Quellcode-Bibliothek Util.thy

Sprache: Isabelle

section ‹Utility Definitions and Properties›

text ‹This file contains various definitions and lemmata not closely related to finite state
machines or testing.›

theory Util
 imports Main "HOL-Library.FSet" "HOL-Library.Sublist" "HOL-Library.Mapping"
begin

subsection ‹Converting Sets to Maps›

text ‹This subsection introduces a function @{text "set_as_map"} that transforms a set of
@{text "('a × 'b)"} tuples to a map mapping each first value @{text "x"} of the contained tuples
to all second values @{text "y"} such that @{text "(x,y)"} is contained in the set.›

definition set_as_map :: "('a × 'c) set ==> ('a ==> 'c set option)" where
 "set_as_map s = (λ x . if (∃ z . (x,z) ∈ s) then Some {z . (x,z) ∈ s} else None)"

lemma set_as_map_code[code] :
 "set_as_map (set xs) = (foldl (λ m (x,z) . case m x of
 None ==> m (x ↦ {z}) |
 Some zs ==> m (x ↦ (insert z zs)))
 Map.empty
 xs)"
proof -
 let ?f = "λ xs . (foldl (λ m (x,z) . case m x of
 None ==> m (x ↦ {z}) |
 Some zs ==> m (x ↦ (insert z zs)))
 Map.empty
 xs)"
 have "(?f xs) = (λ x . if (∃ z . (x,z) ∈ set xs) then Some {z . (x,z) ∈ set xs} else None)"
 proof (induction xs rule: rev_induct)
 case Nil
 then show ?case by auto
 next
 case (snoc xz xs)
 then obtain x z where "xz = (x,z)"
 by force

 have *: "(?f (xs@[(x,z)])) = (case (?f xs) x of
 None ==> (?f xs) (x ↦ {z}) |
 Some zs ==> (?f xs) (x ↦ (insert z zs)))"
 by auto

 then show ?case proof (cases "(?f xs) x")
 case None
 then have **: "(?f (xs@[(x,z)])) = (?f xs) (x ↦ {z})" using * by auto

 have scheme: "∧ m k v . (m(k ↦ v)) = (λk' . if k' = k then Some v else m k')"
 by auto

 have m1: "(?f (xs@[(x,z)])) = (λ x' . if x' = x then Some {z} else (?f xs) x')"
 unfolding **
 unfolding scheme by force

 have "(λ x . if (∃ z . (x,z) ∈ set xs) then Some {z . (x,z) ∈ set xs} else None) x = None"
 using None snoc by auto
 then have "¬(∃ z . (x,z) ∈ set xs)"
 by (metis (mono_tags, lifting) option.distinct(1))
 then have "(∃ z . (x,z) ∈ set (xs@[(x,z)]))" and "{z' . (x,z') ∈ set (xs@[(x,z)])} = {z}"
 by auto
 then have m2: "(λ x' . if (∃ z' . (x',z') ∈ set (xs@[(x,z)]))
 then Some {z' . (x',z') ∈ set (xs@[(x,z)])}
 else None)
 = (λ x' . if x' = x
 then Some {z} else (λ x . if (∃ z . (x,z) ∈ set xs)
 then Some {z . (x,z) ∈ set xs}
 else None) x')"
 by force

 show ?thesis using m1 m2 snoc
 using ‹xz = (x, z)› by presburger
 next
 case (Some zs)
 then have **: "(?f (xs@[(x,z)])) = (?f xs) (x ↦ (insert z zs))" using * by auto
 have scheme: "∧ m k v . (m(k ↦ v)) = (λk' . if k' = k then Some v else m k')"
 by auto

 have m1: "(?f (xs@[(x,z)])) = (λ x' . if x' = x then Some (insert z zs) else (?f xs) x')"
 unfolding **
 unfolding scheme by force

 have "(λ x . if (∃ z . (x,z) ∈ set xs) then Some {z . (x,z) ∈ set xs} else None) x = Some zs"
 using Some snoc by auto
 then have "(∃ z . (x,z) ∈ set xs)"
 unfolding case_prod_conv using option.distinct(2) by metis
 then have "(∃ z . (x,z) ∈ set (xs@[(x,z)]))" by simp

 have "{z' . (x,z') ∈ set (xs@[(x,z)])} = insert z zs"
 proof -
 have "Some {z . (x,z) ∈ set xs} = Some zs"
 using ‹(λ x . if (∃ z . (x,z) ∈ set xs) then Some {z . (x,z) ∈ set xs} else None) x
= Some zs›
 unfolding case_prod_conv using option.distinct(2) by metis
 then have "{z . (x,z) ∈ set xs} = zs" by auto
 then show ?thesis by auto
 qed

 have "∧ a . (λ x' . if (∃ z' . (x',z') ∈ set (xs@[(x,z)]))
 then Some {z' . (x',z') ∈ set (xs@[(x,z)])} else None) a
 = (λ x' . if x' = x
 then Some (insert z zs)
 else (λ x . if (∃ z . (x,z) ∈ set xs)
 then Some {z . (x,z) ∈ set xs} else None) x') a"
 proof -
 fix a show "(λ x' . if (∃ z' . (x',z') ∈ set (xs@[(x,z)]))
 then Some {z' . (x',z') ∈ set (xs@[(x,z)])} else None) a
 = (λ x' . if x' = x
 then Some (insert z zs)
 else (λ x . if (∃ z . (x,z) ∈ set xs)
 then Some {z . (x,z) ∈ set xs} else None) x') a"
 using ‹{z' . (x,z') ∈ set (xs@[(x,z)])} = insert z zs› ‹(∃ z . (x,z) ∈ set (xs@[(x,z)]))›
 by (cases "a = x"; auto)
 qed

 then have m2: "(λ x' . if (∃ z' . (x',z') ∈ set (xs@[(x,z)]))
 then Some {z' . (x',z') ∈ set (xs@[(x,z)])} else None)
 = (λ x' . if x' = x
 then Some (insert z zs)
 else (λ x . if (∃ z . (x,z) ∈ set xs)
 then Some {z . (x,z) ∈ set xs} else None) x')"
 by auto

 show ?thesis using m1 m2 snoc
 using ‹xz = (x, z)› by presburger
 qed
 qed

 then show ?thesis
 unfolding set_as_map_def by simp
qed

abbreviation "member_option x ms ≡ (case ms of None ==> False | Some xs ==> x ∈ xs)"
notation member_option (‹(_∈_{o_})› [1000] 1000)

abbreviation(input) "lookup_with_default f d ≡ (λ x . case f x of None ==> d | Some xs ==> xs)"
abbreviation(input) "m2f f ≡ lookup_with_default f {}"

abbreviation(input) "lookup_with_default_by f g d ≡ (λ x . case f x of None ==> g d | Some xs ==> g xs)"
abbreviation(input) "m2f_by g f ≡ lookup_with_default_by f g {}"

lemma m2f_by_from_m2f :
 "(m2f_by g f xs) = g (m2f f xs)"
 by (simp add: option.case_eq_if)

lemma set_as_map_containment :
 assumes "(x,y) ∈ zs"
 shows "y ∈ (m2f (set_as_map zs)) x"
 using assms unfolding set_as_map_def
 by auto

lemma set_as_map_elem :
 assumes "y ∈ m2f (set_as_map xs) x"
shows "(x,y) ∈ xs"
using assms unfolding set_as_map_def
proof -
 assume a1: "y ∈ (case if ∃z. (x, z) ∈ xs then Some {z. (x, z) ∈ xs} else None of None ==> {} | Some xs ==> xs)"
 then have "∃a. (x, a) ∈ xs"
 using all_not_in_conv by fastforce
 then show ?thesis
 using a1 by simp
qed

subsection ‹Utility Lemmata for existing functions on lists›

subsubsection ‹Utility Lemmata for @{text "find"}›

lemma find_result_props :
 assumes "find P xs = Some x"
 shows "x ∈ set xs" and "P x"
proof -
 show "x ∈ set xs" using assms by (metis find_Some_iff nth_mem)
 show "P x" using assms by (metis find_Some_iff)
qed

lemma find_set :
 assumes "find P xs = Some x"
 shows "x ∈ set xs"
using assms proof(induction xs)
 case Nil
 then show ?case by auto
next
 case (Cons a xs)
 then show ?case
 by (metis find.simps(2) list.set_intros(1) list.set_intros(2) option.inject)
qed

lemma find_condition :
 assumes "find P xs = Some x"
 shows "P x"
using assms proof(induction xs)
 case Nil
 then show ?case by auto
next
 case (Cons a xs)
 then show ?case
 by (metis find.simps(2) option.inject)
qed

lemma find_from :
 assumes "∃ x ∈ set xs . P x"
 shows "find P xs ≠ None"
 by (metis assms find_None_iff)

lemma find_sort_containment :
 assumes "find P (sort xs) = Some x"
shows "x ∈ set xs"
 using assms find_set by force

lemma find_sort_index :
 assumes "find P xs = Some x"
 shows "∃ i < length xs . xs ! i = x ∧ (∀ j < i . ¬ P (xs ! j))"
using assms proof (induction xs arbitrary: x)
 case Nil
 then show ?case by auto
next
 case (Cons a xs)
 show ?case proof (cases "P a")
 case True
 then show ?thesis
 using Cons.prems unfolding find.simps by auto
 next
 case False
 then have "find P (a#xs) = find P xs"
 unfolding find.simps by auto
 then have "find P xs = Some x"
 using Cons.prems by auto
 then show ?thesis
 using Cons.IH False
 by (metis Cons.prems find_Some_iff)
 qed
qed

lemma find_sort_least :
 assumes "find P (sort xs) = Some x"
 shows "∀ x' ∈ set xs . x ≤ x' ∨ ¬ P x'"
 and "x = (LEAST x' ∈ set xs . P x')"
proof -
 obtain i where "i < length (sort xs)"
 and "(sort xs) ! i = x"
 and "(∀ j i ==> j < length xs ==> (sort xs) ! i ≤ (sort xs) ! j"
 by (simp add: sorted_nth_mono)
 then have "∧ j . j < length xs ==> (sort xs) ! i ≤ (sort xs) ! j ∨ ¬ P ((sort xs) ! j)"
 using ‹(∀ j < i . ¬ P ((sort xs) ! j))›
 by (metis not_less_iff_gr_or_eq order_refl)
 then show "∀ x' ∈ set xs . x ≤ x' ∨ ¬ P x'"
 by (metis ‹sort xs ! i = x› in_set_conv_nth length_sort set_sort)
 then show "x = (LEAST x' ∈ set xs . P x')"
 using find_set[OF assms] find_condition[OF assms]
 by (metis (mono_tags, lifting) Least_equality set_sort)
qed

subsubsection ‹Utility Lemmata for @{text "filter"}›

lemma filter_take_length :
 "length (filter P (take i xs)) ≤ length (filter P xs)"
 by (metis append_take_drop_id filter_append le0 le_add_same_cancel1 length_append)

lemma filter_double :
 assumes "x ∈ set (filter P1 xs)"
 and "P2 x"
shows "x ∈ set (filter P2 (filter P1 xs))"
 using assms by simp

lemma filter_list_set :
 assumes "x ∈ set xs"
 and "P x"
shows "x ∈ set (filter P xs)"
 by (simp add: assms(1) assms(2))

lemma filter_list_set_not_contained :
 assumes "x ∈ set xs"
 and "¬ P x"
shows "x ∉ set (filter P xs)"
 by (simp add: assms(1) assms(2))

lemma filter_map_elem : "t ∈ set (map g (filter f xs)) ==> ∃ x ∈ set xs . f x ∧ t = g x"
 by auto

subsubsection ‹Utility Lemmata for @{text "concat"}›

lemma concat_map_elem :
 assumes "y ∈ set (concat (map f xs))"
 obtains x where "x ∈ set xs"
 and "y ∈ set (f x)"
using assms proof (induction xs)
 case Nil
 then show ?case by auto
next
 case (Cons a xs)
 then show ?case
 proof (cases "y ∈ set (f a)")
 case True
 then show ?thesis
 using Cons.prems(1) by auto
 next
 case False
 then have "y ∈ set (concat (map f xs))"
 using Cons by auto
 have "∃ x . x ∈ set xs ∧ y ∈ set (f x)"
 proof (rule ccontr)
 assume "¬(∃x. x ∈ set xs ∧ y ∈ set (f x))"
 then have "¬(y ∈ set (concat (map f xs)))"
 by auto
 then show False
 using ‹y ∈ set (concat (map f xs))› by auto
 qed
 then show ?thesis
 using Cons.prems(1) by auto
 qed
qed

lemma set_concat_map_sublist :
 assumes "x ∈ set (concat (map f xs))"
 and "set xs ⊆ set xs'"
shows "x ∈ set (concat (map f xs'))"
using assms by (induction xs) (auto)

lemma set_concat_map_elem :
 assumes "x ∈ set (concat (map f xs))"
 shows "∃ x' ∈ set xs . x ∈ set (f x')"
using assms by auto

lemma concat_replicate_length : "length (concat (replicate n xs)) = n * (length xs)"
 by (induction n; simp)

subsection ‹Enumerating Lists›

fun lists_of_length :: "'a list ==> nat ==> 'a list list" where
 "lists_of_length T 0 = [[]]" |
 "lists_of_length T (Suc n) = concat (map (λ xs . map (λ x . x#xs) T ) (lists_of_length T n))"

lemma lists_of_length_containment :
 assumes "set xs ⊆ set T"
 and "length xs = n"
shows "xs ∈ set (lists_of_length T n)"
using assms proof (induction xs arbitrary: n)
 case Nil
 then show ?case by auto
next
 case (Cons a xs)
 then obtain k where "n = Suc k"
 by auto
 then have "xs ∈ set (lists_of_length T k)"
 using Cons by auto
 moreover have "a ∈ set T"
 using Cons by auto
 ultimately show ?case
 using ‹n = Suc k› by auto
qed

lemma lists_of_length_length :
 assumes "xs ∈ set (lists_of_length T n)"
 shows "length xs = n"
proof -
 have "∀ xs ∈ set (lists_of_length T n) . length xs = n"
 by (induction n; simp)
 then show ?thesis using assms by blast
qed

lemma lists_of_length_elems :
 assumes "xs ∈ set (lists_of_length T n)"
 shows "set xs ⊆ set T"
proof -
 have "∀ xs ∈ set (lists_of_length T n) . set xs ⊆ set T"
 by (induction n; simp)
 then show ?thesis using assms by blast
qed

lemma lists_of_length_list_set :
 "set (lists_of_length xs k) = {xs' . length xs' = k ∧ set xs' ⊆ set xs}"
 using lists_of_length_containment[of _ xs k]
 lists_of_length_length[of _ xs k]
 lists_of_length_elems[of _ xs k]
 by blast


subsubsection ‹Enumerating List Subsets›

fun generate_selector_lists :: "nat ==> bool list list" where
 "generate_selector_lists k = lists_of_length [False,True] k"


lemma generate_selector_lists_set :
 "set (generate_selector_lists k) = {(bs :: bool list) . length bs = k}"
 using lists_of_length_list_set by auto

lemma selector_list_index_set:
 assumes "length ms = length bs"
 shows "set (map fst (filter snd (zip ms bs))) = { ms ! i | i . i < length bs ∧ bs ! i}"
using assms proof (induction bs arbitrary: ms rule: rev_induct)
 case Nil
 then show ?case by auto
next
 case (snoc b bs)
 let ?ms = "butlast ms"
 let ?m = "last ms"

 have "length ?ms = length bs" using snoc.prems by auto

 have "map fst (filter snd (zip ms (bs @ [b])))
 = (map fst (filter snd (zip ?ms bs))) @ (map fst (filter snd (zip [?m] [b])))"
 by (metis ‹length (butlast ms) = length bs› append_eq_conv_conj filter_append length_0_conv
 map_append snoc.prems snoc_eq_iff_butlast zip_append2)
 then have *: "set (map fst (filter snd (zip ms (bs @ [b]))))
 = set (map fst (filter snd (zip ?ms bs))) ∪ set (map fst (filter snd (zip [?m] [b])))"
 by simp


 have "{ms ! i |i. i < length (bs @ [b]) ∧ (bs @ [b]) ! i}
 = {ms ! i |i. i ≤ (length bs) ∧ (bs @ [b]) ! i}"
 by auto
 moreover have "{ms ! i |i. i ≤ (length bs) ∧ (bs @ [b]) ! i}
 = {ms ! i |i. i < length bs ∧ (bs @ [b]) ! i}
 ∪ {ms ! i |i. i = length bs ∧ (bs @ [b]) ! i}"
 by fastforce
 moreover have "{ms ! i |i. i < length bs ∧ (bs @ [b]) ! i} = {?ms ! i |i. i < length bs ∧ bs ! i}"
 using ‹length ?ms = length bs› by (metis butlast_snoc nth_butlast)
 ultimately have **: "{ms ! i |i. i < length (bs @ [b]) ∧ (bs @ [b]) ! i}
 = {?ms ! i |i. i < length bs ∧ bs ! i}
 ∪ {ms ! i |i. i = length bs ∧ (bs @ [b]) ! i}"
 by simp


 have "set (map fst (filter snd (zip [?m] [b]))) = {ms ! i |i. i = length bs ∧ (bs @ [b]) ! i}"
 proof (cases b)
 case True
 then have "set (map fst (filter snd (zip [?m] [b]))) = {?m}" by fastforce
 moreover have "{ms ! i |i. i = length bs ∧ (bs @ [b]) ! i} = {?m}"
 proof -
 have "(bs @ [b]) ! length bs"
 by (simp add: True)
 moreover have "ms ! length bs = ?m"
 by (metis last_conv_nth length_0_conv length_butlast snoc.prems snoc_eq_iff_butlast)
 ultimately show ?thesis by fastforce
 qed
 ultimately show ?thesis by auto
 next
 case False
 then show ?thesis by auto
 qed

 then have "set (map fst (filter snd (zip (butlast ms) bs)))
 ∪ set (map fst (filter snd (zip [?m] [b])))
 = {butlast ms ! i |i. i < length bs ∧ bs ! i}
 ∪ {ms ! i |i. i = length bs ∧ (bs @ [b]) ! i}"
 using snoc.IH[OF ‹length ?ms = length bs›] by blast

 then show ?case using * **
 by simp
qed

lemma selector_list_ex :
 assumes "set xs ⊆ set ms"
 shows "∃ bs . length bs = length ms ∧ set xs = set (map fst (filter snd (zip ms bs)))"
using assms proof (induction xs rule: rev_induct)
 case Nil
 let ?bs = "replicate (length ms) False"
 have "set [] = set (map fst (filter snd (zip ms ?bs)))"
 by (metis filter_False in_set_zip length_replicate list.simps(8) nth_replicate)
 moreover have "length ?bs = length ms" by auto
 ultimately show ?case by blast
next
 case (snoc a xs)
 then have "set xs ⊆ set ms" and "a ∈ set ms" by auto
 then obtain bs where "length bs = length ms" and "set xs = set (map fst (filter snd (zip ms bs)))"
 using snoc.IH by auto

 from ‹a ∈ set ms› obtain i where "i < length ms" and "ms ! i = a"
 by (meson in_set_conv_nth)

 let ?bs = "list_update bs i True"
 have "length ms = length ?bs" using ‹length bs = length ms› by auto
 have "length ?bs = length bs" by auto

 have "set (map fst (filter snd (zip ms ?bs))) = {ms ! i |i. i < length ?bs ∧ ?bs ! i}"
 using selector_list_index_set[OF ‹length ms = length ?bs›] by assumption

 have "∧ j . j < length ?bs ==> j ≠ i ==> ?bs ! j = bs ! j"
 by auto
 then have "{ms ! j |j. j < length bs ∧ j ≠ i ∧ bs ! j}
 = {ms ! j |j. j < length ?bs ∧ j ≠ i ∧ ?bs ! j}"
 using ‹length ?bs = length bs› by fastforce



 have "{ms ! j |j. j < length ?bs ∧ j = i ∧ ?bs ! j} = {a}"
 using ‹length bs = length ms› ‹i < length ms› ‹ms ! i = a› by auto
 then have "{ms ! i |i. i < length ?bs ∧ ?bs ! i}
 = insert a {ms ! j |j. j < length ?bs ∧ j ≠ i ∧ ?bs ! j}"
 by fastforce


