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author | nipkow |

Mon, 13 May 2002 15:27:28 +0200 | |

changeset 13145 | 59bc43b51aa2 |

parent 13144 | c5ae1522fb82 |

child 13146 | f43153b63361 |

*** empty log message ***

--- a/src/HOL/Lex/RegExp2NA.ML Mon May 13 13:22:15 2002 +0200 +++ b/src/HOL/Lex/RegExp2NA.ML Mon May 13 15:27:28 2002 +0200 @@ -348,7 +348,7 @@ \ (? us v. w = concat us @ v & \ \ (!u:set us. accepts A u) & \ \ (start A,r) : steps A v)"; -by (rev_induct_tac "w" 1); +by (res_inst_tac [("xs","w")] rev_induct 1); by (Simp_tac 1); by (res_inst_tac [("x","[]")] exI 1); by (Simp_tac 1); @@ -368,7 +368,7 @@ Goal "us ~= [] --> (!u : set us. accepts A u) --> accepts (plus A) (concat us)"; by (simp_tac (simpset() addsimps [accepts_conv_steps]) 1); -by (rev_induct_tac "us" 1); +by (res_inst_tac [("xs","us")] rev_induct 1); by (Simp_tac 1); by (rename_tac "u us" 1); by (Simp_tac 1);

--- a/src/HOL/Lex/RegExp2NAe.ML Mon May 13 13:22:15 2002 +0200 +++ b/src/HOL/Lex/RegExp2NAe.ML Mon May 13 15:27:28 2002 +0200 @@ -500,7 +500,7 @@ \ (? us v. w = concat us @ v & \ \ (!u:set us. accepts A u) & \ \ (? r. (start A,r) : steps A v & rr = True#r))"; -by (rev_induct_tac "w" 1); +by (res_inst_tac [("xs","w")] rev_induct 1); by (Asm_full_simp_tac 1); by (Clarify_tac 1); by (res_inst_tac [("x","[]")] exI 1);

--- a/src/HOL/List.thy Mon May 13 13:22:15 2002 +0200 +++ b/src/HOL/List.thy Mon May 13 15:27:28 2002 +0200 @@ -1,7 +1,7 @@ -(* Title: HOL/List.thy - ID: $Id$ - Author: Tobias Nipkow - Copyright 1994 TU Muenchen +(*Title:HOL/List.thy +ID: $Id$ +Author: Tobias Nipkow +Copyright 1994 TU Muenchen *) header {* The datatype of finite lists *} @@ -9,171 +9,171 @@ theory List = PreList: datatype 'a list = - Nil ("[]") - | Cons 'a "'a list" (infixr "#" 65) +Nil("[]") +| Cons 'a"'a list"(infixr "#" 65) consts - "@" :: "'a list => 'a list => 'a list" (infixr 65) - filter :: "('a => bool) => 'a list => 'a list" - concat :: "'a list list => 'a list" - foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b" - foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b" - hd :: "'a list => 'a" - tl :: "'a list => 'a list" - last :: "'a list => 'a" - butlast :: "'a list => 'a list" - set :: "'a list => 'a set" - list_all :: "('a => bool) => ('a list => bool)" - list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" - map :: "('a=>'b) => ('a list => 'b list)" - mem :: "'a => 'a list => bool" (infixl 55) - nth :: "'a list => nat => 'a" (infixl "!" 100) - list_update :: "'a list => nat => 'a => 'a list" - take :: "nat => 'a list => 'a list" - drop :: "nat => 'a list => 'a list" - takeWhile :: "('a => bool) => 'a list => 'a list" - dropWhile :: "('a => bool) => 'a list => 'a list" - rev :: "'a list => 'a list" - zip :: "'a list => 'b list => ('a * 'b) list" - upt :: "nat => nat => nat list" ("(1[_../_'(])") - remdups :: "'a list => 'a list" - null :: "'a list => bool" - "distinct" :: "'a list => bool" - replicate :: "nat => 'a => 'a list" +"@" :: "'a list => 'a list => 'a list"(infixr 65) +filter:: "('a => bool) => 'a list => 'a list" +concat:: "'a list list => 'a list" +foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b" +foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b" +hd:: "'a list => 'a" +tl:: "'a list => 'a list" +last:: "'a list => 'a" +butlast :: "'a list => 'a list" +set :: "'a list => 'a set" +list_all:: "('a => bool) => ('a list => bool)" +list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" +map :: "('a=>'b) => ('a list => 'b list)" +mem :: "'a => 'a list => bool"(infixl 55) +nth :: "'a list => nat => 'a" (infixl "!" 100) +list_update :: "'a list => nat => 'a => 'a list" +take:: "nat => 'a list => 'a list" +drop:: "nat => 'a list => 'a list" +takeWhile :: "('a => bool) => 'a list => 'a list" +dropWhile :: "('a => bool) => 'a list => 'a list" +rev :: "'a list => 'a list" +zip :: "'a list => 'b list => ('a * 'b) list" +upt :: "nat => nat => nat list" ("(1[_../_'(])") +remdups :: "'a list => 'a list" +null:: "'a list => bool" +"distinct":: "'a list => bool" +replicate :: "nat => 'a => 'a list" nonterminals - lupdbinds lupdbind +lupdbindslupdbind syntax - -- {* list Enumeration *} - "@list" :: "args => 'a list" ("[(_)]") +-- {* list Enumeration *} +"@list" :: "args => 'a list"("[(_)]") - -- {* Special syntax for filter *} - "@filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_:_./ _])") +-- {* Special syntax for filter *} +"@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_:_./ _])") - -- {* list update *} - "_lupdbind" :: "['a, 'a] => lupdbind" ("(2_ :=/ _)") - "" :: "lupdbind => lupdbinds" ("_") - "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds" ("_,/ _") - "_LUpdate" :: "['a, lupdbinds] => 'a" ("_/[(_)]" [900,0] 900) +-- {* list update *} +"_lupdbind":: "['a, 'a] => lupdbind"("(2_ :=/ _)") +"" :: "lupdbind => lupdbinds" ("_") +"_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds" ("_,/ _") +"_LUpdate" :: "['a, lupdbinds] => 'a" ("_/[(_)]" [900,0] 900) - upto :: "nat => nat => nat list" ("(1[_../_])") +upto:: "nat => nat => nat list" ("(1[_../_])") translations - "[x, xs]" == "x#[xs]" - "[x]" == "x#[]" - "[x:xs . P]" == "filter (%x. P) xs" +"[x, xs]" == "x#[xs]" +"[x]" == "x#[]" +"[x:xs . P]"== "filter (%x. P) xs" - "_LUpdate xs (_lupdbinds b bs)" == "_LUpdate (_LUpdate xs b) bs" - "xs[i:=x]" == "list_update xs i x" +"_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs" +"xs[i:=x]" == "list_update xs i x" - "[i..j]" == "[i..(Suc j)(]" +"[i..j]" == "[i..(Suc j)(]" syntax (xsymbols) - "@filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_\<in>_ ./ _])") +"@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])") text {* - Function @{text size} is overloaded for all datatypes. Users may - refer to the list version as @{text length}. *} +Function @{text size} is overloaded for all datatypes.Users may +refer to the list version as @{text length}. *} syntax length :: "'a list => nat" translations "length" => "size :: _ list => nat" typed_print_translation {* - let - fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] = - Syntax.const "length" $ t - | size_tr' _ _ _ = raise Match; - in [("size", size_tr')] end +let +fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] = +Syntax.const "length" $ t +| size_tr' _ _ _ = raise Match; +in [("size", size_tr')] end *} primrec - "hd(x#xs) = x" +"hd(x#xs) = x" primrec - "tl([]) = []" - "tl(x#xs) = xs" +"tl([]) = []" +"tl(x#xs) = xs" primrec - "null([]) = True" - "null(x#xs) = False" +"null([]) = True" +"null(x#xs) = False" primrec - "last(x#xs) = (if xs=[] then x else last xs)" +"last(x#xs) = (if xs=[] then x else last xs)" primrec - "butlast [] = []" - "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)" +"butlast []= []" +"butlast(x#xs) = (if xs=[] then [] else x#butlast xs)" primrec - "x mem [] = False" - "x mem (y#ys) = (if y=x then True else x mem ys)" +"x mem [] = False" +"x mem (y#ys) = (if y=x then True else x mem ys)" primrec - "set [] = {}" - "set (x#xs) = insert x (set xs)" +"set [] = {}" +"set (x#xs) = insert x (set xs)" primrec - list_all_Nil: "list_all P [] = True" - list_all_Cons: "list_all P (x#xs) = (P(x) \<and> list_all P xs)" +list_all_Nil:"list_all P [] = True" +list_all_Cons: "list_all P (x#xs) = (P(x) \<and> list_all P xs)" primrec - "map f [] = []" - "map f (x#xs) = f(x)#map f xs" +"map f [] = []" +"map f (x#xs) = f(x)#map f xs" primrec - append_Nil: "[] @ys = ys" - append_Cons: "(x#xs)@ys = x#(xs@ys)" +append_Nil:"[]@ys = ys" +append_Cons: "(x#xs)@ys = x#(xs@ys)" primrec - "rev([]) = []" - "rev(x#xs) = rev(xs) @ [x]" +"rev([]) = []" +"rev(x#xs) = rev(xs) @ [x]" primrec - "filter P [] = []" - "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)" +"filter P [] = []" +"filter P (x#xs) = (if P x then x#filter P xs else filter P xs)" primrec - foldl_Nil: "foldl f a [] = a" - foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs" +foldl_Nil:"foldl f a [] = a" +foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs" primrec - "foldr f [] a = a" - "foldr f (x#xs) a = f x (foldr f xs a)" +"foldr f [] a = a" +"foldr f (x#xs) a = f x (foldr f xs a)" primrec - "concat([]) = []" - "concat(x#xs) = x @ concat(xs)" +"concat([]) = []" +"concat(x#xs) = x @ concat(xs)" primrec - drop_Nil: "drop n [] = []" - drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)" - -- {* Warning: simpset does not contain this definition *} - -- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} +drop_Nil:"drop n [] = []" +drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)" +-- {* Warning: simpset does not contain this definition *} +-- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} primrec - take_Nil: "take n [] = []" - take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)" - -- {* Warning: simpset does not contain this definition *} - -- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} +take_Nil:"take n [] = []" +take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)" +-- {* Warning: simpset does not contain this definition *} +-- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} primrec - nth_Cons: "(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)" - -- {* Warning: simpset does not contain this definition *} - -- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} +nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)" +-- {* Warning: simpset does not contain this definition *} +-- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} primrec - "[][i:=v] = []" - "(x#xs)[i:=v] = - (case i of 0 => v # xs - | Suc j => x # xs[j:=v])" +"[][i:=v] = []" +"(x#xs)[i:=v] = +(case i of 0 => v # xs +| Suc j => x # xs[j:=v])" primrec - "takeWhile P [] = []" - "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])" +"takeWhile P [] = []" +"takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])" primrec - "dropWhile P [] = []" - "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)" +"dropWhile P [] = []" +"dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)" primrec - "zip xs [] = []" - zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)" - -- {* Warning: simpset does not contain this definition *} - -- {* but separate theorems for @{text "xs = []"} and @{text "xs = z # zs"} *} +"zip xs [] = []" +zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)" +-- {* Warning: simpset does not contain this definition *} +-- {* but separate theorems for @{text "xs = []"} and @{text "xs = z # zs"} *} primrec - upt_0: "[i..0(] = []" - upt_Suc: "[i..(Suc j)(] = (if i <= j then [i..j(] @ [j] else [])" +upt_0: "[i..0(] = []" +upt_Suc: "[i..(Suc j)(] = (if i <= j then [i..j(] @ [j] else [])" primrec - "distinct [] = True" - "distinct (x#xs) = (x ~: set xs \<and> distinct xs)" +"distinct [] = True" +"distinct (x#xs) = (x ~: set xs \<and> distinct xs)" primrec - "remdups [] = []" - "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)" +"remdups [] = []" +"remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)" primrec - replicate_0: "replicate 0 x = []" - replicate_Suc: "replicate (Suc n) x = x # replicate n x" +replicate_0: "replicate0x = []" +replicate_Suc: "replicate (Suc n) x = x # replicate n x" defs list_all2_def: "list_all2 P xs ys == length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)" @@ -182,181 +182,181 @@ subsection {* Lexicographic orderings on lists *} consts - lexn :: "('a * 'a)set => nat => ('a list * 'a list)set" +lexn :: "('a * 'a)set => nat => ('a list * 'a list)set" primrec - "lexn r 0 = {}" - "lexn r (Suc n) = - (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int - {(xs,ys). length xs = Suc n \<and> length ys = Suc n}" +"lexn r 0 = {}" +"lexn r (Suc n) = +(prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int +{(xs,ys). length xs = Suc n \<and> length ys = Suc n}" constdefs - lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" - "lex r == \<Union>n. lexn r n" +lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" +"lex r == \<Union>n. lexn r n" - lexico :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" - "lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))" +lexico :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" +"lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))" - sublist :: "'a list => nat set => 'a list" - "sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..size xs(]))" +sublist :: "'a list => nat set => 'a list" +"sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..size xs(]))" lemma not_Cons_self [simp]: "xs \<noteq> x # xs" - by (induct xs) auto +by (induct xs) auto lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric] lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)" - by (induct xs) auto +by (induct xs) auto lemma length_induct: - "(!!xs. \<forall>ys. length ys < length xs --> P ys ==> P xs) ==> P xs" - by (rule measure_induct [of length]) rules +"(!!xs. \<forall>ys. length ys < length xs --> P ys ==> P xs) ==> P xs" +by (rule measure_induct [of length]) rules subsection {* @{text lists}: the list-forming operator over sets *} consts lists :: "'a set => 'a list set" inductive "lists A" - intros - Nil [intro!]: "[]: lists A" - Cons [intro!]: "[| a: A; l: lists A |] ==> a#l : lists A" +intros +Nil [intro!]: "[]: lists A" +Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A" inductive_cases listsE [elim!]: "x#l : lists A" lemma lists_mono: "A \<subseteq> B ==> lists A \<subseteq> lists B" - by (unfold lists.defs) (blast intro!: lfp_mono) +by (unfold lists.defs) (blast intro!: lfp_mono) lemma lists_IntI [rule_format]: - "l: lists A ==> l: lists B --> l: lists (A Int B)" - apply (erule lists.induct) - apply blast+ - done +"l: lists A ==> l: lists B --> l: lists (A Int B)" +apply (erule lists.induct) +apply blast+ +done lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B" - apply (rule mono_Int [THEN equalityI]) - apply (simp add: mono_def lists_mono) - apply (blast intro!: lists_IntI) - done +apply (rule mono_Int [THEN equalityI]) +apply (simp add: mono_def lists_mono) +apply (blast intro!: lists_IntI) +done lemma append_in_lists_conv [iff]: - "(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)" - by (induct xs) auto +"(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)" +by (induct xs) auto subsection {* @{text length} *} text {* - Needs to come before @{text "@"} because of theorem @{text - append_eq_append_conv}. +Needs to come before @{text "@"} because of theorem @{text +append_eq_append_conv}. *} lemma length_append [simp]: "length (xs @ ys) = length xs + length ys" - by (induct xs) auto +by (induct xs) auto lemma length_map [simp]: "length (map f xs) = length xs" - by (induct xs) auto +by (induct xs) auto lemma length_rev [simp]: "length (rev xs) = length xs" - by (induct xs) auto +by (induct xs) auto lemma length_tl [simp]: "length (tl xs) = length xs - 1" - by (cases xs) auto +by (cases xs) auto lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])" - by (induct xs) auto +by (induct xs) auto lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])" - by (induct xs) auto +by (induct xs) auto lemma length_Suc_conv: - "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)" - by (induct xs) auto +"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)" +by (induct xs) auto subsection {* @{text "@"} -- append *} lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)" - by (induct xs) auto +by (induct xs) auto lemma append_Nil2 [simp]: "xs @ [] = xs" - by (induct xs) auto +by (induct xs) auto lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])" - by (induct xs) auto +by (induct xs) auto lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])" - by (induct xs) auto +by (induct xs) auto lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])" - by (induct xs) auto +by (induct xs) auto lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])" - by (induct xs) auto +by (induct xs) auto lemma append_eq_append_conv [rule_format, simp]: "\<forall>ys. length xs = length ys \<or> length us = length vs - --> (xs@us = ys@vs) = (xs=ys \<and> us=vs)" - apply (induct_tac xs) - apply(rule allI) - apply (case_tac ys) - apply simp - apply force - apply (rule allI) - apply (case_tac ys) - apply force - apply simp - done + --> (xs@us = ys@vs) = (xs=ys \<and> us=vs)" +apply (induct_tac xs) + apply(rule allI) + apply (case_tac ys) +apply simp + apply force +apply (rule allI) +apply (case_tac ys) + apply force +apply simp +done lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)" - by simp +by simp lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)" - by simp +by simp lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)" - by simp +by simp lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])" - using append_same_eq [of _ _ "[]"] by auto +using append_same_eq [of _ _ "[]"] by auto lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])" - using append_same_eq [of "[]"] by auto +using append_same_eq [of "[]"] by auto lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs" - by (induct xs) auto +by (induct xs) auto lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)" - by (induct xs) auto +by (induct xs) auto lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs" - by (simp add: hd_append split: list.split) +by (simp add: hd_append split: list.split) lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)" - by (simp split: list.split) +by (simp split: list.split) lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys" - by (simp add: tl_append split: list.split) +by (simp add: tl_append split: list.split) text {* Trivial rules for solving @{text "@"}-equations automatically. *} lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys" - by simp +by simp lemma Cons_eq_appendI: - "[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs" - by (drule sym) simp +"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs" +by (drule sym) simp lemma append_eq_appendI: - "[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us" - by (drule sym) simp +"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us" +by (drule sym) simp text {* - Simplification procedure for all list equalities. - Currently only tries to rearrange @{text "@"} to see if - - both lists end in a singleton list, - - or both lists end in the same list. +Simplification procedure for all list equalities. +Currently only tries to rearrange @{text "@"} to see if +- both lists end in a singleton list, +- or both lists end in the same list. *} ML_setup {* @@ -369,47 +369,47 @@ val append_same_eq = thm "append_same_eq"; val list_eq_pattern = - Thm.read_cterm (Theory.sign_of (the_context ())) ("(xs::'a list) = ys",HOLogic.boolT) +Thm.read_cterm (Theory.sign_of (the_context ())) ("(xs::'a list) = ys",HOLogic.boolT) fun last (cons as Const("List.list.Cons",_) $ _ $ xs) = - (case xs of Const("List.list.Nil",_) => cons | _ => last xs) - | last (Const("List.op @",_) $ _ $ ys) = last ys - | last t = t +(case xs of Const("List.list.Nil",_) => cons | _ => last xs) +| last (Const("List.op @",_) $ _ $ ys) = last ys +| last t = t fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true - | list1 _ = false +| list1 _ = false fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) = - (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs) - | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys - | butlast xs = Const("List.list.Nil",fastype_of xs) +(case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs) +| butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys +| butlast xs = Const("List.list.Nil",fastype_of xs) val rearr_tac = - simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons]) +simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons]) fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) = - let - val lastl = last lhs and lastr = last rhs - fun rearr conv = - let val lhs1 = butlast lhs and rhs1 = butlast rhs - val Type(_,listT::_) = eqT - val appT = [listT,listT] ---> listT - val app = Const("List.op @",appT) - val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr) - val ct = cterm_of sg (HOLogic.mk_Trueprop(HOLogic.mk_eq(F,F2))) - val thm = prove_goalw_cterm [] ct (K [rearr_tac 1]) - handle ERROR => - error("The error(s) above occurred while trying to prove " ^ - string_of_cterm ct) - in Some((conv RS (thm RS trans)) RS eq_reflection) end +let +val lastl = last lhs and lastr = last rhs +fun rearr conv = +let val lhs1 = butlast lhs and rhs1 = butlast rhs +val Type(_,listT::_) = eqT +val appT = [listT,listT] ---> listT +val app = Const("List.op @",appT) +val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr) +val ct = cterm_of sg (HOLogic.mk_Trueprop(HOLogic.mk_eq(F,F2))) +val thm = prove_goalw_cterm [] ct (K [rearr_tac 1]) +handle ERROR => +error("The error(s) above occurred while trying to prove " ^ +string_of_cterm ct) +in Some((conv RS (thm RS trans)) RS eq_reflection) end - in if list1 lastl andalso list1 lastr - then rearr append1_eq_conv - else - if lastl aconv lastr - then rearr append_same_eq - else None - end +in if list1 lastl andalso list1 lastr + then rearr append1_eq_conv + else + if lastl aconv lastr + then rearr append_same_eq + else None +end in val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq end; @@ -421,944 +421,945 @@ subsection {* @{text map} *} lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs" - by (induct xs) simp_all +by (induct xs) simp_all lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)" - by (rule ext, induct_tac xs) auto +by (rule ext, induct_tac xs) auto lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys" - by (induct xs) auto +by (induct xs) auto lemma map_compose: "map (f o g) xs = map f (map g xs)" - by (induct xs) (auto simp add: o_def) +by (induct xs) (auto simp add: o_def) lemma rev_map: "rev (map f xs) = map f (rev xs)" - by (induct xs) auto +by (induct xs) auto lemma map_cong: - "xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys" - -- {* a congruence rule for @{text map} *} - by (clarify, induct ys) auto +"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys" +-- {* a congruence rule for @{text map} *} +by (clarify, induct ys) auto lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])" - by (cases xs) auto +by (cases xs) auto lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])" - by (cases xs) auto +by (cases xs) auto lemma map_eq_Cons: - "(map f xs = y # ys) = (\<exists>x xs'. xs = x # xs' \<and> f x = y \<and> map f xs' = ys)" - by (cases xs) auto +"(map f xs = y # ys) = (\<exists>x xs'. xs = x # xs' \<and> f x = y \<and> map f xs' = ys)" +by (cases xs) auto lemma map_injective: - "!!xs. map f xs = map f ys ==> (\<forall>x y. f x = f y --> x = y) ==> xs = ys" - by (induct ys) (auto simp add: map_eq_Cons) +"!!xs. map f xs = map f ys ==> (\<forall>x y. f x = f y --> x = y) ==> xs = ys" +by (induct ys) (auto simp add: map_eq_Cons) lemma inj_mapI: "inj f ==> inj (map f)" - by (rules dest: map_injective injD intro: injI) +by (rules dest: map_injective