Theory Lattices_Big

theory Lattices_Big
imports Option
(*  Title:      HOL/Lattices_Big.thy
    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
                with contributions by Jeremy Avigad
*)

section ‹Big infimum (minimum) and supremum (maximum) over finite (non-empty) sets›

theory Lattices_Big
  imports Option
begin

subsection ‹Generic lattice operations over a set›

subsubsection ‹Without neutral element›

locale semilattice_set = semilattice
begin

interpretation comp_fun_idem f
  by standard (simp_all add: fun_eq_iff left_commute)

definition F :: "'a set ⇒ 'a"
where
  eq_fold': "F A = the (Finite_Set.fold (λx y. Some (case y of None ⇒ x | Some z ⇒ f x z)) None A)"

lemma eq_fold:
  assumes "finite A"
  shows "F (insert x A) = Finite_Set.fold f x A"
proof (rule sym)
  let ?f = "λx y. Some (case y of None ⇒ x | Some z ⇒ f x z)"
  interpret comp_fun_idem "?f"
    by standard (simp_all add: fun_eq_iff commute left_commute split: option.split)
  from assms show "Finite_Set.fold f x A = F (insert x A)"
  proof induct
    case empty then show ?case by (simp add: eq_fold')
  next
    case (insert y B) then show ?case by (simp add: insert_commute [of x] eq_fold')
  qed
qed

lemma singleton [simp]:
  "F {x} = x"
  by (simp add: eq_fold)

lemma insert_not_elem:
  assumes "finite A" and "x ∉ A" and "A ≠ {}"
  shows "F (insert x A) = x * F A"
proof -
  from ‹A ≠ {}› obtain b where "b ∈ A" by blast
  then obtain B where *: "A = insert b B" "b ∉ B" by (blast dest: mk_disjoint_insert)
  with ‹finite A› and ‹x ∉ A›
    have "finite (insert x B)" and "b ∉ insert x B" by auto
  then have "F (insert b (insert x B)) = x * F (insert b B)"
    by (simp add: eq_fold)
  then show ?thesis by (simp add: * insert_commute)
qed

lemma in_idem:
  assumes "finite A" and "x ∈ A"
  shows "x * F A = F A"
proof -
  from assms have "A ≠ {}" by auto
  with ‹finite A› show ?thesis using ‹x ∈ A›
    by (induct A rule: finite_ne_induct) (auto simp add: ac_simps insert_not_elem)
qed

lemma insert [simp]:
  assumes "finite A" and "A ≠ {}"
  shows "F (insert x A) = x * F A"
  using assms by (cases "x ∈ A") (simp_all add: insert_absorb in_idem insert_not_elem)

lemma union:
  assumes "finite A" "A ≠ {}" and "finite B" "B ≠ {}"
  shows "F (A ∪ B) = F A * F B"
  using assms by (induct A rule: finite_ne_induct) (simp_all add: ac_simps)

lemma remove:
  assumes "finite A" and "x ∈ A"
  shows "F A = (if A - {x} = {} then x else x * F (A - {x}))"
proof -
  from assms obtain B where "A = insert x B" and "x ∉ B" by (blast dest: mk_disjoint_insert)
  with assms show ?thesis by simp
qed

lemma insert_remove:
  assumes "finite A"
  shows "F (insert x A) = (if A - {x} = {} then x else x * F (A - {x}))"
  using assms by (cases "x ∈ A") (simp_all add: insert_absorb remove)

lemma subset:
  assumes "finite A" "B ≠ {}" and "B ⊆ A"
  shows "F B * F A = F A"
proof -
  from assms have "A ≠ {}" and "finite B" by (auto dest: finite_subset)
  with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
qed

lemma closed:
  assumes "finite A" "A ≠ {}" and elem: "⋀x y. x * y ∈ {x, y}"
  shows "F A ∈ A"
using ‹finite A› ‹A ≠ {}› proof (induct rule: finite_ne_induct)
  case singleton then show ?case by simp
next
  case insert with elem show ?case by force
qed

lemma hom_commute:
  assumes hom: "⋀x y. h (x * y) = h x * h y"
  and N: "finite N" "N ≠ {}"
  shows "h (F N) = F (h ` N)"
using N proof (induct rule: finite_ne_induct)
  case singleton thus ?case by simp
next
  case (insert n N)
  then have "h (F (insert n N)) = h (n * F N)" by simp
  also have "… = h n * h (F N)" by (rule hom)
  also have "h (F N) = F (h ` N)" by (rule insert)
  also have "h n * … = F (insert (h n) (h ` N))"
    using insert by simp
  also have "insert (h n) (h ` N) = h ` insert n N" by simp
  finally show ?case .
qed

lemma infinite: "¬ finite A ⟹ F A = the None"
  unfolding eq_fold' by (cases "finite (UNIV::'a set)") (auto intro: finite_subset fold_infinite)

end

locale semilattice_order_set = binary?: semilattice_order + semilattice_set
begin

lemma bounded_iff:
  assumes "finite A" and "A ≠ {}"
  shows "x  F A ⟷ (∀a∈A. x  a)"
  using assms by (induct rule: finite_ne_induct) (simp_all add: bounded_iff)

