(* Title: HOL/Library/Graphs.thy
ID: $Id$
Author: Alexander Krauss, TU Muenchen
*)
header ""
theory Graphs
imports Main SCT_Misc Kleene_Algebras ExecutableSet
begin
subsection {* Basic types, Size Change Graphs *}
datatype ('a, 'b) graph =
Graph "('a \<times> 'b \<times> 'a) set"
fun dest_graph :: "('a, 'b) graph \<Rightarrow> ('a \<times> 'b \<times> 'a) set"
where "dest_graph (Graph G) = G"
lemma graph_dest_graph[simp]:
"Graph (dest_graph G) = G"
by (cases G) simp
lemma split_graph_all:
"(\<And>gr. PROP P gr) \<equiv> (\<And>set. PROP P (Graph set))"
proof
fix set
assume "\<And>gr. PROP P gr"
then show "PROP P (Graph set)" .
next
fix gr
assume "\<And>set. PROP P (Graph set)"
then have "PROP P (Graph (dest_graph gr))" .
then show "PROP P gr" by simp
qed
definition
has_edge :: "('n,'e) graph \<Rightarrow> 'n \<Rightarrow> 'e \<Rightarrow> 'n \<Rightarrow> bool"
("_ \<turnstile> _ \<leadsto>\<^bsup>_\<^esup> _")
where
"has_edge G n e n' = ((n, e, n') \<in> dest_graph G)"
subsection {* Graph composition *}
fun grcomp :: "('n, 'e::times) graph \<Rightarrow> ('n, 'e) graph \<Rightarrow> ('n, 'e) graph"
where
"grcomp (Graph G) (Graph H) =
Graph {(p,b,q) | p b q.
(\<exists>k e e'. (p,e,k)\<in>G \<and> (k,e',q)\<in>H \<and> b = e * e')}"
declare grcomp.simps[code del]
lemma graph_ext:
assumes "\<And>n e n'. has_edge G n e n' = has_edge H n e n'"
shows "G = H"
using prems
by (cases G, cases H, auto simp:split_paired_all has_edge_def)
instance graph :: (type, type) "{comm_monoid_add}"
graph_zero_def: "0 == Graph {}"
graph_plus_def: "G + H == Graph (dest_graph G \<union> dest_graph H)"
proof
fix x y z :: "('a,'b) graph"
show "x + y + z = x + (y + z)"
and "x + y = y + x"
and "0 + x = x"
unfolding graph_plus_def graph_zero_def
by auto
qed
lemmas [code func del] = graph_plus_def
instance graph :: (type, type) "{distrib_lattice, complete_lattice}"
graph_leq_def: "G \<le> H \<equiv> dest_graph G \<subseteq> dest_graph H"
graph_less_def: "G < H \<equiv> dest_graph G \<subset> dest_graph H"
"inf G H \<equiv> Graph (dest_graph G \<inter> dest_graph H)"
"sup G H \<equiv> G + H"
Inf_graph_def: "Inf \<equiv> \<lambda>Gs. Graph (\<Inter>(dest_graph ` Gs))"
proof
fix x y z :: "('a,'b) graph"
fix A :: "('a, 'b) graph set"
show "(x < y) = (x \<le> y \<and> x \<noteq> y)"
unfolding graph_leq_def graph_less_def
by (cases x, cases y) auto
show "x \<le> x" unfolding graph_leq_def ..
