Theory Lattice

(*  Title:      HOL/Algebra/Lattice.thy
    Author:     Clemens Ballarin, started 7 November 2003
    Copyright:  Clemens Ballarin

Most congruence rules by Stephan Hohe.
With additional contributions from Alasdair Armstrong and Simon Foster.
*)

theory Lattice
imports Order
begin

section ‹Lattices›
  
subsection ‹Supremum and infimum›

definition
  sup :: "[_, 'a set] => 'a" (‹(‹open_block notation=‹prefix ⨆››⨆ı_)› [90] 90)
  where "⨆⇘L⇙A = (SOME x. least L x (Upper L A))"

definition
  inf :: "[_, 'a set] => 'a" (‹(‹open_block notation=‹prefix ⨅››⨅ı_)› [90] 90)
  where "⨅⇘L⇙A = (SOME x. greatest L x (Lower L A))"

definition supr :: 
  "('a, 'b) gorder_scheme ⇒ 'c set ⇒ ('c ⇒ 'a) ⇒ 'a "
  where "supr L A f = ⨆⇘L⇙(f ` A)"

definition infi :: 
  "('a, 'b) gorder_scheme ⇒ 'c set ⇒ ('c ⇒ 'a) ⇒ 'a "
  where "infi L A f = ⨅⇘L⇙(f ` A)"

syntax
  "_inf1"     :: "('a, 'b) gorder_scheme ⇒ pttrns ⇒ 'a ⇒ 'a"
    (‹(‹indent=3 notation=‹binder IINF››IINFı _./ _)› [0, 10] 10)
  "_inf"      :: "('a, 'b) gorder_scheme ⇒ pttrn ⇒ 'c set ⇒ 'a ⇒ 'a"
    (‹(‹indent=3 notation=‹binder IINF››IINFı _:_./ _)› [0, 0, 10] 10)
  "_sup1"     :: "('a, 'b) gorder_scheme ⇒ pttrns ⇒ 'a ⇒ 'a"
    (‹(‹indent=3 notation=‹binder SSUP››SSUPı _./ _)› [0, 10] 10)
  "_sup"      :: "('a, 'b) gorder_scheme ⇒ pttrn ⇒ 'c set ⇒ 'a ⇒ 'a"
    (‹(‹indent=3 notation=‹binder SSUP››SSUPı _:_./ _)› [0, 0, 10] 10)
syntax_consts
  "_inf1" "_inf" == infi and
  "_sup1" "_sup" == supr
translations
  "IINF⇘L⇙ x. B"     == "CONST infi L CONST UNIV (%x. B)"
  "IINF⇘L⇙ x:A. B"   == "CONST infi L A (%x. B)"
  "SSUP⇘L⇙ x. B"     == "CONST supr L CONST UNIV (%x. B)"
  "SSUP⇘L⇙ x:A. B"   == "CONST supr L A (%x. B)"

definition
  join :: "[_, 'a, 'a] => 'a" (infixl ‹⊔ı› 65)
  where "x ⊔⇘L⇙ y = ⨆⇘L⇙{x, y}"

definition
  meet :: "[_, 'a, 'a] => 'a" (infixl ‹⊓ı› 70)
  where "x ⊓⇘L⇙ y = ⨅⇘L⇙{x, y}"

definition
  LEAST_FP :: "('a, 'b) gorder_scheme ⇒ ('a ⇒ 'a) ⇒ 'a" (‹LFPı›) where
  "LEAST_FP L f = ⨅⇘L⇙ {u ∈ carrier L. f u ⊑⇘L⇙ u}"    ― ‹least fixed point›

definition
  GREATEST_FP:: "('a, 'b) gorder_scheme ⇒ ('a ⇒ 'a) ⇒ 'a" (‹GFPı›) where
  "GREATEST_FP L f = ⨆⇘L⇙ {u ∈ carrier L. u ⊑⇘L⇙ f u}"    ― ‹greatest fixed point›


subsection ‹Dual operators›

lemma sup_dual [simp]: 
  "⨆⇘inv_gorder L⇙A = ⨅⇘L⇙A"
  by (simp add: sup_def inf_def)

lemma inf_dual [simp]: 
  "⨅⇘inv_gorder L⇙A = ⨆⇘L⇙A"
  by (simp add: sup_def inf_def)

lemma join_dual [simp]:
  "p ⊔⇘inv_gorder L⇙ q = p ⊓⇘L⇙ q"
  by (simp add:join_def meet_def)

