summary |
shortlog |
changelog |
graph |
tags |
bookmarks |
branches |
files |
changeset |
file |
latest |
revisions |
annotate |
diff |
comparison |
raw |
help

src/HOL/List.thy

author | blanchet |

Fri, 18 Oct 2013 10:43:20 +0200 | |

changeset 54147 | 97a8ff4e4ac9 |

parent 53954 | ccfd22f937be |

child 54295 | 45a5523d4a63 |

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

killed most "no_atp", to make Sledgehammer more complete

(* Title: HOL/List.thy Author: Tobias Nipkow *) header {* The datatype of finite lists *} theory List imports Presburger Code_Numeral Quotient ATP Lifting_Set Lifting_Option Lifting_Product begin datatype 'a list = Nil ("[]") | Cons 'a "'a list" (infixr "#" 65) syntax -- {* list Enumeration *} "_list" :: "args => 'a list" ("[(_)]") translations "[x, xs]" == "x#[xs]" "[x]" == "x#[]" subsection {* Basic list processing functions *} primrec hd :: "'a list \<Rightarrow> 'a" where "hd (x # xs) = x" primrec tl :: "'a list \<Rightarrow> 'a list" where "tl [] = []" | "tl (x # xs) = xs" primrec 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)" primrec set :: "'a list \<Rightarrow> 'a set" where "set [] = {}" | "set (x # xs) = insert x (set xs)" definition coset :: "'a list \<Rightarrow> 'a set" where [simp]: "coset xs = - set xs" primrec map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list" where "map f [] = []" | "map f (x # xs) = f x # map f 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)" syntax -- {* Special syntax for filter *} "_filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_<-_./ _])") translations "[x<-xs . P]"== "CONST filter (%x. P) xs" syntax (xsymbols) "_filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])") syntax (HTML output) "_filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])") 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" definition (in monoid_add) listsum :: "'a list \<Rightarrow> 'a" where "listsum xs = foldr plus xs 0" 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)" -- {*Warning: simpset does not contain this definition, but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} 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)" -- {*Warning: simpset does not contain this definition, but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} primrec nth :: "'a list => nat => 'a" (infixl "!" 100) where nth_Cons: "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)" -- {*Warning: simpset does not contain this definition, but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} 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" ("_/[(_)]" [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 \<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 [] => [] | z # zs => (z, y) # zip zs ys)" -- {*Warning: simpset does not contain this definition, but separate theorems for @{text "xs = []"} and @{text "xs = z # zs"} *} 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 <= 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)" hide_const (open) insert hide_fact (open) insert_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)" hide_const (open) find primrec those :: "'a option list \<Rightarrow> 'a list option" where "those [] = Some []" | "those (x # xs) = (case x of None \<Rightarrow> None | Some y \<Rightarrow> Option.map (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" 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 {* Function @{text size} is overloaded for all datatypes. Users may refer to the list version as @{text length}. *} 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 list_all2 :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool" where "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))" definition sublist :: "'a list => nat set => 'a list" where "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))" primrec sublists :: "'a list \<Rightarrow> 'a list list" where "sublists [] = [[]]" | "sublists (x#xs) = (let xss = sublists 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 fun splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where "splice [] ys = ys" | "splice xs [] = xs" | "splice (x#xs) (y#ys) = x # y # splice xs ys" 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 (\<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 "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 "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.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 "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 "sublist [a,b,c,d,e] {0,2,3} = [a,c,d]" by (simp add:sublist_def)}\\ @{lemma "sublists [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 "listsum [1,2,3::nat] = 6" by (simp add: listsum_def)} \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. *} context linorder begin inductive sorted :: "'a list \<Rightarrow> bool" where Nil [iff]: "sorted []" | Cons: "\<forall>y\<in>set xs. x \<le> y \<Longrightarrow> sorted xs \<Longrightarrow> sorted (x # xs)" lemma sorted_single [iff]: "sorted [x]" by (rule sorted.Cons) auto lemma sorted_many: "x \<le> y \<Longrightarrow> sorted (y # zs) \<Longrightarrow> sorted (x # y # zs)" by (rule sorted.Cons) (cases "y # zs" rule: sorted.cases, auto) lemma sorted_many_eq [simp, code]: "sorted (x # y # zs) \<longleftrightarrow> x \<le> y \<and> sorted (y # zs)" by (auto intro: sorted_many elim: sorted.cases) lemma [code]: "sorted [] \<longleftrightarrow> True" "sorted [x] \<longleftrightarrow> True" by simp_all 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)" end subsubsection {* List comprehension *} text{* Input syntax for Haskell-like list comprehension notation. Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"}, the list of all pairs of distinct elements from @{text xs} and @{text ys}. The syntax is as in Haskell, except that @{text"|"} becomes a dot (like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than \verb![e| x <- xs, ...]!. The qualifiers after the dot are \begin{description} \item[generators] @{text"p \<leftarrow> xs"}, where @{text p} is a pattern and @{text xs} an expression of list type, or \item[guards] @{text"b"}, where @{text 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 @{text"[e. x \<leftarrow> 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 \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list" ("[_ . __") "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ <- _") "_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" (", __") "_lc_abs" :: "'a => 'b list => 'b list" (* These are easier than ML code but cannot express the optimized translation of [e. p<-xs] translations "[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)" "_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)" => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)" "[e. P]" => "if P then [e] else []" "_listcompr e (_lc_test P) (_lc_quals Q Qs)" => "if P then (_listcompr e Q Qs) else []" "_listcompr e (_lc_let b) (_lc_quals Q Qs)" => "_Let b (_listcompr e Q Qs)" *) syntax (xsymbols) "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _") syntax (HTML output) "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _") 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}; 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"} $ Syntax.const @{const_syntax dummy_pattern} $ 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 abs_tr ctxt p e opti = (case Term_Position.strip_positions p of Free (s, T) => let val thy = Proof_Context.theory_of ctxt; val s' = Proof_Context.intern_const ctxt s; in if Sign.declared_const thy s' then (pat_tr ctxt p e opti, false) else (Syntax_Trans.abs_tr [p, e], true) end | _ => (pat_tr ctxt p e opti, false)); 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}; fun check s1 s2 = read s1 aconv read s2 orelse error ("Check failed: " ^ quote s1); in check "[(x,y,z). b]" "if b then [(x, y, z)] else []"; check "[(x,y,z). x\<leftarrow>xs]" "map (\<lambda>x. (x, y, z)) xs"; check "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]" "concat (map (\<lambda>x. map (\<lambda>y. e x y) ys) xs)"; check "[(x,y,z). x<a, x>b]" "if x < a then if b < x then [(x, y, z)] else [] else []"; check "[(x,y,z). x\<leftarrow>xs, x>b]" "concat (map (\<lambda>x. if b < x then [(x, y, z)] else []) xs)"; check "[(x,y,z). x<a, x\<leftarrow>xs]" "if x < a then map (\<lambda>x. (x, y, z)) xs else []"; check "[(x,y). Cons True x \<leftarrow> xs]" "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | True # x \<Rightarrow> [(x, y)] | False # x \<Rightarrow> []) xs)"; check "[(x,y,z). Cons x [] \<leftarrow> xs]" "concat (map (\<lambda>xa. case xa of [] \<Rightarrow> [] | [x] \<Rightarrow> [(x, y, z)] | x # aa # lista \<Rightarrow> []) xs)"; check "[(x,y,z). x<a, x>b, x=d]" "if x < a then if b < x then if x = d then [(x, y, z)] else [] else [] else []"; check "[(x,y,z). x<a, x>b, y\<leftarrow>ys]" "if x < a then if b < x then map (\<lambda>y. (x, y, z)) ys else [] else []"; check "[(x,y,z). x<a, x\<leftarrow>xs,y>b]" "if x < a then concat (map (\<lambda>x. if b < y then [(x, y, z)] else []) xs) else []"; check "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]" "if x < a then concat (map (\<lambda>x. map (\<lambda>y. (x, y, z)) ys) xs) else []"; check "[(x,y,z). x\<leftarrow>xs, x>b, y<a]" "concat (map (\<lambda>x. if b < x then if y < a then [(x, y, z)] else [] else []) xs)"; check "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]" "concat (map (\<lambda>x. if b < x then map (\<lambda>y. (x, y, z)) ys else []) xs)"; check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]" "concat (map (\<lambda>x. concat (map (\<lambda>y. if x < y then [(x, y, z)] else []) ys)) xs)"; check "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]" "concat (map (\<lambda>x. concat (map (\<lambda>y. map (\<lambda>z. (x, y, z)) zs) ys)) xs)" end; *} (* term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]" *) 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 HOL.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 HOL.Ex}, _) $ Abs (_, _, Const (@{const_name HOL.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 Set.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) fun simproc ctxt redex = let val thy = Proof_Context.theory_of ctxt val set_Nil_I = @{thm trans} OF [@{thm set.simps(1)}, @{thm empty_def}] 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 List.set}, HOLogic.listT T --> HOLogic.mk_setT T) fun dest_set (Const (@{const_name List.set}, _) $ xs) = xs fun dest_singleton_list (Const (@{const_name List.Cons}, _) $ t $ (Const (@{const_name List.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 List.Nil}, _)) => s | _ => (case s of SOME NONE => SOME (SOME i) | _ => NONE)) in fold_index check cases (SOME NONE) |> the_default NONE end (* returns (case_expr type index chosen_case) option *) fun dest_case case_term = let val (case_const, args) = strip_comb case_term in (case try dest_Const case_const of SOME (c, T) => (case Datatype.info_of_case thy c of SOME _ => (case possible_index_of_singleton_case (fst (split_last args)) of SOME i => let val (Ts, _) = strip_type T val T' = List.last Ts in SOME (List.last args, T', i, nth args i) end | NONE => NONE) | NONE => NONE) | NONE => 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 fun tac _ [] = rtac set_singleton 1 ORELSE rtac inst_Collect_mem_eq 1 | tac ctxt (If :: cont) = Splitter.split_tac [@{thm split_if}] 1 THEN rtac @{thm conjI} 1 THEN rtac @{thm impI} 1 THEN Subgoal.FOCUS (fn {prems, context, ...} => 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})) context) 1) ctxt 1 THEN tac ctxt cont THEN rtac @{thm impI} 1 THEN Subgoal.FOCUS (fn {prems, context, ...} => 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})) context) 1) ctxt 1 THEN rtac set_Nil_I 1 | tac ctxt (Case (T, i) :: cont) = let val info = Datatype.the_info thy (fst (dest_Type T)) in (* do case distinction *) Splitter.split_tac [#split info] 1 THEN EVERY (map_index (fn (i', _) => (if i' < length (#case_rewrites info) - 1 then rtac @{thm conjI} 1 else all_tac) THEN REPEAT_DETERM (rtac @{thm allI} 1) THEN rtac @{thm impI} 1 THEN (if i' = i then (* continue recursively *) Subgoal.FOCUS (fn {prems, context, ...} => 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 (#inject info))))) Conv.all_conv) then_conv (Conv.try_conv (Conv.rewr_conv del_refl_eq)) then_conv conjunct_assoc_conv)) context 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 "(EX x. x = t & P x) = P t" by simp})) ctxt)) context)))) 1) ctxt 1 THEN tac ctxt cont else Subgoal.FOCUS (fn {prems, context, ...