{ assume "x \<le> y" "y \<le> z"
with order_trans show "x \<le> z"
unfolding graph_leq_def . }
{ assume "x \<le> y" "y \<le> x" thus "x = y"
unfolding graph_leq_def
by (cases x, cases y) simp }
show "inf x y \<le> x" "inf x y \<le> y"
unfolding inf_graph_def graph_leq_def
by auto
{ assume "x \<le> y" "x \<le> z" thus "x \<le> inf y z"
unfolding inf_graph_def graph_leq_def
by auto }
show "x \<le> sup x y" "y \<le> sup x y"
unfolding sup_graph_def graph_leq_def graph_plus_def by auto
{ assume "y \<le> x" "z \<le> x" thus "sup y z \<le> x"
unfolding sup_graph_def graph_leq_def graph_plus_def by auto }
show "sup x (inf y z) = inf (sup x y) (sup x z)"
unfolding inf_graph_def sup_graph_def graph_leq_def graph_plus_def by auto
{ assume "x \<in> A" thus "Inf A \<le> x"
unfolding Inf_graph_def graph_leq_def by auto }
{ assume "\<And>x. x \<in> A \<Longrightarrow> z \<le> x" thus "z \<le> Inf A"
unfolding Inf_graph_def graph_leq_def by auto }
qed
lemmas [code func del] = graph_leq_def graph_less_def
inf_graph_def sup_graph_def Inf_graph_def
lemma in_grplus:
"has_edge (G + H) p b q = (has_edge G p b q \<or> has_edge H p b q)"
by (cases G, cases H, auto simp:has_edge_def graph_plus_def)
lemma in_grzero:
"has_edge 0 p b q = False"
by (simp add:graph_zero_def has_edge_def)
subsubsection {* Multiplicative Structure *}
instance graph :: (type, times) mult_zero
graph_mult_def: "G * H == grcomp G H"
proof
fix a :: "('a, 'b) graph"
show "0 * a = 0"
unfolding graph_mult_def graph_zero_def
by (cases a) (simp add:grcomp.simps)
show "a * 0 = 0"
unfolding graph_mult_def graph_zero_def
by (cases a) (simp add:grcomp.simps)
qed
lemmas [code func del] = graph_mult_def
instance graph :: (type, one) one
graph_one_def: "1 == Graph { (x, 1, x) |x. True}" ..
lemma in_grcomp:
"has_edge (G * H) p b q
= (\<exists>k e e'. has_edge G p e k \<and> has_edge H k e' q \<and> b = e * e')"
by (cases G, cases H) (auto simp:graph_mult_def has_edge_def image_def)
lemma in_grunit:
"has_edge 1 p b q = (p = q \<and> b = 1)"
by (auto simp:graph_one_def has_edge_def)
instance graph :: (type, semigroup_mult) semigroup_mult
proof
fix G1 G2 G3 :: "('a,'b) graph"
show "G1 * G2 * G3 = G1 * (G2 * G3)"
proof (rule graph_ext, rule trans)
fix p J q
show "has_edge ((G1 * G2) * G3) p J q =
(\<exists>G i H j I.
has_edge G1 p G i
\<and> has_edge G2 i H j
\<and> has_edge G3 j I q
\<and> J = (G * H) * I)"
by (simp only:in_grcomp) blast
show "\<dots> = has_edge (G1 * (G2 * G3)) p J q"
by (simp only:in_grcomp mult_assoc) blast
qed
qed
fun grpow :: "nat \<Rightarrow> ('a::type, 'b::monoid_mult) graph \<Rightarrow> ('a, 'b) graph"
where
"grpow 0 A = 1"
| "grpow (Suc n) A = A * (grpow n A)"
instance graph :: (type, monoid_mult)
"{semiring_1,idem_add,recpower,star}"
graph_pow_def: "A ^ n == grpow n A"
graph_star_def: "star G == (SUP n. G ^ n)"
proof
fix a b c :: "('a, 'b) graph"
show "1 * a = a"
by (rule graph_ext) (auto simp:in_grcomp in_grunit)
show "a * 1 = a"
by (rule graph_ext) (auto simp:in_grcomp in_grunit)
show "(a + b) * c = a * c + b * c"
by (rule graph_ext, simp add:in_grcomp in_grplus) blast
show "a * (b + c) = a * b + a * c"
by (rule graph_ext, simp add:in_grcomp in_grplus) blast
show "(0::('a,'b) graph) \<noteq> 1" unfolding graph_zero_def graph_one_def
by simp
show "a + a = a" unfolding graph_plus_def by simp
show "a ^ 0 = 1" "\<And>n. a ^ (Suc n) = a * a ^ n"
