)"
  by (iprover intro: INF_lower INF_greatest order_trans order.antisym)

lemma SUP_commute: "(iA. jB. f i j) = (jB. iA. f i j)"
  by (iprover intro: SUP_upper SUP_least order_trans order.antisym)

lemma INF_absorb:
  assumes "k  I"
  shows "A k  (iI. A i) = (iI. A i)"
proof -
  from assms obtain J where "I = insert k J" by blast
  then show ?thesis by simp
qed

lemma SUP_absorb:
  assumes "k  I"
  shows "A k  (iI. A i) = (iI. A i)"
proof -
  from assms obtain J where "I = insert k J" by blast
  then show ?thesis by simp
qed

lemma INF_inf_const1: "I  {}  (iI. inf x (f i)) = inf x (iI. f i)"
  by (intro order.antisym INF_greatest inf_mono order_refl INF_lower)
     (auto intro: INF_lower2 le_infI2 intro!: INF_mono)

lemma INF_inf_const2: "I  {}  (iI. inf (f i) x) = inf (iI. f i) x"
  using INF_inf_const1[of I x f] by (simp add: inf_commute)

lemma less_INF_D:
  assumes "y < (iA. f i)" "i  A"
  shows "y < f i"
proof -
  note y < (iA. f i)
  also have "(iA. f i)  f i" using i  A
    by (rule INF_lower)
  finally show "y < f i" .
qed

lemma SUP_lessD:
  assumes "(iA. f i) < y" "i  A"
  shows "f i < y"
proof -
  have "f i  (iA. f i)"
    using i  A by (rule SUP_upper)
  also note (iA. f i) < y
  finally show "f i < y" .
qed

lemma INF_UNIV_bool_expand: "(b. A b) = A True  A False"
  by (simp add: UNIV_bool inf_commute)

lemma SUP_UNIV_bool_expand: "(b. A b) = A True  A False"
  by (simp add: UNIV_bool sup_commute)

lemma Inf_le_Sup: "A  {}  Inf A  Sup A"
  by (blast intro: Sup_upper2 Inf_lower ex_in_conv)

lemma INF_le_SUP: "A  {}  (f ` A)  (f ` A)"
  using Inf_le_Sup [of "f ` A"] by simp

lemma INF_eq_const: "I  {}  (i. i  I  f i = x)  (f ` I) = x"
  by (auto intro: INF_eqI)

lemma SUP_eq_const: "I  {}  (i. i  I  f i = x)  (f ` I) = x"
  by (auto intro: SUP_eqI)

lemma INF_eq_iff: "I  {}  (i. i  I  f i  c)  (f ` I) = c  (iI. f i = c)"
  by (auto intro: INF_eq_const INF_lower order.antisym)

lemma SUP_eq_iff: "I  {}  (i. i  I  c  f i)  (f ` I) = c  (iI. f i = c)"
  by (auto intro: SUP_eq_const SUP_upper order.antisym)

end

context complete_lattice
begin
lemma Sup_Inf_le: "Sup (Inf ` {f ` A | f . ( Y  A . f Y  Y)})  Inf (Sup ` A)"
  by (rule SUP_least, clarify, rule INF_greatest, simp add: INF_lower2 Sup_upper)
end 

class complete_distrib_lattice = complete_lattice +
  assumes Inf_Sup_le: "Inf (Sup ` A)  Sup (Inf ` {f ` A | f . ( Y  A . f Y  Y)})"
begin
  
lemma Inf_Sup: "Inf (Sup ` A) = Sup (Inf ` {f ` A | f . ( Y  A . f Y  Y)})"
  by (rule order.antisym, rule Inf_Sup_le, rule Sup_Inf_le)

subclass distrib_lattice
proof
  fix a b c
  show "a  b  c = (a  b)  (a  c)"
  proof (rule order.antisym, simp_all, safe)
    show "b  c  a  b"
      by (rule le_infI1, simp)
    show "b  c  a  c"
      by (rule le_infI2, simp)
    have [simp]: "a  c  a  b  c"
      by (rule le_infI1, simp)
    have [simp]: "b  a  a  b  c"
      by (rule le_infI2, simp)
    have "(Sup ` {{a, b}, {a, c}}) =
      (Inf ` {f ` {{a, b}, {a, c}} | f. Y{{a, b}, {a, c}}. f Y  Y})"
      by (rule Inf_Sup)
    from this show "(a  b)  (a  c)  a  b  c"
      apply simp
      by (rule SUP_least, safe, simp_all)
  qed
qed
end

context complete_lattice
begin
context
  fixes f :: "'a  'b::complete_lattice"
  assumes "mono f"
begin