 have "{ms ! j |j. j < length bs ∧ j = i ∧ bs ! j} ⊆ {ms ! j |j. j < length ?bs ∧ j = i ∧ ?bs ! j}"
 by (simp add: Collect_mono)
 then have "{ms ! j |j. j < length bs ∧ j = i ∧ bs ! j} ⊆ {a}"
 using ‹{ms ! j |j. j < length ?bs ∧ j = i ∧ ?bs ! j} = {a}›
 by auto
 moreover have "{ms ! j |j. j < length bs ∧ bs ! j}
 = {ms ! j |j. j < length bs ∧ j = i ∧ bs ! j}
 ∪ {ms ! j |j. j < length bs ∧ j ≠ i ∧ bs ! j}"
 by fastforce

 ultimately have "{ms ! i |i. i < length ?bs ∧ ?bs ! i}
 = insert a {ms ! i |i. i < length bs ∧ bs ! i}"
 using ‹{ms ! j |j. j < length bs ∧ j ≠ i ∧ bs ! j}
= {ms ! j |j. j < length ?bs ∧ j ≠ i ∧ ?bs ! j}›
 using ‹{ms ! ia |ia. ia < length (bs[i := True])
∧ bs[i := True] ! ia}
= insert a {ms ! j |j. j < length (bs[i := True])
∧ j ≠ i ∧ bs[i := True] ! j}›
 by auto

 moreover have "set (map fst (filter snd (zip ms bs))) = {ms ! i |i. i < length bs ∧ bs ! i}"
 using selector_list_index_set[of ms bs] ‹length bs = length ms› by auto

 ultimately have "set (a#xs) = set (map fst (filter snd (zip ms ?bs)))"
 using ‹set (map fst (filter snd (zip ms ?bs))) = {ms ! i |i. i < length ?bs ∧ ?bs ! i}›
 ‹set xs = set (map fst (filter snd (zip ms bs)))›
 by auto
 then show ?case
 using ‹length ms = length ?bs›
 by (metis Un_commute insert_def list.set(1) list.simps(15) set_append singleton_conv)
qed

subsubsection ‹Enumerating Choices from Lists of Lists›

fun generate_choices :: "('a × ('b list)) list ==> ('a × 'b option) list list" where
 "generate_choices [] = [[]]" |
 "generate_choices (xys#xyss) =
 concat (map (λ xy' . map (λ xys' . xy' # xys') (generate_choices xyss))
 ((fst xys, None) # (map (λ y . (fst xys, Some y)) (snd xys))))"

lemma concat_map_hd_tl_elem:
 assumes "hd cs ∈ set P1"
 and "tl cs ∈ set P2"
 and "length cs > 0"
shows "cs ∈ set (concat (map (λ xy' . map (λ xys' . xy' # xys') P2) P1))"
proof -
 have "hd cs # tl cs = cs" using assms(3) by auto
 moreover have "hd cs # tl cs ∈ set (concat (map (λ xy' . map (λ xys' . xy' # xys') P2) P1))"
 using assms(1,2) by auto
 ultimately show ?thesis
 by auto
qed

lemma generate_choices_hd_tl :
 "cs ∈ set (generate_choices (xys#xyss))
 = (length cs = length (xys#xyss)
 ∧ fst (hd cs) = fst xys
 ∧ ((snd (hd cs) = None ∨ (snd (hd cs) ≠ None ∧ the (snd (hd cs)) ∈ set (snd xys))))
 ∧ (tl cs ∈ set (generate_choices xyss)))"
proof (induction xyss arbitrary: cs xys)
 case Nil
 have "(cs ∈ set (generate_choices [xys]))
 = (cs ∈ set ([(fst xys, None)] # map (λy. [(fst xys, Some y)]) (snd xys)))"
 unfolding generate_choices.simps by auto
 moreover have "(cs ∈ set ([(fst xys, None)] # map (λy. [(fst xys, Some y)]) (snd xys)))
 ==> (length cs = length [xys] ∧
 fst (hd cs) = fst xys ∧
 (snd (hd cs) = None ∨ snd (hd cs) ≠ None ∧ the (snd (hd cs)) ∈ set (snd xys)) ∧
 tl cs ∈ set (generate_choices []))"
 by auto
 moreover have "(length cs = length [xys] ∧
 fst (hd cs) = fst xys ∧
 (snd (hd cs) = None ∨ snd (hd cs) ≠ None ∧ the (snd (hd cs)) ∈ set (snd xys)) ∧
 tl cs ∈ set (generate_choices []))
 ==> (cs ∈ set ([(fst xys, None)] # map (λy. [(fst xys, Some y)]) (snd xys)))"
 unfolding generate_choices.simps(1)
 proof -
 assume a1: "length cs = length [xys]
 ∧ fst (hd cs) = fst xys
 ∧ (snd (hd cs) = None ∨ snd (hd cs) ≠ None ∧ the (snd (hd cs)) ∈ set (snd xys))
 ∧ tl cs ∈ set [[]]"
 have f2: "∀ps. ps = [] ∨ ps = (hd ps::'a × 'b option) # tl ps"
 by (meson list.exhaust_sel)
 have f3: "cs ≠ []"
 using a1 by fastforce
 have "snd (hd cs) = None ⟶ (fst xys, None) = hd cs"
 using a1 by (metis prod.exhaust_sel)
 moreover
 { assume "hd cs # tl cs ≠ [(fst xys, Some (the (snd (hd cs))))]"
 then have "snd (hd cs) = None"
 using a1 by (metis (no_types) length_0_conv length_tl list.sel(3)
 option.collapse prod.exhaust_sel) }
 ultimately have "cs ∈ insert [(fst xys, None)] ((λb. [(fst xys, Some b)]) ` set (snd xys))"
 using f3 f2 a1 by fastforce
 then show ?thesis
 by simp
 qed
 ultimately show ?case by blast
next
 case (Cons a xyss)

 have "length cs = length (xys#a#xyss)
 ==> fst (hd cs) = fst xys
 ==> (snd (hd cs) = None ∨ (snd (hd cs) ≠ None ∧ the (snd (hd cs)) ∈ set (snd xys)))
 ==> (tl cs ∈ set (generate_choices (a#xyss)))
 ==> cs ∈ set (generate_choices (xys#a#xyss))"
 proof -
 assume "length cs = length (xys#a#xyss)"
 and "fst (hd cs) = fst xys"
 and "(snd (hd cs) = None ∨ (snd (hd cs) ≠ None ∧ the (snd (hd cs)) ∈ set (snd xys)))"
 and "(tl cs ∈ set (generate_choices (a#xyss)))"
 then have "length cs > 0" by auto

 have "(hd cs) ∈ set ((fst xys, None) # (map (λ y . (fst xys, Some y)) (snd xys)))"
 using ‹fst (hd cs) = fst xys›
 ‹(snd (hd cs) = None ∨ (snd (hd cs) ≠ None ∧ the (snd (hd cs)) ∈ set (snd xys)))›
 by (metis (no_types, lifting) image_eqI list.set_intros(1) list.set_intros(2)
 option.collapse prod.collapse set_map)

 show "cs ∈ set (generate_choices ((xys#(a#xyss))))"
 using generate_choices.simps(2)[of xys "a#xyss"]
 concat_map_hd_tl_elem[OF ‹(hd cs) ∈ set ((fst xys, None) # (map (λ y . (fst xys, Some y)) (snd xys)))›
 ‹(tl cs ∈ set (generate_choices (a#xyss)))›
 ‹length cs > 0›]
 by auto
 qed

 moreover have "cs ∈ set (generate_choices (xys#a#xyss))
 ==> length cs = length (xys#a#xyss)
 ∧ fst (hd cs) = fst xys
 ∧ ((snd (hd cs) = None ∨ (snd (hd cs) ≠ None
 ∧ the (snd (hd cs)) ∈ set (snd xys))))
 ∧ (tl cs ∈ set (generate_choices (a#xyss)))"
 proof -
 assume "cs ∈ set (generate_choices (xys#a#xyss))"
 then have p3: "tl cs ∈ set (generate_choices (a#xyss))"
 using generate_choices.simps(2)[of xys "a#xyss"] by fastforce
 then have "length (tl cs) = length (a # xyss)" using Cons.IH[of "tl cs" "a"] by simp
 then have p1: "length cs = length (xys#a#xyss)" by auto

 have p2 : "fst (hd cs) = fst xys ∧ ((snd (hd cs) = None ∨ (snd (hd cs) ≠ None
 ∧ the (snd (hd cs)) ∈ set (snd xys))))"
 using ‹cs ∈ set (generate_choices (xys#a#xyss))› generate_choices.simps(2)[of xys "a#xyss"]
 by fastforce

 show ?thesis using p1 p2 p3 by simp
 qed

 ultimately show ?case by blast
qed

lemma list_append_idx_prop :
 "(∀ i . (i < length xs ⟶ P (xs ! i)))
 = (∀ j . ((j < length (ys@xs) ∧ j ≥ length ys) ⟶ P ((ys@xs) ! j)))"
proof -
 have "∧ j . ∀i<length xs. P (xs ! i) ==> j < length (ys @ xs)
 ==> length ys ≤ j ⟶ P ((ys @ xs) ! j)"
 by (simp add: nth_append)
 moreover have "∧ i . (∀ j . ((j < length (ys@xs) ∧ j ≥ length ys) ⟶ P ((ys@xs) ! j)))
 ==> i < length xs ==> P (xs ! i)"
 proof -
 fix i assume "(∀ j . ((j < length (ys@xs) ∧ j ≥ length ys) ⟶ P ((ys@xs) ! j)))"
 and "i < length xs"
 then have "P ((ys@xs) ! (length ys + i))"
 by (metis add_strict_left_mono le_add1 length_append)
 moreover have "P (xs ! i) = P ((ys@xs) ! (length ys + i))"
 by simp
 ultimately show "P (xs ! i)" by blast
 qed
 ultimately show ?thesis by blast
qed

lemma list_append_idx_prop2 :
 assumes "length xs' = length xs"
 and "length ys' = length ys"
 shows "(∀ i . (i < length xs ⟶ P (xs ! i) (xs' ! i)))
 = (∀ j . ((j < length (ys@xs) ∧ j ≥ length ys) ⟶ P ((ys@xs) ! j) ((ys'@xs') ! j)))"
proof -
 have "∀i<length xs. P (xs ! i) (xs' ! i) ==>
 ∀j. j < length (ys @ xs) ∧ length ys ≤ j ⟶ P ((ys @ xs) ! j) ((ys' @ xs') ! j)"
 using assms
 proof -
 assume a1: "∀i<length xs. P (xs ! i) (xs' ! i)"
 { fix nn :: nat
 have ff1: "∀n na. (na::nat) + n - n = na"
 by simp
 have ff2: "∀n na. (na::nat) ≤ n + na"
 by auto
 then have ff3: "∀as n. (ys' @ as) ! n = as ! (n - length ys) ∨ ¬ length ys ≤ n"
 using ff1 by (metis (no_types) add.commute assms(2) eq_diff_iff nth_append_length_plus)
 have ff4: "∀n bs bsa. ((bsa @ bs) ! n::'b) = bs ! (n - length bsa) ∨ ¬ length bsa ≤ n"
 using ff2 ff1 by (metis (no_types) add.commute eq_diff_iff nth_append_length_plus)
 have "∀n na nb. ((n::nat) + nb ≤ na ∨ ¬ n ≤ na - nb) ∨ ¬ nb ≤ na"
 using ff2 ff1 by (metis le_diff_iff)
 then have "(¬ nn < length (ys @ xs) ∨ ¬ length ys ≤ nn)
 ∨ P ((ys @ xs) ! nn) ((ys' @ xs') ! nn)"
 using ff4 ff3 a1 by (metis add.commute length_append not_le) }
 then show ?thesis
 by blast
 qed

 moreover have "(∀j. j < length (ys @ xs) ∧ length ys ≤ j ⟶ P ((ys @ xs) ! j) ((ys' @ xs') ! j))
 ==> ∀i<length xs. P (xs ! i) (xs' ! i)"
 using assms
 by (metis le_add1 length_append nat_add_left_cancel_less nth_append_length_plus)

 ultimately show ?thesis by blast
qed

lemma generate_choices_idx :
 "cs ∈ set (generate_choices xyss)
 = (length cs = length xyss
 ∧ (∀ i < length cs . (fst (cs ! i)) = (fst (xyss ! i))
 ∧ ((snd (cs ! i)) = None
 ∨ ((snd (cs ! i)) ≠ None ∧ the (snd (cs ! i)) ∈ set (snd (xyss ! i))))))"
proof (induction xyss arbitrary: cs)
 case Nil
 then show ?case by auto
next
 case (Cons xys xyss)

 have "cs ∈ set (generate_choices (xys#xyss))
 = (length cs = length (xys#xyss)
 ∧ fst (hd cs) = fst xys
 ∧ ((snd (hd cs) = None ∨ (snd (hd cs) ≠ None ∧ the (snd (hd cs)) ∈ set (snd xys))))
 ∧ (tl cs ∈ set (generate_choices xyss)))"
 using generate_choices_hd_tl by metis

 then have "cs ∈ set (generate_choices (xys#xyss))
 = (length cs = length (xys#xyss)
 ∧ fst (hd cs) = fst xys
 ∧ ((snd (hd cs) = None ∨ (snd (hd cs) ≠ None ∧ the (snd (hd cs)) ∈ set (snd xys))))
 ∧ (length (tl cs) = length xyss ∧
 (∀i<length (tl cs).
 fst (tl cs ! i) = fst (xyss ! i) ∧
 (snd (tl cs ! i) = None
 ∨ snd (tl cs ! i) ≠ None ∧ the (snd (tl cs ! i)) ∈ set (snd (xyss ! i))))))"
 using Cons.IH[of "tl cs"] by blast
 then have *: "cs ∈ set (generate_choices (xys#xyss))
 = (length cs = length (xys#xyss)
 ∧ fst (hd cs) = fst xys
 ∧ ((snd (hd cs) = None ∨ (snd (hd cs) ≠ None ∧ the (snd (hd cs)) ∈ set (snd xys))))
 ∧ (∀i<length (tl cs).
 fst (tl cs ! i) = fst (xyss ! i) ∧
 (snd (tl cs ! i) = None
 ∨ snd (tl cs ! i) ≠ None ∧ the (snd (tl cs ! i)) ∈ set (snd (xyss ! i)))))"
 by auto

 have "cs ∈ set (generate_choices (xys#xyss)) ==> (length cs = length (xys # xyss) ∧
 (∀i<length cs.
 fst (cs ! i) = fst ((xys # xyss) ! i) ∧
 (snd (cs ! i) = None ∨
 snd (cs ! i) ≠ None ∧ the (snd (cs ! i)) ∈ set (snd ((xys # xyss) ! i)))))"
 proof -
 assume "cs ∈ set (generate_choices (xys#xyss))"
 then have p1: "length cs = length (xys#xyss)"
 and p2: "fst (hd cs) = fst xys "
 and p3: "((snd (hd cs) = None
 ∨ (snd (hd cs) ≠ None ∧ the (snd (hd cs)) ∈ set (snd xys))))"
 and p4: "(∀i<length (tl cs).
 fst (tl cs ! i) = fst (xyss ! i) ∧
 (snd (tl cs ! i) = None
 ∨ snd (tl cs ! i) ≠ None ∧ the (snd (tl cs ! i)) ∈ set (snd (xyss ! i))))"
 using * by blast+
 then have "length xyss = length (tl cs)" and "length (xys # xyss) = length ([hd cs] @ tl cs)"
 by auto

 have "[hd cs]@(tl cs) = cs"
 by (metis (no_types) p1 append.left_neutral append_Cons length_greater_0_conv
 list.collapse list.simps(3))
 then have p4b: "(∀i<length cs. i > 0 ⟶
 (fst (cs ! i) = fst ((xys#xyss) ! i) ∧
 (snd (cs ! i) = None
 ∨ snd (cs ! i) ≠ None ∧ the (snd (cs ! i)) ∈ set (snd ((xys#xyss) ! i)))))"
 using p4 list_append_idx_prop2[of xyss "tl cs" "xys#xyss" "[hd cs]@(tl cs)"
 "λ x y . fst x = fst y
 ∧ (snd x = None
 ∨ snd x ≠ None ∧ the (snd x) ∈ set (snd y))",
 OF ‹length xyss = length (tl cs)›
 ‹length (xys # xyss) = length ([hd cs] @ tl cs)›]
 by (metis (no_types, lifting) One_nat_def Suc_pred
 ‹length (xys # xyss) = length ([hd cs] @ tl cs)› ‹length xyss = length (tl cs)›
 length_Cons list.size(3) not_less_eq nth_Cons_pos nth_append)

 have p4a :"(fst (cs ! 0) = fst ((xys#xyss) ! 0) ∧ (snd (cs ! 0) = None
 ∨ snd (cs ! 0) ≠ None ∧ the (snd (cs ! 0)) ∈ set (snd ((xys#xyss) ! 0))))"
 using p1 p2 p3 by (metis hd_conv_nth length_greater_0_conv list.simps(3) nth_Cons_0)

 show ?thesis using p1 p4a p4b by fastforce
 qed

 moreover have "(length cs = length (xys # xyss) ∧
 (∀i<length cs.
 fst (cs ! i) = fst ((xys # xyss) ! i) ∧
 (snd (cs ! i) = None ∨
 snd (cs ! i) ≠ None ∧ the (snd (cs ! i)) ∈ set (snd ((xys # xyss) ! i)))))
 ==> cs ∈ set (generate_choices (xys#xyss))"
 using *
 by (metis (no_types, lifting) Nitpick.size_list_simp(2) Suc_mono hd_conv_nth
 length_greater_0_conv length_tl list.sel(3) list.simps(3) nth_Cons_0 nth_tl)

 ultimately show ?case by blast
qed

subsection ‹Finding the Index of the First Element of a List Satisfying a Property›

fun find_index :: "('a ==> bool) ==> 'a list ==> nat option" where
 "find_index f [] = None" |
 "find_index f (x#xs) = (if f x
 then Some 0
 else (case find_index f xs of Some k ==> Some (Suc k) | None ==> None))"

lemma find_index_index :
 assumes "find_index f xs = Some k"
 shows "k < length xs" and "f (xs ! k)" and "∧ j . j < k ==> ¬ f (xs ! j)"
proof -
 have "(k < length xs) ∧ (f (xs ! k)) ∧ (∀ j < k . ¬ (f (xs ! j)))"
 using assms proof (induction xs arbitrary: k)
 case Nil
 then show ?case by auto
 next
 case (Cons x xs)

 show ?case proof (cases "f x")
 case True
 then show ?thesis using Cons.prems by auto
 next
 case False
 then have "find_index f (x#xs)
 = (case find_index f xs of Some k ==> Some (Suc k) | None ==> None)"
 by auto
 then have "(case find_index f xs of Some k ==> Some (Suc k) | None ==> None) = Some k"
 using Cons.prems by auto
 then obtain k' where "find_index f xs = Some k'" and "k = Suc k'"
 by (metis option.case_eq_if option.collapse option.distinct(1) option.sel)

 have "k < length (x # xs) ∧ f ((x # xs) ! k)"
 using Cons.IH[OF ‹find_index f xs = Some k'›] ‹k = Suc k'›
 by auto
 moreover have "(∀j<k. ¬ f ((x # xs) ! j))"
 using Cons.IH[OF ‹find_index f xs = Some k'›] ‹k = Suc k'› False less_Suc_eq_0_disj
 by auto
 ultimately show ?thesis by presburger
 qed
 qed
 then show "k < length xs" and "f (xs ! k)" and "∧ j . j < k ==> ¬ f (xs ! j)" by simp+
qed

lemma find_index_exhaustive :
 assumes "∃ x ∈ set xs . f x"
 shows "find_index f xs ≠ None"
 using assms proof (induction xs)
case Nil
 then show ?case by auto
next
 case (Cons x xs)
 then show ?case by (cases "f x"; auto)
qed

subsection ‹List Distinctness from Sorting›

lemma non_distinct_repetition_indices :
 assumes "¬ distinct xs"
 shows "∃ i j . i < j ∧ j < length xs ∧ xs ! i = xs ! j"
 by (metis assms distinct_conv_nth le_neq_implies_less not_le)

lemma non_distinct_repetition_indices_rev :
 assumes "i < j" and "j < length xs" and "xs ! i = xs ! j"
 shows "¬ distinct xs"
 using assms nth_eq_iff_index_eq by fastforce