unfolding graph_pow_def by simp_all
qed
lemma graph_leqI:
assumes "\<And>n e n'. has_edge G n e n' \<Longrightarrow> has_edge H n e n'"
shows "G \<le> H"
using prems
unfolding graph_leq_def has_edge_def
by auto
lemma in_graph_plusE:
assumes "has_edge (G + H) n e n'"
assumes "has_edge G n e n' \<Longrightarrow> P"
assumes "has_edge H n e n' \<Longrightarrow> P"
shows P
using prems
by (auto simp: in_grplus)
lemma
assumes "x \<in> S k"
shows "x \<in> (\<Union>k. S k)"
using prems by blast
lemma graph_union_least:
assumes "\<And>n. Graph (G n) \<le> C"
shows "Graph (\<Union>n. G n) \<le> C"
using prems unfolding graph_leq_def
by auto
lemma Sup_graph_eq:
"(SUP n. Graph (G n)) = Graph (\<Union>n. G n)"
proof (rule order_antisym)
show "(SUP n. Graph (G n)) \<le> Graph (\<Union>n. G n)"
by (rule SUP_leI) (auto simp add: graph_leq_def)
show "Graph (\<Union>n. G n) \<le> (SUP n. Graph (G n))"
by (rule graph_union_least, rule le_SUPI', rule)
qed
lemma has_edge_leq: "has_edge G p b q = (Graph {(p,b,q)} \<le> G)"
unfolding has_edge_def graph_leq_def
by (cases G) simp
lemma Sup_graph_eq2:
"(SUP n. G n) = Graph (\<Union>n. dest_graph (G n))"
using Sup_graph_eq[of "\<lambda>n. dest_graph (G n)", simplified]
by simp
lemma in_SUP:
"has_edge (SUP x. Gs x) p b q = (\<exists>x. has_edge (Gs x) p b q)"
unfolding Sup_graph_eq2 has_edge_leq graph_leq_def
by simp
instance graph :: (type, monoid_mult) kleene_by_complete_lattice
proof
fix a b c :: "('a, 'b) graph"
show "a \<le> b \<longleftrightarrow> a + b = b" unfolding graph_leq_def graph_plus_def
by (cases a, cases b) auto
from order_less_le show "a < b \<longleftrightarrow> a \<le> b \<and> a \<noteq> b" .
show "a * star b * c = (SUP n. a * b ^ n * c)"
unfolding graph_star_def
by (rule graph_ext) (force simp:in_SUP in_grcomp)
qed
lemma in_star:
"has_edge (star G) a x b = (\<exists>n. has_edge (G ^ n) a x b)"
by (auto simp:graph_star_def in_SUP)
lemma tcl_is_SUP:
"tcl (G::('a::type, 'b::monoid_mult) graph) =
(SUP n. G ^ (Suc n))"
unfolding tcl_def
using star_cont[of 1 G G]
by (simp add:power_Suc power_commutes)
lemma in_tcl:
"has_edge (tcl G) a x b = (\<exists>n>0. has_edge (G ^ n) a x b)"
apply (auto simp: tcl_is_SUP in_SUP)
apply (rule_tac x = "n - 1" in exI, auto)
done
subsection {* Infinite Paths *}
types ('n, 'e) ipath = "('n \<times> 'e) sequence"
definition has_ipath :: "('n, 'e) graph \<Rightarrow> ('n, 'e) ipath \<Rightarrow> bool"
where
"has_ipath G p =
(\<forall>i. has_edge G (fst (p i)) (snd (p i)) (fst (p (Suc i))))"
subsection {* Finite Paths *}
types ('n, 'e) fpath = "('n \<times> ('e \<times> 'n) list)"
inductive2 has_fpath :: "('n, 'e) graph \<Rightarrow> ('n, 'e) fpath \<Rightarrow> bool"
for G :: "('n, 'e) graph"
where
has_fpath_empty: "has_fpath G (n, [])"
| has_fpath_join: "\<lbrakk>G \<turnstile> n \<leadsto>\<^bsup>e\<^esup> n'; has_fpath G (n', es)\<rbrakk> \<Longrightarrow> has_fpath G (n, (e, n')#es)"
definition
"end_node p =
(if snd p = [] then fst p else snd (snd p ! (length (snd p) - 1)))"
definition path_nth :: "('n, 'e) fpath \<Rightarrow> nat \<Rightarrow> ('n \<times> 'e \<times> 'n)"
where
"path_nth p k = (if k = 0 then fst p else snd (snd p ! (k - 1)), snd p ! k)"
lemma endnode_nth:
assumes "length (snd p) = Suc k"
shows "end_node p = snd (snd (path_nth p k))"
using prems unfolding end_node_def path_nth_def
by auto
