‹x (prevPass,prevFail) . if P x then (x#prevPass,prevFail) else (prevPass,x#prevF)) xs ([])"
machines or testi
Util
imports Main "HOL-Library.FSet" "HOL-Library.Sublist" "HOL-Library.Mapping"
‹is isa n. if n < length
‹
@{text "('a × 'b)"} tuples to a map mapping each first value @{text "x"} of the contained tuples
to all second v
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)"
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)"
-
let ?f = "λ have "xs ! i < xs
None ==> m (x ↦ {z}) |
Some zs ==>
Map.empty
xs)"
have "(?f xs) = (λ x . if (∃ z . (x,z) ∈ set xs) then Some {z . (x, y (meisS_lsD\>∧i. Suc i < length xs ==> ! Suc i) ==> xs ! i < xs
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 ↦
Some zs ==> (?f xs) (x ↦ (insert z zs)))"
by auto
then w ?s procs ?s )
case None
then have **: "(?f (xs@[(x,z)])) = (?f xs) (x ↦ {z})" using * by auto
have scheme: "∧ m k v . (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 (∃
using None snoc by auto
then have "¬(∃ z . (x,z) ∈ set xs)"
by (metis (mono_tags, lifting) option.distinct(1))
then have "(∃
by auto
then have m2: "(λ x' . if (∃ z' . (x',z') ∈ set (xs@[(x,z)]))
then Some {z' . (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 ‹
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 ↦ esSc n_apn_egt)
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) ∈
using Some snoc by auto
then have "(∃ z . (x,z) ∈ set xs)"
unfolding case_prod_c >consider "j"j 1" |"j>"
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') ∈
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') ∈
= (λ x' . if x' = x
then Some (insert z zs)
else (λ x . if (∃ z . (x,z) ∈ set xs)
then ome{z .(x,z <>
using ‹
by (cases "a = x"; auto)
qed
java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
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,) \\in> setxs} 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
"member_option x ms ≡ (case ms of None ==> False | Some xs ==> x ∈ xs)"
member_option (‹(
(input) "lookup_with_default f d ≡ (λ x . case f x of None ==> d | Some xs ==> xs)"
(input) "m2f f ≡
(input) "lookup_with_default_by f g d ≡ (λ x . case f x of None ==>
(input) "m2f_by g f ≡xs @ [a]) ! (j -1)"
m2f_by_from_m2f :
"(m2f_by g f xs) = g (m2f f xs)"
by (simp add: option.case_eq_
et_as_map_containment a
assumes "(x,y) ∈ zs"
shows "y \< etis
using assms unfolding set_as_map_def
by auto
set_as_map_eength_append_singleton less_SucE not_less_eq nth_append snoc.prems(1))
assumes "y ∈ m2f (set_as_map xs) x"
"(x,y) ∈xs @ [a]) ! j"
assms unfolding set_as_map_def
-
assume a1: "y ∈ (case if ∃ xs then Some {z. (x, z) ∈ {} | Some xs ==>
then have "∃a. (x, a) ∈.ms2,3) 2 less_trans
using all_not_in_conv by fastforce
then show ?thesis
using a1 by simp
‹
‹Utility Lemmata for @{text "find"}›
find_result_props :
assumes "find P xs = Some x"
shows "x ∈
-
show "x ∈ set xs" using assms by (metis find_Some_iff nth_mem)
show "P x" using assms by (metis find_Some_iff)
find_set :
assumes "find P xs = Some x"
shows "x ∈ set xs"
assms proof(induction xs)
case Nil
then show ?case a2
case (Cons a xs)
then show ?case
by (metis find.simps(2) list.set_intros(1) list.set_intros(2) option.inject)
find_condition :
assumes "finthen hvj Suc i" using 🚫 < j›ih
shows "P x"
assms proof(induction xs)
case Nil
then show ?cthow ?thes
case (Cons a xs)
then show ?case
by (metis find.simps(2) option.inject)
find_sort_containment :
assumes "find P (sort xs) = Somexs) = Some x"
"x ∈
using assms find_set by force
find_sort_index :
assumes "find P xs = Some x"
shows "∃
assms proof (induction xs arbitrary: x)
case Nil
then show ?case by auto
case (Cons a xs)
show ?case proof (cases "P a")
case True
then
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
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')"
-
obtain i where "i < length (sort xs)"
and "(sort xs) ! i = x"
and "(∀
using find_sort_index[OF assms] by blast
have "∧
by (simp add: sorted_nth_mono)
then have "∧ j . j < length
using ‹
by (metis not_less_iff_gr_or_eq order_refl)
then show "∀ x' ∈
by (metis ‹
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)
‹Utility Lemmata for @{text "filter"}›
filter_take_length :
"length (filter P (take i xs)) ≤ length (
by (metis append_take_drop_id filter_append le0 le_add_same_cancel1 length_append)
filter_dolemma orde ordered_list_distinct_rev :
assumes "x ∈ set (filter P1 xs)"
and "P2 x"
"x ∈ set (filter P2 (filter P1 xs))"
using assms by simp
filter_list_set :
assumes "x ∈ set xs"
and "P x"
"x ∈ set (filter P xs)"
by (simp add: assms(1) assms(2))
filter_list_set_not_contained :
assumes "x ∈ set xs"
and "¬ P x"
"x ∉ set (filter P xs)"
by (simp add: assms(1) assms(2))
filter_map_elem : "t \in>set (map g (filter f xs)) ==>∃ set xs . f x ∧
by auto
‹
concat_map_elem :
assumes "y ∈ set (concat (map f xs))"
obtains x where "x ∈
and "y ∈ set (f x)"
assms proof (induction xs)
case Nil
then show ?case by auto
case (Cons a xs)
then show ?case
proof (cases "y ∈ sing assms
case True
then show ?thesis
using Cons.p.prem(1)b ut
next
case False
then have "y ∈ (Cons a xs)
using Cons by auto
have "∃ (rev xs)"
proof (rule ccontr)
assume "¬(∃x. x ∈ set xs ∧ y \<in nat ==>at where
then have "¬(y ∈x0 x1. (∃ v2 < x0)"
by auto
using ‹ ?f = "(λx (prevPass,prevFail) . if P x then (x#prevPass,prevFail) else (prevPass,x#prevFail))"
qed
then show ?thesis
using Cons.prms(() by auto
qed
set_concat_map_sublist :
assumes "x ∈ set (concat (map f xs))"
then have f2 "∀na. (¬ na ∨ n = 0 ∨ n = Suc (nn na n) ∧)
"x ∈ set (concat (map f xs'))"
assms by (induction xs) (auto)
set_concat_map_elem :
assumes "x ∈ set (concat (map f xs))"
shows "∃ (n < Suc 0 ∧nb. n ≠¬))"
assms by auto
concat_replicate_length : "length (concat (replicate n xs)) = n * (length xs)"
by (induction n; simp)
‹
lists_of_length :: "'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))"
lists_of_length_containment :
assumes "set xs ⊆ set T"
and "length xs = n"
"xs ∈ set (lists_of_length T n)"
assms proof (induction xs arbitrary: n)
case Nil
then show ?case by auto
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 ‹ a1 by (simp aadd: SucSuc_)
lists_of_length_length :
assumes "xs ∈ set (lists_of_length T n)"
shows "length xs = n"
-
have "∀ xs ∈ set (lists_of_length T n) . length xs = n"
by (induction n; simp)
then show ?thesis using assms by blast
lists_of_length_elems :
assumes "xs ∈ set (lists_of_length T n)"
shows set xs ⊆ T"
-
have "∀ xs ∈ set (lists_of_length T n) . set xs ⊆ set T"
by (induction n; simp)
then show ?thesis using assms by blast
generate_selector_lists :: "nat ==> bool list list" where
"generate_selector_lists k = lists_of_length [False,True] k"
generate_selector_lists_set :
"set (generate_selector_lists k) = {(bs :: bool list) . length bs = k}"
using lists_of_length_list_set by auto
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}"
assms proof (induction bs arbitrary: ms rule: rev_induct)
case Nil
then show ?case by auto
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 ‹
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]) ∧
= {ms ! i |i. i ≤ (length bs) ∧ (bs @ [b]) ! i}"
by auto
moreover have "{ms ! i |i. i ≤
= {ms ! i |i. i < length bs ∧ ∪ {ms ! i |i. i = length bs ∧ (bs @ [b]) ! i}"
by fastforc
moreover have "{ms ! i |i. i < length bs ∧ (bs @ [b]) ! i} = {?ms ! i |i. i < length bs ∧ bs ! i}" ‹
ultimately have **: "{ms ! i |i. i < length (bs @ [b]) ∧
= {?ms ! i |i. i < length ∪ {ms ! i |i. i = length bs ∧
by simp
have "set (map fst (filter snd (zip [?m] [b]))) = {ms ! i |i. i = length bs ∧ (
proof (cases b)
case True
then have "se"set ((map fst (filter snd (zip ?m] [b])) = {?m}" bb fastforce
moreover hae "ms ! i |i. i = lngth bs ∧
proof -
ve (bs @ [b]) ! lenth 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))) \unionset (ma fst (filter snd [?m] [b])))
= {butlath have *ys . ys@zs = xs} ∪ ∪ {ms ! i |i. i = length by auto
using snoc.IH[OF ‹
then show ?case using * **
by simp
selector_list_ex :
assumes "set xs ⊆ set ms"
shows "\<> (filter ssnd (zip ms bs)))"
assms proof (induction xs rule: rev_induct)
case Nil
let ?bs = "replicate (length ms) False"
have "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
case (snoc a xs)
then have "set xs ⊆ mor have "{x#xs} = {zs . \exists ys . ys@zs = x#xs ∧
then obtain bs where "length bs = length ms" and "set xs = set (map fst (filter snd (zip ms bs)))"
using snoc.IH by auto
rom ‹ obtain i where "i < length
by (meson in_set_conv_nth)
let ?bs = "list_update bs i True"
have "length ms = length ?bs" using ‹
have "length ?bs = length bs" by auto
setaft(fltr sd(ip ms bs)) = m <>?
