wenzelm@13462: (*  Title:      HOL/List.thy
wenzelm@13462:     ID:         $Id$
wenzelm@13462:     Author:     Tobias Nipkow
clasohm@923: *)
clasohm@923: 
wenzelm@13114: header {* The datatype of finite lists *}
wenzelm@13122: 
nipkow@15131: theory List
haftmann@25591: imports ATP_Linkup
wenzelm@21754: uses "Tools/string_syntax.ML"
nipkow@15131: begin
clasohm@923: 
wenzelm@13142: datatype 'a list =
wenzelm@13366:     Nil    ("[]")
wenzelm@13366:   | Cons 'a  "'a list"    (infixr "#" 65)
clasohm@923: 
nipkow@15392: subsection{*Basic list processing functions*}
nipkow@15302: 
clasohm@923: consts
wenzelm@13366:   filter:: "('a => bool) => 'a list => 'a list"
wenzelm@13366:   concat:: "'a list list => 'a list"
wenzelm@13366:   foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"
wenzelm@13366:   foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"
wenzelm@13366:   hd:: "'a list => 'a"
wenzelm@13366:   tl:: "'a list => 'a list"
wenzelm@13366:   last:: "'a list => 'a"
wenzelm@13366:   butlast :: "'a list => 'a list"
wenzelm@13366:   set :: "'a list => 'a set"
wenzelm@13366:   map :: "('a=>'b) => ('a list => 'b list)"
nipkow@23096:   listsum ::  "'a list => 'a::monoid_add"
wenzelm@13366:   nth :: "'a list => nat => 'a"    (infixl "!" 100)
wenzelm@13366:   list_update :: "'a list => nat => 'a => 'a list"
wenzelm@13366:   take:: "nat => 'a list => 'a list"
wenzelm@13366:   drop:: "nat => 'a list => 'a list"
wenzelm@13366:   takeWhile :: "('a => bool) => 'a list => 'a list"
wenzelm@13366:   dropWhile :: "('a => bool) => 'a list => 'a list"
wenzelm@13366:   rev :: "'a list => 'a list"
wenzelm@13366:   zip :: "'a list => 'b list => ('a * 'b) list"
nipkow@15425:   upt :: "nat => nat => nat list" ("(1[_..</_'])")
wenzelm@13366:   remdups :: "'a list => 'a list"
nipkow@15110:   remove1 :: "'a => 'a list => 'a list"
wenzelm@13366:   "distinct":: "'a list => bool"
wenzelm@13366:   replicate :: "nat => 'a => 'a list"
nipkow@19390:   splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
nipkow@15302: 
clasohm@923: 
nipkow@13146: nonterminals lupdbinds lupdbind
nipkow@5077: 
clasohm@923: syntax
wenzelm@13366:   -- {* list Enumeration *}
wenzelm@13366:   "@list" :: "args => 'a list"    ("[(_)]")
clasohm@923: 
wenzelm@13366:   -- {* Special syntax for filter *}
nipkow@23279:   "@filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_<-_./ _])")
clasohm@923: 
wenzelm@13366:   -- {* list update *}
wenzelm@13366:   "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
wenzelm@13366:   "" :: "lupdbind => lupdbinds"    ("_")
wenzelm@13366:   "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
wenzelm@13366:   "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
nipkow@5077: 
clasohm@923: translations
wenzelm@13366:   "[x, xs]" == "x#[xs]"
wenzelm@13366:   "[x]" == "x#[]"
nipkow@23279:   "[x<-xs . P]"== "filter (%x. P) xs"
clasohm@923: 
wenzelm@13366:   "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"
wenzelm@13366:   "xs[i:=x]" == "list_update xs i x"
nipkow@5077: 
nipkow@5427: 
wenzelm@12114: syntax (xsymbols)
nipkow@23279:   "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
kleing@14565: syntax (HTML output)
nipkow@23279:   "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
paulson@3342: 
paulson@3342: 
wenzelm@13142: text {*
wenzelm@14589:   Function @{text size} is overloaded for all datatypes. Users may
wenzelm@13366:   refer to the list version as @{text length}. *}
wenzelm@13142: 
wenzelm@19363: abbreviation
wenzelm@21404:   length :: "'a list => nat" where
wenzelm@19363:   "length == size"
nipkow@15302: 
berghofe@5183: primrec
paulson@15307:   "hd(x#xs) = x"
paulson@15307: 
berghofe@5183: primrec
paulson@15307:   "tl([]) = []"
paulson@15307:   "tl(x#xs) = xs"
paulson@15307: 
berghofe@5183: primrec
paulson@15307:   "last(x#xs) = (if xs=[] then x else last xs)"
paulson@15307: 
berghofe@5183: primrec
paulson@15307:   "butlast []= []"
paulson@15307:   "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
paulson@15307: 
berghofe@5183: primrec
paulson@15307:   "set [] = {}"
paulson@15307:   "set (x#xs) = insert x (set xs)"
paulson@15307: 
berghofe@5183: primrec
paulson@15307:   "map f [] = []"
paulson@15307:   "map f (x#xs) = f(x)#map f xs"
paulson@15307: 
wenzelm@25221: primrec
haftmann@25559:   append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65)
haftmann@25559: where
haftmann@25559:   append_Nil:"[] @ ys = ys"
haftmann@25559:   | append_Cons: "(x#xs) @ ys = x # xs @ ys"
paulson@15307: 
berghofe@5183: primrec
paulson@15307:   "rev([]) = []"
paulson@15307:   "rev(x#xs) = rev(xs) @ [x]"
paulson@15307: 
berghofe@5183: primrec
paulson@15307:   "filter P [] = []"
paulson@15307:   "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
paulson@15307: 
berghofe@5183: primrec
paulson@15307:   foldl_Nil:"foldl f a [] = a"
paulson@15307:   foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
paulson@15307: 
paulson@8000: primrec
paulson@15307:   "foldr f [] a = a"
paulson@15307:   "foldr f (x#xs) a = f x (foldr f xs a)"
paulson@15307: 
berghofe@5183: primrec
paulson@15307:   "concat([]) = []"
paulson@15307:   "concat(x#xs) = x @ concat(xs)"
paulson@15307: 
berghofe@5183: primrec
nipkow@23096: "listsum [] = 0"
nipkow@23096: "listsum (x # xs) = x + listsum xs"
nipkow@23096: 
nipkow@23096: primrec
paulson@15307:   drop_Nil:"drop n [] = []"
paulson@15307:   drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
paulson@15307:   -- {*Warning: simpset does not contain this definition, but separate
paulson@15307:        theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
paulson@15307: 
berghofe@5183: primrec
paulson@15307:   take_Nil:"take n [] = []"
paulson@15307:   take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
paulson@15307:   -- {*Warning: simpset does not contain this definition, but separate
paulson@15307:        theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
paulson@15307: 
wenzelm@13142: primrec
paulson@15307:   nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
paulson@15307:   -- {*Warning: simpset does not contain this definition, but separate
paulson@15307:        theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
paulson@15307: 
berghofe@5183: primrec
paulson@15307:   "[][i:=v] = []"
paulson@15307:   "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])"
paulson@15307: 
paulson@15307: primrec
paulson@15307:   "takeWhile P [] = []"
paulson@15307:   "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
paulson@15307: 
berghofe@5183: primrec
paulson@15307:   "dropWhile P [] = []"
paulson@15307:   "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
paulson@15307: 
berghofe@5183: primrec
paulson@15307:   "zip xs [] = []"
paulson@15307:   zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
paulson@15307:   -- {*Warning: simpset does not contain this definition, but separate
paulson@15307:        theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
paulson@15307: 
nipkow@5427: primrec
nipkow@15425:   upt_0: "[i..<0] = []"
nipkow@15425:   upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
paulson@15307: 
berghofe@5183: primrec
paulson@15307:   "distinct [] = True"
paulson@15307:   "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
paulson@15307: 
berghofe@5183: primrec
paulson@15307:   "remdups [] = []"
paulson@15307:   "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
paulson@15307: 
berghofe@5183: primrec
paulson@15307:   "remove1 x [] = []"
paulson@15307:   "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)"
paulson@15307: 
nipkow@15110: primrec
paulson@15307:   replicate_0: "replicate 0 x = []"
paulson@15307:   replicate_Suc: "replicate (Suc n) x = x # replicate n x"
paulson@15307: 
haftmann@21061: definition
wenzelm@21404:   rotate1 :: "'a list \<Rightarrow> 'a list" where
wenzelm@21404:   "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"
wenzelm@21404: 
wenzelm@21404: definition
wenzelm@21404:   rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
wenzelm@21404:   "rotate n = rotate1 ^ n"
wenzelm@21404: 
wenzelm@21404: definition
wenzelm@21404:   list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
haftmann@25966:   [code func del]: "list_all2 P xs ys =
haftmann@21061:     (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"
wenzelm@21404: 
wenzelm@21404: definition
wenzelm@21404:   sublist :: "'a list => nat set => 'a list" where
wenzelm@21404:   "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"
nipkow@17086: 
nipkow@17086: primrec
haftmann@21061:   "splice [] ys = ys"
haftmann@21061:   "splice (x#xs) ys = (if ys=[] then x#xs else x # hd ys # splice xs (tl ys))"
haftmann@21061:     -- {*Warning: simpset does not contain the second eqn but a derived one. *}
haftmann@21061: 
nipkow@24616: text{* The following simple sort functions are intended for proofs,
nipkow@24616: not for efficient implementations. *}
nipkow@24616: 
wenzelm@25221: context linorder
wenzelm@25221: begin
wenzelm@25221: 
wenzelm@25221: fun sorted :: "'a list \<Rightarrow> bool" where
nipkow@24697: "sorted [] \<longleftrightarrow> True" |
nipkow@24697: "sorted [x] \<longleftrightarrow> True" |
haftmann@25062: "sorted (x#y#zs) \<longleftrightarrow> x <= y \<and> sorted (y#zs)"
nipkow@24697: 
haftmann@25559: primrec insort :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
nipkow@24697: "insort x [] = [x]" |
haftmann@25062: "insort x (y#ys) = (if x <= y then (x#y#ys) else y#(insort x ys))"
nipkow@24697: 
haftmann@25559: primrec sort :: "'a list \<Rightarrow> 'a list" where
nipkow@24697: "sort [] = []" |
nipkow@24697: "sort (x#xs) = insort x (sort xs)"
nipkow@24616: 
wenzelm@25221: end
wenzelm@25221: 
nipkow@24616: 
wenzelm@23388: subsubsection {* List comprehension *}
nipkow@23192: 
nipkow@24349: text{* Input syntax for Haskell-like list comprehension notation.
nipkow@24349: Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
nipkow@24349: the list of all pairs of distinct elements from @{text xs} and @{text ys}.
nipkow@24349: The syntax is as in Haskell, except that @{text"|"} becomes a dot
nipkow@24349: (like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
nipkow@24349: \verb![e| x <- xs, ...]!.
nipkow@24349: 
nipkow@24349: The qualifiers after the dot are
nipkow@24349: \begin{description}
nipkow@24349: \item[generators] @{text"p \<leftarrow> xs"},
nipkow@24476:  where @{text p} is a pattern and @{text xs} an expression of list type, or
nipkow@24476: \item[guards] @{text"b"}, where @{text b} is a boolean expression.
nipkow@24476: %\item[local bindings] @ {text"let x = e"}.
nipkow@24349: \end{description}
nipkow@23240: 
nipkow@24476: Just like in Haskell, list comprehension is just a shorthand. To avoid
nipkow@24476: misunderstandings, the translation into desugared form is not reversed
nipkow@24476: upon output. Note that the translation of @{text"[e. x \<leftarrow> xs]"} is
nipkow@24476: optmized to @{term"map (%x. e) xs"}.
nipkow@23240: 
nipkow@24349: It is easy to write short list comprehensions which stand for complex
nipkow@24349: expressions. During proofs, they may become unreadable (and
nipkow@24349: mangled). In such cases it can be advisable to introduce separate
nipkow@24349: definitions for the list comprehensions in question.  *}
nipkow@24349: 
nipkow@23209: (*
nipkow@23240: Proper theorem proving support would be nice. For example, if
nipkow@23192: @{text"set[f x y. x \<leftarrow> xs, y \<leftarrow> ys, P x y]"}
nipkow@23192: produced something like
nipkow@23209: @{term"{z. EX x: set xs. EX y:set ys. P x y \<and> z = f x y}"}.
nipkow@23209: *)
nipkow@23209: 
nipkow@23240: nonterminals lc_qual lc_quals
nipkow@23192: 
nipkow@23192: syntax
nipkow@23240: "_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
nipkow@24349: "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ <- _")
nipkow@23240: "_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
nipkow@24476: (*"_lc_let" :: "letbinds => lc_qual"  ("let _")*)
nipkow@23240: "_lc_end" :: "lc_quals" ("]")
nipkow@23240: "_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals" (", __")
nipkow@24349: "_lc_abs" :: "'a => 'b list => 'b list"
nipkow@23192: 
nipkow@24476: (* These are easier than ML code but cannot express the optimized
nipkow@24476:    translation of [e. p<-xs]
nipkow@23192: translations
nipkow@24349: "[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
nipkow@23240: "_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
nipkow@24349:  => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
nipkow@23240: "[e. P]" => "if P then [e] else []"
nipkow@23240: "_listcompr e (_lc_test P) (_lc_quals Q Qs)"
nipkow@23240:  => "if P then (_listcompr e Q Qs) else []"
nipkow@24349: "_listcompr e (_lc_let b) (_lc_quals Q Qs)"
nipkow@24349:  => "_Let b (_listcompr e Q Qs)"
nipkow@24476: *)
nipkow@23240: 
nipkow@23279: syntax (xsymbols)
nipkow@24349: "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
nipkow@23279: syntax (HTML output)
nipkow@24349: "_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
nipkow@24349: 
nipkow@24349: parse_translation (advanced) {*
nipkow@24349: let
nipkow@24476:   val NilC = Syntax.const @{const_name Nil};
nipkow@24476:   val ConsC = Syntax.const @{const_name Cons};
nipkow@24476:   val mapC = Syntax.const @{const_name map};
nipkow@24476:   val concatC = Syntax.const @{const_name concat};
nipkow@24476:   val IfC = Syntax.const @{const_name If};
nipkow@24476:   fun singl x = ConsC $ x $ NilC;
nipkow@24476: 
nipkow@24476:    fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
nipkow@24349:     let
nipkow@24476:       val x = Free (Name.variant (add_term_free_names (p$e, [])) "x", dummyT);
nipkow@24476:       val e = if opti then singl e else e;
nipkow@24476:       val case1 = Syntax.const "_case1" $ p $ e;
nipkow@24349:       val case2 = Syntax.const "_case1" $ Syntax.const Term.dummy_patternN
nipkow@24476:                                         $ NilC;
nipkow@24349:       val cs = Syntax.const "_case2" $ case1 $ case2
nipkow@24349:       val ft = DatatypeCase.case_tr false DatatypePackage.datatype_of_constr
nipkow@24349:                  ctxt [x, cs]
nipkow@24349:     in lambda x ft end;
nipkow@24349: 
nipkow@24476:   fun abs_tr ctxt (p as Free(s,T)) e opti =
nipkow@24349:         let val thy = ProofContext.theory_of ctxt;
nipkow@24349:             val s' = Sign.intern_const thy s
nipkow@24476:         in if Sign.declared_const thy s'
nipkow@24476:            then (pat_tr ctxt p e opti, false)
nipkow@24476:            else (lambda p e, true)
nipkow@24349:         end
nipkow@24476:     | abs_tr ctxt p e opti = (pat_tr ctxt p e opti, false);
nipkow@24476: 
nipkow@24476:   fun lc_tr ctxt [e, Const("_lc_test",_)$b, qs] =
nipkow@24476:         let val res = case qs of Const("_lc_end",_) => singl e
nipkow@24476:                       | Const("_lc_quals",_)$q$qs => lc_tr ctxt [e,q,qs];
nipkow@24476:         in IfC $ b $ res $ NilC end
nipkow@24476:     | lc_tr ctxt [e, Const("_lc_gen",_) $ p $ es, Const("_lc_end",_)] =
nipkow@24476:         (case abs_tr ctxt p e true of
nipkow@24476:            (f,true) => mapC $ f $ es
nipkow@24476:          | (f, false) => concatC $ (mapC $ f $ es))
nipkow@24476:     | lc_tr ctxt [e, Const("_lc_gen",_) $ p $ es, Const("_lc_quals",_)$q$qs] =
nipkow@24476:         let val e' = lc_tr ctxt [e,q,qs];
nipkow@24476:         in concatC $ (mapC $ (fst(abs_tr ctxt p e' false)) $ es) end
nipkow@24476: 
nipkow@24476: in [("_listcompr", lc_tr)] end
nipkow@24349: *}
nipkow@23279: 
nipkow@23240: (*
nipkow@23240: term "[(x,y,z). b]"
nipkow@24476: term "[(x,y,z). x\<leftarrow>xs]"
nipkow@24476: term "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]"
nipkow@24476: term "[(x,y,z). x<a, x>b]"
nipkow@24476: term "[(x,y,z). x\<leftarrow>xs, x>b]"
nipkow@24476: term "[(x,y,z). x<a, x\<leftarrow>xs]"
nipkow@24349: term "[(x,y). Cons True x \<leftarrow> xs]"
nipkow@24349: term "[(x,y,z). Cons x [] \<leftarrow> xs]"
nipkow@23240: term "[(x,y,z). x<a, x>b, x=d]"
nipkow@23240: term "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
nipkow@23240: term "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
nipkow@23240: term "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
nipkow@23240: term "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
nipkow@23240: term "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
nipkow@23240: term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
nipkow@23240: term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
nipkow@24349: term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
nipkow@23192: *)
nipkow@23192: 
haftmann@21061: subsubsection {* @{const Nil} and @{const Cons} *}
haftmann@21061: 
haftmann@21061: lemma not_Cons_self [simp]:
haftmann@21061:   "xs \<noteq> x # xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
wenzelm@13114: 
wenzelm@13142: lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma length_induct:
haftmann@21061:   "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
nipkow@17589: by (rule measure_induct [of length]) iprover
wenzelm@13114: 
wenzelm@13114: 
haftmann@21061: subsubsection {* @{const length} *}
wenzelm@13114: 
wenzelm@13142: text {*
haftmann@21061:   Needs to come before @{text "@"} because of theorem @{text
haftmann@21061:   append_eq_append_conv}.