lemma path_nth_graph:
assumes "k < length (snd p)"
assumes "has_fpath G p"
shows "(\<lambda>(n,e,n'). has_edge G n e n') (path_nth p k)"
using prems
proof (induct k arbitrary:p)
case 0 thus ?case
unfolding path_nth_def by (auto elim:has_fpath.cases)
next
case (Suc k p)
from `has_fpath G p` show ?case
proof (rule has_fpath.cases)
case goal1 with Suc show ?case by simp
next
fix n e n' es
assume st: "p = (n, (e, n') # es)"
"G \<turnstile> n \<leadsto>\<^bsup>e\<^esup> n'"
"has_fpath G (n', es)"
with Suc
have "(\<lambda>(n, b, a). G \<turnstile> n \<leadsto>\<^bsup>b\<^esup> a) (path_nth (n', es) k)" by simp
with st show ?thesis by (cases k, auto simp:path_nth_def)
qed
qed
lemma path_nth_connected:
assumes "Suc k < length (snd p)"
shows "fst (path_nth p (Suc k)) = snd (snd (path_nth p k))"
using prems
unfolding path_nth_def
by auto
definition path_loop :: "('n, 'e) fpath \<Rightarrow> ('n, 'e) ipath" ("omega")
where
"omega p \<equiv> (\<lambda>i. (\<lambda>(n,e,n'). (n,e)) (path_nth p (i mod (length (snd p)))))"
lemma fst_p0: "fst (path_nth p 0) = fst p"
unfolding path_nth_def by simp
lemma path_loop_connect:
assumes "fst p = end_node p"
and "0 < length (snd p)" (is "0 < ?l")
shows "fst (path_nth p (Suc i mod (length (snd p))))
= snd (snd (path_nth p (i mod length (snd p))))"
(is "\<dots> = snd (snd (path_nth p ?k))")
proof -
from `0 < ?l` have "i mod ?l < ?l" (is "?k < ?l")
by simp
show ?thesis
proof (cases "Suc ?k < ?l")
case True
hence "Suc ?k \<noteq> ?l" by simp
with path_nth_connected[OF True]
show ?thesis
by (simp add:mod_Suc)
next
case False
with `?k < ?l` have wrap: "Suc ?k = ?l" by simp
hence "fst (path_nth p (Suc i mod ?l)) = fst (path_nth p 0)"
by (simp add: mod_Suc)
also from fst_p0 have "\<dots> = fst p" .
also have "\<dots> = end_node p" .
also have "\<dots> = snd (snd (path_nth p ?k))"
by (auto simp:endnode_nth wrap)
finally show ?thesis .
qed
qed
lemma path_loop_graph:
assumes "has_fpath G p"
and loop: "fst p = end_node p"
and nonempty: "0 < length (snd p)" (is "0 < ?l")
shows "has_ipath G (omega p)"
proof (auto simp:has_ipath_def)
fix i
from `0 < ?l` have "i mod ?l < ?l" (is "?k < ?l")
by simp
with path_nth_graph
have pk_G: "(\<lambda>(n,e,n'). has_edge G n e n') (path_nth p ?k)" .
from path_loop_connect[OF loop nonempty] pk_G
show "has_edge G (fst (omega p i)) (snd (omega p i)) (fst (omega p (Suc i)))"
unfolding path_loop_def has_edge_def split_def
by simp
qed
definition prod :: "('n, 'e::monoid_mult) fpath \<Rightarrow> 'e"
where
"prod p = foldr (op *) (map fst (snd p)) 1"
lemma prod_simps[simp]:
"prod (n, []) = 1"
"prod (n, (e,n')#es) = e * (prod (n',es))"
unfolding prod_def
by simp_all
lemma power_induces_path:
assumes a: "has_edge (A ^ k) n G m"
obtains p
where "has_fpath A p"
and "n = fst p" "m = end_node p"
and "G = prod p"
and "k = length (snd p)"
using a
proof (induct k arbitrary:m n G thesis)
case (0 m n G)
let ?p = "(n, [])"
from 0 have "has_fpath A ?p" "m = end_node ?p" "G = prod ?p"
by (auto simp:in_grunit end_node_def intro:has_fpath.intros)
thus ?case using 0 by (auto simp:end_node_def)
next
case (Suc k m n G)
hence "has_edge (A * A ^ k) n G m"
by (simp add:power_Suc power_commutes)
then obtain G' H j where
a_A: "has_edge A n G' j"
and H_pow: "has_edge (A ^ k) j H m"
and [simp]: "G = G' * H"
by (auto simp:in_grcomp)
from H_pow and Suc
obtain p
where p_path: "has_fpath A p"
