(* Title: HOL/Algebra/Complete_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 Complete_Lattice
imports Lattice
begin
section \<open>Complete Lattices\<close>
locale weak_complete_lattice = weak_partial_order +
assumes sup_exists:
"[| A \<subseteq> carrier L |] ==> \<exists>s. least L s (Upper L A)"
and inf_exists:
"[| A \<subseteq> carrier L |] ==> \<exists>i. greatest L i (Lower L A)"
sublocale weak_complete_lattice \<subseteq> weak_lattice
proof
fix x y
assume a: "x \<in> carrier L" "y \<in> carrier L"
thus "\<exists>s. is_lub L s {x, y}"
by (rule_tac sup_exists[of "{x, y}"], auto)
from a show "\<exists>s. is_glb L s {x, y}"
by (rule_tac inf_exists[of "{x, y}"], auto)
qed
text \<open>Introduction rule: the usual definition of complete lattice\<close>
lemma (in weak_partial_order) weak_complete_latticeI:
assumes sup_exists:
"!!A. [| A \<subseteq> carrier L |] ==> \<exists>s. least L s (Upper L A)"
and inf_exists:
"!!A. [| A \<subseteq> carrier L |] ==> \<exists>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
apply (unfold_locales)
apply (simp_all add:inf_exists sup_exists)
done
qed
lemma (in weak_complete_lattice) supI:
"[| !!l. least L l (Upper L A) ==> P l; A \<subseteq> carrier L |]
==> P (\<Squnion>A)"
proof (unfold sup_def)
assume L: "A \<subseteq> 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 \<subseteq> carrier L ==> \<Squnion>A \<in> carrier L"
by (rule supI) simp_all
lemma (in weak_complete_lattice) sup_cong:
assumes "A \<subseteq> carrier L" "B \<subseteq> carrier L" "A {.=} B"
shows "\<Squnion> A .= \<Squnion> B"
proof -
have "\<And> x. is_lub L x A \<longleftrightarrow> is_lub L x B"
by (rule least_Upper_cong_r, simp_all add: assms)
moreover have "\<Squnion> B \<in> carrier L"
by (simp add: assms(2))
ultimately show ?thesis
by (simp add: sup_def)
qed
sublocale weak_complete_lattice \<subseteq> 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 \<subseteq> carrier L |]
==> P (\<Sqinter>A)"
proof (unfold inf_def)
assume L: "A \<subseteq> 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 \<subseteq> carrier L ==> \<Sqinter>A \<in> carrier L"
by (rule infI) simp_all
lemma (in weak_complete_lattice) inf_cong:
assumes "A \<subseteq> carrier L" "B \<subseteq> carrier L" "A {.=} B"
shows "\<Sqinter> A .= \<Sqinter> B"
proof -
have "\<And> x. is_glb L x A \<longleftrightarrow> is_glb L x B"
by (rule greatest_Lower_cong_r, simp_all add: assms)
moreover have "\<Sqinter> B \<in> 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: "\<exists>g. greatest L g (carrier L)"
and inf_exists:
"\<And>A. [| A \<subseteq> carrier L; A \<noteq> {} |] ==> \<exists>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 \<subseteq> carrier L"
let ?B = "Upper L A"
from L top have "top \<in> ?B" by (fast intro!: Upper_memI intro: greatest_le)
then have B_non_empty: "?B \<noteq> {}" by fast
have B_L: "?B \<subseteq> 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)" ..
have "least L b (Upper L A)"
apply (rule least_UpperI)
apply (rule greatest_le [where A = "Lower L ?B"])
apply (rule b_inf_B)
apply (rule Lower_memI)
apply (erule Upper_memD [THEN conjunct1])
apply assumption
apply (rule L)
apply (fast intro: L [THEN subsetD])
apply (erule greatest_Lower_below [OF b_inf_B])
apply simp
apply (rule L)
apply (rule greatest_closed [OF b_inf_B])
done
then show "\<exists>s. least L s (Upper L A)" ..
next
fix A
assume L: "A \<subseteq> carrier L"
show "\<exists>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
text \<open>Supremum\<close>
declare (in partial_order) weak_sup_of_singleton [simp del]
lemma (in partial_order) sup_of_singleton [simp]:
"x \<in> carrier L ==> \<Squnion>{x} = x"
using weak_sup_of_singleton unfolding eq_is_equal .
lemma (in upper_semilattice) join_assoc_lemma:
assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
shows "x \<squnion> (y \<squnion> z) = \<Squnion>{x, y, z}"
using weak_join_assoc_lemma L unfolding eq_is_equal .
lemma (in upper_semilattice) join_assoc:
assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
shows "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
using weak_join_assoc L unfolding eq_is_equal .
text \<open>Infimum\<close>
declare (in partial_order) weak_inf_of_singleton [simp del]
lemma (in partial_order) inf_of_singleton [simp]:
"x \<in> carrier L ==> \<Sqinter>{x} = x"
using weak_inf_of_singleton unfolding eq_is_equal .
text \<open>Condition on \<open>A\<close>: infimum exists.\<close>
lemma (in lower_semilattice) meet_assoc_lemma:
assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
shows "x \<sqinter> (y \<sqinter> z) = \<Sqinter>{x, y, z}"
using weak_meet_assoc_lemma L unfolding eq_is_equal .
lemma (in lower_semilattice) meet_assoc:
assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
shows "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
using weak_meet_assoc L unfolding eq_is_equal .
