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src/HOL/List.thy

author | haftmann |

Sun, 07 Mar 2021 08:26:02 +0100 | |

changeset 73395 | 6a96e9406e53 |

parent 73379 | b867b436f372 |

permissions | -rw-r--r-- |

reduced dependencies on List_Permutation

(* Title: HOL/List.thy Author: Tobias Nipkow; proofs tidied by LCP *) section \<open>The datatype of finite lists\<close> theory List imports Sledgehammer Code_Numeral Lifting_Set begin datatype (set: 'a) list = Nil ("[]") | Cons (hd: 'a) (tl: "'a list") (infixr "#" 65) for map: map rel: list_all2 pred: list_all where "tl [] = []" datatype_compat list lemma [case_names Nil Cons, cases type: list]: \<comment> \<open>for backward compatibility -- names of variables differ\<close> "(y = [] \<Longrightarrow> P) \<Longrightarrow> (\<And>a list. y = a # list \<Longrightarrow> P) \<Longrightarrow> P" by (rule list.exhaust) lemma [case_names Nil Cons, induct type: list]: \<comment> \<open>for backward compatibility -- names of variables differ\<close> "P [] \<Longrightarrow> (\<And>a list. P list \<Longrightarrow> P (a # list)) \<Longrightarrow> P list" by (rule list.induct) text \<open>Compatibility:\<close> setup \<open>Sign.mandatory_path "list"\<close> lemmas inducts = list.induct lemmas recs = list.rec lemmas cases = list.case setup \<open>Sign.parent_path\<close> lemmas set_simps = list.set (* legacy *) syntax \<comment> \<open>list Enumeration\<close> "_list" :: "args => 'a list" ("[(_)]") translations "[x, xs]" == "x#[xs]" "[x]" == "x#[]" subsection \<open>Basic list processing functions\<close> primrec (nonexhaustive) last :: "'a list \<Rightarrow> 'a" where "last (x # xs) = (if xs = [] then x else last xs)" primrec butlast :: "'a list \<Rightarrow> 'a list" where "butlast [] = []" | "butlast (x # xs) = (if xs = [] then [] else x # butlast xs)" lemma set_rec: "set xs = rec_list {} (\<lambda>x _. insert x) xs" by (induct xs) auto definition coset :: "'a list \<Rightarrow> 'a set" where [simp]: "coset xs = - set xs" primrec append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65) where append_Nil: "[] @ ys = ys" | append_Cons: "(x#xs) @ ys = x # xs @ ys" primrec rev :: "'a list \<Rightarrow> 'a list" where "rev [] = []" | "rev (x # xs) = rev xs @ [x]" primrec filter:: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where "filter P [] = []" | "filter P (x # xs) = (if P x then x # filter P xs else filter P xs)" text \<open>Special input syntax for filter:\<close> syntax (ASCII) "_filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_<-_./ _])") syntax "_filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_\<leftarrow>_ ./ _])") translations "[x<-xs . P]" \<rightharpoonup> "CONST filter (\<lambda>x. P) xs" primrec fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where fold_Nil: "fold f [] = id" | fold_Cons: "fold f (x # xs) = fold f xs \<circ> f x" primrec foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where foldr_Nil: "foldr f [] = id" | foldr_Cons: "foldr f (x # xs) = f x \<circ> foldr f xs" primrec foldl :: "('b \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b" where foldl_Nil: "foldl f a [] = a" | foldl_Cons: "foldl f a (x # xs) = foldl f (f a x) xs" primrec concat:: "'a list list \<Rightarrow> 'a list" where "concat [] = []" | "concat (x # xs) = x @ concat xs" primrec drop:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where drop_Nil: "drop n [] = []" | drop_Cons: "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)" \<comment> \<open>Warning: simpset does not contain this definition, but separate theorems for \<open>n = 0\<close> and \<open>n = Suc k\<close>\<close> primrec take:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where take_Nil:"take n [] = []" | take_Cons: "take n (x # xs) = (case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # take m xs)" \<comment> \<open>Warning: simpset does not contain this definition, but separate theorems for \<open>n = 0\<close> and \<open>n = Suc k\<close>\<close> primrec (nonexhaustive) nth :: "'a list => nat => 'a" (infixl "!" 100) where nth_Cons: "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)" \<comment> \<open>Warning: simpset does not contain this definition, but separate theorems for \<open>n = 0\<close> and \<open>n = Suc k\<close>\<close> primrec list_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a list" where "list_update [] i v = []" | "list_update (x # xs) i v = (case i of 0 \<Rightarrow> v # xs | Suc j \<Rightarrow> x # list_update xs j v)" nonterminal lupdbinds and lupdbind syntax "_lupdbind":: "['a, 'a] => lupdbind" ("(2_ :=/ _)") "" :: "lupdbind => lupdbinds" ("_") "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds" ("_,/ _") "_LUpdate" :: "['a, lupdbinds] => 'a" ("_/[(_)]" [1000,0] 900) translations "_LUpdate xs (_lupdbinds b bs)" == "_LUpdate (_LUpdate xs b) bs" "xs[i:=x]" == "CONST list_update xs i x" primrec takeWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where "takeWhile P [] = []" | "takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])" primrec dropWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where "dropWhile P [] = []" | "dropWhile P (x # xs) = (if P x then dropWhile P xs else x # xs)" primrec zip :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where "zip xs [] = []" | zip_Cons: "zip xs (y # ys) = (case xs of [] \<Rightarrow> [] | z # zs \<Rightarrow> (z, y) # zip zs ys)" \<comment> \<open>Warning: simpset does not contain this definition, but separate theorems for \<open>xs = []\<close> and \<open>xs = z # zs\<close>\<close> abbreviation map2 :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'c list" where "map2 f xs ys \<equiv> map (\<lambda>(x,y). f x y) (zip xs ys)" primrec product :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where "product [] _ = []" | "product (x#xs) ys = map (Pair x) ys @ product xs ys" hide_const (open) product primrec product_lists :: "'a list list \<Rightarrow> 'a list list" where "product_lists [] = [[]]" | "product_lists (xs # xss) = concat (map (\<lambda>x. map (Cons x) (product_lists xss)) xs)" primrec upt :: "nat \<Rightarrow> nat \<Rightarrow> nat list" ("(1[_..</_'])") where upt_0: "[i..<0] = []" | upt_Suc: "[i..<(Suc j)] = (if i \<le> j then [i..<j] @ [j] else [])" definition insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where "insert x xs = (if x \<in> set xs then xs else x # xs)" definition union :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where "union = fold insert" hide_const (open) insert union hide_fact (open) insert_def union_def primrec find :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a option" where "find _ [] = None" | "find P (x#xs) = (if P x then Some x else find P xs)" text \<open>In the context of multisets, \<open>count_list\<close> is equivalent to \<^term>\<open>count \<circ> mset\<close> and it it advisable to use the latter.\<close> primrec count_list :: "'a list \<Rightarrow> 'a \<Rightarrow> nat" where "count_list [] y = 0" | "count_list (x#xs) y = (if x=y then count_list xs y + 1 else count_list xs y)" definition "extract" :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> ('a list * 'a * 'a list) option" where "extract P xs = (case dropWhile (Not \<circ> P) xs of [] \<Rightarrow> None | y#ys \<Rightarrow> Some(takeWhile (Not \<circ> P) xs, y, ys))" hide_const (open) "extract" primrec those :: "'a option list \<Rightarrow> 'a list option" where "those [] = Some []" | "those (x # xs) = (case x of None \<Rightarrow> None | Some y \<Rightarrow> map_option (Cons y) (those xs))" primrec remove1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where "remove1 x [] = []" | "remove1 x (y # xs) = (if x = y then xs else y # remove1 x xs)" primrec removeAll :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where "removeAll x [] = []" | "removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)" primrec distinct :: "'a list \<Rightarrow> bool" where "distinct [] \<longleftrightarrow> True" | "distinct (x # xs) \<longleftrightarrow> x \<notin> set xs \<and> distinct xs" fun successively :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where "successively P [] = True" | "successively P [x] = True" | "successively P (x # y # xs) = (P x y \<and> successively P (y#xs))" definition distinct_adj where "distinct_adj = successively (\<noteq>)" primrec remdups :: "'a list \<Rightarrow> 'a list" where "remdups [] = []" | "remdups (x # xs) = (if x \<in> set xs then remdups xs else x # remdups xs)" fun remdups_adj :: "'a list \<Rightarrow> 'a list" where "remdups_adj [] = []" | "remdups_adj [x] = [x]" | "remdups_adj (x # y # xs) = (if x = y then remdups_adj (x # xs) else x # remdups_adj (y # xs))" primrec replicate :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" where replicate_0: "replicate 0 x = []" | replicate_Suc: "replicate (Suc n) x = x # replicate n x" text \<open> Function \<open>size\<close> is overloaded for all datatypes. Users may refer to the list version as \<open>length\<close>.\<close> abbreviation length :: "'a list \<Rightarrow> nat" where "length \<equiv> size" definition enumerate :: "nat \<Rightarrow> 'a list \<Rightarrow> (nat \<times> 'a) list" where enumerate_eq_zip: "enumerate n xs = zip [n..<n + length xs] xs" primrec rotate1 :: "'a list \<Rightarrow> 'a list" where "rotate1 [] = []" | "rotate1 (x # xs) = xs @ [x]" definition rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where "rotate n = rotate1 ^^ n" definition nths :: "'a list => nat set => 'a list" where "nths xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))" primrec subseqs :: "'a list \<Rightarrow> 'a list list" where "subseqs [] = [[]]" | "subseqs (x#xs) = (let xss = subseqs xs in map (Cons x) xss @ xss)" primrec n_lists :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list list" where "n_lists 0 xs = [[]]" | "n_lists (Suc n) xs = concat (map (\<lambda>ys. map (\<lambda>y. y # ys) xs) (n_lists n xs))" hide_const (open) n_lists function splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where "splice [] ys = ys" | "splice (x#xs) ys = x # splice ys xs" by pat_completeness auto termination by(relation "measure(\<lambda>(xs,ys). size xs + size ys)") auto function shuffles where "shuffles [] ys = {ys}" | "shuffles xs [] = {xs}" | "shuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \<union> (#) y ` shuffles (x # xs) ys" by pat_completeness simp_all termination by lexicographic_order text\<open>Use only if you cannot use \<^const>\<open>Min\<close> instead:\<close> fun min_list :: "'a::ord list \<Rightarrow> 'a" where "min_list (x # xs) = (case xs of [] \<Rightarrow> x | _ \<Rightarrow> min x (min_list xs))" text\<open>Returns first minimum:\<close> fun arg_min_list :: "('a \<Rightarrow> ('b::linorder)) \<Rightarrow> 'a list \<Rightarrow> 'a" where "arg_min_list f [x] = x" | "arg_min_list f (x#y#zs) = (let m = arg_min_list f (y#zs) in if f x \<le> f m then x else m)" text\<open> \begin{figure}[htbp] \fbox{ \begin{tabular}{l} @{lemma "[a,b]@[c,d] = [a,b,c,d]" by simp}\\ @{lemma "length [a,b,c] = 3" by simp}\\ @{lemma "set [a,b,c] = {a,b,c}" by simp}\\ @{lemma "map f [a,b,c] = [f a, f b, f c]" by simp}\\ @{lemma "rev [a,b,c] = [c,b,a]" by simp}\\ @{lemma "hd [a,b,c,d] = a" by simp}\\ @{lemma "tl [a,b,c,d] = [b,c,d]" by simp}\\ @{lemma "last [a,b,c,d] = d" by simp}\\ @{lemma "butlast [a,b,c,d] = [a,b,c]" by simp}\\ @{lemma[source] "filter (\<lambda>n::nat. n<2) [0,2,1] = [0,1]" by simp}\\ @{lemma "concat [[a,b],[c,d,e],[],[f]] = [a,b,c,d,e,f]" by simp}\\ @{lemma "fold f [a,b,c] x = f c (f b (f a x))" by simp}\\ @{lemma "foldr f [a,b,c] x = f a (f b (f c x))" by simp}\\ @{lemma "foldl f x [a,b,c] = f (f (f x a) b) c" by simp}\\ @{lemma "successively (\<noteq>) [True,False,True,False]" by simp}\\ @{lemma "zip [a,b,c] [x,y,z] = [(a,x),(b,y),(c,z)]" by simp}\\ @{lemma "zip [a,b] [x,y,z] = [(a,x),(b,y)]" by simp}\\ @{lemma "enumerate 3 [a,b,c] = [(3,a),(4,b),(5,c)]" by normalization}\\ @{lemma "List.product [a,b] [c,d] = [(a, c), (a, d), (b, c), (b, d)]" by simp}\\ @{lemma "product_lists [[a,b], [c], [d,e]] = [[a,c,d], [a,c,e], [b,c,d], [b,c,e]]" by simp}\\ @{lemma "splice [a,b,c] [x,y,z] = [a,x,b,y,c,z]" by simp}\\ @{lemma "splice [a,b,c,d] [x,y] = [a,x,b,y,c,d]" by simp}\\ @{lemma "shuffles [a,b] [c,d] = {[a,b,c,d],[a,c,b,d],[a,c,d,b],[c,a,b,d],[c,a,d,b],[c,d,a,b]}" by (simp add: insert_commute)}\\ @{lemma "take 2 [a,b,c,d] = [a,b]" by simp}\\ @{lemma "take 6 [a,b,c,d] = [a,b,c,d]" by simp}\\ @{lemma "drop 2 [a,b,c,d] = [c,d]" by simp}\\ @{lemma "drop 6 [a,b,c,d] = []" by simp}\\ @{lemma "takeWhile (%n::nat. n<3) [1,2,3,0] = [1,2]" by simp}\\ @{lemma "dropWhile (%n::nat. n<3) [1,2,3,0] = [3,0]" by simp}\\ @{lemma "distinct [2,0,1::nat]" by simp}\\ @{lemma "remdups [2,0,2,1::nat,2] = [0,1,2]" by simp}\\ @{lemma "remdups_adj [2,2,3,1,1::nat,2,1] = [2,3,1,2,1]" by simp}\\ @{lemma "List.insert 2 [0::nat,1,2] = [0,1,2]" by (simp add: List.insert_def)}\\ @{lemma "List.insert 3 [0::nat,1,2] = [3,0,1,2]" by (simp add: List.insert_def)}\\ @{lemma "List.union [2,3,4] [0::int,1,2] = [4,3,0,1,2]" by (simp add: List.insert_def List.union_def)}\\ @{lemma "List.find (%i::int. i>0) [0,0] = None" by simp}\\ @{lemma "List.find (%i::int. i>0) [0,1,0,2] = Some 1" by simp}\\ @{lemma "count_list [0,1,0,2::int] 0 = 2" by (simp)}\\ @{lemma "List.extract (%i::int. i>0) [0,0] = None" by(simp add: extract_def)}\\ @{lemma "List.extract (%i::int. i>0) [0,1,0,2] = Some([0], 1, [0,2])" by(simp add: extract_def)}\\ @{lemma "remove1 2 [2,0,2,1::nat,2] = [0,2,1,2]" by simp}\\ @{lemma "removeAll 2 [2,0,2,1::nat,2] = [0,1]" by simp}\\ @{lemma "nth [a,b,c,d] 2 = c" by simp}\\ @{lemma "[a,b,c,d][2 := x] = [a,b,x,d]" by simp}\\ @{lemma "nths [a,b,c,d,e] {0,2,3} = [a,c,d]" by (simp add:nths_def)}\\ @{lemma "subseqs [a,b] = [[a, b], [a], [b], []]" by simp}\\ @{lemma "List.n_lists 2 [a,b,c] = [[a, a], [b, a], [c, a], [a, b], [b, b], [c, b], [a, c], [b, c], [c, c]]" by (simp add: eval_nat_numeral)}\\ @{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by simp}\\ @{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate_def eval_nat_numeral)}\\ @{lemma "replicate 4 a = [a,a,a,a]" by (simp add:eval_nat_numeral)}\\ @{lemma "[2..<5] = [2,3,4]" by (simp add:eval_nat_numeral)}\\ @{lemma "min_list [3,1,-2::int] = -2" by (simp)}\\ @{lemma "arg_min_list (\<lambda>i. i*i) [3,-1,1,-2::int] = -1" by (simp)} \end{tabular}} \caption{Characteristic examples} \label{fig:Characteristic} \end{figure} Figure~\ref{fig:Characteristic} shows characteristic examples that should give an intuitive understanding of the above functions. \<close> text\<open>The following simple sort(ed) functions are intended for proofs, not for efficient implementations.\<close> text \<open>A sorted predicate w.r.t. a relation:\<close> fun sorted_wrt :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where "sorted_wrt P [] = True" | "sorted_wrt P (x # ys) = ((\<forall>y \<in> set ys. P x y) \<and> sorted_wrt P ys)" text \<open>A class-based sorted predicate:\<close> context linorder begin fun sorted :: "'a list \<Rightarrow> bool" where "sorted [] = True" | "sorted (x # ys) = ((\<forall>y \<in> set ys. x \<le> y) \<and> sorted ys)" fun strict_sorted :: "'a list \<Rightarrow> bool" where "strict_sorted [] = True" | "strict_sorted (x # ys) = ((\<forall>y \<in> List.set ys. x < y) \<and> strict_sorted ys)" lemma sorted_sorted_wrt: "sorted = sorted_wrt (\<le>)" proof (rule ext) fix xs show "sorted xs = sorted_wrt (\<le>) xs" by(induction xs rule: sorted.induct) auto qed lemma strict_sorted_sorted_wrt: "strict_sorted = sorted_wrt (<)" proof (rule ext) fix xs show "strict_sorted xs = sorted_wrt (<) xs" by(induction xs rule: strict_sorted.induct) auto qed primrec insort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where "insort_key f x [] = [x]" | "insort_key f x (y#ys) = (if f x \<le> f y then (x#y#ys) else y#(insort_key f x ys))" definition sort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b list \<Rightarrow> 'b list" where "sort_key f xs = foldr (insort_key f) xs []" definition insort_insert_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where "insort_insert_key f x xs = (if f x \<in> f ` set xs then xs else insort_key f x xs)" abbreviation "sort \<equiv> sort_key (\<lambda>x. x)" abbreviation "insort \<equiv> insort_key (\<lambda>x. x)" abbreviation "insort_insert \<equiv> insort_insert_key (\<lambda>x. x)" definition stable_sort_key :: "(('b \<Rightarrow> 'a) \<Rightarrow> 'b list \<Rightarrow> 'b list) \<Rightarrow> bool" where "stable_sort_key sk = (\<forall>f xs k. filter (\<lambda>y. f y = k) (sk f xs) = filter (\<lambda>y. f y = k) xs)" lemma strict_sorted_iff: "strict_sorted l \<longleftrightarrow> sorted l \<and> distinct l" by (induction l) (auto iff: antisym_conv1) lemma strict_sorted_imp_sorted: "strict_sorted xs \<Longrightarrow> sorted xs" by (auto simp: strict_sorted_iff) end subsubsection \<open>List comprehension\<close> text\<open>Input syntax for Haskell-like list comprehension notation. Typical example: \<open>[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]\<close>, the list of all pairs of distinct elements from \<open>xs\<close> and \<open>ys\<close>. The syntax is as in Haskell, except that \<open>|\<close> becomes a dot (like in Isabelle's set comprehension): \<open>[e. x \<leftarrow> xs, \<dots>]\<close> rather than \verb![e| x <- xs, ...]!. The qualifiers after the dot are \begin{description} \item[generators] \<open>p \<leftarrow> xs\<close>, where \<open>p\<close> is a pattern and \<open>xs\<close> an expression of list type, or \item[guards] \<open>b\<close>, where \<open>b\<close> is a boolean expression. %\item[local bindings] @ {text"let x = e"}. \end{description} Just like in Haskell, list comprehension is just a shorthand. To avoid misunderstandings, the translation into desugared form is not reversed upon output. Note that the translation of \<open>[e. x \<leftarrow> xs]\<close> is optmized to \<^term>\<open>map (%x. e) xs\<close>. It is easy to write short list comprehensions which stand for complex expressions. During proofs, they may become unreadable (and mangled). In such cases it can be advisable to introduce separate definitions for the list comprehensions in question.\<close> nonterminal lc_qual and lc_quals syntax "_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list" ("[_ . __") "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _") "_lc_test" :: "bool \<Rightarrow> lc_qual" ("_") (*"_lc_let" :: "letbinds => lc_qual" ("let _")*) "_lc_end" :: "lc_quals" ("]") "_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals" (", __") syntax (ASCII) "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ <- _") parse_translation \<open> let val NilC = Syntax.const \<^const_syntax>\<open>Nil\<close>; val ConsC = Syntax.const \<^const_syntax>\<open>Cons\<close>; val mapC = Syntax.const \<^const_syntax>\<open>map\<close>; val concatC = Syntax.const \<^const_syntax>\<open>concat\<close>; val IfC = Syntax.const \<^const_syntax>\<open>If\<close>; val dummyC = Syntax.const \<^const_syntax>\<open>Pure.dummy_pattern\<close> fun single x = ConsC $ x $ NilC; fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *) let (* FIXME proper name context!? *) val x = Free (singleton (Name.variant_list (fold Term.add_free_names [p, e] [])) "x", dummyT); val e = if opti then single e else e; val case1 = Syntax.const \<^syntax_const>\<open>_case1\<close> $ p $ e; val case2 = Syntax.const \<^syntax_const>\<open>_case1\<close> $ dummyC $ NilC; val cs = Syntax.const \<^syntax_const>\<open>_case2\<close> $ case1 $ case2; in Syntax_Trans.abs_tr [x, Case_Translation.case_tr false ctxt [x, cs]] end; fun pair_pat_tr (x as Free _) e = Syntax_Trans.abs_tr [x, e] | pair_pat_tr (_ $ p1 $ p2) e = Syntax.const \<^const_syntax>\<open>case_prod\<close> $ pair_pat_tr p1 (pair_pat_tr p2 e) | pair_pat_tr dummy e = Syntax_Trans.abs_tr [Syntax.const "_idtdummy", e] fun pair_pat ctxt (Const (\<^const_syntax>\<open>Pair\<close>,_) $ s $ t) = pair_pat ctxt s andalso pair_pat ctxt t | pair_pat ctxt (Free (s,_)) = let val thy = Proof_Context.theory_of ctxt; val s' = Proof_Context.intern_const ctxt s; in not (Sign.declared_const thy s') end | pair_pat _ t = (t = dummyC); fun abs_tr ctxt p e opti = let val p = Term_Position.strip_positions p in if pair_pat ctxt p then (pair_pat_tr p e, true) else (pat_tr ctxt p e opti, false) end fun lc_tr ctxt [e, Const (\<^syntax_const>\<open>_lc_test\<close>, _) $ b, qs] = let val res = (case qs of Const (\<^syntax_const>\<open>_lc_end\<close>, _) => single e | Const (\<^syntax_const>\<open>_lc_quals\<close>, _) $ q $ qs => lc_tr ctxt [e, q, qs]); in IfC $ b $ res $ NilC end | lc_tr ctxt [e, Const (\<^syntax_const>\<open>_lc_gen\<close>, _) $ p $ es, Const(\<^syntax_const>\<open>_lc_end\<close>, _)] = (case abs_tr ctxt p e true of (f, true) => mapC $ f $ es | (f, false) => concatC $ (mapC $ f $ es)) | lc_tr ctxt [e, Const (\<^syntax_const>\<open>_lc_gen\<close>, _) $ p $ es, Const (\<^syntax_const>\<open>_lc_quals\<close>, _) $ q $ qs] = let val e' = lc_tr ctxt [e, q, qs]; in concatC $ (mapC $ (fst (abs_tr ctxt p e' false)) $ es) end; in [(\<^syntax_const>\<open>_listcompr\<close>, lc_tr)] end \<close> ML_val \<open> let val read = Syntax.read_term \<^context> o Syntax.implode_input; fun check s1 s2 = read s1 aconv read s2 orelse error ("Check failed: " ^ quote (#1 (Input.source_content s1)) ^ Position.here_list [Input.pos_of s1, Input.pos_of s2]); in check \<open>[(x,y,z). b]\<close> \<open>if b then [(x, y, z)] else []\<close>; check \<open>[(x,y,z). (x,_,y)\<leftarrow>xs]\<close> \<open>map (\<lambda>(x,_,y). (x, y, z)) xs\<close>; check \<open>[e x y. (x,_)\<leftarrow>xs, y\<leftarrow>ys]\<close> \<open>concat (map (\<lambda>(x,_). map (\<lambda>y. e x y) ys) xs)\<close>; check \<open>[(x,y,z). x<a, x>b]\<close> \<open>if x < a then if b < x then [(x, y, z)] else [] else []\<close>; check \<open>[(x,y,z). x\<leftarrow>xs, x>b]\<close> \<open>concat (map (\<lambda>x. if b < x then [(x, y, z)] else []) xs)\<close>; check \<open>[(x,y,z). x<a, x\<leftarrow>xs]\<close> \<open>if x < a then map (\<lambda>x. (x, y, z)) xs else []\<close>; check \<open>[(x,y). Cons True x \<leftarrow> xs]\<close> \<open>concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | True # x \<Rightarrow> [(x, y)] | False # x \<Rightarrow> []) xs)\<close>; check \<open>[(x,y,z). Cons x [] \<leftarrow> xs]\<close> \<open>concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | [x] \<Rightarrow> [(x, y, z)] | x # aa # lista \<Rightarrow> []) xs)\<close>; check \<open>[(x,y,z). x<a, x>b, x=d]\<close> \<open>if x < a then if b < x then if x = d then [(x, y, z)] else [] else [] else []\<close>; check \<open>[(x,y,z). x<a, x>b, y\<leftarrow>ys]\<close> \<open>if x < a then if b < x then map (\<lambda>y. (x, y, z)) ys else [] else []\<close>; check \<open>[(x,y,z). x<a, (_,x)\<leftarrow>xs,y>b]\<close> \<open>if x < a then concat (map (\<lambda>(_,x). if b < y then [(x, y, z)] else []) xs) else []\<close>; check \<open>[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]\<close> \<open>if x < a then concat (map (\<lambda>x. map (\<lambda>y. (x, y, z)) ys) xs) else []\<close>; check \<open>[(x,y,z). x\<leftarrow>xs, x>b, y<a]\<close> \<open>concat (map (\<lambda>x. if b < x then if y < a then [(x, y, z)] else [] else []) xs)\<close>; check \<open>[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]\<close> \<open>concat (map (\<lambda>x. if b < x then map (\<lambda>y. (x, y, z)) ys else []) xs)\<close>; check \<open>[(x,y,z). x\<leftarrow>xs, (y,_)\<leftarrow>ys,y>x]\<close> \<open>concat (map (\<lambda>x. concat (map (\<lambda>(y,_). if x < y then [(x, y, z)] else []) ys)) xs)\<close>; check \<open>[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]\<close> \<open>concat (map (\<lambda>x. concat (map (\<lambda>y. map (\<lambda>z. (x, y, z)) zs) ys)) xs)\<close> end; \<close> ML \<open> (* Simproc for rewriting list comprehensions applied to List.set to set comprehension. *) signature LIST_TO_SET_COMPREHENSION = sig val simproc : Proof.context -> cterm -> thm option end structure List_to_Set_Comprehension : LIST_TO_SET_COMPREHENSION = struct (* conversion *) fun all_exists_conv cv ctxt ct = (case Thm.term_of ct of Const (\<^const_name>\<open>Ex\<close>, _) $ Abs _ => Conv.arg_conv (Conv.abs_conv (all_exists_conv cv o #2) ctxt) ct | _ => cv ctxt ct) fun all_but_last_exists_conv cv ctxt ct = (case Thm.term_of ct of Const (\<^const_name>\<open>Ex\<close>, _) $ Abs (_, _, Const (\<^const_name>\<open>Ex\<close>, _) $ _) => Conv.arg_conv (Conv.abs_conv (all_but_last_exists_conv cv o #2) ctxt) ct | _ => cv ctxt ct) fun Collect_conv cv ctxt ct = (case Thm.term_of ct of Const (\<^const_name>\<open>Collect\<close>, _) $ Abs _ => Conv.arg_conv (Conv.abs_conv cv ctxt) ct | _ => raise CTERM ("Collect_conv", [ct])) fun rewr_conv' th = Conv.rewr_conv (mk_meta_eq th) fun conjunct_assoc_conv ct = Conv.try_conv (rewr_conv' @{thm conj_assoc} then_conv HOLogic.conj_conv Conv.all_conv conjunct_assoc_conv) ct fun right_hand_set_comprehension_conv conv ctxt = HOLogic.Trueprop_conv (HOLogic.eq_conv Conv.all_conv (Collect_conv (all_exists_conv conv o #2) ctxt)) (* term abstraction of list comprehension patterns *) datatype termlets = If | Case of typ * int local val set_Nil_I = @{lemma "set [] = {x. False}" by (simp add: empty_def [symmetric])} val set_singleton = @{lemma "set [a] = {x. x = a}" by simp} val inst_Collect_mem_eq = @{lemma "set A = {x. x \<in> set A}" by simp} val del_refl_eq = @{lemma "(t = t \<and> P) \<equiv> P" by simp} fun mk_set T = Const (\<^const_name>\<open>set\<close>, HOLogic.listT T --> HOLogic.mk_setT T) fun dest_set (Const (\<^const_name>\<open>set\<close>, _) $ xs) = xs fun dest_singleton_list (Const (\<^const_name>\<open>Cons\<close>, _) $ t $ (Const (\<^const_name>\<open>Nil\<close>, _))) = t | dest_singleton_list t = raise TERM ("dest_singleton_list", [t]) (*We check that one case returns a singleton list and all other cases return [], and return the index of the one singleton list case.*) fun possible_index_of_singleton_case cases = let fun check (i, case_t) s = (case strip_abs_body case_t of (Const (\<^const_name>\<open>Nil\<close>, _)) => s | _ => (case s of SOME NONE => SOME (SOME i) | _ => NONE)) in fold_index check cases (SOME NONE) |> the_default NONE end (*returns condition continuing term option*) fun dest_if (Const (\<^const_name>\<open>If\<close>, _) $ cond $ then_t $ Const (\<^const_name>\<open>Nil\<close>, _)) = SOME (cond, then_t) | dest_if _ = NONE (*returns (case_expr type index chosen_case constr_name) option*) fun dest_case ctxt case_term = let val (case_const, args) = strip_comb case_term in (case try dest_Const case_const of SOME (c, T) => (case Ctr_Sugar.ctr_sugar_of_case ctxt c of SOME {ctrs, ...} => (case possible_index_of_singleton_case (fst (split_last args)) of SOME i => let val constr_names = map (fst o dest_Const) ctrs val (Ts, _) = strip_type T val T' = List.last Ts in SOME (List.last args, T', i, nth args i, nth constr_names i) end | NONE => NONE) | NONE => NONE) | NONE => NONE) end fun tac ctxt [] = resolve_tac ctxt [set_singleton] 1 ORELSE resolve_tac ctxt [inst_Collect_mem_eq] 1 | tac ctxt (If :: cont) = Splitter.split_tac ctxt @{thms if_split} 1 THEN resolve_tac ctxt @{thms conjI} 1 THEN resolve_tac ctxt @{thms impI} 1 THEN Subgoal.FOCUS (fn {prems, context = ctxt', ...} => CONVERSION (right_hand_set_comprehension_conv (K (HOLogic.conj_conv (Conv.rewr_conv (List.last prems RS @{thm Eq_TrueI})) Conv.all_conv then_conv rewr_conv' @{lemma "(True \<and> P) = P" by simp})) ctxt') 1) ctxt 1 THEN tac ctxt cont THEN resolve_tac ctxt @{thms impI} 1 THEN Subgoal.FOCUS (fn {prems, context = ctxt', ...} => CONVERSION (right_hand_set_comprehension_conv (K (HOLogic.conj_conv (Conv.rewr_conv (List.last prems RS @{thm Eq_FalseI})) Conv.all_conv then_conv rewr_conv' @{lemma "(False \<and> P) = False" by simp})) ctxt') 1) ctxt 1 THEN resolve_tac ctxt [set_Nil_I] 1 | tac ctxt (Case (T, i) :: cont) = let val SOME {injects, distincts, case_thms, split, ...} = Ctr_Sugar.ctr_sugar_of ctxt (fst (dest_Type T)) in (* do case distinction *) Splitter.split_tac ctxt [split] 1 THEN EVERY (map_index (fn (i', _) => (if i' < length case_thms - 1 then resolve_tac ctxt @{thms conjI} 1 else all_tac) THEN REPEAT_DETERM (resolve_tac ctxt @{thms allI} 1) THEN resolve_tac ctxt @{thms impI} 1 THEN (if i' = i then (* continue recursively *) Subgoal.FOCUS (fn {prems, context = ctxt', ...} => CONVERSION (Thm.eta_conversion then_conv right_hand_set_comprehension_conv (K ((HOLogic.conj_conv (HOLogic.eq_conv Conv.all_conv (rewr_conv' (List.last prems)) then_conv (Conv.try_conv (Conv.rewrs_conv (map mk_meta_eq injects)))) Conv.all_conv) then_conv (Conv.try_conv (Conv.rewr_conv del_refl_eq)) then_conv conjunct_assoc_conv)) ctxt' then_conv (HOLogic.Trueprop_conv (HOLogic.eq_conv Conv.all_conv (Collect_conv (fn (_, ctxt'') => Conv.repeat_conv (all_but_last_exists_conv (K (rewr_conv' @{lemma "(\<exists>x. x = t \<and> P x) = P t" by simp})) ctxt'')) ctxt')))) 1) ctxt 1 THEN tac ctxt cont else Subgoal.FOCUS (fn {prems, context = ctxt', ...} => CONVERSION (right_hand_set_comprehension_conv (K (HOLogic.conj_conv ((HOLogic.eq_conv Conv.all_conv (rewr_conv' (List.last prems))) then_conv (Conv.rewrs_conv (map (fn th => th RS @{thm Eq_FalseI}) distincts))) Conv.all_conv then_conv (rewr_conv' @{lemma "(False \<and> P) = False" by simp}))) ctxt' then_conv HOLogic.Trueprop_conv (HOLogic.eq_conv Conv.all_conv (Collect_conv (fn (_, ctxt'') => Conv.repeat_conv (Conv.bottom_conv (K (rewr_conv' @{lemma "(\<exists>x. P) = P" by simp})) ctxt'')) ctxt'))) 1) ctxt 1 THEN resolve_tac ctxt [set_Nil_I] 1)) case_thms) end in fun simproc ctxt redex = let fun make_inner_eqs bound_vs Tis eqs t = (case dest_case ctxt t of SOME (x, T, i, cont, constr_name) => let val (vs, body) = strip_abs (Envir.eta_long (map snd bound_vs) cont) val x' = incr_boundvars (length vs) x val eqs' = map (incr_boundvars (length vs)) eqs val constr_t = list_comb (Const (constr_name, map snd vs ---> T), map Bound (((length vs) - 1) downto 0)) val constr_eq = Const (\<^const_name>\<open>HOL.eq\<close>, T --> T --> \<^typ>\<open>bool\<close>) $ constr_t $ x' in make_inner_eqs (rev vs @ bound_vs) (Case (T, i) :: Tis) (constr_eq :: eqs') body end | NONE => (case dest_if t of SOME (condition, cont) => make_inner_eqs bound_vs (If :: Tis) (condition :: eqs) cont | NONE => if null eqs then NONE (*no rewriting, nothing to be done*) else let val Type (\<^type_name>\<open>list\<close>, [rT]) = fastype_of1 (map snd bound_vs, t) val pat_eq = (case try dest_singleton_list t of SOME t' => Const (\<^const_name>\<open>HOL.eq\<close>, rT --> rT --> \<^typ>\<open>bool\<close>) $ Bound (length bound_vs) $ t' | NONE => Const (\<^const_name>\<open>Set.member\<close>, rT --> HOLogic.mk_setT rT --> \<^typ>\<open>bool\<close>) $ Bound (length bound_vs) $ (mk_set rT $ t)) val reverse_bounds = curry subst_bounds ((map Bound ((length bound_vs - 1) downto 0)) @ [Bound (length bound_vs)]) val eqs' = map reverse_bounds eqs val pat_eq' = reverse_bounds pat_eq val inner_t = fold (fn (_, T) => fn t => HOLogic.exists_const T $ absdummy T t) (rev bound_vs) (fold (curry HOLogic.mk_conj) eqs' pat_eq') val lhs = Thm.term_of redex val rhs = HOLogic.mk_Collect ("x", rT, inner_t) val rewrite_rule_t = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)) in SOME ((Goal.prove ctxt [] [] rewrite_rule_t (fn {context = ctxt', ...} => tac ctxt' (rev Tis))) RS @{thm eq_reflection}) end)) in make_inner_eqs [] [] [] (dest_set (Thm.term_of redex)) end end end \<close> simproc_setup list_to_set_comprehension ("set xs") = \<open>K List_to_Set_Comprehension.simproc\<close> code_datatype set coset hide_const (open) coset subsubsection \<open>\<^const>\<open>Nil\<close> and \<^const>\<open>Cons\<close>\<close> lemma not_Cons_self [simp]: "xs \<noteq> x # xs" by (induct xs) auto lemma not_Cons_self2 [simp]: "x # xs \<noteq> xs" by (rule not_Cons_self [symmetric]) lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)" by (induct xs) auto lemma tl_Nil: "tl xs = [] \<longleftrightarrow> xs = [] \<or> (\<exists>x. xs = [x])" by (cases xs) auto lemma Nil_tl: "[] = tl xs \<longleftrightarrow> xs = [] \<or> (\<exists>x. xs = [x])" by (cases xs) auto lemma length_induct: "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs" by (fact measure_induct) lemma induct_list012: "\<lbrakk>P []; \<And>x. P [x]; \<And>x y zs. \<lbrakk> P zs; P (y # zs) \<rbrakk> \<Longrightarrow> P (x # y # zs)\<rbrakk> \<Longrightarrow> P xs" by induction_schema (pat_completeness, lexicographic_order) lemma list_nonempty_induct [consumes 1, case_names single cons]: "\<lbrakk> xs \<noteq> []; \<And>x. P [x]; \<And>x xs. xs \<noteq> [] \<Longrightarrow> P xs \<Longrightarrow> P (x # xs)\<rbrakk> \<Longrightarrow> P xs" by(induction xs rule: induct_list012) auto lemma inj_split_Cons: "inj_on (\<lambda>(xs, n). n#xs) X" by (auto intro!: inj_onI) lemma inj_on_Cons1 [simp]: "inj_on ((#) x) A" by(simp add: inj_on_def) subsubsection \<open>\<^const>\<open>length\<close>\<close> text \<open> Needs to come before \<open>@\<close> because of theorem \<open>append_eq_append_conv\<close>. \<close> lemma length_append [simp]: "length (xs @ ys) = length xs + length ys" by (induct xs) auto lemma length_map [simp]: "length (map f xs) = length xs" by (induct xs) auto lemma length_rev [simp]: "length (rev xs) = length xs" by (induct xs) auto lemma length_tl [simp]: "length (tl xs) = length xs - 1" by (cases xs) auto lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])" by (induct xs) auto lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])" by (induct xs) auto lemma length_pos_if_in_set: "x \<in> set xs \<Longrightarrow> length xs > 0" by auto lemma length_Suc_conv: "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)" by (induct xs) auto lemma Suc_length_conv: "(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)" by (induct xs; simp; blast) lemma Suc_le_length_iff: "(Suc n \<le> length xs) = (\<exists>x ys. xs = x # ys \<and> n \<le> length ys)" by (metis Suc_le_D[of n] Suc_le_mono[of n] Suc_length_conv[of _ xs]) lemma impossible_Cons: "length xs \<le> length ys \<Longrightarrow> xs = x # ys = False" by (induct xs) auto lemma list_induct2 [consumes 1, case_names Nil Cons]: "length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow> (\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys)) \<Longrightarrow> P xs ys" proof (induct xs arbitrary: ys) case (Cons x xs ys) then show ?case by (cases ys) simp_all qed simp lemma list_induct3 [consumes 2, case_names Nil Cons]: "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow> (\<And>x xs y ys z zs. length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P xs ys zs \<Longrightarrow> P (x#xs) (y#ys) (z#zs)) \<Longrightarrow> P xs ys zs" proof (induct xs arbitrary: ys zs) case Nil then show ?case by simp next case (Cons x xs ys zs) then show ?case by (cases ys, simp_all) (cases zs, simp_all) qed lemma list_induct4 [consumes 3, case_names Nil Cons]: "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow> P [] [] [] [] \<Longrightarrow> (\<And>x xs y ys z zs w ws. length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow> P xs ys zs ws \<Longrightarrow> P (x#xs) (y#ys) (z#zs) (w#ws)) \<Longrightarrow> P xs ys zs ws" proof (induct xs arbitrary: ys zs ws) case Nil then show ?case by simp next case (Cons x xs ys zs ws) then show ?case by ((cases ys, simp_all), (cases zs,simp_all)) (cases ws, simp_all) qed lemma list_induct2': "\<lbrakk> P [] []; \<And>x xs. P (x#xs) []; \<And>y ys. P [] (y#ys); \<And>x xs y ys. P xs ys \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk> \<Longrightarrow> P xs ys" by (induct xs arbitrary: ys) (case_tac x, auto)+ lemma list_all2_iff: "list_all2 P xs ys \<longleftrightarrow> length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)" by (induct xs ys rule: list_induct2') auto lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False" by (rule Eq_FalseI) auto subsubsection \<open>\<open>@\<close> -- append\<close> global_interpretation append: monoid append Nil proof fix xs ys zs :: "'a list" show "(xs @ ys) @ zs = xs @ (ys @ zs)" by (induct xs) simp_all show "xs @ [] = xs" by (induct xs) simp_all qed simp lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)" by (fact append.assoc) lemma append_Nil2: "xs @ [] = xs" by (fact append.right_neutral) lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])" by (induct xs) auto lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])" by (induct xs) auto lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])" by (induct xs) auto lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])" by (induct xs) auto lemma append_eq_append_conv [simp]: "length xs = length ys \<or> length us = length vs \<Longrightarrow> (xs@us = ys@vs) = (xs=ys \<and> us=vs)" by (induct xs arbitrary: ys; case_tac ys; force) lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) = (\<exists>us. xs = zs @ us \<and> us @ ys = ts \<or> xs @ us = zs \<and> ys = us @ ts)" proof (induct xs arbitrary: ys zs ts) case (Cons x xs) then show ?case by (cases zs) auto qed fastforce lemma same_append_eq [iff, induct_simp]: "(xs @ ys = xs @ zs) = (ys = zs)" by simp lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)" by simp lemma append_same_eq [iff, induct_simp]: "(ys @ xs = zs @ xs) = (ys = zs)" by simp lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])" using append_same_eq [of _ _ "[]"] by auto lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])" using append_same_eq [of "[]"] by auto lemma hd_Cons_tl: "xs \<noteq> [] \<Longrightarrow> hd xs # tl xs = xs" by (fact list.collapse) lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)" by (induct xs) auto lemma hd_append2 [simp]: "xs \<noteq> [] \<Longrightarrow> hd (xs @ ys) = hd xs" by (simp add: hd_append split: list.split) lemma tl_append: "tl (xs @ ys) = (case xs of [] \<Rightarrow> tl ys | z#zs \<Rightarrow> zs @ ys)" by (simp split: list.split) lemma tl_append2 [simp]: "xs \<noteq> [] \<Longrightarrow> tl (xs @ ys) = tl xs @ ys" by (simp add: tl_append split: list.split) lemma Cons_eq_append_conv: "x#xs = ys@zs = (ys = [] \<and> x#xs = zs \<or> (\<exists>ys'. x#ys' = ys \<and> xs = ys'@zs))" by(cases ys) auto lemma append_eq_Cons_conv: "(ys@zs = x#xs) = (ys = [] \<and> zs = x#xs \<or> (\<exists>ys'. ys = x#ys' \<and> ys'@zs = xs))" by(cases ys) auto lemma longest_common_prefix: "\<exists>ps xs' ys'. xs = ps @ xs' \<and> ys = ps @ ys' \<and> (xs' = [] \<or> ys' = [] \<or> hd xs' \<noteq> hd ys')" by (induct xs ys rule: list_induct2') (blast, blast, blast, metis (no_types, hide_lams) append_Cons append_Nil list.sel(1)) text \<open>Trivial rules for solving \<open>@\<close>-equations automatically.