paulson@33269: (*  Author: Amine Chaieb and L C Paulson, University of Cambridge *)
paulson@33269: 
paulson@33269: header {*Sup and Inf Operators on Sets of Reals.*}
paulson@33269: 
paulson@33269: theory SupInf
paulson@33269: imports RComplete
paulson@33269: begin
paulson@33269: 
paulson@33269: instantiation real :: Sup 
paulson@33269: begin
paulson@33269: definition
paulson@33269:   Sup_real_def [code del]: "Sup X == (LEAST z::real. \<forall>x\<in>X. x\<le>z)"
paulson@33269: 
paulson@33269: instance ..
paulson@33269: end
paulson@33269: 
paulson@33269: instantiation real :: Inf 
paulson@33269: begin
paulson@33269: definition
paulson@33269:   Inf_real_def [code del]: "Inf (X::real set) == - (Sup (uminus ` X))"
paulson@33269: 
paulson@33269: instance ..
paulson@33269: end
paulson@33269: 
paulson@33269: subsection{*Supremum of a set of reals*}
paulson@33269: 
paulson@33269: lemma Sup_upper [intro]: (*REAL_SUP_UBOUND in HOL4*)
paulson@33269:   fixes x :: real
paulson@33269:   assumes x: "x \<in> X"
paulson@33269:       and z: "!!x. x \<in> X \<Longrightarrow> x \<le> z"
paulson@33269:   shows "x \<le> Sup X"
paulson@33269: proof (auto simp add: Sup_real_def) 
paulson@33269:   from reals_complete2
paulson@33269:   obtain s where s: "(\<forall>y\<in>X. y \<le> s) & (\<forall>z. ((\<forall>y\<in>X. y \<le> z) --> s \<le> z))"
paulson@33269:     by (blast intro: x z)
paulson@33269:   hence "x \<le> s"
paulson@33269:     by (blast intro: x z)
paulson@33269:   also with s have "... = (LEAST z. \<forall>x\<in>X. x \<le> z)"
paulson@33269:     by (fast intro: Least_equality [symmetric])  
paulson@33269:   finally show "x \<le> (LEAST z. \<forall>x\<in>X. x \<le> z)" .
paulson@33269: qed
paulson@33269: 
paulson@33269: lemma Sup_least [intro]: (*REAL_IMP_SUP_LE in HOL4*)
paulson@33269:   fixes z :: real
paulson@33269:   assumes x: "X \<noteq> {}"
paulson@33269:       and z: "\<And>x. x \<in> X \<Longrightarrow> x \<le> z"
paulson@33269:   shows "Sup X \<le> z"
paulson@33269: proof (auto simp add: Sup_real_def) 
paulson@33269:   from reals_complete2 x
paulson@33269:   obtain s where s: "(\<forall>y\<in>X. y \<le> s) & (\<forall>z. ((\<forall>y\<in>X. y \<le> z) --> s \<le> z))"
paulson@33269:     by (blast intro: z)
paulson@33269:   hence "(LEAST z. \<forall>x\<in>X. x \<le> z) = s"
paulson@33269:     by (best intro: Least_equality)  
paulson@33269:   also with s z have "... \<le> z"
paulson@33269:     by blast
paulson@33269:   finally show "(LEAST z. \<forall>x\<in>X. x \<le> z) \<le> z" .