injD intro: injI) lemma inj_mapD: "inj (map f) ==> inj f" - apply (unfold inj_on_def) - apply clarify - apply (erule_tac x = "[x]" in ballE) - apply (erule_tac x = "[y]" in ballE) - apply simp - apply blast - apply blast - done +apply (unfold inj_on_def) +apply clarify +apply (erule_tac x = "[x]" in ballE) + apply (erule_tac x = "[y]" in ballE) +apply simp + apply blast +apply blast +done lemma inj_map: "inj (map f) = inj f" - by (blast dest: inj_mapD intro: inj_mapI) +by (blast dest: inj_mapD intro: inj_mapI) subsection {* @{text rev} *} lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs" - by (induct xs) auto +by (induct xs) auto lemma rev_rev_ident [simp]: "rev (rev xs) = xs" - by (induct xs) auto +by (induct xs) auto lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])" - by (induct xs) auto +by (induct xs) auto lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])" - by (induct xs) auto +by (induct xs) auto lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)" - apply (induct xs) - apply force - apply (case_tac ys) - apply simp - apply force - done +apply (induct xs) + apply force +apply (case_tac ys) + apply simp +apply force +done lemma rev_induct: "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs" - apply(subst rev_rev_ident[symmetric]) - apply(rule_tac list = "rev xs" in list.induct, simp_all) - done +apply(subst rev_rev_ident[symmetric]) +apply(rule_tac list = "rev xs" in list.induct, simp_all) +done -ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *} -- "compatibility" +ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility" -lemma rev_exhaust: "(xs = [] ==> P) ==> (!!ys y. xs = ys @ [y] ==> P) ==> P" - by (induct xs rule: rev_induct) auto +lemma rev_exhaust: "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P" +by (induct xs rule: rev_induct) auto subsection {* @{text set} *} lemma finite_set [iff]: "finite (set xs)" - by (induct xs) auto +by (induct xs) auto lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)" - by (induct xs) auto +by (induct xs) auto lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)" - by auto +by auto lemma set_empty [iff]: "(set xs = {}) = (xs = [])" - by (induct xs) auto +by (induct xs) auto lemma set_rev [simp]: "set (rev xs) = set xs" - by (induct xs) auto +by (induct xs) auto lemma set_map [simp]: "set (map f xs) = f`(set xs)" - by (induct xs) auto +by (induct xs) auto lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}" - by (induct xs) auto +by (induct xs) auto lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> k < j}" - apply (induct j) - apply simp_all - apply(erule ssubst) - apply auto - apply arith - done +apply (induct j) + apply simp_all +apply(erule ssubst) +apply auto +apply arith +done lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)" - apply (induct xs) - apply simp - apply simp - apply (rule iffI) - apply (blast intro: eq_Nil_appendI Cons_eq_appendI) - apply (erule exE)+ - apply (case_tac ys) - apply auto - done +apply (induct xs) + apply simp +apply simp +apply (rule iffI) + apply (blast intro: eq_Nil_appendI Cons_eq_appendI) +apply (erule exE)+ +apply (case_tac ys) +apply auto +done lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)" - -- {* eliminate @{text lists} in favour of @{text set} *} - by (induct xs) auto +-- {* eliminate @{text lists} in favour of @{text set} *} +by (induct xs) auto lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A" - by (rule in_lists_conv_set [THEN iffD1]) +by (rule in_lists_conv_set [THEN iffD1]) lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A" - by (rule in_lists_conv_set [THEN iffD2]) +by (rule in_lists_conv_set [THEN iffD2]) subsection {* @{text mem} *} lemma set_mem_eq: "(x mem xs) = (x : set xs)" - by (induct xs) auto +by (induct xs) auto subsection {* @{text list_all} *} lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)" - by (induct xs) auto +by (induct xs) auto lemma list_all_append [simp]: - "list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)" - by (induct xs) auto +"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)" +by (induct xs) auto subsection {* @{text filter} *} lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys" - by (induct xs) auto +by (induct xs) auto lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs" - by (induct xs) auto +by (induct xs) auto lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs" - by (induct xs) auto +by (induct xs) auto lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []" - by (induct xs) auto +by (induct xs) auto lemma length_filter [simp]: "length (filter P xs) \<le> length xs" - by (induct xs) (auto simp add: le_SucI) +by (induct xs) (auto simp add: le_SucI) lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs" - by auto +by auto subsection {* @{text concat} *} lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys" - by (induct xs) auto +by (induct xs) auto lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])" - by (induct xss) auto +by (induct xss) auto lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])" - by (induct xss) auto +by (induct xss) auto lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)" - by (induct xs) auto +by (induct xs) auto lemma map_concat: "map f (concat xs) = concat (map (map f) xs)" - by (induct xs) auto +by (induct xs) auto lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)" - by (induct xs) auto +by (induct xs) auto lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))" - by (induct xs) auto +by (induct xs) auto subsection {* @{text nth} *} lemma nth_Cons_0 [simp]: "(x # xs)!0 = x" - by auto +by auto lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n" - by auto +by auto declare nth.simps [simp del] lemma nth_append: - "!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))" - apply(induct "xs") - apply simp - apply (case_tac n) - apply auto - done +"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))" +apply(induct "xs") + apply simp +apply (case_tac n) + apply auto +done lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)" - apply(induct xs) - apply simp - apply (case_tac n) - apply auto - done +apply(induct xs) + apply simp +apply (case_tac n) + apply auto +done lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}" - apply (induct_tac xs) - apply simp - apply simp - apply safe - apply (rule_tac x = 0 in exI) - apply simp - apply (rule_tac x = "Suc i" in exI) - apply simp - apply (case_tac i) - apply simp - apply (rename_tac j) - apply (rule_tac x = j in exI) - apply simp - done +apply (induct_tac xs) + apply simp +apply simp +apply safe +apply (rule_tac x = 0 in exI) +apply simp + apply (rule_tac x = "Suc i" in exI) + apply simp +apply (case_tac i) + apply simp +apply (rename_tac j) +apply (rule_tac x = j in exI) +apply simp +done -lemma list_ball_nth: "[| n < length xs; !x : set xs. P x |] ==> P(xs!n)" - by (auto simp add: set_conv_nth) +lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)" +by (auto simp add: set_conv_nth) lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs" - by (auto simp add: set_conv_nth) +by (auto simp add: set_conv_nth) lemma all_nth_imp_all_set: - "[| !i < length xs. P(xs!i); x : set xs |] ==> P x" - by (auto simp add: set_conv_nth) +"[| !i < length xs. P(xs!i); x : set xs|] ==> P x" +by (auto simp add: set_conv_nth) lemma all_set_conv_all_nth: - "(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))" - by (auto simp add: set_conv_nth) +"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))" +by (auto simp add: set_conv_nth) subsection {* @{text list_update} *} lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs" - by (induct xs) (auto split: nat.split) +by (induct xs) (auto split: nat.split) lemma nth_list_update: - "!!i j. i < length xs ==> (xs[i:=x])!j = (if i = j then x else xs!j)" - by (induct xs) (auto simp add: nth_Cons split: nat.split) +"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)" +by (induct xs) (auto simp add: nth_Cons split: nat.split) lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x" - by (simp add: nth_list_update) +by (simp add: nth_list_update) lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j" - by (induct xs) (auto simp add: nth_Cons split: nat.split) +by (induct xs) (auto simp add: nth_Cons split: nat.split) lemma list_update_overwrite [simp]: - "!