theory RTree
imports "~~/src/HOL/Library/Transitive_Closure_Table"
begin
section {* A theory of relational trees *}
inductive_cases path_nilE [elim!]: "rtrancl_path r x [] y"
inductive_cases path_consE [elim!]: "rtrancl_path r x (z#zs) y"
subsection {* Definitions *}
text {*
In this theory, we are giving to give a notion of of `Relational Graph` and
its derived notion `Relational Tree`. Given a binary relation @{text "r"},
the `Relational Graph of @{text "r"}` is the graph, the edges of which
are those in @{text "r"}. In this way, any binary relation can be viewed
as a `Relational Graph`. Note, this notion of graph includes infinite graphs.
A `Relation Graph` @{text "r"} is said to be a `Relational Tree` if it is both
{\em single valued} and {\em acyclic}.
*}
text {*
The following @{text "sgv"} specifies that relation @{text "r"} is {\em single valued}.
*}
locale sgv =
fixes r
assumes sgv: "single_valued r"
text {*
The following @{text "rtree"} specifies that @{text "r"} is a
{\em Relational Tree}.
*}
locale rtree = sgv +
assumes acl: "acyclic r"
text {*
The following two auxiliary functions @{text "rel_of"} and @{text "pred_of"}
transfer between the predicate and set representation of binary relations.
*}
definition "rel_of r = {(x, y) | x y. r x y}"
definition "pred_of r = (\<lambda> x y. (x, y) \<in> r)"
text {*
To reason about {\em Relational Graph}, a notion of path is
needed, which is given by the following @{text "rpath"} (short
for `relational path`).
The path @{text "xs"} in proposition @{text "rpath r x xs y"} is
a path leading from @{text "x"} to @{text "y"}, which serves as a
witness of the fact @{text "(x, y) \<in> r^*"}.
@{text "rpath"}
is simply a wrapper of the @{text "rtrancl_path"} defined in the imported
theory @{text "Transitive_Closure_Table"}, which defines
a notion of path for the predicate form of binary relations.
*}
definition "rpath r x xs y = rtrancl_path (pred_of r) x xs y"
text {*
Given a path @{text "ps"}, @{text "edges_on ps"} is the
set of edges along the path, which is defined as follows:
*}
definition "edges_on ps = {(a,b) | a b. \<exists> xs ys. ps = xs@[a,b]@ys}"
text {*
The following @{text "indep"} defines a notion of independence.
Two nodes @{text "x"} and @{text "y"} are said to be independent
(expressed as @{text "indep x y"}), if neither one is reachable
from the other in relational graph @{text "r"}.
*}
definition "indep r x y = (((x, y) \<notin> r^*) \<and> ((y, x) \<notin> r^*))"
text {*
In relational tree @{text "r"}, the sub tree of node @{text "x"} is written
@{text "subtree r x"}, which is defined to be the set of nodes (including itself)
which can reach @{text "x"} by following some path in @{text "r"}:
*}
definition "subtree r x = {y . (y, x) \<in> r^*}"
text {*
The following @{text "edge_in r x"} is the set of edges
contained in the sub-tree of @{text "x"}, with @{text "r"} as the underlying graph.
*}
definition "edges_in r x = {(a, b) | a b. (a, b) \<in> r \<and> b \<in> subtree r x}"
text {*
The following lemma @{text "edges_in_meaning"} shows the intuitive meaning
of `an edge @{text "(a, b)"} is in the sub-tree of @{text "x"}`,
i.e., both @{text "a"} and @{text "b"} are in the sub-tree.
*}
lemma edges_in_meaning:
"edges_in r x = {(a, b) | a b. (a, b) \<in> r \<and> a \<in> subtree r x \<and> b \<in> subtree r x}"
proof -
{ fix a b
assume h: "(a, b) \<in> r" "b \<in> subtree r x"
moreover have "a \<in> subtree r x"
proof -
from h(2)[unfolded subtree_def] have "(b, x) \<in> r^*" by simp
with h(1) have "(a, x) \<in> r^*" by auto
thus ?thesis by (auto simp:subtree_def)
qed
ultimately have "((a, b) \<in> r \<and> a \<in> subtree r x \<and> b \<in> subtree r x)"
by (auto)
} thus ?thesis by (auto simp:edges_in_def)
qed
text {*
The following lemma shows the means of @{term "edges_in"} from the other side,
which says to for the edge @{text "(a,b)"} to be outside of the sub-tree of @{text "x"},
it is sufficient if @{text "b"} is.
