author | traytel |
Thu, 25 Apr 2013 10:31:10 +0200 | |
changeset 51772 | d2b265ebc1fa |
parent 50530 | 6266e44b3396 |
child 52992 | abd760a19e22 |
permissions | -rw-r--r-- |
50530 | 1 |
(* Title: HOL/BNF/Examples/Koenig.thy |
50517 | 2 |
Author: Dmitriy Traytel, TU Muenchen |
3 |
Author: Andrei Popescu, TU Muenchen |
|
4 |
Copyright 2012 |
|
5 |
||
6 |
Koenig's lemma. |
|
7 |
*) |
|
8 |
||
9 |
header {* Koenig's lemma *} |
|
10 |
||
11 |
theory Koenig |
|
50518 | 12 |
imports TreeFI Stream |
50517 | 13 |
begin |
14 |
||
15 |
(* selectors for streams *) |
|
16 |
lemma shd_def': "shd as = fst (stream_dtor as)" |
|
17 |
unfolding shd_def stream_case_def fst_def by (rule refl) |
|
18 |
||
19 |
lemma stl_def': "stl as = snd (stream_dtor as)" |
|
20 |
unfolding stl_def stream_case_def snd_def by (rule refl) |
|
21 |
||
22 |
lemma unfold_pair_fun_shd[simp]: "shd (stream_dtor_unfold (f \<odot> g) t) = f t" |
|
23 |
unfolding shd_def' pair_fun_def stream.dtor_unfold by simp |
|
24 |
||
25 |
lemma unfold_pair_fun_stl[simp]: "stl (stream_dtor_unfold (f \<odot> g) t) = |
|
26 |
stream_dtor_unfold (f \<odot> g) (g t)" |
|
27 |
unfolding stl_def' pair_fun_def stream.dtor_unfold by simp |
|
28 |
||
29 |
(* infinite trees: *) |
|
30 |
coinductive infiniteTr where |
|
31 |
"\<lbrakk>tr' \<in> listF_set (sub tr); infiniteTr tr'\<rbrakk> \<Longrightarrow> infiniteTr tr" |
|
32 |
||
33 |
lemma infiniteTr_strong_coind[consumes 1, case_names sub]: |
|
34 |
assumes *: "phi tr" and |
|
35 |
**: "\<And> tr. phi tr \<Longrightarrow> \<exists> tr' \<in> listF_set (sub tr). phi tr' \<or> infiniteTr tr'" |
|
36 |
shows "infiniteTr tr" |
|
37 |
using assms by (elim infiniteTr.coinduct) blast |
|
38 |
||
39 |
lemma infiniteTr_coind[consumes 1, case_names sub, induct pred: infiniteTr]: |
|
40 |
assumes *: "phi tr" and |
|
41 |
**: "\<And> tr. phi tr \<Longrightarrow> \<exists> tr' \<in> listF_set (sub tr). phi tr'" |
|
42 |
shows "infiniteTr tr" |
|
43 |
using assms by (elim infiniteTr.coinduct) blast |
|
44 |
||
45 |
lemma infiniteTr_sub[simp]: |
|
46 |
"infiniteTr tr \<Longrightarrow> (\<exists> tr' \<in> listF_set (sub tr). infiniteTr tr')" |
|
47 |
by (erule infiniteTr.cases) blast |
|
48 |
||
49 |
definition "konigPath \<equiv> stream_dtor_unfold |
|
50 |
(lab \<odot> (\<lambda>tr. SOME tr'. tr' \<in> listF_set (sub tr) \<and> infiniteTr tr'))" |
|
51 |
||
52 |
lemma konigPath_simps[simp]: |
|
53 |
"shd (konigPath t) = lab t" |
|
54 |
"stl (konigPath t) = konigPath (SOME tr. tr \<in> listF_set (sub t) \<and> infiniteTr tr)" |
|
55 |
unfolding konigPath_def by simp+ |
|
56 |
||
57 |
(* proper paths in trees: *) |
|
58 |
coinductive properPath where |
|
59 |
"\<lbrakk>shd as = lab tr; tr' \<in> listF_set (sub tr); properPath (stl as) tr'\<rbrakk> \<Longrightarrow> |
|
60 |
properPath as tr" |
|
61 |
||
62 |
lemma properPath_strong_coind[consumes 1, case_names shd_lab sub]: |
|
63 |
assumes *: "phi as tr" and |
|
64 |
**: "\<And> as tr. phi as tr \<Longrightarrow> shd as = lab tr" and |
|
65 |
***: "\<And> as tr. |
|
66 |
phi as tr \<Longrightarrow> |
|
67 |
\<exists> tr' \<in> listF_set (sub tr). phi (stl as) tr' \<or> properPath (stl as) tr'" |
|
68 |
shows "properPath as tr" |
|
69 |
using assms by (elim properPath.coinduct) blast |
|
70 |
||
71 |
lemma properPath_coind[consumes 1, case_names shd_lab sub, induct pred: properPath]: |
|
72 |
assumes *: "phi as tr" and |
|
73 |
**: "\<And> as tr. phi as tr \<Longrightarrow> shd as = lab tr" and |
|
74 |
***: "\<And> as tr. |
|
75 |
phi as tr \<Longrightarrow> |
|
76 |
\<exists> tr' \<in> listF_set (sub tr). phi (stl as) tr'" |
|
77 |
shows "properPath as tr" |
|
78 |
using properPath_strong_coind[of phi, OF * **] *** by blast |
|
79 |
||
80 |
lemma properPath_shd_lab: |
|
81 |
"properPath as tr \<Longrightarrow> shd as = lab tr" |
|
82 |
by (erule properPath.cases) blast |
|
83 |
||
84 |
lemma properPath_sub: |