lemma ordered_list_distinct :
 fixes xs :: "('a::preorder) list"
 assumes "∧ i . Suc i < length xs ==> (xs ! i) < (xs ! (Suc i))"
 shows "distinct xs"
proof -
 have "∧ i j . i < j ==> j < length xs ==> (xs ! i) < (xs ! j)"
 proof -
 fix i j assume "i < j" and "j < length xs"
 then show "xs ! i < xs ! j"
 using assms proof (induction xs arbitrary: i j rule: rev_induct)
 case Nil
 then show ?case by auto
 next
 case (snoc a xs)
 show ?case proof (cases "j < length xs")
 case True
 show ?thesis using snoc.IH[OF snoc.prems(1) True] snoc.prems(3)
 proof -
 have f1: "i < length xs"
 using True less_trans snoc.prems(1) by blast
 have f2: "∀is isa n. if n < length is then (is @ isa) ! n
 = (is ! n::integer) else (is @ isa) ! n = isa ! (n - length is)"
 by (meson nth_append)
 then have f3: "(xs @ [a]) ! i = xs ! i"
 using f1
 by (simp add: nth_append)
 have "xs ! i < xs ! j"
 using f2
 by (metis Suc_lessD ‹(∧i. Suc i < length xs ==> xs ! i < xs ! Suc i) ==> xs ! i < xs ! j›
 butlast_snoc length_append_singleton less_SucI nth_butlast snoc.prems(3))
 then show ?thesis
 using f3 f2 True
 by (simp add: nth_append)
 qed
 next
 case False
 then have "(xs @ [a]) ! j = a"
 using snoc.prems(2)
 by (metis length_append_singleton less_SucE nth_append_length)

 consider "j = 1" | "j > 1"
 using ‹i < j›
 by linarith
 then show ?thesis proof cases
 case 1
 then have "i = 0" and "j = Suc i" using ‹i < j› by linarith+
 then show ?thesis
 using snoc.prems(3)
 using snoc.prems(2) by blast
 next
 case 2
 then consider "i < j - 1" | "i = j - 1" using ‹i < j› by linarith+
 then show ?thesis proof cases
 case 1

 have "(∧i. Suc i < length xs ==> xs ! i < xs ! Suc i) ==> xs ! i < xs ! (j - 1)"
 using snoc.IH[OF 1] snoc.prems(2) 2 by simp
 then have le1: "(xs @ [a]) ! i < (xs @ [a]) ! (j -1)"
 using snoc.prems(2)
 by (metis "2" False One_nat_def Suc_diff_Suc Suc_lessD diff_zero snoc.prems(3)
 length_append_singleton less_SucE not_less_eq nth_append snoc.prems(1))
 moreover have le2: "(xs @ [a]) ! (j -1) < (xs @ [a]) ! j"
 using snoc.prems(2,3) 2 less_trans
 by (metis (full_types) One_nat_def Suc_diff_Suc diff_zero less_numeral_extra(1))
 ultimately show ?thesis
 using less_trans by blast
 next
 case 2
 then have "j = Suc i" using ‹1 < j› by linarith
 then show ?thesis
 using snoc.prems(3)
 using snoc.prems(2) by blast
 qed
 qed
 qed
 qed
 qed

 then show ?thesis
 by (metis less_asym non_distinct_repetition_indices)
qed

lemma ordered_list_distinct_rev :
 fixes xs :: "('a::preorder) list"
 assumes "∧ i . Suc i < length xs ==> (xs ! i) > (xs ! (Suc i))"
 shows "distinct xs"
proof -
 have "∧ i . Suc i < length (rev xs) ==> ((rev xs) ! i) < ((rev xs) ! (Suc i))"
 using assms
 proof -
 fix i :: nat
 assume a1: "Suc i < length (rev xs)"
 obtain nn :: "nat ==> nat ==> nat" where
 "∀x0 x1. (∃v2. x1 = Suc v2 ∧ v2 < x0) = (x1 = Suc (nn x0 x1) ∧ nn x0 x1 < x0)"
 by moura
 then have f2: "∀n na. (¬ n < Suc na ∨ n = 0 ∨ n = Suc (nn na n) ∧ nn na n < na)
 ∧ (n < Suc na ∨ n ≠ 0 ∧ (∀nb. n ≠ Suc nb ∨ ¬ nb < na))"
 by (meson less_Suc_eq_0_disj)
 have f3: "Suc (length xs - Suc (Suc i)) = length (rev xs) - Suc i"
 using a1 by (simp add: Suc_diff_Suc)
 have "i < length (rev xs)"
 using a1 by (meson Suc_lessD)
 then have "i < length xs"
 by simp
 then show "rev xs ! i < rev xs ! Suc i"
 using f3 f2 a1 by (metis (no_types) assms diff_less length_rev not_less_iff_gr_or_eq rev_nth)
 qed
 then have "distinct (rev xs)"
 using ordered_list_distinct[of "rev xs"] by blast
 then show ?thesis by auto
qed

subsection ‹Calculating Prefixes and Suffixes›

fun suffixes :: "'a list ==> 'a list list" where
 "suffixes [] = [[]]" |
 "suffixes (x#xs) = (suffixes xs) @ [x#xs]"

lemma suffixes_set :
 "set (suffixes xs) = {zs . ∃ ys . ys@zs = xs}"
proof (induction xs)
 case Nil
 then show ?case by auto
next
 case (Cons x xs)
 then have *: "set (suffixes (x#xs)) = {zs . ∃ ys . ys@zs = xs} ∪ {x#xs}"
 by auto

 have "{zs . ∃ ys . ys@zs = xs} = {zs . ∃ ys . x#ys@zs = x#xs}"
 by force
 then have "{zs . ∃ ys . ys@zs = xs} = {zs . ∃ ys . ys@zs = x#xs ∧ ys ≠ []}"
 by (metis Cons_eq_append_conv list.distinct(1))
 moreover have "{x#xs} = {zs . ∃ ys . ys@zs = x#xs ∧ ys = []}"
 by force

 ultimately show ?case using * by force
qed

lemma prefixes_set : "set (prefixes xs) = {xs' . ∃ xs'' . xs'@xs'' = xs}"
proof (induction xs)
 case Nil
 then show ?case by auto
next
 case (Cons x xs)
 moreover have "prefixes (x#xs) = [] # map ((#) x) (prefixes xs)"
 by auto
 ultimately have *: "set (prefixes (x#xs)) = insert [] (((#) x) ` {xs'. ∃xs''. xs' @ xs'' = xs})"
 by auto
 also have "… = {xs' . ∃ xs'' . xs'@xs'' = (x#xs)}"
 proof
 show "insert [] ((#) x ` {xs'. ∃xs''. xs' @ xs'' = xs}) ⊆ {xs'. ∃xs''. xs' @ xs'' = x # xs}"
 by auto
 show "{xs'. ∃xs''. xs' @ xs'' = x # xs} ⊆ insert [] ((#) x ` {xs'. ∃xs''. xs' @ xs'' = xs})"
 proof
 fix y assume "y ∈ {xs'. ∃xs''. xs' @ xs'' = x # xs}"
 then obtain y' where "y@y' = x # xs"
 by blast
 then show "y ∈ insert [] ((#) x ` {xs'. ∃xs''. xs' @ xs'' = xs})"
 by (cases y; auto)
 qed
 qed
 finally show ?case .
qed

fun is_prefix :: "'a list ==> 'a list ==> bool" where
 "is_prefix [] _ = True" |
 "is_prefix (x#xs) [] = False" |
 "is_prefix (x#xs) (y#ys) = (x = y ∧ is_prefix xs ys)"

lemma is_prefix_prefix : "is_prefix xs ys = (∃ xs' . ys = xs@xs')"
proof (induction xs arbitrary: ys)
 case Nil
 then show ?case by auto
next
 case (Cons x xs)
 show ?case proof (cases "is_prefix (x#xs) ys")
 case True
 then show ?thesis using Cons.IH
 by (metis append_Cons is_prefix.simps(2) is_prefix.simps(3) neq_Nil_conv)
 next
 case False
 then show ?thesis
 using Cons.IH by auto
 qed
qed

fun add_prefixes :: "'a list list ==> 'a list list" where
 "add_prefixes xs = concat (map prefixes xs)"

lemma add_prefixes_set : "set (add_prefixes xs) = {xs' . ∃ xs'' . xs'@xs'' ∈ set xs}"
proof -
 have "set (add_prefixes xs) = {xs' . ∃ x ∈ set xs . xs' ∈ set (prefixes x)}"
 unfolding add_prefixes.simps by auto
 also have "… = {xs' . ∃ xs'' . xs'@xs'' ∈ set xs}"
 proof (induction xs)
 case Nil
 then show ?case using prefixes_set by auto
 next
 case (Cons a xs)
 then show ?case
 proof -
 have "∧ xs' . xs' ∈ {xs'. ∃x∈set (a # xs). xs' ∈ set (prefixes x)}
 ⟷ xs' ∈ {xs'. ∃xs''. xs' @ xs'' ∈ set (a # xs)}"
 proof -
 fix xs'
 show "xs' ∈ {xs'. ∃x∈set (a # xs). xs' ∈ set (prefixes x)}
 ⟷ xs' ∈ {xs'. ∃xs''. xs' @ xs'' ∈ set (a # xs)}"
 unfolding prefixes_set by force
 qed
 then show ?thesis by blast
 qed
 qed
 finally show ?thesis by blast
qed

lemma prefixes_set_ob :
 assumes "xs ∈ set (prefixes xss)"
 obtains xs' where "xss = xs@xs'"
 using assms unfolding prefixes_set
 by auto

lemma prefixes_finite : "finite { x ∈ set (prefixes xs) . P x}"
 by (metis Collect_mem_eq List.finite_set finite_Collect_conjI)


lemma prefixes_set_Cons_insert: "set (prefixes (w' @ [xy])) = Set.insert (w'@[xy]) (set (prefixes (w')))"
 unfolding prefixes_set
proof (induction w' arbitrary: xy rule: rev_induct)
 case Nil
 then show ?case
 by (auto; simp add: append_eq_Cons_conv)
 next
 case (snoc x xs)
 then show ?case
 by (auto; metis (no_types, opaque_lifting) butlast.simps(2) butlast_append butlast_snoc)
 qed

lemma prefixes_set_subset:
 "set (prefixes xs) ⊆ set (prefixes (xs@ys))"
 unfolding prefixes_set by auto

lemma prefixes_prefix_subset :
 assumes "xs ∈ set (prefixes ys)"
 shows "set (prefixes xs) ⊆ set (prefixes ys)"
 using assms unfolding prefixes_set by auto

lemma prefixes_butlast_is_prefix :
 "butlast xs ∈ set (prefixes xs)"
 unfolding prefixes_set
 by (metis (mono_tags, lifting) append_butlast_last_id butlast.simps(1) mem_Collect_eq self_append_conv2)

lemma prefixes_take_iff :
 "xs ∈ set (prefixes ys) ⟷ take (length xs) ys = xs"
proof
 show "xs ∈ set (prefixes ys) ==> take (length xs) ys = xs"
 unfolding prefixes_set
 by (simp add: append_eq_conv_conj)

 show "take (length xs) ys = xs ==> xs ∈ set (prefixes ys)"
 unfolding prefixes_set
 by (metis (mono_tags, lifting) append_take_drop_id mem_Collect_eq)
qed

lemma prefixes_set_Nil : "[] ∈ list.set (prefixes xs)"
 by (metis append.left_neutral list.set_intros(1) prefixes.simps(1) prefixes_set_subset subset_iff)

lemma prefixes_prefixes :
 assumes "ys ∈ list.set (prefixes xs)"
 "zs ∈ list.set (prefixes xs)"
 shows "ys ∈ list.set (prefixes zs) ∨ zs ∈ list.set (prefixes ys)"
proof (rule ccontr)
 let ?ys = "take (length ys) zs"
 let ?zs = "take (length zs) ys"

 assume "¬ (ys ∈ list.set (prefixes zs) ∨ zs ∈ list.set (prefixes ys))"
 then have "?ys ≠ ys" and "?zs ≠ zs"
 using prefixes_take_iff by blast+
 moreover have "?ys = ys ∨ ?zs = zs"
 using assms
 by (metis linear min.commute prefixes_take_iff take_all_iff take_take)
 ultimately show False
 by simp
qed

subsubsection ‹Pairs of Distinct Prefixes›

fun prefix_pairs :: "'a list ==> ('a list × 'a list) list"
 where "prefix_pairs [] = []" |
 "prefix_pairs xs = prefix_pairs (butlast xs) @ (map (λ ys. (ys,xs)) (butlast (prefixes xs)))"

lemma prefixes_butlast :
 "set (butlast (prefixes xs)) = {ys . ∃ zs . ys@zs = xs ∧ zs ≠ []}"
proof (induction "length xs" arbitrary: xs)
 case 0
 then show ?case by auto
next
 case (Suc k)

 then obtain x xs' where "xs = x#xs'" and "k = length xs' "
 by (metis length_Suc_conv)

 then have "prefixes xs = [] # map ((#) x) (prefixes xs')"
 by auto
 then have "butlast (prefixes xs) = [] # map ((#) x) (butlast (prefixes xs'))"
 by (simp add: map_butlast)
 then have "set (butlast (prefixes xs)) = insert [] (((#) x) ` {ys . ∃ zs . ys@zs = xs' ∧ zs ≠ []})"
 using Suc.hyps(1)[OF ‹k = length xs'›]
 by auto
 also have "… = {ys . ∃ zs . ys@zs = (x#xs') ∧ zs ≠ []}"
 proof
 show "insert [] ((#) x ` {ys. ∃zs. ys @ zs = xs' ∧ zs ≠ []}) ⊆ {ys. ∃zs. ys @ zs = x # xs' ∧ zs ≠ []}"
 by auto
 show "{ys. ∃zs. ys @ zs = x # xs' ∧ zs ≠ []} ⊆ insert [] ((#) x ` {ys. ∃zs. ys @ zs = xs' ∧ zs ≠ []})"
 proof
 fix ys assume "ys ∈ {ys. ∃zs. ys @ zs = x # xs' ∧ zs ≠ []}"
 then show "ys ∈ insert [] ((#) x ` {ys. ∃zs. ys @ zs = xs' ∧ zs ≠ []})"
 by (cases ys; auto)
 qed
 qed
 finally show ?case
 unfolding ‹xs = x#xs'› .
qed

lemma prefix_pairs_set :
 "set (prefix_pairs xs) = {(zs,ys) | zs ys . ∃ xs1 xs2 . zs@xs1 = ys ∧ ys@xs2 = xs ∧ xs1 ≠ []}"
proof (induction xs rule: rev_induct)
 case Nil
 then show ?case by auto
next
 case (snoc x xs)
 have "prefix_pairs (xs @ [x]) = prefix_pairs (butlast (xs @ [x])) @ (map (λ ys. (ys,(xs @ [x]))) (butlast (prefixes (xs @ [x]))))"
 by (cases "(xs @ [x])"; auto)
 then have *: "prefix_pairs (xs @ [x]) = prefix_pairs xs @ (map (λ ys. (ys,(xs @ [x]))) (butlast (prefixes (xs @ [x]))))"
 by auto

 have "set (prefix_pairs xs) = {(zs, ys) |zs ys. ∃xs1 xs2. zs @ xs1 = ys ∧ ys @ xs2 = xs ∧ xs1 ≠ []}"
 using snoc.IH by assumption
 then have "set (prefix_pairs xs) = {(zs, ys) |zs ys. ∃xs1 xs2. zs @ xs1 = ys ∧ ys @ xs2 @ [x] = xs@[x] ∧ xs1 ≠ []}"
 by auto
 also have "... = {(zs, ys) |zs ys. ∃xs1 xs2. zs @ xs1 = ys ∧ ys @ xs2 = xs @[x] ∧ xs1 ≠ [] ∧ xs2 ≠ []}"
 proof -
 let ?P1 = "λ zs ys . (∃xs1 xs2. zs @ xs1 = ys ∧ ys @ xs2 @ [x] = xs@[x] ∧ xs1 ≠ [])"
 let ?P2 = "λ zs ys . (∃xs1 xs2. zs @ xs1 = ys ∧ ys @ xs2 = xs @[x] ∧ xs1 ≠ [] ∧ xs2 ≠ [])"

 have "∧ ys zs . ?P2 zs ys ==> ?P1 zs ys"
 by (metis append_assoc butlast_append butlast_snoc)
 then have "∧ ys zs . ?P1 ys zs = ?P2 ys zs"
 by blast
 then show ?thesis by force
 qed
 finally have "set (prefix_pairs xs) = {(zs, ys) |zs ys. ∃xs1 xs2. zs @ xs1 = ys ∧ ys @ xs2 = xs @ [x] ∧ xs1 ≠ [] ∧ xs2 ≠ []}"
 by assumption

 moreover have "set (map (λ ys. (ys,(xs @ [x]))) (butlast (prefixes (xs @ [x])))) = {(zs, ys) |zs ys. ∃xs1 xs2. zs @ xs1 = ys ∧ ys @ xs2 = xs @ [x] ∧ xs1 ≠ [] ∧ xs2 = []}"
 using prefixes_butlast[of "xs@[x]"] by force

 ultimately show ?case using * by force
qed

lemma prefix_pairs_set_alt :
 "set (prefix_pairs xs) = {(xs1,xs1@xs2) | xs1 xs2 . xs2 ≠ [] ∧ (∃ xs3 . xs1@xs2@xs3 = xs)}"
 unfolding prefix_pairs_set by auto

lemma prefixes_Cons :
 assumes "(x#xs) ∈ set (prefixes (y#ys))"
 shows "x = y" and "xs ∈ set (prefixes ys)"
proof -
 show "x = y"
 by (metis Cons_eq_appendI assms nth_Cons_0 prefixes_set_ob)

 show "xs ∈ set (prefixes ys)"
 proof -
 obtain xs' xs'' where "(x#xs) = xs'" and "(y#ys) = xs'@xs''"
 by (meson assms prefixes_set_ob)
 then have "xs' = x#tl xs'"
 by auto
 then have "xs = tl xs'"
 using ‹(x#xs) = xs'› by auto
 moreover have "ys = (tl xs')@xs''"
 using ‹(y#ys) = xs'@xs''› ‹xs' = x#tl xs'›
 by (metis append_Cons list.inject)
 ultimately show ?thesis
 unfolding prefixes_set by blast
 qed
qed

lemma prefixes_prepend :
 assumes "xs' ∈ set (prefixes xs)"
 shows "ys@xs' ∈ set (prefixes (ys@xs))"
proof -
 obtain xs'' where "xs = xs'@xs''"
 using assms
 using prefixes_set_ob by auto
 then have "(ys@xs) = (ys@xs')@xs''"
 by auto
 then show ?thesis
 unfolding prefixes_set by auto
qed

lemma prefixes_prefix_suffix_ob :
 assumes "a ∈ set (prefixes (b@c))"
 and "a ∉ set (prefixes b)"
obtains c' c'' where "c = c'@c''"
 and "a = b@c'"
 and "c' ≠ []"
proof -
 have "∃ c' c'' . c = c'@c'' ∧ a = b@c' ∧ c' ≠ []"
 using assms
 proof (induction b arbitrary: a)
 case Nil
 then show ?case
 unfolding prefixes_set
 by fastforce
 next
 case (Cons x xs)
 show ?case proof (cases a)
 case Nil
 then show ?thesis
 by (metis Cons.prems(2) list.size(3) prefixes_take_iff take_eq_Nil)
 next
 case (Cons a' as)
 then have "a' # as ∈ set (prefixes (x #(xs@c)))"
 using Cons.prems(1) by auto

 have "a' = x" and "as ∈ set (prefixes (xs@c))"
 using prefixes_Cons[OF ‹a' # as ∈ set (prefixes (x #(xs@c)))›]
 by auto
 moreover have "as ∉ set (prefixes xs)"
 using ‹a ∉ set (prefixes (x # xs))› unfolding Cons ‹a' = x› prefixes_set by auto

 ultimately obtain c' c'' where "c = c'@c''"
 and "as = xs@c'"
 and "c' ≠ []"
 using Cons.IH by blast
 then have "c = c'@c''" and "a = (x#xs)@c'" and "c' ≠ []"
 unfolding Cons ‹a' = x› by auto
 then show ?thesis
 using that by blast
 qed
 qed
 then show ?thesis using that by blast
qed

fun list_ordered_pairs :: "'a list ==> ('a × 'a) list" where
 "list_ordered_pairs [] = []" |
 "list_ordered_pairs (x#xs) = (map (Pair x) xs) @ (list_ordered_pairs xs)"

lemma list_ordered_pairs_set_containment :
 assumes "x ∈ list.set xs"
 and "y ∈ list.set xs"
 and "x ≠ y"
shows "(x,y) ∈ list.set (list_ordered_pairs xs) ∨ (y,x) ∈ list.set (list_ordered_pairs xs)"
 using assms by (induction xs; auto)

subsection ‹Calculating Distinct Non-Reflexive Pairs over List Elements›

fun non_sym_dist_pairs' :: "'a list ==> ('a × 'a) list" where
 "non_sym_dist_pairs' [] = []" |
 "non_sym_dist_pairs' (x#xs) = (map (λ y. (x,y)) xs) @ non_sym_dist_pairs' xs"