wenzelm@13142: *}
wenzelm@13114: 
wenzelm@13142: lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma length_map [simp]: "length (map f xs) = length xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma length_rev [simp]: "length (rev xs) = length xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma length_tl [simp]: "length (tl xs) = length xs - 1"
nipkow@13145: by (cases xs) auto
wenzelm@13114: 
wenzelm@13142: lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
nipkow@23479: lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
nipkow@23479: by auto
nipkow@23479: 
wenzelm@13114: lemma length_Suc_conv:
nipkow@13145: "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
nipkow@13145: by (induct xs) auto
wenzelm@13142: 
nipkow@14025: lemma Suc_length_conv:
nipkow@14025: "(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
paulson@14208: apply (induct xs, simp, simp)
nipkow@14025: apply blast
nipkow@14025: done
nipkow@14025: 
wenzelm@25221: lemma impossible_Cons: "length xs <= length ys ==> xs = x # ys = False"
wenzelm@25221:   by (induct xs) auto
wenzelm@25221: 
haftmann@26442: lemma list_induct2 [consumes 1, case_names Nil Cons]:
haftmann@26442:   "length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
haftmann@26442:    (\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
haftmann@26442:    \<Longrightarrow> P xs ys"
haftmann@26442: proof (induct xs arbitrary: ys)
haftmann@26442:   case Nil then show ?case by simp
haftmann@26442: next
haftmann@26442:   case (Cons x xs ys) then show ?case by (cases ys) simp_all
haftmann@26442: qed
haftmann@26442: 
haftmann@26442: lemma list_induct3 [consumes 2, case_names Nil Cons]:
haftmann@26442:   "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
haftmann@26442:    (\<And>x xs y ys z zs. length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P xs ys zs \<Longrightarrow> P (x#xs) (y#ys) (z#zs))
haftmann@26442:    \<Longrightarrow> P xs ys zs"
haftmann@26442: proof (induct xs arbitrary: ys zs)
haftmann@26442:   case Nil then show ?case by simp
haftmann@26442: next
haftmann@26442:   case (Cons x xs ys zs) then show ?case by (cases ys, simp_all)
haftmann@26442:     (cases zs, simp_all)
haftmann@26442: qed
wenzelm@13114: 
krauss@22493: lemma list_induct2': 
krauss@22493:   "\<lbrakk> P [] [];
krauss@22493:   \<And>x xs. P (x#xs) [];
krauss@22493:   \<And>y ys. P [] (y#ys);
krauss@22493:    \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
krauss@22493:  \<Longrightarrow> P xs ys"
krauss@22493: by (induct xs arbitrary: ys) (case_tac x, auto)+
krauss@22493: 
nipkow@22143: lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
nipkow@24349: by (rule Eq_FalseI) auto
wenzelm@24037: 
wenzelm@24037: simproc_setup list_neq ("(xs::'a list) = ys") = {*
nipkow@22143: (*
nipkow@22143: Reduces xs=ys to False if xs and ys cannot be of the same length.
nipkow@22143: This is the case if the atomic sublists of one are a submultiset
nipkow@22143: of those of the other list and there are fewer Cons's in one than the other.
nipkow@22143: *)
wenzelm@24037: 
wenzelm@24037: let
nipkow@22143: 
nipkow@22143: fun len (Const("List.list.Nil",_)) acc = acc
nipkow@22143:   | len (Const("List.list.Cons",_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
haftmann@23029:   | len (Const("List.append",_) $ xs $ ys) acc = len xs (len ys acc)
nipkow@22143:   | len (Const("List.rev",_) $ xs) acc = len xs acc
nipkow@22143:   | len (Const("List.map",_) $ _ $ xs) acc = len xs acc
nipkow@22143:   | len t (ts,n) = (t::ts,n);
nipkow@22143: 
wenzelm@24037: fun list_neq _ ss ct =
nipkow@22143:   let
wenzelm@24037:     val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
nipkow@22143:     val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
nipkow@22143:     fun prove_neq() =
nipkow@22143:       let
nipkow@22143:         val Type(_,listT::_) = eqT;
haftmann@22994:         val size = HOLogic.size_const listT;
nipkow@22143:         val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
nipkow@22143:         val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
nipkow@22143:         val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
haftmann@22633:           (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
haftmann@22633:       in SOME (thm RS @{thm neq_if_length_neq}) end
nipkow@22143:   in
wenzelm@23214:     if m < n andalso submultiset (op aconv) (ls,rs) orelse
wenzelm@23214:        n < m andalso submultiset (op aconv) (rs,ls)
nipkow@22143:     then prove_neq() else NONE
nipkow@22143:   end;
wenzelm@24037: in list_neq end;
nipkow@22143: *}
nipkow@22143: 
nipkow@22143: 
nipkow@15392: subsubsection {* @{text "@"} -- append *}
wenzelm@13114: 
wenzelm@13142: lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma append_Nil2 [simp]: "xs @ [] = xs"
nipkow@13145: by (induct xs) auto
nipkow@3507: 
nipkow@24449: interpretation semigroup_append: semigroup_add ["op @"]
nipkow@24449: by unfold_locales simp
nipkow@24449: interpretation monoid_append: monoid_add ["[]" "op @"]
nipkow@24449: by unfold_locales (simp+)
nipkow@24449: 
wenzelm@13142: lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@25221: lemma append_eq_append_conv [simp, noatp]:
nipkow@24526:  "length xs = length ys \<or> length us = length vs
berghofe@13883:  ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
nipkow@24526: apply (induct xs arbitrary: ys)
paulson@14208:  apply (case_tac ys, simp, force)
paulson@14208: apply (case_tac ys, force, simp)
nipkow@13145: done
wenzelm@13142: 
nipkow@24526: lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
nipkow@24526:   (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
nipkow@24526: apply (induct xs arbitrary: ys zs ts)
nipkow@14495:  apply fastsimp
nipkow@14495: apply(case_tac zs)
nipkow@14495:  apply simp
nipkow@14495: apply fastsimp
nipkow@14495: done
nipkow@14495: 
wenzelm@13142: lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
nipkow@13145: by simp
wenzelm@13142: 
wenzelm@13142: lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
nipkow@13145: by simp
wenzelm@13114: 
wenzelm@13142: lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
nipkow@13145: by simp
wenzelm@13114: 
wenzelm@13142: lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
nipkow@13145: using append_same_eq [of _ _ "[]"] by auto
nipkow@3507: 
wenzelm@13142: lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
nipkow@13145: using append_same_eq [of "[]"] by auto
wenzelm@13114: 
paulson@24286: lemma hd_Cons_tl [simp,noatp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
nipkow@13145: by (simp add: hd_append split: list.split)
wenzelm@13114: 
wenzelm@13142: lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145: by (simp split: list.split)
wenzelm@13114: 
wenzelm@13142: lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145: by (simp add: tl_append split: list.split)
wenzelm@13114: 
wenzelm@13114: 
nipkow@14300: lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300:  (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300: by(cases ys) auto
nipkow@14300: 
nipkow@15281: lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281:  (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281: by(cases ys) auto
nipkow@15281: 
nipkow@14300: 
wenzelm@13142: text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114: 
wenzelm@13114: lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145: by simp
wenzelm@13114: 
wenzelm@13142: lemma Cons_eq_appendI:
nipkow@13145: "[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145: by (drule sym) simp
wenzelm@13114: 
wenzelm@13142: lemma append_eq_appendI:
nipkow@13145: "[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145: by (drule sym) simp
wenzelm@13114: 
wenzelm@13114: 
wenzelm@13142: text {*
nipkow@13145: Simplification procedure for all list equalities.
nipkow@13145: Currently only tries to rearrange @{text "@"} to see if
nipkow@13145: - both lists end in a singleton list,
nipkow@13145: - or both lists end in the same list.
wenzelm@13142: *}
wenzelm@13142: 
wenzelm@26480: ML {*
nipkow@3507: local
nipkow@3507: 
wenzelm@13114: fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462:   (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
haftmann@23029:   | last (Const("List.append",_) $ _ $ ys) = last ys
wenzelm@13462:   | last t = t;
wenzelm@13114: 
wenzelm@13114: fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462:   | list1 _ = false;
wenzelm@13114: 
wenzelm@13114: fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462:   (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
haftmann@23029:   | butlast ((app as Const("List.append",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462:   | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114: 
haftmann@22633: val rearr_ss = HOL_basic_ss addsimps [@{thm append_assoc},
haftmann@22633:   @{thm append_Nil}, @{thm append_Cons}];
wenzelm@16973: 
wenzelm@20044: fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462:   let
wenzelm@13462:     val lastl = last lhs and lastr = last rhs;
wenzelm@13462:     fun rearr conv =
wenzelm@13462:       let
wenzelm@13462:         val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462:         val Type(_,listT::_) = eqT
wenzelm@13462:         val appT = [listT,listT] ---> listT
haftmann@23029:         val app = Const("List.append",appT)
wenzelm@13462:         val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480:         val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@20044:         val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
wenzelm@17877:           (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
skalberg@15531:       in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114: 
wenzelm@13462:   in
haftmann@22633:     if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
haftmann@22633:     else if lastl aconv lastr then rearr @{thm append_same_eq}
skalberg@15531:     else NONE
wenzelm@13462:   end;
wenzelm@13462: 
wenzelm@13114: in
wenzelm@13462: 
wenzelm@13462: val list_eq_simproc =
haftmann@22633:   Simplifier.simproc @{theory} "list_eq" ["(xs::'a list) = ys"] (K list_eq);
wenzelm@13462: 
wenzelm@13114: end;
wenzelm@13114: 
wenzelm@13114: Addsimprocs [list_eq_simproc];
wenzelm@13114: *}
wenzelm@13114: 
wenzelm@13114: 
nipkow@15392: subsubsection {* @{text map} *}
wenzelm@13114: 
wenzelm@13142: lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145: by (induct xs) simp_all
wenzelm@13114: 
wenzelm@13142: lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145: by (rule ext, induct_tac xs) auto
wenzelm@13114: 
wenzelm@13142: lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145: by (induct xs) (auto simp add: o_def)
wenzelm@13114: 
wenzelm@13142: lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
nipkow@13737: lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737: by (induct xs) auto
nipkow@13737: 
krauss@19770: lemma map_cong [fundef_cong, recdef_cong]:
nipkow@13145: "xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145: -- {* a congruence rule for @{text map} *}
nipkow@13737: by simp
wenzelm@13114: 
wenzelm@13142: lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145: by (cases xs) auto
wenzelm@13114: 
wenzelm@13142: lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145: by (cases xs) auto
wenzelm@13114: 
paulson@18447: lemma map_eq_Cons_conv:
nipkow@14025:  "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145: by (cases xs) auto
wenzelm@13114: 
paulson@18447: lemma Cons_eq_map_conv:
nipkow@14025:  "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025: by (cases ys) auto
nipkow@14025: 
paulson@18447: lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
paulson@18447: lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
paulson@18447: declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
paulson@18447: 
nipkow@14111: lemma ex_map_conv:
nipkow@14111:   "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
paulson@18447: by(induct ys, auto simp add: Cons_eq_map_conv)
nipkow@14111: 
nipkow@15110: lemma map_eq_imp_length_eq:
nipkow@24526:   "map f xs = map f ys ==> length xs = length ys"
nipkow@24526: apply (induct ys arbitrary: xs)
nipkow@15110:  apply simp
paulson@24632: apply (metis Suc_length_conv length_map)
nipkow@15110: done
nipkow@15110: 
nipkow@15110: lemma map_inj_on:
nipkow@15110:  "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110:   ==> xs = ys"
nipkow@15110: apply(frule map_eq_imp_length_eq)
nipkow@15110: apply(rotate_tac -1)
nipkow@15110: apply(induct rule:list_induct2)
nipkow@15110:  apply simp
nipkow@15110: apply(simp)
nipkow@15110: apply (blast intro:sym)
nipkow@15110: done
nipkow@15110: 
nipkow@15110: lemma inj_on_map_eq_map:
nipkow@15110:  "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110: by(blast dest:map_inj_on)
nipkow@15110: 
wenzelm@13114: lemma map_injective:
nipkow@24526:  "map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@24526: by (induct ys arbitrary: xs) (auto dest!:injD)
wenzelm@13114: 
nipkow@14339: lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339: by(blast dest:map_injective)
nipkow@14339: 
wenzelm@13114: lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@17589: by (iprover dest: map_injective injD intro: inj_onI)
wenzelm@13114: 
wenzelm@13114: lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208: apply (unfold inj_on_def, clarify)
nipkow@13145: apply (erule_tac x = "[x]" in ballE)
paulson@14208:  apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145: apply blast
nipkow@13145: done
wenzelm@13114: 
nipkow@14339: lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145: by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114: 
nipkow@15303: lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303: apply(rule inj_onI)
nipkow@15303: apply(erule map_inj_on)
nipkow@15303: apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303: done
nipkow@15303: 
kleing@14343: lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343: by (induct xs, auto)
wenzelm@13114: 
nipkow@14402: lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402: by (induct xs) auto
nipkow@14402: 
nipkow@15110: lemma map_fst_zip[simp]:
nipkow@15110:   "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110: by (induct rule:list_induct2, simp_all)
nipkow@15110: 
nipkow@15110: lemma map_snd_zip[simp]:
nipkow@15110:   "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110: by (induct rule:list_induct2, simp_all)
nipkow@15110: 
nipkow@15110: 
nipkow@15392: subsubsection {* @{text rev} *}
wenzelm@13114: 
wenzelm@13142: lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
kleing@15870: lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870: by auto
kleing@15870: 
wenzelm@13142: lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
kleing@15870: lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870: by (cases xs) auto
kleing@15870: 
kleing@15870: lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870: by (cases xs) auto
kleing@15870: 
haftmann@21061: lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
haftmann@21061: apply (induct xs arbitrary: ys, force)
paulson@14208: apply (case_tac ys, simp, force)
nipkow@13145: done
wenzelm@13114: 
nipkow@15439: lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439: by(simp add:inj_on_def)
nipkow@15439: 
wenzelm@13366: lemma rev_induct [case_names Nil snoc]:
wenzelm@13366:   "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489: apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145: apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13366: lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366:   "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145: by (induct xs rule: rev_induct) auto
wenzelm@13114: 
wenzelm@13366: lemmas rev_cases = rev_exhaust
wenzelm@13366: 
nipkow@18423: lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
nipkow@18423: by(rule rev_cases[of xs]) auto
nipkow@18423: 
wenzelm@13114: 
nipkow@15392: subsubsection {* @{text set} *}
wenzelm@13114: 
wenzelm@13142: lemma finite_set [iff]: "finite (set xs)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
nipkow@17830: lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
nipkow@17830: by(cases xs) auto
oheimb@14099: 
wenzelm@13142: lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145: by auto
wenzelm@13114: 
oheimb@14099: lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099: by auto
oheimb@14099: 
wenzelm@13142: lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
nipkow@15245: lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245: by(induct xs) auto
nipkow@15245: 
wenzelm@13142: lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
nipkow@15425: lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
paulson@14208: apply (induct j, simp_all)
paulson@14208: apply (erule ssubst, auto)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13142: 
wenzelm@25221: lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs"
nipkow@18049: proof (induct xs)
nipkow@26073:   case Nil thus ?case by simp
nipkow@26073: next
nipkow@26073:   case Cons thus ?case by (auto intro: Cons_eq_appendI)
nipkow@26073: qed
nipkow@26073: 
nipkow@26073: lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
nipkow@26073: by (metis Un_upper2 insert_subset set.simps(2) set_append split_list)
nipkow@26073: 
nipkow@26073: lemma split_list_first: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@26073: proof (induct xs)
nipkow@26073:   case Nil thus ?case by simp
nipkow@18049: next
nipkow@18049:   case (Cons a xs)
nipkow@18049:   show ?case
nipkow@18049:   proof cases
wenzelm@25221:     assume "x = a" thus ?case using Cons by fastsimp
nipkow@18049:   next
nipkow@26073:     assume "x \<noteq> a" thus ?case using Cons by(fastsimp intro!: Cons_eq_appendI)
nipkow@26073:   qed
nipkow@26073: qed
nipkow@26073: 
nipkow@26073: lemma in_set_conv_decomp_first:
nipkow@26073:   "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
nipkow@26073: by (metis in_set_conv_decomp split_list_first)
nipkow@26073: 
nipkow@26073: lemma split_list_last: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
nipkow@26073: proof (induct xs rule:rev_induct)
nipkow@26073:   case Nil thus ?case by simp
nipkow@26073: next
nipkow@26073:   case (snoc a xs)
nipkow@26073:   show ?case
nipkow@26073:   proof cases
nipkow@26073:     assume "x = a" thus ?case using snoc by simp (metis ex_in_conv set_empty2)
nipkow@26073:   next
nipkow@26073:     assume "x \<noteq> a" thus ?case using snoc by fastsimp
nipkow@18049:   qed
nipkow@18049: qed
nipkow@18049: 
nipkow@26073: lemma in_set_conv_decomp_last:
nipkow@26073:   "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
nipkow@26073: by (metis in_set_conv_decomp split_list_last)
nipkow@26073: 
nipkow@26073: lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs & P x"
nipkow@26073: proof (induct xs)
nipkow@26073:   case Nil thus ?case by simp
nipkow@26073: next
nipkow@26073:   case Cons thus ?case
nipkow@26073:     by(simp add:Bex_def)(metis append_Cons append.simps(1))
nipkow@26073: qed
nipkow@26073: 
nipkow@26073: lemma split_list_propE:
nipkow@26073: assumes "\<exists>x \<in> set xs. P x"
nipkow@26073: obtains ys x zs where "xs = ys @ x # zs" and "P x"
nipkow@26073: by(metis split_list_prop[OF assms])
nipkow@26073: 
nipkow@26073: 
nipkow@26073: lemma split_list_first_prop:
nipkow@26073:   "\<exists>x \<in> set xs. P x \<Longrightarrow>
nipkow@26073:    \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)"
nipkow@26073: proof(induct xs)
nipkow@26073:   case Nil thus ?case by simp
nipkow@26073: next
nipkow@26073:   case (Cons x xs)
nipkow@26073:   show ?case
nipkow@26073:   proof cases
nipkow@26073:     assume "P x"
nipkow@26073:     thus ?thesis by simp (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
nipkow@26073:   next
nipkow@26073:     assume "\<not> P x"
nipkow@26073:     hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
nipkow@26073:     thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
nipkow@26073:   qed
nipkow@26073: qed
nipkow@26073: 
nipkow@26073: lemma split_list_first_propE:
nipkow@26073: assumes "\<exists>x \<in> set xs. P x"
nipkow@26073: obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y"
nipkow@26073: by(metis split_list_first_prop[OF assms])
nipkow@26073: 
nipkow@26073: lemma split_list_first_prop_iff:
nipkow@26073:   "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
nipkow@26073:    (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))"
nipkow@26073: by(metis split_list_first_prop[where P=P] in_set_conv_decomp)
nipkow@26073: 
nipkow@26073: 
nipkow@26073: lemma split_list_last_prop:
nipkow@26073:   "\<exists>x \<in> set xs. P x \<Longrightarrow>
nipkow@26073:    \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)"
nipkow@26073: proof(induct xs rule:rev_induct)
nipkow@26073:   case Nil thus ?case by simp
nipkow@26073: next
nipkow@26073:   case (snoc x xs)
nipkow@26073:   show ?case
nipkow@26073:   proof cases
nipkow@26073:     assume "P x" thus ?thesis by (metis emptyE set_empty)
nipkow@26073:   next
nipkow@26073:     assume "\<not> P x"
nipkow@26073:     hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
nipkow@26073:     thus ?thesis using `\<not> P x` snoc(1) by fastsimp
nipkow@26073:   qed
nipkow@26073: qed
nipkow@26073: 
nipkow@26073: lemma split_list_last_propE:
nipkow@26073: assumes "\<exists>x \<in> set xs. P x"
nipkow@26073: obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z"
nipkow@26073: by(metis split_list_last_prop[OF assms])
nipkow@26073: 
nipkow@26073: lemma split_list_last_prop_iff:
nipkow@26073:   "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
nipkow@26073:    (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
nipkow@26073: by(metis split_list_last_prop[where P=P] in_set_conv_decomp)
nipkow@26073: 
nipkow@26073: 
nipkow@26073: lemma finite_list: "finite A ==> EX xs. set xs = A"
paulson@13508: apply (erule finite_induct, auto)
nipkow@26073: apply (metis set.simps(2))
paulson@13508: done
paulson@13508: 
kleing@14388: lemma card_length: "card (set xs) \<le> length xs"
kleing@14388: by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114: 
haftmann@26442: lemma set_minus_filter_out:
haftmann@26442:   "set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)"
haftmann@26442:   by (induct xs) auto
paulson@15168: 
nipkow@15392: subsubsection {* @{text filter} *}
wenzelm@13114: 
wenzelm@13142: lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
nipkow@15305: lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305: by (induct xs) simp_all
nipkow@15305: 
wenzelm@13142: lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
nipkow@16998: lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998: by (induct xs) (auto simp add: le_SucI)
nipkow@16998: 
nipkow@18423: lemma sum_length_filter_compl:
nipkow@18423:   "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
nipkow@18423: by(induct xs) simp_all
nipkow@18423: 
wenzelm@13142: lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
nipkow@16998: lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@24349: by (induct xs) simp_all
nipkow@16998: 
nipkow@16998: lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998: apply (induct xs)
nipkow@16998:  apply auto
nipkow@16998: apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998: apply simp
nipkow@16998: done
wenzelm@13114: 
nipkow@16965: lemma filter_map:
nipkow@16965:   "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965: by (induct xs) simp_all
nipkow@16965: 
nipkow@16965: lemma length_filter_map[simp]:
nipkow@16965:   "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965: by (simp add:filter_map)
nipkow@16965: 
wenzelm@13142: lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145: by auto
wenzelm@13114: 
nipkow@15246: lemma length_filter_less:
nipkow@15246:   "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246: proof (induct xs)
nipkow@15246:   case Nil thus ?case by simp
nipkow@15246: next
nipkow@15246:   case (Cons x xs) thus ?case
nipkow@15246:     apply (auto split:split_if_asm)
nipkow@15246:     using length_filter_le[of P xs] apply arith
nipkow@15246:   done
nipkow@15246: qed
wenzelm@13114: 
nipkow@15281: lemma length_filter_conv_card:
nipkow@15281:  "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281: proof (induct xs)
nipkow@15281:   case Nil thus ?case by simp
nipkow@15281: next
nipkow@15281:   case (Cons x xs)
nipkow@15281:   let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281:   have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281:   show ?case (is "?l = card ?S'")
nipkow@15281:   proof (cases)
nipkow@15281:     assume "p x"
nipkow@15281:     hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@25162:       by(auto simp: image_def split:nat.split dest:gr0_implies_Suc)
nipkow@15281:     have "length (filter p (x # xs)) = Suc(card ?S)"
wenzelm@23388:       using Cons `p x` by simp
nipkow@15281:     also have "\<dots> = Suc(card(Suc ` ?S))" using fin
nipkow@15281:       by (simp add: card_image inj_Suc)
nipkow@15281:     also have "\<dots> = card ?S'" using eq fin
nipkow@15281:       by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281:     finally show ?thesis .
