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src/HOL/List.thy

author | nipkow |

Mon, 05 Nov 2007 18:18:39 +0100 | |

changeset 25287 | 094dab519ff5 |

parent 25277 | 95128fcdd7e8 |

child 25296 | c187b7422156 |

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

added lemmas

(* Title: HOL/List.thy ID: $Id$ Author: Tobias Nipkow *) header {* The datatype of finite lists *} theory List imports PreList uses "Tools/string_syntax.ML" begin datatype 'a list = Nil ("[]") | Cons 'a "'a list" (infixr "#" 65) subsection{*Basic list processing functions*} consts filter:: "('a => bool) => 'a list => 'a list" concat:: "'a list list => 'a list" foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b" foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b" hd:: "'a list => 'a" tl:: "'a list => 'a list" last:: "'a list => 'a" butlast :: "'a list => 'a list" set :: "'a list => 'a set" map :: "('a=>'b) => ('a list => 'b list)" listsum :: "'a list => 'a::monoid_add" nth :: "'a list => nat => 'a" (infixl "!" 100) list_update :: "'a list => nat => 'a => 'a list" take:: "nat => 'a list => 'a list" drop:: "nat => 'a list => 'a list" takeWhile :: "('a => bool) => 'a list => 'a list" dropWhile :: "('a => bool) => 'a list => 'a list" rev :: "'a list => 'a list" zip :: "'a list => 'b list => ('a * 'b) list" upt :: "nat => nat => nat list" ("(1[_..</_'])") remdups :: "'a list => 'a list" remove1 :: "'a => 'a list => 'a list" "distinct":: "'a list => bool" replicate :: "nat => 'a => 'a list" splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" nonterminals lupdbinds lupdbind syntax -- {* list Enumeration *} "@list" :: "args => 'a list" ("[(_)]") -- {* Special syntax for filter *} "@filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_<-_./ _])") -- {* list update *} "_lupdbind":: "['a, 'a] => lupdbind" ("(2_ :=/ _)") "" :: "lupdbind => lupdbinds" ("_") "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds" ("_,/ _") "_LUpdate" :: "['a, lupdbinds] => 'a" ("_/[(_)]" [900,0] 900) translations "[x, xs]" == "x#[xs]" "[x]" == "x#[]" "[x<-xs . P]"== "filter (%x. P) xs" "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs" "xs[i:=x]" == "list_update xs i x" syntax (xsymbols) "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])") syntax (HTML output) "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])") text {* Function @{text size} is overloaded for all datatypes. Users may refer to the list version as @{text length}. *} abbreviation length :: "'a list => nat" where "length == size" primrec "hd(x#xs) = x" primrec "tl([]) = []" "tl(x#xs) = xs" primrec "last(x#xs) = (if xs=[] then x else last xs)" primrec "butlast []= []" "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)" primrec "set [] = {}" "set (x#xs) = insert x (set xs)" primrec "map f [] = []" "map f (x#xs) = f(x)#map f xs" setup {* snd o Sign.declare_const [] (*authentic syntax*) ("append", @{typ "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"}, InfixrName ("@", 65)) *} primrec append_Nil:"[]@ys = ys" append_Cons: "(x#xs)@ys = x#(xs@ys)" primrec "rev([]) = []" "rev(x#xs) = rev(xs) @ [x]" primrec "filter P [] = []" "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)" primrec foldl_Nil:"foldl f a [] = a" foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs" primrec "foldr f [] a = a" "foldr f (x#xs) a = f x (foldr f xs a)" primrec "concat([]) = []" "concat(x#xs) = x @ concat(xs)" primrec "listsum [] = 0" "listsum (x # xs) = x + listsum xs" primrec drop_Nil:"drop n [] = []" drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)" -- {*Warning: simpset does not contain this definition, but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} primrec take_Nil:"take n [] = []" take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)" -- {*Warning: simpset does not contain this definition, but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} primrec nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)" -- {*Warning: simpset does not contain this definition, but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} primrec "[][i:=v] = []" "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])" primrec "takeWhile P [] = []" "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])" primrec "dropWhile P [] = []" "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)" primrec "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 upt_0: "[i..