subsection \<open>Infimum Laws\<close>
context weak_complete_lattice
begin
lemma inf_glb:
assumes "A \<subseteq> carrier L"
shows "greatest L (\<Sqinter>A) (Lower L A)"
proof -
obtain i where "greatest L i (Lower L A)"
by (metis assms inf_exists)
thus ?thesis
apply (simp add: inf_def)
apply (rule someI2[of _ "i"])
apply (auto)
done
qed
lemma inf_lower:
assumes "A \<subseteq> carrier L" "x \<in> A"
shows "\<Sqinter>A \<sqsubseteq> x"
by (metis assms greatest_Lower_below inf_glb)
lemma inf_greatest:
assumes "A \<subseteq> carrier L" "z \<in> carrier L"
"(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x)"
shows "z \<sqsubseteq> \<Sqinter>A"
by (metis Lower_memI assms greatest_le inf_glb)
lemma weak_inf_empty [simp]: "\<Sqinter>{} .= \<top>"
by (metis Lower_empty empty_subsetI inf_glb top_greatest weak_greatest_unique)
lemma weak_inf_carrier [simp]: "\<Sqinter>carrier L .= \<bottom>"
by (metis bottom_weak_eq inf_closed inf_lower subset_refl)
lemma weak_inf_insert [simp]:
"\<lbrakk> a \<in> carrier L; A \<subseteq> carrier L \<rbrakk> \<Longrightarrow> \<Sqinter>insert a A .= a \<sqinter> \<Sqinter>A"
apply (rule weak_le_antisym)
apply (force intro: meet_le inf_greatest inf_lower inf_closed)
apply (rule inf_greatest)
apply (force)
apply (force intro: inf_closed)
apply (auto)
apply (metis inf_closed meet_left)
apply (force intro: le_trans inf_closed meet_right meet_left inf_lower)
done
subsection \<open>Supremum Laws\<close>
lemma sup_lub:
assumes "A \<subseteq> carrier L"
shows "least L (\<Squnion>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 \<subseteq> carrier L" "x \<in> A"
shows "x \<sqsubseteq> \<Squnion>A"
by (metis assms least_Upper_above supI)
lemma sup_least:
assumes "A \<subseteq> carrier L" "z \<in> carrier L"
"(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z)"
shows "\<Squnion>A \<sqsubseteq> z"
by (metis Upper_memI assms least_le sup_lub)
lemma weak_sup_empty [simp]: "\<Squnion>{} .= \<bottom>"
by (metis Upper_empty bottom_least empty_subsetI sup_lub weak_least_unique)
lemma weak_sup_carrier [simp]: "\<Squnion>carrier L .= \<top>"
by (metis Lower_closed Lower_empty sup_closed sup_upper top_closed top_higher weak_le_antisym)
lemma weak_sup_insert [simp]:
"\<lbrakk> a \<in> carrier L; A \<subseteq> carrier L \<rbrakk> \<Longrightarrow> \<Squnion>insert a A .= a \<squnion> \<Squnion>A"
apply (rule weak_le_antisym)
apply (rule sup_least)
apply (auto)
apply (metis join_left sup_closed)
apply (rule le_trans) defer
apply (rule join_right)
apply (auto)
apply (rule join_le)
apply (auto intro: sup_upper sup_least sup_closed)
done
end
subsection \<open>Fixed points of a lattice\<close>
definition "fps L f = {x \<in> carrier L. f x .=\<^bsub>L\<^esub> x}"
abbreviation "fpl L f \<equiv> L\<lparr>carrier := fps L f\<rparr>"
lemma (in weak_partial_order)
use_fps: "x \<in> fps L f \<Longrightarrow> f x .= x"
by (simp add: fps_def)
lemma fps_carrier [simp]:
"fps L f \<subseteq> carrier L"
by (auto simp add: fps_def)
lemma (in weak_complete_lattice) fps_sup_image:
assumes "f \<in> carrier L \<rightarrow> carrier L" "A \<subseteq> fps L f"
shows "\<Squnion> (f ` A) .= \<Squnion> A"
proof -
from assms(2) have AL: "A \<subseteq> 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 \<subseteq> carrier L"
by (auto)
from assms(2) show "f ` A {.=} A"
apply (auto simp add: fps_def)
apply (rule set_eqI2)
apply blast
apply (rename_tac b)
apply (rule_tac x="f b" in bexI)
apply (metis (mono_tags, lifting) Ball_Collect assms(1) Pi_iff local.sym)
apply (auto)
done
qed
qed
lemma (in weak_complete_lattice) fps_idem:
"\<lbrakk> f \<in> carrier L \<rightarrow> carrier L; Idem f \<rbrakk> \<Longrightarrow> fps L f {.=} f ` carrier L"
apply (rule set_eqI2)
apply (auto simp add: idempotent_def fps_def)
apply (metis Pi_iff local.sym)
apply force
done
context weak_complete_lattice
begin
lemma weak_sup_pre_fixed_point:
assumes "f \<in> carrier L \<rightarrow> carrier L" "isotone L L f" "A \<subseteq> fps L f"
shows "(\<Squnion>\<^bsub>L\<^esub> A) \<sqsubseteq>\<^bsub>L\<^esub> f (\<Squnion>\<^bsub>L\<^esub> A)"
proof (rule sup_least)
from assms(3) show AL: "A \<subseteq> carrier L"
by (auto simp add: fps_def)
thus fA: "f (\<Squnion>A) \<in> carrier L"
by (simp add: assms funcset_carrier[of f L L])
fix x
assume xA: "x \<in> A"
hence "x \<in> fps L f"
using assms subsetCE by blast
hence "f x .=\<^bsub>L\<^esub> x"
by (auto simp add: fps_def)
moreover have "f x \<sqsubseteq>\<^bsub>L\<^esub> f (\<Squnion>\<^bsub>L\<^esub>A)"