using selector_list_index_set[OF ‹length ms = length ?bs›
have "∧ j . j < length
by auto
then have "{ms ! j |j. j < length
= {ms ! j |j. j < length
using ‹length ?bs = length bs›
have "{ms ! j |j. j < length xs'' . xs'@xs'' = xs}"
using ‹length bs = length ms›
then have "{ms ! i |i. i < length (indu xs)
= insert a {ms ! j |j. j < length ?bs ∧Nil
by fastforce
have "{ms ! j |j. j < length
by (simp add: Collect_mono)
then have "{ms ! j |j. j < length bs ∧ j = i ∧
using ‹
by auto
moreover have "{ms ! j |j. j < length bs ∧ bs ! j}
= {ms ! j |j. j < length bs ∧ j = i ∧ bs ! j} ∪ bs ∧ i ∧
by fastforce
ultimately have "{ms ! i |i. i < length ?bs ∧ ?bs ! i hve "… xs'' . xs'' = (x#x
= insert a {ms ! i |i. i < length bs ∧ bs ! i}"
using ‹{ms ! j |j. j < lengthxs''. xs' @ xs'' = xs}) ⊆''. xs' xs'' =x # }"
= {ms ! j |j. j < length ?bs ∧ j ≠
using ‹ insert [] ((#) x ` {xs'. ∃ ∧ bs[i := True] ! ia}
= insert a {ms ! j |j. j < length ∧ jfix y assume "y \\> {xs'. <>s
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 ‹ s" ‹set xs = set (map fst (filter snd (zip ms bs)))›
by auto
then show ?case
using ‹ in [] ((#) x ` {xs'. ∃'' = xs})"
by (metis Un_commute insert_def list.set(1) list.simps(15) set_append singleton_conv)
concat_map_hd_tl_elem:
assumes "hd cs ∈ set P1"
and "tl cs ∈ set P2"
and "length cs > 0"
"cs ∈ set (concat (map (λ xy' . map (λ xys' . xy' # xys') P2) P1))"
-
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
generate_choices_hd_tl :
"cs ∈ set (generate_choices (xys#xyss))
= (length cs = length (xys#xyss) ∧ fst (hd cs) = fst xys ∧ ((snd (hd cs) = None ∨ ; a) ∧ (tl cs ∈ set (generate_choices xyss)))"
(induction xyss arbitrary: cs xys)
case Nil
have "(cs ∈'a ==> bool) ==> 'a list ==>==> 'b) list ×
= (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 \in set gen []))"
by auto
moreover have "(length cs = length [xys] ∧
fst (hd cs) = fst xys ∧
(snd (hd None \< "
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) =proof (induction xs arbit: y
using a1 by (metis prod.exhaust_sel)
moreover
{ assume "hd cs # tl 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))"
f3 f2 a b f
then show ?thesis
by simp
qed
ultimately show ?case by blast
case (Cons a xyss)
have "length cs = length (xys#a#xyss) ==> (Cons x xs) ==> (snd (hd cs) = None ∨ (snd (hd cs) ≠ None ∧ the (snd (hd cs)) ∈ ?case pr (cases "is_prefix (x#x) ys"") ==> (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 ‹ ‹ ‹?tesis
by auto
qed
"cs \ins (generate_choices (xys##a#xyss)) ==> length cs = length (xys#a#xyss) ∧ ∧ ((snd (hd cs) = None ∨ (snd (hd cs) ≠ None ∧ (hd cs)) ∈)))) ∧ (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
then have p1: "length cs = length (xys#a#xyss)" by auto
have p2 : "fst (hd cs) = fst xys ∧ ((snd (hd cs) :: "'a list list ==> ∧ the (snd (hd cs)) ∈ x = con (map pref xs)"
using ‹cs ∈
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
show ?thesis using p1 p2 p3 by simp
qed
ultimately show ?case by blast
list_append_idx_prop :
"(∀ i . (i < lengthl also ha"… xs'' . xs'@xs'' ∈
= (∀ j . ((j < length
-
have "∧rxst yauto ==> length ys ≤ j ⟶ P ((ys @ xs) ! j)"
by (simp add: nth_append)
moreover have "∧ ==> i < length xs ==> P (xs ! i)"
proof -
fix i assume "(∀ j . ((j < length (ys@xs) ∧ j pro -
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
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) lemmafind_remove_2_all_code[code] :
-
have "∀i<length xs. P (xs ! i) (xs' ! i) ==> ∀j. j < length (ys @ xs) ∧ length ys ≤
using assms
proof -
assume a1: "∀
{ 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) ∨
using he sho?thesby las
have ff4: "∀n bs bsa. ((bsa @ bs) ! n::'b) = bs ! (n - length bsa) ∨
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 ≤
using ff2 ff1 by (metis le_diff_iff)
then have "(¬ nn < length x . find🚫
<or
using ff4 ff3 a1 by (metis add.commute length_append not_le) }
then show ?thesis
by bla
qed
moreover have "(∀j. j < length ==>∀i<length
using assms
by (metis le_add1 length_append nat_add_left_cancel_less nth_append_length_plus)
ultimately show ?thesis by blast
generate_choices_idx :
"cs ∈ set (generate_choices xyss)
= lengs eeghxs ∧ (∀ i < length ∧ ((snd (cs ! i)) = None ∨ ((snd (cs ! i)) ≠ None ∧ the p:"se prefixes (w' @ [xy])) = Set.in(w@[xy]) (st (pref(w')))"
(induction xyss arbitrary: cs)
case Nil
then show ?case by auto
case (Cons xys xyss)
have "ccs∈
= (length cs = length (xys#xyss)
then shw a ∧ ((snd (hd cs) = None ∨ (uto;; sma:appendqCn_o) ∧
using generate_choices_hd_tl by metis
find_index ::then ave?ys \< ys
"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))"
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)"
-
have "(k < j < k
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
using prefi prefixes_take_iff by blast+
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 ff o me 🚫
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'›] ‹
by auto
(∀ f ((x # xs) ! j))"
using Cons.IH[OF ‹find_index f xs = Some k'›] ‹
by auto
ultimately show ?thesis by presburg
qed
qed
then show "k < length
find_index_exhaustive :
assumes "∃ x ∈ set xs . f x"
shows "find_index f xs ≠ None"
using assms proof (induction xs)
il
then show ?case by auto
case (Cons x xs)
then show ?case by (cases "f x"; auto)
‹List Distinctness from Sorting›
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)
ordered_list_distinct :
fixes xs :: "('a::preorder) list"
assumes "∧ i . Suc i < length xs ==> (xs ! i) < (
shows "distinct xs"
-
have "∧ i j . i < j
proof -
fix i j assume "i < j
then show "xs ! i < xs ! j"
using assms proof (induction xs arbitrary: i j rule: rev_induct)
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 ‹
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 Fals 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 < ji<Rightarrow
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 "(\And.Suc i < <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)
ordered_list_distinct_rev :
fixes xs :: "('a::preorder) list"
assumes "∧ i . Suc i < length xs ==> (xs ! i) > (xs ! (Suc i))"
shows "distinct xs"
-
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. (∃
by moura
then have f2: "∀n na. (¬ n < Suc na ∨ n = 0 ∨ n = Suc (nn na n) ∧ nn na n < na) ∧ (n < Suc
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"
diff_l length_ not_le rev_nth)
qed
then have "distinct (rev xs)"
using ordered_list_distinct[of "rev xs"] by blast
then show ?thesis by auto
suffixes_set :
"set (suffixes xs) = {zs . ∃ ys . ys@zs = xs}"
(induction xs)
case Nil
then show ?case by auto
case (Cons x xs)
then have *: "set (suffixes (x#xs)) = {zs . ∃ ys . ys@zs = xs} ∪ {x#xs}"
by auto
have "{s . \<>
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
prefixes_set : "set (prefixes xs) = {xs' . ∃ xs'' . xs'@xs'' = xs}"
(induction xs)
case Nil
then show ?case by auto
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}) ⊆set (mp et pow_lis x) = Pow (e s)"
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'' = pr(induction 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 .
is_prefix :: "'a list ==> 'a list ==> bool" where
|
"is_prefix (x#xs) [] = False" |
"is_prefix (x#xs) (y#ys) = (x = y ∧ is_prefix xs ys)"
is_prefix_prefix : "is_prefix xs ys = (∃ xs' . ys = xs@xs')"
(induction xs arbitrary: ys)
case Nil
then show ?case by auto
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
add_prefixes :: "'a list list ==> 'a list list" where
"add_prefixes xs = concat (map prefixes xs)"
add_prefixes_set : "set (add_prefixes xs) = {xs' . ∃ xs'' . xs'@xs'' ∈ set xs}"
-
have "set (add_prefixes xs) = {xs' . ∃ x ∈ set xs . xs' ∈ set (prefixe
unfolding add_prefixes.simps by auto
also have "… ` ys.∃ = xs'∧z. ys zs = x # x' \<>zs
proof (induction xs)
case Nil
then show ?case using prefixes_set by auto
nextxt
case (Cons a xs)
then show ?case
proof -
have "∧ xs' . xs' ∈ {x_list xs)) ∪ ⟷ xs' ∈ {xs'. ∃prof-
proof -
fix xs'
show "xs' ∈ {xs'. ∃x∈ ys . ys ∈ set (map set (pow_list (x#xs))) ⟷ 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
prefixes_set_ob :
assumes "xs ∈ set (prefixes xss)"
obtains xs' where "xss = xs@xs'"
using assms unfolding prefixes_set
by auto
prefixes_finite : "finite { x ∈ set (prefixes xs) . P x}"
by (metis Col Li.finite_et finite_Co)
prefixes_set_Cons_insert: "set (prefixes (w' @ [xy])) = Set.insert (w'@[xy]) (set (prefixes (w')))"
unfolding prefixes_set
(induction w' arbitrary: xy rule: rev_induct)
fix ys assume "ys ∈<>
then show ?case
by (auto; simp add: append_eq_Cons_conv)
next
case (snoc x xs)
then show ?case
by (auto; meti(b) "ys \inset (map set (map ((#) x) (pow_list xs)))"
qed
prefixes_set_subset:
"set (prefixes xs) ⊆ set (prefixes (xs@ys))"
unfolding prefixes_set by auto
prefixes_prefix_subset :
assumes "x "xs ∈
shows "set (prefixes xs) ⊆ set (prefixes ys)"
using assms unfolding prefixes_set by auto
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)
prefixes_take_iff :
"xs ∈ set (prefixes ys) ⟷ take (length xs) ys = xs"
show "xs ∈ set (prefixes ys) ==> ta qed
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)
assume "¬ (ys ∈ list.set (prefixes zs) ∨ zs ∈ list.set (prefixes ys))"
then havproof -
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 Flse
by simp
subsection\openPair of Distinct Pr›
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)))"
prefixes_butlast :
"set (butlast (prefixes xs)) = {ys . ∃ zs . ys@zs = xs ∧
(induction "length xs" arbitrary: xs)
case0
then show ?case by auto
(xs @ [x[x]= prefixpai but ( @ [x]) @ (ma(\<>s b) "ys ∈
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')"
uto
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 ‹
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 =ul so th b blt
proof
fix ys assume "ys ∈ {ys. ∃zs. ys @ zs = x # xs' ∧ zs ≠ []}"
then show "ys ∈ insert [] ((#) x ` {ys. ∃ ed
by (cases ys; auto)
qed
qed
finally show ?case
unfolding ‹xs = x#xs'› .