lemma meet_dual [simp]:
  "p ⊓⇘inv_gorder L⇙ q = p ⊔⇘L⇙ q"
  by (simp add:join_def meet_def)

lemma top_dual [simp]:
  "⊤⇘inv_gorder L⇙ = ⊥⇘L⇙"
  by (simp add: top_def bottom_def)

lemma bottom_dual [simp]:
  "⊥⇘inv_gorder L⇙ = ⊤⇘L⇙"
  by (simp add: top_def bottom_def)

lemma LFP_dual [simp]:
  "LEAST_FP (inv_gorder L) f = GREATEST_FP L f"
  by (simp add:LEAST_FP_def GREATEST_FP_def)

lemma GFP_dual [simp]:
  "GREATEST_FP (inv_gorder L) f = LEAST_FP L f"
  by (simp add:LEAST_FP_def GREATEST_FP_def)


subsection ‹Lattices›

locale weak_upper_semilattice = weak_partial_order +
  assumes sup_of_two_exists:
    "[| x ∈ carrier L; y ∈ carrier L |] ==> ∃s. least L s (Upper L {x, y})"

locale weak_lower_semilattice = weak_partial_order +
  assumes inf_of_two_exists:
    "[| x ∈ carrier L; y ∈ carrier L |] ==> ∃s. greatest L s (Lower L {x, y})"

locale weak_lattice = weak_upper_semilattice + weak_lower_semilattice

lemma (in weak_lattice) dual_weak_lattice:
  "weak_lattice (inv_gorder L)"
proof -
  interpret dual: weak_partial_order "inv_gorder L"
    by (metis dual_weak_order)
  show ?thesis
  proof qed (simp_all add: inf_of_two_exists sup_of_two_exists)
qed


subsubsection ‹Supremum›

lemma (in weak_upper_semilattice) joinI:
  "[| !!l. least L l (Upper L {x, y}) ==> P l; x ∈ carrier L; y ∈ carrier L |]
  ==> P (x ⊔ y)"
proof (unfold join_def sup_def)
  assume L: "x ∈ carrier L"  "y ∈ carrier L"
    and P: "!!l. least L l (Upper L {x, y}) ==> P l"
  with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
  with L show "P (SOME l. least L l (Upper L {x, y}))"
    by (fast intro: someI2 P)
qed

lemma (in weak_upper_semilattice) join_closed [simp]:
  "[| x ∈ carrier L; y ∈ carrier L |] ==> x ⊔ y ∈ carrier L"
  by (rule joinI) (rule least_closed)

lemma (in weak_upper_semilattice) join_cong_l:
  assumes carr: "x ∈ carrier L" "x' ∈ carrier L" "y ∈ carrier L"
    and xx': "x .= x'"
  shows "x ⊔ y .= x' ⊔ y"
proof (rule joinI, rule joinI)
  fix a b
  from xx' carr
      have seq: "{x, y} {.=} {x', y}" by (rule set_eq_pairI)

  assume leasta: "least L a (Upper L {x, y})"
  assume "least L b (Upper L {x', y})"
  with carr
      have leastb: "least L b (Upper L {x, y})"
      by (simp add: least_Upper_cong_r[OF _ _ seq])

  from leasta leastb
      show "a .= b" by (rule weak_least_unique)
qed (rule carr)+

lemma (in weak_upper_semilattice) join_cong_r:
  assumes carr: "x ∈ carrier L" "y ∈ carrier L" "y' ∈ carrier L"
    and yy': "y .= y'"
  shows "x ⊔ y .= x ⊔ y'"
proof (rule joinI, rule joinI)
  fix a b
  have "{x, y} = {y, x}" by fast
  also from carr yy'
      have "{y, x} {.=} {y', x}" by (intro set_eq_pairI)
  also have "{y', x} = {x, y'}" by fast
  finally
      have seq: "{x, y} {.=} {x, y'}" .

