Theory Conditionally_Complete_Lattices

(*  Title:      HOL/Conditionally_Complete_Lattices.thy
    Author:     Amine Chaieb and L C Paulson, University of Cambridge
    Author:     Johannes Hölzl, TU München
    Author:     Luke S. Serafin, Carnegie Mellon University
*)

section Conditionally-complete Lattices

theory Conditionally_Complete_Lattices
imports Finite_Set Lattices_Big Set_Interval
begin

locale preordering_bdd = preordering
begin

definition bdd :: 'a set  bool
  where unfold: bdd A  (M. x  A. x  M)

lemma empty [simp, intro]:
  bdd {}
  by (simp add: unfold)

lemma I [intro]:
  bdd A if x. x  A  x  M
  using that by (auto simp add: unfold)

lemma E:
  assumes bdd A
  obtains M where x. x  A  x  M
  using assms that by (auto simp add: unfold)

lemma I2:
  bdd (f ` A) if x. x  A  f x  M
  using that by (auto simp add: unfold)

lemma mono:
  bdd A if bdd B A  B
  using that by (auto simp add: unfold)

lemma Int1 [simp]:
  bdd (A  B) if bdd A
  using mono that by auto

lemma Int2 [simp]:
  bdd (A  B) if bdd B
  using mono that by auto

end

subsection Preorders

context preorder
begin

sublocale bdd_above: preordering_bdd (≤) (<)
  defines bdd_above_primitive_def: bdd_above = bdd_above.bdd ..

sublocale bdd_below: preordering_bdd (≥) (>)
  defines bdd_below_primitive_def: bdd_below = bdd_below.bdd ..

lemma bdd_above_def: bdd_above A  (M. x  A. x  M)
  by (fact bdd_above.unfold)

lemma bdd_below_def: bdd_below A  (M. x  A. M  x)
  by (fact bdd_below.unfold)

lemma bdd_aboveI: "(x. x  A  x  M)  bdd_above A"
  by (fact bdd_above.I)

lemma bdd_belowI: "(x. x  A  m  x)  bdd_below A"
  by (fact bdd_below.I)

lemma bdd_aboveI2: "(x. x  A  f x  M)  bdd_above (f`A)"
  by (fact bdd_above.I2)

lemma bdd_belowI2: "(x. x  A  m  f x)  bdd_below (f`A)"
  by (fact bdd_below.I2)

lemma bdd_above_empty: "bdd_above {}"
  by (fact bdd_above.empty)

lemma bdd_below_empty: "bdd_below {}"
  by (fact bdd_below.empty)

lemma bdd_above_mono: "bdd_above B  A  B  bdd_above A"
  by (fact bdd_above.mono)

lemma bdd_below_mono: "bdd_below B  A  B  bdd_below A"
  by (fact bdd_below.mono)

lemma bdd_above_Int1: "bdd_above A  bdd_above (A  B)"
  by (fact bdd_above.Int1)

lemma bdd_above_Int2: "bdd_above B  bdd_above (A  B)"
  by (fact bdd_above.Int2)

lemma bdd_below_Int1: "bdd_below A  bdd_below (A  B)"
  by (fact bdd_below.Int1)

lemma bdd_below_Int2: "bdd_below B  bdd_below (A  B)"
  by (fact bdd_below.Int2)

lemma bdd_above_Ioo [simp, intro]: "bdd_above {a <..< b}"
  by (auto simp add: bdd_above_def intro!: exI[of _ b] less_imp_le)

lemma bdd_above_Ico [simp, intro]: "bdd_above {a ..< b}"
  by (auto simp add: bdd_above_def intro!: exI[of _ b] less_imp_le)

lemma bdd_above_Iio [simp, intro]: "bdd_above {..< b}"
  by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)

lemma bdd_above_Ioc [simp, intro]: "bdd_above {a <.. b}"
  by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)

lemma bdd_above_Icc [simp, intro]: "bdd_above {a .. b}"
  by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)

lemma bdd_above_Iic [simp, intro]: "bdd_above {.. b}"
  by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)

lemma bdd_below_Ioo [simp, intro]: "bdd_below {a <..< b}"
  by (auto simp add: bdd_below_def intro!: exI[of _ a] less_imp_le)

lemma bdd_below_Ioc [simp, intro]: "bdd_below {a <.. b}"
  by (auto simp add: bdd_below_def intro!: exI[of _ a] less_imp_le)

lemma bdd_below_Ioi [simp, intro]: "bdd_below {a <..}"
  by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)

lemma bdd_below_Ico [simp, intro]: "bdd_below {a ..< b}"
  by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)

lemma bdd_below_Icc [simp, intro]: "bdd_below {a .. b}"
  by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)

lemma bdd_below_Ici [simp, intro]: "bdd_below {a ..}"
  by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)

end

context order_top
begin

lemma bdd_above_top [simp, intro!]: "bdd_above A"
  by (rule bdd_aboveI [of _ top]) simp