and [simp]: "j = fst p" "m = end_node p" "H = prod p"
"k = length (snd p)"
by blast
let ?p' = "(n, (G', j)#snd p)"
from a_A and p_path
have "has_fpath A ?p'" "m = end_node ?p'" "G = prod ?p'"
by (auto simp:end_node_def nth.simps intro:has_fpath.intros split:nat.split)
thus ?case using Suc by auto
qed
subsection {* Sub-Paths *}
definition sub_path :: "('n, 'e) ipath \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> ('n, 'e) fpath"
("(_\<langle>_,_\<rangle>)")
where
"p\<langle>i,j\<rangle> =
(fst (p i), map (\<lambda>k. (snd (p k), fst (p (Suc k)))) [i ..< j])"
lemma sub_path_is_path:
assumes ipath: "has_ipath G p"
assumes l: "i \<le> j"
shows "has_fpath G (p\<langle>i,j\<rangle>)"
using l
proof (induct i rule:inc_induct)
case base show ?case by (auto simp:sub_path_def intro:has_fpath.intros)
next
case (step i)
with ipath upt_rec[of i j]
show ?case
by (auto simp:sub_path_def has_ipath_def intro:has_fpath.intros)
qed
lemma sub_path_start[simp]:
"fst (p\<langle>i,j\<rangle>) = fst (p i)"
by (simp add:sub_path_def)
lemma nth_upto[simp]: "k < j - i \<Longrightarrow> [i ..< j] ! k = i + k"
by (induct k) auto
lemma sub_path_end[simp]:
"i < j \<Longrightarrow> end_node (p\<langle>i,j\<rangle>) = fst (p j)"
by (auto simp:sub_path_def end_node_def)
lemma foldr_map: "foldr f (map g xs) = foldr (f o g) xs"
by (induct xs) auto
lemma upto_append[simp]:
assumes "i \<le> j" "j \<le> k"
shows "[ i ..< j ] @ [j ..< k] = [i ..< k]"
using prems and upt_add_eq_append[of i j "k - j"]
by simp
lemma foldr_monoid: "foldr (op *) xs 1 * foldr (op *) ys 1
= foldr (op *) (xs @ ys) (1::'a::monoid_mult)"
by (induct xs) (auto simp:mult_assoc)
lemma sub_path_prod:
assumes "i < j"
assumes "j < k"
shows "prod (p\<langle>i,k\<rangle>) = prod (p\<langle>i,j\<rangle>) * prod (p\<langle>j,k\<rangle>)"
using prems
unfolding prod_def sub_path_def
by (simp add:map_compose[symmetric] comp_def)
(simp only:foldr_monoid map_append[symmetric] upto_append)
lemma path_acgpow_aux:
assumes "length es = l"
assumes "has_fpath G (n,es)"
shows "has_edge (G ^ l) n (prod (n,es)) (end_node (n,es))"
using prems
proof (induct l arbitrary:n es)
case 0 thus ?case
by (simp add:in_grunit end_node_def)
next
case (Suc l n es)
hence "es \<noteq> []" by auto
let ?n' = "snd (hd es)"
let ?es' = "tl es"
let ?e = "fst (hd es)"
from Suc have len: "length ?es' = l" by auto
from Suc
have [simp]: "end_node (n, es) = end_node (?n', ?es')"
by (cases es) (auto simp:end_node_def nth.simps split:nat.split)
from `has_fpath G (n,es)`
have "has_fpath G (?n', ?es')"
by (rule has_fpath.cases) (auto intro:has_fpath.intros)
with Suc len
have "has_edge (G ^ l) ?n' (prod (?n', ?es')) (end_node (?n', ?es'))"
by auto
moreover
from `es \<noteq> []`
have "prod (n, es) = ?e * (prod (?n', ?es'))"
by (cases es) auto
moreover
from `has_fpath G (n,es)` have c:"has_edge G n ?e ?n'"
by (rule has_fpath.cases) (insert `es \<noteq> []`, auto)
ultimately
show ?case
unfolding power_Suc
by (auto simp:in_grcomp)
qed
lemma path_acgpow:
"has_fpath G p
\<Longrightarrow> has_edge (G ^ length (snd p)) (fst p) (prod p) (end_node p)"
by (cases p)
(rule path_acgpow_aux[of "snd p" "length (snd p)" _ "fst p", simplified])
lemma star_paths:
"has_edge (star G) a x b =
(\<exists>p. has_fpath G p \<and> a = fst p \<and> b = end_node p \<and> x = prod p)"
proof
assume "has_edge (star G) a x b"
