61 lines
1.5 KiB
Plaintext
61 lines
1.5 KiB
Plaintext
(*
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* Copyright 2014, NICTA
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*
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* This software may be distributed and modified according to the terms of
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* the BSD 2-Clause license. Note that NO WARRANTY is provided.
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* See "LICENSE_BSD2.txt" for details.
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*
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* @TAG(NICTA_BSD)
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*)
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(*
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Miscellaneous library definitions and lemmas.
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*)
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header "Distinct Proposition"
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theory DistinctProp
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imports
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"../lib/Lib"
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"~~/src/HOL/Library/Sublist"
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begin
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text {* distinct\_prop *}
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primrec
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distinct_prop :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a list \<Rightarrow> bool)"
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where
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"distinct_prop P [] = True"
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| "distinct_prop P (x # xs) = ((\<forall>y\<in>set xs. P x y) \<and> distinct_prop P xs)"
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lemma distinct_prop_map:
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"distinct_prop P (map f xs)
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= distinct_prop (\<lambda>x y. P (f x) (f y)) xs"
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apply (induct xs)
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apply simp
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apply simp
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done
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lemma distinct_prop_append:
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"distinct_prop P (xs @ ys) =
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(distinct_prop P xs \<and> distinct_prop P ys \<and> (\<forall>x \<in> set xs. \<forall>y \<in> set ys. P x y))"
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apply (induct xs arbitrary: ys)
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apply simp
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apply (simp add: conj_ac ball_Un)
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done
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lemma distinct_prop_distinct:
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"\<lbrakk> distinct xs; \<And>x y. \<lbrakk> x \<in> set xs; y \<in> set xs; x \<noteq> y \<rbrakk> \<Longrightarrow> P x y \<rbrakk>
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\<Longrightarrow> distinct_prop P xs"
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apply (induct xs)
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apply simp
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apply clarsimp
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apply blast
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done
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lemma distinct_prop_True [simp]:
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"distinct_prop (\<lambda>x y. True) xs"
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by (induct xs, auto)
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end
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