end

context order_bot
begin

lemma bdd_below_bot [simp, intro!]: "bdd_below A"
  by (rule bdd_belowI [of _ bot]) simp

end

lemma bdd_above_image_mono: "mono f  bdd_above A  bdd_above (f`A)"
  by (auto simp: bdd_above_def mono_def)

lemma bdd_below_image_mono: "mono f  bdd_below A  bdd_below (f`A)"
  by (auto simp: bdd_below_def mono_def)

lemma bdd_above_image_antimono: "antimono f  bdd_below A  bdd_above (f`A)"
  by (auto simp: bdd_above_def bdd_below_def antimono_def)

lemma bdd_below_image_antimono: "antimono f  bdd_above A  bdd_below (f`A)"
  by (auto simp: bdd_above_def bdd_below_def antimono_def)

lemma
  fixes X :: "'a::ordered_ab_group_add set"
  shows bdd_above_uminus[simp]: "bdd_above (uminus ` X)  bdd_below X"
    and bdd_below_uminus[simp]: "bdd_below (uminus ` X)  bdd_above X"
  using bdd_above_image_antimono[of uminus X] bdd_below_image_antimono[of uminus "uminus`X"]
  using bdd_below_image_antimono[of uminus X] bdd_above_image_antimono[of uminus "uminus`X"]
  by (auto simp: antimono_def image_image)

subsection Lattices

context lattice
begin

lemma bdd_above_insert [simp]: "bdd_above (insert a A) = bdd_above A"
  by (auto simp: bdd_above_def intro: le_supI2 sup_ge1)

lemma bdd_below_insert [simp]: "bdd_below (insert a A) = bdd_below A"
  by (auto simp: bdd_below_def intro: le_infI2 inf_le1)

lemma bdd_finite [simp]:
  assumes "finite A" shows bdd_above_finite: "bdd_above A" and bdd_below_finite: "bdd_below A"
  using assms by (induct rule: finite_induct, auto)

lemma bdd_above_Un [simp]: "bdd_above (A  B) = (bdd_above A  bdd_above B)"
proof
  assume "bdd_above (A  B)"
  thus "bdd_above A  bdd_above B" unfolding bdd_above_def by auto
next
  assume "bdd_above A  bdd_above B"
  then obtain a b where "xA. x  a" "xB. x  b" unfolding bdd_above_def by auto
  hence "x  A  B. x  sup a b" by (auto intro: Un_iff le_supI1 le_supI2)
  thus "bdd_above (A  B)" unfolding bdd_above_def ..
qed

lemma bdd_below_Un [simp]: "bdd_below (A  B) = (bdd_below A  bdd_below B)"
proof
  assume "bdd_below (A  B)"
  thus "bdd_below A  bdd_below B" unfolding bdd_below_def by auto
next
  assume "bdd_below A  bdd_below B"
  then obtain a b where "xA. a  x" "xB. b  x" unfolding bdd_below_def by auto
  hence "x  A  B. inf a b  x" by (auto intro: Un_iff le_infI1 le_infI2)
  thus "bdd_below (A  B)" unfolding bdd_below_def ..
qed

lemma bdd_above_image_sup[simp]:
  "bdd_above ((λx. sup (f x) (g x)) ` A)  bdd_above (f`A)  bdd_above (g`A)"
by (auto simp: bdd_above_def intro: le_supI1 le_supI2)

lemma bdd_below_image_inf[simp]:
  "bdd_below ((λx. inf (f x) (g x)) ` A)  bdd_below (f`A)  bdd_below (g`A)"
by (auto simp: bdd_below_def intro: le_infI1 le_infI2)

lemma bdd_below_UN[simp]: "finite I  bdd_below (iI. A i) = (i  I. bdd_below (A i))"
by (induction I rule: finite.induct) auto

lemma bdd_above_UN[simp]: "finite I  bdd_above (iI. A i) = (i  I. bdd_above (A i))"
by (induction I rule: finite.induct) auto

end


text 

To avoid name classes with the classcomplete_lattice-class we prefix constSup and
constInf in theorem names with c.



subsection Conditionally complete lattices

class conditionally_complete_lattice = lattice + Sup + Inf +
  assumes cInf_lower: "x  X  bdd_below X  Inf X  x"
    and cInf_greatest: "X  {}  (x. x  X  z  x)  z  Inf X"
  assumes cSup_upper: "x  X  bdd_above X  x  Sup X"
    and cSup_least: "X  {}  (x. x  X  x  z)  Sup X  z"
begin

lemma cSup_upper2: "x  X  y  x  bdd_above X  y  Sup X"
  by (metis cSup_upper order_trans)

lemma cInf_lower2: "x  X  x  y  bdd_below X  Inf X  y"
  by (metis cInf_lower order_trans)

lemma cSup_mono: "B  {}  bdd_above A  (b. b  B  aA. b  a)  Sup B  Sup A"
  by (metis cSup_least cSup_upper2)

