header {* Sublist Ordering *}
theory Sublist_Order
imports Main
begin
text {*
This theory defines sublist ordering on lists.
A list @{text ys} is a sublist of a list @{text xs},
iff one obtains @{text ys} by erasing some elements from @{text xs}.
*}
subsection {* Definitions and basic lemmas *}
instantiation list :: (type) order
begin
inductive less_eq_list where
empty [simp, intro!]: "[] ≤ xs"
| drop: "ys ≤ xs ==> ys ≤ x # xs"
| take: "ys ≤ xs ==> x # ys ≤ x # xs"
lemmas ileq_empty = empty
lemmas ileq_drop = drop
lemmas ileq_take = take
lemma ileq_cases [cases set, case_names empty drop take]:
assumes "xs ≤ ys"
and "xs = [] ==> P"
and "!!z zs. ys = z # zs ==> xs ≤ zs ==> P"
and "!!x zs ws. xs = x # zs ==> ys = x # ws ==> zs ≤ ws ==> P"
shows P
using assms by (blast elim: less_eq_list.cases)
lemma ileq_induct [induct set, case_names empty drop take]:
assumes "xs ≤ ys"
and "!!zs. P [] zs"
and "!!z zs ws. ws ≤ zs ==> P ws zs ==> P ws (z # zs)"
and "!!z zs ws. ws ≤ zs ==> P ws zs ==> P (z # ws) (z # zs)"
shows "P xs ys"
using assms by (induct rule: less_eq_list.induct) blast+
definition
[code del]: "(xs :: 'a list) < ys <-> xs ≤ ys ∧ ¬ ys ≤ xs"
lemma ileq_length: "xs ≤ ys ==> length xs ≤ length ys"
by (induct rule: ileq_induct) auto
lemma ileq_below_empty [simp]: "xs ≤ [] <-> xs = []"
by (auto dest: ileq_length)
instance proof
fix xs ys :: "'a list"
show "xs < ys <-> xs ≤ ys ∧ ¬ ys ≤ xs" unfolding less_list_def ..
next
fix xs :: "'a list"
show "xs ≤ xs" by (induct xs) (auto intro!: ileq_empty ileq_drop ileq_take)
next
fix xs ys :: "'a list"
{ fix n
have "!!l l'. [|l≤l'; l'≤l; n=length l + length l'|] ==> l=l'"
proof (induct n rule: nat_less_induct)
case (1 n l l') from "1.prems"(1) show ?case proof (cases rule: ileq_cases)
case empty with "1.prems"(2) show ?thesis by auto
next
case (drop a l2') with "1.prems"(2) have "length l'≤length l" "length l ≤ length l2'" "1+length l2' = length l'" by (auto dest: ileq_length)
hence False by simp thus ?thesis ..
next
case (take a l1' l2') hence LEN': "length l1' + length l2' < length l + length l'" by simp
from "1.prems" have LEN: "length l' = length l" by (auto dest!: ileq_length)
from "1.prems"(2) show ?thesis proof (cases rule: ileq_cases[case_names empty' drop' take'])
case empty' with take LEN show ?thesis by simp
next
case (drop' ah l2h) with take LEN have "length l1' ≤ length l2h" "1+length l2h = length l2'" "length l2' = length l1'" by (auto dest: ileq_length)
hence False by simp thus ?thesis ..