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]" - by (induct xs) (auto split: nat.split) +"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]" +by (induct xs) (auto split: nat.split) lemma list_update_same_conv: - "!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)" - by (induct xs) (auto split: nat.split) +"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)" +by (induct xs) (auto split: nat.split) lemma update_zip: - "!!i xy xs. length xs = length ys ==> - (zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])" - by (induct ys) (auto, case_tac xs, auto split: nat.split) +"!!i xy xs. length xs = length ys ==> +(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])" +by (induct ys) (auto, case_tac xs, auto split: nat.split) lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)" - by (induct xs) (auto split: nat.split) +by (induct xs) (auto split: nat.split) lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A" - by (blast dest!: set_update_subset_insert [THEN subsetD]) +by (blast dest!: set_update_subset_insert [THEN subsetD]) subsection {* @{text last} and @{text butlast} *} lemma last_snoc [simp]: "last (xs @ [x]) = x" - by (induct xs) auto +by (induct xs) auto lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs" - by (induct xs) auto +by (induct xs) auto lemma length_butlast [simp]: "length (butlast xs) = length xs - 1" - by (induct xs rule: rev_induct) auto +by (induct xs rule: rev_induct) auto lemma butlast_append: - "!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)" - by (induct xs) auto +"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)" +by (induct xs) auto lemma append_butlast_last_id [simp]: - "xs \<noteq> [] ==> butlast xs @ [last xs] = xs" - by (induct xs) auto +"xs \<noteq> [] ==> butlast xs @ [last xs] = xs" +by (induct xs) auto lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs" - by (induct xs) (auto split: split_if_asm) +by (induct xs) (auto split: split_if_asm) lemma in_set_butlast_appendI: - "x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))" - by (auto dest: in_set_butlastD simp add: butlast_append) +"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))" +by (auto dest: in_set_butlastD simp add: butlast_append) subsection {* @{text take} and @{text drop} *} lemma take_0 [simp]: "take 0 xs = []" - by (induct xs) auto +by (induct xs) auto lemma drop_0 [simp]: "drop 0 xs = xs" - by (induct xs) auto +by (induct xs) auto lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs" - by simp +by simp lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs" - by simp +by simp declare take_Cons [simp del] and drop_Cons [simp del] lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n" - by (induct n) (auto, case_tac xs, auto) +by (induct n) (auto, case_tac xs, auto) lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)" - by (induct n) (auto, case_tac xs, auto) +by (induct n) (auto, case_tac xs, auto) lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs" - by (induct n) (auto, case_tac xs, auto) +by (induct n) (auto, case_tac xs, auto) lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []" - by (induct n) (auto, case_tac xs, auto) +by (induct n) (auto, case_tac xs, auto) lemma take_append [simp]: - "!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)" - by (induct n) (auto, case_tac xs, auto) +"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)" +by (induct n) (auto, case_tac xs, auto) lemma drop_append [simp]: - "!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys" - by (induct n) (auto, case_tac xs, auto) +"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys" +by (induct n) (auto, case_tac xs, auto) lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs" - apply (induct m) - apply auto - apply (case_tac xs) - apply auto - apply (case_tac na) - apply auto - done +apply (induct m) + apply auto +apply (case_tac xs) + apply auto +apply (case_tac na) + apply auto +done lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs" - apply (induct m) - apply auto - apply (case_tac xs) - apply auto - done +apply (induct m) + apply auto +apply (case_tac xs) + apply auto +done lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)" - apply (induct m) - apply auto - apply (case_tac xs) - apply auto - done +apply (induct m) + apply auto +apply (case_tac xs) + apply auto +done lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs" - apply (induct n) - apply auto - apply (case_tac xs) - apply auto - done +apply (induct n) + apply auto +apply (case_tac xs) + apply auto +done lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)" - apply (induct n) - apply auto - apply (case_tac xs) - apply auto - done +apply (induct n) + apply auto +apply (case_tac xs) + apply auto +done lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)" - apply (induct n) - apply auto - apply (case_tac xs) - apply auto - done +apply (induct n) + apply auto +apply (case_tac xs) + apply auto +done lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)" - apply (induct xs) - apply auto - apply (case_tac i) - apply auto - done +apply (induct xs) + apply auto +apply (case_tac i) + apply auto +done lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)" - apply (induct xs) - apply auto - apply (case_tac i) - apply auto - done +apply (induct xs) + apply auto +apply (case_tac i) + apply auto +done lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i" - apply (induct xs) - apply auto - apply (case_tac n) - apply(blast ) - apply (case_tac i) - apply auto - done +apply (induct xs) + apply auto +apply (case_tac n) + apply(blast ) +apply (case_tac i) + apply auto +done lemma nth_drop [simp]: - "!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)" - apply (induct n) - apply auto - apply (case_tac xs) - apply auto - done +"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)" +apply (induct n) + apply auto +apply (case_tac xs) + apply auto +done lemma append_eq_conv_conj: - "!