lemma cInf_mono: "B  {}  bdd_below A  (b. b  B  aA. a  b)  Inf A  Inf B"
  by (metis cInf_greatest cInf_lower2)

lemma cSup_subset_mono: "A  {}  bdd_above B  A  B  Sup A  Sup B"
  by (metis cSup_least cSup_upper subsetD)

lemma cInf_superset_mono: "A  {}  bdd_below B  A  B  Inf B  Inf A"
  by (metis cInf_greatest cInf_lower subsetD)

lemma cSup_eq_maximum: "z  X  (x. x  X  x  z)  Sup X = z"
  by (intro order.antisym cSup_upper[of z X] cSup_least[of X z]) auto

lemma cInf_eq_minimum: "z  X  (x. x  X  z  x)  Inf X = z"
  by (intro order.antisym cInf_lower[of z X] cInf_greatest[of X z]) auto

lemma cSup_le_iff: "S  {}  bdd_above S  Sup S  a  (xS. x  a)"
  by (metis order_trans cSup_upper cSup_least)

lemma le_cInf_iff: "S  {}  bdd_below S  a  Inf S  (xS. a  x)"
  by (metis order_trans cInf_lower cInf_greatest)

lemma cSup_eq_non_empty:
  assumes 1: "X  {}"
  assumes 2: "x. x  X  x  a"
  assumes 3: "y. (x. x  X  x  y)  a  y"
  shows "Sup X = a"
  by (intro 3 1 order.antisym cSup_least) (auto intro: 2 1 cSup_upper)

lemma cInf_eq_non_empty:
  assumes 1: "X  {}"
  assumes 2: "x. x  X  a  x"
  assumes 3: "y. (x. x  X  y  x)  y  a"
  shows "Inf X = a"
  by (intro 3 1 order.antisym cInf_greatest) (auto intro: 2 1 cInf_lower)

lemma cInf_cSup: "S  {}  bdd_below S  Inf S = Sup {x. sS. x  s}"
  by (rule cInf_eq_non_empty) (auto intro!: cSup_upper cSup_least simp: bdd_below_def)

lemma cSup_cInf: "S  {}  bdd_above S  Sup S = Inf {x. sS. s  x}"
  by (rule cSup_eq_non_empty) (auto intro!: cInf_lower cInf_greatest simp: bdd_above_def)

lemma cSup_insert: "X  {}  bdd_above X  Sup (insert a X) = sup a (Sup X)"
  by (intro cSup_eq_non_empty) (auto intro: le_supI2 cSup_upper cSup_least)

lemma cInf_insert: "X  {}  bdd_below X  Inf (insert a X) = inf a (Inf X)"
  by (intro cInf_eq_non_empty) (auto intro: le_infI2 cInf_lower cInf_greatest)

lemma cSup_singleton [simp]: "Sup {x} = x"
  by (intro cSup_eq_maximum) auto

lemma cInf_singleton [simp]: "Inf {x} = x"
  by (intro cInf_eq_minimum) auto

lemma cSup_insert_If:  "bdd_above X  Sup (insert a X) = (if X = {} then a else sup a (Sup X))"
  using cSup_insert[of X] by simp

lemma cInf_insert_If: "bdd_below X  Inf (insert a X) = (if X = {} then a else inf a (Inf X))"
  using cInf_insert[of X] by simp

lemma le_cSup_finite: "finite X  x  X  x  Sup X"
proof (induct X arbitrary: x rule: finite_induct)
  case (insert x X y) then show ?case
    by (cases "X = {}") (auto simp: cSup_insert intro: le_supI2)
qed simp

lemma cInf_le_finite: "finite X  x  X  Inf X  x"
proof (induct X arbitrary: x rule: finite_induct)
  case (insert x X y) then show ?case
    by (cases "X = {}") (auto simp: cInf_insert intro: le_infI2)
qed simp

lemma cSup_eq_Sup_fin: "finite X  X  {}  Sup X = Sup_fin X"
  by (induct X rule: finite_ne_induct) (simp_all add: cSup_insert)

lemma cInf_eq_Inf_fin: "finite X  X  {}  Inf X = Inf_fin X"
  by (induct X rule: finite_ne_induct) (simp_all add: cInf_insert)

lemma cSup_atMost[simp]: "Sup {..x} = x"
  by (auto intro!: cSup_eq_maximum)

lemma cSup_greaterThanAtMost[simp]: "y < x  Sup {y<..x} = x"
  by (auto intro!: cSup_eq_maximum)

lemma cSup_atLeastAtMost[simp]: "y  x  Sup {y..x} = x"
  by (auto intro!: cSup_eq_maximum)

lemma cInf_atLeast[simp]: "Inf {x..} = x"
  by (auto intro!: cInf_eq_minimum)

lemma cInf_atLeastLessThan[simp]: "y < x  Inf {y..<x} = y"
  by (auto intro!: cInf_eq_minimum)