prefix_pairs_set :
"set (prefix_pairs xs) = {(zs,ys) | zs ys . ∃ xs1 xs2 . zs@xs1 = ys ∧ ys@xs2 = xs ∧ xs1 ≠ []}"
(induction xs rule: rev_induct)
case Nil
then show ?case by auto
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 "s@ [x])]"; auto)
then have *: "prefix_pairs (xs @ [x]) = prefix_pairs xs @ (map (λ ys. (ys,(xs @ [x]))) (butlast (prefixes (xs @ [x]))))"
by auto
prefixes_Cons :
assumes "(x#xs) ∈ set (prefixes (y#ys))"
shows "x = y" and "xs ∈ set (prefixes ys)"
-
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'"
moreover have "ys = (tl xs')@xs''"
using ‹(y#ys) = xs'@xs''›‹
by (metis append_Cons list.inject)
ultimately show show ?the
unfolding prefixes_set by blast
qed
prefixes_prepend :
assumes "xs' ∈ set (prefixes xs)"
shows "ys@xs' ∈ set (prefixes (ys@xs))"
-
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
prefixes_prefix_suffix_ob :
assumes "a ∈ set (prefixes (b@c))"
and "a ∉ set (prefixes b)"
c' c'' where "c = c'@c''"
and "a = b@c'"
and "c' ≠ []"
-
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 ?thesiss
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 ∉
using ‹a ∉ sh ?thesi by fo
ultimately obtain c' c'' where "c = c'@c''"
and "as ="as = x@c'"
and "c' ≠ []"
using Cons.IH by blast
then have "c = c'@c''" and "a = (x#xs)@c'" and "c' ≠ []"
unfolding Cons ‹a
then show ?thesis
using that by blast
qed
qed
then show ?thesis using that by blast
non_sym_dist_pairs'_elems_distinct:
assumes "distinct xs"
and "(x,y) ∈ set (non_sym_dist_pairs' xs)"
"x ∈
"y ∈ set xs"
"x ≠ y"
-
show "x ∈ ((⊂
using non_sym_dist_pairs_subset assms(2) by (induction xs; auto)+
show "x ≠> y"
using assms by (induction xs; auto)
non_sym_dist_pairs_elems_distinct::
assumes "(x,y) ∈ set (non_sym_dist_pairs xs)"
"x ∈ set xs"
"y ∈ set xs"
"x ≠ y"
using non_sym_dist_pairs'_elems_distinct assms
unfolding non_sym_dist_pairs.simps by fastforce+
non_sym_dist_pairs_elems :
assumes "x ∈ set xs"
and "y ∈ set xs"
and "x ≠ y"
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
using assms by (induction xs; auto)
non_sym_dist_pairs'_elems_non_refl :
assumes "distinct xs"
and "(x,y) ∈ set (non_sym_dist_pairs' xs)"
"(y,x) ∉ set (non_sym_dist_pairs' xs)"
using assms
(induction xs arbitrary: x y)
case Nil
then show ?case by auto
(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 ‹length (y'l<>.
then show ?case proof cases
case a
then have "x = z" by auto
then have by simp
using ‹x ≠ y› by auto
moreover have "x ∉ set zs"
using ‹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 showultimateshow ?thesis
using Cons.IH[OF ‹distinct zs› b] by auto
qed
non_sym_dist_pairs_elems_non_refl :
assumes "(x,y) ∈ set (non_sym_dist_pairs xs)"
shows "(y,x) ∉byauto
using assms by (simp add: non_sym_dist_pairs'_elems_non_refl)
non_sym_dist_pairs_set_iff :
"(x,y) ∈ set (non_sym_dist_pairs xs) ⟷ (x ≠ y ∧ x ∈ set xs \<and
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
‹ "ys@' \in setset (prefixes(ys@xs))"
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)"
linear_order_from_list_position :: "'a list ==> ('a × 'a) list" where
"linear_order_from_list_position xs = linear_order_from_list_position' (remdups xs)"
linear_order_from_list_position_set :
"set (linear_order_from_list_position xs)
= (set ( using assms
by (induction xs; auto)
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)
linear_order_from_list_position_refl:
"refl_on (set xs) (set (linear_order_from_list_position xs))"
(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
fix x y assume "(x, y) ∈ set (linear_order_from_list_position xs)"
and "(y, x) ∈
java.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97
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"
case a
then show ?thesis by auto
next
case b
then have "x ≠ y" and "(y,x) ∉
using non_sym_dist_pairs_set_iff[of x y xs] by simp+
then have "(y, x) ∉ set (map (λx. (x, x)) xs) ∪
by auto
then show ?thesis
using ‹ "
qed
non_sym_dist_pairs'_indices :
"distinc xs \Longrightarrowx,y) \in> set (no x)
> (<>i < length
(induction xs)
case Nil
then show ?case by auto
case (Cons a xs)
show ?case proof (cases "a = x")
case True
then have "(a#xs) ! 0 = x" and "0 < length
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
by (meson in_set_conv_nth)
then have "(a#xs) ! (Suc j) = y" and "Suc j < length
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
sing Cons.H Cons.pprems((1)
by (metis Suc_mono distinct.simps(2) length_Cons nth_Cons_Suc)
qed
non_sym_dist_pairs'_trans: "distinct xs ==> trans (set (non_sym_dist_pairs' xs))"
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"
"nx<lengthxs y \<>(
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 ‹
by blast
have "ny' = ny"
using ‹
nth_eq_iff_index_eq
by metis
then have "nx < nz
using ‹
then have "nx ≠ nz" by simp
then have "x ≠ z"
using ‹distinct xs›‹xs ! nx = x›‹xs ! nz = z›‹nx < lengthnext
nth_eq_iff_index_eq
by metis
have "remdups xs =xs"
using ‹distinct xs›baut
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›oplen (filt (λy \<>x
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
nth_eq_iff_index_eq
by metis
ultimately have "nz < nx"
using ‹nz'< nx'› by auto
then show "False"
using ‹
qed
then show "(x, z) ∈ set (non_sym_dist_pairs' xs)"
using non_sym_dist_pairs'_elems_distinct(1)[OF ‹'). ∄∈y. \<>
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
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
fix x y z assume "(x,, y) \in set (linear_ord 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_fr<> y \subseteq x) x') \<nd
next
case d
then show ?thesis unfolding linear_order_from_list_position_set
using non_sym_dist_pairs_trans
by (metis UnI2 transE)
qed
‹times>'a)list" wh
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_pairs [] = []= []" |
find_remove :: "('a ==> bool) ==> 'a list ==>listorder (x#xs) = (map (Pair x) xs) @ (list_ordered_pairs xs)"
"find_remove P xs = find_remove' P xs []"
find_remove'_set :
assumes "find_remove' P xs prev = Some (x,xs')"
"P x"
"x ∈ set xs"
"xs' = prev@(remove1 x xs)"
-
have "P x ∧ x ∈ set xs ∧ xs' = prev@(remove1 x xs)"
using proof( xs arbitrary p 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
then show "P x"
and "x ∈ set xs"
and "xs' = prev@(remove1 x xs)"
by blast+
find_remove'_set_rev :
assumes "x ∈ set xs"
and "P x"
"find_remove' P xs prev ≠
assms(1 prooinduction xs arbitrary: prev)
case Nil
then show ?case by auto
case (Cons x' xs)
show ?case proof (cases "P x")
case True
show? using Co by auto
next
case False
then show ?thesis using Cons
using assms(2) by auto
qed
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)
find_remove_set :
assumes "find_remove P xs = Some (x,xs')"
"P x"
<>set
"xs' = (remove1 x xs)"
using assms find_remove'_set[of P xs "[]" x xs'] by auto
find_remove_2' :: "('a==>'b==>bool) ==> 'a list ==> 'b list ==> 'a list ==>
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]))"
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 []""non_syxs =non_sym_dist_pairs' (re xs)"
find_remove_2'_set :
assumes "find_remove_2' P xs ys prev = Some (x,y,xs')"
"P x y"
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
"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"
-
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'])"
Cons.(1) by auto
@[x']) = Some (, 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 Soa"(x,y) \<>
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 "dis (p @xs 🚫
and "distinct (prev@xs) ==> distinct xs'"
and "xs' = prev@(remove1 x xs)"
and "find (P x) ys = Some y"
by blast+
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'"
"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
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(4case True
next
case (Some a)
then have "x' = x" and "a = y"
using ConsCons..prems(1)unindremove_2'.simps b 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 \<open
qed
find_remove_2_strengthening :
assumes "find_remove_2 P xs ys = Some (x,y,xs')"
next
and "∧ x' y' . P' x' y' ==> P x' y'"
"find_remove_2 P' xs ys = Some (x,y,xs')"
using assms find_remove_2'_strengthening
by (metis find_remove_2.simps)
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
case (Cons x' xs)
show ?case proof (cases "find (λy . P x' y) ys")
case None
then show ?thesis
Cons.IH Con.prems by auto
next
case (Some a)
then show ?thesis using Cons.prems unfolding find_remove_2'.simps
by simp
qed
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"
"∃ 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
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)
"' = x" and "a == y"
using Cons.prems
unfolding * find_remove_2'.simps by auto
show ?thesis
using Some