  assume leasta: "least L a (Upper L {x, y})"
  assume "least L b (Upper L {x, y'})"
  with carr
      have leastb: "least L b (Upper L {x, y})"
      by (simp add: least_Upper_cong_r[OF _ _ seq])

  from leasta leastb
      show "a .= b" by (rule weak_least_unique)
qed (rule carr)+

lemma (in weak_partial_order) sup_of_singletonI:      (* only reflexivity needed ? *)
  "x ∈ carrier L ==> least L x (Upper L {x})"
  by (rule least_UpperI) auto

lemma (in weak_partial_order) weak_sup_of_singleton [simp]:
  "x ∈ carrier L ==> ⨆{x} .= x"
  unfolding sup_def
  by (rule someI2) (auto intro: weak_least_unique sup_of_singletonI)

lemma (in weak_partial_order) sup_of_singleton_closed [simp]:
  "x ∈ carrier L ⟹ ⨆{x} ∈ carrier L"
  unfolding sup_def
  by (rule someI2) (auto intro: sup_of_singletonI)

text ‹Condition on ‹A›: supremum exists.›

lemma (in weak_upper_semilattice) sup_insertI:
  "[| !!s. least L s (Upper L (insert x A)) ==> P s;
  least L a (Upper L A); x ∈ carrier L; A ⊆ carrier L |]
  ==> P (⨆(insert x A))"
proof (unfold sup_def)
  assume L: "x ∈ carrier L"  "A ⊆ carrier L"
    and P: "!!l. least L l (Upper L (insert x A)) ==> P l"
    and least_a: "least L a (Upper L A)"
  from L least_a have La: "a ∈ carrier L" by simp
  from L sup_of_two_exists least_a
  obtain s where least_s: "least L s (Upper L {a, x})" by blast
  show "P (SOME l. least L l (Upper L (insert x A)))"
  proof (rule someI2)
    show "least L s (Upper L (insert x A))"
    proof (rule least_UpperI)
      fix z
      assume "z ∈ insert x A"
      then show "z ⊑ s"
      proof
        assume "z = x" then show ?thesis
          by (simp add: least_Upper_above [OF least_s] L La)
      next
        assume "z ∈ A"
        with L least_s least_a show ?thesis
          by (rule_tac le_trans [where y = a]) (auto dest: least_Upper_above)
      qed
    next
      fix y
      assume y: "y ∈ Upper L (insert x A)"
      show "s ⊑ y"
      proof (rule least_le [OF least_s], rule Upper_memI)
        fix z
        assume z: "z ∈ {a, x}"
        then show "z ⊑ y"
        proof
          have y': "y ∈ Upper L A"
            by (meson Upper_antimono in_mono subset_insertI y)
          assume "z = a"
          with y' least_a show ?thesis by (fast dest: least_le)
        next
          assume "z ∈ {x}"
          with y L show ?thesis by blast
        qed
      qed (rule Upper_closed [THEN subsetD, OF y])
    next
      from L show "insert x A ⊆ carrier L" by simp
      from least_s show "s ∈ carrier L" by simp
    qed
  qed (rule P)
qed

lemma (in weak_upper_semilattice) finite_sup_least:
  "[| finite A; A ⊆ carrier L; A ≠ {} |] ==> least L (⨆A) (Upper L A)"
proof (induct set: finite)
  case empty
  then show ?case by simp
next
  case (insert x A)
  show ?case
  proof (cases "A = {}")
    case True
    with insert show ?thesis
      by simp (simp add: least_cong [OF weak_sup_of_singleton] sup_of_singletonI)
        (* The above step is hairy; least_cong can make simp loop.
        Would want special version of simp to apply least_cong. *)
  next
    case False
    with insert have "least L (⨆A) (Upper L A)" by simp
    with _ show ?thesis
      by (rule sup_insertI) (simp_all add: insert [simplified])
  qed
qed

lemma (in weak_upper_semilattice) finite_sup_insertI:
  assumes P: "!!l. least L l (Upper L (insert x A)) ==> P l"
    and xA: "finite A"  "x ∈ carrier L"  "A ⊆ carrier L"
  shows "P (⨆ (insert x A))"
proof (cases "A = {}")
  case True with P and xA show ?thesis
    by (simp add: finite_sup_least)
next
  case False with P and xA show ?thesis
    by (simp add: sup_insertI finite_sup_least)
qed

lemma (in weak_upper_semilattice) finite_sup_closed [simp]:
  "[| finite A; A ⊆ carrier L; A ≠ {} |] ==> ⨆A ∈ carrier L"
proof (induct set: finite)
  case empty then show ?case by simp
next
  case insert then show ?case
    by - (rule finite_sup_insertI, simp_all)
qed

lemma (in weak_upper_semilattice) join_left:
  "[| x ∈ carrier L; y ∈ carrier L |] ==> x ⊑ x ⊔ y"
  by (rule joinI [folded join_def]) (blast dest: least_mem)