nipkow@15281:   next
nipkow@15281:     assume "\<not> p x"
nipkow@15281:     hence eq: "?S' = Suc ` ?S"
nipkow@25162:       by(auto simp add: image_def split:nat.split elim:lessE)
nipkow@15281:     have "length (filter p (x # xs)) = card ?S"
wenzelm@23388:       using Cons `\<not> p x` by simp
nipkow@15281:     also have "\<dots> = card(Suc ` ?S)" using fin
nipkow@15281:       by (simp add: card_image inj_Suc)
nipkow@15281:     also have "\<dots> = card ?S'" using eq fin
nipkow@15281:       by (simp add:card_insert_if)
nipkow@15281:     finally show ?thesis .
nipkow@15281:   qed
nipkow@15281: qed
nipkow@15281: 
nipkow@17629: lemma Cons_eq_filterD:
nipkow@17629:  "x#xs = filter P ys \<Longrightarrow>
nipkow@17629:   \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
wenzelm@19585:   (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
nipkow@17629: proof(induct ys)
nipkow@17629:   case Nil thus ?case by simp
nipkow@17629: next
nipkow@17629:   case (Cons y ys)
nipkow@17629:   show ?case (is "\<exists>x. ?Q x")
nipkow@17629:   proof cases
nipkow@17629:     assume Py: "P y"
nipkow@17629:     show ?thesis
nipkow@17629:     proof cases
wenzelm@25221:       assume "x = y"
wenzelm@25221:       with Py Cons.prems have "?Q []" by simp
wenzelm@25221:       then show ?thesis ..
nipkow@17629:     next
wenzelm@25221:       assume "x \<noteq> y"
wenzelm@25221:       with Py Cons.prems show ?thesis by simp
nipkow@17629:     qed
nipkow@17629:   next
wenzelm@25221:     assume "\<not> P y"
wenzelm@25221:     with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastsimp
wenzelm@25221:     then have "?Q (y#us)" by simp
wenzelm@25221:     then show ?thesis ..
nipkow@17629:   qed
nipkow@17629: qed
nipkow@17629: 
nipkow@17629: lemma filter_eq_ConsD:
nipkow@17629:  "filter P ys = x#xs \<Longrightarrow>
nipkow@17629:   \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
nipkow@17629: by(rule Cons_eq_filterD) simp
nipkow@17629: 
nipkow@17629: lemma filter_eq_Cons_iff:
nipkow@17629:  "(filter P ys = x#xs) =
nipkow@17629:   (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
nipkow@17629: by(auto dest:filter_eq_ConsD)
nipkow@17629: 
nipkow@17629: lemma Cons_eq_filter_iff:
nipkow@17629:  "(x#xs = filter P ys) =
nipkow@17629:   (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
nipkow@17629: by(auto dest:Cons_eq_filterD)
nipkow@17629: 
krauss@19770: lemma filter_cong[fundef_cong, recdef_cong]:
nipkow@17501:  "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
nipkow@17501: apply simp
nipkow@17501: apply(erule thin_rl)
nipkow@17501: by (induct ys) simp_all
nipkow@17501: 
nipkow@15281: 
haftmann@26442: subsubsection {* List partitioning *}
haftmann@26442: 
haftmann@26442: primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where
haftmann@26442:   "partition P [] = ([], [])"
haftmann@26442:   | "partition P (x # xs) = 
haftmann@26442:       (let (yes, no) = partition P xs
haftmann@26442:       in if P x then (x # yes, no) else (yes, x # no))"
haftmann@26442: 
haftmann@26442: lemma partition_filter1:
haftmann@26442:     "fst (partition P xs) = filter P xs"
haftmann@26442: by (induct xs) (auto simp add: Let_def split_def)
haftmann@26442: 
haftmann@26442: lemma partition_filter2:
haftmann@26442:     "snd (partition P xs) = filter (Not o P) xs"
haftmann@26442: by (induct xs) (auto simp add: Let_def split_def)
haftmann@26442: 
haftmann@26442: lemma partition_P:
haftmann@26442:   assumes "partition P xs = (yes, no)"
haftmann@26442:   shows "(\<forall>p \<in> set yes.  P p) \<and> (\<forall>p  \<in> set no. \<not> P p)"
haftmann@26442: proof -
haftmann@26442:   from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
haftmann@26442:     by simp_all
haftmann@26442:   then show ?thesis by (simp_all add: partition_filter1 partition_filter2)
haftmann@26442: qed
haftmann@26442: 
haftmann@26442: lemma partition_set:
haftmann@26442:   assumes "partition P xs = (yes, no)"
haftmann@26442:   shows "set yes \<union> set no = set xs"
haftmann@26442: proof -
haftmann@26442:   from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
haftmann@26442:     by simp_all
haftmann@26442:   then show ?thesis by (auto simp add: partition_filter1 partition_filter2) 
haftmann@26442: qed
haftmann@26442: 
haftmann@26442: 
nipkow@15392: subsubsection {* @{text concat} *}
wenzelm@13114: 
wenzelm@13142: lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
paulson@18447: lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145: by (induct xss) auto
wenzelm@13114: 
paulson@18447: lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145: by (induct xss) auto
wenzelm@13114: 
nipkow@24308: lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
nipkow@24476: lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs"
nipkow@24349: by (induct xs) auto
nipkow@24349: 
wenzelm@13142: lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13114: 
nipkow@15392: subsubsection {* @{text nth} *}
wenzelm@13114: 
wenzelm@13142: lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145: by auto
wenzelm@13114: 
wenzelm@13142: lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145: by auto
wenzelm@13114: 
wenzelm@13142: declare nth.simps [simp del]
wenzelm@13114: 
wenzelm@13114: lemma nth_append:
nipkow@24526:   "(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
nipkow@24526: apply (induct xs arbitrary: n, simp)
paulson@14208: apply (case_tac n, auto)
nipkow@13145: done
wenzelm@13114: 
nipkow@14402: lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
wenzelm@25221: by (induct xs) auto
nipkow@14402: 
nipkow@14402: lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
wenzelm@25221: by (induct xs) auto
nipkow@14402: 
nipkow@24526: lemma nth_map [simp]: "n < length xs ==> (map f xs)!n = f(xs!n)"
nipkow@24526: apply (induct xs arbitrary: n, simp)
paulson@14208: apply (case_tac n, auto)
nipkow@13145: done
wenzelm@13114: 
nipkow@18423: lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
nipkow@18423: by(cases xs) simp_all
nipkow@18423: 
nipkow@18049: 
nipkow@18049: lemma list_eq_iff_nth_eq:
nipkow@24526:  "(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
nipkow@24526: apply(induct xs arbitrary: ys)
paulson@24632:  apply force
nipkow@18049: apply(case_tac ys)
nipkow@18049:  apply simp
nipkow@18049: apply(simp add:nth_Cons split:nat.split)apply blast
nipkow@18049: done
nipkow@18049: 
wenzelm@13142: lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251: apply (induct xs, simp, simp)
nipkow@13145: apply safe
paulson@24632: apply (metis nat_case_0 nth.simps zero_less_Suc)
paulson@24632: apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
paulson@14208: apply (case_tac i, simp)
paulson@24632: apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff)
nipkow@13145: done
wenzelm@13114: 
nipkow@17501: lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
nipkow@17501: by(auto simp:set_conv_nth)
nipkow@17501: 
nipkow@13145: lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145: by (auto simp add: set_conv_nth)
wenzelm@13114: 
wenzelm@13142: lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145: by (auto simp add: set_conv_nth)
wenzelm@13114: 
wenzelm@13114: lemma all_nth_imp_all_set:
nipkow@13145: "[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145: by (auto simp add: set_conv_nth)
wenzelm@13114: 
wenzelm@13114: lemma all_set_conv_all_nth:
nipkow@13145: "(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145: by (auto simp add: set_conv_nth)
wenzelm@13114: 
kleing@25296: lemma rev_nth:
kleing@25296:   "n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)"
kleing@25296: proof (induct xs arbitrary: n)
kleing@25296:   case Nil thus ?case by simp
kleing@25296: next
kleing@25296:   case (Cons x xs)
kleing@25296:   hence n: "n < Suc (length xs)" by simp
kleing@25296:   moreover
kleing@25296:   { assume "n < length xs"
kleing@25296:     with n obtain n' where "length xs - n = Suc n'"
kleing@25296:       by (cases "length xs - n", auto)
kleing@25296:     moreover
kleing@25296:     then have "length xs - Suc n = n'" by simp
kleing@25296:     ultimately
kleing@25296:     have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp
kleing@25296:   }
kleing@25296:   ultimately
kleing@25296:   show ?case by (clarsimp simp add: Cons nth_append)
kleing@25296: qed
wenzelm@13114: 
nipkow@15392: subsubsection {* @{text list_update} *}
wenzelm@13114: 
nipkow@24526: lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
nipkow@24526: by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114: 
wenzelm@13114: lemma nth_list_update:
nipkow@24526: "i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@24526: by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114: 
wenzelm@13142: lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145: by (simp add: nth_list_update)
wenzelm@13114: 
nipkow@24526: lemma nth_list_update_neq [simp]: "i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@24526: by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114: 
nipkow@24526: lemma list_update_id[simp]: "xs[i := xs!i] = xs"
nipkow@24526: by (induct xs arbitrary: i) (simp_all split:nat.splits)
nipkow@24526: 
nipkow@24526: lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
nipkow@24526: apply (induct xs arbitrary: i)
nipkow@17501:  apply simp
nipkow@17501: apply (case_tac i)
nipkow@17501: apply simp_all
nipkow@17501: done
nipkow@17501: 
wenzelm@13114: lemma list_update_same_conv:
nipkow@24526: "i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@24526: by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114: 
nipkow@14187: lemma list_update_append1:
nipkow@24526:  "i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
nipkow@24526: apply (induct xs arbitrary: i, simp)
nipkow@14187: apply(simp split:nat.split)
nipkow@14187: done
nipkow@14187: 
kleing@15868: lemma list_update_append:
nipkow@24526:   "(xs @ ys) [n:= x] = 
kleing@15868:   (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
nipkow@24526: by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868: 
nipkow@14402: lemma list_update_length [simp]:
nipkow@14402:  "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402: by (induct xs, auto)
nipkow@14402: 
wenzelm@13114: lemma update_zip:
nipkow@24526:   "length xs = length ys ==>
nipkow@24526:   (zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@24526: by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split)
nipkow@24526: 
nipkow@24526: lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)"
nipkow@24526: by (induct xs arbitrary: i) (auto split: nat.split)
wenzelm@13114: 
wenzelm@13114: lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145: by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114: 
nipkow@24526: lemma set_update_memI: "n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
nipkow@24526: by (induct xs arbitrary: n) (auto split:nat.splits)
kleing@15868: 
haftmann@24796: lemma list_update_overwrite:
haftmann@24796:   "xs [i := x, i := y] = xs [i := y]"
haftmann@24796: apply (induct xs arbitrary: i)
haftmann@24796: apply simp
haftmann@24796: apply (case_tac i)
haftmann@24796: apply simp_all
haftmann@24796: done
haftmann@24796: 
haftmann@24796: lemma list_update_swap:
haftmann@24796:   "i \<noteq> i' \<Longrightarrow> xs [i := x, i' := x'] = xs [i' := x', i := x]"
haftmann@24796: apply (induct xs arbitrary: i i')
haftmann@24796: apply simp
haftmann@24796: apply (case_tac i, case_tac i')
haftmann@24796: apply auto
haftmann@24796: apply (case_tac i')
haftmann@24796: apply auto
haftmann@24796: done
haftmann@24796: 
wenzelm@13114: 
nipkow@15392: subsubsection {* @{text last} and @{text butlast} *}
wenzelm@13114: 
wenzelm@13142: lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
nipkow@14302: lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302: by(simp add:last.simps)
nipkow@14302: 
nipkow@14302: lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302: by(simp add:last.simps)
nipkow@14302: 
nipkow@14302: lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302: by (induct xs) (auto)
nipkow@14302: 
nipkow@14302: lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302: by(simp add:last_append)
nipkow@14302: 
nipkow@14302: lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302: by(simp add:last_append)
nipkow@14302: 
nipkow@17762: lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
nipkow@17762: by(rule rev_exhaust[of xs]) simp_all
nipkow@17762: 
nipkow@17762: lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
nipkow@17762: by(cases xs) simp_all
nipkow@17762: 
nipkow@17765: lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
nipkow@17765: by (induct as) auto
nipkow@17762: 
wenzelm@13142: lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145: by (induct xs rule: rev_induct) auto
wenzelm@13114: 
wenzelm@13114: lemma butlast_append:
nipkow@24526:   "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@24526: by (induct xs arbitrary: ys) auto
wenzelm@13114: 
wenzelm@13142: lemma append_butlast_last_id [simp]:
nipkow@13145: "xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145: by (induct xs) (auto split: split_if_asm)
wenzelm@13114: 
wenzelm@13114: lemma in_set_butlast_appendI:
nipkow@13145: "x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145: by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114: 
nipkow@24526: lemma last_drop[simp]: "n < length xs \<Longrightarrow> last (drop n xs) = last xs"
nipkow@24526: apply (induct xs arbitrary: n)
nipkow@17501:  apply simp
nipkow@17501: apply (auto split:nat.split)
nipkow@17501: done
nipkow@17501: 
nipkow@17589: lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
nipkow@17589: by(induct xs)(auto simp:neq_Nil_conv)
nipkow@17589: 
huffman@26584: lemma butlast_conv_take: "butlast xs = take (length xs - 1) xs"
huffman@26584: by (induct xs, simp, case_tac xs, simp_all)
huffman@26584: 
haftmann@24796: 
nipkow@15392: subsubsection {* @{text take} and @{text drop} *}
wenzelm@13114: 
wenzelm@13142: lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145: by simp
wenzelm@13114: 
wenzelm@13142: lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145: by simp
wenzelm@13114: 
wenzelm@13142: declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114: 
nipkow@15110: lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110: by(clarsimp simp add:neq_Nil_conv)
nipkow@15110: 
nipkow@14187: lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187: by(cases xs, simp_all)
nipkow@14187: 
huffman@26584: lemma take_tl: "take n (tl xs) = tl (take (Suc n) xs)"
huffman@26584: by (induct xs arbitrary: n) simp_all
huffman@26584: 
nipkow@24526: lemma drop_tl: "drop n (tl xs) = tl(drop n xs)"
nipkow@24526: by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@24526: 
huffman@26584: lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)"
huffman@26584: by (cases n, simp, cases xs, auto)
huffman@26584: 
huffman@26584: lemma tl_drop: "tl (drop n xs) = drop n (tl xs)"
huffman@26584: by (simp only: drop_tl)
huffman@26584: 
nipkow@24526: lemma nth_via_drop: "drop n xs = y#ys \<Longrightarrow> xs!n = y"
nipkow@24526: apply (induct xs arbitrary: n, simp)
nipkow@14187: apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187: done
nipkow@14187: 
nipkow@13913: lemma take_Suc_conv_app_nth:
nipkow@24526:   "i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
nipkow@24526: apply (induct xs arbitrary: i, simp)
paulson@14208: apply (case_tac i, auto)
nipkow@13913: done
nipkow@13913: 
mehta@14591: lemma drop_Suc_conv_tl:
nipkow@24526:   "i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
nipkow@24526: apply (induct xs arbitrary: i, simp)
mehta@14591: apply (case_tac i, auto)
mehta@14591: done
mehta@14591: 
nipkow@24526: lemma length_take [simp]: "length (take n xs) = min (length xs) n"
nipkow@24526: by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526: 
nipkow@24526: lemma length_drop [simp]: "length (drop n xs) = (length xs - n)"
nipkow@24526: by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526: 
nipkow@24526: lemma take_all [simp]: "length xs <= n ==> take n xs = xs"
nipkow@24526: by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526: 
nipkow@24526: lemma drop_all [simp]: "length xs <= n ==> drop n xs = []"
nipkow@24526: by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114: 
wenzelm@13142: lemma take_append [simp]:
nipkow@24526:   "take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@24526: by (induct n arbitrary: xs) (auto, case_tac xs, auto)
wenzelm@13114: 
wenzelm@13142: lemma drop_append [simp]:
nipkow@24526:   "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@24526: by (induct n arbitrary: xs) (auto, case_tac xs, auto)
nipkow@24526: 
nipkow@24526: lemma take_take [simp]: "take n (take m xs) = take (min n m) xs"
nipkow@24526: apply (induct m arbitrary: xs n, auto)
paulson@14208: apply (case_tac xs, auto)
nipkow@15236: apply (case_tac n, auto)
nipkow@13145: done
wenzelm@13114: 
nipkow@24526: lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs"
nipkow@24526: apply (induct m arbitrary: xs, auto)
paulson@14208: apply (case_tac xs, auto)
nipkow@13145: done
wenzelm@13114: 
nipkow@24526: lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)"
nipkow@24526: apply (induct m arbitrary: xs n, auto)
paulson@14208: apply (case_tac xs, auto)
nipkow@13145: done
wenzelm@13114: 
nipkow@24526: lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@24526: apply(induct xs arbitrary: m n)
nipkow@14802:  apply simp
nipkow@14802: apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802: done
nipkow@14802: 
nipkow@24526: lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs"
nipkow@24526: apply (induct n arbitrary: xs, auto)
paulson@14208: apply (case_tac xs, auto)
nipkow@13145: done
wenzelm@13114: 
nipkow@24526: lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@24526: apply(induct xs arbitrary: n)
nipkow@15110:  apply simp
nipkow@15110: apply(simp add:take_Cons split:nat.split)
nipkow@15110: done
nipkow@15110: 
nipkow@24526: lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)"
nipkow@24526: apply(induct xs arbitrary: n)
nipkow@15110: apply simp
nipkow@15110: apply(simp add:drop_Cons split:nat.split)
nipkow@15110: done
nipkow@15110: 
nipkow@24526: lemma take_map: "take n (map f xs) = map f (take n xs)"
nipkow@24526: apply (induct n arbitrary: xs, auto)
paulson@14208: apply (case_tac xs, auto)
nipkow@13145: done
wenzelm@13114: 
nipkow@24526: lemma drop_map: "drop n (map f xs) = map f (drop n xs)"
nipkow@24526: apply (induct n arbitrary: xs, auto)
paulson@14208: apply (case_tac xs, auto)
nipkow@13145: done
wenzelm@13114: 
nipkow@24526: lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)"
nipkow@24526: apply (induct xs arbitrary: i, auto)
paulson@14208: apply (case_tac i, auto)
nipkow@13145: done
wenzelm@13114: 
nipkow@24526: lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)"
nipkow@24526: apply (induct xs arbitrary: i, auto)
paulson@14208: apply (case_tac i, auto)
nipkow@13145: done
wenzelm@13114: 
nipkow@24526: lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i"
nipkow@24526: apply (induct xs arbitrary: i n, auto)
paulson@14208: apply (case_tac n, blast)
paulson@14208: apply (case_tac i, auto)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13142: lemma nth_drop [simp]:
nipkow@24526:   "n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
nipkow@24526: apply (induct n arbitrary: xs i, auto)
paulson@14208: apply (case_tac xs, auto)
nipkow@13145: done
nipkow@3507: 
huffman@26584: lemma butlast_take:
huffman@26584:   "n <= length xs ==> butlast (take n xs) = take (n - 1) xs"
huffman@26584: by (simp add: butlast_conv_take min_max.inf_absorb1 min_max.inf_absorb2)
huffman@26584: 
huffman@26584: lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)"
