doc-src/TutorialI/Sets/Examples.thy

+theory Examples = Main:
+
+ML "reset eta_contract"
+ML "Pretty.setmargin 64"
+
+text{*membership, intersection *}
+text{*difference and empty set*}
+text{*complement, union and universal set*}
+
+lemma "(x \<in> A \<inter> B) = (x \<in> A \<and> x \<in> B)"
+  apply (blast)
+  done
+
+text{*
+@{thm[display] IntI[no_vars]}
+\rulename{IntI}
+
+@{thm[display] IntD1[no_vars]}
+\rulename{IntD1}
+
+@{thm[display] IntD2[no_vars]}
+\rulename{IntD2}
+*}
+
+lemma "(x \<in> -A) = (x \<notin> A)"
+  apply (blast)
+  done
+
+text{*
+@{thm[display] Compl_iff[no_vars]}
+\rulename{Compl_iff}
+*}
+
+lemma "- (A \<union> B) = -A \<inter> -B"
+  apply (blast)
+  done
+
+text{*
+@{thm[display] Compl_Un[no_vars]}
+\rulename{Compl_Un}
+*}
+
+lemma "A-A = {}"
+  apply (blast)
+  done
+
+text{*
+@{thm[display] Diff_disjoint[no_vars]}
+\rulename{Diff_disjoint}
+*}
+
+  
+
+lemma "A \<union> -A = UNIV"
+  apply (blast)
+  done
+
+text{*
+@{thm[display] Compl_partition[no_vars]}
+\rulename{Compl_partition}
+*}
+
+text{*subset relation*}
+
+
+text{*
+@{thm[display] subsetI[no_vars]}
+\rulename{subsetI}
+
+@{thm[display] subsetD[no_vars]}
+\rulename{subsetD}
+*}
+
+lemma "((A \<union> B) \<subseteq> C) = (A \<subseteq> C \<and> B \<subseteq> C)"
+  apply (blast)
+  done
+
+text{*
+@{thm[display] Un_subset_iff[no_vars]}
+\rulename{Un_subset_iff}
+*}
+
+lemma "(A \<subseteq> -B) = (B \<subseteq> -A)"
+  apply (blast)
+  done
+
+lemma "(A <= -B) = (B <= -A)"
+  oops
+
+text{*ASCII version: blast fails because of overloading because
+ it doesn't have to be sets*}
+
+lemma "((A:: 'a set) <= -B) = (B <= -A)"
+  apply (blast)
+  done
+
+text{*A type constraint lets it work*}
+
+text{*An issue here: how do we discuss the distinction between ASCII and
+X-symbol notation?  Here the latter disambiguates.*}
+
+
+text{*
+set extensionality
+
+@{thm[display] set_ext[no_vars]}
+\rulename{set_ext}
+
+@{thm[display] equalityI[no_vars]}
+\rulename{equalityI}
+
+@{thm[display] equalityE[no_vars]}
+\rulename{equalityE}
+*}
+
+
+text{*finite sets: insertion and membership relation*}
+text{*finite set notation*}
+
+lemma "insert x A = {x} \<union> A"
+  apply (blast)
+  done
+
+text{*
+@{thm[display] insert_is_Un[no_vars]}
+\rulename{insert_is_Un}
+*}
+
+lemma "{a,b} \<union> {c,d} = {a,b,c,d}"
+  apply (blast)
+  done
+
+lemma "{a,b} \<inter> {b,c} = {b}"
+  apply (auto)
+  oops
+text{*fails because it isn't valid*}
+
+lemma "{a,b} \<inter> {b,c} = (if a=c then {a,b} else {b})"
+  apply (simp)
+  apply (blast)
+  done
+
+text{*or just force or auto.  blast alone can't handle the if-then-else*}
+
+text{*next: some comprehension examples*}
+
+lemma "(a \<in> {z. P z}) = P a"
+  apply (blast)
+  done
+
+text{*
+@{thm[display] mem_Collect_eq[no_vars]}
+\rulename{mem_Collect_eq}
+*}
+
+lemma "{x. x \<in> A} = A"
+  apply (blast)
+  done
+  
+text{*
+@{thm[display] Collect_mem_eq[no_vars]}
+\rulename{Collect_mem_eq}
+*}
+
+lemma "{x. P x \<or> x \<in> A} = {x. P x} \<union> A"
+  apply (blast)
+  done
+
+lemma "{x. P x \<longrightarrow> Q x} = -{x. P x} \<union> {x. Q x}"
+  apply (blast)
+  done
+
+constdefs
+  prime   :: "nat set"
+    "prime == {p. 1<p & (ALL m. m dvd p --> m=1 | m=p)}"
+
+lemma "{p*q | p q. p\<in>prime \<and> q\<in>prime} = 
+       {z. \<exists>p q. z = p*q \<and> p\<in>prime \<and> q\<in>prime}"
+  apply (blast)
+  done
+
+text{*binders*}
+
+text{*bounded quantifiers*}
+
+lemma "(\<exists>x\<in>A. P x) = (\<exists>x. x\<in>A \<and> P x)"
+  apply (blast)
+  done
+
+text{*
+@{thm[display] bexI[no_vars]}
+\rulename{bexI}
+*}
+
+text{*
+@{thm[display] bexE[no_vars]}
+\rulename{bexE}
+*}
+
+lemma "(\<forall>x\<in>A. P x) = (\<forall>x. x\<in>A \<longrightarrow> P x)"
+  apply (blast)
+  done
+
+text{*
+@{thm[display] ballI[no_vars]}
+\rulename{ballI}
+*}
+
+text{*
+@{thm[display] bspec[no_vars]}
+\rulename{bspec}
+*}
+
+text{*indexed unions and variations*}
+
+lemma "(\<Union>x. B x) = (\<Union>x\<in>UNIV. B x)"
+  apply (blast)
+  done
+
+text{*
+@{thm[display] UN_iff[no_vars]}
+\rulename{UN_iff}
+*}
+
+text{*
+@{thm[display] Union_iff[no_vars]}
+\rulename{Union_iff}
+*}
+
+lemma "(\<Union>x\<in>A. B x) = {y. \<exists>x\<in>A. y \<in> B x}"
+  apply (blast)
+  done
+
+lemma "\<Union>S = (\<Union>x\<in>S. x)"
+  apply (blast)
+  done
+
+text{*
+@{thm[display] UN_I[no_vars]}
+\rulename{UN_I}
+*}
+
+text{*
+@{thm[display] UN_E[no_vars]}
+\rulename{UN_E}
+*}
+
+text{*indexed intersections*}
+
+lemma "(\<Inter>x. B x) = {y. \<forall>x. y \<in> B x}"
+  apply (blast)
+  done
+
+text{*
+@{thm[display] INT_iff[no_vars]}
+\rulename{INT_iff}
+*}
+
+text{*
+@{thm[display] Inter_iff[no_vars]}
+\rulename{Inter_iff}
+*}
+
+text{*mention also card, Pow, etc.*}
+
+
+text{*
+@{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}
+*}
+
+end