# Theory AuxLemmas

section ‹Auxiliary lemmas›

theory AuxLemmas imports Main begin

abbreviation "arbitrary == undefined"

text ‹Lemmas about left- and rightmost elements in lists›

lemma leftmost_element_property:
assumes "x  set xs. P x"
obtains zs x' ys where "xs = zs@x'#ys" and "P x'" and "z  set zs. ¬ P z"
proof(atomize_elim)
from x  set xs. P x
show "zs x' ys. xs = zs @ x' # ys  P x'  (zset zs. ¬ P z)"
proof(induct xs)
case Nil thus ?case by simp
next
case (Cons x' xs')
note IH = aset xs'. P a
zs x' ys. xs' = zs@x'#ys  P x'  (zset zs. ¬ P z)
show ?case
proof (cases "P x'")
case True
then have "(ys. x' # xs' = [] @ x' # ys)  P x'  (xset []. ¬ P x)" by simp
then show ?thesis by blast
next
case False
with yset (x'#xs'). P y have "yset xs'. P y" by simp
from IH[OF this] obtain y ys zs where "xs' = zs@y#ys"
and "P y" and "zset zs. ¬ P z" by blast
from zset zs. ¬ P z False have "zset (x'#zs). ¬ P z" by simp
with xs' = zs@y#ys P y show ?thesis by (metis Cons_eq_append_conv)
qed
qed
qed

lemma rightmost_element_property:
assumes "x  set xs. P x"
obtains ys x' zs where "xs = ys@x'#zs" and "P x'" and "z  set zs. ¬ P z"
proof(atomize_elim)
from x  set xs. P x
show "ys x' zs. xs = ys @ x' # zs  P x'  (zset zs. ¬ P z)"
proof(induct xs)
case Nil thus ?case by simp
next
case (Cons x' xs')
note IH = aset xs'. P a
ys x' zs. xs' = ys @ x' # zs  P x'  (zset zs. ¬ P z)
show ?case
proof(cases "yset xs'. P y")
case True
from IH[OF this] obtain y ys zs where "xs' = ys @ y # zs"
and "P y" and "zset zs. ¬ P z" by blast
thus ?thesis by (metis Cons_eq_append_conv)
next
case False
with yset (x'#xs'). P y have "P x'" by simp
with False show ?thesis by (metis eq_Nil_appendI)
qed
qed
qed

text ‹Lemma concerning maps and @›

lemma map_append_append_maps:
assumes map:"map f xs = ys@zs"
obtains xs' xs'' where "map f xs' = ys" and "map f xs'' = zs" and "xs=xs'@xs''"
by (metis append_eq_conv_conj append_take_drop_id assms drop_map take_map that)

text ‹Lemma concerning splitting of @{term list}s›

lemma  path_split_general:
assumes all:"zs. xs  ys@zs"
obtains j zs where "xs = (take j ys)@zs" and "j < length ys"
and "k > j. zs'. xs  (take k ys)@zs'"
proof(atomize_elim)
from zs. xs  ys@zs
show "j zs. xs = take j ys @ zs  j < length ys
(k>j. zs'. xs  take k ys @ zs')"
proof(induct ys arbitrary:xs)
case Nil thus ?case by auto
next
case (Cons y' ys')
note IH = xs. zs. xs  ys' @ zs
j zs. xs = take j ys' @ zs  j < length ys'
(k. j < k  (zs'. xs  take k ys' @ zs'))
show ?case
proof(cases xs)
case Nil thus ?thesis by simp
next
case (Cons x' xs')
with zs. xs  (y' # ys') @ zs have "x'  y'  (zs. xs'  ys' @ zs)"
by simp
show ?thesis
proof(cases "x' = y'")
case True
with x'  y'  (zs. xs'  ys' @ zs) have "zs. xs'  ys' @ zs" by simp
from IH[OF this] have "j zs. xs' = take j ys' @ zs  j < length ys'
(k. j < k  (zs'. xs'  take k ys' @ zs'))" .
then obtain j zs where "xs' = take j ys' @ zs"
and "j < length ys'"
and all_sub:"k. j < k  (zs'. xs'  take k ys' @ zs')"
by blast
from xs' = take j ys' @ zs True
have "(x'#xs') = take (Suc j) (y' # ys') @ zs"
by simp
from all_sub True have all_imp:"k. j < k
(zs'. (x'#xs')  take (Suc k) (y' # ys') @ zs')"
by auto
{ fix l assume "(Suc j) < l"
then obtain k where [simp]:"l = Suc k" by(cases l) auto
with (Suc j) < l have "j < k" by simp
with all_imp
have "zs'. (x'#xs')  take (Suc k) (y' # ys') @ zs'"
by simp
hence "zs'. (x'#xs')  take l (y' # ys') @ zs'"
by simp }
with (x'#xs') = take (Suc j) (y' # ys') @ zs j < length ys' Cons
show ?thesis by (metis Suc_length_conv less_Suc_eq_0_disj)
next
case False
with Cons have "i zs'. i > 0  xs  take i (y' # ys') @ zs'"
by auto(case_tac i,auto)
moreover
have "zs. xs = take 0 (y' # ys') @ zs" by simp
ultimately show ?thesis by(rule_tac x="0" in exI,auto)
qed
qed
qed
qed

end