lemma (in weak_upper_semilattice) join_right:
  "[| x ∈ carrier L; y ∈ carrier L |] ==> y ⊑ x ⊔ y"
  by (rule joinI [folded join_def]) (blast dest: least_mem)

lemma (in weak_upper_semilattice) sup_of_two_least:
  "[| x ∈ carrier L; y ∈ carrier L |] ==> least L (⨆{x, y}) (Upper L {x, y})"
proof (unfold sup_def)
  assume L: "x ∈ carrier L"  "y ∈ carrier L"
  with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
  with L show "least L (SOME z. least L z (Upper L {x, y})) (Upper L {x, y})"
  by (fast intro: someI2 weak_least_unique)  (* blast fails *)
qed

lemma (in weak_upper_semilattice) join_le:
  assumes sub: "x ⊑ z"  "y ⊑ z"
    and x: "x ∈ carrier L" and y: "y ∈ carrier L" and z: "z ∈ carrier L"
  shows "x ⊔ y ⊑ z"
proof (rule joinI [OF _ x y])
  fix s
  assume "least L s (Upper L {x, y})"
  with sub z show "s ⊑ z" by (fast elim: least_le intro: Upper_memI)
qed

lemma (in weak_lattice) weak_le_iff_meet:
  assumes "x ∈ carrier L" "y ∈ carrier L"
  shows "x ⊑ y ⟷ (x ⊔ y) .= y"
  by (meson assms(1) assms(2) join_closed join_le join_left join_right le_cong_r local.le_refl weak_le_antisym)
  
lemma (in weak_upper_semilattice) weak_join_assoc_lemma:
  assumes L: "x ∈ carrier L"  "y ∈ carrier L"  "z ∈ carrier L"
  shows "x ⊔ (y ⊔ z) .= ⨆{x, y, z}"
proof (rule finite_sup_insertI)
  ― ‹The textbook argument in Jacobson I, p 457›
  fix s
  assume sup: "least L s (Upper L {x, y, z})"
  show "x ⊔ (y ⊔ z) .= s"
  proof (rule weak_le_antisym)
    from sup L show "x ⊔ (y ⊔ z) ⊑ s"
      by (fastforce intro!: join_le elim: least_Upper_above)
  next
    from sup L show "s ⊑ x ⊔ (y ⊔ z)"
    by (erule_tac least_le)
      (blast intro!: Upper_memI intro: le_trans join_left join_right join_closed)
  qed (simp_all add: L least_closed [OF sup])
qed (simp_all add: L)

text ‹Commutativity holds for ‹=›.›

lemma join_comm:
  fixes L (structure)
  shows "x ⊔ y = y ⊔ x"
  by (unfold join_def) (simp add: insert_commute)

lemma (in weak_upper_semilattice) weak_join_assoc:
  assumes L: "x ∈ carrier L"  "y ∈ carrier L"  "z ∈ carrier L"
  shows "(x ⊔ y) ⊔ z .= x ⊔ (y ⊔ z)"
proof -
  (* FIXME: could be simplified by improved simp: uniform use of .=,
     omit [symmetric] in last step. *)
  have "(x ⊔ y) ⊔ z = z ⊔ (x ⊔ y)" by (simp only: join_comm)
  also from L have "... .= ⨆{z, x, y}" by (simp add: weak_join_assoc_lemma)
  also from L have "... = ⨆{x, y, z}" by (simp add: insert_commute)
  also from L have "... .= x ⊔ (y ⊔ z)" by (simp add: weak_join_assoc_lemma [symmetric])
  finally show ?thesis by (simp add: L)
qed


subsubsection ‹Infimum›

lemma (in weak_lower_semilattice) meetI:
  "[| !!i. greatest L i (Lower L {x, y}) ==> P i;
  x ∈ carrier L; y ∈ carrier L |]
  ==> P (x ⊓ y)"
proof (unfold meet_def inf_def)
  assume L: "x ∈ carrier L"  "y ∈ carrier L"
    and P: "!!g. greatest L g (Lower L {x, y}) ==> P g"
  with inf_of_two_exists obtain i where "greatest L i (Lower L {x, y})" by fast
  with L show "P (SOME g. greatest L g (Lower L {x, y}))"
  by (fast intro: someI2 weak_greatest_unique P)
qed

lemma (in weak_lower_semilattice) meet_closed [simp]:
  "[| x ∈ carrier L; y ∈ carrier L |] ==> x ⊓ y ∈ carrier L"
  by (rule meetI) (rule greatest_closed)