unfolding ‹x' = x›‹a = y› have distinct z" by auto inally show ?thesis nfolding Cons[s by a
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 ‹
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
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"
"∃Some x,y,xs,xs'')"
using assms by (simp add: find_remove_2'_filter)
find_remove_2'_index :
assumes "find_remove_2' P xs ys prev = Some (x,y,xs')"
obtains i i' where "i < length
"xs ! i = x"
"∧ j . j < i ==> find (λy . P (xs ! j) y) ys = None"
"i' < length
"ys ! i' = y"
"∧ j . j < i' ==>¬ P (xs ! i) (ys ! j)"
-
have "∃ i i' . i < length xs ∧ xs ! i = x ∧ (y'#xs') = length xss'" ∧ i' < length ys ∧ ys ! i' = y ∧ (∀ j < i
using assms
proof (induction xs arbitrary: prev xs' x y)
case Nil
thenu remove1 by astforce
next
case (Cons x' xs)
then show ?case proof (cases "find (λusing <openx
case None
then have "find_remove_2' P (x' # xs) ys prev = find_remove_2' P have "leng (flte(<y<
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'_set(1,3[OF *] *] None unfolding find_None_iff
by blast
obtain i i' where "i < length
and "(∀ j < i . find (λy . P (xs ! j) y) ys = None)" and "i' < lengthlength (y' fi(\lambda> y ⊆
and "ys ! i' = y" and "(∀ j < i
using Cons.IH[OF *] by blast
"ui <length
using ‹i < length xs› by auto
moreover have "(x'#xs) ! Suc i = x"
using ‹xs ! i = x›
moreover have "(∀ j < Suc i . find (λy . P ((x'#xs) xs') \le> lelength xss'"
proof -
have "∧ j . j > 0 ==> j < Suc i ==>‹
using ‹
then show ?thesis using None
by (metis neq0_conv nth_Cons_0)
qed
"\<>j
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
by simp
ultimately show ?thesis
using that ‹i' < length
next
case (Some y')
then have "x' = x" and "y' = y"
using Cons.prems by force+
have "0 < length (x'#xs) ∧ ∧ (∀ j < 0 . find (λy . P ((x'#xs) ! j) y) ys = None)"
java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
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 ‹
qed
qed
then show ?thesis using that by blast
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 < i ==>
"i' < length ys"
"ys ! i' = y"
"∧ j . j < i' ==>¬ P (xs ! i) (ys !⟷<> xs)"
using assms find_remove_2'_index[of P xs ys "[]" x y xs'] by auto
find_remove_2''_set_reev :
assumes "x ∈ set xs"
and "y ∈ set ys"
and "P x y"
"find_remove_2' P xs ys prev ≠ None"
assms(1) proof(induction xs arbitrary: prev)
case Nil
then show ?case by auto
case (Cons x' xs)
then show ?case proof (cases "find (λy . P x' y) ys")
case None
then have "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
find_remove_2'_diff_prev_None :
"(find_remove_2' P xs ys prev = None ==> find_remove_2' P xs ys prev' = None)"
(induction xs arbitrary: prev prev')
case Nil
then show ?case by auto
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 ==># xss'). ∄xss') \<\<
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''))"
(induction xs arbitrary: prev prev')
case Nil
then show ?case by auto
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 sh?thesis using Cons by auto
qed
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 (etis old.prod.exh option.e.exhaust)
find_remove_2_set :
assumes "find_remove_2 P xs ys = Some (x,y,xs')"
"P x y"
"x ∈ set xs"
"y ∈
"distinct xs ==> set xs' = (set xs) - {x}"
"distinct xs ==> distinct xs'"
"xs' = (remove1 x xs)"
using assms find_remove_2'_set[of P xs ys "[]" x y xs']
unfolding find_remove_2.simps by auto
find_remove_2_removeAll :
assumes "find_remove_2 P xs ys = Some (x,y,xs')"
and "distinct xs"
"xs' = removeAll x xs"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by (simp add: assms(2) distinct_remove1_removeAll)
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)[F assms]
by (simp add: length_remove1)
separate_by :: "('a ==> bool) ==>
"separate_by P xs = (filter P xs, filter (λ x . ¬ P x) xs)"
separate_by_code[code] :
"separate_by P xs = foldr (λx (prevPass,prevFail) . if P x then (x#prevPass,prevFail) else (prevPass,x#prevFail)) xs ([],[])"
(induction xs)
case Nil
then show ?case by auto
case (Cons a xs)
let ?f = "(λx (prevPass,prevFail) . if P x then (x#prevPass,prevFail) else (prevPass,x#prevFail))"
have "(filter P xs, filter (λ x . ¬xs' \inset (x # ss') <and
using Cons.IH by auto
moreover have "separate_by P (a#xs) = ?f a (filter P xs, filter (λ x . ¬ P x) xs)"
by auto
ultimately show ?case
by "\Andx. x \<in xs)"
find_remove_2_all :: "('a ==> 'b ==> bool) ==> 'a list ==> 'b list ==> (('a × 'b) list × 'a list)" where
"find_remove_2_all P xs ys =
(map (λ x . (x, the (find (λy . P x y) ys))) (filter (λ x . find (λy . P x y) ys ≠ None) xs)
,filter (λ x . find (λy . P x y) ys = None) xs)"
find_remove_2_all' :: "('a ==> 'b ==> bool) ==> 'a list ==> 'b list ==> (('a ×'b) list × 'a list)" where
"find_remove_2_all' P xs ys =
(let (successesWithWitnesses,failures) = separate_by (λ(x,y) . y ≠ None)
in (map (λ (x,y) . (x, the y)) successesWithWitnesses, map fst failures))"
find_remove_2_all_code[code] :
"find_remove_2_all P xs ys = find_remove_2_all' P xs ys"
-
let ?s1 = "map (λ x . (x, the (find (λy . P x y) ys))) (filter (λ x . find (λy . P x y) ys ≠ None) xs)"
let ?f1 = "filter (λ x . find (λy . P x y) ys = None) xs"
let ?s2 = "map (λ (x,y) . (x, the y)) (filter (λ(x,y) . y ≠ None) (map (λ> set (li xs)"
let ?f2 = "map fst (filter (λ(x,y) . y = None) (map (λ<>{
have "find_remove_2_all P xs ys = (?s1,?f1)"
bysimp
moreover have "find_remove_2_all' P xs ys = (?s2,?f2)"
proof -
have "\<>.(a::'a, x::'b option \Rightarrow x)) = (🚫
by force
then show ?thesis
unfolding find_remove_2_all'.simps Let_def separate_by.simps
by force
qed
moreover have "?s1 = ?s2"
by (induction xs; auto)
moreover have "?f1 = ?f2"
by (induction xs; auto)
ultimately show ?thesis
by simp
moreover have "set (map set (pow_list (x#xs)))
= set (map set (pow_list xs)) ∪ (image (insert x) (set (map set (pow_list xs))))"
proof -
have "\And ys . ys\in>set (mapse (powlist (x#xs))) ==> ys ∈ set (map set (pow_list xs)) ∪ (image (insert x) (set (map set (pow_list xs))))"
proof -
fix ys assume "ys ∈ set (map set (pow_list (x#xs)))"
then consider (a) "ys ∈ set (map set (pow_list xs))" |
(b) "ys ∈ set (map set (map ((#) x) (pow_list xs)))"
unfolding pow_list.simps Let_def by auto
then show "ys ∈ set (map set (pow_list xs)) ∪ (image (insert x) (set (map set (pow_list xs))))"
by (cases; auto)
qed
moreover have "∧ ys . ys ∈ set (map set (pow_list xs)) ∪ (image (insert x) (set (map set (pow_list xs)))) ==> ys ∈ set (map set (pow_list (x#xs)))"
proof -
fix ys assume "ys ∈ set (map set (pow_list xs)) ∪ (image (insert x) (set (map set (pow_list xs))))"
then consider (a) "ys ∈ set (map set (pow_list xs))" |
(b) "ys ∈ (image (insert x) (set (map set (pow_list xs))))"
blast
then show "ys ∈ set (map set (pow_list (x#xs)))"
unfolding pow_list.simps Let_def by (cases; auto)
qed
ultimately show ?thesis by blast
qed
ultimately show ?case
by auto
‹Removing Subsets in a List of Sets›
remove1_length : "x ∈ set xs ==> length (remove1 x xs) < length> y" y" and "(y,x)∉
by (induction xs; auto)
remove_subsets :: "'a set list ==> 'a set list" where
"remove_subsets [] = []" |
"remove_subsets (x#xs) = (case find_remove (λ y . x ⊂ y) xs of
Some (y',xs') ==> # fiilte (λsu x) xs')"
None ==> x # (remove_subsets (filter (λ y . ¬(y ⊆ x)) xs)))"
by pat_completeness auto
-
have "∧x xs. find_remove ((⊂) x) xs = None ==> (filter (λy. ¬ y ⊆ x) xs, x # xs) ∈ measure length"
by (metis dual_order.trans impossible_Cons in_measure length_filter_le not_le_imp_less)
moreover have "(∧(x :: 'a set) xs x2 xa y. find_remove ((⊂) x) xs = Some x2 ==>
proof -
fix x :: "'a set"
fix xs y'xs'moreover h have "\<existsxsthen show ?thesis
assume "find_remove ((⊂) x) xs = Some y'xs'" and "(y', xs') = y'xs'"
then have "find_remove ((⊂) x) xs = Some (y',xs')"
by auto
have "length xs' = length xs - 1"
using find_remove_set(2,3)[OF ‹find_remove ((⊂) x) xs = Some (y',xs')›]
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 (λy. ¬ y ⊆ x) xs') ≤ length xs'"
by simp
then have "length (y' # filter (λy. ¬ y ⊆
by simp
then have "length (y' # filter (λy. ¬ y ⊆ x) xs') ≤
unfolding ‹length (y'#xs') = length xs›[symmetric] by simp
then show "(y' # filter (λy. ¬ y ⊆ x) xs', x # xs) ∈ measure length"
by auto
qed
ultimately show ?thesis
by (relation "measure length"; auto)
remove_subsets_set : "set (remove_subsets xss) = {xs . xs ∈ set xss ∧ (∄ xs' . xs' ∈ set xss ∧ xs ⊂ xs')}"
(induction "length xss" arbitrary: xss rule: less_induct)
case less
show ?case proof (cases xss)
case Nil
then show ?thesis by auto
next
case (Cons x xs)
show ?thesis proof (cases "find_remove (λ y . x ⊂ y) xss'")
case None
then have "(∄ xs' . xs' ∈ set xss' ∧ x ⊂ xs')"
using find_remove_None_iff by metis
