author | blanchet |
Fri, 21 Sep 2012 16:34:40 +0200 | |
changeset 49509 | 163914705f8d |
parent 49508 | 1e205327f059 |
permissions | -rw-r--r-- |
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(* Title: HOL/BNF/Examples/Stream.thy |
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Author: Dmitriy Traytel, TU Muenchen |
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Author: Andrei Popescu, TU Muenchen |
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Copyright 2012 |
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Infinite streams. |
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*) |
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header {* Infinite Streams *} |
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theory Stream |
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imports TreeFI |
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begin |
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hide_const (open) Quotient_Product.prod_rel |
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hide_fact (open) Quotient_Product.prod_rel_def |
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codata_raw stream: 's = "'a \<times> 's" |
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(* selectors for streams *) |
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definition "hdd as \<equiv> fst (stream_dtor as)" |
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definition "tll as \<equiv> snd (stream_dtor as)" |
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lemma unfold_pair_fun_hdd[simp]: "hdd (stream_dtor_unfold (f \<odot> g) t) = f t" |
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unfolding hdd_def pair_fun_def stream.dtor_unfolds by simp |
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lemma unfold_pair_fun_tll[simp]: "tll (stream_dtor_unfold (f \<odot> g) t) = |
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stream_dtor_unfold (f \<odot> g) (g t)" |
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unfolding tll_def pair_fun_def stream.dtor_unfolds by simp |
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(* infinite trees: *) |
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coinductive infiniteTr where |
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"\<lbrakk>tr' \<in> listF_set (sub tr); infiniteTr tr'\<rbrakk> \<Longrightarrow> infiniteTr tr" |
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lemma infiniteTr_strong_coind[consumes 1, case_names sub]: |
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assumes *: "phi tr" and |
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**: "\<And> tr. phi tr \<Longrightarrow> \<exists> tr' \<in> listF_set (sub tr). phi tr' \<or> infiniteTr tr'" |
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shows "infiniteTr tr" |
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using assms by (elim infiniteTr.coinduct) blast |
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lemma infiniteTr_coind[consumes 1, case_names sub, induct pred: infiniteTr]: |
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assumes *: "phi tr" and |
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**: "\<And> tr. phi tr \<Longrightarrow> \<exists> tr' \<in> listF_set (sub tr). phi tr'" |
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shows "infiniteTr tr" |
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using assms by (elim infiniteTr.coinduct) blast |
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lemma infiniteTr_sub[simp]: |
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"infiniteTr tr \<Longrightarrow> (\<exists> tr' \<in> listF_set (sub tr). infiniteTr tr')" |
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by (erule infiniteTr.cases) blast |
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definition "konigPath \<equiv> stream_dtor_unfold |
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(lab \<odot> (\<lambda>tr. SOME tr'. tr' \<in> listF_set (sub tr) \<and> infiniteTr tr'))" |
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lemma hdd_simps1[simp]: "hdd (konigPath t) = lab t" |
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unfolding konigPath_def by simp |
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lemma tll_simps2[simp]: "tll (konigPath t) = |
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konigPath (SOME tr. tr \<in> listF_set (sub t) \<and> infiniteTr tr)" |
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unfolding konigPath_def by simp |
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(* proper paths in trees: *) |
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coinductive properPath where |
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"\<lbrakk>hdd as = lab tr; tr' \<in> listF_set (sub tr); properPath (tll as) tr'\<rbrakk> \<Longrightarrow> |
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properPath as tr" |
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lemma properPath_strong_coind[consumes 1, case_names hdd_lab sub]: |
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assumes *: "phi as tr" and |
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**: "\<And> as tr. phi as tr \<Longrightarrow> hdd as = lab tr" and |
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***: "\<And> as tr. |
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phi as tr \<Longrightarrow> |
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\<exists> tr' \<in> listF_set (sub tr). phi (tll as) tr' \<or> properPath (tll as) tr'" |
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shows "properPath as tr" |
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using assms by (elim properPath.coinduct) blast |
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lemma properPath_coind[consumes 1, case_names hdd_lab sub, induct pred: properPath]: |
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assumes *: "phi as tr" and |
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**: "\<And> as tr. phi as tr \<Longrightarrow> hdd as = lab tr" and |
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***: "\<And> as tr. |
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phi as tr \<Longrightarrow> |
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\<exists> tr' \<in> listF_set (sub tr). phi (tll as) tr'" |
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shows "properPath as tr" |
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using properPath_strong_coind[of phi, OF * **] *** by blast |