<0] = []" upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])" primrec "distinct [] = True" "distinct (x#xs) = (x ~: set xs \<and> distinct xs)" primrec "remdups [] = []" "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)" primrec "remove1 x [] = []" "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)" primrec replicate_0: "replicate 0 x = []" replicate_Suc: "replicate (Suc n) x = x # replicate n x" definition rotate1 :: "'a list \<Rightarrow> 'a list" where "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])" definition rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where "rotate n = rotate1 ^ n" definition list_all2 :: "('a => 'b => bool) => 'a list => 'b list => 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 "splice [] ys = ys" "splice (x#xs) ys = (if ys=[] then x#xs else x # hd ys # splice xs (tl ys))" -- {*Warning: simpset does not contain the second eqn but a derived one. *} text{* The following simple sort functions are intended for proofs, not for efficient implementations. *} context linorder begin fun sorted :: "'a list \<Rightarrow> bool" where "sorted [] \<longleftrightarrow> True" | "sorted [x] \<longleftrightarrow> True" | "sorted (x#y#zs) \<longleftrightarrow> x <= y \<and> sorted (y#zs)" fun insort :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where "insort x [] = [x]" | "insort x (y#ys) = (if x <= y then (x#y#ys) else y#(insort x ys))" fun sort :: "'a list \<Rightarrow> 'a list" where "sort [] = []" | "sort (x#xs) = insort x (sort xs)" 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. *} (* Proper theorem proving support would be nice. For example, if @{text"set[f x y. x \<leftarrow> xs, y \<leftarrow> ys, P x y]"} produced something like @{term"{z. EX x: set xs. EX y:set ys. P x y \<and> z = f x y}"}. *) nonterminals lc_qual 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 (advanced) {* let val NilC = Syntax.const @{const_name Nil}; val ConsC = Syntax.const @{const_name Cons}; val mapC = Syntax.const @{const_name map}; val concatC = Syntax.const @{const_name concat}; val IfC = Syntax.const @{const_name If}; fun singl x = ConsC $ x $ NilC; fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *) let val x = Free (Name.variant (add_term_free_names (p$e, [])) "x", dummyT); val e = if opti then singl e else e; val case1 = Syntax.const "_case1" $ p $ e; val case2 = Syntax.const "_case1" $ Syntax.const Term.dummy_patternN $ NilC; val cs = Syntax.const "_case2" $ case1 $ case2 val ft = DatatypeCase.case_tr false DatatypePackage.datatype_of_constr ctxt [x, cs] in lambda x ft end; fun abs_tr ctxt (p as Free(s,T)) e opti = let val thy = ProofContext.theory_of ctxt; val s' = Sign.intern_const thy s in if Sign.declared_const thy s' then (pat_tr ctxt p e opti, false) else (lambda p e, true) end | abs_tr ctxt p e opti = (pat_tr ctxt p e opti, false); fun lc_tr ctxt [e, Const("_lc_test",_)$b, qs] = let val res = case qs of Const("_lc_end",_) => singl e | Const("_lc_quals",_)$q$qs => lc_tr ctxt [e,q,qs]; in IfC $ b $ res $ NilC end | lc_tr ctxt [e, Const("_lc_gen",_) $ p $ es, 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("_lc_gen",_) $ p $ es, 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 [("_listcompr", lc_tr)] end *} (* term "[(x,y,z). b]" term "[(x,y,z). x\<leftarrow>xs]" term "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]" term "[(x,y,z). x<a, x>b]" term "[(x,y,z). x\<leftarrow>xs, x>b]" term "[(x,y,z). x<a, x\<leftarrow>xs]" term "[(x,y). Cons True x \<leftarrow> xs]" term "[(x,y,z). Cons x [] \<leftarrow> xs]" term "[(x,y,z). x<a, x>b, x=d]" term "[(x,y,z). x<a, x>b, y\<leftarrow>ys]" term "[(x,y,z). x<a, x\<leftarrow>xs,y>b]" term "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]" term "[(x,y,z). x\<leftarrow>xs, x>b, y<a]" term "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]" term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]" term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]" term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]" *) subsubsection {* @{const Nil} and @{const Cons} *} lemma not_Cons_self [simp]: "xs \<noteq> x # xs" by (induct xs) auto lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric] lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)" by (induct xs) auto lemma length_induct: "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs" by (rule measure_induct [of length]) iprover 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]: "\<lbrakk> length xs = length ys; P [] []; \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk> \<Longrightarrow> P xs ys" apply(induct xs arbitrary: ys) apply simp apply(case_tac ys) apply simp apply simp done 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("List.list.Nil",_)) acc = acc | len (Const("List.list.Cons",_) $ _ $ xs) (ts,n) = len xs (ts,n+1) | len (Const("List.append",_) $ xs $ ys) acc = len xs (len ys acc) | len (Const("List.rev",_) $ xs) acc = len xs acc | len (Const("List.map",_) $ _ $ xs) acc = len xs acc | len t (ts,n) = (t::ts,n); fun list_neq _ ss 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 (Simplifier.the_context ss) [] [] neq_len (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 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 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 interpretation semigroup_append: semigroup_add ["op @"] by unfold_locales simp interpretation monoid_append: monoid_add ["[]" "op @"] by unfold_locales (simp+) 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, noatp]: "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 fastsimp apply(case_tac zs) apply simp apply fastsimp done lemma same_append_eq [iff]: "(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]: "(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,noatp]: "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. *} ML_setup {* local fun last (cons as Const("List.list.Cons",_) $ _ $ xs) = (case xs of Const("List.list.Nil",_) => cons | _ => last xs) | last (Const("List.append",_) $ _ $ ys) = last ys | last t = t; fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true | list1 _ = false; fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) = (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs) | butlast ((app as Const("List.append",_) $ xs) $ ys) = app $ butlast ys | butlast xs = Const("List.list.Nil",fastype_of xs); val rearr_ss = HOL_basic_ss addsimps [@{thm append_assoc}, @{thm append_Nil}, @{thm append_Cons}]; fun list_eq ss (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("List.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 (Simplifier.the_context ss) [] [] eq (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 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 val list_eq_simproc = Simplifier.simproc @{theory} "list_eq" ["(xs::'a list) = ys"] (K list_eq); end; Addsimprocs [list_eq_simproc]; *} subsubsection {* @{text map} *} 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_compose: "map (f o g) xs = map f (map g xs)" by (induct xs) (auto simp add: o_def) 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, recdef_cong]: "xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys" -- {* a congruence rule for @{text map} *} 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: "map f xs = map f ys ==> length xs = length ys" apply (induct ys arbitrary: xs) apply simp apply (metis Suc_length_conv length_map) done 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) subsubsection {* @{text 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 {* @{text set} *} 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] = {k. i \<le> k \<and> k < j}" apply (induct j, simp_all) apply (erule ssubst, auto) done lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)" proof (induct xs) case Nil show ?case by simp next case (Cons a xs) show ?case proof assume "x \<in> set (a # xs)" with Cons show "\<exists>ys zs. a # xs = ys @ x # zs" by (auto intro: Cons_eq_appendI) next assume "\<exists>ys zs. a # xs = ys @ x # zs" then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast show "x \<in> set (a # xs)" by (cases ys) (auto simp add: eq) qed qed lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs" by (rule in_set_conv_decomp [THEN iffD1]) lemma in_set_conv_decomp_first: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)" proof (induct xs) case Nil show ?case by simp next case (Cons a xs) show ?case proof cases assume "x = a" thus ?case using Cons by fastsimp next assume "x \<noteq> a" show ?case proof assume "x \<in> set (a # xs)" with Cons and `x \<noteq> a` show "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys" by (fastsimp intro!: Cons_eq_appendI) next assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys" then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast show "x \<in> set (a # xs)" by (cases ys) (auto simp add: eq) qed qed qed lemma split_list_first: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys" by (rule in_set_conv_decomp_first [THEN iffD1]) lemma finite_list: "finite A ==> EX l. set l = A" apply (erule finite_induct, auto) apply (rule_tac x="x#l" in exI, auto) done lemma card_length: "card (set xs) \<le> length xs" by (induct xs) (auto simp add: card_insert_if) subsubsection {* @{text 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 inj_Suc) 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 inj_Suc) 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 fastsimp 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, recdef_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 {* @{text 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 subsubsection {* @{text nth} *} lemma nth_Cons_0 [simp]: "(x # xs)!0 = x" by auto lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n" by auto declare nth.simps [simp del] 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 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 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) subsubsection {* @{text 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_overwrite [simp]: "i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]" by (induct xs arbitrary: i) (auto 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_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 update_zip: "length xs = length ys ==> (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: "xs [i := x, i := y] = xs [i := y]" apply (induct xs arbitrary: i) apply simp apply (case_tac i) apply 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 subsubsection {* @{text last} and @{text 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 add:last.simps) lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs" by(simp add:last.simps) 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 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 last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)" by(induct