(*  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