# Theory Complete_Lattice

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

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

theory Complete_Lattice
imports Lattice
begin

section ‹Complete Lattices›

locale weak_complete_lattice = weak_partial_order +
assumes sup_exists:
"[| A ⊆ carrier L |] ==> ∃s. least L s (Upper L A)"
and inf_exists:
"[| A ⊆ carrier L |] ==> ∃i. greatest L i (Lower L A)"

sublocale weak_complete_lattice ⊆ weak_lattice
proof
fix x y
assume a: "x ∈ carrier L" "y ∈ carrier L"
thus "∃s. is_lub L s {x, y}"
by (rule_tac sup_exists[of "{x, y}"], auto)
from a show "∃s. is_glb L s {x, y}"
by (rule_tac inf_exists[of "{x, y}"], auto)
qed

text ‹Introduction rule: the usual definition of complete lattice›

lemma (in weak_partial_order) weak_complete_latticeI:
assumes sup_exists:
"!!A. [| A ⊆ carrier L |] ==> ∃s. least L s (Upper L A)"
and inf_exists:
"!!A. [| A ⊆ carrier L |] ==> ∃i. greatest L i (Lower L A)"
shows "weak_complete_lattice L"
by standard (auto intro: sup_exists inf_exists)

lemma (in weak_complete_lattice) dual_weak_complete_lattice:
"weak_complete_lattice (inv_gorder L)"
proof -
interpret dual: weak_lattice "inv_gorder L"
by (metis dual_weak_lattice)
show ?thesis
by (unfold_locales) (simp_all add:inf_exists sup_exists)
qed

lemma (in weak_complete_lattice) supI:
"[| !!l. least L l (Upper L A) ==> P l; A ⊆ carrier L |]
==> P (⨆A)"
proof (unfold sup_def)
assume L: "A ⊆ carrier L"
and P: "!!l. least L l (Upper L A) ==> P l"
with sup_exists obtain s where "least L s (Upper L A)" by blast
with L show "P (SOME l. least L l (Upper L A))"
by (fast intro: someI2 weak_least_unique P)
qed

lemma (in weak_complete_lattice) sup_closed [simp]:
"A ⊆ carrier L ==> ⨆A ∈ carrier L"
by (rule supI) simp_all

lemma (in weak_complete_lattice) sup_cong:
assumes "A ⊆ carrier L" "B ⊆ carrier L" "A {.=} B"
shows "⨆ A .= ⨆ B"
proof -
have "⋀ x. is_lub L x A ⟷ is_lub L x B"
by (rule least_Upper_cong_r, simp_all add: assms)
moreover have "⨆ B ∈ carrier L"
by (simp add: assms(2))
ultimately show ?thesis
by (simp add: sup_def)
qed

sublocale weak_complete_lattice ⊆ weak_bounded_lattice
apply (unfold_locales)
apply (metis Upper_empty empty_subsetI sup_exists)
apply (metis Lower_empty empty_subsetI inf_exists)
done

lemma (in weak_complete_lattice) infI:
"[| !!i. greatest L i (Lower L A) ==> P i; A ⊆ carrier L |]
==> P (⨅A)"
proof (unfold inf_def)
assume L: "A ⊆ carrier L"
and P: "!!l. greatest L l (Lower L A) ==> P l"
with inf_exists obtain s where "greatest L s (Lower L A)" by blast
with L show "P (SOME l. greatest L l (Lower L A))"
by (fast intro: someI2 weak_greatest_unique P)
qed

lemma (in weak_complete_lattice) inf_closed [simp]:
"A ⊆ carrier L ==> ⨅A ∈ carrier L"
by (rule infI) simp_all

lemma (in weak_complete_lattice) inf_cong:
assumes "A ⊆ carrier L" "B ⊆ carrier L" "A {.=} B"
shows "⨅ A .= ⨅ B"
proof -
have "⋀ x. is_glb L x A ⟷ is_glb L x B"
by (rule greatest_Lower_cong_r, simp_all add: assms)
moreover have "⨅ B ∈ carrier L"
by (simp add: assms(2))
ultimately show ?thesis
by (simp add: inf_def)
qed

theorem (in weak_partial_order) weak_complete_lattice_criterion1:
assumes top_exists: "∃g. greatest L g (carrier L)"
and inf_exists:
"⋀A. [| A ⊆ carrier L; A ≠ {} |] ==> ∃i. greatest L i (Lower L A)"
shows "weak_complete_lattice L"
proof (rule weak_complete_latticeI)
from top_exists obtain top where top: "greatest L top (carrier L)" ..
fix A
assume L: "A ⊆ carrier L"
let ?B = "Upper L A"
from L top have "top ∈ ?B" by (fast intro!: Upper_memI intro: greatest_le)
then have B_non_empty: "?B ≠ {}" by fast
have B_L: "?B ⊆ carrier L" by simp
from inf_exists [OF B_L B_non_empty]
obtain b where b_inf_B: "greatest L b (Lower L ?B)" ..
then have bcarr: "b ∈ carrier L"
by auto
have "least L b (Upper L A)"
proof (rule least_UpperI)
show "⋀x. x ∈ A ⟹ x ⊑ b"
by (meson L Lower_memI Upper_memD b_inf_B greatest_le set_mp)
show "⋀y. y ∈ Upper L A ⟹ b ⊑ y"
by (meson B_L b_inf_B greatest_Lower_below)
qed (use bcarr L in auto)
then show "∃s. least L s (Upper L A)" ..
next
fix A
assume L: "A ⊆ carrier L"
show "∃i. greatest L i (Lower L A)"
by (metis L Lower_empty inf_exists top_exists)
qed

text ‹Supremum›

declare (in partial_order) weak_sup_of_singleton [simp del]

lemma (in partial_order) sup_of_singleton [simp]:
"x ∈ carrier L ==> ⨆{x} = x"
using weak_sup_of_singleton unfolding eq_is_equal .

lemma (in upper_semilattice) join_assoc_lemma:
assumes L: "x ∈ carrier L"  "y ∈ carrier L"  "z ∈ carrier L"
shows "x ⊔ (y ⊔ z) = ⨆{x, y, z}"
using weak_join_assoc_lemma L unfolding eq_is_equal .

lemma (in upper_semilattice) join_assoc:
assumes L: "x ∈ carrier L"  "y ∈ carrier L"  "z ∈ carrier L"
shows "(x ⊔ y) ⊔ z = x ⊔ (y ⊔ z)"
using weak_join_assoc L unfolding eq_is_equal .

text ‹Infimum›

declare (in partial_order) weak_inf_of_singleton [simp del]

lemma (in partial_order) inf_of_singleton [simp]:
"x ∈ carrier L ==> ⨅{x} = x"
using weak_inf_of_singleton unfolding eq_is_equal .

