author | haftmann |
Fri, 08 Jul 2016 23:43:11 +0200 | |
changeset 63415 | 8f91c2f447a0 |
parent 63377 | 64adf4ba9526 |
child 63462 | c1fe30f2bc32 |
permissions | -rw-r--r-- |
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combinator to build partial equivalence relations from a predicate and an equivalenc relation
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(* Author: Florian Haftmann, TU Muenchen *) |
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section \<open>A combinator to build partial equivalence relations from a predicate and an equivalence relation\<close> |
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theory Combine_PER |
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imports Main "~~/src/HOL/Library/Lattice_Syntax" |
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begin |
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definition combine_per :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" |
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where |
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"combine_per P R = (\<lambda>x y. P x \<and> P y) \<sqinter> R" |
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lemma combine_per_simp [simp]: |
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fixes R (infixl "\<approx>" 50) |
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shows "combine_per P R x y \<longleftrightarrow> P x \<and> P y \<and> x \<approx> y" |
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by (simp add: combine_per_def) |
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lemma combine_per_top [simp]: |
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"combine_per \<top> R = R" |
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by (simp add: fun_eq_iff) |
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lemma combine_per_eq [simp]: |
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"combine_per P HOL.eq = HOL.eq \<sqinter> (\<lambda>x y. P x)" |
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by (auto simp add: fun_eq_iff) |
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lemma symp_combine_per: |
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"symp R \<Longrightarrow> symp (combine_per P R)" |
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by (auto simp add: symp_def sym_def combine_per_def) |
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lemma transp_combine_per: |
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"transp R \<Longrightarrow> transp (combine_per P R)" |
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by (auto simp add: transp_def trans_def combine_per_def) |
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lemma combine_perI: |
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fixes R (infixl "\<approx>" 50) |
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shows "P x \<Longrightarrow> P y \<Longrightarrow> x \<approx> y \<Longrightarrow> combine_per P R x y" |
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by (simp add: combine_per_def) |
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lemma symp_combine_per_symp: |
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"symp R \<Longrightarrow> symp (combine_per P R)" |
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by (auto intro!: sympI elim: sympE) |
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lemma transp_combine_per_transp: |
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"transp R \<Longrightarrow> transp (combine_per P R)" |
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by (auto intro!: transpI elim: transpE) |
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lemma equivp_combine_per_part_equivp [intro?]: |
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fixes R (infixl "\<approx>" 50) |
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assumes "\<exists>x. P x" and "equivp R" |
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shows "part_equivp (combine_per P R)" |
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proof - |
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from \<open>\<exists>x. P x\<close> obtain x where "P x" .. |
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moreover from \<open>equivp R\<close> have "x \<approx> x" by (rule equivp_reflp) |
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ultimately have "\<exists>x. P x \<and> x \<approx> x" by blast |
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with \<open>equivp R\<close> show ?thesis |
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by (auto intro!: part_equivpI symp_combine_per_symp transp_combine_per_transp |
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elim: equivpE) |
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qed |
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end |