lemma (in weak_lower_semilattice) meet_cong_l:
  assumes carr: "x ∈ carrier L" "x' ∈ carrier L" "y ∈ carrier L"
    and xx': "x .= x'"
  shows "x ⊓ y .= x' ⊓ y"
proof (rule meetI, rule meetI)
  fix a b
  from xx' carr
      have seq: "{x, y} {.=} {x', y}" by (rule set_eq_pairI)

  assume greatesta: "greatest L a (Lower L {x, y})"
  assume "greatest L b (Lower L {x', y})"
  with carr
      have greatestb: "greatest L b (Lower L {x, y})"
      by (simp add: greatest_Lower_cong_r[OF _ _ seq])

  from greatesta greatestb
      show "a .= b" by (rule weak_greatest_unique)
qed (rule carr)+

lemma (in weak_lower_semilattice) meet_cong_r:
  assumes carr: "x ∈ carrier L" "y ∈ carrier L" "y' ∈ carrier L"
    and yy': "y .= y'"
  shows "x ⊓ y .= x ⊓ y'"
proof (rule meetI, rule meetI)
  fix a b
  have "{x, y} = {y, x}" by fast
  also from carr yy'
      have "{y, x} {.=} {y', x}" by (intro set_eq_pairI)
  also have "{y', x} = {x, y'}" by fast
  finally
      have seq: "{x, y} {.=} {x, y'}" .

  assume greatesta: "greatest L a (Lower L {x, y})"
  assume "greatest L b (Lower L {x, y'})"
  with carr
      have greatestb: "greatest L b (Lower L {x, y})"
      by (simp add: greatest_Lower_cong_r[OF _ _ seq])

  from greatesta greatestb
      show "a .= b" by (rule weak_greatest_unique)
qed (rule carr)+

lemma (in weak_partial_order) inf_of_singletonI:      (* only reflexivity needed ? *)
  "x ∈ carrier L ==> greatest L x (Lower L {x})"
  by (rule greatest_LowerI) auto

lemma (in weak_partial_order) weak_inf_of_singleton [simp]:
  "x ∈ carrier L ==> ⨅{x} .= x"
  unfolding inf_def
  by (rule someI2) (auto intro: weak_greatest_unique inf_of_singletonI)

lemma (in weak_partial_order) inf_of_singleton_closed:
  "x ∈ carrier L ==> ⨅{x} ∈ carrier L"
  unfolding inf_def
  by (rule someI2) (auto intro: inf_of_singletonI)

text ‹Condition on ‹A›: infimum exists.›

lemma (in weak_lower_semilattice) inf_insertI:
  "[| !!i. greatest L i (Lower L (insert x A)) ==> P i;
  greatest L a (Lower L A); x ∈ carrier L; A ⊆ carrier L |]
  ==> P (⨅(insert x A))"
proof (unfold inf_def)
  assume L: "x ∈ carrier L"  "A ⊆ carrier L"
    and P: "!!g. greatest L g (Lower L (insert x A)) ==> P g"
    and greatest_a: "greatest L a (Lower L A)"
  from L greatest_a have La: "a ∈ carrier L" by simp
  from L inf_of_two_exists greatest_a
  obtain i where greatest_i: "greatest L i (Lower L {a, x})" by blast
  show "P (SOME g. greatest L g (Lower L (insert x A)))"
  proof (rule someI2)
    show "greatest L i (Lower L (insert x A))"
    proof (rule greatest_LowerI)
      fix z
      assume "z ∈ insert x A"
      then show "i ⊑ z"
      proof
        assume "z = x" then show ?thesis
          by (simp add: greatest_Lower_below [OF greatest_i] L La)
      next
        assume "z ∈ A"
        with L greatest_i greatest_a show ?thesis
          by (rule_tac le_trans [where y = a]) (auto dest: greatest_Lower_below)
      qed
    next
      fix y
      assume y: "y ∈ Lower L (insert x A)"
      show "y ⊑ i"
      proof (rule greatest_le [OF greatest_i], rule Lower_memI)
        fix z
        assume z: "z ∈ {a, x}"
        then show "y ⊑ z"
        proof
          have y': "y ∈ Lower L A"
            by (meson Lower_antimono in_mono subset_insertI y)
          assume "z = a"
          with y' greatest_a show ?thesis by (fast dest: greatest_le)
        next
          assume "z ∈ {x}"
          with y L show ?thesis by blast
        qed
      qed (rule Lower_closed [THEN subsetD, OF y])
    next
      from L show "insert x A ⊆ carrier L" by simp
      from greatest_i show "i ∈ carrier L" by simp
    qed
  qed (rule P)
qed