} => 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}) (#distinct info)))) Conv.all_conv then_conv (rewr_conv' @{lemma "(False & P) = False" by simp}))) context 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 "(EX x. P) = P" by simp})) ctxt)) context))) 1) ctxt 1 THEN rtac set_Nil_I 1)) (#case_rewrites info)) end fun make_inner_eqs bound_vs Tis eqs t = (case dest_case t of SOME (x, T, i, cont) => 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_name, _) = nth (the (Datatype.get_constrs thy (fst (dest_Type T)))) i 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 eqs = [] then NONE (* no rewriting, nothing to be done *) else let val Type (@{type_name List.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 = 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, ...} => tac context (rev Tis))) RS @{thm eq_reflection}) end)) in make_inner_eqs [] [] [] (dest_set (term_of redex)) 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 \<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> (EX x. xs = [x])" by (cases xs) auto lemma Nil_tl: "[] = tl xs \<longleftrightarrow> xs = [] \<or> (EX 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 list_nonempty_induct [consumes 1, case_names single cons]: assumes "xs \<noteq> []" assumes single: "\<And>x. P [x]" assumes cons: "\<And>x xs. xs \<noteq> [] \<Longrightarrow> P xs \<Longrightarrow> P (x # xs)" shows "P xs" using `xs \<noteq> []` proof (induct xs) case Nil then show ?case by simp next case (Cons x xs) show ?case proof (cases xs) case Nil with single show ?thesis by simp next case Cons show ?thesis proof (rule cons) from Cons show "xs \<noteq> []" by simp with Cons.hyps show "P xs" . qed qed qed lemma inj_split_Cons: "inj_on (\<lambda>(xs, n). n#xs) X" by (auto intro!: inj_onI) subsubsection {* @{const length} *} text {* Needs to come before @{text "@"} because of theorem @{text 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 \<noteq> [])" by (induct xs) auto lemma length_pos_if_in_set: "x : 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)" apply (induct xs, simp, simp) apply blast done 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 \<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 Nil then show ?case by simp next case (Cons x xs ys) then show ?case by (cases ys) simp_all qed 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 neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (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 (op aconv) (ls,rs) orelse n < m andalso submultiset (op aconv) (rs,ls) then prove_neq() else NONE end; in K list_neq end; *} subsubsection {* @{text "@"} -- append *} lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)" by (induct xs) auto lemma append_Nil2 [simp]: "xs @ [] = xs" by (induct xs) auto 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 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)" apply (induct xs arbitrary: ys) apply (case_tac ys, simp, force) apply (case_tac ys, force, simp) done lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) = (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)" apply (induct xs arbitrary: ys zs ts) apply fastforce apply(case_tac zs) apply simp apply fastforce done 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 [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs" by (induct xs) auto lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)" by (induct xs) auto lemma hd_append2 [simp]: "xs \<noteq> [] ==> 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 \<noteq> [] ==> 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 | (EX ys'. x#ys' = ys & xs = ys'@zs))" by(cases ys) auto lemma append_eq_Cons_conv: "(ys@zs = x#xs) = (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))" by(cases ys) auto text {* Trivial rules for solving @{text "@"}-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 @{text "@"} 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 (term_of ct) end; *} subsubsection {* @{const map} *} 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: "(!!x. x : set xs --> f x = g x) ==> 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) o (map g)) = map(f o g)" apply(rule ext) apply(simp) done 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 \<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: "(EX xs. ys = map f xs) = (ALL y : set ys. EX 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: "[| map f xs = map f ys; inj_on f (set xs Un set ys) |] ==> xs = ys" apply(frule map_eq_imp_length_eq) apply(rotate_tac -1) apply(induct rule:list_induct2) apply simp apply(simp) apply (blast intro:sym) done 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 ==> inj f ==> 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 ==> inj (map f)" by (iprover dest: map_injective injD intro: inj_onI) lemma inj_mapD: "inj (map f) ==> inj f" apply (unfold inj_on_def, clarify) apply (erule_tac x = "[x]" in ballE) apply (erule_tac x = "[y]" in ballE, simp, blast) apply blast done 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" apply(rule inj_onI) apply(erule map_inj_on) apply(blast intro:inj_onI dest:inj_onD) done 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) enriched_type 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)" apply (induct xs arbitrary: ys, force) apply (case_tac ys, simp, force) done 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_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])" by(rule rev_cases[of xs]) auto subsubsection {* @{const set} *} declare set.simps [code_post] --"pretty output" 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 : 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 : 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 : 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 : 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 : 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 (metis List.set.simps(1) emptyE) next assume "x \<noteq> a" thus ?case using snoc by fastforce qed qed lemma in_set_conv_decomp_last: "(x : 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 & 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" thus ?thesis by simp (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append) next assume "\<not> P x" hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp thus ?thesis using `\<not> P x` 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 (metis emptyE set_empty) next assume "\<not> P x" hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp thus ?thesis using `\<not> P x` 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 (metis split_list_last_prop [where P=P] in_set_conv_decomp) lemma finite_list: "finite A ==> EX xs. set xs = A" by (erule finite_induct) (auto simp add: set.simps(2) [symmetric] simp del: set.simps(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 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 (\<lambda>x. Q x \<and> P x) 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 (%x. ~P x) xs) = length xs" by(induct xs) simp_all lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs" by (induct xs) auto lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> 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)" apply (induct xs) apply auto apply(cut_tac P=P and xs=xs in length_filter_le) apply simp done lemma filter_map: "filter P (map f xs) = map f (filter (P o f) xs)" by (induct xs) simp_all lemma length_filter_map[simp]: "length (filter P (map f xs)) = length(filter (P o 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 : set xs; ~ 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 apply (auto split:split_if_asm) using length_filter_le[of P xs] apply arith done 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 "\<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) (simp add:image_def) 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 `\<not> p x` 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 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" apply simp apply(erule thin_rl) by (induct ys) simp_all subsubsection {* List partitioning *} 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 o 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 o 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 = []) = (\<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) = (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: "\<forall>(x, y) \<in> 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 ==> \<forall>(x, y) \<in> 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 \<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))" apply (induct xs arbitrary: n, simp) apply (case_tac n, auto) done 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)" apply (induct xs arbitrary: n, simp) apply (case_tac n, auto) done lemma nth_tl: assumes "n < length (tl x)" shows "tl x ! n = x ! Suc n" using assms by (induct x) 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> (ALL i<length xs. xs!i = ys!i))" apply(induct xs arbitrary: ys) apply force apply(case_tac ys) apply simp apply(simp add:nth_Cons split:nat.split)apply blast done lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}" apply (induct xs, simp, simp) apply safe apply (metis nat_case_0 nth.simps zero_less_Suc) apply (metis less_Suc_eq_0_disj nth_Cons_Suc) apply (case_tac i, simp) apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff) done 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 `?lhs` have "xs ! (n - 1) = x" by simp moreover from `n > 0` `n \<le> length xs` have "n - 1 < length xs" by simp ultimately have "\<exists>i<length xs. xs ! i = x" by auto with `x \<notin> 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 \<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 `?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: "(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> 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: "(ALL i<k. EX x. P i x) = (EX xs. size xs = k & (ALL i<k. P i (xs!i)))" (is "_ = (EX xs. ?P k xs)") proof(induct k) case 0 show ?case by simp next case (Suc k) show ?case (is "?L = ?R" is "_ = (EX 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 \<noteq> 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 \<le> i \<Longrightarrow> xs[i:=x] = xs" apply (induct xs arbitrary: i) apply simp apply (case_tac i) apply simp_all done lemma list_update_nonempty[simp]: "xs[k:=x] = [] \<longleftrightarrow> xs=[]" by(metis length_0_conv 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 \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys" apply (induct xs arbitrary: i, simp) apply(simp split:nat.split) done 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]) <= 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 \<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]" apply (induct xs arbitrary: i) apply simp apply (case_tac i, simp_all) done lemma list_update_swap: "i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]" apply (induct xs arbitrary: i i') apply simp apply (case_tac i, case_tac i') apply auto apply (case_tac i') apply auto done 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 = [] \<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> [] ==> 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: split_if_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 \<Longrightarrow> last (drop n xs) = last xs" apply (induct xs arbitrary: n) apply simp apply (auto split:nat.split) done 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 (induct xs, simp, case_tac xs, 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])" apply(cases xs rule:rev_cases) apply simp apply(simp add:list_update_append split:nat.splits) done 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> [] & butlast ys = xs & last ys = x)" by (metis append_butlast_last_id append_is_Nil_conv butlast_snoc last_snoc not_Cons_self) subsubsection {* @{const take} and @{const drop} *} lemma take_0 [simp]: "take 0 xs = []" by (induct xs) auto lemma drop_0 [simp]: "drop 0 xs = xs" by (induct xs) auto 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_1_Cons [simp]: "take 1 (x # xs) = [x]" unfolding One_nat_def by simp lemma drop_1_Cons [simp]: "drop 1 (x # xs) = xs" unfolding One_nat_def by simp 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 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" apply (induct xs arbitrary: n, simp) apply(simp add:drop_Cons nth_Cons split:nat.splits) done lemma take_Suc_conv_app_nth: "i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]" apply (induct xs arbitrary: i, simp) apply (case_tac i, auto) done lemma drop_Suc_conv_tl: "i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs" apply (induct xs arbitrary: i, simp) apply (case_tac i, auto) done 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" apply (induct m arbitrary: xs n, auto) apply (case_tac xs, auto) apply (case_tac n, auto) done lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs" apply (induct m arbitrary: xs, auto) apply (case_tac xs, auto) done lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)" apply (induct m arbitrary: xs n, auto) apply (case_tac xs, auto) done lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)" apply(induct xs arbitrary: m n) apply simp apply(simp add: take_Cons drop_Cons split:nat.split) done lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs" apply (induct n arbitrary: xs, auto) apply (case_tac xs, auto) done lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])" apply(induct xs arbitrary: n) apply simp apply(simp add:take_Cons split:nat.split) done lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)" apply(induct xs arbitrary: n) apply simp apply(simp add:drop_Cons split:nat.split) done lemma take_map: "take n (map f xs) = map f (take n xs)" apply (induct n arbitrary: xs, auto) apply (case_tac xs, auto) done lemma drop_map: "drop n (map f xs) = map f (drop n xs)" apply (induct n arbitrary: xs, auto) apply (case_tac xs, auto) done lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)" apply (induct xs arbitrary: i, auto) apply (case_tac i, auto) done lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)" apply (induct xs arbitrary: i, auto) apply (case_tac i, auto) done lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i" apply (induct xs arbitrary: i n, auto) apply (case_tac n, blast) apply (case_tac i, auto) done lemma nth_drop [simp]: "n + i <= length xs ==> (drop n xs)!i = xs!