lemma boundedI:
  assumes "finite A"
  assumes "A ≠ {}"
  assumes "⋀a. a ∈ A ⟹ x  a"
  shows "x  F A"
  using assms by (simp add: bounded_iff)

lemma boundedE:
  assumes "finite A" and "A ≠ {}" and "x  F A"
  obtains "⋀a. a ∈ A ⟹ x  a"
  using assms by (simp add: bounded_iff)

lemma coboundedI:
  assumes "finite A"
    and "a ∈ A"
  shows "F A  a"
proof -
  from assms have "A ≠ {}" by auto
  from ‹finite A› ‹A ≠ {}› ‹a ∈ A› show ?thesis
  proof (induct rule: finite_ne_induct)
    case singleton thus ?case by (simp add: refl)
  next
    case (insert x B)
    from insert have "a = x ∨ a ∈ B" by simp
    then show ?case using insert by (auto intro: coboundedI2)
  qed
qed

lemma antimono:
  assumes "A ⊆ B" and "A ≠ {}" and "finite B"
  shows "F B  F A"
proof (cases "A = B")
  case True then show ?thesis by (simp add: refl)
next
  case False
  have B: "B = A ∪ (B - A)" using ‹A ⊆ B› by blast
  then have "F B = F (A ∪ (B - A))" by simp
  also have "… = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
  also have "…  F A" by simp
  finally show ?thesis .
qed

end


subsubsection ‹With neutral element›

locale semilattice_neutr_set = semilattice_neutr
begin

interpretation comp_fun_idem f
  by standard (simp_all add: fun_eq_iff left_commute)

definition F :: "'a set ⇒ 'a"
where
  eq_fold: "F A = Finite_Set.fold f 1 A"

lemma infinite [simp]:
  "¬ finite A ⟹ F A = 1"
  by (simp add: eq_fold)

lemma empty [simp]:
  "F {} = 1"
  by (simp add: eq_fold)

lemma insert [simp]:
  assumes "finite A"
  shows "F (insert x A) = x * F A"
  using assms by (simp add: eq_fold)

lemma in_idem:
  assumes "finite A" and "x ∈ A"
  shows "x * F A = F A"
proof -
  from assms have "A ≠ {}" by auto
  with ‹finite A› show ?thesis using ‹x ∈ A›
    by (induct A rule: finite_ne_induct) (auto simp add: ac_simps)
qed

lemma union:
  assumes "finite A" and "finite B"
  shows "F (A ∪ B) = F A * F B"
  using assms by (induct A) (simp_all add: ac_simps)

lemma remove:
  assumes "finite A" and "x ∈ A"
  shows "F A = x * F (A - {x})"
proof -
  from assms obtain B where "A = insert x B" and "x ∉ B" by (blast dest: mk_disjoint_insert)
  with assms show ?thesis by simp
qed

lemma insert_remove:
  assumes "finite A"
  shows "F (insert x A) = x * F (A - {x})"
  using assms by (cases "x ∈ A") (simp_all add: insert_absorb remove)

lemma subset:
  assumes "finite A" and "B ⊆ A"
  shows "F B * F A = F A"
proof -
  from assms have "finite B" by (auto dest: finite_subset)
  with assms show ?thesis by (simp add: union [symmetric] Un_absorb1)
qed

lemma closed:
  assumes "finite A" "A ≠ {}" and elem: "⋀x y. x * y ∈ {x, y}"
  shows "F A ∈ A"
using ‹finite A› ‹A ≠ {}› proof (induct rule: finite_ne_induct)
  case singleton then show ?case by simp
next
  case insert with elem show ?case by force
qed

end

locale semilattice_order_neutr_set = binary?: semilattice_neutr_order + semilattice_neutr_set
begin

lemma bounded_iff:
  assumes "finite A"
  shows "x  F A ⟷ (∀a∈A. x  a)"
  using assms by (induct A) (simp_all add: bounded_iff)

lemma boundedI:
  assumes "finite A"
  assumes "⋀a. a ∈ A ⟹ x  a"
  shows "x  F A"
  using assms by (simp add: bounded_iff)

lemma boundedE:
  assumes "finite A" and "x  F A"
  obtains "⋀a. a ∈ A ⟹ x  a"
  using assms by (simp add: bounded_iff)

lemma coboundedI:
  assumes "finite A"
    and "a ∈ A"
  shows "F A  a"
proof -
  from assms have "A ≠ {}" by auto
  from ‹finite A› ‹A ≠ {}› ‹a ∈ A› show ?thesis
  proof (induct rule: finite_ne_induct)
    case singleton thus ?case by (simp add: refl)
  next
    case (insert x B)
    from insert have "a = x ∨ a ∈ B" by simp
    then show ?case using insert by (auto intro: coboundedI2)
  qed
qed

lemma antimono:
  assumes "A ⊆ B" and "finite B"
  shows "F B  F A"
proof (cases "A = B")
  case True then show ?thesis by (simp add: refl)
next
  case False
  have B: "B = A ∪ (B - A)" using ‹A ⊆ B› by blast
  then have "F B = F (A ∪ (B - A))" by simp
  also have "… = F A * F (B - A)" using False assms by (subst union) (auto intro: finite_subset)
  also have "…  F A" by simp
  finally show ?thesis .
qed

end


subsection ‹Lattice operations on finite sets›

context semilattice_inf
begin

sublocale Inf_fin: semilattice_order_set inf less_eq less
defines
  Inf_fin ("⨅fin_" [900] 900) = Inf_fin.F ..

