--- 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