(n + i)" apply (induct n arbitrary: xs i, auto) apply (case_tac xs, auto) done lemma butlast_take: "n <= length xs ==> butlast (take n xs) = take (n - 1) xs" by (simp add: butlast_conv_take min_max.inf_absorb1 min_max.inf_absorb2) lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)" by (simp add: butlast_conv_take drop_take add_ac) lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs" by (simp add: butlast_conv_take min_max.inf_absorb1) lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)" by (simp add: butlast_conv_take drop_take add_ac) 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 <= n \<Longrightarrow> set(take m xs) <= set(take n xs)" apply (induct xs arbitrary: m n) apply simp apply (case_tac n) apply (auto simp: take_Cons) done 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 >= n \<Longrightarrow> set(drop m xs) <= set(drop n xs)" apply(induct xs arbitrary: m n) apply(auto simp:drop_Cons split:nat.split) apply (metis set_drop_subset subset_iff) done lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs" using set_take_subset by fast lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : 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)" apply (induct xs arbitrary: zs, simp, clarsimp) apply (case_tac zs, auto) done lemma take_add: "take (i+j) xs = take i xs @ take j (drop i xs)" apply (induct xs arbitrary: i, auto) apply (case_tac i, simp_all) done 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)" apply(induct xs\<^sub>1 arbitrary: ys\<^sub>1) apply simp apply(case_tac ys\<^sub>1) apply simp_all done lemma take_hd_drop: "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (Suc n) xs" apply(induct xs arbitrary: n) apply simp apply(simp add:drop_Cons split:nat.split) done 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]" apply (rule_tac take_Suc_conv_app_nth) by arith ultimately show ?thesis by auto qed 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 nth_drop': "i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs" apply (induct i arbitrary: xs) apply (simp add: neq_Nil_conv) apply (erule exE)+ apply simp apply (case_tac xs) apply simp_all done subsubsection {* @{const takeWhile} and @{const dropWhile} *} 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]: "[| 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: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs" by (induct xs) auto lemma takeWhile_nth: "j < length (takeWhile P xs) \<Longrightarrow> takeWhile P xs ! j = xs ! j" apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto lemma dropWhile_nth: "j < length (dropWhile P xs) \<Longrightarrow> dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))" apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto lemma length_dropWhile_le: "length (dropWhile P xs) \<le> 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: "\<not> P y \<Longrightarrow>dropWhile P (xs @ y # ys) = dropWhile P xs @ y # ys" by (induct xs) 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: split_if_asm) lemma set_takeWhileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x" by (induct xs) (auto split: split_if_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 & \<not> 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 \<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))" using assms 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 (Cons x xs) thus ?case proof (cases n) case (Suc n') note this[simp] have "P x" using Cons.prems(1)[of 0] by simp moreover have "takeWhile P xs = take n' xs" proof (rule Cons.hyps) case goal1 thus "P (xs ! i)" using Cons.prems(1)[of "Suc i"] by simp next case goal2 thus ?case using Cons by auto qed ultimately show ?thesis by simp qed simp qed simp 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 `\<not> ?thesis` nth_length_takeWhile[of P xs] by auto qed text{* The following two lemmmas could be generalized to an arbitrary property. *} 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)" apply(induct xs) apply simp apply auto apply(subst dropWhile_append2) apply auto done lemma takeWhile_not_last: "distinct xs \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs" apply(induct xs) apply simp apply(case_tac xs) apply(auto) done 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 [] \<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]: "[| 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 (\<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)" unfolding zip_map1 by auto lemma map_zip_map2: "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)" unfolding zip_map2 by auto text{* Courtesy of Andreas Lochbihler: *} lemma zip_same_conv_map: "zip xs xs = map (\<lambda>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)" apply (induct ys arbitrary: i xs, simp) apply (case_tac xs) apply (simp_all add: nth.simps split: nat.split) done 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(rule sym, simp add: update_zip) lemma zip_replicate [simp]: "zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)" apply (induct i arbitrary: j, auto) apply (case_tac j, auto) done lemma take_zip: "take n (zip xs ys) = zip (take n xs) (take n ys)" apply (induct n arbitrary: xs ys) apply simp apply (case_tac xs, simp) apply (case_tac ys, simp_all) done lemma drop_zip: "drop n (zip xs ys) = zip (drop n xs) (drop n ys)" apply (induct n arbitrary: xs ys) apply simp apply (case_tac xs, simp) apply (case_tac ys, simp_all) done lemma zip_takeWhile_fst: "zip (takeWhile P xs) ys = takeWhile (P \<circ> 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 \<circ> 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)\<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) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : 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 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_def) lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])" by (simp add: list_all2_def) lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])" by (simp add: list_all2_def) 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_def) 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_def 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 = (EX 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)" apply (simp add: list_all2_def zip_append1) apply (rule iffI) 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: min_def, clarify) apply (simp add: ball_Un) done lemma list_all2_append2: "list_all2 P xs (ys @ zs) = (EX 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)" apply (simp add: list_all2_def zip_append2) apply (rule iffI) 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: min_def, clarify) apply (simp add: ball_Un) done 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_def 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 \<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). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b" by (simp add: list_all2_def) 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)" apply (induct xs arbitrary: n ys) apply simp apply (clarsimp simp add: list_all2_Cons1) apply (case_tac n) apply auto done lemma list_all2_dropI [simp,intro?]: "list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)" apply (induct as arbitrary: n bs, simp) apply (clarsimp simp add: list_all2_Cons1) apply (case_tac n, simp, simp) done 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" apply (induct xs arbitrary: ys, simp) apply (case_tac ys, auto) done lemma list_all2_eq: "xs = ys \<longleftrightarrow> list_all2 (op =) 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) subsubsection {* @{const List.product} and @{const product_lists} *} lemma product_list_set: "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 `xs \<in> ?L` 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 {* @{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: assumes f: "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f x \<circ> f y = f y \<circ> f x" and x: "x \<in> set xs" shows "fold f xs = fold f (remove1 x xs) \<circ> f x" using assms 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: assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x = id" shows "fold f xs = id" using assms by (induct xs) simp_all lemma fold_commute: assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h" shows "h \<circ> fold g xs = fold f xs \<circ> h" using assms 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: assumes "\<And>x. x \<in> set xs \<Longrightarrow> Q x" and "P s" and "\<And>x s. Q x \<Longrightarrow> P s \<Longrightarrow> P (f x s)" shows "P (fold f xs s)" using assms 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 o f) xs" by (induct xs) simp_all lemma fold_rev: assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y" shows "fold f (rev xs) = fold f xs" using assms 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 \<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: "F (set (x # xs)) = fold f xs x" proof - interpret comp_fun_idem f by default (simp_all add: fun_eq_iff left_commute) show ?thesis by (simp add: eq_fold fold_set_fold) qed declare Inf_fin.set_eq_fold [code] declare Sup_fin.set_eq_fold [code] declare Min.set_eq_fold [code] declare Max.set_eq_fold [code] 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: "INFI (set xs) f = fold (inf \<circ> f) xs top" unfolding INF_def set_map [symmetric] Inf_set_fold fold_map .. declare INF_set_fold [code] lemma (in complete_lattice) SUP_set_fold: "SUPR (set xs) f = fold (sup \<circ> f) xs bot" unfolding SUP_def set_map [symmetric] Sup_set_fold fold_map .. declare SUP_set_fold [code] 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 (\<lambda>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 (\<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: assumes "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f y \<circ> f x = f x \<circ> f y" shows "foldr f xs = fold f xs" using assms 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 o f) xs a" by (simp add: foldr_conv_fold fold_map rev_map) 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 {* @{const upt} *} lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])" -- {* simp does not terminate! *} 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 <= i ==> [i..<j] = []" by (subst upt_rec) simp lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)" by(induct j)simp_all lemma upt_eq_Cons_conv: "([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)" apply(induct j arbitrary: x xs) apply simp apply(clarsimp simp add: append_eq_Cons_conv) apply arith done lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]" -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *} by simp lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]" by (simp add: upt_rec) lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]" -- {* LOOPS as a simprule, since @{text "j <= j"}. *} 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 ==> [i..<j] ! k = i + k" apply (induct j) apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split) done lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i" by(simp add:upt_conv_Cons) lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1" apply(cases j) apply simp by(simp add:upt_Suc_append) lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]" apply (induct m arbitrary: i, simp) apply (subst upt_rec) apply (rule sym) apply (subst upt_rec) apply (simp del: upt.simps) done lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]" apply(induct j) apply auto done lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]" by (induct n) auto lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)" apply (induct n m arbitrary: i rule: diff_induct) prefer 3 apply (subst map_Suc_upt[symmetric]) apply (auto simp add: less_diff_conv) done lemma map_decr_upt: "map (\<lambda>n. n - Suc 0) [Suc m..