# seL4/isabelle

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 (* Title: HOL/List.thy Author: Tobias Nipkow; proofs tidied by LCP *) section \The datatype of finite lists\ 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]: \ \for backward compatibility -- names of variables differ\ "(y = [] \ P) \ (\a list. y = a # list \ P) \ P" by (rule list.exhaust) lemma [case_names Nil Cons, induct type: list]: \ \for backward compatibility -- names of variables differ\ "P [] \ (\a list. P list \ P (a # list)) \ P list" by (rule list.induct) text \Compatibility:\ setup \Sign.mandatory_path "list"\ lemmas inducts = list.induct lemmas recs = list.rec lemmas cases = list.case setup \Sign.parent_path\ lemmas set_simps = list.set (* legacy *) syntax \ \list Enumeration\ "_list" :: "args => 'a list" ("[(_)]") translations "[x, xs]" == "x#[xs]" "[x]" == "x#[]" subsection \Basic list processing functions\ primrec (nonexhaustive) last :: "'a list \ 'a" where "last (x # xs) = (if xs = [] then x else last xs)" primrec butlast :: "'a list \ 'a list" where "butlast [] = []" | "butlast (x # xs) = (if xs = [] then [] else x # butlast xs)" lemma set_rec: "set xs = rec_list {} (\x _. insert x) xs" by (induct xs) auto definition coset :: "'a list \ 'a set" where [simp]: "coset xs = - set xs" primrec append :: "'a list \ 'a list \ 'a list" (infixr "@" 65) where append_Nil: "[] @ ys = ys" | append_Cons: "(x#xs) @ ys = x # xs @ ys" primrec rev :: "'a list \ 'a list" where "rev [] = []" | "rev (x # xs) = rev xs @ [x]" primrec filter:: "('a \ bool) \ 'a list \ 'a list" where "filter P [] = []" | "filter P (x # xs) = (if P x then x # filter P xs else filter P xs)" text \Special input syntax for filter:\ syntax (ASCII) "_filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_<-_./ _])") syntax "_filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_\_ ./ _])") translations "[x<-xs . P]" \ "CONST filter (\x. P) xs" primrec fold :: "('a \ 'b \ 'b) \ 'a list \ 'b \ 'b" where fold_Nil: "fold f [] = id" | fold_Cons: "fold f (x # xs) = fold f xs \ f x" primrec foldr :: "('a \ 'b \ 'b) \ 'a list \ 'b \ 'b" where foldr_Nil: "foldr f [] = id" | foldr_Cons: "foldr f (x # xs) = f x \ foldr f xs" primrec foldl :: "('b \ 'a \ 'b) \ 'b \ 'a list \ '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 \ 'a list" where "concat [] = []" | "concat (x # xs) = x @ concat xs" primrec drop:: "nat \ 'a list \ 'a list" where drop_Nil: "drop n [] = []" | drop_Cons: "drop n (x # xs) = (case n of 0 \ x # xs | Suc m \ drop m xs)" \ \Warning: simpset does not contain this definition, but separate theorems for \n = 0\ and \n = Suc k\\ primrec take:: "nat \ 'a list \ 'a list" where take_Nil:"take n [] = []" | take_Cons: "take n (x # xs) = (case n of 0 \ [] | Suc m \ x # take m xs)" \ \Warning: simpset does not contain this definition, but separate theorems for \n = 0\ and \n = Suc k\\ primrec (nonexhaustive) nth :: "'a list => nat => 'a" (infixl "!" 100) where nth_Cons: "(x # xs) ! n = (case n of 0 \ x | Suc k \ xs ! k)" \ \Warning: simpset does not contain this definition, but separate theorems for \n = 0\ and \n = Suc k\\ primrec list_update :: "'a list \ nat \ 'a \ 'a list" where "list_update [] i v = []" | "list_update (x # xs) i v = (case i of 0 \ v # xs | Suc j \ 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" ("_/[(_)]" [900,0] 900) translations "_LUpdate xs (_lupdbinds b bs)" == "_LUpdate (_LUpdate xs b) bs" "xs[i:=x]" == "CONST list_update xs i x" primrec takeWhile :: "('a \ bool) \ 'a list \ 'a list" where "takeWhile P [] = []" | "takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])" primrec dropWhile :: "('a \ bool) \ 'a list \ 'a list" where "dropWhile P [] = []" | "dropWhile P (x # xs) = (if P x then dropWhile P xs else x # xs)" primrec zip :: "'a list \ 'b list \ ('a \ 'b) list" where "zip xs [] = []" | zip_Cons: "zip xs (y # ys) = (case xs of [] \ [] | z # zs \ (z, y) # zip zs ys)" \ \Warning: simpset does not contain this definition, but separate theorems for \xs = []\ and \xs = z # zs\\ abbreviation map2 :: "('a \ 'b \ 'c) \ 'a list \ 'b list \ 'c list" where "map2 f xs ys \ map ($$x,y). f x y) (zip xs ys)" primrec product :: "'a list \ 'b list \ ('a \ '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 \ 'a list list" where "product_lists [] = [[]]" | "product_lists (xs # xss) = concat (map (\x. map (Cons x) (product_lists xss)) xs)" primrec upt :: "nat \ nat \ nat list" ("(1[_.. j then [i.. 'a list \ 'a list" where "insert x xs = (if x \ set xs then xs else x # xs)" definition union :: "'a list \ 'a list \ 'a list" where "union = fold insert" hide_const (open) insert union hide_fact (open) insert_def union_def primrec find :: "('a \ bool) \ 'a list \ 'a option" where "find _ [] = None" | "find P (x#xs) = (if P x then Some x else find P xs)" text \In the context of multisets, \count_list\ is equivalent to @{term "count \ mset"} and it it advisable to use the latter.\ primrec count_list :: "'a list \ 'a \ 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 \ bool) \ 'a list \ ('a list * 'a * 'a list) option" where "extract P xs = (case dropWhile (Not \ P) xs of [] \ None | y#ys \ Some(takeWhile (Not \ P) xs, y, ys))" hide_const (open) "extract" primrec those :: "'a option list \ 'a list option" where "those [] = Some []" | "those (x # xs) = (case x of None \ None | Some y \ map_option (Cons y) (those xs))" primrec remove1 :: "'a \ 'a list \ 'a list" where "remove1 x [] = []" | "remove1 x (y # xs) = (if x = y then xs else y # remove1 x xs)" primrec removeAll :: "'a \ 'a list \ '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 \ bool" where "distinct [] \ True" | "distinct (x # xs) \ x \ set xs \ distinct xs" primrec remdups :: "'a list \ 'a list" where "remdups [] = []" | "remdups (x # xs) = (if x \ set xs then remdups xs else x # remdups xs)" fun remdups_adj :: "'a list \ '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 \ 'a \ 'a list" where replicate_0: "replicate 0 x = []" | replicate_Suc: "replicate (Suc n) x = x # replicate n x" text \ Function \size\ is overloaded for all datatypes. Users may refer to the list version as \length\.\ abbreviation length :: "'a list \ nat" where "length \ size" definition enumerate :: "nat \ 'a list \ (nat \ 'a) list" where enumerate_eq_zip: "enumerate n xs = zip [n.. 'a list" where "rotate1 [] = []" | "rotate1 (x # xs) = xs @ [x]" definition rotate :: "nat \ 'a list \ 'a list" where "rotate n = rotate1 ^^ n" definition nths :: "'a list => nat set => 'a list" where "nths xs A = map fst (filter (\p. snd p \ A) (zip xs [0.. 