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

lemma (in lower_semilattice) meet_assoc_lemma:
assumes L: "x ∈ carrier L"  "y ∈ carrier L"  "z ∈ carrier L"
shows "x ⊓ (y ⊓ z) = ⨅{x, y, z}"
using weak_meet_assoc_lemma L unfolding eq_is_equal .

lemma (in lower_semilattice) meet_assoc:
assumes L: "x ∈ carrier L"  "y ∈ carrier L"  "z ∈ carrier L"
shows "(x ⊓ y) ⊓ z = x ⊓ (y ⊓ z)"
using weak_meet_assoc L unfolding eq_is_equal .

subsection ‹Infimum Laws›

context weak_complete_lattice
begin

lemma inf_glb:
assumes "A ⊆ carrier L"
shows "greatest L (⨅A) (Lower L A)"
proof -
obtain i where "greatest L i (Lower L A)"
by (metis assms inf_exists)
thus ?thesis
by (metis inf_def someI_ex)
qed

lemma inf_lower:
assumes "A ⊆ carrier L" "x ∈ A"
shows "⨅A ⊑ x"
by (metis assms greatest_Lower_below inf_glb)

lemma inf_greatest:
assumes "A ⊆ carrier L" "z ∈ carrier L"
"(⋀x. x ∈ A ⟹ z ⊑ x)"
shows "z ⊑ ⨅A"
by (metis Lower_memI assms greatest_le inf_glb)

lemma weak_inf_empty [simp]: "⨅{} .= ⊤"
by (metis Lower_empty empty_subsetI inf_glb top_greatest weak_greatest_unique)

lemma weak_inf_carrier [simp]: "⨅carrier L .= ⊥"
by (metis bottom_weak_eq inf_closed inf_lower subset_refl)

lemma weak_inf_insert [simp]:
assumes "a ∈ carrier L" "A ⊆ carrier L"
shows "⨅insert a A .= a ⊓ ⨅A"
proof (rule weak_le_antisym)
show "⨅insert a A ⊑ a ⊓ ⨅A"
by (simp add: assms inf_lower local.inf_greatest meet_le)
show aA: "a ⊓ ⨅A ∈ carrier L"
using assms by simp
show "a ⊓ ⨅A ⊑ ⨅insert a A"
apply (rule inf_greatest)
using assms apply (simp_all add: aA)
by (meson aA inf_closed inf_lower local.le_trans meet_left meet_right subsetCE)
show "⨅insert a A ∈ carrier L"
using assms by (force intro: le_trans inf_closed meet_right meet_left inf_lower)
qed

subsection ‹Supremum Laws›

lemma sup_lub:
assumes "A ⊆ carrier L"
shows "least L (⨆A) (Upper L A)"
by (metis Upper_is_closed assms least_closed least_cong supI sup_closed sup_exists weak_least_unique)

lemma sup_upper:
assumes "A ⊆ carrier L" "x ∈ A"
shows "x ⊑ ⨆A"
by (metis assms least_Upper_above supI)

lemma sup_least:
assumes "A ⊆ carrier L" "z ∈ carrier L"
"(⋀x. x ∈ A ⟹ x ⊑ z)"
shows "⨆A ⊑ z"
by (metis Upper_memI assms least_le sup_lub)

lemma weak_sup_empty [simp]: "⨆{} .= ⊥"
by (metis Upper_empty bottom_least empty_subsetI sup_lub weak_least_unique)

lemma weak_sup_carrier [simp]: "⨆carrier L .= ⊤"
by (metis Lower_closed Lower_empty sup_closed sup_upper top_closed top_higher weak_le_antisym)

lemma weak_sup_insert [simp]:
assumes "a ∈ carrier L" "A ⊆ carrier L"
shows "⨆insert a A .= a ⊔ ⨆A"
proof (rule weak_le_antisym)
show aA: "a ⊔ ⨆A ∈ carrier L"
using assms by simp
show "⨆insert a A ⊑ a ⊔ ⨆A"
apply (rule sup_least)
using assms apply (simp_all add: aA)
by (meson aA join_left join_right local.le_trans subsetCE sup_closed sup_upper)
show "a ⊔ ⨆A ⊑ ⨆insert a A"
by (simp add: assms join_le local.sup_least sup_upper)
show "⨆insert a A ∈ carrier L"
using assms by (force intro: le_trans inf_closed meet_right meet_left inf_lower)
qed

end

subsection ‹Fixed points of a lattice›

definition "fps L f = {x ∈ carrier L. f x .=⇘L⇙ x}"

abbreviation "fpl L f ≡ L⦇carrier := fps L f⦈"

lemma (in weak_partial_order)
use_fps: "x ∈ fps L f ⟹ f x .= x"
by (simp add: fps_def)

lemma fps_carrier [simp]:
"fps L f ⊆ carrier L"
by (auto simp add: fps_def)

lemma (in weak_complete_lattice) fps_sup_image:
assumes "f ∈ carrier L → carrier L" "A ⊆ fps L f"
shows "⨆ (f  A) .= ⨆ A"
proof -
from assms(2) have AL: "A ⊆ carrier L"
by (auto simp add: fps_def)
show ?thesis
proof (rule sup_cong, simp_all add: AL)
from assms(1) AL show "f  A ⊆ carrier L"
by auto
then have *: "⋀b. ⟦A ⊆ {x ∈ carrier L. f x .= x}; b ∈ A⟧ ⟹ ∃a∈f  A. b .= a"
by (meson AL assms(2) image_eqI local.sym subsetCE use_fps)
from assms(2) show "f  A {.=} A"
by (auto simp add: fps_def intro: set_eqI2 [OF _ *])
qed
qed

lemma (in weak_complete_lattice) fps_idem:
assumes "f ∈ carrier L → carrier L" "Idem f"
shows "fps L f {.=} f  carrier L"
proof (rule set_eqI2)
show "⋀a. a ∈ fps L f ⟹ ∃b∈f  carrier L. a .= b"
using assms by (force simp add: fps_def intro: local.sym)
show "⋀b. b ∈ f  carrier L ⟹ ∃a∈fps L f. b .= a"
using assms by (force simp add: idempotent_def fps_def)
qed

context weak_complete_lattice
begin

lemma weak_sup_pre_fixed_point:
assumes "f ∈ carrier L → carrier L" "isotone L L f" "A ⊆ fps L f"
shows "(⨆⇘L⇙ A) ⊑⇘L⇙ f (⨆⇘L⇙ A)"
proof (rule sup_least)
from assms(3) show AL: "A ⊆ carrier L"
by (auto simp add: fps_def)
thus fA: "f (⨆A) ∈ carrier L"
by (simp add: assms funcset_carrier[of f L L])
fix x
assume xA: "x ∈ A"
hence "x ∈ fps L f"
using assms subsetCE by blast
hence "f x .=⇘L⇙ x"
by (auto simp add: fps_def)
moreover have "f x ⊑⇘L⇙ f (⨆⇘L⇙A)"
by (meson AL assms(2) subsetCE sup_closed sup_upper use_iso1 xA)
ultimately show "x ⊑⇘L⇙ f (⨆⇘L⇙A)"
by (meson AL fA assms(1) funcset_carrier le_cong local.refl subsetCE xA)
qed