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)" - apply(induct xs) - apply simp - apply clarsimp - apply (case_tac zs) - apply auto - done +"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)" +apply(induct xs) + apply simp +apply clarsimp +apply (case_tac zs) +apply auto +done subsection {* @{text takeWhile} and @{text dropWhile} *} lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs" - by (induct xs) auto +by (induct xs) auto lemma takeWhile_append1 [simp]: - "[| x:set xs; ~P(x) |] ==> takeWhile P (xs @ ys) = takeWhile P xs" - by (induct xs) auto +"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs" +by (induct xs) auto lemma takeWhile_append2 [simp]: - "(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys" - by (induct xs) auto +"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys" +by (induct xs) auto lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs" - by (induct xs) auto +by (induct xs) auto lemma dropWhile_append1 [simp]: - "[| x : set xs; ~P(x) |] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys" - by (induct xs) auto +"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys" +by (induct xs) auto lemma dropWhile_append2 [simp]: - "(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys" - by (induct xs) auto +"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys" +by (induct xs) auto lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x" - by (induct xs) (auto split: split_if_asm) +by (induct xs) (auto split: split_if_asm) subsection {* @{text zip} *} lemma zip_Nil [simp]: "zip [] ys = []" - by (induct ys) auto +by (induct ys) auto lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys" - by simp +by simp declare zip_Cons [simp del] lemma length_zip [simp]: - "!!xs. length (zip xs ys) = min (length xs) (length ys)" - apply(induct ys) - apply simp - apply (case_tac xs) - apply auto - done +"!!xs. length (zip xs ys) = min (length xs) (length ys)" +apply(induct ys) + apply simp +apply (case_tac xs) + apply auto +done lemma zip_append1: - "!!xs. zip (xs @ ys) zs = - zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)" - apply (induct zs) - apply simp - apply (case_tac xs) - apply simp_all - done +"!!xs. zip (xs @ ys) zs = +zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)" +apply (induct zs) + apply simp +apply (case_tac xs) + apply simp_all +done lemma zip_append2: - "!!ys. zip xs (ys @ zs) = - zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs" - apply (induct xs) - apply simp - apply (case_tac ys) - apply simp_all - done +"!!ys. zip xs (ys @ zs) = +zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs" +apply (induct xs) + apply simp +apply (case_tac ys) + apply simp_all +done lemma zip_append [simp]: "[| length xs = length us; length ys = length vs |] ==> - zip (xs@ys) (us@vs) = zip xs us @ zip ys vs" - by (simp add: zip_append1) +zip (xs@ys) (us@vs) = zip xs us @ zip ys vs" +by (simp add: zip_append1) lemma zip_rev: - "!!xs. length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)" - apply(induct ys) - apply simp - apply (case_tac xs) - apply simp_all - done +"!!xs. length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)" +apply(induct ys) + apply simp +apply (case_tac xs) + apply simp_all +done lemma nth_zip [simp]: - "!!i xs. [| i < length xs; i < length ys |] ==> (zip xs ys)!i = (xs!i, ys!i)" - apply (induct ys) - apply simp - apply (case_tac xs) - apply (simp_all add: nth.simps split: nat.split) - done +"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)" +apply (induct ys) + apply simp +apply (case_tac xs) + apply (simp_all add: nth.simps split: nat.split) +done lemma set_zip: - "set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}" - by (simp add: set_conv_nth cong: rev_conj_cong) +"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}" +by (simp add: set_conv_nth cong: rev_conj_cong) lemma zip_update: - "length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]" - by (rule sym, simp add: update_zip) +"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]" +by (rule sym, simp add: update_zip) lemma zip_replicate [simp]: - "!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)" - apply (induct i) - apply auto - apply (case_tac j) - apply auto - done +"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)" +apply (induct i) + apply auto +apply (case_tac j) + apply auto +done subsection {* @{text list_all2} *} lemma list_all2_lengthD: "list_all2 P xs ys ==> length xs = length ys" - by (simp add: list_all2_def) +by (simp add: list_all2_def) lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])" - by (simp add: list_all2_def) +by (simp add: list_all2_def) lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])" - by (simp add: list_all2_def) +by (simp add: list_all2_def) lemma list_all2_Cons [iff]: - "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)" - by (auto simp add: list_all2_def) +"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)" +by (auto simp add: list_all2_def) lemma list_all2_Cons1: - "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)" - by (cases ys) auto +"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)" +by (cases ys) auto lemma list_all2_Cons2: - "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)" - by (cases xs) auto +"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)" +by (cases xs) auto lemma list_all2_rev [iff]: - "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys" - by (simp add: list_all2_def zip_rev cong: conj_cong) +"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys" +by (simp add: list_all2_def zip_rev cong: conj_cong) lemma list_all2_append1: - "list_all2 P (xs @ ys) zs = - (EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and> - list_all2 P xs us \<and> list_all2 P ys vs)" - apply (simp add: list_all2_def zip_append1) - apply (rule iffI) - apply (rule_tac x = "take (length xs) zs" in exI) - apply (rule_tac x = "drop (length xs) zs" in exI) - apply (force split: nat_diff_split simp add: min_def) - apply clarify - apply (simp add: ball_Un) - done +"list_all2 P (xs @ ys) zs = +(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and> +list_all2 P xs us \<and> list_all2 P ys vs)" +apply (simp add: list_all2_def zip_append1) +apply (rule iffI) + apply (rule_tac x = "take (length xs) zs" in exI) + apply (rule_tac x = "drop (length xs) zs" in exI) + apply (force split: nat_diff_split simp add: min_def) +apply clarify +apply (simp add: ball_Un) +done lemma list_all2_append2: - "list_all2 P xs (ys @ zs) = - (EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and> - list_all2 P us ys \<and> list_all2 P vs zs)" - apply (simp add: list_all2_def zip_append2) - apply (rule iffI) - apply (rule_tac x = "take (length ys) xs" in exI) - apply (rule_tac x = "drop (length ys) xs" in exI) - apply (force split: nat_diff_split simp add: min_def) - apply clarify - apply (simp add: ball_Un) - done +"list_all2 P xs (ys @ zs) = +(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and> +list_all2 P us ys \<and> list_all2 P vs zs)" +apply (simp add: list_all2_def zip_append2) +apply (rule iffI) + apply (rule_tac x = "take (length ys) xs" in exI) + apply (rule_tac x = "drop (length ys) xs" in exI) + apply (force split: nat_diff_split simp add: min_def) +apply clarify +apply (simp add: ball_Un) +done lemma list_all2_conv_all_nth: - "list_all2 P xs ys = - (length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))" - by (force simp add: list_all2_def set_zip) +"list_all2 P xs ys = +(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))" +by (force simp add: list_all2_def set_zip) lemma list_all2_trans[rule_format]: - "\<forall>a b c. P1 a b --> P2 b c --> P3 a c ==> - \<forall>bs cs. list_all2 P1 as bs --> list_all2 P2 bs cs --> list_all2 P3 as cs" - apply(induct_tac as) - apply simp - apply(rule allI) - apply(induct_tac bs) - apply simp - apply(rule allI) - apply(induct_tac cs) - apply auto - done +"\<forall>a b c. P1 a b --> P2 b c --> P3 a c ==> +\<forall>bs cs. list_all2 P1 as bs --> list_all2 P2 bs cs --> list_all2 P3 as cs" +apply(induct_tac as) + apply simp +apply(rule allI) +apply(induct_tac bs) + apply simp +apply(rule allI) +apply(induct_tac cs) + apply auto +done subsection {* @{text foldl} *} lemma foldl_append [simp]: - "!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys" - by (induct xs) auto +"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys" +by (induct xs) auto text {* - Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more - difficult to use because it requires an additional transitivity step. +Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more +difficult to use because it requires an additional transitivity step. *} lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns" - by (induct ns) auto +by (induct ns) auto lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns" - by (force intro: start_le_sum simp add: in_set_conv_decomp) +by (force intro: start_le_sum simp add: in_set_conv_decomp) lemma sum_eq_0_conv [iff]: - "!