*}
lemma edges_in_refutation:
assumes "b \<notin> subtree r x"
shows "(a, b) \<notin> edges_in r x"
using assms by (unfold edges_in_def subtree_def, auto)
subsection {* Auxiliary lemmas *}
lemma index_minimize:
assumes "P (i::nat)"
obtains j where "P j" and "\<forall> k < j. \<not> P k"
proof -
have "\<exists> j. P j \<and> (\<forall> k < j. \<not> P k)"
using assms
proof(induct i rule:less_induct)
case (less t)
show ?case
proof(cases "\<forall> j < t. \<not> P j")
case True
with less (2) show ?thesis by blast
next
case False
then obtain j where "j < t" "P j" by auto
from less(1)[OF this]
show ?thesis .
qed
qed
with that show ?thesis by metis
qed
subsection {* Properties of Relational Graphs and Relational Trees *}
subsubsection {* Properties of @{text "rel_of"} and @{text "pred_of"} *}
text {* The following lemmas establish bijectivity of the two functions *}
lemma pred_rel_eq: "pred_of (rel_of r) = r" by (auto simp:rel_of_def pred_of_def)
lemma rel_pred_eq: "rel_of (pred_of r) = r" by (auto simp:rel_of_def pred_of_def)
lemma rel_of_star: "rel_of (r^**) = (rel_of r)^*"
by (unfold rel_of_def rtranclp_rtrancl_eq, auto)
lemma pred_of_star: "pred_of (r^*) = (pred_of r)^**"
proof -
{ fix x y
have "pred_of (r^*) x y = (pred_of r)^** x y"
by (unfold pred_of_def rtranclp_rtrancl_eq, auto)
} thus ?thesis by auto
qed
lemma star_2_pstar: "(x, y) \<in> r^* = (pred_of (r^*)) x y"
by (simp add: pred_of_def)
subsubsection {* Properties of @{text "rpath"} *}
text {* Induction rule for @{text "rpath"}: *}
print_statement rtrancl_path.induct
lemma rpath_induct [consumes 1, case_names rbase rstep, induct pred: rpath]:
assumes "rpath r x1 x2 x3"
and "\<And>x. P x [] x"
and "\<And>x y ys z. (x, y) \<in> r \<Longrightarrow> rpath r y ys z \<Longrightarrow> P y ys z \<Longrightarrow> P x (y # ys) z"
shows "P x1 x2 x3"
using assms[unfolded rpath_def]
by (induct, auto simp:pred_of_def rpath_def)
text {* Introduction rule for empty path *}
lemma rbaseI [intro!]:
assumes "x = y"
shows "rpath r x [] y"
by (unfold rpath_def assms,
rule Transitive_Closure_Table.rtrancl_path.base)
text {* Introduction rule for non-empty path *}
lemma rstepI [intro!]:
assumes "(x, y) \<in> r"
and "rpath r y ys z"
shows "rpath r x (y#ys) z"
proof(unfold rpath_def, rule Transitive_Closure_Table.rtrancl_path.step)
from assms(1) show "pred_of r x y" by (auto simp:pred_of_def)
next
from assms(2) show "rtrancl_path (pred_of r) y ys z"
by (auto simp:pred_of_def rpath_def)
qed
text {* Introduction rule for @{text "@"}-path *}
lemma rpath_appendI [intro]:
assumes "rpath r x xs a" and "rpath r a ys y"
shows "rpath r x (xs @ ys) y"
using assms
by (unfold rpath_def, auto intro:rtrancl_path_trans)
text {* Elimination rule for empty path *}
lemma rpath_cases [cases pred:rpath]:
assumes "rpath r a1 a2 a3"
obtains (rbase) "a1 = a3" and "a2 = []"
| (rstep) y :: "'a" and ys :: "'a list"
where "(a1, y) \<in> r" and "a2 = y # ys" and "rpath r y ys a3"
using assms [unfolded rpath_def]
by (cases, auto simp:rpath_def pred_of_def)
lemma rpath_nilE [elim!, cases pred:rpath]:
assumes "rpath r x [] y"
obtains "y = x"
using assms[unfolded rpath_def] by auto
-- {* This is a auxiliary lemmas used only in the proof of @{text "rpath_nnl_lastE"} *}
lemma rpath_nnl_last:
assumes "rtrancl_path r x xs y"
and "xs \<noteq> []"
obtains xs' where "xs = xs'@[y]"
proof -
from append_butlast_last_id[OF `xs \<noteq> []`, symmetric]
obtain xs' y' where eq_xs: "xs = (xs' @ y' # [])" by simp
with assms(1)
have "rtrancl_path r x ... y" by simp
hence "y = y'" by (rule rtrancl_path_appendE, auto)
with eq_xs have "xs = xs'@[y]" by simp
from that[OF this] show ?thesis .
qed
text {*
Elimination rule for non-empty paths constructed with @{text "#"}.
*}
lemma rpath_ConsE [elim!, cases pred:rpath]:
assumes "rpath r x (y # ys) x2"
obtains (rstep) "(x, y) \<in> r" and "rpath r y ys x2"
using assms[unfolded rpath_def]
by (cases, auto simp:rpath_def pred_of_def)
text {*
Elimination rule for non-empty path, where the destination node
@{text "y"} is shown to be at the end of the path.