xs)(auto simp:neq_Nil_conv) subsubsection {* @{text take} and @{text 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_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 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 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 hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n" by(simp add: hd_conv_nth) 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 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: "i+j \<le> length(xs) \<Longrightarrow> 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\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) = (if size xs\<^isub>1 \<le> size ys\<^isub>1 then xs\<^isub>1 = take (size xs\<^isub>1) ys\<^isub>1 \<and> xs\<^isub>2 = drop (size xs\<^isub>1) ys\<^isub>1 @ ys\<^isub>2 else take (size ys\<^isub>1) xs\<^isub>1 = ys\<^isub>1 \<and> drop (size ys\<^isub>1) xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>2)" apply(induct xs\<^isub>1 arbitrary: ys\<^isub>1) apply simp apply(case_tac ys\<^isub>1) apply simp_all done lemma take_hd_drop: "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) 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 {* @{text takeWhile} and @{text dropWhile} *} 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 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 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) 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: "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<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, recdef_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, recdef_cong]: "[| l = k; !!x. x : set l ==> P x = Q x |] ==> dropWhile P l = dropWhile Q k" by (induct k arbitrary: l, simp_all) subsubsection {* @{text 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 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_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; length ys = length vs |] ==> 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 map_zip_map: "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)" apply(induct xs arbitrary:ys) apply simp apply(case_tac ys) apply simp_all done lemma map_zip_map2: "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)" apply(induct xs arbitrary:ys) apply simp apply(case_tac ys) apply simp_all done 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_update: "length xs = length ys ==> 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 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) subsubsection {* @{text 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_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> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])" by (simp add: list_all2_conv_all_nth nth_list_update) lemma list_all2_update_cong2: "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])" by (simp add: list_all2_lengthD list_all2_update_cong) 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 subsubsection {* @{text foldl} and @{text foldr} *} lemma foldl_append [simp]: "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys" by (induct xs arbitrary: a) auto lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)" by (induct xs) auto lemma foldr_map: "foldr g (map f xs) a = foldr (g o f) xs a" by(induct xs) simp_all text{* For efficient code generation: avoid intermediate list. *} lemma foldl_map[code unfold]: "foldl g a (map f xs) = foldl (%a x. g a (f x)) a xs" by(induct xs arbitrary:a) simp_all lemma foldl_cong [fundef_cong, recdef_cong]: "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] ==> foldl f a l = foldl g b k" by (induct k arbitrary: a b l) simp_all lemma foldr_cong [fundef_cong, recdef_cong]: "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] ==> foldr f l a = foldr g k b" by (induct k arbitrary: a b l) simp_all lemma (in semigroup_add) foldl_assoc: shows "foldl op+ (x+y) zs = x + (foldl op+ y zs)" by (induct zs arbitrary: y) (simp_all add:add_assoc) lemma (in monoid_add) foldl_absorb0: shows "x + (foldl op+ 0 zs) = foldl op+ x zs" by (induct zs) (simp_all add:foldl_assoc) text{* The ``First Duality Theorem'' in Bird \& Wadler: *} lemma foldl_foldr1_lemma: "foldl op + a xs = a + foldr op + xs (0\<Colon>'a::monoid_add)" by (induct xs arbitrary: a) (auto simp:add_assoc) corollary foldl_foldr1: "foldl op + 0 xs = foldr op + xs (0\<Colon>'a::monoid_add)" by (simp add:foldl_foldr1_lemma) text{* The ``Third Duality Theorem'' in Bird \& Wadler: *} lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)" by (induct xs) auto lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a" by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"]) lemma (in ab_semigroup_add) foldr_conv_foldl: "foldr op + xs a = foldl op + a xs" by (induct xs, auto simp add: foldl_assoc add_commute) text {* Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more difficult to use because it requires an additional transitivity step. *} lemma start_le_sum: "(m::nat) <= n ==> m <= foldl (op +) n ns" by (induct ns arbitrary: n) auto lemma elem_le_sum: "(n::nat) : set ns ==> n <= foldl (op +) 0 ns" by (force intro: start_le_sum simp add: in_set_conv_decomp) lemma sum_eq_0_conv [iff]: "(foldl (op +) (m::nat) ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))" by (induct ns arbitrary: m) auto lemma foldr_invariant: "\<lbrakk>Q x ; \<forall> x\<in> set xs. P x; \<forall> x y. P x \<and> Q y \<longrightarrow> Q (f x y) \<rbrakk> \<Longrightarrow> Q (foldr f xs x)" by (induct xs, simp_all) lemma foldl_invariant: "\<lbrakk>Q x ; \<forall> x\<in> set xs. P x; \<forall> x y. P x \<and> Q y \<longrightarrow> Q (f y x) \<rbrakk> \<Longrightarrow> Q (foldl f x xs)" by (induct xs arbitrary: x, simp_all) text{* @{const foldl} and @{text concat} *} lemma concat_conv_foldl: "concat xss = foldl op@ [] xss" by (induct xss) (simp_all add:monoid_append.foldl_absorb0) lemma foldl_conv_concat: "foldl (op @) xs xxs = xs @ (concat xxs)" by(simp add:concat_conv_foldl monoid_append.foldl_absorb0) subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*} lemma listsum_append[simp]: "listsum (xs @ ys) = listsum xs + listsum ys" by (induct xs) (simp_all add:add_assoc) lemma listsum_rev[simp]: fixes xs :: "'a::comm_monoid_add list" shows "listsum (rev xs) = listsum xs" by (induct xs) (simp_all add:add_ac) lemma listsum_foldr: "listsum xs = foldr (op +) xs 0" by(induct xs) auto text{* For efficient code generation --- @{const listsum} is not tail recursive but @{const foldl} is. *} lemma listsum[code unfold]: "listsum xs = foldl (op +) 0 xs" by(simp add:listsum_foldr foldl_foldr1) 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 (map (%x. b) xs)" "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (map (%x. b) xs)" lemma listsum_0 [simp]: "(\<Sum>x\<leftarrow>xs. 0) = 0" by (induct xs) simp_all text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *} lemma uminus_listsum_map: "- listsum (map f xs) = (listsum (map (uminus o f) xs) :: 'a::ab_group_add)" by(induct xs) simp_all subsubsection {* @{text upt} *} lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])" -- {* simp does not terminate! *} by (induct j) auto 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 (metis 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 nth_upt) done 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" apply (frule nth_take_lemma [OF le_refl eq_imp_le]) apply (simp_all add: take_all) done 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 take_all) 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 lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard] lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard] lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard] declare take_Cons_number_of [simp] drop_Cons_number_of [simp] nth_Cons_number_of [simp] subsubsection {* @{text "distinct"} and @{text remdups} *} 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) = 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 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_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 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 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 length_remdups_concat: "length(remdups(concat xss)) = card(\<Union>xs \<in> set xss. set xs)" by(simp add: set_concat distinct_card[symmetric]) subsubsection {* @{text 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 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 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 subsubsection {* @{text replicate} *} lemma length_replicate [simp]: "length (replicate n x) = n" by (induct n) auto lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)" by (induct n) 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 lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x" by (induct n) auto lemma tl_replicate [simp]: "n \<noteq> 0 ==> 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_replicateD: "x : set (replicate n y) ==> x = y" by (simp add: set_replicate_conv_if split: split_if_asm) 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) subsubsection{*@{text rotate1} and @{text rotate}*} lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]" by(simp add:rotate1_def) 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(simp add:rotate1_def split:list.split) 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(simp add:rotate1_def split:list.split) 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(simp add:rotate1_def split:list.split) blast 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(simp add:rotate1_def split:list.split) 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(simp add:rotate1_def split:list.split) 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 {* @{text sublist} --- a generalization of @{text 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] zip_append 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) moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto ultimately show ?case by(simp add: sublist_Cons cong:filter_cong) qed subsubsection {* @{const splice} *} lemma splice_Nil2 [simp, code]: "splice xs [] = xs" by (cases xs) simp_all lemma