lemma cInf_atLeastAtMost[simp]: "y  x  Inf {y..x} = y"
  by (auto intro!: cInf_eq_minimum)

lemma cINF_lower: "bdd_below (f ` A)  x  A  (f ` A)  f x"
  using cInf_lower [of _ "f ` A"] by simp

lemma cINF_greatest: "A  {}  (x. x  A  m  f x)  m  (f ` A)"
  using cInf_greatest [of "f ` A"] by auto

lemma cSUP_upper: "x  A  bdd_above (f ` A)  f x  (f ` A)"
  using cSup_upper [of _ "f ` A"] by simp

lemma cSUP_least: "A  {}  (x. x  A  f x  M)  (f ` A)  M"
  using cSup_least [of "f ` A"] by auto

lemma cINF_lower2: "bdd_below (f ` A)  x  A  f x  u  (f ` A)  u"
  by (auto intro: cINF_lower order_trans)

lemma cSUP_upper2: "bdd_above (f ` A)  x  A  u  f x  u  (f ` A)"
  by (auto intro: cSUP_upper order_trans)

lemma cSUP_const [simp]: "A  {}  (xA. c) = c"
  by (intro order.antisym cSUP_least) (auto intro: cSUP_upper)

lemma cINF_const [simp]: "A  {}  (xA. c) = c"
  by (intro order.antisym cINF_greatest) (auto intro: cINF_lower)

lemma le_cINF_iff: "A  {}  bdd_below (f ` A)  u  (f ` A)  (xA. u  f x)"
  by (metis cINF_greatest cINF_lower order_trans)

lemma cSUP_le_iff: "A  {}  bdd_above (f ` A)  (f ` A)  u  (xA. f x  u)"
  by (metis cSUP_least cSUP_upper order_trans)

lemma less_cINF_D: "bdd_below (f`A)  y < (iA. f i)  i  A  y < f i"
  by (metis cINF_lower less_le_trans)

lemma cSUP_lessD: "bdd_above (f`A)  (iA. f i) < y  i  A  f i < y"
  by (metis cSUP_upper le_less_trans)

lemma cINF_insert: "A  {}  bdd_below (f ` A)  (f ` insert a A) = inf (f a) ((f ` A))"
  by (simp add: cInf_insert)

lemma cSUP_insert: "A  {}  bdd_above (f ` A)  (f ` insert a A) = sup (f a) ((f ` A))"
  by (simp add: cSup_insert)

lemma cINF_mono: "B  {}  bdd_below (f ` A)  (m. m  B  nA. f n  g m)  (f ` A)  (g ` B)"
  using cInf_mono [of "g ` B" "f ` A"] by auto

lemma cSUP_mono: "A  {}  bdd_above (g ` B)  (n. n  A  mB. f n  g m)  (f ` A)  (g ` B)"
  using cSup_mono [of "f ` A" "g ` B"] by auto

lemma cINF_superset_mono: "A  {}  bdd_below (g ` B)  A  B  (x. x  B  g x  f x)  (g ` B)  (f ` A)"
  by (rule cINF_mono) auto

lemma cSUP_subset_mono: 
  "A  {}; bdd_above (g ` B); A  B; x. x  A  f x  g x   (f ` A)   (g ` B)"
  by (rule cSUP_mono) auto

lemma less_eq_cInf_inter: "bdd_below A  bdd_below B  A  B  {}  inf (Inf A) (Inf B)  Inf (A  B)"
  by (metis cInf_superset_mono lattice_class.inf_sup_ord(1) le_infI1)

lemma cSup_inter_less_eq: "bdd_above A  bdd_above B  A  B  {}  Sup (A  B)  sup (Sup A) (Sup B) "
  by (metis cSup_subset_mono lattice_class.inf_sup_ord(1) le_supI1)

lemma cInf_union_distrib: "A  {}  bdd_below A  B  {}  bdd_below B  Inf (A  B) = inf (Inf A) (Inf B)"
  by (intro order.antisym le_infI cInf_greatest cInf_lower) (auto intro: le_infI1 le_infI2 cInf_lower)

lemma cINF_union: "A  {}  bdd_below (f ` A)  B  {}  bdd_below (f ` B)   (f ` (A  B)) =  (f ` A)   (f ` B)"
  using cInf_union_distrib [of "f ` A" "f ` B"] by (simp add: image_Un)

lemma cSup_union_distrib: "A  {}  bdd_above A  B  {}  bdd_above B  Sup (A  B) = sup (Sup A) (Sup B)"
  by (intro order.antisym le_supI cSup_least cSup_upper) (auto intro: le_supI1 le_supI2 cSup_upper)

lemma cSUP_union: "A  {}  bdd_above (f ` A)  B  {}  bdd_above (f ` B)   (f ` (A  B)) =  (f ` A)   (f ` B)"
  using cSup_union_distrib [of "f ` A" "f ` B"] by (simp add: image_Un)