*}
lemma rpath_nnl_lastE:
assumes "rpath r x xs y"
and "xs \<noteq> []"
obtains xs' where "xs = xs'@[y]"
using assms[unfolded rpath_def]
by (rule rpath_nnl_last, auto)
text {* Other elimination rules of @{text "rpath"} *}
lemma rpath_appendE:
assumes "rpath r x (xs @ [a] @ ys) y"
obtains "rpath r x (xs @ [a]) a" and "rpath r a ys y"
using rtrancl_path_appendE[OF assms[unfolded rpath_def, simplified], folded rpath_def]
by auto
lemma rpath_subE:
assumes "rpath r x (xs @ [a] @ ys @ [b] @ zs) y"
obtains "rpath r x (xs @ [a]) a" and "rpath r a (ys @ [b]) b" and "rpath r b zs y"
using assms
by (elim rpath_appendE, auto)
text {* Every path has a unique end point. *}
lemma rpath_dest_eq:
assumes "rpath r x xs x1"
and "rpath r x xs x2"
shows "x1 = x2"
using assms
by (induct, auto)
subsubsection {* Properites of @{text "edges_on"} *}
lemma edges_on_len:
assumes "(a,b) \<in> edges_on l"
shows "length l \<ge> 2"
using assms
by (unfold edges_on_def, auto)
text {* Elimination of @{text "edges_on"} for non-empty path *}
lemma edges_on_consE [elim, cases set:edges_on]:
assumes "(a,b) \<in> edges_on (x#xs)"
obtains (head) xs' where "x = a" and "xs = b#xs'"
| (tail) "(a,b) \<in> edges_on xs"
proof -
from assms obtain l1 l2
where h: "(x#xs) = l1 @ [a,b] @ l2" by (unfold edges_on_def, blast)
have "(\<exists> xs'. x = a \<and> xs = b#xs') \<or> ((a,b) \<in> edges_on xs)"
proof(cases "l1")
case Nil with h
show ?thesis by auto
next
case (Cons e el)
from h[unfolded this]
have "xs = el @ [a,b] @ l2" by auto
thus ?thesis
by (unfold edges_on_def, auto)
qed
thus ?thesis
proof
assume "(\<exists>xs'. x = a \<and> xs = b # xs')"
then obtain xs' where "x = a" "xs = b#xs'" by blast
from that(1)[OF this] show ?thesis .
next
assume "(a, b) \<in> edges_on xs"
from that(2)[OF this] show ?thesis .
qed
qed
text {*
Every edges on the path is a graph edges:
*}
lemma rpath_edges_on:
assumes "rpath r x xs y"
shows "(edges_on (x#xs)) \<subseteq> r"
using assms
proof(induct arbitrary:y)
case (rbase x)
thus ?case by (unfold edges_on_def, auto)
next
case (rstep x y ys z)
show ?case
proof -
{ fix a b
assume "(a, b) \<in> edges_on (x # y # ys)"
hence "(a, b) \<in> r" by (cases, insert rstep, auto)
} thus ?thesis by auto
qed
qed
text {* @{text "edges_on"} is mono with respect to @{text "#"}-operation: *}
lemma edges_on_Cons_mono:
shows "edges_on xs \<subseteq> edges_on (x#xs)"
proof -
{ fix a b
assume "(a, b) \<in> edges_on xs"
then obtain l1 l2 where "xs = l1 @ [a,b] @ l2"
by (auto simp:edges_on_def)
hence "x # xs = (x#l1) @ [a, b] @ l2" by auto
hence "(a, b) \<in> edges_on (x#xs)"
by (unfold edges_on_def, blast)
} thus ?thesis by auto
qed
text {*
The following rule @{text "rpath_transfer"} is used to show
that one path is intact as long as all the edges on it are intact
with the change of graph.
If @{text "x#xs"} is path in graph @{text "r1"} and
every edges along the path is also in @{text "r2"},
then @{text "x#xs"} is also a edge in graph @{text "r2"}:
*}
lemma rpath_transfer:
assumes "rpath r1 x xs y"
and "edges_on (x#xs) \<subseteq> r2"
shows "rpath r2 x xs y"
using assms
proof(induct)
case (rstep x y ys z)
show ?case
proof(rule rstepI)
show "(x, y) \<in> r2"
proof -
have "(x, y) \<in> edges_on (x # y # ys)"
by (unfold edges_on_def, auto)
with rstep(4) show ?thesis by auto
qed
next
show "rpath r2 y ys z"
using rstep edges_on_Cons_mono[of "y#ys" "x"] by (auto)
qed
qed (unfold rpath_def, auto intro!:Transitive_Closure_Table.rtrancl_path.base)
text {*
The following lemma extracts the path from @{text "x"} to @{text "y"}
from proposition @{text "(x, y) \<in> r^*"}
*}
lemma star_rpath:
assumes "(x, y) \<in> r^*"
obtains xs where "rpath r x xs y"
proof -
have "\<exists> xs. rpath r x xs y"
proof(unfold rpath_def, rule iffD1[OF rtranclp_eq_rtrancl_path])
from assms
show "(pred_of r)\<^sup>*\<^sup>* x y"
apply (fold pred_of_star)
by (auto simp:pred_of_def)
qed
from that and this show ?thesis by blast
qed
text {*
The following lemma uses the path @{text "xs"} from @{text "x"} to @{text "y"}
as a witness to show @{text "(x, y) \<in> r^*"}.