\<close> lemma eq_Nil_appendI: "xs = ys \<Longrightarrow> xs = [] @ ys" by simp lemma Cons_eq_appendI: "\<lbrakk>x # xs1 = ys; xs = xs1 @ zs\<rbrakk> \<Longrightarrow> x # xs = ys @ zs" by auto lemma append_eq_appendI: "\<lbrakk>xs @ xs1 = zs; ys = xs1 @ us\<rbrakk> \<Longrightarrow> xs @ ys = zs @ us" by auto text \<open> Simplification procedure for all list equalities. Currently only tries to rearrange \<open>@\<close> to see if - both lists end in a singleton list, - or both lists end in the same list. \<close> simproc_setup list_eq ("(xs::'a list) = ys") = \<open> let fun last (cons as Const (\<^const_name>\<open>Cons\<close>, _) $ _ $ xs) = (case xs of Const (\<^const_name>\<open>Nil\<close>, _) => cons | _ => last xs) | last (Const(\<^const_name>\<open>append\<close>,_) $ _ $ ys) = last ys | last t = t; fun list1 (Const(\<^const_name>\<open>Cons\<close>,_) $ _ $ Const(\<^const_name>\<open>Nil\<close>,_)) = true | list1 _ = false; fun butlast ((cons as Const(\<^const_name>\<open>Cons\<close>,_) $ x) $ xs) = (case xs of Const (\<^const_name>\<open>Nil\<close>, _) => xs | _ => cons $ butlast xs) | butlast ((app as Const (\<^const_name>\<open>append\<close>, _) $ xs) $ ys) = app $ butlast ys | butlast xs = Const(\<^const_name>\<open>Nil\<close>, fastype_of xs); val rearr_ss = simpset_of (put_simpset HOL_basic_ss \<^context> addsimps [@{thm append_assoc}, @{thm append_Nil}, @{thm append_Cons}]); fun list_eq ctxt (F as (eq as Const(_,eqT)) $ lhs $ rhs) = let val lastl = last lhs and lastr = last rhs; fun rearr conv = let val lhs1 = butlast lhs and rhs1 = butlast rhs; val Type(_,listT::_) = eqT val appT = [listT,listT] ---> listT val app = Const(\<^const_name>\<open>append\<close>,appT) val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr) val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2)); val thm = Goal.prove ctxt [] [] eq (K (simp_tac (put_simpset rearr_ss ctxt) 1)); in SOME ((conv RS (thm RS trans)) RS eq_reflection) end; in if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv} else if lastl aconv lastr then rearr @{thm append_same_eq} else NONE end; in fn _ => fn ctxt => fn ct => list_eq ctxt (Thm.term_of ct) end \<close> subsubsection \<open>\<^const>\<open>map\<close>\<close> lemma hd_map: "xs \<noteq> [] \<Longrightarrow> hd (map f xs) = f (hd xs)" by (cases xs) simp_all lemma map_tl: "map f (tl xs) = tl (map f xs)" by (cases xs) simp_all lemma map_ext: "(\<And>x. x \<in> set xs \<longrightarrow> f x = g x) \<Longrightarrow> map f xs = map g xs" by (induct xs) simp_all lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)" by (rule ext, induct_tac xs) auto lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys" by (induct xs) auto lemma map_map [simp]: "map f (map g xs) = map (f \<circ> g) xs" by (induct xs) auto lemma map_comp_map[simp]: "((map f) \<circ> (map g)) = map(f \<circ> g)" by (rule ext) simp lemma rev_map: "rev (map f xs) = map f (rev xs)" by (induct xs) auto lemma map_eq_conv[simp]: "(map f xs = map g xs) = (\<forall>x \<in> set xs. f x = g x)" by (induct xs) auto lemma map_cong [fundef_cong]: "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> map f xs = map g ys" by simp lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])" by (cases xs) auto lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])" by (cases xs) auto lemma map_eq_Cons_conv: "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)" by (cases xs) auto lemma Cons_eq_map_conv: "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)" by (cases ys) auto lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1] lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1] declare map_eq_Cons_D [dest!] Cons_eq_map_D [dest!] lemma ex_map_conv: "(\<exists>xs. ys = map f xs) = (\<forall>y \<in> set ys. \<exists>x. y = f x)" by(induct ys, auto simp add: Cons_eq_map_conv) lemma map_eq_imp_length_eq: assumes "map f xs = map g ys" shows "length xs = length ys" using assms proof (induct ys arbitrary: xs) case Nil then show ?case by simp next case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto from Cons xs have "map f zs = map g ys" by simp with Cons have "length zs = length ys" by blast with xs show ?case by simp qed lemma map_inj_on: assumes map: "map f xs = map f ys" and inj: "inj_on f (set xs Un set ys)" shows "xs = ys" using map_eq_imp_length_eq [OF map] assms proof (induct rule: list_induct2) case (Cons x xs y ys) then show ?case by (auto intro: sym) qed auto lemma inj_on_map_eq_map: "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)" by(blast dest:map_inj_on) lemma map_injective: "map f xs = map f ys \<Longrightarrow> inj f \<Longrightarrow> xs = ys" by (induct ys arbitrary: xs) (auto dest!:injD) lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)" by(blast dest:map_injective) lemma inj_mapI: "inj f \<Longrightarrow> inj (map f)" by (iprover dest: map_injective injD intro: inj_onI) lemma inj_mapD: "inj (map f) \<Longrightarrow> inj f" by (metis (no_types, hide_lams) injI list.inject list.simps(9) the_inv_f_f) lemma inj_map[iff]: "inj (map f) = inj f" by (blast dest: inj_mapD intro: inj_mapI) lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A" by (blast intro:inj_onI dest:inj_onD map_inj_on) lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs" by (induct xs, auto) lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs" by (induct xs) auto lemma map_fst_zip[simp]: "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs" by (induct rule:list_induct2, simp_all) lemma map_snd_zip[simp]: "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys" by (induct rule:list_induct2, simp_all) lemma map_fst_zip_take: "map fst (zip xs ys) = take (min (length xs) (length ys)) xs" by (induct xs ys rule: list_induct2') simp_all lemma map_snd_zip_take: "map snd (zip xs ys) = take (min (length xs) (length ys)) ys" by (induct xs ys rule: list_induct2') simp_all lemma map2_map_map: "map2 h (map f xs) (map g xs) = map (\<lambda>x. h (f x) (g x)) xs" by (induction xs) (auto) functor map: map by (simp_all add: id_def) declare map.id [simp] subsubsection \<open>\<^const>\<open>rev\<close>\<close> lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs" by (induct xs) auto lemma rev_rev_ident [simp]: "rev (rev xs) = xs" by (induct xs) auto lemma rev_swap: "(rev xs = ys) = (xs = rev ys)" by auto lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])" by (induct xs) auto lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])" by (induct xs) auto lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])" by (cases xs) auto lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])" by (cases xs) auto lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)" proof (induct xs arbitrary: ys) case Nil then show ?case by force next case Cons then show ?case by (cases ys) auto qed lemma inj_on_rev[iff]: "inj_on rev A" by(simp add:inj_on_def) lemma rev_induct [case_names Nil snoc]: assumes "P []" and "\<And>x xs. P xs \<Longrightarrow> P (xs @ [x])" shows "P xs" proof - have "P (rev (rev xs))" by (rule_tac list = "rev xs" in list.induct, simp_all add: assms) then show ?thesis by simp qed lemma rev_exhaust [case_names Nil snoc]: "(xs = [] \<Longrightarrow> P) \<Longrightarrow>(\<And>ys y. xs = ys @ [y] \<Longrightarrow> P) \<Longrightarrow> P" by (induct xs rule: rev_induct) auto lemmas rev_cases = rev_exhaust lemma rev_nonempty_induct [consumes 1, case_names single snoc]: assumes "xs \<noteq> []" and single: "\<And>x. P [x]" and snoc': "\<And>x xs. xs \<noteq> [] \<Longrightarrow> P xs \<Longrightarrow> P (xs@[x])" shows "P xs" using \<open>xs \<noteq> []\<close> proof (induct xs rule: rev_induct) case (snoc x xs) then show ?case proof (cases xs) case Nil thus ?thesis by (simp add: single) next case Cons with snoc show ?thesis by (fastforce intro!: snoc') qed qed simp lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])" by(rule rev_cases[of xs]) auto subsubsection \<open>\<^const>\<open>set\<close>\<close> declare list.set[code_post] \<comment> \<open>pretty output\<close> lemma finite_set [iff]: "finite (set xs)" by (induct xs) auto lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)" by (induct xs) auto lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs \<in> set xs" by(cases xs) auto lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)" by auto lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" by auto lemma set_empty [iff]: "(set xs = {}) = (xs = [])" by (induct xs) auto lemma set_empty2[iff]: "({} = set xs) = (xs = [])" by(induct xs) auto lemma set_rev [simp]: "set (rev xs) = set xs" by (induct xs) auto lemma set_map [simp]: "set (map f xs) = f`(set xs)" by (induct xs) auto lemma set_filter [simp]: "set (filter P xs) = {x. x \<in> set xs \<and> P x}" by (induct xs) auto lemma set_upt [simp]: "set[i..<j] = {i..<j}" by (induct j) auto lemma split_list: "x \<in> set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs" proof (induct xs) case Nil thus ?case by simp next case Cons thus ?case by (auto intro: Cons_eq_appendI) qed lemma in_set_conv_decomp: "x \<in> set xs \<longleftrightarrow> (\<exists>ys zs. xs = ys @ x # zs)" by (auto elim: split_list) lemma split_list_first: "x \<in> set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys" proof (induct xs) case Nil thus ?case by simp next case (Cons a xs) show ?case proof cases assume "x = a" thus ?case using Cons by fastforce next assume "x \<noteq> a" thus ?case using Cons by(fastforce intro!: Cons_eq_appendI) qed qed lemma in_set_conv_decomp_first: "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)" by (auto dest!: split_list_first) lemma split_list_last: "x \<in> set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs" proof (induct xs rule: rev_induct) case Nil thus ?case by simp next case (snoc a xs) show ?case proof cases assume "x = a" thus ?case using snoc by (auto intro!: exI) next assume "x \<noteq> a" thus ?case using snoc by fastforce qed qed lemma in_set_conv_decomp_last: "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)" by (auto dest!: split_list_last) lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs \<and> P x" proof (induct xs) case Nil thus ?case by simp next case Cons thus ?case by(simp add:Bex_def)(metis append_Cons append.simps(1)) qed lemma split_list_propE: assumes "\<exists>x \<in> set xs. P x" obtains ys x zs where "xs = ys @ x # zs" and "P x" using split_list_prop [OF assms] by blast lemma split_list_first_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)" proof (induct xs) case Nil thus ?case by simp next case (Cons x xs) show ?case proof cases assume "P x" hence "x # xs = [] @ x # xs \<and> P x \<and> (\<forall>y\<in>set []. \<not> P y)" by simp thus ?thesis by fast next assume "\<not> P x" hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp thus ?thesis using \<open>\<not> P x\<close> Cons(1) by (metis append_Cons set_ConsD) qed qed lemma split_list_first_propE: assumes "\<exists>x \<in> set xs. P x" obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y" using split_list_first_prop [OF assms] by blast lemma split_list_first_prop_iff: "(\<exists>x \<in> set xs. P x) \<longleftrightarrow> (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))" by (rule, erule split_list_first_prop) auto lemma split_list_last_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)" proof(induct xs rule:rev_induct) case Nil thus ?case by simp next case (snoc x xs) show ?case proof cases assume "P x" thus ?thesis by (auto intro!: exI) next assume "\<not> P x" hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp thus ?thesis using \<open>\<not> P x\<close> snoc(1) by fastforce qed qed lemma split_list_last_propE: assumes "\<exists>x \<in> set xs. P x" obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z" using split_list_last_prop [OF assms] by blast lemma split_list_last_prop_iff: "(\<exists>x \<in> set xs. P x) \<longleftrightarrow> (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))" by rule (erule split_list_last_prop, auto) lemma finite_list: "finite A \<Longrightarrow> \<exists>xs. set xs = A" by (erule finite_induct) (auto simp add: list.set(2)[symmetric] simp del: list.set(2)) lemma card_length: "card (set xs) \<le> length xs" by (induct xs) (auto simp add: card_insert_if) lemma set_minus_filter_out: "set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)" by (induct xs) auto lemma append_Cons_eq_iff: "\<lbrakk> x \<notin> set xs; x \<notin> set ys \<rbrakk> \<Longrightarrow> xs @ x # ys = xs' @ x # ys' \<longleftrightarrow> (xs = xs' \<and> ys = ys')" by(auto simp: append_eq_Cons_conv Cons_eq_append_conv append_eq_append_conv2) subsubsection \<open>\<^const>\<open>concat\<close>\<close> lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys" by (induct xs) auto lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])" by (induct xss) auto lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])" by (induct xss) auto lemma set_concat [simp]: "set (concat xs) = (\<Union>x\<in>set xs. set x)" by (induct xs) auto lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs" by (induct xs) auto lemma map_concat: "map f (concat xs) = concat (map (map f) xs)" by (induct xs) auto lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))" by (induct xs) auto lemma length_concat_rev[simp]: "length (concat (rev xs)) = length (concat xs)" by (induction xs) auto lemma concat_eq_concat_iff: "\<forall>(x, y) \<in> set (zip xs ys). length x = length y \<Longrightarrow> length xs = length ys \<Longrightarrow> (concat xs = concat ys) = (xs = ys)" proof (induct xs arbitrary: ys) case (Cons x xs ys) thus ?case by (cases ys) auto qed (auto) lemma concat_injective: "concat xs = concat ys \<Longrightarrow> length xs = length ys \<Longrightarrow> \<forall>(x, y) \<in> set (zip xs ys). length x = length y \<Longrightarrow> xs = ys" by (simp add: concat_eq_concat_iff) lemma concat_eq_appendD: assumes "concat xss = ys @ zs" "xss \<noteq> []" shows "\<exists>xss1 xs xs' xss2. xss = xss1 @ (xs @ xs') # xss2 \<and> ys = concat xss1 @ xs \<and> zs = xs' @ concat xss2" using assms proof(induction xss arbitrary: ys) case (Cons xs xss) from Cons.prems consider us where "xs @ us = ys" "concat xss = us @ zs" | us where "xs = ys @ us" "us @ concat xss = zs" by(auto simp add: append_eq_append_conv2) then show ?case proof cases case 1 then show ?thesis using Cons.IH[OF 1(2)] by(cases xss)(auto intro: exI[where x="[]"], metis append.assoc append_Cons concat.simps(2)) qed(auto intro: exI[where x="[]"]) qed simp lemma concat_eq_append_conv: "concat xss = ys @ zs \<longleftrightarrow> (if xss = [] then ys = [] \<and> zs = [] else \<exists>xss1 xs xs' xss2. xss = xss1 @ (xs @ xs') # xss2 \<and> ys = concat xss1 @ xs \<and> zs = xs' @ concat xss2)" by(auto dest: concat_eq_appendD) lemma hd_concat: "\<lbrakk>xs \<noteq> []; hd xs \<noteq> []\<rbrakk> \<Longrightarrow> hd (concat xs) = hd (hd xs)" by (metis concat.simps(2) hd_Cons_tl hd_append2) simproc_setup list_neq ("(xs::'a list) = ys") = \<open> (* Reduces xs=ys to False if xs and ys cannot be of the same length. This is the case if the atomic sublists of one are a submultiset of those of the other list and there are fewer Cons's in one than the other. *) let fun len (Const(\<^const_name>\<open>Nil\<close>,_)) acc = acc | len (Const(\<^const_name>\<open>Cons\<close>,_) $ _ $ xs) (ts,n) = len xs (ts,n+1) | len (Const(\<^const_name>\<open>append\<close>,_) $ xs $ ys) acc = len xs (len ys acc) | len (Const(\<^const_name>\<open>rev\<close>,_) $ xs) acc = len xs acc | len (Const(\<^const_name>\<open>map\<close>,_) $ _ $ xs) acc = len xs acc | len (Const(\<^const_name>\<open>concat\<close>,T) $ (Const(\<^const_name>\<open>rev\<close>,_) $ xss)) acc = len (Const(\<^const_name>\<open>concat\<close>,T) $ xss) acc | len t (ts,n) = (t::ts,n); val ss = simpset_of \<^context>; fun list_neq ctxt ct = let val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct; val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0); fun prove_neq() = let val Type(_,listT::_) = eqT; val size = HOLogic.size_const listT; val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs); val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len); val thm = Goal.prove ctxt [] [] neq_len (K (simp_tac (put_simpset ss ctxt) 1)); in SOME (thm RS @{thm neq_if_length_neq}) end in if m < n andalso submultiset (op aconv) (ls,rs) orelse n < m andalso submultiset (op aconv) (rs,ls) then prove_neq() else NONE end; in K list_neq end \<close> subsubsection \<open>\<^const>\<open>filter\<close>\<close> lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys" by (induct xs) auto lemma rev_filter: "rev (filter P xs) = filter P (rev xs)" by (induct xs) simp_all lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs" by (induct xs) auto lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)" by (induct xs) auto lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs" by (induct xs) (auto simp add: le_SucI) lemma sum_length_filter_compl: "length(filter P xs) + length(filter (\<lambda>x. \<not>P x) xs) = length xs" by(induct xs) simp_all lemma filter_True [simp]: "\<forall>x \<in> set xs. P x \<Longrightarrow> filter P xs = xs" by (induct xs) auto lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x \<Longrightarrow> filter P xs = []" by (induct xs) auto lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" by (induct xs) simp_all lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)" proof (induct xs) case (Cons x xs) then show ?case using length_filter_le by (simp add: impossible_Cons) qed auto lemma filter_map: "filter P (map f xs) = map f (filter (P \<circ> f) xs)" by (induct xs) simp_all lemma length_filter_map[simp]: "length (filter P (map f xs)) = length(filter (P \<circ> f) xs)" by (simp add:filter_map) lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs" by auto lemma length_filter_less: "\<lbrakk> x \<in> set xs; \<not> P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs" proof (induct xs) case Nil thus ?case by simp next case (Cons x xs) thus ?case using Suc_le_eq by fastforce qed lemma length_filter_conv_card: "length(filter p xs) = card{i. i < length xs \<and> p(xs!i)}" proof (induct xs) case Nil thus ?case by simp next case (Cons x xs) let ?S = "{i. i < length xs \<and> p(xs!i)}" have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite) show ?case (is "?l = card ?S'") proof (cases) assume "p x" hence eq: "?S' = insert 0 (Suc ` ?S)" by(auto simp: image_def split:nat.split dest:gr0_implies_Suc) have "length (filter p (x # xs)) = Suc(card ?S)" using Cons \<open>p x\<close> by simp also have "\<dots> = Suc(card(Suc ` ?S))" using fin by (simp add: card_image) also have "\<dots> = card ?S'" using eq fin by (simp add:card_insert_if) finally show ?thesis . next assume "\<not> p x" hence eq: "?S' = Suc ` ?S" by(auto simp add: image_def split:nat.split elim:lessE) have "length (filter p (x # xs)) = card ?S" using Cons \<open>\<not> p x\<close> by simp also have "\<dots> = card(Suc ` ?S)" using fin by (simp add: card_image) also have "\<dots> = card ?S'" using eq fin by (simp add:card_insert_if) finally show ?thesis . qed qed lemma Cons_eq_filterD: "x#xs = filter P ys \<Longrightarrow> \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs" (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs") proof(induct ys) case Nil thus ?case by simp next case (Cons y ys) show ?case (is "\<exists>x. ?Q x") proof cases assume Py: "P y" show ?thesis proof cases assume "x = y" with Py Cons.prems have "?Q []" by simp then show ?thesis .. next assume "x \<noteq> y" with Py Cons.prems show ?thesis by simp qed next assume "\<not> P y" with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastforce then have "?Q (y#us)" by simp then show ?thesis .. qed qed lemma filter_eq_ConsD: "filter P ys = x#xs \<Longrightarrow> \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs" by(rule Cons_eq_filterD) simp lemma filter_eq_Cons_iff: "(filter P ys = x#xs) = (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)" by(auto dest:filter_eq_ConsD) lemma Cons_eq_filter_iff: "(x#xs = filter P ys) = (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)" by(auto dest:Cons_eq_filterD) lemma inj_on_filter_key_eq: assumes "inj_on f (insert y (set xs))" shows "filter (\<lambda>x. f y = f x) xs = filter (HOL.eq y) xs" using assms by (induct xs) auto lemma filter_cong[fundef_cong]: "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys" by (induct ys arbitrary: xs) auto subsubsection \<open>List partitioning\<close> primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where "partition P [] = ([], [])" | "partition P (x # xs) = (let (yes, no) = partition P xs in if P x then (x # yes, no) else (yes, x # no))" lemma partition_filter1: "fst (partition P xs) = filter P xs" by (induct xs) (auto simp add: Let_def split_def) lemma partition_filter2: "snd (partition P xs) = filter (Not \<circ> P) xs" by (induct xs) (auto simp add: Let_def split_def) lemma partition_P: assumes "partition P xs = (yes, no)" shows "(\<forall>p \<in> set yes. P p) \<and> (\<forall>p \<in> set no. \<not> P p)" proof - from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)" by simp_all then show ?thesis by (simp_all add: partition_filter1 partition_filter2) qed lemma partition_set: assumes "partition P xs = (yes, no)" shows "set yes \<union> set no = set xs" proof - from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)" by simp_all then show ?thesis by (auto simp add: partition_filter1 partition_filter2) qed lemma partition_filter_conv[simp]: "partition f xs = (filter f xs,filter (Not \<circ> f) xs)" unfolding partition_filter2[symmetric] unfolding partition_filter1[symmetric] by simp declare partition.simps[simp del] subsubsection \<open>\<^const>\<open>nth\<close>\<close> lemma nth_Cons_0 [simp, code]: "(x # xs)!0 = x" by auto lemma nth_Cons_Suc [simp, code]: "(x # xs)!(Suc n) = xs!n" by auto declare nth.simps [simp del] lemma nth_Cons_pos[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)" by(auto simp: Nat.gr0_conv_Suc) lemma nth_append: "(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))" proof (induct xs arbitrary: n) case (Cons x xs) then show ?case using less_Suc_eq_0_disj by auto qed simp lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x" by (induct xs) auto lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n" by (induct xs) auto lemma nth_map [simp]: "n < length xs \<Longrightarrow> (map f xs)!n = f(xs!n)" proof (induct xs arbitrary: n) case (Cons x xs) then show ?case using less_Suc_eq_0_disj by auto qed simp lemma nth_tl: "n < length (tl xs) \<Longrightarrow> tl xs ! n = xs ! Suc n" by (induction xs) auto lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0" by(cases xs) simp_all lemma list_eq_iff_nth_eq: "(xs = ys) = (length xs = length ys \<and> (\<forall>i<length xs. xs!i = ys!i))" proof (induct xs arbitrary: ys) case (Cons x xs ys) show ?case proof (cases ys) case (Cons y ys) then show ?thesis using Cons.hyps by fastforce qed simp qed force lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}" proof (induct xs) case (Cons x xs) have "insert x {xs ! i |i. i < length xs} = {(x # xs) ! i |i. i < Suc (length xs)}" (is "?L=?R") proof show "?L \<subseteq> ?R" by force show "?R \<subseteq> ?L" using less_Suc_eq_0_disj by auto qed with Cons show ?case by simp qed simp lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)" by(auto simp:set_conv_nth) lemma nth_equal_first_eq: assumes "x \<notin> set xs" assumes "n \<le> length xs" shows "(x # xs) ! n = x \<longleftrightarrow> n = 0" (is "?lhs \<longleftrightarrow> ?rhs") proof assume ?lhs show ?rhs proof (rule ccontr) assume "n \<noteq> 0" then have "n > 0" by simp with \<open>?lhs\<close> have "xs ! (n - 1) = x" by simp moreover from \<open>n > 0\<close> \<open>n \<le> length xs\<close> have "n - 1 < length xs" by simp ultimately have "\<exists>i<length xs. xs ! i = x" by auto with \<open>x \<notin> set xs\<close> in_set_conv_nth [of x xs] show False by simp qed next assume ?rhs then show ?lhs by simp qed lemma nth_non_equal_first_eq: assumes "x \<noteq> y" shows "(x # xs) ! n = y \<longleftrightarrow> xs ! (n - 1) = y \<and> n > 0" (is "?lhs \<longleftrightarrow> ?rhs") proof assume "?lhs" with assms have "n > 0" by (cases n) simp_all with \<open>?lhs\<close> show ?rhs by simp next assume "?rhs" then show "?lhs" by simp qed lemma list_ball_nth: "\<lbrakk>n < length xs; \<forall>x \<in> set xs. P x\<rbrakk> \<Longrightarrow> P(xs!n)" by (auto simp add: set_conv_nth) lemma nth_mem [simp]: "n < length xs \<Longrightarrow> xs!n \<in> set xs" by (auto simp add: set_conv_nth) lemma all_nth_imp_all_set: "\<lbrakk>\<forall>i < length xs. P(xs!i); x \<in> set xs\<rbrakk> \<Longrightarrow> P x" by (auto simp add: set_conv_nth) lemma all_set_conv_all_nth: "(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs \<longrightarrow> P (xs ! i))" by (auto simp add: set_conv_nth) lemma rev_nth: "n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)" proof (induct xs arbitrary: n) case Nil thus ?case by simp next case (Cons x xs) hence n: "n < Suc (length xs)" by simp moreover { assume "n < length xs" with n obtain n' where n': "length xs - n = Suc n'" by (cases "length xs - n", auto) moreover from n' have "length xs - Suc n = n'" by simp ultimately have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp } ultimately show ?case by (clarsimp simp add: Cons nth_append) qed lemma Skolem_list_nth: "(\<forall>i<k. \<exists>x. P i x) = (\<exists>xs. size xs = k \<and> (\<forall>i<k. P i (xs!i)))" (is "_ = (\<exists>xs. ?P k xs)") proof(induct k) case 0 show ?case by simp next case (Suc k) show ?case (is "?L = ?R" is "_ = (\<exists>xs. ?P' xs)") proof assume "?R" thus "?L" using Suc by auto next assume "?L" with Suc obtain x xs where "?P k xs \<and> P k x" by (metis less_Suc_eq) hence "?P'(xs@[x])" by(simp add:nth_append less_Suc_eq) thus "?R" .. qed qed subsubsection \<open>\<^const>\<open>list_update\<close>\<close> lemma length_list_update [simp]: "length(xs[i:=x]) = length xs" by (induct xs arbitrary: i) (auto split: nat.split) lemma nth_list_update: "i < length xs\<Longrightarrow> (xs[i:=x])!j = (if i = j then x else xs!j)" by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split) lemma nth_list_update_eq [simp]: "i < length xs \<Longrightarrow> (xs[i:=x])!i = x" by (simp add: nth_list_update) lemma nth_list_update_neq [simp]: "i \<noteq> j \<Longrightarrow> xs[i:=x]!j = xs!j" by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split) lemma list_update_id[simp]: "xs[i := xs!i] = xs" by (induct xs arbitrary: i) (simp_all split:nat.splits) lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs" proof (induct xs arbitrary: i) case (Cons x xs i) then show ?case by (metis leD length_list_update list_eq_iff_nth_eq nth_list_update_neq) qed simp lemma list_update_nonempty[simp]: "xs[k:=x] = [] \<longleftrightarrow> xs=[]" by (simp only: length_0_conv[symmetric] length_list_update) lemma list_update_same_conv: "i < length xs \<Longrightarrow> (xs[i := x] = xs) = (xs!i = x)" by (induct xs arbitrary: i) (auto split: nat.split) lemma list_update_append1: "i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys" by (induct xs arbitrary: i)(auto split:nat.split) lemma list_update_append: "(xs @ ys) [n:= x] = (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))" by (induct xs arbitrary: n) (auto split:nat.splits) lemma list_update_length [simp]: "(xs @ x # ys)[length xs := y] = (xs @ y # ys)" by (induct xs, auto) lemma map_update: "map f (xs[k:= y]) = (map f xs)[k := f y]" by(induct xs arbitrary: k)(auto split:nat.splits) lemma rev_update: "k < length xs \<Longrightarrow> rev (xs[k:= y]) = (rev xs)[length xs - k - 1 := y]" by (induct xs arbitrary: k) (auto simp: list_update_append split:nat.splits) lemma update_zip: "(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])" by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split) lemma set_update_subset_insert: "set(xs[i:=x]) \<le> insert x (set xs)" by (induct xs arbitrary: i) (auto split: nat.split) lemma set_update_subsetI: "\<lbrakk>set xs \<subseteq> A; x \<in> A\<rbrakk> \<Longrightarrow> set(xs[i := x]) \<subseteq> A" by (blast dest!: set_update_subset_insert [THEN subsetD]) lemma set_update_memI: "n < length xs \<Longrightarrow> x \<in> set (xs[n := x])" by (induct xs arbitrary: n) (auto split:nat.splits) lemma list_update_overwrite[simp]: "xs [i := x, i := y] = xs [i := y]" by (induct xs arbitrary: i) (simp_all split: nat.split) lemma list_update_swap: "i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]" by (induct xs arbitrary: i i') (simp_all split: nat.split) lemma list_update_code [code]: "[][i := y] = []" "(x # xs)[0 := y] = y # xs" "(x # xs)[Suc i := y] = x # xs[i := y]" by simp_all subsubsection \<open>\<^const>\<open>last\<close> and \<^const>\<open>butlast\<close>\<close> lemma last_snoc [simp]: "last (xs @ [x]) = x" by (induct xs) auto lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs" by (induct xs) auto lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x" by simp lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs" by simp lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)" by (induct xs) (auto) lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs" by(simp add:last_append) lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys" by(simp add:last_append) lemma last_tl: "xs = [] \<or> tl xs \<noteq> [] \<Longrightarrow>last (tl xs) = last xs" by (induct xs) simp_all lemma butlast_tl: "butlast (tl xs) = tl (butlast xs)" by (induct xs) simp_all lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs" by(rule rev_exhaust[of xs]) simp_all lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs" by(cases xs) simp_all lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as" by (induct as) auto lemma length_butlast [simp]: "length (butlast xs) = length xs - 1" by (induct xs rule: rev_induct) auto lemma butlast_append: "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)" by (induct xs arbitrary: ys) auto lemma append_butlast_last_id [simp]: "xs \<noteq> [] \<Longrightarrow> butlast xs @ [last xs] = xs" by (induct xs) auto lemma in_set_butlastD: "x \<in> set (butlast xs) \<Longrightarrow> x \<in> set xs" by (induct xs) (auto split: if_split_asm) lemma in_set_butlast_appendI: "x \<in> set (butlast xs) \<or> x \<in> set (butlast ys) \<Longrightarrow> x \<in> set (butlast (xs @ ys))" by (auto dest: in_set_butlastD simp add: butlast_append) lemma last_drop[simp]: "n < length xs \<Longrightarrow> last (drop n xs) = last xs" by (induct xs arbitrary: n)(auto split:nat.split) lemma nth_butlast: assumes "n < length (butlast xs)" shows "butlast xs ! n = xs ! n" proof (cases xs) case (Cons y ys) moreover from assms have "butlast xs ! n = (butlast xs @ [last xs]) ! n" by (simp add: nth_append) ultimately show ?thesis using append_butlast_last_id by simp qed simp lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)" by(induct xs)(auto simp:neq_Nil_conv) lemma butlast_conv_take: "butlast xs = take (length xs - 1) xs" by (induction xs rule: induct_list012) simp_all lemma last_list_update: "xs \<noteq> [] \<Longrightarrow> last(xs[k:=x]) = (if k = size xs - 1 then x else last xs)" by (auto simp: last_conv_nth) lemma butlast_list_update: "butlast(xs[k:=x]) = (if k = size xs - 1 then butlast xs else (butlast xs)[k:=x])" by(cases xs rule:rev_cases)(auto simp: list_update_append split: nat.splits) lemma last_map: "xs \<noteq> [] \<Longrightarrow> last (map f xs) = f (last xs)" by (cases xs rule: rev_cases) simp_all lemma map_butlast: "map f (butlast xs) = butlast (map f xs)" by (induct xs) simp_all lemma snoc_eq_iff_butlast: "xs @ [x] = ys \<longleftrightarrow> (ys \<noteq> [] \<and> butlast ys = xs \<and> last ys = x)" by fastforce corollary longest_common_suffix: "\<exists>ss xs' ys'. xs = xs' @ ss \<and> ys = ys' @ ss \<and> (xs' = [] \<or> ys' = [] \<or> last xs' \<noteq> last ys')" using longest_common_prefix[of "rev xs" "rev ys"] unfolding rev_swap rev_append by (metis last_rev rev_is_Nil_conv) lemma butlast_rev [simp]: "butlast (rev xs) = rev (tl xs)" by (cases xs) simp_all subsubsection \<open>\<^const>\<open>take\<close> and \<^const>\<open>drop\<close>\<close> lemma take_0: "take 0 xs = []" by (induct xs) auto lemma drop_0: "drop 0 xs = xs" by (induct xs) auto lemma take0[simp]: "take 0 = (\<lambda>xs. [])" by(rule ext) (rule take_0) lemma drop0[simp]: "drop 0 = (\<lambda>x. x)" by(rule ext) (rule drop_0) lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs" by simp lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs" by simp declare take_Cons [simp del] and drop_Cons [simp del] lemma take_Suc: "xs \<noteq> [] \<Longrightarrow> take (Suc n) xs = hd xs # take n (tl xs)" by(clarsimp simp add:neq_Nil_conv) lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)" by(cases xs, simp_all) lemma hd_take[simp]: "j > 0 \<Longrightarrow> hd (take j xs) = hd xs" by (metis gr0_conv_Suc list.sel(1) take.simps(1) take_Suc) lemma take_tl: "take n (tl xs) = tl (take (Suc n) xs)" by (induct xs arbitrary: n) simp_all lemma drop_tl: "drop n (tl xs) = tl(drop n xs)" by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split) lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)" by (cases n, simp, cases xs, auto) lemma tl_drop: "tl (drop n xs) = drop n (tl xs)" by (simp only: drop_tl) lemma nth_via_drop: "drop n xs = y#ys \<Longrightarrow> xs!n = y" by (induct xs arbitrary: n, simp)(auto simp: drop_Cons nth_Cons split: nat.splits) lemma take_Suc_conv_app_nth: "i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]" proof (induct xs arbitrary: i) case Nil then show ?case by simp next case Cons then show ?case by (cases i) auto qed lemma Cons_nth_drop_Suc: "i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs" proof (induct xs arbitrary: i) case Nil then show ?case by simp next case Cons then show ?case by (cases i) auto qed lemma length_take [simp]: "length (take n xs) = min (length xs) n" by (induct n arbitrary: xs) (auto, case_tac xs, auto) lemma length_drop [simp]: "length (drop n xs) = (length xs - n)" by (induct n arbitrary: xs) (auto, case_tac xs, auto) lemma take_all [simp]: "length xs \<le> n \<Longrightarrow> take n xs = xs" by (induct n arbitrary: xs) (auto, case_tac xs, auto) lemma drop_all [simp]: "length xs \<le> n \<Longrightarrow> drop n xs = []" by (induct n arbitrary: xs) (auto, case_tac xs, auto) lemma take_all_iff [simp]: "take n xs = xs \<longleftrightarrow> length xs \<le> n" by (metis length_take min.order_iff take_all) lemma drop_all_iff [simp]: "drop n xs = [] \<longleftrightarrow> length xs \<le> n" by (metis diff_is_0_eq drop_all length_drop list.size(3)) lemma take_append [simp]: "take n (xs @ ys) = (take n xs @ take (n - length xs) ys)" by (induct n arbitrary: xs) (auto, case_tac xs, auto) lemma drop_append [simp]: "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys" by (induct n arbitrary: xs) (auto, case_tac xs, auto) lemma take_take [simp]: "take n (take m xs) = take (min n m) xs" proof (induct m arbitrary: xs n) case 0 then show ?case by simp next case Suc then show ?case by (cases xs; cases n) simp_all qed lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs" proof (induct m arbitrary: xs) case 0 then show ?case by simp next case Suc then show ?case by (cases xs) simp_all qed lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)" proof (induct m arbitrary: xs n) case 0 then show ?case by simp next case Suc then show ?case by (cases xs; cases n) simp_all qed lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)" by(induct xs arbitrary: m n)(auto simp: take_Cons drop_Cons split: nat.split) lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs" proof (induct n arbitrary: xs) case 0 then show ?case by simp next case Suc then show ?case by (cases xs) simp_all qed lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])" by(induct xs arbitrary: n)(auto simp: take_Cons split:nat.split) lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs \<le> n)" by (induct xs arbitrary: n) (auto simp: drop_Cons split:nat.split) lemma take_map: "take n (map f xs) = map f (take n xs)" proof (induct n arbitrary: xs) case 0 then show ?case by simp next case Suc then show ?case by (cases xs) simp_all qed lemma drop_map: "drop n (map f xs) = map f (drop n xs)" proof (induct n arbitrary: xs) case 0 then show ?case by simp next case Suc then show ?case by (cases xs) simp_all qed lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)" proof (induct xs arbitrary: i) case Nil then show ?case by simp next case Cons then show ?case by (cases i) auto qed lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)" proof (induct xs arbitrary: i) case Nil then show ?case by simp next case Cons then show ?case by (cases i) auto qed lemma drop_rev: "drop n (rev xs) = rev (take (length xs - n) xs)" by (cases "length xs < n") (auto simp: rev_take) lemma take_rev: "take n (rev xs) = rev (drop (length xs - n) xs)" by (cases "length xs < n") (auto simp: rev_drop) lemma nth_take [simp]: "i < n \<Longrightarrow> (take n xs)!i = xs!i" proof (induct xs arbitrary: i n) case Nil then show ?case by simp next case Cons then show ?case by (cases n; cases i) simp_all qed lemma nth_drop [simp]: "n \<le> length xs \<Longrightarrow> (drop n xs)!i = xs!