fun non_sym_dist_pairs :: "'a list ==> ('a × 'a) list" where
 "non_sym_dist_pairs xs = non_sym_dist_pairs' (remdups xs)"

lemma non_sym_dist_pairs_subset : "set (non_sym_dist_pairs xs) ⊆ (set xs) × (set xs)"
 by (induction xs; auto)

lemma non_sym_dist_pairs'_elems_distinct:
 assumes "distinct xs"
 and "(x,y) ∈ set (non_sym_dist_pairs' xs)"
shows "x ∈ set xs"
and "y ∈ set xs"
and "x ≠ y"
proof -
 show "x ∈ set xs" and "y ∈ set xs"
 using non_sym_dist_pairs_subset assms(2) by (induction xs; auto)+
 show "x ≠ y"
 using assms by (induction xs; auto)
qed

lemma non_sym_dist_pairs_elems_distinct:
 assumes "(x,y) ∈ set (non_sym_dist_pairs xs)"
shows "x ∈ set xs"
and "y ∈ set xs"
and "x ≠ y"
 using non_sym_dist_pairs'_elems_distinct assms
 unfolding non_sym_dist_pairs.simps by fastforce+

lemma non_sym_dist_pairs_elems :
 assumes "x ∈ set xs"
 and "y ∈ set xs"
 and "x ≠ y"
shows "(x,y) ∈ set (non_sym_dist_pairs xs) ∨ (y,x) ∈ set (non_sym_dist_pairs xs)"
 using assms by (induction xs; auto)

lemma non_sym_dist_pairs'_elems_non_refl :
 assumes "distinct xs"
 and "(x,y) ∈ set (non_sym_dist_pairs' xs)"
shows "(y,x) ∉ set (non_sym_dist_pairs' xs)"
 using assms
proof (induction xs arbitrary: x y)
 case Nil
 then show ?case by auto
next
 case (Cons z zs)
 then have "distinct zs" by auto

 have "x ≠ y"
 using non_sym_dist_pairs'_elems_distinct[OF Cons.prems] by simp

 consider (a) "(x,y) ∈ set (map (Pair z) zs)" |
 (b) "(x,y) ∈ set (non_sym_dist_pairs' zs)"
 using ‹(x,y) ∈ set (non_sym_dist_pairs' (z#zs))› unfolding non_sym_dist_pairs'.simps by auto
 then show ?case proof cases
 case a
 then have "x = z" by auto
 then have "(y,x) ∉ set (map (Pair z) zs)"
 using ‹x ≠ y› by auto
 moreover have "x ∉ set zs"
 using ‹x = z› ‹distinct (z#zs)› by auto
 ultimately show ?thesis
 using ‹distinct zs› non_sym_dist_pairs'_elems_distinct(2) by fastforce
 next
 case b
 then have "x ≠ z" and "y ≠ z"
 using Cons.prems unfolding non_sym_dist_pairs'.simps
 by (meson distinct.simps(2) non_sym_dist_pairs'_elems_distinct(1,2))+

 then show ?thesis
 using Cons.IH[OF ‹distinct zs› b] by auto
 qed
qed

lemma non_sym_dist_pairs_elems_non_refl :
 assumes "(x,y) ∈ set (non_sym_dist_pairs xs)"
 shows "(y,x) ∉ set (non_sym_dist_pairs xs)"
 using assms by (simp add: non_sym_dist_pairs'_elems_non_refl)

lemma non_sym_dist_pairs_set_iff :
 "(x,y) ∈ set (non_sym_dist_pairs xs)
 ⟷ (x ≠ y ∧ x ∈ set xs ∧ y ∈ set xs ∧ (y,x) ∉ set (non_sym_dist_pairs xs))"
 using non_sym_dist_pairs_elems_non_refl[of x y xs]
 non_sym_dist_pairs_elems[of x xs y]
 non_sym_dist_pairs_elems_distinct[of x y xs] by blast

subsection ‹Finite Linear Order From List Positions›

fun linear_order_from_list_position' :: "'a list ==> ('a × 'a) list" where
 "linear_order_from_list_position' [] = []" |
 "linear_order_from_list_position' (x#xs)
 = (x,x) # (map (λ y . (x,y)) xs) @ (linear_order_from_list_position' xs)"

fun linear_order_from_list_position :: "'a list ==> ('a × 'a) list" where
 "linear_order_from_list_position xs = linear_order_from_list_position' (remdups xs)"

lemma linear_order_from_list_position_set :
 "set (linear_order_from_list_position xs)
 = (set (map (λ x . (x,x)) xs)) ∪ set (non_sym_dist_pairs xs)"
 by (induction xs; auto)

lemma linear_order_from_list_position_total:
 "total_on (set xs) (set (linear_order_from_list_position xs))"
 unfolding linear_order_from_list_position_set
 using non_sym_dist_pairs_elems[of _ xs]
 by (meson UnI2 total_onI)

lemma linear_order_from_list_position_refl:
 "refl_on (set xs) (set (linear_order_from_list_position xs))"
proof (rule refl_onI)
 show "∧x. x ∈ set xs ==> (x, x) ∈ set (linear_order_from_list_position xs)"
 unfolding linear_order_from_list_position_set
 using non_sym_dist_pairs_subset[of xs] by auto
qed

lemma linear_order_from_list_position_antisym:
 "antisym (set (linear_order_from_list_position xs))"
proof
 fix x y assume "(x, y) ∈ set (linear_order_from_list_position xs)"
 and "(y, x) ∈ set (linear_order_from_list_position xs)"
 then have "(x, y) ∈ set (map (λx. (x, x)) xs) ∪ set (non_sym_dist_pairs xs)"
 and "(y, x) ∈ set (map (λx. (x, x)) xs) ∪ set (non_sym_dist_pairs xs)"
 unfolding linear_order_from_list_position_set by blast+
 then consider (a) "(x, y) ∈ set (map (λx. (x, x)) xs)" |
 (b) "(x, y) ∈ set (non_sym_dist_pairs xs)"
 by blast
 then show "x = y"
 proof cases
 case a
 then show ?thesis by auto
 next
 case b
 then have "x ≠ y" and "(y,x) ∉ set (non_sym_dist_pairs xs)"
 using non_sym_dist_pairs_set_iff[of x y xs] by simp+
 then have "(y, x) ∉ set (map (λx. (x, x)) xs) ∪ set (non_sym_dist_pairs xs)"
 by auto
 then show ?thesis
 using ‹(y, x) ∈ set (map (λx. (x, x)) xs) ∪ set (non_sym_dist_pairs xs)› by blast
 qed
qed

lemma non_sym_dist_pairs'_indices :
 "distinct xs ==> (x,y) ∈ set (non_sym_dist_pairs' xs)
 ==> (∃ i j . xs ! i = x ∧ xs ! j = y ∧ i < j ∧ i < length xs ∧ j < length xs)"
proof (induction xs)
 case Nil
 then show ?case by auto
next
 case (Cons a xs)
 show ?case proof (cases "a = x")
 case True
 then have "(a#xs) ! 0 = x" and "0 < length (a#xs)"
 by auto

 have "y ∈ set xs"
 using non_sym_dist_pairs'_elems_distinct(2,3)[OF Cons.prems(1,2)] True by auto
 then obtain j where "xs ! j = y" and "j < length xs"
 by (meson in_set_conv_nth)
 then have "(a#xs) ! (Suc j) = y" and "Suc j < length (a#xs)"
 by auto

 then show ?thesis
 using ‹(a#xs) ! 0 = x› ‹0 < length (a#xs)› by blast
 next
 case False
 then have "(x,y) ∈ set (non_sym_dist_pairs' xs)"
 using Cons.prems(2) by auto
 then show ?thesis
 using Cons.IH Cons.prems(1)
 by (metis Suc_mono distinct.simps(2) length_Cons nth_Cons_Suc)
 qed
qed

lemma non_sym_dist_pairs'_trans: "distinct xs ==> trans (set (non_sym_dist_pairs' xs))"
proof
 fix x y z assume "distinct xs"
 and "(x, y) ∈ set (non_sym_dist_pairs' xs)"
 and "(y, z) ∈ set (non_sym_dist_pairs' xs)"

 obtain nx ny where "xs ! nx = x" and "xs ! ny = y" and "nx < ny"
 and "nx < length xs" and "ny < length xs"
 using non_sym_dist_pairs'_indices[OF ‹distinct xs› ‹(x, y) ∈ set (non_sym_dist_pairs' xs)›]
 by blast

 obtain ny' nz where "xs ! ny' = y" and "xs ! nz = z" and "ny'< nz"
 and "ny' < length xs" and "nz < length xs"
 using non_sym_dist_pairs'_indices[OF ‹distinct xs› ‹(y, z) ∈ set (non_sym_dist_pairs' xs)›]
 by blast

 have "ny' = ny"
 using ‹distinct xs› ‹xs ! ny = y› ‹xs ! ny' = y› ‹ny < length xs› ‹ny' < length xs›
 nth_eq_iff_index_eq
 by metis
 then have "nx < nz"
 using ‹nx < ny› ‹ny' < nz› by auto

 then have "nx ≠ nz" by simp
 then have "x ≠ z"
 using ‹distinct xs› ‹xs ! nx = x› ‹xs ! nz = z› ‹nx < length xs› ‹nz < length xs›
 nth_eq_iff_index_eq
 by metis

 have "remdups xs = xs"
 using ‹distinct xs› by auto

 have "¬(z, x) ∈ set (non_sym_dist_pairs' xs)"
 proof
 assume "(z, x) ∈ set (non_sym_dist_pairs' xs)"
 then obtain nz' nx' where "xs ! nx' = x" and "xs ! nz' = z" and "nz'< nx'"
 and "nx' < length xs" and "nz' < length xs"
 using non_sym_dist_pairs'_indices[OF ‹distinct xs›, of z x] by metis

 have "nx' = nx"
 using ‹distinct xs› ‹xs ! nx = x› ‹xs ! nx' = x› ‹nx < length xs› ‹nx' < length xs›
 nth_eq_iff_index_eq
 by metis
 moreover have "nz' = nz"
 using ‹distinct xs› ‹xs ! nz = z› ‹xs ! nz' = z› ‹nz < length xs› ‹nz' < length xs›
 nth_eq_iff_index_eq
 by metis
 ultimately have "nz < nx"
 using ‹nz'< nx'› by auto
 then show "False"
 using ‹nx < nz› by simp
 qed
 then show "(x, z) ∈ set (non_sym_dist_pairs' xs)"
 using non_sym_dist_pairs'_elems_distinct(1)[OF ‹distinct xs› ‹(x, y) ∈ set (non_sym_dist_pairs' xs)›]
 non_sym_dist_pairs'_elems_distinct(2)[OF ‹distinct xs› ‹(y, z) ∈ set (non_sym_dist_pairs' xs)›]
 ‹x ≠ z›
 non_sym_dist_pairs_elems[of x xs z]
 unfolding non_sym_dist_pairs.simps ‹remdups xs = xs›
 by blast
qed

lemma non_sym_dist_pairs_trans: "trans (set (non_sym_dist_pairs xs))"
 using non_sym_dist_pairs'_trans[of "remdups xs", OF distinct_remdups]
 unfolding non_sym_dist_pairs.simps
 by assumption

lemma linear_order_from_list_position_trans: "trans (set (linear_order_from_list_position xs))"
proof
 fix x y z assume "(x, y) ∈ set (linear_order_from_list_position xs)"
 and "(y, z) ∈ set (linear_order_from_list_position xs)"
 then consider (a) "(x, y) ∈ set (map (λx. (x, x)) xs) ∧ (y, z) ∈ set (map (λx. (x, x)) xs)" |
 (b) "(x, y) ∈ set (map (λx. (x, x)) xs) ∧ (y, z) ∈ set (non_sym_dist_pairs xs)" |
 (c) "(x, y) ∈ set (non_sym_dist_pairs xs) ∧ (y, z) ∈ set (map (λx. (x, x)) xs)" |
 (d) "(x, y) ∈ set (non_sym_dist_pairs xs) ∧ (y, z) ∈ set (non_sym_dist_pairs xs)"
 unfolding linear_order_from_list_position_set by blast+
 then show "(x, z) ∈ set (linear_order_from_list_position xs)"
 proof cases
 case a
 then show ?thesis unfolding linear_order_from_list_position_set by auto
 next
 case b
 then show ?thesis unfolding linear_order_from_list_position_set by auto
 next
 case c
 then show ?thesis unfolding linear_order_from_list_position_set by auto
 next
 case d
 then show ?thesis unfolding linear_order_from_list_position_set
 using non_sym_dist_pairs_trans
 by (metis UnI2 transE)
 qed
qed

subsection ‹Find And Remove in a Single Pass›

fun find_remove' :: "('a ==> bool) ==> 'a list ==> 'a list ==> ('a × 'a list) option" where
 "find_remove' P [] _ = None" |
 "find_remove' P (x#xs) prev = (if P x
 then Some (x,prev@xs)
 else find_remove' P xs (prev@[x]))"

fun find_remove :: "('a ==> bool) ==> 'a list ==> ('a × 'a list) option" where
 "find_remove P xs = find_remove' P xs []"

lemma find_remove'_set :
 assumes "find_remove' P xs prev = Some (x,xs')"
shows "P x"
and "x ∈ set xs"
and "xs' = prev@(remove1 x xs)"
proof -
 have "P x ∧ x ∈ set xs ∧ xs' = prev@(remove1 x xs)"
 using assms proof (induction xs arbitrary: prev xs')
 case Nil
 then show ?case by auto
 next
 case (Cons x xs)
 show ?case proof (cases "P x")
 case True
 then show ?thesis using Cons by auto
 next
 case False
 then show ?thesis using Cons by fastforce
 qed
 qed
 then show "P x"
 and "x ∈ set xs"
 and "xs' = prev@(remove1 x xs)"
 by blast+
qed

lemma find_remove'_set_rev :
 assumes "x ∈ set xs"
 and "P x"
shows "find_remove' P xs prev ≠ None"
using assms(1) proof(induction xs arbitrary: prev)
 case Nil
 then show ?case by auto
next
 case (Cons x' xs)
 show ?case proof (cases "P x")
 case True
 then show ?thesis using Cons by auto
 next
 case False
 then show ?thesis using Cons
 using assms(2) by auto
 qed
qed

lemma find_remove_None_iff :
 "find_remove P xs = None ⟷ ¬ (∃x . x ∈ set xs ∧ P x)"
 unfolding find_remove.simps
 using find_remove'_set(1,2)
 find_remove'_set_rev
 by (metis old.prod.exhaust option.exhaust)

lemma find_remove_set :
 assumes "find_remove P xs = Some (x,xs')"
shows "P x"
and "x ∈ set xs"
and "xs' = (remove1 x xs)"
 using assms find_remove'_set[of P xs "[]" x xs'] by auto

fun find_remove_2' :: "('a==>'b==>bool) ==> 'a list ==> 'b list ==> 'a list ==> ('a × 'b × 'a list) option"
 where
 "find_remove_2' P [] _ _ = None" |
 "find_remove_2' P (x#xs) ys prev = (case find (λy . P x y) ys of
 Some y ==> Some (x,y,prev@xs) |
 None ==> find_remove_2' P xs ys (prev@[x]))"

fun find_remove_2 :: "('a ==> 'b ==> bool) ==> 'a list ==> 'b list ==> ('a × 'b × 'a list) option" where
 "find_remove_2 P xs ys = find_remove_2' P xs ys []"

lemma find_remove_2'_set :
 assumes "find_remove_2' P xs ys prev = Some (x,y,xs')"
shows "P x y"
and "x ∈ set xs"
and "y ∈ set ys"
and "distinct (prev@xs) ==> set xs' = (set prev ∪ set xs) - {x}"
and "distinct (prev@xs) ==> distinct xs'"
and "xs' = prev@(remove1 x xs)"
and "find (P x) ys = Some y"
proof -
 have "P x y
 ∧ x ∈ set xs
 ∧ y ∈ set ys
 ∧ (distinct (prev@xs) ⟶ set xs' = (set prev ∪ set xs) - {x})
 ∧ (distinct (prev@xs) ⟶ distinct xs')
 ∧ (xs' = prev@(remove1 x xs))
 ∧ find (P x) ys = Some y"
 using assms
 proof (induction xs arbitrary: prev xs' x y)
 case Nil
 then show ?case by auto
 next
 case (Cons x' xs)
 then show ?case proof (cases "find (λy . P x' y) ys")
 case None
 then have "find_remove_2' P (x' # xs) ys prev = find_remove_2' P xs ys (prev@[x'])"
 using Cons.prems(1) by auto
 hence *: "find_remove_2' P xs ys (prev@[x']) = Some (x, y, xs')"
 using Cons.prems(1) by simp

 have "x' ≠ x"
 by (metis "*" Cons.IH None find_from)
 moreover have "distinct (prev @ x' # xs) ⟶ distinct ((x' # prev) @ xs)"
 by auto
 ultimately show ?thesis using Cons.IH[OF *]
 by auto
 next
 case (Some y')
 then have "find_remove_2' P (x' # xs) ys prev = Some (x',y',prev@xs)"
 by auto
 then show ?thesis using Some
 using Cons.prems(1) find_condition find_set by fastforce
 qed
 qed
 then show "P x y"
 and "x ∈ set xs"
 and "y ∈ set ys"
 and "distinct (prev @ xs) ==> set xs' = (set prev ∪ set xs) - {x}"
 and "distinct (prev@xs) ==> distinct xs'"
 and "xs' = prev@(remove1 x xs)"
 and "find (P x) ys = Some y"
 by blast+
qed

lemma find_remove_2'_strengthening :
 assumes "find_remove_2' P xs ys prev = Some (x,y,xs')"
 and "P' x y"
 and "∧ x' y' . P' x' y' ==> P x' y'"
shows "find_remove_2' P' xs ys prev = Some (x,y,xs')"
 using assms proof (induction xs arbitrary: prev)
 case Nil
 then show ?case by auto
next
 case (Cons x' xs)
 then show ?case proof (cases "find (λy . P x' y) ys")
 case None
 then show ?thesis using Cons
 by (metis (mono_tags, lifting) find_None_iff find_remove_2'.simps(2) option.simps(4))
 next
 case (Some a)
 then have "x' = x" and "a = y"
 using Cons.prems(1) unfolding find_remove_2'.simps by auto
 then have "find (λy . P x y) ys = Some y"
 using find_remove_2'_set[OF Cons.prems(1)] by auto
 then have "find (λy . P' x y) ys = Some y"
 using Cons.prems(3) proof (induction ys)
 case Nil
 then show ?case by auto
 next
 case (Cons y' ys)
 then show ?case
 by (metis assms(2) find.simps(2) option.inject)
 qed

 then show ?thesis
 using find_remove_2'_set(6)[OF Cons.prems(1)]
 unfolding ‹x' = x› find_remove_2'.simps by auto
 qed
qed

lemma find_remove_2_strengthening :
 assumes "find_remove_2 P xs ys = Some (x,y,xs')"
 and "P' x y"
 and "∧ x' y' . P' x' y' ==> P x' y'"
shows "find_remove_2 P' xs ys = Some (x,y,xs')"
 using assms find_remove_2'_strengthening
 by (metis find_remove_2.simps)

lemma find_remove_2'_prev_independence :
 assumes "find_remove_2' P xs ys prev = Some (x,y,xs')"
 shows "∃ xs'' . find_remove_2' P xs ys prev' = Some (x,y,xs'')"
 using assms proof (induction xs arbitrary: prev prev' xs')
 case Nil
 then show ?case by auto
next
 case (Cons x' xs)
 show ?case proof (cases "find (λy . P x' y) ys")
 case None
 then show ?thesis
 using Cons.IH Cons.prems by auto

 next
 case (Some a)
 then show ?thesis using Cons.prems unfolding find_remove_2'.simps
 by simp
 qed
qed

lemma find_remove_2'_filter :
 assumes "find_remove_2' P (filter P' xs) ys prev = Some (x,y,xs')"
 and "∧ x y . ¬ P' x ==> ¬ P x y"
shows "∃ xs'' . find_remove_2' P xs ys prev = Some (x,y,xs'')"
 using assms(1) proof (induction xs arbitrary: prev prev xs')
 case Nil
 then show ?case by auto
next
 case (Cons x' xs)
 then show ?case proof (cases "P' x'")
 case True
 then have *:"find_remove_2' P (filter P' (x' # xs)) ys prev
 = find_remove_2' P (x' # filter P' xs) ys prev"
 by auto

