Quelle Hull.thy Sprache: Isabelle

(*  Title:      HOL/Hull.thy
    Author:     Amine Chaieb, University of Cambridge
    Author:     Jose Divasón <jose.divasonm at unirioja.es>
    Author:     Jesús Aransay <jesus-maria.aransay at unirioja.es>
    Author:     Johannes Hölzl, VU Amsterdam
*)

theory Hull
  imports Main
begin

subsection \<open>A generic notion of the convex, affine, conic hull, or closed "hull".\<close>

definition hull :: "('a set \ bool) \ 'a set \ 'a set" (infixl \hull\ 75)
  where "S hull s = \{t. S t \ s \ t}"

lemma hull_same: "S s \ S hull s = s"
  unfolding hull_def by auto

lemma hull_in: "(\T. Ball T S \ S (\T)) \ S (S hull s)"
  unfolding hull_def Ball_def by auto

lemma hull_eq: "(\T. Ball T S \ S (\T)) \ (S hull s) = s \ S s"
  using hull_same[of S s] hull_in[of S s] by metis

lemma hull_hull [simp]: "S hull (S hull s) = S hull s"
  unfolding hull_def by blast

lemma hull_subset[intro]: "s \ (S hull s)"
  unfolding hull_def by blast

lemma hull_mono: "s \ t \ (S hull s) \ (S hull t)"
  unfolding hull_def by blast

lemma hull_antimono: "\x. S x \ T x \ (T hull s) \ (S hull s)"
  unfolding hull_def by blast

lemma hull_minimal: "s \ t \ S t \ (S hull s) \ t"
  unfolding hull_def by blast

lemma subset_hull: "S t \ S hull s \ t \ s \ t"
  unfolding hull_def by blast

lemma hull_UNIV [simp]: "S hull UNIV = UNIV"
  unfolding hull_def by auto

lemma hull_unique: "s \ t \ S t \ (\t'. s \ t' \ S t' \ t \ t') \ (S hull s = t)"
  unfolding hull_def by auto

lemma hull_induct: "\a \ Q hull S; \x. x\ S \ P x; Q {x. P x}\ \ P a"
  using hull_minimal[of S "{x. P x}" Q]
  by (auto simp add: subset_eq)

lemma hull_inc: "x \ S \ x \ P hull S"
  by (metis hull_subset subset_eq)

lemma hull_Un_subset: "(S hull s) \ (S hull t) \ (S hull (s \ t))"
  unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)

lemma hull_Un:
  assumes T: "\T. Ball T S \ S (\T)"
  shows "S hull (s \ t) = S hull (S hull s \ S hull t)"
  apply (rule equalityI)
  apply (meson hull_mono hull_subset sup.mono)
  by (metis hull_Un_subset hull_hull hull_mono)

lemma hull_Un_left: "P hull (S \ T) = P hull (P hull S \ T)"
  apply (rule equalityI)
   apply (simp add: Un_commute hull_mono hull_subset sup.coboundedI2)
  by (metis Un_subset_iff hull_hull hull_mono hull_subset)

lemma hull_Un_right: "P hull (S \ T) = P hull (S \ P hull T)"
  by (metis hull_Un_left sup.commute)

lemma hull_insert:
   "P hull (insert a S) = P hull (insert a (P hull S))"
  by (metis hull_Un_right insert_is_Un)

lemma hull_redundant_eq: "a \ (S hull s) \ S hull (insert a s) = S hull s"
  unfolding hull_def by blast

lemma hull_redundant: "a \ (S hull s) \ S hull (insert a s) = S hull s"
  by (metis hull_redundant_eq)

end

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Beweissystem der NASA

Beweissystem Isabelle

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Cephes Mathematical Library

Wiener Entwicklungsmethode

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