# Theory More_List

```(* Author: Andreas Lochbihler, ETH Zürich
Author: Florian Haftmann, TU Muenchen  *)

section ‹Less common functions on lists›

theory More_List
imports Main
begin

definition strip_while :: "('a ⇒ bool) ⇒ 'a list ⇒ 'a list"
where
"strip_while P = rev ∘ dropWhile P ∘ rev"

lemma strip_while_rev [simp]:
"strip_while P (rev xs) = rev (dropWhile P xs)"

lemma strip_while_Nil [simp]:
"strip_while P [] = []"

lemma strip_while_append [simp]:
"¬ P x ⟹ strip_while P (xs @ [x]) = xs @ [x]"

lemma strip_while_append_rec [simp]:
"P x ⟹ strip_while P (xs @ [x]) = strip_while P xs"

lemma strip_while_Cons [simp]:
"¬ P x ⟹ strip_while P (x # xs) = x # strip_while P xs"
by (induct xs rule: rev_induct) (simp_all add: strip_while_def)

lemma strip_while_eq_Nil [simp]:
"strip_while P xs = [] ⟷ (∀x∈set xs. P x)"

lemma strip_while_eq_Cons_rec:
"strip_while P (x # xs) = x # strip_while P xs ⟷ ¬ (P x ∧ (∀x∈set xs. P x))"
by (induct xs rule: rev_induct) (simp_all add: strip_while_def)

lemma split_strip_while_append:
fixes xs :: "'a list"
obtains ys zs :: "'a list"
where "strip_while P xs = ys" and "∀x∈set zs. P x" and "xs = ys @ zs"
proof (rule that)
show "strip_while P xs = strip_while P xs" ..
show "∀x∈set (rev (takeWhile P (rev xs))). P x" by (simp add: takeWhile_eq_all_conv [symmetric])
have "rev xs = rev (strip_while P xs @ rev (takeWhile P (rev xs)))"
then show "xs = strip_while P xs @ rev (takeWhile P (rev xs))"
by (simp only: rev_is_rev_conv)
qed

lemma strip_while_snoc [simp]:
"strip_while P (xs @ [x]) = (if P x then strip_while P xs else xs @ [x])"

lemma strip_while_map:
"strip_while P (map f xs) = map f (strip_while (P ∘ f) xs)"
by (simp add: strip_while_def rev_map dropWhile_map)

lemma strip_while_dropWhile_commute:
"strip_while P (dropWhile Q xs) = dropWhile Q (strip_while P xs)"
proof (induct xs)
case Nil
then show ?case
by simp
next
case (Cons x xs)
show ?case
proof (cases "∀y∈set xs. P y")
case True
with dropWhile_append2 [of "rev xs"] show ?thesis
by (auto simp add: strip_while_def dest: set_dropWhileD)
next
case False
then obtain y where "y ∈ set xs" and "¬ P y"
by blast
with Cons dropWhile_append3 [of P y "rev xs"] show ?thesis
qed
qed

lemma dropWhile_strip_while_commute:
"dropWhile P (strip_while Q xs) = strip_while Q (dropWhile P xs)"

definition no_leading :: "('a ⇒ bool) ⇒ 'a list ⇒ bool"
where
"no_leading P xs ⟷ (xs ≠ [] ⟶ ¬ P (hd xs))"

"no_leading P (x # xs) ⟷ ¬ P x"

"no_leading P (xs @ ys) ⟷ no_leading P xs ∧ (xs = [] ⟶ no_leading P ys)"
by (induct xs) simp_all

by (induct xs) simp_all

assumes "dropWhile P xs = ys"
obtains zs where "xs = zs @ ys" and "⋀z. z ∈ set zs ⟹ P z" and "no_leading P ys"
proof -
from assms have "∃zs. xs = zs @ ys ∧ (∀z ∈ set zs. P z) ∧ no_leading P ys"
proof (induct xs arbitrary: ys)
case Nil then show ?case by simp
next
case (Cons x xs ys)
show ?case proof (cases "P x")
case True with Cons.hyps [of ys] Cons.prems
have "∃zs. xs = zs @ ys ∧ (∀a∈set zs. P a) ∧ no_leading P ys"
by simp
then obtain zs where "xs = zs @ ys" and "⋀z. z ∈ set zs ⟹ P z"
by blast
with True have "x # xs = (x # zs) @ ys" and "⋀z. z ∈ set (x # zs) ⟹ P z"
by auto
with * show ?thesis
by blast    next
case False
with Cons show ?thesis by (cases ys) simp_all
qed
qed
with that show thesis
by blast
qed

lemma dropWhile_idem_iff:
"dropWhile P xs = xs ⟷ no_leading P xs"
by (cases xs) (auto elim: dropWhile_eq_obtain_leading)

abbreviation no_trailing :: "('a ⇒ bool) ⇒ 'a list ⇒ bool"
where
"no_trailing P xs ≡ no_leading P (rev xs)"

lemma no_trailing_unfold:
"no_trailing P xs ⟷ (xs ≠ [] ⟶ ¬ P (last xs))"