lemma mono_Inf: "f (A)  (xA. f x)"
  using mono f by (auto intro: complete_lattice_class.INF_greatest Inf_lower dest: monoD)

lemma mono_Sup: "(xA. f x)  f (A)"
  using mono f by (auto intro: complete_lattice_class.SUP_least Sup_upper dest: monoD)

lemma mono_INF: "f (iI. A i)  (xI. f (A x))"
  by (intro complete_lattice_class.INF_greatest monoD[OF mono f] INF_lower)

lemma mono_SUP: "(xI. f (A x))  f (iI. A i)"
  by (intro complete_lattice_class.SUP_least monoD[OF mono f] SUP_upper)

end

end

class complete_boolean_algebra = boolean_algebra + complete_distrib_lattice
begin

lemma uminus_Inf: "- (A) = (uminus ` A)"
proof (rule order.antisym)
  show "- A  (uminus ` A)"
    by (rule compl_le_swap2, rule Inf_greatest, rule compl_le_swap2, rule Sup_upper) simp
  show "(uminus ` A)  - A"
    by (rule Sup_least, rule compl_le_swap1, rule Inf_lower) auto
qed

lemma uminus_INF: "- (xA. B x) = (xA. - B x)"
  by (simp add: uminus_Inf image_image)

lemma uminus_Sup: "- (A) = (uminus ` A)"
proof -
  have "A = - (uminus ` A)"
    by (simp add: image_image uminus_INF)
  then show ?thesis by simp
qed

lemma uminus_SUP: "- (xA. B x) = (xA. - B x)"
  by (simp add: uminus_Sup image_image)

end

class complete_linorder = linorder + complete_lattice
begin

lemma dual_complete_linorder:
  "class.complete_linorder Sup Inf sup (≥) (>) inf  "
  by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)

lemma complete_linorder_inf_min: "inf = min"
  by (auto intro: order.antisym simp add: min_def fun_eq_iff)

lemma complete_linorder_sup_max: "sup = max"
  by (auto intro: order.antisym simp add: max_def fun_eq_iff)

lemma Inf_less_iff: "S < a  (xS. x < a)"
  by (simp add: not_le [symmetric] le_Inf_iff)

lemma INF_less_iff: "(iA. f i) < a  (xA. f x < a)"
  by (simp add: Inf_less_iff [of "f ` A"])

lemma less_Sup_iff: "a < S  (xS. a < x)"
  by (simp add: not_le [symmetric] Sup_le_iff)

lemma less_SUP_iff: "a < (iA. f i)  (xA. a < f x)"
  by (simp add: less_Sup_iff [of _ "f ` A"])

lemma Sup_eq_top_iff [simp]: "A =   (x<. iA. x < i)"
proof
  assume *: "A = "
  show "(x<. iA. x < i)"
    unfolding * [symmetric]
  proof (intro allI impI)
    fix x
    assume "x < A"
    then show "iA. x < i"
      by (simp add: less_Sup_iff)
  qed
next
  assume *: "x<. iA. x < i"
  show "A = "
  proof (rule ccontr)
    assume "A  "
    with top_greatest [of "A"] have "A < "
      unfolding le_less by auto
    with * have "A < A"
      unfolding less_Sup_iff by auto
    then show False by auto
  qed
qed

lemma SUP_eq_top_iff [simp]: "(iA. f i) =   (x<. iA. x < f i)"
  using Sup_eq_top_iff [of "f ` A"] by simp

lemma Inf_eq_bot_iff [simp]: "A =   (x>. iA. i < x)"
  using dual_complete_linorder
  by (rule complete_linorder.Sup_eq_top_iff)

lemma INF_eq_bot_iff [simp]: "(iA. f i) =   (x>. iA. f i < x)"
  using Inf_eq_bot_iff [of "f ` A"] by simp

lemma Inf_le_iff: "A  x  (y>x. aA. y > a)"
proof safe
  fix y
  assume "x  A" "y > x"
  then have "y > A" by auto
  then show "aA. y > a"
    unfolding Inf_less_iff .
qed (auto elim!: allE[of _ "A"] simp add: not_le[symmetric] Inf_lower)

lemma INF_le_iff: "(f ` A)  x  (y>x. iA. y > f i)"
  using Inf_le_iff [of "f ` A"] by simp

lemma le_Sup_iff: "x  A  (y<x. aA. y < a)"
proof safe
  fix y
  assume "x  A" "y < x"
  then have "y < A" by auto
  then show "aA. y < a"
    unfolding less_Sup_iff .
qed (auto elim!: allE[of _ "A"] simp add: not_le[symmetric] Sup_upper)