'a list list" where "subseqs [] = [[]]" | "subseqs (x#xs) = (let xss = subseqs xs in map (Cons x) xss @ xss)" primrec n_lists :: "nat \ 'a list \ 'a list list" where "n_lists 0 xs = [[]]" | "n_lists (Suc n) xs = concat (map (\ys. map (\y. y # ys) xs) (n_lists n xs))" hide_const (open) n_lists fun splice :: "'a list \ 'a list \ 'a list" where "splice [] ys = ys" | "splice xs [] = xs" | "splice (x#xs) (y#ys) = x # y # splice xs ys" function shuffle where "shuffle [] ys = {ys}" | "shuffle xs [] = {xs}" | "shuffle (x # xs) (y # ys) = (#) x  shuffle xs (y # ys) \ (#) y  shuffle (x # xs) ys" by pat_completeness simp_all termination by lexicographic_order text\Use only if you cannot use @{const Min} instead:\ fun min_list :: "'a::ord list \ 'a" where "min_list (x # xs) = (case xs of [] \ x | _ \ min x (min_list xs))" text\Returns first minimum:\ fun arg_min_list :: "('a \ ('b::linorder)) \ 'a list \ '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 \ f m then x else m)" text\ \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 (\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 "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 "shuffle [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 (\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. \ text\The following simple sort functions are intended for proofs, not for efficient implementations.\ text \A sorted predicate w.r.t. a relation:\ fun sorted_wrt :: "('a \ 'a \ bool) \ 'a list \ bool" where "sorted_wrt P [] = True" | "sorted_wrt P (x # ys) = ((\y \ set ys. P x y) \ sorted_wrt P ys)" (* FIXME: define sorted in terms of sorted_wrt *) text \A class-based sorted predicate:\ context linorder begin fun sorted :: "'a list \ bool" where "sorted [] = True" | "sorted (x # ys) = ((\y \ set ys. x \ y) \ sorted ys)" lemma sorted_sorted_wrt: "sorted = sorted_wrt ($$" proof (rule ext) fix xs show "sorted xs = sorted_wrt (\) xs" by(induction xs rule: sorted.induct) auto qed primrec insort_key :: "('b \ 'a) \ 'b \ 'b list \ 'b list" where "insort_key f x [] = [x]" | "insort_key f x (y#ys) = (if f x \ f y then (x#y#ys) else y#(insort_key f x ys))" definition sort_key :: "('b \ 'a) \ 'b list \ 'b list" where "sort_key f xs = foldr (insort_key f) xs []" definition insort_insert_key :: "('b \ 'a) \ 'b \ 'b list \ 'b list" where "insort_insert_key f x xs = (if f x \ f  set xs then xs else insort_key f x xs)" abbreviation "sort \ sort_key (\x. x)" abbreviation "insort \ insort_key (\x. x)" abbreviation "insort_insert \ insort_insert_key (\x. x)" definition stable_sort_key :: "(('b \ 'a) \ 'b list \ 'b list) \ bool" where "stable_sort_key sk = (\f xs k. filter (\y. f y = k) (sk f xs) = filter (\y. f y = k) xs)" end subsubsection \List comprehension\ text\Input syntax for Haskell-like list comprehension notation. Typical example: $(x,y). x \ xs, y \ ys, x \ y]\, the list of all pairs of distinct elements from \xs\ and \ys\. The syntax is as in Haskell, except that \|\ becomes a dot (like in Isabelle's set comprehension): \[e. x \ xs,$\ rather than \verb![e| x <- xs, ...]!. The qualifiers after the dot are \begin{description} \item[generators] \p \ xs\, where \p\ is a pattern and \xs\ an expression of list type, or \item[guards] \b\, where \b\ 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 \[e. x \ xs]\ is optmized to @{term"map (%x. e) xs"}. 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.\ nonterminal lc_qual and lc_quals syntax "_listcompr" :: "'a \ lc_qual \ lc_quals \ 'a list" ("[_ . __") "_lc_gen" :: "'a \ 'a list \ lc_qual" ("_ \ _") "_lc_test" :: "bool \ lc_qual" ("_") (*"_lc_let" :: "letbinds => lc_qual" ("let _")*) "_lc_end" :: "lc_quals" ("]") "_lc_quals" :: "lc_qual \ lc_quals \ lc_quals" (", __") syntax (ASCII) "_lc_gen" :: "'a \ 'a list \ lc_qual" ("_ <- _") parse_translation \ let val NilC = Syntax.const @{const_syntax Nil}; val ConsC = Syntax.const @{const_syntax Cons}; val mapC = Syntax.const @{const_syntax map}; val concatC = Syntax.const @{const_syntax concat}; val IfC = Syntax.const @{const_syntax If}; val dummyC = Syntax.const @{const_syntax Pure.dummy_pattern} 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 "_case1"} $p$ e; val case2 = Syntax.const @{syntax_const "_case1"} $dummyC$ NilC; val cs = Syntax.const @{syntax_const "_case2"} $case1$ 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 case_prod} $pair_pat_tr p1 (pair_pat_tr p2 e) | pair_pat_tr dummy e = Syntax_Trans.abs_tr [Syntax.const "_idtdummy", e] fun pair_pat ctxt (Const (@{const_syntax "Pair"},_)$ s $t) = pair_pat ctxt s andalso pair_pat ctxt t | pair_pat ctxt (Free (s,_)) = let val thy = Proof_Context.theory_of ctxt; val s' = Proof_Context.intern_const ctxt s; in not (Sign.declared_const thy s') end | pair_pat _ t = (t = dummyC); fun abs_tr ctxt p e opti = let val p = Term_Position.strip_positions p in if pair_pat ctxt p then (pair_pat_tr p e, true) else (pat_tr ctxt p e opti, false) end fun lc_tr ctxt [e, Const (@{syntax_const "_lc_test"}, _)$ b, qs] = let val res = (case qs of Const (@{syntax_const "_lc_end"}, _) => single e | Const (@{syntax_const "_lc_quals"}, _) $q$ qs => lc_tr ctxt [e, q, qs]); in IfC $b$ res $NilC end | lc_tr ctxt [e, Const (@{syntax_const "_lc_gen"}, _)$ p $es, Const(@{syntax_const "_lc_end"}, _)] = (case abs_tr ctxt p e true of (f, true) => mapC$ f $es | (f, false) => concatC$ (mapC $f$ es)) | lc_tr ctxt [e, Const (@{syntax_const "_lc_gen"}, _) $p$ es, Const (@{syntax_const "_lc_quals"}, _) $q$ 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 "_listcompr"}, lc_tr)] end \ ML_val \ 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 (Input.source_content s1) ^ Position.here_list [Input.pos_of s1, Input.pos_of s2]); in check \[(x,y,z). b]\ \if b then [(x, y, z)] else []\; check \[(x,y,z). (x,_,y)\xs]\ \map (\(x,_,y). (x, y, z)) xs\; check \[e x y. (x,_)\xs, y\ys]\ \concat (map (\(x,_). map (\y. e x y) ys) xs)\; check \[(x,y,z). xb]\ \if x < a then if b < x then [(x, y, z)] else [] else []\; check \[(x,y,z). x\xs, x>b]\ \concat (map (\x. if b < x then [(x, y, z)] else []) xs)\; check \[(x,y,z). xxs]\ \if x < a then map (\x. (x, y, z)) xs else []\; check \[(x,y). Cons True x \ xs]\ \concat (map (\xa. case xa of [] \ [] | True # x \ [(x, y)] | False # x \ []) xs)\; check \[(x,y,z). Cons x [] \ xs]\ \concat (map (\xa. case xa of [] \ [] | [x] \ [(x, y, z)] | x # aa # lista \ []) xs)\; check \[(x,y,z). xb, x=d]\ \if x < a then if b < x then if x = d then [(x, y, z)] else [] else [] else []\; check \[(x,y,z). xb, y\ys]\ \if x < a then if b < x then map (\y. (x, y, z)) ys else [] else []\; check \[(x,y,z). xxs,y>b]\ \if x < a then concat (map (\(_,x). if b < y then [(x, y, z)] else []) xs) else []\; check \[(x,y,z). xxs, y\ys]\ \if x < a then concat (map (\x. map (\y. (x, y, z)) ys) xs) else []\; check \[(x,y,z). x\xs, x>b, y \concat (map (\x. if b < x then if y < a then [(x, y, z)] else [] else []) xs)\; check \[(x,y,z). x\xs, x>b, y\ys]\ \concat (map (\x. if b < x then map (\y. (x, y, z)) ys else []) xs)\; check \[(x,y,z). x\xs, (y,_)\ys,y>x]\ \concat (map (\x. concat (map (\(y,_). if x < y then [(x, y, z)] else []) ys)) xs)\; check \[(x,y,z). x\xs, y\ys,z\zs]\ \concat (map (\x. concat (map (\y. map (\z. (x, y, z)) zs) ys)) xs)\ end; \ ML \ (* 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 Ex}, _)$ 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 Ex}, _) $Abs (_, _, Const (@{const_name Ex}, _)$ _) => 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 Collect}, _) $Abs _ => Conv.arg_conv (Conv.abs_conv cv ctxt) ct | _ => raise CTERM ("Collect_conv", [ct])) fun rewr_conv' th = Conv.rewr_conv (mk_meta_eq th) fun conjunct_assoc_conv ct = Conv.try_conv (rewr_conv' @{thm conj_assoc} then_conv HOLogic.conj_conv Conv.all_conv conjunct_assoc_conv) ct fun right_hand_set_comprehension_conv conv ctxt = HOLogic.Trueprop_conv (HOLogic.eq_conv Conv.all_conv (Collect_conv (all_exists_conv conv o #2) ctxt)) (* term abstraction of list comprehension patterns *) datatype termlets = If | Case of typ * int local val set_Nil_I = @{lemma "set [] = {x. False}" by (simp add: empty_def [symmetric])} val set_singleton = @{lemma "set [a] = {x. x = a}" by simp} val inst_Collect_mem_eq = @{lemma "set A = {x. x \ set A}" by simp} val del_refl_eq = @{lemma "(t = t \ P) \ P" by simp} fun mk_set T = Const (@{const_name set}, HOLogic.listT T --> HOLogic.mk_setT T) fun dest_set (Const (@{const_name set}, _)$ xs) = xs fun dest_singleton_list (Const (@{const_name Cons}, _) $t$ (Const (@{const_name Nil}, _))) = 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 Nil}, _)) => 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 If}, _) $cond$ then_t $Const (@{const_name Nil}, _)) = 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 \ 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 \ 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 "(\x. x = t \ 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 \ 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 "(\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 HOL.eq}, T --> T --> @{typ bool})$ constr_t $x' in make_inner_eqs (rev vs @ bound_vs) (Case (T, i) :: Tis) (constr_eq :: eqs') body end | NONE => (case dest_if t of SOME (condition, cont) => make_inner_eqs bound_vs (If :: Tis) (condition :: eqs) cont | NONE => if null eqs then NONE (*no rewriting, nothing to be done*) else let val Type (@{type_name list}, [rT]) = fastype_of1 (map snd bound_vs, t) val pat_eq = (case try dest_singleton_list t of SOME t' => Const (@{const_name HOL.eq}, rT --> rT --> @{typ bool})$ Bound (length bound_vs) $t' | NONE => Const (@{const_name Set.member}, rT --> HOLogic.mk_setT rT --> @{typ bool})$ 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 \ simproc_setup list_to_set_comprehension ("set xs") = \K List_to_Set_Comprehension.simproc\ code_datatype set coset hide_const (open) coset subsubsection \@{const Nil} and @{const Cons}\ lemma not_Cons_self [simp]: "xs \ x # xs" by (induct xs) auto lemma not_Cons_self2 [simp]: "x # xs \ xs" by (rule not_Cons_self [symmetric]) lemma neq_Nil_conv: "(xs \ []) = (\y ys. xs = y # ys)" by (induct xs) auto lemma tl_Nil: "tl xs = [] \ xs = [] \ (\x. xs = [x])" by (cases xs) auto lemma Nil_tl: "[] = tl xs \ xs = [] \ (\x. xs = [x])" by (cases xs) auto lemma length_induct: "(\xs. \ys. length ys < length xs \ P ys \ P xs) \ P xs" by (fact measure_induct) lemma induct_list012: "\P []; \x. P [x]; \x y zs. P (y # zs) \ P (x # y # zs)\ \ P xs" by induction_schema (pat_completeness, lexicographic_order) lemma list_nonempty_induct [consumes 1, case_names single cons]: "\ xs \ []; \x. P [x]; \x xs. xs \ [] \ P xs \ P (x # xs)\ \ P xs" by(induction xs rule: induct_list012) auto lemma inj_split_Cons: "inj_on (\(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 \@{const length}\ text \ Needs to come before \@\ because of theorem \append_eq_append_conv\. \ 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 \ [])" by (induct xs) auto lemma length_pos_if_in_set: "x \ set xs \ length xs > 0" by auto lemma length_Suc_conv: "(length xs = Suc n) = (\y ys. xs = y # ys \ length ys = n)" by (induct xs) auto lemma Suc_length_conv: "(Suc n = length xs) = (\y ys. xs = y # ys \ length ys = n)" by (induct xs; simp; blast) lemma impossible_Cons: "length xs \ length ys ==> xs = x # ys = False" by (induct xs) auto lemma list_induct2 [consumes 1, case_names Nil Cons]: "length xs = length ys \ P [] [] \ (\x xs y ys. length xs = length ys \ P xs ys \ P (x#xs) (y#ys)) \ 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 \ length ys = length zs \ P [] [] [] \ (\x xs y ys z zs. length xs = length ys \ length ys = length zs \ P xs ys zs \ P (x#xs) (y#ys) (z#zs)) \ 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 \ length ys = length zs \ length zs = length ws \ P [] [] [] [] \ (\x xs y ys z zs w ws. length xs = length ys \ length ys = length zs \ length zs = length ws \ P xs ys zs ws \ P (x#xs) (y#ys) (z#zs) (w#ws)) \ 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': "\ P [] []; \x xs. P (x#xs) []; \y ys. P [] (y#ys); \x xs y ys. P xs ys \ P (x#xs) (y#ys) \ \ P xs ys" by (induct xs arbitrary: ys) (case_tac x, auto)+ lemma list_all2_iff: "list_all2 P xs ys \ length xs = length ys \ (\(x, y) \ set (zip xs ys). P x y)" by (induct xs ys rule: list_induct2') auto lemma neq_if_length_neq: "length xs \ length ys \ (xs = ys) == False" by (rule Eq_FalseI) auto simproc_setup list_neq ("(xs::'a list) = ys") = \ (* 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 Nil},_)) acc = acc | len (Const(@{const_name Cons},_)$ _ $xs) (ts,n) = len xs (ts,n+1) | len (Const(@{const_name append},_)$ xs $ys) acc = len xs (len ys acc) | len (Const(@{const_name rev},_)$ xs) acc = len xs acc | len (Const(@{const_name map},_) $_$ xs) acc = len xs 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 (aconv) (ls,rs) orelse n < m andalso submultiset (aconv) (rs,ls) then prove_neq() else NONE end; in K list_neq end; \ subsubsection \\@\ -- append\ 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 = [] \ ys = [])" by (induct xs) auto lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \ 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 \ length us = length vs ==> (xs@us = ys@vs) = (xs=ys \ us=vs)" by (induct xs arbitrary: ys; case_tac ys; force) lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) = (\us. xs = zs @ us \ us @ ys = ts \ xs @ us = zs \ ys = us @ ts)" proof (induct xs arbitrary: ys zs ts) case (Cons x xs) then show ?case by (case_tac 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 \ 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 \ [] ==> 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 \ [] ==> hd (xs @ ys) = hd xs" by (simp add: hd_append split: list.split) lemma tl_append: "tl (xs @ ys) = (case xs of [] \ tl ys | z#zs \ zs @ ys)" by (simp split: list.split) lemma tl_append2 [simp]: "xs \ [] ==> tl (xs @ ys) = tl xs @ ys" by (simp add: tl_append split: list.split) lemma Cons_eq_append_conv: "x#xs = ys@zs = (ys = [] \ x#xs = zs \ (\ys'. x#ys' = ys \ xs = ys'@zs))" by(cases ys) auto lemma append_eq_Cons_conv: "(ys@zs = x#xs) = (ys = [] \ zs = x#xs \ (\ys'. ys = x#ys' \ ys'@zs = xs))" by(cases ys) auto lemma longest_common_prefix: "\ps xs' ys'. xs = ps @ xs' \ ys = ps @ ys' \ (xs' = [] \ ys' = [] \ hd xs' \ 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 \Trivial rules for solving \@\-equations automatically.\ lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys" by simp lemma Cons_eq_appendI: "[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs" by (drule sym) simp lemma append_eq_appendI: "[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us" by (drule sym) simp text \ Simplification procedure for all list equalities. Currently only tries to rearrange \@\ to see if - both lists end in a singleton list, - or both lists end in the same list. \ simproc_setup list_eq ("(xs::'a list) = ys") = \ let fun last (cons as Const (@{const_name Cons}, _)$ _ $xs) = (case xs of Const (@{const_name Nil}, _) => cons | _ => last xs) | last (Const(@{const_name append},_)$ _ $ys) = last ys | last t = t; fun list1 (Const(@{const_name Cons},_)$ _ $Const(@{const_name Nil},_)) = true | list1 _ = false; fun butlast ((cons as Const(@{const_name Cons},_)$ x) $xs) = (case xs of Const (@{const_name Nil}, _) => xs | _ => cons$ butlast xs) | butlast ((app as Const (@{const_name append}, _) $xs)$ ys) = app $butlast ys | butlast xs = Const(@{const_name Nil}, fastype_of xs); val rearr_ss = simpset_of (put_simpset HOL_basic_ss @{context} addsimps [@{thm append_assoc}, @{thm append_Nil}, @{thm append_Cons}]); fun list_eq ctxt (F as (eq as Const(_,eqT))$ lhs $rhs) = let val lastl = last lhs and lastr = last rhs; fun rearr conv = let val lhs1 = butlast lhs and rhs1 = butlast rhs; val Type(_,listT::_) = eqT val appT = [listT,listT] ---> listT val app = Const(@{const_name append},appT) val F2 = eq$ (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; \ subsubsection \@{const map}\ lemma hd_map: "xs \ [] \ 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: "(\x. x \ set xs \ f x = g x) ==> map f xs = map g xs" by (induct xs) simp_all lemma map_ident [simp]: "map (\x. x) = (\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 \ g) xs" by (induct xs) auto lemma map_comp_map[simp]: "((map f) \ (map g)) = map(f \ 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) = (\x \ set xs. f x = g x)" by (induct xs) auto lemma map_cong [fundef_cong]: "xs = ys \ (\x. x \ set ys \ f x = g x) \ 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) = (\z zs. xs = z#zs \ f z = y \ map f zs = ys)" by (cases xs) auto lemma Cons_eq_map_conv: "(x#xs = map f ys) = (\z zs. ys = z#zs \ x = f z \ 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: "(\xs. ys = map f xs) = (\y \ set ys. \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) \ (map f xs = map f ys) = (xs = ys)" by(blast dest:map_inj_on) lemma map_injective: "map f xs = map f ys ==> inj f ==> xs = ys" by (induct ys arbitrary: xs) (auto dest!:injD) lemma inj_map_eq_map[simp]: "inj f \ (map f xs = map f ys) = (xs = ys)" by(blast dest:map_injective) lemma inj_mapI: "inj f ==> inj (map f)" by (iprover dest: map_injective injD intro: inj_onI) lemma inj_mapD: "inj (map f) ==> 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 (\(set  A)) \ inj_on (map f) A" by (blast intro:inj_onI dest:inj_onD map_inj_on) lemma map_idI: "(\x. x \ set xs \ f x = x) \ map f xs = xs" by (induct xs, auto) lemma map_fun_upd [simp]: "y \ set xs \ map (f(y:=v)) xs = map f xs" by (induct xs) auto lemma map_fst_zip[simp]: "length xs = length ys \ map fst (zip xs ys) = xs" by (induct rule:list_induct2, simp_all) lemma map_snd_zip[simp]: "length xs = length ys \ 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 (\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 \@{const rev}\ 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 (Cons a xs) then show ?case by (case_tac ys) auto qed force lemma inj_on_rev[iff]: "inj_on rev A" by(simp add:inj_on_def) lemma rev_induct [case_names Nil snoc]: "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs" apply(simplesubst rev_rev_ident[symmetric]) apply(rule_tac list = "rev xs" in list.induct, simp_all) done lemma rev_exhaust [case_names Nil snoc]: "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> 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 \ []" and single: "\x. P [x]" and snoc': "\x xs. xs \ [] \ P xs \ P (xs@[x])" shows "P xs" using \xs \ []\ 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 \@{const set}\ declare list.set[code_post] \ \pretty output\ lemma finite_set [iff]: "finite (set xs)" by (induct xs) auto lemma set_append [simp]: "set (xs @ ys) = (set xs \ set ys)" by (induct xs) auto lemma hd_in_set[simp]: "xs \ [] \ hd xs \ set xs" by(cases xs) auto lemma set_subset_Cons: "set xs \ set (x # xs)" by auto lemma set_ConsD: "y \ set (x # xs) \ y=x \ y \ 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 \ set xs \ P x}" by (induct xs) auto lemma set_upt [simp]: "set[i.. set xs \ \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 \ set xs \ (\ys zs. xs = ys @ x # zs)" by (auto elim: split_list) lemma split_list_first: "x \ set xs \ \ys zs. xs = ys @ x # zs \ x \ 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 \ a" thus ?case using Cons by(fastforce intro!: Cons_eq_appendI) qed qed lemma in_set_conv_decomp_first: "(x \ set xs) = (\ys zs. xs = ys @ x # zs \ x \ set ys)" by (auto dest!: split_list_first) lemma split_list_last: "x \ set xs \ \ys zs. xs = ys @ x # zs \ x \ 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 \ a" thus ?case using snoc by fastforce qed qed lemma in_set_conv_decomp_last: "(x \ set xs) = (\ys zs. xs = ys @ x # zs \ x \ set zs)" by (auto dest!: split_list_last) lemma split_list_prop: "\x \ set xs. P x \ \ys x zs. xs = ys @ x # zs \ 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 "\x \ 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: "\x \ set xs. P x \ \ys x zs. xs = ys@x#zs \ P x \ (\y \ set ys. \ 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 \ P x \ (\y\set []. \ P y)" by simp thus ?thesis by fast next assume "\ P x" hence "\x\set xs. P x" using Cons(2) by simp thus ?thesis using \\ P x\ Cons(1) by (metis append_Cons set_ConsD) qed qed lemma split_list_first_propE: assumes "\x \ set xs. P x" obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\y \ set ys. \ P y" using split_list_first_prop [OF assms] by blast lemma split_list_first_prop_iff: "(\x \ set xs. P x) \ (\ys x zs. xs = ys@x#zs \ P x \ (\y \ set ys. \ P y))" by (rule, erule split_list_first_prop) auto lemma split_list_last_prop: "\x \ set xs. P x \ \ys x zs. xs = ys@x#zs \ P x \ (\z \ set zs. \ 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 "\ P x" hence "\x\set xs. P x" using snoc(2) by simp thus ?thesis using \\ P x\ snoc(1) by fastforce qed qed lemma split_list_last_propE: assumes "\x \ set xs. P x" obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\z \ set zs. \ P z" using split_list_last_prop [OF assms] by blast lemma split_list_last_prop_iff: "(\x \ set xs. P x) \ (\ys x zs. xs = ys@x#zs \ P x \ (\z \ set zs. \ P z))" by rule (erule split_list_last_prop, auto) lemma finite_list: "finite A \ \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) \ length xs" by (induct xs) (auto simp add: card_insert_if) lemma set_minus_filter_out: "set xs - {y} = set (filter (\x. \ (x = y)) xs)" by (induct xs) auto lemma append_Cons_eq_iff: "\ x \ set xs; x \ set ys \ \ xs @ x # ys = xs' @ x # ys' \ (xs = xs' \ ys = ys')" by(auto simp: append_eq_Cons_conv Cons_eq_append_conv append_eq_append_conv2) subsubsection \@{const filter}\ 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 (\x. Q x \ P x) xs" by (induct xs) auto lemma length_filter_le [simp]: "length (filter P xs) \ length xs" by (induct xs) (auto simp add: le_SucI) lemma sum_length_filter_compl: "length(filter P xs) + length(filter (\x. \P x) xs) = length xs" by(induct xs) simp_all lemma filter_True [simp]: "\x \ set xs. P x ==> filter P xs = xs" by (induct xs) auto lemma filter_False [simp]: "\x \ set xs. \ P x ==> filter P xs = []" by (induct xs) auto lemma filter_empty_conv: "(filter P xs = []) = (\x\set xs. \ P x)" by (induct xs) simp_all lemma filter_id_conv: "(filter P xs = xs) = (\x\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 \ f) xs)" by (induct xs) simp_all lemma length_filter_map[simp]: "length (filter P (map f xs)) = length(filter (P \ f) xs)" by (simp add:filter_map) lemma filter_is_subset [simp]: "set (filter P xs) \ set xs" by auto lemma length_filter_less: "\ x \ set xs; \ P x \ \ 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 \ p(xs!i)}" proof (induct xs) case Nil thus ?case by simp next case (Cons x xs) let ?S = "{i. i < length xs \ 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 \p x\ by simp also have "\ = Suc(card(Suc  ?S))" using fin by (simp add: card_image) also have "\ = card ?S'" using eq fin by (simp add:card_insert_if) (simp add:image_def) finally show ?thesis . next assume "\ 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 \\ p x\ by simp also have "\ = card(Suc  ?S)" using fin by (simp add: card_image) also have "\ = 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 \ \us vs. ys = us @ x # vs \ (\u\set us. \ P u) \ P x \ xs = filter P vs" (is "_ \ \us vs. ?P ys us vs") proof(induct ys) case Nil thus ?case by simp next case (Cons y ys) show ?case (is "\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 \ y" with Py Cons.prems show ?thesis by simp qed next assume "\ 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 \ \us vs. ys = us @ x # vs \ (\u\set us. \ P u) \ P x \ xs = filter P vs" by(rule Cons_eq_filterD) simp lemma filter_eq_Cons_iff: "(filter P ys = x#xs) = (\us vs. ys = us @ x # vs \ (\u\set us. \ P u) \ P x \ xs = filter P vs)" by(auto dest:filter_eq_ConsD) lemma Cons_eq_filter_iff: "(x#xs = filter P ys) = (\us vs. ys = us @ x # vs \ (\u\set us. \ P u) \ P x \ 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 (\x. f y = f x) xs = filter (HOL.eq y) xs" using assms by (induct xs) auto lemma filter_cong[fundef_cong]: "xs = ys \ (\x. x \ set ys \ P x = Q x) \ filter P xs = filter Q ys" by (induct ys arbitrary: xs) auto subsubsection \List partitioning\ primrec partition :: "('a \ bool) \'a list \ 'a list \ '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 \ P) xs" by (induct xs) (auto simp add: Let_def split_def) lemma partition_P: assumes "partition P xs = (yes, no)" shows "(\p \ set yes. P p) \ (\p \ set no. \ 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 \ 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 \ f) xs)" unfolding partition_filter2[symmetric] unfolding partition_filter1[symmetric] by simp declare partition.simps[simp del] subsubsection \@{const concat}\ lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys" by (induct xs) auto lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\xs \ set xss. xs = [])" by (induct xss) auto lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\xs \ set xss. xs = [])" by (induct xss) auto lemma set_concat [simp]: "set (concat xs) = (UN x: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 filter_concat: "filter p (concat xs) = concat (map (filter p) xs)" by (induct xs) auto lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))" by (induct xs) auto lemma concat_eq_concat_iff: "\(x, y) \ set (zip xs ys). length x = length y ==> length xs = length ys ==> (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 ==> length xs = length ys ==> \(x, y) \ set (zip xs ys). length x = length y ==> xs = ys" by (simp add: concat_eq_concat_iff) subsubsection \@{const nth}\ 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 \ (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 ==> (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) \ tl xs ! n = xs ! Suc n" by (induction xs) auto lemma hd_conv_nth: "xs \ [] \ hd xs = xs!0" by(cases xs) simp_all lemma list_eq_iff_nth_eq: "(xs = ys) = (length xs = length ys \ (\i ?R" by force show "?R \ ?L" using less_Suc_eq_0_disj by auto qed with Cons show ?case by simp qed simp lemma in_set_conv_nth: "(x \ set xs) = (\i < length xs. xs!i = x)" by(auto simp:set_conv_nth) lemma nth_equal_first_eq: assumes "x \ set xs" assumes "n \ length xs" shows "(x # xs) ! n = x \ n = 0" (is "?lhs \ ?rhs") proof assume ?lhs show ?rhs proof (rule ccontr) assume "n \ 0" then have "n > 0" by simp with \?lhs\ have "xs ! (n - 1) = x" by simp moreover from \n > 0\ \n \ length xs\ have "n - 1 < length xs" by simp ultimately have "\ix \ set xs\ 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 \ y" shows "(x # xs) ! n = y \ xs ! (n - 1) = y \ n > 0" (is "?lhs \ ?rhs") proof assume "?lhs" with assms have "n > 0" by (cases n) simp_all with \?lhs\ show ?rhs by simp next assume "?rhs" then show "?lhs" by simp qed lemma list_ball_nth: "\n < length xs; \x \ set xs. P x\ \ P(xs!n)" by (auto simp add: set_conv_nth) lemma nth_mem [simp]: "n < length xs \ xs!n \ set xs" by (auto simp add: set_conv_nth) lemma all_nth_imp_all_set: "\\i < length xs. P(xs!i); x \ set xs\ \ P x" by (auto simp add: set_conv_nth) lemma all_set_conv_all_nth: "(\x \ set xs. P x) = (\i. i < length xs \ P (xs ! i))" by (auto simp add: set_conv_nth) lemma rev_nth: "n < size xs \ 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: "(\ix. P i x) = (\xs. size xs = k \ (\ixs. ?P k xs)") proof(induct k) case 0 show ?case by