author | traytel |
Fri, 21 Sep 2012 12:27:56 +0200 | |
changeset 49495 | 675b9df572df |
parent 49325 | 340844cbf7af |
child 49509 | 163914705f8d |
permissions | -rw-r--r-- |
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(* Title: HOL/Codatatype/BNF_Def.thy |
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Author: Dmitriy Traytel, TU Muenchen |
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Copyright 2012 |
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Definition of bounded natural functors. |
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*) |
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header {* Definition of Bounded Natural Functors *} |
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theory BNF_Def |
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imports BNF_Util |
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keywords |
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"print_bnfs" :: diag and |
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"bnf_def" :: thy_goal |
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begin |
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lemma collect_o: "collect F o g = collect ((\<lambda>f. f o g) ` F)" |
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by (rule ext) (auto simp only: o_apply collect_def) |
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||
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lemma converse_mono: |
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"R1 ^-1 \<subseteq> R2 ^-1 \<longleftrightarrow> R1 \<subseteq> R2" |
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unfolding converse_def by auto |
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lemma converse_shift: |
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"R1 \<subseteq> R2 ^-1 \<Longrightarrow> R1 ^-1 \<subseteq> R2" |
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unfolding converse_def by auto |
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||
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definition convol ("<_ , _>") where |
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"<f , g> \<equiv> %a. (f a, g a)" |
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lemma fst_convol: |
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"fst o <f , g> = f" |
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apply(rule ext) |
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unfolding convol_def by simp |
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lemma snd_convol: |
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"snd o <f , g> = g" |
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apply(rule ext) |
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unfolding convol_def by simp |
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lemma convol_memI: |
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"\<lbrakk>f x = f' x; g x = g' x; P x\<rbrakk> \<Longrightarrow> <f , g> x \<in> {(f' a, g' a) |a. P a}" |
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unfolding convol_def by auto |
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||
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definition csquare where |
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"csquare A f1 f2 p1 p2 \<longleftrightarrow> (\<forall> a \<in> A. f1 (p1 a) = f2 (p2 a))" |
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(* The pullback of sets *) |
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definition thePull where |
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"thePull B1 B2 f1 f2 = {(b1,b2). b1 \<in> B1 \<and> b2 \<in> B2 \<and> f1 b1 = f2 b2}" |
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lemma wpull_thePull: |
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"wpull (thePull B1 B2 f1 f2) B1 B2 f1 f2 fst snd" |
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unfolding wpull_def thePull_def by auto |
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lemma wppull_thePull: |
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assumes "wppull A B1 B2 f1 f2 e1 e2 p1 p2" |
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shows |
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"\<exists> j. \<forall> a' \<in> thePull B1 B2 f1 f2. |
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j a' \<in> A \<and> |
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e1 (p1 (j a')) = e1 (fst a') \<and> e2 (p2 (j a')) = e2 (snd a')" |
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(is "\<exists> j. \<forall> a' \<in> ?A'. ?phi a' (j a')") |
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proof(rule bchoice[of ?A' ?phi], default) |
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fix a' assume a': "a' \<in> ?A'" |
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hence "fst a' \<in> B1" unfolding thePull_def by auto |
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moreover |
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from a' have "snd a' \<in> B2" unfolding thePull_def by auto |
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moreover have "f1 (fst a') = f2 (snd a')" |
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using a' unfolding csquare_def thePull_def by auto |
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ultimately show "\<exists> ja'. ?phi a' ja'" |
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using assms unfolding wppull_def by blast |
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qed |
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lemma wpull_wppull: |
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assumes wp: "wpull A' B1 B2 f1 f2 p1' p2'" and |