(n + i)" proof (induct n arbitrary: xs) case 0 then show ?case by simp next case Suc then show ?case by (cases xs) simp_all qed lemma butlast_take: "n \<le> length xs \<Longrightarrow> butlast (take n xs) = take (n - 1) xs" by (simp add: butlast_conv_take min.absorb1 min.absorb2) lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)" by (simp add: butlast_conv_take drop_take ac_simps) lemma take_butlast: "n < length xs \<Longrightarrow> take n (butlast xs) = take n xs" by (simp add: butlast_conv_take min.absorb1) lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)" by (simp add: butlast_conv_take drop_take ac_simps) lemma butlast_power: "(butlast ^^ n) xs = take (length xs - n) xs" by (induct n) (auto simp: butlast_take) lemma hd_drop_conv_nth: "n < length xs \<Longrightarrow> hd(drop n xs) = xs!n" by(simp add: hd_conv_nth) lemma set_take_subset_set_take: "m \<le> n \<Longrightarrow> set(take m xs) \<le> set(take n xs)" proof (induct xs arbitrary: m n) case (Cons x xs m n) then show ?case by (cases n) (auto simp: take_Cons) qed simp lemma set_take_subset: "set(take n xs) \<subseteq> set xs" by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split) lemma set_drop_subset: "set(drop n xs) \<subseteq> set xs" by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split) lemma set_drop_subset_set_drop: "m \<ge> n \<Longrightarrow> set(drop m xs) \<le> set(drop n xs)" proof (induct xs arbitrary: m n) case (Cons x xs m n) then show ?case by (clarsimp simp: drop_Cons split: nat.split) (metis set_drop_subset subset_iff) qed simp lemma in_set_takeD: "x \<in> set(take n xs) \<Longrightarrow> x \<in> set xs" using set_take_subset by fast lemma in_set_dropD: "x \<in> set(drop n xs) \<Longrightarrow> x \<in> set xs" using set_drop_subset by fast lemma append_eq_conv_conj: "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)" proof (induct xs arbitrary: zs) case (Cons x xs zs) then show ?case by (cases zs, auto) qed auto lemma map_eq_append_conv: "map f xs = ys @ zs \<longleftrightarrow> (\<exists>us vs. xs = us @ vs \<and> ys = map f us \<and> zs = map f vs)" proof - have "map f xs \<noteq> ys @ zs \<and> map f xs \<noteq> ys @ zs \<or> map f xs \<noteq> ys @ zs \<or> map f xs = ys @ zs \<and> (\<exists>bs bsa. xs = bs @ bsa \<and> ys = map f bs \<and> zs = map f bsa)" by (metis append_eq_conv_conj append_take_drop_id drop_map take_map) then show ?thesis using map_append by blast qed lemma append_eq_map_conv: "ys @ zs = map f xs \<longleftrightarrow> (\<exists>us vs. xs = us @ vs \<and> ys = map f us \<and> zs = map f vs)" by (metis map_eq_append_conv) lemma take_add: "take (i+j) xs = take i xs @ take j (drop i xs)" proof (induct xs arbitrary: i) case (Cons x xs i) then show ?case by (cases i, auto) qed auto lemma append_eq_append_conv_if: "(xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>1 @ ys\<^sub>2) = (if size xs\<^sub>1 \<le> size ys\<^sub>1 then xs\<^sub>1 = take (size xs\<^sub>1) ys\<^sub>1 \<and> xs\<^sub>2 = drop (size xs\<^sub>1) ys\<^sub>1 @ ys\<^sub>2 else take (size ys\<^sub>1) xs\<^sub>1 = ys\<^sub>1 \<and> drop (size ys\<^sub>1) xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>2)" proof (induct xs\<^sub>1 arbitrary: ys\<^sub>1) case (Cons a xs\<^sub>1 ys\<^sub>1) then show ?case by (cases ys\<^sub>1, auto) qed auto lemma take_hd_drop: "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (Suc n) xs" by (induct xs arbitrary: n) (simp_all add:drop_Cons split:nat.split) lemma id_take_nth_drop: "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" proof - assume si: "i < length xs" hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto moreover from si have "take (Suc i) xs = take i xs @ [xs!i]" using take_Suc_conv_app_nth by blast ultimately show ?thesis by auto qed lemma take_update_cancel[simp]: "n \<le> m \<Longrightarrow> take n (xs[m := y]) = take n xs" by(simp add: list_eq_iff_nth_eq) lemma drop_update_cancel[simp]: "n < m \<Longrightarrow> drop m (xs[n := x]) = drop m xs" by(simp add: list_eq_iff_nth_eq) lemma upd_conv_take_nth_drop: "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs" proof - assume i: "i < length xs" have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]" by(rule arg_cong[OF id_take_nth_drop[OF i]]) also have "\<dots> = take i xs @ a # drop (Suc i) xs" using i by (simp add: list_update_append) finally show ?thesis . qed lemma take_update_swap: "take m (xs[n := x]) = (take m xs)[n := x]" proof (cases "n \<ge> length xs") case False then show ?thesis by (simp add: upd_conv_take_nth_drop take_Cons drop_take min_def diff_Suc split: nat.split) qed auto lemma drop_update_swap: assumes "m \<le> n" shows "drop m (xs[n := x]) = (drop m xs)[n-m := x]" proof (cases "n \<ge> length xs") case False with assms show ?thesis by (simp add: upd_conv_take_nth_drop drop_take) qed auto lemma nth_image: "l \<le> size xs \<Longrightarrow> nth xs ` {0..<l} = set(take l xs)" by(auto simp: set_conv_nth image_def) (metis Suc_le_eq nth_take order_trans) lemma list_all_take_drop_conv: "list_all P (take n xs) \<and> list_all P (drop n xs) \<longleftrightarrow> list_all P xs" proof (induction xs arbitrary: n) case Nil then show ?case by simp next case (Cons a xs) then show ?case by (cases n) simp_all qed lemmas list_all_takeI = list_all_take_drop_conv[THEN iffD2, THEN conjunct1] lemmas list_all_dropI = list_all_take_drop_conv[THEN iffD2, THEN conjunct2] lemmas list_all_take_dropD = conjI[THEN list_all_take_drop_conv[THEN iffD1]] subsubsection \<open>\<^const>\<open>takeWhile\<close> and \<^const>\<open>dropWhile\<close>\<close> lemma length_takeWhile_le: "length (takeWhile P xs) \<le> length xs" by (induct xs) auto lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs" by (induct xs) auto lemma takeWhile_append1 [simp]: "\<lbrakk>x \<in> set xs; \<not>P(x)\<rbrakk> \<Longrightarrow> takeWhile P (xs @ ys) = takeWhile P xs" by (induct xs) auto lemma takeWhile_append2 [simp]: "(\<And>x. x \<in> set xs \<Longrightarrow> P x) \<Longrightarrow> takeWhile P (xs @ ys) = xs @ takeWhile P ys" by (induct xs) auto lemma takeWhile_append: "takeWhile P (xs @ ys) = (if \<forall>x\<in>set xs. P x then xs @ takeWhile P ys else takeWhile P xs)" using takeWhile_append1[of _ xs P ys] takeWhile_append2[of xs P ys] by auto lemma takeWhile_tail: "\<not> P x \<Longrightarrow> takeWhile P (xs @ (x#l)) = takeWhile P xs" by (induct xs) auto lemma takeWhile_eq_Nil_iff: "takeWhile P xs = [] \<longleftrightarrow> xs = [] \<or> \<not>P (hd xs)" by (cases xs) auto lemma takeWhile_nth: "j < length (takeWhile P xs) \<Longrightarrow> takeWhile P xs ! j = xs ! j" by (metis nth_append takeWhile_dropWhile_id) lemma dropWhile_nth: "j < length (dropWhile P xs) \<Longrightarrow> dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))" by (metis add.commute nth_append_length_plus takeWhile_dropWhile_id) lemma length_dropWhile_le: "length (dropWhile P xs) \<le> length xs" by (induct xs) auto lemma dropWhile_append1 [simp]: "\<lbrakk>x \<in> set xs; \<not>P(x)\<rbrakk> \<Longrightarrow> dropWhile P (xs @ ys) = (dropWhile P xs)@ys" by (induct xs) auto lemma dropWhile_append2 [simp]: "(\<And>x. x \<in> set xs \<Longrightarrow> P(x)) \<Longrightarrow> dropWhile P (xs @ ys) = dropWhile P ys" by (induct xs) auto lemma dropWhile_append3: "\<not> P y \<Longrightarrow>dropWhile P (xs @ y # ys) = dropWhile P xs @ y # ys" by (induct xs) auto lemma dropWhile_append: "dropWhile P (xs @ ys) = (if \<forall>x\<in>set xs. P x then dropWhile P ys else dropWhile P xs @ ys)" using dropWhile_append1[of _ xs P ys] dropWhile_append2[of xs P ys] by auto lemma dropWhile_last: "x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> last (dropWhile P xs) = last xs" by (auto simp add: dropWhile_append3 in_set_conv_decomp) lemma set_dropWhileD: "x \<in> set (dropWhile P xs) \<Longrightarrow> x \<in> set xs" by (induct xs) (auto split: if_split_asm) lemma set_takeWhileD: "x \<in> set (takeWhile P xs) \<Longrightarrow> x \<in> set xs \<and> P x" by (induct xs) (auto split: if_split_asm) lemma takeWhile_eq_all_conv[simp]: "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)" by(induct xs, auto) lemma dropWhile_eq_Nil_conv[simp]: "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)" by(induct xs, auto) lemma dropWhile_eq_Cons_conv: "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys \<and> \<not> P y)" by(induct xs, auto) lemma dropWhile_eq_self_iff: "dropWhile P xs = xs \<longleftrightarrow> xs = [] \<or> \<not>P (hd xs)" by (cases xs) (auto simp: dropWhile_eq_Cons_conv) lemma distinct_takeWhile[simp]: "distinct xs \<Longrightarrow> distinct (takeWhile P xs)" by (induct xs) (auto dest: set_takeWhileD) lemma distinct_dropWhile[simp]: "distinct xs \<Longrightarrow> distinct (dropWhile P xs)" by (induct xs) auto lemma takeWhile_map: "takeWhile P (map f xs) = map f (takeWhile (P \<circ> f) xs)" by (induct xs) auto lemma dropWhile_map: "dropWhile P (map f xs) = map f (dropWhile (P \<circ> f) xs)" by (induct xs) auto lemma takeWhile_eq_take: "takeWhile P xs = take (length (takeWhile P xs)) xs" by (induct xs) auto lemma dropWhile_eq_drop: "dropWhile P xs = drop (length (takeWhile P xs)) xs" by (induct xs) auto lemma hd_dropWhile: "dropWhile P xs \<noteq> [] \<Longrightarrow> \<not> P (hd (dropWhile P xs))" by (induct xs) auto lemma takeWhile_eq_filter: assumes "\<And> x. x \<in> set (dropWhile P xs) \<Longrightarrow> \<not> P x" shows "takeWhile P xs = filter P xs" proof - have A: "filter P xs = filter P (takeWhile P xs @ dropWhile P xs)" by simp have B: "filter P (dropWhile P xs) = []" unfolding filter_empty_conv using assms by blast have "filter P xs = takeWhile P xs" unfolding A filter_append B by (auto simp add: filter_id_conv dest: set_takeWhileD) thus ?thesis .. qed lemma takeWhile_eq_take_P_nth: "\<lbrakk> \<And> i. \<lbrakk> i < n ; i < length xs \<rbrakk> \<Longrightarrow> P (xs ! i) ; n < length xs \<Longrightarrow> \<not> P (xs ! n) \<rbrakk> \<Longrightarrow> takeWhile P xs = take n xs" proof (induct xs arbitrary: n) case Nil thus ?case by simp next case (Cons x xs) show ?case proof (cases n) case 0 with Cons show ?thesis by simp next case [simp]: (Suc n') have "P x" using Cons.prems(1)[of 0] by simp moreover have "takeWhile P xs = take n' xs" proof (rule Cons.hyps) fix i assume "i < n'" "i < length xs" thus "P (xs ! i)" using Cons.prems(1)[of "Suc i"] by simp next assume "n' < length xs" thus "\<not> P (xs ! n')" using Cons by auto qed ultimately show ?thesis by simp qed qed lemma nth_length_takeWhile: "length (takeWhile P xs) < length xs \<Longrightarrow> \<not> P (xs ! length (takeWhile P xs))" by (induct xs) auto lemma length_takeWhile_less_P_nth: assumes all: "\<And> i. i < j \<Longrightarrow> P (xs ! i)" and "j \<le> length xs" shows "j \<le> length (takeWhile P xs)" proof (rule classical) assume "\<not> ?thesis" hence "length (takeWhile P xs) < length xs" using assms by simp thus ?thesis using all \<open>\<not> ?thesis\<close> nth_length_takeWhile[of P xs] by auto qed lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))" by(induct xs) (auto simp: takeWhile_tail[where l="[]"]) lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow> dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)" proof (induct xs) case (Cons a xs) then show ?case by(auto, subst dropWhile_append2, auto) qed simp lemma takeWhile_not_last: "distinct xs \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs" by(induction xs rule: induct_list012) auto lemma takeWhile_cong [fundef_cong]: "\<lbrakk>l = k; \<And>x. x \<in> set l \<Longrightarrow> P x = Q x\<rbrakk> \<Longrightarrow> takeWhile P l = takeWhile Q k" by (induct k arbitrary: l) (simp_all) lemma dropWhile_cong [fundef_cong]: "\<lbrakk>l = k; \<And>x. x \<in> set l \<Longrightarrow> P x = Q x\<rbrakk> \<Longrightarrow> dropWhile P l = dropWhile Q k" by (induct k arbitrary: l, simp_all) lemma takeWhile_idem [simp]: "takeWhile P (takeWhile P xs) = takeWhile P xs" by (induct xs) auto lemma dropWhile_idem [simp]: "dropWhile P (dropWhile P xs) = dropWhile P xs" by (induct xs) auto lemma list_all_takeWhile_dropWhile_conv: "list_all P (takeWhile Q xs) \<and> list_all P (dropWhile Q xs) \<longleftrightarrow> list_all P xs" by (induction xs; simp) lemmas list_all_takeWhileI = list_all_takeWhile_dropWhile_conv[THEN iffD2, THEN conjunct1] lemmas list_all_dropWhileI = list_all_takeWhile_dropWhile_conv[THEN iffD2, THEN conjunct2] lemmas list_all_takeWhile_dropWhileD = conjI[THEN list_all_takeWhile_dropWhile_conv[THEN iffD1]] subsubsection \<open>\<^const>\<open>zip\<close>\<close> lemma zip_Nil [simp]: "zip [] ys = []" by (induct ys) auto lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys" by simp declare zip_Cons [simp del] lemma [code]: "zip [] ys = []" "zip xs [] = []" "zip (x # xs) (y # ys) = (x, y) # zip xs ys" by (fact zip_Nil zip.simps(1) zip_Cons_Cons)+ lemma zip_Cons1: "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)" by(auto split:list.split) lemma length_zip [simp]: "length (zip xs ys) = min (length xs) (length ys)" by (induct xs ys rule:list_induct2') auto lemma zip_obtain_same_length: assumes "\<And>zs ws n. length zs = length ws \<Longrightarrow> n = min (length xs) (length ys) \<Longrightarrow> zs = take n xs \<Longrightarrow> ws = take n ys \<Longrightarrow> P (zip zs ws)" shows "P (zip xs ys)" proof - let ?n = "min (length xs) (length ys)" have "P (zip (take ?n xs) (take ?n ys))" by (rule assms) simp_all moreover have "zip xs ys = zip (take ?n xs) (take ?n ys)" proof (induct xs arbitrary: ys) case Nil then show ?case by simp next case (Cons x xs) then show ?case by (cases ys) simp_all qed ultimately show ?thesis by simp qed lemma zip_append1: "zip (xs @ ys) zs = zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)" by (induct xs zs rule:list_induct2') auto lemma zip_append2: "zip xs (ys @ zs) = zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs" by (induct xs ys rule:list_induct2') auto lemma zip_append [simp]: "\<lbrakk>length xs = length us\<rbrakk> \<Longrightarrow> zip (xs@ys) (us@vs) = zip xs us @ zip ys vs" by (simp add: zip_append1) lemma zip_rev: "length xs = length ys \<Longrightarrow> zip (rev xs) (rev ys) = rev (zip xs ys)" by (induct rule:list_induct2, simp_all) lemma zip_map_map: "zip (map f xs) (map g ys) = map (\<lambda> (x, y). (f x, g y)) (zip xs ys)" proof (induct xs arbitrary: ys) case (Cons x xs) note Cons_x_xs = Cons.hyps show ?case proof (cases ys) case (Cons y ys') show ?thesis unfolding Cons using Cons_x_xs by simp qed simp qed simp lemma zip_map1: "zip (map f xs) ys = map (\<lambda>(x, y). (f x, y)) (zip xs ys)" using zip_map_map[of f xs "\<lambda>x. x" ys] by simp lemma zip_map2: "zip xs (map f ys) = map (\<lambda>(x, y). (x, f y)) (zip xs ys)" using zip_map_map[of "\<lambda>x. x" xs f ys] by simp lemma map_zip_map: "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)" by (auto simp: zip_map1) lemma map_zip_map2: "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)" by (auto simp: zip_map2) text\<open>Courtesy of Andreas Lochbihler:\<close> lemma zip_same_conv_map: "zip xs xs = map (\<lambda>x. (x, x)) xs" by(induct xs) auto lemma nth_zip [simp]: "\<lbrakk>i < length xs; i < length ys\<rbrakk> \<Longrightarrow> (zip xs ys)!i = (xs!i, ys!i)" proof (induct ys arbitrary: i xs) case (Cons y ys) then show ?case by (cases xs) (simp_all add: nth.simps split: nat.split) qed auto lemma set_zip: "set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}" by(simp add: set_conv_nth cong: rev_conj_cong) lemma zip_same: "((a,b) \<in> set (zip xs xs)) = (a \<in> set xs \<and> a = b)" by(induct xs) auto lemma zip_update: "zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]" by (simp add: update_zip) lemma zip_replicate [simp]: "zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)" proof (induct i arbitrary: j) case (Suc i) then show ?case by (cases j, auto) qed auto lemma zip_replicate1: "zip (replicate n x) ys = map (Pair x) (take n ys)" by(induction ys arbitrary: n)(case_tac [2] n, simp_all) lemma take_zip: "take n (zip xs ys) = zip (take n xs) (take n ys)" proof (induct n arbitrary: xs ys) case 0 then show ?case by simp next case Suc then show ?case by (cases xs; cases ys) simp_all qed lemma drop_zip: "drop n (zip xs ys) = zip (drop n xs) (drop n ys)" proof (induct n arbitrary: xs ys) case 0 then show ?case by simp next case Suc then show ?case by (cases xs; cases ys) simp_all qed lemma zip_takeWhile_fst: "zip (takeWhile P xs) ys = takeWhile (P \<circ> fst) (zip xs ys)" proof (induct xs arbitrary: ys) case Nil then show ?case by simp next case Cons then show ?case by (cases ys) auto qed lemma zip_takeWhile_snd: "zip xs (takeWhile P ys) = takeWhile (P \<circ> snd) (zip xs ys)" proof (induct xs arbitrary: ys) case Nil then show ?case by simp next case Cons then show ?case by (cases ys) auto qed lemma set_zip_leftD: "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs" by (induct xs ys rule:list_induct2') auto lemma set_zip_rightD: "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys" by (induct xs ys rule:list_induct2') auto lemma in_set_zipE: "(x,y) \<in> set(zip xs ys) \<Longrightarrow> (\<lbrakk> x \<in> set xs; y \<in> set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R" by(blast dest: set_zip_leftD set_zip_rightD) lemma zip_map_fst_snd: "zip (map fst zs) (map snd zs) = zs" by (induct zs) simp_all lemma zip_eq_conv: "length xs = length ys \<Longrightarrow> zip xs ys = zs \<longleftrightarrow> map fst zs = xs \<and> map snd zs = ys" by (auto simp add: zip_map_fst_snd) lemma in_set_zip: "p \<in> set (zip xs ys) \<longleftrightarrow> (\<exists>n. xs ! n = fst p \<and> ys ! n = snd p \<and> n < length xs \<and> n < length ys)" by (cases p) (auto simp add: set_zip) lemma in_set_impl_in_set_zip1: assumes "length xs = length ys" assumes "x \<in> set xs" obtains y where "(x, y) \<in> set (zip xs ys)" proof - from assms have "x \<in> set (map fst (zip xs ys))" by simp from this that show ?thesis by fastforce qed lemma in_set_impl_in_set_zip2: assumes "length xs = length ys" assumes "y \<in> set ys" obtains x where "(x, y) \<in> set (zip xs ys)" proof - from assms have "y \<in> set (map snd (zip xs ys))" by simp from this that show ?thesis by fastforce qed lemma zip_eq_Nil_iff: "zip xs ys = [] \<longleftrightarrow> xs = [] \<or> ys = []" by (cases xs; cases ys) simp_all lemma zip_eq_ConsE: assumes "zip xs ys = xy # xys" obtains x xs' y ys' where "xs = x # xs'" and "ys = y # ys'" and "xy = (x, y)" and "xys = zip xs' ys'" proof - from assms have "xs \<noteq> []" and "ys \<noteq> []" using zip_eq_Nil_iff [of xs ys] by simp_all then obtain x xs' y ys' where xs: "xs = x # xs'" and ys: "ys = y # ys'" by (cases xs; cases ys) auto with assms have "xy = (x, y)" and "xys = zip xs' ys'" by simp_all with xs ys show ?thesis .. qed lemma semilattice_map2: "semilattice (map2 (\<^bold>*))" if "semilattice (\<^bold>*)" for f (infixl "\<^bold>*" 70) proof - from that interpret semilattice f . show ?thesis proof show "map2 (\<^bold>*) (map2 (\<^bold>*) xs ys) zs = map2 (\<^bold>*) xs (map2 (\<^bold>*) ys zs)" for xs ys zs :: "'a list" proof (induction "zip xs (zip ys zs)" arbitrary: xs ys zs) case Nil from Nil [symmetric] show ?case by (auto simp add: zip_eq_Nil_iff) next case (Cons xyz xyzs) from Cons.hyps(2) [symmetric] show ?case by (rule zip_eq_ConsE) (erule zip_eq_ConsE, auto intro: Cons.hyps(1) simp add: ac_simps) qed show "map2 (\<^bold>*) xs ys = map2 (\<^bold>*) ys xs" for xs ys :: "'a list" proof (induction "zip xs ys" arbitrary: xs ys) case Nil then show ?case by (auto simp add: zip_eq_Nil_iff dest: sym) next case (Cons xy xys) from Cons.hyps(2) [symmetric] show ?case by (rule zip_eq_ConsE) (auto intro: Cons.hyps(1) simp add: ac_simps) qed show "map2 (\<^bold>*) xs xs = xs" for xs :: "'a list" by (induction xs) simp_all qed qed lemma pair_list_eqI: assumes "map fst xs = map fst ys" and "map snd xs = map snd ys" shows "xs = ys" proof - from assms(1) have "length xs = length ys" by (rule map_eq_imp_length_eq) from this assms show ?thesis by (induct xs ys rule: list_induct2) (simp_all add: prod_eqI) qed lemma hd_zip: \<open>hd (zip xs ys) = (hd xs, hd ys)\<close> if \<open>xs \<noteq> []\<close> and \<open>ys \<noteq> []\<close> using that by (cases xs; cases ys) simp_all lemma last_zip: \<open>last (zip xs ys) = (last xs, last ys)\<close> if \<open>xs \<noteq> []\<close> and \<open>ys \<noteq> []\<close> and \<open>length xs = length ys\<close> using that by (cases xs rule: rev_cases; cases ys rule: rev_cases) simp_all subsubsection \<open>\<^const>\<open>list_all2\<close>\<close> lemma list_all2_lengthD [intro?]: "list_all2 P xs ys \<Longrightarrow> length xs = length ys" by (simp add: list_all2_iff) lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])" by (simp add: list_all2_iff) lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])" by (simp add: list_all2_iff) lemma list_all2_Cons [iff, code]: "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)" by (auto simp add: list_all2_iff) lemma list_all2_Cons1: "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)" by (cases ys) auto lemma list_all2_Cons2: "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)" by (cases xs) auto lemma list_all2_induct [consumes 1, case_names Nil Cons, induct set: list_all2]: assumes P: "list_all2 P xs ys" assumes Nil: "R [] []" assumes Cons: "\<And>x xs y ys. \<lbrakk>P x y; list_all2 P xs ys; R xs ys\<rbrakk> \<Longrightarrow> R (x # xs) (y # ys)" shows "R xs ys" using P by (induct xs arbitrary: ys) (auto simp add: list_all2_Cons1 Nil Cons) lemma list_all2_rev [iff]: "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys" by (simp add: list_all2_iff zip_rev cong: conj_cong) lemma list_all2_rev1: "list_all2 P (rev xs) ys = list_all2 P xs (rev ys)" by (subst list_all2_rev [symmetric]) simp lemma list_all2_append1: "list_all2 P (xs @ ys) zs = (\<exists>us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and> list_all2 P xs us \<and> list_all2 P ys vs)" (is "?lhs = ?rhs") proof assume ?lhs then show ?rhs apply (rule_tac x = "take (length xs) zs" in exI) apply (rule_tac x = "drop (length xs) zs" in exI) apply (force split: nat_diff_split simp add: list_all2_iff zip_append1) done next assume ?rhs then show ?lhs by (auto simp add: list_all2_iff) qed lemma list_all2_append2: "list_all2 P xs (ys @ zs) = (\<exists>us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and> list_all2 P us ys \<and> list_all2 P vs zs)" (is "?lhs = ?rhs") proof assume ?lhs then show ?rhs apply (rule_tac x = "take (length ys) xs" in exI) apply (rule_tac x = "drop (length ys) xs" in exI) apply (force split: nat_diff_split simp add: list_all2_iff zip_append2) done next assume ?rhs then show ?lhs by (auto simp add: list_all2_iff) qed lemma list_all2_append: "length xs = length ys \<Longrightarrow> list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)" by (induct rule:list_induct2, simp_all) lemma list_all2_appendI [intro?, trans]: "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)" by (simp add: list_all2_append list_all2_lengthD) lemma list_all2_conv_all_nth: "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))" by (force simp add: list_all2_iff set_zip) lemma list_all2_trans: assumes tr: "!!a b c. P1 a b \<Longrightarrow> P2 b c \<Longrightarrow> P3 a c" shows "!!bs cs. list_all2 P1 as bs \<Longrightarrow> list_all2 P2 bs cs \<Longrightarrow> list_all2 P3 as cs" (is "!!bs cs. PROP ?Q as bs cs") proof (induct as) fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs" show "!!cs. PROP ?Q (x # xs) bs cs" proof (induct bs) fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs" show "PROP ?Q (x # xs) (y # ys) cs" by (induct cs) (auto intro: tr I1 I2) qed simp qed simp lemma list_all2_all_nthI [intro?]: "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b" by (simp add: list_all2_conv_all_nth) lemma list_all2I: "\<forall>x \<in> set (zip a b). case_prod P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b" by (simp add: list_all2_iff) lemma list_all2_nthD: "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)" by (simp add: list_all2_conv_all_nth) lemma list_all2_nthD2: "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)" by (frule list_all2_lengthD) (auto intro: list_all2_nthD) lemma list_all2_map1: "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs" by (simp add: list_all2_conv_all_nth) lemma list_all2_map2: "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs" by (auto simp add: list_all2_conv_all_nth) lemma list_all2_refl [intro?]: "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs" by (simp add: list_all2_conv_all_nth) lemma list_all2_update_cong: "\<lbrakk> list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])" by (cases "i < length ys") (auto simp add: list_all2_conv_all_nth nth_list_update) lemma list_all2_takeI [simp,intro?]: "list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)" proof (induct xs arbitrary: n ys) case (Cons x xs) then show ?case by (cases n) (auto simp: list_all2_Cons1) qed auto lemma list_all2_dropI [simp,intro?]: "list_all2 P xs ys \<Longrightarrow> list_all2 P (drop n xs) (drop n ys)" proof (induct xs arbitrary: n ys) case (Cons x xs) then show ?case by (cases n) (auto simp: list_all2_Cons1) qed auto lemma list_all2_mono [intro?]: "list_all2 P xs ys \<Longrightarrow> (\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys) \<Longrightarrow> list_all2 Q xs ys" by (rule list.rel_mono_strong) lemma list_all2_eq: "xs = ys \<longleftrightarrow> list_all2 (=) xs ys" by (induct xs ys rule: list_induct2') auto lemma list_eq_iff_zip_eq: "xs = ys \<longleftrightarrow> length xs = length ys \<and> (\<forall>(x,y) \<in> set (zip xs ys). x = y)" by(auto simp add: set_zip list_all2_eq list_all2_conv_all_nth cong: conj_cong) lemma list_all2_same: "list_all2 P xs xs \<longleftrightarrow> (\<forall>x\<in>set xs. P x x)" by(auto simp add: list_all2_conv_all_nth set_conv_nth) lemma zip_assoc: "zip xs (zip ys zs) = map (\<lambda>((x, y), z). (x, y, z)) (zip (zip xs ys) zs)" by(rule list_all2_all_nthI[where P="(=)", unfolded list.rel_eq]) simp_all lemma zip_commute: "zip xs ys = map (\<lambda>(x, y). (y, x)) (zip ys xs)" by(rule list_all2_all_nthI[where P="(=)", unfolded list.rel_eq]) simp_all lemma zip_left_commute: "zip xs (zip ys zs) = map (\<lambda>(y, (x, z)). (x, y, z)) (zip ys (zip xs zs))" by(rule list_all2_all_nthI[where P="(=)", unfolded list.rel_eq]) simp_all lemma zip_replicate2: "zip xs (replicate n y) = map (\<lambda>x. (x, y)) (take n xs)" by(subst zip_commute)(simp add: zip_replicate1) subsubsection \<open>\<^const>\<open>List.product\<close> and \<^const>\<open>product_lists\<close>\<close> lemma product_concat_map: "List.product xs ys = concat (map (\<lambda>x. map (\<lambda>y. (x,y)) ys) xs)" by(induction xs) (simp)+ lemma set_product[simp]: "set (List.product xs ys) = set xs \<times> set ys" by (induct xs) auto lemma length_product [simp]: "length (List.product xs ys) = length xs * length ys" by (induct xs) simp_all lemma product_nth: assumes "n < length xs * length ys" shows "List.product xs ys ! n = (xs ! (n div length ys), ys ! (n mod length ys))" using assms proof (induct xs arbitrary: n) case Nil then show ?case by simp next case (Cons x xs n) then have "length ys > 0" by auto with Cons show ?case by (auto simp add: nth_append not_less le_mod_geq le_div_geq) qed lemma in_set_product_lists_length: "xs \<in> set (product_lists xss) \<Longrightarrow> length xs = length xss" by (induct xss arbitrary: xs) auto lemma product_lists_set: "set (product_lists xss) = {xs. list_all2 (\<lambda>x ys. x \<in> set ys) xs xss}" (is "?L = Collect ?R") proof (intro equalityI subsetI, unfold mem_Collect_eq) fix xs assume "xs \<in> ?L" then have "length xs = length xss" by (rule in_set_product_lists_length) from this \<open>xs \<in> ?L\<close> show "?R xs" by (induct xs xss rule: list_induct2) auto next fix xs assume "?R xs" then show "xs \<in> ?L" by induct auto qed subsubsection \<open>\<^const>\<open>fold\<close> with natural argument order\<close> lemma fold_simps [code]: \<comment> \<open>eta-expanded variant for generated code -- enables tail-recursion optimisation in Scala\<close> "fold f [] s = s" "fold f (x # xs) s = fold f xs (f x s)" by simp_all lemma fold_remove1_split: "\<lbrakk> \<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f x \<circ> f y = f y \<circ> f x; x \<in> set xs \<rbrakk> \<Longrightarrow> fold f xs = fold f (remove1 x xs) \<circ> f x" by (induct xs) (auto simp add: comp_assoc) lemma fold_cong [fundef_cong]: "a = b \<Longrightarrow> xs = ys \<Longrightarrow> (\<And>x. x \<in> set xs \<Longrightarrow> f x = g x) \<Longrightarrow> fold f xs a = fold g ys b" by (induct ys arbitrary: a b xs) simp_all lemma fold_id: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = id) \<Longrightarrow> fold f xs = id" by (induct xs) simp_all lemma fold_commute: "(\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h) \<Longrightarrow> h \<circ> fold g xs = fold f xs \<circ> h" by (induct xs) (simp_all add: fun_eq_iff) lemma fold_commute_apply: assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h" shows "h (fold g xs s) = fold f xs (h s)" proof - from assms have "h \<circ> fold g xs = fold f xs \<circ> h" by (rule fold_commute) then show ?thesis by (simp add: fun_eq_iff) qed lemma fold_invariant: "\<lbrakk> \<And>x. x \<in> set xs \<Longrightarrow> Q x; P s; \<And>x s. Q x \<Longrightarrow> P s \<Longrightarrow> P (f x s) \<rbrakk> \<Longrightarrow> P (fold f xs s)" by (induct xs arbitrary: s) simp_all lemma fold_append [simp]: "fold f (xs @ ys) = fold f ys \<circ> fold f xs" by (induct xs) simp_all lemma fold_map [code_unfold]: "fold g (map f xs) = fold (g \<circ> f) xs" by (induct xs) simp_all lemma fold_filter: "fold f (filter P xs) = fold (\<lambda>x. if P x then f x else id) xs" by (induct xs) simp_all lemma fold_rev: "(\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y) \<Longrightarrow> fold f (rev xs) = fold f xs" by (induct xs) (simp_all add: fold_commute_apply fun_eq_iff) lemma fold_Cons_rev: "fold Cons xs = append (rev xs)" by (induct xs) simp_all lemma rev_conv_fold [code]: "rev xs = fold Cons xs []" by (simp add: fold_Cons_rev) lemma fold_append_concat_rev: "fold append xss = append (concat (rev xss))" by (induct xss) simp_all text \<open>\<^const>\<open>Finite_Set.fold\<close> and \<^const>\<open>fold\<close>\<close> lemma (in comp_fun_commute) fold_set_fold_remdups: "Finite_Set.fold f y (set xs) = fold f (remdups xs) y" by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_left_comm insert_absorb) lemma (in comp_fun_idem) fold_set_fold: "Finite_Set.fold f y (set xs) = fold f xs y" by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_left_comm) lemma union_set_fold [code]: "set xs \<union> A = fold Set.insert xs A" proof - interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert) show ?thesis by (simp add: union_fold_insert fold_set_fold) qed lemma union_coset_filter [code]: "List.coset xs \<union> A = List.coset (List.filter (\<lambda>x. x \<notin> A) xs)" by auto lemma minus_set_fold [code]: "A - set xs = fold Set.remove xs A" proof - interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove) show ?thesis by (simp add: minus_fold_remove [of _ A] fold_set_fold) qed lemma minus_coset_filter [code]: "A - List.coset xs = set (List.filter (\<lambda>x. x \<in> A) xs)" by auto lemma inter_set_filter [code]: "A \<inter> set xs = set (List.filter (\<lambda>x. x \<in> A) xs)" by auto lemma inter_coset_fold [code]: "A \<inter> List.coset xs = fold Set.remove xs A" by (simp add: Diff_eq [symmetric] minus_set_fold) lemma (in semilattice_set) set_eq_fold [code]: "F (set (x # xs)) = fold f xs x" proof - interpret comp_fun_idem f by standard (simp_all add: fun_eq_iff left_commute) show ?thesis by (simp add: eq_fold fold_set_fold) qed lemma (in complete_lattice) Inf_set_fold: "Inf (set xs) = fold inf xs top" proof - interpret comp_fun_idem "inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a" by (fact comp_fun_idem_inf) show ?thesis by (simp add: Inf_fold_inf fold_set_fold inf_commute) qed declare Inf_set_fold [where 'a = "'a set", code] lemma (in complete_lattice) Sup_set_fold: "Sup (set xs) = fold sup xs bot" proof - interpret comp_fun_idem "sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a" by (fact comp_fun_idem_sup) show ?thesis by (simp add: Sup_fold_sup fold_set_fold sup_commute) qed declare Sup_set_fold [where 'a = "'a set", code] lemma (in complete_lattice) INF_set_fold: "\<Sqinter>(f ` set xs) = fold (inf \<circ> f) xs top" using Inf_set_fold [of "map f xs"] by (simp add: fold_map) lemma (in complete_lattice) SUP_set_fold: "\<Squnion>(f ` set xs) = fold (sup \<circ> f) xs bot" using Sup_set_fold [of "map f xs"] by (simp add: fold_map) subsubsection \<open>Fold variants: \<^const>\<open>foldr\<close> and \<^const>\<open>foldl\<close>\<close> text \<open>Correspondence\<close> lemma foldr_conv_fold [code_abbrev]: "foldr f xs = fold f (rev xs)" by (induct xs) simp_all lemma foldl_conv_fold: "foldl f s xs = fold (\<lambda>x s. f s x) xs s" by (induct xs arbitrary: s) simp_all lemma foldr_conv_foldl: \<comment> \<open>The ``Third Duality Theorem'' in Bird \& Wadler:\<close> "foldr f xs a = foldl (\<lambda>x y. f y x) a (rev xs)" by (simp add: foldr_conv_fold foldl_conv_fold) lemma foldl_conv_foldr: "foldl f a xs = foldr (\<lambda>x y. f y x) (rev xs) a" by (simp add: foldr_conv_fold foldl_conv_fold) lemma foldr_fold: "(\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y) \<Longrightarrow> foldr f xs = fold f xs" unfolding foldr_conv_fold by (rule fold_rev) lemma foldr_cong [fundef_cong]: "a = b \<Longrightarrow> l = k \<Longrightarrow> (\<And>a x. x \<in> set l \<Longrightarrow> f x a = g x a) \<Longrightarrow> foldr f l a = foldr g k b" by (auto simp add: foldr_conv_fold intro!: fold_cong) lemma foldl_cong [fundef_cong]: "a = b \<Longrightarrow> l = k \<Longrightarrow> (\<And>a x. x \<in> set l \<Longrightarrow> f a x = g a x) \<Longrightarrow> foldl f a l = foldl g b k" by (auto simp add: foldl_conv_fold intro!: fold_cong) lemma foldr_append [simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)" by (simp add: foldr_conv_fold) lemma foldl_append [simp]: "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys" by (simp add: foldl_conv_fold) lemma foldr_map [code_unfold]: "foldr g (map f xs) a = foldr (g \<circ> f) xs a" by (simp add: foldr_conv_fold fold_map rev_map) lemma foldr_filter: "foldr f (filter P xs) = foldr (\<lambda>x. if P x then f x else id) xs" by (simp add: foldr_conv_fold rev_filter fold_filter) lemma foldl_map [code_unfold]: "foldl g a (map f xs) = foldl (\<lambda>a x. g a (f x)) a xs" by (simp add: foldl_conv_fold fold_map comp_def) lemma concat_conv_foldr [code]: "concat xss = foldr append xss []" by (simp add: fold_append_concat_rev foldr_conv_fold) subsubsection \<open>\<^const>\<open>upt\<close>\<close> lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])" \<comment> \<open>simp does not terminate!