lemma weak_sup_post_fixed_point:
assumes "f ∈ carrier L → carrier L" "isotone L L f" "A ⊆ fps L f"
shows "f (⨅⇘L⇙ A) ⊑⇘L⇙ (⨅⇘L⇙ A)"
proof (rule inf_greatest)
from assms(3) show AL: "A ⊆ carrier L"
by (auto simp add: fps_def)
thus fA: "f (⨅A) ∈ carrier L"
by (simp add: assms funcset_carrier[of f L L])
fix x
assume xA: "x ∈ A"
hence "x ∈ fps L f"
using assms subsetCE by blast
hence "f x .=⇘L⇙ x"
by (auto simp add: fps_def)
moreover have "f (⨅⇘L⇙A) ⊑⇘L⇙ f x"
by (meson AL assms(2) inf_closed inf_lower subsetCE use_iso1 xA)
ultimately show "f (⨅⇘L⇙A) ⊑⇘L⇙ x"
by (meson AL assms(1) fA funcset_carrier le_cong_r subsetCE xA)
qed

subsubsection ‹Least fixed points›

lemma LFP_closed [intro, simp]:
"LFP f ∈ carrier L"
by (metis (lifting) LEAST_FP_def inf_closed mem_Collect_eq subsetI)

lemma LFP_lowerbound:
assumes "x ∈ carrier L" "f x ⊑ x"
shows "LFP f ⊑ x"
by (auto intro:inf_lower assms simp add:LEAST_FP_def)

lemma LFP_greatest:
assumes "x ∈ carrier L"
"(⋀u. ⟦ u ∈ carrier L; f u ⊑ u ⟧ ⟹ x ⊑ u)"
shows "x ⊑ LFP f"
by (auto simp add:LEAST_FP_def intro:inf_greatest assms)

lemma LFP_lemma2:
assumes "Mono f" "f ∈ carrier L → carrier L"
shows "f (LFP f) ⊑ LFP f"
proof (rule LFP_greatest)
have f: "⋀x. x ∈ carrier L ⟹ f x ∈ carrier L"
using assms by (auto simp add: Pi_def)
with assms show "f (LFP f) ∈ carrier L"
by blast
show "⋀u. ⟦u ∈ carrier L; f u ⊑ u⟧ ⟹ f (LFP f) ⊑ u"
by (meson LFP_closed LFP_lowerbound assms(1) f local.le_trans use_iso1)
qed

lemma LFP_lemma3:
assumes "Mono f" "f ∈ carrier L → carrier L"
shows "LFP f ⊑ f (LFP f)"
using assms by (simp add: Pi_def) (metis LFP_closed LFP_lemma2 LFP_lowerbound assms(2) use_iso2)

lemma LFP_weak_unfold:
"⟦ Mono f; f ∈ carrier L → carrier L ⟧ ⟹ LFP f .= f (LFP f)"
by (auto intro: LFP_lemma2 LFP_lemma3 funcset_mem)

lemma LFP_fixed_point [intro]:
assumes "Mono f" "f ∈ carrier L → carrier L"
shows "LFP f ∈ fps L f"
proof -
have "f (LFP f) ∈ carrier L"
using assms(2) by blast
with assms show ?thesis
by (simp add: LFP_weak_unfold fps_def local.sym)
qed

lemma LFP_least_fixed_point:
assumes "Mono f" "f ∈ carrier L → carrier L" "x ∈ fps L f"
shows "LFP f ⊑ x"
using assms by (force intro: LFP_lowerbound simp add: fps_def)

lemma LFP_idem:
assumes "f ∈ carrier L → carrier L" "Mono f" "Idem f"
shows "LFP f .= (f ⊥)"
proof (rule weak_le_antisym)
from assms(1) show fb: "f ⊥ ∈ carrier L"
by (rule funcset_mem, simp)
from assms show mf: "LFP f ∈ carrier L"
by blast
show "LFP f ⊑ f ⊥"
proof -
have "f (f ⊥) .= f ⊥"
by (auto simp add: fps_def fb assms(3) idempotent)
moreover have "f (f ⊥) ∈ carrier L"
by (rule funcset_mem[of f "carrier L"], simp_all add: assms fb)
ultimately show ?thesis
by (auto intro: LFP_lowerbound simp add: fb)
qed
show "f ⊥ ⊑ LFP f"
proof -
have "f ⊥ ⊑ f (LFP f)"
by (auto intro: use_iso1[of _ f] simp add: assms)
moreover have "... .= LFP f"
using assms(1) assms(2) fps_def by force
moreover from assms(1) have "f (LFP f) ∈ carrier L"
by (auto)
ultimately show ?thesis
using fb by blast
qed
qed

subsubsection ‹Greatest fixed points›

lemma GFP_closed [intro, simp]:
"GFP f ∈ carrier L"
by (auto intro:sup_closed simp add:GREATEST_FP_def)

lemma GFP_upperbound:
assumes "x ∈ carrier L" "x ⊑ f x"
shows "x ⊑ GFP f"
by (auto intro:sup_upper assms simp add:GREATEST_FP_def)

lemma GFP_least:
assumes "x ∈ carrier L"
"(⋀u. ⟦ u ∈ carrier L; u ⊑ f u ⟧ ⟹ u ⊑ x)"
shows "GFP f ⊑ x"
by (auto simp add:GREATEST_FP_def intro:sup_least assms)

lemma GFP_lemma2:
assumes "Mono f" "f ∈ carrier L → carrier L"
shows "GFP f ⊑ f (GFP f)"
proof (rule GFP_least)
have f: "⋀x. x ∈ carrier L ⟹ f x ∈ carrier L"
using assms by (auto simp add: Pi_def)
with assms show "f (GFP f) ∈ carrier L"
by blast
show "⋀u. ⟦u ∈ carrier L; u ⊑ f u⟧ ⟹ u ⊑ f (GFP f)"
by (meson GFP_closed GFP_upperbound le_trans assms(1) f local.le_trans use_iso1)
qed

lemma GFP_lemma3:
assumes "Mono f" "f ∈ carrier L → carrier L"
shows "f (GFP f) ⊑ GFP f"
by (metis GFP_closed GFP_lemma2 GFP_upperbound assms funcset_mem use_iso2)

lemma GFP_weak_unfold:
"⟦ Mono f; f ∈ carrier L → carrier L ⟧ ⟹ GFP f .= f (GFP f)"
by (auto intro: GFP_lemma2 GFP_lemma3 funcset_mem)

lemma (in weak_complete_lattice) GFP_fixed_point [intro]:
assumes "Mono f" "f ∈ carrier L → carrier L"
shows "GFP f ∈ fps L f"
using assms
proof -
have "f (GFP f) ∈ carrier L"
using assms(2) by blast
with assms show ?thesis
by (simp add: GFP_weak_unfold fps_def local.sym)
qed