lemma le_SUP_iff: "x  (f ` A)  (y<x. iA. y < f i)"
  using le_Sup_iff [of _ "f ` A"] by simp

end

subsection ‹Complete lattice on typbool

instantiation bool :: complete_lattice
begin

definition [simp, code]: "A  False  A"

definition [simp, code]: "A  True  A"

instance
  by standard (auto intro: bool_induct)

end

lemma not_False_in_image_Ball [simp]: "False  P ` A  Ball A P"
  by auto

lemma True_in_image_Bex [simp]: "True  P ` A  Bex A P"
  by auto

lemma INF_bool_eq [simp]: "(λA f. (f ` A)) = Ball"
  by (simp add: fun_eq_iff)

lemma SUP_bool_eq [simp]: "(λA f. (f ` A)) = Bex"
  by (simp add: fun_eq_iff)

instance bool :: complete_boolean_algebra
  by (standard, fastforce)

subsection ‹Complete lattice on typ_  _

instantiation "fun" :: (type, Inf) Inf
begin

definition "A = (λx. fA. f x)"

lemma Inf_apply [simp, code]: "(A) x = (fA. f x)"
  by (simp add: Inf_fun_def)

instance ..

end

instantiation "fun" :: (type, Sup) Sup
begin

definition "A = (λx. fA. f x)"

lemma Sup_apply [simp, code]: "(A) x = (fA. f x)"
  by (simp add: Sup_fun_def)

instance ..

end

instantiation "fun" :: (type, complete_lattice) complete_lattice
begin

instance
  by standard (auto simp add: le_fun_def intro: INF_lower INF_greatest SUP_upper SUP_least)

end

lemma INF_apply [simp]: "(yA. f y) x = (yA. f y x)"
  by (simp add: image_comp)

lemma SUP_apply [simp]: "(yA. f y) x = (yA. f y x)"
  by (simp add: image_comp)

subsection ‹Complete lattice on unary and binary predicates›

lemma Inf1_I: "(P. P  A  P a)  (A) a"
  by auto

lemma INF1_I: "(x. x  A  B x b)  (xA. B x) b"
  by simp

lemma INF2_I: "(x. x  A  B x b c)  (xA. B x) b c"
  by simp

lemma Inf2_I: "(r. r  A  r a b)  (A) a b"
  by auto

lemma Inf1_D: "(A) a  P  A  P a"
  by auto

lemma INF1_D: "(xA. B x) b  a  A  B a b"
  by simp

lemma Inf2_D: "(A) a b  r  A  r a b"
  by auto

lemma INF2_D: "(xA. B x) b c  a  A  B a b c"
  by simp

lemma Inf1_E:
  assumes "(A) a"
  obtains "P a" | "P  A"
  using assms by auto

lemma INF1_E:
  assumes "(xA. B x) b"
  obtains "B a b" | "a  A"
  using assms by auto

lemma Inf2_E:
  assumes "(A) a b"
  obtains "r a b" | "r  A"
  using assms by auto

lemma INF2_E:
  assumes "(xA. B x) b c"
  obtains "B a b c" | "a  A"
  using assms by auto

lemma Sup1_I: "P  A  P a  (A) a"
  by auto

lemma SUP1_I: "a  A  B a b  (xA. B x) b"
  by auto

lemma Sup2_I: "r  A  r a b  (A) a b"
  by auto

lemma SUP2_I: "a  A  B a b c  (xA. B x) b c"
  by auto

lemma Sup1_E:
  assumes "(A) a"
  obtains P where "P  A" and "P a"
  using assms by auto

lemma SUP1_E:
  assumes "(xA. B x) b"
  obtains x where "x  A" and "B x b"
  using assms by auto

lemma Sup2_E:
  assumes "(A) a b"
  obtains r where "r  A" "r a b"
  using assms by auto

lemma SUP2_E:
  assumes "(xA. B x) b c"
  obtains x where "x  A" "B x b c"
  using assms by auto


subsection ‹Complete lattice on typ_ set

instantiation "set" :: (type) complete_lattice
begin

definition "A = {x. ((λB. x  B) ` A)}"

definition "A = {x. ((λB. x  B) ` A)}"

instance
  by standard (auto simp add: less_eq_set_def Inf_set_def Sup_set_def le_fun_def)

end

subsubsection ‹Inter›

abbreviation Inter :: "'a set set  'a set"  ("")
  where "S  S"

lemma Inter_eq: "A = {x. B  A. x  B}"
proof (rule set_eqI)
  fix x
  have "(Q{P. BA. P  x  B}. Q)  (BA. x  B)"
    by auto
  then show "x  A  x  {x. B  A. x  B}"
    by (simp add: Inf_set_def image_def)
qed