\<close> by (induct j) auto lemmas upt_rec_numeral[simp] = upt_rec[of "numeral m" "numeral n"] for m n lemma upt_conv_Nil [simp]: "j \<le> i \<Longrightarrow> [i..<j] = []" by (subst upt_rec) simp lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j \<le> i)" by(induct j)simp_all lemma upt_eq_Cons_conv: "([i..<j] = x#xs) = (i < j \<and> i = x \<and> [i+1..<j] = xs)" proof (induct j arbitrary: x xs) case (Suc j) then show ?case by (simp add: upt_rec) qed simp lemma upt_Suc_append: "i \<le> j \<Longrightarrow> [i..<(Suc j)] = [i..<j]@[j]" \<comment> \<open>Only needed if \<open>upt_Suc\<close> is deleted from the simpset.\<close> by simp lemma upt_conv_Cons: "i < j \<Longrightarrow> [i..<j] = i # [Suc i..<j]" by (simp add: upt_rec) lemma upt_conv_Cons_Cons: \<comment> \<open>no precondition\<close> "m # n # ns = [m..<q] \<longleftrightarrow> n # ns = [Suc m..<q]" proof (cases "m < q") case False then show ?thesis by simp next case True then show ?thesis by (simp add: upt_conv_Cons) qed lemma upt_add_eq_append: "i<=j \<Longrightarrow> [i..<j+k] = [i..<j]@[j..<j+k]" \<comment> \<open>LOOPS as a simprule, since \<open>j \<le> j\<close>.\<close> by (induct k) auto lemma length_upt [simp]: "length [i..<j] = j - i" by (induct j) (auto simp add: Suc_diff_le) lemma nth_upt [simp]: "i + k < j \<Longrightarrow> [i..<j] ! k = i + k" by (induct j) (auto simp add: less_Suc_eq nth_append split: nat_diff_split) lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i" by(simp add:upt_conv_Cons) lemma tl_upt [simp]: "tl [m..<n] = [Suc m..<n]" by (simp add: upt_rec) lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1" by(cases j)(auto simp: upt_Suc_append) lemma take_upt [simp]: "i+m \<le> n \<Longrightarrow> take m [i..<n] = [i..<i+m]" proof (induct m arbitrary: i) case (Suc m) then show ?case by (subst take_Suc_conv_app_nth) auto qed simp lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]" by(induct j) auto lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]" by (induct n) auto lemma map_add_upt: "map (\<lambda>i. i + n) [0..<m] = [n..<m + n]" by (induct m) simp_all lemma nth_map_upt: "i < n-m \<Longrightarrow> (map f [m..<n]) ! i = f(m+i)" proof (induct n m arbitrary: i rule: diff_induct) case (3 x y) then show ?case by (metis add.commute length_upt less_diff_conv nth_map nth_upt) qed auto lemma map_decr_upt: "map (\<lambda>n. n - Suc 0) [Suc m..<Suc n] = [m..<n]" by (induct n) simp_all lemma map_upt_Suc: "map f [0 ..< Suc n] = f 0 # map (\<lambda>i. f (Suc i)) [0 ..< n]" by (induct n arbitrary: f) auto lemma nth_take_lemma: "k \<le> length xs \<Longrightarrow> k \<le> length ys \<Longrightarrow> (\<And>i. i < k \<longrightarrow> xs!i = ys!i) \<Longrightarrow> take k xs = take k ys" proof (induct k arbitrary: xs ys) case (Suc k) then show ?case apply (simp add: less_Suc_eq_0_disj) by (simp add: Suc.prems(3) take_Suc_conv_app_nth) qed simp lemma nth_equalityI: "\<lbrakk>length xs = length ys; \<And>i. i < length xs \<Longrightarrow> xs!i = ys!i\<rbrakk> \<Longrightarrow> xs = ys" by (frule nth_take_lemma [OF le_refl eq_imp_le]) simp_all lemma map_nth: "map (\<lambda>i. xs ! i) [0..<length xs] = xs" by (rule nth_equalityI, auto) lemma list_all2_antisym: "\<lbrakk> (\<And>x y. \<lbrakk>P x y; Q y x\<rbrakk> \<Longrightarrow> x = y); list_all2 P xs ys; list_all2 Q ys xs \<rbrakk> \<Longrightarrow> xs = ys" by (simp add: list_all2_conv_all_nth nth_equalityI) lemma take_equalityI: "(\<forall>i. take i xs = take i ys) \<Longrightarrow> xs = ys" \<comment> \<open>The famous take-lemma.\<close> by (metis length_take min.commute order_refl take_all) lemma take_Cons': "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)" by (cases n) simp_all lemma drop_Cons': "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)" by (cases n) simp_all lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))" by (cases n) simp_all lemma take_Cons_numeral [simp]: "take (numeral v) (x # xs) = x # take (numeral v - 1) xs" by (simp add: take_Cons') lemma drop_Cons_numeral [simp]: "drop (numeral v) (x # xs) = drop (numeral v - 1) xs" by (simp add: drop_Cons') lemma nth_Cons_numeral [simp]: "(x # xs) ! numeral v = xs ! (numeral v - 1)" by (simp add: nth_Cons') lemma map_upt_eqI: \<open>map f [m..<n] = xs\<close> if \<open>length xs = n - m\<close> \<open>\<And>i. i < length xs \<Longrightarrow> xs ! i = f (m + i)\<close> proof (rule nth_equalityI) from \<open>length xs = n - m\<close> show \<open>length (map f [m..<n]) = length xs\<close> by simp next fix i assume \<open>i < length (map f [m..<n])\<close> then have \<open>i < n - m\<close> by simp with that have \<open>xs ! i = f (m + i)\<close> by simp with \<open>i < n - m\<close> show \<open>map f [m..<n] ! i = xs ! i\<close> by simp qed subsubsection \<open>\<open>upto\<close>: interval-list on \<^typ>\<open>int\<close>\<close> function upto :: "int \<Rightarrow> int \<Rightarrow> int list" ("(1[_../_])") where "upto i j = (if i \<le> j then i # [i+1..j] else [])" by auto termination by(relation "measure(%(i::int,j). nat(j - i + 1))") auto declare upto.simps[simp del] lemmas upto_rec_numeral [simp] = upto.simps[of "numeral m" "numeral n"] upto.simps[of "numeral m" "- numeral n"] upto.simps[of "- numeral m" "numeral n"] upto.simps[of "- numeral m" "- numeral n"] for m n lemma upto_empty[simp]: "j < i \<Longrightarrow> [i..j] = []" by(simp add: upto.simps) lemma upto_single[simp]: "[i..i] = [i]" by(simp add: upto.simps) lemma upto_Nil[simp]: "[i..j] = [] \<longleftrightarrow> j < i" by (simp add: upto.simps) lemma upto_Nil2[simp]: "[] = [i..j] \<longleftrightarrow> j < i" by (simp add: upto.simps) lemma upto_rec1: "i \<le> j \<Longrightarrow> [i..j] = i#[i+1..j]" by(simp add: upto.simps) lemma upto_rec2: "i \<le> j \<Longrightarrow> [i..j] = [i..j - 1]@[j]" proof(induct "nat(j-i)" arbitrary: i j) case 0 thus ?case by(simp add: upto.simps) next case (Suc n) hence "n = nat (j - (i + 1))" "i < j" by linarith+ from this(2) Suc.hyps(1)[OF this(1)] Suc(2,3) upto_rec1 show ?case by simp qed lemma length_upto[simp]: "length [i..j] = nat(j - i + 1)" by(induction i j rule: upto.induct) (auto simp: upto.simps) lemma set_upto[simp]: "set[i..j] = {i..j}" proof(induct i j rule:upto.induct) case (1 i j) from this show ?case unfolding upto.simps[of i j] by auto qed lemma nth_upto[simp]: "i + int k \<le> j \<Longrightarrow> [i..j] ! k = i + int k" proof(induction i j arbitrary: k rule: upto.induct) case (1 i j) then show ?case by (auto simp add: upto_rec1 [of i j] nth_Cons') qed lemma upto_split1: "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> [i..k] = [i..j-1] @ [j..k]" proof (induction j rule: int_ge_induct) case base thus ?case by (simp add: upto_rec1) next case step thus ?case using upto_rec1 upto_rec2 by simp qed lemma upto_split2: "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> [i..k] = [i..j] @ [j+1..k]" using upto_rec1 upto_rec2 upto_split1 by auto lemma upto_split3: "\<lbrakk> i \<le> j; j \<le> k \<rbrakk> \<Longrightarrow> [i..k] = [i..j-1] @ j # [j+1..k]" using upto_rec1 upto_split1 by auto text\<open>Tail recursive version for code generation:\<close> definition upto_aux :: "int \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> int list" where "upto_aux i j js = [i..j] @ js" lemma upto_aux_rec [code]: "upto_aux i j js = (if j<i then js else upto_aux i (j - 1) (j#js))" by (simp add: upto_aux_def upto_rec2) lemma upto_code[code]: "[i..j] = upto_aux i j []" by(simp add: upto_aux_def) subsubsection \<open>\<^const>\<open>successively\<close>\<close> lemma successively_Cons: "successively P (x # xs) \<longleftrightarrow> xs = [] \<or> P x (hd xs) \<and> successively P xs" by (cases xs) auto lemma successively_cong [cong]: assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> P x y \<longleftrightarrow> Q x y" "xs = ys" shows "successively P xs \<longleftrightarrow> successively Q ys" unfolding assms(2) [symmetric] using assms(1) by (induction xs) (auto simp: successively_Cons) lemma successively_append_iff: "successively P (xs @ ys) \<longleftrightarrow> successively P xs \<and> successively P ys \<and> (xs = [] \<or> ys = [] \<or> P (last xs) (hd ys))" by (induction xs) (auto simp: successively_Cons) lemma successively_if_sorted_wrt: "sorted_wrt P xs \<Longrightarrow> successively P xs" by (induction xs rule: induct_list012) auto lemma successively_iff_sorted_wrt_strong: assumes "\<And>x y z. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> z \<in> set xs \<Longrightarrow> P x y \<Longrightarrow> P y z \<Longrightarrow> P x z" shows "successively P xs \<longleftrightarrow> sorted_wrt P xs" proof assume "successively P xs" from this and assms show "sorted_wrt P xs" proof (induction xs rule: induct_list012) case (3 x y xs) from "3.prems" have "P x y" by auto have IH: "sorted_wrt P (y # xs)" using "3.prems" by(intro "3.IH"(2) list.set_intros(2))(simp, blast intro: list.set_intros(2)) have "P x z" if asm: "z \<in> set xs" for z proof - from IH and asm have "P y z" by auto with \<open>P x y\<close> show "P x z" using "3.prems" asm by auto qed with IH and \<open>P x y\<close> show ?case by auto qed auto qed (use successively_if_sorted_wrt in blast) lemma successively_conv_sorted_wrt: assumes "transp P" shows "successively P xs \<longleftrightarrow> sorted_wrt P xs" using assms unfolding transp_def by (intro successively_iff_sorted_wrt_strong) blast lemma successively_rev [simp]: "successively P (rev xs) \<longleftrightarrow> successively (\<lambda>x y. P y x) xs" by (induction xs rule: remdups_adj.induct) (auto simp: successively_append_iff successively_Cons) lemma successively_map: "successively P (map f xs) \<longleftrightarrow> successively (\<lambda>x y. P (f x) (f y)) xs" by (induction xs rule: induct_list012) auto lemma successively_mono: assumes "successively P xs" assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> P x y \<Longrightarrow> Q x y" shows "successively Q xs" using assms by (induction Q xs rule: successively.induct) auto lemma successively_altdef: "successively = (\<lambda>P. rec_list True (\<lambda>x xs b. case xs of [] \<Rightarrow> True | y # _ \<Rightarrow> P x y \<and> b))" proof (intro ext) fix P and xs :: "'a list" show "successively P xs = rec_list True (\<lambda>x xs b. case xs of [] \<Rightarrow> True | y # _ \<Rightarrow> P x y \<and> b) xs" by (induction xs) (auto simp: successively_Cons split: list.splits) qed subsubsection \<open>\<^const>\<open>distinct\<close> and \<^const>\<open>remdups\<close> and \<^const>\<open>remdups_adj\<close>\<close> lemma distinct_tl: "distinct xs \<Longrightarrow> distinct (tl xs)" by (cases xs) simp_all lemma distinct_append [simp]: "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})" by (induct xs) auto lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs" by(induct xs) auto lemma set_remdups [simp]: "set (remdups xs) = set xs" by (induct xs) (auto simp add: insert_absorb) lemma distinct_remdups [iff]: "distinct (remdups xs)" by (induct xs) auto lemma distinct_remdups_id: "distinct xs \<Longrightarrow> remdups xs = xs" by (induct xs, auto) lemma remdups_id_iff_distinct [simp]: "remdups xs = xs \<longleftrightarrow> distinct xs" by (metis distinct_remdups distinct_remdups_id) lemma finite_distinct_list: "finite A \<Longrightarrow> \<exists>xs. set xs = A \<and> distinct xs" by (metis distinct_remdups finite_list set_remdups) lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])" by (induct x, auto) lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])" by (induct x, auto) lemma length_remdups_leq[iff]: "length(remdups xs) \<le> length xs" by (induct xs) auto lemma length_remdups_eq[iff]: "(length (remdups xs) = length xs) = (remdups xs = xs)" proof (induct xs) case (Cons a xs) then show ?case by simp (metis Suc_n_not_le_n impossible_Cons length_remdups_leq) qed auto lemma remdups_filter: "remdups(filter P xs) = filter P (remdups xs)" by (induct xs) auto lemma distinct_map: "distinct(map f xs) = (distinct xs \<and> inj_on f (set xs))" by (induct xs) auto lemma distinct_map_filter: "distinct (map f xs) \<Longrightarrow> distinct (map f (filter P xs))" by (induct xs) auto lemma distinct_filter [simp]: "distinct xs \<Longrightarrow> distinct (filter P xs)" by (induct xs) auto lemma distinct_upt[simp]: "distinct[i..<j]" by (induct j) auto lemma distinct_upto[simp]: "distinct[i..j]" proof (induction i j rule: upto.induct) case (1 i j) then show ?case by (simp add: upto.simps [of i]) qed lemma distinct_take[simp]: "distinct xs \<Longrightarrow> distinct (take i xs)" proof (induct xs arbitrary: i) case (Cons a xs) then show ?case by (metis Cons.prems append_take_drop_id distinct_append) qed auto lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)" proof (induct xs arbitrary: i) case (Cons a xs) then show ?case by (metis Cons.prems append_take_drop_id distinct_append) qed auto lemma distinct_list_update: assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}" shows "distinct (xs[i:=a])" proof (cases "i < length xs") case True with a have anot: "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}" by simp (metis in_set_dropD in_set_takeD) show ?thesis proof (cases "a = xs!i") case True with d show ?thesis by auto next case False have "set (take i xs) \<inter> set (drop (Suc i) xs) = {}" by (metis True d disjoint_insert(1) distinct_append id_take_nth_drop list.set(2)) then show ?thesis using d False anot \<open>i < length xs\<close> by (simp add: upd_conv_take_nth_drop) qed next case False with d show ?thesis by auto qed lemma distinct_concat: "\<lbrakk> distinct xs; \<And> ys. ys \<in> set xs \<Longrightarrow> distinct ys; \<And> ys zs. \<lbrakk> ys \<in> set xs ; zs \<in> set xs ; ys \<noteq> zs \<rbrakk> \<Longrightarrow> set ys \<inter> set zs = {} \<rbrakk> \<Longrightarrow> distinct (concat xs)" by (induct xs) auto text \<open>An iff-version of @{thm distinct_concat} is available further down as \<open>distinct_concat_iff\<close>.\<close> text \<open>It is best to avoid the following indexed version of distinct, but sometimes it is useful.\<close> lemma distinct_conv_nth: "distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j \<longrightarrow> xs!i \<noteq> xs!j)" proof (induct xs) case (Cons x xs) show ?case apply (auto simp add: Cons nth_Cons split: nat.split_asm) apply (metis Suc_less_eq2 in_set_conv_nth less_not_refl zero_less_Suc)+ done qed auto lemma nth_eq_iff_index_eq: "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)" by(auto simp: distinct_conv_nth) lemma distinct_Ex1: "distinct xs \<Longrightarrow> x \<in> set xs \<Longrightarrow> (\<exists>!i. i < length xs \<and> xs ! i = x)" by (auto simp: in_set_conv_nth nth_eq_iff_index_eq) lemma inj_on_nth: "distinct xs \<Longrightarrow> \<forall>i \<in> I. i < length xs \<Longrightarrow> inj_on (nth xs) I" by (rule inj_onI) (simp add: nth_eq_iff_index_eq) lemma bij_betw_nth: assumes "distinct xs" "A = {..<length xs}" "B = set xs" shows "bij_betw ((!) xs) A B" using assms unfolding bij_betw_def by (auto intro!: inj_on_nth simp: set_conv_nth) lemma set_update_distinct: "\<lbrakk> distinct xs; n < length xs \<rbrakk> \<Longrightarrow> set(xs[n := x]) = insert x (set xs - {xs!n})" by(auto simp: set_eq_iff in_set_conv_nth nth_list_update nth_eq_iff_index_eq) lemma distinct_swap[simp]: "\<lbrakk> i < size xs; j < size xs\<rbrakk> \<Longrightarrow> distinct(xs[i := xs!j, j := xs!i]) = distinct xs" apply (simp add: distinct_conv_nth nth_list_update) apply (safe; metis) done lemma set_swap[simp]: "\<lbrakk> i < size xs; j < size xs \<rbrakk> \<Longrightarrow> set(xs[i := xs!j, j := xs!i]) = set xs" by(simp add: set_conv_nth nth_list_update) metis lemma distinct_card: "distinct xs \<Longrightarrow> card (set xs) = size xs" by (induct xs) auto lemma card_distinct: "card (set xs) = size xs \<Longrightarrow> distinct xs" proof (induct xs) case (Cons x xs) show ?case proof (cases "x \<in> set xs") case False with Cons show ?thesis by simp next case True with Cons.prems have "card (set xs) = Suc (length xs)" by (simp add: card_insert_if split: if_split_asm) moreover have "card (set xs) \<le> length xs" by (rule card_length) ultimately have False by simp thus ?thesis .. qed qed simp lemma distinct_length_filter: "distinct xs \<Longrightarrow> length (filter P xs) = card ({x. P x} Int set xs)" by (induct xs) (auto) lemma not_distinct_decomp: "\<not> distinct ws \<Longrightarrow> \<exists>xs ys zs y. ws = xs@[y]@ys@[y]@zs" proof (induct n == "length ws" arbitrary:ws) case (Suc n ws) then show ?case using length_Suc_conv [of ws n] apply (auto simp: eq_commute) apply (metis append_Nil in_set_conv_decomp_first) by (metis append_Cons) qed simp lemma not_distinct_conv_prefix: defines "dec as xs y ys \<equiv> y \<in> set xs \<and> distinct xs \<and> as = xs @ y # ys" shows "\<not>distinct as \<longleftrightarrow> (\<exists>xs y ys. dec as xs y ys)" (is "?L = ?R") proof assume "?L" then show "?R" proof (induct "length as" arbitrary: as rule: less_induct) case less obtain xs ys zs y where decomp: "as = (xs @ y # ys) @ y # zs" using not_distinct_decomp[OF less.prems] by auto show ?case proof (cases "distinct (xs @ y # ys)") case True with decomp have "dec as (xs @ y # ys) y zs" by (simp add: dec_def) then show ?thesis by blast next case False with less decomp obtain xs' y' ys' where "dec (xs @ y # ys) xs' y' ys'" by atomize_elim auto with decomp have "dec as xs' y' (ys' @ y # zs)" by (simp add: dec_def) then show ?thesis by blast qed qed qed (auto simp: dec_def) lemma distinct_product: "distinct xs \<Longrightarrow> distinct ys \<Longrightarrow> distinct (List.product xs ys)" by (induct xs) (auto intro: inj_onI simp add: distinct_map) lemma distinct_product_lists: assumes "\<forall>xs \<in> set xss. distinct xs" shows "distinct (product_lists xss)" using assms proof (induction xss) case (Cons xs xss) note * = this then show ?case proof (cases "product_lists xss") case Nil then show ?thesis by (induct xs) simp_all next case (Cons ps pss) with * show ?thesis by (auto intro!: inj_onI distinct_concat simp add: distinct_map) qed qed simp lemma length_remdups_concat: "length (remdups (concat xss)) = card (\<Union>xs\<in>set xss. set xs)" by (simp add: distinct_card [symmetric]) lemma remdups_append2: "remdups (xs @ remdups ys) = remdups (xs @ ys)" by(induction xs) auto lemma length_remdups_card_conv: "length(remdups xs) = card(set xs)" proof - have xs: "concat[xs] = xs" by simp from length_remdups_concat[of "[xs]"] show ?thesis unfolding xs by simp qed lemma remdups_remdups: "remdups (remdups xs) = remdups xs" by (induct xs) simp_all lemma distinct_butlast: assumes "distinct xs" shows "distinct (butlast xs)" proof (cases "xs = []") case False from \<open>xs \<noteq> []\<close> obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto with \<open>distinct xs\<close> show ?thesis by simp qed (auto) lemma remdups_map_remdups: "remdups (map f (remdups xs)) = remdups (map f xs)" by (induct xs) simp_all lemma distinct_zipI1: assumes "distinct xs" shows "distinct (zip xs ys)" proof (rule zip_obtain_same_length) fix xs' :: "'a list" and ys' :: "'b list" and n assume "length xs' = length ys'" assume "xs' = take n xs" with assms have "distinct xs'" by simp with \<open>length xs' = length ys'\<close> show "distinct (zip xs' ys')" by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE) qed lemma distinct_zipI2: assumes "distinct ys" shows "distinct (zip xs ys)" proof (rule zip_obtain_same_length) fix xs' :: "'b list" and ys' :: "'a list" and n assume "length xs' = length ys'" assume "ys' = take n ys" with assms have "distinct ys'" by simp with \<open>length xs' = length ys'\<close> show "distinct (zip xs' ys')" by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE) qed lemma set_take_disj_set_drop_if_distinct: "distinct vs \<Longrightarrow> i \<le> j \<Longrightarrow> set (take i vs) \<inter> set (drop j vs) = {}" by (auto simp: in_set_conv_nth distinct_conv_nth) (* The next two lemmas help Sledgehammer. *) lemma distinct_singleton: "distinct [x]" by simp lemma distinct_length_2_or_more: "distinct (a # b # xs) \<longleftrightarrow> (a \<noteq> b \<and> distinct (a # xs) \<and> distinct (b # xs))" by force lemma remdups_adj_altdef: "(remdups_adj xs = ys) \<longleftrightarrow> (\<exists>f::nat => nat. mono f \<and> f ` {0 ..< size xs} = {0 ..< size ys} \<and> (\<forall>i < size xs. xs!i = ys!(f i)) \<and> (\<forall>i. i + 1 < size xs \<longrightarrow> (xs!i = xs!(i+1) \<longleftrightarrow> f i = f(i+1))))" (is "?L \<longleftrightarrow> (\<exists>f. ?p f xs ys)") proof assume ?L then show "\<exists>f. ?p f xs ys" proof (induct xs arbitrary: ys rule: remdups_adj.induct) case (1 ys) thus ?case by (intro exI[of _ id]) (auto simp: mono_def) next case (2 x ys) thus ?case by (intro exI[of _ id]) (auto simp: mono_def) next case (3 x1 x2 xs ys) let ?xs = "x1 # x2 # xs" let ?cond = "x1 = x2" define zs where "zs = remdups_adj (x2 # xs)" from 3(1-2)[of zs] obtain f where p: "?p f (x2 # xs) zs" unfolding zs_def by (cases ?cond) auto then have f0: "f 0 = 0" by (intro mono_image_least[where f=f]) blast+ from p have mono: "mono f" and f_xs_zs: "f ` {0..<length (x2 # xs)} = {0..<length zs}" by auto have ys: "ys = (if x1 = x2 then zs else x1 # zs)" unfolding 3(3)[symmetric] zs_def by auto have zs0: "zs ! 0 = x2" unfolding zs_def by (induct xs) auto have zsne: "zs \<noteq> []" unfolding zs_def by (induct xs) auto let ?Succ = "if ?cond then id else Suc" let ?x1 = "if ?cond then id else Cons x1" let ?f = "\<lambda> i. if i = 0 then 0 else ?Succ (f (i - 1))" have ys: "ys = ?x1 zs" unfolding ys by (cases ?cond, auto) have mono: "mono ?f" using \<open>mono f\<close> unfolding mono_def by auto show ?case unfolding ys proof (intro exI[of _ ?f] conjI allI impI) show "mono ?f" by fact next fix i assume i: "i < length ?xs" with p show "?xs ! i = ?x1 zs ! (?f i)" using zs0 by auto next fix i assume i: "i + 1 < length ?xs" with p show "(?xs ! i = ?xs ! (i + 1)) = (?f i = ?f (i + 1))" by (cases i) (auto simp: f0) next have id: "{0 ..< length (?x1 zs)} = insert 0 (?Succ ` {0 ..< length zs})" using zsne by (cases ?cond, auto) { fix i assume "i < Suc (length xs)" hence "Suc i \<in> {0..<Suc (Suc (length xs))} \<inter> Collect ((<) 0)" by auto from imageI[OF this, of "\<lambda>i. ?Succ (f (i - Suc 0))"] have "?Succ (f i) \<in> (\<lambda>i. ?Succ (f (i - Suc 0))) ` ({0..<Suc (Suc (length xs))} \<inter> Collect ((<) 0))" by auto } then show "?f ` {0 ..< length ?xs} = {0 ..< length (?x1 zs)}" unfolding id f_xs_zs[symmetric] by auto qed qed next assume "\<exists> f. ?p f xs ys" then show ?L proof (induct xs arbitrary: ys rule: remdups_adj.induct) case 1 then show ?case by auto next case (2 x) then obtain f where f_img: "f ` {0 ..< size [x]} = {0 ..< size ys}" and f_nth: "\<And>i. i < size [x] \<Longrightarrow> [x]!i = ys!(f i)" by blast have "length ys = card (f ` {0 ..< size [x]})" using f_img by auto then have *: "length ys = 1" by auto then have "f 0 = 0" using f_img by auto with * show ?case using f_nth by (cases ys) auto next case (3 x1 x2 xs) from "3.prems" obtain f where f_mono: "mono f" and f_img: "f ` {0..<length (x1 # x2 # xs)} = {0..<length ys}" and f_nth: "\<And>i. i < length (x1 # x2 # xs) \<Longrightarrow> (x1 # x2 # xs) ! i = ys ! f i" "\<And>i. i + 1 < length (x1 # x2 #xs) \<Longrightarrow> ((x1 # x2 # xs) ! i = (x1 # x2 # xs) ! (i + 1)) = (f i = f (i + 1))" by blast show ?case proof cases assume "x1 = x2" let ?f' = "f \<circ> Suc" have "remdups_adj (x1 # xs) = ys" proof (intro "3.hyps" exI conjI impI allI) show "mono ?f'" using f_mono by (simp add: mono_iff_le_Suc) next have "?f' ` {0 ..< length (x1 # xs)} = f ` {Suc 0 ..< length (x1 # x2 # xs)}" using less_Suc_eq_0_disj by auto also have "\<dots> = f ` {0 ..< length (x1 # x2 # xs)}" proof - have "f 0 = f (Suc 0)" using \<open>x1 = x2\<close> f_nth[of 0] by simp then show ?thesis using less_Suc_eq_0_disj by auto qed also have "\<dots> = {0 ..< length ys}" by fact finally show "?f' ` {0 ..< length (x1 # xs)} = {0 ..< length ys}" . qed (insert f_nth[of "Suc i" for i], auto simp: \<open>x1 = x2\<close>) then show ?thesis using \<open>x1 = x2\<close> by simp next assume "x1 \<noteq> x2" have two: "Suc (Suc 0) \<le> length ys" proof - have "2 = card {f 0, f 1}" using \<open>x1 \<noteq> x2\<close> f_nth[of 0] by auto also have "\<dots> \<le> card (f ` {0..< length (x1 # x2 # xs)})" by (rule card_mono) auto finally show ?thesis using f_img by simp qed have "f 0 = 0" using f_mono f_img by (rule mono_image_least) simp have "f (Suc 0) = Suc 0" proof (rule ccontr) assume "f (Suc 0) \<noteq> Suc 0" then have "Suc 0 < f (Suc 0)" using f_nth[of 0] \<open>x1 \<noteq> x2\<close> \<open>f 0 = 0\<close> by auto then have "\<And>i. Suc 0 < f (Suc i)" using f_mono by (meson Suc_le_mono le0 less_le_trans monoD) then have "Suc 0 \<noteq> f i" for i using \<open>f 0 = 0\<close> by (cases i) fastforce+ then have "Suc 0 \<notin> f ` {0 ..< length (x1 # x2 # xs)}" by auto then show False using f_img two by auto qed obtain ys' where "ys = x1 # x2 # ys'" using two f_nth[of 0] f_nth[of 1] by (auto simp: Suc_le_length_iff \<open>f 0 = 0\<close> \<open>f (Suc 0) = Suc 0\<close>) have Suc0_le_f_Suc: "Suc 0 \<le> f (Suc i)" for i by (metis Suc_le_mono \<open>f (Suc 0) = Suc 0\<close> f_mono le0 mono_def) define f' where "f' x = f (Suc x) - 1" for x have f_Suc: "f (Suc i) = Suc (f' i)" for i using Suc0_le_f_Suc[of i] by (auto simp: f'_def) have "remdups_adj (x2 # xs) = (x2 # ys')" proof (intro "3.hyps" exI conjI impI allI) show "mono f'" using Suc0_le_f_Suc f_mono by (auto simp: f'_def mono_iff_le_Suc le_diff_iff) next have "f' ` {0 ..< length (x2 # xs)} = (\<lambda>x. f x - 1) ` {0 ..< length (x1 # x2 #xs)}" by (auto simp: f'_def \<open>f 0 = 0\<close> \<open>f (Suc 0) = Suc 0\<close> image_def Bex_def less_Suc_eq_0_disj) also have "\<dots> = (\<lambda>x. x - 1) ` f ` {0 ..< length (x1 # x2 #xs)}" by (auto simp: image_comp) also have "\<dots> = (\<lambda>x. x - 1) ` {0 ..< length ys}" by (simp only: f_img) also have "\<dots> = {0 ..< length (x2 # ys')}" using \<open>ys = _\<close> by (fastforce intro: rev_image_eqI) finally show "f' ` {0 ..< length (x2 # xs)} = {0 ..< length (x2 # ys')}" . qed (insert f_nth[of "Suc i" for i] \<open>x1 \<noteq> x2\<close>, auto simp add: f_Suc \<open>ys = _\<close>) then show ?case using \<open>ys = _\<close> \<open>x1 \<noteq> x2\<close> by simp qed qed qed lemma hd_remdups_adj[simp]: "hd (remdups_adj xs) = hd xs" by (induction xs rule: remdups_adj.induct) simp_all lemma remdups_adj_Cons: "remdups_adj (x # xs) = (case remdups_adj xs of [] \<Rightarrow> [x] | y # xs \<Rightarrow> if x = y then y # xs else x # y # xs)" by (induct xs arbitrary: x) (auto split: list.splits) lemma remdups_adj_append_two: "remdups_adj (xs @ [x,y]) = remdups_adj (xs @ [x]) @ (if x = y then [] else [y])" by (induct xs rule: remdups_adj.induct, simp_all) lemma remdups_adj_adjacent: "Suc i < length (remdups_adj xs) \<Longrightarrow> remdups_adj xs ! i \<noteq> remdups_adj xs ! Suc i" proof (induction xs arbitrary: i rule: remdups_adj.induct) case (3 x y xs i) thus ?case by (cases i, cases "x = y") (simp, auto simp: hd_conv_nth[symmetric]) qed simp_all lemma remdups_adj_rev[simp]: "remdups_adj (rev xs) = rev (remdups_adj xs)" by (induct xs rule: remdups_adj.induct, simp_all add: remdups_adj_append_two) lemma remdups_adj_length[simp]: "length (remdups_adj xs) \<le> length xs" by (induct xs rule: remdups_adj.induct, auto) lemma remdups_adj_length_ge1[simp]: "xs \<noteq> [] \<Longrightarrow> length (remdups_adj xs) \<ge> Suc 0" by (induct xs rule: remdups_adj.induct, simp_all) lemma remdups_adj_Nil_iff[simp]: "remdups_adj xs = [] \<longleftrightarrow> xs = []" by (induct xs rule: remdups_adj.induct, simp_all) lemma remdups_adj_set[simp]: "set (remdups_adj xs) = set xs" by (induct xs rule: remdups_adj.induct, simp_all) lemma last_remdups_adj [simp]: "last (remdups_adj xs) = last xs" by (induction xs rule: remdups_adj.induct) auto lemma remdups_adj_Cons_alt[simp]: "x # tl (remdups_adj (x # xs)) = remdups_adj (x # xs)" by (induct xs rule: remdups_adj.induct, auto) lemma remdups_adj_distinct: "distinct xs \<Longrightarrow> remdups_adj xs = xs" by (induct xs rule: remdups_adj.induct, simp_all) lemma remdups_adj_append: "remdups_adj (xs\<^sub>1 @ x # xs\<^sub>2) = remdups_adj (xs\<^sub>1 @ [x]) @ tl (remdups_adj (x # xs\<^sub>2))" by (induct xs\<^sub>1 rule: remdups_adj.induct, simp_all) lemma remdups_adj_singleton: "remdups_adj xs = [x] \<Longrightarrow> xs = replicate (length xs) x" by (induct xs rule: remdups_adj.induct, auto split: if_split_asm) lemma remdups_adj_map_injective: assumes "inj f" shows "remdups_adj (map f xs) = map f (remdups_adj xs)" by (induct xs rule: remdups_adj.induct) (auto simp add: injD[OF assms]) lemma remdups_adj_replicate: "remdups_adj (replicate n x) = (if n = 0 then [] else [x])" by (induction n) (auto simp: remdups_adj_Cons) lemma remdups_upt [simp]: "remdups [m..<n] = [m..<n]" proof (cases "m \<le> n") case False then show ?thesis by simp next case True then obtain q where "n = m + q" by (auto simp add: le_iff_add) moreover have "remdups [m..<m + q] = [m..