author | blanchet |
Sat, 08 Sep 2012 21:04:27 +0200 | |
changeset 49220 | a6260b4fb410 |
parent 49128 | 1a86ef0a0210 |
child 49222 | cbe8c859817c |
permissions | -rw-r--r-- |
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(* Title: Codatatype_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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4 |
Copyright 2012 |
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|
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Infinite streams. |
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7 |
*) |
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|
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header {* Infinite Streams *} |
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|
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theory Stream |
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imports TreeFI |
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13 |
begin |
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codata_raw stream: 's = "'a \<times> 's" |
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|
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(* selectors for streams *) |
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definition "hdd as \<equiv> fst (stream_unf as)" |
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definition "tll as \<equiv> snd (stream_unf as)" |
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lemma coiter_pair_fun_hdd[simp]: "hdd (stream_unf_coiter (f \<odot> g) t) = f t" |
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unfolding hdd_def pair_fun_def stream.unf_coiter by simp |
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|
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lemma coiter_pair_fun_tll[simp]: "tll (stream_unf_coiter (f \<odot> g) t) = |
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stream_unf_coiter (f \<odot> g) (g t)" |
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unfolding tll_def pair_fun_def stream.unf_coiter by simp |
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|
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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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|
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lemma infiniteTr_coind_upto[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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|
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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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|
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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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|
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definition "konigPath \<equiv> stream_unf_coiter |
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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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|
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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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|
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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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|
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lemma properPath_coind_upto[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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|
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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_coind_upto[of phi, OF * **] *** by blast |
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|
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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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|
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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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|
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90 |
(* prove the following by coinduction *) |
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91 |
theorem Konig: |
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assumes "infiniteTr tr" |
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93 |
shows "properPath (konigPath tr) tr" |
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94 |
proof- |
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95 |
{fix as |
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96 |
assume "infiniteTr tr \<and> as = konigPath tr" hence "properPath as tr" |
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97 |
proof (induct rule: properPath_coind, safe) |
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98 |
fix t |
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99 |
let ?t = "SOME t'. t' \<in> listF_set (sub t) \<and> infiniteTr t'" |
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100 |
assume "infiniteTr t" |
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101 |
hence "\<exists>t' \<in> listF_set (sub t). infiniteTr t'" by simp |
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102 |
hence "\<exists>t'. t' \<in> listF_set (sub t) \<and> infiniteTr t'" by blast |
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103 |
hence "?t \<in> listF_set (sub t) \<and> infiniteTr ?t" by (elim someI_ex) |
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104 |
moreover have "tll (konigPath t) = konigPath ?t" by simp |
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105 |
ultimately show "\<exists>t' \<in> listF_set (sub t). |
7f79f94a432c
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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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|
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(* some more stream theorems *) |
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|
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lemma stream_map[simp]: "stream_map f = stream_unf_coiter (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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49220 | 118 |
lemma pre_stream_pred[simp]: "pre_stream_pred \<phi>1 \<phi>2 a b = (\<phi>1 (fst a) (fst b) \<and> \<phi>2 (snd a) (snd b))" |
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by (auto simp: pre_stream.pred_unfold) |
|
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lemmas stream_coind = mp[OF stream.pred_coinduct, unfolded pre_stream_pred[abs_def], |
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folded hdd_def tll_def] |
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|
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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_unf_coiter ((%(xs, ys). hdd xs + hdd ys) \<odot> (%(xs, ys). (tll xs, tll ys))) (xs, ys)" |
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|
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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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130 |
|
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definition ones :: "nat stream" where [simp]: "ones = stream_unf_coiter ((%x. 1) \<odot> id) ()" |
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definition twos :: "nat stream" where [simp]: "twos = stream_unf_coiter ((%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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|
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lemma "ones \<oplus> ones = twos" |
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136 |
by (intro stream_coind[where phi="%x1 x2. \<exists>x. x1 = ones \<oplus> ones \<and> x2 = twos"]) |
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137 |
auto |
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138 |
|
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lemma "n \<cdot> twos = ns (2 * n)" |
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by (intro stream_coind[where phi="%x1 x2. \<exists>n. x1 = n \<cdot> twos \<and> x2 = ns (2 * n)"]) |
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141 |
force+ |
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142 |
|
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143 |
lemma prod_scalar: "(n * m) \<cdot> xs = n \<cdot> m \<cdot> xs" |
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144 |
by (intro stream_coind[where phi="%x1 x2. \<exists>n m xs. x1 = (n * m) \<cdot> xs \<and> x2 = n \<cdot> m \<cdot> xs"]) |
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145 |
force+ |
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146 |
|
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lemma scalar_plus: "n \<cdot> (xs \<oplus> ys) = n \<cdot> xs \<oplus> n \<cdot> ys" |
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148 |
by (intro stream_coind[where phi="%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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149 |
(force simp: add_mult_distrib2)+ |
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150 |
|
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151 |
lemma plus_comm: "xs \<oplus> ys = ys \<oplus> xs" |
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by (intro stream_coind[where phi="%x1 x2. \<exists>xs ys. x1 = xs \<oplus> ys \<and> x2 = ys \<oplus> xs"]) |
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153 |
force+ |
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154 |
|
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lemma plus_assoc: "(xs \<oplus> ys) \<oplus> zs = xs \<oplus> ys \<oplus> zs" |
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156 |
by (intro stream_coind[where phi="%x1 x2. \<exists>xs ys zs. x1 = (xs \<oplus> ys) \<oplus> zs \<and> x2 = xs \<oplus> ys \<oplus> zs"]) |
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157 |
force+ |
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158 |
|
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159 |
end |