end

context semilattice_sup
begin

sublocale Sup_fin: semilattice_order_set sup greater_eq greater
defines
  Sup_fin ("⨆fin_" [900] 900) = Sup_fin.F ..

end


subsection ‹Infimum and Supremum over non-empty sets›

context lattice
begin

lemma Inf_fin_le_Sup_fin [simp]: 
  assumes "finite A" and "A ≠ {}"
  shows "⨅finA ≤ ⨆finA"
proof -
  from ‹A ≠ {}› obtain a where "a ∈ A" by blast
  with ‹finite A› have "⨅finA ≤ a" by (rule Inf_fin.coboundedI)
  moreover from ‹finite A› ‹a ∈ A› have "a ≤ ⨆finA" by (rule Sup_fin.coboundedI)
  ultimately show ?thesis by (rule order_trans)
qed

lemma sup_Inf_absorb [simp]:
  "finite A ⟹ a ∈ A ⟹ ⨅finA ⊔ a = a"
  by (rule sup_absorb2) (rule Inf_fin.coboundedI)

lemma inf_Sup_absorb [simp]:
  "finite A ⟹ a ∈ A ⟹ a ⊓ ⨆finA = a"
  by (rule inf_absorb1) (rule Sup_fin.coboundedI)

end

context distrib_lattice
begin

lemma sup_Inf1_distrib:
  assumes "finite A"
    and "A ≠ {}"
  shows "sup x (⨅finA) = ⨅fin{sup x a|a. a ∈ A}"
using assms by (simp add: image_def Inf_fin.hom_commute [where h="sup x", OF sup_inf_distrib1])
  (rule arg_cong [where f="Inf_fin"], blast)

lemma sup_Inf2_distrib:
  assumes A: "finite A" "A ≠ {}" and B: "finite B" "B ≠ {}"
  shows "sup (⨅finA) (⨅finB) = ⨅fin{sup a b|a b. a ∈ A ∧ b ∈ B}"
using A proof (induct rule: finite_ne_induct)
  case singleton then show ?case
    by (simp add: sup_Inf1_distrib [OF B])
next
  case (insert x A)
  have finB: "finite {sup x b |b. b ∈ B}"
    by (rule finite_surj [where f = "sup x", OF B(1)], auto)
  have finAB: "finite {sup a b |a b. a ∈ A ∧ b ∈ B}"
  proof -
    have "{sup a b |a b. a ∈ A ∧ b ∈ B} = (UN a:A. UN b:B. {sup a b})"
      by blast
    thus ?thesis by(simp add: insert(1) B(1))
  qed
  have ne: "{sup a b |a b. a ∈ A ∧ b ∈ B} ≠ {}" using insert B by blast
  have "sup (⨅fin(insert x A)) (⨅finB) = sup (inf x (⨅finA)) (⨅finB)"
    using insert by simp
  also have "… = inf (sup x (⨅finB)) (sup (⨅finA) (⨅finB))" by(rule sup_inf_distrib2)
  also have "… = inf (⨅fin{sup x b|b. b ∈ B}) (⨅fin{sup a b|a b. a ∈ A ∧ b ∈ B})"
    using insert by(simp add:sup_Inf1_distrib[OF B])
  also have "… = ⨅fin({sup x b |b. b ∈ B} ∪ {sup a b |a b. a ∈ A ∧ b ∈ B})"
    (is "_ = ⨅fin?M")
    using B insert
    by (simp add: Inf_fin.union [OF finB _ finAB ne])
  also have "?M = {sup a b |a b. a ∈ insert x A ∧ b ∈ B}"
    by blast
  finally show ?case .
qed

lemma inf_Sup1_distrib:
  assumes "finite A" and "A ≠ {}"
  shows "inf x (⨆finA) = ⨆fin{inf x a|a. a ∈ A}"
using assms by (simp add: image_def Sup_fin.hom_commute [where h="inf x", OF inf_sup_distrib1])
  (rule arg_cong [where f="Sup_fin"], blast)