<m + q]" by (induct q) simp_all ultimately show ?thesis by simp qed lemma successively_remdups_adjI: "successively P xs \<Longrightarrow> successively P (remdups_adj xs)" by (induction xs rule: remdups_adj.induct) (auto simp: successively_Cons) lemma successively_remdups_adj_iff: "(\<And>x. x \<in> set xs \<Longrightarrow> P x x) \<Longrightarrow> successively P (remdups_adj xs) \<longleftrightarrow> successively P xs" by (induction xs rule: remdups_adj.induct)(auto simp: successively_Cons) lemma remdups_adj_Cons': "remdups_adj (x # xs) = x # remdups_adj (dropWhile (\<lambda>y. y = x) xs)" by (induction xs) auto lemma remdups_adj_singleton_iff: "length (remdups_adj xs) = Suc 0 \<longleftrightarrow> xs \<noteq> [] \<and> xs = replicate (length xs) (hd xs)" proof safe assume *: "xs = replicate (length xs) (hd xs)" and [simp]: "xs \<noteq> []" show "length (remdups_adj xs) = Suc 0" by (subst *) (auto simp: remdups_adj_replicate) next assume "length (remdups_adj xs) = Suc 0" thus "xs = replicate (length xs) (hd xs)" by (induction xs rule: remdups_adj.induct) (auto split: if_splits) qed auto lemma tl_remdups_adj: "ys \<noteq> [] \<Longrightarrow> tl (remdups_adj ys) = remdups_adj (dropWhile (\<lambda>x. x = hd ys) (tl ys))" by (cases ys) (simp_all add: remdups_adj_Cons') lemma remdups_adj_append_dropWhile: "remdups_adj (xs @ y # ys) = remdups_adj (xs @ [y]) @ remdups_adj (dropWhile (\<lambda>x. x = y) ys)" by (subst remdups_adj_append) (simp add: tl_remdups_adj) lemma remdups_adj_append': assumes "xs = [] \<or> ys = [] \<or> last xs \<noteq> hd ys" shows "remdups_adj (xs @ ys) = remdups_adj xs @ remdups_adj ys" proof - have ?thesis if [simp]: "xs \<noteq> []" "ys \<noteq> []" and "last xs \<noteq> hd ys" proof - obtain x xs' where xs: "xs = xs' @ [x]" by (cases xs rule: rev_cases) auto have "remdups_adj (xs' @ x # ys) = remdups_adj (xs' @ [x]) @ remdups_adj ys" using \<open>last xs \<noteq> hd ys\<close> unfolding xs by (metis (full_types) dropWhile_eq_self_iff last_snoc remdups_adj_append_dropWhile) thus ?thesis by (simp add: xs) qed thus ?thesis using assms by (cases "xs = []"; cases "ys = []") auto qed lemma remdups_adj_append'': "xs \<noteq> [] \<Longrightarrow> remdups_adj (xs @ ys) = remdups_adj xs @ remdups_adj (dropWhile (\<lambda>y. y = last xs) ys)" by (induction xs rule: remdups_adj.induct) (auto simp: remdups_adj_Cons') subsection \<open>@{const distinct_adj}\<close> lemma distinct_adj_Nil [simp]: "distinct_adj []" and distinct_adj_singleton [simp]: "distinct_adj [x]" and distinct_adj_Cons_Cons [simp]: "distinct_adj (x # y # xs) \<longleftrightarrow> x \<noteq> y \<and> distinct_adj (y # xs)" by (auto simp: distinct_adj_def) lemma distinct_adj_Cons: "distinct_adj (x # xs) \<longleftrightarrow> xs = [] \<or> x \<noteq> hd xs \<and> distinct_adj xs" by (cases xs) auto lemma distinct_adj_ConsD: "distinct_adj (x # xs) \<Longrightarrow> distinct_adj xs" by (cases xs) auto lemma distinct_adj_remdups_adj[simp]: "distinct_adj (remdups_adj xs)" by (induction xs rule: remdups_adj.induct) (auto simp: distinct_adj_Cons) lemma distinct_adj_altdef: "distinct_adj xs \<longleftrightarrow> remdups_adj xs = xs" proof assume "remdups_adj xs = xs" with distinct_adj_remdups_adj[of xs] show "distinct_adj xs" by simp next assume "distinct_adj xs" thus "remdups_adj xs = xs" by (induction xs rule: induct_list012) auto qed lemma distinct_adj_rev [simp]: "distinct_adj (rev xs) \<longleftrightarrow> distinct_adj xs" by (simp add: distinct_adj_def eq_commute) lemma distinct_adj_append_iff: "distinct_adj (xs @ ys) \<longleftrightarrow> distinct_adj xs \<and> distinct_adj ys \<and> (xs = [] \<or> ys = [] \<or> last xs \<noteq> hd ys)" by (auto simp: distinct_adj_def successively_append_iff) lemma distinct_adj_appendD1 [dest]: "distinct_adj (xs @ ys) \<Longrightarrow> distinct_adj xs" and distinct_adj_appendD2 [dest]: "distinct_adj (xs @ ys) \<Longrightarrow> distinct_adj ys" by (auto simp: distinct_adj_append_iff) lemma distinct_adj_mapI: "distinct_adj xs \<Longrightarrow> inj_on f (set xs) \<Longrightarrow> distinct_adj (map f xs)" unfolding distinct_adj_def successively_map by (erule successively_mono) (auto simp: inj_on_def) lemma distinct_adj_mapD: "distinct_adj (map f xs) \<Longrightarrow> distinct_adj xs" unfolding distinct_adj_def successively_map by (erule successively_mono) auto lemma distinct_adj_map_iff: "inj_on f (set xs) \<Longrightarrow> distinct_adj (map f xs) \<longleftrightarrow> distinct_adj xs" using distinct_adj_mapD distinct_adj_mapI by blast subsubsection \<open>\<^const>\<open>insert\<close>\<close> lemma in_set_insert [simp]: "x \<in> set xs \<Longrightarrow> List.insert x xs = xs" by (simp add: List.insert_def) lemma not_in_set_insert [simp]: "x \<notin> set xs \<Longrightarrow> List.insert x xs = x # xs" by (simp add: List.insert_def) lemma insert_Nil [simp]: "List.insert x [] = [x]" by simp lemma set_insert [simp]: "set (List.insert x xs) = insert x (set xs)" by (auto simp add: List.insert_def) lemma distinct_insert [simp]: "distinct (List.insert x xs) = distinct xs" by (simp add: List.insert_def) lemma insert_remdups: "List.insert x (remdups xs) = remdups (List.insert x xs)" by (simp add: List.insert_def) subsubsection \<open>\<^const>\<open>List.union\<close>\<close> text\<open>This is all one should need to know about union:\<close> lemma set_union[simp]: "set (List.union xs ys) = set xs \<union> set ys" unfolding List.union_def by(induct xs arbitrary: ys) simp_all lemma distinct_union[simp]: "distinct(List.union xs ys) = distinct ys" unfolding List.union_def by(induct xs arbitrary: ys) simp_all subsubsection \<open>\<^const>\<open>List.find\<close>\<close> lemma find_None_iff: "List.find P xs = None \<longleftrightarrow> \<not> (\<exists>x. x \<in> set xs \<and> P x)" proof (induction xs) case Nil thus ?case by simp next case (Cons x xs) thus ?case by (fastforce split: if_splits) qed lemma find_Some_iff: "List.find P xs = Some x \<longleftrightarrow> (\<exists>i<length xs. P (xs!i) \<and> x = xs!i \<and> (\<forall>j<i. \<not> P (xs!j)))" proof (induction xs) case Nil thus ?case by simp next case (Cons x xs) thus ?case apply(auto simp: nth_Cons' split: if_splits) using diff_Suc_1[unfolded One_nat_def] less_Suc_eq_0_disj by fastforce qed lemma find_cong[fundef_cong]: assumes "xs = ys" and "\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x" shows "List.find P xs = List.find Q ys" proof (cases "List.find P xs") case None thus ?thesis by (metis find_None_iff assms) next case (Some x) hence "List.find Q ys = Some x" using assms by (auto simp add: find_Some_iff) thus ?thesis using Some by auto qed lemma find_dropWhile: "List.find P xs = (case dropWhile (Not \<circ> P) xs of [] \<Rightarrow> None | x # _ \<Rightarrow> Some x)" by (induct xs) simp_all subsubsection \<open>\<^const>\<open>count_list\<close>\<close> lemma count_notin[simp]: "x \<notin> set xs \<Longrightarrow> count_list xs x = 0" by (induction xs) auto lemma count_le_length: "count_list xs x \<le> length xs" by (induction xs) auto lemma sum_count_set: "set xs \<subseteq> X \<Longrightarrow> finite X \<Longrightarrow> sum (count_list xs) X = length xs" proof (induction xs arbitrary: X) case (Cons x xs) then show ?case using sum.remove [of X x "count_list xs"] by (auto simp: sum.If_cases simp flip: diff_eq) qed simp subsubsection \<open>\<^const>\<open>List.extract\<close>\<close> lemma extract_None_iff: "List.extract P xs = None \<longleftrightarrow> \<not> (\<exists> x\<in>set xs. P x)" by(auto simp: extract_def dropWhile_eq_Cons_conv split: list.splits) (metis in_set_conv_decomp) lemma extract_SomeE: "List.extract P xs = Some (ys, y, zs) \<Longrightarrow> xs = ys @ y # zs \<and> P y \<and> \<not> (\<exists> y \<in> set ys. P y)" by(auto simp: extract_def dropWhile_eq_Cons_conv split: list.splits) lemma extract_Some_iff: "List.extract P xs = Some (ys, y, zs) \<longleftrightarrow> xs = ys @ y # zs \<and> P y \<and> \<not> (\<exists> y \<in> set ys. P y)" by(auto simp: extract_def dropWhile_eq_Cons_conv dest: set_takeWhileD split: list.splits) lemma extract_Nil_code[code]: "List.extract P [] = None" by(simp add: extract_def) lemma extract_Cons_code[code]: "List.extract P (x # xs) = (if P x then Some ([], x, xs) else (case List.extract P xs of None \<Rightarrow> None | Some (ys, y, zs) \<Rightarrow> Some (x#ys, y, zs)))" by(auto simp add: extract_def comp_def split: list.splits) (metis dropWhile_eq_Nil_conv list.distinct(1)) subsubsection \<open>\<^const>\<open>remove1\<close>\<close> lemma remove1_append: "remove1 x (xs @ ys) = (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)" by (induct xs) auto lemma remove1_commute: "remove1 x (remove1 y zs) = remove1 y (remove1 x zs)" by (induct zs) auto lemma in_set_remove1[simp]: "a \<noteq> b \<Longrightarrow> a \<in> set(remove1 b xs) = (a \<in> set xs)" by (induct xs) auto lemma set_remove1_subset: "set(remove1 x xs) \<subseteq> set xs" by (induct xs) auto lemma set_remove1_eq [simp]: "distinct xs \<Longrightarrow> set(remove1 x xs) = set xs - {x}" by (induct xs) auto lemma length_remove1: "length(remove1 x xs) = (if x \<in> set xs then length xs - 1 else length xs)" by (induct xs) (auto dest!:length_pos_if_in_set) lemma remove1_filter_not[simp]: "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs" by(induct xs) auto lemma filter_remove1: "filter Q (remove1 x xs) = remove1 x (filter Q xs)" by (induct xs) auto lemma notin_set_remove1[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(remove1 y xs)" by(insert set_remove1_subset) fast lemma distinct_remove1[simp]: "distinct xs \<Longrightarrow> distinct(remove1 x xs)" by (induct xs) simp_all lemma remove1_remdups: "distinct xs \<Longrightarrow> remove1 x (remdups xs) = remdups (remove1 x xs)" by (induct xs) simp_all lemma remove1_idem: "x \<notin> set xs \<Longrightarrow> remove1 x xs = xs" by (induct xs) simp_all subsubsection \<open>\<^const>\<open>removeAll\<close>\<close> lemma removeAll_filter_not_eq: "removeAll x = filter (\<lambda>y. x \<noteq> y)" proof fix xs show "removeAll x xs = filter (\<lambda>y. x \<noteq> y) xs" by (induct xs) auto qed lemma removeAll_append[simp]: "removeAll x (xs @ ys) = removeAll x xs @ removeAll x ys" by (induct xs) auto lemma set_removeAll[simp]: "set(removeAll x xs) = set xs - {x}" by (induct xs) auto lemma removeAll_id[simp]: "x \<notin> set xs \<Longrightarrow> removeAll x xs = xs" by (induct xs) auto (* Needs count:: 'a \<Rightarrow> 'a list \<Rightarrow> nat lemma length_removeAll: "length(removeAll x xs) = length xs - count x xs" *) lemma removeAll_filter_not[simp]: "\<not> P x \<Longrightarrow> removeAll x (filter P xs) = filter P xs" by(induct xs) auto lemma distinct_removeAll: "distinct xs \<Longrightarrow> distinct (removeAll x xs)" by (simp add: removeAll_filter_not_eq) lemma distinct_remove1_removeAll: "distinct xs \<Longrightarrow> remove1 x xs = removeAll x xs" by (induct xs) simp_all lemma map_removeAll_inj_on: "inj_on f (insert x (set xs)) \<Longrightarrow> map f (removeAll x xs) = removeAll (f x) (map f xs)" by (induct xs) (simp_all add:inj_on_def) lemma map_removeAll_inj: "inj f \<Longrightarrow> map f (removeAll x xs) = removeAll (f x) (map f xs)" by (rule map_removeAll_inj_on, erule subset_inj_on, rule subset_UNIV) lemma length_removeAll_less_eq [simp]: "length (removeAll x xs) \<le> length xs" by (simp add: removeAll_filter_not_eq) lemma length_removeAll_less [termination_simp]: "x \<in> set xs \<Longrightarrow> length (removeAll x xs) < length xs" by (auto dest: length_filter_less simp add: removeAll_filter_not_eq) lemma distinct_concat_iff: "distinct (concat xs) \<longleftrightarrow> distinct (removeAll [] xs) \<and> (\<forall>ys. ys \<in> set xs \<longrightarrow> distinct ys) \<and> (\<forall>ys zs. ys \<in> set xs \<and> zs \<in> set xs \<and> ys \<noteq> zs \<longrightarrow> set ys \<inter> set zs = {})" apply (induct xs) apply(simp_all, safe, auto) by (metis Int_iff UN_I empty_iff equals0I set_empty) subsubsection \<open>\<^const>\<open>replicate\<close>\<close> lemma length_replicate [simp]: "length (replicate n x) = n" by (induct n) auto lemma replicate_eqI: assumes "length xs = n" and "\<And>y. y \<in> set xs \<Longrightarrow> y = x" shows "xs = replicate n x" using assms proof (induct xs arbitrary: n) case Nil then show ?case by simp next case (Cons x xs) then show ?case by (cases n) simp_all qed lemma Ex_list_of_length: "\<exists>xs. length xs = n" by (rule exI[of _ "replicate n undefined"]) simp lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)" by (induct n) auto lemma map_replicate_const: "map (\<lambda> x. k) lst = replicate (length lst) k" by (induct lst) auto lemma replicate_app_Cons_same: "(replicate n x) @ (x # xs) = x # replicate n x @ xs" by (induct n) auto lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x" by (induct n) (auto simp: replicate_app_Cons_same) lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x" by (induct n) auto text\<open>Courtesy of Matthias Daum:\<close> lemma append_replicate_commute: "replicate n x @ replicate k x = replicate k x @ replicate n x" by (metis add.commute replicate_add) text\<open>Courtesy of Andreas Lochbihler:\<close> lemma filter_replicate: "filter P (replicate n x) = (if P x then replicate n x else [])" by(induct n) auto lemma hd_replicate [simp]: "n \<noteq> 0 \<Longrightarrow> hd (replicate n x) = x" by (induct n) auto lemma tl_replicate [simp]: "tl (replicate n x) = replicate (n - 1) x" by (induct n) auto lemma last_replicate [simp]: "n \<noteq> 0 \<Longrightarrow> last (replicate n x) = x" by (atomize (full), induct n) auto lemma nth_replicate[simp]: "i < n \<Longrightarrow> (replicate n x)!i = x" by (induct n arbitrary: i)(auto simp: nth_Cons split: nat.split) text\<open>Courtesy of Matthias Daum (2 lemmas):\<close> lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x" proof (cases "k \<le> i") case True then show ?thesis by (simp add: min_def) next case False then have "replicate k x = replicate i x @ replicate (k - i) x" by (simp add: replicate_add [symmetric]) then show ?thesis by (simp add: min_def) qed lemma drop_replicate[simp]: "drop i (replicate k x) = replicate (k-i) x" proof (induct k arbitrary: i) case (Suc k) then show ?case by (simp add: drop_Cons') qed simp lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}" by (induct n) auto lemma set_replicate [simp]: "n \<noteq> 0 \<Longrightarrow> set (replicate n x) = {x}" by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc) lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})" by auto lemma in_set_replicate[simp]: "(x \<in> set (replicate n y)) = (x = y \<and> n \<noteq> 0)" by (simp add: set_replicate_conv_if) lemma Ball_set_replicate[simp]: "(\<forall>x \<in> set(replicate n a). P x) = (P a \<or> n=0)" by(simp add: set_replicate_conv_if) lemma Bex_set_replicate[simp]: "(\<exists>x \<in> set(replicate n a). P x) = (P a \<and> n\<noteq>0)" by(simp add: set_replicate_conv_if) lemma replicate_append_same: "replicate i x @ [x] = x # replicate i x" by (induct i) simp_all lemma map_replicate_trivial: "map (\<lambda>i. x) [0..<i] = replicate i x" by (induct i) (simp_all add: replicate_append_same) lemma concat_replicate_trivial[simp]: "concat (replicate i []) = []" by (induct i) (auto simp add: map_replicate_const) lemma replicate_empty[simp]: "(replicate n x = []) \<longleftrightarrow> n=0" by (induct n) auto lemma empty_replicate[simp]: "([] = replicate n x) \<longleftrightarrow> n=0" by (induct n) auto lemma replicate_eq_replicate[simp]: "(replicate m x = replicate n y) \<longleftrightarrow> (m=n \<and> (m\<noteq>0 \<longrightarrow> x=y))" proof (induct m arbitrary: n) case (Suc m n) then show ?case by (induct n) auto qed simp lemma takeWhile_replicate[simp]: "takeWhile P (replicate n x) = (if P x then replicate n x else [])" using takeWhile_eq_Nil_iff by fastforce lemma dropWhile_replicate[simp]: "dropWhile P (replicate n x) = (if P x then [] else replicate n x)" using dropWhile_eq_self_iff by fastforce lemma replicate_length_filter: "replicate (length (filter (\<lambda>y. x = y) xs)) x = filter (\<lambda>y. x = y) xs" by (induct xs) auto lemma comm_append_are_replicate: "xs @ ys = ys @ xs \<Longrightarrow> \<exists>m n zs. concat (replicate m zs) = xs \<and> concat (replicate n zs) = ys" proof (induction "length (xs @ ys) + length xs" arbitrary: xs ys rule: less_induct) case less consider (1) "length ys < length xs" | (2) "xs = []" | (3) "length xs \<le> length ys \<and> xs \<noteq> []" by linarith then show ?case proof (cases) case 1 then show ?thesis using less.hyps[OF _ less.prems[symmetric]] nat_add_left_cancel_less by auto next case 2 then have "concat (replicate 0 ys) = xs \<and> concat (replicate 1 ys) = ys" by simp then show ?thesis by blast next case 3 then have "length xs \<le> length ys" and "xs \<noteq> []" by blast+ from \<open>length xs \<le> length ys\<close> and \<open>xs @ ys = ys @ xs\<close> obtain ws where "ys = xs @ ws" by (auto simp: append_eq_append_conv2) from this and \<open>xs \<noteq> []\<close> have "length ws < length ys" by simp from \<open>xs @ ys = ys @ xs\<close>[unfolded \<open>ys = xs @ ws\<close>] have "xs @ ws = ws @ xs" by simp from less.hyps[OF _ this] \<open>length ws < length ys\<close> obtain m n' zs where "concat (replicate m zs) = xs" and "concat (replicate n' zs) = ws" by auto then have "concat (replicate (m+n') zs) = ys" using \<open>ys = xs @ ws\<close> by (simp add: replicate_add) then show ?thesis using \<open>concat (replicate m zs) = xs\<close> by blast qed qed lemma comm_append_is_replicate: fixes xs ys :: "'a list" assumes "xs \<noteq> []" "ys \<noteq> []" assumes "xs @ ys = ys @ xs" shows "\<exists>n zs. n > 1 \<and> concat (replicate n zs) = xs @ ys" proof - obtain m n zs where "concat (replicate m zs) = xs" and "concat (replicate n zs) = ys" using comm_append_are_replicate[OF assms(3)] by blast then have "m + n > 1" and "concat (replicate (m+n) zs) = xs @ ys" using \<open>xs \<noteq> []\<close> and \<open>ys \<noteq> []\<close> by (auto simp: replicate_add) then show ?thesis by blast qed lemma Cons_replicate_eq: "x # xs = replicate n y \<longleftrightarrow> x = y \<and> n > 0 \<and> xs = replicate (n - 1) x" by (induct n) auto lemma replicate_length_same: "(\<forall>y\<in>set xs. y = x) \<Longrightarrow> replicate (length xs) x = xs" by (induct xs) simp_all lemma foldr_replicate [simp]: "foldr f (replicate n x) = f x ^^ n" by (induct n) (simp_all) lemma fold_replicate [simp]: "fold f (replicate n x) = f x ^^ n" by (subst foldr_fold [symmetric]) simp_all subsubsection \<open>\<^const>\<open>enumerate\<close>\<close> lemma enumerate_simps [simp, code]: "enumerate n [] = []" "enumerate n (x # xs) = (n, x) # enumerate (Suc n) xs" by (simp_all add: enumerate_eq_zip upt_rec) lemma length_enumerate [simp]: "length (enumerate n xs) = length xs" by (simp add: enumerate_eq_zip) lemma map_fst_enumerate [simp]: "map fst (enumerate n xs) = [n..<n + length xs]" by (simp add: enumerate_eq_zip) lemma map_snd_enumerate [simp]: "map snd (enumerate n xs) = xs" by (simp add: enumerate_eq_zip) lemma in_set_enumerate_eq: "p \<in> set (enumerate n xs) \<longleftrightarrow> n \<le> fst p \<and> fst p < length xs + n \<and> nth xs (fst p - n) = snd p" proof - { fix m assume "n \<le> m" moreover assume "m < length xs + n" ultimately have "[n..<n + length xs] ! (m - n) = m \<and> xs ! (m - n) = xs ! (m - n) \<and> m - n < length xs" by auto then have "\<exists>q. [n..<n + length xs] ! q = m \<and> xs ! q = xs ! (m - n) \<and> q < length xs" .. } then show ?thesis by (cases p) (auto simp add: enumerate_eq_zip in_set_zip) qed lemma nth_enumerate_eq: "m < length xs \<Longrightarrow> enumerate n xs ! m = (n + m, xs ! m)" by (simp add: enumerate_eq_zip) lemma enumerate_replicate_eq: "enumerate n (replicate m a) = map (\<lambda>q. (q, a)) [n..<n + m]" by (rule pair_list_eqI) (simp_all add: enumerate_eq_zip comp_def map_replicate_const) lemma enumerate_Suc_eq: "enumerate (Suc n) xs = map (apfst Suc) (enumerate n xs)" by (rule pair_list_eqI) (simp_all add: not_le, simp del: map_map add: map_Suc_upt map_map [symmetric]) lemma distinct_enumerate [simp]: "distinct (enumerate n xs)" by (simp add: enumerate_eq_zip distinct_zipI1) lemma enumerate_append_eq: "enumerate n (xs @ ys) = enumerate n xs @ enumerate (n + length xs) ys" by (simp add: enumerate_eq_zip add.assoc zip_append2) lemma enumerate_map_upt: "enumerate n (map f [n..<m]) = map (\<lambda>k. (k, f k)) [n..<m]" by (cases "n \<le> m") (simp_all add: zip_map2 zip_same_conv_map enumerate_eq_zip) subsubsection \<open>\<^const>\<open>rotate1\<close> and \<^const>\<open>rotate\<close>\<close> lemma rotate0[simp]: "rotate 0 = id" by(simp add:rotate_def) lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)" by(simp add:rotate_def) lemma rotate_add: "rotate (m+n) = rotate m \<circ> rotate n" by(simp add:rotate_def funpow_add) lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs" by(simp add:rotate_add) lemma rotate1_map: "rotate1 (map f xs) = map f (rotate1 xs)" by(cases xs) simp_all lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)" by(simp add:rotate_def funpow_swap1) lemma rotate1_length01[simp]: "length xs \<le> 1 \<Longrightarrow> rotate1 xs = xs" by(cases xs) simp_all lemma rotate_length01[simp]: "length xs \<le> 1 \<Longrightarrow> rotate n xs = xs" by (induct n) (simp_all add:rotate_def) lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]" by (cases xs) simp_all lemma rotate_drop_take: "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs" proof (induct n) case (Suc n) show ?case proof (cases "xs = []") case False then show ?thesis proof (cases "n mod length xs = 0") case True then show ?thesis by (auto simp add: mod_Suc False Suc.hyps drop_Suc rotate1_hd_tl take_Suc Suc_length_conv) next case False with \<open>xs \<noteq> []\<close> Suc show ?thesis by (simp add: rotate_def mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric] take_hd_drop linorder_not_le) qed qed simp qed simp lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs" by(simp add:rotate_drop_take) lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs" by(simp add:rotate_drop_take) lemma length_rotate1[simp]: "length(rotate1 xs) = length xs" by (cases xs) simp_all lemma length_rotate[simp]: "length(rotate n xs) = length xs" by (induct n arbitrary: xs) (simp_all add:rotate_def) lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs" by (cases xs) auto lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs" by (induct n) (simp_all add:rotate_def) lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)" by(simp add:rotate_drop_take take_map drop_map) lemma set_rotate1[simp]: "set(rotate1 xs) = set xs" by (cases xs) auto lemma set_rotate[simp]: "set(rotate n xs) = set xs" by (induct n) (simp_all add:rotate_def) lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])" by (cases xs) auto lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])" by (induct n) (simp_all add:rotate_def) lemma rotate_rev: "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)" proof (cases "length xs = 0 \<or> n mod length xs = 0") case False then show ?thesis by(simp add:rotate_drop_take rev_drop rev_take) qed force lemma hd_rotate_conv_nth: assumes "xs \<noteq> []" shows "hd(rotate n xs) = xs!(n mod length xs)" proof - have "n mod length xs < length xs" using assms by simp then show ?thesis by (metis drop_eq_Nil hd_append2 hd_drop_conv_nth leD rotate_drop_take) qed lemma rotate_append: "rotate (length l) (l @ q) = q @ l" by (induct l arbitrary: q) (auto simp add: rotate1_rotate_swap) lemma nth_rotate: \<open>rotate m xs ! n = xs ! ((m + n) mod length xs)\<close> if \<open>n < length xs\<close> using that apply (auto simp add: rotate_drop_take nth_append not_less less_diff_conv ac_simps dest!: le_Suc_ex) apply (metis add.commute mod_add_right_eq mod_less) apply (metis (no_types, lifting) Nat.diff_diff_right add.commute add_diff_cancel_right' diff_le_self dual_order.strict_trans2 length_greater_0_conv less_nat_zero_code list.size(3) mod_add_right_eq mod_add_self2 mod_le_divisor mod_less) done lemma nth_rotate1: \<open>rotate1 xs ! n = xs ! (Suc n mod length xs)\<close> if \<open>n < length xs\<close> using that nth_rotate [of n xs 1] by simp subsubsection \<open>\<^const>\<open>nths\<close> --- a generalization of \<^const>\<open>nth\<close> to sets\<close> lemma nths_empty [simp]: "nths xs {} = []" by (auto simp add: nths_def) lemma nths_nil [simp]: "nths [] A = []" by (auto simp add: nths_def) lemma nths_all: "\<forall>i < length xs. i \<in> I \<Longrightarrow> nths xs I = xs" apply (simp add: nths_def) apply (subst filter_True) apply (auto simp: in_set_zip subset_iff) done lemma length_nths: "length (nths xs I) = card{i. i < length xs \<and> i \<in> I}" by(simp add: nths_def length_filter_conv_card cong:conj_cong) lemma nths_shift_lemma_Suc: "map fst (filter (\<lambda>p. P(Suc(snd p))) (zip xs is)) = map fst (filter (\<lambda>p. P(snd p)) (zip xs (map Suc is)))" proof (induct xs arbitrary: "is") case (Cons x xs "is") show ?case by (cases "is") (auto simp add: Cons.hyps) qed simp lemma nths_shift_lemma: "map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [i..<i + length xs])) = map fst (filter (\<lambda>p. snd p + i \<in> A) (zip xs [0..<length xs]))" by (induct xs rule: rev_induct) (simp_all add: add.commute) lemma nths_append: "nths (l @ l') A = nths l A @ nths l' {j. j + length l \<in> A}" unfolding nths_def proof (induct l' rule: rev_induct) case (snoc x xs) then show ?case by (simp add: upt_add_eq_append[of 0] nths_shift_lemma add.commute) qed auto lemma nths_Cons: "nths (x # l) A = (if 0 \<in> A then [x] else []) @ nths l {j. Suc j \<in> A}" proof (induct l rule: rev_induct) case (snoc x xs) then show ?case by (simp flip: append_Cons add: nths_append) qed (auto simp: nths_def) lemma nths_map: "nths (map f xs) I = map f (nths xs I)" by(induction xs arbitrary: I) (simp_all add: nths_Cons) lemma set_nths: "set(nths xs I) = {xs!i|i. i<size xs \<and> i \<in> I}" by (induct xs arbitrary: I) (auto simp: nths_Cons nth_Cons split:nat.split dest!: gr0_implies_Suc) lemma set_nths_subset: "set(nths xs I) \<subseteq> set xs" by(auto simp add:set_nths) lemma notin_set_nthsI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(nths xs I)" by(auto simp add:set_nths) lemma in_set_nthsD: "x \<in> set(nths xs I) \<Longrightarrow> x \<in> set xs" by(auto simp add:set_nths) lemma nths_singleton [simp]: "nths [x] A = (if 0 \<in> A then [x] else [])" by (simp add: nths_Cons) lemma distinct_nthsI[simp]: "distinct xs \<Longrightarrow> distinct (nths xs I)" by (induct xs arbitrary: I) (auto simp: nths_Cons) lemma nths_upt_eq_take [simp]: "nths l {..<n} = take n l" by (induct l rule: rev_induct) (simp_all split: nat_diff_split add: nths_append) lemma nths_nths: "nths (nths xs A) B = nths xs {i \<in> A. \<exists>j \<in> B. card {i' \<in> A. i' < i} = j}" by (induction xs arbitrary: A B) (auto simp add: nths_Cons card_less_Suc card_less_Suc2) lemma drop_eq_nths: "drop n xs = nths xs {i. i \<ge> n}" by (induction xs arbitrary: n) (auto simp add: nths_Cons nths_all drop_Cons' intro: arg_cong2[where f=nths, OF refl]) lemma nths_drop: "nths (drop n xs) I = nths xs ((+) n ` I)" by(force simp: drop_eq_nths nths_nths simp flip: atLeastLessThan_iff intro: arg_cong2[where f=nths, OF refl]) lemma filter_eq_nths: "filter P xs = nths xs {i. i<length xs \<and> P(xs!i)}" by(induction xs) (auto simp: nths_Cons) lemma filter_in_nths: "distinct xs \<Longrightarrow> filter (%x. x \<in> set (nths xs s)) xs = nths xs s" proof (induct xs arbitrary: s) case Nil thus ?case by simp next case (Cons a xs) then have "\<forall>x. x \<in> set xs \<longrightarrow> x \<noteq> a" by auto with Cons show ?case by(simp add: nths_Cons cong:filter_cong) qed subsubsection \<open>\<^const>\<open>subseqs\<close> and \<^const>\<open>List.n_lists\<close>\<close> lemma length_subseqs: "length (subseqs xs) = 2 ^ length xs" by (induct xs) (simp_all add: Let_def) lemma subseqs_powset: "set ` set (subseqs xs) = Pow (set xs)" proof - have aux: "\<And>x A. set ` Cons x ` A = insert x ` set ` A" by (auto simp add: image_def) have "set (map set (subseqs xs)) = Pow (set xs)" by (induct xs) (simp_all add: aux Let_def Pow_insert Un_commute comp_def del: map_map) then show ?thesis by simp qed lemma distinct_set_subseqs: assumes "distinct xs" shows "distinct (map set (subseqs xs))" proof (rule card_distinct) have "finite (set xs)" .. then have "card (Pow (set xs)) = 2 ^ card (set xs)" by (rule card_Pow) with assms distinct_card [of xs] have "card (Pow (set xs)) = 2 ^ length xs" by simp then show "card (set (map set (subseqs xs))) = length (map set (subseqs xs))" by (simp add: subseqs_powset length_subseqs) qed lemma n_lists_Nil [simp]: "List.n_lists n [] = (if n = 0 then [[]] else [])" by (induct n) simp_all lemma length_n_lists_elem: "ys \<in> set (List.n_lists n xs) \<Longrightarrow> length ys = n" by (induct n arbitrary: ys) auto lemma set_n_lists: "set (List.n_lists n xs) = {ys. length ys = n \<and> set ys \<subseteq> set xs}" proof (rule set_eqI) fix ys :: "'a list" show "ys \<in> set (List.n_lists n xs) \<longleftrightarrow> ys \<in> {ys. length ys = n \<and> set ys \<subseteq> set xs}" proof - have "ys \<in> set (List.n_lists n xs) \<Longrightarrow> length ys = n" by (induct n arbitrary: ys) auto moreover have "\<And>x. ys \<in> set (List.n_lists n xs) \<Longrightarrow> x \<in> set ys \<Longrightarrow> x \<in> set xs" by (induct n arbitrary: ys) auto moreover have "set ys \<subseteq> set xs \<Longrightarrow> ys \<in> set (List.n_lists (length ys) xs)" by (induct ys) auto ultimately show ?thesis by auto qed qed lemma subseqs_refl: "xs \<in> set (subseqs xs)" by (induct xs) (simp_all add: Let_def) lemma subset_subseqs: "X \<subseteq> set xs \<Longrightarrow> X \<in> set ` set (subseqs xs)" unfolding subseqs_powset by simp lemma Cons_in_subseqsD: "y # ys \<in> set (subseqs xs) \<Longrightarrow> ys \<in> set (subseqs xs)" by (induct xs) (auto simp: Let_def) lemma subseqs_distinctD: "\<lbrakk> ys \<in> set (subseqs xs); distinct xs \<rbrakk> \<Longrightarrow> distinct ys" proof (induct xs arbitrary: ys) case (Cons x xs ys) then show ?case by (auto simp: Let_def) (metis Pow_iff contra_subsetD image_eqI subseqs_powset) qed simp subsubsection \<open>\<^const>\<open>splice\<close>\<close> lemma splice_Nil2 [simp]: "splice xs [] = xs" by (cases xs) simp_all lemma length_splice[simp]: "length(splice xs ys) = length xs + length ys" by (induct xs ys rule: splice.induct) auto lemma split_Nil_iff[simp]: "splice xs ys = [] \<longleftrightarrow> xs = [] \<and> ys = []" by (induct xs ys rule: splice.induct) auto lemma splice_replicate[simp]: "splice (replicate m x) (replicate n x) = replicate (m+n) x" proof (induction "replicate m x" "replicate n x" arbitrary: m n rule: splice.induct) case (2 x xs) then show ?case by (auto simp add: Cons_replicate_eq dest: gr0_implies_Suc) qed auto subsubsection \<open>\<^const>\<open>shuffles\<close>\<close> lemma shuffles_commutes: "shuffles xs ys = shuffles ys xs" by (induction xs ys rule: shuffles.induct) (simp_all add: Un_commute) lemma Nil_in_shuffles[simp]: "[] \<in> shuffles xs ys \<longleftrightarrow> xs = [] \<and> ys = []" by (induct xs ys rule: shuffles.induct) auto lemma shufflesE: "zs \<in> shuffles xs ys \<Longrightarrow> (zs = xs \<Longrightarrow> ys = [] \<Longrightarrow> P) \<Longrightarrow> (zs = ys \<Longrightarrow> xs = [] \<Longrightarrow> P) \<Longrightarrow> (\<And>x xs' z zs'. xs = x # xs' \<Longrightarrow> zs = z # zs' \<Longrightarrow> x = z \<Longrightarrow> zs' \<in> shuffles xs' ys \<Longrightarrow> P) \<Longrightarrow> (\<And>y ys' z zs'. ys = y # ys' \<Longrightarrow> zs = z # zs' \<Longrightarrow> y = z \<Longrightarrow> zs' \<in> shuffles xs ys' \<Longrightarrow> P) \<Longrightarrow> P" by (induct xs ys rule: shuffles.induct) auto lemma Cons_in_shuffles_iff: "z # zs \<in> shuffles xs ys \<longleftrightarrow> (xs \<noteq> [] \<and> hd xs = z \<and> zs \<in> shuffles (tl xs) ys \<or> ys \<noteq> [] \<and> hd ys = z \<and> zs \<in> shuffles xs (tl ys))" by (induct xs ys rule: shuffles.induct) auto lemma splice_in_shuffles [simp, intro]: "splice xs ys \<in> shuffles xs ys" by (induction xs ys rule: splice.induct) (simp_all add: Cons_in_shuffles_iff shuffles_commutes) lemma Nil_in_shufflesI: "xs = [] \<Longrightarrow> ys = [] \<Longrightarrow> [] \<in> shuffles xs ys" by simp lemma Cons_in_shuffles_leftI: "zs \<in> shuffles xs ys \<Longrightarrow> z # zs \<in> shuffles (z # xs) ys" by (cases ys) auto lemma Cons_in_shuffles_rightI: "zs \<in> shuffles xs ys \<Longrightarrow> z # zs \<in> shuffles xs (z # ys)" by (cases xs) auto lemma finite_shuffles [simp, intro]: "finite (shuffles xs ys)" by (induction xs ys rule: shuffles.induct) simp_all lemma length_shuffles: "zs \<in> shuffles xs ys \<Longrightarrow> length zs = length xs + length ys" by (induction xs ys arbitrary: zs rule: shuffles.induct) auto lemma set_shuffles: "zs \<in> shuffles xs ys \<Longrightarrow> set zs = set xs \<union> set ys" by (induction xs ys arbitrary: zs rule: shuffles.induct) auto lemma distinct_disjoint_shuffles: assumes "distinct xs" "distinct ys" "set xs \<inter> set ys = {}" "zs \<in> shuffles xs ys" shows "distinct zs" using assms proof (induction xs ys arbitrary: zs rule: shuffles.induct) case (3 x xs y ys) show ?case proof (cases zs) case (Cons z zs') with "3.prems" and "3.IH"[of zs'] show ?thesis by (force dest: set_shuffles) qed simp_all qed simp_all lemma Cons_shuffles_subset1: "(#) x ` shuffles xs ys \<subseteq> shuffles (x # xs) ys" by (cases ys) auto lemma Cons_shuffles_subset2: "(#) y ` shuffles xs ys \<subseteq> shuffles xs (y # ys)" by (cases xs) auto lemma filter_shuffles: "filter P ` shuffles xs ys = shuffles (filter P xs) (filter P ys)" proof - have *: "filter P ` (#) x ` A = (if P x then (#) x ` filter P ` A else filter P ` A)" for x A by (auto simp: image_image) show ?thesis by (induction xs ys rule: shuffles.induct) (simp_all split: if_splits add: image_Un * Un_absorb1 Un_absorb2 Cons_shuffles_subset1 Cons_shuffles_subset2) qed lemma filter_shuffles_disjoint1: assumes "set xs \<inter> set ys = {}" "zs \<in> shuffles xs ys" shows "filter (\<lambda>x. x \<in> set xs) zs = xs" (is "filter ?P _ = _") and "filter (\<lambda>x. x \<notin> set xs) zs = ys" (is "filter ?Q _ = _") using assms proof - from assms have "filter ?P zs \<in> filter ?P ` shuffles xs ys" by blast also have "filter ?P ` shuffles xs ys = shuffles (filter ?P xs) (filter ?P ys)" by (rule filter_shuffles) also have "filter ?P xs = xs" by (rule filter_True) simp_all also have "filter ?P ys = []" by (rule filter_False) (insert assms(1), auto) also have "shuffles xs [] = {xs}" by simp finally show "filter ?P zs = xs" by simp next from assms have "filter ?Q zs \<in> filter ?Q ` shuffles xs ys" by blast also have "filter ?Q ` shuffles xs ys = shuffles (filter ?Q xs) (filter ?Q ys)" by (rule filter_shuffles) also have "filter ?Q ys = ys" by (rule filter_True) (insert assms(1), auto) also have "filter ?Q xs = []" by (rule filter_False) (insert assms(1), auto) also have "shuffles [] ys = {ys}" by simp finally show "filter ?Q zs = ys" by simp qed lemma filter_shuffles_disjoint2: assumes "set xs \<inter> set ys = {}" "zs \<in> shuffles xs ys" shows "filter (\<lambda>x. x \<in> set ys) zs = ys" "filter (\<lambda>x. x \<notin> set ys) zs = xs" using filter_shuffles_disjoint1[of ys xs zs] assms by (simp_all add: shuffles_commutes Int_commute) lemma partition_in_shuffles: "xs \<in> shuffles (filter P xs) (filter (\<lambda>x. \<not>P x) xs)" proof (induction xs) case (Cons x xs) show ?case proof (cases "P x") case True hence "x # xs \<in> (#) x ` shuffles (filter P xs) (filter (\<lambda>x. \<not>P x) xs)" by (intro imageI Cons.IH) also have "\<dots> \<subseteq> shuffles (filter P (x # xs)) (filter (\<lambda>x. \<not>P x) (x # xs))" by (simp add: True Cons_shuffles_subset1) finally show ?thesis . next case False hence "x # xs \<in> (#) x ` shuffles (filter P xs) (filter (\<lambda>x. \<not>P x) xs)" by (intro imageI Cons.IH) also have "\<dots> \<subseteq> shuffles (filter P (x # xs)) (filter (\<lambda>x. \<not>P x) (x # xs))" by (simp add: False Cons_shuffles_subset2) finally show ?thesis . qed qed auto lemma inv_image_partition: assumes "\<And>x. x \<in> set xs \<Longrightarrow> P x" "\<And>y. y \<in> set ys \<Longrightarrow> \<not>P y" shows "partition P -` {(xs, ys)} = shuffles xs ys" proof (intro equalityI subsetI) fix zs assume zs: "zs \<in> shuffles xs ys" hence [simp]: "set zs = set xs \<union> set ys" by (rule set_shuffles) from assms have "filter P zs = filter (\<lambda>x. x \<in> set xs) zs" "filter (\<lambda>x. \<not>P x) zs = filter (\<lambda>x. x \<in> set ys) zs" by (intro filter_cong refl; force)+ moreover from assms have "set xs \<inter> set ys = {}" by auto ultimately show "zs \<in> partition P -` {(xs, ys)}" using zs by (simp add: o_def filter_shuffles_disjoint1 filter_shuffles_disjoint2) next fix zs assume "zs \<in> partition P -` {(xs, ys)}" thus "zs \<in> shuffles xs ys" using partition_in_shuffles[of zs] by (auto simp: o_def) qed subsubsection \<open>Transpose\<close> function transpose where "transpose [] = []" | "transpose ([] # xss) = transpose xss" | "transpose ((x#xs) # xss) = (x # [h. (h#t) \<leftarrow> xss]) # transpose (xs # [t. (h#t) \<leftarrow> xss])" by pat_completeness auto lemma transpose_aux_filter_head: "concat (map (case_list [] (\<lambda>h t. [h])) xss) = map (\<lambda>xs. hd xs) (filter (\<lambda>ys. ys \<noteq> []) xss)" by (induct xss) (auto split: list.split) lemma transpose_aux_filter_tail: "concat (map (case_list [] (\<lambda>h t. [t])) xss) = map (\<lambda>xs. tl xs) (filter (\<lambda>ys. ys \<noteq> []) xss)" by (induct xss) (auto split: list.split) lemma transpose_aux_max: "max (Suc (length xs)) (foldr (\<lambda>xs. max (length xs)) xss 0) = Suc (max (length xs) (foldr (\<lambda>x. max (length x - Suc 0)) (filter (\<lambda>ys. ys \<noteq> []) xss) 0))" (is "max _ ?foldB = Suc (max _ ?foldA)") proof (cases "(filter (\<lambda>ys. ys \<noteq> []) xss) = []") case True hence "foldr (\<lambda>xs. max (length xs)) xss 0 = 0" proof (induct xss) case (Cons x xs) then have "x = []" by (cases x) auto with Cons show ?case by auto qed simp thus ?thesis using True by simp next case False have foldA: "?foldA = foldr (\<lambda>x. max (length x)) (filter (\<lambda>ys. ys \<noteq> []) xss) 0 - 1" by (induct xss) auto have foldB: "?foldB = foldr (\<lambda>x. max (length x)) (filter (\<lambda>ys. ys \<noteq> []) xss) 0" by (induct xss) auto have "0 < ?foldB" proof - from False obtain z zs where zs: "(filter (\<lambda>ys. ys \<noteq> []) xss) = z#zs" by (auto simp: neq_Nil_conv) hence "z \<in> set (filter (\<lambda>ys. ys \<noteq> []) xss)" by auto hence "z \<noteq> []" by auto thus ?thesis unfolding foldB zs by (auto simp: max_def intro: less_le_trans) qed thus ?thesis unfolding foldA foldB max_Suc_Suc[symmetric] by simp qed termination transpose by (relation "measure (\<lambda>xs. foldr (\<lambda>xs. max (length xs)) xs 0 + length xs)") (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max less_Suc_eq_le) lemma transpose_empty: "(transpose xs = []) \<longleftrightarrow> (\<forall>x \<in> set xs. x = [])" by (induct rule: transpose.induct) simp_all lemma length_transpose: fixes xs :: "'a list list" shows "length (transpose xs) = foldr (\<lambda>xs. max (length xs)) xs 0" by (induct rule: transpose.induct) (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max max_Suc_Suc[symmetric] simp del: max_Suc_Suc) lemma nth_transpose: fixes xs :: "'a list list" assumes "i < length (transpose xs)" shows "transpose xs ! i = map (\<lambda>xs. xs ! i) (filter (\<lambda>ys. i < length ys) xs)" using assms proof (induct arbitrary: i rule: transpose.induct) case (3 x xs xss) define XS where "XS = (x # xs) # xss" hence [simp]: "XS \<noteq> []" by auto thus ?case proof (cases i) case 0 thus ?thesis by (simp add: transpose_aux_filter_head hd_conv_nth) next case (Suc j) have *: "\<And>xss. xs # map tl xss = map tl ((x#xs)#xss)" by simp have **: "\<And>xss. (x#xs) # filter (\<lambda>ys. ys \<noteq> []) xss = filter (\<lambda>ys. ys \<noteq> []) ((x#xs)#xss)" by simp { fix x have "Suc j < length x \<longleftrightarrow> x \<noteq> [] \<and> j < length x - Suc 0" by (cases x) simp_all } note *** = this have j_less: "j < length (transpose (xs # concat (map (case_list [] (\<lambda>h t. [t])) xss)))" using "3.prems" by (simp add: transpose_aux_filter_tail length_transpose Suc) show ?thesis unfolding transpose.simps \<open>i = Suc j\<close> nth_Cons_Suc "3.hyps"[OF j_less] apply (auto simp: transpose_aux_filter_tail filter_map comp_def length_transpose * ** *** XS_def[symmetric]) by (simp add: nth_tl) qed qed simp_all lemma transpose_map_map: "transpose (map (map f) xs) = map (map f) (transpose xs)" proof (rule nth_equalityI) have [simp]: "length (transpose (map (map f) xs)) = length (transpose xs)" by (simp add: length_transpose foldr_map comp_def) show "length (transpose (map (map f) xs)) = length (map (map f) (transpose xs))" by simp fix i assume "i < length (transpose (map (map f) xs))" thus "transpose (map (map f) xs) ! i = map (map f) (transpose xs) ! i" by (simp add: nth_transpose filter_map comp_def) qed subsubsection \<open>\<^const>\<open>min\<close> and \<^const>\<open>arg_min\<close>\<close> lemma min_list_Min: "xs \<noteq> [] \<Longrightarrow> min_list xs = Min (set xs)" by (induction xs rule: induct_list012)(auto) lemma f_arg_min_list_f: "xs \<noteq> [] \<Longrightarrow> f (arg_min_list f xs) = Min (f ` (set xs))" by(induction f xs rule: arg_min_list.induct) (auto simp: min_def intro!: antisym) lemma arg_min_list_in: "xs \<noteq> [] \<Longrightarrow> arg_min_list f xs \<in> set xs" by(induction xs rule: induct_list012) (auto simp: Let_def) subsubsection \<open>(In)finiteness\<close> lemma finite_maxlen: "finite (M::'a list set) \<Longrightarrow> \<exists>n. \<forall>s\<in>M. size s < n" proof (induct rule: finite.induct) case emptyI show ?case by simp next case (insertI M xs) then obtain n where "\<forall>s\<in>M. length s < n" by blast hence "\<forall>s\<in>insert xs M. size s < max n (size xs) + 1" by auto thus ?case .. qed lemma lists_length_Suc_eq: "{xs. set xs \<subseteq> A \<and> length xs = Suc n} = (\<lambda>(xs, n). n#xs) ` ({xs. set xs \<subseteq> A \<and> length xs = n} \<times> A)" by (auto simp: length_Suc_conv) lemma assumes "finite A" shows finite_lists_length_eq: "finite {xs. set xs \<subseteq> A \<and> length xs = n}" and card_lists_length_eq: "card {xs. set xs \<subseteq> A \<and> length xs = n} = (card A)^n" using \<open>finite A\<close> by (induct n) (auto simp: card_image inj_split_Cons lists_length_Suc_eq cong: conj_cong) lemma finite_lists_length_le: assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs \<le> n}" (is "finite ?S") proof- have "?S = (\<Union>n\<in>{0..n}. {xs. set xs \<subseteq> A \<and> length xs = n})" by auto thus ?thesis by (auto intro!: finite_lists_length_eq[OF \<open>finite A\<close>] simp only:) qed lemma card_lists_length_le: assumes "finite A" shows "card {xs. set xs \<subseteq> A \<and> length xs \<le> n} = (\<Sum>i\<le>n. card A^i)" proof - have "(\<Sum>i\<le>n. card A^i) = card (\<Union>i\<le>n. {xs. set xs \<subseteq> A \<and> length xs = i})" using \<open>finite A\<close> by (subst card_UN_disjoint) (auto simp add: card_lists_length_eq finite_lists_length_eq) also have "(\<Union>i\<le>n. {xs. set xs \<subseteq> A \<and> length xs = i}) = {xs. set xs \<subseteq> A \<and> length xs \<le> n}" by auto finally show ?thesis by simp qed lemma finite_lists_distinct_length_eq [intro]: assumes "finite A" shows "finite {xs. length xs = n \<and> distinct xs \<and> set xs \<subseteq> A}" (is "finite ?S") proof - have "finite {xs. set xs \<subseteq> A \<and> length xs = n}" using \<open>finite A\<close> by (rule finite_lists_length_eq) moreover have "?S \<subseteq> {xs. set xs \<subseteq> A \<and> length xs = n}" by auto ultimately show ?thesis using finite_subset by auto qed lemma card_lists_distinct_length_eq: assumes "finite A" "k \<le> card A" shows "card {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A} = \<Prod>{card A - k + 1 .. card A}" using assms proof (induct k) case 0 then have "{xs. length xs = 0 \<and> distinct xs \<and> set xs \<subseteq> A} = {[]}" by auto then show ?case by simp next case (Suc k) let "?k_list" = "\<lambda>k xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A" have inj_Cons: "\<And>A. inj_on (\<lambda>(xs, n). n # xs) A" by (rule inj_onI) auto from Suc have "k \<le> card A" by simp moreover note \<open>finite A\<close> moreover have "finite {xs. ?k_list k xs}" by (rule finite_subset) (use finite_lists_length_eq[OF \<open>finite A\<close>, of k] in auto) moreover have "\<And>i j. i \<noteq> j \<longrightarrow> {i} \<times> (A - set i) \<inter> {j} \<times> (A - set j) = {}" by auto moreover have "\<And>i. i \<in> {xs. ?k_list k xs} \<Longrightarrow> card (A - set i) = card A - k" by (simp add: card_Diff_subset distinct_card) moreover have "{xs. ?k_list (Suc k) xs} = (\<lambda>(xs, n). n#xs) ` \<Union>((\<lambda>xs. {xs} \<times> (A - set xs)) ` {xs. ?k_list k xs})" by (auto simp: length_Suc_conv) moreover have "Suc (card A - Suc k) = card A - k" using Suc.prems by simp then have "(card A - k) * \<Prod>{Suc (card A - k)..card A} = \<Prod>{Suc (card A - Suc k)..card A}" by (subst prod.insert[symmetric]) (simp add: atLeastAtMost_insertL)+ ultimately show ?case by (simp add: card_image inj_Cons card_UN_disjoint Suc.hyps algebra_simps) qed lemma card_lists_distinct_length_eq': assumes "k < card A" shows "card {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A} = \<Prod>{card A - k + 1 .. card A}" proof - from \<open>k < card A\<close> have "finite A" and "k \<le> card A" using card.infinite by force+ from this show ?thesis by (rule card_lists_distinct_length_eq) qed lemma infinite_UNIV_listI: "\<not> finite(UNIV::'a list set)" by (metis UNIV_I finite_maxlen length_replicate less_irrefl) lemma same_length_different: assumes "xs \<noteq> ys" and "length xs = length ys" shows "\<exists>pre x xs' y ys'. x\<noteq>y \<and> xs = pre @ [x] @ xs' \<and> ys = pre @ [y] @ ys'" using assms proof (induction xs arbitrary: ys) case Nil then show ?case by auto next case (Cons x xs) then obtain z zs where ys: "ys = Cons z zs" by (metis length_Suc_conv) show ?case proof (cases "x=z") case True then have "xs \<noteq> zs" "length xs = length zs" using Cons.prems ys by auto then obtain pre u xs' v ys' where "u\<noteq>v" and xs: "xs = pre @ [u] @ xs'" and zs: "zs = pre @ [v] @ys'" using Cons.IH by meson then have "x # xs = (z#pre) @ [u] @ xs' \<and> ys = (z#pre) @ [v] @ ys'" by (simp add: True ys) with \<open>u\<noteq>v\<close> show ?thesis by blast next case False then have "x # xs = [] @ [x] @ xs \<and> ys = [] @ [z] @ zs" by (simp add: ys) then show ?thesis using False by blast qed qed subsection \<open>Sorting\<close> subsubsection \<open>\<^const>\<open>sorted_wrt\<close>\<close> text \<open>Sometimes the second equation in the definition of \<^const>\<open>sorted_wrt\<close> is too aggressive because it relates each list element to \emph{all} its successors. Then this equation should be removed and \<open>sorted_wrt2_simps\<close> should be added instead.\<close> lemma sorted_wrt1: "sorted_wrt P [x] = True" by(simp) lemma sorted_wrt2: "transp P \<Longrightarrow> sorted_wrt P (x # y # zs) = (P x y \<and> sorted_wrt P (y # zs))" proof (induction zs arbitrary: x y) case (Cons z zs) then show ?case by simp (meson transpD)+ qed auto lemmas sorted_wrt2_simps = sorted_wrt1 sorted_wrt2 lemma sorted_wrt_true [simp]: "sorted_wrt (\<lambda>_ _. True) xs" by (induction xs) simp_all lemma sorted_wrt_append: "sorted_wrt P (xs @ ys) \<longleftrightarrow> sorted_wrt P xs \<and> sorted_wrt P ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. P x y)" by (induction xs) auto lemma sorted_wrt_map: "sorted_wrt R (map f xs) = sorted_wrt (\<lambda>x y. R (f x) (f y)) xs" by (induction xs) simp_all lemma assumes "sorted_wrt f xs" shows sorted_wrt_take: "sorted_wrt f (take n xs)" and sorted_wrt_drop: "sorted_wrt f (drop n xs)" proof - from assms have "sorted_wrt f (take n xs @ drop n xs)" by simp thus "sorted_wrt f (take n xs)" and "sorted_wrt f (drop n xs)" unfolding sorted_wrt_append by simp_all qed lemma sorted_wrt_filter: "sorted_wrt f xs \<Longrightarrow> sorted_wrt f (filter P xs)" by (induction xs) auto lemma sorted_wrt_rev: "sorted_wrt P (rev xs) = sorted_wrt (\<lambda>x y. P y x) xs" by (induction xs) (auto simp add: sorted_wrt_append) lemma sorted_wrt_mono_rel: "(\<And>x y. \<lbrakk> x \<in> set xs; y \<in> set xs; P x y \<rbrakk> \<Longrightarrow> Q x y) \<Longrightarrow> sorted_wrt P xs \<Longrightarrow> sorted_wrt Q xs" by(induction xs)(auto) lemma sorted_wrt01: "length xs \<le> 1 \<Longrightarrow> sorted_wrt P xs" by(auto simp: le_Suc_eq length_Suc_conv) lemma sorted_wrt_iff_nth_less: "sorted_wrt P xs = (\<forall>i j. i < j \<longrightarrow> j < length xs \<longrightarrow> P (xs ! i) (xs ! j))" by (induction xs) (auto simp add: in_set_conv_nth Ball_def nth_Cons split: nat.split) lemma sorted_wrt_nth_less: "\<lbrakk> sorted_wrt P xs; i < j; j < length xs \<rbrakk> \<Longrightarrow> P (xs ! i) (xs ! j)" by(auto simp: sorted_wrt_iff_nth_less) lemma sorted_wrt_iff_nth_Suc_transp: assumes "transp P" shows "sorted_wrt P xs \<longleftrightarrow> (\<forall>i. Suc i < length xs \<longrightarrow> P (xs!i) (xs!(Suc i)))" (is "?L = ?R") proof assume ?L thus ?R by (simp add: sorted_wrt_iff_nth_less) next assume ?R have "i < j \<Longrightarrow> j < length xs \<Longrightarrow> P (xs ! i) (xs ! j)" for i j by(induct i j rule: less_Suc_induct)(simp add: \<open>?R\<close>, meson assms transpE transp_less) thus ?L by (simp add: sorted_wrt_iff_nth_less) qed lemma sorted_wrt_upt[simp]: "sorted_wrt (<) [m..<n]" by(induction n) (auto simp: sorted_wrt_append) lemma sorted_wrt_upto[simp]: "sorted_wrt (<) [i..j]" proof(induct i j rule:upto.induct) case (1 i j) from this show ?case unfolding upto.simps[of i j] by auto qed text \<open>Each element is greater or equal to its index:\<close> lemma sorted_wrt_less_idx: "sorted_wrt (<) ns \<Longrightarrow> i < length ns \<Longrightarrow> i \<le> ns!i" proof (induction ns arbitrary: i rule: rev_induct) case Nil thus ?case by simp next case snoc thus ?case by (auto simp: nth_append sorted_wrt_append) (metis less_antisym not_less nth_mem) qed subsubsection \<open>\<^const>\<open>sorted\<close>\<close> context linorder begin text \<open>Sometimes the second equation in the definition of \<^const>\<open>sorted\<close> is too aggressive because it relates each list element to \emph{all} its successors. Then this equation should be removed and \<open>sorted2_simps\<close> should be added instead. Executable code is one such use case.\<close> lemma sorted1: "sorted [x] = True" by simp lemma sorted2: "sorted (x # y # zs) = (x \<le> y \<and> sorted (y # zs))" by(induction zs) auto lemmas sorted2_simps = sorted1 sorted2 lemmas [code] = sorted.simps(1) sorted2_simps lemma sorted_append: "sorted (xs@ys) = (sorted xs \<and> sorted ys \<and> (\<forall>x \<in> set xs. \<forall>y \<in> set ys. x\<le>y))" by (simp add: sorted_sorted_wrt sorted_wrt_append) lemma sorted_map: "sorted (map f xs) = sorted_wrt (\<lambda>x y. f x \<le> f y) xs" by (simp add: sorted_sorted_wrt sorted_wrt_map) lemma sorted01: "length xs \<le> 1 \<Longrightarrow> sorted xs" by (simp add: sorted_sorted_wrt sorted_wrt01) lemma sorted_tl: "sorted xs \<Longrightarrow> sorted (tl xs)" by (cases xs) (simp_all) lemma sorted_iff_nth_mono_less: "sorted xs = (\<forall>i j. i < j \<longrightarrow> j < length xs \<longrightarrow> xs ! i \<le> xs ! j)" by (simp add: sorted_sorted_wrt sorted_wrt_iff_nth_less) lemma sorted_iff_nth_mono: "sorted xs = (\<forall>i j. i \<le> j \<longrightarrow> j < length xs \<longrightarrow> xs ! i \<le> xs ! j)" by (auto simp: sorted_iff_nth_mono_less nat_less_le) lemma sorted_nth_mono: "sorted xs \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!i \<le> xs!j" by (auto simp: sorted_iff_nth_mono) lemma sorted_iff_nth_Suc: "sorted xs \<longleftrightarrow> (\<forall>i. Suc i < length xs \<longrightarrow> xs!i \<le> xs!(Suc i))" by(simp add: sorted_sorted_wrt sorted_wrt_iff_nth_Suc_transp) lemma sorted_rev_nth_mono: "sorted (rev xs) \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!j \<le> xs!i" using sorted_nth_mono[ of "rev xs" "length xs - j - 1" "length xs - i - 1"] rev_nth[of "length xs - i - 1" "xs"] rev_nth[of "length xs - j - 1" "xs"] by auto lemma sorted_rev_iff_nth_mono: "sorted (rev xs) \<longleftrightarrow> (\<forall> i j. i \<le> j \<longrightarrow> j < length xs \<longrightarrow> xs!j \<le> xs!i)" (is "?L = ?R") proof assume ?L thus ?R by (blast intro: sorted_rev_nth_mono) next assume ?R have "rev xs ! k \<le> rev xs ! l" if asms: "k \<le> l" "l < length(rev xs)" for k l proof - have "k < length xs" "l < length xs" "length xs - Suc l \<le> length xs - Suc k" "length xs - Suc k < length xs" using asms by auto thus "rev xs ! k \<le> rev xs ! l" using \<open>?R\<close> \<open>k \<le> l\<close> unfolding rev_nth[OF \<open>k < length xs\<close>] rev_nth[OF \<open>l < length xs\<close>] by blast qed thus ?L by (simp add: sorted_iff_nth_mono) qed lemma sorted_rev_iff_nth_Suc: "sorted (rev xs) \<longleftrightarrow> (\<forall>i. Suc i < length xs \<longrightarrow> xs!(Suc i) \<le> xs!i)" proof- interpret dual: linorder "(\<lambda>x y. y \<le> x)" "(\<lambda>x y. y < x)" using dual_linorder . show ?thesis using dual_linorder dual.sorted_iff_nth_Suc dual.sorted_iff_nth_mono unfolding sorted_rev_iff_nth_mono by simp qed lemma sorted_map_remove1: "sorted (map f xs) \<Longrightarrow> sorted (map f (remove1 x xs))" by (induct xs) (auto) lemma sorted_remove1: "sorted xs \<Longrightarrow> sorted (remove1 a xs)" using sorted_map_remove1 [of "\<lambda>x. x"] by simp lemma sorted_butlast: assumes "xs \<noteq> []" and "sorted xs" shows "sorted (butlast xs)" proof - from \<open>xs \<noteq> []\<close> obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto with \<open>sorted xs\<close> show ?thesis by (simp add: sorted_append) qed lemma sorted_replicate [simp]: "sorted(replicate n x)" by(induction n) (auto) lemma sorted_remdups[simp]: "sorted xs \<Longrightarrow> sorted (remdups xs)" by (induct xs) (auto) lemma sorted_remdups_adj[simp]: "sorted xs \<Longrightarrow> sorted (remdups_adj xs)" by (induct xs rule: remdups_adj.induct, simp_all split: if_split_asm) lemma sorted_nths: "sorted xs \<Longrightarrow> sorted (nths xs I)" by(induction xs arbitrary: I)(auto simp: nths_Cons) lemma sorted_distinct_set_unique: assumes "sorted xs" "distinct xs" "sorted ys" "distinct ys" "set xs = set ys" shows "xs = ys" proof - from assms have 1: "length xs = length ys" by (auto dest!: distinct_card) from assms show ?thesis proof(induct rule:list_induct2[OF 1]) case 1 show ?case by simp next case 2 thus ?case by simp (metis Diff_insert_absorb antisym insertE insert_iff) qed qed lemma map_sorted_distinct_set_unique: assumes "inj_on f (set xs \<union> set ys)" assumes "sorted (map f xs)" "distinct (map f xs)" "sorted (map f ys)" "distinct (map f ys)" assumes "set xs = set ys" shows "xs = ys" proof - from assms have "map f xs = map f ys" by (simp add: sorted_distinct_set_unique) with \<open>inj_on f (set xs \<union> set ys)\<close> show "xs = ys" by (blast intro: map_inj_on) qed lemma assumes "sorted xs" shows sorted_take: "sorted (take n xs)" and sorted_drop: "sorted (drop n xs)" proof - from assms have "sorted (take n xs @ drop n xs)" by simp then show "sorted (take n xs)" and "sorted (drop n xs)" unfolding sorted_append by simp_all qed lemma sorted_dropWhile: "sorted xs \<Longrightarrow> sorted (dropWhile P xs)" by (auto dest: sorted_drop simp add: dropWhile_eq_drop) lemma sorted_takeWhile: "sorted xs \<Longrightarrow> sorted (takeWhile P xs)" by (subst takeWhile_eq_take) (auto dest: sorted_take) lemma sorted_filter: "sorted (map f xs) \<Longrightarrow> sorted (map f (filter P xs))" by (induct xs) simp_all lemma foldr_max_sorted: assumes "sorted (rev xs)" shows "foldr max xs y = (if xs = [] then y else max (xs ! 0) y)" using assms proof (induct xs) case (Cons x xs) then have "sorted (rev xs)" using sorted_append by auto with Cons show ?case by (cases xs) (auto simp add: sorted_append max_def) qed simp lemma filter_equals_takeWhile_sorted_rev: assumes sorted: "sorted (rev (map f xs))" shows "filter (\<lambda>x. t < f x) xs = takeWhile (\<lambda> x. t < f x) xs" (is "filter ?P xs = ?tW") proof (rule takeWhile_eq_filter[symmetric]) let "?dW" = "dropWhile ?P xs" fix x assume "x \<in> set ?dW" then obtain i where i: "i < length ?dW" and nth_i: "x = ?dW ! i" unfolding in_set_conv_nth by auto hence "length ?tW + i < length (?tW @ ?dW)" unfolding length_append by simp hence i': "length (map f ?tW) + i < length (map f xs)" by simp have "(map f ?tW @ map f ?dW) ! (length (map f ?tW) + i) \<le> (map f ?tW @ map f ?dW) ! (length (map f ?tW) + 0)" using sorted_rev_nth_mono[OF sorted _ i', of "length ?tW"] unfolding map_append[symmetric] by simp hence "f x \<le> f (?dW ! 0)" unfolding nth_append_length_plus nth_i using i preorder_class.le_less_trans[OF le0 i] by simp also have "... \<le> t" using hd_dropWhile[of "?P" xs] le0[THEN preorder_class.le_less_trans, OF i] using hd_conv_nth[of "?dW"] by simp finally show "\<not> t < f x" by simp qed lemma sorted_map_same: "sorted (map f (filter (\<lambda>x. f x = g xs) xs))" proof (induct xs arbitrary: g) case Nil then show ?case by simp next case (Cons x xs) then have "sorted (map f (filter (\<lambda>y. f y = (\<lambda>xs. f x) xs) xs))" . moreover from Cons have "sorted (map f (filter (\<lambda>y. f y = (g \<circ> Cons x) xs) xs))" . ultimately show ?case by simp_all qed lemma sorted_same: "sorted (filter (\<lambda>x. x = g xs) xs)" using sorted_map_same [of "\<lambda>x. x"] by simp end lemma sorted_upt[simp]: "sorted [m..<n]" by(simp add: sorted_sorted_wrt sorted_wrt_mono_rel[OF _ sorted_wrt_upt]) lemma sorted_upto[simp]: "sorted [m..n]" by(simp add: sorted_sorted_wrt sorted_wrt_mono_rel[OF _ sorted_wrt_upto]) subsubsection \<open>Sorting functions\<close> text\<open>Currently it is not shown that \<^const>\<open>sort\<close> returns a permutation of its input because the nicest proof is via multisets, which are not part of Main. Alternatively one could define a function that counts the number of occurrences of an element in a list and use that instead of multisets to state the correctness property.\<close> context linorder begin lemma set_insort_key: "set (insort_key f x xs) = insert x (set xs)" by (induct xs) auto lemma length_insort [simp]: "length (insort_key f x xs) = Suc (length xs)" by (induct xs) simp_all lemma insort_key_left_comm: assumes "f x \<noteq> f y" shows "insort_key f y (insort_key f x xs) = insort_key f x (insort_key f y xs)" by (induct xs) (auto simp add: assms dest: antisym) lemma insort_left_comm: "insort x (insort y xs) = insort y (insort x xs)" by (cases "x = y") (auto intro: insort_key_left_comm) lemma comp_fun_commute_insort: "comp_fun_commute insort" proof qed (simp add: insort_left_comm fun_eq_iff) lemma sort_key_simps [simp]: "sort_key f [] = []" "sort_key f (x#xs) = insort_key f x (sort_key f xs)" by (simp_all add: sort_key_def) lemma sort_key_conv_fold: assumes "inj_on f (set xs)" shows "sort_key f xs = fold (insort_key f) xs []" proof - have "fold (insort_key f) (rev xs) = fold (insort_key f) xs" proof (rule fold_rev, rule ext) fix zs fix x y assume "x \<in> set xs" "y \<in> set xs" with assms have *: "f y = f x \<Longrightarrow> y = x" by (auto dest: inj_onD) have **: "x = y \<longleftrightarrow> y = x" by auto show "(insort_key f y \<circ> insort_key f x) zs = (insort_key f x \<circ> insort_key f y) zs" by (induct zs) (auto intro: * simp add: **) qed then show ?thesis by (simp add: sort_key_def foldr_conv_fold) qed lemma sort_conv_fold: "sort xs = fold insort xs []" by (rule sort_key_conv_fold) simp lemma length_sort[simp]: "length (sort_key f xs) = length xs" by (induct xs, auto) lemma set_sort[simp]: "set(sort_key f xs) = set xs" by (induct xs) (simp_all add: set_insort_key) lemma distinct_insort: "distinct (insort_key f x xs) = (x \<notin> set xs \<and> distinct xs)" by(induct xs)(auto simp: set_insort_key) lemma distinct_sort[simp]: "distinct (sort_key f xs) = distinct xs" by (induct xs) (simp_all add: distinct_insort) lemma sorted_insort_key: "sorted (map f (insort_key f x xs)) = sorted (map f xs)" by (induct xs) (auto simp: set_insort_key) lemma sorted_insort: "sorted (insort x xs) = sorted xs" using sorted_insort_key [where f="\<lambda>x. x"] by simp theorem sorted_sort_key [simp]: "sorted (map f (sort_key f xs))" by (induct xs) (auto simp:sorted_insort_key) theorem sorted_sort [simp]: "sorted (sort xs)" using sorted_sort_key [where f="\<lambda>x. x"] by simp lemma insort_not_Nil [simp]: "insort_key f a xs \<noteq> []" by (induction xs) simp_all lemma insort_is_Cons: "\<forall>x\<in>set xs. f a \<le> f x \<Longrightarrow> insort_key f a xs = a # xs" by (cases xs) auto lemma sorted_sort_id: "sorted xs \<Longrightarrow> sort xs = xs" by (induct xs) (auto simp add: insort_is_Cons) lemma insort_key_remove1: assumes "a \<in> set xs" and "sorted (map f xs)" and "hd (filter (\<lambda>x. f a = f x) xs) = a" shows "insort_key f a (remove1 a xs) = xs" using assms proof (induct xs) case (Cons x xs) then show ?case proof (cases "x = a") case False then have "f x \<noteq> f a" using Cons.prems by auto then have "f x < f a" using Cons.prems by auto with \<open>f x \<noteq> f a\<close> show ?thesis using Cons by (auto simp: insort_is_Cons) qed (auto simp: insort_is_Cons) qed simp lemma insort_remove1: assumes "a \<in> set xs" and "sorted xs" shows "insort a (remove1 a xs) = xs" proof (rule insort_key_remove1) define n where "n = length (filter ((=) a) xs) - 1" from \<open>a \<in> set xs\<close> show "a \<in> set xs" . from \<open>sorted xs\<close> show "sorted (map (\<lambda>x. x) xs)" by simp from \<open>a \<in> set xs\<close> have "a \<in> set (filter ((=) a) xs)" by auto then have "set (filter ((=) a) xs) \<noteq> {}" by auto then have "filter ((=) a) xs \<noteq> []" by (auto simp only: set_empty) then have "length (filter ((=) a) xs) > 0" by simp then have n: "Suc n = length (filter ((=) a) xs)" by (simp add: n_def) moreover have "replicate (Suc n) a = a # replicate n a" by simp ultimately show "hd (filter ((=) a) xs) = a" by (simp add: replicate_length_filter) qed lemma finite_sorted_distinct_unique: assumes "finite A" shows "\<exists>!xs. set xs = A \<and> sorted xs \<and> distinct xs" proof - obtain xs where "distinct xs" "A = set xs" using finite_distinct_list [OF assms] by metis then show ?thesis by (rule_tac a="sort xs" in ex1I) (auto simp: sorted_distinct_set_unique) qed lemma insort_insert_key_triv: "f x \<in> f ` set xs \<Longrightarrow> insort_insert_key f x xs = xs" by (simp add: insort_insert_key_def) lemma insort_insert_triv: "x \<in> set xs \<Longrightarrow> insort_insert x xs = xs" using insort_insert_key_triv [of "\<lambda>x. x"] by simp lemma insort_insert_insort_key: "f x \<notin> f ` set xs \<Longrightarrow> insort_insert_key f x xs = insort_key f x xs" by (simp add: insort_insert_key_def) lemma insort_insert_insort: "x \<notin> set xs \<Longrightarrow> insort_insert x xs = insort x xs" using insort_insert_insort_key [of "\<lambda>x. x"] by simp lemma set_insort_insert: "set (insort_insert x xs) = insert x (set xs)" by (auto simp add: insort_insert_key_def set_insort_key) lemma distinct_insort_insert: assumes "distinct xs" shows "distinct (insort_insert_key f x xs)" using assms by (induct xs) (auto simp add: insort_insert_key_def set_insort_key) lemma sorted_insort_insert_key: assumes "sorted (map f xs)" shows "sorted (map f (insort_insert_key f x xs))" using assms by (simp add: insort_insert_key_def sorted_insort_key) lemma sorted_insort_insert: assumes "sorted xs" shows "sorted (insort_insert x xs)" using assms sorted_insort_insert_key [of "\<lambda>x. x"] by simp lemma filter_insort_triv: "\<not> P x \<Longrightarrow> filter P (insort_key f x xs) = filter P xs" by (induct xs) simp_all lemma filter_insort: "sorted (map f xs) \<Longrightarrow> P x \<Longrightarrow> filter P (insort_key f x xs) = insort_key f x (filter P xs)" by (induct xs) (auto, subst insort_is_Cons, auto) lemma filter_sort: "filter P (sort_key f xs) = sort_key f (filter P xs)" by (induct xs) (simp_all add: filter_insort_triv filter_insort) lemma remove1_insort [simp]: "remove1 x (insort x xs) = xs" by (induct xs) simp_all end lemma sort_upt [simp]: "sort [m..<n] = [m..<n]" by (rule sorted_sort_id) simp lemma sort_upto [simp]: "sort [i..j] = [i..j]" by (rule sorted_sort_id) simp lemma sorted_find_Min: "sorted xs \<Longrightarrow> \<exists>x \<in> set xs. P x \<Longrightarrow> List.find P xs = Some (Min {x\<in>set xs. P x})" proof (induct xs) case Nil then show ?case by simp next case (Cons x xs) show ?case proof (cases "P x") case True with Cons show ?thesis by (auto intro: Min_eqI [symmetric]) next case False then have "{y. (y = x \<or> y \<in> set xs) \<and> P y} = {y \<in> set xs. P y}" by auto with Cons False show ?thesis by (simp_all) qed qed lemma sorted_enumerate [simp]: "sorted (map fst (enumerate n xs))" by (simp add: enumerate_eq_zip) text \<open>Stability of \<^const>\<open>sort_key\<close>:\<close> lemma sort_key_stable: "filter (\<lambda>y. f y = k) (sort_key f xs) = filter (\<lambda>y. f y = k) xs" by (induction xs) (auto simp: filter_insort insort_is_Cons filter_insort_triv) corollary stable_sort_key_sort_key: "stable_sort_key sort_key" by(simp add: stable_sort_key_def sort_key_stable) lemma sort_key_const: "sort_key (\<lambda>x. c) xs = xs" by (metis (mono_tags) filter_True sort_key_stable) subsubsection \<open>\<^const>\<open>transpose\<close> on sorted lists\<close> lemma sorted_transpose[simp]: "sorted (rev (map length (transpose xs)))" by (auto simp: sorted_iff_nth_mono rev_nth nth_transpose length_filter_conv_card intro: card_mono) lemma transpose_max_length: "foldr (\<lambda>xs. max (length xs)) (transpose xs) 0 = length (filter (\<lambda>x. x \<noteq> []) xs)" (is "?L = ?R") proof (cases "transpose xs = []") case False have "?L = foldr max (map length (transpose xs)) 0" by (simp add: foldr_map comp_def) also have "... = length (transpose xs ! 0)" using False sorted_transpose by (simp add: foldr_max_sorted) finally show ?thesis using False by (simp add: nth_transpose) next case True hence "filter (\<lambda>x. x \<noteq> []) xs = []" by (auto intro!: filter_False simp: transpose_empty) thus ?thesis by (simp add: transpose_empty True) qed lemma length_transpose_sorted: fixes xs :: "'a list list" assumes sorted: "sorted (rev (map length xs))" shows "length (transpose xs) = (if xs = [] then 0 else length (xs ! 0))" proof (cases "xs = []") case False thus ?thesis using foldr_max_sorted[OF sorted] False unfolding length_transpose foldr_map comp_def by simp qed simp lemma nth_nth_transpose_sorted[simp]: fixes xs :: "'a list list" assumes sorted: "sorted (rev (map length xs))" and i: "i < length (transpose xs)" and j: "j < length (filter (\<lambda>ys. i < length ys) xs)" shows "transpose xs ! i ! j = xs ! j ! i" using j filter_equals_takeWhile_sorted_rev[OF sorted, of i] nth_transpose[OF i] nth_map[OF j] by (simp add: takeWhile_nth) lemma transpose_column_length: fixes xs :: "'a list list" assumes sorted: "sorted (rev (map length xs))" and "i < length xs" shows "length (filter (\<lambda>ys. i < length ys) (transpose xs)) = length (xs ! i)" proof - have "xs \<noteq> []" using \<open>i < length xs\<close> by auto note filter_equals_takeWhile_sorted_rev[OF sorted, simp] { fix j assume "j \<le> i" note sorted_rev_nth_mono[OF sorted, of j i, simplified, OF this \<open>i < length xs\<close>] } note sortedE = this[consumes 1] have "{j. j < length (transpose xs) \<and> i < length (transpose xs ! j)} = {..< length (xs ! i)}" proof safe fix j assume "j < length (transpose xs)" and "i < length (transpose xs ! j)" with this(2) nth_transpose[OF this(1)] have "i < length (takeWhile (\<lambda>ys. j < length ys) xs)" by simp from nth_mem[OF this] takeWhile_nth[OF this] show "j < length (xs ! i)" by (auto dest: set_takeWhileD) next fix j assume "j < length (xs ! i)" thus "j < length (transpose xs)" using foldr_max_sorted[OF sorted] \<open>xs \<noteq> []\<close> sortedE[OF le0] by (auto simp: length_transpose comp_def foldr_map) have "Suc i \<le> length (takeWhile (\<lambda>ys. j < length ys) xs)" using \<open>i < length xs\<close> \<open>j < length (xs ! i)\<close> less_Suc_eq_le by (auto intro!: length_takeWhile_less_P_nth dest!: sortedE) with nth_transpose[OF \<open>j < length (transpose xs)\<close>] show "i < length (transpose xs ! j)" by simp qed thus ?thesis by (simp add: length_filter_conv_card) qed lemma transpose_column: fixes xs :: "'a list list" assumes sorted: "sorted (rev (map length xs))" and "i < length xs" shows "map (\<lambda>ys. ys ! i) (filter (\<lambda>ys. i < length ys) (transpose xs)) = xs ! i" (is "?R = _") proof (rule nth_equalityI) show length: "length ?R = length (xs ! i)" using transpose_column_length[OF assms] by simp fix j assume j: "j < length ?R" note * = less_le_trans[OF this, unfolded length_map, OF length_filter_le] from j have j_less: "j < length (xs ! i)" using length by simp have i_less_tW: "Suc i \<le> length (takeWhile (\<lambda>ys. Suc j \<le> length ys) xs)" proof (rule length_takeWhile_less_P_nth) show "Suc i \<le> length xs" using \<open>i < length xs\<close> by simp fix k assume "k < Suc i" hence "k \<le> i" by auto with sorted_rev_nth_mono[OF sorted this] \<open>i < length xs\<close> have "length (xs ! i) \<le> length (xs ! k)" by simp thus "Suc j \<le> length (xs ! k)" using j_less by simp qed have i_less_filter: "i < length (filter (\<lambda>ys. j < length ys) xs) " unfolding filter_equals_takeWhile_sorted_rev[OF sorted, of j] using i_less_tW by (simp_all add: Suc_le_eq) from j show "?R ! j = xs ! i ! j" unfolding filter_equals_takeWhile_sorted_rev[OF sorted_transpose, of i] by (simp add: takeWhile_nth nth_nth_transpose_sorted[OF sorted * i_less_filter]) qed lemma transpose_transpose: fixes xs :: "'a list list" assumes sorted: "sorted (rev (map length xs))" shows "transpose (transpose xs) = takeWhile (\<lambda>x. x \<noteq> []) xs" (is "?L = ?R") proof - have len: "length ?L = length ?R" unfolding length_transpose transpose_max_length using filter_equals_takeWhile_sorted_rev[OF sorted, of 0] by simp { fix i assume "i < length ?R" with less_le_trans[OF _ length_takeWhile_le[of _ xs]] have "i < length xs" by simp } note * = this show ?thesis by (rule nth_equalityI) (simp_all add: len nth_transpose transpose_column[OF sorted] * takeWhile_nth) qed theorem transpose_rectangle: assumes "xs = [] \<Longrightarrow> n = 0" assumes rect: "\<And> i. i < length xs \<Longrightarrow> length (xs ! i) = n" shows "transpose xs = map (\<lambda> i. map (\<lambda> j. xs ! j ! i) [0..<length xs]) [0..<n]" (is "?trans = ?map") proof (rule nth_equalityI) have "sorted (rev (map length xs))" by (auto simp: rev_nth rect sorted_iff_nth_mono) from foldr_max_sorted[OF this] assms show len: "length ?trans = length ?map" by (simp_all add: length_transpose foldr_map comp_def) moreover { fix i assume "i < n" hence "filter (\<lambda>ys. i < length ys) xs = xs" using rect by (auto simp: in_set_conv_nth intro!: filter_True) } ultimately show "\<And>i. i < length (transpose xs) \<Longrightarrow> ?trans ! i = ?map ! i" by (auto simp: nth_transpose intro: nth_equalityI) qed subsubsection \<open>\<open>sorted_list_of_set\<close>\<close> text\<open>This function maps (finite) linearly ordered sets to sorted lists. Warning: in most cases it is not a good idea to convert from sets to lists but one should convert in the other direction (via \<^const>\<open>set\<close>).