lemma cINF_inf_distrib: "A  {}  bdd_below (f`A)  bdd_below (g`A)   (f ` A)   (g ` A) = (aA. inf (f a) (g a))"
  by (intro order.antisym le_infI cINF_greatest cINF_lower2)
     (auto intro: le_infI1 le_infI2 cINF_greatest cINF_lower le_infI)

lemma SUP_sup_distrib: "A  {}  bdd_above (f`A)  bdd_above (g`A)   (f ` A)   (g ` A) = (aA. sup (f a) (g a))"
  by (intro order.antisym le_supI cSUP_least cSUP_upper2)
     (auto intro: le_supI1 le_supI2 cSUP_least cSUP_upper le_supI)

lemma cInf_le_cSup:
  "A  {}  bdd_above A  bdd_below A  Inf A  Sup A"
  by (auto intro!: cSup_upper2[of "SOME a. a  A"] intro: someI cInf_lower)

  context
    fixes f :: "'a  'b::conditionally_complete_lattice"
    assumes "mono f"
  begin
  
  lemma mono_cInf: "bdd_below A; A{}  f (Inf A)  (INF xA. f x)"
    by (simp add: mono f conditionally_complete_lattice_class.cINF_greatest cInf_lower monoD)
  
  lemma mono_cSup: "bdd_above A; A{}  (SUP xA. f x)  f (Sup A)"
    by (simp add: mono f conditionally_complete_lattice_class.cSUP_least cSup_upper monoD)
  
  lemma mono_cINF: "bdd_below (A`I); I{}  f (INF iI. A i)  (INF xI. f (A x))"
    by (simp add: mono f conditionally_complete_lattice_class.cINF_greatest cINF_lower monoD)
  
  lemma mono_cSUP: "bdd_above (A`I); I{}  (SUP xI. f (A x))  f (SUP iI. A i)"
    by (simp add: mono f conditionally_complete_lattice_class.cSUP_least cSUP_upper monoD)
  
  end

end

text The special case of well-orderings

lemma wellorder_InfI:
  fixes k :: "'a::{wellorder,conditionally_complete_lattice}"
  assumes "k  A" shows "Inf A  A" 
  using wellorder_class.LeastI [of "λx. x  A" k]
  by (simp add: Least_le assms cInf_eq_minimum)

lemma wellorder_Inf_le1:
  fixes k :: "'a::{wellorder,conditionally_complete_lattice}"
  assumes "k  A" shows "Inf A  k"
  by (meson Least_le assms bdd_below.I cInf_lower)

subsection Complete lattices

instance complete_lattice  conditionally_complete_lattice
  by standard (auto intro: Sup_upper Sup_least Inf_lower Inf_greatest)

lemma cSup_eq:
  fixes a :: "'a :: {conditionally_complete_lattice, no_bot}"
  assumes upper: "x. x  X  x  a"
  assumes least: "y. (x. x  X  x  y)  a  y"
  shows "Sup X = a"
proof cases
  assume "X = {}" with lt_ex[of a] least show ?thesis by (auto simp: less_le_not_le)
qed (intro cSup_eq_non_empty assms)

lemma cInf_eq:
  fixes a :: "'a :: {conditionally_complete_lattice, no_top}"
  assumes upper: "x. x  X  a  x"
  assumes least: "y. (x. x  X  y  x)  y  a"
  shows "Inf X = a"
proof cases
  assume "X = {}" with gt_ex[of a] least show ?thesis by (auto simp: less_le_not_le)
qed (intro cInf_eq_non_empty assms)

class conditionally_complete_linorder = conditionally_complete_lattice + linorder
begin

lemma less_cSup_iff:
  "X  {}  bdd_above X  y < Sup X  (xX. y < x)"
  by (rule iffI) (metis cSup_least not_less, metis cSup_upper less_le_trans)

lemma cInf_less_iff: "X  {}  bdd_below X  Inf X < y  (xX. x < y)"
  by (rule iffI) (metis cInf_greatest not_less, metis cInf_lower le_less_trans)

lemma cINF_less_iff: "A  {}  bdd_below (f`A)  (iA. f i) < a  (xA. f x < a)"
  using cInf_less_iff[of "f`A"] by auto

lemma less_cSUP_iff: "A  {}  bdd_above (f`A)  a < (iA. f i)  (xA. a < f x)"
  using less_cSup_iff[of "f`A"] by auto

lemma less_cSupE:
  assumes "y < Sup X" "X  {}" obtains x where "x  X" "y < x"
  by (metis cSup_least assms not_le that)

lemma less_cSupD:
  "X  {}  z < Sup X  xX. z < x"
  by (metis less_cSup_iff not_le_imp_less bdd_above_def)

lemma cInf_lessD:
  "X  {}  Inf X < z  xX. x < z"
  by (metis cInf_less_iff not_le_imp_less bdd_below_def)