by (induct xs) simp_all

lemma no_trailing_Nil [iff]:
"no_trailing P []"
by simp

lemma no_trailing_Cons [simp]:
"no_trailing P (x # xs) ⟷ no_trailing P xs ∧ (xs = [] ⟶ ¬ P x)"
by simp

lemma no_trailing_append:
"no_trailing P (xs @ ys) ⟷ no_trailing P ys ∧ (ys = [] ⟶ no_trailing P xs)"
by (induct xs) simp_all

lemma no_trailing_append_Cons [simp]:
"no_trailing P (xs @ y # ys) ⟷ no_trailing P (y # ys)"
by simp

lemma no_trailing_strip_while [simp]:
"no_trailing P (strip_while P xs)"
by (induct xs rule: rev_induct) simp_all

lemma strip_while_idem [simp]:
"no_trailing P xs ⟹ strip_while P xs = xs"
by (cases xs rule: rev_cases) simp_all

lemma strip_while_eq_obtain_trailing:
assumes "strip_while P xs = ys"
obtains zs where "xs = ys @ zs" and "⋀z. z ∈ set zs ⟹ P z" and "no_trailing P ys"
proof -
from assms have "rev (rev (dropWhile P (rev xs))) = rev ys"
then have "dropWhile P (rev xs) = rev ys"
by simp
then obtain zs where A: "rev xs = zs @ rev ys" and B: "⋀z. z ∈ set zs ⟹ P z"
and C: "no_trailing P ys"
from A have "rev (rev xs) = rev (zs @ rev ys)"
by simp
then have "xs = ys @ rev zs"
by simp
moreover from B have "⋀z. z ∈ set (rev zs) ⟹ P z"
by simp
ultimately show thesis using that C by blast
qed

lemma strip_while_idem_iff:
"strip_while P xs = xs ⟷ no_trailing P xs"
proof -
define ys where "ys = rev xs"
moreover have "strip_while P (rev ys) = rev ys ⟷ no_trailing P (rev ys)"
ultimately show ?thesis by simp
qed

lemma no_trailing_map:
"no_trailing P (map f xs) ⟷ no_trailing (P ∘ f) xs"

lemma no_trailing_drop [simp]:
"no_trailing P (drop n xs)" if "no_trailing P xs"
proof -
from that have "no_trailing P (take n xs @ drop n xs)"
by simp
then show ?thesis
by (simp only: no_trailing_append)
qed

lemma no_trailing_upt [simp]:
"no_trailing P [n..<m] ⟷ (n < m ⟶ ¬ P (m - 1))"

definition nth_default :: "'a ⇒ 'a list ⇒ nat ⇒ 'a"
where
"nth_default dflt xs n = (if n < length xs then xs ! n else dflt)"

lemma nth_default_nth:
"n < length xs ⟹ nth_default dflt xs n = xs ! n"

lemma nth_default_beyond:
"length xs ≤ n ⟹ nth_default dflt xs n = dflt"

lemma nth_default_Nil [simp]:
"nth_default dflt [] n = dflt"

lemma nth_default_Cons:
"nth_default dflt (x # xs) n = (case n of 0 ⇒ x | Suc n' ⇒ nth_default dflt xs n')"
by (simp add: nth_default_def split: nat.split)

lemma nth_default_Cons_0 [simp]:
"nth_default dflt (x # xs) 0 = x"

lemma nth_default_Cons_Suc [simp]:
"nth_default dflt (x # xs) (Suc n) = nth_default dflt xs n"

lemma nth_default_replicate_dflt [simp]:
"nth_default dflt (replicate n dflt) m = dflt"

lemma nth_default_append:
"nth_default dflt (xs @ ys) n =
(if n < length xs then nth xs n else nth_default dflt ys (n - length xs))"
by (auto simp add: nth_default_def nth_append)

lemma nth_default_append_trailing [simp]:
"nth_default dflt (xs @ replicate n dflt) = nth_default dflt xs"

lemma nth_default_snoc_default [simp]:
"nth_default dflt (xs @ [dflt]) = nth_default dflt xs"
by (auto simp add: nth_default_def fun_eq_iff nth_append)

lemma nth_default_eq_dflt_iff:
"nth_default dflt xs k = dflt ⟷ (k < length xs ⟶ xs ! k = dflt)"

lemma nth_default_take_eq:
"nth_default dflt (take m xs) n =
(if n < m then nth_default dflt xs n else dflt)"

lemma in_enumerate_iff_nth_default_eq:
"x ≠ dflt ⟹ (n, x) ∈ set (enumerate 0 xs) ⟷ nth_default dflt xs n = x"
by (auto simp add: nth_default_def in_set_conv_nth enumerate_eq_zip)

lemma last_conv_nth_default:
assumes "xs ≠ []"
shows "last xs = nth_default dflt xs (length xs - 1)"
using assms by (simp add: nth_default_def last_conv_nth)

lemma nth_default_map_eq:
"f dflt' = dflt ⟹ nth_default dflt (map f xs) n = f (nth_default dflt' xs n)"

lemma finite_nth_default_neq_default [simp]:
"finite {k. nth_default dflt xs k ≠ dflt}"

lemma sorted_list_of_set_nth_default:
"sorted_list_of_set {k. nth_default dflt xs k ≠ dflt} = map fst (filter (λ(_, x). x ≠ dflt) (enumerate 0 xs))"
by (rule sorted_distinct_set_unique) (auto simp add: nth_default_def in_set_conv_nth
sorted_filter distinct_map_filter enumerate_eq_zip intro: rev_image_eqI)

lemma map_nth_default:
"map (nth_default x xs) [0..<length xs] = xs"
proof -
have *: "map (nth_default x xs) [0..<length xs] = map (List.nth xs) [0..<length xs]"
by (rule map_cong) (simp_all add: nth_default_nth)
show ?thesis by (simp add: * map_nth)
qed

lemma range_nth_default [simp]:
"range (nth_default dflt xs) = insert dflt (set xs)"
by (auto simp add: nth_default_def [abs_def] in_set_conv_nth)

lemma nth_strip_while:
assumes "n < length (strip_while P xs)"
shows "strip_while P xs ! n = xs ! n"
proof -
have "length (dropWhile P (rev xs)) + length (takeWhile P (rev xs)) = length xs"
(simp add: arg_cong [where f=length, OF takeWhile_dropWhile_id, unfolded length_append])
then show ?thesis using assms
by (simp add: strip_while_def rev_nth dropWhile_nth)
qed

lemma length_strip_while_le:
"length (strip_while P xs) ≤ length xs"
unfolding strip_while_def o_def length_rev
by (subst (2) length_rev[symmetric])
(simp add: strip_while_def length_dropWhile_le del: length_rev)

lemma nth_default_strip_while_dflt [simp]:
"nth_default dflt (strip_while ((=) dflt) xs) = nth_default dflt xs"
by (induct xs rule: rev_induct) auto

lemma nth_default_eq_iff:
"nth_default dflt xs = nth_default dflt ys
⟷ strip_while (HOL.eq dflt) xs = strip_while (HOL.eq dflt) ys" (is "?P ⟷ ?Q")
proof
let ?xs = "strip_while (HOL.eq dflt) xs" and ?ys = "strip_while (HOL.eq dflt) ys"
assume ?P
then have eq: "nth_default dflt ?xs = nth_default dflt ?ys"
by simp
have len: "length ?xs = length ?ys"
proof (rule ccontr)
assume len: "length ?xs ≠ length ?ys"
{ fix xs ys :: "'a list"
let ?xs = "strip_while (HOL.eq dflt) xs" and ?ys = "strip_while (HOL.eq dflt) ys"
assume eq: "nth_default dflt ?xs = nth_default dflt ?ys"
assume len: "length ?xs < length ?ys"
then have "length ?ys > 0" by arith
then have "?ys ≠ []" by simp
with last_conv_nth_default [of ?ys dflt]
have "last ?ys = nth_default dflt ?ys (length ?ys - 1)"
by auto
moreover from ‹?ys ≠ []› no_trailing_strip_while [of "HOL.eq dflt" ys]
have "last ?ys ≠ dflt" by (simp add: no_trailing_unfold)
ultimately have "nth_default dflt ?xs (length ?ys - 1) ≠ dflt"
using eq by simp
moreover from len have "length ?ys - 1 ≥ length ?xs" by simp
ultimately have False by (simp only: nth_default_beyond) simp
}
from this [of xs ys] this [of ys xs] len eq show False
by (auto simp only: linorder_class.neq_iff)
qed
then show ?Q
proof (rule nth_equalityI [rule_format])
fix n
assume n: "n < length ?xs"
with len have "n < length ?ys"
by simp
with n have xs: "nth_default dflt ?xs n = ?xs ! n"
and ys: "nth_default dflt ?ys n = ?ys ! n"
by (simp_all only: nth_default_nth)
with eq show "?xs ! n = ?ys ! n"
by simp
qed
next
assume ?Q
then have "nth_default dflt (strip_while (HOL.eq dflt) xs) = nth_default dflt (strip_while (HOL.eq dflt) ys)"
by simp
then show ?P
by simp
qed

lemma nth_default_map2:
‹nth_default d (map2 f xs ys) n = f (nth_default d1 xs n) (nth_default d2 ys n)›
if ‹length xs = length ys› and ‹f d1 d2 = d› for bs cs
using that proof (induction xs ys arbitrary: n rule: list_induct2)
case Nil
then show ?case
by simp
next
case (Cons x xs y ys)
then show ?case
by (cases n) simp_all
qed

end

```