huffman@26584: by (simp add: butlast_conv_take drop_take)
huffman@26584: 
huffman@26584: lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs"
huffman@26584: by (simp add: butlast_conv_take min_max.inf_absorb1)
huffman@26584: 
huffman@26584: lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)"
huffman@26584: by (simp add: butlast_conv_take drop_take)
huffman@26584: 
nipkow@18423: lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
nipkow@18423: by(simp add: hd_conv_nth)
nipkow@18423: 
nipkow@24526: lemma set_take_subset: "set(take n xs) \<subseteq> set xs"
nipkow@24526: by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split)
nipkow@24526: 
nipkow@24526: lemma set_drop_subset: "set(drop n xs) \<subseteq> set xs"
nipkow@24526: by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split)
nipkow@14025: 
nipkow@14187: lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187: using set_take_subset by fast
nipkow@14187: 
nipkow@14187: lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187: using set_drop_subset by fast
nipkow@14187: 
wenzelm@13114: lemma append_eq_conv_conj:
nipkow@24526:   "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
nipkow@24526: apply (induct xs arbitrary: zs, simp, clarsimp)
paulson@14208: apply (case_tac zs, auto)
nipkow@13145: done
wenzelm@13142: 
nipkow@24526: lemma take_add: 
nipkow@24526:   "i+j \<le> length(xs) \<Longrightarrow> take (i+j) xs = take i xs @ take j (drop i xs)"
nipkow@24526: apply (induct xs arbitrary: i, auto) 
nipkow@24526: apply (case_tac i, simp_all)
paulson@14050: done
paulson@14050: 
nipkow@14300: lemma append_eq_append_conv_if:
nipkow@24526:  "(xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300:   (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300:    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
nipkow@14300:    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)"
nipkow@24526: apply(induct xs\<^isub>1 arbitrary: ys\<^isub>1)
nipkow@14300:  apply simp
nipkow@14300: apply(case_tac ys\<^isub>1)
nipkow@14300: apply simp_all
nipkow@14300: done
nipkow@14300: 
nipkow@15110: lemma take_hd_drop:
nipkow@24526:   "n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
nipkow@24526: apply(induct xs arbitrary: n)
nipkow@15110: apply simp
nipkow@15110: apply(simp add:drop_Cons split:nat.split)
nipkow@15110: done
nipkow@15110: 
nipkow@17501: lemma id_take_nth_drop:
nipkow@17501:  "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
nipkow@17501: proof -
nipkow@17501:   assume si: "i < length xs"
nipkow@17501:   hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
nipkow@17501:   moreover
nipkow@17501:   from si have "take (Suc i) xs = take i xs @ [xs!i]"
nipkow@17501:     apply (rule_tac take_Suc_conv_app_nth) by arith
nipkow@17501:   ultimately show ?thesis by auto
nipkow@17501: qed
nipkow@17501:   
nipkow@17501: lemma upd_conv_take_nth_drop:
nipkow@17501:  "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
nipkow@17501: proof -
nipkow@17501:   assume i: "i < length xs"
nipkow@17501:   have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
nipkow@17501:     by(rule arg_cong[OF id_take_nth_drop[OF i]])
nipkow@17501:   also have "\<dots> = take i xs @ a # drop (Suc i) xs"
nipkow@17501:     using i by (simp add: list_update_append)
nipkow@17501:   finally show ?thesis .
nipkow@17501: qed
nipkow@17501: 
haftmann@24796: lemma nth_drop':
haftmann@24796:   "i < length xs \<Longrightarrow> xs ! i # drop (Suc i) xs = drop i xs"
haftmann@24796: apply (induct i arbitrary: xs)
haftmann@24796: apply (simp add: neq_Nil_conv)
haftmann@24796: apply (erule exE)+
haftmann@24796: apply simp
haftmann@24796: apply (case_tac xs)
haftmann@24796: apply simp_all
haftmann@24796: done
haftmann@24796: 
wenzelm@13114: 
nipkow@15392: subsubsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114: 
wenzelm@13142: lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma takeWhile_append1 [simp]:
nipkow@13145: "[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma takeWhile_append2 [simp]:
nipkow@13145: "(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma dropWhile_append1 [simp]:
nipkow@13145: "[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
wenzelm@13142: lemma dropWhile_append2 [simp]:
nipkow@13145: "(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
krauss@23971: lemma set_takeWhileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145: by (induct xs) (auto split: split_if_asm)
wenzelm@13114: 
nipkow@13913: lemma takeWhile_eq_all_conv[simp]:
nipkow@13913:  "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913: by(induct xs, auto)
nipkow@13913: 
nipkow@13913: lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913:  "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913: by(induct xs, auto)
nipkow@13913: 
nipkow@13913: lemma dropWhile_eq_Cons_conv:
nipkow@13913:  "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913: by(induct xs, auto)
nipkow@13913: 
nipkow@17501: text{* The following two lemmmas could be generalized to an arbitrary
nipkow@17501: property. *}
nipkow@17501: 
nipkow@17501: lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501:  takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
nipkow@17501: by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
nipkow@17501: 
nipkow@17501: lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501:   dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
nipkow@17501: apply(induct xs)
nipkow@17501:  apply simp
nipkow@17501: apply auto
nipkow@17501: apply(subst dropWhile_append2)
nipkow@17501: apply auto
nipkow@17501: done
nipkow@17501: 
nipkow@18423: lemma takeWhile_not_last:
nipkow@18423:  "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
nipkow@18423: apply(induct xs)
nipkow@18423:  apply simp
nipkow@18423: apply(case_tac xs)
nipkow@18423: apply(auto)
nipkow@18423: done
nipkow@18423: 
krauss@19770: lemma takeWhile_cong [fundef_cong, recdef_cong]:
krauss@18336:   "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336:   ==> takeWhile P l = takeWhile Q k"
nipkow@24349: by (induct k arbitrary: l) (simp_all)
krauss@18336: 
krauss@19770: lemma dropWhile_cong [fundef_cong, recdef_cong]:
krauss@18336:   "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336:   ==> dropWhile P l = dropWhile Q k"
nipkow@24349: by (induct k arbitrary: l, simp_all)
krauss@18336: 
wenzelm@13114: 
nipkow@15392: subsubsection {* @{text zip} *}
wenzelm@13114: 
wenzelm@13142: lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145: by (induct ys) auto
wenzelm@13114: 
wenzelm@13142: lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145: by simp
wenzelm@13114: 
wenzelm@13142: declare zip_Cons [simp del]
wenzelm@13114: 
nipkow@15281: lemma zip_Cons1:
nipkow@15281:  "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281: by(auto split:list.split)
nipkow@15281: 
wenzelm@13142: lemma length_zip [simp]:
krauss@22493: "length (zip xs ys) = min (length xs) (length ys)"
krauss@22493: by (induct xs ys rule:list_induct2') auto
wenzelm@13114: 
wenzelm@13114: lemma zip_append1:
krauss@22493: "zip (xs @ ys) zs =
nipkow@13145: zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
krauss@22493: by (induct xs zs rule:list_induct2') auto
wenzelm@13114: 
wenzelm@13114: lemma zip_append2:
krauss@22493: "zip xs (ys @ zs) =
nipkow@13145: zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
krauss@22493: by (induct xs ys rule:list_induct2') auto
wenzelm@13114: 
wenzelm@13142: lemma zip_append [simp]:
wenzelm@13142:  "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145: zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145: by (simp add: zip_append1)
wenzelm@13114: 
wenzelm@13114: lemma zip_rev:
nipkow@14247: "length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247: by (induct rule:list_induct2, simp_all)
wenzelm@13114: 
nipkow@23096: lemma map_zip_map:
nipkow@23096:  "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)"
nipkow@23096: apply(induct xs arbitrary:ys) apply simp
nipkow@23096: apply(case_tac ys)
nipkow@23096: apply simp_all
nipkow@23096: done
nipkow@23096: 
nipkow@23096: lemma map_zip_map2:
nipkow@23096:  "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)"
nipkow@23096: apply(induct xs arbitrary:ys) apply simp
nipkow@23096: apply(case_tac ys)
nipkow@23096: apply simp_all
nipkow@23096: done
nipkow@23096: 
wenzelm@13142: lemma nth_zip [simp]:
nipkow@24526: "[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
nipkow@24526: apply (induct ys arbitrary: i xs, simp)
nipkow@13145: apply (case_tac xs)
nipkow@13145:  apply (simp_all add: nth.simps split: nat.split)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13114: lemma set_zip:
nipkow@13145: "set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145: by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114: 
wenzelm@13114: lemma zip_update:
nipkow@13145: "length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145: by (rule sym, simp add: update_zip)
wenzelm@13114: 
wenzelm@13142: lemma zip_replicate [simp]:
nipkow@24526:   "zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
nipkow@24526: apply (induct i arbitrary: j, auto)
paulson@14208: apply (case_tac j, auto)
nipkow@13145: done
wenzelm@13114: 
nipkow@19487: lemma take_zip:
nipkow@24526:   "take n (zip xs ys) = zip (take n xs) (take n ys)"
nipkow@24526: apply (induct n arbitrary: xs ys)
nipkow@19487:  apply simp
nipkow@19487: apply (case_tac xs, simp)
nipkow@19487: apply (case_tac ys, simp_all)
nipkow@19487: done
nipkow@19487: 
nipkow@19487: lemma drop_zip:
nipkow@24526:   "drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
nipkow@24526: apply (induct n arbitrary: xs ys)
nipkow@19487:  apply simp
nipkow@19487: apply (case_tac xs, simp)
nipkow@19487: apply (case_tac ys, simp_all)
nipkow@19487: done
nipkow@19487: 
krauss@22493: lemma set_zip_leftD:
krauss@22493:   "(x,y)\<in> set (zip xs ys) \<Longrightarrow> x \<in> set xs"
krauss@22493: by (induct xs ys rule:list_induct2') auto
krauss@22493: 
krauss@22493: lemma set_zip_rightD:
krauss@22493:   "(x,y)\<in> set (zip xs ys) \<Longrightarrow> y \<in> set ys"
krauss@22493: by (induct xs ys rule:list_induct2') auto
wenzelm@13142: 
nipkow@23983: lemma in_set_zipE:
nipkow@23983:   "(x,y) : set(zip xs ys) \<Longrightarrow> (\<lbrakk> x : set xs; y : set ys \<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
nipkow@23983: by(blast dest: set_zip_leftD set_zip_rightD)
nipkow@23983: 
nipkow@15392: subsubsection {* @{text list_all2} *}
wenzelm@13114: 
kleing@14316: lemma list_all2_lengthD [intro?]: 
kleing@14316:   "list_all2 P xs ys ==> length xs = length ys"
nipkow@24349: by (simp add: list_all2_def)
haftmann@19607: 
haftmann@19787: lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
nipkow@24349: by (simp add: list_all2_def)
haftmann@19607: 
haftmann@19787: lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
nipkow@24349: by (simp add: list_all2_def)
haftmann@19607: 
haftmann@19607: lemma list_all2_Cons [iff, code]:
haftmann@19607:   "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
nipkow@24349: by (auto simp add: list_all2_def)
wenzelm@13114: 
wenzelm@13114: lemma list_all2_Cons1:
nipkow@13145: "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145: by (cases ys) auto
wenzelm@13114: 
wenzelm@13114: lemma list_all2_Cons2:
nipkow@13145: "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145: by (cases xs) auto
wenzelm@13114: 
wenzelm@13142: lemma list_all2_rev [iff]:
nipkow@13145: "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145: by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114: 
kleing@13863: lemma list_all2_rev1:
kleing@13863: "list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863: by (subst list_all2_rev [symmetric]) simp
kleing@13863: 
wenzelm@13114: lemma list_all2_append1:
nipkow@13145: "list_all2 P (xs @ ys) zs =
nipkow@13145: (EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145: list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145: apply (simp add: list_all2_def zip_append1)
nipkow@13145: apply (rule iffI)
nipkow@13145:  apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145:  apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208:  apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145: apply (simp add: ball_Un)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13114: lemma list_all2_append2:
nipkow@13145: "list_all2 P xs (ys @ zs) =
nipkow@13145: (EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145: list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145: apply (simp add: list_all2_def zip_append2)
nipkow@13145: apply (rule iffI)
nipkow@13145:  apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145:  apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208:  apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145: apply (simp add: ball_Un)
nipkow@13145: done
wenzelm@13114: 
kleing@13863: lemma list_all2_append:
nipkow@14247:   "length xs = length ys \<Longrightarrow>
nipkow@14247:   list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247: by (induct rule:list_induct2, simp_all)
kleing@13863: 
kleing@13863: lemma list_all2_appendI [intro?, trans]:
kleing@13863:   "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
nipkow@24349: by (simp add: list_all2_append list_all2_lengthD)
kleing@13863: 
wenzelm@13114: lemma list_all2_conv_all_nth:
nipkow@13145: "list_all2 P xs ys =
nipkow@13145: (length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145: by (force simp add: list_all2_def set_zip)
wenzelm@13114: 
berghofe@13883: lemma list_all2_trans:
berghofe@13883:   assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883:   shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883:         (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883: proof (induct as)
berghofe@13883:   fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883:   show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883:   proof (induct bs)
berghofe@13883:     fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883:     show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883:       by (induct cs) (auto intro: tr I1 I2)
berghofe@13883:   qed simp
berghofe@13883: qed simp
berghofe@13883: 
kleing@13863: lemma list_all2_all_nthI [intro?]:
kleing@13863:   "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
nipkow@24349: by (simp add: list_all2_conv_all_nth)
kleing@13863: 
paulson@14395: lemma list_all2I:
paulson@14395:   "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
nipkow@24349: by (simp add: list_all2_def)
paulson@14395: 
kleing@14328: lemma list_all2_nthD:
kleing@13863:   "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@24349: by (simp add: list_all2_conv_all_nth)
kleing@13863: 
nipkow@14302: lemma list_all2_nthD2:
nipkow@14302:   "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@24349: by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302: 
kleing@13863: lemma list_all2_map1: 
kleing@13863:   "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
nipkow@24349: by (simp add: list_all2_conv_all_nth)
kleing@13863: 
kleing@13863: lemma list_all2_map2: 
kleing@13863:   "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
nipkow@24349: by (auto simp add: list_all2_conv_all_nth)
kleing@13863: 
kleing@14316: lemma list_all2_refl [intro?]:
kleing@13863:   "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
nipkow@24349: by (simp add: list_all2_conv_all_nth)
kleing@13863: 
kleing@13863: lemma list_all2_update_cong:
kleing@13863:   "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
nipkow@24349: by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863: 
kleing@13863: lemma list_all2_update_cong2:
kleing@13863:   "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
nipkow@24349: by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863: 
nipkow@14302: lemma list_all2_takeI [simp,intro?]:
nipkow@24526:   "list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@24526: apply (induct xs arbitrary: n ys)
nipkow@24526:  apply simp
nipkow@24526: apply (clarsimp simp add: list_all2_Cons1)
nipkow@24526: apply (case_tac n)
nipkow@24526: apply auto
nipkow@24526: done
nipkow@14302: 
nipkow@14302: lemma list_all2_dropI [simp,intro?]:
nipkow@24526:   "list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
nipkow@24526: apply (induct as arbitrary: n bs, simp)
nipkow@24526: apply (clarsimp simp add: list_all2_Cons1)
nipkow@24526: apply (case_tac n, simp, simp)
nipkow@24526: done
kleing@13863: 
kleing@14327: lemma list_all2_mono [intro?]:
nipkow@24526:   "list_all2 P xs ys \<Longrightarrow> (\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys) \<Longrightarrow> list_all2 Q xs ys"
nipkow@24526: apply (induct xs arbitrary: ys, simp)
nipkow@24526: apply (case_tac ys, auto)
nipkow@24526: done
kleing@13863: 
haftmann@22551: lemma list_all2_eq:
haftmann@22551:   "xs = ys \<longleftrightarrow> list_all2 (op =) xs ys"
nipkow@24349: by (induct xs ys rule: list_induct2') auto
haftmann@22551: 
wenzelm@13142: 
nipkow@15392: subsubsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142: 
wenzelm@13142: lemma foldl_append [simp]:
nipkow@24526:   "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@24526: by (induct xs arbitrary: a) auto
wenzelm@13142: 
nipkow@14402: lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402: by (induct xs) auto
nipkow@14402: 
nipkow@23096: lemma foldr_map: "foldr g (map f xs) a = foldr (g o f) xs a"
nipkow@23096: by(induct xs) simp_all
nipkow@23096: 
nipkow@24449: text{* For efficient code generation: avoid intermediate list. *}
nipkow@24449: lemma foldl_map[code unfold]:
nipkow@24449:   "foldl g a (map f xs) = foldl (%a x. g a (f x)) a xs"
nipkow@23096: by(induct xs arbitrary:a) simp_all
nipkow@23096: 
krauss@19770: lemma foldl_cong [fundef_cong, recdef_cong]:
krauss@18336:   "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] 
krauss@18336:   ==> foldl f a l = foldl g b k"
nipkow@24349: by (induct k arbitrary: a b l) simp_all
krauss@18336: 
krauss@19770: lemma foldr_cong [fundef_cong, recdef_cong]:
krauss@18336:   "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] 
krauss@18336:   ==> foldr f l a = foldr g k b"
nipkow@24349: by (induct k arbitrary: a b l) simp_all
krauss@18336: 
nipkow@24449: lemma (in semigroup_add) foldl_assoc:
haftmann@25062: shows "foldl op+ (x+y) zs = x + (foldl op+ y zs)"
nipkow@24449: by (induct zs arbitrary: y) (simp_all add:add_assoc)
nipkow@24449: 
nipkow@24449: lemma (in monoid_add) foldl_absorb0:
haftmann@25062: shows "x + (foldl op+ 0 zs) = foldl op+ x zs"
nipkow@24449: by (induct zs) (simp_all add:foldl_assoc)
nipkow@24449: 
nipkow@24449: 
nipkow@23096: text{* The ``First Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096: 
nipkow@23096: lemma foldl_foldr1_lemma:
nipkow@23096:  "foldl op + a xs = a + foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096: by (induct xs arbitrary: a) (auto simp:add_assoc)
nipkow@23096: 
nipkow@23096: corollary foldl_foldr1:
nipkow@23096:  "foldl op + 0 xs = foldr op + xs (0\<Colon>'a::monoid_add)"
nipkow@23096: by (simp add:foldl_foldr1_lemma)
nipkow@23096: 
nipkow@23096: 
nipkow@23096: text{* The ``Third Duality Theorem'' in Bird \& Wadler: *}
nipkow@23096: 
nipkow@14402: lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402: by (induct xs) auto
nipkow@14402: 
nipkow@14402: lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402: by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402: 
haftmann@25062: lemma (in ab_semigroup_add) foldr_conv_foldl: "foldr op + xs a = foldl op + a xs"
chaieb@24471:   by (induct xs, auto simp add: foldl_assoc add_commute)
chaieb@24471: 
wenzelm@13142: text {*
nipkow@13145: Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145: difficult to use because it requires an additional transitivity step.