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))" - by (induct ns) auto +"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))" +by (induct ns) auto subsection {* @{text upto} *} lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])" - -- {* Does not terminate! *} - by (induct j) auto +-- {* Does not terminate! *} +by (induct j) auto lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []" - by (subst upt_rec) simp +by (subst upt_rec) simp lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]" - -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *} - by simp +-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *} +by simp lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]" - apply(rule trans) - apply(subst upt_rec) - prefer 2 apply(rule refl) - apply simp - done +apply(rule trans) +apply(subst upt_rec) + prefer 2 apply(rule refl) +apply simp +done lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]" - -- {* LOOPS as a simprule, since @{text "j <= j"}. *} - by (induct k) auto +-- {* LOOPS as a simprule, since @{text "j <= j"}. *} +by (induct k) auto lemma length_upt [simp]: "length [i..j(] = j - i" - by (induct j) (auto simp add: Suc_diff_le) +by (induct j) (auto simp add: Suc_diff_le) lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k" - apply (induct j) - apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split) - done +apply (induct j) +apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split) +done lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]" - apply (induct m) - apply simp - apply (subst upt_rec) - apply (rule sym) - apply (subst upt_rec) - apply (simp del: upt.simps) - done +apply (induct m) + apply simp +apply (subst upt_rec) +apply (rule sym) +apply (subst upt_rec) +apply (simp del: upt.simps) +done lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]" - by (induct n) auto +by (induct n) auto lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..n(]) ! i = f(m+i)" - apply (induct n m rule: diff_induct) - prefer 3 apply (subst map_Suc_upt[symmetric]) - apply (auto simp add: less_diff_conv nth_upt) - done +apply (induct n m rule: diff_induct) +prefer 3 apply (subst map_Suc_upt[symmetric]) +apply (auto simp add: less_diff_conv nth_upt) +done lemma nth_take_lemma [rule_format]: - "ALL xs ys. k <= length xs --> k <= length ys - --> (ALL i. i < k --> xs!i = ys!i) - --> take k xs = take k ys" - apply (induct k) - apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib) - apply clarify - txt {* Both lists must be non-empty *} - apply (case_tac xs) - apply simp - apply (case_tac ys) - apply clarify - apply (simp (no_asm_use)) - apply clarify - txt {* prenexing's needed, not miniscoping *} - apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps) - apply blast - done +"ALL xs ys. k <= length xs --> k <= length ys +--> (ALL i. i < k --> xs!i = ys!i) +--> take k xs = take k ys" +apply (induct k) +apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib) +apply clarify +txt {* Both lists must be non-empty *} +apply (case_tac xs) + apply simp +apply (case_tac ys) + apply clarify + apply (simp (no_asm_use)) +apply clarify +txt {* prenexing's needed, not miniscoping *} +apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps) +apply blast +done lemma nth_equalityI: "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys" - apply (frule nth_take_lemma [OF le_refl eq_imp_le]) - apply (simp_all add: take_all) - done +apply (frule nth_take_lemma [OF le_refl eq_imp_le]) +apply (simp_all add: take_all) +done lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys" - -- {* The famous take-lemma. *} - apply (drule_tac x = "max (length xs) (length ys)" in spec) - apply (simp add: le_max_iff_disj take_all) - done +-- {* The famous take-lemma. *} +apply (drule_tac x = "max (length xs) (length ys)" in spec) +apply (simp add: le_max_iff_disj take_all) +done subsection {* @{text "distinct"} and @{text remdups} *} lemma distinct_append [simp]: - "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})" - by (induct xs) auto +"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})" +by (induct xs) auto lemma set_remdups [simp]: "set (remdups xs) = set xs" - by (induct xs) (auto simp add: insert_absorb) +by (induct xs) (auto simp add: insert_absorb) lemma distinct_remdups [iff]: "distinct (remdups xs)" - by (induct xs) auto +by (induct xs) auto lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)" - by (induct xs) auto +by (induct xs) auto text {* - It is best to avoid this indexed version of distinct, but sometimes - it is useful. *} +It is best to avoid this indexed version of distinct, but sometimes +it is useful. *} lemma distinct_conv_nth: - "distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)" - apply (induct_tac xs) - apply simp - apply simp - apply (rule iffI) - apply clarsimp - apply (case_tac i) - apply (case_tac j) - apply simp - apply (simp add: set_conv_nth) - apply (case_tac j) - apply (clarsimp simp add: set_conv_nth) - apply simp - apply (rule conjI) - apply (clarsimp simp add: set_conv_nth) - apply (erule_tac x = 0 in allE) - apply (erule_tac x = "Suc i" in allE) - apply simp - apply clarsimp - apply (erule_tac x = "Suc i" in allE) - apply (erule_tac x = "Suc j" in allE) - apply simp - done +"distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)" +apply (induct_tac xs) + apply simp +apply simp +apply (rule iffI) + apply clarsimp + apply (case_tac i) +apply (case_tac j) + apply simp +apply (simp add: set_conv_nth) + apply (case_tac j) +apply (clarsimp simp add: set_conv_nth) + apply simp +apply (rule conjI) + apply (clarsimp simp add: set_conv_nth) + apply (erule_tac x = 0 in allE) + apply (erule_tac x = "Suc i" in allE) + apply simp +apply clarsimp +apply (erule_tac x = "Suc i" in allE) +apply (erule_tac x = "Suc j" in allE) +apply simp +done subsection {* @{text replicate} *} lemma length_replicate [simp]: "length (replicate n x) = n" - by (induct n) auto +by (induct n) auto lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)" - by (induct n) auto +by (induct n) auto lemma replicate_app_Cons_same: - "(replicate n x) @ (x # xs) = x # replicate n x @ xs" - by (induct n) auto +"(replicate n x) @ (x # xs) = x # replicate n x @ xs" +by (induct n) auto lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x" - apply(induct n) - apply simp - apply (simp add: replicate_app_Cons_same) - done +apply(induct n) + apply simp +apply (simp add: replicate_app_Cons_same) +done lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x" - by (induct n) auto +by (induct n) auto lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x" - by (induct n) auto +by (induct n) auto lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x" - by (induct n) auto +by (induct n) auto lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x" - by (atomize (full), induct n) auto +by (atomize (full), induct n) auto lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x" - apply(induct n) - apply simp - apply (simp add: nth_Cons split: nat.split) - done +apply(induct n) + apply simp +apply (simp add: nth_Cons split: nat.split) +done lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}" - by (induct n) auto +by (induct n) auto lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}" - by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc) +by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc) lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})" - by auto +by auto lemma in_set_replicateD: "x : set (replicate n y) ==> x = y" - by (simp add: set_replicate_conv_if split: split_if_asm) +by (simp add: set_replicate_conv_if split: split_if_asm) subsection {* Lexcicographic orderings on lists *} lemma wf_lexn: "wf r ==> wf (lexn r n)" - apply (induct_tac n) - apply simp - apply simp - apply(rule wf_subset) - prefer 2 apply (rule Int_lower1) - apply(rule wf_prod_fun_image) - prefer 2 apply (rule injI) - apply auto - done +apply (induct_tac n) + apply simp +apply simp +apply(rule wf_subset) + prefer 2 apply (rule Int_lower1) +apply(rule wf_prod_fun_image) + prefer 2 apply (rule injI) +apply auto +done lemma lexn_length: - "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n" - by (induct n) auto +"!