<Suc n] = [m..<n]" by (induct n) simp_all lemma nth_take_lemma: "k <= length xs ==> k <= length ys ==> (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys" apply (atomize, induct k arbitrary: xs ys) apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify) txt {* Both lists must be non-empty *} apply (case_tac xs, simp) apply (case_tac ys, clarify) apply (simp (no_asm_use)) apply clarify txt {* prenexing's needed, not miniscoping *} apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps) apply blast done lemma nth_equalityI: "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> 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) (* needs nth_equalityI *) 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" apply (simp add: list_all2_conv_all_nth) apply (rule nth_equalityI, blast, simp) done lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys" -- {* The famous take-lemma. *} apply (drule_tac x = "max (length xs) (length ys)" in spec) apply (simp add: le_max_iff_disj) done 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') subsubsection {* @{text upto}: interval-list on @{typ int} *} 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" "neg_numeral n"] upto.simps[of "neg_numeral m" "numeral n"] upto.simps[of "neg_numeral m" "neg_numeral n"] for m n lemma upto_empty[simp]: "j < i \<Longrightarrow> [i..j] = []" 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 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] simp_from_to[of i j] by auto qed text{* Tail recursive version for code generation: *} 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 {* @{const distinct} and @{const remdups} and @{const remdups_adj} *} 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 ==> 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> EX xs. set xs = A & 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) <= length xs" by (induct xs) auto lemma length_remdups_eq[iff]: "(length (remdups xs) = length xs) = (remdups xs = xs)" apply(induct xs) apply auto apply(subgoal_tac "length (remdups xs) <= length xs") apply arith apply(rule length_remdups_leq) done lemma remdups_filter: "remdups(filter P xs) = filter P (remdups xs)" apply(induct xs) apply auto done lemma distinct_map: "distinct(map f xs) = (distinct xs & inj_on f (set xs))" by (induct xs) auto lemma distinct_filter [simp]: "distinct xs ==> 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]" apply(induct i j rule:upto.induct) apply(subst upto.simps) apply(simp) done lemma distinct_take[simp]: "distinct xs \<Longrightarrow> distinct (take i xs)" apply(induct xs arbitrary: i) apply simp apply (case_tac i) apply simp_all apply(blast dest:in_set_takeD) done lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)" apply(induct xs arbitrary: i) apply simp apply (case_tac i) apply simp_all done 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 "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}" apply (drule_tac id_take_nth_drop) by simp with d True show ?thesis apply (simp add: upd_conv_take_nth_drop) apply (drule subst [OF id_take_nth_drop]) apply assumption apply simp apply (cases "a = xs!i") apply simp by blast next case False with d show ?thesis by auto qed lemma distinct_concat: assumes "distinct xs" and "\<And> ys. ys \<in> set xs \<Longrightarrow> distinct ys" and "\<And> ys zs. \<lbrakk> ys \<in> set xs ; zs \<in> set xs ; ys \<noteq> zs \<rbrakk> \<Longrightarrow> set ys \<inter> set zs = {}" shows "distinct (concat xs)" using assms by (induct xs) auto text {* It is best to avoid this indexed version of distinct, but sometimes it is useful. *} lemma distinct_conv_nth: "distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)" apply (induct xs, simp, simp) apply (rule iffI, clarsimp) apply (case_tac i) apply (case_tac j, simp) apply (simp add: set_conv_nth) apply (case_tac j) apply (clarsimp simp add: set_conv_nth, simp) apply (rule conjI) (*TOO SLOW apply (metis Zero_neq_Suc gr0_conv_Suc in_set_conv_nth lessI less_trans_Suc nth_Cons' nth_Cons_Suc) *) apply (clarsimp simp add: set_conv_nth) apply (erule_tac x = 0 in allE, simp) apply (erule_tac x = "Suc i" in allE, simp, clarsimp) (*TOO SLOW apply (metis Suc_Suc_eq lessI less_trans_Suc nth_Cons_Suc) *) apply (erule_tac x = "Suc i" in allE, simp) apply (erule_tac x = "Suc j" in allE, simp) done 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_card: "distinct xs ==> card (set xs) = size xs" by (induct xs) auto lemma card_distinct: "card (set xs) = size xs ==> distinct xs" proof (induct xs) case Nil thus ?case by simp next 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: split_if_asm) moreover have "card (set xs) \<le> length xs" by (rule card_length) ultimately have False by simp thus ?thesis .. qed qed 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: "~ distinct ws ==> EX xs ys zs y. ws = xs@[y]@ys@[y]@zs" apply (induct n == "length ws" arbitrary:ws) apply simp apply(case_tac ws) apply simp apply (simp split:split_if_asm) apply (metis Cons_eq_appendI eq_Nil_appendI split_list) done 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: assumes "distinct xs" and "distinct ys" shows "distinct (List.product xs ys)" using assms by (induct xs) (auto intro: inj_onI simp add: product_list_set 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 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 `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto with `distinct xs` 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 `length xs' = length ys'` 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 `length xs' = length ys'` 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 (metis distinct.simps(2) hd.simps hd_in_set list.simps(2) set_ConsD set_rev_mp set_subset_Cons) 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_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 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: split_if_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]) subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*} lemma (in monoid_add) listsum_simps [simp]: "listsum [] = 0" "listsum (x # xs) = x + listsum xs" by (simp_all add: listsum_def) lemma (in monoid_add) listsum_append [simp]: "listsum (xs @ ys) = listsum xs + listsum ys" by (induct xs) (simp_all add: add.assoc) lemma (in comm_monoid_add) listsum_rev [simp]: "listsum (rev xs) = listsum xs" by (simp add: listsum_def foldr_fold fold_rev fun_eq_iff add_ac) lemma (in monoid_add) fold_plus_listsum_rev: "fold plus xs = plus (listsum (rev xs))" proof fix x have "fold plus xs x = fold plus xs (x + 0)" by simp also have "\<dots> = fold plus (x # xs) 0" by simp also have "\<dots> = foldr plus (rev xs @ [x]) 0" by (simp add: foldr_conv_fold) also have "\<dots> = listsum (rev xs @ [x])" by (simp add: listsum_def) also have "\<dots> = listsum (rev xs) + listsum [x]" by simp finally show "fold plus xs x = listsum (rev xs) + x" by simp qed text{* Some syntactic sugar for summing a function over a list: *} syntax "_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3SUM _<-_. _)" [0, 51, 10] 10) syntax (xsymbols) "_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10) syntax (HTML output) "_listsum" :: "pttrn => 'a list => 'b => 'b" ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10) translations -- {* Beware of argument permutation! *} "SUM x<-xs. b" == "CONST listsum (CONST map (%x. b) xs)" "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (CONST map (%x. b) xs)" lemma (in comm_monoid_add) listsum_map_remove1: "x \<in> set xs \<Longrightarrow> listsum (map f xs) = f x + listsum (map f (remove1 x xs))" by (induct xs) (auto simp add: ac_simps) lemma (in monoid_add) list_size_conv_listsum: "list_size f xs = listsum (map f xs) + size xs" by (induct xs) auto lemma (in monoid_add) length_concat: "length (concat xss) = listsum (map length xss)" by (induct xss) simp_all lemma (in monoid_add) length_product_lists: "length (product_lists xss) = foldr op * (map length xss) 1" proof (induct xss) case (Cons xs xss) then show ?case by (induct xs) (auto simp: length_concat o_def) qed simp lemma (in monoid_add) listsum_map_filter: assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0" shows "listsum (map f (filter P xs)) = listsum (map f xs)" using assms by (induct xs) auto lemma (in comm_monoid_add) distinct_listsum_conv_Setsum: "distinct xs \<Longrightarrow> listsum xs = Setsum (set xs)" by (induct xs) simp_all lemma listsum_eq_0_nat_iff_nat [simp]: "listsum ns = (0::nat) \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)" by (induct ns) simp_all lemma member_le_listsum_nat: "(n :: nat) \<in> set ns \<Longrightarrow> n \<le> listsum ns" by (induct ns) auto lemma elem_le_listsum_nat: "k < size ns \<Longrightarrow> ns ! k \<le> listsum (ns::nat list)" by (rule member_le_listsum_nat) simp lemma listsum_update_nat: "k < size ns \<Longrightarrow> listsum (ns[k := (n::nat)]) = listsum ns + n - ns ! k" apply(induct ns arbitrary:k) apply (auto split:nat.split) apply(drule elem_le_listsum_nat) apply arith done lemma (in monoid_add) listsum_triv: "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r" by (induct xs) (simp_all add: distrib_right) lemma (in monoid_add) listsum_0 [simp]: "(\<Sum>x\<leftarrow>xs. 0) = 0" by (induct xs) (simp_all add: distrib_right) text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *} lemma (in ab_group_add) uminus_listsum_map: "- listsum (map f xs) = listsum (map (uminus \<circ> f) xs)" by (induct xs) simp_all lemma (in comm_monoid_add) listsum_addf: "(\<Sum>x\<leftarrow>xs. f x + g x) = listsum (map f xs) + listsum (map g xs)" by (induct xs) (simp_all add: algebra_simps) lemma (in ab_group_add) listsum_subtractf: "(\<Sum>x\<leftarrow>xs. f x - g x) = listsum (map f xs) - listsum (map g xs)" by (induct xs) (simp_all add: algebra_simps) lemma (in semiring_0) listsum_const_mult: "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)" by (induct xs) (simp_all add: algebra_simps) lemma (in semiring_0) listsum_mult_const: "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c" by (induct xs) (simp_all add: algebra_simps) lemma (in ordered_ab_group_add_abs) listsum_abs: "\<bar>listsum xs\<bar> \<le> listsum (map abs xs)" by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq]) lemma listsum_mono: fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}" shows "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (\<Sum>x\<leftarrow>xs. f x) \<le> (\<Sum>x\<leftarrow>xs. g x)" by (induct xs) (simp, simp add: add_mono) lemma (in monoid_add) listsum_distinct_conv_setsum_set: "distinct xs \<Longrightarrow> listsum (map f xs) = setsum f (set xs)" by (induct xs) simp_all lemma (in monoid_add) interv_listsum_conv_setsum_set_nat: "listsum (map f [m..<n]) = setsum f (set [m..<n])" by (simp add: listsum_distinct_conv_setsum_set) lemma (in monoid_add) interv_listsum_conv_setsum_set_int: "listsum (map f [k..l]) = setsum f (set [k..l])" by (simp add: listsum_distinct_conv_setsum_set) text {* General equivalence between @{const listsum} and @{const setsum} *} lemma (in monoid_add) listsum_setsum_nth: "listsum xs = (\<Sum> i = 0 ..