wenzelm@13142: *}
wenzelm@13142: 
nipkow@24526: lemma start_le_sum: "(m::nat) <= n ==> m <= foldl (op +) n ns"
nipkow@24526: by (induct ns arbitrary: n) auto
nipkow@24526: 
nipkow@24526: lemma elem_le_sum: "(n::nat) : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145: by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142: 
wenzelm@13142: lemma sum_eq_0_conv [iff]:
nipkow@24526:   "(foldl (op +) (m::nat) ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@24526: by (induct ns arbitrary: m) auto
wenzelm@13114: 
chaieb@24471: lemma foldr_invariant: 
chaieb@24471:   "\<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)"
chaieb@24471:   by (induct xs, simp_all)
chaieb@24471: 
chaieb@24471: lemma foldl_invariant: 
chaieb@24471:   "\<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)"
chaieb@24471:   by (induct xs arbitrary: x, simp_all)
chaieb@24471: 
nipkow@24449: text{* @{const foldl} and @{text concat} *}
nipkow@24449: 
nipkow@24449: lemma concat_conv_foldl: "concat xss = foldl op@ [] xss"
nipkow@24449: by (induct xss) (simp_all add:monoid_append.foldl_absorb0)
nipkow@24449: 
nipkow@24449: lemma foldl_conv_concat:
nipkow@24449:   "foldl (op @) xs xxs = xs @ (concat xxs)"
nipkow@24449: by(simp add:concat_conv_foldl monoid_append.foldl_absorb0)
nipkow@24449: 
nipkow@23096: subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}
nipkow@23096: 
haftmann@26442: lemma listsum_append [simp]: "listsum (xs @ ys) = listsum xs + listsum ys"
nipkow@24449: by (induct xs) (simp_all add:add_assoc)
nipkow@24449: 
haftmann@26442: lemma listsum_rev [simp]:
haftmann@26442:   fixes xs :: "'a\<Colon>comm_monoid_add list"
haftmann@26442:   shows "listsum (rev xs) = listsum xs"
nipkow@24449: by (induct xs) (simp_all add:add_ac)
nipkow@24449: 
haftmann@26442: lemma listsum_foldr: "listsum xs = foldr (op +) xs 0"
haftmann@26442: by (induct xs) auto
haftmann@26442: 
haftmann@26442: lemma length_concat: "length (concat xss) = listsum (map length xss)"
haftmann@26442: by (induct xss) simp_all
nipkow@23096: 
nipkow@24449: text{* For efficient code generation ---
nipkow@24449:        @{const listsum} is not tail recursive but @{const foldl} is. *}
nipkow@24449: lemma listsum[code unfold]: "listsum xs = foldl (op +) 0 xs"
nipkow@23096: by(simp add:listsum_foldr foldl_foldr1)
nipkow@23096: 
nipkow@24449: 
nipkow@23096: text{* Some syntactic sugar for summing a function over a list: *}
nipkow@23096: 
nipkow@23096: syntax
nipkow@23096:   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
nipkow@23096: syntax (xsymbols)
nipkow@23096:   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096: syntax (HTML output)
nipkow@23096:   "_listsum" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
nipkow@23096: 
nipkow@23096: translations -- {* Beware of argument permutation! *}
nipkow@23096:   "SUM x<-xs. b" == "CONST listsum (map (%x. b) xs)"
nipkow@23096:   "\<Sum>x\<leftarrow>xs. b" == "CONST listsum (map (%x. b) xs)"
nipkow@23096: 
haftmann@26442: lemma listsum_triv: "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
haftmann@26442:   by (induct xs) (simp_all add: left_distrib)
haftmann@26442: 
nipkow@23096: lemma listsum_0 [simp]: "(\<Sum>x\<leftarrow>xs. 0) = 0"
haftmann@26442:   by (induct xs) (simp_all add: left_distrib)
nipkow@23096: 
nipkow@23096: text{* For non-Abelian groups @{text xs} needs to be reversed on one side: *}
nipkow@23096: lemma uminus_listsum_map:
haftmann@26442:   fixes f :: "'a \<Rightarrow> 'b\<Colon>ab_group_add"
haftmann@26442:   shows "- listsum (map f xs) = (listsum (map (uminus o f) xs))"
haftmann@26442: by (induct xs) simp_all
nipkow@23096: 
wenzelm@13114: 
nipkow@24645: subsubsection {* @{text upt} *}
wenzelm@13114: 
nipkow@17090: lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
nipkow@17090: -- {* simp does not terminate! *}
nipkow@13145: by (induct j) auto
wenzelm@13142: 
nipkow@15425: lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
nipkow@13145: by (subst upt_rec) simp
wenzelm@13114: 
nipkow@15425: lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
nipkow@15281: by(induct j)simp_all
nipkow@15281: 
nipkow@15281: lemma upt_eq_Cons_conv:
nipkow@24526:  "([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
nipkow@24526: apply(induct j arbitrary: x xs)
nipkow@15281:  apply simp
nipkow@15281: apply(clarsimp simp add: append_eq_Cons_conv)
nipkow@15281: apply arith
nipkow@15281: done
nipkow@15281: 
nipkow@15425: lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
nipkow@13145: -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145: by simp
wenzelm@13114: 
nipkow@15425: lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
paulson@24632: by (metis upt_rec)
wenzelm@13114: 
nipkow@15425: lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
nipkow@13145: -- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145: by (induct k) auto
wenzelm@13114: 
nipkow@15425: lemma length_upt [simp]: "length [i..<j] = j - i"
nipkow@13145: by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114: 
nipkow@15425: lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
nipkow@13145: apply (induct j)
nipkow@13145: apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145: done
wenzelm@13114: 
nipkow@17906: 
nipkow@17906: lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
nipkow@17906: by(simp add:upt_conv_Cons)
nipkow@17906: 
nipkow@17906: lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
nipkow@17906: apply(cases j)
nipkow@17906:  apply simp
nipkow@17906: by(simp add:upt_Suc_append)
nipkow@17906: 
nipkow@24526: lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]"
nipkow@24526: apply (induct m arbitrary: i, simp)
nipkow@13145: apply (subst upt_rec)
nipkow@13145: apply (rule sym)
nipkow@13145: apply (subst upt_rec)
nipkow@13145: apply (simp del: upt.simps)
nipkow@13145: done
nipkow@3507: 
nipkow@17501: lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
nipkow@17501: apply(induct j)
nipkow@17501: apply auto
nipkow@17501: done
nipkow@17501: 
nipkow@24645: lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]"
nipkow@13145: by (induct n) auto
wenzelm@13114: 
nipkow@24526: lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
nipkow@24526: apply (induct n m  arbitrary: i rule: diff_induct)
nipkow@13145: prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145: apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145: done
wenzelm@13114: 
berghofe@13883: lemma nth_take_lemma:
nipkow@24526:   "k <= length xs ==> k <= length ys ==>
berghofe@13883:      (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
nipkow@24526: apply (atomize, induct k arbitrary: xs ys)
paulson@14208: apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145: txt {* Both lists must be non-empty *}
paulson@14208: apply (case_tac xs, simp)
paulson@14208: apply (case_tac ys, clarify)
nipkow@13145:  apply (simp (no_asm_use))
nipkow@13145: apply clarify
nipkow@13145: txt {* prenexing's needed, not miniscoping *}
nipkow@13145: apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145: apply blast
nipkow@13145: done
wenzelm@13114: 
wenzelm@13114: lemma nth_equalityI:
wenzelm@13114:  "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145: apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145: apply (simp_all add: take_all)
nipkow@13145: done
wenzelm@13142: 
haftmann@24796: lemma map_nth:
haftmann@24796:   "map (\<lambda>i. xs ! i) [0..<length xs] = xs"
haftmann@24796:   by (rule nth_equalityI, auto)
haftmann@24796: 
kleing@13863: (* needs nth_equalityI *)
kleing@13863: lemma list_all2_antisym:
kleing@13863:   "\<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> 
kleing@13863:   \<Longrightarrow> xs = ys"
kleing@13863:   apply (simp add: list_all2_conv_all_nth) 
paulson@14208:   apply (rule nth_equalityI, blast, simp)
kleing@13863:   done
kleing@13863: 
wenzelm@13142: lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145: -- {* The famous take-lemma. *}
nipkow@13145: apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145: apply (simp add: le_max_iff_disj take_all)
nipkow@13145: done
wenzelm@13142: 
wenzelm@13142: 
nipkow@15302: lemma take_Cons':
nipkow@15302:      "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@15302: by (cases n) simp_all
nipkow@15302: 
nipkow@15302: lemma drop_Cons':
nipkow@15302:      "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@15302: by (cases n) simp_all
nipkow@15302: 
nipkow@15302: lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@15302: by (cases n) simp_all
nipkow@15302: 
paulson@18622: lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard]
paulson@18622: lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard]
paulson@18622: lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard]
paulson@18622: 
paulson@18622: declare take_Cons_number_of [simp] 
paulson@18622:         drop_Cons_number_of [simp] 
paulson@18622:         nth_Cons_number_of [simp] 
nipkow@15302: 
nipkow@15302: 
nipkow@15392: subsubsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142: 
wenzelm@13142: lemma distinct_append [simp]:
nipkow@13145: "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145: by (induct xs) auto
wenzelm@13142: 
nipkow@15305: lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
nipkow@15305: by(induct xs) auto
nipkow@15305: 
wenzelm@13142: lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145: by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142: 
wenzelm@13142: lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145: by (induct xs) auto
wenzelm@13142: 
nipkow@25287: lemma distinct_remdups_id: "distinct xs ==> remdups xs = xs"
nipkow@25287: by (induct xs, auto)
nipkow@25287: 
nipkow@25287: lemma remdups_id_iff_distinct[simp]: "(remdups xs = xs) = distinct xs"
nipkow@25287: by(metis distinct_remdups distinct_remdups_id)
nipkow@25287: 
nipkow@24566: lemma finite_distinct_list: "finite A \<Longrightarrow> EX xs. set xs = A & distinct xs"
paulson@24632: by (metis distinct_remdups finite_list set_remdups)
nipkow@24566: 
paulson@15072: lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
nipkow@24349: by (induct x, auto) 
paulson@15072: 
paulson@15072: lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
nipkow@24349: by (induct x, auto)
paulson@15072: 
nipkow@15245: lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245: by (induct xs) auto
nipkow@15245: 
nipkow@15245: lemma length_remdups_eq[iff]:
nipkow@15245:   "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245: apply(induct xs)
nipkow@15245:  apply auto
nipkow@15245: apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245:  apply arith
nipkow@15245: apply(rule length_remdups_leq)
nipkow@15245: done
nipkow@15245: 
nipkow@18490: 
nipkow@18490: lemma distinct_map:
nipkow@18490:   "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
nipkow@18490: by (induct xs) auto
nipkow@18490: 
nipkow@18490: 
wenzelm@13142: lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145: by (induct xs) auto
wenzelm@13114: 
nipkow@17501: lemma distinct_upt[simp]: "distinct[i..<j]"
nipkow@17501: by (induct j) auto
nipkow@17501: 
nipkow@24526: lemma distinct_take[simp]: "distinct xs \<Longrightarrow> distinct (take i xs)"
nipkow@24526: apply(induct xs arbitrary: i)
nipkow@17501:  apply simp
nipkow@17501: apply (case_tac i)
nipkow@17501:  apply simp_all
nipkow@17501: apply(blast dest:in_set_takeD)
nipkow@17501: done
nipkow@17501: 
nipkow@24526: lemma distinct_drop[simp]: "distinct xs \<Longrightarrow> distinct (drop i xs)"
nipkow@24526: apply(induct xs arbitrary: i)
nipkow@17501:  apply simp
nipkow@17501: apply (case_tac i)
nipkow@17501:  apply simp_all
nipkow@17501: done
nipkow@17501: 
nipkow@17501: lemma distinct_list_update:
nipkow@17501: assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
nipkow@17501: shows "distinct (xs[i:=a])"
nipkow@17501: proof (cases "i < length xs")
nipkow@17501:   case True
nipkow@17501:   with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
nipkow@17501:     apply (drule_tac id_take_nth_drop) by simp
nipkow@17501:   with d True show ?thesis
nipkow@17501:     apply (simp add: upd_conv_take_nth_drop)
nipkow@17501:     apply (drule subst [OF id_take_nth_drop]) apply assumption
nipkow@17501:     apply simp apply (cases "a = xs!i") apply simp by blast
nipkow@17501: next
nipkow@17501:   case False with d show ?thesis by auto
nipkow@17501: qed
nipkow@17501: 
nipkow@17501: 
nipkow@17501: text {* It is best to avoid this indexed version of distinct, but
nipkow@17501: sometimes it is useful. *}
nipkow@17501: 
wenzelm@13142: lemma distinct_conv_nth:
nipkow@17501: "distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@15251: apply (induct xs, simp, simp)
paulson@14208: apply (rule iffI, clarsimp)
nipkow@13145:  apply (case_tac i)
paulson@14208: apply (case_tac j, simp)
nipkow@13145: apply (simp add: set_conv_nth)
nipkow@13145:  apply (case_tac j)
paulson@24648: apply (clarsimp simp add: set_conv_nth, simp) 
nipkow@13145: apply (rule conjI)
paulson@24648: (*TOO SLOW
paulson@24632: apply (metis Zero_neq_Suc gr0_conv_Suc in_set_conv_nth lessI less_trans_Suc nth_Cons' nth_Cons_Suc)
paulson@24648: *)
paulson@24648:  apply (clarsimp simp add: set_conv_nth)
paulson@24648:  apply (erule_tac x = 0 in allE, simp)
paulson@24648:  apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
wenzelm@25130: (*TOO SLOW
paulson@24632: apply (metis Suc_Suc_eq lessI less_trans_Suc nth_Cons_Suc)
wenzelm@25130: *)
wenzelm@25130: apply (erule_tac x = "Suc i" in allE, simp)
wenzelm@25130: apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145: done
wenzelm@13114: 
nipkow@18490: lemma nth_eq_iff_index_eq:
nipkow@18490:  "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
nipkow@18490: by(auto simp: distinct_conv_nth)
nipkow@18490: 
nipkow@15110: lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
nipkow@24349: by (induct xs) auto
kleing@14388: 
nipkow@15110: lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388: proof (induct xs)
kleing@14388:   case Nil thus ?case by simp
kleing@14388: next
kleing@14388:   case (Cons x xs)
kleing@14388:   show ?case
kleing@14388:   proof (cases "x \<in> set xs")
kleing@14388:     case False with Cons show ?thesis by simp
kleing@14388:   next
kleing@14388:     case True with Cons.prems
kleing@14388:     have "card (set xs) = Suc (length xs)" 
kleing@14388:       by (simp add: card_insert_if split: split_if_asm)
kleing@14388:     moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388:     ultimately have False by simp
kleing@14388:     thus ?thesis ..
