(* Title: HOL/Library/Lubs_Glbs.thy
Author: Jacques D. Fleuriot, University of Cambridge
Author: Amine Chaieb, University of Cambridge *)
header {* Definitions of Least Upper Bounds and Greatest Lower Bounds *}
theory Lubs_Glbs
imports Complex_Main
begin
text {* Thanks to suggestions by James Margetson *}
definition setle :: "'a set \<Rightarrow> 'a::ord \<Rightarrow> bool" (infixl "*<=" 70)
where "S *<= x = (ALL y: S. y \<le> x)"
definition setge :: "'a::ord \<Rightarrow> 'a set \<Rightarrow> bool" (infixl "<=*" 70)
where "x <=* S = (ALL y: S. x \<le> y)"
subsection {* Rules for the Relations @{text "*<="} and @{text "<=*"} *}
lemma setleI: "ALL y: S. y \<le> x \<Longrightarrow> S *<= x"
by (simp add: setle_def)
lemma setleD: "S *<= x \<Longrightarrow> y: S \<Longrightarrow> y \<le> x"
by (simp add: setle_def)
lemma setgeI: "ALL y: S. x \<le> y \<Longrightarrow> x <=* S"
by (simp add: setge_def)
lemma setgeD: "x <=* S \<Longrightarrow> y: S \<Longrightarrow> x \<le> y"
by (simp add: setge_def)
definition leastP :: "('a \<Rightarrow> bool) \<Rightarrow> 'a::ord \<Rightarrow> bool"
where "leastP P x = (P x \<and> x <=* Collect P)"
definition isUb :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
where "isUb R S x = (S *<= x \<and> x: R)"
definition isLub :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
where "isLub R S x = leastP (isUb R S) x"
definition ubs :: "'a set \<Rightarrow> 'a::ord set \<Rightarrow> 'a set"
where "ubs R S = Collect (isUb R S)"
subsection {* Rules about the Operators @{term leastP}, @{term ub} and @{term lub} *}
lemma leastPD1: "leastP P x \<Longrightarrow> P x"
by (simp add: leastP_def)
lemma leastPD2: "leastP P x \<Longrightarrow> x <=* Collect P"
by (simp add: leastP_def)
lemma leastPD3: "leastP P x \<Longrightarrow> y: Collect P \<Longrightarrow> x \<le> y"
by (blast dest!: leastPD2 setgeD)
lemma isLubD1: "isLub R S x \<Longrightarrow> S *<= x"
by (simp add: isLub_def isUb_def leastP_def)
lemma isLubD1a: "isLub R S x \<Longrightarrow> x: R"
by (simp add: isLub_def isUb_def leastP_def)
lemma isLub_isUb: "isLub R S x \<Longrightarrow> isUb R S x"
unfolding isUb_def by (blast dest: isLubD1 isLubD1a)
lemma isLubD2: "isLub R S x \<Longrightarrow> y : S \<Longrightarrow> y \<le> x"
by (blast dest!: isLubD1 setleD)
lemma isLubD3: "isLub R S x \<Longrightarrow> leastP (isUb R S) x"
by (simp add: isLub_def)
lemma isLubI1: "leastP(isUb R S) x \<Longrightarrow> isLub R S x"
by (simp add: isLub_def)
lemma isLubI2: "isUb R S x \<Longrightarrow> x <=* Collect (isUb R S) \<Longrightarrow> isLub R S x"
by (simp add: isLub_def leastP_def)
lemma isUbD: "isUb R S x \<Longrightarrow> y : S \<Longrightarrow> y \<le> x"
by (simp add: isUb_def setle_def)
lemma isUbD2: "isUb R S x \<Longrightarrow> S *<= x"
by (simp add: isUb_def)
lemma isUbD2a: "isUb R S x \<Longrightarrow> x: R"
by (simp add: isUb_def)
lemma isUbI: "S *<= x \<Longrightarrow> x: R \<Longrightarrow> isUb R S x"
by (simp add: isUb_def)
lemma isLub_le_isUb: "isLub R S x \<Longrightarrow> isUb R S y \<Longrightarrow> x \<le> y"
unfolding isLub_def by (blast intro!: leastPD3)
lemma isLub_ubs: "isLub R S x \<Longrightarrow> x <=* ubs R S"