 show ?thesis proof (cases "find (λy . P x' y) ys")
 case None
 then show ?thesis
 by (metis Cons.IH Cons.prems find_remove_2'.simps(2) option.simps(4) *)
 next
 case (Some a)
 then have "x' = x" and "a = y"
 using Cons.prems
 unfolding * find_remove_2'.simps by auto

 show ?thesis
 using Some
 unfolding ‹x' = x› ‹a = y› find_remove_2'.simps
 by simp
 qed
 next
 case False
 then have "find_remove_2' P (filter P' xs) ys prev = Some (x,y,xs')"
 using Cons.prems by auto

 from False assms(2) have "find (λy . P x' y) ys = None"
 by (simp add: find_None_iff)
 then have "find_remove_2' P (x'#xs) ys prev = find_remove_2' P xs ys (prev@[x'])"
 by auto

 show ?thesis
 using Cons.IH[OF ‹find_remove_2' P (filter P' xs) ys prev = Some (x,y,xs')›]
 unfolding ‹find_remove_2' P (x'#xs) ys prev = find_remove_2' P xs ys (prev@[x'])›
 using find_remove_2'_prev_independence by metis
 qed
qed

lemma find_remove_2_filter :
 assumes "find_remove_2 P (filter P' xs) ys = Some (x,y,xs')"
 and "∧ x y . ¬ P' x ==> ¬ P x y"
shows "∃ xs'' . find_remove_2 P xs ys = Some (x,y,xs'')"
 using assms by (simp add: find_remove_2'_filter)

lemma find_remove_2'_index :
 assumes "find_remove_2' P xs ys prev = Some (x,y,xs')"
 obtains i i' where "i < length xs"
 "xs ! i = x"
 "∧ j . j find (λy . P (xs ! j) y) ys = None"
 "i' < length ys"
 "ys ! i' = y"
 "∧ j . j < i' ==> ¬ P (xs ! i) (ys ! j)"
proof -
 have "∃ i i' . i < length xs
 ∧ xs ! i = x
 ∧ (∀ j < i . find (λy . P (xs ! j) y) ys = None)
 ∧ i' < length ys ∧ ys ! i' = y
 ∧ (∀ j < i' . ¬ P (xs ! i) (ys ! j))"
 using assms
 proof (induction xs arbitrary: prev xs' x y)
 case Nil
 then show ?case by auto
 next
 case (Cons x' xs)
 then show ?case proof (cases "find (λy . P x' y) ys")
 case None
 then have "find_remove_2' P (x' # xs) ys prev = find_remove_2' P xs ys (prev@[x'])"
 using Cons.prems(1) by auto
 hence *: "find_remove_2' P xs ys (prev@[x']) = Some (x, y, xs')"
 using Cons.prems(1) by simp

 have "x' ≠ x"
 using find_remove_2'_set(1,3)[OF *] None unfolding find_None_iff
 by blast

 obtain i i' where "i < length xs" and "xs ! i = x"
 and "(∀ j < i . find (λy . P (xs ! j) y) ys = None)" and "i' < length ys"
 and "ys ! i' = y" and "(∀ j < i' . ¬ P (xs ! i) (ys ! j))"
 using Cons.IH[OF *] by blast

 have "Suc i < length (x'#xs)"
 using ‹i < length xs› by auto
 moreover have "(x'#xs) ! Suc i = x"
 using ‹xs ! i = x› by auto
 moreover have "(∀ j < Suc i . find (λy . P ((x'#xs) ! j) y) ys = None)"
 proof -
 have "∧ j . j > 0 ==> j < Suc i ==> find (λy . P ((x'#xs) ! j) y) ys = None"
 using ‹(∀ j < i . find (λy . P (xs ! j) y) ys = None)› by auto
 then show ?thesis using None
 by (metis neq0_conv nth_Cons_0)
 qed
 moreover have "(∀ j < i' . ¬ P ((x'#xs) ! Suc i) (ys ! j))"
 using ‹(∀ j < i' . ¬ P (xs ! i) (ys ! j))›
 by simp

 ultimately show ?thesis
 using that ‹i' < length ys› ‹ys ! i' = y› by blast
 next
 case (Some y')
 then have "x' = x" and "y' = y"
 using Cons.prems by force+

 have "0 < length (x'#xs) ∧ (x'#xs) ! 0 = x'
 ∧ (∀ j < 0 . find (λy . P ((x'#xs) ! j) y) ys = None)"
 by auto
 moreover obtain i' where "i' < length ys" and "ys ! i' = y'"
 and "(∀ j < i' . ¬ P ((x'#xs) ! 0) (ys ! j))"
 using find_sort_index[OF Some] by auto
 ultimately show ?thesis
 unfolding ‹x' = x› ‹y' = y› by blast
 qed
 qed
 then show ?thesis using that by blast
qed

lemma find_remove_2_index :
 assumes "find_remove_2 P xs ys = Some (x,y,xs')"
 obtains i i' where "i < length xs"
 "xs ! i = x"
 "∧ j . j find (λy . P (xs ! j) y) ys = None"
 "i' < length ys"
 "ys ! i' = y"
 "∧ j . j < i' ==> ¬ P (xs ! i) (ys ! j)"
 using assms find_remove_2'_index[of P xs ys "[]" x y xs'] by auto

lemma find_remove_2'_set_rev :
 assumes "x ∈ set xs"
 and "y ∈ set ys"
 and "P x y"
shows "find_remove_2' P xs ys prev ≠ None"
using assms(1) proof(induction xs arbitrary: prev)
 case Nil
 then show ?case by auto
next
 case (Cons x' xs)
 then show ?case proof (cases "find (λy . P x' y) ys")
 case None
 then have "x ≠ x'"
 using assms(2,3) by (metis find_None_iff)
 then have "x ∈ set xs"
 using Cons.prems by auto
 then show ?thesis
 using Cons.IH unfolding find_remove_2'.simps None by auto
 next
 case (Some a)
 then show ?thesis by auto
 qed
qed

lemma find_remove_2'_diff_prev_None :
 "(find_remove_2' P xs ys prev = None ==> find_remove_2' P xs ys prev' = None)"
proof (induction xs arbitrary: prev prev')
 case Nil
 then show ?case by auto
next
 case (Cons x xs)
 show ?case proof (cases "find (λy . P x y) ys")
 case None
 then have "find_remove_2' P (x#xs) ys prev = find_remove_2' P xs ys (prev@[x])"
 and "find_remove_2' P (x#xs) ys prev' = find_remove_2' P xs ys (prev'@[x])"
 by auto
 then show ?thesis using Cons by auto
 next
 case (Some a)
 then show ?thesis using Cons by auto
 qed
qed

lemma find_remove_2'_diff_prev_Some :
 "(find_remove_2' P xs ys prev = Some (x,y,xs')
 ==> ∃ xs'' . find_remove_2' P xs ys prev' = Some (x,y,xs''))"
proof (induction xs arbitrary: prev prev')
 case Nil
 then show ?case by auto
next
 case (Cons x xs)
 show ?case proof (cases "find (λy . P x y) ys")
 case None
 then have "find_remove_2' P (x#xs) ys prev = find_remove_2' P xs ys (prev@[x])"
 and "find_remove_2' P (x#xs) ys prev' = find_remove_2' P xs ys (prev'@[x])"
 by auto
 then show ?thesis using Cons by auto
 next
 case (Some a)
 then show ?thesis using Cons by auto
 qed
qed

lemma find_remove_2_None_iff :
 "find_remove_2 P xs ys = None ⟷ ¬ (∃x y . x ∈ set xs ∧ y ∈ set ys ∧ P x y)"
 unfolding find_remove_2.simps
 using find_remove_2'_set(1-3) find_remove_2'_set_rev
 by (metis old.prod.exhaust option.exhaust)

lemma find_remove_2_set :
 assumes "find_remove_2 P xs ys = Some (x,y,xs')"
shows "P x y"
and "x ∈ set xs"
and "y ∈ set ys"
and "distinct xs ==> set xs' = (set xs) - {x}"
and "distinct xs ==> distinct xs'"
and "xs' = (remove1 x xs)"
 using assms find_remove_2'_set[of P xs ys "[]" x y xs']
 unfolding find_remove_2.simps by auto

lemma find_remove_2_removeAll :
 assumes "find_remove_2 P xs ys = Some (x,y,xs')"
 and "distinct xs"
shows "xs' = removeAll x xs"
 using find_remove_2_set(6)[OF assms(1)]
 by (simp add: assms(2) distinct_remove1_removeAll)

lemma find_remove_2_length :
 assumes "find_remove_2 P xs ys = Some (x,y,xs')"
 shows "length xs' = length xs - 1"
 using find_remove_2_set(2,6)[OF assms]
 by (simp add: length_remove1)

fun separate_by :: "('a ==> bool) ==> 'a list ==> ('a list × 'a list)" where
 "separate_by P xs = (filter P xs, filter (λ x . ¬ P x) xs)"

lemmaext
 "separate_by P xs = foldr (λeil([],])"
proof
 case
 then show ?case x,) <set
next cme
 casexs

 letby ura yuto

 haveforce
 using Cons.IH by auto
 moreover have "separate_by P (a#xs) = ?f a (filter P xs, filter (λ
 moreoverv"=and (bs @ [b]) ! i} = {?m}"
 ultimately show ?case
 by esP a" )
qed

fun find_remove_2_all
 "find_remove_2_all P xs ys =
 (map (λ
 ,filter (λ

fun find_remove_2_all' :: "(>'b ==> 'b list (('a ×

 by
 in (map (<dots

emma
 by
proof -
 let ?s1 = "map ed
 let ?f1 = "filter (λd (<>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 let ?
 let ?f2 = java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null

 have "find_remove_2_all P xs ys = (?s1 snd
 imp
 moreover have find_remove_2_all
 proof snd (cs ! 0) < ne
 have= "s snd (cs ! i) ≠ set (snd ((xys # xyss) ! i)))))
 by force
 then show ?thesis
 unfolding find_remove_2_all'.simps Let_def separ
 k<length xs) ∧ (f (xs ! k)) ∧ (∀ . ¬ (f (xs ! j)))"
 moreoverhave then"casefid_idx f s o Some \ightarrow (Suc k) | None ==>
 moreover have "?s1 = ?s2"
 by
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by (induction xs; auto)
 ultimately show ?thesis
 by simp
qed


subsection ‹

fun pow_list :: " a list \>'a list list" where
"pow_list [] = [[]]" |
"pow_list (x#xs) = (let pxs = pow_list xs in pxs @ map (λ

"sps(ow__lit xs)= Powo(stx)"
java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
Nil
then show ?case by aue" |

case (Cons x xs)

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 29

moreover ex
\ (image (insert x) (set (map set (pow_list xs))))"
proof -o
have "∧
==>
proof -
fix ys assume "ys ∈
then consider (a) "ys ∈ set (map set (pow_list xs))" |
ys<>
unfolding pow_list.simps Let_def by auto
then show "ys ∈ set (map set (pow_list xs)) ∪
by (cases; auto)

moreover have "∧
∪(fixs)
==>le ? "ae(leg sz"

fix ys assume "ys ∈ase
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
thenc
(image (insert x) (set (map set (pow_list xs))))"

then show "ys ∈ set (map set (pow_list (x#xs)))"
unfolding pow_list.simps Let_def by (cases; auto)
qed
ltimatelysow?thesis byblast

ultimately show ?case
(ca es"(@ [x]]"; autto)

Removing Subsets in a List of Sets›

lemma remove1_length : "x ∈ set xs ==>by auto
 by (induction xs; auto)

function remove_subsets :: "'a set list ==>
 "remove_subsets [] = []" |
 java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 Somet ?hsis
 None ==>
 by pat_completeness auto
termination
java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 7
 have "java.lang.NullPointerException
 by (metis list.set (list_ordered_pairsxs (y,x) ∈
 moreover have "(∧
 f -
 fix x :: "'a set"
 fix xs y'xs' y' xs'
 assume "find_removeprefixes_Cons
 then have "ed
 by auto

 have "length xs' = length xs - 1"
 using find_remove_set(23[O\
 by (simp add: length_remove1)
 then have "length (y'#xs') = length xs"
 using find_remove_set(2)[OF ‹find_remove ((⊂) x) xs = Some (y',xs')›]
 using remove1_length by fastforce

 have "length (filter (<lambdacase

 then have "length'# filer (\lambday ¬ y ⊆
 imp
 then have "length (y' # filter (λx = z› ‹
 unfolding \<opencase
 then show "(y' # filter (\<lambda>y. \<not ultimatelyshowthesisis
 to
 qed
 ultimately show ?thesis
 by
qed

lemma remove_subsets_set : "set (remove_subsets xss) = {xs . xs \<in> set xss \<and> (\<nexists> xs' . xs' \<in> set xss \<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
proof then "y \in> set (map (\<lambda>x. (x, x)) xs) \<union> set (non_sym_dist_pairs xs)"
 case less

 show ?case proof (cases xss)

 "inct<Longrightarrow> (xy\>ton_sym_dist_pairss)
 then show ?thesis by auto
 next
 case (Cons x xss')

 show ?thesis proof (cases "find_remove (\<lambda> y . x \<subset> y) xss'")
 case None
 then have "sing .Hssjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
 using find_remove_None_iff by metis

 have "length (filter (\<lambda>y.not(y \<subseteq> x)) xss') < length xss"
 using Cons
 by (meson dual_order.trans impossible_Cons leI length_filter_le)

 have "remove_subsets (x#xss') = x # (remove_subsets (filter (\<lambda> y . \<not>(y \<subseteq> x)) xss'))"
 using None y uto
 then have "set (remove_subsets (x#xss')) = insert x {xs \<in> set (filter (\<lambda>y. \<not>proof
 using less[OF \<penengther(lambda> y . \<not>(\subseteq )) xss') < length xss\<close>]
 by auto
 have \dots>= xs. xs<innth_eq_iff_index_eqjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
 proof -
 have "\<And> xs . xs \<in> insert x {xs \<in> set (filter (\<lambda>y. \<not> y \<subseteq> x) xss'<nexistsxs'. xs' \ set (filter (\<lambda>not y \<subseteq> x) xss') \<and> xs \<subset> xs'}
 \<Longrightarrow> xs \<in> {xs \yauto
 proof -
\in set (filter (\<lambda>y. \<not> y \<subseteq> x) xss'). \<nexists>xs'. xs' \<in> set (filter (\<lambda>y. \<not> <subseteq>ss<nd>xs \<subset> xs'}"
 then consider "xs = x" | "xs \<in> set (filter (\<lambda>y. \<proof-
 next
 then show "xs \<in> {xs \<in> set (x # xss'). \<nexists>xs'. qed
 )proofctionarbitraryrev
 qed
 moreover have "\<And> xs . xs \<in> {xs \<in> set (x # xss'). \<nexists>xs'. xs' \< then thesisons
 \<Longrightarrow> xs \<in> insert x {xs \<in>nd "x\in set xs"
 proof -
 fix xs assume "xs \<in> {xs \<in> set (x # xss'). \<nexists>xs'. xs' \<in> set (x # xss') \<and> xs \<subset> xs'}"
 then have "xs \<in> set (x # xss')" and "\<nexists>xs'. xs' \<in> set (x # xss') \<and> xs \<subset> xs'"
 by blast+
 then consider "xs = x" | "xs \<in> set xss'" by auto
 then show "xs \<in> insert x {xs \<in> set (filter (\<lambda>y. \<not> y \<subseteq> x) xss'). \<nexists>xs'. xs' \ using Consprems) java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
 proof cases
 case 1
 then show ?thesis by auto
 next
 case 2
 show ?thesis proof (cases "xs \<subseteq> x")

 then show ?thesis
 using \<open>\<nexists>xs'. xs' \<in> set (x # xss') \<and> xs \<subset> xs'\<close> by auto

 case False
 then have "xs \<in> set (filter (\<lambda>y. \<not> y \<subseteq
 using 2 by auto
 moreover have "\<nexists>xs'. xs' \<in> set (filter (\<lambda>y. \<usingCons java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
 ,\ set(non_sym_dist_pairs xs"
 ultimately show ?thesis by auto
 qed
 qed

 ultimately show ?thesis
 by (meson subset_antisym then have "anda
 qed
 allyyshow ?thesisis nfoldingoldingldingingg Consssymmetric byassumption
 next
 case (Some a)
 then obtain y' xs' where *: "find_remove (\<lambda> y . x \<subset> y) xss' = Some (y',xs')" by force


 have "length xs' = length xss' - 1"
 usingfind_remove_set2,3)[F *
 by (simp add: length_remove1)
 then have "length
 using find_remove_set(2)[OF *]
usingve1_length e

 length(ter(lambda>y. \<ot> y \<subseteq> x) xs') \<le> length xs'"
 by simp
 then have "lengthgthh'filter<>y. \<not \subseteq x) xs') \<le> length xs' + 1"
 by simp
) ngth'
 unfolding \openlength (y'#xs') = length xss'\<close>[symmetric] by simp
 then have "length (y' # filter (\< moreover have "\forall < i' . \<not> P ((x'#xs) ! Suc i) (ys ! j))"
 unfolding Cons by auto

 have "remove_subsets (x#xss') = remove_subsets (y'# (filter (\<lambda> y . \<not>(y \<subseteq> x)) xs' auto
 using * by auto
 then have "set (remove_subsets (x#xss')) = {xs \<in> set "xs java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
 using less[OF \<open>length (y' # filter (\<lambda>y. \<not> y \<subseteq find_remove_2_:
 by auto
 also have "\<dots> = {xs . xs \<in> set (x#xss') \<and> (\<nexists> xs' . xs' \<in> set (x#xss') thenx> x'"
 proof -
 have "\<And> xs to
 xs \<in> {xs \<in> set (x #'\xs'. xs' \<in> set (x # sss <and> xs \<subset> xs'}"
 proof -
 fix xs assume "xs \<in> {xs \<in> set (y' # filter (\<lambda>y. \<not> y \<subseteq
 then have "xs \<in> set (y' # filter (\<lambda>y. \<not> y \<subseteq(.haustnustjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
 by blast+

 have java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 using \<open>xs \<in> set (y' # filter (\<lambda>y. \<not> y \<subseteq> x) xs')\<close> find_remove_set(2using_2[s
 by auto
 moreover have "\<nexists>xs'. s<> set >xs \<subset> xs'"
 using \<open>\<nexists>xs'a. xs'a \<in> set (y' # filter (\<lambda>y. \<not> y \<subseteq> x) xs') \<andjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by (metis dual_order.strict_trans filter_list_set in_set_remove1 list.set_intros(1) list.set_intros(2) psubsetI set_ConsD)
 ultimately show "xs \in {s \<in> set (x # xss'). \<nexists>xs'. xs' \<in> set (x # xss') \<and> xs \<subset> xs'}"
 by blast
 qed
 moreover have "\<And> xs . xs \<in> {xs \<in
 xs \<in> {xs \<in> set 'ter<>y <ot y \<subseteq> x) xs'). \<nexists>xs'a. xs'a \<in> set (y' # filter (\<lambda>y. \<not> y \<subseteq> x) xs') \<and> xs \<subset> xs'a}"
 proof -
 fix xs assume "xs \byblast
 then have "xs \<in> set (x # xss')" and "\<nexists>xs'. xs' \<in> set (x # xss') a
 by blast+

 then have "xs \<in> set (y' lter<>. \<not> y \<subseteq)xs
 using find_remove_set[OF *]
 by (metis filter_list_set in_set_remove1 list.
 overrave<exists>xs'a. xs'a \<in> set (y' # filter (\<lambda>y wisjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
 using \<open>xs \<in> set (x # xss')\<close> \<open>\<nexists>xs'. xs' \<in> set (x # xss') \<and> xs \<subset> xs'\<close> find_remove_set[OF *]
 by (metis filter_is_subset list.set_intros(2) notin_set_remove1 set_ConsD subset_iff)
 ultimately show "xs \<in> {xs \<in> set (y' #casesxss
 by blast
 qed
 ultimately show ?thesis by blast
 qed
 finally show ?thesis unfolding Cons by assumption
 qed
 qed
qed

subsection \<open>ar Order onnSumumm<>

instantiation sum :: (ord,ord) ord
begin

'Rightarrow 'a + 'b \<Rightarrow> bool" ere
 "less_eq_sum (Inl a) (Inl b) = (a \<le> b)" |
 "less_eq_sum (Inl a) (Inr bcaseue
 "less_eq_sum(a# !!Suc)= y and" j length (axs)"
 "less_eq_sum (Inr a) (Inr b) = (a \<le> b)"

fun less_sum :: "'a + 'b \<Rightarrow> java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 "less_sum a b = (a \<le> b \<and> a \<noteq> b)"