*}
lemma rpath_star:
assumes "rpath r x xs y"
shows "(x, y) \<in> r^*"
proof -
from iffD2[OF rtranclp_eq_rtrancl_path] and assms[unfolded rpath_def]
have "(pred_of r)\<^sup>*\<^sup>* x y" by metis
thus ?thesis by (simp add: pred_of_star star_2_pstar)
qed
text {*
The following lemmas establishes a relation from pathes in @{text "r"}
to @{text "r^+"} relation.
*}
lemma rpath_plus:
assumes "rpath r x xs y"
and "xs \<noteq> []"
shows "(x, y) \<in> r^+"
proof -
from assms(2) obtain e es where "xs = e#es" by (cases xs, auto)
from assms(1)[unfolded this]
show ?thesis
proof(cases)
case rstep
show ?thesis
proof -
from rpath_star[OF rstep(2)] have "(e, y) \<in> r\<^sup>*" .
with rstep(1) show "(x, y) \<in> r^+" by auto
qed
qed
qed
subsubsection {* Properties of @{text "subtree"} *}
text {*
@{text "subtree"} is mono with respect to the underlying graph.
*}
lemma subtree_mono:
assumes "r1 \<subseteq> r2"
shows "subtree r1 x \<subseteq> subtree r2 x"
proof
fix c
assume "c \<in> subtree r1 x"
hence "(c, x) \<in> r1^*" by (auto simp:subtree_def)
from star_rpath[OF this] obtain xs
where rp:"rpath r1 c xs x" by metis
hence "rpath r2 c xs x"
proof(rule rpath_transfer)
from rpath_edges_on[OF rp] have "edges_on (c # xs) \<subseteq> r1" .
with assms show "edges_on (c # xs) \<subseteq> r2" by auto
qed
thus "c \<in> subtree r2 x"
by (rule rpath_star[elim_format], auto simp:subtree_def)
qed
text {*
The following lemma characterizes the change of sub-tree of @{text "x"}
with the removal of an outside edge @{text "(a,b)"}.
Note that, according to lemma @{thm edges_in_refutation}, the assumption
@{term "b \<notin> subtree r x"} amounts to saying @{text "(a, b)"}
is outside the sub-tree of @{text "x"}.
*}
lemma subtree_del_outside: (* ddd *)
assumes "b \<notin> subtree r x"
shows "subtree (r - {(a, b)}) x = (subtree r x)"
proof -
{ fix c
assume "c \<in> (subtree r x)"
hence "(c, x) \<in> r^*" by (auto simp:subtree_def)
hence "c \<in> subtree (r - {(a, b)}) x"
proof(rule star_rpath)
fix xs
assume rp: "rpath r c xs x"
show ?thesis
proof -
from rp
have "rpath (r - {(a, b)}) c xs x"
proof(rule rpath_transfer)
from rpath_edges_on[OF rp] have "edges_on (c # xs) \<subseteq> r" .
moreover have "(a, b) \<notin> edges_on (c#xs)"
proof
assume "(a, b) \<in> edges_on (c # xs)"
then obtain l1 l2 where h: "c#xs = l1@[a,b]@l2" by (auto simp:edges_on_def)
hence "tl (c#xs) = tl (l1@[a,b]@l2)" by simp
then obtain l1' where eq_xs_b: "xs = l1'@[b]@l2" by (cases l1, auto)
from rp[unfolded this]
show False
proof(rule rpath_appendE)
assume "rpath r b l2 x"
thus ?thesis
by(rule rpath_star[elim_format], insert assms(1), auto simp:subtree_def)
qed
qed
ultimately show "edges_on (c # xs) \<subseteq> r - {(a,b)}" by auto
qed
thus ?thesis by (rule rpath_star[elim_format], auto simp:subtree_def)
qed
qed
} moreover {
fix c
assume "c \<in> subtree (r - {(a, b)}) x"
moreover have "... \<subseteq> (subtree r x)" by (rule subtree_mono, auto)
ultimately have "c \<in> (subtree r x)" by auto
} ultimately show ?thesis by auto
qed
lemma subtree_insert_ext:
assumes "b \<in> subtree r x"
shows "subtree (r \<union> {(a, b)}) x = (subtree r x) \<union> (subtree r a)"
using assms by (auto simp:subtree_def rtrancl_insert)
lemma subtree_insert_next:
assumes "b \<notin> subtree r x"
shows "subtree (r \<union> {(a, b)}) x = (subtree r x)"
using assms
by (auto simp:subtree_def rtrancl_insert)
subsubsection {* Properties about relational trees *}
context rtree
begin
lemma rpath_overlap_oneside: (* ddd *)
assumes "rpath r x xs1 x1"
and "rpath r x xs2 x2"
and "length xs1 \<le> length xs2"
obtains xs3 where "xs2 = xs1 @ xs3"
proof(cases "xs1 = []")
case True
with that show ?thesis by auto
next
case False
have "\<forall> i \<le> length xs1. take i xs1 = take i xs2"
proof -
{ assume "\<not> (\<forall> i \<le> length xs1. take i xs1 = take i xs2)"