lemma inf_Sup2_distrib:
  assumes A: "finite A" "A ≠ {}" and B: "finite B" "B ≠ {}"
  shows "inf (⨆finA) (⨆finB) = ⨆fin{inf a b|a b. a ∈ A ∧ b ∈ B}"
using A proof (induct rule: finite_ne_induct)
  case singleton thus ?case
    by(simp add: inf_Sup1_distrib [OF B])
next
  case (insert x A)
  have finB: "finite {inf x b |b. b ∈ B}"
    by(rule finite_surj[where f = "%b. inf x b", OF B(1)], auto)
  have finAB: "finite {inf a b |a b. a ∈ A ∧ b ∈ B}"
  proof -
    have "{inf a b |a b. a ∈ A ∧ b ∈ B} = (UN a:A. UN b:B. {inf a b})"
      by blast
    thus ?thesis by(simp add: insert(1) B(1))
  qed
  have ne: "{inf a b |a b. a ∈ A ∧ b ∈ B} ≠ {}" using insert B by blast
  have "inf (⨆fin(insert x A)) (⨆finB) = inf (sup x (⨆finA)) (⨆finB)"
    using insert by simp
  also have "… = sup (inf x (⨆finB)) (inf (⨆finA) (⨆finB))" by(rule inf_sup_distrib2)
  also have "… = sup (⨆fin{inf x b|b. b ∈ B}) (⨆fin{inf a b|a b. a ∈ A ∧ b ∈ B})"
    using insert by(simp add:inf_Sup1_distrib[OF B])
  also have "… = ⨆fin({inf x b |b. b ∈ B} ∪ {inf a b |a b. a ∈ A ∧ b ∈ B})"
    (is "_ = ⨆fin?M")
    using B insert
    by (simp add: Sup_fin.union [OF finB _ finAB ne])
  also have "?M = {inf a b |a b. a ∈ insert x A ∧ b ∈ B}"
    by blast
  finally show ?case .
qed

end

context complete_lattice
begin

lemma Inf_fin_Inf:
  assumes "finite A" and "A ≠ {}"
  shows "⨅finA = ⨅A"
proof -
  from assms obtain b B where "A = insert b B" and "finite B" by auto
  then show ?thesis
    by (simp add: Inf_fin.eq_fold inf_Inf_fold_inf inf.commute [of b])
qed

lemma Sup_fin_Sup:
  assumes "finite A" and "A ≠ {}"
  shows "⨆finA = ⨆A"
proof -
  from assms obtain b B where "A = insert b B" and "finite B" by auto
  then show ?thesis
    by (simp add: Sup_fin.eq_fold sup_Sup_fold_sup sup.commute [of b])
qed

end


subsection ‹Minimum and Maximum over non-empty sets›

context linorder
begin

sublocale Min: semilattice_order_set min less_eq less
  + Max: semilattice_order_set max greater_eq greater
defines
  Min = Min.F and Max = Max.F ..

end

text ‹An aside: @{const Min}/@{const Max} on linear orders as special case of @{const Inf_fin}/@{const Sup_fin}›

lemma Inf_fin_Min:
  "Inf_fin = (Min :: 'a::{semilattice_inf, linorder} set ⇒ 'a)"
  by (simp add: Inf_fin_def Min_def inf_min)

lemma Sup_fin_Max:
  "Sup_fin = (Max :: 'a::{semilattice_sup, linorder} set ⇒ 'a)"
  by (simp add: Sup_fin_def Max_def sup_max)

context linorder
begin

lemma dual_min:
  "ord.min greater_eq = max"
  by (auto simp add: ord.min_def max_def fun_eq_iff)

lemma dual_max:
  "ord.max greater_eq = min"
  by (auto simp add: ord.max_def min_def fun_eq_iff)

lemma dual_Min:
  "linorder.Min greater_eq = Max"
proof -
  interpret dual: linorder greater_eq greater by (fact dual_linorder)
  show ?thesis by (simp add: dual.Min_def dual_min Max_def)
qed

lemma dual_Max:
  "linorder.Max greater_eq = Min"
proof -
  interpret dual: linorder greater_eq greater by (fact dual_linorder)
  show ?thesis by (simp add: dual.Max_def dual_max Min_def)
qed

lemmas Min_singleton = Min.singleton
lemmas Max_singleton = Max.singleton
lemmas Min_insert = Min.insert
lemmas Max_insert = Max.insert
lemmas Min_Un = Min.union
lemmas Max_Un = Max.union
lemmas hom_Min_commute = Min.hom_commute
lemmas hom_Max_commute = Max.hom_commute

lemma Min_in [simp]:
  assumes "finite A" and "A ≠ {}"
  shows "Min A ∈ A"
  using assms by (auto simp add: min_def Min.closed)

lemma Max_in [simp]:
  assumes "finite A" and "A ≠ {}"
  shows "Max A ∈ A"
  using assms by (auto simp add: max_def Max.closed)

lemma Min_insert2:
  assumes "finite A" and min: "⋀b. b ∈ A ⟹ a ≤ b"
  shows "Min (insert a A) = a"
proof (cases "A = {}")
  case True
  then show ?thesis by simp
next
  case False
  with ‹finite A› have "Min (insert a A) = min a (Min A)"
    by simp
  moreover from ‹finite A› ‹A ≠ {}› min have "a ≤ Min A" by simp
  ultimately show ?thesis by (simp add: min.absorb1)
qed

lemma Max_insert2:
  assumes "finite A" and max: "⋀b. b ∈ A ⟹ b ≤ a"
  shows "Max (insert a A) = a"
proof (cases "A = {}")
  case True
  then show ?thesis by simp
next
  case False
  with ‹finite A› have "Max (insert a A) = max a (Max A)"
    by simp
  moreover from ‹finite A› ‹A ≠ {}› max have "Max A ≤ a" by simp
  ultimately show ?thesis by (simp add: max.absorb1)
qed