instanceo_classessses
end

nstantiationm : ((linorderinorder,linorder)linorder
begin

lemma less_le_not_le_sum
 fixes x :: "'a + 'b"
 and y :: "'a + 'b"
shows "(x < y) = (x \<le> y \<and> \<not> yusing<>s \<in> set (y' # filter (\<lambda>y. \<not> y \<subseteq> x) xs')\<close> find_remove_set(2java.lang.StringIndexOutOfBoundsException: Index 127 out of bounds for length 127
 by (cases x; cases y; auto)

lemma order_refl_sum :
 fixes x :: "'a + 'b"
 shows "x \<le> x"
 by (cases x; auto)

lemma order_trans_sum :
 fixes x :: "'a + 'b"
 fixes y :: "'a + 'b"
 fixes z :: "'a + 'b"
 shows "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
 by (cases x; cases y; cases z; auto)

lemma antisym_sum :
 fixes x :: "'a + 'b"
 fixes y :: "'a + 'b"
 <e>y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
 by (cases x; cases y; auto)

lemma linear_sum :
 fixes x :: "'a + 'b"
 fixes y :: "'a + 'b"
 shows "x \<le> y byro_classeslassestis)
 by (sesxases;auto

instance
 using less_le_not_le_sum order_refl_sum order_trans_sum antisym_sum linear_sum
 by (intro_classes; metis+)
end

subsection definition as_list_helper"et<> 'a "java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65

definition remove_proper_prefixes :: "'a list set \<Rightarrow> 'a list set" where
 "remove_proper_prefixes xs = {x . x \<in> xs \<and> (\<nexists> x' . x' \<noteq> [] \<and> x@x' \<in> xs)}"

lemma remove_proper_prefixes_code[code] :
 "remove_proper_prefixes (set xs) = set (filter (\<lambda>x . (\<forall> y \<in> set xs . is_prefix x y \by
proof-

 have *: "remove_proper_prefixes (set xs) = Set.filter (\<lambda> zs . \<nexists>ys . ys \<noteq> [] \< "y"
 unfolding remove_proper_prefixes_def by force

 have "\<And> zs . (\<nexists>ys . ys \<noteq> [] \<and> zs @ ys \<in> (set xs)) = (\<forall> ys \<in> set xs . is_prefix zs ys \<longrightarrow> zs = ysbyjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
 unfolding is_prefix_prefix by auto

 then show ?thesis
 unfolding * filter_set by auto
qed

subsection \<open>Underspecified List Representations of Sets\<close>

definition as_list_helper :: "'a set \<Rightarrow> 'a list" where
 "as_list_helper X = (SOME xs . set xs = X \<and> distinct xs)"

lemma as_list_helper_props :
 assumes "finiteX"
 shows "set (as_list_helper X) = X"
 " ( X)"
 using finite_distinct_list[OF assms]
 using someI[of "\<lambda> xs . set xs = X \<and> distinct xs"]
 metis as_list_helper_defand snoteq>"

subsection \<open>Assigning indices to elements of a finite set\<close>

fun assign_indices :: "using insertIHyast
 "assign_indices xs = (\<lambda> x . the (find_index ((=)x) (sorted_list_of_set xs)))"

lemma assign_indices_bij:
 assumes "finite xs"
 shows "bij_betw (assign_indices xs) xs {..<card xs}"
proof -

 have *:"set (sorted_list_of_set xs) = xs"
 by (simp add: assms)


 have "<Andxy case(Sucjjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
 proof -
 fix x y assume "x\<in>xs" and "y\<in>xs" and "assign_indices xs x = assign_indices xs y"

 obtain i where "find_index ((=)x) (sorted_list_of_set xs) = Some i"
 using find_index_exhaustive[ofassumes<> et"
 using \<open>x\< by (metisSuc_leIssms drop_conv_nth__nv_nththlast_snocnoctake_hd_dropero_less_Sucthenhowthesisisfoldinglinear_order_from_list_position_set
 by blast
 then have "assign_indices xs x = i"
java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 13

 obtain j where "find_index ((=)y) (sorted_list_of_set xs) = Some j"
 using find_index_exhaustive[of "sorted_list_of_set xs" "((=) y)"
 using \<open>y\<in>xs\<close> unfolding *
 by blast
 then have "assign_indices xs y = j"
 by auto
 then have "i = j"
 using \<open>assign_indices xs x = assign_indices xs y\<close> \<open>assign_indices xs x = i\<close>
 by auto
 then have "find_index ((=)y) (sorted_list_of_set xs) = Some i"
 using \<open>find_index ((=)y) (sorted_list_of_set xs) = Some jlemma list_prefix_subset<xists ys . ts = xs@ys \<Longrightarrow> set xs \<subseteq> set ts" by auto
 by auto

 show "x = y"
 using find_index_index(2)[OF \<open>find_index ((=)x) (sorted_list_of_set xs) = Some i\<close>]
 singng(F<openfind_index()sorted_list_of_set =Some i\close]
 by auto
 qed
 moreover have "(assign_indices xs=< xs}"
roof
 show "assign_indices xs ` xs \<subseteq> {..<card xs}"
 proof
 fix i assume "i \
 then obtain x where "x \<in> xs" and "i = assign_indices xs x"
 by blast
 moreover obtain j where "find_index ((=)x) (sorted_list_of_set xs) = Some j"
 using find_index_exhaustive[of "sorted_list_of_set xs" case java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 \>\<in>xs\<close> unfolding *
 by blast
 ultimately have "find_index ((=)x) (sorted_list_of_set xs) = Some i"
 by auto
 >{..<card xs}"
 using find_index_index_ndexx)OFopenfind_index ((=)x) (sorted_list_of_set xs) = Some i\<close>]
 by auto
 qed
 show "{..<card xs} \<subseteq> assign_indices xs ` xs"
 proof
 fix i assume "i \<in> {..<card xs}fromprems\foralli<length ss - 1. ss ! i \<subset> ss ! Suc i"
 then have "i < ngthsorted_list_of_set
 by auto
 then have "sorted_list_of_set xs ! ithenshow? java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
 using "*" nth_mem by blast
 then obtain j where "find_index ((=) (sorted_list_of_set xs ! i)) (sorted_list_of_set xs) = Some j"

 unfolding *
 by blast
 have "i = j"
 using find_index_index(1,2)[OF \<open>find_index ((=) (sorted_list_of_set xs ! i)) (sorted_list_of_set xs) = Some j<>]
 using \<open>i < length (sorted_list_of_set xs)\<close> distinct_sorted_list_of_set nth_eq_iff_index_eq st
 then show "i \<in> assign_indices xs
 using \<open>find_index ((=) (sorted_list_of_set xs ! i)) (sorted_list_of_set xs) = Some j\<close>
 by (metis \<open> " have "\not Other Lemmata\<close>

lemma foldr_elem_check:
 assumes "list.set xs \<subseteq> A"
 shows "foldr (\<lambda> x y . if x \<notin> A then y else f x y) xs v = foldr f xs v"
 using assms by (induction xs; auto)

lemma foldl_elem_check:
 assumes "list.set xs \<subseteq> A"
 shows "foldl (\<lambda> y x . if x \<notin> A then y else f y x) v xs = foldl f v xs"
 using assms by (induction xs rule: rev_induct; auto)

lemmath_helper
 assumes "length xs = length ys"
 shows "foldr (\<lambda>_ x . f x) xs b = foldr (\<lambda>a x . f x) ys b"
 usingbyuto

lemma list_append_subset3 : lemma singngnsert

lemma subset_filter : "set xs \<subseteq> set ys \<Longrightarrow> set xs = set take_le
 by auto

lemma map_filter_elem :
 assumes "y \<in> set (List.map_filter f xs)"
 btains where <in> set "
 and "f x = Some y"
 using assms unfolding List.map_filter_def
byto

lemma filter_length_weakening :
 assumes "\<And> q . f1 q \<Longrightarrow> f2 q"
 shows "length (filter f1 p) \<le> length (filter f2 p)"
proof (induction p)
 case Nil
 nhow?aseby uto
next
 case (Cons a p)
 then show ?case using assms by (cases "f1 a"; auto)
qed

lemma max_length_elem :
 fixesultimatelyshow ?hesisusing IHOF*java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
mes "finitexs"
 and "xs \<noteq> {}"
shows "\<exists> x \<in> xs . \<not>(\<exists> y \<in> xs . length y > length x)"
using assms proof (induction xs)
 case empty
 then show ?case by auto
next
( xF
 then showcaseproof cases"F= }"
 case True
 then show ?thesis by blast
 next
 False
 then obtain y where "y \<in> F" and "\<not> and "find (P x)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 using insert.IH by blast
 then show ?thesis using dual_order.strict_trans by (cases "length x > length y"; auto)
 qed
qed

lemma length_elem
 fixes xs :: "'a list set"

 and "xs \<noteq> {}"
shows "\<exists> x \<in> xs . \<not>(\<exists> y \<in> xs . length y < length x)"
using assms proof (induction xs)
 case empty
then show ?case by auto
next
 case (insert x F)
 then show ?case proof (cases "F = {}")
 case True
 then show ?thesis by blast
 next
 case False
 then obtain y where "y \<in> F" and "\<not>(\<exists> y' \<in> F . length y' < length y)"
 using insert.IH by blast
 then show ?thesis using dual_order.strict_trans by (cases "length x < length y"; auto)
 qed
qed

lemma list_property_from_index_property :
 assumes "\<And> i . i < length xs \<Longrightarrow> P (xs ! i)"
 shows "\<And> x . x \<in> set xs \<Longrightarrow> P x"
 by (metis assms in_set_conv_nth)

lemma list_distinct_prefix :
<> i.i length xs \<Longrightarrow> xs ! i \<notin> set (take i xs)"
 shows "distinct xs"
proof -
 have "\<And> j . distinct (take j
 proof -
 fix j
 show "distinct (take j xs)"
 proof (induction j)
 0
 then show ?case by auto
 next
 case (Suc j)
 then show ?case proof (cases "Suc j \<le> length xs")
 case True
 thenve Sucjs(kexs s
 by (simp add: Suc_le_eq take_Suc_conv_app_nth)
 then show ?thesis using Suc.IH assms[of j] True by auto
 next
 case False
 then have "take (Suc j) xs \<And> x' . x' \<ge> Suc x \<Longrightarrow> x' < ( : prev'xs'java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
 then show ?thesis using Suc.IH by auto
 qed
 qed
 java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 then? .premsunfoldingfind_remove_2
 by blast
 then show ?thesis by auto
qed

lemma concat_pair_set :
 "set (concat (map (\<lambda>x. map (Pair x) ys) xs)) = {xy . fst xy \<in> set xs \<and> snd xy _
 by auto

lemma list_set_sym :
 t(@ y@)" byyuto

lemma list_contains_last_take :
 assumes "x \<in> set xs"
 shows "\<exists> i . 0 i \<le> length xs \<and> last (take i xs) =card `=c"
 by metisc_leIsd_drop_conv_nthv_nth in_set_conv_nthonv_nththlast_snoccke_hd_drops_Suc

lemma take_last_index :
 assumes "i < length xs"
 shows "last (take (Suc i) xs) = xs ! i"
 by (simp add: assms take_Suc_conv_app_nth)

lemma integer_singleton_least :
 assumes "{x . P x} = {a::integer}"
 shows "a = (LEAST x . P x)"
 by (metis Collect_empty_eqeast_equality ert_not_emptylect_eqder_reflsingletonDetonDnD

lemma sort_list_split :
 "\<forall> x \<in> set (takeake i(sorttxs.\forall> y \<in> set (drop i (sort xs)) . x \<le> y"
 using sorted_append by fastforce

lemma set_map_subset :
 assumes "x \<in> set xs"
 and "t \<in> set moreover X" and thenhave find_remove_2 (filterP')prev=me(y'
shows "t \<in> set (map f xs)"
 using assms by auto

lemma rev_induct2[consumes 1, case_names Nil snoc]:
 '> pj
 and "P [] []"
and "(\<And>x xs y ys.find_remove_2 P(ilter 'xs Some(,,s)>
 shows "P xs ys"
arbitrary
 case Nil
 showase uto
next
 case (snoc x xs)
 then show ?case proof (cases ys)
 case Nil
 then show ?thesis
 using snoc.prems(1) by auto
 next
 case (Cons a list)
 then show ?thesis
 by (metis append_butlast_last_id diff_Suc_1 length_append_singleton list.distinct(1) snoc.hyps snoc.prems)
 qed
qed

lemma finite_set_min_param_ex :
 assumes "finite XS"
 and "\<And> x . x \<in> XS \<Longrightarrow> \<exists> k . \<forall> k' . k \<le> k' \<longrightarrow> P x k'"
shows "\<exists> (k::nat) . \<forall> x \<in> XS . P x k"
proof -
 obtain f where f_def : "\<And> x . x \<in> XS \<Longrightarrow> \<forall> k' . (f x) \<le> k' \<longrightarrow> P x k'"
 using assms(2) by meson
 let ?k = "Max (image f XS)"
 have "\<forall> x \<in> XS . P x ?k"
 using f_def by (simp add: assms(1))
 then show ?thesis by blast
qed

fun list_max :: "nat list \<Rightarrow insertnsertrems o
 "list_max [] = 0" |
 "list_max xs = Max (set xs)"

lemma list_max_is_max : "q \<in> set xs \<Longrightarrow> q \<le> list_max xs"
 byy (metisListt.nite_set_tax_gength_greater_0_convpos_if_in_setst_max)

lemma list_prefix_subset : "\<exists> ys . ts = xs@ys \<Longrightarrow> set xs \<subseteq> set ts" by auto
lemma list_map_set_prop : "x \<in> set (map f xs) \<Longrightarrow> \<forall> y . P (f y) \<Longrightarrow> P x" by auto
lemma list_concat_non_elem : "x \<notin> set xs \< proof (induction xs arbitrary:prevxs xy
lemma list_prefix_elem : "x \<in> set (xs@ys) \<Longrightarrow> x \<notin> set ys \<Longrightarrow> x \<in> set xs" by auto
lemma list_map_source_elem : "x \ then show ? by autojava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

lemma maximal_set_cover :
 fixes X :: "'a set set"
 assumes "finite X"
 and "S \<in> X"
shows "\<exists> S' \<in> X . S \<subseteq> S' \<and> (\<forall> S'' \<in> X . \<not>(S' \<subset> S''))"
proof (rule ccontr)
 assume "\<not> (\<exists>S'\<in>X. S \<subseteq> S' \<and> (\<forall>S''\<in>X. \<not> S' \<subset> S''))"
 then have add: Suc.prems(1) card_UN_disjoint
 by auto

 have "\<And> k . \<exists> ss . (length ss = Suc k) \<and> (hd ss = S) \<and> (\<forall> i < k . ss ! i \<subset> ss ! (Suc i)) \<and> (set ss \<subseteq> X)"
 proof -
 fix k show "\<exists> ss . (length ss = Suc k) \<and> (hd ss = S) \<and> (\<forall> i < k . ss ! i \<subset> ss ! (Suc i)) \<and> (set ss \<subseteq> X)"
 proof (induction k)
 case 0
 have "length [S] = Suc 0 \<and> hd [S] = S \<and> (\<forall> i < 0 . [S] ! i \<subset> [S] ! (Suc i)) \<and> (setthen showhesisng insertrt blast
 then show ?case by blast
 next
 case (Suc k)
 then obtain ssere"lengths=
 and "hd ss = S"
 and "(\<forall>i<k. ss ! i \<subset> ss ! Suc i)"
 and "set ss \<subseteq> X"
 by blast
 then have "ss ! k \<in> Xjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
 by auto
 moreover have "S \<subseteq> (ss ! k)"
 proof -
 have "\<And> i . i < Suc k \<Longrightarrow> S \<subseteq> (
 proofjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 fix i assume "i < Suc k"
 then show "S \<subseteq> (ss ! i)"
 proof (induction i)
 case 0
 then show ?case using \<open>hd ss = S\<close> \<open>length ss = Suc k\<close>
 by (metis hd_conv_nth list.size(3) nat.distinct(1) order_refl)
 next
 case (Suc i)
 then have "S \<subseteq> ss ! i" and "i < k" by auto
 then have "ss ! i \<subset> ss ! Suc i" using \<open>(\<forall>i<k. ss ! i \<subset> ss ! Suc i)\<close> by blast
 then show ?case using \<open>S \<subseteq> ss !\<> by uto
 qed
 qed
 show?esisusing<open>length ss=Suck<close> auto
 qed
 ultimately obtain T' where "T' \<in> X" and "ss ! k \<subset> T'"
 using * by meson

 let ?ss = "ss@[T']"

 havelength =Suc(Suc kjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
 using \<open>length ss = Suc k\<close> by auto
 moreover have "hd ?ss = S"
 using \<open different_lists_shared_prefix
 moreover have "(\<forall>i < Suc k. ?ss ! i \<subset> ?ss ! Suc i)"
 using \<open>(\<forall>i<k. ss ! i \<subset> ss ! Suc i)\<close> \<open>ss ! k \<subset> T'\<close>
 by (metis Suc_lessI \<open>length ss = Suc k\<close> diff_Suc_1 less_SucE nth_append nth_append_length) then show ?thesis
 moreover have "set ?ss \<subseteq> X" next
 using \<open>set ss \<subseteq> X\<close> \<open>T' \<in> X\<close> by auto
 ultimately show ?case by blast
 qed
 qed

 foldr_funion_fempty:"oldr\union>|) xs fempty = ffUnion (fset_of_list xs)"
 and "(hd ss = S)"
 and "(\<forall> i < card X . ss ! i \<subset> ss ! (Suc i))"
 and "(set ss \<subseteq> X)"
 by blast
 then have "(\<forall> i < length ss - 1 . ss ! i \<subset> ss ! (Suc i))"
 by auto

 have **: "\<And> i (ss :: 'a set list) . (\<forall> i < by (induction uto
 proof -
 fix i
 fix ss :: "'a set list"
 assume "i < length ss " and "(\<forall> i < length ss - 1 . ss ! i \<subset> ss ! (Suc i))"
 then show "\<forall> s \<in> set (take i ss) . s \<subset> ss ! i"
 proof (induction i)
 case 0
 then show ?case by auto
 next
 case moreover have "\Andy. y \in> s<Longrightarrow length y < Suc (Max (length ` xs))"
 then have "\<forall>s\<in>set (take i qed
 then have "\<forall>s\<in>set (take i ss). s \<subset> ss ! (Suc i)" using Suc.prems
 by (metis One_nat_def Suc_diff_Suc Suc_lessE diff_zero dual_order.strict_trans nat.inject zero_less_Suc)
 moreover have "ss ! i \<> ss ! (Suc i)" using Suc.prems by auto
 moreover have "(take (Suc i) ss) = (take i ss)@[ss ! i]" using Suc.prems(1 proof
 by (simp add: take_Suc_conv_app_nth)
 ultimately show ?case by auto
 qed
 qed

 have "distinct ss"
 using \<open\forall> i<lengthss - 1 . ss ! i \<subset> ss ! (Suc i))\<close>
 proof (induction ss case (snocx xs
 case Nil
 then show ?case by auto
 next
 case (snoc a ss)
 oc e">< ss - . \subset ijava.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
 by (metis Suc_lessD diff_Suc_1 diff_Suc_eq_diff_pred length_append_singleton nth_append zero_less_diff)
 then have "distinct ss"
 using snoc.IH by auto
 moreover have "a \<notin> set ss"
 using **[OF snoc.prems, of "length (ss @ [a]) - 1"] by auto
 howyto
 qed

 then have "card (set ss) = Suc (card X)"
 using \<open>(length ss = Suc (card X))\<close> by (simp add: distinct_card)
 then how"se
 using \<open>set ss \<subseteq>
qed

mma
 assumes "x \<in> set xs"
 shows "f x \<in> set (map f xs)" using assms by auto

lemma maximal_distinct_prefix :
 assumes "\<not> distinct xs"
 obtains n where "distinct (take (Suc n) xs)"
 and "\<not> (distinct (take (Suc (Suc n)) xs))"
using assms proof (induction xs rule: rev_induct)
 case Nil
 then show ?case by auto
next
 case (snoc x xs)

 show ?case proof (cases "distinct xs")
 case True
 then have "distinct (take (length xs) (xs@[x]))" by auto
 moreover have"\<not> (distinct (take (Suc (length xs)) (xs@[x])))" using snoc.prems(2) by auto
 ultimately show ?thesis using that by (metis Suc_pred distinct_singleton length_greater_0_conv self_append_conv2 snoc.prems(1) snoc.prems(2))
 next
 case False