kleing@14388:   qed
kleing@14388: qed
kleing@14388: 
nipkow@25287: lemma not_distinct_decomp: "~ distinct ws ==> EX xs ys zs y. ws = xs@[y]@ys@[y]@zs"
nipkow@25287: apply (induct n == "length ws" arbitrary:ws) apply simp
nipkow@25287: apply(case_tac ws) apply simp
nipkow@25287: apply (simp split:split_if_asm)
nipkow@25287: apply (metis Cons_eq_appendI eq_Nil_appendI split_list)
nipkow@25287: done
nipkow@18490: 
nipkow@18490: lemma length_remdups_concat:
nipkow@18490:  "length(remdups(concat xss)) = card(\<Union>xs \<in> set xss. set xs)"
nipkow@24308: by(simp add: set_concat distinct_card[symmetric])
nipkow@17906: 
nipkow@17906: 
nipkow@15392: subsubsection {* @{text remove1} *}
nipkow@15110: 
nipkow@18049: lemma remove1_append:
nipkow@18049:   "remove1 x (xs @ ys) =
nipkow@18049:   (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
nipkow@18049: by (induct xs) auto
nipkow@18049: 
nipkow@23479: lemma in_set_remove1[simp]:
nipkow@23479:   "a \<noteq> b \<Longrightarrow> a : set(remove1 b xs) = (a : set xs)"
nipkow@23479: apply (induct xs)
nipkow@23479: apply auto
nipkow@23479: done
nipkow@23479: 
nipkow@15110: lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110: apply(induct xs)
nipkow@15110:  apply simp
nipkow@15110: apply simp
nipkow@15110: apply blast
nipkow@15110: done
nipkow@15110: 
paulson@17724: lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110: apply(induct xs)
nipkow@15110:  apply simp
nipkow@15110: apply simp
nipkow@15110: apply blast
nipkow@15110: done
nipkow@15110: 
nipkow@23479: lemma length_remove1:
nipkow@23479:   "length(remove1 x xs) = (if x : set xs then length xs - 1 else length xs)"
nipkow@23479: apply (induct xs)
nipkow@23479:  apply (auto dest!:length_pos_if_in_set)
nipkow@23479: done
nipkow@23479: 
nipkow@18049: lemma remove1_filter_not[simp]:
nipkow@18049:   "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
nipkow@18049: by(induct xs) auto
nipkow@18049: 
nipkow@15110: lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110: apply(insert set_remove1_subset)
nipkow@15110: apply fast
nipkow@15110: done
nipkow@15110: 
nipkow@15110: lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110: by (induct xs) simp_all
nipkow@15110: 
wenzelm@13114: 
nipkow@15392: subsubsection {* @{text replicate} *}
wenzelm@13114: 
wenzelm@13142: lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145: by (induct n) auto
nipkow@13124: 
wenzelm@13142: lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145: by (induct n) auto
wenzelm@13114: 
wenzelm@13114: lemma replicate_app_Cons_same:
nipkow@13145: "(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145: by (induct n) auto
wenzelm@13114: 
wenzelm@13142: lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208: apply (induct n, simp)
nipkow@13145: apply (simp add: replicate_app_Cons_same)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13142: lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145: by (induct n) auto
wenzelm@13114: 
nipkow@16397: text{* Courtesy of Matthias Daum: *}
nipkow@16397: lemma append_replicate_commute:
nipkow@16397:   "replicate n x @ replicate k x = replicate k x @ replicate n x"
nipkow@16397: apply (simp add: replicate_add [THEN sym])
nipkow@16397: apply (simp add: add_commute)
nipkow@16397: done
nipkow@16397: 
wenzelm@13142: lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145: by (induct n) auto
wenzelm@13114: 
wenzelm@13142: lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145: by (induct n) auto
wenzelm@13114: 
wenzelm@13142: lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145: by (atomize (full), induct n) auto
wenzelm@13114: 
nipkow@24526: lemma nth_replicate[simp]: "i < n ==> (replicate n x)!i = x"
nipkow@24526: apply (induct n arbitrary: i, simp)
nipkow@13145: apply (simp add: nth_Cons split: nat.split)
nipkow@13145: done
wenzelm@13114: 
nipkow@16397: text{* Courtesy of Matthias Daum (2 lemmas): *}
nipkow@16397: lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
nipkow@16397: apply (case_tac "k \<le> i")
nipkow@16397:  apply  (simp add: min_def)
nipkow@16397: apply (drule not_leE)
nipkow@16397: apply (simp add: min_def)
nipkow@16397: apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
nipkow@16397:  apply  simp
nipkow@16397: apply (simp add: replicate_add [symmetric])
nipkow@16397: done
nipkow@16397: 
nipkow@24526: lemma drop_replicate[simp]: "drop i (replicate k x) = replicate (k-i) x"
nipkow@24526: apply (induct k arbitrary: i)
nipkow@16397:  apply simp
nipkow@16397: apply clarsimp
nipkow@16397: apply (case_tac i)
nipkow@16397:  apply simp
nipkow@16397: apply clarsimp
nipkow@16397: done
nipkow@16397: 
nipkow@16397: 
wenzelm@13142: lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145: by (induct n) auto
wenzelm@13114: 
wenzelm@13142: lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145: by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114: 
wenzelm@13142: lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145: by auto
wenzelm@13114: 
wenzelm@13142: lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145: by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114: 
haftmann@24796: lemma replicate_append_same:
haftmann@24796:   "replicate i x @ [x] = x # replicate i x"
haftmann@24796:   by (induct i) simp_all
haftmann@24796: 
haftmann@24796: lemma map_replicate_trivial:
haftmann@24796:   "map (\<lambda>i. x) [0..<i] = replicate i x"
haftmann@24796:   by (induct i) (simp_all add: replicate_append_same)
haftmann@24796: 
wenzelm@13114: 
nipkow@15392: subsubsection{*@{text rotate1} and @{text rotate}*}
nipkow@15302: 
nipkow@15302: lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
nipkow@15302: by(simp add:rotate1_def)
nipkow@15302: 
nipkow@15302: lemma rotate0[simp]: "rotate 0 = id"
nipkow@15302: by(simp add:rotate_def)
nipkow@15302: 
nipkow@15302: lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
nipkow@15302: by(simp add:rotate_def)
nipkow@15302: 
nipkow@15302: lemma rotate_add:
nipkow@15302:   "rotate (m+n) = rotate m o rotate n"
nipkow@15302: by(simp add:rotate_def funpow_add)
nipkow@15302: 
nipkow@15302: lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
nipkow@15302: by(simp add:rotate_add)
nipkow@15302: 
nipkow@18049: lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
nipkow@18049: by(simp add:rotate_def funpow_swap1)
nipkow@18049: 
nipkow@15302: lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
nipkow@15302: by(cases xs) simp_all
nipkow@15302: 
nipkow@15302: lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
nipkow@15302: apply(induct n)
nipkow@15302:  apply simp
nipkow@15302: apply (simp add:rotate_def)
nipkow@13145: done
wenzelm@13114: 
nipkow@15302: lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
nipkow@15302: by(simp add:rotate1_def split:list.split)
nipkow@15302: 
nipkow@15302: lemma rotate_drop_take:
nipkow@15302:   "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
nipkow@15302: apply(induct n)
nipkow@15302:  apply simp
nipkow@15302: apply(simp add:rotate_def)
nipkow@15302: apply(cases "xs = []")
nipkow@15302:  apply (simp)
nipkow@15302: apply(case_tac "n mod length xs = 0")
nipkow@15302:  apply(simp add:mod_Suc)
nipkow@15302:  apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
nipkow@15302: apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
nipkow@15302:                 take_hd_drop linorder_not_le)
nipkow@13145: done
wenzelm@13114: 
nipkow@15302: lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
nipkow@15302: by(simp add:rotate_drop_take)
nipkow@15302: 
nipkow@15302: lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
nipkow@15302: by(simp add:rotate_drop_take)
nipkow@15302: 
nipkow@15302: lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
nipkow@15302: by(simp add:rotate1_def split:list.split)
nipkow@15302: 
nipkow@24526: lemma length_rotate[simp]: "length(rotate n xs) = length xs"
nipkow@24526: by (induct n arbitrary: xs) (simp_all add:rotate_def)
nipkow@15302: 
nipkow@15302: lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
nipkow@15302: by(simp add:rotate1_def split:list.split) blast
nipkow@15302: 
nipkow@15302: lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
nipkow@15302: by (induct n) (simp_all add:rotate_def)
nipkow@15302: 
nipkow@15302: lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
nipkow@15302: by(simp add:rotate_drop_take take_map drop_map)
nipkow@15302: 
nipkow@15302: lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
nipkow@15302: by(simp add:rotate1_def split:list.split)
nipkow@15302: 
nipkow@15302: lemma set_rotate[simp]: "set(rotate n xs) = set xs"
nipkow@15302: by (induct n) (simp_all add:rotate_def)
nipkow@15302: 
nipkow@15302: lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
nipkow@15302: by(simp add:rotate1_def split:list.split)
nipkow@15302: 
nipkow@15302: lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
nipkow@15302: by (induct n) (simp_all add:rotate_def)
wenzelm@13114: 
nipkow@15439: lemma rotate_rev:
nipkow@15439:   "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
nipkow@15439: apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439: apply(cases "length xs = 0")
nipkow@15439:  apply simp
nipkow@15439: apply(cases "n mod length xs = 0")
nipkow@15439:  apply simp
nipkow@15439: apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439: done
nipkow@15439: 
nipkow@18423: lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
nipkow@18423: apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
nipkow@18423: apply(subgoal_tac "length xs \<noteq> 0")
nipkow@18423:  prefer 2 apply simp
nipkow@18423: using mod_less_divisor[of "length xs" n] by arith
nipkow@18423: 
wenzelm@13114: 
nipkow@15392: subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114: 
wenzelm@13142: lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145: by (auto simp add: sublist_def)
wenzelm@13114: 
wenzelm@13142: lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145: by (auto simp add: sublist_def)
wenzelm@13114: 
nipkow@15281: lemma length_sublist:
nipkow@15281:   "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
nipkow@15281: by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
nipkow@15281: 
nipkow@15281: lemma sublist_shift_lemma_Suc:
nipkow@24526:   "map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
nipkow@24526:    map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
nipkow@24526: apply(induct xs arbitrary: "is")
nipkow@15281:  apply simp
nipkow@15281: apply (case_tac "is")
nipkow@15281:  apply simp
nipkow@15281: apply simp
nipkow@15281: done
nipkow@15281: 
wenzelm@13114: lemma sublist_shift_lemma:
nipkow@23279:      "map fst [p<-zip xs [i..<i + length xs] . snd p : A] =
nipkow@23279:       map fst [p<-zip xs [0..<length xs] . snd p + i : A]"
nipkow@13145: by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114: 
wenzelm@13114: lemma sublist_append:
paulson@15168:      "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145: apply (unfold sublist_def)
paulson@14208: apply (induct l' rule: rev_induct, simp)
nipkow@13145: apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145: apply (simp add: add_commute)
nipkow@13145: done
wenzelm@13114: 
wenzelm@13114: lemma sublist_Cons:
nipkow@13145: "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145: apply (induct l rule: rev_induct)
nipkow@13145:  apply (simp add: sublist_def)
nipkow@13145: apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145: done
wenzelm@13114: 
nipkow@24526: lemma set_sublist: "set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
nipkow@24526: apply(induct xs arbitrary: I)
nipkow@25162: apply(auto simp: sublist_Cons nth_Cons split:nat.split dest!: gr0_implies_Suc)
nipkow@15281: done
nipkow@15281: 
nipkow@15281: lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
nipkow@15281: by(auto simp add:set_sublist)
nipkow@15281: 
nipkow@15281: lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
nipkow@15281: by(auto simp add:set_sublist)
nipkow@15281: 
nipkow@15281: lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
nipkow@15281: by(auto simp add:set_sublist)
nipkow@15281: 
wenzelm@13142: lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145: by (simp add: sublist_Cons)
wenzelm@13114: 
nipkow@15281: 
nipkow@24526: lemma distinct_sublistI[simp]: "distinct xs \<Longrightarrow> distinct(sublist xs I)"
nipkow@24526: apply(induct xs arbitrary: I)
nipkow@15281:  apply simp
nipkow@15281: apply(auto simp add:sublist_Cons)
nipkow@15281: done
nipkow@15281: 
nipkow@15281: 
nipkow@15045: lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
paulson@14208: apply (induct l rule: rev_induct, simp)
nipkow@13145: apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145: done
wenzelm@13114: 
nipkow@24526: lemma filter_in_sublist:
nipkow@24526:  "distinct xs \<Longrightarrow> filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
nipkow@24526: proof (induct xs arbitrary: s)
nipkow@17501:   case Nil thus ?case by simp
nipkow@17501: next
nipkow@17501:   case (Cons a xs)
nipkow@17501:   moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
nipkow@17501:   ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
nipkow@17501: qed
nipkow@17501: 
wenzelm@13114: 
nipkow@19390: subsubsection {* @{const splice} *}
nipkow@19390: 
haftmann@19607: lemma splice_Nil2 [simp, code]:
nipkow@19390:  "splice xs [] = xs"
nipkow@19390: by (cases xs) simp_all
nipkow@19390: 
haftmann@19607: lemma splice_Cons_Cons [simp, code]:
nipkow@19390:  "splice (x#xs) (y#ys) = x # y # splice xs ys"
nipkow@19390: by simp
nipkow@19390: 
haftmann@19607: declare splice.simps(2) [simp del, code del]
nipkow@19390: 
nipkow@24526: lemma length_splice[simp]: "length(splice xs ys) = length xs + length ys"
nipkow@24526: apply(induct xs arbitrary: ys) apply simp
nipkow@22793: apply(case_tac ys)
nipkow@22793:  apply auto
nipkow@22793: done
nipkow@22793: 
nipkow@24616: 
nipkow@24616: subsection {*Sorting*}
nipkow@24616: 
nipkow@24617: text{* Currently it is not shown that @{const sort} returns a
nipkow@24617: permutation of its input because the nicest proof is via multisets,
nipkow@24617: which are not yet available. Alternatively one could define a function
nipkow@24617: that counts the number of occurrences of an element in a list and use
nipkow@24617: that instead of multisets to state the correctness property. *}
nipkow@24617: 
nipkow@24616: context linorder
nipkow@24616: begin
nipkow@24616: 
haftmann@25062: lemma sorted_Cons: "sorted (x#xs) = (sorted xs & (ALL y:set xs. x <= y))"
nipkow@24616: apply(induct xs arbitrary: x) apply simp
nipkow@24616: by simp (blast intro: order_trans)
nipkow@24616: 
nipkow@24616: lemma sorted_append:
haftmann@25062:   "sorted (xs@ys) = (sorted xs & sorted ys & (\<forall>x \<in> set xs. \<forall>y \<in> set ys. x\<le>y))"
nipkow@24616: by (induct xs) (auto simp add:sorted_Cons)
nipkow@24616: 
nipkow@24616: lemma set_insort: "set(insort x xs) = insert x (set xs)"
nipkow@24616: by (induct xs) auto
nipkow@24616: 
nipkow@24617: lemma set_sort[simp]: "set(sort xs) = set xs"