paulson@33269: qed
paulson@33269: 
paulson@33609: lemma less_SupE:
paulson@33609:   fixes y :: real
paulson@33609:   assumes "y < Sup X" "X \<noteq> {}"
paulson@33609:   obtains x where "x\<in>X" "y < x"
paulson@33609: by (metis SupInf.Sup_least assms linorder_not_less that)
paulson@33609: 
paulson@33269: lemma Sup_singleton [simp]: "Sup {x::real} = x"
paulson@33269:   by (force intro: Least_equality simp add: Sup_real_def)
paulson@33269:  
paulson@33269: lemma Sup_eq_maximum: (*REAL_SUP_MAX in HOL4*)
paulson@33269:   fixes z :: real
paulson@33269:   assumes X: "z \<in> X" and z: "!!x. x \<in> X \<Longrightarrow> x \<le> z"
paulson@33269:   shows  "Sup X = z"
paulson@33269:   by (force intro: Least_equality X z simp add: Sup_real_def)
paulson@33269:  
paulson@33269: lemma Sup_upper2: (*REAL_IMP_LE_SUP in HOL4*)
paulson@33269:   fixes x :: real
paulson@33269:   shows "x \<in> X \<Longrightarrow> y \<le> x \<Longrightarrow> (!!x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> y \<le> Sup X"
paulson@33269:   by (metis Sup_upper real_le_trans)
paulson@33269: 
paulson@33269: lemma Sup_real_iff : (*REAL_SUP_LE in HOL4*)
paulson@33269:   fixes z :: real
paulson@33269:   shows "X ~= {} ==> (!!x. x \<in> X ==> x \<le> z) ==> (\<exists>x\<in>X. y<x) <-> y < Sup X"
paulson@33269:   by (metis Sup_least Sup_upper linorder_not_le le_less_trans)
paulson@33269: 
paulson@33269: lemma Sup_eq:
paulson@33269:   fixes a :: real
paulson@33269:   shows "(!!x. x \<in> X \<Longrightarrow> x \<le> a) 
paulson@33269:         \<Longrightarrow> (!!y. (!!x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y) \<Longrightarrow> Sup X = a"
paulson@33269:   by (metis Sup_least Sup_upper add_le_cancel_left diff_add_cancel insert_absorb
nipkow@33657:         insert_not_empty real_le_antisym)
paulson@33269: 
paulson@33269: lemma Sup_le:
paulson@33269:   fixes S :: "real set"
paulson@33269:   shows "S \<noteq> {} \<Longrightarrow> S *<= b \<Longrightarrow> Sup S \<le> b"
paulson@33269: by (metis SupInf.Sup_least setle_def)
paulson@33269: 
paulson@33269: lemma Sup_upper_EX: 
paulson@33269:   fixes x :: real
paulson@33269:   shows "x \<in> X \<Longrightarrow> \<exists>z. \<forall>x. x \<in> X \<longrightarrow> x \<le> z \<Longrightarrow>  x \<le> Sup X"
paulson@33269:   by blast
paulson@33269: 
paulson@33269: lemma Sup_insert_nonempty: 
paulson@33269:   fixes x :: real
paulson@33269:   assumes x: "x \<in> X"
paulson@33269:       and z: "!!x. x \<in> X \<Longrightarrow> x \<le> z"
paulson@33269:   shows "Sup (insert a X) = max a (Sup X)"
paulson@33269: proof (cases "Sup X \<le> a")
paulson@33269:   case True
paulson@33269:   thus ?thesis
paulson@33269:     apply (simp add: max_def) 
paulson@33269:     apply (rule Sup_eq_maximum)
paulson@33269:     apply (metis insertCI)
paulson@33269:     apply (metis Sup_upper insertE le_iff_sup real_le_linear real_le_trans sup_absorb1 z)     
paulson@33269:     done
paulson@33269: next
paulson@33269:   case False
paulson@33269:   hence 1:"a < Sup X" by simp
paulson@33269:   have "Sup X \<le> Sup (insert a X)"
paulson@33269:     apply (rule Sup_least)
paulson@33269:     apply (metis empty_psubset_nonempty psubset_eq x)
paulson@33269:     apply (rule Sup_upper_EX) 
paulson@33269:     apply blast
paulson@33269:     apply (metis insert_iff real_le_linear real_le_refl real_le_trans z)
paulson@33269:     done
paulson@33269:   moreover 
paulson@33269:   have "Sup (insert a X) \<le> Sup X"
