# 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 {}

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:
by (fact bdd_above.empty)

lemma bdd_below_empty:
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]:
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]:
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]:
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]:
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 _ ]) simp

end

context order_bot
begin

lemma bdd_below_bot [simp, intro!]: "bdd_below A"
by (rule bdd_belowI [of _ ]) 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
shows bdd_above_uminus[simp]:
and bdd_below_uminus[simp]:
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  unfolding bdd_above_def by auto
next
assume
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  unfolding bdd_below_def by auto
next
assume
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))"

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

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 ::
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 ::
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 cSup_unique:
fixes b :: "'a :: {conditionally_complete_lattice, no_bot}"
assumes "c. (x  s. x  c)  b  c"
shows "Sup s = b"
by (metis assms cSup_eq order.refl)

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)

lemma cInf_unique:
fixes b :: "'a :: {conditionally_complete_lattice, no_top}"
assumes "c. (x  s. x  c)  b  c"
shows "Inf s = b"
by (meson assms cInf_eq order.refl)

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
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]:
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]:
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 ::
shows "finite S  Inf (insert x S) = (if S = {} then x else min x (Inf S))"

lemma Sup_insert_finite:
fixes S ::
shows "finite S  Sup (insert x S) = (if S = {} then x else max x (Sup S))"

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

lemma finite_imp_Sup_less:
fixes a ::
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 ::
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

context
fixes f::
begin

lemma bdd_above_uminus_image: "bdd_above ((λx. - f x) ` A)  bdd_below (f ` A)"
by (metis bdd_above_uminus image_image)

lemma bdd_below_uminus_image: "bdd_below ((λx. - f x) ` A)  bdd_above (f ` A)"
by (metis bdd_below_uminus image_image)

lemma uminus_cSUP:
assumes "bdd_above (f ` A)" "A  {}"
shows  "- (SUP xA. f x) = (INF xA. - f x)"
proof (rule antisym)
show "(INF xA. - f x)  - Sup (f ` A)"
by (metis cINF_lower cSUP_least bdd_below_uminus_image assms le_minus_iff)
have *: "x. x A  f x  Sup (f ` A)"
then show "- Sup (f ` A)  (INF xA. - f x)"
qed

end

context
fixes f::
begin

lemma uminus_cINF:
assumes "bdd_below (f ` A)" "A  {}"
shows  "- (INF xA. f x) = (SUP xA. - f x)"
by (metis (mono_tags, lifting) INF_cong uminus_cSUP assms bdd_above_uminus_image minus_equation_iff)

assumes "bdd_above (f ` A)" "A  {}"
shows  "(SUP xA. a + f x) = a + (SUP xA. f x)" (is "?L=?R")
proof (rule antisym)
have bdd: "bdd_above ((λx. a + f x) ` A)"
by (metis assms bdd_above_image_mono image_image mono_add)
with assms show "?L  ?R"
by (simp add: assms cSup_le_iff cSUP_upper)
have "x. x  A  f x  (SUP xA. a + f x) - a"
by (simp add: bdd cSup_upper le_diff_eq)
with A  {} have " (f ` A)  (xA. a + f x) - a"
then show "?R  ?L"
qed

lemma Inf_add_eq: ―‹you don't get a shorter proof by duality›
assumes "bdd_below (f ` A)" "A  {}"
shows  "(INF xA. a + f x) = a + (INF xA. f x)" (is "?L=?R")
proof (rule antisym)
show "?R  ?L"
using assms mono_add mono_cINF by blast
have bdd: "bdd_below ((λx. a + f x) ` A)"
by (metis add_left_mono assms(1) bdd_below.E bdd_below.I2 imageI)
with assms have "x. x  A  f x  (INF xA. a + f x) - a"
with A  {} have "(xA. a + f x) - a   (f ` A)"
with assms show "?L  ?R"
qed

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
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)"

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 ::
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 ::
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