then obtain i where "i \<le> length xs1 \<and> take i xs1 \<noteq> take i xs2" by auto
from this(1) have "False"
proof(rule index_minimize)
fix j
assume h1: "j \<le> length xs1 \<and> take j xs1 \<noteq> take j xs2"
and h2: " \<forall>k<j. \<not> (k \<le> length xs1 \<and> take k xs1 \<noteq> take k xs2)"
-- {* @{text "j - 1"} is the branch point between @{text "xs1"} and @{text "xs2"} *}
let ?idx = "j - 1"
-- {* A number of inequalities concerning @{text "j - 1"} are derived first *}
have lt_i: "?idx < length xs1" using False h1
by (metis Suc_diff_1 le_neq_implies_less length_greater_0_conv lessI less_imp_diff_less)
have lt_i': "?idx < length xs2" using lt_i and assms(3) by auto
have lt_j: "?idx < j" using h1 by (cases j, auto)
-- {* From thesis inequalities, a number of equations concerning @{text "xs1"}
and @{text "xs2"} are derived *}
have eq_take: "take ?idx xs1 = take ?idx xs2"
using h2[rule_format, OF lt_j] and h1 by auto
have eq_xs1: " xs1 = take ?idx xs1 @ xs1 ! (?idx) # drop (Suc (?idx)) xs1"
using id_take_nth_drop[OF lt_i] .
have eq_xs2: "xs2 = take ?idx xs2 @ xs2 ! (?idx) # drop (Suc (?idx)) xs2"
using id_take_nth_drop[OF lt_i'] .
-- {* The branch point along the path is finally pinpointed *}
have neq_idx: "xs1!?idx \<noteq> xs2!?idx"
proof -
have "take j xs1 = take ?idx xs1 @ [xs1 ! ?idx]"
using eq_xs1 Suc_diff_1 lt_i lt_j take_Suc_conv_app_nth by fastforce
moreover have eq_tk2: "take j xs2 = take ?idx xs2 @ [xs2 ! ?idx]"
using Suc_diff_1 lt_i' lt_j take_Suc_conv_app_nth by fastforce
ultimately show ?thesis using eq_take h1 by auto
qed
show ?thesis
proof(cases " take (j - 1) xs1 = []")
case True
have "(x, xs1!?idx) \<in> r"
proof -
from eq_xs1[unfolded True, simplified, symmetric] assms(1)
have "rpath r x ( xs1 ! ?idx # drop (Suc ?idx) xs1) x1" by simp
from this[unfolded rpath_def]
show ?thesis by (auto simp:pred_of_def)
qed
moreover have "(x, xs2!?idx) \<in> r"
proof -
from eq_xs2[folded eq_take, unfolded True, simplified, symmetric] assms(2)
have "rpath r x ( xs2 ! ?idx # drop (Suc ?idx) xs2) x2" by simp
from this[unfolded rpath_def]
show ?thesis by (auto simp:pred_of_def)
qed
ultimately show ?thesis using neq_idx sgv[unfolded single_valued_def] by metis
next
case False
then obtain e es where eq_es: "take ?idx xs1 = es@[e]"
using rev_exhaust by blast
have "(e, xs1!?idx) \<in> r"
proof -
from eq_xs1[unfolded eq_es]
have "xs1 = es@[e, xs1!?idx]@drop (Suc ?idx) xs1" by simp
hence "(e, xs1!?idx) \<in> edges_on xs1" by (simp add:edges_on_def, metis)
with rpath_edges_on[OF assms(1)] edges_on_Cons_mono[of xs1 x]
show ?thesis by auto
qed moreover have "(e, xs2!?idx) \<in> r"
proof -
from eq_xs2[folded eq_take, unfolded eq_es]
have "xs2 = es@[e, xs2!?idx]@drop (Suc ?idx) xs2" by simp
hence "(e, xs2!?idx) \<in> edges_on xs2" by (simp add:edges_on_def, metis)
with rpath_edges_on[OF assms(2)] edges_on_Cons_mono[of xs2 x]
show ?thesis by auto
qed
ultimately show ?thesis
using sgv[unfolded single_valued_def] neq_idx by metis
qed
qed
} thus ?thesis by auto
qed
from this[rule_format, of "length xs1"]
have "take (length xs1) xs1 = take (length xs1) xs2" by simp
moreover have "xs2 = take (length xs1) xs2 @ drop (length xs1) xs2" by simp
ultimately have "xs2 = xs1 @ drop (length xs1) xs2" by auto
from that[OF this] show ?thesis .
qed
lemma rpath_overlap [consumes 2, cases pred:rpath]:
assumes "rpath r x xs1 x1"
and "rpath r x xs2 x2"
obtains (less_1) xs3 where "xs2 = xs1 @ xs3"
| (less_2) xs3 where "xs1 = xs2 @ xs3"
proof -
have "length xs1 \<le> length xs2 \<or> length xs2 \<le> length xs1" by auto
with assms rpath_overlap_oneside that show ?thesis by metis
qed
text {*
As a corollary of @{thm "rpath_overlap_oneside"},
the following two lemmas gives one important property of relation tree,
i.e. there is at most one path between any two nodes.