< length xs. xs ! i)" using interv_listsum_conv_setsum_set_nat [of "op ! xs" 0 "length xs"] by (simp add: map_nth) subsubsection {* @{const insert} *} 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 xs \<Longrightarrow> distinct (List.insert x 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 {* @{const List.find} *} 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 by(auto simp: nth_Cons' split: if_splits) (metis One_nat_def diff_Suc_1 less_Suc_eq_0_disj) 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 {* @{const remove1} *} 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 : set(remove1 b xs) = (a : set xs)" apply (induct xs) apply auto done lemma set_remove1_subset: "set(remove1 x xs) <= set xs" apply(induct xs) apply simp apply simp apply blast done lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}" apply(induct xs) apply simp apply simp apply blast done lemma length_remove1: "length(remove1 x xs) = (if x : set xs then length xs - 1 else length xs)" apply (induct xs) apply (auto dest!:length_pos_if_in_set) done 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 ~: set xs ==> x ~: set(remove1 y xs)" apply(insert set_remove1_subset) apply fast done lemma distinct_remove1[simp]: "distinct xs ==> 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: assumes "x \<notin> set xs" shows "remove1 x xs = xs" using assms by (induct xs) simp_all subsubsection {* @{const removeAll} *} 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 ==> 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(metis map_removeAll_inj_on subset_inj_on subset_UNIV) subsubsection {* @{const replicate} *} lemma length_replicate [simp]: "length (replicate n x) = n" by (induct n) auto 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" apply (induct n, simp) apply (simp add: replicate_app_Cons_same) done lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x" by (induct n) auto text{* Courtesy of Matthias Daum: *} lemma append_replicate_commute: "replicate n x @ replicate k x = replicate k x @ replicate n x" apply (simp add: replicate_add [THEN sym]) apply (simp add: add_commute) done text{* Courtesy of Andreas Lochbihler: *} 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 ==> 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 ==> last (replicate n x) = x" by (atomize (full), induct n) auto lemma nth_replicate[simp]: "i < n ==> (replicate n x)!i = x" apply (induct n arbitrary: i, simp) apply (simp add: nth_Cons split: nat.split) done text{* Courtesy of Matthias Daum (2 lemmas): *} lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x" apply (case_tac "k \<le> i") apply (simp add: min_def) apply (drule not_leE) apply (simp add: min_def) apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x") apply simp apply (simp add: replicate_add [symmetric]) done lemma drop_replicate[simp]: "drop i (replicate k x) = replicate (k-i) x" apply (induct k arbitrary: i) apply simp apply clarsimp apply (case_tac i) apply simp apply clarsimp done lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}" by (induct n) auto lemma set_replicate [simp]: "n \<noteq> 0 ==> 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 : set (replicate n y)) = (x = y & n \<noteq> 0)" by (simp add: set_replicate_conv_if) lemma Ball_set_replicate[simp]: "(ALL x : set(replicate n a). P x) = (P a | n=0)" by(simp add: set_replicate_conv_if) lemma Bex_set_replicate[simp]: "(EX x : set(replicate n a). P x) = (P a & 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 & (m\<noteq>0 \<longrightarrow> x=y))" apply(induct m arbitrary: n) apply simp apply(induct_tac n) apply auto done 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: fixes xs ys :: "'a list" assumes "xs \<noteq> []" "ys \<noteq> []" assumes "xs @ ys = ys @ xs" shows "\<exists>m n zs. concat (replicate m zs) = xs \<and> concat (replicate n zs) = ys" using assms proof (induct "length (xs @ ys)" arbitrary: xs ys rule: less_induct) case less def xs' \<equiv> "if (length xs \<le> length ys) then xs else ys" and ys' \<equiv> "if (length xs \<le> length ys) then ys else xs" then have prems': "length xs' \<le> length ys'" "xs' @ ys' = ys' @ xs'" and "xs' \<noteq> []" and len: "length (xs @ ys) = length (xs' @ ys')" using less by (auto intro: less.hyps) from prems' obtain ws where "ys' = xs' @ ws" by (auto simp: append_eq_append_conv2) have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ys'" proof (cases "ws = []") case True then have "concat (replicate 1 xs') = xs'" and "concat (replicate 1 xs') = ys'" using `ys' = xs' @ ws` by auto then show ?thesis by blast next case False from `ys' = xs' @ ws` and `xs' @ ys' = ys' @ xs'` have "xs' @ ws = ws @ xs'" by simp then have "\<exists>m n zs. concat (replicate m zs) = xs' \<and> concat (replicate n zs) = ws" using False and `xs' \<noteq> []` and `ys' = xs' @ ws` and len by (intro less.hyps) auto then obtain m n zs where *: "concat (replicate m zs) = xs'" and "concat (replicate n zs) = ws" by blast then have "concat (replicate (m + n) zs) = ys'" using `ys' = xs' @ ws` by (simp add: replicate_add) with * show ?thesis by blast qed then show ?case using xs'_def ys'_def by metis 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 assms by (metis comm_append_are_replicate) then have "m + n > 1" and "concat (replicate (m+n) zs) = xs @ ys" using `xs \<noteq> []` and `ys \<noteq> []` 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 {* @{const enumerate} *} lemma enumerate_simps [simp, code]: "enumerate n [] = []" "enumerate n (x # xs) = (n, x) # enumerate (Suc n) xs" apply (auto simp add: enumerate_eq_zip not_le) apply (cases "n < n + length xs") apply (auto simp add: upt_conv_Cons) done 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: assumes "m < length xs" shows "enumerate n xs ! m = (n + m, xs ! m)" using assms 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 [simp del] add: map_Suc_upt map_map [symmetric]) lemma distinct_enumerate [simp]: "distinct (enumerate n xs)" by (simp add: enumerate_eq_zip distinct_zipI1) subsubsection {* @{const rotate1} and @{const rotate} *} 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 o 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_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)" by(simp add:rotate_def funpow_swap1) lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs" by(cases xs) simp_all lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs" apply(induct n) apply simp apply (simp add:rotate_def) done 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" apply(induct n) apply simp apply(simp add:rotate_def) apply(cases "xs = []") apply (simp) apply(case_tac "n mod length xs = 0") apply(simp add:mod_Suc) apply(simp add: rotate1_hd_tl drop_Suc take_Suc) apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric] take_hd_drop linorder_not_le) done 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)" apply(simp add:rotate_drop_take rev_drop rev_take) apply(cases "length xs = 0") apply simp apply(cases "n mod length xs = 0") apply simp apply(simp add:rotate_drop_take rev_drop rev_take) done lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)" apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth) apply(subgoal_tac "length xs \<noteq> 0") prefer 2 apply simp using mod_less_divisor[of "length xs" n] by arith subsubsection {* @{const sublist} --- a generalization of @{const nth} to sets *} lemma sublist_empty [simp]: "sublist xs {} = []" by (auto simp add: sublist_def) lemma sublist_nil [simp]: "sublist [] A = []" by (auto simp add: sublist_def) lemma length_sublist: "length(sublist xs I) = card{i. i < length xs \<and> i : I}" by(simp add: sublist_def length_filter_conv_card cong:conj_cong) lemma sublist_shift_lemma_Suc: "map fst (filter (%p. P(Suc(snd p))) (zip xs is)) = map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))" apply(induct xs arbitrary: "is") apply simp apply (case_tac "is") apply simp apply simp done lemma sublist_shift_lemma: "map fst [p<-zip xs [i..<i + length xs] . snd p : A] = map fst [p<-zip xs [0..<length xs] . snd p + i : A]" by (induct xs rule: rev_induct) (simp_all add: add_commute) lemma sublist_append: "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}" apply (unfold sublist_def) apply (induct l' rule: rev_induct, simp) apply (simp add: upt_add_eq_append[of 0] sublist_shift_lemma) apply (simp add: add_commute) done lemma sublist_Cons: "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}" apply (induct l rule: rev_induct) apply (simp add: sublist_def) apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append) done lemma set_sublist: "set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}" apply(induct xs arbitrary: I) apply(auto simp: sublist_Cons nth_Cons split:nat.split dest!: gr0_implies_Suc) done lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs" by(auto simp add:set_sublist) lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)" by(auto simp add:set_sublist) lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs" by(auto simp add:set_sublist) lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])" by (simp add: sublist_Cons) lemma distinct_sublistI[simp]: "distinct xs \<Longrightarrow> distinct(sublist xs I)" apply(induct xs arbitrary: I) apply simp apply(auto simp add:sublist_Cons) done lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l" apply (induct l rule: rev_induct, simp) apply (simp split: nat_diff_split add: sublist_append) done lemma filter_in_sublist: "distinct xs \<Longrightarrow> filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s" proof (induct xs arbitrary: s) case Nil thus ?case by simp next case (Cons a xs) then have "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto with Cons show ?case by(simp add: sublist_Cons cong:filter_cong) qed subsubsection {* @{const sublists} and @{const List.n_lists} *} lemma length_sublists: "length (sublists xs) = 2 ^ length xs" by (induct xs) (simp_all add: Let_def) lemma sublists_powset: "set ` set (sublists 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 (sublists 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_sublists: assumes "distinct xs" shows "distinct (map set (sublists xs))" proof (rule card_distinct) have "finite (set xs)" by rule 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 (sublists xs))) = length (map set (sublists xs))" by (simp add: sublists_powset length_sublists) qed lemma n_lists_Nil [simp]: "List.n_lists n [] = (if n = 0 then [[]] else [])" by (induct n) simp_all lemma length_n_lists: "length (List.n_lists n xs) = length xs ^ n" by (induct n) (auto simp add: length_concat o_def listsum_triv) 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 distinct_n_lists: assumes "distinct xs" shows "distinct (List.n_lists n xs)" proof (rule card_distinct) from assms have card_length: "card (set xs) = length xs" by (rule distinct_card) have "card (set (List.n_lists n xs)) = card (set xs) ^ n" proof (induct n) case 0 then show ?case by simp next case (Suc n) moreover have "card (\<Union>ys\<in>set (List.n_lists n xs). (\<lambda>y. y # ys) ` set xs) = (\<Sum>ys\<in>set (List.n_lists n xs). card ((\<lambda>y. y # ys) ` set xs))" by (rule card_UN_disjoint) auto moreover have "\<And>ys. card ((\<lambda>y. y # ys) ` set xs) = card (set xs)" by (rule card_image) (simp add: inj_on_def) ultimately show ?case by auto qed also have "\<dots> = length xs ^ n" by (simp add: card_length) finally show "card (set (List.n_lists n xs)) = length (List.n_lists n xs)" by (simp add: length_n_lists) qed subsubsection {* @{const splice} *} lemma splice_Nil2 [simp, code]: "splice xs [] = xs" by (cases xs) simp_all declare splice.simps(1,3)[code] declare splice.simps(2)[simp del] lemma length_splice[simp]: "length(splice xs ys) = length xs + length ys" by (induct xs ys rule: splice.induct) auto subsubsection {* Transpose *} 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 (list_case [] (\<lambda>h t. [h])) xss) = map (\<lambda>xs. hd xs) [ys\<leftarrow>xss . ys \<noteq> []]" by (induct xss) (auto split: list.split) lemma transpose_aux_filter_tail: "concat (map (list_case [] (\<lambda>h t. [t])) xss) = map (\<lambda>xs. tl xs) [ys\<leftarrow>xss . ys \<noteq> []]" 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)) [ys\<leftarrow>xss . ys\<noteq>[]] 0))" (is "max _ ?foldB = Suc (max _ ?foldA)") proof (cases "[ys\<leftarrow>xss . ys\<noteq>[]] = []") 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)) [ys\<leftarrow>xss . ys \<noteq> []] 0 - 1" by (induct xss) auto have foldB: "?foldB = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0" by (induct xss) auto have "0 < ?foldB" proof - from False obtain z zs where zs: "[ys\<leftarrow>xss . ys \<noteq> []] = z#zs" by (auto simp: neq_Nil_conv) hence "z \<in> set ([ys\<leftarrow>xss . ys \<noteq> []])" 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) [ys \<leftarrow> xs. i < length ys]" using assms proof (induct arbitrary: i rule: transpose.induct) case (3 x xs xss) def 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 (list_case [] (\<lambda>h t. [t])) xss)))" using "3.prems" by (simp add: transpose_aux_filter_tail length_transpose Suc) show ?thesis unfolding transpose.simps `i = Suc j` nth_Cons_Suc "3.hyps"[OF j_less] apply (auto simp: transpose_aux_filter_tail filter_map comp_def length_transpose * ** *** XS_def[symmetric]) apply (rule_tac y=x in list.exhaust) by auto qed qed simp_all lemma transpose_map_map: "transpose (map (map f) xs) = map (map f) (transpose xs)" proof (rule nth_equalityI, safe) 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 {* (In)finiteness *} lemma finite_maxlen: "finite (M::'a list set) ==> EX n. ALL s: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 "ALL s: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 `finite A` 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 `finite A`] 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 `finite A` 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 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}" 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 < card A" by simp moreover have "finite A" using assms by (simp add: card_ge_0_finite) moreover have "finite {xs. ?k_list k xs}" using finite_lists_length_eq[OF `finite A`, of k] by - (rule finite_subset, 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>Collect (?k_list k) \<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 setprod_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 infinite_UNIV_listI: "~ finite(UNIV::'a list set)" apply(rule notI) apply(drule finite_maxlen) apply (metis UNIV_I length_replicate less_not_refl) done subsection {* Sorting *} text{* Currently it is not shown that @{const sort} returns a permutation of its input because the nicest proof is via multisets, which are not yet available. 