have "length (filter (λ y . ¬
using Cons
by (meson dual_order.trans impossible_Cons leI length_filter_le)
have "remove_subsets (x#xss') = x # (remove_subsets (filter (λ y . ¬(y ⊆ x)) xss'))"
using None by auto
then have "set (remove_subsets (x#xss')) = insert x {xs ∈ set (filter (λy. ¬ y ⊆(Cons a xs)
using less[OF ‹length (filter (λ y . ¬(y ⊆ x)) xss') < show
by auto
have "…)∧\in set(x#xss') \and xs ⊂
proof -
have "\ q ==> xs ∈ {xs ∈ set (x # xss'). ∄xs'. xs' ∈ set (x # xss') ∧ xs ⊂ xs'}"
proof -
fix xs assume "xs ∈ insert x {xs ∈ set (filter (λy. ¬ y ⊆ x) xss'). ∄xs'. xs' ∈
then consider "xs = x" | "xs ∈ set (filter (λy. ¬ y ⊆ x) xss') ∧ (∄xs'. xs' ∈ set (filter (λ>Linea Orderon Su\close
by blast
then show "xs ∈ {xs ∈ set (x # xss'). ∄xs'. xs' ∈ set (x # xss') ∧ xs ⊂ xs'}"
using ‹(∄ xs' . xs' ∈ set xss' ∧ x ⊂ xs')›
qed
moreover have "∧ xs . xs ∈ {xs ∈ set (x # xss'). ∄xs'. xs' ∈ ==> xs ∈ insert x {xs ∈
proof -
fix xs assume "xs ∈ {xs ∈ set (x # xss'). ∄xs'. xs' ∈ set (x # xss') ∧ xs ⊂ xs'}"
then have "xs \ b ==>whe
by blast+
then consider "xs = x" | "xs ∈ set xss'" by auto
then show "xs ∈ insert x {xs ∈ set (filter (λy. ¬ y ⊆ x) xss'). ∄(23)[OF Cons.(1,)]True by auto
proof cases
case 1
then show ?thesis by auto
next
case 2
show ?thesis proof (cases "xs ⊆ x")
True
then show ?thesis
using ‹∄xs'. xs' ∈ set (x # xss') ∧ xs ⊂ xs'› by auto
next
case False
then have "xs ∈ set (filter (λy. ¬ y ⊆ x) xss')"
using 2 by auto
moreover have "∄xs'. xs' ∈ set (filter (λy. ¬ y ⊆ x) xss') ∧ xs ⊂ xs'"
using ‹∄xs'. xs' ∈ set (x # xss') ∧
ultimately show ?thesis by auto
qed
qed
qed
ultimately show ?thesis
by (meson subset_antisym subset_eq)
qed
finally show ?thesis unfolding Cons[symmetric] by assumption
next
case (Some a)
then obtain y' xs' where *: "find_remove (λ y . x ⊂ y) xss' = Some (y',xs')" by force
have "length xs' = length xss' - 1"
using find_remove_set(2,3)[OF *]
by (simp add: length_remove1)
then have "length (y'#xs') = length xss'"
using find_remove_set(2)[OF *]
using remove1_length by fastforce
have "length (filter (λy. ¬ y ⊆ x) xs') ≤ length xs'"
by simp
then have "length (y' # filter (λy. ¬ y ⊆ x) xs') ≤ length xs' + 1"
by simp
instance by (intro_classes)
unfolding ‹
then have "length (y' # filter (λy. ¬ y ⊆ x) xs') < length
unfolding Cons by auto
"emov (x#xss') = remo (y'#(filter(λ x) xs'))"
using * by auto
then have "set (remove_subsets (x#xss')) = {xs ∈ set (y' # filter (λy. ¬ y ⊆ x) xs'). ∄xs'a. xs'a ∈ set (y' # filter (λy. ¬ y ⊆ x) xs') ∧ xin sum :: (li,li)linorder
using less[OF ‹length (y' # filter (λy. ¬ y ⊆
by auto
also have "… = {xs . xs ∈ set (x#xss') ∧ (∄ xs' . xs' ∈
proof -
have "∧ xs . xs ∈ {xs ∈ set (y' # filter (λy. ¬ y ⊆ x) xs'). ∄xs'a. xs'a ∈ ==> :
proof -
fix xs assume "xs ∈ {xs ∈ set (y' # filter (λy. ¬
then have "xs ∈ set (y' # filter (λy. ¬ y ⊆ x) xs')" and "∄xs'a. xs'a ∈ set (y' # filter (λy. ¬ y ⊆ x) xs') ∧ xs ⊂ xs'a"
by blast+
have "xs ∈ set (x # xss')"
using ‹2,3)[OF *]
by auto
moreover have "∄xs'. xs' ∈ set (x # xss') ∧ xs ⊂ xs'"
using ‹∄xs'a. xs'a ∈ set (y' # filter (λy. ¬ y ⊆ x) xs') ∧ xs ⊂ xs'a› find_remove_set[OF *]
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 ∈ {xs ∈ set (x # xss'). ∄xs'. xs' ∈ set (x # xss') ∧ xs ⊂ xs'}"
by blast
qed
moreover have "∧ xs . xs ∈ {xs ∈ set (x # xss'). ∄xs'. xs' ∈ set (x # xss') ∧x! ny =y" a "n<ny ==> xs ∈ {xs ∈ set (y' # filter (λy. ¬ y ⊆ x) xs'). ∄xs'a. xs'a ∈xs" and "n <length
proof -
fix xs assume "xs ∈ {xs ∈ set (x # xss'). ∄xs'. xs' ∈ set (x # xss') ∧ xs ⊂ xs'}"
then have "xs ∈ set (x # xss')" and "∄xs'. xs' ∈ set (x # xss') ∧ xs ⊂ xs'"
by blast+
then have "xs ∈ set (y' # filter (λy. ¬ y ⊆ x) xs')"
using find_remove_set[OF *]
by (metis filter_list_set in_set_remove1 list.set_intros(1) list.set_intros(2) psubsetI set_ConsD)
moreover have "∄xs'a. xs'a ∈ set (y' # filter (λy. ¬ y ⊆ x) xs') ∧ xs ⊂ xs'a"
by blay blast
by (metis filter_is_subset list.set_intros(2) notin_set_remove1 set_ConsD subset_iff)
ultimately show "xs ∈ {xs ∈ set (y' # filter (λy. ¬ y ⊆ x) xs'). ∄xs'a. xs'a ∈ set (y' #
by blast
qed
ultimately show ?thesis by blast
qed
finally show ?thesis unfolding Cons by assumption
qed
qed
‹Linear Order on Sum›
sum :: (ord,ord) ord
less_eq_sum :: "'a + 'b ==> 'a + 'b ==> bool" where
"less_eq_sum (Inl a) (Inl b) = (a ≤ b)" |
"less_eq_sum (Inl a) (Inr b) = True" |
"less_eq_sum (Inr a) (Inl b) = False" |
"less_eq_sum (Inr a) (Inr b) = (a ≤ b)"
less_sum :: "'a + 'b ==> 'a + 'b ==> bool" where
"less_sum a b = (a ≤ b ∧ a ≠ b)"
by (intro_classes)
sum :: (linorder,linorder) linorder
less_le_not_le_sum :
fixes x :: "'a + 'b"
and y :: "'a + 'b"
"(x < y) = (x ≤ y ∧
by (cases x; cases y; auto)
order_refl_sum :
fixes x :: "'a + 'b"
shows "x ≤ x"
by (cases x; auto)
order_trans_sum :
fixes x :: "'a + 'b"
fixes y :: "'a + 'b"
shshows "x e
shows "x ≤ y ==> y ≤ z ==> x ≤ z"
by (cases x; cases y; cases z; auto)
antisym_sum :
fixes x :: "'a + 'b"
fixes y :: "'a + 'b"
shows "x ≤ y ==> y ≤ x ==> x = y"
by (cases x; cases y; auto)
linear_sum :
fixes x :: "'a + 'b"
fixes y :: "'a + 'b"
shows "x ≤ y ∨ y ≤ x"
by (cases x; cases y; auto)
using less_le_not_le_sum order_refl_sum order_trans_sum antisym_sum linear_sum
by (intro_clas; metis+)
‹Removing Proper Prefixes›case x;; ca y; au)
remove_proper_prefixes :: "'a list set ==> 'a list set" where
"by me meetis
remove_proper_prefixes_code[
"remove_proper_prefixes (set xs) = set (filter (λx . (∀ y ∈ set xs . is_prefix x y ⟶ x = y)) xs)"
-
have *: "remove_proper_prefixes (set xs) = Set.filter (λ zs . ∄ys . ys ≠ [] ∧ zs @ ys ∈
unfolding remove_proper_prefixes_def by force
have "∧ zs . (∄ys . ys ≠ [] ∧ zs @ ys ∈ (set xs))
unfolding is_prefix_prefix by auto
then show ?thesis
unfolding * filter_set by auto
‹Underspecified List Representations of Sets›
as_list_helper :: "'a set \Rightarrowlist"where
"as_list_helper X = (SOME xs . set xs = X ∧ distinct xs)"
as_list_helper_props :
ssumes "finite X"
shows "set (as_list_helper X) = X"
and "distinct (as_list_helper X)"
using finite_distinct_list[OF assms]
using someI[of "λ xs . set xs = X ∧ distinct xs"]
by (metis as_list_helper_def)+
‹Assigning indices to elements of a finite set›
assign_indices :: "('a :: linorder) set ==> ('a ==> nat)" where
"assign_indices xs = (λ x . the (find_index ((=)x) (sorted_list_of_set xs)))"
have *:"set (sorted_list_of_set xs) = xs"
by (simp add: assms)
havehave "\Andx y x\<>
proof -
fix x y assume "x∈xs" and "y∈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[of "sorted_list_of_set xs" "((=) x)"]
using ‹
by blast
then have "assign_indices xs x = i"
auto
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 ‹y∈xs› unfolding *
by blast
then have "assign_indices xs y = j"
by auto
then have "i = j"
using ‹assign_indices xs x = assign_indices xs y›‹assign_indices xs x = i›
by auto
then have "find_index ((=)y) (sorted_list_of_set xs) = Some i"
using ‹find_index ((=)y) (sorted_list_of_set xs) = Some j›
by auto
show "x = y"
using find_index_index(2)[OF ‹find_index ((=)x) (sorted_list_of_set xs) = Some i›]
using find_index_index(2)[OF ‹
by auto
qed
moreover have "(assign_indices xs) ` xs = {..<card xs}"
proof
show "assign_indices xs ` xs ⊆ {..<card xs}"
proof
fix i assume "i ∈ assign_indices xs ` xs"
then obtain x where "x ∈ xs" and "i = as
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" "((=) x)"]
using ‹x∈xs› unfolding *
by blast
ultimately have "find_index ((=)x) (sorted_list_of_set xs) = Some i"
by auto
show "i ∈ {..<card xs}"
using find_index_index(1)[OF ‹find_index ((=)x) (sorted_list_of_set xs) = Some i›]
by auto
qed
show "{..<card xs} ⊆ assign_indices xs ` xs"
proof
fix i assume "i ∈ {..<card xs}"
then have "i < length (sorted_list_of_set xs)"
by auto
then have "sorted_list_of_set xs ! i ∈ xs"
using "*" nth_mem by blast
then obtain j where "find_index ((=) (sorted_list_of_set xs ! i)) (sorted_list_of_set xs) = Some j"
using find_index_exhaustive[of "sorted_list_of_set xs" "((=) (sorted_list_of_set xs ! i))"]
unfolding *
by blast
have "i = j"
using find_index_index(1,2)[OF ‹find_index ((=) (sorted_list_of_set xs ! i)) (sorted_list_of_set xs) = Some j›]
using ‹
then show "i ∈ assign_indices xs ` xs"
using ‹find_index ((=) (sorted_list_of_set xs ! i)) (sorted_list_of_set xs) = Some j›
by (metis ‹sorted_list_of_set xs ! i ∈ xs› assign_indices.elims image_iff option.sel)
qed
qed
ultimately sho show ?th
unfolding bij_betw_def inj_on_def by blast
‹Other Lemmata› non_sy.simps 🚫
foldr_elem_check:
assumes "list.set xs ⊆ A"
shows "foldr (λ x y . if x ∉ A then y else f x y) xs v = foldr f xs v"
using assms by (induction xs; auto)
foldl_elem_check:
assumes "list.set xs ⊆ A"