Similar to the proof of @{thm rpath_overlap}, we starts with
the one side version first.
*}
lemma rpath_unique_oneside:
assumes "rpath r x xs1 y"
and "rpath r x xs2 y"
and "length xs1 \<le> length xs2"
shows "xs1 = xs2"
proof -
from rpath_overlap_oneside[OF assms]
obtain xs3 where less_1: "xs2 = xs1 @ xs3" by blast
show ?thesis
proof(cases "xs3 = []")
case True
from less_1[unfolded this] show ?thesis by simp
next
case False
note FalseH = this
show ?thesis
proof(cases "xs1 = []")
case True
have "(x, x) \<in> r^+"
proof(rule rpath_plus)
from assms(1)[unfolded True]
have "y = x" by (cases rule:rpath_nilE, simp)
from assms(2)[unfolded this] show "rpath r x xs2 x" .
next
from less_1 and False show "xs2 \<noteq> []" by simp
qed
with acl show ?thesis by (unfold acyclic_def, auto)
next
case False
then obtain e es where eq_xs1: "xs1 = es@[e]" using rev_exhaust by auto
from assms(2)[unfolded less_1 this]
have "rpath r x (es @ [e] @ xs3) y" by simp
thus ?thesis
proof(cases rule:rpath_appendE)
case 1
from rpath_dest_eq [OF 1(1)[folded eq_xs1] assms(1)]
have "e = y" .
from rpath_plus [OF 1(2)[unfolded this] FalseH]
have "(y, y) \<in> r^+" .
with acl show ?thesis by (unfold acyclic_def, auto)
qed
qed
qed
qed
text {*
The following is the full version of path uniqueness.
*}
lemma rpath_unique:
assumes "rpath r x xs1 y"
and "rpath r x xs2 y"
shows "xs1 = xs2"
proof(cases "length xs1 \<le> length xs2")
case True
from rpath_unique_oneside[OF assms this] show ?thesis .
next
case False
hence "length xs2 \<le> length xs1" by simp
from rpath_unique_oneside[OF assms(2,1) this]
show ?thesis by simp
qed
text {*
The following lemma shows that the `independence` relation is symmetric.
It is an obvious auxiliary lemma which will be used later.
*}
lemma sym_indep: "indep r x y \<Longrightarrow> indep r y x"
by (unfold indep_def, auto)
text {*
This is another `obvious` lemma about trees, which says trees rooted at
independent nodes are disjoint.
*}
lemma subtree_disjoint:
assumes "indep r x y"
shows "subtree r x \<inter> subtree r y = {}"
proof -
{ fix z x y xs1 xs2 xs3
assume ind: "indep r x y"
and rp1: "rpath r z xs1 x"
and rp2: "rpath r z xs2 y"
and h: "xs2 = xs1 @ xs3"
have False
proof(cases "xs1 = []")
case True
from rp1[unfolded this] have "x = z" by auto
from rp2[folded this] rpath_star ind[unfolded indep_def]
show ?thesis by metis
next
case False
then obtain e es where eq_xs1: "xs1 = es@[e]" using rev_exhaust by blast
from rp2[unfolded h this]
have "rpath r z (es @ [e] @ xs3) y" by simp
thus ?thesis
proof(cases rule:rpath_appendE)
case 1
have "e = x" using 1(1)[folded eq_xs1] rp1 rpath_dest_eq by metis
from rpath_star[OF 1(2)[unfolded this]] ind[unfolded indep_def]
show ?thesis by auto
qed
qed
} note my_rule = this
{ fix z
assume h: "z \<in> subtree r x" "z \<in> subtree r y"
from h(1) have "(z, x) \<in> r^*" by (unfold subtree_def, auto)
then obtain xs1 where rp1: "rpath r z xs1 x" using star_rpath by metis
from h(2) have "(z, y) \<in> r^*" by (unfold subtree_def, auto)
then obtain xs2 where rp2: "rpath r z xs2 y" using star_rpath by metis
from rp1 rp2
have False
by (cases, insert my_rule[OF sym_indep[OF assms(1)] rp2 rp1]
my_rule[OF assms(1) rp1 rp2], auto)
} thus ?thesis by auto
qed
text {*
The following lemma @{text "subtree_del"} characterizes the change of sub-tree of
@{text "x"} with the removal of an inside edge @{text "(a, b)"}.
