author | blanchet |
Tue, 21 Jan 2014 01:14:49 +0100 | |
changeset 55091 | c43394c2e5ec |
parent 55075 | b3d0a02a756d |
child 55129 | 26bd1cba3ab5 |
permissions | -rw-r--r-- |
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(* Title: HOL/BNF_Examples/Derivation_Trees/Prelim.thy |
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Author: Andrei Popescu, TU Muenchen |
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Copyright 2012 |
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Preliminaries. |
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*) |
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header {* Preliminaries *} |
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theory Prelim |
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imports "~~/src/HOL/Library/More_BNFs" |
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begin |
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notation BNF_Def.convol ("<_ , _>") |
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declare fset_to_fset[simp] |
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lemma fst_snd_convol_o[simp]: "<fst o s, snd o s> = s" |
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apply(rule ext) by (simp add: convol_def) |
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abbreviation sm_abbrev (infix "\<oplus>" 60) |
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where "f \<oplus> g \<equiv> Sum_Type.sum_map f g" |
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lemma sum_map_InlD: "(f \<oplus> g) z = Inl x \<Longrightarrow> \<exists>y. z = Inl y \<and> f y = x" |
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by (cases z) auto |
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lemma sum_map_InrD: "(f \<oplus> g) z = Inr x \<Longrightarrow> \<exists>y. z = Inr y \<and> g y = x" |
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by (cases z) auto |
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abbreviation sum_case_abbrev ("[[_,_]]" 800) |
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where "[[f,g]] \<equiv> Sum_Type.sum_case f g" |
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lemma Inl_oplus_elim: |
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assumes "Inl tr \<in> (id \<oplus> f) ` tns" |
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shows "Inl tr \<in> tns" |
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using assms apply clarify by (case_tac x, auto) |
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lemma Inl_oplus_iff[simp]: "Inl tr \<in> (id \<oplus> f) ` tns \<longleftrightarrow> Inl tr \<in> tns" |
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using Inl_oplus_elim |
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by (metis id_def image_iff sum_map.simps(1)) |
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lemma Inl_m_oplus[simp]: "Inl -` (id \<oplus> f) ` tns = Inl -` tns" |
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using Inl_oplus_iff unfolding vimage_def by auto |
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lemma Inr_oplus_elim: |
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assumes "Inr tr \<in> (id \<oplus> f) ` tns" |
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shows "\<exists> n. Inr n \<in> tns \<and> f n = tr" |
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using assms apply clarify by (case_tac x, auto) |
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lemma Inr_oplus_iff[simp]: |
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"Inr tr \<in> (id \<oplus> f) ` tns \<longleftrightarrow> (\<exists> n. Inr n \<in> tns \<and> f n = tr)" |
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apply (rule iffI) |
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apply (metis Inr_oplus_elim) |
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by (metis image_iff sum_map.simps(2)) |
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lemma Inr_m_oplus[simp]: "Inr -` (id \<oplus> f) ` tns = f ` (Inr -` tns)" |
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using Inr_oplus_iff unfolding vimage_def by auto |
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lemma Inl_Inr_image_cong: |
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assumes "Inl -` A = Inl -` B" and "Inr -` A = Inr -` B" |
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shows "A = B" |
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apply safe using assms apply(case_tac x, auto) by(case_tac x, auto) |
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end |