lemma complete_interval:
  assumes "a < b" and "P a" and "¬ P b"
  shows "c. a  c  c  b  (x. a  x  x < c  P x) 
             (d. (x. a  x  x < d  P x)  d  c)"
proof (rule exI [where x = "Sup {d. x. a  x  x < d  P x}"], safe)
  show "a  Sup {d. c. a  c  c < d  P c}"
    by (rule cSup_upper, auto simp: bdd_above_def)
       (metis a < b ¬ P b linear less_le)
next
  show "Sup {d. c. a  c  c < d  P c}  b"
    by (rule cSup_least)
       (use a<b ¬ P b in auto simp add: less_le_not_le)
next
  fix x
  assume x: "a  x" and lt: "x < Sup {d. c. a  c  c < d  P c}"
  show "P x"
    by (rule less_cSupE [OF lt]) (use less_le_not_le x in auto)
next
  fix d
  assume 0: "x. a  x  x < d  P x"
  then have "d  {d. c. a  c  c < d  P c}"
    by auto
  moreover have "bdd_above {d. c. a  c  c < d  P c}"
    unfolding bdd_above_def using a<b ¬ P b linear
    by (simp add: less_le) blast
  ultimately show "d  Sup {d. c. a  c  c < d  P c}"
    by (auto simp: cSup_upper)
qed

end

subsection Instances

instance complete_linorder < conditionally_complete_linorder
  ..

lemma cSup_eq_Max: "finite (X::'a::conditionally_complete_linorder set)  X  {}  Sup X = Max X"
  using cSup_eq_Sup_fin[of X] by (simp add: Sup_fin_Max)

lemma cInf_eq_Min: "finite (X::'a::conditionally_complete_linorder set)  X  {}  Inf X = Min X"
  using cInf_eq_Inf_fin[of X] by (simp add: Inf_fin_Min)

lemma cSup_lessThan[simp]: "Sup {..<x::'a::{conditionally_complete_linorder, no_bot, dense_linorder}} = x"
  by (auto intro!: cSup_eq_non_empty intro: dense_le)

lemma cSup_greaterThanLessThan[simp]: "y < x  Sup {y<..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x"
  by (auto intro!: cSup_eq_non_empty intro: dense_le_bounded)

lemma cSup_atLeastLessThan[simp]: "y < x  Sup {y..<x::'a::{conditionally_complete_linorder, dense_linorder}} = x"
  by (auto intro!: cSup_eq_non_empty intro: dense_le_bounded)

lemma cInf_greaterThan[simp]: "Inf {x::'a::{conditionally_complete_linorder, no_top, dense_linorder} <..} = x"
  by (auto intro!: cInf_eq_non_empty intro: dense_ge)

lemma cInf_greaterThanAtMost[simp]: "y < x  Inf {y<..x::'a::{conditionally_complete_linorder, dense_linorder}} = y"
  by (auto intro!: cInf_eq_non_empty intro: dense_ge_bounded)

lemma cInf_greaterThanLessThan[simp]: "y < x  Inf {y<..<x::'a::{conditionally_complete_linorder, dense_linorder}} = y"
  by (auto intro!: cInf_eq_non_empty intro: dense_ge_bounded)

lemma Inf_insert_finite:
  fixes S :: "'a::conditionally_complete_linorder set"
  shows "finite S  Inf (insert x S) = (if S = {} then x else min x (Inf S))"
  by (simp add: cInf_eq_Min)

lemma Sup_insert_finite:
  fixes S :: "'a::conditionally_complete_linorder set"
  shows "finite S  Sup (insert x S) = (if S = {} then x else max x (Sup S))"
  by (simp add: cSup_insert sup_max)

lemma finite_imp_less_Inf:
  fixes a :: "'a::conditionally_complete_linorder"
  shows "finite X; x  X; x. xX  a < x  a < Inf X"
  by (induction X rule: finite_induct) (simp_all add: cInf_eq_Min Inf_insert_finite)

lemma finite_less_Inf_iff:
  fixes a :: "'a :: conditionally_complete_linorder"
  shows "finite X; X  {}  a < Inf X  (x  X. a < x)"
  by (auto simp: cInf_eq_Min)

lemma finite_imp_Sup_less:
  fixes a :: "'a::conditionally_complete_linorder"
  shows "finite X; x  X; x. xX  a > x  a > Sup X"
  by (induction X rule: finite_induct) (simp_all add: cSup_eq_Max Sup_insert_finite)

lemma finite_Sup_less_iff:
  fixes a :: "'a :: conditionally_complete_linorder"
  shows "finite X; X  {}  a > Sup X  (x  X. a > x)"
  by (auto simp: cSup_eq_Max)

class linear_continuum = conditionally_complete_linorder + dense_linorder +
  assumes UNIV_not_singleton: "a b::'a. a  b"
begin

lemma ex_gt_or_lt: "b. a < b  b < a"
  by (metis UNIV_not_singleton neq_iff)