lemma GFP_greatest_fixed_point:
assumes "Mono f" "f ∈ carrier L → carrier L" "x ∈ fps L f"
shows "x ⊑ GFP f"
using assms
by (rule_tac GFP_upperbound, auto simp add: fps_def, meson PiE local.sym weak_refl)

lemma GFP_idem:
assumes "f ∈ carrier L → carrier L" "Mono f" "Idem f"
shows "GFP f .= (f ⊤)"
proof (rule weak_le_antisym)
from assms(1) show fb: "f ⊤ ∈ carrier L"
by (rule funcset_mem, simp)
from assms show mf: "GFP f ∈ carrier L"
by blast
show "f ⊤ ⊑ GFP f"
proof -
have "f (f ⊤) .= f ⊤"
by (auto simp add: fps_def fb assms(3) idempotent)
moreover have "f (f ⊤) ∈ carrier L"
by (rule funcset_mem[of f "carrier L"], simp_all add: assms fb)
ultimately show ?thesis
by (rule_tac GFP_upperbound, simp_all add: fb local.sym)
qed
show "GFP f ⊑ f ⊤"
proof -
have "GFP f ⊑ f (GFP f)"
by (simp add: GFP_lemma2 assms(1) assms(2))
moreover have "... ⊑ f ⊤"
by (auto intro: use_iso1[of _ f] simp add: assms)
moreover from assms(1) have "f (GFP f) ∈ carrier L"
by (auto)
ultimately show ?thesis
using fb local.le_trans by blast
qed
qed

end

subsection ‹Complete lattices where ‹eq› is the Equality›

locale complete_lattice = partial_order +
assumes sup_exists:
"[| A ⊆ carrier L |] ==> ∃s. least L s (Upper L A)"
and inf_exists:
"[| A ⊆ carrier L |] ==> ∃i. greatest L i (Lower L A)"

sublocale complete_lattice ⊆ lattice
proof
fix x y
assume a: "x ∈ carrier L" "y ∈ carrier L"
thus "∃s. is_lub L s {x, y}"
by (rule_tac sup_exists[of "{x, y}"], auto)
from a show "∃s. is_glb L s {x, y}"
by (rule_tac inf_exists[of "{x, y}"], auto)
qed

sublocale complete_lattice ⊆ weak?: weak_complete_lattice
by standard (auto intro: sup_exists inf_exists)

lemma complete_lattice_lattice [simp]:
assumes "complete_lattice X"
shows "lattice X"
proof -
interpret c: complete_lattice X
by (simp add: assms)
show ?thesis
by (unfold_locales)
qed

text ‹Introduction rule: the usual definition of complete lattice›

lemma (in partial_order) complete_latticeI:
assumes sup_exists:
"!!A. [| A ⊆ carrier L |] ==> ∃s. least L s (Upper L A)"
and inf_exists:
"!!A. [| A ⊆ carrier L |] ==> ∃i. greatest L i (Lower L A)"
shows "complete_lattice L"
by standard (auto intro: sup_exists inf_exists)

theorem (in partial_order) complete_lattice_criterion1:
assumes top_exists: "∃g. greatest L g (carrier L)"
and inf_exists:
"!!A. [| A ⊆ carrier L; A ≠ {} |] ==> ∃i. greatest L i (Lower L A)"
shows "complete_lattice L"
proof (rule complete_latticeI)
from top_exists obtain top where top: "greatest L top (carrier L)" ..
fix A
assume L: "A ⊆ carrier L"
let ?B = "Upper L A"
from L top have "top ∈ ?B" by (fast intro!: Upper_memI intro: greatest_le)
then have B_non_empty: "?B ≠ {}" by fast
have B_L: "?B ⊆ carrier L" by simp
from inf_exists [OF B_L B_non_empty]
obtain b where b_inf_B: "greatest L b (Lower L ?B)" ..
then have bcarr: "b ∈ carrier L"
by blast
have "least L b (Upper L A)"
proof (rule least_UpperI)
show "⋀x. x ∈ A ⟹ x ⊑ b"
by (meson L Lower_memI Upper_memD b_inf_B greatest_le set_rev_mp)
show "⋀y. y ∈ Upper L A ⟹ b ⊑ y"
by (auto elim: greatest_Lower_below [OF b_inf_B])
qed (use L bcarr in auto)
then show "∃s. least L s (Upper L A)" ..
next
fix A
assume L: "A ⊆ carrier L"
show "∃i. greatest L i (Lower L A)"
proof (cases "A = {}")
case True then show ?thesis
by (simp add: top_exists)
next
case False with L show ?thesis
by (rule inf_exists)
qed
qed

(* TODO: prove dual version *)

subsection ‹Fixed points›

context complete_lattice
begin

lemma LFP_unfold:
"⟦ Mono f; f ∈ carrier L → carrier L ⟧ ⟹ LFP f = f (LFP f)"
using eq_is_equal weak.LFP_weak_unfold by auto

lemma LFP_const:
"t ∈ carrier L ⟹ LFP (λ x. t) = t"
by (simp add: local.le_antisym weak.LFP_greatest weak.LFP_lowerbound)

lemma LFP_id:
"LFP id = ⊥"
by (simp add: local.le_antisym weak.LFP_lowerbound)

lemma GFP_unfold:
"⟦ Mono f; f ∈ carrier L → carrier L ⟧ ⟹ GFP f = f (GFP f)"
using eq_is_equal weak.GFP_weak_unfold by auto

lemma GFP_const:
"t ∈ carrier L ⟹ GFP (λ x. t) = t"
by (simp add: local.le_antisym weak.GFP_least weak.GFP_upperbound)

lemma GFP_id:
"GFP id = ⊤"
using weak.GFP_upperbound by auto

end

subsection ‹Interval complete lattices›

context weak_complete_lattice
begin

lemma at_least_at_most_Sup: "⟦ a ∈ carrier L; b ∈ carrier L; a ⊑ b ⟧ ⟹ ⨆ ⦃a..b⦄ .= b"
by (rule weak_le_antisym [OF sup_least sup_upper]) (auto simp add: at_least_at_most_closed)

lemma at_least_at_most_Inf: "⟦ a ∈ carrier L; b ∈ carrier L; a ⊑ b ⟧ ⟹ ⨅ ⦃a..b⦄ .= a"
by (rule weak_le_antisym [OF inf_lower inf_greatest]) (auto simp add: at_least_at_most_closed)

end

lemma weak_complete_lattice_interval:
assumes "weak_complete_lattice L" "a ∈ carrier L" "b ∈ carrier L" "a ⊑⇘L⇙ b"
shows "weak_complete_lattice (L ⦇ carrier := ⦃a..b⦄⇘L⇙ ⦈)"
proof -
interpret L: weak_complete_lattice L
by (simp add: assms)
interpret weak_partial_order "L ⦇ carrier := ⦃a..b⦄⇘L⇙ ⦈"
proof -
have "⦃a..b⦄⇘L⇙ ⊆ carrier L"
by (auto simp add: at_least_at_most_def)
thus "weak_partial_order (L⦇carrier := ⦃a..b⦄⇘L⇙⦈)"
by (simp add: L.weak_partial_order_axioms weak_partial_order_subset)
qed

