# Theory AuxLemmas

```section ‹Auxiliary lemmas›

theory AuxLemmas imports Main begin

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
```