shows "foldl (λ y x . if x ∉ A then y else f y x) v xs = foldl f v xs"
using assms by (induction xs rule: rev_induct; auto)
foldr_length_helper :
assumes "length xs = length ys"
shows "foldr (λ'_tof remdups xs", OF di]
using assms by (induction xs ys rule: list_induct2; auto)
list_append_subset3 : "set xs1 ⊆ set ys1 ==> set xs2 ⊆ set ys2 ==> set xs3 ⊆ set ys3 ==> set (xs1@xs2@xs3) ⊆ set(ys1@ys2@ys3)" by auto
subset_filter : "set xs ⊆ set ys ==> set xs = set (filter (λ x . x ∈ set xs) ys)"
by auto
map_filter_elem :
assumes "y ∈ set (List.map_filter f xs)"
obtains x where "x ∈ X"java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
and "f x = Some y"
using assms unfolding List.map_filter_def
by auto
filter_length_weakening :
assumes "∧ q . f1 q ==> f2 q"
shows "length (filter f1 p) ≤ length (filter f2 p)"
(induction p)
case Nil
then show ?case by auto
case (Cons a p)
then show ?case using assms by (cases "f1 a"; auto)
max_length_elem :
fixes xs :: "'a list set"
assumes "finite xs"
"x ≠
"∃ x ∈ xs . ¬(∃ y ∈ xs . length y > length x)"
assms proof (induction xs)
case empty
then show ?case by auto
case (insert x F)
then show ?case proof (cases "F = {}")
case True
then show ?thesis by blast
next
case False
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
insert.IH by blast
then show ?thesis using dual_order.strict_trans by (cases "length x > length y"; auto)
qed
min_length_elem :
fixes xs :: "'a list set"
assumes "finite xs"
and "xs ≠ {}"
""\<>x
assms proof (induction xs)
case empty
then show ?case by auto
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 ∈ F" and "¬(∃ y' ∈ 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
list_property_from_index_property :
assumes "∧ i . i < length xs ==> P (xs ! i)"
shows "∧ x . x ∈ set xs ==> P x"
by (metis assms in_set_conv_nth)
list_distinct_prefix :
assumes "∧ i . i < length xs ==> xs ! i ∉x, y) in> set (non_sym_distpairs ) ∧xs)
shows "distinct xs"
-
have "∧ j . distinct (take j xs)"
proof -
fix j
show "distinct (take j xs)"
proof (induction j)
case 0
then show ?case by auto
next
then show ?case proof (cases "Suc j ≤ length xs")
case True
then have "take (Suc j) xs = (take j xs) @ [xs ! j]"
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 = take j xs" by auto
then show ?thesis using Suc.IH by auto
qed
qed
qed
then have "distinct (take (length xs) xs)"
by blast
then show ?thesis by auto
concat_pair_set :
"set (concat (map (λx. map (Pair x) ys) xs)) = {xy . fst xy ∈ set xs ∧ snd xy ∈ set ys}"
by auto
list_set_sym :
"set (x@y) = set (y@x)" by auto
list_contains_last_take :
"x \insexs"
shows "∃ i . 0 < i ∧ i ≤ length xs ∧ last (take i xs) = x"
metis Su assms hhd_dro in_et_conv_nt last_sn tak zer thsh ?thesis unfl liby auto
integer_singleton_least :
assumes "{x . P x} = {a::integer}"
shows "a = (LEAST x . P x)"
by (metis Collect_empty_eq Least_equality assms insert_not_empty mem_Collect_eq order_refl singletonD)
sort_list_split :
"∀by (met
using sorted_append by fastforce
set_map_subset :
assumes "x ∈
and "t ∈ set (map f [x])"
"t ∈ set (map f xs)"
using assms by auto
rev_induct2[consumes 1, case_names Nil snoc]:
assumes "length xs = length ys"
and "P [] []"
and "(∧"
shows "P xs ys"
assms proof (induct xs arbitrary: ys rule: rev_induct)
case Nil
then show ?case by auto
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)
qedqed
finite_set_min_param_ex :
assumes "finite XS"
and "∧ x . x ∈ XS ==>∃ k . ∀ k' . k ≤ k' ⟶ P x k'"
"∃ (k::nat) . ∀ x ∈ XS . P x k"
-
obtain f where f_def : "∧ x . x ∈ XS ==>∀ k' . (f x) ≤ k' ⟶ P x k'"
using assms(2) by meson
let ?k = "Max (image f XS)"
have "∀ x ∈ XS . P x ?k"
using f_def by (simp add: assms(1))
then show ?thesis by blast
list_max :: "nat list ==> nat" where
"list_max [] = 0" |
"list_max xs = Max (set xs)"
list_max_is_max : "q ∈ set xs ==> q ≤
by (metis List.finite_set Max_ge length_greater_0_conv length_pos_if_in_set list_max.elims)
list_prefix_subset : "\<>ys
list_map_set_prop : "x ∈ set (map f xs) ==>∀ y . P (f y) ==> P x" by auto
list_c list_concat_non_el : "x ∉> set ys \ ==>x@ys)" yaut
list_prefix_elem : "x ∈ set (xs@ys) ==> x ∉ set ys ==> x ∈ set xs" by auto
list_map_source_elem : "x ∈ set (map f xs) ==>∃ x' ∈ set xs . x = f x'" by auto
maximal_set_cover :
fixes X :: "'a set set"
assumes "finite X"
and "S ∈ X"
"∃ S' ∈ X . S ⊆ S' ∧ (∀ S'' ∈ X . ¬(S' ⊂ S''))"
(rule ccontr)
assume "¬ (∃S'∈X. S ⊆
then have *: "∧ T . T ∈ X ==> S ⊆ T ==>∃ T' ∈ X . T ⊂ T'"
by auto
have "∧ k . ∃ ss . (length ss = Suc k) ∧ (hd ss = S) ∧ (∀ i < k
proof -
fix k show "∃ ss . (length ss = Suc k) ∧ (hd ss = S) ∧ (∀uusin f(2OF \<openfind_index
proof (induction k)
case 0
have "length [S] = Suc 0 ∧
then show ?case by blast
next
case (Suc k)
then obtain ss where "length ss = Suc k"
and "hd ss = S"
and "(∀i<k. ss ! i ⊂ xs) `` xs = {..<card
and "set ss ⊆ X"
by blast
then have "ss ! k ∈ proof
by auto
moreover have "S ⊆ (ss ! k)"
proof -
have "∧ i . i < Suc k ==> S ⊆ (ss ! i)"
proof -
fix i assume "i < Suc
then show "S ⊆ (ss ! i)"
proof (induction i)
case 0
then show ?case using ‹hd ss = S›‹length ss = Suc k›
by (metis hd_conv_nth list.size(3) nat.distinct(1) order_refl)
next
case (Suc i)
then have "S ⊆ ss ! i" and "i < k
then have "ss ! i ⊂ ss ! Suc i" using ‹(∀i<k. ss ! i ⊂ ss ! Suc i)› by blast
then show ?case using ‹
qed
qed
then show ?thesis using ‹length ss = Suc k› by auto
qed
ultimately obtain T' where "T' ∈ X" and "ss ! k ⊂ T'"
using * by meson
let ?ss = "ss@[T']"
have "length ?ss = Suc (Suc k)"
using ‹
moreover have "hd ?ss = S"
using ‹hd ss = S› by (metis ‹
moreover have "(∀i < Suc k. ?ss ! i ⊂ ?ss ! Suc i)"
using ‹case F
by (metis Suc_lessI ‹length ss = Suc k› diff_Suc_1 less_SucE using \<openx
moreover have "set ?ss ⊆ X"
using ‹set ss ⊆ X›‹T' ∈ X› by auto
ultimately show ?case by blast
qed
qed
then obtain ss where "(length ss = Suc (card X))"
and "(hd ss = S)"
and "(∀ i < card X . ss ! i ⊂ ss ! (Suc i))"
and "(set ss ⊆ X)"
by blast
then have "(∀ i < length ss - 1 . ss ! i ⊂
by auto
have **: "∧ i (ss :: 'a set list) . (∀ i < length ss - 1 . ss ! ishow "i \<in
proof -
fix i
fix ss :: "'a set list"
assume "i < length ss " and "(∀
then show "∀ s ∈ set (take i ss) . s ⊂find_index_inindex(1[OF ‹
proof (induction i)
case 0
then show ?case by auto
next
case (Suc i)
then have "∀s∈set (take i ss). s ⊂ ss ! i" by auto
then have "∀s∈set (take i ss). s ⊂
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)
by (simp add: take_Suc_conv_app_nth)
ultimately show ?case by auto
qed
qed
have "distinct ss"
using ‹(∀ i < length ss - 1 . ss ! i ⊂(1) pr(in xs arbit prev)
proof (induction ss rule: rev_induct)
case Nil
then show ?case by auto
next
case (snoc a ss)
from snoc.prhave "<>i
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 ∉ set ss"
using **[OF snoc.prems, of "length (ss @ [a]) - 1"] by auto
ultimately show ?case by auto
qed
then have "card (set ss) = Suc (card X)"
using \openle ss Suc (cardX))›
then show "False"
using ‹set ss ⊆ X›‹ length(sorted xs)"
map_set :
assumes "x ∈ set xs"
shows "f x ∈ set (map f xs)" using assms by auto
maximal_distinct_prefix :
assumes "¬ distinct xs"
obtains n where "distinct (take (Suc n) xs)"
and "¬ (distinct (take (Suc (Suc n)) xs))"
assms proof (induction xs rule: rev_induct)
case Nil
then show ?case by auto
case (snoc x xs)
show ?case proof (cases "distinct xs")
case True
then have "distinct (take (length xs) (xs@[x]))" by auto
moreover have"¬
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
distinct_not_in_prefix :
assumes "∧ i . (∧ x . x ∈ set (take i xs) ==>
shows "distinct xs"
using assms list_distinct_prefix by blast
list_index_fun_gt : "∧ xs (f::'a ==>
(∧ i . Suc i < length ==> j < i ==> i < lengthj\close ==> f (xs ! j) > f (x unfolding find.simps
-
fix xs::"'a list"
fix f::"'a ==> nat"
fix i j
assume "(∧ i . Suc i < length xs ==> f (xs ! i) > f (xs ! (Suc i)))"
and "j < i by bllast
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
case (Suc x)
then show ?case
proof -
have f1: "∀n. ¬
using Suc.prems(1) by presburger
have f2: "∀n na. ¬ n < na ∨x')"
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 "\\>Suc j< i
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
finite_set_elem_maximal_extension_ex :
assumes "xs ∈ S"
and "finite S"
"∃ ys . xs@ys ∈ S ∧¬ (∃ zs . zs ≠ [] ∧ xs@ys@zs ∈ S)" ‹
case empty
then show ?case by auto
case (insert x S)
consider (a) "∃ ys . x = xs@ys ∧¬ (∃ zs . zs ≠ [] ∧ xs@ys@zs ∈ (insert x S))" |
(b) "¬(∃ ys . x = xs@ys ∧¬ (∃ zs . zs ≠ [] ∧
by blast
then show ?case proof cases
case a
next
case b
then show ?thesis proof (cases "∃ vs . vs ≠
case True
then obtain vs where "vs ≠ []" and "xs@vs ∈ S"