by (meson AL assms(2) subsetCE sup_closed sup_upper use_iso1 xA)
ultimately show "x \<sqsubseteq>\<^bsub>L\<^esub> f (\<Squnion>\<^bsub>L\<^esub>A)"
by (meson AL fA assms(1) funcset_carrier le_cong local.refl subsetCE xA)
qed
lemma weak_sup_post_fixed_point:
assumes "f \<in> carrier L \<rightarrow> carrier L" "isotone L L f" "A \<subseteq> fps L f"
shows "f (\<Sqinter>\<^bsub>L\<^esub> A) \<sqsubseteq>\<^bsub>L\<^esub> (\<Sqinter>\<^bsub>L\<^esub> A)"
proof (rule inf_greatest)
from assms(3) show AL: "A \<subseteq> carrier L"
by (auto simp add: fps_def)
thus fA: "f (\<Sqinter>A) \<in> carrier L"
by (simp add: assms funcset_carrier[of f L L])
fix x
assume xA: "x \<in> A"
hence "x \<in> fps L f"
using assms subsetCE by blast
hence "f x .=\<^bsub>L\<^esub> x"
by (auto simp add: fps_def)
moreover have "f (\<Sqinter>\<^bsub>L\<^esub>A) \<sqsubseteq>\<^bsub>L\<^esub> f x"
by (meson AL assms(2) inf_closed inf_lower subsetCE use_iso1 xA)
ultimately show "f (\<Sqinter>\<^bsub>L\<^esub>A) \<sqsubseteq>\<^bsub>L\<^esub> x"
by (meson AL assms(1) fA funcset_carrier le_cong_r subsetCE xA)
qed
subsubsection \<open>Least fixed points\<close>
lemma LFP_closed [intro, simp]:
"LFP f \<in> carrier L"
by (metis (lifting) LEAST_FP_def inf_closed mem_Collect_eq subsetI)
lemma LFP_lowerbound:
assumes "x \<in> carrier L" "f x \<sqsubseteq> x"
shows "LFP f \<sqsubseteq> x"
by (auto intro:inf_lower assms simp add:LEAST_FP_def)
lemma LFP_greatest:
assumes "x \<in> carrier L"
"(\<And>u. \<lbrakk> u \<in> carrier L; f u \<sqsubseteq> u \<rbrakk> \<Longrightarrow> x \<sqsubseteq> u)"
shows "x \<sqsubseteq> LFP f"
by (auto simp add:LEAST_FP_def intro:inf_greatest assms)
lemma LFP_lemma2:
assumes "Mono f" "f \<in> carrier L \<rightarrow> carrier L"
shows "f (LFP f) \<sqsubseteq> LFP f"
using assms
apply (auto simp add:Pi_def)
apply (rule LFP_greatest)
apply (metis LFP_closed)
apply (metis LFP_closed LFP_lowerbound le_trans use_iso1)
done
lemma LFP_lemma3:
assumes "Mono f" "f \<in> carrier L \<rightarrow> carrier L"
shows "LFP f \<sqsubseteq> f (LFP f)"
using assms
apply (auto simp add:Pi_def)
apply (metis LFP_closed LFP_lemma2 LFP_lowerbound assms(2) use_iso2)
done
lemma LFP_weak_unfold:
"\<lbrakk> Mono f; f \<in> carrier L \<rightarrow> carrier L \<rbrakk> \<Longrightarrow> LFP f .= f (LFP f)"
by (auto intro: LFP_lemma2 LFP_lemma3 funcset_mem)
lemma LFP_fixed_point [intro]:
assumes "Mono f" "f \<in> carrier L \<rightarrow> carrier L"
shows "LFP f \<in> fps L f"
proof -
have "f (LFP f) \<in> 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 \<in> carrier L \<rightarrow> carrier L" "x \<in> fps L f"
shows "LFP f \<sqsubseteq> x"
using assms by (force intro: LFP_lowerbound simp add: fps_def)
lemma LFP_idem:
assumes "f \<in> carrier L \<rightarrow> carrier L" "Mono f" "Idem f"
shows "LFP f .= (f \<bottom>)"
proof (rule weak_le_antisym)
from assms(1) show fb: "f \<bottom> \<in> carrier L"
by (rule funcset_mem, simp)
from assms show mf: "LFP f \<in> carrier L"
by blast
show "LFP f \<sqsubseteq> f \<bottom>"
proof -
have "f (f \<bottom>) .= f \<bottom>"
by (auto simp add: fps_def fb assms(3) idempotent)
moreover have "f (f \<bottom>) \<in> 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 \<bottom> \<sqsubseteq> LFP f"
proof -
have "f \<bottom> \<sqsubseteq> 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) \<in> carrier L"
by (auto)
ultimately show ?thesis
using fb by blast
qed
qed
subsubsection \<open>Greatest fixed points\<close>
lemma GFP_closed [intro, simp]:
"GFP f \<in> carrier L"
by (auto intro:sup_closed simp add:GREATEST_FP_def)
lemma GFP_upperbound:
assumes "x \<in> carrier L" "x \<sqsubseteq> f x"
shows "x \<sqsubseteq> GFP f"
by (auto intro:sup_upper assms simp add:GREATEST_FP_def)
lemma GFP_least:
assumes "x \<in> carrier L"
"(\<And>u. \<lbrakk> u \<in> carrier L; u \<sqsubseteq> f u \<rbrakk> \<Longrightarrow> u \<sqsubseteq> x)"
shows "GFP f \<sqsubseteq> x"
by (auto simp add:GREATEST_FP_def intro:sup_least assms)
lemma GFP_lemma2:
assumes "Mono f" "f \<in> carrier L \<rightarrow> carrier L"
shows "GFP f \<sqsubseteq> f (GFP f)"
using assms
apply (auto simp add:Pi_def)
apply (rule GFP_least)
apply (metis GFP_closed)
apply (metis GFP_closed GFP_upperbound le_trans use_iso2)
done
lemma GFP_lemma3:
assumes "Mono f" "f \<in> carrier L \<rightarrow> carrier L"
shows "f (GFP f) \<sqsubseteq> GFP f"
by (metis GFP_closed GFP_lemma2 GFP_upperbound assms funcset_mem use_iso2)
lemma GFP_weak_unfold:
"\<lbrakk> Mono f; f \<in> carrier L \<rightarrow> carrier L \<rbrakk> \<Longrightarrow> 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 \<in> carrier L \<rightarrow> carrier L"
shows "GFP f \<in> fps L f"
using assms
proof -
have "f (GFP f) \<in> 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 \<in> carrier L \<rightarrow> carrier L" "x \<in> fps L f"
shows "x \<sqsubseteq> 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 \<in> carrier L \<rightarrow> carrier L" "Mono f" "Idem f"
shows "GFP f .= (f \<top>)"
proof (rule weak_le_antisym)
from assms(1) show fb: "f \<top> \<in> carrier L"
by (rule funcset_mem, simp)
from assms show mf: "GFP f \<in> carrier L"
by blast
show "f \<top> \<sqsubseteq> GFP f"
proof -
have "f (f \<top>) .= f \<top>"
by (auto simp add: fps_def fb assms(3) idempotent)
moreover have "f (f \<top>) \<in> 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 \<sqsubseteq> f \<top>"
proof -
have "GFP f \<sqsubseteq> f (GFP f)"
by (simp add: GFP_lemma2 assms(1) assms(2))
moreover have "... \<sqsubseteq> f \<top>"
by (auto intro: use_iso1[of _ f] simp add: assms)
moreover from assms(1) have "f (GFP f) \<in> carrier L"
by (auto)
ultimately show ?thesis
using fb local.le_trans by blast
qed
qed
end
subsection \<open>Complete lattices where @{text eq} is the Equality\<close>
locale complete_lattice = partial_order +
assumes sup_exists:
"[| A \<subseteq> carrier L |] ==> \<exists>s. least L s (Upper L A)"
and inf_exists:
"[| A \<subseteq> carrier L |] ==> \<exists>i. greatest L i (Lower L A)"
sublocale complete_lattice \<subseteq> lattice
proof
fix x y
assume a: "x \<in> carrier L" "y \<in> carrier L"
thus "\<exists>s. is_lub L s {x, y}"
by (rule_tac sup_exists[of "{x, y}"], auto)
from a show "\<exists>s. is_glb L s {x, y}"
by (rule_tac inf_exists[of "{x, y}"], auto)
qed
sublocale complete_lattice \<subseteq> 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 \<open>Introduction rule: the usual definition of complete lattice\<close>
lemma (in partial_order) complete_latticeI:
assumes sup_exists:
"!!A. [| A \<subseteq> carrier L |] ==> \<exists>s. least L s (Upper L A)"
and inf_exists:
"!!A. [| A \<subseteq> carrier L |] ==> \<exists>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: "\<exists>g. greatest L g (carrier L)"
and inf_exists:
"!!A. [| A \<subseteq> carrier L; A \<noteq> {} |] ==> \<exists>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 \<subseteq> carrier L"
let ?B = "Upper L A"
from L top have "top \<in> ?B" by (fast intro!: Upper_memI intro: greatest_le)
then have B_non_empty: "?B \<noteq> {}" by fast
have B_L: "?B \<subseteq> 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)" ..
have "least L b (Upper L A)"
apply (rule least_UpperI)
apply (rule greatest_le [where A = "Lower L ?B"])
apply (rule b_inf_B)
apply (rule Lower_memI)
apply (erule Upper_memD [THEN conjunct1])
apply assumption
apply (rule L)
apply (fast intro: L [THEN subsetD])
apply (erule greatest_Lower_below [OF b_inf_B])
apply simp
apply (rule L)
apply (rule greatest_closed [OF b_inf_B])
done
then show "\<exists>s. least L s (Upper L A)" ..
next
fix A
assume L: "A \<subseteq> carrier L"
show "\<exists>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 \<open>Fixed points\<close>
context complete_lattice
begin
lemma LFP_unfold:
"\<lbrakk> Mono f; f \<in> carrier L \<rightarrow> carrier L \<rbrakk> \<Longrightarrow> LFP f = f (LFP f)"
using eq_is_equal weak.LFP_weak_unfold by auto
lemma LFP_const:
"t \<in> carrier L \<Longrightarrow> LFP (\<lambda> x. t) = t"
by (simp add: local.le_antisym weak.LFP_greatest weak.LFP_lowerbound)
lemma LFP_id:
"LFP id = \<bottom>"
by (simp add: local.le_antisym weak.LFP_lowerbound)
lemma GFP_unfold:
"\<lbrakk> Mono f; f \<in> carrier L \<rightarrow> carrier L \<rbrakk> \<Longrightarrow> GFP f = f (GFP f)"
using eq_is_equal weak.GFP_weak_unfold by auto
lemma GFP_const:
"t \<in> carrier L \<Longrightarrow> GFP (\<lambda> x. t) = t"
by (simp add: local.le_antisym weak.GFP_least weak.GFP_upperbound)
lemma GFP_id:
"GFP id = \<top>"
using weak.GFP_upperbound by auto
end
subsection \<open>Interval complete lattices\<close>
context weak_complete_lattice
begin
lemma at_least_at_most_Sup:
"\<lbrakk> a \<in> carrier L; b \<in> carrier L; a \<sqsubseteq> b \<rbrakk> \<Longrightarrow> \<Squnion> \<lbrace>a..b\<rbrace> .= b"
apply (rule weak_le_antisym)
apply (rule sup_least)
apply (auto simp add: at_least_at_most_closed)
apply (rule sup_upper)
apply (auto simp add: at_least_at_most_closed)
done
lemma at_least_at_most_Inf:
"\<lbrakk> a \<in> carrier L; b \<in> carrier L; a \<sqsubseteq> b \<rbrakk> \<Longrightarrow> \<Sqinter> \<lbrace>a..b\<rbrace> .= a"
apply (rule weak_le_antisym)
apply (rule inf_lower)
apply (auto simp add: at_least_at_most_closed)
apply (rule inf_greatest)
apply (auto simp add: at_least_at_most_closed)
done
end
lemma weak_complete_lattice_interval:
assumes "weak_complete_lattice L" "a \<in> carrier L" "b \<in> carrier L" "a \<sqsubseteq>\<^bsub>L\<^esub> b"
shows "weak_complete_lattice (L \<lparr> carrier := \<lbrace>a..b\<rbrace>\<^bsub>L\<^esub> \<rparr>)"
proof -
interpret L: weak_complete_lattice L
by (simp add: assms)
interpret weak_partial_order "L \<lparr> carrier := \<lbrace>a..b\<rbrace>\<^bsub>L\<^esub> \<rparr>"
proof -
have "\<lbrace>a..b\<rbrace>\<^bsub>L\<^esub> \<subseteq> carrier L"
by (auto, simp add: at_least_at_most_def)
thus "weak_partial_order (L\<lparr>carrier := \<lbrace>a..b\<rbrace>\<^bsub>L\<^esub>\<rparr>)"
by (simp add: L.weak_partial_order_axioms weak_partial_order_subset)
qed
show ?thesis
proof
fix A
assume a: "A \<subseteq> carrier (L\<lparr>carrier := \<lbrace>a..b\<rbrace>\<^bsub>L\<^esub>\<rparr>)"
show "\<exists>s. is_lub (L\<lparr>carrier := \<lbrace>a..b\<rbrace>\<^bsub>L\<^esub>\<rparr>) 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="\<Squnion>\<^bsub>L\<^esub> A" in exI, rule least_UpperI, simp_all)
show b:"\<And> x. x \<in> A \<Longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> \<Squnion>\<^bsub>L\<^esub>A"
using a by (auto intro: L.sup_upper, meson L.at_least_at_most_closed L.sup_upper subset_trans)
show "\<And>y. y \<in> Upper (L\<lparr>carrier := \<lbrace>a..b\<rbrace>\<^bsub>L\<^esub>\<rparr>) A \<Longrightarrow> \<Squnion>\<^bsub>L\<^esub>A \<sqsubseteq>\<^bsub>L\<^esub> 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 \<subseteq> \<lbrace>a..b\<rbrace>\<^bsub>L\<^esub>"
by (auto)
from a show "\<Squnion>\<^bsub>L\<^esub>A \<in> \<lbrace>a..b\<rbrace>\<^bsub>L\<^esub>"
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 "\<exists>s. is_glb (L\<lparr>carrier := \<lbrace>a..b\<rbrace>\<^bsub>L\<^esub>\<rparr>) 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="\<Sqinter>\<^bsub>L\<^esub> A" in exI, rule greatest_LowerI, simp_all)
show b:"\<And>x. x \<in> A \<Longrightarrow> \<Sqinter>\<^bsub>L\<^esub>A \<sqsubseteq>\<^bsub>L\<^esub> x"
using a L.at_least_at_most_closed by (force intro!: L.inf_lower)
show "\<And>y. y \<in> Lower (L\<lparr>carrier := \<lbrace>a..b\<rbrace>\<^bsub>L\<^esub>\<rparr>) A \<Longrightarrow> y \<sqsubseteq>\<^bsub>L\<^esub> \<Sqinter>\<^bsub>L\<^esub>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 \<subseteq> \<lbrace>a..b\<rbrace>\<^bsub>L\<^esub>"
by (auto)
from a show "\<Sqinter>\<^bsub>L\<^esub>A \<in> \<lbrace>a..b\<rbrace>\<^bsub>L\<^esub>"
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 \<open>Knaster-Tarski theorem and variants\<close>
text \<open>The set of fixed points of a complete lattice is itself a complete lattice\<close>
theorem Knaster_Tarski:
assumes "weak_complete_lattice L" "f \<in> carrier L \<rightarrow> 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 \<in> carrier L. f x .=\<^bsub>L\<^esub> x} \<subseteq> 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 \<subseteq> fps L f"
show "\<exists>s. is_lub (fpl L f) s A"
proof
from A have AL: "A \<subseteq> carrier L"
by (meson fps_carrier subset_eq)
let ?w = "\<Squnion>\<^bsub>L\<^esub> A"
have w: "f (\<Squnion>\<^bsub>L\<^esub>A) \<in> carrier L"
by (rule funcset_mem[of f "carrier L"], simp_all add: AL assms(2))
have pf_w: "(\<Squnion>\<^bsub>L\<^esub> A) \<sqsubseteq>\<^bsub>L\<^esub> f (\<Squnion>\<^bsub>L\<^esub> A)"
by (simp add: A L.weak_sup_pre_fixed_point assms(2) assms(3))
have f_top_chain: "f ` \<lbrace>?w..\<top>\<^bsub>L\<^esub>\<rbrace>\<^bsub>L\<^esub> \<subseteq> \<lbrace>?w..\<top>\<^bsub>L\<^esub>\<rbrace>\<^bsub>L\<^esub>"
proof (auto simp add: at_least_at_most_def)
fix x
assume b: "x \<in> carrier L" "\<Squnion>\<^bsub>L\<^esub>A \<sqsubseteq>\<^bsub>L\<^esub> x"
from b show fx: "f x \<in> carrier L"
using assms(2) by blast
show "\<Squnion>\<^bsub>L\<^esub>A \<sqsubseteq>\<^bsub>L\<^esub> f x"
proof -
have "?w \<sqsubseteq>\<^bsub>L\<^esub> f ?w"
proof (rule_tac L.sup_least, simp_all add: AL w)
fix y
assume c: "y \<in> A"
hence y: "y \<in> fps L f"
using A subsetCE by blast
with assms have "y .=\<^bsub>L\<^esub> f y"
proof -
from y have "y \<in> carrier L"
by (simp add: fps_def)