lemma Inter_iff [simp]: "A  C  (XC. A  X)"
  by (unfold Inter_eq) blast

lemma InterI [intro!]: "(X. X  C  A  X)  A  C"
  by (simp add: Inter_eq)

text  A ``destruct'' rule -- every termX in termC
  contains termA as an element, but propA  X can hold when
  propX  C does not!  This rule is analogous to spec›.
›

lemma InterD [elim, Pure.elim]: "A  C  X  C  A  X"
  by auto

lemma InterE [elim]: "A  C  (X  C  R)  (A  X  R)  R"
  ― ‹``Classical'' elimination rule -- does not require proving
    propX  C.›
  unfolding Inter_eq by blast

lemma Inter_lower: "B  A  A  B"
  by (fact Inf_lower)

lemma Inter_subset: "(X. X  A  X  B)  A  {}  A  B"
  by (fact Inf_less_eq)

lemma Inter_greatest: "(X. X  A  C  X)  C  A"
  by (fact Inf_greatest)

lemma Inter_empty: "{} = UNIV"
  by (fact Inf_empty) (* already simp *)

lemma Inter_UNIV: "UNIV = {}"
  by (fact Inf_UNIV) (* already simp *)

lemma Inter_insert: "(insert a B) = a  B"
  by (fact Inf_insert) (* already simp *)

lemma Inter_Un_subset: "A  B  (A  B)"
  by (fact less_eq_Inf_inter)

lemma Inter_Un_distrib: "(A  B) = A  B"
  by (fact Inf_union_distrib)

lemma Inter_UNIV_conv [simp]:
  "A = UNIV  (xA. x = UNIV)"
  "UNIV = A  (xA. x = UNIV)"
  by (fact Inf_top_conv)+

lemma Inter_anti_mono: "B  A  A  B"
  by (fact Inf_superset_mono)


subsubsection ‹Intersections of families›

syntax (ASCII)
  "_INTER1"     :: "pttrns  'b set  'b set"           ("(3INT _./ _)" [0, 10] 10)
  "_INTER"      :: "pttrn  'a set  'b set  'b set"  ("(3INT _:_./ _)" [0, 0, 10] 10)

syntax
  "_INTER1"     :: "pttrns  'b set  'b set"           ("(3_./ _)" [0, 10] 10)
  "_INTER"      :: "pttrn  'a set  'b set  'b set"  ("(3__./ _)" [0, 0, 10] 10)

syntax (latex output)
  "_INTER1"     :: "pttrns  'b set  'b set"           ("(3(‹unbreakable›_)/ _)" [0, 10] 10)
  "_INTER"      :: "pttrn  'a set  'b set  'b set"  ("(3(‹unbreakable›__)/ _)" [0, 0, 10] 10)

translations
  "x y. f"   "x. y. f"
  "x. f"     "(CONST range (λx. f))"
  "xA. f"   "CONST Inter ((λx. f) ` A)"

lemma INTER_eq: "(xA. B x) = {y. xA. y  B x}"
  by (auto intro!: INF_eqI)

lemma INT_iff [simp]: "b  (xA. B x)  (xA. b  B x)"
  using Inter_iff [of _ "B ` A"] by simp

lemma INT_I [intro!]: "(x. x  A  b  B x)  b  (xA. B x)"
  by auto

lemma INT_D [elim, Pure.elim]: "b  (xA. B x)  a  A  b  B a"
  by auto

lemma INT_E [elim]: "b  (xA. B x)  (b  B a  R)  (a  A  R)  R"
  ― ‹"Classical" elimination -- by the Excluded Middle on propaA.›
  by auto

lemma Collect_ball_eq: "{x. yA. P x y} = (yA. {x. P x y})"
  by blast

lemma Collect_all_eq: "{x. y. P x y} = (y. {x. P x y})"
  by blast

lemma INT_lower: "a  A  (xA. B x)  B a"
  by (fact INF_lower)

lemma INT_greatest: "(x. x  A  C  B x)  C  (xA. B x)"
  by (fact INF_greatest)

lemma INT_empty: "(x{}. B x) = UNIV"
  by (fact INF_empty)

lemma INT_absorb: "k  I  A k  (iI. A i) = (iI. A i)"
  by (fact INF_absorb)

lemma INT_subset_iff: "B  (iI. A i)  (iI. B  A i)"
  by (fact le_INF_iff)

lemma INT_insert [simp]: "(x  insert a A. B x) = B a   (B ` A)"
  by (fact INF_insert)

lemma INT_Un: "(i  A  B. M i) = (i  A. M i)  (iB. M i)"
  by (fact INF_union)

lemma INT_insert_distrib: "u  A  (xA. insert a (B x)) = insert a (xA. B x)"
  by blast