lemma Min_le [simp]:
  assumes "finite A" and "x ∈ A"
  shows "Min A ≤ x"
  using assms by (fact Min.coboundedI)

lemma Max_ge [simp]:
  assumes "finite A" and "x ∈ A"
  shows "x ≤ Max A"
  using assms by (fact Max.coboundedI)

lemma Min_eqI:
  assumes "finite A"
  assumes "⋀y. y ∈ A ⟹ y ≥ x"
    and "x ∈ A"
  shows "Min A = x"
proof (rule antisym)
  from ‹x ∈ A› have "A ≠ {}" by auto
  with assms show "Min A ≥ x" by simp
next
  from assms show "x ≥ Min A" by simp
qed

lemma Max_eqI:
  assumes "finite A"
  assumes "⋀y. y ∈ A ⟹ y ≤ x"
    and "x ∈ A"
  shows "Max A = x"
proof (rule antisym)
  from ‹x ∈ A› have "A ≠ {}" by auto
  with assms show "Max A ≤ x" by simp
next
  from assms show "x ≤ Max A" by simp
qed

lemma eq_Min_iff:
  "⟦ finite A; A ≠ {} ⟧ ⟹ m = Min A  ⟷  m ∈ A ∧ (∀a ∈ A. m ≤ a)"
by (meson Min_in Min_le antisym)

lemma Min_eq_iff:
  "⟦ finite A; A ≠ {} ⟧ ⟹ Min A = m  ⟷  m ∈ A ∧ (∀a ∈ A. m ≤ a)"
by (meson Min_in Min_le antisym)

lemma eq_Max_iff:
  "⟦ finite A; A ≠ {} ⟧ ⟹ m = Max A  ⟷  m ∈ A ∧ (∀a ∈ A. a ≤ m)"
by (meson Max_in Max_ge antisym)

lemma Max_eq_iff:
  "⟦ finite A; A ≠ {} ⟧ ⟹ Max A = m  ⟷  m ∈ A ∧ (∀a ∈ A. a ≤ m)"
by (meson Max_in Max_ge antisym)

context
  fixes A :: "'a set"
  assumes fin_nonempty: "finite A" "A ≠ {}"
begin

lemma Min_ge_iff [simp]:
  "x ≤ Min A ⟷ (∀a∈A. x ≤ a)"
  using fin_nonempty by (fact Min.bounded_iff)

lemma Max_le_iff [simp]:
  "Max A ≤ x ⟷ (∀a∈A. a ≤ x)"
  using fin_nonempty by (fact Max.bounded_iff)

lemma Min_gr_iff [simp]:
  "x < Min A ⟷ (∀a∈A. x < a)"
  using fin_nonempty  by (induct rule: finite_ne_induct) simp_all

lemma Max_less_iff [simp]:
  "Max A < x ⟷ (∀a∈A. a < x)"
  using fin_nonempty by (induct rule: finite_ne_induct) simp_all

lemma Min_le_iff:
  "Min A ≤ x ⟷ (∃a∈A. a ≤ x)"
  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_le_iff_disj)

lemma Max_ge_iff:
  "x ≤ Max A ⟷ (∃a∈A. x ≤ a)"
  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: le_max_iff_disj)

lemma Min_less_iff:
  "Min A < x ⟷ (∃a∈A. a < x)"
  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: min_less_iff_disj)

lemma Max_gr_iff:
  "x < Max A ⟷ (∃a∈A. x < a)"
  using fin_nonempty by (induct rule: finite_ne_induct) (simp_all add: less_max_iff_disj)

end

lemma Max_eq_if:
  assumes "finite A"  "finite B"  "∀a∈A. ∃b∈B. a ≤ b"  "∀b∈B. ∃a∈A. b ≤ a"
  shows "Max A = Max B"
proof cases
  assume "A = {}" thus ?thesis using assms by simp
next
  assume "A ≠ {}" thus ?thesis using assms
    by(blast intro: antisym Max_in Max_ge_iff[THEN iffD2])
qed

lemma Min_antimono:
  assumes "M ⊆ N" and "M ≠ {}" and "finite N"
  shows "Min N ≤ Min M"
  using assms by (fact Min.antimono)

lemma Max_mono:
  assumes "M ⊆ N" and "M ≠ {}" and "finite N"
  shows "Max M ≤ Max N"
  using assms by (fact Max.antimono)

end

context linorder  (* FIXME *)
begin

lemma mono_Min_commute:
  assumes "mono f"
  assumes "finite A" and "A ≠ {}"
  shows "f (Min A) = Min (f ` A)"
proof (rule linorder_class.Min_eqI [symmetric])
  from ‹finite A› show "finite (f ` A)" by simp
  from assms show "f (Min A) ∈ f ` A" by simp
  fix x
  assume "x ∈ f ` A"
  then obtain y where "y ∈ A" and "x = f y" ..
  with assms have "Min A ≤ y" by auto
  with ‹mono f› have "f (Min A) ≤ f y" by (rule monoE)
  with ‹x = f y› show "f (Min A) ≤ x" by simp
qed