end

instantiation nat :: conditionally_complete_linorder
begin

definition "Sup (X::nat set) = (if X={} then 0 else Max X)"
definition "Inf (X::nat set) = (LEAST n. n  X)"

lemma bdd_above_nat: "bdd_above X  finite (X::nat set)"
proof
  assume "bdd_above X"
  then obtain z where "X  {.. z}"
    by (auto simp: bdd_above_def)
  then show "finite X"
    by (rule finite_subset) simp
qed simp

instance
proof
  fix x :: nat
  fix X :: "nat set"
  show "Inf X  x" if "x  X" "bdd_below X"
    using that by (simp add: Inf_nat_def Least_le)
  show "x  Inf X" if "X  {}" "y. y  X  x  y"
    using that unfolding Inf_nat_def ex_in_conv[symmetric] by (rule LeastI2_ex)
  show "x  Sup X" if "x  X" "bdd_above X"
    using that by (auto simp add: Sup_nat_def bdd_above_nat)
  show "Sup X  x" if "X  {}" "y. y  X  y  x"
  proof -
    from that have "bdd_above X"
      by (auto simp: bdd_above_def)
    with that show ?thesis 
      by (simp add: Sup_nat_def bdd_above_nat)
  qed
qed

end

lemma Inf_nat_def1:
  fixes K::"nat set"
  assumes "K  {}"
  shows "Inf K  K"
by (auto simp add: Min_def Inf_nat_def) (meson LeastI assms bot.extremum_unique subsetI)

lemma Sup_nat_empty [simp]: "Sup {} = (0::nat)"
  by (auto simp add: Sup_nat_def) 



instantiation int :: conditionally_complete_linorder
begin

definition "Sup (X::int set) = (THE x. x  X  (yX. y  x))"
definition "Inf (X::int set) = - (Sup (uminus ` X))"

instance
proof
  { fix x :: int and X :: "int set" assume "X  {}" "bdd_above X"
    then obtain x y where "X  {..y}" "x  X"
      by (auto simp: bdd_above_def)
    then have *: "finite (X  {x..y})" "X  {x..y}  {}" and "x  y"
      by (auto simp: subset_eq)
    have "∃!xX. (yX. y  x)"
    proof
      { fix z assume "z  X"
        have "z  Max (X  {x..y})"
        proof cases
          assume "x  z" with z  X X  {..y} *(1) show ?thesis
            by (auto intro!: Max_ge)
        next
          assume "¬ x  z"
          then have "z < x" by simp
          also have "x  Max (X  {x..y})"
            using x  X *(1) x  y by (intro Max_ge) auto
          finally show ?thesis by simp
        qed }
      note le = this
      with Max_in[OF *] show ex: "Max (X  {x..y})  X  (zX. z  Max (X  {x..y}))" by auto

      fix z assume *: "z  X  (yX. y  z)"
      with le have "z  Max (X  {x..y})"
        by auto
      moreover have "Max (X  {x..y})  z"
        using * ex by auto
      ultimately show "z = Max (X  {x..y})"
        by auto
    qed
    then have "Sup X  X  (yX. y  Sup X)"
      unfolding Sup_int_def by (rule theI') }
  note Sup_int = this

  { fix x :: int and X :: "int set" assume "x  X" "bdd_above X" then show "x  Sup X"
      using Sup_int[of X] by auto }
  note le_Sup = this
  { fix x :: int and X :: "int set" assume "X  {}" "y. y  X  y  x" then show "Sup X  x"
      using Sup_int[of X] by (auto simp: bdd_above_def) }
  note Sup_le = this

  { fix x :: int and X :: "int set" assume "x  X" "bdd_below X" then show "Inf X  x"
      using le_Sup[of "-x" "uminus ` X"] by (auto simp: Inf_int_def) }
  { fix x :: int and X :: "int set" assume "X  {}" "y. y  X  x  y" then show "x  Inf X"
      using Sup_le[of "uminus ` X" "-x"] by (force simp: Inf_int_def) }
qed
end