moreover hence "f y \<in> 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 \<sqsubseteq>\<^bsub>L\<^esub> \<Squnion>\<^bsub>L\<^esub>A"
by (simp add: AL L.sup_upper c(1))
ultimately show "y \<sqsubseteq>\<^bsub>L\<^esub> f (\<Squnion>\<^bsub>L\<^esub>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 \<sqsubseteq>\<^bsub>L\<^esub> \<top>\<^bsub>L\<^esub>"
by (simp add: fx)
qed
let ?L' = "L\<lparr> carrier := \<lbrace>?w..\<top>\<^bsub>L\<^esub>\<rbrace>\<^bsub>L\<^esub> \<rparr>"
interpret L': weak_complete_lattice ?L'
by (auto intro: weak_complete_lattice_interval simp add: L.weak_complete_lattice_axioms AL)
let ?L'' = "L\<lparr> carrier := fps L f \<rparr>"
show "is_lub ?L'' (LFP\<^bsub>?L'\<^esub> f) A"
proof (rule least_UpperI, simp_all)
fix x
assume "x \<in> Upper ?L'' A"
hence "LFP\<^bsub>?L'\<^esub> f \<sqsubseteq>\<^bsub>?L'\<^esub> 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\<^bsub>?L'\<^esub> f \<sqsubseteq>\<^bsub>L\<^esub> x"
by (simp)
next
fix x
assume xA: "x \<in> A"
show "x \<sqsubseteq>\<^bsub>L\<^esub> LFP\<^bsub>?L'\<^esub> f"
proof -
have "LFP\<^bsub>?L'\<^esub> f \<in> 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 \<subseteq> fps L f"
by (simp add: A)
next
show "LFP\<^bsub>?L'\<^esub> f \<in> fps L f"
proof (auto simp add: fps_def)
have "LFP\<^bsub>?L'\<^esub> f \<in> carrier ?L'"
by (rule L'.LFP_closed)
thus c:"LFP\<^bsub>?L'\<^esub> f \<in> carrier L"
by (auto simp add: at_least_at_most_def)
have "LFP\<^bsub>?L'\<^esub> f .=\<^bsub>?L'\<^esub> f (LFP\<^bsub>?L'\<^esub> f)"
proof (rule "L'.LFP_weak_unfold", simp_all)
show "f \<in> \<lbrace>\<Squnion>\<^bsub>L\<^esub>A..\<top>\<^bsub>L\<^esub>\<rbrace>\<^bsub>L\<^esub> \<rightarrow> \<lbrace>\<Squnion>\<^bsub>L\<^esub>A..\<top>\<^bsub>L\<^esub>\<rbrace>\<^bsub>L\<^esub>"
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\<^bsub>L\<lparr>carrier := \<lbrace>\<Squnion>\<^bsub>L\<^esub>A..\<top>\<^bsub>L\<^esub>\<rbrace>\<^bsub>L\<^esub>\<rparr>\<^esub> 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\<^bsub>?L'\<^esub> f) .=\<^bsub>L\<^esub> LFP\<^bsub>?L'\<^esub> f"
by (simp add: L.equivalence_axioms funcset_carrier' c assms(2) equivalence.sym)
qed
qed
qed
show "\<exists>i. is_glb (L\<lparr>carrier := fps L f\<rparr>) i A"
proof
from A have AL: "A \<subseteq> carrier L"
by (meson fps_carrier subset_eq)
let ?w = "\<Sqinter>\<^bsub>L\<^esub> A"
have w: "f (\<Sqinter>\<^bsub>L\<^esub>A) \<in> carrier L"
by (simp add: AL funcset_carrier' assms(2))
have pf_w: "f (\<Sqinter>\<^bsub>L\<^esub> A) \<sqsubseteq>\<^bsub>L\<^esub> (\<Sqinter>\<^bsub>L\<^esub> A)"
by (simp add: A L.weak_sup_post_fixed_point assms(2) assms(3))
have f_bot_chain: "f ` \<lbrace>\<bottom>\<^bsub>L\<^esub>..?w\<rbrace>\<^bsub>L\<^esub> \<subseteq> \<lbrace>\<bottom>\<^bsub>L\<^esub>..?w\<rbrace>\<^bsub>L\<^esub>"
proof (auto simp add: at_least_at_most_def)
fix x
assume b: "x \<in> carrier L" "x \<sqsubseteq>\<^bsub>L\<^esub> \<Sqinter>\<^bsub>L\<^esub>A"
from b show fx: "f x \<in> carrier L"
using assms(2) by blast
show "f x \<sqsubseteq>\<^bsub>L\<^esub> \<Sqinter>\<^bsub>L\<^esub>A"
proof -
have "f ?w \<sqsubseteq>\<^bsub>L\<^esub> ?w"
proof (rule_tac L.inf_greatest, simp_all add: AL w)
fix y
assume c: "y \<in> A"
with assms have "y .=\<^bsub>L\<^esub> f y"
by (metis (no_types, lifting) A funcset_carrier'[OF assms(2)] L.sym fps_def mem_Collect_eq subset_eq)
moreover have "\<Sqinter>\<^bsub>L\<^esub>A \<sqsubseteq>\<^bsub>L\<^esub> y"
by (simp add: AL L.inf_lower c)
ultimately show "f (\<Sqinter>\<^bsub>L\<^esub>A) \<sqsubseteq>\<^bsub>L\<^esub> 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 "\<bottom>\<^bsub>L\<^esub> \<sqsubseteq>\<^bsub>L\<^esub> f x"
by (simp add: fx)
qed
let ?L' = "L\<lparr> carrier := \<lbrace>\<bottom>\<^bsub>L\<^esub>..?w\<rbrace>\<^bsub>L\<^esub> \<rparr>"
interpret L': weak_complete_lattice ?L'
by (auto intro!: weak_complete_lattice_interval simp add: L.weak_complete_lattice_axioms AL)
let ?L'' = "L\<lparr> carrier := fps L f \<rparr>"
show "is_glb ?L'' (GFP\<^bsub>?L'\<^esub> f) A"
proof (rule greatest_LowerI, simp_all)
fix x
assume "x \<in> Lower ?L'' A"
hence "x \<sqsubseteq>\<^bsub>?L'\<^esub> GFP\<^bsub>?L'\<^esub> 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 \<sqsubseteq>\<^bsub>L\<^esub> GFP\<^bsub>?L'\<^esub> f"
by (simp)
next
fix x
assume xA: "x \<in> A"
show "GFP\<^bsub>?L'\<^esub> f \<sqsubseteq>\<^bsub>L\<^esub> x"
proof -
have "GFP\<^bsub>?L'\<^esub> f \<in> 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 \<subseteq> fps L f"
by (simp add: A)
next