lemma (in weak_lower_semilattice) finite_inf_greatest:
  "[| finite A; A ⊆ carrier L; A ≠ {} |] ==> greatest L (⨅A) (Lower L A)"
proof (induct set: finite)
  case empty then show ?case by simp
next
  case (insert x A)
  show ?case
  proof (cases "A = {}")
    case True
    with insert show ?thesis
      by simp (simp add: greatest_cong [OF weak_inf_of_singleton]
        inf_of_singleton_closed inf_of_singletonI)
  next
    case False
    from insert show ?thesis
    proof (rule_tac inf_insertI)
      from False insert show "greatest L (⨅A) (Lower L A)" by simp
    qed simp_all
  qed
qed

lemma (in weak_lower_semilattice) finite_inf_insertI:
  assumes P: "!!i. greatest L i (Lower L (insert x A)) ==> P i"
    and xA: "finite A"  "x ∈ carrier L"  "A ⊆ carrier L"
  shows "P (⨅ (insert x A))"
proof (cases "A = {}")
  case True with P and xA show ?thesis
    by (simp add: finite_inf_greatest)
next
  case False with P and xA show ?thesis
    by (simp add: inf_insertI finite_inf_greatest)
qed

lemma (in weak_lower_semilattice) finite_inf_closed [simp]:
  "[| finite A; A ⊆ carrier L; A ≠ {} |] ==> ⨅A ∈ carrier L"
proof (induct set: finite)
  case empty then show ?case by simp
next
  case insert then show ?case
    by (rule_tac finite_inf_insertI) (simp_all)
qed

lemma (in weak_lower_semilattice) meet_left:
  "[| x ∈ carrier L; y ∈ carrier L |] ==> x ⊓ y ⊑ x"
  by (rule meetI [folded meet_def]) (blast dest: greatest_mem)

lemma (in weak_lower_semilattice) meet_right:
  "[| x ∈ carrier L; y ∈ carrier L |] ==> x ⊓ y ⊑ y"
  by (rule meetI [folded meet_def]) (blast dest: greatest_mem)

lemma (in weak_lower_semilattice) inf_of_two_greatest:
  "[| x ∈ carrier L; y ∈ carrier L |] ==>
  greatest L (⨅{x, y}) (Lower L {x, y})"
proof (unfold inf_def)
  assume L: "x ∈ carrier L"  "y ∈ carrier L"
  with inf_of_two_exists obtain s where "greatest L s (Lower L {x, y})" by fast
  with L
  show "greatest L (SOME z. greatest L z (Lower L {x, y})) (Lower L {x, y})"
  by (fast intro: someI2 weak_greatest_unique)  (* blast fails *)
qed

lemma (in weak_lower_semilattice) meet_le:
  assumes sub: "z ⊑ x"  "z ⊑ y"
    and x: "x ∈ carrier L" and y: "y ∈ carrier L" and z: "z ∈ carrier L"
  shows "z ⊑ x ⊓ y"
proof (rule meetI [OF _ x y])
  fix i
  assume "greatest L i (Lower L {x, y})"
  with sub z show "z ⊑ i" by (fast elim: greatest_le intro: Lower_memI)
qed

lemma (in weak_lattice) weak_le_iff_join:
  assumes "x ∈ carrier L" "y ∈ carrier L"
  shows "x ⊑ y ⟷ x .= (x ⊓ y)"
  by (meson assms(1) assms(2) local.le_refl local.le_trans meet_closed meet_le meet_left meet_right weak_le_antisym weak_refl)
  
lemma (in weak_lower_semilattice) weak_meet_assoc_lemma:
  assumes L: "x ∈ carrier L"  "y ∈ carrier L"  "z ∈ carrier L"
  shows "x ⊓ (y ⊓ z) .= ⨅{x, y, z}"
proof (rule finite_inf_insertI)
  txt ‹The textbook argument in Jacobson I, p 457›
  fix i
  assume inf: "greatest L i (Lower L {x, y, z})"
  show "x ⊓ (y ⊓ z) .= i"
  proof (rule weak_le_antisym)
    from inf L show "i ⊑ x ⊓ (y ⊓ z)"
      by (fastforce intro!: meet_le elim: greatest_Lower_below)
  next
    from inf L show "x ⊓ (y ⊓ z) ⊑ i"
    by (erule_tac greatest_le)
      (blast intro!: Lower_memI intro: le_trans meet_left meet_right meet_closed)
  qed (simp_all add: L greatest_closed [OF inf])
qed (simp_all add: L)