paulson@33269:     apply (rule Sup_least)
paulson@33269:     apply blast
paulson@33269:     apply (metis False Sup_upper insertE real_le_linear z) 
paulson@33269:     done
paulson@33269:   ultimately have "Sup (insert a X) = Sup X"
paulson@33269:     by (blast intro:  antisym )
paulson@33269:   thus ?thesis
paulson@33269:     by (metis 1 min_max.le_iff_sup real_less_def)
paulson@33269: qed
paulson@33269: 
paulson@33269: lemma Sup_insert_if: 
paulson@33269:   fixes X :: "real set"
paulson@33269:   assumes z: "!!x. x \<in> X \<Longrightarrow> x \<le> z"
paulson@33269:   shows "Sup (insert a X) = (if X={} then a else max a (Sup X))"
paulson@33269: by auto (metis Sup_insert_nonempty z) 
paulson@33269: 
paulson@33269: lemma Sup: 
paulson@33269:   fixes S :: "real set"
paulson@33269:   shows "S \<noteq> {} \<Longrightarrow> (\<exists>b. S *<= b) \<Longrightarrow> isLub UNIV S (Sup S)"
paulson@33269: by  (auto simp add: isLub_def setle_def leastP_def isUb_def intro!: setgeI) 
paulson@33269: 
paulson@33269: lemma Sup_finite_Max: 
paulson@33269:   fixes S :: "real set"
paulson@33269:   assumes fS: "finite S" and Se: "S \<noteq> {}"
paulson@33269:   shows "Sup S = Max S"
paulson@33269: using fS Se
paulson@33269: proof-
paulson@33269:   let ?m = "Max S"
paulson@33269:   from Max_ge[OF fS] have Sm: "\<forall> x\<in> S. x \<le> ?m" by metis
paulson@33269:   with Sup[OF Se] have lub: "isLub UNIV S (Sup S)" by (metis setle_def)
paulson@33269:   from Max_in[OF fS Se] lub have mrS: "?m \<le> Sup S"
paulson@33269:     by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def)
paulson@33269:   moreover
paulson@33269:   have "Sup S \<le> ?m" using Sm lub
paulson@33269:     by (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
paulson@33269:   ultimately  show ?thesis by arith
paulson@33269: qed
paulson@33269: 
paulson@33269: lemma Sup_finite_in:
paulson@33269:   fixes S :: "real set"
paulson@33269:   assumes fS: "finite S" and Se: "S \<noteq> {}"
paulson@33269:   shows "Sup S \<in> S"
paulson@33269:   using Sup_finite_Max[OF fS Se] Max_in[OF fS Se] by metis
paulson@33269: 
paulson@33269: lemma Sup_finite_ge_iff: 
paulson@33269:   fixes S :: "real set"
paulson@33269:   assumes fS: "finite S" and Se: "S \<noteq> {}"
paulson@33269:   shows "a \<le> Sup S \<longleftrightarrow> (\<exists> x \<in> S. a \<le> x)"
paulson@33269: by (metis Max_ge Se Sup_finite_Max Sup_finite_in fS linorder_not_le less_le_trans)
paulson@33269: 
paulson@33269: lemma Sup_finite_le_iff: 
paulson@33269:   fixes S :: "real set"
paulson@33269:   assumes fS: "finite S" and Se: "S \<noteq> {}"
paulson@33269:   shows "a \<ge> Sup S \<longleftrightarrow> (\<forall> x \<in> S. a \<ge> x)"
paulson@33269: by (metis Max_ge Se Sup_finite_Max Sup_finite_in fS le_iff_sup real_le_trans) 
paulson@33269: 
paulson@33269: lemma Sup_finite_gt_iff: 
paulson@33269:   fixes S :: "real set"
paulson@33269:   assumes fS: "finite S" and Se: "S \<noteq> {}"
paulson@33269:   shows "a < Sup S \<longleftrightarrow> (\<exists> x \<in> S. a < x)"
paulson@33269: by (metis Se Sup_finite_le_iff fS linorder_not_less)
paulson@33269: 
paulson@33269: lemma Sup_finite_lt_iff: 
paulson@33269:   fixes S :: "real set"
paulson@33269:   assumes fS: "finite S" and Se: "S \<noteq> {}"
paulson@33269:   shows "a > Sup S \<longleftrightarrow> (\<forall> x \<in> S. a > x)"
paulson@33269: by (metis Se Sup_finite_ge_iff fS linorder_not_less)
paulson@33269: 
paulson@33269: lemma Sup_unique:
paulson@33269:   fixes S :: "real set"