show "GFP\<^bsub>?L'\<^esub> f \<in> fps L f"
proof (auto simp add: fps_def)
have "GFP\<^bsub>?L'\<^esub> f \<in> carrier ?L'"
by (rule L'.GFP_closed)
thus c:"GFP\<^bsub>?L'\<^esub> f \<in> carrier L"
by (auto simp add: at_least_at_most_def)
have "GFP\<^bsub>?L'\<^esub> f .=\<^bsub>?L'\<^esub> f (GFP\<^bsub>?L'\<^esub> f)"
proof (rule "L'.GFP_weak_unfold", simp_all)
show "f \<in> \<lbrace>\<bottom>\<^bsub>L\<^esub>..?w\<rbrace>\<^bsub>L\<^esub> \<rightarrow> \<lbrace>\<bottom>\<^bsub>L\<^esub>..?w\<rbrace>\<^bsub>L\<^esub>"
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\<^bsub>L\<lparr>carrier := \<lbrace>\<bottom>\<^bsub>L\<^esub>..?w\<rbrace>\<^bsub>L\<^esub>\<rparr>\<^esub> 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\<^bsub>?L'\<^esub> f) .=\<^bsub>L\<^esub> GFP\<^bsub>?L'\<^esub> 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 \<in> carrier L \<rightarrow> carrier L"
shows "\<top>\<^bsub>fpl L f\<^esub> .=\<^bsub>L\<^esub> GFP\<^bsub>L\<^esub> 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 "\<top>\<^bsub>fpl L f\<^esub> \<sqsubseteq>\<^bsub>L\<^esub> GFP\<^bsub>L\<^esub> f"
by (rule L.GFP_greatest_fixed_point, simp_all add: assms L'.top_closed[simplified])
show "GFP\<^bsub>L\<^esub> f \<sqsubseteq>\<^bsub>L\<^esub> \<top>\<^bsub>fpl L f\<^esub>"
proof -
have "GFP\<^bsub>L\<^esub> f \<in> fps L f"
by (rule L.GFP_fixed_point, simp_all add: assms)
hence "GFP\<^bsub>L\<^esub> f \<in> carrier (fpl L f)"
by simp
hence "GFP\<^bsub>L\<^esub> f \<sqsubseteq>\<^bsub>fpl L f\<^esub> \<top>\<^bsub>fpl L f\<^esub>"
by (rule L'.top_higher)
thus ?thesis
by simp
qed
show "\<top>\<^bsub>fpl L f\<^esub> \<in> carrier L"
proof -
have "carrier (fpl L f) \<subseteq> 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 \<in> carrier L \<rightarrow> carrier L"
shows "\<bottom>\<^bsub>fpl L f\<^esub> .=\<^bsub>L\<^esub> LFP\<^bsub>L\<^esub> 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\<^bsub>L\<^esub> f \<sqsubseteq>\<^bsub>L\<^esub> \<bottom>\<^bsub>fpl L f\<^esub>"
by (rule L.LFP_least_fixed_point, simp_all add: assms L'.bottom_closed[simplified])
show "\<bottom>\<^bsub>fpl L f\<^esub> \<sqsubseteq>\<^bsub>L\<^esub> LFP\<^bsub>L\<^esub> f"
proof -
have "LFP\<^bsub>L\<^esub> f \<in> fps L f"
by (rule L.LFP_fixed_point, simp_all add: assms)
hence "LFP\<^bsub>L\<^esub> f \<in> carrier (fpl L f)"
by simp
hence "\<bottom>\<^bsub>fpl L f\<^esub> \<sqsubseteq>\<^bsub>fpl L f\<^esub> LFP\<^bsub>L\<^esub> f"
by (rule L'.bottom_lower)
thus ?thesis
by simp
qed
show "\<bottom>\<^bsub>fpl L f\<^esub> \<in> carrier L"
proof -
have "carrier (fpl L f) \<subseteq> carrier L"
by (auto simp add: fps_def)
with L'.bottom_closed show ?thesis
by blast
qed
qed
qed
text \<open>If a function is both idempotent and isotone then the image of the function forms a complete lattice\<close>
theorem Knaster_Tarski_idem:
assumes "complete_lattice L" "f \<in> carrier L \<rightarrow> carrier L" "isotone L L f" "idempotent L f"
shows "complete_lattice (L\<lparr>carrier := f ` carrier L\<rparr>)"
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\<lparr>carrier := f ` carrier L\<rparr>)"
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 \<in> carrier L \<rightarrow> carrier L"
shows "\<top>\<^bsub>fpl L f\<^esub> .=\<^bsub>L\<^esub> f (\<top>\<^bsub>L\<^esub>)" "\<bottom>\<^bsub>fpl L f\<^esub> .=\<^bsub>L\<^esub> f (\<bottom>\<^bsub>L\<^esub>)"
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 \<subseteq> carrier L"
by (auto simp add: fps_def)
show "\<top>\<^bsub>fpl L f\<^esub> .=\<^bsub>L\<^esub> f (\<top>\<^bsub>L\<^esub>)"
proof -
from FA have "\<top>\<^bsub>fpl L f\<^esub> \<in> carrier L"
proof -
have "\<top>\<^bsub>fpl L f\<^esub> \<in> fps L f"
using L'.top_closed by auto
thus ?thesis
using FA by blast
qed
moreover with assms have "f \<top>\<^bsub>L\<^esub> \<in> 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 "\<bottom>\<^bsub>fpl L f\<^esub> .=\<^bsub>L\<^esub> f (\<bottom>\<^bsub>L\<^esub>)"
proof -
from FA have "\<bottom>\<^bsub>fpl L f\<^esub> \<in> carrier L"
proof -
have "\<bottom>\<^bsub>fpl L f\<^esub> \<in> fps L f"
using L'.bottom_closed by auto
thus ?thesis
using FA by blast
qed
moreover with assms have "f \<bottom>\<^bsub>L\<^esub> \<in> 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 \<in> carrier L \<rightarrow> carrier L"
"A \<subseteq> fps L f"
shows "\<Sqinter>\<^bsub>fpl L f\<^esub> A .=\<^bsub>L\<^esub> f (\<Sqinter>\<^bsub>L\<^esub> 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 \<subseteq> carrier L"