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. *} 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 (in linorder) 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 (in linorder) 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 sorted_Cons: "sorted (x#xs) = (sorted xs & (ALL y:set xs. x <= y))" apply(induct xs arbitrary: x) apply simp by simp (blast intro: order_trans) lemma sorted_tl: "sorted xs \<Longrightarrow> sorted (tl xs)" by (cases xs) (simp_all add: sorted_Cons) lemma sorted_append: "sorted (xs@ys) = (sorted xs & sorted ys & (\<forall>x \<in> set xs. \<forall>y \<in> set ys. x\<le>y))" by (induct xs) (auto simp add:sorted_Cons) lemma sorted_nth_mono: "sorted xs \<Longrightarrow> i \<le> j \<Longrightarrow> j < length xs \<Longrightarrow> xs!i \<le> xs!j" by (induct xs arbitrary: i j) (auto simp:nth_Cons' sorted_Cons) 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_nth_monoI: "(\<And> i j. \<lbrakk> i \<le> j ; j < length xs \<rbrakk> \<Longrightarrow> xs ! i \<le> xs ! j) \<Longrightarrow> sorted xs" proof (induct xs) case (Cons x xs) have "sorted xs" proof (rule Cons.hyps) fix i j assume "i \<le> j" and "j < length xs" with Cons.prems[of "Suc i" "Suc j"] show "xs ! i \<le> xs ! j" by auto qed moreover { fix y assume "y \<in> set xs" then obtain j where "j < length xs" and "xs ! j = y" unfolding in_set_conv_nth by blast with Cons.prems[of 0 "Suc j"] have "x \<le> y" by auto } ultimately show ?case unfolding sorted_Cons by auto qed simp lemma sorted_equals_nth_mono: "sorted xs = (\<forall>j < length xs. \<forall>i \<le> j. xs ! i \<le> xs ! j)" by (auto intro: sorted_nth_monoI sorted_nth_mono) lemma set_insort: "set(insort_key f x xs) = insert x (set xs)" by (induct xs) auto lemma set_sort[simp]: "set(sort_key f xs) = set xs" by (induct xs) (simp_all add:set_insort) lemma distinct_insort: "distinct (insort_key f x xs) = (x \<notin> set xs \<and> distinct xs)" by(induct xs)(auto simp:set_insort) 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:sorted_Cons set_insort) 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 sorted_butlast: assumes "xs \<noteq> []" and "sorted xs" shows "sorted (butlast xs)" proof - from `xs \<noteq> []` obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto with `sorted xs` show ?thesis by (simp add: sorted_append) qed lemma insort_not_Nil [simp]: "insort_key f a xs \<noteq> []" by (induct 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: sorted_Cons insort_is_Cons) lemma sorted_map_remove1: "sorted (map f xs) \<Longrightarrow> sorted (map f (remove1 x xs))" by (induct xs) (auto simp add: sorted_Cons) lemma sorted_remove1: "sorted xs \<Longrightarrow> sorted (remove1 a xs)" using sorted_map_remove1 [of "\<lambda>x. x"] by simp 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 simp: sorted_Cons) with `f x \<noteq> f a` show ?thesis using Cons by (auto simp: sorted_Cons insort_is_Cons) qed (auto simp: sorted_Cons 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) from `a \<in> set xs` show "a \<in> set xs" . from `sorted xs` show "sorted (map (\<lambda>x. x) xs)" by simp from `a \<in> set xs` have "a \<in> set (filter (op = a) xs)" by auto then have "set (filter (op = a) xs) \<noteq> {}" by auto then have "filter (op = a) xs \<noteq> []" by (auto simp only: set_empty) then have "length (filter (op = a) xs) > 0" by simp then obtain n where n: "Suc n = length (filter (op = a) xs)" by (cases "length (filter (op = a) xs)") simp_all moreover have "replicate (Suc n) a = a # replicate n a" by simp ultimately show "hd (filter (op = a) xs) = a" by (simp add: replicate_length_filter) qed lemma sorted_remdups[simp]: "sorted l \<Longrightarrow> sorted (remdups l)" by (induct l) (auto simp: sorted_Cons) lemma sorted_remdups_adj[simp]: "sorted xs \<Longrightarrow> sorted (remdups_adj xs)" by (induct xs rule: remdups_adj.induct, simp_all split: split_if_asm add: sorted_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 add:sorted_Cons) (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 `inj_on f (set xs \<union> set ys)` show "xs = ys" by (blast intro: map_inj_on) qed lemma finite_sorted_distinct_unique: shows "finite A \<Longrightarrow> EX! xs. set xs = A & sorted xs & distinct xs" apply(drule finite_distinct_list) apply clarify apply(rule_tac a="sort xs" in ex1I) apply (auto simp: sorted_distinct_set_unique) done 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 add: sorted_Cons) 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 "[x \<leftarrow> xs. t < f x] = 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 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) 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) 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)" using assms by (induct xs) (auto simp add: sorted_Cons, 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 sorted_map_same: "sorted (map f [x\<leftarrow>xs. f x = g xs])" proof (induct xs arbitrary: g) case Nil then show ?case by simp next case (Cons x xs) then have "sorted (map f [y\<leftarrow>xs . f y = (\<lambda>xs. f x) xs])" . moreover from Cons have "sorted (map f [y\<leftarrow>xs . f y = (g \<circ> Cons x) xs])" . ultimately show ?case by (simp_all add: sorted_Cons) qed lemma sorted_same: "sorted [x\<leftarrow>xs. x = g xs]" using sorted_map_same [of "\<lambda>x. x"] by simp lemma remove1_insort [simp]: "remove1 x (insort x xs) = xs" by (induct xs) simp_all end lemma sorted_upt[simp]: "sorted[i..<j]" by (induct j) (simp_all add:sorted_append) lemma sorted_upto[simp]: "sorted[i..j]" apply(induct i j rule:upto.induct) apply(subst upto.simps) apply(simp add:sorted_Cons) done lemma sorted_find_Min: assumes "sorted xs" assumes "\<exists>x \<in> set xs. P x" shows "List.find P xs = Some (Min {x\<in>set xs. P x})" using assms proof (induct xs rule: sorted.induct) case Nil then show ?case by simp next case (Cons xs x) 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 subsubsection {* @{const transpose} on sorted lists *} lemma sorted_transpose[simp]: shows "sorted (rev (map length (transpose xs)))" by (auto simp: sorted_equals_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 [x \<leftarrow> xs. x \<noteq> []]" (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 "[x \<leftarrow> xs. x \<noteq> []] = []" 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 [ys \<leftarrow> xs. i < length ys]" 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 `i < length xs` 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 `i < length xs`] } 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] `xs \<noteq> []` 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 `i < length xs` `j < length (xs ! i)` less_Suc_eq_le by (auto intro!: length_takeWhile_less_P_nth dest!: sortedE) with nth_transpose[OF `j < length (transpose xs)`] 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, safe) 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 `i < length xs` by simp fix k assume "k < Suc i" hence "k \<le> i" by auto with sorted_rev_nth_mono[OF sorted this] `i < length xs` 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 [ys\<leftarrow>xs . j < length ys]" 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 intro!: sorted_nth_monoI) 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 "[ys\<leftarrow>xs . i < length ys] = xs" using rect by (auto simp: in_set_conv_nth intro!: filter_True) } ultimately show "\<forall>i < length ?trans. ?trans ! i = ?map ! i" by (auto simp: nth_transpose intro: nth_equalityI) qed subsubsection {* @{text sorted_list_of_set} *} text{* 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 set}). *} subsubsection {* @{text sorted_list_of_set} *} text{* 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 set}). *} definition (in linorder) sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where "sorted_list_of_set = folding.F insort []" sublocale linorder < sorted_list_of_set!: folding insort Nil where "folding.F insort [] = sorted_list_of_set" proof - interpret comp_fun_commute insort by (fact comp_fun_commute_insort) show "folding insort" by default (fact comp_fun_commute) show "folding.F insort [] = sorted_list_of_set" by (simp only: sorted_list_of_set_def) qed context linorder begin lemma sorted_list_of_set_empty: "sorted_list_of_set {} = []" by (fact sorted_list_of_set.empty) lemma sorted_list_of_set_insert [simp]: assumes "finite A" shows "sorted_list_of_set (insert x A) = insort x (sorted_list_of_set (A - {x}))" using assms 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 = {}" using assms by (auto simp: sorted_list_of_set.remove) lemma sorted_list_of_set [simp]: "finite A \<Longrightarrow> set (sorted_list_of_set A) = A \<and> sorted (sorted_list_of_set A) \<and> distinct (sorted_list_of_set A)" by (induct A rule: finite_induct) (simp_all add: set_insort sorted_insort distinct_insort) lemma distinct_sorted_list_of_set: "distinct (sorted_list_of_set A)" using sorted_list_of_set by (cases "finite A") auto 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 end lemma sorted_list_of_set_range [simp]: "sorted_list_of_set {m..<n} = [m..<n]" by (rule sorted_distinct_set_unique) simp_all subsubsection {* @{text lists}: the list-forming operator over sets *} inductive_set lists :: "'a set => 'a list set" for A :: "'a set" where Nil [intro!, simp]: "[]: lists A" | Cons [intro!, simp]: "[| a: A; l: lists A|] ==> a#l : lists A" inductive_cases listsE [elim!]: "x#l : 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 ==> 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 ==> 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 : lists A \<longleftrightarrow> x:A \<and> xs : 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)" -- {* eliminate @{text listsp} in favour of @{text set} *} by (induct xs) auto lemmas in_lists_conv_set [code_unfold] = in_listsp_conv_set [to_set] lemma in_listspD [dest!]: "listsp A xs ==> \<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 ==> 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 <= 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: map.simps simp add: map.simps[symmetric] intro!: imageI) } then show ?thesis by auto qed subsubsection {* Inductive definition for membership *} 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)" apply (rule iffI) apply (induct set: ListMem) apply auto apply (induct xs) apply (auto intro: ListMem.intros) done subsubsection {* Lists as Cartesian products *} text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from @{term A} and tail drawn from @{term Xs}.*} 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{*Yields the set of lists, all of the same length as the argument and with elements drawn from the corresponding element of the argument.*} primrec listset :: "'a set list \<Rightarrow> 'a list set" where "listset [] = {[]}" | "listset (A # As) = set_Cons A (listset As)" subsection {* Relations on Lists *} subsubsection {* Length Lexicographic Ordering *} text{*These orderings preserve well-foundedness: shorter lists precede longer lists. These ordering are not used in dictionaries.