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lemma properPath_hdd_lab: |
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"properPath as tr \<Longrightarrow> hdd as = lab tr" |
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by (erule properPath.cases) blast |
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lemma properPath_sub: |
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"properPath as tr \<Longrightarrow> |
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\<exists> tr' \<in> listF_set (sub tr). phi (tll as) tr' \<or> properPath (tll as) tr'" |
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by (erule properPath.cases) blast |
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(* prove the following by coinduction *) |
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theorem Konig: |
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assumes "infiniteTr tr" |
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shows "properPath (konigPath tr) tr" |
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proof- |
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{fix as |
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assume "infiniteTr tr \<and> as = konigPath tr" hence "properPath as tr" |
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proof (induct rule: properPath_coind, safe) |
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fix t |
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let ?t = "SOME t'. t' \<in> listF_set (sub t) \<and> infiniteTr t'" |
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assume "infiniteTr t" |
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hence "\<exists>t' \<in> listF_set (sub t). infiniteTr t'" by simp |
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hence "\<exists>t'. t' \<in> listF_set (sub t) \<and> infiniteTr t'" by blast |
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hence "?t \<in> listF_set (sub t) \<and> infiniteTr ?t" by (elim someI_ex) |
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moreover have "tll (konigPath t) = konigPath ?t" by simp |
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ultimately show "\<exists>t' \<in> listF_set (sub t). |
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infiniteTr t' \<and> tll (konigPath t) = konigPath t'" by blast |
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qed simp |
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} |
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thus ?thesis using assms by blast |
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qed |
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(* some more stream theorems *) |
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lemma stream_map[simp]: "stream_map f = stream_dtor_unfold (f o hdd \<odot> tll)" |
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unfolding stream_map_def pair_fun_def hdd_def[abs_def] tll_def[abs_def] |
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map_pair_def o_def prod_case_beta by simp |
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lemma prod_rel[simp]: "prod_rel \<phi>1 \<phi>2 a b = (\<phi>1 (fst a) (fst b) \<and> \<phi>2 (snd a) (snd b))" |
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unfolding prod_rel_def by auto |
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lemmas stream_coind = |
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mp[OF stream.rel_coinduct, unfolded prod_rel[abs_def], folded hdd_def tll_def] |
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definition plus :: "nat stream \<Rightarrow> nat stream \<Rightarrow> nat stream" (infixr "\<oplus>" 66) where |
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[simp]: "plus xs ys = |
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stream_dtor_unfold ((%(xs, ys). hdd xs + hdd ys) \<odot> (%(xs, ys). (tll xs, tll ys))) (xs, ys)" |
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definition scalar :: "nat \<Rightarrow> nat stream \<Rightarrow> nat stream" (infixr "\<cdot>" 68) where |
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[simp]: "scalar n = stream_map (\<lambda>x. n * x)" |
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definition ones :: "nat stream" where [simp]: "ones = stream_dtor_unfold ((%x. 1) \<odot> id) ()" |
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definition twos :: "nat stream" where [simp]: "twos = stream_dtor_unfold ((%x. 2) \<odot> id) ()" |
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definition ns :: "nat \<Rightarrow> nat stream" where [simp]: "ns n = scalar n ones" |
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lemma "ones \<oplus> ones = twos" |
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by (intro stream_coind[where P="%x1 x2. \<exists>x. x1 = ones \<oplus> ones \<and> x2 = twos"]) auto |
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lemma "n \<cdot> twos = ns (2 * n)" |
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by (intro stream_coind[where P="%x1 x2. \<exists>n. x1 = n \<cdot> twos \<and> x2 = ns (2 * n)"]) force+ |
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lemma prod_scalar: "(n * m) \<cdot> xs = n \<cdot> m \<cdot> xs" |
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by (intro stream_coind[where P="%x1 x2. \<exists>n m xs. x1 = (n * m) \<cdot> xs \<and> x2 = n \<cdot> m \<cdot> xs"]) force+ |
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lemma scalar_plus: "n \<cdot> (xs \<oplus> ys) = n \<cdot> xs \<oplus> n \<cdot> ys" |
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by (intro stream_coind[where P="%x1 x2. \<exists>n xs ys. x1 = n \<cdot> (xs \<oplus> ys) \<and> x2 = n \<cdot> xs \<oplus> n \<cdot> ys"]) |
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(force simp: add_mult_distrib2)+ |
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lemma plus_comm: "xs \<oplus> ys = ys \<oplus> xs" |
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by (intro stream_coind[where P="%x1 x2. \<exists>xs ys. x1 = xs \<oplus> ys \<and> x2 = ys \<oplus> xs"]) force+ |
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lemma plus_assoc: "(xs \<oplus> ys) \<oplus> zs = xs \<oplus> ys \<oplus> zs" |
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by (intro stream_coind[where P="%x1 x2. \<exists>xs ys zs. x1 = (xs \<oplus> ys) \<oplus> zs \<and> x2 = xs \<oplus> ys \<oplus> zs"]) force+ |
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end |