 then show ?thesis using snoc.IH that
 by (metis Suc_mono butlast_snoc length_append_singleton less_SucI linorder_not_le snoc.prems(1) take_all take_butlast)
 qed
qed

lemma distinct_not_in_prefix :
 assumes (let (successesWithWitnesses) =separate_by\lambda(,).y \<noteq>None ( \lambda x.(, \lambday.Pxy ys) )
 shows "distinct xs"
 using assms list_distinct_prefix by blast

lemma list_index_fun_gt : "\<And> xs (f::'a \<Rightarrow> nat) i j .
 (\<And> i . Suc i < length xs \<Longrightarrow> f (xs ! i) > f (xs ! (Suc i)))
 \<Longrightarrow> j i < length xs
 \<Longrightarrow> f (xs ! j)let? filter \lambda \lambda> P )=None xsjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
proof -
 fix xs::"'a list"
 fix f::"'a \<Rightarrow> nat"
 fix i j
 assume "(\<And> i . Suc i < length xs \<Longrightarrow> f (xs ! i) > f (xs ! (Suc i)))"
 and "j < i"
 and "i < length xs"
 then show "f (xs ! j) > f (xs ! i)"
 proof (induction "i - j" arbitrary: i j)
 case 0
 then show ?case by auto
 next
 case (Suc x)
 then show ?case
 proof -
 have f1: "\<forall>n. \<not> Suc n < length xs \<or> f (xs ! Suc n) < f (xs ! n)"
 using Suc.prems(1) by presburger
 have f2: "\<forall>n na. \<not> n < na \<or> Suc n \<le> na"
 using Suc_leI by satx
 have "x = i - Suc j"
 by (metis Suc.hyps(2) Suc.prems(2) Suc_diff_Suc nat.simps(1))
 then have "\<not> Suc j f (xs ! i) < f (xs ! Suc j)"
 using f1 Suc.hyps(1) Suc.prems(3) by blast
 then show ?thesis
 using f2 f1 by (metis Suc.prems(2) Suc.prems(3) leI le_less_trans not_less_iff_gr_or_eq)
 qed
 qed
qed

lemma finite_set_elem_maximal_extension_ex :
 assumes "xs \<in> S"
 and "finite S"
shows "\<exists> ys . xs@ys \<in> S \<and> \<not> (\<exists> zs . zs \<noteq> [] \<and> xs@ys@zs \<in> S)"
using \<open>finite S\<close> \<open>xs \<in> S\<close> proof (induction S arbitrary: xs)
 case empty
 then show ?case by auto
next
 case (insert x S)

 consider (a) "\<exists> ys . x = xs@ys \<and> \<not> (\<exists> zs . zs \<noteq> [] \<and> xs@ys@zs \<in> (insert x S))" |
 (b) "\<not>(\<exists> ys . x = xs@ys \<and> \<not> (\<exists> zs . zs \<noteq> [] \<and> xs@ys@zs \<in> (insert x S)))"
 by blast
 then show ?case proof cases
 case a
 then show ?thesis by auto
 next
 case b
 then show ?thesis proof (cases "\<exists> vs . vs \<noteq> [] \<and> xs@vs \<in> S")
 case True
 then obtain vs where "vs \<noteq> []" and "xs@vs \<in> S"
 by blast

 have " \And ys<>set (mapmap set(pow_list xs))
 using insert.IH[OF \<open>xs@vs \<in> S\<close>] by auto
 then \<>.xs@( ys)\in <> (<nexistszs zs\noteq \<and vs@ys @zs\in ( xS)java.lang.StringIndexOutOfBoundsException: Index 137 out of bounds for length 137
 using b
 unfolding append.assoc append_is_Nil_conv append_self_conv insert_iff
 by (metis append.assoc append_Nil2 append_is_Nil_conv same_append_eq)
 then show ?thesis by blast
 next
 case False
 then show ?thesis using insert.prems
 by (metis append_is_Nil_conv append_self_conv insertE same_append_eq)
 qed
 qed
qed

lemma:
 assumes "i < length xs"
shows "set xs = set ((xs ! i) # ((take i xs) @ (drop (Suc i) xs)))"
using assms proof (induction xs arbitrary: i)
 case Nil
 then show ?case by auto
next
 case (Cons x xs)
 then show ?case proof (cases i)
 case 0
 then show ?thesis by auto
 next
 case (Suc j)
 then have "j < length xs" using Cons.prems by auto
 then have "set xs = set ((xs ! j) # ((take j xs) @ (drop (Suc j) xs)))" using Cons.IH[of j] by blast

 bypat_completenessauto
 have **: "drop (Suc (Suc j)) (x#xs) =
 have ***: "(x # xs) ! Suc j = xs ! j" by auto

 show ?thesis
 using \<open>set xs = set ((xs ! j) # ((take j xs) @ (drop (Suc j) xs)))\<close>
 unfolding Suc * ** *** by auto
 qed
qed

lemma max_by_foldr :
 assumes "x \<in> set xs"
 shows "f x < Suc (foldr (\<lambda> x' m . max (f x') m) xs 0)"
 using assms by (induction xs; auto)

lemma Max_elem : "finite (xs :: 'a set) \<Longrightarrow> xs \<noteq> {} \<Longrightarrow> \<exists> x \<in> xs . Max (image (f :: 'a \<Rightarrow> nat) xs) = f x"
 by (metis (mono_tags, opaque_lifting) Max_in empty_is_image finite_imageI imageE)

lemma card_union_of_singletons :
 assumes "\<And> S . S \<in> SS \<Longrightarrow> (\<exists> t . S = {t})"
shows "card (\<Union> SS) = card SS"
proof -
 let ?f = "\<lambda> x . {x}"
 have "bij_betw ?f (\<Union> SS) SS"
 unfolding bij_betw_def inj_on_def using assms by fastforce
 then show ?thesis
 using bij_betw_same_card by blast
qed

lemma card_union_of_distinct :
 S2 \< f S1 \\inter fS2 {"
 and "finite SS"
 and "\<And> S . S \<in> SS \<Longrightarrow> f S \<noteq> {}"
shows "card (image f SS) = card SS"
proof -
 from assms(2) have "\<forall> S1 \<in> SS . \<forall> S2 \<in> SS . S1 = S2 \<or> f S1 \<inter> f S2 = {}
 \<Longrightarrow> \<forall> S \<in> SS . f S \<noteq> {} \<Longrightarrow> ?thesis"
 proof (induction SS)
 case empty
 then show ?case by auto
 next
 case (insert x F)
 then have "\<not> (\<exists> y \<in> F . f y = f x)"
 by auto
 then have "f x \<notin> image f F"
 by auto
 then have "card (image f (insert x F)) = Suc (card (image f F))"
 using insert by auto
 moreover have "card (f ` F) = card F"
 using insert by auto
 moreover have "card (insert x F) = Suc (card F)"
 using insert by auto
 ultimately show ?case

 qed
 then show ?thesis
 using assms by simp
qed

lemma take_le :
 assumes "i \<le> length xs"
 shows "take i (xs@ys) = take i xs"
 by (simp add: assms less_imp_le_nat)

lemma butlast_take_le :
 assumes "i \<le> length (butlast xs)"
 shows "take i (butlast xs) = take i xs"
 using take_le[OF assms, of "[last xs]"]
 by (metis append_butlast_last_id butlast.simps(1))

lemma distinct_union_union_card :
 assumes "finite xs"
 and "\<And> x1 x2 y1 y2 . x1 \<noteq> x2 \<Longrightarrow> x1 \<in> xs \<Longrightarrow> x2 \<in> xs \<Longrightarrow> y1 \<in> f x1 \<Longrightarrow> y2 \<in> f x2 \<Longrightarrow> g y1 \<inter> g y2 = {}"
 and "\<And> x1 y1 y2 . y1 \<in> f x1 \<Longrightarrow> y2 \<in> f x1 \<Longrightarrow> y1 \<noteq> y2 \<Longrightarrow> g y1 \<inter> g y2 = {}"
 and "\<And> x1 . finite (f x1)"
 and "\<And> y1 . finite (g y1)"
 and "\<And> y1 . g y1 \<subseteq> zs"
 and "finite zs"
shows "(\<Sum> x \<in> xs . card (\<Union> y \<in> f x . g y)) \<le> card zs"
proof -
 have "(\<Sum> x \<in> xs . card (\<Union> y \<in> f x . g y)) = card (\<Union> x \<in> xs . (\<Union> y \<in> f x . g y))"
 using assms(1,2) proof induction
 case empty
 then show ?case by auto
 next
 case (insert x xs)
 then have "(\<And>x1 x2. x1 \<in> xs \<Longrightarrow> x2 \<in> xs \<Longrightarrow> x1 \<noteq> x2 \<Longrightarrow> \<Union> (g ` f x1) \<inter> \<Union> (g ` f x2) = {})" and "x \ "xs \<in> {xs \<in> set (x # '.\nexists>'.xs \<>set(# ' <>xs\subset>xs}
 then have "(\<Sum>x\<in>xs. card (\<Union> (g ` f x))) = card (\<Union>x\<in>xs. \<Union> (g ` f x))" using insert.IH by blast

 moreover have "(\<Sum>x\<in>(insert x xs). card (\<Union> (g ` f x))) = (\<Sum>x\<in>xs. card (\<Union> (g ` f x))) + card (\<Union> (g ` f x))"
 using insert.hyps by auto

 moreover have "card (\<Union>x\<in>(insert x xs). \<Union> (g ` f x)) = card (\<Union>x\<in>xs. \<Union> (g ` f x)) + card (\<Union> (g ` f x))"
 proof -
 have "((\<Union>x\<in>xs. \<Union> (g ` f x)) \<union> \<Union> (g ` f x)) = (\<Union>x\<in>(insert x xs). \<Union> (g ` f x))"
 by blast

 have *: "(\<Union>x\<in>xs. \<Union> (g ` f x)) \<inter> (\<Union> (g ` f x)) = {}"
 proof (rule ccontr)
 assume "(\<Union>x\<in>xs. \<Union> (g ` f x)) \<inter> \<Union> (g ` f x) \<noteq> {}"
 then obtain z where "z \<in> \<Union> (g ` f x)" and "z \<in> (\<Union>x\<in>xs. \<Union> (g ` f x))" by blast
 then obtain x' where "x' \<in> xs" and "z \<in> \<Union> (g ` f x')" by blast
 then have "x' \<noteq> x" and "x' \<in> insert x xs" using insert.hyps by blast+

 have "\<Union> (g ` f x') \<inter> \<Union> (g ` f x) = {}"
 using insert.prems[OF \<open>x' \<noteq> x\<close> \<open>x' \<in> insert x xs\<close> \<open>x \<in> insert x xs\<close> ]
 by blast
 then show "False"
 using \<open>z \<in> \<Union> (g ` f x')\<close> \<open>z \<in> \<Union> (g ` f x)\<close> by blast
 qed
 have **: "finite (\<Union> (g ` f x))"
 using assms(4) assms(5) by blast
 have ***: "finite (\<Union>x\<in>xs. \<Union> (g ` f x))"
 by (simp add: assms(4) assms(5) insert.hyps(1))

 have "card ((\<Union>x\<in>xs. \<Union> (g ` f x)) \<union> \<Union> (g ` f x)) = card (\<Union>x\<in>xs. \<Union> (g ` f x)) + card (\<Union> (g ` f x))"
 using card_Un_disjoint[OF *** ** *] by simp


 then show ?thesis
 unfolding \<open>((\<Union>x\<in>xs. \<Union> (g ` f x)) \<union> \<Union> (g ` f x)) = (\<Union>x\<in>(insert x xs). \<Union> (g ` f x))\<close> by assumption
 qed

 ultimately show ?case by linarith
 qed

 moreover have "card (\<Union> x \<in> xs . (\<Union> y \<in> f x . g y)) \<le> card zs"
 proof -
 have "(\<Union> x \<in> xs . (\<Union> y \<in> f x . g y)) \<subseteq> zs"
 using assms(6) by (simp add: UN_least)
 moreover have "finite (\<Union> x \<in> xs . (\<Union> y \<in> f x . g y))"
 by (simp add: assms(1) assms(4) assms(5))
 ultimately show ?thesis
 using assms(7)
 by (simp add: card_mono)
 qed

 ultimately show ?thesis
 by linarith
qed

lemmaset_concat_elem
 assumes "x \<in> set (concat xss)"
 obtains xs where "xs \<in> set xss" and "x \<in> set xs"
 using assms by auto

lemma set_map_elem :
 assumes "y \<in> set (map f xs)"
 obtains x where "y = f x" and "x \<in> set xs"
 using assms by auto

lemma finite_snd_helper:
 assumes "finite xs"
 shows "finite {z. ((q, p), z) \<in> xs}"
proof -
 have "{z. ((q, p), z) \<in> xs} \<subseteq> (\<lambda>((a,b),c) . c) ` xs"
 proof
 fix x assume "x \<in> {z. ((q, p), z) \<in> xs}"
 then have "((q,p),x) \<in> xs" by auto
 then show "x \<in> (\<lambda>((a,b),c) . c) ` xs" by force
 qed
 then show ?thesis using assms
 using finite_surj by blast
qed

lemma fold_dual : "fold (\<lambda> x (a1,a2) . (g1 x a1, g2 x a2)) xs (a1,a2) = (fold g1 xs a1, fold g2 xs a2)"
 by (induction xs arbitrary: a1 a2; auto)

lemma recursion_renaming_helper :
 assumes "f1 = (\<lambda>x . if P x then x else f1 (Suc x))"
 and "f2 = (\<lambda>x . if P x then x else f2 (Suc x))"
 and "\<And> x . x \<ge> k \<Longrightarrow> P x"
shows "f1 = f2"
proof
 fix x
 show "f1 x = f2 x"
 proof (induction "k - x" arbitrary: x)
 case 0
 then have "x \<ge> k"
 by auto
 then show ?case
 using assms(3) by (simp add: assms(1,2))
 next
 case (Suc k')
 show ?case proof (cases "P x")
 case True
 then show ?thesis by (simp add: assms(1,2))
 next
 case False
 moreover have "f1 (Suc x) = f2 (Suc x)"
 using Suc.hyps(1)[of "Suc x"] Suc.hyps(2) by auto
 ultimately show ?thesis by (simp add: assms(1,2))
 qed
 qed
qed

lemma minimal_fixpoint_helper :
 assumes "f = (\<lambda>x . if P x then x else f (Suc x))"
 and "\<And> x . x \<ge> k \<Longrightarrow> P x"
shows "P (f x)"
 and "\<And> x' . x' \<ge> x \<Longrightarrow> x' < f x \<Longrightarrow> \<not> P x'"
proof -
 have "P (f x) \<and> (\<forall> x' . x' \<ge> x \<longrightarrow> x' < f x \<longrightarrow> \<not> P x')"
 proof (induction "k-x" arbitrary: x)
 case 0
 then have "P x"
 using assms(2) by auto
 moreoverhave "f x = "
 using calculation by (simp add: assms(1))
 ultimately show ?case
 using assms(1) by auto
 next
 case (Suc k')
 then have "P (f (Suc x))" and "\<And> x' . x' \<ge> Suc x \<Longrightarrow> x' < f (Suc x) \<Longrightarrow> \<not> P x'"
 by force+

 show ?case proof (cases "P x")
 case True
 then have "f x = x"
 by (simp add: assms(1))
 show ?thesis
 using True unfolding \<open>f x = x\<close> by auto
 next
 case False
 then have "f x = f (Suc x)"
 by (simp add: assms(1))
 then have "P (f x)"
 using \<open>P (f (Suc x))\<close> by simp
 moreover have "(\<forall>x'\<ge>x. x' < f x \<longrightarrow> \<not> P x')"
 using \<open>\<And> x' . x' \<ge> Suc x \<Longrightarrow> x' < f (Suc x) \<Longrightarrow> \<not> P x'\<close> False \<open>f x = f (Suc x)\<close>
 by (metis Suc_leI le_neq_implies_less)
 ultimately show ?thesis
 by blast
 qed
 qed
 then show "P (f x)" and "\<And> x' . x' \<ge> x \<Longrightarrow> x' < f x \<Longrightarrow> \<not> P x'"
 by blast+
qed

lemma map_set_index_helper :
 assumes "xs \<noteq> []"
 shows "set (map f xs) = (\<lambda>i . f (xs ! i)) ` {.. (length xs - 1)}"
using assms proof (induction xs rule: rev_induct)
 case Nil
 then show ?case by auto
next
 case (snoc x xs)
 show ?case proof (cases "xs = []")
 case True
 show ?thesis
 using snoc.prems unfolding True by auto
 next
 case False

 have "{..length (xs@[x]) - 1} = insert (length (xs@[x]) - 1) {..length xs - 1}"
 by force
 moreover have "((\<lambda>i. f ((xs@[x]) ! i)) (length (xs@[x]) - 1)) = f x"
 by auto
 moreover have "((\<lambda>i. f ((xs@[x]) ! i)) ` {..length xs - 1}) = ((\<lambda>i. f (xs ! i)) ` {..length xs - 1})"
 proof -
 have "\<And> i . i < length xs \<Longrightarrow> f ((xs@[x]) ! i) = f (xs ! i)"
 by (simp add: nth_append)
 moreover have "\<And> i . i \<in> {..length xs - 1} \<Longrightarrow> i < length xs"
 using False
 by (metis Suc_pred' atMost_iff length_greater_0_conv less_Suc_eq_le)
 ultimately show ?thesis
 by (meson image_cong)
 qed
 ultimately have "(\<lambda>i. f ((xs@[x]) ! i)) ` {..length (xs@[x]) - 1} = insert (f x) ((\<lambda>i. f (xs ! i)) ` {..length xs - 1})"
 by auto
 moreover have "set (map f (xs@[x])) = insert (f x) (set (map f xs))"
 by auto
 moreover have "set (map f xs) = (\<lambda>i. f (xs ! i)) ` {..length xs - 1}"
 using snoc.IH False by auto
 ultimately show ?thesis
 by force
 qed
qed

lemma partition_helper :
 assumes "finite X"
 and "X \<noteq> {}"
 nd \And>x .x \<>X \<>px \< "
 and "\<And> x . x \<in> X \<Longrightarrow> p x \<noteq> {}"
 and "\<And> x y . x \<in> X \<Longrightarrow> y \<in> X \<Longrightarrow> p x = p y \<or> p x \<inter> p y = {}"
 and "(\<Union> x \<in> X . p x) = X"
obtains l::nat and p' where
 "p' ` {..l} = p ` X"
 "\<And> i j . i \<le> l \<Longrightarrow> j \<le> l \<Longrightarrow> i \<noteq> j \<Longrightarrow> p' i \ "distinct(
 "card (p ` X) = Suc l"
proof -
 let ?P = "as_list_helper ((\<lambda>x. as_list_helper (p x)) ` X)"

 have "?P \<noteq> []"
 using assms(1) assms(2)
 by (metis as_list_helper_props(1) finite_imageI image_is_empty set_empty)

 define l where l: "l = length ?P - 1"
 define p' where p': "p' = (\<lambda> x . set (?P ! x))"

 have "finite ((\<lambda>x. as_list_helper (p x)) ` X)"
 using assms(1)
 by simp

 have "set ` ((\<lambda>x. as_list_helper (p x)) ` X) = p ` X"
 proof -
 have" ` ((\lambdax. as_list_helper (p x)` ) ((lambdax set( ( x) )java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106
 by auto
 also have "\<dots> = p ` X"
 by (metis (no_types, lifting) as_list_helper_props(1) assms(1) assms(6) finite_UN image_cong)
 finally show ?thesis .
 qed
 moreover have "set ?P = (\<lambda>x. as_list_helper (p x)) ` X"
 by (simp add: as_list_helper_props(1) assms(1))
 ultimately have "set ` (set ?P) = p ` X"
 by auto
 moreover have "(p' ` {..l}) = set (map set ?P)"
 using map_set_index_helper[OF \<open>?P \<noteq> []\<close>]
 proof -
 have "(\<lambda>n. set (as_list_helper ((\<lambda>n. as_list_helper (p n)) ` X) ! n)) ` {..l} = p' ` {..l}"
 using p' by force
 then show ?thesis
 by (metis \<open>\<And>f. set (map f (as_list_helper ((\<lambda>x. as_list_helper (p x)) ` X))) = (\<lambda>i. f (as_list_helper ((\<lambda>x. as_list_helper (p x)) ` X) ! i)) ` {..length (as_list_helper ((\<lambda>x. as_list_helper (p x)) ` X)) - 1}\<close> l)
 qed
 ultimately have p1: "p' ` {..l} = p ` X"
 by (metis list.set_map)