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n" +by (induct n) auto lemma wf_lex [intro!]: "wf r ==> wf (lex r)" - apply (unfold lex_def) - apply (rule wf_UN) - apply (blast intro: wf_lexn) - apply clarify - apply (rename_tac m n) - apply (subgoal_tac "m \<noteq> n") - prefer 2 apply blast - apply (blast dest: lexn_length not_sym) - done +apply (unfold lex_def) +apply (rule wf_UN) +apply (blast intro: wf_lexn) +apply clarify +apply (rename_tac m n) +apply (subgoal_tac "m \<noteq> n") + prefer 2 apply blast +apply (blast dest: lexn_length not_sym) +done lemma lexn_conv: - "lexn r n = - {(xs,ys). length xs = n \<and> length ys = n \<and> - (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}" - apply (induct_tac n) - apply simp - apply blast - apply (simp add: image_Collect lex_prod_def) - apply auto - apply blast - apply (rename_tac a xys x xs' y ys') - apply (rule_tac x = "a # xys" in exI) - apply simp - apply (case_tac xys) - apply simp_all - apply blast - done +"lexn r n = +{(xs,ys). length xs = n \<and> length ys = n \<and> +(\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}" +apply (induct_tac n) + apply simp + apply blast +apply (simp add: image_Collect lex_prod_def) +apply auto +apply blast + apply (rename_tac a xys x xs' y ys') + apply (rule_tac x = "a # xys" in exI) + apply simp +apply (case_tac xys) + apply simp_all +apply blast +done lemma lex_conv: - "lex r = - {(xs,ys). length xs = length ys \<and> - (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}" - by (force simp add: lex_def lexn_conv) +"lex r = +{(xs,ys). length xs = length ys \<and> +(\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}" +by (force simp add: lex_def lexn_conv) lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)" - by (unfold lexico_def) blast +by (unfold lexico_def) blast lemma lexico_conv: - "lexico r = {(xs,ys). length xs < length ys | - length xs = length ys \<and> (xs, ys) : lex r}" - by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def) +"lexico r = {(xs,ys). length xs < length ys | +length xs = length ys \<and> (xs, ys) : lex r}" +by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def) lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r" - by (simp add: lex_conv) +by (simp add: lex_conv) lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r" - by (simp add:lex_conv) +by (simp add:lex_conv) lemma Cons_in_lex [iff]: - "((x # xs, y # ys) : lex r) = - ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)" - apply (simp add: lex_conv) - apply (rule iffI) - prefer 2 apply (blast intro: Cons_eq_appendI) - apply clarify - apply (case_tac xys) - apply simp - apply simp - apply blast - done +"((x # xs, y # ys) : lex r) = +((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)" +apply (simp add: lex_conv) +apply (rule iffI) + prefer 2 apply (blast intro: Cons_eq_appendI) +apply clarify +apply (case_tac xys) + apply simp +apply simp +apply blast +done subsection {* @{text sublist} --- a generalization of @{text nth} to sets *} lemma sublist_empty [simp]: "sublist xs {} = []" - by (auto simp add: sublist_def) +by (auto simp add: sublist_def) lemma sublist_nil [simp]: "sublist [] A = []" - by (auto simp add: sublist_def) +by (auto simp add: sublist_def) lemma sublist_shift_lemma: - "map fst [p:zip xs [i..i + length xs(] . snd p : A] = - map fst [p:zip xs [0..length xs(] . snd p + i : A]" - by (induct xs rule: rev_induct) (simp_all add: add_commute) +"map fst [p:zip xs [i..i + length xs(] . snd p : A] = +map fst [p:zip xs [0..length xs(] . snd p + i : A]" +by (induct xs rule: rev_induct) (simp_all add: add_commute) lemma sublist_append: - "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}" - apply (unfold sublist_def) - apply (induct l' rule: rev_induct) - apply simp - apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma) - apply (simp add: add_commute) - done +"sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}" +apply (unfold sublist_def) +apply (induct l' rule: rev_induct) + apply simp +apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma) +apply (simp add: add_commute) +done lemma sublist_Cons: - "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}" - apply (induct l rule: rev_induct) - apply (simp add: sublist_def) - apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append) - done +"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}" +apply (induct l rule: rev_induct) + apply (simp add: sublist_def) +apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append) +done lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])" - by (simp add: sublist_Cons) +by (simp add: sublist_Cons) lemma sublist_upt_eq_take [simp]: "sublist l {..n(} = take n l" - apply (induct l rule: rev_induct) - apply simp - apply (simp split: nat_diff_split add: sublist_append) - done +apply (induct l rule: rev_induct) + apply simp +apply (simp split: nat_diff_split add: sublist_append) +done lemma take_Cons': - "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)" - by (cases n) simp_all +"take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)" +by (cases n) simp_all lemma drop_Cons': - "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)" - by (cases n) simp_all +"drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)" +by (cases n) simp_all lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))" - by (cases n) simp_all +by (cases n) simp_all -lemmas [of "number_of v", standard, simp] = - take_Cons' drop_Cons' nth_Cons' +lemmas [simp] = take_Cons'[of "number_of v",standard] + drop_Cons'[of "number_of v",standard] + nth_Cons'[of _ _ "number_of v",standard] end

--- a/src/HOL/UNITY/Comp/AllocBase.ML Mon May 13 13:22:15 2002 +0200 +++ b/src/HOL/UNITY/Comp/AllocBase.ML Mon May 13 15:27:28 2002 +0200 @@ -56,7 +56,7 @@ Goal "bag_of (sublist l A) = \ \ (\\<Sum>i: A Int lessThan (length l). {# l!i #})"; -by (rev_induct_tac "l" 1); +by (res_inst_tac [("xs","l")] rev_induct 1); by (Simp_tac 1); by (asm_simp_tac (simpset() addsimps [sublist_append, Int_insert_right, lessThan_Suc,

--- a/src/HOL/UNITY/GenPrefix.ML Mon May 13 13:22:15 2002 +0200 +++ b/src/HOL/UNITY/GenPrefix.ML Mon May 13 15:27:28 2002 +0200 @@ -341,7 +341,7 @@ Addsimps [prefix_snoc]; Goal "(xs <= ys@zs) = (xs <= ys | (? us. xs = ys@us & us <= zs))"; -by (rev_induct_tac "zs" 1); +by (res_inst_tac [("xs","zs")] rev_induct 1); by (Force_tac 1); by (asm_full_simp_tac (simpset() delsimps [append_assoc] addsimps [append_assoc RS sym])1); @@ -351,7 +351,7 @@ (*Although the prefix ordering is not linear, the prefixes of a list are linearly ordered.*) Goal "!!zs::'a list. xs <= zs --> ys <= zs --> xs <= ys | ys <= xs"; -by (rev_induct_tac "zs" 1); +by (res_inst_tac [("xs","zs")] rev_induct 1); by Auto_tac; qed_spec_mp "common_prefix_linear";