paulson@33269:   shows "S *<= b \<Longrightarrow> (\<forall>b' < b. \<exists>x \<in> S. b' < x) \<Longrightarrow> Sup S = b"
paulson@33269: unfolding setle_def
paulson@33269: apply (rule Sup_eq, auto) 
paulson@33269: apply (metis linorder_not_less) 
paulson@33269: done
paulson@33269: 
paulson@33269: lemma Sup_abs_le:
paulson@33269:   fixes S :: "real set"
paulson@33269:   shows "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Sup S\<bar> \<le> a"
paulson@33269: by (auto simp add: abs_le_interval_iff) (metis Sup_upper2) 
paulson@33269: 
paulson@33269: lemma Sup_bounds:
paulson@33269:   fixes S :: "real set"
paulson@33269:   assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"
paulson@33269:   shows "a \<le> Sup S \<and> Sup S \<le> b"
paulson@33269: proof-
paulson@33269:   from Sup[OF Se] u have lub: "isLub UNIV S (Sup S)" by blast
paulson@33269:   hence b: "Sup S \<le> b" using u 
paulson@33269:     by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def) 
paulson@33269:   from Se obtain y where y: "y \<in> S" by blast
paulson@33269:   from lub l have "a \<le> Sup S"
paulson@33269:     by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def)
paulson@33269:        (metis le_iff_sup le_sup_iff y)
paulson@33269:   with b show ?thesis by blast
paulson@33269: qed
paulson@33269: 
paulson@33269: lemma Sup_asclose: 
paulson@33269:   fixes S :: "real set"
paulson@33269:   assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>Sup S - l\<bar> \<le> e"
paulson@33269: proof-
paulson@33269:   have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" by arith
paulson@33269:   thus ?thesis using S b Sup_bounds[of S "l - e" "l+e"] unfolding th
paulson@33269:     by  (auto simp add: setge_def setle_def)
paulson@33269: qed
paulson@33269: 
paulson@33269: 
paulson@33269: subsection{*Infimum of a set of reals*}
paulson@33269: 
paulson@33269: lemma Inf_lower [intro]: 
paulson@33269:   fixes z :: real
paulson@33269:   assumes x: "x \<in> X"
paulson@33269:       and z: "!!x. x \<in> X \<Longrightarrow> z \<le> x"
paulson@33269:   shows "Inf X \<le> x"
paulson@33269: proof -
paulson@33269:   have "-x \<le> Sup (uminus ` X)"
paulson@33269:     by (rule Sup_upper [where z = "-z"]) (auto simp add: image_iff x z)
paulson@33269:   thus ?thesis 
paulson@33269:     by (auto simp add: Inf_real_def)
paulson@33269: qed
paulson@33269: 
paulson@33269: lemma Inf_greatest [intro]: 
paulson@33269:   fixes z :: real
paulson@33269:   assumes x: "X \<noteq> {}"
paulson@33269:       and z: "\<And>x. x \<in> X \<Longrightarrow> z \<le> x"
paulson@33269:   shows "z \<le> Inf X"
paulson@33269: proof -
huffman@35216:   have "Sup (uminus ` X) \<le> -z" using x z by force
paulson@33269:   hence "z \<le> - Sup (uminus ` X)"
paulson@33269:     by simp
paulson@33269:   thus ?thesis 
paulson@33269:     by (auto simp add: Inf_real_def)
paulson@33269: qed
paulson@33269: 
paulson@33269: lemma Inf_singleton [simp]: "Inf {x::real} = x"
paulson@33269:   by (simp add: Inf_real_def) 
paulson@33269:  
paulson@33269: lemma Inf_eq_minimum: (*REAL_INF_MIN in HOL4*)
paulson@33269:   fixes z :: real
paulson@33269:   assumes x: "z \<in> X" and z: "!!x. x \<in> X \<Longrightarrow> z \<le> x"
paulson@33269:   shows  "Inf X = z"
paulson@33269: proof -
paulson@33269:   have "Sup (uminus ` X) = -z" using x z
paulson@33269:     by (force intro: Sup_eq_maximum x z)
paulson@33269:   thus ?thesis
paulson@33269:     by (simp add: Inf_real_def) 
paulson@33269: qed