nipkow@24616: by (induct xs) (simp_all add:set_insort)
nipkow@24616: 
nipkow@24616: lemma distinct_insort: "distinct (insort x xs) = (x \<notin> set xs \<and> distinct xs)"
nipkow@24616: by(induct xs)(auto simp:set_insort)
nipkow@24616: 
nipkow@24617: lemma distinct_sort[simp]: "distinct (sort xs) = distinct xs"
nipkow@24616: by(induct xs)(simp_all add:distinct_insort set_sort)
nipkow@24616: 
nipkow@24616: lemma sorted_insort: "sorted (insort x xs) = sorted xs"
nipkow@24616: apply (induct xs)
nipkow@24650:  apply(auto simp:sorted_Cons set_insort)
nipkow@24616: done
nipkow@24616: 
nipkow@24616: theorem sorted_sort[simp]: "sorted (sort xs)"
nipkow@24616: by (induct xs) (auto simp:sorted_insort)
nipkow@24616: 
bulwahn@26143: lemma insort_is_Cons: "\<forall>x\<in>set xs. a \<le> x \<Longrightarrow> insort a xs = a # xs"
bulwahn@26143: by (cases xs) auto
bulwahn@26143: 
bulwahn@26143: lemma sorted_remove1: "sorted xs \<Longrightarrow> sorted (remove1 a xs)"
bulwahn@26143: by (induct xs, auto simp add: sorted_Cons)
bulwahn@26143: 
bulwahn@26143: lemma insort_remove1: "\<lbrakk> a \<in> set xs; sorted xs \<rbrakk> \<Longrightarrow> insort a (remove1 a xs) = xs"
bulwahn@26143: by (induct xs, auto simp add: sorted_Cons insort_is_Cons)
bulwahn@26143: 
bulwahn@26143: lemma sorted_remdups[simp]:
bulwahn@26143:   "sorted l \<Longrightarrow> sorted (remdups l)"
bulwahn@26143: by (induct l) (auto simp: sorted_Cons)
bulwahn@26143: 
nipkow@24645: lemma sorted_distinct_set_unique:
nipkow@24645: assumes "sorted xs" "distinct xs" "sorted ys" "distinct ys" "set xs = set ys"
nipkow@24645: shows "xs = ys"
nipkow@24645: proof -
nipkow@24645:   from assms have 1: "length xs = length ys" by (metis distinct_card)
nipkow@24645:   from assms show ?thesis
nipkow@24645:   proof(induct rule:list_induct2[OF 1])
nipkow@24645:     case 1 show ?case by simp
nipkow@24645:   next
nipkow@24645:     case 2 thus ?case by (simp add:sorted_Cons)
nipkow@24645:        (metis Diff_insert_absorb antisym insertE insert_iff)
nipkow@24645:   qed
nipkow@24645: qed
nipkow@24645: 
nipkow@24645: lemma finite_sorted_distinct_unique:
nipkow@24645: shows "finite A \<Longrightarrow> EX! xs. set xs = A & sorted xs & distinct xs"
nipkow@24645: apply(drule finite_distinct_list)
nipkow@24645: apply clarify
nipkow@24645: apply(rule_tac a="sort xs" in ex1I)
nipkow@24645: apply (auto simp: sorted_distinct_set_unique)
nipkow@24645: done
nipkow@24645: 
nipkow@24616: end
nipkow@24616: 
nipkow@25277: lemma sorted_upt[simp]: "sorted[i..<j]"
nipkow@25277: by (induct j) (simp_all add:sorted_append)
nipkow@25277: 
nipkow@24616: 
nipkow@25069: subsubsection {* @{text sorted_list_of_set} *}
nipkow@25069: 
nipkow@25069: text{* This function maps (finite) linearly ordered sets to sorted
nipkow@25069: lists. Warning: in most cases it is not a good idea to convert from
nipkow@25069: sets to lists but one should convert in the other direction (via
nipkow@25069: @{const set}). *}
nipkow@25069: 
nipkow@25069: 
nipkow@25069: context linorder
nipkow@25069: begin
nipkow@25069: 
nipkow@25069: definition
nipkow@25069:  sorted_list_of_set :: "'a set \<Rightarrow> 'a list" where
nipkow@25069: "sorted_list_of_set A == THE xs. set xs = A & sorted xs & distinct xs"
nipkow@25069: 
nipkow@25069: lemma sorted_list_of_set[simp]: "finite A \<Longrightarrow>
nipkow@25069:   set(sorted_list_of_set A) = A &
nipkow@25069:   sorted(sorted_list_of_set A) & distinct(sorted_list_of_set A)"
nipkow@25069: apply(simp add:sorted_list_of_set_def)
nipkow@25069: apply(rule the1I2)
nipkow@25069:  apply(simp_all add: finite_sorted_distinct_unique)
nipkow@25069: done
nipkow@25069: 
nipkow@25069: lemma sorted_list_of_empty[simp]: "sorted_list_of_set {} = []"
nipkow@25069: unfolding sorted_list_of_set_def
nipkow@25069: apply(subst the_equality[of _ "[]"])
nipkow@25069: apply simp_all
nipkow@25069: done
nipkow@25069: 
nipkow@25069: end
nipkow@25069: 
nipkow@25069: 
nipkow@24645: subsubsection {* @{text upto}: the generic interval-list *}
nipkow@24645: 
nipkow@24697: class finite_intvl_succ = linorder +
nipkow@24697: fixes successor :: "'a \<Rightarrow> 'a"
nipkow@25069: assumes finite_intvl: "finite{a..b}"
haftmann@25062: and successor_incr: "a < successor a"
haftmann@25062: and ord_discrete: "\<not>(\<exists>x. a < x & x < successor a)"
nipkow@24697: 
nipkow@24697: context finite_intvl_succ
nipkow@24697: begin
nipkow@24697: 
nipkow@24697: definition
haftmann@25062:  upto :: "'a \<Rightarrow> 'a \<Rightarrow> 'a list" ("(1[_../_])") where
nipkow@25069: "upto i j == sorted_list_of_set {i..j}"
nipkow@25069: 
nipkow@25069: lemma upto[simp]: "set[a..b] = {a..b} & sorted[a..b] & distinct[a..b]"
nipkow@25069: by(simp add:upto_def finite_intvl)
nipkow@24697: 
haftmann@25062: lemma insert_intvl: "i \<le> j \<Longrightarrow> insert i {successor i..j} = {i..j}"
nipkow@24697: apply(insert successor_incr[of i])
nipkow@24697: apply(auto simp: atLeastAtMost_def atLeast_def atMost_def)
nipkow@24697: apply (metis ord_discrete less_le not_le)
nipkow@24645: done
nipkow@24645: 
nipkow@25069: lemma sorted_list_of_set_rec: "i \<le> j \<Longrightarrow>
nipkow@25069:   sorted_list_of_set {i..j} = i # sorted_list_of_set {successor i..j}"
nipkow@25069: apply(simp add:sorted_list_of_set_def upto_def)
nipkow@25069: apply (rule the1_equality[OF finite_sorted_distinct_unique])
nipkow@25069:  apply (simp add:finite_intvl)
nipkow@25069: apply(rule the1I2[OF finite_sorted_distinct_unique])
nipkow@25069:  apply (simp add:finite_intvl)
nipkow@25069: apply (simp add: sorted_Cons insert_intvl Ball_def)
nipkow@25069: apply (metis successor_incr leD less_imp_le order_trans)
nipkow@25069: done
nipkow@25069: 
haftmann@25062: lemma upto_rec[code]: "[i..j] = (if i \<le> j then i # [successor i..j] else [])"
nipkow@25069: by(simp add: upto_def sorted_list_of_set_rec)
nipkow@24697: 
nipkow@24697: end
nipkow@24697: 
nipkow@24697: text{* The integers are an instance of the above class: *}
nipkow@24697: 
haftmann@25571: instantiation int:: finite_intvl_succ
haftmann@25571: begin
haftmann@25571: 
haftmann@25571: definition
haftmann@25571:   successor_int_def: "successor = (%i\<Colon>int. i+1)"
haftmann@25571: 
haftmann@25571: instance
haftmann@25571:   by intro_classes (simp_all add: successor_int_def)
haftmann@25571: 
haftmann@25571: end
nipkow@24645: 
nipkow@24697: text{* Now @{term"[i..j::int]"} is defined for integers. *}
nipkow@24697: 
nipkow@24698: hide (open) const successor
nipkow@24698: 
nipkow@24645: 
nipkow@15392: subsubsection {* @{text lists}: the list-forming operator over sets *}
nipkow@15302: 
berghofe@23740: inductive_set
berghofe@22262:   lists :: "'a set => 'a list set"
berghofe@23740:   for A :: "'a set"
berghofe@23740: where
berghofe@23740:     Nil [intro!]: "[]: lists A"
paulson@24286:   | Cons [intro!,noatp]: "[| a: A;l: lists A|] ==> a#l : lists A"
paulson@24286: 
paulson@24286: inductive_cases listsE [elim!,noatp]: "x#l : lists A"
paulson@24286: inductive_cases listspE [elim!,noatp]: "listsp A (x # l)"
berghofe@23740: 
berghofe@23740: lemma listsp_mono [mono]: "A \<le> B ==> listsp A \<le> listsp B"
nipkow@24349: by (clarify, erule listsp.induct, blast+)
berghofe@22262: 
berghofe@23740: lemmas lists_mono = listsp_mono [to_set]
berghofe@22262: 
haftmann@22422: lemma listsp_infI:
haftmann@22422:   assumes l: "listsp A l" shows "listsp B l ==> listsp (inf A B) l" using l
nipkow@24349: by induct blast+
nipkow@15302: 
haftmann@22422: lemmas lists_IntI = listsp_infI [to_set]
haftmann@22422: 
haftmann@22422: lemma listsp_inf_eq [simp]: "listsp (inf A B) = inf (listsp A) (listsp B)"
haftmann@22422: proof (rule mono_inf [where f=listsp, THEN order_antisym])
berghofe@22262:   show "mono listsp" by (simp add: mono_def listsp_mono)
haftmann@22422:   show "inf (listsp A) (listsp B) \<le> listsp (inf A B)" by (blast intro: listsp_infI)
kleing@14388: qed
kleing@14388: 
haftmann@22422: lemmas listsp_conj_eq [simp] = listsp_inf_eq [simplified inf_fun_eq inf_bool_eq]
haftmann@22422: 
haftmann@22422: lemmas lists_Int_eq [simp] = listsp_inf_eq [to_set]
berghofe@22262: 
berghofe@22262: lemma append_in_listsp_conv [iff]:
berghofe@22262:      "(listsp A (xs @ ys)) = (listsp A xs \<and> listsp A ys)"
nipkow@15302: by (induct xs) auto
nipkow@15302: 
berghofe@22262: lemmas append_in_lists_conv [iff] = append_in_listsp_conv [to_set]
berghofe@22262: 
berghofe@22262: lemma in_listsp_conv_set: "(listsp A xs) = (\<forall>x \<in> set xs. A x)"
berghofe@22262: -- {* eliminate @{text listsp} in favour of @{text set} *}
nipkow@15302: by (induct xs) auto
nipkow@15302: 
berghofe@22262: lemmas in_lists_conv_set = in_listsp_conv_set [to_set]
berghofe@22262: 
paulson@24286: lemma in_listspD [dest!,noatp]: "listsp A xs ==> \<forall>x\<in>set xs. A x"
berghofe@22262: by (rule in_listsp_conv_set [THEN iffD1])
berghofe@22262: 
paulson@24286: lemmas in_listsD [dest!,noatp] = in_listspD [to_set]
paulson@24286: 
paulson@24286: lemma in_listspI [intro!,noatp]: "\<forall>x\<in>set xs. A x ==> listsp A xs"
berghofe@22262: by (rule in_listsp_conv_set [THEN iffD2])
berghofe@22262: 
paulson@24286: lemmas in_listsI [intro!,noatp] = in_listspI [to_set]
nipkow@15302: 
nipkow@15302: lemma lists_UNIV [simp]: "lists UNIV = UNIV"
nipkow@15302: by auto
nipkow@15302: 
nipkow@17086: 
nipkow@17086: 
nipkow@17086: subsubsection{* Inductive definition for membership *}
nipkow@17086: 
berghofe@23740: inductive ListMem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
berghofe@22262: where
berghofe@22262:     elem:  "ListMem x (x # xs)"
berghofe@22262:   | insert:  "ListMem x xs \<Longrightarrow> ListMem x (y # xs)"
berghofe@22262: 
berghofe@22262: lemma ListMem_iff: "(ListMem x xs) = (x \<in> set xs)"
nipkow@17086: apply (rule iffI)
nipkow@17086:  apply (induct set: ListMem)
nipkow@17086:   apply auto
nipkow@17086: apply (induct xs)
nipkow@17086:  apply (auto intro: ListMem.intros)
nipkow@17086: done
nipkow@17086: 
nipkow@17086: 
nipkow@17086: 
nipkow@15392: subsubsection{*Lists as Cartesian products*}
nipkow@15302: 
nipkow@15302: text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
nipkow@15302: @{term A} and tail drawn from @{term Xs}.*}
nipkow@15302: 
nipkow@15302: constdefs
nipkow@15302:   set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
nipkow@15302:   "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"
nipkow@15302: 
paulson@17724: lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
nipkow@15302: by (auto simp add: set_Cons_def)
nipkow@15302: 
nipkow@15302: text{*Yields the set of lists, all of the same length as the argument and
nipkow@15302: with elements drawn from the corresponding element of the argument.*}
nipkow@15302: 
nipkow@15302: consts  listset :: "'a set list \<Rightarrow> 'a list set"
nipkow@15302: primrec
nipkow@15302:    "listset []    = {[]}"
nipkow@15302:    "listset(A#As) = set_Cons A (listset As)"
nipkow@15302: 
nipkow@15302: 
paulson@15656: subsection{*Relations on Lists*}
paulson@15656: 
paulson@15656: subsubsection {* Length Lexicographic Ordering *}
paulson@15656: 
paulson@15656: text{*These orderings preserve well-foundedness: shorter lists 
paulson@15656:   precede longer lists. These ordering are not used in dictionaries.*}
paulson@15656: 
paulson@15656: consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
paulson@15656:         --{*The lexicographic ordering for lists of the specified length*}
nipkow@15302: primrec
paulson@15656:   "lexn r 0 = {}"
paulson@15656:   "lexn r (Suc n) =
paulson@15656:     (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
paulson@15656:     {(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
nipkow@15302: 
nipkow@15302: constdefs
paulson@15656:   lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
paulson@15656:     "lex r == \<Union>n. lexn r n"
paulson@15656:         --{*Holds only between lists of the same length*}
paulson@15656: 
nipkow@15693:   lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@15693:     "lenlex r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
paulson@15656:         --{*Compares lists by their length and then lexicographically*}
nipkow@15302: 
nipkow@15302: 
nipkow@15302: lemma wf_lexn: "wf r ==> wf (lexn r n)"
nipkow@15302: apply (induct n, simp, simp)
nipkow@15302: apply(rule wf_subset)
nipkow@15302:  prefer 2 apply (rule Int_lower1)
nipkow@15302: apply(rule wf_prod_fun_image)
nipkow@15302:  prefer 2 apply (rule inj_onI, auto)
nipkow@15302: done
nipkow@15302: 
nipkow@15302: lemma lexn_length:
nipkow@24526:   "(xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@24526: by (induct n arbitrary: xs ys) auto
nipkow@15302: 
nipkow@15302: lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@15302: apply (unfold lex_def)
nipkow@15302: apply (rule wf_UN)
nipkow@15302: apply (blast intro: wf_lexn, clarify)
nipkow@15302: apply (rename_tac m n)
nipkow@15302: apply (subgoal_tac "m \<noteq> n")
nipkow@15302:  prefer 2 apply blast
nipkow@15302: apply (blast dest: lexn_length not_sym)
nipkow@15302: done
nipkow@15302: 
nipkow@15302: lemma lexn_conv:
paulson@15656:   "lexn r n =
paulson@15656:     {(xs,ys). length xs = n \<and> length ys = n \<and>
paulson@15656:     (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
nipkow@18423: apply (induct n, simp)
nipkow@15302: apply (simp add: image_Collect lex_prod_def, safe, blast)
nipkow@15302:  apply (rule_tac x = "ab # xys" in exI, simp)
nipkow@15302: apply (case_tac xys, simp_all, blast)
nipkow@15302: done
nipkow@15302: 
nipkow@15302: lemma lex_conv:
paulson@15656:   "lex r =
paulson@15656:     {(xs,ys). length xs = length ys \<and>
paulson@15656:     (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@15302: by (force simp add: lex_def lexn_conv)
nipkow@15302: 
nipkow@15693: lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
nipkow@15693: by (unfold lenlex_def) blast
nipkow@15693: 
nipkow@15693: lemma lenlex_conv:
nipkow@15693:     "lenlex r = {(xs,ys). length xs < length ys |
paulson@15656:                  length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@19623: by (simp add: lenlex_def diag_def lex_prod_def inv_image_def)
nipkow@15302: 
nipkow@15302: lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@15302: by (simp add: lex_conv)
nipkow@15302: 
nipkow@15302: lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@15302: by (simp add:lex_conv)
nipkow@15302: 
paulson@18447: lemma Cons_in_lex [simp]:
paulson@15656:     "((x # xs, y # ys) : lex r) =
paulson@15656:       ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@15302: apply (simp add: lex_conv)
nipkow@15302: apply (rule iffI)
nipkow@15302:  prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
nipkow@15302: apply (case_tac xys, simp, simp)
nipkow@15302: apply blast
nipkow@15302: done
nipkow@15302: 
nipkow@15302: 
paulson@15656: subsubsection {* Lexicographic Ordering *}
paulson@15656: 
paulson@15656: text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
paulson@15656:     This ordering does \emph{not} preserve well-foundedness.