simp next case (Suc k) show ?case (is "?L = ?R" is "_ = (\xs. ?P' xs)") proof assume "?R" thus "?L" using Suc by auto next assume "?L" with Suc obtain x xs where "?P k xs \ 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 \@{const list_update}\ 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==> (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 ==> (xs[i:=x])!i = x" by (simp add: nth_list_update) lemma nth_list_update_neq [simp]: "i \ j ==> 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 \ i \ 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] = [] \ xs=[]" by (simp only: length_0_conv[symmetric] length_list_update) lemma list_update_same_conv: "i < length xs ==> (xs[i := x] = xs) = (xs!i = x)" by (induct xs arbitrary: i) (auto split: nat.split) lemma list_update_append1: "i < size xs \ (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 \ 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]) \ insert x (set xs)" by (induct xs arbitrary: i) (auto split: nat.split) lemma set_update_subsetI: "\set xs \ A; x \ A\ \ set(xs[i := x]) \ A" by (blast dest!: set_update_subset_insert [THEN subsetD]) lemma set_update_memI: "n < length xs \ x \ 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 \ i' \ 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 \@{const last} and @{const butlast}\ 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 = [] \ last(x#xs) = x" by simp lemma last_ConsR: "xs \ [] \ 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 = [] \ last(xs @ ys) = last xs" by(simp add:last_append) lemma last_appendR[simp]: "ys \ [] \ last(xs @ ys) = last ys" by(simp add:last_append) lemma last_tl: "xs = [] \ tl xs \ [] \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 \ [] \ hd(rev xs) = last xs" by(rule rev_exhaust[of xs]) simp_all lemma last_rev: "xs \ [] \ last(rev xs) = hd xs" by(cases xs) simp_all lemma last_in_set[simp]: "as \ [] \ last as \ 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 \ [] \ butlast xs @ [last xs] = xs" by (induct xs) auto lemma in_set_butlastD: "x \ set (butlast xs) \ x \ set xs" by (induct xs) (auto split: if_split_asm) lemma in_set_butlast_appendI: "x \ set (butlast xs) \ x \ set (butlast ys) \ x \ set (butlast (xs @ ys))" by (auto dest: in_set_butlastD simp add: butlast_append) lemma last_drop[simp]: "n < length xs \ 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\[] \ 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 \ [] \ 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 \ [] \ 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 \ (ys \ [] \ butlast ys = xs \ last ys = x)" by fastforce corollary longest_common_suffix: "\ss xs' ys'. xs = xs' @ ss \ ys = ys' @ ss \ (xs' = [] \ ys' = [] \ last xs' \ last ys')" using longest_common_prefix[of "rev xs" "rev ys"] unfolding rev_swap rev_append by (metis last_rev rev_is_Nil_conv) subsubsection \@{const take} and @{const drop}\ 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 = (\xs. [])" by(rule ext) (rule take_0) lemma drop0[simp]: "drop 0 = (\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 \ [] \ 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 \ 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 \ 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 \ take (Suc i) xs = take i xs @ [xs!i]" proof (induct xs arbitrary: i) case (Cons x xs) then show ?case by (case_tac i, auto) qed simp lemma Cons_nth_drop_Suc: "i < length xs \ (xs!i) # (drop (Suc i) xs) = drop i xs" proof (induct xs arbitrary: i) case (Cons x xs) then show ?case by (case_tac i, auto) qed simp 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 \ n ==> take n xs = xs" by (induct n arbitrary: xs) (auto, case_tac xs, auto) lemma drop_all [simp]: "length xs \ n ==> drop n xs = []" by (induct n arbitrary: xs) (auto, case_tac xs, auto) 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 (Suc m) then show ?case by (case_tac xs; case_tac n; simp) qed auto lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs" proof (induct m arbitrary: xs) case (Suc m) then show ?case by (case_tac xs; simp) qed auto lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)" proof (induct m arbitrary: xs n) case (Suc m) then show ?case by (case_tac xs; case_tac n; simp) qed auto 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 (Suc n) then show ?case by (case_tac xs; simp) qed auto lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \ xs = [])" by(induct xs arbitrary: n)(auto simp: take_Cons split:nat.split) lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs \ 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 (Suc n) then show ?case by (case_tac xs; simp) qed auto lemma drop_map: "drop n (map f xs) = map f (drop n xs)" proof (induct n arbitrary: xs) case (Suc n) then show ?case by (case_tac xs; simp) qed auto lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)" proof (induct xs arbitrary: i) case (Cons x xs) then show ?case by (case_tac i, auto) qed simp lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)" proof (induct xs arbitrary: i) case (Cons x xs) then show ?case by (case_tac i, auto) qed simp 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 ==> (take n xs)!i = xs!i" proof (induct xs arbitrary: i n) case (Cons x xs) then show ?case by (case_tac n; case_tac i; simp) qed auto lemma nth_drop [simp]: "n \ length xs ==> (drop n xs)!i = xs!(n + i)" proof (induct n arbitrary: xs) case (Suc n) then show ?case by (case_tac xs; simp) qed auto lemma butlast_take: "n \ length xs ==> 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 ==> 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 hd_drop_conv_nth: "n < length xs \ hd(drop n xs) = xs!n" by(simp add: hd_conv_nth) lemma set_take_subset_set_take: "m \ n \ set(take m xs) \ 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) \ set xs" by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split) lemma set_drop_subset: "set(drop n xs) \ set xs" by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split) lemma set_drop_subset_set_drop: "m \ n \ set(drop m xs) \ 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 \ set(take n xs) \ x \ set xs" using set_take_subset by fast lemma in_set_dropD: "x \ set(drop n xs) \ x \ set xs" using set_drop_subset by fast lemma append_eq_conv_conj: "(xs @ ys = zs) = (xs = take (length xs) zs \ 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 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 \ size ys\<^sub>1 then xs\<^sub>1 = take (size xs\<^sub>1) ys\<^sub>1 \ 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 \ 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 \ 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 \ 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 \ m \ take n (xs[m := y]) = take n xs" by(simp add: list_eq_iff_nth_eq) lemma drop_update_cancel[simp]: "n < m \ 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 \ 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 "\ = 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 \ 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 \ n" shows "drop m (xs[n := x]) = (drop m xs)[n-m := x]" proof (cases "n \ length xs") case False with assms show ?thesis by (simp add: upd_conv_take_nth_drop drop_take) qed auto lemma nth_image: "l \ size xs \ nth xs  {0..