lemma INT_constant [simp]: "(yA. c) = (if A = {} then UNIV else c)"
  by (fact INF_constant)

lemma INTER_UNIV_conv:
  "(UNIV = (xA. B x)) = (xA. B x = UNIV)"
  "((xA. B x) = UNIV) = (xA. B x = UNIV)"
  by (fact INF_top_conv)+ (* already simp *)

lemma INT_bool_eq: "(b. A b) = A True  A False"
  by (fact INF_UNIV_bool_expand)

lemma INT_anti_mono: "A  B  (x. x  A  f x  g x)  (xB. f x)  (xA. g x)"
  ― ‹The last inclusion is POSITIVE!›
  by (fact INF_superset_mono)

lemma Pow_INT_eq: "Pow (xA. B x) = (xA. Pow (B x))"
  by blast

lemma vimage_INT: "f -` (xA. B x) = (xA. f -` B x)"
  by blast


subsubsection ‹Union›

abbreviation Union :: "'a set set  'a set"  ("")
  where "S  S"

lemma Union_eq: "A = {x. B  A. x  B}"
proof (rule set_eqI)
  fix x
  have "(Q{P. BA. P  x  B}. Q)  (BA. x  B)"
    by auto
  then show "x  A  x  {x. BA. x  B}"
    by (simp add: Sup_set_def image_def)
qed

lemma Union_iff [simp]: "A  C  (XC. AX)"
  by (unfold Union_eq) blast

lemma UnionI [intro]: "X  C  A  X  A  C"
  ― ‹The order of the premises presupposes that termC is rigid;
    termA may be flexible.›
  by auto

lemma UnionE [elim!]: "A  C  (X. A  X  X  C  R)  R"
  by auto

lemma Union_upper: "B  A  B  A"
  by (fact Sup_upper)

lemma Union_least: "(X. X  A  X  C)  A  C"
  by (fact Sup_least)

lemma Union_empty: "{} = {}"
  by (fact Sup_empty) (* already simp *)

lemma Union_UNIV: "UNIV = UNIV"
  by (fact Sup_UNIV) (* already simp *)

lemma Union_insert: "(insert a B) = a  B"
  by (fact Sup_insert) (* already simp *)

lemma Union_Un_distrib [simp]: "(A  B) = A  B"
  by (fact Sup_union_distrib)

lemma Union_Int_subset: "(A  B)  A  B"
  by (fact Sup_inter_less_eq)

lemma Union_empty_conv: "(A = {})  (xA. x = {})"
  by (fact Sup_bot_conv) (* already simp *)

lemma empty_Union_conv: "({} = A)  (xA. x = {})"
  by (fact Sup_bot_conv) (* already simp *)

lemma subset_Pow_Union: "A  Pow (A)"
  by blast

lemma Union_Pow_eq [simp]: "(Pow A) = A"
  by blast

lemma Union_mono: "A  B  A  B"
  by (fact Sup_subset_mono)

lemma Union_subsetI: "(x. x  A  y. y  B  x  y)  A  B"
  by blast

lemma disjnt_inj_on_iff:
 "inj_on f (𝒜); X  𝒜; Y  𝒜  disjnt (f ` X) (f ` Y)  disjnt X Y"
  unfolding disjnt_def
  by safe (use inj_on_eq_iff in fastforce+)

lemma disjnt_Union1 [simp]: "disjnt (𝒜) B  (A  𝒜. disjnt A B)"
  by (auto simp: disjnt_def)

lemma disjnt_Union2 [simp]: "disjnt B (𝒜)  (A  𝒜. disjnt B A)"
  by (auto simp: disjnt_def)


subsubsection ‹Unions of families›

syntax (ASCII)
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3UN _./ _)" [0, 10] 10)
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3UN _:_./ _)" [0, 0, 10] 10)

syntax
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3_./ _)" [0, 10] 10)
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3__./ _)" [0, 0, 10] 10)

syntax (latex output)
  "_UNION1"     :: "pttrns => 'b set => 'b set"           ("(3(‹unbreakable›_)/ _)" [0, 10] 10)
  "_UNION"      :: "pttrn => 'a set => 'b set => 'b set"  ("(3(‹unbreakable›__)/ _)" [0, 0, 10] 10)

translations
  "x y. f"    "x. y. f"
  "x. f"      "(CONST range (λx. f))"
  "xA. f"    "CONST Union ((λx. f) ` A)"

text ‹
  Note the difference between ordinary syntax of indexed
  unions and intersections (e.g.\ ⋃a1∈A1. B›)
  and their \LaTeX\ rendition: terma1A1. B.
›

