Quellcode-Bibliothek

^© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: Examples.thy Sprache: Isabelle

Original von: Isabelle^©

theory Examples imports Complex_Main begin

declare [[eta_contract = false]]

text\<open>membership, intersection\<close>
text\<open>difference and empty set\<close>
text\<open>complement, union and universal set\<close>

lemma "(x \ A \ B) = (x \ A \ x \ B)"
by blast

text\<open>
@{thm[display] IntI[no_vars]}
\rulename{IntI}

@{thm[display] IntD1[no_vars]}
\rulename{IntD1}

@{thm[display] IntD2[no_vars]}
\rulename{IntD2}
\<close>

lemma "(x \ -A) = (x \ A)"
by blast

text\<open>
@{thm[display] Compl_iff[no_vars]}
\rulename{Compl_iff}
\<close>

lemma "- (A \ B) = -A \ -B"
by blast

text\<open>
@{thm[display] Compl_Un[no_vars]}
\rulename{Compl_Un}
\<close>

lemma "A-A = {}"
by blast

text\<open>
@{thm[display] Diff_disjoint[no_vars]}
\rulename{Diff_disjoint}
\<close>

lemma "A \ -A = UNIV"
by blast

text\<open>
@{thm[display] Compl_partition[no_vars]}
\rulename{Compl_partition}
\<close>

text\<open>subset relation\<close>

text\<open>
@{thm[display] subsetI[no_vars]}
\rulename{subsetI}

@{thm[display] subsetD[no_vars]}
\rulename{subsetD}
\<close>

lemma "((A \ B) \ C) = (A \ C \ B \ C)"
by blast

text\<open>
@{thm[display] Un_subset_iff[no_vars]}
\rulename{Un_subset_iff}
\<close>

lemma "(A \ -B) = (B \ -A)"
by blast

lemma "(A <= -B) = (B <= -A)"
  oops

text\<open>ASCII version: blast fails because of overloading because
it doesn't have to be sets\

lemma "((A:: 'a set) <= -B) = (B <= -A)"
by blast

text\<open>A type constraint lets it work\<close>

text\<open>An issue here: how do we discuss the distinction between ASCII and
symbol notation?  Here the latter disambiguates.\<close>

text\<open>
set extensionality

@{thm[display] set_eqI[no_vars]}
\rulename{set_eqI}

@{thm[display] equalityI[no_vars]}
\rulename{equalityI}

@{thm[display] equalityE[no_vars]}
\rulename{equalityE}
\<close>

text\<open>finite sets: insertion and membership relation\<close>
text\<open>finite set notation\<close>

lemma "insert x A = {x} \ A"
by blast

text\<open>
@{thm[display] insert_is_Un[no_vars]}
\rulename{insert_is_Un}
\<close>

lemma "{a,b} \ {c,d} = {a,b,c,d}"
by blast

lemma "{a,b} \ {b,c} = {b}"
apply auto
oops
text\<open>fails because it isn't valid\<close>

lemma "{a,b} \ {b,c} = (if a=c then {a,b} else {b})"
apply simp
by blast

text\<open>or just force or auto.  blast alone can't handle the if-then-else\<close>

text\<open>next: some comprehension examples\<close>

lemma "(a \ {z. P z}) = P a"
by blast

text\<open>
@{thm[display] mem_Collect_eq[no_vars]}
\rulename{mem_Collect_eq}
\<close>

lemma "{x. x \ A} = A"
by blast

text\<open>
@{thm[display] Collect_mem_eq[no_vars]}
\rulename{Collect_mem_eq}
\<close>

lemma "{x. P x \ x \ A} = {x. P x} \ A"
by blast

lemma "{x. P x \ Q x} = -{x. P x} \ {x. Q x}"
by blast

definition prime :: "nat set" where
    "prime == {p. 1

m. m dvd p \ m=1 \ m=p)}"

lemma "{p*q | p q. p\prime \ q\prime} =
{z. \<exists>p q. z = p*q \<and> p\<in>prime \<and> q\<in>prime}"
by (rule refl)

text\<open>binders\<close>

text\<open>bounded quantifiers\<close>

lemma "(\x\A. P x) = (\x. x\A \ P x)"
by blast

text\<open>
@{thm[display] bexI[no_vars]}
\rulename{bexI}
\<close>

text\<open>
@{thm[display] bexE[no_vars]}
\rulename{bexE}
\<close>

lemma "(\x\A. P x) = (\x. x\A \ P x)"
by blast

text\<open>
@{thm[display] ballI[no_vars]}
\rulename{ballI}
\<close>

text\<open>
@{thm[display] bspec[no_vars]}
\rulename{bspec}
\<close>

text\<open>indexed unions and variations\<close>

lemma "(\x. B x) = (\x\UNIV. B x)"
by blast

text\<open>
@{thm[display] UN_iff[no_vars]}
\rulename{UN_iff}
\<close>

text\<open>
@{thm[display] Union_iff[no_vars]}
\rulename{Union_iff}
\<close>

lemma "(\x\A. B x) = {y. \x\A. y \ B x}"
by blast

lemma "\S = (\x\S. x)"
by blast

text\<open>
@{thm[display] UN_I[no_vars]}
\rulename{UN_I}
\<close>

text\<open>
@{thm[display] UN_E[no_vars]}
\rulename{UN_E}
\<close>

text\<open>indexed intersections\<close>

lemma "(\x. B x) = {y. \x. y \ B x}"
by blast

text\<open>
@{thm[display] INT_iff[no_vars]}
\rulename{INT_iff}
\<close>

text\<open>
@{thm[display] Inter_iff[no_vars]}
\rulename{Inter_iff}
\<close>

text\<open>mention also card, Pow, etc.\<close>

text\<open>
@{thm[display] card_Un_Int[no_vars]}
\rulename{card_Un_Int}

@{thm[display] card_Pow[no_vars]}
\rulename{card_Pow}

@{thm[display] n_subsets[no_vars]}
\rulename{n_subsets}
\<close>

end

¤ Dauer der Verarbeitung: 0.20 Sekunden (vorverarbeitet) ¤

Download des Quellennavigators
Download des sprechenden Kalenders
in der Quellcodebibliothek suchen

Haftungshinweis

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.