show ?thesis
proof
fix A
assume a: "A ⊆ carrier (L⦇carrier := ⦃a..b⦄⇘L⇙⦈)"
show "∃s. is_lub (L⦇carrier := ⦃a..b⦄⇘L⇙⦈) s A"
proof (cases "A = {}")
case True
thus ?thesis
by (rule_tac x="a" in exI, auto simp add: least_def assms)
next
case False
show ?thesis
proof (rule_tac x="⨆⇘L⇙ A" in exI, rule least_UpperI, simp_all)
show b:"⋀ x. x ∈ A ⟹ x ⊑⇘L⇙ ⨆⇘L⇙A"
using a by (auto intro: L.sup_upper, meson L.at_least_at_most_closed L.sup_upper subset_trans)
show "⋀y. y ∈ Upper (L⦇carrier := ⦃a..b⦄⇘L⇙⦈) A ⟹ ⨆⇘L⇙A ⊑⇘L⇙ y"
using a L.at_least_at_most_closed by (rule_tac L.sup_least, auto intro: funcset_mem simp add: Upper_def)
from a show "A ⊆ ⦃a..b⦄⇘L⇙"
by (auto)
from a show "⨆⇘L⇙A ∈ ⦃a..b⦄⇘L⇙"
apply (rule_tac L.at_least_at_most_member)
apply (auto)
apply (meson L.at_least_at_most_closed L.sup_closed subset_trans)
apply (meson False L.at_least_at_most_closed L.at_least_at_most_lower L.le_trans L.sup_closed b all_not_in_conv assms(2) contra_subsetD subset_trans)
apply (rule L.sup_least)
apply (auto simp add: assms)
using L.at_least_at_most_closed apply blast
done
qed
qed
show "∃s. is_glb (L⦇carrier := ⦃a..b⦄⇘L⇙⦈) s A"
proof (cases "A = {}")
case True
thus ?thesis
by (rule_tac x="b" in exI, auto simp add: greatest_def assms)
next
case False
show ?thesis
proof (rule_tac x="⨅⇘L⇙ A" in exI, rule greatest_LowerI, simp_all)
show b:"⋀x. x ∈ A ⟹ ⨅⇘L⇙A ⊑⇘L⇙ x"
using a L.at_least_at_most_closed by (force intro!: L.inf_lower)
show "⋀y. y ∈ Lower (L⦇carrier := ⦃a..b⦄⇘L⇙⦈) A ⟹ y ⊑⇘L⇙ ⨅⇘L⇙A"
using a L.at_least_at_most_closed by (rule_tac L.inf_greatest, auto intro: funcset_carrier' simp add: Lower_def)
from a show "A ⊆ ⦃a..b⦄⇘L⇙"
by (auto)
from a show "⨅⇘L⇙A ∈ ⦃a..b⦄⇘L⇙"
apply (rule_tac L.at_least_at_most_member)
apply (auto)
apply (meson L.at_least_at_most_closed L.inf_closed subset_trans)
apply (meson L.at_least_at_most_closed L.at_least_at_most_lower L.inf_greatest assms(2) set_rev_mp subset_trans)
apply (meson False L.at_least_at_most_closed L.at_least_at_most_upper L.inf_closed L.le_trans b all_not_in_conv assms(3) contra_subsetD subset_trans)
done
qed
qed
qed
qed

subsection ‹Knaster-Tarski theorem and variants›

text ‹The set of fixed points of a complete lattice is itself a complete lattice›

theorem Knaster_Tarski:
assumes "weak_complete_lattice L" "f ∈ carrier L → carrier L" "isotone L L f"
shows "weak_complete_lattice (fpl L f)" (is "weak_complete_lattice ?L'")
proof -
interpret L: weak_complete_lattice L
by (simp add: assms)
interpret weak_partial_order ?L'
proof -
have "{x ∈ carrier L. f x .=⇘L⇙ x} ⊆ carrier L"
by (auto)
thus "weak_partial_order ?L'"
by (simp add: L.weak_partial_order_axioms weak_partial_order_subset)
qed
show ?thesis
proof (unfold_locales, simp_all)
fix A
assume A: "A ⊆ fps L f"
show "∃s. is_lub (fpl L f) s A"
proof
from A have AL: "A ⊆ carrier L"
by (meson fps_carrier subset_eq)

let ?w = "⨆⇘L⇙ A"
have w: "f (⨆⇘L⇙A) ∈ carrier L"
by (rule funcset_mem[of f "carrier L"], simp_all add: AL assms(2))

have pf_w: "(⨆⇘L⇙ A) ⊑⇘L⇙ f (⨆⇘L⇙ A)"
by (simp add: A L.weak_sup_pre_fixed_point assms(2) assms(3))

have f_top_chain: "f  ⦃?w..⊤⇘L⇙⦄⇘L⇙ ⊆ ⦃?w..⊤⇘L⇙⦄⇘L⇙"
proof (auto simp add: at_least_at_most_def)
fix x
assume b: "x ∈ carrier L" "⨆⇘L⇙A ⊑⇘L⇙ x"
from b show fx: "f x ∈ carrier L"
using assms(2) by blast
show "⨆⇘L⇙A ⊑⇘L⇙ f x"
proof -
have "?w ⊑⇘L⇙ f ?w"
proof (rule_tac L.sup_least, simp_all add: AL w)
fix y
assume c: "y ∈ A"
hence y: "y ∈ fps L f"
using A subsetCE by blast
with assms have "y .=⇘L⇙ f y"
proof -
from y have "y ∈ carrier L"
by (simp add: fps_def)
moreover hence "f y ∈ carrier L"
by (rule_tac funcset_mem[of f "carrier L"], simp_all add: assms)
ultimately show ?thesis using y
by (rule_tac L.sym, simp_all add: L.use_fps)
qed
moreover have "y ⊑⇘L⇙ ⨆⇘L⇙A"
by (simp add: AL L.sup_upper c(1))
ultimately show "y ⊑⇘L⇙ f (⨆⇘L⇙A)"
by (meson fps_def AL funcset_mem L.refl L.weak_complete_lattice_axioms assms(2) assms(3) c(1) isotone_def rev_subsetD weak_complete_lattice.sup_closed weak_partial_order.le_cong)
qed
thus ?thesis
by (meson AL funcset_mem L.le_trans L.sup_closed assms(2) assms(3) b(1) b(2) use_iso2)
qed

show "f x ⊑⇘L⇙ ⊤⇘L⇙"
by (simp add: fx)
qed

let ?L' = "L⦇ carrier := ⦃?w..⊤⇘L⇙⦄⇘L⇙ ⦈"