lemma meet_comm:
  fixes L (structure)
  shows "x ⊓ y = y ⊓ x"
  by (unfold meet_def) (simp add: insert_commute)

lemma (in weak_lower_semilattice) weak_meet_assoc:
  assumes L: "x ∈ carrier L"  "y ∈ carrier L"  "z ∈ carrier L"
  shows "(x ⊓ y) ⊓ z .= x ⊓ (y ⊓ z)"
proof -
  (* FIXME: improved simp, see weak_join_assoc above *)
  have "(x ⊓ y) ⊓ z = z ⊓ (x ⊓ y)" by (simp only: meet_comm)
  also from L have "... .= ⨅ {z, x, y}" by (simp add: weak_meet_assoc_lemma)
  also from L have "... = ⨅ {x, y, z}" by (simp add: insert_commute)
  also from L have "... .= x ⊓ (y ⊓ z)" by (simp add: weak_meet_assoc_lemma [symmetric])
  finally show ?thesis by (simp add: L)
qed

text ‹Total orders are lattices.›

sublocale weak_total_order ⊆ weak?: weak_lattice
proof
  fix x y
  assume L: "x ∈ carrier L"  "y ∈ carrier L"
  show "∃s. least L s (Upper L {x, y})"
  proof -
    note total L
    moreover
    {
      assume "x ⊑ y"
      with L have "least L y (Upper L {x, y})"
        by (rule_tac least_UpperI) auto
    }
    moreover
    {
      assume "y ⊑ x"
      with L have "least L x (Upper L {x, y})"
        by (rule_tac least_UpperI) auto
    }
    ultimately show ?thesis by blast
  qed
next
  fix x y
  assume L: "x ∈ carrier L"  "y ∈ carrier L"
  show "∃i. greatest L i (Lower L {x, y})"
  proof -
    note total L
    moreover
    {
      assume "y ⊑ x"
      with L have "greatest L y (Lower L {x, y})"
        by (rule_tac greatest_LowerI) auto
    }
    moreover
    {
      assume "x ⊑ y"
      with L have "greatest L x (Lower L {x, y})"
        by (rule_tac greatest_LowerI) auto
    }
    ultimately show ?thesis by blast
  qed
qed


subsection ‹Weak Bounded Lattices›

locale weak_bounded_lattice = 
  weak_lattice + 
  weak_partial_order_bottom + 
  weak_partial_order_top
begin

lemma bottom_meet: "x ∈ carrier L ⟹ ⊥ ⊓ x .= ⊥"
  by (metis bottom_least least_def meet_closed meet_left weak_le_antisym)

lemma bottom_join: "x ∈ carrier L ⟹ ⊥ ⊔ x .= x"
  by (metis bottom_least join_closed join_le join_right le_refl least_def weak_le_antisym)

lemma bottom_weak_eq:
  "⟦ b ∈ carrier L; ⋀ x. x ∈ carrier L ⟹ b ⊑ x ⟧ ⟹ b .= ⊥"
  by (metis bottom_closed bottom_lower weak_le_antisym)

lemma top_join: "x ∈ carrier L ⟹ ⊤ ⊔ x .= ⊤"
  by (metis join_closed join_left top_closed top_higher weak_le_antisym)

lemma top_meet: "x ∈ carrier L ⟹ ⊤ ⊓ x .= x"
  by (metis le_refl meet_closed meet_le meet_right top_closed top_higher weak_le_antisym)

lemma top_weak_eq:  "⟦ t ∈ carrier L; ⋀ x. x ∈ carrier L ⟹ x ⊑ t ⟧ ⟹ t .= ⊤"
  by (metis top_closed top_higher weak_le_antisym)

end

sublocale weak_bounded_lattice ⊆ weak_partial_order ..


subsection ‹Lattices where ‹eq› is the Equality›

locale upper_semilattice = partial_order +
  assumes sup_of_two_exists:
    "[| x ∈ carrier L; y ∈ carrier L |] ==> ∃s. least L s (Upper L {x, y})"

sublocale upper_semilattice ⊆ weak?: weak_upper_semilattice
  by unfold_locales (rule sup_of_two_exists)

locale lower_semilattice = partial_order +
  assumes inf_of_two_exists:
    "[| x ∈ carrier L; y ∈ carrier L |] ==> ∃s. greatest L s (Lower L {x, y})"

sublocale lower_semilattice ⊆ weak?: weak_lower_semilattice
  by unfold_locales (rule inf_of_two_exists)

locale lattice = upper_semilattice + lower_semilattice

sublocale lattice ⊆ weak_lattice ..