unfolding ubs_def isLub_def by (rule leastPD2)
lemma isLub_unique: "[| isLub R S x; isLub R S y |] ==> x = (y::'a::linorder)"
apply (frule isLub_isUb)
apply (frule_tac x = y in isLub_isUb)
apply (blast intro!: order_antisym dest!: isLub_le_isUb)
done
lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"
by (simp add: isUbI setleI)
definition greatestP :: "('a \<Rightarrow> bool) \<Rightarrow> 'a::ord \<Rightarrow> bool"
where "greatestP P x = (P x \<and> Collect P *<= x)"
definition isLb :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
where "isLb R S x = (x <=* S \<and> x: R)"
definition isGlb :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
where "isGlb R S x = greatestP (isLb R S) x"
definition lbs :: "'a set \<Rightarrow> 'a::ord set \<Rightarrow> 'a set"
where "lbs R S = Collect (isLb R S)"
subsection {* Rules about the Operators @{term greatestP}, @{term isLb} and @{term isGlb} *}
lemma greatestPD1: "greatestP P x \<Longrightarrow> P x"
by (simp add: greatestP_def)
lemma greatestPD2: "greatestP P x \<Longrightarrow> Collect P *<= x"
by (simp add: greatestP_def)
lemma greatestPD3: "greatestP P x \<Longrightarrow> y: Collect P \<Longrightarrow> x \<ge> y"
by (blast dest!: greatestPD2 setleD)
lemma isGlbD1: "isGlb R S x \<Longrightarrow> x <=* S"
by (simp add: isGlb_def isLb_def greatestP_def)
lemma isGlbD1a: "isGlb R S x \<Longrightarrow> x: R"
by (simp add: isGlb_def isLb_def greatestP_def)
lemma isGlb_isLb: "isGlb R S x \<Longrightarrow> isLb R S x"
unfolding isLb_def by (blast dest: isGlbD1 isGlbD1a)
lemma isGlbD2: "isGlb R S x \<Longrightarrow> y : S \<Longrightarrow> y \<ge> x"
by (blast dest!: isGlbD1 setgeD)
lemma isGlbD3: "isGlb R S x \<Longrightarrow> greatestP (isLb R S) x"
by (simp add: isGlb_def)
lemma isGlbI1: "greatestP (isLb R S) x \<Longrightarrow> isGlb R S x"
by (simp add: isGlb_def)
lemma isGlbI2: "isLb R S x \<Longrightarrow> Collect (isLb R S) *<= x \<Longrightarrow> isGlb R S x"
by (simp add: isGlb_def greatestP_def)
lemma isLbD: "isLb R S x \<Longrightarrow> y : S \<Longrightarrow> y \<ge> x"
by (simp add: isLb_def setge_def)
lemma isLbD2: "isLb R S x \<Longrightarrow> x <=* S "
by (simp add: isLb_def)
lemma isLbD2a: "isLb R S x \<Longrightarrow> x: R"
by (simp add: isLb_def)
lemma isLbI: "x <=* S \<Longrightarrow> x: R \<Longrightarrow> isLb R S x"
by (simp add: isLb_def)
lemma isGlb_le_isLb: "isGlb R S x \<Longrightarrow> isLb R S y \<Longrightarrow> x \<ge> y"
unfolding isGlb_def by (blast intro!: greatestPD3)
lemma isGlb_ubs: "isGlb R S x \<Longrightarrow> lbs R S *<= x"
unfolding lbs_def isGlb_def by (rule greatestPD2)
lemma isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::'a::linorder)"
apply (frule isGlb_isLb)
apply (frule_tac x = y in isGlb_isLb)
apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
done
lemma bdd_above_setle: "bdd_above A \<longleftrightarrow> (\<exists>a. A *<= a)"
by (auto simp: bdd_above_def setle_def)
lemma bdd_below_setge: "bdd_below A \<longleftrightarrow> (\<exists>a. a <=* A)"
by (auto simp: bdd_below_def setge_def)
lemma isLub_cSup:
"(S::'a :: conditionally_complete_lattice set) \<noteq> {} \<Longrightarrow> (\<exists>b. S *<= b) \<Longrightarrow> isLub UNIV S (Sup S)"