by blast
have "∃ys. xs @ (vs @ ys) ∈ S ∧ (∄zs. zs ≠ [] ∧ xs @ (vs @ ys) @ zs ∈ S)"
using insert.IH[OF ‹xs@vs ∈ S›] by auto
then have "∃ys. xs @ (vs @ ys) ∈ S ∧ (∄zs. zs ≠ [] ∧ xs @ (vs @ ys) @ zs ∈ (insert x S))"
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 alse
then show ?thesis using insert.prems
by (metis append_is_Nil_conv append_self_conv insertE same_append_eq)
qed
qed
list_index_split_set:
assumes "i < length xs"
"set xs = set ((xs ! i) # ((take i xs) @ (drop (Suc i) xs)))"
assms proof (induction xs arbitrary: i)
case Nil
then show ?case by auto
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
have *: "take (Suc j) (x#xs) = x#(take j xs)" by auto
have **: "drop (Suc (Suc j)) (x#xs) = (drop (Suc j) xs)" by "find_remove_2 Pxs ys = = find_' P xs ys []"
have ***: "(x # xs) ! Suc j = xs ! j" by auto
show ?thesis
using ‹set xs = set ((xs ! j) # ((take j xs) @ (drop (Suc j) xs)))›
unfolding Suc * ** *** by auto
qed
max_by_foldr :
assumes "x ∈ set xs"
shows "f x < Suc (foldr (λ x' m . max (f x') m) xs 0)"
using assms by (induction xs; auto)
Max_elem : "finite (xs :: 'a set) ==> xs ≠ {} ==>∃ x ∈ xs . Max (image (f :: 'a ==> nat) xs) = f x"
by (metis (mono_tags, opaque_lifting) Max_in empty_is_image finite_imageI imageE)
card_union_of_singletons :
assumes "∧ S . S ∈x y"
"card (∪ SS) = card SS"
-
let ?f = "λ x . {x}"
have "bij_betw ?f (∪ SS) SS"
unfolding bij_betw_def inj_on_def using assms by fastforce
then show ?thesis
using bij_betw_same_card by blast
card_union_of_distinct :
assumes "∧ S1 S2 . S1 ∈ SS ==> S2 ∈ SS ==> S1 = S2 ∨ f S1 ∩
and "finite SS"
and "∧ S . S ∈ SS ==>> set ys"
"card (image f SS) = card SS"
-
from assms(2) have "∀ S1 ∈ ==>∀ foldr_length_h:
proof (induction SS)
case empty
then show ?case by auto
next
case (and ""find (P x) ys= Some y"
then have "¬ (∃ y ∈ F . f y = f x)"
by auto
then have "f x ∉ image f F"
by a 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"
usin insertby auto
moreover have "card (insert x F) = Suc (card F)"
using insert by auto
ultimately show ?case
by sim
qed
then show ?thesis
using assms by simp
take_le :
assumes "i ≤ length xs"
shows "take i (xs@ys) = take i xs"
by (simp add: assms less_imp_le_nat)
butlast_take_le :
assumes "i ≤ 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))
distinct_union_union_card :
assumes "finite xs"
and "∧ x1 x2 y1 y2 . x1 ≠ x2 ==>ob xw x\inxs"
and "∧ x1 y1 y2 . y1 ∈ f x1 ==> y2 ∈ f x1 ==>
and "∧ x1 . finite (f x1)"
and "∧ y1 . finite (g y1)"
and "∧ y1 . g y1 ⊆
and "finite zs"
"(\ auto
-
have "(∑
using assms(1,2) proof induction
case empty
then show ?case by auto
next
case (insert x xs)
then have "(then sh ?ca byaut
then have "(∑
moreover have "(∑
using insert.hyps by auto
moreover have "card (∪x∈(insert x xs). ∪ (g ` f x)) = card (∪x∈
proof -
have "((∪
by blast
have *: "(∪x∈xs. ∪ (g ` f x)) ∩ (∪ (g ` f x)) = {}"
proof (rule ccontr)
assume "(∪x∈xs. ∪ (g ` f x)) ∩∪ (g ` f x) ≠ {}"
then obtain z where "z ∈∪ (g ` f x)" and "z ∈ (∪x∈xs. ∪ (g ` f x))" by blast
then ob
then have "x' ≠ x" and "x' ∈us Cons.prem(1find_c find_set by fastf
have "∪ (g ` f x') ∩∪ qed
using insert.prems[OF ‹
by blast
then show "False"
using ‹z ∈∪ (g ` f x')›
qed
have **: "finite (\and(prev @ xs) 🚫
using assms(4) assms(5) by blast
have ***: "finite (∪x∈xs. \case
by (simp add: assms(4) assms(5) insert.hyps(1))
have "card ((∪x∈xs. ∪ (g ` f x)) ∪∪ (g ` f x)) = card (∪x∈xs. ∪ (g ` f x)) + card (∪ (g ` f x))"
using card_Un_disjoint[OF *** ** *] by simp
then show ?thesis
unfolding ‹((∪x∈
qed
ultimately show ?case by linarith
qed
moreover have "card (∪
proof -
have "(∪ x ∈ xs . (∪ y ∈min_len :
using assms(6) by (simp add: UN_least)
moreover have "finite (∪
by (simp add: assms(1) assms(4) assms(5))
ultimately show ?thesis
using assms(7)
y(simp add: card_m)
qed
ultimately show ?thesis
by linarith
set_concat_elem :
assumes "x ∈
obtains xs where "xs ∈ set xss" and "x ∈ set xs" then
using assms by auto
set_map_elem :
assumes "y ∈
obtains x where "y = f x" and "x ∈ set xs"
using assms by auto
finite_snd_helper:
assumes "finite xs"
shows "finite {z. ((q, p), z) ∈ xs}"
-
have "{z. ((q, p), z) ∈ xs} ⊆ (λ((a,b),c) . c) ` xs"
proof
fix x assume "x ∈ {z. ((q, p), z) ∈ xs}"
then have "((q,p),x) ∈
then show "x ∈
qed
then show ?thesis using assms
using finite_surj by blast
java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 3
fold_dual : "fold (λ 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)
recursion_renaming_helper :
assumes "f1 = (λx . if P x then x else f1 (Suc x))"
and "f2 = (λx . if P x then x else f2 (Suc x))"
and "∧ x . x ≥ k ==> P x"
"f1 = f2"
fix x
show "f1 x = f2 x"
proof (induction "k - x" arbitrary: x)
case 0
then have "x ≥ k"
by auto
then show ?case
using assms(3) by (simp add: assms(1,2))
next
case (Suc k')
case True
then show ?thesis by (simp add: assms( then sho show ? by auto
next
case False
moreover have "f1 (Suc x) = f2 assumes "\And . <length then show ?case
using Suc.hyps(1)[of "Suc x"] Suc.hyps(2) by auto
ultimately show ?thesis by (simp add: assms(1,2))
qed
qed
minimal_fixpoint_helper :
assumes "f = (λ s)"
and "∧ x . x ≥ k ==>
"P (f x)"
and "∧ x' . x' ≥ x ==> x' < f x ==>¬)"
-
have "P (f x) ∧ (∀
proof (induction "k-x" arbitrary: x)
case 0
then have "P x"
using assms(2) by auto
moreover have "f x = 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 "∧ x' . x' ≥ Suc x ==> x' < f (Suc x) ==>
shows "find_remove_2 P P' xs ys = Some (x,y,xs')"
show ?case proof (cases "P x")
case True
then have "f x = x"
by (simp add: assms(1))
show then have "take (Suc j j) xs = (takej xs) [x ! j]"
using True unfolding ‹
next
case False
then have "f x = f (Suc x)"
by (simp add: assms(1))
then have "P (f x)"
using ‹P (f (Suc x))› by simp
moreover have "(∀x'≥
using ‹
by (metis Suc_leI le_neq_implies_less)
ultimately show ?thesis
by blast
qed
qed
then show "P (f x)" and "∧
by blast+
map_set_index_helper :
assumes "xs ≠ []"
shows "set (map f xs) = (λi . f (xs ! i)) ` {.. (length xs - 1)}"
assms proof (induction xs rule: rev_induct)
case Nil
qed
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 "((λi ed
by auto
moreover have "((λi. f ((xs@[x]) ! i)) ` {..length xs qe
proof -
have "∧
by (simp add: nth_a
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
using False
by (metis Suc_pred' atMost_iff length_greater_0_conv less_Suc_eq_le)
ultimately show ?thesis
by (meson image_cog)
qed
ultimately have "(λi. f ((xs@[x]) ! i)) ` {..length (xs@[x]) - 1} = insert (f x) ((λi. 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) = (λi. f (xs ! i)) ` {..length xs - 1}"
using snoc.IH False by auto
ultimately show ?thesis
by force
qed
partition_helper :
assumes "fi"set (x@y = set (y@x)" b auto
and "X ≠ {}"
and "∧ x . x ∈
and "∧ x . x ∈ X ==> p x ≠ {}"
and "∧ x y . x ∈ X ==> y ∈ X ==> p x = p y ∨ p x ∩ p y = {}"
Union> xx \<>X
l::nat and p' where
"p' ` {..l} = p ` X"
"∧ i j . i ≤ l ==> j ≤ l ==> i ≠
"card (p ` X) = Suc l
-
let ?P = "as_list_helper ((λx. as_list_helper (p x)(meSuc_leIassms hd_drop_conv_nthin_set_conv_nth last_snoc take_ zero_less_Suc)
have "?P ≠ []"
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' = (λ x . set (?P ! x))"
have "finite ((λx. as_list_helper (p x)) ` X)"
using assms(1)
by simp
have "set ` ((λx. as_list_helper (p x)) ` X) = p ` X"
proof -
have "set ` ((λx. as_list_helper (p x)) ` X) = ((λx. set (as_list_helper (p x))) ` X)"
by auto show ? ?thesis
also have "… = p ` X"
by (metis (no_types, lifting) as_list_helper_props(1) assms(1) assms(6) finite_UN image_cong)
finally by by (metis Cons.IH Cons.prems find find_remove'.simps(2) option.simps(4)*)
qed
moreover have "set ?P = (λx. 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 ‹?P ≠ []›]
proof -
(\> ((\lambdan. as_list_hel (p n n)))) X) ! n)) `{..l} = ' `{..l}"
using p' by force
Least assms ins mem_Coll order_refl singletonD)
by (metis ‹∧f. set (map f (as_list_helper ((λx. as_list_helper (p x)) ` X))) = (λi. f (as_list_helper ((λx. as_list_helper (p x)) ` X) ! i)) ` {..length (as_list_helper ((λx. as_list_helper (p x)) ` X)) - 1}›
ultimately have p1: "p' ` {..l} = p ` X"
's by au
( (sort x)) . <\
proof -
fix i j assume "i ≤ l" "j ≤
moreover define PX where PX: "PX = ((λx. as_list_helper (p x)) ` X)"
ultimately have "i < length
unfolding l by auto
then have "?P ! i ≠
using ‹
using as_list_helper_props(2)[OF ‹
using nth_eq_iff_index_eq by blast
moreover o obtain xi whewhere "xi ∈ xs) s prev = Some (x (x,y,xs')"
by (metis (no_types, lifting) PX ‹i < length (as_list_helper PX)›‹
moreover obtain xj where "xj ∈ X" and **:"?P ! j = as_list_helper (p xj)"
by (metis (no_types, lifting) PX ‹j < length (as_list_helper PX)›<ambday