by (auto simp add: fps_def)
have A: "A \<subseteq> carrier L"
using FA assms(5) by blast
have fA: "f (\<Sqinter>\<^bsub>L\<^esub>A) \<in> 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: "\<Sqinter>\<^bsub>fpl L f\<^esub>A \<in> fps L f"
by (rule L'.inf_closed[simplified], simp add: assms)
show ?thesis
proof (rule L.weak_le_antisym)
show ic: "\<Sqinter>\<^bsub>fpl L f\<^esub>A \<in> carrier L"
using FA infA by blast
show fc: "f (\<Sqinter>\<^bsub>L\<^esub>A) \<in> carrier L"
using FA fA by blast
show "f (\<Sqinter>\<^bsub>L\<^esub>A) \<sqsubseteq>\<^bsub>L\<^esub> \<Sqinter>\<^bsub>fpl L f\<^esub>A"
proof -
have "\<And>x. x \<in> A \<Longrightarrow> f (\<Sqinter>\<^bsub>L\<^esub>A) \<sqsubseteq>\<^bsub>L\<^esub> 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 (\<Sqinter>\<^bsub>L\<^esub>A) \<sqsubseteq>\<^bsub>fpl L f\<^esub> \<Sqinter>\<^bsub>fpl L f\<^esub>A"
by (rule_tac L'.inf_greatest, simp_all add: fA assms(3,5))
thus ?thesis
by (simp)
qed
show "\<Sqinter>\<^bsub>fpl L f\<^esub>A \<sqsubseteq>\<^bsub>L\<^esub> f (\<Sqinter>\<^bsub>L\<^esub>A)"
proof -
have "\<And>x. x \<in> A \<Longrightarrow> \<Sqinter>\<^bsub>fpl L f\<^esub>A \<sqsubseteq>\<^bsub>fpl L f\<^esub> x"
by (rule L'.inf_lower, simp_all add: assms)
hence "\<Sqinter>\<^bsub>fpl L f\<^esub>A \<sqsubseteq>\<^bsub>L\<^esub> (\<Sqinter>\<^bsub>L\<^esub>A)"
apply (rule_tac L.inf_greatest, simp_all add: A)
using FA infA apply blast
done
hence 1:"f(\<Sqinter>\<^bsub>fpl L f\<^esub>A) \<sqsubseteq>\<^bsub>L\<^esub> f(\<Sqinter>\<^bsub>L\<^esub>A)"
by (metis (no_types, lifting) A FA L.inf_closed assms(2) infA subsetCE use_iso1)
have 2:"\<Sqinter>\<^bsub>fpl L f\<^esub>A \<sqsubseteq>\<^bsub>L\<^esub> f (\<Sqinter>\<^bsub>fpl L f\<^esub>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 \<open>Examples\<close>
subsubsection \<open>The Powerset of a Set is a Complete Lattice\<close>
theorem powerset_is_complete_lattice:
"complete_lattice \<lparr>carrier = Pow A, eq = op =, le = op \<subseteq>\<rparr>"
(is "complete_lattice ?L")
proof (rule partial_order.complete_latticeI)
show "partial_order ?L"
by standard auto
next
fix B
assume "B \<subseteq> carrier ?L"
then have "least ?L (\<Union> B) (Upper ?L B)"
by (fastforce intro!: least_UpperI simp: Upper_def)
then show "\<exists>s. least ?L s (Upper ?L B)" ..
next
fix B
assume "B \<subseteq> carrier ?L"
then have "greatest ?L (\<Inter> B \<inter> A) (Lower ?L B)"
txt \<open>@{term "\<Inter> B"} is not the infimum of @{term B}:
@{term "\<Inter> {} = UNIV"} which is in general bigger than @{term "A"}! \<close>
by (fastforce intro!: greatest_LowerI simp: Lower_def)
then show "\<exists>i. greatest ?L i (Lower ?L B)" ..
qed
text \<open>Another example, that of the lattice of subgroups of a group,
can be found in Group theory (Section~\ref{sec:subgroup-lattice}).\<close>
subsection \<open>Limit preserving functions\<close>
definition weak_sup_pres :: "('a, 'c) gorder_scheme \<Rightarrow> ('b, 'd) gorder_scheme \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool" where
"weak_sup_pres X Y f \<equiv> complete_lattice X \<and> complete_lattice Y \<and> (\<forall> A \<subseteq> carrier X. A \<noteq> {} \<longrightarrow> f (\<Squnion>\<^bsub>X\<^esub> A) = (\<Squnion>\<^bsub>Y\<^esub> (f ` A)))"
definition sup_pres :: "('a, 'c) gorder_scheme \<Rightarrow> ('b, 'd) gorder_scheme \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool" where
"sup_pres X Y f \<equiv> complete_lattice X \<and> complete_lattice Y \<and> (\<forall> A \<subseteq> carrier X. f (\<Squnion>\<^bsub>X\<^esub> A) = (\<Squnion>\<^bsub>Y\<^esub> (f ` A)))"
definition weak_inf_pres :: "('a, 'c) gorder_scheme \<Rightarrow> ('b, 'd) gorder_scheme \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool" where
"weak_inf_pres X Y f \<equiv> complete_lattice X \<and> complete_lattice Y \<and> (\<forall> A \<subseteq> carrier X. A \<noteq> {} \<longrightarrow> f (\<Sqinter>\<^bsub>X\<^esub> A) = (\<Sqinter>\<^bsub>Y\<^esub> (f ` A)))"
definition inf_pres :: "('a, 'c) gorder_scheme \<Rightarrow> ('b, 'd) gorder_scheme \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool" where
"inf_pres X Y f \<equiv> complete_lattice X \<and> complete_lattice Y \<and> (\<forall> A \<subseteq> carrier X. f (\<Sqinter>\<^bsub>X\<^esub> A) = (\<Sqinter>\<^bsub>Y\<^esub> (f ` A)))"
lemma weak_sup_pres:
"sup_pres X Y f \<Longrightarrow> 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 \<Longrightarrow> 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