interpret L': weak_complete_lattice ?L'
by (auto intro: weak_complete_lattice_interval simp add: L.weak_complete_lattice_axioms AL)

let ?L'' = "L⦇ carrier := fps L f ⦈"

show "is_lub ?L'' (LFP⇘?L'⇙ f) A"
proof (rule least_UpperI, simp_all)
fix x
assume "x ∈ Upper ?L'' A"
hence "LFP⇘?L'⇙ f ⊑⇘?L'⇙ x"
apply (rule_tac L'.LFP_lowerbound)
apply (auto simp add: Upper_def)
apply (simp add: A AL L.at_least_at_most_member L.sup_least set_rev_mp)
apply (simp add: Pi_iff assms(2) fps_def, rule_tac L.weak_refl)
apply (auto)
apply (rule funcset_mem[of f "carrier L"], simp_all add: assms(2))
done
thus " LFP⇘?L'⇙ f ⊑⇘L⇙ x"
by (simp)
next
fix x
assume xA: "x ∈ A"
show "x ⊑⇘L⇙ LFP⇘?L'⇙ f"
proof -
have "LFP⇘?L'⇙ f ∈ carrier ?L'"
by blast
thus ?thesis
by (simp, meson AL L.at_least_at_most_closed L.at_least_at_most_lower L.le_trans L.sup_closed L.sup_upper xA subsetCE)
qed
next
show "A ⊆ fps L f"
by (simp add: A)
next
show "LFP⇘?L'⇙ f ∈ fps L f"
proof (auto simp add: fps_def)
have "LFP⇘?L'⇙ f ∈ carrier ?L'"
by (rule L'.LFP_closed)
thus c:"LFP⇘?L'⇙ f ∈ carrier L"
by (auto simp add: at_least_at_most_def)
have "LFP⇘?L'⇙ f .=⇘?L'⇙ f (LFP⇘?L'⇙ f)"
proof (rule "L'.LFP_weak_unfold", simp_all)
show "f ∈ ⦃⨆⇘L⇙A..⊤⇘L⇙⦄⇘L⇙ → ⦃⨆⇘L⇙A..⊤⇘L⇙⦄⇘L⇙"
apply (auto simp add: Pi_def at_least_at_most_def)
using assms(2) apply blast
apply (meson AL funcset_mem L.le_trans L.sup_closed assms(2) assms(3) pf_w use_iso2)
using assms(2) apply blast
done
from assms(3) show "Mono⇘L⦇carrier := ⦃⨆⇘L⇙A..⊤⇘L⇙⦄⇘L⇙⦈⇙ f"
apply (auto simp add: isotone_def)
using L'.weak_partial_order_axioms apply blast
apply (meson L.at_least_at_most_closed subsetCE)
done
qed
thus "f (LFP⇘?L'⇙ f) .=⇘L⇙ LFP⇘?L'⇙ f"
by (simp add: L.equivalence_axioms funcset_carrier' c assms(2) equivalence.sym)
qed
qed
qed
show "∃i. is_glb (L⦇carrier := fps L f⦈) i A"
proof
from A have AL: "A ⊆ carrier L"
by (meson fps_carrier subset_eq)

let ?w = "⨅⇘L⇙ A"
have w: "f (⨅⇘L⇙A) ∈ carrier L"
by (simp add: AL funcset_carrier' assms(2))

have pf_w: "f (⨅⇘L⇙ A) ⊑⇘L⇙ (⨅⇘L⇙ A)"
by (simp add: A L.weak_sup_post_fixed_point assms(2) assms(3))

have f_bot_chain: "f  ⦃⊥⇘L⇙..?w⦄⇘L⇙ ⊆ ⦃⊥⇘L⇙..?w⦄⇘L⇙"
proof (auto simp add: at_least_at_most_def)
fix x
assume b: "x ∈ carrier L" "x ⊑⇘L⇙ ⨅⇘L⇙A"
from b show fx: "f x ∈ carrier L"
using assms(2) by blast
show "f x ⊑⇘L⇙ ⨅⇘L⇙A"
proof -
have "f ?w ⊑⇘L⇙ ?w"
proof (rule_tac L.inf_greatest, simp_all add: AL w)
fix y
assume c: "y ∈ A"
with assms have "y .=⇘L⇙ f y"
by (metis (no_types, lifting) A funcset_carrier'[OF assms(2)] L.sym fps_def mem_Collect_eq subset_eq)
moreover have "⨅⇘L⇙A ⊑⇘L⇙ y"
by (simp add: AL L.inf_lower c)
ultimately show "f (⨅⇘L⇙A) ⊑⇘L⇙ y"
by (meson AL L.inf_closed L.le_trans c pf_w set_rev_mp w)
qed
thus ?thesis
by (meson AL L.inf_closed L.le_trans assms(3) b(1) b(2) fx use_iso2 w)
qed

show "⊥⇘L⇙ ⊑⇘L⇙ f x"
by (simp add: fx)
qed

let ?L' = "L⦇ carrier := ⦃⊥⇘L⇙..?w⦄⇘L⇙ ⦈"

interpret L': weak_complete_lattice ?L'
by (auto intro!: weak_complete_lattice_interval simp add: L.weak_complete_lattice_axioms AL)

let ?L'' = "L⦇ carrier := fps L f ⦈"

show "is_glb ?L'' (GFP⇘?L'⇙ f) A"
proof (rule greatest_LowerI, simp_all)
fix x
assume "x ∈ Lower ?L'' A"
hence "x ⊑⇘?L'⇙ GFP⇘?L'⇙ f"
apply (rule_tac L'.GFP_upperbound)
apply (auto simp add: Lower_def)
apply (meson A AL L.at_least_at_most_member L.bottom_lower L.weak_complete_lattice_axioms fps_carrier subsetCE weak_complete_lattice.inf_greatest)
apply (simp add: funcset_carrier' L.sym assms(2) fps_def)
done
thus "x ⊑⇘L⇙ GFP⇘?L'⇙ f"
by (simp)
next
fix x
assume xA: "x ∈ A"
show "GFP⇘?L'⇙ f ⊑⇘L⇙ x"
proof -
have "GFP⇘?L'⇙ f ∈ carrier ?L'"
by blast
thus ?thesis
by (simp, meson AL L.at_least_at_most_closed L.at_least_at_most_upper L.inf_closed L.inf_lower L.le_trans subsetCE xA)
qed
next
show "A ⊆ fps L f"
by (simp add: A)
next
show "GFP⇘?L'⇙ f ∈ fps L f"
proof (auto simp add: fps_def)
have "GFP⇘?L'⇙ f ∈ carrier ?L'"
by (rule L'.GFP_closed)
thus c:"GFP⇘?L'⇙ f ∈ carrier L"
by (auto simp add: at_least_at_most_def)
have "GFP⇘?L'⇙ f .=⇘?L'⇙ f (GFP⇘?L'⇙ f)"
proof (rule "L'.GFP_weak_unfold", simp_all)
show "f ∈ ⦃⊥⇘L⇙..?w⦄⇘L⇙ → ⦃⊥⇘L⇙..?w⦄⇘L⇙"
apply (auto simp add: Pi_def at_least_at_most_def)
using assms(2) apply blast
apply (simp add: funcset_carrier' assms(2))
apply (meson AL funcset_carrier L.inf_closed L.le_trans assms(2) assms(3) pf_w use_iso2)
done
from assms(3) show "Mono⇘L⦇carrier := ⦃⊥⇘L⇙..?w⦄⇘L⇙⦈⇙ f"
apply (auto simp add: isotone_def)
using L'.weak_partial_order_axioms apply blast
using L.at_least_at_most_closed apply (blast intro: funcset_carrier')
done
qed
thus "f (GFP⇘?L'⇙ f) .=⇘L⇙ GFP⇘?L'⇙ f"
by (simp add: L.equivalence_axioms funcset_carrier' c assms(2) equivalence.sym)
qed
qed
qed
qed
qed