paulson@33269:  
paulson@33269: lemma Inf_lower2:
paulson@33269:   fixes x :: real
paulson@33269:   shows "x \<in> X \<Longrightarrow> x \<le> y \<Longrightarrow> (!!x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf X \<le> y"
paulson@33269:   by (metis Inf_lower real_le_trans)
paulson@33269: 
paulson@33269: lemma Inf_real_iff:
paulson@33269:   fixes z :: real
paulson@33269:   shows "X \<noteq> {} \<Longrightarrow> (!!x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> (\<exists>x\<in>X. x<y) \<longleftrightarrow> Inf X < y"
paulson@33269:   by (metis Inf_greatest Inf_lower less_le_not_le real_le_linear 
paulson@33269:             order_less_le_trans)
paulson@33269: 
paulson@33269: lemma Inf_eq:
paulson@33269:   fixes a :: real
paulson@33269:   shows "(!!x. x \<in> X \<Longrightarrow> a \<le> x) \<Longrightarrow> (!!y. (!!x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a) \<Longrightarrow> Inf X = a"
paulson@33269:   by (metis Inf_greatest Inf_lower add_le_cancel_left diff_add_cancel
nipkow@33657:         insert_absorb insert_not_empty real_le_antisym)
paulson@33269: 
paulson@33269: lemma Inf_ge: 
paulson@33269:   fixes S :: "real set"
paulson@33269:   shows "S \<noteq> {} \<Longrightarrow> b <=* S \<Longrightarrow> Inf S \<ge> b"
paulson@33269: by (metis SupInf.Inf_greatest setge_def)
paulson@33269: 
paulson@33269: lemma Inf_lower_EX: 
paulson@33269:   fixes x :: real
paulson@33269:   shows "x \<in> X \<Longrightarrow> \<exists>z. \<forall>x. x \<in> X \<longrightarrow> z \<le> x \<Longrightarrow> Inf X \<le> x"
paulson@33269:   by blast
paulson@33269: 
paulson@33269: lemma Inf_insert_nonempty: 
paulson@33269:   fixes x :: real
paulson@33269:   assumes x: "x \<in> X"
paulson@33269:       and z: "!!x. x \<in> X \<Longrightarrow> z \<le> x"
paulson@33269:   shows "Inf (insert a X) = min a (Inf X)"
paulson@33269: proof (cases "a \<le> Inf X")
paulson@33269:   case True
paulson@33269:   thus ?thesis
paulson@33269:     by (simp add: min_def)
huffman@35216:        (blast intro: Inf_eq_minimum real_le_refl real_le_trans z)
paulson@33269: next
paulson@33269:   case False
paulson@33269:   hence 1:"Inf X < a" by simp
paulson@33269:   have "Inf (insert a X) \<le> Inf X"
paulson@33269:     apply (rule Inf_greatest)
paulson@33269:     apply (metis empty_psubset_nonempty psubset_eq x)
paulson@33269:     apply (rule Inf_lower_EX) 
paulson@33269:     apply (blast intro: elim:) 
paulson@33269:     apply (metis insert_iff real_le_linear real_le_refl real_le_trans z)
paulson@33269:     done
paulson@33269:   moreover 
paulson@33269:   have "Inf X \<le> Inf (insert a X)"
paulson@33269:     apply (rule Inf_greatest)
paulson@33269:     apply blast
paulson@33269:     apply (metis False Inf_lower insertE real_le_linear z) 
paulson@33269:     done
paulson@33269:   ultimately have "Inf (insert a X) = Inf X"
paulson@33269:     by (blast intro:  antisym )
paulson@33269:   thus ?thesis
paulson@33269:     by (metis False min_max.inf_absorb2 real_le_linear)
paulson@33269: qed
paulson@33269: 
paulson@33269: lemma Inf_insert_if: 
paulson@33269:   fixes X :: "real set"
paulson@33269:   assumes z:  "!!x. x \<in> X \<Longrightarrow> z \<le> x"
paulson@33269:   shows "Inf (insert a X) = (if X={} then a else min a (Inf X))"
paulson@33269: by auto (metis Inf_insert_nonempty z) 
paulson@33269: 
paulson@33269: lemma Inf_greater:
paulson@33269:   fixes z :: real
paulson@33269:   shows "X \<noteq> {} \<Longrightarrow>  Inf X < z \<Longrightarrow> \<exists>x \<in> X. x < z"