lemma disjoint_UN_iff: "disjnt A (iI. B i)  (iI. disjnt A (B i))"
  by (auto simp: disjnt_def)

lemma UNION_eq: "(xA. B x) = {y. xA. y  B x}"
  by (auto intro!: SUP_eqI)

lemma bind_UNION [code]: "Set.bind A f = (f ` A)"
  by (simp add: bind_def UNION_eq)

lemma member_bind [simp]: "x  Set.bind A f  x  (f ` A)"
  by (simp add: bind_UNION)

lemma Union_SetCompr_eq: "{f x| x. P x} = {a. x. P x  a  f x}"
  by blast

lemma UN_iff [simp]: "b  (xA. B x)  (xA. b  B x)"
  using Union_iff [of _ "B ` A"] by simp

lemma UN_I [intro]: "a  A  b  B a  b  (xA. B x)"
  ― ‹The order of the premises presupposes that termA is rigid;
    termb may be flexible.›
  by auto

lemma UN_E [elim!]: "b  (xA. B x)  (x. xA  b  B x  R)  R"
  by auto

lemma UN_upper: "a  A  B a  (xA. B x)"
  by (fact SUP_upper)

lemma UN_least: "(x. x  A  B x  C)  (xA. B x)  C"
  by (fact SUP_least)

lemma Collect_bex_eq: "{x. yA. P x y} = (yA. {x. P x y})"
  by blast

lemma UN_insert_distrib: "u  A  (xA. insert a (B x)) = insert a (xA. B x)"
  by blast

lemma UN_empty: "(x{}. B x) = {}"
  by (fact SUP_empty)

lemma UN_empty2: "(xA. {}) = {}"
  by (fact SUP_bot) (* already simp *)

lemma UN_absorb: "k  I  A k  (iI. A i) = (iI. A i)"
  by (fact SUP_absorb)

lemma UN_insert [simp]: "(xinsert a A. B x) = B a  (B ` A)"
  by (fact SUP_insert)

lemma UN_Un [simp]: "(i  A  B. M i) = (iA. M i)  (iB. M i)"
  by (fact SUP_union)

lemma UN_UN_flatten: "(x  (yA. B y). C x) = (yA. xB y. C x)"
  by blast

lemma UN_subset_iff: "((iI. A i)  B) = (iI. A i  B)"
  by (fact SUP_le_iff)

lemma UN_constant [simp]: "(yA. c) = (if A = {} then {} else c)"
  by (fact SUP_constant)

lemma UNION_singleton_eq_range: "(xA. {f x}) = f ` A"
  by blast

lemma image_Union: "f ` S = (xS. f ` x)"
  by blast

lemma UNION_empty_conv:
  "{} = (xA. B x)  (xA. B x = {})"
  "(xA. B x) = {}  (xA. B x = {})"
  by (fact SUP_bot_conv)+ (* already simp *)

lemma Collect_ex_eq: "{x. y. P x y} = (y. {x. P x y})"
  by blast

lemma ball_UN: "(z  (B ` A). P z)  (xA. z  B x. P z)"
  by blast

lemma bex_UN: "(z  (B ` A). P z)  (xA. zB x. P z)"
  by blast

lemma Un_eq_UN: "A  B = (b. if b then A else B)"
  by safe (auto simp add: if_split_mem2)

lemma UN_bool_eq: "(b. A b) = (A True  A False)"
  by (fact SUP_UNIV_bool_expand)

lemma UN_Pow_subset: "(xA. Pow (B x))  Pow (xA. B x)"
  by blast

lemma UN_mono:
  "A  B  (x. x  A  f x  g x) 
    (xA. f x)  (xB. g x)"
  by (fact SUP_subset_mono)

lemma vimage_Union: "f -` (A) = (XA. f -` X)"
  by blast

lemma vimage_UN: "f -` (xA. B x) = (xA. f -` B x)"
  by blast

lemma vimage_eq_UN: "f -` B = (yB. f -` {y})"
  ― ‹NOT suitable for rewriting›
  by blast

lemma image_UN: "f ` (B ` A) = (xA. f ` B x)"
  by blast

lemma UN_singleton [simp]: "(xA. {x}) = A"
  by blast

lemma inj_on_image: "inj_on f (A)  inj_on ((`) f) A"
  unfolding inj_on_def by blast


subsubsection ‹Distributive laws›

lemma Int_Union: "A  B = (CB. A  C)"
  by blast

lemma Un_Inter: "A  B = (CB. A  C)"
  by blast

lemma Int_Union2: "B  A = (CB. C  A)"
  by blast

lemma INT_Int_distrib: "(iI. A i  B i) = (iI. A i)  (iI. B i)"
  by (rule sym) (rule INF_inf_distrib)