by (auto simp add: isLub_def setle_def leastP_def isUb_def
intro!: setgeI cSup_upper cSup_least)
lemma isGlb_cInf:
"(S::'a :: conditionally_complete_lattice set) \<noteq> {} \<Longrightarrow> (\<exists>b. b <=* S) \<Longrightarrow> isGlb UNIV S (Inf S)"
by (auto simp add: isGlb_def setge_def greatestP_def isLb_def
intro!: setleI cInf_lower cInf_greatest)
lemma cSup_le: "(S::'a::conditionally_complete_lattice set) \<noteq> {} \<Longrightarrow> S *<= b \<Longrightarrow> Sup S \<le> b"
by (metis cSup_least setle_def)
lemma cInf_ge: "(S::'a :: conditionally_complete_lattice set) \<noteq> {} \<Longrightarrow> b <=* S \<Longrightarrow> Inf S \<ge> b"
by (metis cInf_greatest setge_def)
lemma cSup_bounds:
fixes S :: "'a :: conditionally_complete_lattice set"
shows "S \<noteq> {} \<Longrightarrow> a <=* S \<Longrightarrow> S *<= b \<Longrightarrow> a \<le> Sup S \<and> Sup S \<le> b"
using cSup_least[of S b] cSup_upper2[of _ S a]
by (auto simp: bdd_above_setle setge_def setle_def)
lemma cSup_unique: "(S::'a :: {conditionally_complete_linorder, no_bot} set) *<= b \<Longrightarrow> (\<forall>b'<b. \<exists>x\<in>S. b' < x) \<Longrightarrow> Sup S = b"
by (rule cSup_eq) (auto simp: not_le[symmetric] setle_def)
lemma cInf_unique: "b <=* (S::'a :: {conditionally_complete_linorder, no_top} set) \<Longrightarrow> (\<forall>b'>b. \<exists>x\<in>S. b' > x) \<Longrightarrow> Inf S = b"
by (rule cInf_eq) (auto simp: not_le[symmetric] setge_def)
text{* Use completeness of reals (supremum property) to show that any bounded sequence has a least upper bound*}
lemma reals_complete: "\<exists>X. X \<in> S \<Longrightarrow> \<exists>Y. isUb (UNIV::real set) S Y \<Longrightarrow> \<exists>t. isLub (UNIV :: real set) S t"
by (intro exI[of _ "Sup S"] isLub_cSup) (auto simp: setle_def isUb_def intro!: cSup_upper)
lemma Bseq_isUb: "\<And>X :: nat \<Rightarrow> real. Bseq X \<Longrightarrow> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
lemma Bseq_isLub: "\<And>X :: nat \<Rightarrow> real. Bseq X \<Longrightarrow> \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
by (blast intro: reals_complete Bseq_isUb)
lemma isLub_mono_imp_LIMSEQ:
fixes X :: "nat \<Rightarrow> real"
assumes u: "isLub UNIV {x. \<exists>n. X n = x} u" (* FIXME: use 'range X' *)
assumes X: "\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n"
shows "X ----> u"
proof -
have "X ----> (SUP i. X i)"
using u[THEN isLubD1] X
by (intro LIMSEQ_incseq_SUP) (auto simp: incseq_def image_def eq_commute bdd_above_setle)
also have "(SUP i. X i) = u"
using isLub_cSup[of "range X"] u[THEN isLubD1]
by (intro isLub_unique[OF _ u]) (auto simp add: SUP_def image_def eq_commute)
finally show ?thesis .
qed
lemmas real_isGlb_unique = isGlb_unique[where 'a=real]
lemma real_le_inf_subset: "t \<noteq> {} \<Longrightarrow> t \<subseteq> s \<Longrightarrow> \<exists>b. b <=* s \<Longrightarrow> Inf s \<le> Inf (t::real set)"
by (rule cInf_superset_mono) (auto simp: bdd_below_setge)
lemma real_ge_sup_subset: "t \<noteq> {} \<Longrightarrow> t \<subseteq> s \<Longrightarrow> \<exists>b. s *<= b \<Longrightarrow> Sup s \<ge> Sup (t::real set)"
by (rule cSup_subset_mono) (auto simp: bdd_above_setle)
end