paulson@33269:   by (metis Inf_real_iff mem_def not_leE)
paulson@33269: 
paulson@33269: lemma Inf_close:
paulson@33269:   fixes e :: real
paulson@33269:   shows "X \<noteq> {} \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>x \<in> X. x < Inf X + e"
paulson@33269:   by (metis add_strict_increasing comm_monoid_add.mult_commute Inf_greater linorder_not_le pos_add_strict)
paulson@33269: 
paulson@33269: lemma Inf_finite_Min:
paulson@33269:   fixes S :: "real set"
paulson@33269:   shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> Inf S = Min S"
paulson@33269: by (simp add: Inf_real_def Sup_finite_Max image_image) 
paulson@33269: 
paulson@33269: lemma Inf_finite_in: 
paulson@33269:   fixes S :: "real set"
paulson@33269:   assumes fS: "finite S" and Se: "S \<noteq> {}"
paulson@33269:   shows "Inf S \<in> S"
paulson@33269:   using Inf_finite_Min[OF fS Se] Min_in[OF fS Se] by metis
paulson@33269: 
paulson@33269: lemma Inf_finite_ge_iff: 
paulson@33269:   fixes S :: "real set"
paulson@33269:   shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall> x \<in> S. a \<le> x)"
paulson@33269: by (metis Inf_finite_Min Inf_finite_in Min_le real_le_trans)
paulson@33269: 
paulson@33269: lemma Inf_finite_le_iff:
paulson@33269:   fixes S :: "real set"
paulson@33269:   shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<ge> Inf S \<longleftrightarrow> (\<exists> x \<in> S. a \<ge> x)"
paulson@33269: by (metis Inf_finite_Min Inf_finite_ge_iff Inf_finite_in Min_le
nipkow@33657:           real_le_antisym real_le_linear)
paulson@33269: 
paulson@33269: lemma Inf_finite_gt_iff: 
paulson@33269:   fixes S :: "real set"
paulson@33269:   shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a < Inf S \<longleftrightarrow> (\<forall> x \<in> S. a < x)"
paulson@33269: by (metis Inf_finite_le_iff linorder_not_less)
paulson@33269: 
paulson@33269: lemma Inf_finite_lt_iff: 
paulson@33269:   fixes S :: "real set"
paulson@33269:   shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a > Inf S \<longleftrightarrow> (\<exists> x \<in> S. a > x)"
paulson@33269: by (metis Inf_finite_ge_iff linorder_not_less)
paulson@33269: 
paulson@33269: lemma Inf_unique:
paulson@33269:   fixes S :: "real set"
paulson@33269:   shows "b <=* S \<Longrightarrow> (\<forall>b' > b. \<exists>x \<in> S. b' > x) \<Longrightarrow> Inf S = b"
paulson@33269: unfolding setge_def
paulson@33269: apply (rule Inf_eq, auto) 
paulson@33269: apply (metis less_le_not_le linorder_not_less) 
paulson@33269: done
paulson@33269: 
paulson@33269: lemma Inf_abs_ge:
paulson@33269:   fixes S :: "real set"
paulson@33269:   shows "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Inf S\<bar> \<le> a"
paulson@33269: by (simp add: Inf_real_def) (rule Sup_abs_le, auto) 
paulson@33269: 
paulson@33269: lemma Inf_asclose:
paulson@33269:   fixes S :: "real set"
paulson@33269:   assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>Inf S - l\<bar> \<le> e"
paulson@33269: proof -
paulson@33269:   have "\<bar>- Sup (uminus ` S) - l\<bar> =  \<bar>Sup (uminus ` S) - (-l)\<bar>"
paulson@33269:     by auto
paulson@33269:   also have "... \<le> e" 
paulson@33269:     apply (rule Sup_asclose) 
paulson@33269:     apply (auto simp add: S)
paulson@33269:     apply (metis abs_minus_add_cancel b comm_monoid_add.mult_commute real_diff_def) 
paulson@33269:     done
paulson@33269:   finally have "\<bar>- Sup (uminus ` S) - l\<bar> \<le> e" .