lemma interval_cases:
  fixes S :: "'a :: conditionally_complete_linorder set"
  assumes ivl: "a b x. a  S  b  S  a  x  x  b  x  S"
  shows "a b. S = {} 
    S = UNIV 
    S = {..<b} 
    S = {..b} 
    S = {a<..} 
    S = {a..} 
    S = {a<..<b} 
    S = {a<..b} 
    S = {a..<b} 
    S = {a..b}"
proof -
  define lower upper where "lower = {x. sS. s  x}" and "upper = {x. sS. x  s}"
  with ivl have "S = lower  upper"
    by auto
  moreover
  have "a. upper = UNIV  upper = {}  upper = {.. a}  upper = {..< a}"
  proof cases
    assume *: "bdd_above S  S  {}"
    from * have "upper  {.. Sup S}"
      by (auto simp: upper_def intro: cSup_upper2)
    moreover from * have "{..< Sup S}  upper"
      by (force simp add: less_cSup_iff upper_def subset_eq Ball_def)
    ultimately have "upper = {.. Sup S}  upper = {..< Sup S}"
      unfolding ivl_disj_un(2)[symmetric] by auto
    then show ?thesis by auto
  next
    assume "¬ (bdd_above S  S  {})"
    then have "upper = UNIV  upper = {}"
      by (auto simp: upper_def bdd_above_def not_le dest: less_imp_le)
    then show ?thesis
      by auto
  qed
  moreover
  have "b. lower = UNIV  lower = {}  lower = {b ..}  lower = {b <..}"
  proof cases
    assume *: "bdd_below S  S  {}"
    from * have "lower  {Inf S ..}"
      by (auto simp: lower_def intro: cInf_lower2)
    moreover from * have "{Inf S <..}  lower"
      by (force simp add: cInf_less_iff lower_def subset_eq Ball_def)
    ultimately have "lower = {Inf S ..}  lower = {Inf S <..}"
      unfolding ivl_disj_un(1)[symmetric] by auto
    then show ?thesis by auto
  next
    assume "¬ (bdd_below S  S  {})"
    then have "lower = UNIV  lower = {}"
      by (auto simp: lower_def bdd_below_def not_le dest: less_imp_le)
    then show ?thesis
      by auto
  qed
  ultimately show ?thesis
    unfolding greaterThanAtMost_def greaterThanLessThan_def atLeastAtMost_def atLeastLessThan_def
    by (metis inf_bot_left inf_bot_right inf_top.left_neutral inf_top.right_neutral)
qed

lemma cSUP_eq_cINF_D:
  fixes f :: "_  'b::conditionally_complete_lattice"
  assumes eq: "(xA. f x) = (xA. f x)"
     and bdd: "bdd_above (f ` A)" "bdd_below (f ` A)"
     and a: "a  A"
  shows "f a = (xA. f x)"
proof (rule antisym)
  show "f a   (f ` A)"
    by (metis a bdd(1) eq cSUP_upper)
  show " (f ` A)  f a"
    using a bdd by (auto simp: cINF_lower)
qed

lemma cSUP_UNION:
  fixes f :: "_  'b::conditionally_complete_lattice"
  assumes ne: "A  {}" "x. x  A  B(x)  {}"
      and bdd_UN: "bdd_above (xA. f ` B x)"
  shows "(z  xA. B x. f z) = (xA. zB x. f z)"
proof -
  have bdd: "x. x  A  bdd_above (f ` B x)"
    using bdd_UN by (meson UN_upper bdd_above_mono)
  obtain M where "x y. x  A  y  B(x)  f y  M"
    using bdd_UN by (auto simp: bdd_above_def)
  then have bdd2: "bdd_above ((λx. zB x. f z) ` A)"
    unfolding bdd_above_def by (force simp: bdd cSUP_le_iff ne(2))
  have "(z  xA. B x. f z)  (xA. zB x. f z)"
    using assms by (fastforce simp add: intro!: cSUP_least intro: cSUP_upper2 simp: bdd2 bdd)
  moreover have "(xA. zB x. f z)  ( z  xA. B x. f z)"
    using assms by (fastforce simp add: intro!: cSUP_least intro: cSUP_upper simp: image_UN bdd_UN)
  ultimately show ?thesis
    by (rule order_antisym)
qed

lemma cINF_UNION:
  fixes f :: "_  'b::conditionally_complete_lattice"
  assumes ne: "A  {}" "x. x  A  B(x)  {}"
      and bdd_UN: "bdd_below (xA. f ` B x)"
  shows "(z  xA. B x. f z) = (xA. zB x. f z)"
proof -
  have bdd: "x. x  A  bdd_below (f ` B x)"
    using bdd_UN by (meson UN_upper bdd_below_mono)
  obtain M where "x y. x  A  y  B(x)  f y  M"
    using bdd_UN by (auto simp: bdd_below_def)
  then have bdd2: "bdd_below ((λx. zB x. f z) ` A)"
    unfolding bdd_below_def by (force simp: bdd le_cINF_iff ne(2))
  have "(z  xA. B x. f z)  (xA. zB x. f z)"
    using assms by (fastforce simp add: intro!: cINF_greatest intro: cINF_lower simp: bdd2 bdd)
  moreover have "(xA. zB x. f z)  (z  xA. B x. f z)"
    using assms  by (fastforce simp add: intro!: cINF_greatest intro: cINF_lower2  simp: bdd bdd_UN bdd2)
  ultimately show ?thesis
    by (rule order_antisym)
qed

lemma cSup_abs_le:
  fixes S :: "('a::{linordered_idom,conditionally_complete_linorder}) set"
  shows "S  {}  (x. xS  ¦x¦  a)  ¦Sup S¦  a"
  apply (auto simp add: abs_le_iff intro: cSup_least)
  by (metis bdd_aboveI cSup_upper neg_le_iff_le order_trans)

end