(* 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
¤ Dauer der Verarbeitung: 0.14 Sekunden
(vorverarbeitet)
¤
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