Note that, the case for the removal of an outside edge has already been dealt with
in lemma @{text "subtree_del_outside"}).
This lemma is underpinned by the following two `obvious` facts:
\begin{enumearte}
\item
In graph @{text "r"}, for an inside edge @{text "(a,b) \<in> edges_in r x"},
every node @{text "c"} in the sub-tree of @{text "a"} has a path
which goes first from @{text "c"} to @{text "a"}, then through edge @{text "(a, b)"}, and
finally reaches @{text "x"}. By the uniqueness of path in a tree,
all paths from sub-tree of @{text "a"} to @{text "x"} are such constructed, therefore
must go through @{text "(a, b)"}. The consequence is: with the removal of @{text "(a,b)"},
all such paths will be broken.
\item
On the other hand, all paths not originate from within the sub-tree of @{text "a"}
will not be affected by the removal of edge @{text "(a, b)"}.
The reason is simple: if the path is affected by the removal, it must
contain @{text "(a, b)"}, then it must originate from within the sub-tree of @{text "a"}.
\end{enumearte}
*}
lemma subtree_del_inside: (* ddd *)
assumes "(a,b) \<in> edges_in r x"
shows "subtree (r - {(a, b)}) x = (subtree r x) - subtree r a"
proof -
from assms have asm: "b \<in> subtree r x" "(a, b) \<in> r" by (auto simp:edges_in_def)
-- {* The proof follows a common pattern to prove the equality of sets. *}
{ -- {* The `left to right` direction.
*}
fix c
-- {* Assuming @{text "c"} is inside the sub-tree of @{text "x"} in the reduced graph *}
assume h: "c \<in> subtree (r - {(a, b)}) x"
-- {* We are going to show that @{text "c"} can not be in the sub-tree of @{text "a"} in
the original graph. *}
-- {* In other words, all nodes inside the sub-tree of @{text "a"} in the original
graph will be removed from the sub-tree of @{text "x"} in the reduced graph. *}
-- {* The reason, as analyzed before, is that all paths from within the
sub-tree of @{text "a"} are broken with the removal of edge @{text "(a,b)"}.
*}
have "c \<in> (subtree r x) - subtree r a"
proof -
let ?r' = "r - {(a, b)}" -- {* The reduced graph is abbreviated as @{text "?r'"} *}
from h have "(c, x) \<in> ?r'^*" by (auto simp:subtree_def)
-- {* Extract from the reduced graph the path @{text "xs"} from @{text "c"} to @{text "x"}. *}
then obtain xs where rp0: "rpath ?r' c xs x" by (rule star_rpath, auto)
-- {* It is easy to show @{text "xs"} is also a path in the original graph *}
hence rp1: "rpath r c xs x"
proof(rule rpath_transfer)
from rpath_edges_on[OF rp0]
show "edges_on (c # xs) \<subseteq> r" by auto
qed
-- {* @{text "xs"} is used as the witness to show that @{text "c"}
in the sub-tree of @{text "x"} in the original graph. *}
hence "c \<in> subtree r x"
by (rule rpath_star[elim_format], auto simp:subtree_def)
-- {* The next step is to show that @{text "c"} can not be in the sub-tree of @{text "a"}
in the original graph. *}
-- {* We need to use the fact that all paths originate from within sub-tree of @{text "a"}
are broken. *}
moreover have "c \<notin> subtree r a"
proof
-- {* Proof by contradiction, suppose otherwise *}
assume otherwise: "c \<in> subtree r a"
-- {* Then there is a path in original graph leading from @{text "c"} to @{text "a"} *}
obtain xs1 where rp_c: "rpath r c xs1 a"
proof -
from otherwise have "(c, a) \<in> r^*" by (auto simp:subtree_def)
thus ?thesis by (rule star_rpath, auto intro!:that)
qed
-- {* Starting from this path, we are going to construct a fictional
path from @{text "c"} to @{text "x"}, which, as explained before,
is broken, so that contradiction can be derived. *}
-- {* First, there is a path from @{text "b"} to @{text "x"} *}
obtain ys where rp_b: "rpath r b ys x"
proof -
from asm have "(b, x) \<in> r^*" by (auto simp:subtree_def)
thus ?thesis by (rule star_rpath, auto intro!:that)
qed
-- {* The paths @{text "xs1"} and @{text "ys"} can be
tied together using @{text "(a,b)"} to form a path
from @{text "c"} to @{text "x"}: *}
have "rpath r c (xs1 @ b # ys) x"
proof -
from rstepI[OF asm(2) rp_b] have "rpath r a (b # ys) x" .
from rpath_appendI[OF rp_c this]
show ?thesis .
qed
-- {* By the uniqueness of path between two nodes of a tree, we have: *}
from rpath_unique[OF rp1 this] have eq_xs: "xs = xs1 @ b # ys" .