paulson@33269:   thus ?thesis
paulson@33269:     by (simp add: Inf_real_def)
paulson@33269: qed
paulson@33269: 
paulson@33271: subsection{*Relate max and min to Sup and Inf.*}
paulson@33269: 
paulson@33269: lemma real_max_Sup:
paulson@33269:   fixes x :: real
paulson@33269:   shows "max x y = Sup {x,y}"
paulson@33269: proof-
paulson@33269:   have f: "finite {x, y}" "{x,y} \<noteq> {}"  by simp_all
paulson@33269:   from Sup_finite_le_iff[OF f, of "max x y"] have "Sup {x,y} \<le> max x y" by simp
paulson@33269:   moreover
paulson@33269:   have "max x y \<le> Sup {x,y}" using Sup_finite_ge_iff[OF f, of "max x y"]
paulson@33269:     by (simp add: linorder_linear)
paulson@33269:   ultimately show ?thesis by arith
paulson@33269: qed
paulson@33269: 
paulson@33269: lemma real_min_Inf: 
paulson@33269:   fixes x :: real
paulson@33269:   shows "min x y = Inf {x,y}"
paulson@33269: proof-
paulson@33269:   have f: "finite {x, y}" "{x,y} \<noteq> {}"  by simp_all
paulson@33269:   from Inf_finite_le_iff[OF f, of "min x y"] have "Inf {x,y} \<le> min x y"
paulson@33269:     by (simp add: linorder_linear)
paulson@33269:   moreover
paulson@33269:   have "min x y \<le> Inf {x,y}" using Inf_finite_ge_iff[OF f, of "min x y"]
paulson@33269:     by simp
paulson@33269:   ultimately show ?thesis by arith
paulson@33269: qed
paulson@33269: 
paulson@33609: lemma reals_complete_interval:
paulson@33609:   fixes a::real and b::real
paulson@33609:   assumes "a < b" and "P a" and "~P b"
paulson@33609:   shows "\<exists>c. a \<le> c & c \<le> b & (\<forall>x. a \<le> x & x < c --> P x) &
paulson@33609:              (\<forall>d. (\<forall>x. a \<le> x & x < d --> P x) --> d \<le> c)"
paulson@33609: proof (rule exI [where x = "Sup {d. \<forall>x. a \<le> x & x < d --> P x}"], auto)
paulson@33609:   show "a \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
paulson@33609:     by (rule SupInf.Sup_upper [where z=b], auto)
huffman@35216:        (metis `a < b` `\<not> P b` real_le_linear real_less_def)
paulson@33609: next
paulson@33609:   show "Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c} \<le> b"
paulson@33609:     apply (rule SupInf.Sup_least) 
paulson@33609:     apply (auto simp add: )
paulson@33609:     apply (metis less_le_not_le)
paulson@33609:     apply (metis `a<b` `~ P b` real_le_linear real_less_def) 
paulson@33609:     done
paulson@33609: next
paulson@33609:   fix x
paulson@33609:   assume x: "a \<le> x" and lt: "x < Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
paulson@33609:   show "P x"
paulson@33609:     apply (rule less_SupE [OF lt], auto)
paulson@33609:     apply (metis less_le_not_le)
paulson@33609:     apply (metis x) 
paulson@33609:     done
paulson@33609: next
paulson@33609:   fix d
paulson@33609:     assume 0: "\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x"
paulson@33609:     thus "d \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
paulson@33609:       by (rule_tac z="b" in SupInf.Sup_upper, auto) 
paulson@33609:          (metis `a<b` `~ P b` real_le_linear real_less_def) 
paulson@33609: qed
paulson@33609: 
paulson@33269: end