lemma UN_Un_distrib: "(iI. A i  B i) = (iI. A i)  (iI. B i)"
  by (rule sym) (rule SUP_sup_distrib)

lemma Int_Inter_image: "(xC. A x  B x) = (A ` C)  (B ` C)"  (* FIXME drop *)
  by (simp add: INT_Int_distrib)

lemma Int_Inter_eq: "A   = (if ={} then A else (B. A  B))"
                    "  A = (if ={} then A else (B. B  A))"
  by auto

lemma Un_Union_image: "(xC. A x  B x) = (A ` C)  (B ` C)"  (* FIXME drop *)
  ― ‹Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5:›
  ― ‹Union of a family of unions›
  by (simp add: UN_Un_distrib)

lemma Un_INT_distrib: "B  (iI. A i) = (iI. B  A i)"
  by blast

lemma Int_UN_distrib: "B  (iI. A i) = (iI. B  A i)"
  ― ‹Halmos, Naive Set Theory, page 35.›
  by blast

lemma Int_UN_distrib2: "(iI. A i)  (jJ. B j) = (iI. jJ. A i  B j)"
  by blast

lemma Un_INT_distrib2: "(iI. A i)  (jJ. B j) = (iI. jJ. A i  B j)"
  by blast

lemma Union_disjoint: "(C  A = {})  (BC. B  A = {})"
  by blast

lemma SUP_UNION: "(x(yA. g y). f x) = (yA. xg y. f x :: _ :: complete_lattice)"
  by (rule order_antisym) (blast intro: SUP_least SUP_upper2)+


subsection ‹Injections and bijections›

lemma inj_on_Inter: "S  {}  (A. A  S  inj_on f A)  inj_on f (S)"
  unfolding inj_on_def by blast

lemma inj_on_INTER: "I  {}  (i. i  I  inj_on f (A i))  inj_on f (i  I. A i)"
  unfolding inj_on_def by safe simp

lemma inj_on_UNION_chain:
  assumes chain: "i j. i  I  j  I  A i  A j  A j  A i"
    and inj: "i. i  I  inj_on f (A i)"
  shows "inj_on f (i  I. A i)"
proof -
  have "x = y"
    if *: "i  I" "j  I"
    and **: "x  A i" "y  A j"
    and ***: "f x = f y"
    for i j x y
    using chain [OF *]
  proof
    assume "A i  A j"
    with ** have "x  A j" by auto
    with inj * ** *** show ?thesis
      by (auto simp add: inj_on_def)
  next
    assume "A j  A i"
    with ** have "y  A i" by auto
    with inj * ** *** show ?thesis
      by (auto simp add: inj_on_def)
  qed
  then show ?thesis
    by (unfold inj_on_def UNION_eq) auto
qed

lemma bij_betw_UNION_chain:
  assumes chain: "i j. i  I  j  I  A i  A j  A j  A i"
    and bij: "i. i  I  bij_betw f (A i) (A' i)"
  shows "bij_betw f (i  I. A i) (i  I. A' i)"
  unfolding bij_betw_def
proof safe
  have "i. i  I  inj_on f (A i)"
    using bij bij_betw_def[of f] by auto
  then show "inj_on f ((A ` I))"
    using chain inj_on_UNION_chain[of I A f] by auto
next
  fix i x
  assume *: "i  I" "x  A i"
  with bij have "f x  A' i"
    by (auto simp: bij_betw_def)
  with * show "f x  (A' ` I)" by blast
next
  fix i x'
  assume *: "i  I" "x'  A' i"
  with bij have "x  A i. x' = f x"
    unfolding bij_betw_def by blast
  with * have "j  I. x  A j. x' = f x"
    by blast
  then show "x'  f ` (A ` I)"
    by blast
qed

(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
lemma image_INT: "inj_on f C  xA. B x  C  j  A  f ` ((B ` A)) = (xA. f ` B x)"
  by (auto simp add: inj_on_def) blast

lemma bij_image_INT: "bij f  f ` ((B ` A)) = (xA. f ` B x)"
  by (auto simp: bij_def inj_def surj_def) blast

lemma UNION_fun_upd: "(A(i := B) ` J) = (A ` (J - {i}))  (if i  J then B else {})"
  by (auto simp add: set_eq_iff)