lemma mono_Max_commute:
  assumes "mono f"
  assumes "finite A" and "A ≠ {}"
  shows "f (Max A) = Max (f ` A)"
proof (rule linorder_class.Max_eqI [symmetric])
  from ‹finite A› show "finite (f ` A)" by simp
  from assms show "f (Max A) ∈ f ` A" by simp
  fix x
  assume "x ∈ f ` A"
  then obtain y where "y ∈ A" and "x = f y" ..
  with assms have "y ≤ Max A" by auto
  with ‹mono f› have "f y ≤ f (Max A)" by (rule monoE)
  with ‹x = f y› show "x ≤ f (Max A)" by simp
qed

lemma finite_linorder_max_induct [consumes 1, case_names empty insert]:
  assumes fin: "finite A"
  and empty: "P {}" 
  and insert: "⋀b A. finite A ⟹ ∀a∈A. a < b ⟹ P A ⟹ P (insert b A)"
  shows "P A"
using fin empty insert
proof (induct rule: finite_psubset_induct)
  case (psubset A)
  have IH: "⋀B. ⟦B < A; P {}; (⋀A b. ⟦finite A; ∀a∈A. a<b; P A⟧ ⟹ P (insert b A))⟧ ⟹ P B" by fact 
  have fin: "finite A" by fact 
  have empty: "P {}" by fact
  have step: "⋀b A. ⟦finite A; ∀a∈A. a < b; P A⟧ ⟹ P (insert b A)" by fact
  show "P A"
  proof (cases "A = {}")
    assume "A = {}" 
    then show "P A" using ‹P {}› by simp
  next
    let ?B = "A - {Max A}" 
    let ?A = "insert (Max A) ?B"
    have "finite ?B" using ‹finite A› by simp
    assume "A ≠ {}"
    with ‹finite A› have "Max A : A" by auto
    then have A: "?A = A" using insert_Diff_single insert_absorb by auto
    then have "P ?B" using ‹P {}› step IH [of ?B] by blast
    moreover 
    have "∀a∈?B. a < Max A" using Max_ge [OF ‹finite A›] by fastforce
    ultimately show "P A" using A insert_Diff_single step [OF ‹finite ?B›] by fastforce
  qed
qed

lemma finite_linorder_min_induct [consumes 1, case_names empty insert]:
  "⟦finite A; P {}; ⋀b A. ⟦finite A; ∀a∈A. b < a; P A⟧ ⟹ P (insert b A)⟧ ⟹ P A"
  by (rule linorder.finite_linorder_max_induct [OF dual_linorder])

lemma Least_Min:
  assumes "finite {a. P a}" and "∃a. P a"
  shows "(LEAST a. P a) = Min {a. P a}"
proof -
  { fix A :: "'a set"
    assume A: "finite A" "A ≠ {}"
    have "(LEAST a. a ∈ A) = Min A"
    using A proof (induct A rule: finite_ne_induct)
      case singleton show ?case by (rule Least_equality) simp_all
    next
      case (insert a A)
      have "(LEAST b. b = a ∨ b ∈ A) = min a (LEAST a. a ∈ A)"
        by (auto intro!: Least_equality simp add: min_def not_le Min_le_iff insert.hyps dest!: less_imp_le)
      with insert show ?case by simp
    qed
  } from this [of "{a. P a}"] assms show ?thesis by simp
qed

lemma infinite_growing:
  assumes "X ≠ {}"
  assumes *: "⋀x. x ∈ X ⟹ ∃y∈X. y > x"
  shows "¬ finite X"
proof
  assume "finite X"
  with ‹X ≠ {}› have "Max X ∈ X" "∀x∈X. x ≤ Max X"
    by auto
  with *[of "Max X"] show False
    by auto
qed

end

context linordered_ab_semigroup_add
begin

lemma add_Min_commute:
  fixes k
  assumes "finite N" and "N ≠ {}"
  shows "k + Min N = Min {k + m | m. m ∈ N}"
proof -
  have "⋀x y. k + min x y = min (k + x) (k + y)"
    by (simp add: min_def not_le)
      (blast intro: antisym less_imp_le add_left_mono)
  with assms show ?thesis
    using hom_Min_commute [of "plus k" N]
    by simp (blast intro: arg_cong [where f = Min])
qed

lemma add_Max_commute:
  fixes k
  assumes "finite N" and "N ≠ {}"
  shows "k + Max N = Max {k + m | m. m ∈ N}"
proof -
  have "⋀x y. k + max x y = max (k + x) (k + y)"
    by (simp add: max_def not_le)
      (blast intro: antisym less_imp_le add_left_mono)
  with assms show ?thesis
    using hom_Max_commute [of "plus k" N]
    by simp (blast intro: arg_cong [where f = Max])
qed

end

context linordered_ab_group_add
begin

lemma minus_Max_eq_Min [simp]:
  "finite S ⟹ S ≠ {} ⟹ - Max S = Min (uminus ` S)"
  by (induct S rule: finite_ne_induct) (simp_all add: minus_max_eq_min)