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1: "\<forall> a' \<in> A'. j a' \<in> A \<and> e1 (p1 (j a')) = e1 (p1' a') \<and> e2 (p2 (j a')) = e2 (p2' a')" |
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shows "wppull A B1 B2 f1 f2 e1 e2 p1 p2" |
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unfolding wppull_def proof safe |
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fix b1 b2 |
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assume b1: "b1 \<in> B1" and b2: "b2 \<in> B2" and f: "f1 b1 = f2 b2" |
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then obtain a' where a': "a' \<in> A'" and b1: "b1 = p1' a'" and b2: "b2 = p2' a'" |
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using wp unfolding wpull_def by blast |
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show "\<exists>a\<in>A. e1 (p1 a) = e1 b1 \<and> e2 (p2 a) = e2 b2" |
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apply (rule bexI[of _ "j a'"]) unfolding b1 b2 using a' 1 by auto |
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qed |
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lemma wppull_id: "\<lbrakk>wpull UNIV UNIV UNIV f1 f2 p1 p2; e1 = id; e2 = id\<rbrakk> \<Longrightarrow> |
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wppull UNIV UNIV UNIV f1 f2 e1 e2 p1 p2" |
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by (erule wpull_wppull) auto |
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lemma Id_alt: "Id = Gr UNIV id" |
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unfolding Gr_def by auto |
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lemma Gr_UNIV_id: "f = id \<Longrightarrow> (Gr UNIV f)^-1 O Gr UNIV f = Gr UNIV f" |
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unfolding Gr_def by auto |
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lemma Gr_mono: "A \<subseteq> B \<Longrightarrow> Gr A f \<subseteq> Gr B f" |
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unfolding Gr_def by auto |
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lemma wpull_Gr: |
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"wpull (Gr A f) A (f ` A) f id fst snd" |
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unfolding wpull_def Gr_def by auto |
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definition "pick_middle P Q a c = (SOME b. (a,b) \<in> P \<and> (b,c) \<in> Q)" |
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lemma pick_middle: |
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"(a,c) \<in> P O Q \<Longrightarrow> (a, pick_middle P Q a c) \<in> P \<and> (pick_middle P Q a c, c) \<in> Q" |
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unfolding pick_middle_def apply(rule someI_ex) |
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using assms unfolding relcomp_def by auto |
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definition fstO where "fstO P Q ac = (fst ac, pick_middle P Q (fst ac) (snd ac))" |
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definition sndO where "sndO P Q ac = (pick_middle P Q (fst ac) (snd ac), snd ac)" |
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lemma fstO_in: "ac \<in> P O Q \<Longrightarrow> fstO P Q ac \<in> P" |
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unfolding fstO_def |
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by (subst (asm) surjective_pairing) (rule pick_middle[THEN conjunct1]) |
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lemma fst_fstO: "fst bc = (fst \<circ> fstO P Q) bc" |
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unfolding comp_def fstO_def by simp |
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lemma snd_sndO: "snd bc = (snd \<circ> sndO P Q) bc" |
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unfolding comp_def sndO_def by simp |
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lemma sndO_in: "ac \<in> P O Q \<Longrightarrow> sndO P Q ac \<in> Q" |
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unfolding sndO_def |
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by (subst (asm) surjective_pairing) (rule pick_middle[THEN conjunct2]) |
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lemma csquare_fstO_sndO: |
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"csquare (P O Q) snd fst (fstO P Q) (sndO P Q)" |
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unfolding csquare_def fstO_def sndO_def using pick_middle by simp |
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lemma wppull_fstO_sndO: |
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shows "wppull (P O Q) P Q snd fst fst snd (fstO P Q) (sndO P Q)" |
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using pick_middle unfolding wppull_def fstO_def sndO_def relcomp_def by auto |
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lemma snd_fst_flip: "snd xy = (fst o (%(x, y). (y, x))) xy" |
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by (simp split: prod.split) |
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lemma fst_snd_flip: "fst xy = (snd o (%(x, y). (y, x))) xy" |
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by (simp split: prod.split) |
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lemma flip_rel: "A \<subseteq> (R ^-1) \<Longrightarrow> (%(x, y). (y, x)) ` A \<subseteq> R" |
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by auto |
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lemma pointfreeE: "f o g = f' o g' \<Longrightarrow> f (g x) = f' (g' x)" |
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unfolding o_def fun_eq_iff by simp |
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ML_file "Tools/bnf_def_tactics.ML" |
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ML_file"Tools/bnf_def.ML" |
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end |