nipkow@17090:      Author: N. Voelker, March 2005. *} 
paulson@15656: 
paulson@15656: constdefs 
paulson@15656:   lexord :: "('a * 'a)set \<Rightarrow> ('a list * 'a list) set" 
paulson@15656:   "lexord  r == {(x,y). \<exists> a v. y = x @ a # v \<or> 
paulson@15656:             (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
paulson@15656: 
paulson@15656: lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
nipkow@24349: by (unfold lexord_def, induct_tac y, auto) 
paulson@15656: 
paulson@15656: lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
nipkow@24349: by (unfold lexord_def, induct_tac x, auto)
paulson@15656: 
paulson@15656: lemma lexord_cons_cons[simp]:
paulson@15656:      "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
paulson@15656:   apply (unfold lexord_def, safe, simp_all)
paulson@15656:   apply (case_tac u, simp, simp)
paulson@15656:   apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
paulson@15656:   apply (erule_tac x="b # u" in allE)
paulson@15656:   by force
paulson@15656: 
paulson@15656: lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
paulson@15656: 
paulson@15656: lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
nipkow@24349: by (induct_tac x, auto)  
paulson@15656: 
paulson@15656: lemma lexord_append_left_rightI:
paulson@15656:      "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
nipkow@24349: by (induct_tac u, auto)
paulson@15656: 
paulson@15656: lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
nipkow@24349: by (induct x, auto)
paulson@15656: 
paulson@15656: lemma lexord_append_leftD:
paulson@15656:      "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
nipkow@24349: by (erule rev_mp, induct_tac x, auto)
paulson@15656: 
paulson@15656: lemma lexord_take_index_conv: 
paulson@15656:    "((x,y) : lexord r) = 
paulson@15656:     ((length x < length y \<and> take (length x) y = x) \<or> 
paulson@15656:      (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
paulson@15656:   apply (unfold lexord_def Let_def, clarsimp) 
paulson@15656:   apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
paulson@15656:   apply auto 
paulson@15656:   apply (rule_tac x="hd (drop (length x) y)" in exI)
paulson@15656:   apply (rule_tac x="tl (drop (length x) y)" in exI)
paulson@15656:   apply (erule subst, simp add: min_def) 
paulson@15656:   apply (rule_tac x ="length u" in exI, simp) 
paulson@15656:   apply (rule_tac x ="take i x" in exI) 
paulson@15656:   apply (rule_tac x ="x ! i" in exI) 
paulson@15656:   apply (rule_tac x ="y ! i" in exI, safe) 
paulson@15656:   apply (rule_tac x="drop (Suc i) x" in exI)
paulson@15656:   apply (drule sym, simp add: drop_Suc_conv_tl) 
paulson@15656:   apply (rule_tac x="drop (Suc i) y" in exI)
paulson@15656:   by (simp add: drop_Suc_conv_tl) 
paulson@15656: 
paulson@15656: -- {* lexord is extension of partial ordering List.lex *} 
paulson@15656: lemma lexord_lex: " (x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
paulson@15656:   apply (rule_tac x = y in spec) 
paulson@15656:   apply (induct_tac x, clarsimp) 
paulson@15656:   by (clarify, case_tac x, simp, force)
paulson@15656: 
paulson@15656: lemma lexord_irreflexive: "(! x. (x,x) \<notin> r) \<Longrightarrow> (y,y) \<notin> lexord r"
paulson@15656:   by (induct y, auto)
paulson@15656: 
paulson@15656: lemma lexord_trans: 
paulson@15656:     "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
paulson@15656:    apply (erule rev_mp)+
paulson@15656:    apply (rule_tac x = x in spec) 
paulson@15656:   apply (rule_tac x = z in spec) 
paulson@15656:   apply ( induct_tac y, simp, clarify)
paulson@15656:   apply (case_tac xa, erule ssubst) 
paulson@15656:   apply (erule allE, erule allE) -- {* avoid simp recursion *} 
paulson@15656:   apply (case_tac x, simp, simp) 
paulson@24632:   apply (case_tac x, erule allE, erule allE, simp)
paulson@15656:   apply (erule_tac x = listb in allE) 
paulson@15656:   apply (erule_tac x = lista in allE, simp)
paulson@15656:   apply (unfold trans_def)
paulson@15656:   by blast
paulson@15656: 
paulson@15656: lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
nipkow@24349: by (rule transI, drule lexord_trans, blast) 
paulson@15656: 
paulson@15656: 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"
paulson@15656:   apply (rule_tac x = y in spec) 
paulson@15656:   apply (induct_tac x, rule allI) 
paulson@15656:   apply (case_tac x, simp, simp) 
paulson@15656:   apply (rule allI, case_tac x, simp, simp) 
paulson@15656:   by blast
paulson@15656: 
paulson@15656: 
krauss@21103: subsection {* Lexicographic combination of measure functions *}
krauss@21103: 
krauss@21103: text {* These are useful for termination proofs *}
krauss@21103: 
krauss@21103: definition
krauss@21103:   "measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)"
krauss@21103: 
krauss@21106: lemma wf_measures[recdef_wf, simp]: "wf (measures fs)"
nipkow@24349: unfolding measures_def
nipkow@24349: by blast
krauss@21103: 
krauss@21103: lemma in_measures[simp]: 
krauss@21103:   "(x, y) \<in> measures [] = False"
krauss@21103:   "(x, y) \<in> measures (f # fs)
krauss@21103:          = (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))"  
nipkow@24349: unfolding measures_def
nipkow@24349: by auto
krauss@21103: 
krauss@21103: lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)"
nipkow@24349: by simp
krauss@21103: 
krauss@21103: lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)"
nipkow@24349: by auto
krauss@21103: 
krauss@21103: 
nipkow@15392: subsubsection{*Lifting a Relation on List Elements to the Lists*}
nipkow@15302: 
berghofe@23740: inductive_set
berghofe@23740:   listrel :: "('a * 'a)set => ('a list * 'a list)set"
berghofe@23740:   for r :: "('a * 'a)set"
berghofe@22262: where
berghofe@23740:     Nil:  "([],[]) \<in> listrel r"
berghofe@23740:   | Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
berghofe@23740: 
berghofe@23740: inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
berghofe@23740: inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
berghofe@23740: inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
berghofe@23740: inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
nipkow@15302: 
nipkow@15302: 
nipkow@15302: lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
nipkow@15302: apply clarify  
berghofe@23740: apply (erule listrel.induct)
berghofe@23740: apply (blast intro: listrel.intros)+
nipkow@15302: done
nipkow@15302: 
nipkow@15302: lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
nipkow@15302: apply clarify 
berghofe@23740: apply (erule listrel.induct, auto) 
nipkow@15302: done
nipkow@15302: 
nipkow@15302: lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" 
nipkow@15302: apply (simp add: refl_def listrel_subset Ball_def)
nipkow@15302: apply (rule allI) 
nipkow@15302: apply (induct_tac x) 
berghofe@23740: apply (auto intro: listrel.intros)
nipkow@15302: done
nipkow@15302: 
nipkow@15302: lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
nipkow@15302: apply (auto simp add: sym_def)
berghofe@23740: apply (erule listrel.induct) 
berghofe@23740: apply (blast intro: listrel.intros)+
nipkow@15302: done
nipkow@15302: 
nipkow@15302: lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
nipkow@15302: apply (simp add: trans_def)
nipkow@15302: apply (intro allI) 
nipkow@15302: apply (rule impI) 
berghofe@23740: apply (erule listrel.induct) 
berghofe@23740: apply (blast intro: listrel.intros)+
nipkow@15302: done
nipkow@15302: 
nipkow@15302: theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
nipkow@15302: by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) 
nipkow@15302: 
nipkow@15302: lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
berghofe@23740: by (blast intro: listrel.intros)
nipkow@15302: 
nipkow@15302: lemma listrel_Cons:
nipkow@15302:      "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})";
berghofe@23740: by (auto simp add: set_Cons_def intro: listrel.intros) 
nipkow@15302: 
nipkow@15302: 
nipkow@15392: subsection{*Miscellany*}
nipkow@15392: 
nipkow@15392: subsubsection {* Characters and strings *}
wenzelm@13366: 
wenzelm@13366: datatype nibble =
wenzelm@13366:     Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
wenzelm@13366:   | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
wenzelm@13366: 
haftmann@26148: lemma UNIV_nibble:
haftmann@26148:   "UNIV = {Nibble0, Nibble1, Nibble2, Nibble3, Nibble4, Nibble5, Nibble6, Nibble7,
haftmann@26148:     Nibble8, Nibble9, NibbleA, NibbleB, NibbleC, NibbleD, NibbleE, NibbleF}" (is "_ = ?A")
haftmann@26148: proof (rule UNIV_eq_I)
haftmann@26148:   fix x show "x \<in> ?A" by (cases x) simp_all
haftmann@26148: qed
haftmann@26148: 
haftmann@26148: instance nibble :: finite
haftmann@26148:   by default (simp add: UNIV_nibble)
haftmann@26148: 
wenzelm@13366: datatype char = Char nibble nibble
wenzelm@13366:   -- "Note: canonical order of character encoding coincides with standard term ordering"
wenzelm@13366: 
haftmann@26148: lemma UNIV_char:
haftmann@26148:   "UNIV = image (split Char) (UNIV \<times> UNIV)"
haftmann@26148: proof (rule UNIV_eq_I)
haftmann@26148:   fix x show "x \<in> image (split Char) (UNIV \<times> UNIV)" by (cases x) auto
haftmann@26148: qed
haftmann@26148: 
haftmann@26148: instance char :: finite
haftmann@26148:   by default (simp add: UNIV_char)
haftmann@26148: 
wenzelm@13366: types string = "char list"
wenzelm@13366: 
wenzelm@13366: syntax
wenzelm@13366:   "_Char" :: "xstr => char"    ("CHR _")
wenzelm@13366:   "_String" :: "xstr => string"    ("_")
wenzelm@13366: 
wenzelm@21754: setup StringSyntax.setup
wenzelm@13366: 
haftmann@20453: 
haftmann@21061: subsection {* Code generator *}
haftmann@21061: 
haftmann@21061: subsubsection {* Setup *}
berghofe@15064: 
berghofe@16770: types_code
berghofe@16770:   "list" ("_ list")
berghofe@16770: attach (term_of) {*
wenzelm@21760: fun term_of_list f T = HOLogic.mk_list T o map f;
berghofe@16770: *}
berghofe@16770: attach (test) {*
berghofe@25885: fun gen_list' aG aT i j = frequency
berghofe@25885:   [(i, fn () =>
berghofe@25885:       let
berghofe@25885:         val (x, t) = aG j;
berghofe@25885:         val (xs, ts) = gen_list' aG aT (i-1) j
berghofe@25885:       in (x :: xs, fn () => HOLogic.cons_const aT $ t () $ ts ()) end),
berghofe@25885:    (1, fn () => ([], fn () => HOLogic.nil_const aT))] ()
berghofe@25885: and gen_list aG aT i = gen_list' aG aT i i;
berghofe@16770: *}
berghofe@16770:   "char" ("string")
berghofe@16770: attach (term_of) {*
berghofe@24130: val term_of_char = HOLogic.mk_char o ord;
berghofe@16770: *}
berghofe@16770: attach (test) {*
berghofe@25885: fun gen_char i =
berghofe@25885:   let val j = random_range (ord "a") (Int.min (ord "a" + i, ord "z"))
berghofe@25885:   in (chr j, fn () => HOLogic.mk_char j) end;
berghofe@15064: *}
berghofe@15064: 
berghofe@15064: consts_code "Cons" ("(_ ::/ _)")
berghofe@15064: 
haftmann@20453: code_type list
haftmann@20453:   (SML "_ list")
haftmann@21911:   (OCaml "_ list")
haftmann@21113:   (Haskell "![_]")
haftmann@20453: 
haftmann@22799: code_reserved SML
haftmann@22799:   list
haftmann@22799: 
haftmann@22799: code_reserved OCaml
haftmann@22799:   list
haftmann@22799: 
haftmann@20453: code_const Nil
haftmann@21113:   (SML "[]")
haftmann@21911:   (OCaml "[]")
haftmann@21113:   (Haskell "[]")
haftmann@20453: 
haftmann@21911: setup {*
haftmann@24219:   fold (fn target => CodeTarget.add_pretty_list target
haftmann@22799:     @{const_name Nil} @{const_name Cons}
haftmann@22799:   ) ["SML", "OCaml", "Haskell"]
haftmann@21911: *}
haftmann@21911: 
haftmann@22799: code_instance list :: eq
haftmann@22799:   (Haskell -)
haftmann@20588: 
haftmann@21455: code_const "op = \<Colon> 'a\<Colon>eq list \<Rightarrow> 'a list \<Rightarrow> bool"
haftmann@20588:   (Haskell infixl 4 "==")
haftmann@20588: 
haftmann@20453: setup {*
haftmann@20453: let
haftmann@20453: 
haftmann@20453: fun list_codegen thy defs gr dep thyname b t =
berghofe@24902:   let
berghofe@24902:     val ts = HOLogic.dest_list t;
berghofe@24902:     val (gr', _) = Codegen.invoke_tycodegen thy defs dep thyname false
berghofe@24902:       (gr, fastype_of t);
berghofe@24902:     val (gr'', ps) = foldl_map
berghofe@24902:       (Codegen.invoke_codegen thy defs dep thyname false) (gr', ts)
berghofe@24902:   in SOME (gr'', Pretty.list "[" "]" ps) end handle TERM _ => NONE;
haftmann@20453: 
haftmann@20453: fun char_codegen thy defs gr dep thyname b t =
berghofe@24902:   let
berghofe@24902:     val i = HOLogic.dest_char t;
berghofe@24902:     val (gr', _) = Codegen.invoke_tycodegen thy defs dep thyname false
berghofe@24902:       (gr, fastype_of t)
berghofe@24902:   in SOME (gr', Pretty.str (ML_Syntax.print_string (chr i)))
berghofe@24902:   end handle TERM _ => NONE;
haftmann@20453: 
haftmann@20453: in
haftmann@20453:   Codegen.add_codegen "list_codegen" list_codegen
haftmann@20453:   #> Codegen.add_codegen "char_codegen" char_codegen
haftmann@20453: end;
haftmann@20453: *}
berghofe@15064: 
haftmann@21061: 
haftmann@21061: subsubsection {* Generation of efficient code *}
haftmann@21061: 
wenzelm@25221: primrec
haftmann@25559:   member :: "'a \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "mem" 55)
haftmann@25559: where 
haftmann@25559:   "x mem [] \<longleftrightarrow> False"
haftmann@25559:   | "x mem (y#ys) \<longleftrightarrow> (if y = x then True else x mem ys)"
haftmann@21061: 
haftmann@21061: primrec
haftmann@26442:   null:: "'a list \<Rightarrow> bool"
haftmann@26442: where
haftmann@21061:   "null [] = True"
haftmann@26442:   | "null (x#xs) = False"
haftmann@21061: 
haftmann@21061: primrec
haftmann@26442:   list_inter :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
haftmann@26442: where
haftmann@21061:   "list_inter [] bs = []"
haftmann@26442:   | "list_inter (a#as) bs =
haftmann@21061:      (if a \<in> set bs then a # list_inter as bs else list_inter as bs)"
haftmann@21061: 
haftmann@21061: primrec
haftmann@26442:   list_all :: "('a \<Rightarrow> bool) \<Rightarrow> ('a list \<Rightarrow> bool)"
haftmann@26442: where
haftmann@21061:   "list_all P [] = True"
haftmann@26442:   | "list_all P (x#xs) = (P x \<and> list_all P xs)"
haftmann@21061: 
haftmann@21061: primrec
haftmann@26442:   list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
haftmann@26442: where
haftmann@21061:   "list_ex P [] = False"
haftmann@26442:   | "list_ex P (x#xs) = (P x \<or> list_ex P xs)"
haftmann@21061: 
haftmann@21061: primrec
haftmann@26442:   filtermap :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list"
haftmann@26442: where
haftmann@21061:   "filtermap f [] = []"
haftmann@26442:   | "filtermap f (x#xs) =
haftmann@21061:      (case f x of None \<Rightarrow> filtermap f xs
haftmann@21061:       | Some y \<Rightarrow> y # filtermap f xs)"
haftmann@21061: 
haftmann@21061: primrec
haftmann@26442:   map_filter :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list"
haftmann@26442: where
haftmann@21061:   "map_filter f P [] = []"
haftmann@26442:   | "map_filter f P (x#xs) =
haftmann@21061:      (if P x then f x # map_filter f P xs else map_filter f P xs)"
haftmann@21061: 
haftmann@21061: text {*
wenzelm@21754:   Only use @{text mem} for generating executable code.  Otherwise use
wenzelm@21754:   @{prop "x : set xs"} instead --- it is much easier to reason about.
haftmann@21061:   The same is true for @{const list_all} and @{const list_ex}: write
haftmann@21061:   @{text "\<forall>x\<in>set xs"} and @{text "\<exists>x\<in>set xs"} instead because the HOL
wenzelm@21754:   quantifiers are aleady known to the automatic provers. In fact, the
wenzelm@21754:   declarations in the code subsection make sure that @{text "\<in>"},
wenzelm@21754:   @{text "\<forall>x\<in>set xs"} and @{text "\<exists>x\<in>set xs"} are implemented
wenzelm@21754:   efficiently.
haftmann@21061: 
haftmann@21061:   Efficient emptyness check is implemented by @{const null}.
haftmann@21061: 
haftmann@23060:   The functions @{const filtermap} and @{const map_filter} are just
haftmann@23060:   there to generate efficient code. Do not use
wenzelm@21754:   them for modelling and proving.
haftmann@21061: *}
haftmann@21061: 
haftmann@23060: lemma rev_foldl_cons [code]:
haftmann@23060:   "rev xs = foldl (\<lambda>xs x. x # xs) [] xs"
haftmann@23060: proof (induct xs)
haftmann@23060:   case Nil then show ?case by simp
haftmann@23060: next
haftmann@23060:   case Cons
haftmann@23060:   {
haftmann@23060:     fix x xs ys
haftmann@23060:     have "foldl (\<lambda>xs x. x # xs) ys xs @ [x]
haftmann@23060:       = foldl (\<lambda>xs x. x # xs) (ys @ [x]) xs"
haftmann@23060:     by (induct xs arbitrary: ys) auto
haftmann@23060:   }
haftmann@23060:   note aux = this
haftmann@23060:   show ?case by (induct xs) (auto simp add: Cons aux)
haftmann@23060: qed
haftmann@23060: 
haftmann@24166: lemma mem_iff [code post]:
haftmann@22422:   "x mem xs \<longleftrightarrow> x \<in> set xs"
nipkow@24349: by (induct xs) auto
haftmann@21061: 
haftmann@22799: lemmas in_set_code [code unfold] = mem_iff [symmetric]
haftmann@21061: 
haftmann@21061: lemma empty_null [code inline]:
haftmann@22422:   "xs = [] \<longleftrightarrow> null xs"
nipkow@24349: by (cases xs) simp_all
haftmann@21061: 
haftmann@24166: lemmas null_empty [code post] =
haftmann@21061:   empty_null [symmetric]
haftmann@21061: 
haftmann@21061: lemma list_inter_conv:
haftmann@21061:   "set (list_inter xs ys) = set xs \<inter> set ys"
nipkow@24349: by (induct xs) auto
haftmann@21061: 
haftmann@24166: lemma list_all_iff [code post]:
haftmann@22422:   "list_all P xs \<longleftrightarrow> (\<forall>x \<in> set xs. P x)"
nipkow@24349: by (induct xs) auto
haftmann@21061: 
haftmann@22799: lemmas list_ball_code [code unfold] = list_all_iff [symmetric]
haftmann@21061: 
haftmann@21061: lemma list_all_append [simp]:
haftmann@22422:   "list_all P (xs @ ys) \<longleftrightarrow> (list_all P xs \<and> list_all P ys)"
nipkow@24349: by (induct xs) auto
haftmann@21061: 
haftmann@21061: lemma list_all_rev [simp]:
haftmann@22422:   "list_all P (rev xs) \<longleftrightarrow> list_all P xs"
nipkow@24349: by (simp add: list_all_iff)
haftmann@21061: 
haftmann@22506: lemma list_all_length:
haftmann@22506:   "list_all P xs \<longleftrightarrow> (\<forall>n < length xs. P (xs ! n))"
haftmann@22506:   unfolding list_all_iff by (auto intro: all_nth_imp_all_set)
haftmann@22506: 
haftmann@24166: lemma list_ex_iff [code post]:
haftmann@22422:   "list_ex P xs \<longleftrightarrow> (\<exists>x \<in> set xs. P x)"
nipkow@24349: by (induct xs) simp_all
haftmann@21061: 
haftmann@21061: lemmas list_bex_code [code unfold] =
haftmann@22799:   list_ex_iff [symmetric]
haftmann@21061: 
haftmann@22506: lemma list_ex_length:
haftmann@22506:   "list_ex P xs \<longleftrightarrow> (\<exists>n < length xs. P (xs ! n))"
haftmann@22506:   unfolding list_ex_iff set_conv_nth by auto
haftmann@22506: 
haftmann@21061: lemma filtermap_conv:
haftmann@21061:    "filtermap f xs = map (\<lambda>x. the (f x)) (filter (\<lambda>x. f x \<noteq> None) xs)"
nipkow@24349: by (induct xs) (simp_all split: option.split) 
haftmann@21061: 
haftmann@21061: lemma map_filter_conv [simp]:
haftmann@21061:   "map_filter f P xs = map f (filter P xs)"
nipkow@24349: by (induct xs) auto
haftmann@21061: 
nipkow@24449: 
nipkow@24449: text {* Code for bounded quantification and summation over nats. *}
haftmann@21891: 
haftmann@22799: lemma atMost_upto [code unfold]:
nipkow@24645:   "{..n} = set [0..<Suc n]"
nipkow@24349: by auto
haftmann@22799: 
haftmann@22799: lemma atLeast_upt [code unfold]:
haftmann@21891:   "{..<n} = set [0..<n]"
nipkow@24349: by auto
haftmann@22799: 
nipkow@24449: lemma greaterThanLessThan_upt [code unfold]:
haftmann@21891:   "{n<..<m} = set [Suc n..<m]"
nipkow@24349: by auto
haftmann@22799: 
nipkow@24449: lemma atLeastLessThan_upt [code unfold]:
haftmann@21891:   "{n..<m} = set [n..<m]"
nipkow@24349: by auto
haftmann@22799: 
haftmann@22799: lemma greaterThanAtMost_upto [code unfold]:
nipkow@24645:   "{n<..m} = set [Suc n..<Suc m]"
nipkow@24349: by auto
haftmann@22799: 
haftmann@22799: lemma atLeastAtMost_upto [code unfold]:
nipkow@24645:   "{n..m} = set [n..<Suc m]"
nipkow@24349: by auto
haftmann@22799: 
haftmann@22799: lemma all_nat_less_eq [code unfold]:
haftmann@21891:   "(\<forall>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..<n}. P m)"
nipkow@24349: by auto
haftmann@22799: 
haftmann@22799: lemma ex_nat_less_eq [code unfold]:
haftmann@21891:   "(\<exists>m<n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..<n}. P m)"
nipkow@24349: by auto
haftmann@22799: 
haftmann@22799: lemma all_nat_less [code unfold]:
haftmann@21891:   "(\<forall>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<forall>m \<in> {0..n}. P m)"
nipkow@24349: by auto
haftmann@22799: 
haftmann@22799: lemma ex_nat_less [code unfold]:
haftmann@21891:   "(\<exists>m\<le>n\<Colon>nat. P m) \<longleftrightarrow> (\<exists>m \<in> {0..n}. P m)"
nipkow@24349: by auto
haftmann@22799: 
haftmann@26442: lemma setsum_set_upt_conv_listsum [code unfold]:
haftmann@26442:   "setsum f (set [k..<n]) = listsum (map f [k..<n])"
nipkow@24449: apply(subst atLeastLessThan_upt[symmetric])
nipkow@24449: by (induct n) simp_all
nipkow@24449: 
wenzelm@23388: end