lemma minus_Min_eq_Max [simp]:
  "finite S ⟹ S ≠ {} ⟹ - Min S = Max (uminus ` S)"
  by (induct S rule: finite_ne_induct) (simp_all add: minus_min_eq_max)

end

context complete_linorder
begin

lemma Min_Inf:
  assumes "finite A" and "A ≠ {}"
  shows "Min A = Inf A"
proof -
  from assms obtain b B where "A = insert b B" and "finite B" by auto
  then show ?thesis
    by (simp add: Min.eq_fold complete_linorder_inf_min [symmetric] inf_Inf_fold_inf inf.commute [of b])
qed

lemma Max_Sup:
  assumes "finite A" and "A ≠ {}"
  shows "Max A = Sup A"
proof -
  from assms obtain b B where "A = insert b B" and "finite B" by auto
  then show ?thesis
    by (simp add: Max.eq_fold complete_linorder_sup_max [symmetric] sup_Sup_fold_sup sup.commute [of b])
qed

end


subsection ‹Arg Min›

definition is_arg_min :: "('a ⇒ 'b::ord) ⇒ ('a ⇒ bool) ⇒ 'a ⇒ bool" where
"is_arg_min f P x = (P x ∧ ¬(∃y. P y ∧ f y < f x))"

definition arg_min :: "('a ⇒ 'b::ord) ⇒ ('a ⇒ bool) ⇒ 'a" where
"arg_min f P = (SOME x. is_arg_min f P x)"

abbreviation arg_min_on :: "('a ⇒ 'b::ord) ⇒ 'a set ⇒ 'a" where
"arg_min_on f S ≡ arg_min f (λx. x ∈ S)"

syntax
  "_arg_min" :: "('a ⇒ 'b) ⇒ pttrn ⇒ bool ⇒ 'a"
    ("(3ARG'_MIN _ _./ _)" [0, 0, 10] 10)
translations
  "ARG_MIN f x. P"  "CONST arg_min f (λx. P)"

lemma is_arg_min_linorder: fixes f :: "'a ⇒ 'b :: linorder"
shows "is_arg_min f P x = (P x ∧ (∀y. P y ⟶ f x ≤ f y))"
by(auto simp add: is_arg_min_def)

lemma arg_minI:
  "⟦ P x;
    ⋀y. P y ⟹ ¬ f y < f x;
    ⋀x. ⟦ P x; ∀y. P y ⟶ ¬ f y < f x ⟧ ⟹ Q x ⟧
  ⟹ Q (arg_min f P)"
apply (simp add: arg_min_def is_arg_min_def)
apply (rule someI2_ex)
 apply blast
apply blast
done

lemma arg_min_equality:
  "⟦ P k; ⋀x. P x ⟹ f k ≤ f x ⟧ ⟹ f (arg_min f P) = f k"
  for f :: "_ ⇒ 'a::order"
apply (rule arg_minI)
  apply assumption
 apply (simp add: less_le_not_le)
by (metis le_less)

lemma wf_linord_ex_has_least:
  "⟦ wf r; ∀x y. (x, y) ∈ r+ ⟷ (y, x) ∉ r*; P k ⟧
   ⟹ ∃x. P x ∧ (∀y. P y ⟶ (m x, m y) ∈ r*)"
apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
apply (drule_tac x = "m ` Collect P" in spec)
by force

lemma ex_has_least_nat: "P k ⟹ ∃x. P x ∧ (∀y. P y ⟶ m x ≤ m y)"
  for m :: "'a ⇒ nat"
apply (simp only: pred_nat_trancl_eq_le [symmetric])
apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
 apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le)
by assumption

lemma arg_min_nat_lemma:
  "P k ⟹ P(arg_min m P) ∧ (∀y. P y ⟶ m (arg_min m P) ≤ m y)"
  for m :: "'a ⇒ nat"
apply (simp add: arg_min_def is_arg_min_linorder)
apply (rule someI_ex)
apply (erule ex_has_least_nat)
done

lemmas arg_min_natI = arg_min_nat_lemma [THEN conjunct1]

lemma is_arg_min_arg_min_nat: fixes m :: "'a ⇒ nat"
shows "P x ⟹ is_arg_min m P (arg_min m P)"
by (metis arg_min_nat_lemma is_arg_min_linorder)

lemma arg_min_nat_le: "P x ⟹ m (arg_min m P) ≤ m x"
  for m :: "'a ⇒ nat"
by (rule arg_min_nat_lemma [THEN conjunct2, THEN spec, THEN mp])

lemma ex_min_if_finite:
  "⟦ finite S; S ≠ {} ⟧ ⟹ ∃m∈S. ¬(∃x∈S. x < (m::'a::order))"
by(induction rule: finite.induct) (auto intro: order.strict_trans)

lemma ex_is_arg_min_if_finite: fixes f :: "'a ⇒ 'b :: order"
shows "⟦ finite S; S ≠ {} ⟧ ⟹ ∃x. is_arg_min f (λx. x : S) x"
unfolding is_arg_min_def
using ex_min_if_finite[of "f ` S"]
by auto