theorem Knaster_Tarski_top:
assumes "weak_complete_lattice L" "isotone L L f" "f ∈ carrier L → carrier L"
shows "⊤⇘fpl L f⇙ .=⇘L⇙ GFP⇘L⇙ f"
proof -
interpret L: weak_complete_lattice L
by (simp add: assms)
interpret L': weak_complete_lattice "fpl L f"
by (rule Knaster_Tarski, simp_all add: assms)
show ?thesis
proof (rule L.weak_le_antisym, simp_all)
show "⊤⇘fpl L f⇙ ⊑⇘L⇙ GFP⇘L⇙ f"
by (rule L.GFP_greatest_fixed_point, simp_all add: assms L'.top_closed[simplified])
show "GFP⇘L⇙ f ⊑⇘L⇙ ⊤⇘fpl L f⇙"
proof -
have "GFP⇘L⇙ f ∈ fps L f"
by (rule L.GFP_fixed_point, simp_all add: assms)
hence "GFP⇘L⇙ f ∈ carrier (fpl L f)"
by simp
hence "GFP⇘L⇙ f ⊑⇘fpl L f⇙ ⊤⇘fpl L f⇙"
by (rule L'.top_higher)
thus ?thesis
by simp
qed
show "⊤⇘fpl L f⇙ ∈ carrier L"
proof -
have "carrier (fpl L f) ⊆ carrier L"
by (auto simp add: fps_def)
with L'.top_closed show ?thesis
by blast
qed
qed
qed

theorem Knaster_Tarski_bottom:
assumes "weak_complete_lattice L" "isotone L L f" "f ∈ carrier L → carrier L"
shows "⊥⇘fpl L f⇙ .=⇘L⇙ LFP⇘L⇙ f"
proof -
interpret L: weak_complete_lattice L
by (simp add: assms)
interpret L': weak_complete_lattice "fpl L f"
by (rule Knaster_Tarski, simp_all add: assms)
show ?thesis
proof (rule L.weak_le_antisym, simp_all)
show "LFP⇘L⇙ f ⊑⇘L⇙ ⊥⇘fpl L f⇙"
by (rule L.LFP_least_fixed_point, simp_all add: assms L'.bottom_closed[simplified])
show "⊥⇘fpl L f⇙ ⊑⇘L⇙ LFP⇘L⇙ f"
proof -
have "LFP⇘L⇙ f ∈ fps L f"
by (rule L.LFP_fixed_point, simp_all add: assms)
hence "LFP⇘L⇙ f ∈ carrier (fpl L f)"
by simp
hence "⊥⇘fpl L f⇙ ⊑⇘fpl L f⇙ LFP⇘L⇙ f"
by (rule L'.bottom_lower)
thus ?thesis
by simp
qed
show "⊥⇘fpl L f⇙ ∈ carrier L"
proof -
have "carrier (fpl L f) ⊆ carrier L"
by (auto simp add: fps_def)
with L'.bottom_closed show ?thesis
by blast
qed
qed
qed

text ‹If a function is both idempotent and isotone then the image of the function forms a complete lattice›

theorem Knaster_Tarski_idem:
assumes "complete_lattice L" "f ∈ carrier L → carrier L" "isotone L L f" "idempotent L f"
shows "complete_lattice (L⦇carrier := f  carrier L⦈)"
proof -
interpret L: complete_lattice L
by (simp add: assms)
have "fps L f = f  carrier L"
using L.weak.fps_idem[OF assms(2) assms(4)]
by (simp add: L.set_eq_is_eq)
then interpret L': weak_complete_lattice "(L⦇carrier := f  carrier L⦈)"
by (metis Knaster_Tarski L.weak.weak_complete_lattice_axioms assms(2) assms(3))
show ?thesis
using L'.sup_exists L'.inf_exists
by (unfold_locales, auto simp add: L.eq_is_equal)
qed

theorem Knaster_Tarski_idem_extremes:
assumes "weak_complete_lattice L" "isotone L L f" "idempotent L f" "f ∈ carrier L → carrier L"
shows "⊤⇘fpl L f⇙ .=⇘L⇙ f (⊤⇘L⇙)" "⊥⇘fpl L f⇙ .=⇘L⇙ f (⊥⇘L⇙)"
proof -
interpret L: weak_complete_lattice "L"
by (simp_all add: assms)
interpret L': weak_complete_lattice "fpl L f"
by (rule Knaster_Tarski, simp_all add: assms)
have FA: "fps L f ⊆ carrier L"
by (auto simp add: fps_def)
show "⊤⇘fpl L f⇙ .=⇘L⇙ f (⊤⇘L⇙)"
proof -
from FA have "⊤⇘fpl L f⇙ ∈ carrier L"
proof -
have "⊤⇘fpl L f⇙ ∈ fps L f"
using L'.top_closed by auto
thus ?thesis
using FA by blast
qed
moreover with assms have "f ⊤⇘L⇙ ∈ carrier L"
by (auto)

ultimately show ?thesis
using L.trans[OF Knaster_Tarski_top[of L f] L.GFP_idem[of f]]
by (simp_all add: assms)
qed
show "⊥⇘fpl L f⇙ .=⇘L⇙ f (⊥⇘L⇙)"
proof -
from FA have "⊥⇘fpl L f⇙ ∈ carrier L"
proof -
have "⊥⇘fpl L f⇙ ∈ fps L f"
using L'.bottom_closed by auto
thus ?thesis
using FA by blast
qed
moreover with assms have "f ⊥⇘L⇙ ∈ carrier L"
by (auto)

ultimately show ?thesis
using L.trans[OF Knaster_Tarski_bottom[of L f] L.LFP_idem[of f]]
by (simp_all add: assms)
qed
qed