then obtain n where pow: "has_edge (G ^ n) a x b"
by (auto simp:in_star)
then obtain p where
"has_fpath G p" "a = fst p" "b = end_node p" "x = prod p"
by (rule power_induces_path)
thus "\<exists>p. has_fpath G p \<and> a = fst p \<and> b = end_node p \<and> x = prod p"
by blast
next
assume "\<exists>p. has_fpath G p \<and> a = fst p \<and> b = end_node p \<and> x = prod p"
then obtain p where
"has_fpath G p" "a = fst p" "b = end_node p" "x = prod p"
by blast
hence "has_edge (G ^ length (snd p)) a x b"
by (auto intro:path_acgpow)
thus "has_edge (star G) a x b"
by (auto simp:in_star)
qed
lemma plus_paths:
"has_edge (tcl G) a x b =
(\<exists>p. has_fpath G p \<and> a = fst p \<and> b = end_node p \<and> x = prod p \<and> 0 < length (snd p))"
proof
assume "has_edge (tcl G) a x b"
then obtain n where pow: "has_edge (G ^ n) a x b" and "0 < n"
by (auto simp:in_tcl)
from pow obtain p where
"has_fpath G p" "a = fst p" "b = end_node p" "x = prod p"
"n = length (snd p)"
by (rule power_induces_path)
with `0 < n`
show "\<exists>p. has_fpath G p \<and> a = fst p \<and> b = end_node p \<and> x = prod p \<and> 0 < length (snd p) "
by blast
next
assume "\<exists>p. has_fpath G p \<and> a = fst p \<and> b = end_node p \<and> x = prod p
\<and> 0 < length (snd p)"
then obtain p where
"has_fpath G p" "a = fst p" "b = end_node p" "x = prod p"
"0 < length (snd p)"
by blast
hence "has_edge (G ^ length (snd p)) a x b"
by (auto intro:path_acgpow)
with `0 < length (snd p)`
show "has_edge (tcl G) a x b"
by (auto simp:in_tcl)
qed
definition
"contract s p =
(\<lambda>i. (fst (p (s i)), prod (p\<langle>s i,s (Suc i)\<rangle>)))"
lemma ipath_contract:
assumes [simp]: "increasing s"
assumes ipath: "has_ipath G p"
shows "has_ipath (tcl G) (contract s p)"
unfolding has_ipath_def
proof
fix i
let ?p = "p\<langle>s i,s (Suc i)\<rangle>"
from increasing_strict
have "fst (p (s (Suc i))) = end_node ?p" by simp
moreover
from increasing_strict[of s i "Suc i"] have "snd ?p \<noteq> []"
by (simp add:sub_path_def)
moreover
from ipath increasing_weak[of s] have "has_fpath G ?p"
by (rule sub_path_is_path) auto
ultimately
show "has_edge (tcl G)
(fst (contract s p i)) (snd (contract s p i)) (fst (contract s p (Suc i)))"
unfolding contract_def plus_paths
by (intro exI) auto
qed
lemma prod_unfold:
"i < j \<Longrightarrow> prod (p\<langle>i,j\<rangle>)
= snd (p i) * prod (p\<langle>Suc i, j\<rangle>)"
unfolding prod_def
by (simp add:sub_path_def upt_rec[of "i" j])
lemma sub_path_loop:
assumes "0 < k"
assumes k:"k = length (snd loop)"
assumes loop: "fst loop = end_node loop"
shows "(omega loop)\<langle>k * i,k * Suc i\<rangle> = loop" (is "?\<omega> = loop")
proof (rule prod_eqI)
show "fst ?\<omega> = fst loop"
by (auto simp:path_loop_def path_nth_def split_def k)
show "snd ?\<omega> = snd loop"
proof (rule nth_equalityI[rule_format])
show leneq: "length (snd ?\<omega>) = length (snd loop)"
unfolding sub_path_def k by simp
fix j assume "j < length (snd (?\<omega>))"
with leneq and k have "j < k" by simp
have a: "\<And>i. fst (path_nth loop (Suc i mod k))
= snd (snd (path_nth loop (i mod k)))"
unfolding k
apply (rule path_loop_connect[OF loop])
by (insert prems, auto)
from `j < k`
show "snd ?\<omega> ! j = snd loop ! j"
unfolding sub_path_def
apply (simp add:path_loop_def split_def add_ac)
apply (simp add:a k[symmetric])
by (simp add:path_nth_def)
qed
qed
end