lemma bij_betw_Pow:
  assumes "bij_betw f A B"
  shows "bij_betw (image f) (Pow A) (Pow B)"
proof -
  from assms have "inj_on f A"
    by (rule bij_betw_imp_inj_on)
  then have "inj_on f ((Pow A))"
    by simp
  then have "inj_on (image f) (Pow A)"
    by (rule inj_on_image)
  then have "bij_betw (image f) (Pow A) (image f ` Pow A)"
    by (rule inj_on_imp_bij_betw)
  moreover from assms have "f ` A = B"
    by (rule bij_betw_imp_surj_on)
  then have "image f ` Pow A = Pow B"
    by (rule image_Pow_surj)
  ultimately show ?thesis by simp
qed


subsubsection ‹Complement›

lemma Compl_INT [simp]: "- (xA. B x) = (xA. -B x)"
  by blast

lemma Compl_UN [simp]: "- (xA. B x) = (xA. -B x)"
  by blast

subsubsection ‹Miniscoping and maxiscoping›

text  Miniscoping: pushing in quantifiers and big Unions and Intersections.›

lemma UN_simps [simp]:
  "a B C. (xC. insert a (B x)) = (if C={} then {} else insert a (xC. B x))"
  "A B C. (xC. A x  B) = ((if C={} then {} else (xC. A x)  B))"
  "A B C. (xC. A  B x) = ((if C={} then {} else A  (xC. B x)))"
  "A B C. (xC. A x  B) = ((xC. A x)  B)"
  "A B C. (xC. A  B x) = (A (xC. B x))"
  "A B C. (xC. A x - B) = ((xC. A x) - B)"
  "A B C. (xC. A - B x) = (A - (xC. B x))"
  "A B. (xA. B x) = (yA. xy. B x)"
  "A B C. (z((B ` A)). C z) = (xA. zB x. C z)"
  "A B f. (xf`A. B x) = (aA. B (f a))"
  by auto

lemma INT_simps [simp]:
  "A B C. (xC. A x  B) = (if C={} then UNIV else (xC. A x)  B)"
  "A B C. (xC. A  B x) = (if C={} then UNIV else A (xC. B x))"
  "A B C. (xC. A x - B) = (if C={} then UNIV else (xC. A x) - B)"
  "A B C. (xC. A - B x) = (if C={} then UNIV else A - (xC. B x))"
  "a B C. (xC. insert a (B x)) = insert a (xC. B x)"
  "A B C. (xC. A x  B) = ((xC. A x)  B)"
  "A B C. (xC. A  B x) = (A  (xC. B x))"
  "A B. (xA. B x) = (yA. xy. B x)"
  "A B C. (z((B ` A)). C z) = (xA. zB x. C z)"
  "A B f. (xf`A. B x) = (aA. B (f a))"
  by auto

lemma UN_ball_bex_simps [simp]:
  "A P. (xA. P x)  (yA. xy. P x)"
  "A B P. (x((B ` A)). P x) = (aA. x B a. P x)"
  "A P. (xA. P x)  (yA. xy. P x)"
  "A B P. (x((B ` A)). P x)  (aA. xB a. P x)"
  by auto


text  Maxiscoping: pulling out big Unions and Intersections.›

lemma UN_extend_simps:
  "a B C. insert a (xC. B x) = (if C={} then {a} else (xC. insert a (B x)))"
  "A B C. (xC. A x)  B = (if C={} then B else (xC. A x  B))"
  "A B C. A  (xC. B x) = (if C={} then A else (xC. A  B x))"
  "A B C. ((xC. A x)  B) = (xC. A x  B)"
  "A B C. (A  (xC. B x)) = (xC. A  B x)"
  "A B C. ((xC. A x) - B) = (xC. A x - B)"
  "A B C. (A - (xC. B x)) = (xC. A - B x)"
  "A B. (yA. xy. B x) = (xA. B x)"
  "A B C. (xA. zB x. C z) = (z((B ` A)). C z)"
  "A B f. (aA. B (f a)) = (xf`A. B x)"
  by auto

lemma INT_extend_simps:
  "A B C. (xC. A x)  B = (if C={} then B else (xC. A x  B))"
  "A B C. A  (xC. B x) = (if C={} then A else (xC. A  B x))"
  "A B C. (xC. A x) - B = (if C={} then UNIV - B else (xC. A x - B))"
  "A B C. A - (xC. B x) = (if C={} then A else (xC. A - B x))"
  "a B C. insert a (xC. B x) = (xC. insert a (B x))"
  "A B C. ((xC. A x)  B) = (xC. A x  B)"
  "A B C. A  (xC. B x) = (xC. A  B x)"
  "A B. (yA. xy. B x) = (xA. B x)"
  "A B C. (xA. zB x. C z) = (z((B ` A)). C z)"
  "A B f. (aA. B (f a)) = (xf`A. B x)"
  by auto

text ‹Finally›

lemmas mem_simps =
  insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
  mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
  ― ‹Each of these has ALREADY been added [simp]› above.›

end