theorem Knaster_Tarski_idem_inf_eq:
assumes "weak_complete_lattice L" "isotone L L f" "idempotent L f" "f ∈ carrier L → carrier L"
"A ⊆ fps L f"
shows "⨅⇘fpl L f⇙ A .=⇘L⇙ f (⨅⇘L⇙ A)"
proof -
interpret L: weak_complete_lattice "L"
by (simp_all add: assms)
interpret L': weak_complete_lattice "fpl L f"
by (rule Knaster_Tarski, simp_all add: assms)
have FA: "fps L f ⊆ carrier L"
by (auto simp add: fps_def)
have A: "A ⊆ carrier L"
using FA assms(5) by blast
have fA: "f (⨅⇘L⇙A) ∈ fps L f"
by (metis (no_types, lifting) A L.idempotent L.inf_closed PiE assms(3) assms(4) fps_def mem_Collect_eq)
have infA: "⨅⇘fpl L f⇙A ∈ fps L f"
by (rule L'.inf_closed[simplified], simp add: assms)
show ?thesis
proof (rule L.weak_le_antisym)
show ic: "⨅⇘fpl L f⇙A ∈ carrier L"
using FA infA by blast
show fc: "f (⨅⇘L⇙A) ∈ carrier L"
using FA fA by blast
show "f (⨅⇘L⇙A) ⊑⇘L⇙ ⨅⇘fpl L f⇙A"
proof -
have "⋀x. x ∈ A ⟹ f (⨅⇘L⇙A) ⊑⇘L⇙ x"
by (meson A FA L.inf_closed L.inf_lower L.le_trans L.weak_sup_post_fixed_point assms(2) assms(4) assms(5) fA subsetCE)
hence "f (⨅⇘L⇙A) ⊑⇘fpl L f⇙ ⨅⇘fpl L f⇙A"
by (rule_tac L'.inf_greatest, simp_all add: fA assms(3,5))
thus ?thesis
by (simp)
qed
show "⨅⇘fpl L f⇙A ⊑⇘L⇙ f (⨅⇘L⇙A)"
proof -
have "⋀x. x ∈ A ⟹ ⨅⇘fpl L f⇙A ⊑⇘fpl L f⇙ x"
by (rule L'.inf_lower, simp_all add: assms)
hence "⨅⇘fpl L f⇙A ⊑⇘L⇙ (⨅⇘L⇙A)"
apply (rule_tac L.inf_greatest, simp_all add: A)
using FA infA apply blast
done
hence 1:"f(⨅⇘fpl L f⇙A) ⊑⇘L⇙ f(⨅⇘L⇙A)"
by (metis (no_types, lifting) A FA L.inf_closed assms(2) infA subsetCE use_iso1)
have 2:"⨅⇘fpl L f⇙A ⊑⇘L⇙ f (⨅⇘fpl L f⇙A)"
by (metis (no_types, lifting) FA L.sym L.use_fps L.weak_complete_lattice_axioms PiE assms(4) infA subsetCE weak_complete_lattice_def weak_partial_order.weak_refl)

show ?thesis
using FA fA infA by (auto intro!: L.le_trans[OF 2 1] ic fc, metis FA PiE assms(4) subsetCE)
qed
qed
qed

subsection ‹Examples›

subsubsection ‹The Powerset of a Set is a Complete Lattice›

theorem powerset_is_complete_lattice:
"complete_lattice ⦇carrier = Pow A, eq = (=), le = (⊆)⦈"
(is "complete_lattice ?L")
proof (rule partial_order.complete_latticeI)
show "partial_order ?L"
by standard auto
next
fix B
assume "B ⊆ carrier ?L"
then have "least ?L (⋃ B) (Upper ?L B)"
by (fastforce intro!: least_UpperI simp: Upper_def)
then show "∃s. least ?L s (Upper ?L B)" ..
next
fix B
assume "B ⊆ carrier ?L"
then have "greatest ?L (⋂ B ∩ A) (Lower ?L B)"
txt ‹@{term "⋂ B"} is not the infimum of @{term B}:
@{term "⋂ {} = UNIV"} which is in general bigger than @{term "A"}! ›
by (fastforce intro!: greatest_LowerI simp: Lower_def)
then show "∃i. greatest ?L i (Lower ?L B)" ..
qed

text ‹Another example, that of the lattice of subgroups of a group,
can be found in Group theory (Section~\ref{sec:subgroup-lattice}).›

subsection ‹Limit preserving functions›

definition weak_sup_pres :: "('a, 'c) gorder_scheme ⇒ ('b, 'd) gorder_scheme ⇒ ('a ⇒ 'b) ⇒ bool" where
"weak_sup_pres X Y f ≡ complete_lattice X ∧ complete_lattice Y ∧ (∀ A ⊆ carrier X. A ≠ {} ⟶ f (⨆⇘X⇙ A) = (⨆⇘Y⇙ (f  A)))"

definition sup_pres :: "('a, 'c) gorder_scheme ⇒ ('b, 'd) gorder_scheme ⇒ ('a ⇒ 'b) ⇒ bool" where
"sup_pres X Y f ≡ complete_lattice X ∧ complete_lattice Y ∧ (∀ A ⊆ carrier X. f (⨆⇘X⇙ A) = (⨆⇘Y⇙ (f  A)))"

definition weak_inf_pres :: "('a, 'c) gorder_scheme ⇒ ('b, 'd) gorder_scheme ⇒ ('a ⇒ 'b) ⇒ bool" where
"weak_inf_pres X Y f ≡ complete_lattice X ∧ complete_lattice Y ∧ (∀ A ⊆ carrier X. A ≠ {} ⟶ f (⨅⇘X⇙ A) = (⨅⇘Y⇙ (f  A)))"

definition inf_pres :: "('a, 'c) gorder_scheme ⇒ ('b, 'd) gorder_scheme ⇒ ('a ⇒ 'b) ⇒ bool" where
"inf_pres X Y f ≡ complete_lattice X ∧ complete_lattice Y ∧ (∀ A ⊆ carrier X. f (⨅⇘X⇙ A) = (⨅⇘Y⇙ (f  A)))"

lemma weak_sup_pres:
"sup_pres X Y f ⟹ weak_sup_pres X Y f"
by (simp add: sup_pres_def weak_sup_pres_def)

lemma weak_inf_pres:
"inf_pres X Y f ⟹ weak_inf_pres X Y f"
by (simp add: inf_pres_def weak_inf_pres_def)

lemma sup_pres_is_join_pres:
assumes "weak_sup_pres X Y f"
shows "join_pres X Y f"
using assms
apply (simp add: join_pres_def weak_sup_pres_def, safe)
apply (rename_tac x y)
apply (drule_tac x="{x, y}" in spec)
apply (auto simp add: join_def)
done

lemma inf_pres_is_meet_pres:
assumes "weak_inf_pres X Y f"
shows "meet_pres X Y f"
using assms
apply (simp add: meet_pres_def weak_inf_pres_def, safe)
apply (rename_tac x y)
apply (drule_tac x="{x, y}" in spec)
apply (auto simp add: meet_def)
done

end