lemma (in lattice) dual_lattice:
  "lattice (inv_gorder L)"
proof -
  interpret dual: weak_lattice "inv_gorder L"
    by (metis dual_weak_lattice)

  show ?thesis
    apply (unfold_locales)
    apply (simp_all add: inf_of_two_exists sup_of_two_exists)
    apply (rule eq_is_equal)
  done
qed
  
lemma (in lattice) le_iff_join:
  assumes "x ∈ carrier L" "y ∈ carrier L"
  shows "x ⊑ y ⟷ x = (x ⊓ y)"
  by (simp add: assms(1) assms(2) eq_is_equal weak_le_iff_join)

lemma (in lattice) le_iff_meet:
  assumes "x ∈ carrier L" "y ∈ carrier L"
  shows "x ⊑ y ⟷ (x ⊔ y) = y"
  by (simp add: assms eq_is_equal weak_le_iff_meet)

text ‹ Total orders are lattices. ›

sublocale total_order ⊆ weak?: lattice
  by standard (auto intro: weak.weak.sup_of_two_exists weak.weak.inf_of_two_exists)
    
text ‹Functions that preserve joins and meets›
  
definition join_pres :: "('a, 'c) gorder_scheme ⇒ ('b, 'd) gorder_scheme ⇒ ('a ⇒ 'b) ⇒ bool" where
"join_pres X Y f ≡ lattice X ∧ lattice Y ∧ (∀ x ∈ carrier X. ∀ y ∈ carrier X. f (x ⊔⇘X⇙ y) = f x ⊔⇘Y⇙ f y)"

definition meet_pres :: "('a, 'c) gorder_scheme ⇒ ('b, 'd) gorder_scheme ⇒ ('a ⇒ 'b) ⇒ bool" where
"meet_pres X Y f ≡ lattice X ∧ lattice Y ∧ (∀ x ∈ carrier X. ∀ y ∈ carrier X. f (x ⊓⇘X⇙ y) = f x ⊓⇘Y⇙ f y)"

lemma join_pres_isotone:
  assumes "f ∈ carrier X → carrier Y" "join_pres X Y f"
  shows "isotone X Y f"
proof (rule isotoneI)
  show "weak_partial_order X" "weak_partial_order Y"
    using assms unfolding join_pres_def lattice_def upper_semilattice_def lower_semilattice_def
    by (meson partial_order.axioms(1))+
  show "⋀x y. ⟦x ∈ carrier X; y ∈ carrier X; x ⊑⇘X⇙ y⟧ ⟹ f x ⊑⇘Y⇙ f y"
    by (metis (no_types, lifting) PiE assms join_pres_def lattice.le_iff_meet)
qed

lemma meet_pres_isotone:
  assumes "f ∈ carrier X → carrier Y" "meet_pres X Y f"
  shows "isotone X Y f"
proof (rule isotoneI)
  show "weak_partial_order X" "weak_partial_order Y"
    using assms unfolding meet_pres_def lattice_def upper_semilattice_def lower_semilattice_def
    by (meson partial_order.axioms(1))+
  show "⋀x y. ⟦x ∈ carrier X; y ∈ carrier X; x ⊑⇘X⇙ y⟧ ⟹ f x ⊑⇘Y⇙ f y"
    by (metis (no_types, lifting) PiE assms lattice.le_iff_join meet_pres_def)
qed


subsection ‹Bounded Lattices›

locale bounded_lattice = 
  lattice + 
  weak_partial_order_bottom + 
  weak_partial_order_top

sublocale bounded_lattice ⊆ weak_bounded_lattice ..

context bounded_lattice
begin

lemma bottom_eq:  
  "⟦ b ∈ carrier L; ⋀ x. x ∈ carrier L ⟹ b ⊑ x ⟧ ⟹ b = ⊥"
  by (metis bottom_closed bottom_lower le_antisym)

lemma top_eq:  "⟦ t ∈ carrier L; ⋀ x. x ∈ carrier L ⟹ x ⊑ t ⟧ ⟹ t = ⊤"
  by (metis le_antisym top_closed top_higher)

end

hide_const (open) Lattice.inf
hide_const (open) Lattice.sup

end