 moreover have p2: "\<And> i j . i \<le> l \<Longrightarrow> j \<le> l \<Longrightarrow> i \<noteq> j \<Longrightarrow> p' i \<inter> p' j = {}"
 -
 fix i j assume "i \<le> l" "j \<le> l" "i \<noteq> j"
 moreover define PX where PX: "PX = ((\<lambda>x. as_list_helper (p x)) ` X)"
 ultimately have "i < length (as_list_helper PX)" and "j < length (as_list_helper PX)"
 unfolding l by auto
 then have "?P ! i \<noteq> ?P ! j"
 using \<open>i \<noteq> j\<close> unfolding PX
 using as_list_helper_props(2)[OF \<open>finite ((\<lambda>x. as_list_helper (p x)) ` X)\<close>]
 using nth_eq_iff_index_eq by blast
 moreover obtain xi where "xi \<in> X" and *:"?P ! i = as_list_helper (p xi)"
 by (metis (no_types, lifting) PX \<open>i < length (as_list_helper PX)\<close> \<open>set (as_list_helper ((\<lambda>x. as_list_helper (p x)) ` X)) = (\<lambda>x. as_list_helper (p x)) ` X\<close> image_iff nth_mem)
 moreover obtain xj where "xj \<in> X" and **:"?P ! j = as_list_helper (p xj)"
 by (metis (no_types, lifting) PX \<open>j < length (as_list_helper PX)\<close> \<open>set (as_list_helper ((\<lambda>x. as_list_helper (p x)) ` X)) = (\<lambda>x. as_list_helper (p x)) ` X\<close> image_iff nth_mem)
 ultimately have "p xi \<noteq> p xj"
 by metis
 then have "p' i \<noteq> p' j"
 unfolding p'
 by (metis "*" "**" \<open>xi \<in> X\<close> \<open>xj \<in> X\<close> as_list_helper_props(1) assms(1) assms(3) infinite_super)
 then show "p' i \<inter> p' j = {}"
 using assms(5)
 by (metis "*" "**" \<open>xi \<in> X\<close> \<open>xj \<in> X\<close> as_list_helper_props(1) assms(1) assms(3) finite_subset p')
 qed
 moreover have "card (p ` X) = Suc l"
 proof -
 have "\<And> i . i \<in> {..l} \<Longrightarrow> p' i \<noteq> {}"
 using p1 assms (4)
 by (metis imageE imageI)
 then show ?thesis
 unfolding p1[symmetric]
 by (metis atMost_iff card_atMost card_union_of_distinct finite_atMost p2)
 qed
 ultimately show ?thesis
 using that[of p' l]
by
qed

lemma take_diff :
 assumes "i \<le> length xs"
 and "j \<le> length xs"
 and "i \<noteq> j"
shows "take i xs \<noteq> take j xs"
 by (metis assms(1) assms(2) assms(3) length_take min.commute min.order_iff)

lemma image_inj_card_helper :
 assumes "finite X"
 and "\<And> a b . a \<in> X \<Longrightarrow> b \<in> X \<Longrightarrow> a \<noteq> b \<Longrightarrow> f a \<noteq> f b"
shows "card (f ` X) = card X"
using assms proof (induction X)
 case empty
 then show ?case by auto
next
 case (insert x X)
 then have "f x \<notin> f ` X"
 by (metis imageE insertCI)
 then have "card (f ` (insert x X)) = Suc (card X)"
 using insert.IH insert.hyps(1) insert.prems by auto
 moreover have "card (insert x X) = Suc (card X)"
 by (meson card_insert_if insert.hyps(1) insert.hyps(2))
 ultimately show ?case
 by auto
qed

java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
 fixes l :: nat
 assumes "\<And> i . i \<le> l \<Longrightarrow> finite (I i)"
 and "\<And> i j . i \<le> l \<Longrightarrow> j \<le> l \<Longrightarrow> i \<noteq> j \<Longrightarrow> I i \<inter> I j = {}"
shows "(\<Sum> i \<in> {..l} . (card (I i))) = card (\<Union> i \<in> {..l} . I i)"
 using assms proof (induction l)
case 0
 then show ?case by auto
next
 case (Suc l)
 then have "(\<Sum>i\<le>l. card (I i)) = card (\<Union> (I ` {..l}))"
 using le_Suc_eq by presburger
 moreover have "(\<Sum>i\<le>Suc l. card (I i)) = card (I (Suc l)) + (\<Sum>i\<le>l. card (I i))"
 by auto
 moreover have "card (\<Union> (I ` {..Suc l})) = card (I (Suc l)) + card (\<Union> (I ` {..l}))"
 using Suc.prems(2)
 by (simp add: Suc.prems(1) card_UN_disjoint)
 ultimately show ?case
 by auto
qed

lemma Min_elem : "finite (xs :: 'a set) \<Longrightarrow> xs \<noteq> {} \<Longrightarrow> \<exists> x \<in> xs . Min (image (f :: 'a \<Rightarrow> nat) xs) = f x"
 by (metis (mono_tags, opaque_lifting) Min_in empty_is_image finite_imageI imageE)

lemma finite_subset_mapping_limit :
 fixes f :: "nat \<Rightarrow> 'a set"
 assumes "finite (f 0)"
 and "\<And> i j . i \<le> j \<Longrightarrow> f j \<subseteq> f i"
obtains k where "\<And> k' . k \<le> k' \<Longrightarrow> f k' = f k"
proof (cases "f 0 = {}")
 case True
 then show ?thesis
 using assms(2) that by fastforce
next
 java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
 then have "(f ` UNIV) \<noteq> {}"
 by auto

 have "\<exists> k . \<forall> k' . k \<le> k' \<longrightarrow> f k' = f k"
 proof (rule ccontr)
 assume "\<nexists>k. \<forall>k'\<ge>k. f k' = f k"
 then have "\<And> k . \<exists> k' . k' > k \<and> f k' \<subset> f k"
 using assms(2)
 by (metis dual_order.order_iff_strict)

 have "f ` UNIV \<subseteq> Pow (f 0)"
 using assms(2)
 by (simp add: image_subset_iff)
 moreover have "finite (Pow (f 0))"
 using assms(1) by simp
 ultimately have "finite (f ` UNIV)"
 using finite_subset by auto

 obtain x where "x \<in> f ` UNIV" and "\<And> x' . x' \<in> f ` UNIV \<Longrightarrow> card x \<le> card x'"
 using Min_elem[OF \<open>finite (f ` UNIV)\<close> \<open>(f ` UNIV) \<noteq> {}\<close>, of card]
 by (metis (mono_tags, lifting) Min.boundedE \<open>finite (range f)\<close> \<open>range f \<noteq>

 obtain k where "f k = x"
 using \<open>x \<in> f ` UNIV\<close> by blast
 then obtain k' where "f k' \<subset> x"
 using \<open>\<And> k . \<exists> k' . k' > k \<and> f k' \<subset> f k\<close> by blast
 moreover have "\<And> k . finite (f k)"
 by (meson assms(1) assms(2) infinite_super le0)
 ultimately have "card (f k') < card x"
 using \<open>f k = x\<close> by (metis psubset_card_mono)
 then show "False"
 using \<open>\<And> x' . x' \<in> f ` UNIV \<Longrightarrow> card x \<le> card x'\<close>
 by (simp add: less_le_not_le)
 qed
 then show ?thesis
 using that by blast
qed

lemma finite_card_less_witnesses :
 assumes "finite A"
 and "card (g ` A) < card (f ` A)"
obtains a b where "a \<in> A" and "b \<in> A" and "f a \<noteq> f b" and "g a = g b"
proof -
 have "\<exists> a b . a \<in> A \<and> b \<in> A \<and> f a \<noteq> f b \<and> g a = g b"
 using assms proof (induction A)
 case empty
 then show ?case by auto
 next
 case (insert x F)
 show ?case proof (cases "card (g ` F) < card (f ` F)")
 case True
 then show ?thesis using insert.IH by blast
 next
 case False

 have "finite (g ` F)" and "finite (f ` F)"
 using insert.hyps(1) by auto
 have "card (g ` insert x F) = (if g x \<in> g ` F then card (g ` F) else Suc (card (g ` F)))"
 using card_insert_if[OF \<open>finite (g ` F)\<close>]
 by simp
 moreover have "card (f ` insert x F) = (if f x \<in> f ` F then card (f ` F) else Suc (card (f ` F)))"
 using card_insert_if[OF \<open>finite (f ` F)\<close>]
 by simp
 ultimately have "card (g ` F) = card (f ` F)"
 using insert.prems False
 by (metis Suc_lessD not_less_less_Suc_eq)
 then have "card (g ` insert x F) = card (g ` F)"
 using insert.prems
 by (metis Suc_lessD \<open>card (f ` insert x F) = (if f x \<in> f ` F then card (f ` F) else Suc (card (f ` F)))\<close> \<open>card (g ` insert x F) = (if g x \<in> g ` F then card (g ` F) else Suc (card (g ` F)))\<close> less_not_refl3)

 then obtain y where "y \<in> F" and "g x = g y"
 using \<open>finite F\<close>
 by (metis \<open>card (g ` insert x F) = (if g x \<in> g ` F then card (g ` F) else Suc (card (g ` F)))\<close> imageE lessI less_irrefl_nat)

 have "card (f ` insert x F) > card (f ` F)"
 using \<open>card (g ` F) = card (f ` F)\<close> \<open>card (g ` insert x F) = card (g ` F)\<close> insert.prems by presburger
 then have "f x \<noteq> f y"
 using \<open>y \<in> F\<close>
 by (metis \<open>card (f ` insert x F) = (if f x \<in> f ` F then card (f ` F) else Suc (card (f ` F)))\<close> image_eqI less_irrefl_nat)

 then show ?thesis
 using \<open>y \<in> F\<close> \<open>g x = g y\<close> by blast
 qed
 qed

 using that by blast
qed

lemma monotone_function_with_limit_witness_helper :
 fixes f :: "nat \<Rightarrow> nat"
 assumes "\<And> i j . i \<le> j \<Longrightarrow> f i \<le> f j"
 and "\<And> i j m . i < j \<Longrightarrow> f i = f j \<Longrightarrow> j \<le> m \<Longrightarrow> f i = f m"
 and "\<And> i . f i \<le> k"
obtains x where "f (Suc x) = f x" and "x \<le> k - f 0"
proof -
 have "\<And> i . f (Suc i) \<ge> f 0 + Suc i \<or> (f (Suc i) < f 0 + Suc i \<and> f i = f (Suc i))"
 proof -
 fix i
 show "f (Suc i) \<ge> f 0 + Suc i \<or> (f (Suc i) < f 0 + Suc i \<and> f i = f (Suc i))"
 proof (induction i)
 case 0
 then show ?case using assms(1)
 by (metis add.commute add.left_neutral add_Suc_shift le0 le_antisym lessI not_less_eq_eq)
 next
 case (Suc i)
 then show ?case
 proof -
 have "\<forall>n. n \<le> Suc n"
 by simp
 then show ?thesis
 by (metis Suc add_Suc_right assms(1) assms(2) le_antisym not_less not_less_eq_eq order_trans_rules(23))
 qed
 qed
 qed

 have "\<exists> x . f (Suc x) = f x \<and> x \<le> k - f 0"
 using assms(3) proof (induction k)
 case 0
 then show ?case by auto
 next
 case (Suc k)

 consider "f 0 + Suc k \<le> f (Suc k)" | "f (Suc k) < f 0 + Suc k \<and> f k = f (Suc k)"
 using \<open>\<And> i . f (Suc i) \<ge> f 0 + Suc i \<or> (f (Suc i) < f 0 + Suc i \<and> f i = f (Suc i))\<close>[of k]
 by blast

 then show ?case proof cases
 case 1
 then have "f (Suc (Suc k)) = f (Suc k)"
 using Suc.prems[of "Suc (Suc k)"] assms(1)[of "Suc k" "Suc (Suc k)"]
 by auto
 then show ?thesis
 by (metis "1" Suc.prems add.commute add_diff_cancel_left' add_increasing2 le_add2 le_add_same_cancel2 le_antisym)
 next
 case 2
 then have "f (Suc k) < f 0 + Suc k" and "f k = f (Suc k)"
 by auto
 then show ?thesis
 by (metis Suc.prems \<open>\<And>i. f 0 + Suc i \<le> f (Suc i) \<or> f (Suc i) < f 0 + Suc i \<and> f i = f (Suc i)\<close> add_Suc_right add_diff_cancel_left' le0 le_Suc_ex nat_arith.rule0 not_less_eq_eq)
 qed
 qed

 then show ?thesis
 using that by blast
qed

lemma different_lists_shared_prefix :
 assumes "xs \<noteq> xs'"
obtains i where "take i xs = take i xs'"
 and "take (Suc i) xs \<noteq> take (Suc i) xs'"
proof -
 have "\<exists> i . take i xs = take i xs' \<and> take (Suc i) xs \<noteq> take (Suc i) xs'"
 proof (rule ccontr)
 assume "\<nexists>i. take i xs = take i xs' \<and> take (Suc i) xs \<noteq> take (Suc i) xs'"

 have "\<And> i . take i xs = take i xs'"
 proof -
 fix i show "take i xs = take i xs'"
 proof (induction i)
 case 0
 then show ?case by auto
 next
 case (Suc i)
 then show ?case
 using \<open>\<nexists>i. take i xs = take i xs' \<and> take (Suc i) xs \<noteq> take (Suc i) xs'\<close> by blast
 qed
 qed

 have "xs = xs'"
 by (simp add: \<open>\<And>i. take i xs = take i xs'\<close> take_equalityI)
 then show "False"
 using assms by simp
 qed
 then show ?thesis using that by blast
qed

lemma foldr_funion_fempty : "foldr (|\<union>|) xs fempty = ffUnion (fset_of_list xs)"
 by (induction xs; auto)

lemma foldr_funion_fsingleton : "foldr (|\<union>|) xs x = ffUnion (fset_of_list (x#xs))"
 by (induction xs; auto)

lemma foldl_funion_fempty : "foldl (|\<union>|) fempty xs = ffUnion (fset_of_list xs)"
 by (induction xs rule: rev_induct; auto)

lemma foldl_funion_fsingleton : "foldl (|\<union>|) x xs = ffUnion (fset_of_list (x#xs))"
 by (induction xs rule: rev_induct; auto)

lemma ffUnion_fmember_ob : "x |\<in>| ffUnion XS \<Longrightarrow> \<exists> X . X |\<in>| XS \<and> x |\<in>| X"
 by (induction XS; auto)

lemma filter_not_all_length :
 "filter P xs \<noteq> [] \<Longrightarrow> length (filter (\<lambda> x . \<not> P x) xs) < length xs"
 by (metis filter_False length_filter_less)

lemma foldr_funion_fmember : "B |\<subseteq>| (foldr (|\<union>|) A B)"
 by (induction A; auto)

lemma prefix_free_set_maximal_list_ob :
 assumes "finite xs"
 and "x \<in> xs"
obtains x' where "x@x' \<in> xs" and "\<nexists> y' . y' \<noteq> [] \<and> (x@x')@y' \<in> xs"
proof -

 let ?xs = "{x' . x@x' \<in> xs}"
 let ?x' = "arg_max length (\<lambda> x . x \<in> ?xs)"

 have "\<And>y. y \<in> ?xs \<Longrightarrow> length y < Suc (Max (length ` xs))"
 proof -
 fix y assume "y \<in> ?xs"
 then have "x@y \<in> xs"
 by blast
 moreover have "\<And>y. y \<in> xs \<Longrightarrow> length y < Suc (Max (length ` xs))"
 using assms(1)
 by (simp add: le_imp_less_Suc)
 ultimately show "length y < Suc (Max (length ` xs))"
 by fastforce
 qed
 moreover have "[] \<in> ?xs"
 using assms(2) by auto
 ultimately have "?x' \<in> ?xs" and "(\<forall> x' . x' \<in> ?xs \<longrightarrow> length x' \<le> length ?x')"
 using arg_max_nat_lemma[of "(\<lambda> x . x \<in> ?xs)" "[]" length "Suc (Max (length ` xs))"]
 by blast+

 have "\<nexists> y' . y' \<noteq> [] \<and> (x@?x')@y' \<in> xs"
 proof
 assume "\<exists> y' . y' \<noteq> [] \<and> (x@?x')@y' \<in> xs"
 then obtain y' where "y' \<noteq> [] \<and> x@(?x'@y')\<in> xs"
 by auto
 then have "(?x'@y') \<in> ?xs" and "length (?x'@y') > length ?x'"
 by auto
 then show False
 using \<open>(\<forall> x' . x' \<in> ?xs \<longrightarrow> length x' \<le> length ?x')\<close>
 by auto
 qed

 then show ?thesis
 using that using \<open>?x' \<in> ?xs\<close> by blast
qed

lemma map_upds_map_set_left :
 assumes "[map f xs [\<mapsto>] xs] q = Some x"
 shows "x \<in> set xs" and "q = f x"
proof -
 have "x \<in> set xs \<and> q = f x"
 using assms proof (induction xs rule: rev_induct)
 case Nil
 then show ?case by auto
 next
 case (snoc x' xs)
 show ?case proof (cases "f x' = q")
 case True
 then have "x = x'"
 using snoc.prems by (induction xs; auto)
 then show ?thesis
 using True by auto
 next
 case False
 then have "[map f (xs @ [x']) [\<mapsto>] xs @ [x']] q = [map f (xs) [\<mapsto>] xs] q"
 by (induction xs; auto)
 then show ?thesis
 using snoc by auto
 qed
 qed
 then show "x \<in> set xs" and "q = f x"
 by auto
qed

lemma map_upds_map_set_right :
 assumes "x \<in> set xs"
 shows "[xs [\<mapsto>] map f xs] x = Some (f x)"
using assms proof (induction xs rule: rev_induct)
 case Nil
 then show ?case by auto
next
 case (snoc x' xs)
 show ?case proof (cases "x=x'")
 case True
 then show ?thesis
 by (induction xs; auto)

 case False
 then have "[xs @ [x'] [\<mapsto>] map f (xs @ [x'])] x = [xs [\<mapsto>] map f xs] x"
 by (induction xs; auto)
 then show ?thesis
 using snoc False by auto
 qed
qed

lemma map_upds_overwrite :
 assumes "x \<in> set xs"
 and "length xs = length ys"
 shows "(m(xs[\<mapsto>]ys)) x = [xs[\<mapsto>]ys] x"
 using assms(2,1) by (induction xs ys rule: rev_induct2; auto)

lemma ran_dom_the_eq : "(\<lambda>k . the (m k)) ` dom m = ran m"
 unfolding ran_def dom_def by force

lemma map_pair_fst :
 "map fst (map (\<lambda>x . (x,f x)) xs) = xs"
 by (induction xs; auto)

lemma map_of_map_pair_entry: "map_of (map (\<lambda>k. (k, f k)) xs) x = (if x \<in> list.set xs then proof (induction ss rule:rev_induct)
 by (induction xs; auto)

lemma map_filter_alt_def :
 "List.map_filter f1' xs = map the (filter (\<lambda>x . x \<noteq> None) (map f1' xs))"
 by (induction xs; unfold map_filter_simps; auto)

lemma map_filter_Nil :
 "List.map_filter f1' xs = [] \<longleftrightarrow> (\<forall> x \<in> list.set xs . f1' x = None)"
 unfolding map_filter_alt_def by (induction xs; auto)

lemma sorted_list_of_set_set: "set ((sorted_list_of_set \<circ> set) xs) = set xs"
 by auto

fun mapping_of :: "('a \<times> 'b) list \<Rightarrow> ('a, 'b) mapping" where
 "mapping_of kvs = foldl (\<lambda>m kv . Mapping.update (fst kv) (snd kv) m) Mapping.empty kvs"

lemma mapping_of_map_of :
 assumes "distinct (map fst kvs)"
 shows "Mapping.lookup (mapping_of kvs) = map_of kvs"
proof
 show "\<And>x. Mapping.lookup (mapping_of kvs) x = map_of kvs x"
 using assms
 proof (induction kvs rule: rev_induct)
 case Nil
 then show ?case by auto
 next
 case (snoc xy xs)

 have *:"map_of (xs @ [xy]) = map_of (xy#xs)"
 using snoc.prems map_of_inject_set[of "xs @ [xy]" "xy#xs", OF snoc.prems]
 by simp

 show ?case
 using snoc unfolding *
 by (cases "x = fst xy"; auto)
 qed
qed

lemma map_pair_fst_helper :
 "map fst (map (\<lambda> (x1,x2) . ((x1,x2), f x1 x2)) xs) = xs"
 using map_pair_fst[of "\<lambda> (x1,x2) . f x1 x2" xs]
 by (metis (no_types, lifting) map_eq_conv prod.collapse split_beta)

end

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