*} primrec -- {*The lexicographic ordering for lists of the specified length*} lexn :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> ('a list \<times> 'a list) set" where "lexn r 0 = {}" | "lexn r (Suc n) = (map_pair (%(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)" -- {*Holds only between lists of the same length*} 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))" -- {*Compares lists by their length and then lexicographically*} lemma wf_lexn: "wf r ==> wf (lexn r n)" apply (induct n, simp, simp) apply(rule wf_subset) prefer 2 apply (rule Int_lower1) apply(rule wf_map_pair_image) prefer 2 apply (rule inj_onI, auto) done lemma lexn_length: "(xs, ys) : lexn r n ==> length xs = n \<and> length ys = n" by (induct n arbitrary: xs ys) auto lemma wf_lex [intro!]: "wf r ==> wf (lex r)" apply (unfold lex_def) apply (rule wf_UN) apply (blast intro: wf_lexn, clarify) apply (rename_tac m n) apply (subgoal_tac "m \<noteq> n") prefer 2 apply blast apply (blast dest: lexn_length not_sym) done 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):r)}" apply (induct n, simp) apply (simp add: image_Collect lex_prod_def, safe, blast) apply (rule_tac x = "ab # xys" in exI, simp) apply (case_tac xys, simp_all, blast) done 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):r)}" by (force simp add: lex_def lexn_conv) lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)" by (unfold lenlex_def) blast lemma lenlex_conv: "lenlex r = {(xs,ys). length xs < length ys | length xs = length ys \<and> (xs, ys) : lex r}" by (simp add: lenlex_def Id_on_def lex_prod_def inv_image_def) 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) : lex r) = ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)" apply (simp add: lex_conv) apply (rule iffI) prefer 2 apply (blast intro: Cons_eq_appendI, clarify) apply (case_tac xys, simp, simp) apply blast done subsubsection {* Lexicographic Ordering *} text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b". This ordering does \emph{not} preserve well-foundedness. Author: N. Voelker, March 2005. *} 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) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))" apply (unfold lexord_def, safe, simp_all) apply (case_tac u, simp, simp) apply (case_tac u, simp, clarsimp, blast, blast, clarsimp) apply (erule_tac x="b # u" in allE) by force lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r" by (induct_tac x, auto) lemma lexord_append_left_rightI: "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r" by (induct_tac u, auto) lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r" by (induct x, auto) lemma lexord_append_leftD: "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r" by (erule rev_mp, induct_tac x, auto) lemma lexord_take_index_conv: "((x,y) : lexord r) = ((length x < length y \<and> take (length x) y = x) \<or> (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))" apply (unfold lexord_def Let_def, clarsimp) apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2) apply auto apply (rule_tac x="hd (drop (length x) y)" in exI) apply (rule_tac x="tl (drop (length x) y)" in exI) apply (erule subst, simp add: min_def) apply (rule_tac x ="length u" in exI, simp) apply (rule_tac x ="take i x" in exI) apply (rule_tac x ="x ! i" in exI) apply (rule_tac x ="y ! i" in exI, safe) apply (rule_tac x="drop (Suc i) x" in exI) apply (drule sym, simp add: drop_Suc_conv_tl) apply (rule_tac x="drop (Suc i) y" in exI) by (simp add: drop_Suc_conv_tl) -- {* lexord is extension of partial ordering List.lex *} lemma lexord_lex: "(x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)" apply (rule_tac x = y in spec) apply (induct_tac x, clarsimp) by (clarify, case_tac x, simp, force) lemma lexord_irreflexive: "ALL x. (x,x) \<notin> r \<Longrightarrow> (xs,xs) \<notin> lexord r" by (induct xs) auto text{* By Ren\'e Thiemann: *} 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 (rule transI, drule lexord_trans, blast) lemma lexord_linear: "(! a b. (a,b)\<in> r | a = b | (b,a) \<in> r) \<Longrightarrow> (x,y) : lexord r | x = y | (y,x) : lexord r" apply (rule_tac x = y in spec) apply (induct_tac x, rule allI) apply (case_tac x, simp, simp) apply (rule allI, case_tac x, simp, simp) by blast subsubsection {* Lexicographic combination of measure functions *} text {* These are useful for termination proofs *} 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 ==> (x, y) \<in> measures (f#fs)" by simp lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)" by auto subsubsection {* Lifting Relations to Lists: one element *} 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 (metis Cons_listrel1_Cons) lemma listrel1I2: "(xs, ys) \<in> listrel1 r \<Longrightarrow> (x # xs, x # ys) \<in> listrel1 r" by (metis Cons_listrel1_Cons) 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^-1) = (listrel1 r)^-1" unfolding listrel1_def by blast lemma in_listrel1_converse: "(x,y) : listrel1 (r^-1) \<longleftrightarrow> (x,y) : (listrel1 r)^-1" 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{* Accessible part and wellfoundedness: *} lemma Cons_acc_listrel1I [intro!]: "x \<in> acc r \<Longrightarrow> xs \<in> acc (listrel1 r) \<Longrightarrow> (x # xs) \<in> acc (listrel1 r)" apply (induct arbitrary: xs set: acc) apply (erule thin_rl) apply (erule acc_induct) apply (rule accI) apply (blast) done lemma lists_accD: "xs \<in> lists (acc r) \<Longrightarrow> xs \<in> acc (listrel1 r)" apply (induct set: lists) apply (rule accI) apply simp apply (rule accI) apply (fast dest: acc_downward) done lemma lists_accI: "xs \<in> acc (listrel1 r) \<Longrightarrow> xs \<in> lists (acc r)" apply (induct set: 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(metis wf_acc_iff in_lists_conv_set lists_accI lists_accD Cons_in_lists_iff) subsubsection {* Lifting Relations to Lists: all elements *} 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: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (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) : listrel r \<longleftrightarrow> length xs = length ys & (\<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) : listrel r \<longleftrightarrow> length xs = length ys & (\<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" apply clarify apply (erule listrel.induct) apply (blast intro: listrel.intros)+ done lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A" apply clarify apply (erule listrel.induct, auto) done lemma listrel_refl_on: "refl_on A r \<Longrightarrow> refl_on (lists A) (listrel r)" apply (simp add: refl_on_def listrel_subset Ball_def) apply (rule allI) apply (induct_tac x) apply (auto intro: listrel.intros) done lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" apply (auto simp add: sym_def) apply (erule listrel.induct) apply (blast intro: listrel.intros)+ done lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" apply (simp add: trans_def) apply (intro allI) apply (rule impI) apply (erule listrel.induct) apply (blast intro: listrel.intros)+ done 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) : listrel(r^*)" using listrel_refl_on[of UNIV, OF refl_rtrancl] by(auto simp: refl_on_def) lemma listrel_rtrancl_trans: "\<lbrakk> (xs,ys) : listrel(r^*); (ys,zs) : listrel(r^*) \<rbrakk> \<Longrightarrow> (xs,zs) : listrel(r^*)" 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 {* Relating @{term listrel1}, @{term listrel} and closures: *} lemma listrel1_rtrancl_subset_rtrancl_listrel1: "listrel1 (r^*) \<subseteq> (listrel1 r)^*" proof (rule subrelI) fix xs ys assume 1: "(xs,ys) \<in> listrel1 (r^*)" { fix x y us vs have "(x,y) : r^* \<Longrightarrow> (us @ x # vs, us @ y # vs) : (listrel1 r)^*" 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)^*" using 1 by(blast elim: listrel1E) qed lemma rtrancl_listrel1_eq_len: "(x,y) \<in> (listrel1 r)^* \<Longrightarrow> length x = length y" by (induct rule: rtrancl.induct) (auto intro: listrel1_eq_len) lemma rtrancl_listrel1_ConsI1: "(xs,ys) : (listrel1 r)^* \<Longrightarrow> (x#xs,x#ys) : (listrel1 r)^*" apply(induct rule: rtrancl.induct) apply simp by (metis listrel1I2 rtrancl.rtrancl_into_rtrancl) lemma rtrancl_listrel1_ConsI2: "(x,y) \<in> r^* \<Longrightarrow> (xs, ys) \<in> (listrel1 r)^* \<Longrightarrow> (x # xs, y # ys) \<in> (listrel1 r)^*" by (blast intro: rtrancl_trans rtrancl_listrel1_ConsI1 subsetD[OF listrel1_rtrancl_subset_rtrancl_listrel1 listrel1I1]) 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) : listrel1 r \<Longrightarrow> (xs,ys) : listrel(r^*)" by(erule listrel1E)(auto simp add: listrel_iff_zip set_zip) lemma listrel_rtrancl_eq_rtrancl_listrel1: "listrel (r^*) = (listrel1 r)^*" proof { fix x y assume "(x,y) \<in> listrel (r^*)" then have "(x,y) \<in> (listrel1 r)^*" by induct (auto intro: rtrancl_listrel1_ConsI2) } then show "listrel (r^*) \<subseteq> (listrel1 r)^*" by (rule subrelI) next show "listrel (r^*) \<supseteq> (listrel1 r)^*" proof(rule subrelI) fix xs ys assume "(xs,ys) \<in> (listrel1 r)^*" then show "(xs,ys) \<in> listrel (r^*)" 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) : listrel r \<Longrightarrow> (xs,ys) : (listrel1 r)^*" 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)^*" by(fast intro:rtrancl_listrel1_if_listrel) subsection {* Size function *} lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (list_size f)" by (rule is_measure_trivial) lemma [measure_function]: "is_measure f \<Longrightarrow> is_measure (option_size f)" by (rule is_measure_trivial) lemma list_size_estimation[termination_simp]: "x \<in> set xs \<Longrightarrow> y < f x \<Longrightarrow> y < list_size f xs" by (induct xs) auto lemma list_size_estimation'[termination_simp]: "x \<in> set xs \<Longrightarrow> y \<le> f x \<Longrightarrow> y \<le> list_size f xs" by (induct xs) auto lemma list_size_map[simp]: "list_size f (map g xs) = list_size (f o g) xs" by (induct xs) auto lemma list_size_append[simp]: "list_size f (xs @ ys) = list_size f xs + list_size f ys" by (induct xs, auto) lemma list_size_pointwise[termination_simp]: "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> list_size f xs \<le> list_size g xs" by (induct xs) force+ subsection {* Monad operation *} 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) subsection {* Transfer *} definition embed_list :: "nat list \<Rightarrow> int list" where "embed_list l = map int l" definition nat_list :: "int list \<Rightarrow> bool" where "nat_list l = nat_set (set l)" definition return_list :: "int list \<Rightarrow> nat list" where "return_list l = map nat l" lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow> embed_list (return_list l) = l" unfolding embed_list_def return_list_def nat_list_def nat_set_def apply (induct l) apply auto done lemma transfer_nat_int_list_functions: "l @ m = return_list (embed_list l @ embed_list m)" "[] = return_list []" unfolding return_list_def embed_list_def apply auto apply (induct l, auto) apply (induct m, auto) done (* lemma transfer_nat_int_fold1: "fold f l x = fold (%x. f (nat x)) (embed_list l) x"; *) subsection {* Code generation *} text{* Optional tail recursive version of @{const map}. Can avoid stack overflow in some target languages. *} 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{* Code equation: *} lemma map_eq_map_tailrec: "map = map_tailrec" by(simp add: fun_eq_iff map_tailrec_def map_tailrec_rev) subsubsection {* Counterparts for set-related operations *} definition member :: "'a list \<Rightarrow> 'a \<Rightarrow> bool" where [code_abbrev]: "member xs x \<longleftrightarrow> x \<in> set xs" text {* Use @{text member} only for generating executable code. Otherwise use @{prop "x \<in> set xs"} instead --- it is much easier to reason about. *} 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) definition list_all :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where list_all_iff [code_abbrev]: "list_all P xs \<longleftrightarrow> Ball (set xs) P" 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 {* Usually you should prefer @{text "\<forall>x\<in>set xs"}, @{text "\<exists>x\<in>set xs"} and @{text "\<exists>!x. x\<in>set xs \<and> _"} over @{const list_all}, @{const list_ex} and @{const list_ex1} in specifications. *} lemma list_all_simps [simp, 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) lemma list_all_cong [fundef_cong]: "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> f x = g x) \<Longrightarrow> list_all f xs = list_all g ys" by (simp add: list_all_iff) 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 {* Executable checks for relations on sets *} 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 | (x = y & lexordp r xs ys))" unfolding lexordp_def by auto text {* Bounded quantification and summation over nats. *} 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\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..<n}. P m)" by auto lemma ex_nat_less_eq [code_unfold]: "(\<exists>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..<n}. P m)" by auto lemma all_nat_less [code_unfold]: "(\<forall>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..n}. P m)" by auto lemma ex_nat_less [code_unfold]: "(\<exists>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..n}. P m)" by auto lemma setsum_set_upt_conv_listsum_nat [code_unfold]: "setsum f (set [m..<n]) = listsum (map f [m..