lemma arg_min_SOME_Min:
  "finite S ⟹ arg_min_on f S = (SOME y. y ∈ S ∧ f y = Min(f ` S))"
unfolding arg_min_def is_arg_min_linorder
apply(rule arg_cong[where f = Eps])
apply (auto simp: fun_eq_iff intro: Min_eqI[symmetric])
done

lemma arg_min_if_finite: fixes f :: "'a ⇒ 'b :: order"
assumes "finite S" "S ≠ {}"
shows  "arg_min_on f S ∈ S" and "¬(∃x∈S. f x < f (arg_min_on f S))"
using ex_is_arg_min_if_finite[OF assms, of f]
unfolding arg_min_def is_arg_min_def
by(auto dest!: someI_ex)

lemma arg_min_least: fixes f :: "'a ⇒ 'b :: linorder"
shows "⟦ finite S;  S ≠ {};  y ∈ S ⟧ ⟹ f(arg_min_on f S) ≤ f y"
by(simp add: arg_min_SOME_Min inv_into_def2[symmetric] f_inv_into_f)

lemma arg_min_inj_eq: fixes f :: "'a ⇒ 'b :: order"
shows "⟦ inj_on f {x. P x}; P a; ∀y. P y ⟶ f a ≤ f y ⟧ ⟹ arg_min f P = a"
apply(simp add: arg_min_def is_arg_min_def)
apply(rule someI2[of _ a])
 apply (simp add: less_le_not_le)
by (metis inj_on_eq_iff less_le mem_Collect_eq)


subsection ‹Arg Max›

definition is_arg_max :: "('a ⇒ 'b::ord) ⇒ ('a ⇒ bool) ⇒ 'a ⇒ bool" where
"is_arg_max f P x = (P x ∧ ¬(∃y. P y ∧ f y > f x))"

definition arg_max :: "('a ⇒ 'b::ord) ⇒ ('a ⇒ bool) ⇒ 'a" where
"arg_max f P = (SOME x. is_arg_max f P x)"

abbreviation arg_max_on :: "('a ⇒ 'b::ord) ⇒ 'a set ⇒ 'a" where
"arg_max_on f S ≡ arg_max f (λx. x ∈ S)"

syntax
  "_arg_max" :: "('a ⇒ 'b) ⇒ pttrn ⇒ bool ⇒ 'a"
    ("(3ARG'_MAX _ _./ _)" [0, 0, 10] 10)
translations
  "ARG_MAX f x. P"  "CONST arg_max f (λx. P)"

lemma is_arg_max_linorder: fixes f :: "'a ⇒ 'b :: linorder"
shows "is_arg_max f P x = (P x ∧ (∀y. P y ⟶ f x ≥ f y))"
by(auto simp add: is_arg_max_def)

lemma arg_maxI:
  "P x ⟹
    (⋀y. P y ⟹ ¬ f y > f x) ⟹
    (⋀x. P x ⟹ ∀y. P y ⟶ ¬ f y > f x ⟹ Q x) ⟹
    Q (arg_max f P)"
apply (simp add: arg_max_def is_arg_max_def)
apply (rule someI2_ex)
 apply blast
apply blast
done

lemma arg_max_equality:
  "⟦ P k; ⋀x. P x ⟹ f x ≤ f k ⟧ ⟹ f (arg_max f P) = f k"
  for f :: "_ ⇒ 'a::order"
apply (rule arg_maxI [where f = f])
  apply assumption
 apply (simp add: less_le_not_le)
by (metis le_less)

lemma ex_has_greatest_nat_lemma:
  "P k ⟹ ∀x. P x ⟶ (∃y. P y ∧ ¬ f y ≤ f x) ⟹ ∃y. P y ∧ ¬ f y < f k + n"
  for f :: "'a ⇒ nat"
by (induct n) (force simp: le_Suc_eq)+

lemma ex_has_greatest_nat:
  "P k ⟹ ∀y. P y ⟶ f y < b ⟹ ∃x. P x ∧ (∀y. P y ⟶ f y ≤ f x)"
  for f :: "'a ⇒ nat"
apply (rule ccontr)
apply (cut_tac P = P and n = "b - f k" in ex_has_greatest_nat_lemma)
  apply (subgoal_tac [3] "f k ≤ b")
   apply auto
done

lemma arg_max_nat_lemma:
  "⟦ P k;  ∀y. P y ⟶ f y < b ⟧
  ⟹ P (arg_max f P) ∧ (∀y. P y ⟶ f y ≤ f (arg_max f P))"
  for f :: "'a ⇒ nat"
apply (simp add: arg_max_def is_arg_max_linorder)
apply (rule someI_ex)
apply (erule (1) ex_has_greatest_nat)
done

lemmas arg_max_natI = arg_max_nat_lemma [THEN conjunct1]

lemma arg_max_nat_le: "P x ⟹ ∀y. P y ⟶ f y < b ⟹ f x ≤ f (arg_max f P)"
  for f :: "'a ⇒ nat"
by (blast dest: arg_max_nat_lemma [THEN conjunct2, THEN spec, of P])

end