@{const takeWhile} and @{const dropWhile}\ lemma length_takeWhile_le: "length (takeWhile P xs) \ 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]: "\x \ set xs; \P(x)\ \ takeWhile P (xs @ ys) = takeWhile P xs" by (induct xs) auto lemma takeWhile_append2 [simp]: "(\x. x \ set xs \ P x) \ takeWhile P (xs @ ys) = xs @ takeWhile P ys" by (induct xs) auto lemma takeWhile_tail: "\ P x \ takeWhile P (xs @ (x#l)) = takeWhile P xs" by (induct xs) auto lemma takeWhile_nth: "j < length (takeWhile P xs) \ takeWhile P xs ! j = xs ! j" by (metis nth_append takeWhile_dropWhile_id) lemma dropWhile_nth: "j < length (dropWhile P xs) \ 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) \ length xs" by (induct xs) auto lemma dropWhile_append1 [simp]: "\x \ set xs; \P(x)\ \ dropWhile P (xs @ ys) = (dropWhile P xs)@ys" by (induct xs) auto lemma dropWhile_append2 [simp]: "(\x. x \ set xs \ P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys" by (induct xs) auto lemma dropWhile_append3: "\ P y \dropWhile P (xs @ y # ys) = dropWhile P xs @ y # ys" by (induct xs) auto lemma dropWhile_last: "x \ set xs \ \ P x \ last (dropWhile P xs) = last xs" by (auto simp add: dropWhile_append3 in_set_conv_decomp) lemma set_dropWhileD: "x \ set (dropWhile P xs) \ x \ set xs" by (induct xs) (auto split: if_split_asm) lemma set_takeWhileD: "x \ set (takeWhile P xs) \ x \ set xs \ P x" by (induct xs) (auto split: if_split_asm) lemma takeWhile_eq_all_conv[simp]: "(takeWhile P xs = xs) = (\x \ set xs. P x)" by(induct xs, auto) lemma dropWhile_eq_Nil_conv[simp]: "(dropWhile P xs = []) = (\x \ set xs. P x)" by(induct xs, auto) lemma dropWhile_eq_Cons_conv: "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys \ \ P y)" by(induct xs, auto) lemma distinct_takeWhile[simp]: "distinct xs ==> distinct (takeWhile P xs)" by (induct xs) (auto dest: set_takeWhileD) lemma distinct_dropWhile[simp]: "distinct xs ==> distinct (dropWhile P xs)" by (induct xs) auto lemma takeWhile_map: "takeWhile P (map f xs) = map f (takeWhile (P \ f) xs)" by (induct xs) auto lemma dropWhile_map: "dropWhile P (map f xs) = map f (dropWhile (P \ 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 \ [] \ \ P (hd (dropWhile P xs))" by (induct xs) auto lemma takeWhile_eq_filter: assumes "\ x. x \ set (dropWhile P xs) \ \ 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: "\ \ i. \ i < n ; i < length xs \ \ P (xs ! i) ; n < length xs \ \ P (xs ! n) \ \ 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 "\ 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 \ \ P (xs ! length (takeWhile P xs))" by (induct xs) auto lemma length_takeWhile_less_P_nth: assumes all: "\ i. i < j \ P (xs ! i)" and "j \ length xs" shows "j \ length (takeWhile P xs)" proof (rule classical) assume "\ ?thesis" hence "length (takeWhile P xs) < length xs" using assms by simp thus ?thesis using all \\ ?thesis\ nth_length_takeWhile[of P xs] by auto qed lemma takeWhile_neq_rev: "\distinct xs; x \ set xs\ \ takeWhile (\y. y \ x) (rev xs) = rev (tl (dropWhile (\y. y \ x) xs))" by(induct xs) (auto simp: takeWhile_tail[where l="[]"]) lemma dropWhile_neq_rev: "\distinct xs; x \ set xs\ \ dropWhile (\y. y \ x) (rev xs) = x # rev (takeWhile (\y. y \ 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 \ takeWhile (\y. y \ last xs) xs = butlast xs" by(induction xs rule: induct_list012) auto lemma takeWhile_cong [fundef_cong]: "\l = k; \x. x \ set l \ P x = Q x\ \ takeWhile P l = takeWhile Q k" by (induct k arbitrary: l) (simp_all) lemma dropWhile_cong [fundef_cong]: "\l = k; \x. x \ set l \ P x = Q x\ \ 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 subsubsection \@{const zip}\ 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 [] \ [] | y#ys \ (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 "\zs ws n. length zs = length ws \ n = min (length xs) (length ys) \ zs = take n xs \ ws = take n ys \ 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]: "[| length xs = length us |] ==> zip (xs@ys) (us@vs) = zip xs us @ zip ys vs" by (simp add: zip_append1) lemma zip_rev: "length xs = length ys ==> 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 (\ (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 (\(x, y). (f x, y)) (zip xs ys)" using zip_map_map[of f xs "\x. x" ys] by simp lemma zip_map2: "zip xs (map f ys) = map (\(x, y). (x, f y)) (zip xs ys)" using zip_map_map[of "\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\Courtesy of Andreas Lochbihler:\ lemma zip_same_conv_map: "zip xs xs = map (\x. (x, x)) xs" by(induct xs) auto lemma nth_zip [simp]: "[| i < length xs; i < length ys|] ==> (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) \ set (zip xs xs)) = (a \ set xs \ 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 (Suc n) then show ?case by (case_tac xs; case_tac ys; simp) qed simp lemma drop_zip: "drop n (zip xs ys) = zip (drop n xs) (drop n ys)" proof (induct n arbitrary: xs ys) case (Suc n) then show ?case by (case_tac xs; case_tac ys; simp) qed simp lemma zip_takeWhile_fst: "zip (takeWhile P xs) ys = takeWhile (P \ fst) (zip xs ys)" proof (induct xs arbitrary: ys) case (Cons x xs) thus ?case by (cases ys) auto qed simp lemma zip_takeWhile_snd: "zip xs (takeWhile P ys) = takeWhile (P \ snd) (zip xs ys)" proof (induct xs arbitrary: ys) case (Cons x xs) thus ?case by (cases ys) auto qed simp lemma set_zip_leftD: "(x,y)\ set (zip xs ys) \ x \ set xs" by (induct xs ys rule:list_induct2') auto lemma set_zip_rightD: "(x,y)\ set (zip xs ys) \ y \ set ys" by (induct xs ys rule:list_induct2') auto lemma in_set_zipE: "(x,y) \ set(zip xs ys) \ (\ x \ set xs; y \ set ys \ \ R) \ 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 \ zip xs ys = zs \ map fst zs = xs \ map snd zs = ys" by (auto simp add: zip_map_fst_snd) lemma in_set_zip: "p \ set (zip xs ys) \ (\n. xs ! n = fst p \ ys ! n = snd p \ n < length xs \ 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 \ set xs" obtains y where "(x, y) \ set (zip xs ys)" proof - from assms have "x \ 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 \ set ys" obtains x where "(x, y) \ set (zip xs ys)" proof - from assms have "y \ set (map snd (zip xs ys))" by simp from this that show ?thesis by fastforce 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 subsubsection \@{const list_all2}\ lemma list_all2_lengthD [intro?]: "list_all2 P xs ys ==> 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 \ list_all2 P xs ys)" by (auto simp add: list_all2_iff) lemma list_all2_Cons1: "list_all2 P (x # xs) ys = (\z zs. ys = z # zs \ P x z \ list_all2 P xs zs)" by (cases ys) auto lemma list_all2_Cons2: "list_all2 P xs (y # ys) = (\z zs. xs = z # zs \ P z y \ 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: "\x xs y ys. \P x y; list_all2 P xs ys; R xs ys\ \ 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 = (\us vs. zs = us @ vs \ length us = length xs \ length vs = length ys \ list_all2 P xs us \ 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) = (\us vs. xs = us @ vs \ length us = length ys \ length vs = length zs \ list_all2 P us ys \ 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 \ list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \ list_all2 P us vs)" by (induct rule:list_induct2, simp_all) lemma list_all2_appendI [intro?, trans]: "\ list_all2 P a b; list_all2 P c d \ \ 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 \ (\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 ==> P2 b c ==> P3 a c" shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> 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 \ (\n. n < length a \ P (a!n) (b!n)) \ list_all2 P a b" by (simp add: list_all2_conv_all_nth) lemma list_all2I: "\x \ set (zip a b). case_prod P x \ length a = length b \ list_all2 P a b" by (simp add: list_all2_iff) lemma list_all2_nthD: "\ list_all2 P xs ys; p < size xs \ \ P (xs!p) (ys!p)" by (simp add: list_all2_conv_all_nth) lemma list_all2_nthD2: "\list_all2 P xs ys; p < size ys\ \ 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 (\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 (\x y. P x (f y)) as bs" by (auto simp add: list_all2_conv_all_nth) lemma list_all2_refl [intro?]: "(\x. P x x) \ list_all2 P xs xs" by (simp add: list_all2_conv_all_nth) lemma list_all2_update_cong: "\ list_all2 P xs ys; P x y \ \ 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 \ 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 \ 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 \ (\xs ys. P xs ys \ Q xs ys) \ list_all2 Q xs ys" by (rule list.rel_mono_strong) lemma list_all2_eq: "xs = ys \ list_all2 (=) xs ys" by (induct xs ys rule: list_induct2') auto lemma list_eq_iff_zip_eq: "xs = ys \ length xs = length ys \ (\(x,y) \ 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 \ (\x\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 (\((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 (\(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 (\(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 (\x. (x, y)) (take n xs)" by(subst zip_commute)(simp add: zip_replicate1) subsubsection \@{const List.product} and @{const product_lists}\ lemma product_concat_map: "List.product xs ys = concat (map (\x. map (\y. (x,y)) ys) xs)" by(induction xs) (simp)+ lemma set_product[simp]: "set (List.product xs ys) = set xs \ 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 \ set (product_lists xss) \ length xs = length xss" by (induct xss arbitrary: xs) auto lemma product_lists_set: "set (product_lists xss) = {xs. list_all2 (\x ys. x \ set ys) xs xss}" (is "?L = Collect ?R") proof (intro equalityI subsetI, unfold mem_Collect_eq) fix xs assume "xs \ ?L" then have "length xs = length xss" by (rule in_set_product_lists_length) from this \xs \ ?L\ show "?R xs" by (induct xs xss rule: list_induct2) auto next fix xs assume "?R xs" then show "xs \ ?L" by induct auto qed subsubsection \@{const fold} with natural argument order\ lemma fold_simps [code]: \ \eta-expanded variant for generated code -- enables tail-recursion optimisation in Scala\ "fold f [] s = s" "fold f (x # xs) s = fold f xs (f x s)" by simp_all lemma fold_remove1_split: "\ \x y. x \ set xs \ y \ set xs \ f x \ f y = f y \ f x; x \ set xs \ \ fold f xs = fold f (remove1 x xs) \ f x" by (induct xs) (auto simp add: comp_assoc) lemma fold_cong [fundef_cong]: "a = b \ xs = ys \ (\x. x \ set xs \ f x = g x) \ fold f xs a = fold g ys b" by (induct ys arbitrary: a b xs) simp_all lemma fold_id: "(\x. x \ set xs \ f x = id) \ fold f xs = id" by (induct xs) simp_all lemma fold_commute: "(\x. x \ set xs \ h \ g x = f x \ h) \ h \ fold g xs = fold f xs \ h" by (induct xs) (simp_all add: fun_eq_iff) lemma fold_commute_apply: assumes "\x. x \ set xs \ h \ g x = f x \ h" shows "h (fold g xs s) = fold f xs (h s)" proof - from assms have "h \ fold g xs = fold f xs \ h" by (rule fold_commute) then show ?thesis by (simp add: fun_eq_iff) qed lemma fold_invariant: "\ \x. x \ set xs \ Q x; P s; \x s. Q x \ P s \ P (f x s) \ \ P (fold f xs s)" by (induct xs arbitrary: s) simp_all lemma fold_append [simp]: "fold f (xs @ ys) = fold f ys \ fold f xs" by (induct xs) simp_all lemma fold_map [code_unfold]: "fold g (map f xs) = fold (g \ f) xs" by (induct xs) simp_all lemma fold_filter: "fold f (filter P xs) = fold (\x. if P x then f x else id) xs" by (induct xs) simp_all lemma fold_rev: "(\x y. x \ set xs \ y \ set xs \ f y \ f x = f x \ f y) \ 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 \@{const Finite_Set.fold} and @{const fold}\ 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 \ 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 \ A = List.coset (List.filter (\x. x \ 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 (\x. x \ A) xs)" by auto lemma inter_set_filter [code]: "A \ set xs = set (List.filter (\x. x \ A) xs)" by auto lemma inter_coset_fold [code]: "A \ 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 \ 'a \ '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 \ 'a \ '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: "INFIMUM (set xs) f = fold (inf \ f) xs top" using Inf_set_fold [of "map f xs "] by (simp add: fold_map) lemma (in complete_lattice) SUP_set_fold: "SUPREMUM (set xs) f = fold (sup \ f) xs bot" using Sup_set_fold [of "map f xs "] by (simp add: fold_map) subsubsection \Fold variants: @{const foldr} and @{const foldl}\ text \Correspondence\ 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 (\x s. f s x) xs s" by (induct xs arbitrary: s) simp_all lemma foldr_conv_foldl: \ \The Third Duality Theorem'' in Bird \& Wadler:\ "foldr f xs a = foldl (\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 (\x y. f y x) (rev xs) a" by (simp add: foldr_conv_fold foldl_conv_fold) lemma foldr_fold: "(\x y. x \ set xs \ y \ set xs \ f y \ f x = f x \ f y) \ foldr f xs = fold f xs" unfolding foldr_conv_fold by (rule fold_rev) lemma foldr_cong [fundef_cong]: "a = b \ l = k \ (\a x. x \ set l \ f x a = g x a) \ 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 \ l = k \ (\a x. x \ set l \ f a x = g a x) \ foldl f a l = foldl g b k" by (auto simp add: foldl_conv_fold intro!: fold_cong) lemma