|
85 |
"properPath as tr \<Longrightarrow> |
|
86 |
\<exists> tr' \<in> listF_set (sub tr). phi (stl as) tr' \<or> properPath (stl as) tr'" |
|
87 |
by (erule properPath.cases) blast |
|
88 |
||
89 |
(* prove the following by coinduction *) |
|
90 |
theorem Konig: |
|
91 |
assumes "infiniteTr tr" |
|
92 |
shows "properPath (konigPath tr) tr" |
|
93 |
proof- |
|
94 |
{fix as |
|
95 |
assume "infiniteTr tr \<and> as = konigPath tr" hence "properPath as tr" |
|
96 |
proof (induct rule: properPath_coind, safe) |
|
97 |
fix t |
|
98 |
let ?t = "SOME t'. t' \<in> listF_set (sub t) \<and> infiniteTr t'" |
|
99 |
assume "infiniteTr t" |
|
100 |
hence "\<exists>t' \<in> listF_set (sub t). infiniteTr t'" by simp |
|
101 |
hence "\<exists>t'. t' \<in> listF_set (sub t) \<and> infiniteTr t'" by blast |
|
102 |
hence "?t \<in> listF_set (sub t) \<and> infiniteTr ?t" by (elim someI_ex) |
|
103 |
moreover have "stl (konigPath t) = konigPath ?t" by simp |
|
104 |
ultimately show "\<exists>t' \<in> listF_set (sub t). |
|
105 |
infiniteTr t' \<and> stl (konigPath t) = konigPath t'" by blast |
|
106 |
qed simp |
|
107 |
} |
|
108 |
thus ?thesis using assms by blast |
|
109 |
qed |
|
110 |
||
111 |
(* some more stream theorems *) |
|
112 |
||
51772
d2b265ebc1fa
specify nicer names for map, set and rel in the stream library
traytel
parents:
50530
diff
changeset
|
113 |
lemma stream_map[simp]: "smap f = stream_dtor_unfold (f o shd \<odot> stl)" |
d2b265ebc1fa
specify nicer names for map, set and rel in the stream library
traytel
parents:
50530
diff
changeset
|
114 |
unfolding smap_def pair_fun_def shd_def'[abs_def] stl_def'[abs_def] |
50517 | 115 |
map_pair_def o_def prod_case_beta by simp |
116 |
||
117 |
definition plus :: "nat stream \<Rightarrow> nat stream \<Rightarrow> nat stream" (infixr "\<oplus>" 66) where |
|
118 |
[simp]: "plus xs ys = |
|
119 |
stream_dtor_unfold ((%(xs, ys). shd xs + shd ys) \<odot> (%(xs, ys). (stl xs, stl ys))) (xs, ys)" |
|
120 |
||
121 |
definition scalar :: "nat \<Rightarrow> nat stream \<Rightarrow> nat stream" (infixr "\<cdot>" 68) where |
|
51772
d2b265ebc1fa
specify nicer names for map, set and rel in the stream library
traytel
parents:
50530
diff
changeset
|
122 |
[simp]: "scalar n = smap (\<lambda>x. n * x)" |
50517 | 123 |
|
124 |
definition ones :: "nat stream" where [simp]: "ones = stream_dtor_unfold ((%x. 1) \<odot> id) ()" |
|
125 |
definition twos :: "nat stream" where [simp]: "twos = stream_dtor_unfold ((%x. 2) \<odot> id) ()" |
|
126 |
definition ns :: "nat \<Rightarrow> nat stream" where [simp]: "ns n = scalar n ones" |
|
127 |
||
128 |
lemma "ones \<oplus> ones = twos" |
|
129 |
by (rule stream.coinduct[of "%x1 x2. \<exists>x. x1 = ones \<oplus> ones \<and> x2 = twos"]) auto |
|
130 |
||
131 |
lemma "n \<cdot> twos = ns (2 * n)" |
|
132 |
by (rule stream.coinduct[of "%x1 x2. \<exists>n. x1 = n \<cdot> twos \<and> x2 = ns (2 * n)"]) force+ |
|
133 |
||
134 |
lemma prod_scalar: "(n * m) \<cdot> xs = n \<cdot> m \<cdot> xs" |
|
135 |
by (rule stream.coinduct[of "%x1 x2. \<exists>n m xs. x1 = (n * m) \<cdot> xs \<and> x2 = n \<cdot> m \<cdot> xs"]) force+ |
|
136 |
||
137 |
lemma scalar_plus: "n \<cdot> (xs \<oplus> ys) = n \<cdot> xs \<oplus> n \<cdot> ys" |
|
138 |
by (rule stream.coinduct[of "%x1 x2. \<exists>n xs ys. x1 = n \<cdot> (xs \<oplus> ys) \<and> x2 = n \<cdot> xs \<oplus> n \<cdot> ys"]) |
|
139 |
(force simp: add_mult_distrib2)+ |
|
140 |
||
141 |
lemma plus_comm: "xs \<oplus> ys = ys \<oplus> xs" |
|
142 |
by (rule stream.coinduct[of "%x1 x2. \<exists>xs ys. x1 = xs \<oplus> ys \<and> x2 = ys \<oplus> xs"]) force+ |
|
143 |
||
144 |
lemma plus_assoc: "(xs \<oplus> ys) \<oplus> zs = xs \<oplus> ys \<oplus> zs" |
|
145 |
by (rule stream.coinduct[of "%x1 x2. \<exists>xs ys zs. x1 = (xs \<oplus> ys) \<oplus> zs \<and> x2 = xs \<oplus> ys \<oplus> zs"]) force+ |
|
146 |
||
147 |
end |