-- {* Contradiction can be derived from from this fictional path . *}
show False
proof -
-- {* It can be shown that @{term "(a,b)"} is on this fictional path. *}
have "(a, b) \<in> edges_on (c#xs)"
proof(cases "xs1 = []")
case True
from rp_c[unfolded this] have "rpath r c [] a" .
hence eq_c: "c = a" by (rule rpath_nilE, simp)
hence "c#xs = a#xs" by simp
from this and eq_xs have "c#xs = a # xs1 @ b # ys" by simp
from this[unfolded True] have "c#xs = []@[a,b]@ys" by simp
thus ?thesis by (auto simp:edges_on_def)
next
case False
from rpath_nnl_lastE[OF rp_c this]
obtain xs' where "xs1 = xs'@[a]" by auto
from eq_xs[unfolded this] have "c#xs = (c#xs')@[a,b]@ys" by simp
thus ?thesis by (unfold edges_on_def, blast)
qed
-- {* It can also be shown that @{term "(a,b)"} is not on this fictional path. *}
moreover have "(a, b) \<notin> edges_on (c#xs)"
using rpath_edges_on[OF rp0] by auto
-- {* Contradiction is thus derived. *}
ultimately show False by auto
qed
qed
ultimately show ?thesis by auto
qed
} moreover {
-- {* The `right to left` direction.
*}
fix c
-- {* Assuming that @{text "c"} is in the sub-tree of @{text "x"}, but
outside of the sub-tree of @{text "a"} in the original graph, *}
assume h: "c \<in> (subtree r x) - subtree r a"
-- {* we need to show that in the reduced graph, @{text "c"} is still in
the sub-tree of @{text "x"}. *}
have "c \<in> subtree (r - {(a, b)}) x"
proof -
-- {* The proof goes by showing that the path from @{text "c"} to @{text "x"}
in the original graph is not affected by the removal of @{text "(a,b)"}.
*}
from h have "(c, x) \<in> r^*" by (unfold subtree_def, auto)
-- {* Extract the path @{text "xs"} from @{text "c"} to @{text "x"} in the original graph. *}
from star_rpath[OF this] obtain xs where rp: "rpath r c xs x" by auto
-- {* Show that it is also a path in the reduced graph. *}
hence "rpath (r - {(a, b)}) c xs x"
-- {* The proof goes by using rule @{thm rpath_transfer} *}
proof(rule rpath_transfer)
-- {* We need to show all edges on the path are still in the reduced graph. *}
show "edges_on (c # xs) \<subseteq> r - {(a, b)}"
proof -
-- {* It is easy to show that all the edges are in the original graph. *}
from rpath_edges_on [OF rp] have " edges_on (c # xs) \<subseteq> r" .
-- {* The essential part is to show that @{text "(a, b)"} is not on the path. *}
moreover have "(a,b) \<notin> edges_on (c#xs)"
proof
-- {* Proof by contradiction, suppose otherwise: *}
assume otherwise: "(a, b) \<in> edges_on (c#xs)"
-- {* Then @{text "(a, b)"} is in the middle of the path.
with @{text "l1"} and @{text "l2"} be the nodes in
the front and rear respectively. *}
then obtain l1 l2 where eq_xs:
"c#xs = l1 @ [a, b] @ l2" by (unfold edges_on_def, blast)
-- {* From this, it can be shown that @{text "c"} is
in the sub-tree of @{text "a"} *}
have "c \<in> subtree r a"
proof(cases "l1 = []")
case True
-- {* If @{text "l1"} is null, it can be derived that @{text "c = a"}. *}
with eq_xs have "c = a" by auto
-- {* So, @{text "c"} is obviously in the sub-tree of @{text "a"}. *}
thus ?thesis by (unfold subtree_def, auto)
next
case False
-- {* When @{text "l1"} is not null, it must have a tail @{text "es"}: *}
then obtain e es where "l1 = e#es" by (cases l1, auto)
-- {* The relation of this tail with @{text "xs"} is derived: *}
with eq_xs have "xs = es@[a,b]@l2" by auto
-- {* From this, a path from @{text "c"} to @{text "a"} is made visible: *}
from rp[unfolded this] have "rpath r c (es @ [a] @ (b#l2)) x" by simp
thus ?thesis
proof(cases rule:rpath_appendE)
-- {* The path from @{text "c"} to @{text "a"} is extraced
using @{thm "rpath_appendE"}: *}
case 1
from rpath_star[OF this(1)]
-- {* The extracted path servers as a witness that @{text "c"} is
in the sub-tree of @{text "a"}: *}
show ?thesis by (simp add:subtree_def)
qed
qed with h show False by auto
qed ultimately show ?thesis by auto
qed
qed
-- {* From , it is shown that @{text "c"} is in the sub-tree of @{text "x"}
inthe reduced graph. *}
from rpath_star[OF this] show ?thesis by (auto simp:subtree_def)
qed
}
-- {* The equality of sets is derived from the two directions just proved. *}
ultimately show ?thesis by auto
qed
end
end