<n])" by (simp add: interv_listsum_conv_setsum_set_nat) text{* Bounded @{text LEAST} operator: *} 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)" 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 : set xs \<and> P y" hence "y : set (filter P xs)" by auto thus "x \<le> y" by (metis Cons eq_iff filter_sort set_ConsD set_sort sorted_Cons sorted_sort) qed thus ?thesis using Cons by (simp add: Bleast_def) qed declare Bleast_def[symmetric, code_unfold] text {* Summation over ints. *} 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] lemma setsum_set_upto_conv_listsum_int [code_unfold]: "setsum f (set [i..j::int]) = listsum (map f [i..j])" by (simp add: interv_listsum_conv_setsum_set_int) subsubsection {* Optimizing by rewriting *} definition null :: "'a list \<Rightarrow> bool" where [code_abbrev]: "null xs \<longleftrightarrow> xs = []" text {* Efficient emptyness check is implemented by @{const null}. *} 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 {* Operations @{const maps} and @{const map_filter} avoid intermediate lists on execution -- do not use for proving. *} 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 {* Optimized code for @{text"\<forall>i\<in>{a..b::int}"} and @{text"\<forall>n:{a..<b::nat}"} and similiarly for @{text"\<exists>"}. *} 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 {* optimized code (tail-recursive) for @{term length} *} 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 {* Pretty lists *} ML {* (* Code generation for list literals. *) signature LIST_CODE = sig val implode_list: string -> string -> Code_Thingol.iterm -> Code_Thingol.iterm list option val default_list: int * string -> (Code_Printer.fixity -> Code_Thingol.iterm -> Pretty.T) -> Code_Printer.fixity -> Code_Thingol.iterm -> Code_Thingol.iterm -> Pretty.T val add_literal_list: string -> theory -> theory end; structure List_Code : LIST_CODE = struct open Basic_Code_Thingol; fun implode_list nil' cons' t = let fun dest_cons (IConst { name = c, ... } `$ t1 `$ t2) = if c = cons' then SOME (t1, t2) else NONE | dest_cons _ = NONE; val (ts, t') = Code_Thingol.unfoldr dest_cons t; in case t' of IConst { name = c, ... } => if c = nil' then SOME ts else NONE | _ => NONE end; fun default_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 [nil', cons'] pr thm vars fxy [(t1, _), (t2, _)] = case Option.map (cons t1) (implode_list nil' cons' t2) of SOME ts => Code_Printer.literal_list literals (map (pr vars Code_Printer.NOBR) ts) | NONE => default_list (Code_Printer.infix_cons literals) (pr vars) fxy t1 t2; in Code_Target.set_printings (Code_Symbol.Constant (@{const_name Cons}, [(target, SOME (Code_Printer.complex_const_syntax (2, ([@{const_name Nil}, @{const_name Cons}], pretty))))])) end end; *} 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 {* fold (List_Code.add_literal_list) ["SML", "OCaml", "Haskell", "Scala"] *} code_reserved SML list code_reserved OCaml list subsubsection {* Use convenient predefined operations *} code_printing constant "op @" \<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 {* Implementation of sets by lists *} 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 [] \<le> set [] \<longleftrightarrow> False" by auto text {* A frequent case – avoid intermediate sets *} 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) lemma setsum_code [code]: "setsum f (set xs) = listsum (map f (remdups xs))" by (simp add: listsum_distinct_conv_setsum_set) 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 {* Operations on relations *} 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 (%(x, y). if x : S then Some y else None) R" unfolding map_project_def by (auto split: prod.split split_if_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 simp add: Bex_def) lemma wf_set [code]: "wf (set xs) = acyclic (set xs)" by (simp add: wf_iff_acyclic_if_finite) subsection {* Setup for Lifting/Transfer *} subsubsection {* Relator and predicator properties *} lemma list_all2_eq'[relator_eq]: "list_all2 (op =) = (op =)" by (rule ext)+ (simp add: list_all2_eq) lemma list_all2_mono'[relator_mono]: assumes "A \<le> B" shows "(list_all2 A) \<le> (list_all2 B)" using assms by (auto intro: list_all2_mono) lemma list_all2_OO[relator_distr]: "list_all2 A OO list_all2 B = list_all2 (A OO B)" proof (intro ext iffI) fix xs ys assume "list_all2 (A OO B) xs ys" thus "(list_all2 A OO list_all2 B) xs ys" unfolding OO_def by (induct, simp, simp add: list_all2_Cons1 list_all2_Cons2, fast) next fix xs ys assume "(list_all2 A OO list_all2 B) xs ys" then obtain zs where "list_all2 A xs zs" and "list_all2 B zs ys" .. thus "list_all2 (A OO B) xs ys" by (induct arbitrary: ys, simp, clarsimp simp add: list_all2_Cons1, fast) qed lemma Domainp_list[relator_domain]: assumes "Domainp A = P" shows "Domainp (list_all2 A) = (list_all P)" proof - { fix x have *: "\<And>x. (\<exists>y. A x y) = P x" using assms unfolding Domainp_iff by blast have "(\<exists>y. (list_all2 A x y)) = list_all P x" by (induction x) (simp_all add: * list_all2_Cons1) } then show ?thesis unfolding Domainp_iff[abs_def] by (auto iff: fun_eq_iff) qed lemma reflp_list_all2[reflexivity_rule]: assumes "reflp R" shows "reflp (list_all2 R)" proof (rule reflpI) from assms have *: "\<And>xs. R xs xs" by (rule reflpE) fix xs show "list_all2 R xs xs" by (induct xs) (simp_all add: *) qed lemma left_total_list_all2[reflexivity_rule]: "left_total R \<Longrightarrow> left_total (list_all2 R)" unfolding left_total_def apply safe apply (rename_tac xs, induct_tac xs, simp, simp add: list_all2_Cons1) done lemma left_unique_list_all2 [reflexivity_rule]: "left_unique R \<Longrightarrow> left_unique (list_all2 R)" unfolding left_unique_def apply (subst (2) all_comm, subst (1) all_comm) apply (rule allI, rename_tac zs, induct_tac zs) apply (auto simp add: list_all2_Cons2) done lemma right_total_list_all2 [transfer_rule]: "right_total R \<Longrightarrow> right_total (list_all2 R)" unfolding right_total_def by (rule allI, induct_tac y, simp, simp add: list_all2_Cons2) lemma right_unique_list_all2 [transfer_rule]: "right_unique R \<Longrightarrow> right_unique (list_all2 R)" unfolding right_unique_def apply (rule allI, rename_tac xs, induct_tac xs) apply (auto simp add: list_all2_Cons1) done lemma bi_total_list_all2 [transfer_rule]: "bi_total A \<Longrightarrow> bi_total (list_all2 A)" unfolding bi_total_def apply safe apply (rename_tac xs, induct_tac xs, simp, simp add: list_all2_Cons1) apply (rename_tac ys, induct_tac ys, simp, simp add: list_all2_Cons2) done lemma bi_unique_list_all2 [transfer_rule]: "bi_unique A \<Longrightarrow> bi_unique (list_all2 A)" unfolding bi_unique_def apply (rule conjI) apply (rule allI, rename_tac xs, induct_tac xs) apply (simp, force simp add: list_all2_Cons1) apply (subst (2) all_comm, subst (1) all_comm) apply (rule allI, rename_tac xs, induct_tac xs) apply (simp, force simp add: list_all2_Cons2) done lemma list_invariant_commute [invariant_commute]: "list_all2 (Lifting.invariant P) = Lifting.invariant (list_all P)" apply (simp add: fun_eq_iff list_all2_def list_all_iff Lifting.invariant_def Ball_def) apply (intro allI) apply (induct_tac rule: list_induct2') apply simp_all apply fastforce done subsubsection {* Quotient theorem for the Lifting package *} lemma Quotient_list[quot_map]: assumes "Quotient R Abs Rep T" shows "Quotient (list_all2 R) (map Abs) (map Rep) (list_all2 T)" proof (unfold Quotient_alt_def, intro conjI allI impI) from assms have 1: "\<And>x y. T x y \<Longrightarrow> Abs x = y" unfolding Quotient_alt_def by simp fix xs ys assume "list_all2 T xs ys" thus "map Abs xs = ys" by (induct, simp, simp add: 1) next from assms have 2: "\<And>x. T (Rep x) x" unfolding Quotient_alt_def by simp fix xs show "list_all2 T (map Rep xs) xs" by (induct xs, simp, simp add: 2) next from assms have 3: "\<And>x y. R x y \<longleftrightarrow> T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y" unfolding Quotient_alt_def by simp fix xs ys show "list_all2 R xs ys \<longleftrightarrow> list_all2 T xs (map Abs xs) \<and> list_all2 T ys (map Abs ys) \<and> map Abs xs = map Abs ys" by (induct xs ys rule: list_induct2', simp_all, metis 3) qed subsubsection {* Transfer rules for the Transfer package *} context begin interpretation lifting_syntax . lemma Nil_transfer [transfer_rule]: "(list_all2 A) [] []" by simp lemma Cons_transfer [transfer_rule]: "(A ===> list_all2 A ===> list_all2 A) Cons Cons" unfolding fun_rel_def by simp lemma list_case_transfer [transfer_rule]: "(B ===> (A ===> list_all2 A ===> B) ===> list_all2 A ===> B) list_case list_case" unfolding fun_rel_def by (simp split: list.split) lemma list_rec_transfer [transfer_rule]: "(B ===> (A ===> list_all2 A ===> B ===> B) ===> list_all2 A ===> B) list_rec list_rec" unfolding fun_rel_def by (clarify, erule list_all2_induct, simp_all) lemma tl_transfer [transfer_rule]: "(list_all2 A ===> list_all2 A) tl tl" unfolding tl_def by transfer_prover lemma butlast_transfer [transfer_rule]: "(list_all2 A ===> list_all2 A) butlast butlast" by (rule fun_relI, erule list_all2_induct, auto) lemma set_transfer [transfer_rule]: "(list_all2 A ===> set_rel A) set set" unfolding set_def by transfer_prover lemma map_transfer [transfer_rule]: "((A ===> B) ===> list_all2 A ===> list_all2 B) map map" unfolding List.map_def by transfer_prover 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 ===> op =) ===> 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]: "(op = ===> list_all2 A ===> list_all2 A) drop drop" unfolding List.drop_def by transfer_prover lemma take_transfer [transfer_rule]: "(op = ===> list_all2 A ===> list_all2 A) take take" unfolding List.take_def by transfer_prover lemma list_update_transfer [transfer_rule]: "(list_all2 A ===> op = ===> A ===> list_all2 A) list_update list_update" unfolding list_update_def by transfer_prover lemma takeWhile_transfer [transfer_rule]: "((A ===> op =) ===> list_all2 A ===> list_all2 A) takeWhile takeWhile" unfolding takeWhile_def by transfer_prover lemma dropWhile_transfer [transfer_rule]: "((A ===> op =) ===> 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 (prod_rel A B)) zip zip" unfolding zip_def by transfer_prover lemma product_transfer [transfer_rule]: "(list_all2 A ===> list_all2 B ===> list_all2 (prod_rel 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 ===> op =) ===> list_all2 A ===> option_rel A) List.find List.find" unfolding List.find_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 distinct_transfer [transfer_rule]: assumes [transfer_rule]: "bi_unique A" shows "(list_all2 A ===> op =) distinct distinct" unfolding distinct_def by transfer_prover 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 fun_relI, erule list_all2_induct) qed (auto simp: remdups_adj_Cons assms[unfolded bi_unique_def] split: list.splits) lemma replicate_transfer [transfer_rule]: "(op = ===> A ===> list_all2 A) replicate replicate" unfolding replicate_def by transfer_prover lemma length_transfer [transfer_rule]: "(list_all2 A ===> op =) length length" unfolding list_size_overloaded_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]: "(op = ===> list_all2 A ===> list_all2 A) rotate rotate" unfolding rotate_def [abs_def] by transfer_prover lemma list_all2_transfer [transfer_rule]: "((A ===> B ===> op =) ===> list_all2 A ===> list_all2 B ===> op =) list_all2 list_all2" apply (subst (4) list_all2_def [abs_def]) apply (subst (3) list_all2_def [abs_def]) apply transfer_prover done lemma sublist_transfer [transfer_rule]: "(list_all2 A ===> set_rel (op =) ===> list_all2 A) sublist sublist" unfolding sublist_def [abs_def] by transfer_prover lemma partition_transfer [transfer_rule]: "((A ===> op =) ===> list_all2 A ===> prod_rel (list_all2 A) (list_all2 A)) partition partition" unfolding partition_def by transfer_prover lemma lists_transfer [transfer_rule]: "(set_rel A ===> set_rel (list_all2 A)) lists lists" apply (rule fun_relI, rule set_relI) apply (erule lists.induct, simp) apply (simp only: set_rel_def list_all2_Cons1, metis lists.Cons) apply (erule lists.induct, simp) apply (simp only: set_rel_def list_all2_Cons2, metis lists.Cons) done lemma set_Cons_transfer [transfer_rule]: "(set_rel A ===> set_rel (list_all2 A) ===> set_rel (list_all2 A)) set_Cons set_Cons" unfolding fun_rel_def set_rel_def set_Cons_def apply safe apply (simp add: list_all2_Cons1, fast) apply (simp add: list_all2_Cons2, fast) done lemma listset_transfer [transfer_rule]: "(list_all2 (set_rel A) ===> set_rel (list_all2 A)) listset listset" unfolding listset_def by transfer_prover lemma null_transfer [transfer_rule]: "(list_all2 A ===> op =) List.null List.null" unfolding fun_rel_def List.null_def by auto lemma list_all_transfer [transfer_rule]: "((A ===> op =) ===> list_all2 A ===> op =) list_all list_all" unfolding list_all_iff [abs_def] by transfer_prover lemma list_ex_transfer [transfer_rule]: "((A ===> op =) ===> list_all2 A ===> op =) 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 fun_relI, erule list_all2_induct, simp add: fun_rel_def, simp) apply (rule fun_relI) apply (erule_tac xs=x in list_all2_induct, simp, simp add: fun_rel_def) done lemma listsum_transfer[transfer_rule]: assumes [transfer_rule]: "A 0 0" assumes [transfer_rule]: "(A ===> A ===> A) op + op +" shows "(list_all2 A ===> A) listsum listsum" unfolding listsum_def[abs_def] by transfer_prover end end