\<close> context linorder begin definition sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where "sorted_list_of_set = folding.F insort []" sublocale sorted_list_of_set: folding insort Nil rewrites "folding.F insort [] = sorted_list_of_set" proof - interpret comp_fun_commute insort by (fact comp_fun_commute_insort) show "folding insort" by standard (fact comp_fun_commute) show "folding.F insort [] = sorted_list_of_set" by (simp only: sorted_list_of_set_def) qed lemma sorted_list_of_set_empty: "sorted_list_of_set {} = []" by (fact sorted_list_of_set.empty) lemma sorted_list_of_set_insert [simp]: "finite A \<Longrightarrow> sorted_list_of_set (insert x A) = insort x (sorted_list_of_set (A - {x}))" by (fact sorted_list_of_set.insert_remove) lemma sorted_list_of_set_eq_Nil_iff [simp]: "finite A \<Longrightarrow> sorted_list_of_set A = [] \<longleftrightarrow> A = {}" by (auto simp: sorted_list_of_set.remove) lemma set_sorted_list_of_set [simp]: "finite A \<Longrightarrow> set (sorted_list_of_set A) = A" by(induct A rule: finite_induct) (simp_all add: set_insort_key) lemma sorted_sorted_list_of_set [simp]: "sorted (sorted_list_of_set A)" proof (cases "finite A") case True thus ?thesis by(induction A) (simp_all add: sorted_insort) next case False thus ?thesis by simp qed lemma distinct_sorted_list_of_set [simp]: "distinct (sorted_list_of_set A)" proof (cases "finite A") case True thus ?thesis by(induction A) (simp_all add: distinct_insort) next case False thus ?thesis by simp qed lemma length_sorted_list_of_set [simp]: "length (sorted_list_of_set A) = card A" proof (cases "finite A") case True then show ?thesis by(metis distinct_card distinct_sorted_list_of_set set_sorted_list_of_set) qed auto lemmas sorted_list_of_set = set_sorted_list_of_set sorted_sorted_list_of_set distinct_sorted_list_of_set lemma sorted_list_of_set_sort_remdups [code]: "sorted_list_of_set (set xs) = sort (remdups xs)" proof - interpret comp_fun_commute insort by (fact comp_fun_commute_insort) show ?thesis by (simp add: sorted_list_of_set.eq_fold sort_conv_fold fold_set_fold_remdups) qed lemma sorted_list_of_set_remove: assumes "finite A" shows "sorted_list_of_set (A - {x}) = remove1 x (sorted_list_of_set A)" proof (cases "x \<in> A") case False with assms have "x \<notin> set (sorted_list_of_set A)" by simp with False show ?thesis by (simp add: remove1_idem) next case True then obtain B where A: "A = insert x B" by (rule Set.set_insert) with assms show ?thesis by simp qed lemma strict_sorted_list_of_set [simp]: "strict_sorted (sorted_list_of_set A)" by (simp add: strict_sorted_iff) lemma finite_set_strict_sorted: assumes "finite A" obtains l where "strict_sorted l" "set l = A" "length l = card A" by (metis assms distinct_card distinct_sorted_list_of_set set_sorted_list_of_set strict_sorted_list_of_set) lemma strict_sorted_equal: assumes "strict_sorted xs" and "strict_sorted ys" and "set ys = set xs" shows "ys = xs" using assms proof (induction xs arbitrary: ys) case (Cons x xs) show ?case proof (cases ys) case Nil then show ?thesis using Cons.prems by auto next case (Cons y ys') then have "xs = ys'" by (metis Cons.prems list.inject sorted_distinct_set_unique strict_sorted_iff) moreover have "x = y" using Cons.prems \<open>xs = ys'\<close> local.Cons by fastforce ultimately show ?thesis using local.Cons by blast qed qed auto lemma strict_sorted_equal_Uniq: "\<exists>\<^sub>\<le>\<^sub>1xs. strict_sorted xs \<and> set xs = A" by (simp add: Uniq_def strict_sorted_equal) lemma sorted_list_of_set_inject: assumes "sorted_list_of_set A = sorted_list_of_set B" "finite A" "finite B" shows "A = B" using assms set_sorted_list_of_set by fastforce lemma sorted_list_of_set_unique: assumes "finite A" shows "strict_sorted l \<and> set l = A \<and> length l = card A \<longleftrightarrow> sorted_list_of_set A = l" using assms strict_sorted_equal by force end lemma sorted_list_of_set_range [simp]: "sorted_list_of_set {m..<n} = [m..<n]" by (rule sorted_distinct_set_unique) simp_all lemma sorted_list_of_set_lessThan_Suc [simp]: "sorted_list_of_set {..<Suc k} = sorted_list_of_set {..<k} @ [k]" proof - have "sorted_wrt (<) (sorted_list_of_set {..<k} @ [k])" unfolding sorted_wrt_append by (auto simp flip: strict_sorted_sorted_wrt) then show ?thesis by (simp add: lessThan_atLeast0) qed lemma sorted_list_of_set_atMost_Suc [simp]: "sorted_list_of_set {..Suc k} = sorted_list_of_set {..k} @ [Suc k]" using lessThan_Suc_atMost sorted_list_of_set_lessThan_Suc by fastforce lemma sorted_list_of_set_nonempty: assumes "finite A" "A \<noteq> {}" shows "sorted_list_of_set A = Min A # sorted_list_of_set (A - {Min A})" using assms by (auto simp: less_le simp flip: sorted_list_of_set_unique intro: Min_in) lemma sorted_list_of_set_greaterThanLessThan: assumes "Suc i < j" shows "sorted_list_of_set {i<..<j} = Suc i # sorted_list_of_set {Suc i<..<j}" proof - have "{i<..<j} = insert (Suc i) {Suc i<..<j}" using assms by auto then show ?thesis by (metis assms atLeastSucLessThan_greaterThanLessThan sorted_list_of_set_range upt_conv_Cons) qed lemma sorted_list_of_set_greaterThanAtMost: assumes "Suc i \<le> j" shows "sorted_list_of_set {i<..j} = Suc i # sorted_list_of_set {Suc i<..j}" using sorted_list_of_set_greaterThanLessThan [of i "Suc j"] by (metis assms greaterThanAtMost_def greaterThanLessThan_eq le_imp_less_Suc lessThan_Suc_atMost) lemma nth_sorted_list_of_set_greaterThanLessThan: "n < j - Suc i \<Longrightarrow> sorted_list_of_set {i<..<j} ! n = Suc (i+n)" by (induction n arbitrary: i) (auto simp: sorted_list_of_set_greaterThanLessThan) lemma nth_sorted_list_of_set_greaterThanAtMost: "n < j - i \<Longrightarrow> sorted_list_of_set {i<..j} ! n = Suc (i+n)" using nth_sorted_list_of_set_greaterThanLessThan [of n "Suc j" i] by (simp add: greaterThanAtMost_def greaterThanLessThan_eq lessThan_Suc_atMost) subsubsection \<open>\<open>lists\<close>: the list-forming operator over sets\<close> inductive_set lists :: "'a set => 'a list set" for A :: "'a set" where Nil [intro!, simp]: "[] \<in> lists A" | Cons [intro!, simp]: "\<lbrakk>a \<in> A; l \<in> lists A\<rbrakk> \<Longrightarrow> a#l \<in> lists A" inductive_cases listsE [elim!]: "x#l \<in> lists A" inductive_cases listspE [elim!]: "listsp A (x # l)" inductive_simps listsp_simps[code]: "listsp A []" "listsp A (x # xs)" lemma listsp_mono [mono]: "A \<le> B \<Longrightarrow> listsp A \<le> listsp B" by (rule predicate1I, erule listsp.induct, blast+) lemmas lists_mono = listsp_mono [to_set] lemma listsp_infI: assumes l: "listsp A l" shows "listsp B l \<Longrightarrow> listsp (inf A B) l" using l by induct blast+ lemmas lists_IntI = listsp_infI [to_set] lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)" proof (rule mono_inf [where f=listsp, THEN order_antisym]) show "mono listsp" by (simp add: mono_def listsp_mono) show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro!: listsp_infI) qed lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_def inf_bool_def] lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set] lemma Cons_in_lists_iff[simp]: "x#xs \<in> lists A \<longleftrightarrow> x \<in> A \<and> xs \<in> lists A" by auto lemma append_in_listsp_conv [iff]: "(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)" by (induct xs) auto lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set] lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)" \<comment> \<open>eliminate \<open>listsp\<close> in favour of \<open>set\<close>\<close> by (induct xs) auto lemmas in_lists_conv_set [code_unfold] = in_listsp_conv_set [to_set] lemma in_listspD [dest!]: "listsp A xs \<Longrightarrow> \<forall>x\<in>set xs. A x" by (rule in_listsp_conv_set [THEN iffD1]) lemmas in_listsD [dest!] = in_listspD [to_set] lemma in_listspI [intro!]: "\<forall>x\<in>set xs. A x \<Longrightarrow> listsp A xs" by (rule in_listsp_conv_set [THEN iffD2]) lemmas in_listsI [intro!] = in_listspI [to_set] lemma lists_eq_set: "lists A = {xs. set xs \<le> A}" by auto lemma lists_empty [simp]: "lists {} = {[]}" by auto lemma lists_UNIV [simp]: "lists UNIV = UNIV" by auto lemma lists_image: "lists (f`A) = map f ` lists A" proof - { fix xs have "\<forall>x\<in>set xs. x \<in> f ` A \<Longrightarrow> xs \<in> map f ` lists A" by (induct xs) (auto simp del: list.map simp add: list.map[symmetric] intro!: imageI) } then show ?thesis by auto qed subsubsection \<open>Inductive definition for membership\<close> inductive ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool" where elem: "ListMem x (x # xs)" | insert: "ListMem x xs \<Longrightarrow> ListMem x (y # xs)" lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)" proof show "ListMem x xs \<Longrightarrow> x \<in> set xs" by (induct set: ListMem) auto show "x \<in> set xs \<Longrightarrow> ListMem x xs" by (induct xs) (auto intro: ListMem.intros) qed subsubsection \<open>Lists as Cartesian products\<close> text\<open>\<open>set_Cons A Xs\<close>: the set of lists with head drawn from \<^term>\<open>A\<close> and tail drawn from \<^term>\<open>Xs\<close>.\<close> definition set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set" where "set_Cons A XS = {z. \<exists>x xs. z = x # xs \<and> x \<in> A \<and> xs \<in> XS}" lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A" by (auto simp add: set_Cons_def) text\<open>Yields the set of lists, all of the same length as the argument and with elements drawn from the corresponding element of the argument.\<close> primrec listset :: "'a set list \<Rightarrow> 'a list set" where "listset [] = {[]}" | "listset (A # As) = set_Cons A (listset As)" subsection \<open>Relations on Lists\<close> subsubsection \<open>Length Lexicographic Ordering\<close> text\<open>These orderings preserve well-foundedness: shorter lists precede longer lists. These ordering are not used in dictionaries.\<close> primrec \<comment> \<open>The lexicographic ordering for lists of the specified length\<close> lexn :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> ('a list \<times> 'a list) set" where "lexn r 0 = {}" | "lexn r (Suc n) = (map_prod (%(x, xs). x#xs) (%(x, xs). x#xs) ` (r <*lex*> lexn r n)) Int {(xs, ys). length xs = Suc n \<and> length ys = Suc n}" definition lex :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where "lex r = (\<Union>n. lexn r n)" \<comment> \<open>Holds only between lists of the same length\<close> definition lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" where "lenlex r = inv_image (less_than <*lex*> lex r) (\<lambda>xs. (length xs, xs))" \<comment> \<open>Compares lists by their length and then lexicographically\<close> lemma wf_lexn: assumes "wf r" shows "wf (lexn r n)" proof (induct n) case (Suc n) have inj: "inj (\<lambda>(x, xs). x # xs)" using assms by (auto simp: inj_on_def) have wf: "wf (map_prod (\<lambda>(x, xs). x # xs) (\<lambda>(x, xs). x # xs) ` (r <*lex*> lexn r n))" by (simp add: Suc.hyps assms wf_lex_prod wf_map_prod_image [OF _ inj]) then show ?case by (rule wf_subset) auto qed auto lemma lexn_length: "(xs, ys) \<in> lexn r n \<Longrightarrow> length xs = n \<and> length ys = n" by (induct n arbitrary: xs ys) auto lemma wf_lex [intro!]: assumes "wf r" shows "wf (lex r)" unfolding lex_def proof (rule wf_UN) show "wf (lexn r i)" for i by (simp add: assms wf_lexn) show "\<And>i j. lexn r i \<noteq> lexn r j \<Longrightarrow> Domain (lexn r i) \<inter> Range (lexn r j) = {}" by (metis DomainE Int_emptyI RangeE lexn_length) qed lemma lexn_conv: "lexn r n = {(xs,ys). length xs = n \<and> length ys = n \<and> (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y) \<in> r)}" proof (induction n) case (Suc n) then show ?case apply (simp add: image_Collect lex_prod_def, safe, blast) apply (rule_tac x = "ab # xys" in exI, simp) apply (case_tac xys; force) done qed auto text\<open>By Mathias Fleury:\<close> proposition lexn_transI: assumes "trans r" shows "trans (lexn r n)" unfolding trans_def proof (intro allI impI) fix as bs cs assume asbs: "(as, bs) \<in> lexn r n" and bscs: "(bs, cs) \<in> lexn r n" obtain abs a b as' bs' where n: "length as = n" and "length bs = n" and as: "as = abs @ a # as'" and bs: "bs = abs @ b # bs'" and abr: "(a, b) \<in> r" using asbs unfolding lexn_conv by blast obtain bcs b' c' cs' bs' where n': "length cs = n" and "length bs = n" and bs': "bs = bcs @ b' # bs'" and cs: "cs = bcs @ c' # cs'" and b'c'r: "(b', c') \<in> r" using bscs unfolding lexn_conv by blast consider (le) "length bcs < length abs" | (eq) "length bcs = length abs" | (ge) "length bcs > length abs" by linarith thus "(as, cs) \<in> lexn r n" proof cases let ?k = "length bcs" case le hence "as ! ?k = bs ! ?k" unfolding as bs by (simp add: nth_append) hence "(as ! ?k, cs ! ?k) \<in> r" using b'c'r unfolding bs' cs by auto moreover have "length bcs < length as" using le unfolding as by simp from id_take_nth_drop[OF this] have "as = take ?k as @ as ! ?k # drop (Suc ?k) as" . moreover have "length bcs < length cs" unfolding cs by simp from id_take_nth_drop[OF this] have "cs = take ?k cs @ cs ! ?k # drop (Suc ?k) cs" . moreover have "take ?k as = take ?k cs" using le arg_cong[OF bs, of "take (length bcs)"] unfolding cs as bs' by auto ultimately show ?thesis using n n' unfolding lexn_conv by auto next let ?k = "length abs" case ge hence "bs ! ?k = cs ! ?k" unfolding bs' cs by (simp add: nth_append) hence "(as ! ?k, cs ! ?k) \<in> r" using abr unfolding as bs by auto moreover have "length abs < length as" using ge unfolding as by simp from id_take_nth_drop[OF this] have "as = take ?k as @ as ! ?k # drop (Suc ?k) as" . moreover have "length abs < length cs" using n n' unfolding as by simp from id_take_nth_drop[OF this] have "cs = take ?k cs @ cs ! ?k # drop (Suc ?k) cs" . moreover have "take ?k as = take ?k cs" using ge arg_cong[OF bs', of "take (length abs)"] unfolding cs as bs by auto ultimately show ?thesis using n n' unfolding lexn_conv by auto next let ?k = "length abs" case eq hence *: "abs = bcs" "b = b'" using bs bs' by auto hence "(a, c') \<in> r" using abr b'c'r assms unfolding trans_def by blast with * show ?thesis using n n' unfolding lexn_conv as bs cs by auto qed qed corollary lex_transI: assumes "trans r" shows "trans (lex r)" using lexn_transI [OF assms] by (clarsimp simp add: lex_def trans_def) (metis lexn_length) lemma lex_conv: "lex r = {(xs,ys). length xs = length ys \<and> (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y) \<in> r)}" by (force simp add: lex_def lexn_conv) lemma wf_lenlex [intro!]: "wf r \<Longrightarrow> wf (lenlex r)" by (unfold lenlex_def) blast lemma lenlex_conv: "lenlex r = {(xs,ys). length xs < length ys \<or> length xs = length ys \<and> (xs, ys) \<in> lex r}" by (auto simp add: lenlex_def Id_on_def lex_prod_def inv_image_def) lemma total_lenlex: assumes "total r" shows "total (lenlex r)" proof - have "(xs,ys) \<in> lexn r (length xs) \<or> (ys,xs) \<in> lexn r (length xs)" if "xs \<noteq> ys" and len: "length xs = length ys" for xs ys proof - obtain pre x xs' y ys' where "x\<noteq>y" and xs: "xs = pre @ [x] @ xs'" and ys: "ys = pre @ [y] @ys'" by (meson len \<open>xs \<noteq> ys\<close> same_length_different) then consider "(x,y) \<in> r" | "(y,x) \<in> r" by (meson UNIV_I assms total_on_def) then show ?thesis by cases (use len in \<open>(force simp add: lexn_conv xs ys)+\<close>) qed then show ?thesis by (fastforce simp: lenlex_def total_on_def lex_def) qed lemma lenlex_transI [intro]: "trans r \<Longrightarrow> trans (lenlex r)" unfolding lenlex_def by (meson lex_transI trans_inv_image trans_less_than trans_lex_prod) lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r" by (simp add: lex_conv) lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r" by (simp add:lex_conv) lemma Cons_in_lex [simp]: "(x # xs, y # ys) \<in> lex r \<longleftrightarrow> (x, y) \<in> r \<and> length xs = length ys \<or> x = y \<and> (xs, ys) \<in> lex r" (is "?lhs = ?rhs") proof assume ?lhs then show ?rhs by (simp add: lex_conv) (metis hd_append list.sel(1) list.sel(3) tl_append2) next assume ?rhs then show ?lhs by (simp add: lex_conv) (blast intro: Cons_eq_appendI) qed lemma Nil_lenlex_iff1 [simp]: "([], ns) \<in> lenlex r \<longleftrightarrow> ns \<noteq> []" and Nil_lenlex_iff2 [simp]: "(ns,[]) \<notin> lenlex r" by (auto simp: lenlex_def) lemma Cons_lenlex_iff: "((m # ms, n # ns) \<in> lenlex r) \<longleftrightarrow> length ms < length ns \<or> length ms = length ns \<and> (m,n) \<in> r \<or> (m = n \<and> (ms,ns) \<in> lenlex r)" by (auto simp: lenlex_def) lemma lenlex_irreflexive: "(\<And>x. (x,x) \<notin> r) \<Longrightarrow> (xs,xs) \<notin> lenlex r" by (induction xs) (auto simp add: Cons_lenlex_iff) lemma lenlex_trans: "\<lbrakk>(x,y) \<in> lenlex r; (y,z) \<in> lenlex r; trans r\<rbrakk> \<Longrightarrow> (x,z) \<in> lenlex r" by (meson lenlex_transI transD) lemma lenlex_length: "(ms, ns) \<in> lenlex r \<Longrightarrow> length ms \<le> length ns" by (auto simp: lenlex_def) lemma lex_append_rightI: "(xs, ys) \<in> lex r \<Longrightarrow> length vs = length us \<Longrightarrow> (xs @ us, ys @ vs) \<in> lex r" by (fastforce simp: lex_def lexn_conv) lemma lex_append_leftI: "(ys, zs) \<in> lex r \<Longrightarrow> (xs @ ys, xs @ zs) \<in> lex r" by (induct xs) auto lemma lex_append_leftD: "\<forall>x. (x,x) \<notin> r \<Longrightarrow> (xs @ ys, xs @ zs) \<in> lex r \<Longrightarrow> (ys, zs) \<in> lex r" by (induct xs) auto lemma lex_append_left_iff: "\<forall>x. (x,x) \<notin> r \<Longrightarrow> (xs @ ys, xs @ zs) \<in> lex r \<longleftrightarrow> (ys, zs) \<in> lex r" by(metis lex_append_leftD lex_append_leftI) lemma lex_take_index: assumes "(xs, ys) \<in> lex r" obtains i where "i < length xs" and "i < length ys" and "take i xs = take i ys" and "(xs ! i, ys ! i) \<in> r" proof - obtain n us x xs' y ys' where "(xs, ys) \<in> lexn r n" and "length xs = n" and "length ys = n" and "xs = us @ x # xs'" and "ys = us @ y # ys'" and "(x, y) \<in> r" using assms by (fastforce simp: lex_def lexn_conv) then show ?thesis by (intro that [of "length us"]) auto qed lemma irrefl_lex: "irrefl r \<Longrightarrow> irrefl (lex r)" by (meson irrefl_def lex_take_index) lemma lexl_not_refl [simp]: "irrefl r \<Longrightarrow> (x,x) \<notin> lex r" by (meson irrefl_def lex_take_index) subsubsection \<open>Lexicographic Ordering\<close> text \<open>Classical lexicographic ordering on lists, ie. "a" < "ab" < "b". This ordering does \emph{not} preserve well-foundedness. Author: N. Voelker, March 2005.\<close> definition lexord :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where "lexord r = {(x,y). \<exists> a v. y = x @ a # v \<or> (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}" lemma lexord_Nil_left[simp]: "([],y) \<in> lexord r = (\<exists> a x. y = a # x)" by (unfold lexord_def, induct_tac y, auto) lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r" by (unfold lexord_def, induct_tac x, auto) lemma lexord_cons_cons[simp]: "(a # x, b # y) \<in> lexord r \<longleftrightarrow> (a,b)\<in> r \<or> (a = b \<and> (x,y)\<in> lexord r)" (is "?lhs = ?rhs") proof assume ?lhs then show ?rhs apply (simp add: lexord_def) apply (metis hd_append list.sel(1) list.sel(3) tl_append2) done qed (auto simp add: lexord_def; (blast | meson Cons_eq_appendI)) lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons lemma lexord_same_pref_iff: "(xs @ ys, xs @ zs) \<in> lexord r \<longleftrightarrow> (\<exists>x \<in> set xs. (x,x) \<in> r) \<or> (ys, zs) \<in> lexord r" by(induction xs) auto lemma lexord_same_pref_if_irrefl[simp]: "irrefl r \<Longrightarrow> (xs @ ys, xs @ zs) \<in> lexord r \<longleftrightarrow> (ys, zs) \<in> lexord r" by (simp add: irrefl_def lexord_same_pref_iff) lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r" by (metis append_Nil2 lexord_Nil_left lexord_same_pref_iff) lemma lexord_append_left_rightI: "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r" by (simp add: lexord_same_pref_iff) lemma lexord_append_leftI: "(u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r" by (simp add: lexord_same_pref_iff) lemma lexord_append_leftD: "\<lbrakk>(x @ u, x @ v) \<in> lexord r; (\<forall>a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r" by (simp add: lexord_same_pref_iff) lemma lexord_take_index_conv: "((x,y) \<in> lexord r) = ((length x < length y \<and> take (length x) y = x) \<or> (\<exists>i. i < min(length x)(length y) \<and> take i x = take i y \<and> (x!i,y!i) \<in> r))" proof - have "(\<exists>a v. y = x @ a # v) = (length x < length y \<and> take (length x) y = x)" by (metis Cons_nth_drop_Suc append_eq_conv_conj drop_all list.simps(3) not_le) moreover have "(\<exists>u a b. (a, b) \<in> r \<and> (\<exists>v. x = u @ a # v) \<and> (\<exists>w. y = u @ b # w)) = (\<exists>i<length x. i < length y \<and> take i x = take i y \<and> (x ! i, y ! i) \<in> r)" apply safe using less_iff_Suc_add apply auto[1] by (metis id_take_nth_drop) ultimately show ?thesis by (auto simp: lexord_def Let_def) qed \<comment> \<open>lexord is extension of partial ordering List.lex\<close> lemma lexord_lex: "(x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)" proof (induction x arbitrary: y) case (Cons a x y) then show ?case by (cases y) (force+) qed auto lemma lexord_sufI: assumes "(u,w) \<in> lexord r" "length w \<le> length u" shows "(u@v,w@z) \<in> lexord r" proof- from leD[OF assms(2)] assms(1)[unfolded lexord_take_index_conv[of u w r] min_absorb2[OF assms(2)]] obtain i where "take i u = take i w" and "(u!i,w!i) \<in> r" and "i < length w" by blast hence "((u@v)!i, (w@z)!i) \<in> r" unfolding nth_append using less_le_trans[OF \<open>i < length w\<close> assms(2)] \<open>(u!i,w!i) \<in> r\<close> by presburger moreover have "i < min (length (u@v)) (length (w@z))" using assms(2) \<open>i < length w\<close> by simp moreover have "take i (u@v) = take i (w@z)" using assms(2) \<open>i < length w\<close> \<open>take i u = take i w\<close> by simp ultimately show ?thesis using lexord_take_index_conv by blast qed lemma lexord_sufE: assumes "(xs@zs,ys@qs) \<in> lexord r" "xs \<noteq> ys" "length xs = length ys" "length zs = length qs" shows "(xs,ys) \<in> lexord r" proof- obtain i where "i < length (xs@zs)" and "i < length (ys@qs)" and "take i (xs@zs) = take i (ys@qs)" and "((xs@zs) ! i, (ys@qs) ! i) \<in> r" using assms(1) lex_take_index[unfolded lexord_lex,of "xs @ zs" "ys @ qs" r] length_append[of xs zs, unfolded assms(3,4), folded length_append[of ys qs]] by blast have "length (take i xs) = length (take i ys)" by (simp add: assms(3)) have "i < length xs" using assms(2,3) le_less_linear take_all[of xs i] take_all[of ys i] \<open>take i (xs @ zs) = take i (ys @ qs)\<close> append_eq_append_conv take_append by metis hence "(xs ! i, ys ! i) \<in> r" using \<open>((xs @ zs) ! i, (ys @ qs) ! i) \<in> r\<close> assms(3) by (simp add: nth_append) moreover have "take i xs = take i ys" using assms(3) \<open>take i (xs @ zs) = take i (ys @ qs)\<close> by auto ultimately show ?thesis unfolding lexord_take_index_conv using \<open>i < length xs\<close> assms(3) by fastforce qed lemma lexord_irreflexive: "\<forall>x. (x,x) \<notin> r \<Longrightarrow> (xs,xs) \<notin> lexord r" by (induct xs) auto text\<open>By Ren\'e Thiemann:\<close> lemma lexord_partial_trans: "(\<And>x y z. x \<in> set xs \<Longrightarrow> (x,y) \<in> r \<Longrightarrow> (y,z) \<in> r \<Longrightarrow> (x,z) \<in> r) \<Longrightarrow> (xs,ys) \<in> lexord r \<Longrightarrow> (ys,zs) \<in> lexord r \<Longrightarrow> (xs,zs) \<in> lexord r" proof (induct xs arbitrary: ys zs) case Nil from Nil(3) show ?case unfolding lexord_def by (cases zs, auto) next case (Cons x xs yys zzs) from Cons(3) obtain y ys where yys: "yys = y # ys" unfolding lexord_def by (cases yys, auto) note Cons = Cons[unfolded yys] from Cons(3) have one: "(x,y) \<in> r \<or> x = y \<and> (xs,ys) \<in> lexord r" by auto from Cons(4) obtain z zs where zzs: "zzs = z # zs" unfolding lexord_def by (cases zzs, auto) note Cons = Cons[unfolded zzs] from Cons(4) have two: "(y,z) \<in> r \<or> y = z \<and> (ys,zs) \<in> lexord r" by auto { assume "(xs,ys) \<in> lexord r" and "(ys,zs) \<in> lexord r" from Cons(1)[OF _ this] Cons(2) have "(xs,zs) \<in> lexord r" by auto } note ind1 = this { assume "(x,y) \<in> r" and "(y,z) \<in> r" from Cons(2)[OF _ this] have "(x,z) \<in> r" by auto } note ind2 = this from one two ind1 ind2 have "(x,z) \<in> r \<or> x = z \<and> (xs,zs) \<in> lexord r" by blast thus ?case unfolding zzs by auto qed lemma lexord_trans: "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r" by(auto simp: trans_def intro:lexord_partial_trans) lemma lexord_transI: "trans r \<Longrightarrow> trans (lexord r)" by (meson lexord_trans transI) lemma total_lexord: "total r \<Longrightarrow> total (lexord r)" unfolding total_on_def proof clarsimp fix x y assume "\<forall>x y. x \<noteq> y \<longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r" and "(x::'a list) \<noteq> y" and "(y, x) \<notin> lexord r" then show "(x, y) \<in> lexord r" proof (induction x arbitrary: y) case Nil then show ?case by (metis lexord_Nil_left list.exhaust) next case (Cons a x y) then show ?case by (cases y) (force+) qed qed corollary lexord_linear: "(\<forall>a b. (a,b) \<in> r \<or> a = b \<or> (b,a) \<in> r) \<Longrightarrow> (x,y) \<in> lexord r \<or> x = y \<or> (y,x) \<in> lexord r" using total_lexord by (metis UNIV_I total_on_def) lemma lexord_irrefl: "irrefl R \<Longrightarrow> irrefl (lexord R)" by (simp add: irrefl_def lexord_irreflexive) lemma lexord_asym: assumes "asym R" shows "asym (lexord R)" proof fix xs ys assume "(xs, ys) \<in> lexord R" then show "(ys, xs) \<notin> lexord R" proof (induct xs arbitrary: ys) case Nil then show ?case by simp next case (Cons x xs) then obtain z zs where ys: "ys = z # zs" by (cases ys) auto with assms Cons show ?case by (auto elim: asym.cases) qed qed lemma lexord_asymmetric: assumes "asym R" assumes hyp: "(a, b) \<in> lexord R" shows "(b, a) \<notin> lexord R" proof - from \<open>asym R\<close> have "asym (lexord R)" by (rule lexord_asym) then show ?thesis by (rule asym.cases) (auto simp add: hyp) qed lemma asym_lex: "asym R \<Longrightarrow> asym (lex R)" by (meson asym.simps irrefl_lex lexord_asym lexord_lex) lemma asym_lenlex: "asym R \<Longrightarrow> asym (lenlex R)" by (simp add: lenlex_def asym_inv_image asym_less_than asym_lex asym_lex_prod) lemma lenlex_append1: assumes len: "(us,xs) \<in> lenlex R" and eq: "length vs = length ys" shows "(us @ vs, xs @ ys) \<in> lenlex R" using len proof (induction us) case Nil then show ?case by (simp add: lenlex_def eq) next case (Cons u us) with lex_append_rightI show ?case by (fastforce simp add: lenlex_def eq) qed lemma lenlex_append2 [simp]: assumes "irrefl R" shows "(us @ xs, us @ ys) \<in> lenlex R \<longleftrightarrow> (xs, ys) \<in> lenlex R" proof (induction us) case Nil then show ?case by (simp add: lenlex_def) next case (Cons u us) with assms show ?case by (auto simp: lenlex_def irrefl_def) qed text \<open> Predicate version of lexicographic order integrated with Isabelle's order type classes. Author: Andreas Lochbihler \<close> context ord begin context notes [[inductive_internals]] begin inductive lexordp :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where Nil: "lexordp [] (y # ys)" | Cons: "x < y \<Longrightarrow> lexordp (x # xs) (y # ys)" | Cons_eq: "\<lbrakk> \<not> x < y; \<not> y < x; lexordp xs ys \<rbrakk> \<Longrightarrow> lexordp (x # xs) (y # ys)" end lemma lexordp_simps [simp]: "lexordp [] ys = (ys \<noteq> [])" "lexordp xs [] = False" "lexordp (x # xs) (y # ys) \<longleftrightarrow> x < y \<or> \<not> y < x \<and> lexordp xs ys" by(subst lexordp.simps, fastforce simp add: neq_Nil_conv)+ inductive lexordp_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where Nil: "lexordp_eq [] ys" | Cons: "x < y \<Longrightarrow> lexordp_eq (x # xs) (y # ys)" | Cons_eq: "\<lbrakk> \<not> x < y; \<not> y < x; lexordp_eq xs ys \<rbrakk> \<Longrightarrow> lexordp_eq (x # xs) (y # ys)" lemma lexordp_eq_simps [simp]: "lexordp_eq [] ys = True" "lexordp_eq xs [] \<longleftrightarrow> xs = []" "lexordp_eq (x # xs) [] = False" "lexordp_eq (x # xs) (y # ys) \<longleftrightarrow> x < y \<or> \<not> y < x \<and> lexordp_eq xs ys" by(subst lexordp_eq.simps, fastforce)+ lemma lexordp_append_rightI: "ys \<noteq> Nil \<Longrightarrow> lexordp xs (xs @ ys)" by(induct xs)(auto simp add: neq_Nil_conv) lemma lexordp_append_left_rightI: "x < y \<Longrightarrow> lexordp (us @ x # xs) (us @ y # ys)" by(induct us) auto lemma lexordp_eq_refl: "lexordp_eq xs xs" by(induct xs) simp_all lemma lexordp_append_leftI: "lexordp us vs \<Longrightarrow> lexordp (xs @ us) (xs @ vs)" by(induct xs) auto lemma lexordp_append_leftD: "\<lbrakk> lexordp (xs @ us) (xs @ vs); \<forall>a. \<not> a < a \<rbrakk> \<Longrightarrow> lexordp us vs" by(induct xs) auto lemma lexordp_irreflexive: assumes irrefl: "\<forall>x. \<not> x < x" shows "\<not> lexordp xs xs" proof assume "lexordp xs xs" thus False by(induct xs ys\<equiv>xs)(simp_all add: irrefl) qed lemma lexordp_into_lexordp_eq: "lexordp xs ys \<Longrightarrow> lexordp_eq xs ys" by (induction rule: lexordp.induct) simp_all lemma lexordp_eq_pref: "lexordp_eq u (u @ v)" by (metis append_Nil2 lexordp_append_rightI lexordp_eq_refl lexordp_into_lexordp_eq) end declare ord.lexordp_simps [simp, code] declare ord.lexordp_eq_simps [code, simp] context order begin lemma lexordp_antisym: assumes "lexordp xs ys" "lexordp ys xs" shows False using assms by induct auto lemma lexordp_irreflexive': "\<not> lexordp xs xs" by(rule lexordp_irreflexive) simp end context linorder begin lemma lexordp_cases [consumes 1, case_names Nil Cons Cons_eq, cases pred: lexordp]: assumes "lexordp xs ys" obtains (Nil) y ys' where "xs = []" "ys = y # ys'" | (Cons) x xs' y ys' where "xs = x # xs'" "ys = y # ys'" "x < y" | (Cons_eq) x xs' ys' where "xs = x # xs'" "ys = x # ys'" "lexordp xs' ys'" using assms by cases (fastforce simp add: not_less_iff_gr_or_eq)+ lemma lexordp_induct [consumes 1, case_names Nil Cons Cons_eq, induct pred: lexordp]: assumes major: "lexordp xs ys" and Nil: "\<And>y ys. P [] (y # ys)" and Cons: "\<And>x xs y ys. x < y \<Longrightarrow> P (x # xs) (y # ys)" and Cons_eq: "\<And>x xs ys. \<lbrakk> lexordp xs ys; P xs ys \<rbrakk> \<Longrightarrow> P (x # xs) (x # ys)" shows "P xs ys" using major by induct (simp_all add: Nil Cons not_less_iff_gr_or_eq Cons_eq) lemma lexordp_iff: "lexordp xs ys \<longleftrightarrow> (\<exists>x vs. ys = xs @ x # vs) \<or> (\<exists>us a b vs ws. a < b \<and> xs = us @ a # vs \<and> ys = us @ b # ws)" (is "?lhs = ?rhs") proof assume ?lhs thus ?rhs proof induct case Cons_eq thus ?case by simp (metis append.simps(2)) qed(fastforce intro: disjI2 del: disjCI intro: exI[where x="[]"])+ next assume ?rhs thus ?lhs by(auto intro: lexordp_append_leftI[where us="[]", simplified] lexordp_append_leftI) qed lemma lexordp_conv_lexord: "lexordp xs ys \<longleftrightarrow> (xs, ys) \<in> lexord {(x, y). x < y}" by(simp add: lexordp_iff lexord_def) lemma lexordp_eq_antisym: assumes "lexordp_eq xs ys" "lexordp_eq ys xs" shows "xs = ys" using assms by induct simp_all lemma lexordp_eq_trans: assumes "lexordp_eq xs ys" and "lexordp_eq ys zs" shows "lexordp_eq xs zs" using assms by (induct arbitrary: zs) (case_tac zs; auto)+ lemma lexordp_trans: assumes "lexordp xs ys" "lexordp ys zs" shows "lexordp xs zs" using assms by (induct arbitrary: zs) (case_tac zs; auto)+ lemma lexordp_linear: "lexordp xs ys \<or> xs = ys \<or> lexordp ys xs" by(induct xs arbitrary: ys; case_tac ys; fastforce) lemma lexordp_conv_lexordp_eq: "lexordp xs ys \<longleftrightarrow> lexordp_eq xs ys \<and> \<not> lexordp_eq ys xs" (is "?lhs \<longleftrightarrow> ?rhs") proof assume ?lhs hence "\<not> lexordp_eq ys xs" by induct simp_all with \<open>?lhs\<close> show ?rhs by (simp add: lexordp_into_lexordp_eq) next assume ?rhs hence "lexordp_eq xs ys" "\<not> lexordp_eq ys xs" by simp_all thus ?lhs by induct simp_all qed lemma lexordp_eq_conv_lexord: "lexordp_eq xs ys \<longleftrightarrow> xs = ys \<or> lexordp xs ys" by(auto simp add: lexordp_conv_lexordp_eq lexordp_eq_refl dest: lexordp_eq_antisym) lemma lexordp_eq_linear: "lexordp_eq xs ys \<or> lexordp_eq ys xs" by (induct xs arbitrary: ys) (case_tac ys; auto)+ lemma lexordp_linorder: "class.linorder lexordp_eq lexordp" by unfold_locales (auto simp add: lexordp_conv_lexordp_eq lexordp_eq_refl lexordp_eq_antisym intro: lexordp_eq_trans del: disjCI intro: lexordp_eq_linear) end lemma sorted_insort_is_snoc: "sorted xs \<Longrightarrow> \<forall>x \<in> set xs. a \<ge> x \<Longrightarrow> insort a xs = xs @ [a]" by (induct xs) (auto dest!: insort_is_Cons) subsubsection \<open>Lexicographic combination of measure functions\<close> text \<open>These are useful for termination proofs\<close> definition "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)" lemma wf_measures[simp]: "wf (measures fs)" unfolding measures_def by blast lemma in_measures[simp]: "(x, y) \<in> measures [] = False" "(x, y) \<in> measures (f # fs) = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))" unfolding measures_def by auto lemma measures_less: "f x < f y \<Longrightarrow> (x, y) \<in> measures (f#fs)" by simp lemma measures_lesseq: "f x \<le> f y \<Longrightarrow> (x, y) \<in> measures fs \<Longrightarrow> (x, y) \<in> measures (f#fs)" by auto subsubsection \<open>Lifting Relations to Lists: one element\<close> definition listrel1 :: "('a \<times> 'a) set \<Rightarrow> ('a list \<times> 'a list) set" where "listrel1 r = {(xs,ys). \<exists>us z z' vs. xs = us @ z # vs \<and> (z,z') \<in> r \<and> ys = us @ z' # vs}" lemma listrel1I: "\<lbrakk> (x, y) \<in> r; xs = us @ x # vs; ys = us @ y # vs \<rbrakk> \<Longrightarrow> (xs, ys) \<in> listrel1 r" unfolding listrel1_def by auto lemma listrel1E: "\<lbrakk> (xs, ys) \<in> listrel1 r; !!x y us vs. \<lbrakk> (x, y) \<in> r; xs = us @ x # vs; ys = us @ y # vs \<rbrakk> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P" unfolding listrel1_def by auto lemma not_Nil_listrel1 [iff]: "([], xs) \<notin> listrel1 r" unfolding listrel1_def by blast lemma not_listrel1_Nil [iff]: "(xs, []) \<notin> listrel1 r" unfolding listrel1_def by blast lemma Cons_listrel1_Cons [iff]: "(x # xs, y # ys) \<in> listrel1 r \<longleftrightarrow> (x,y) \<in> r \<and> xs = ys \<or> x = y \<and> (xs, ys) \<in> listrel1 r" by (simp add: listrel1_def Cons_eq_append_conv) (blast) lemma listrel1I1: "(x,y) \<in> r \<Longrightarrow> (x # xs, y # xs) \<in> listrel1 r" by fast lemma listrel1I2: "(xs, ys) \<in> listrel1 r \<Longrightarrow> (x # xs, x # ys) \<in> listrel1 r" by fast lemma append_listrel1I: "(xs, ys) \<in> listrel1 r \<and> us = vs \<or> xs = ys \<and> (us, vs) \<in> listrel1 r \<Longrightarrow> (xs @ us, ys @ vs) \<in> listrel1 r" unfolding listrel1_def by auto (blast intro: append_eq_appendI)+ lemma Cons_listrel1E1[elim!]: assumes "(x # xs, ys) \<in> listrel1 r" and "\<And>y. ys = y # xs \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R" and "\<And>zs. ys = x # zs \<Longrightarrow> (xs, zs) \<in> listrel1 r \<Longrightarrow> R" shows R using assms by (cases ys) blast+ lemma Cons_listrel1E2[elim!]: assumes "(xs, y # ys) \<in> listrel1 r" and "\<And>x. xs = x # ys \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> R" and "\<And>zs. xs = y # zs \<Longrightarrow> (zs, ys) \<in> listrel1 r \<Longrightarrow> R" shows R using assms by (cases xs) blast+ lemma snoc_listrel1_snoc_iff: "(xs @ [x], ys @ [y]) \<in> listrel1 r \<longleftrightarrow> (xs, ys) \<in> listrel1 r \<and> x = y \<or> xs = ys \<and> (x,y) \<in> r" (is "?L \<longleftrightarrow> ?R") proof assume ?L thus ?R by (fastforce simp: listrel1_def snoc_eq_iff_butlast butlast_append) next assume ?R then show ?L unfolding listrel1_def by force qed lemma listrel1_eq_len: "(xs,ys) \<in> listrel1 r \<Longrightarrow> length xs = length ys" unfolding listrel1_def by auto lemma listrel1_mono: "r \<subseteq> s \<Longrightarrow> listrel1 r \<subseteq> listrel1 s" unfolding listrel1_def by blast lemma listrel1_converse: "listrel1 (r\<inverse>) = (listrel1 r)\<inverse>" unfolding listrel1_def by blast lemma in_listrel1_converse: "(x,y) \<in> listrel1 (r\<inverse>) \<longleftrightarrow> (x,y) \<in> (listrel1 r)\<inverse>" unfolding listrel1_def by blast lemma listrel1_iff_update: "(xs,ys) \<in> (listrel1 r) \<longleftrightarrow> (\<exists>y n. (xs ! n, y) \<in> r \<and> n < length xs \<and> ys = xs[n:=y])" (is "?L \<longleftrightarrow> ?R") proof assume "?L" then obtain x y u v where "xs = u @ x # v" "ys = u @ y # v" "(x,y) \<in> r" unfolding listrel1_def by auto then have "ys = xs[length u := y]" and "length u < length xs" and "(xs ! length u, y) \<in> r" by auto then show "?R" by auto next assume "?R" then obtain x y n where "(xs!n, y) \<in> r" "n < size xs" "ys = xs[n:=y]" "x = xs!n" by auto then obtain u v where "xs = u @ x # v" and "ys = u @ y # v" and "(x, y) \<in> r" by (auto intro: upd_conv_take_nth_drop id_take_nth_drop) then show "?L" by (auto simp: listrel1_def) qed text\<open>Accessible part and wellfoundedness:\<close> lemma Cons_acc_listrel1I [intro!]: "x \<in> Wellfounded.acc r \<Longrightarrow> xs \<in> Wellfounded.acc (listrel1 r) \<Longrightarrow> (x # xs) \<in> Wellfounded.acc (listrel1 r)" apply (induct arbitrary: xs set: Wellfounded.acc) apply (erule thin_rl) apply (erule acc_induct) apply (rule accI) apply (blast) done lemma lists_accD: "xs \<in> lists (Wellfounded.acc r) \<Longrightarrow> xs \<in> Wellfounded.acc (listrel1 r)" proof (induct set: lists) case Nil then show ?case by (meson acc.intros not_listrel1_Nil) next case (Cons a l) then show ?case by blast qed lemma lists_accI: "xs \<in> Wellfounded.acc (listrel1 r) \<Longrightarrow> xs \<in> lists (Wellfounded.acc r)" apply (induct set: Wellfounded.acc) apply clarify apply (rule accI) apply (fastforce dest!: in_set_conv_decomp[THEN iffD1] simp: listrel1_def) done lemma wf_listrel1_iff[simp]: "wf(listrel1 r) = wf r" by (auto simp: wf_acc_iff intro: lists_accD lists_accI[THEN Cons_in_lists_iff[THEN iffD1, THEN conjunct1]]) subsubsection \<open>Lifting Relations to Lists: all elements\<close> inductive_set listrel :: "('a \<times> 'b) set \<Rightarrow> ('a list \<times> 'b list) set" for r :: "('a \<times> 'b) set" where Nil: "([],[]) \<in> listrel r" | Cons: "\<lbrakk>(x,y) \<in> r; (xs,ys) \<in> listrel r\<rbrakk> \<Longrightarrow> (x#xs, y#ys) \<in> listrel r" inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r" inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r" inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r" inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r" lemma listrel_eq_len: "(xs, ys) \<in> listrel r \<Longrightarrow> length xs = length ys" by(induct rule: listrel.induct) auto lemma listrel_iff_zip [code_unfold]: "(xs,ys) \<in> listrel r \<longleftrightarrow> length xs = length ys \<and> (\<forall>(x,y) \<in> set(zip xs ys). (x,y) \<in> r)" (is "?L \<longleftrightarrow> ?R") proof assume ?L thus ?R by induct (auto intro: listrel_eq_len) next assume ?R thus ?L apply (clarify) by (induct rule: list_induct2) (auto intro: listrel.intros) qed lemma listrel_iff_nth: "(xs,ys) \<in> listrel r \<longleftrightarrow> length xs = length ys \<and> (\<forall>n < length xs. (xs!n, ys!n) \<in> r)" (is "?L \<longleftrightarrow> ?R") by (auto simp add: all_set_conv_all_nth listrel_iff_zip) lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s" by (meson listrel_iff_nth subrelI subset_eq) lemma listrel_subset: assumes "r \<subseteq> A \<times> A" shows "listrel r \<subseteq> lists A \<times> lists A" proof clarify show "a \<in> lists A \<and> b \<in> lists A" if "(a, b) \<in> listrel r" for a b using that assms by (induction rule: listrel.induct, auto) qed lemma listrel_refl_on: assumes "refl_on A r" shows "refl_on (lists A) (listrel r)" proof - have "l \<in> lists A \<Longrightarrow> (l, l) \<in> listrel r" for l using assms unfolding refl_on_def by (induction l, auto intro: listrel.intros) then show ?thesis by (meson assms listrel_subset refl_on_def) qed lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" by (simp add: listrel_iff_nth sym_def) lemma listrel_trans: assumes "trans r" shows "trans (listrel r)" proof - have "(x, z) \<in> listrel r" if "(x, y) \<in> listrel r" "(y, z) \<in> listrel r" for x y z using that proof induction case (Cons x y xs ys) then show ?case by clarsimp (metis assms listrel.Cons listrel_iff_nth transD) qed auto then show ?thesis using transI by blast qed theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)" by (simp add: equiv_def listrel_refl_on listrel_sym listrel_trans) lemma listrel_rtrancl_refl[iff]: "(xs,xs) \<in> listrel(r\<^sup>*)" using listrel_refl_on[of UNIV, OF refl_rtrancl] by(auto simp: refl_on_def) lemma listrel_rtrancl_trans: "\<lbrakk>(xs,ys) \<in> listrel(r\<^sup>*); (ys,zs) \<in> listrel(r\<^sup>*)\<rbrakk> \<Longrightarrow> (xs,zs) \<in> listrel(r\<^sup>*)" by (metis listrel_trans trans_def trans_rtrancl) lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}" by (blast intro: listrel.intros) lemma listrel_Cons: "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})" by (auto simp add: set_Cons_def intro: listrel.intros) text \<open>Relating \<^term>\<open>listrel1\<close>, \<^term>\<open>listrel\<close> and closures:\<close> lemma listrel1_rtrancl_subset_rtrancl_listrel1: "listrel1 (r\<^sup>*) \<subseteq> (listrel1 r)\<^sup>*" proof (rule subrelI) fix xs ys assume 1: "(xs,ys) \<in> listrel1 (r\<^sup>*)" { fix x y us vs have "(x,y) \<in> r\<^sup>* \<Longrightarrow> (us @ x # vs, us @ y # vs) \<in> (listrel1 r)\<^sup>*" proof(induct rule: rtrancl.induct) case rtrancl_refl show ?case by simp next case rtrancl_into_rtrancl thus ?case by (metis listrel1I rtrancl.rtrancl_into_rtrancl) qed } thus "(xs,ys) \<in> (listrel1 r)\<^sup>*" using 1 by(blast elim: listrel1E) qed lemma rtrancl_listrel1_eq_len: "(x,y) \<in> (listrel1 r)\<^sup>* \<Longrightarrow> length x = length y" by (induct rule: rtrancl.induct) (auto intro: listrel1_eq_len) lemma rtrancl_listrel1_ConsI1: "(xs,ys) \<in> (listrel1 r)\<^sup>* \<Longrightarrow> (x#xs,x#ys) \<in> (listrel1 r)\<^sup>*" proof (induction rule: rtrancl.induct) case (rtrancl_into_rtrancl a b c) then show ?case by (metis listrel1I2 rtrancl.rtrancl_into_rtrancl) qed auto lemma rtrancl_listrel1_ConsI2: "(x,y) \<in> r\<^sup>* \<Longrightarrow> (xs, ys) \<in> (listrel1 r)\<^sup>* \<Longrightarrow> (x # xs, y # ys) \<in> (listrel1 r)\<^sup>*" by (meson in_mono listrel1I1 listrel1_rtrancl_subset_rtrancl_listrel1 rtrancl_listrel1_ConsI1 rtrancl_trans) lemma listrel1_subset_listrel: "r \<subseteq> r' \<Longrightarrow> refl r' \<Longrightarrow> listrel1 r \<subseteq> listrel(r')" by(auto elim!: listrel1E simp add: listrel_iff_zip set_zip refl_on_def) lemma listrel_reflcl_if_listrel1: "(xs,ys) \<in> listrel1 r \<Longrightarrow> (xs,ys) \<in> listrel(r\<^sup>*)" by(erule listrel1E)(auto simp add: listrel_iff_zip set_zip) lemma listrel_rtrancl_eq_rtrancl_listrel1: "listrel (r\<^sup>*) = (listrel1 r)\<^sup>*" proof { fix x y assume "(x,y) \<in> listrel (r\<^sup>*)" then have "(x,y) \<in> (listrel1 r)\<^sup>*" by induct (auto intro: rtrancl_listrel1_ConsI2) } then show "listrel (r\<^sup>*) \<subseteq> (listrel1 r)\<^sup>*" by (rule subrelI) next show "listrel (r\<^sup>*) \<supseteq> (listrel1 r)\<^sup>*" proof(rule subrelI) fix xs ys assume "(xs,ys) \<in> (listrel1 r)\<^sup>*" then show "(xs,ys) \<in> listrel (r\<^sup>*)" proof induct case base show ?case by(auto simp add: listrel_iff_zip set_zip) next case (step ys zs) thus ?case by (metis listrel_reflcl_if_listrel1 listrel_rtrancl_trans) qed qed qed lemma rtrancl_listrel1_if_listrel: "(xs,ys) \<in> listrel r \<Longrightarrow> (xs,ys) \<in> (listrel1 r)\<^sup>*" by(metis listrel_rtrancl_eq_rtrancl_listrel1 subsetD[OF listrel_mono] r_into_rtrancl subsetI) lemma listrel_subset_rtrancl_listrel1: "listrel r \<subseteq> (listrel1 r)\<^sup>*" by(fast intro:rtrancl_listrel1_if_listrel) subsection \<open>Size function\<close> lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (size_list f)" by (rule is_measure_trivial) lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (size_option f)" by (rule is_measure_trivial) lemma size_list_estimation[termination_simp]: "x \<in> set xs \<Longrightarrow> y < f x \<Longrightarrow> y < size_list f xs" by (induct xs) auto lemma size_list_estimation'[termination_simp]: "x \<in> set xs \<Longrightarrow> y \<le> f x \<Longrightarrow> y \<le> size_list f xs" by (induct xs) auto lemma size_list_map[simp]: "size_list f (map g xs) = size_list (f \<circ> g) xs" by (induct xs) auto lemma size_list_append[simp]: "size_list f (xs @ ys) = size_list f xs + size_list f ys" by (induct xs, auto) lemma size_list_pointwise[termination_simp]: "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> size_list f xs \<le> size_list g xs" by (induct xs) force+ subsection \<open>Monad operation\<close> definition bind :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b list) \<Rightarrow> 'b list" where "bind xs f = concat (map f xs)" hide_const (open) bind lemma bind_simps [simp]: "List.bind [] f = []" "List.bind (x # xs) f = f x @ List.bind xs f" by (simp_all add: bind_def) lemma list_bind_cong [fundef_cong]: assumes "xs = ys" "(\<And>x. x \<in> set xs \<Longrightarrow> f x = g x)" shows "List.bind xs f = List.bind ys g" proof - from assms(2) have "List.bind xs f = List.bind xs g" by (induction xs) simp_all with assms(1) show ?thesis by simp qed lemma set_list_bind: "set (List.bind xs f) = (\<Union>x\<in>set xs. set (f x))" by (induction xs) simp_all subsection \<open>Code generation\<close> text\<open>Optional tail recursive version of \<^const>\<open>map\<close>. Can avoid stack overflow in some target languages.\<close> fun map_tailrec_rev :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'b list" where "map_tailrec_rev f [] bs = bs" | "map_tailrec_rev f (a#as) bs = map_tailrec_rev f as (f a # bs)" lemma map_tailrec_rev: "map_tailrec_rev f as bs = rev(map f as) @ bs" by(induction as arbitrary: bs) simp_all definition map_tailrec :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list" where "map_tailrec f as = rev (map_tailrec_rev f as [])" text\<open>Code equation:\<close> lemma map_eq_map_tailrec: "map = map_tailrec" by(simp add: fun_eq_iff map_tailrec_def map_tailrec_rev) subsubsection \<open>Counterparts for set-related operations\<close> definition member :: "'a list \<Rightarrow> 'a \<Rightarrow> bool" where [code_abbrev]: "member xs x \<longleftrightarrow> x \<in> set xs" text \<open> Use \<open>member\<close> only for generating executable code. Otherwise use \<^prop>\<open>x \<in> set xs\<close> instead --- it is much easier to reason about. \<close> lemma member_rec [code]: "member (x # xs) y \<longleftrightarrow> x = y \<or> member xs y" "member [] y \<longleftrightarrow> False" by (auto simp add: member_def) lemma in_set_member (* FIXME delete candidate *): "x \<in> set xs \<longleftrightarrow> member xs x" by (simp add: member_def) lemmas list_all_iff [code_abbrev] = fun_cong[OF list.pred_set] definition list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where list_ex_iff [code_abbrev]: "list_ex P xs \<longleftrightarrow> Bex (set xs) P" definition list_ex1 :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where list_ex1_iff [code_abbrev]: "list_ex1 P xs \<longleftrightarrow> (\<exists>! x. x \<in> set xs \<and> P x)" text \<open> Usually you should prefer \<open>\<forall>x\<in>set xs\<close>, \<open>\<exists>x\<in>set xs\<close> and \<open>\<exists>!x. x\<in>set xs \<and> _\<close> over \<^const>\<open>list_all\<close>, \<^const>\<open>list_ex\<close> and \<^const>\<open>list_ex1\<close> in specifications. \<close> lemma list_all_simps [code]: "list_all P (x # xs) \<longleftrightarrow> P x \<and> list_all P xs" "list_all P [] \<longleftrightarrow> True" by (simp_all add: list_all_iff) lemma list_ex_simps [simp, code]: "list_ex P (x # xs) \<longleftrightarrow> P x \<or> list_ex P xs" "list_ex P [] \<longleftrightarrow> False" by (simp_all add: list_ex_iff) lemma list_ex1_simps [simp, code]: "list_ex1 P [] = False" "list_ex1 P (x # xs) = (if P x then list_all (\<lambda>y. \<not> P y \<or> x = y) xs else list_ex1 P xs)" by (auto simp add: list_ex1_iff list_all_iff) lemma Ball_set_list_all: (* FIXME delete candidate *) "Ball (set xs) P \<longleftrightarrow> list_all P xs" by (simp add: list_all_iff) lemma Bex_set_list_ex: (* FIXME delete candidate *) "Bex (set xs) P \<longleftrightarrow> list_ex P xs" by (simp add: list_ex_iff) lemma list_all_append [simp]: "list_all P (xs @ ys) \<longleftrightarrow> list_all P xs \<and> list_all P ys" by (auto simp add: list_all_iff) lemma list_ex_append [simp]: "list_ex P (xs @ ys) \<longleftrightarrow> list_ex P xs \<or> list_ex P ys" by (auto simp add: list_ex_iff) lemma list_all_rev [simp]: "list_all P (rev xs) \<longleftrightarrow> list_all P xs" by (simp add: list_all_iff) lemma list_ex_rev [simp]: "list_ex P (rev xs) \<longleftrightarrow> list_ex P xs" by (simp add: list_ex_iff) lemma list_all_length: "list_all P xs \<longleftrightarrow> (\<forall>n < length xs. P (xs ! n))" by (auto simp add: list_all_iff set_conv_nth) lemma list_ex_length: "list_ex P xs \<longleftrightarrow> (\<exists>n < length xs. P (xs ! n))" by (auto simp add: list_ex_iff set_conv_nth) lemmas list_all_cong [fundef_cong] = list.pred_cong lemma list_ex_cong [fundef_cong]: "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_ex f xs = list_ex g ys" by (simp add: list_ex_iff) definition can_select :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" where [code_abbrev]: "can_select P A = (\<exists>!x\<in>A. P x)" lemma can_select_set_list_ex1 [code]: "can_select P (set A) = list_ex1 P A" by (simp add: list_ex1_iff can_select_def) text \<open>Executable checks for relations on sets\<close> definition listrel1p :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where "listrel1p r xs ys = ((xs, ys) \<in> listrel1 {(x, y). r x y})" lemma [code_unfold]: "(xs, ys) \<in> listrel1 r = listrel1p (\<lambda>x y. (x, y) \<in> r) xs ys" unfolding listrel1p_def by auto lemma [code]: "listrel1p r [] xs = False" "listrel1p r xs [] = False" "listrel1p r (x # xs) (y # ys) \<longleftrightarrow> r x y \<and> xs = ys \<or> x = y \<and> listrel1p r xs ys" by (simp add: listrel1p_def)+ definition lexordp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" where "lexordp r xs ys = ((xs, ys) \<in> lexord {(x, y). r x y})" lemma [code_unfold]: "(xs, ys) \<in> lexord r = lexordp (\<lambda>x y. (x, y) \<in> r) xs ys" unfolding lexordp_def by auto lemma [code]: "lexordp r xs [] = False" "lexordp r [] (y#ys) = True" "lexordp r (x # xs) (y # ys) = (r x y \<or> (x = y \<and> lexordp r xs ys))" unfolding lexordp_def by auto text \<open>Bounded quantification and summation over nats.\<close> lemma atMost_upto [code_unfold]: "{..n} = set [0..<Suc n]" by auto lemma atLeast_upt [code_unfold]: "{..<n} = set [0..<n]" by auto lemma greaterThanLessThan_upt [code_unfold]: "{n<..<m} = set [Suc n..<m]" by auto lemmas atLeastLessThan_upt [code_unfold] = set_upt [symmetric] lemma greaterThanAtMost_upt [code_unfold]: "{n<..m} = set [Suc n..<Suc m]" by auto lemma atLeastAtMost_upt [code_unfold]: "{n..m} = set [n..<Suc m]" by auto lemma all_nat_less_eq [code_unfold]: "(\<forall>m<n::nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..<n}. P m)" by auto lemma ex_nat_less_eq [code_unfold]: "(\<exists>m<n::nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..<n}. P m)" by auto lemma all_nat_less [code_unfold]: "(\<forall>m\<le>n::nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..n}. P m)" by auto lemma ex_nat_less [code_unfold]: "(\<exists>m\<le>n::nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..n}. P m)" by auto text\<open>Bounded \<open>LEAST\<close> operator:\<close> definition "Bleast S P = (LEAST x. x \<in> S \<and> P x)" definition "abort_Bleast S P = (LEAST x. x \<in> S \<and> P x)" declare [[code abort: abort_Bleast]] lemma Bleast_code [code]: "Bleast (set xs) P = (case filter P (sort xs) of x#xs \<Rightarrow> x | [] \<Rightarrow> abort_Bleast (set xs) P)" proof (cases "filter P (sort xs)") case Nil thus ?thesis by (simp add: Bleast_def abort_Bleast_def) next case (Cons x ys) have "(LEAST x. x \<in> set xs \<and> P x) = x" proof (rule Least_equality) show "x \<in> set xs \<and> P x" by (metis Cons Cons_eq_filter_iff in_set_conv_decomp set_sort) next fix y assume "y \<in> set xs \<and> P y" hence "y \<in> set (filter P xs)" by auto thus "x \<le> y" by (metis Cons eq_iff filter_sort set_ConsD set_sort sorted.simps(2) sorted_sort) qed thus ?thesis using Cons by (simp add: Bleast_def) qed declare Bleast_def[symmetric, code_unfold] text \<open>Summation over ints.\<close> lemma greaterThanLessThan_upto [code_unfold]: "{i<..<j::int} = set [i+1..j - 1]" by auto lemma atLeastLessThan_upto [code_unfold]: "{i..<j::int} = set [i..j - 1]" by auto lemma greaterThanAtMost_upto [code_unfold]: "{i<..j::int} = set [i+1..j]" by auto lemmas atLeastAtMost_upto [code_unfold] = set_upto [symmetric] subsubsection \<open>Optimizing by rewriting\<close> definition null :: "'a list \<Rightarrow> bool" where [code_abbrev]: "null xs \<longleftrightarrow> xs = []" text \<open> Efficient emptyness check is implemented by \<^const>\<open>null\<close>. \<close> lemma null_rec [code]: "null (x # xs) \<longleftrightarrow> False" "null [] \<longleftrightarrow> True" by (simp_all add: null_def) lemma eq_Nil_null: (* FIXME delete candidate *) "xs = [] \<longleftrightarrow> null xs" by (simp add: null_def) lemma equal_Nil_null [code_unfold]: "HOL.equal xs [] \<longleftrightarrow> null xs" "HOL.equal [] = null" by (auto simp add: equal null_def) definition maps :: "('a \<Rightarrow> 'b list) \<Rightarrow> 'a list \<Rightarrow> 'b list" where [code_abbrev]: "maps f xs = concat (map f xs)" definition map_filter :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list" where [code_post]: "map_filter f xs = map (the \<circ> f) (filter (\<lambda>x. f x \<noteq> None) xs)" text \<open> Operations \<^const>\<open>maps\<close> and \<^const>\<open>map_filter\<close> avoid intermediate lists on execution -- do not use for proving. \<close> lemma maps_simps [code]: "maps f (x # xs) = f x @ maps f xs" "maps f [] = []" by (simp_all add: maps_def) lemma map_filter_simps [code]: "map_filter f (x # xs) = (case f x of None \<Rightarrow> map_filter f xs | Some y \<Rightarrow> y # map_filter f xs)" "map_filter f [] = []" by (simp_all add: map_filter_def split: option.split) lemma concat_map_maps: (* FIXME delete candidate *) "concat (map f xs) = maps f xs" by (simp add: maps_def) lemma map_filter_map_filter [code_unfold]: "map f (filter P xs) = map_filter (\<lambda>x. if P x then Some (f x) else None) xs" by (simp add: map_filter_def) text \<open>Optimized code for \<open>\<forall>i\<in>{a..b::int}\<close> and \<open>\<forall>n:{a..<b::nat}\<close> and similiarly for \<open>\<exists>\<close>.\<close> definition all_interval_nat :: "(nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where "all_interval_nat P i j \<longleftrightarrow> (\<forall>n \<in> {i..<j}. P n)" lemma [code]: "all_interval_nat P i j \<longleftrightarrow> i \<ge> j \<or> P i \<and> all_interval_nat P (Suc i) j" proof - have *: "\<And>n. P i \<Longrightarrow> \<forall>n\<in>{Suc i..<j}. P n \<Longrightarrow> i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n" proof - fix n assume "P i" "\<forall>n\<in>{Suc i..<j}. P n" "i \<le> n" "n < j" then show "P n" by (cases "n = i") simp_all qed show ?thesis by (auto simp add: all_interval_nat_def intro: *) qed lemma list_all_iff_all_interval_nat [code_unfold]: "list_all P [i..<j] \<longleftrightarrow> all_interval_nat P i j" by (simp add: list_all_iff all_interval_nat_def) lemma list_ex_iff_not_all_inverval_nat [code_unfold]: "list_ex P [i..<j] \<longleftrightarrow> \<not> (all_interval_nat (Not \<circ> P) i j)" by (simp add: list_ex_iff all_interval_nat_def) definition all_interval_int :: "(int \<Rightarrow> bool) \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool" where "all_interval_int P i j \<longleftrightarrow> (\<forall>k \<in> {i..j}. P k)" lemma [code]: "all_interval_int P i j \<longleftrightarrow> i > j \<or> P i \<and> all_interval_int P (i + 1) j" proof - have *: "\<And>k. P i \<Longrightarrow> \<forall>k\<in>{i+1..j}. P k \<Longrightarrow> i \<le> k \<Longrightarrow> k \<le> j \<Longrightarrow> P k" proof - fix k assume "P i" "\<forall>k\<in>{i+1..j}. P k" "i \<le> k" "k \<le> j" then show "P k" by (cases "k = i") simp_all qed show ?thesis by (auto simp add: all_interval_int_def intro: *) qed lemma list_all_iff_all_interval_int [code_unfold]: "list_all P [i..j] \<longleftrightarrow> all_interval_int P i j" by (simp add: list_all_iff all_interval_int_def) lemma list_ex_iff_not_all_inverval_int [code_unfold]: "list_ex P [i..j] \<longleftrightarrow> \<not> (all_interval_int (Not \<circ> P) i j)" by (simp add: list_ex_iff all_interval_int_def) text \<open>optimized code (tail-recursive) for \<^term>\<open>length\<close>\<close> definition gen_length :: "nat \<Rightarrow> 'a list \<Rightarrow> nat" where "gen_length n xs = n + length xs" lemma gen_length_code [code]: "gen_length n [] = n" "gen_length n (x # xs) = gen_length (Suc n) xs" by(simp_all add: gen_length_def) declare list.size(3-4)[code del] lemma length_code [code]: "length = gen_length 0" by(simp add: gen_length_def fun_eq_iff) hide_const (open) member null maps map_filter all_interval_nat all_interval_int gen_length subsubsection \<open>Pretty lists\<close> ML \<open> (* Code generation for list literals. *) signature LIST_CODE = sig val add_literal_list: string -> theory -> theory end; structure List_Code : LIST_CODE = struct open Basic_Code_Thingol; fun implode_list t = let fun dest_cons (IConst { sym = Code_Symbol.Constant \<^const_name>\<open>Cons\<close>, ... } `$ t1 `$ t2) = SOME (t1, t2) | dest_cons _ = NONE; val (ts, t') = Code_Thingol.unfoldr dest_cons t; in case t' of IConst { sym = Code_Symbol.Constant \<^const_name>\<open>Nil\<close>, ... } => SOME ts | _ => NONE end; fun print_list (target_fxy, target_cons) pr fxy t1 t2 = Code_Printer.brackify_infix (target_fxy, Code_Printer.R) fxy ( pr (Code_Printer.INFX (target_fxy, Code_Printer.X)) t1, Code_Printer.str target_cons, pr (Code_Printer.INFX (target_fxy, Code_Printer.R)) t2 ); fun add_literal_list target = let fun pretty literals pr _ vars fxy [(t1, _), (t2, _)] = case Option.map (cons t1) (implode_list t2) of SOME ts => Code_Printer.literal_list literals (map (pr vars Code_Printer.NOBR) ts) | NONE => print_list (Code_Printer.infix_cons literals) (pr vars) fxy t1 t2; in Code_Target.set_printings (Code_Symbol.Constant (\<^const_name>\<open>Cons\<close>, [(target, SOME (Code_Printer.complex_const_syntax (2, pretty)))])) end end; \<close> code_printing type_constructor list \<rightharpoonup> (SML) "_ list" and (OCaml) "_ list" and (Haskell) "![(_)]" and (Scala) "List[(_)]" | constant Nil \<rightharpoonup> (SML) "[]" and (OCaml) "[]" and (Haskell) "[]" and (Scala) "!Nil" | class_instance list :: equal \<rightharpoonup> (Haskell) - | constant "HOL.equal :: 'a list \<Rightarrow> 'a list \<Rightarrow> bool" \<rightharpoonup> (Haskell) infix 4 "==" setup \<open>fold (List_Code.add_literal_list) ["SML", "OCaml", "Haskell", "Scala"]\<close> code_reserved SML list code_reserved OCaml list subsubsection \<open>Use convenient predefined operations\<close> code_printing constant "(@)" \<rightharpoonup> (SML) infixr 7 "@" and (OCaml) infixr 6 "@" and (Haskell) infixr 5 "++" and (Scala) infixl 7 "++" | constant map \<rightharpoonup> (Haskell) "map" | constant filter \<rightharpoonup> (Haskell) "filter" | constant concat \<rightharpoonup> (Haskell) "concat" | constant List.maps \<rightharpoonup> (Haskell) "concatMap" | constant rev \<rightharpoonup> (Haskell) "reverse" | constant zip \<rightharpoonup> (Haskell) "zip" | constant List.null \<rightharpoonup> (Haskell) "null" | constant takeWhile \<rightharpoonup> (Haskell) "takeWhile" | constant dropWhile \<rightharpoonup> (Haskell) "dropWhile" | constant list_all \<rightharpoonup> (Haskell) "all" | constant list_ex \<rightharpoonup> (Haskell) "any" subsubsection \<open>Implementation of sets by lists\<close> lemma is_empty_set [code]: "Set.is_empty (set xs) \<longleftrightarrow> List.null xs" by (simp add: Set.is_empty_def null_def) lemma empty_set [code]: "{} = set []" by simp lemma UNIV_coset [code]: "UNIV = List.coset []" by simp lemma compl_set [code]: "- set xs = List.coset xs" by simp lemma compl_coset [code]: "- List.coset xs = set xs" by simp lemma [code]: "x \<in> set xs \<longleftrightarrow> List.member xs x" "x \<in> List.coset xs \<longleftrightarrow> \<not> List.member xs x" by (simp_all add: member_def) lemma insert_code [code]: "insert x (set xs) = set (List.insert x xs)" "insert x (List.coset xs) = List.coset (removeAll x xs)" by simp_all lemma remove_code [code]: "Set.remove x (set xs) = set (removeAll x xs)" "Set.remove x (List.coset xs) = List.coset (List.insert x xs)" by (simp_all add: remove_def Compl_insert) lemma filter_set [code]: "Set.filter P (set xs) = set (filter P xs)" by auto lemma image_set [code]: "image f (set xs) = set (map f xs)" by simp lemma subset_code [code]: "set xs \<le> B \<longleftrightarrow> (\<forall>x\<in>set xs. x \<in> B)" "A \<le> List.coset ys \<longleftrightarrow> (\<forall>y\<in>set ys. y \<notin> A)" "List.coset [] \<subseteq> set [] \<longleftrightarrow> False" by auto text \<open>A frequent case -- avoid intermediate sets\<close> lemma [code_unfold]: "set xs \<subseteq> set ys \<longleftrightarrow> list_all (\<lambda>x. x \<in> set ys) xs" by (auto simp: list_all_iff) lemma Ball_set [code]: "Ball (set xs) P \<longleftrightarrow> list_all P xs" by (simp add: list_all_iff) lemma Bex_set [code]: "Bex (set xs) P \<longleftrightarrow> list_ex P xs" by (simp add: list_ex_iff) lemma card_set [code]: "card (set xs) = length (remdups xs)" proof - have "card (set (remdups xs)) = length (remdups xs)" by (rule distinct_card) simp then show ?thesis by simp qed lemma the_elem_set [code]: "the_elem (set [x]) = x" by simp lemma Pow_set [code]: "Pow (set []) = {{}}" "Pow (set (x # xs)) = (let A = Pow (set xs) in A \<union> insert x ` A)" by (simp_all add: Pow_insert Let_def) definition map_project :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a set \<Rightarrow> 'b set" where "map_project f A = {b. \<exists> a \<in> A. f a = Some b}" lemma [code]: "map_project f (set xs) = set (List.map_filter f xs)" by (auto simp add: map_project_def map_filter_def image_def) hide_const (open) map_project text \<open>Operations on relations\<close> lemma product_code [code]: "Product_Type.product (set xs) (set ys) = set [(x, y). x \<leftarrow> xs, y \<leftarrow> ys]" by (auto simp add: Product_Type.product_def) lemma Id_on_set [code]: "Id_on (set xs) = set [(x, x). x \<leftarrow> xs]" by (auto simp add: Id_on_def) lemma [code]: "R `` S = List.map_project (\<lambda>(x, y). if x \<in> S then Some y else None) R" unfolding map_project_def by (auto split: prod.split if_split_asm) lemma trancl_set_ntrancl [code]: "trancl (set xs) = ntrancl (card (set xs) - 1) (set xs)" by (simp add: finite_trancl_ntranl) lemma set_relcomp [code]: "set xys O set yzs = set ([(fst xy, snd yz). xy \<leftarrow> xys, yz \<leftarrow> yzs, snd xy = fst yz])" by auto (auto simp add: Bex_def image_def) lemma wf_set [code]: "wf (set xs) = acyclic (set xs)" by (simp add: wf_iff_acyclic_if_finite) subsection \<open>Setup for Lifting/Transfer\<close> subsubsection \<open>Transfer rules for the Transfer package\<close> context includes lifting_syntax begin lemma tl_transfer [transfer_rule]: "(list_all2 A ===> list_all2 A) tl tl" unfolding tl_def[abs_def] by transfer_prover lemma butlast_transfer [transfer_rule]: "(list_all2 A ===> list_all2 A) butlast butlast" by (rule rel_funI, erule list_all2_induct, auto) lemma map_rec: "map f xs = rec_list Nil (%x _ y. Cons (f x) y) xs" by (induct xs) auto lemma append_transfer [transfer_rule]: "(list_all2 A ===> list_all2 A ===> list_all2 A) append append" unfolding List.append_def by transfer_prover lemma rev_transfer [transfer_rule]: "(list_all2 A ===> list_all2 A) rev rev" unfolding List.rev_def by transfer_prover lemma filter_transfer [transfer_rule]: "((A ===> (=)) ===> list_all2 A ===> list_all2 A) filter filter" unfolding List.filter_def by transfer_prover lemma fold_transfer [transfer_rule]: "((A ===> B ===> B) ===> list_all2 A ===> B ===> B) fold fold" unfolding List.fold_def by transfer_prover lemma foldr_transfer [transfer_rule]: "((A ===> B ===> B) ===> list_all2 A ===> B ===> B) foldr foldr" unfolding List.foldr_def by transfer_prover lemma foldl_transfer [transfer_rule]: "((B ===> A ===> B) ===> B ===> list_all2 A ===> B) foldl foldl" unfolding List.foldl_def by transfer_prover lemma concat_transfer [transfer_rule]: "(list_all2 (list_all2 A) ===> list_all2 A) concat concat" unfolding List.concat_def by transfer_prover lemma drop_transfer [transfer_rule]: "((=) ===> list_all2 A ===> list_all2 A) drop drop" unfolding List.drop_def by transfer_prover lemma take_transfer [transfer_rule]: "((=) ===> list_all2 A ===> list_all2 A) take take" unfolding List.take_def by transfer_prover lemma list_update_transfer [transfer_rule]: "(list_all2 A ===> (=) ===> A ===> list_all2 A) list_update list_update" unfolding list_update_def by transfer_prover lemma takeWhile_transfer [transfer_rule]: "((A ===> (=)) ===> list_all2 A ===> list_all2 A) takeWhile takeWhile" unfolding takeWhile_def by transfer_prover lemma dropWhile_transfer [transfer_rule]: "((A ===> (=)) ===> list_all2 A ===> list_all2 A) dropWhile dropWhile" unfolding dropWhile_def by transfer_prover lemma zip_transfer [transfer_rule]: "(list_all2 A ===> list_all2 B ===> list_all2 (rel_prod A B)) zip zip" unfolding zip_def by transfer_prover lemma product_transfer [transfer_rule]: "(list_all2 A ===> list_all2 B ===> list_all2 (rel_prod A B)) List.product List.product" unfolding List.product_def by transfer_prover lemma product_lists_transfer [transfer_rule]: "(list_all2 (list_all2 A) ===> list_all2 (list_all2 A)) product_lists product_lists" unfolding product_lists_def by transfer_prover lemma insert_transfer [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(A ===> list_all2 A ===> list_all2 A) List.insert List.insert" unfolding List.insert_def [abs_def] by transfer_prover lemma find_transfer [transfer_rule]: "((A ===> (=)) ===> list_all2 A ===> rel_option A) List.find List.find" unfolding List.find_def by transfer_prover lemma those_transfer [transfer_rule]: "(list_all2 (rel_option P) ===> rel_option (list_all2 P)) those those" unfolding List.those_def by transfer_prover lemma remove1_transfer [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(A ===> list_all2 A ===> list_all2 A) remove1 remove1" unfolding remove1_def by transfer_prover lemma removeAll_transfer [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(A ===> list_all2 A ===> list_all2 A) removeAll removeAll" unfolding removeAll_def by transfer_prover lemma successively_transfer [transfer_rule]: "((A ===> A ===> (=)) ===> list_all2 A ===> (=)) successively successively" unfolding successively_altdef by transfer_prover lemma distinct_transfer [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(list_all2 A ===> (=)) distinct distinct" unfolding distinct_def by transfer_prover lemma distinct_adj_transfer [transfer_rule]: assumes "bi_unique A" shows "(list_all2 A ===> (=)) distinct_adj distinct_adj" unfolding rel_fun_def proof (intro allI impI) fix xs ys assume "list_all2 A xs ys" thus "distinct_adj xs \<longleftrightarrow> distinct_adj ys" proof (induction rule: list_all2_induct) case (Cons x xs y ys) note * = this show ?case proof (cases xs) case [simp]: (Cons x' xs') with * obtain y' ys' where [simp]: "ys = y' # ys'" by (cases ys) auto from * show ?thesis using assms by (auto simp: distinct_adj_Cons bi_unique_def) qed (use * in auto) qed auto qed lemma remdups_transfer [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(list_all2 A ===> list_all2 A) remdups remdups" unfolding remdups_def by transfer_prover lemma remdups_adj_transfer [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(list_all2 A ===> list_all2 A) remdups_adj remdups_adj" proof (rule rel_funI, erule list_all2_induct) qed (auto simp: remdups_adj_Cons assms[unfolded bi_unique_def] split: list.splits) lemma replicate_transfer [transfer_rule]: "((=) ===> A ===> list_all2 A) replicate replicate" unfolding replicate_def by transfer_prover lemma length_transfer [transfer_rule]: "(list_all2 A ===> (=)) length length" unfolding size_list_overloaded_def size_list_def by transfer_prover lemma rotate1_transfer [transfer_rule]: "(list_all2 A ===> list_all2 A) rotate1 rotate1" unfolding rotate1_def by transfer_prover lemma rotate_transfer [transfer_rule]: "((=) ===> list_all2 A ===> list_all2 A) rotate rotate" unfolding rotate_def [abs_def] by transfer_prover lemma nths_transfer [transfer_rule]: "(list_all2 A ===> rel_set (=) ===> list_all2 A) nths nths" unfolding nths_def [abs_def] by transfer_prover lemma subseqs_transfer [transfer_rule]: "(list_all2 A ===> list_all2 (list_all2 A)) subseqs subseqs" unfolding subseqs_def [abs_def] by transfer_prover lemma partition_transfer [transfer_rule]: "((A ===> (=)) ===> list_all2 A ===> rel_prod (list_all2 A) (list_all2 A)) partition partition" unfolding partition_def by transfer_prover lemma lists_transfer [transfer_rule]: "(rel_set A ===> rel_set (list_all2 A)) lists lists" proof (rule rel_funI, rule rel_setI) show "\<lbrakk>l \<in> lists X; rel_set A X Y\<rbrakk> \<Longrightarrow> \<exists>y\<in>lists Y. list_all2 A l y" for X Y l proof (induction l rule: lists.induct) case (Cons a l) then show ?case by (simp only: rel_set_def list_all2_Cons1, metis lists.Cons) qed auto show "\<lbrakk>l \<in> lists Y; rel_set A X Y\<rbrakk> \<Longrightarrow> \<exists>x\<in>lists X. list_all2 A x l" for X Y l proof (induction l rule: lists.induct) case (Cons a l) then show ?case by (simp only: rel_set_def list_all2_Cons2, metis lists.Cons) qed auto qed lemma set_Cons_transfer [transfer_rule]: "(rel_set A ===> rel_set (list_all2 A) ===> rel_set (list_all2 A)) set_Cons set_Cons" unfolding rel_fun_def rel_set_def set_Cons_def by (fastforce simp add: list_all2_Cons1 list_all2_Cons2) lemma listset_transfer [transfer_rule]: "(list_all2 (rel_set A) ===> rel_set (list_all2 A)) listset listset" unfolding listset_def by transfer_prover lemma null_transfer [transfer_rule]: "(list_all2 A ===> (=)) List.null List.null" unfolding rel_fun_def List.null_def by auto lemma list_all_transfer [transfer_rule]: "((A ===> (=)) ===> list_all2 A ===> (=)) list_all list_all" unfolding list_all_iff [abs_def] by transfer_prover lemma list_ex_transfer [transfer_rule]: "((A ===> (=)) ===> list_all2 A ===> (=)) list_ex list_ex" unfolding list_ex_iff [abs_def] by transfer_prover lemma splice_transfer [transfer_rule]: "(list_all2 A ===> list_all2 A ===> list_all2 A) splice splice" apply (rule rel_funI, erule list_all2_induct, simp add: rel_fun_def, simp) apply (rule rel_funI) apply (erule_tac xs=x in list_all2_induct, simp, simp add: rel_fun_def) done lemma shuffles_transfer [transfer_rule]: "(list_all2 A ===> list_all2 A ===> rel_set (list_all2 A)) shuffles shuffles" proof (intro rel_funI, goal_cases) case (1 xs xs' ys ys') thus ?case proof (induction xs ys arbitrary: xs' ys' rule: shuffles.induct) case (3 x xs y ys xs' ys') from "3.prems" obtain x' xs'' where xs': "xs' = x' # xs''" by (cases xs') auto from "3.prems" obtain y' ys'' where ys': "ys' = y' # ys''" by (cases ys') auto have [transfer_rule]: "A x x'" "A y y'" "list_all2 A xs xs''" "list_all2 A ys ys''" using "3.prems" by (simp_all add: xs' ys') have [transfer_rule]: "rel_set (list_all2 A) (shuffles xs (y # ys)) (shuffles xs'' ys')" and [transfer_rule]: "rel_set (list_all2 A) (shuffles (x # xs) ys) (shuffles xs' ys'')" using "3.prems" by (auto intro!: "3.IH" simp: xs' ys') have "rel_set (list_all2 A) ((#) x ` shuffles xs (y # ys) \<union> (#) y ` shuffles (x # xs) ys) ((#) x' ` shuffles xs'' ys' \<union> (#) y' ` shuffles xs' ys'')" by transfer_prover thus ?case by (simp add: xs' ys') qed (auto simp: rel_set_def) qed lemma rtrancl_parametric [transfer_rule]: assumes [transfer_rule]: "bi_unique A" "bi_total A" shows "(rel_set (rel_prod A A) ===> rel_set (rel_prod A A)) rtrancl rtrancl" unfolding rtrancl_def by transfer_prover lemma monotone_parametric [transfer_rule]: assumes [transfer_rule]: "bi_total A" shows "((A ===> A ===> (=)) ===> (B ===> B ===> (=)) ===> (A ===> B) ===> (=)) monotone monotone" unfolding monotone_def[abs_def] by transfer_prover lemma fun_ord_parametric [transfer_rule]: assumes [transfer_rule]: "bi_total C" shows "((A ===> B ===> (=)) ===> (C ===> A) ===> (C ===> B) ===> (=)) fun_ord fun_ord" unfolding fun_ord_def[abs_def] by transfer_prover lemma fun_lub_parametric [transfer_rule]: assumes [transfer_rule]: "bi_total A" "bi_unique A" shows "((rel_set A ===> B) ===> rel_set (C ===> A) ===> C ===> B) fun_lub fun_lub" unfolding fun_lub_def[abs_def] by transfer_prover end end