next
case (take' ah l1h l2h)
with take have 2: "ah=a" "l1h=l2'" "l2h=l1'" "l1' ≤ l2'" "l2' ≤ l1'" by auto
with LEN' "1.hyps" "1.prems"(3) have "l1'=l2'" by blast
with take 2 show ?thesis by simp
qed
qed
qed
}
moreover assume "xs ≤ ys" "ys ≤ xs"
ultimately show "xs = ys" by blast
next
fix xs ys zs :: "'a list"
{
fix n
have "!!x y z. [|x ≤ y; y ≤ z; n=length x + length y + length z|] ==> x ≤ z"
proof (induct rule: nat_less_induct[case_names I])
case (I n x y z)
from I.prems(2) show ?case proof (cases rule: ileq_cases)
case empty with I.prems(1) show ?thesis by auto
next
case (drop a z') hence "length x + length y + length z' < length x + length y + length z" by simp
with I.hyps I.prems(3,1) drop(2) have "x≤z'" by blast
with drop(1) show ?thesis by (auto intro: ileq_drop)
next
case (take a y' z') from I.prems(1) show ?thesis proof (cases rule: ileq_cases[case_names empty' drop' take'])
case empty' thus ?thesis by auto
next
case (drop' ah y'h) with take have "x≤y'" "y'≤z'" "length x + length y' + length z' < length x + length y + length z" by auto
with I.hyps I.prems(3) have "x≤z'" by (blast)
with take(2) show ?thesis by (auto intro: ileq_drop)
next
case (take' ah x' y'h) with take have 2: "x=a#x'" "x'≤y'" "y'≤z'" "length x' + length y' + length z' < length x + length y + length z" by auto
with I.hyps I.prems(3) have "x'≤z'" by blast
with 2 take(2) show ?thesis by (auto intro: ileq_take)
qed
qed
qed
}
moreover assume "xs ≤ ys" "ys ≤ zs"
ultimately show "xs ≤ zs" by blast
qed
end
lemmas ileq_intros = ileq_empty ileq_drop ileq_take
lemma ileq_drop_many: "xs ≤ ys ==> xs ≤ zs @ ys"
by (induct zs) (auto intro: ileq_drop)
lemma ileq_take_many: "xs ≤ ys ==> zs @ xs ≤ zs @ ys"
by (induct zs) (auto intro: ileq_take)
lemma ileq_same_length: "xs ≤ ys ==> length xs = length ys ==> xs = ys"
by (induct rule: ileq_induct) (auto dest: ileq_length)
lemma ileq_same_append [simp]: "x # xs ≤ xs <-> False"
by (auto dest: ileq_length)
lemma ilt_length [intro]:
assumes "xs < ys"
shows "length xs < length ys"
proof -
from assms have "xs ≤ ys" and "xs ≠ ys" by (simp_all add: less_le)
moreover with ileq_length have "length xs ≤ length ys" by auto
ultimately show ?thesis by (auto intro: ileq_same_length)
qed
lemma ilt_empty [simp]: "[] < xs <-> xs ≠ []"
by (unfold less_list_def, auto)
lemma ilt_emptyI: "xs ≠ [] ==> [] < xs"
by (unfold less_list_def, auto)
lemma ilt_emptyD: "[] < xs ==> xs ≠ []"
by (unfold less_list_def, auto)
lemma ilt_below_empty[simp]: "xs < [] ==> False"
by (auto dest: ilt_length)
lemma ilt_drop: "xs < ys ==> xs < x # ys"
by (unfold less_le) (auto intro: ileq_intros)
lemma ilt_take: "xs < ys ==> x # xs < x # ys"
by (unfold less_le) (auto intro: ileq_intros)
lemma ilt_drop_many: "xs < ys ==> xs < zs @ ys"
by (induct zs) (auto intro: ilt_drop)
lemma ilt_take_many: "xs < ys ==> zs @ xs < zs @ ys"
by (induct zs) (auto intro: ilt_take)
subsection {* Appending elements *}
lemma ileq_rev_take: "xs ≤ ys ==> xs @ [x] ≤ ys @ [x]"
by (induct rule: ileq_induct) (auto intro: ileq_intros ileq_drop_many)
lemma ilt_rev_take: "xs < ys ==> xs @ [x] < ys @ [x]"
by (unfold less_le) (auto dest: ileq_rev_take)
lemma ileq_rev_drop: "xs ≤ ys ==> xs ≤ ys @ [x]"
by (induct rule: ileq_induct) (auto intro: ileq_intros)
lemma ileq_rev_drop_many: "xs ≤ ys ==> xs ≤ ys @ zs"
by (induct zs rule: rev_induct) (auto dest: ileq_rev_drop)
subsection {* Relation to standard list operations *}
lemma ileq_map: "xs ≤ ys ==> map f xs ≤ map f ys"
by (induct rule: ileq_induct) (auto intro: ileq_intros)
lemma ileq_filter_left[simp]: "filter f xs ≤ xs"
by (induct xs) (auto intro: ileq_intros)
lemma ileq_filter: "xs ≤ ys ==> filter f xs ≤ filter f ys"
by (induct rule: ileq_induct) (auto intro: ileq_intros)
end