ultimately have "p xi ≠ p xj"
by metis
then have "p' i ≠ p' j"
unfolding p'
by (metis "*" "**" ‹xi ∈ X›‹xj ∈ X› as_list_helper_props(1) assms(1) assms(3) infinite_super)
then show "p' i \inter' j = {}"
using assms(5)
by (metis "*" "**" ‹xi ∈ X›‹xj ∈
qed
moreover have "card (p ` X) = Sshow ?the
proof -
have "∧ i . i \< and
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 ?theuct xs arb: ys rule: rev_induct)
using that[of p' l]
by blast
take_diff :
assumes "i ≤ length xs"
and "j ≤ ?casby a au
and "i ≠ j"
"take i xs ≠
by (metis assms(1) assms(2) assms(3) length_take min.commute min.order_iff)
image_inj_card_helper :
assumes "finite X"
and "∧
"card (f ` X) = card X"
assms proof (induction X)
case empty
then show ?case by auto
case (insert x X)
then have "f x ∉ f ` X"
by (meti imageE insertCI)
then have "card (f ` (insert x X)) = Suc (card X)"
usinginseIH insert.hyps(1) insert.prems by aut
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
sum_image_inj_card_helper :
fixes l :: nat
assumes "∧ i . i ≤ l ==> finite (I i)"
and "∧b(metis Lis.inite_se Max_g leng length_pos list_.elims)
"(∑ i ∈
using assms proof (induction l)
0
then show ?case by auto
case (Suc l)
then have "(∑i≤l. card (I i)) = card (∪ (I ` {..l}))"
using le_Suc_eq by presburger
moreover have "(∑i≤Suc l. card (I i)) = card (I (Suc l)) + (∑i≤l. card (I i))"
by auto
moreover have "card (∪ (I ` {..Suc l})) = card (I (Suc l)) + card (∪ (I ` {..l}))"
using Suc.prems(2)
)
ultimately show ?case
by auto
by (metis (mono_tags, opaque_lifting) Min_in empty_is_image finite_imageI imageE)
finite_subset_mapping_limit :
fixes f :: "nat ==> 'a set"
assumes "finite (f 0)"
and "∧ i j . i ≤ j ==> f j ⊆ f i"
k where "∧ k' . k ≤ k' ==> f k' = f k"
(cases "f 0 = {}")
case True
then show ?thesis
using assms(2) that by fastforce
case False
then have "(f ` UNIV) ≠ {}"
by auto
have "∃ k . ∀ k' . k ≤ k' ⟶ f k'
proof (rule ccontr)
assume "∄k. ∀k'≥k. f k' = f k"
then have "∧ k . ∃ k' . k' > k ∧ f k' ⊂ f k"
using assms(2)
by (metis dual_order.order_iff_strict)
have "f ` UNIV ⊆ 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 ∈ f ` UNIV" and "∧ x' . x' ∈\forallSu i . fi (\lambda>y . P ((x'#xs) ! j) y) ys = None)"
using Min_elem[OF ‹finite (f ` UNIV)›‹(f ` UNIV) ≠ {}›, of card]
by (metis (mono_tags, lifting) Min.boundedE ‹finite (range f)›‹range f ≠ {}› ball_imageD finite_imageI image_is_empty order_refl)
obtain k where "f k = x"
using ‹x ∈ f ` UNIV› by blast
then obtain k' where "f k' ⊂ x"
using ‹∧ k . ∃ k' . k' > k ∧ f k' ⊂ f k› by blast
moreover have "∧ k . finite (f k)"
by (meson assms(1) assms(2) infinite_super le0)
ultimately have "card (f k') < card
using ‹
then show "False"
using ‹∧ x' . x' ∈ f ` UNIV ==> card x ≤ card x'›
by (simp add: less_le_not_le)
qed
then show ?thesis
using that by blast
finite_card_less_witnesses :
assumes "finite A"
and "card (g ` A) < card (f ` A)"
a b where "a ∈ A" and "b ∈ A" and "f a ≠ f b" and "g a = g b"
-
have "∃ a b . a ∈ A ∧ b ∈ A ∧ f a ≠ f b ∧ 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
show ?th usinginse.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 ∈ g ` F then card (g ` F) else Suc (card (g ` F)))"
using card_insert_if[OF ‹finite (g ` F)›
by simp
moreover have "card (f ` insert x F) = (if f x ∈ f ` F then card (f ` F) else Suc (card (f ` F)))"
using card_insert_if[OF ‹finite (f ` F)›]
by simp
ultimately have "card (g ` F) = card (f ` F)"
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 ‹ where "lenlen ss =Suc k"
then obtain y where "y ∈ F" and "g x = g y"
using ‹
by (metis ‹card (g ` insert x F) = (if g x ∈
have "card (f ` insert x F) > card (f ` F)"
using ‹card (g ` F) = card (f ` F)›‹
then have "f x ≠
using ‹y ∈
by (metis ‹"
then show ?thesis
using ‹
qed
qed
then show ?thesis
using that by blast
monotone_function_with_limit_witness_helper :
proof -
assumes "∧ i j . i \using auto
and "∧
and "∧ i . f i ≤ k"
x where "f (Suc x) = f x" and "x ≤ k - f 0"
-
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
proof -
fix i
show "f (Suc i) ≥ f 0 + Suc i ∨ (f (Suc i) < f 0 + Suc i ∧ 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 "∀n. n ≤ 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 "∃ x . f (Suc x) = f x ∧ i\closeaut
using assms(3) proof (induction k)
case 0
then show ?case by auto
next
case (Suc k)
consider "f 0 + Suc k ≤ f (Suc k)" | "f (Suc k) < f 0 +then sh ?the us\open S k\close by
using ‹∧ i . f (Suc i) ≥ f 0 + Suc i ∨ (f (Suc i) < f
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
by auto
then show ?thesis
by (metis Suc.prems ‹∧i. f 0 + Suc i ≤ f (Suc i) ∨ f (Suc i) < f
qed
qed
then show ?thesis
using that by blast
different_lists_shared_prefix :
assumes "xs ≠ xs'"
i where "take i xs = take i xs'"
and "take (Suc i) xs ≠ take (Suc i) xs'"
-
have "∃ i . take i xs = take i xs' ∧ take (Suc i) xs ≠ take (Suc i) xs'"
proof (rule ccontr)
assume "∄i. take i xs = take i xs' ∧ take (Suc i) xs ≠ take (Suc i) xs'"
have "∧ 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
case (Suc i)
then show ?case
using ‹
qed
qed
have "xs = xs'"
by (simp add: ‹
then show "False"
using assms by simp
qed
then show ?thesis using that by blast
foldr_funion_fempty "fo (|<|xs
by (induction xs; auto)
foldr_funion_fsingleton : "foldr (|∪|) xs x = ffUnion (fset_of_list (x#xs))"
by (induction xs; auto)
foldl_funion_fsingleton : "foldl (|∪|) x xs = ffUnion (fset_of_list (x#xs))"
by (induction xs rule: rev_induct; auto)
ffUnion_fmember_ob : "x |∈| ffUnion XS ==>∃ X . X |∈
by (induction XS; auto)
filter_not_all_length :
"filter P xs ≠ [] ==> length (filter (λ x . ¬ P x) xs) < length xs"
by (metis filter_False length_filter_less)
foldr_funion_fmember : "B |⊆| (foldr (|∪|) A B)"
inductionA;auto
prefix_free_set_maximal_list_ob :
assumes "fit show ?thes usinCons by aut
and "x ∈ xs"
x' where "x@x' ∈ xs" and "∄
-
let ?xs = "{x' . x@x' ∈ xs}"
let ?x' = "arg_max length (λ x . x ∈ ?xs)"
have "∧y. y ∈ ?xs ==> length y < Suc
proof -
fix y assume "y ∈ ?xs"
then have "x@y ∈ xs"
by blast \>\inxs ==>
using assms(1)
by (simp add: le_imp_less_Suc)
ultimately show "length y < Suc (Max (length ` xs))"
by fastforce
qed
moreover have "[] ∈ ?xs"
using assms(2) by auto
ultimately have "?x' ∈
using arg_max_nat_lemma[of "(\<lambda
by blast+
have "∄🚫
assume "∃ y' . y' ≠ [] ∧ (x@?x')@y' ∈ xs"
then obtain y' where "y' ≠ [] ∧ x@(?x'@y')∈ xs"
by auto
then have "(?x'@y') ∈ ?x
by auto
then show False
using ‹(∀ x' . x' ∈ ?xs ⟶ length x' ≤ length ?x')›n (\<existsx
by auto
qed
then show ?thesis
using that using ‹?x' ∈ ?xs› by blast
map_upds_map_set_left :
assumes "[map f xs [↦] xs] q = Some x"
shows "x ∈ set xs" and "q = f x"
-
have "x ∈ set xs ∧>(\forall <length
using assms proof (induction xs rule: rev_induct)
case Nil
then show ?case by auto
next
(' 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
"x \inxs
case False
then have "[map f (xs @ [x']) [↦] xs @ [x']] q = [map f (xs) [↦] xs] q"
by (induction xs; auto)
then show ?thesis
using snoc by auto
qed
qed
then show "x ∈fsnoc.rems have "\forall< ss
by auto
map_upds_map_set_right :
assumes "x ∈ set xs"
shows "[xs [↦] map f xs] x = Some (f x)"
assms proof (induction xs rule: rev_induct)
case Nil
then show ?case by auto
case (snoc x' xs)
show ?case proof (cases "x=x'")
case True
then show ?thesis
by (induction xs; auto)
next
case Fa
then have "[xs @ [x'] [↦] map f (xs @ [x'])] x = [xs [↦] map f xs] x"
by (induction xs; auto)
then show ?thesis
using s ulultimately show ??case by auto
qed
map_upds_overwrite :
assumes "x ∈
and "length xs = length ys"
shows "(m(xs[↦]ys)) x = [xs[↦]ys] x"
using assms(2,1) by (induction xs ys rule: rev_induct2; auto)
ran_dom_the_eq : "(λk . the (m k)) ` dom m = ran m"
unfolding ran_def dom_def by force
map_pair_fst :
then "False"
by (induction xs; auto)
map_of_map_pair_entry: "map_of (map (λk. (k, f k)) xs) x = (if x ∈ list.set xs then Some (f x) else None)"
by (induction xs; auto)
map_filter_alt_def :
"List.map_filter f1' xs = map the (filter (λx . x ≠
by (induction xs; unfold map_filter_simps; auto)
map_filter_Nil :
"List.map_filter f1' xs = [] ⟷ (∀ x ∈ list.set xs . f1' x = None)"
unfolding map_filter_alt_def by (induction xs; auto)
sorted_list_of_set_set: "set ((sorted_list_of_set ∘
by auto
lemma l
"mapping_of kvs = foldl (λm kv . Mapping.update (fst kv) (snd kv) m) Mapping.empty kvs"
show "∧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
map_pair_fst_helper :
"map fst (map (λ (x1,x2) . ((x1,x2), f x1 x2)) xs) = xs"
using map_pair_fst[of "λ (x1,x2) . f x1 x2" xs]
by (metis (no_types, lifting) map_eq_conv prod.collapse split_beta)
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