splice_Cons_Cons [simp, code]: "splice (x#xs) (y#ys) = x # y # splice xs ys" by simp declare splice.simps(2) [simp del, code del] lemma length_splice[simp]: "length(splice xs ys) = length xs + length ys" apply(induct xs arbitrary: ys) apply simp apply(case_tac ys) apply auto 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 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_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 set_insort: "set(insort x xs) = insert x (set xs)" by (induct xs) auto lemma set_sort[simp]: "set(sort xs) = set xs" by (induct xs) (simp_all add:set_insort) lemma distinct_insort: "distinct (insort x xs) = (x \<notin> set xs \<and> distinct xs)" by(induct xs)(auto simp:set_insort) lemma distinct_sort[simp]: "distinct (sort xs) = distinct xs" by(induct xs)(simp_all add:distinct_insort set_sort) lemma sorted_insort: "sorted (insort x xs) = sorted xs" apply (induct xs) apply(auto simp:sorted_Cons set_insort) done theorem sorted_sort[simp]: "sorted (sort xs)" by (induct xs) (auto simp:sorted_insort) 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 (metis 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 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 end lemma sorted_upt[simp]: "sorted[i..<j]" by (induct j) (simp_all add:sorted_append) 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}). *} context linorder begin definition sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where "sorted_list_of_set A == THE xs. set xs = A & sorted xs & distinct xs" lemma sorted_list_of_set[simp]: "finite A \<Longrightarrow> set(sorted_list_of_set A) = A & sorted(sorted_list_of_set A) & distinct(sorted_list_of_set A)" apply(simp add:sorted_list_of_set_def) apply(rule the1I2) apply(simp_all add: finite_sorted_distinct_unique) done lemma sorted_list_of_empty[simp]: "sorted_list_of_set {} = []" unfolding sorted_list_of_set_def apply(subst the_equality[of _ "[]"]) apply simp_all done end subsubsection {* @{text upto}: the generic interval-list *} class finite_intvl_succ = linorder + fixes successor :: "'a \<Rightarrow> 'a" assumes finite_intvl: "finite{a..b}" and successor_incr: "a < successor a" and ord_discrete: "\<not>(\<exists>x. a < x & x < successor a)" context finite_intvl_succ begin definition upto :: "'a \<Rightarrow> 'a \<Rightarrow> 'a list" ("(1[_../_])") where "upto i j == sorted_list_of_set {i..j}" lemma upto[simp]: "set[a..b] = {a..b} & sorted[a..b] & distinct[a..b]" by(simp add:upto_def finite_intvl) lemma insert_intvl: "i \<le> j \<Longrightarrow> insert i {successor i..j} = {i..j}" apply(insert successor_incr[of i]) apply(auto simp: atLeastAtMost_def atLeast_def atMost_def) apply (metis ord_discrete less_le not_le) done lemma sorted_list_of_set_rec: "i \<le> j \<Longrightarrow> sorted_list_of_set {i..j} = i # sorted_list_of_set {successor i..j}" apply(simp add:sorted_list_of_set_def upto_def) apply (rule the1_equality[OF finite_sorted_distinct_unique]) apply (simp add:finite_intvl) apply(rule the1I2[OF finite_sorted_distinct_unique]) apply (simp add:finite_intvl) apply (simp add: sorted_Cons insert_intvl Ball_def) apply (metis successor_incr leD less_imp_le order_trans) done lemma upto_rec[code]: "[i..j] = (if i \<le> j then i # [successor i..j] else [])" by(simp add: upto_def sorted_list_of_set_rec) end text{* The integers are an instance of the above class: *} instance int:: finite_intvl_succ successor_int_def: "successor == (%i. i+1)" by intro_classes (simp_all add: successor_int_def) text{* Now @{term"[i..j::int]"} is defined for integers. *} hide (open) const successor subsubsection {* @{text lists}: the list-forming operator over sets *} inductive_set lists :: "'a set => 'a list set" for A :: "'a set" where Nil [intro!]: "[]: lists A" | Cons [intro!,noatp]: "[| a: A;l: lists A|] ==> a#l : lists A" inductive_cases listsE [elim!,noatp]: "x#l : lists A" inductive_cases listspE [elim!,noatp]: "listsp A (x # l)" lemma listsp_mono [mono]: "A \<le> B ==> listsp A \<le> listsp B" by (clarify, 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_eq inf_bool_eq] lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set] 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 = in_listsp_conv_set [to_set] lemma in_listspD [dest!,noatp]: "listsp A xs ==> \<forall>x\<in>set xs. A x" by (rule in_listsp_conv_set [THEN iffD1]) lemmas in_listsD [dest!,noatp] = in_listspD [to_set] lemma in_listspI [intro!,noatp]: "\<forall>x\<in>set xs. A x ==> listsp A xs" by (rule in_listsp_conv_set [THEN iffD2]) lemmas in_listsI [intro!,noatp] = in_listspI [to_set] lemma lists_UNIV [simp]: "lists UNIV = UNIV" by auto 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}.*} constdefs set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set" "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & 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.*} consts listset :: "'a set list \<Rightarrow> 'a list set" primrec "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.*} consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set" --{*The lexicographic ordering for lists of the specified length*} primrec "lexn r 0 = {}" "lexn r (Suc n) = (prod_fun (%(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}" constdefs lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" "lex r == \<Union>n. lexn r n" --{*Holds only between lists of the same length*} lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" "lenlex r == inv_image (less_than <*lex*> lex r) (%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_prod_fun_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 diag_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. *} constdefs lexord :: "('a * 'a)set \<Rightarrow> ('a list * 'a list) set" "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: "(! x. (x,x) \<notin> r) \<Longrightarrow> (y,y) \<notin> lexord r" by (induct y, auto) lemma lexord_trans: "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r" apply (erule rev_mp)+ apply (rule_tac x = x in spec) apply (rule_tac x = z in spec) apply ( induct_tac y, simp, clarify) apply (case_tac xa, erule ssubst) apply (erule allE, erule allE) -- {* avoid simp recursion *} apply (case_tac x, simp, simp) apply (case_tac x, erule allE, erule allE, simp) apply (erule_tac x = listb in allE) apply (erule_tac x = lista in allE, simp) apply (unfold trans_def) by blast 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 subsection {* 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[recdef_wf, 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 a Relation on List Elements to the Lists*} inductive_set listrel :: "('a * 'a)set => ('a list * 'a list)set" for r :: "('a * 'a)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_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: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" apply (simp add: refl_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 listrel_sym listrel_trans) 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) subsection{*Miscellany*} subsubsection {* Characters and strings *} datatype nibble = Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7 | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF datatype char = Char nibble nibble -- "Note: canonical order of character encoding coincides with standard term ordering" types string = "char list" syntax "_Char" :: "xstr => char" ("CHR _") "_String" :: "xstr => string" ("_") setup StringSyntax.setup subsection {* Code generator *} subsubsection {* Setup *} types_code "list" ("_ list") attach (term_of) {* fun term_of_list f T = HOLogic.mk_list T o map f; *} attach (test) {* fun gen_list' aG i j = frequency [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] () and gen_list aG i = gen_list' aG i i; *} "char" ("string") attach (term_of) {* val term_of_char = HOLogic.mk_char o ord; *} attach (test) {* fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z"))); *} consts_code "Cons" ("(_ ::/ _)") code_type list (SML "_ list") (OCaml "_ list") (Haskell "![_]") code_reserved SML list code_reserved OCaml list code_const Nil (SML "[]") (OCaml "[]") (Haskell "[]") setup {* fold (fn target => CodeTarget.add_pretty_list target @{const_name Nil} @{const_name Cons} ) ["SML", "OCaml", "Haskell"] *} code_instance list :: eq (Haskell -) code_const "op = \<Colon> 'a\<Colon>eq list \<Rightarrow> 'a list \<Rightarrow> bool" (Haskell infixl 4 "==") setup {* let fun list_codegen thy defs gr dep thyname b t = let val ts = HOLogic.dest_list t; val (gr', _) = Codegen.invoke_tycodegen thy defs dep thyname false (gr, fastype_of t); val (gr'', ps) = foldl_map (Codegen.invoke_codegen thy defs dep thyname false) (gr', ts) in SOME (gr'', Pretty.list "[" "]" ps) end handle TERM _ => NONE; fun char_codegen thy defs gr dep thyname b t = let val i = HOLogic.dest_char t; val (gr', _) = Codegen.invoke_tycodegen thy defs dep thyname false (gr, fastype_of t) in SOME (gr', Pretty.str (ML_Syntax.print_string (chr i))) end handle TERM _ => NONE; in Codegen.add_codegen "list_codegen" list_codegen #> Codegen.add_codegen "char_codegen" char_codegen end; *} subsubsection {* Generation of efficient code *} consts null:: "'a list \<Rightarrow> bool" list_inter :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" list_all :: "('a \<Rightarrow> bool) \<Rightarrow> ('a list \<Rightarrow> bool)" filtermap :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list" map_filter :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list" setup {* snd o Sign.declare_const [] (*authentic syntax*) ("member", @{typ "'a \<Rightarrow> 'a list \<Rightarrow> bool"}, InfixlName ("mem", 55)) *} primrec "x mem [] = False" "x mem (y#ys) = (if y=x then True else x mem ys)" primrec "null [] = True" "null (x#xs) = False" primrec "list_inter [] bs = []" "list_inter (a#as) bs = (if a \<in> set bs then a # list_inter as bs else list_inter as bs)" primrec "list_all P [] = True" "list_all P (x#xs) = (P x \<and> list_all P xs)" primrec "list_ex P [] = False" "list_ex P (x#xs) = (P x \<or> list_ex P xs)" primrec "filtermap f [] = []" "filtermap f (x#xs) = (case f x of None \<Rightarrow> filtermap f xs | Some y \<Rightarrow> y # filtermap f xs)" primrec "map_filter f P [] = []" "map_filter f P (x#xs) = (if P x then f x # map_filter f P xs else map_filter f P xs)" text {* Only use @{text mem} for generating executable code. Otherwise use @{prop "x : set xs"} instead --- it is much easier to reason about. The same is true for @{const list_all} and @{const list_ex}: write @{text "\<forall>x\<in>set xs"} and @{text "\<exists>x\<in>set xs"} instead because the HOL quantifiers are aleady known to the automatic provers. In fact, the declarations in the code subsection make sure that @{text "\<in>"}, @{text "\<forall>x\<in>set xs"} and @{text "\<exists>x\<in>set xs"} are implemented efficiently. Efficient emptyness check is implemented by @{const null}. The functions @{const filtermap} and @{const map_filter} are just there to generate efficient code. Do not use them for modelling and proving. *} lemma rev_foldl_cons [code]: "rev xs = foldl (\<lambda>xs x. x # xs) [] xs" proof (induct xs) case Nil then show ?case by simp next case Cons { fix x xs ys have "foldl (\<lambda>xs x. x # xs) ys xs @ [x] = foldl (\<lambda>xs x. x # xs) (ys @ [x]) xs" by (induct xs arbitrary: ys) auto } note aux = this show ?case by (induct xs) (auto simp add: Cons aux) qed lemma mem_iff [code post]: "x mem xs \<longleftrightarrow> x \<in> set xs" by (induct xs) auto lemmas in_set_code [code unfold] = mem_iff [symmetric] lemma empty_null [code inline]: "xs = [] \<longleftrightarrow> null xs" by (cases xs) simp_all lemmas null_empty [code post] = empty_null [symmetric] lemma list_inter_conv: "set (list_inter xs ys) = set xs \<inter> set ys" by (induct xs) auto lemma list_all_iff [code post]: "list_all P xs \<longleftrightarrow> (\<forall>x \<in> set xs. P x)" by (induct xs) auto lemmas list_ball_code [code unfold] = list_all_iff [symmetric] lemma list_all_append [simp]: "list_all P (xs @ ys) \<longleftrightarrow> (list_all P xs \<and> list_all P ys)" by (induct xs) auto lemma list_all_rev [simp]: "list_all P (rev xs) \<longleftrightarrow> list_all P xs" by (simp add: list_all_iff) lemma list_all_length: "list_all P xs \<longleftrightarrow> (\<forall>n < length xs. P (xs ! n))" unfolding list_all_iff by (auto intro: all_nth_imp_all_set) lemma list_ex_iff [code post]: "list_ex P xs \<longleftrightarrow> (\<exists>x \<in> set xs. P x)" by (induct xs) simp_all lemmas list_bex_code [code unfold] = list_ex_iff [symmetric] lemma list_ex_length: "list_ex P xs \<longleftrightarrow> (\<exists>n < length xs. P (xs ! n))" unfolding list_ex_iff set_conv_nth by auto lemma filtermap_conv: "filtermap f xs = map (\<lambda>x. the (f x)) (filter (\<lambda>x. f x \<noteq> None) xs)" by (induct xs) (simp_all split: option.split) lemma map_filter_conv [simp]: "map_filter f P xs = map f (filter P xs)" by (induct xs) auto text {* Code for 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 lemma atLeastLessThan_upt [code unfold]: "{n..<m} = set [n..<m]" by auto lemma greaterThanAtMost_upto [code unfold]: "{n<..m} = set [Suc n..<Suc m]" by auto lemma atLeastAtMost_upto [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[code unfold]: "setsum f (set[k..<n]) = listsum (map f [k..<n])" apply(subst atLeastLessThan_upt[symmetric]) by (induct n) simp_all subsubsection {* List partitioning *} consts partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" primrec "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_P: "partition P xs = (yes, no) \<Longrightarrow> (\<forall>p\<in> set yes. P p) \<and> (\<forall>p\<in> set no. \<not> P p)" proof (induct xs arbitrary: yes no rule: partition.induct) case Nil then show ?case by simp next case (Cons a as) let ?p = "partition P as" let ?p' = "partition P (a # as)" note prem = `?p' = (yes, no)` show ?case proof (cases "P a") case True with prem have yes: "yes = a # fst ?p" and no: "no = snd ?p" by (simp_all add: Let_def split_def) have "(\<forall>p\<in> set (fst ?p). P p) \<and> (\<forall>p\<in> set no. \<not> P p)" by (rule Cons.hyps) (simp add: yes no) with True yes show ?thesis by simp next case False with prem have yes: "yes = fst ?p" and no: "no = a # snd ?p" by (simp_all add: Let_def split_def) have "(\<forall>p\<in> set yes. P p) \<and> (\<forall>p\<in> set (snd ?p). \<not> P p)" by (rule Cons.hyps) (simp add: yes no) with False no show ?thesis by simp qed qed lemma partition_filter1: "fst (partition P xs) = filter P xs" by (induct xs rule: partition.induct) (auto simp add: Let_def split_def) lemma partition_filter2: "snd (partition P xs) = filter (Not o P) xs" by (induct xs rule: partition.induct) (auto simp add: Let_def split_def) lemma partition_set: assumes "partition P xs = (yes, no)" shows "set yes \<union> set no = set xs" proof - have "set xs = {x. x \<in> set xs \<and> P x} \<union> {x. x \<in> set xs \<and> \<not> P x}" by blast also have "\<dots> = set (List.filter P xs) Un (set (List.filter (Not o P) xs))" by simp also have "\<dots> = set (fst (partition P xs)) \<union> set (snd (partition P xs))" using partition_filter1 [of P xs] partition_filter2 [of P xs] by simp finally show "set yes Un set no = set xs" using assms by simp qed end