--- a/src/HOL/List.thy Mon May 13 10:40:59 2002 +0200
+++ b/src/HOL/List.thy Mon May 13 11:05:27 2002 +0200
@@ -8,54 +8,56 @@
theory List = PreList:
-datatype 'a list = Nil ("[]") | Cons 'a "'a list" (infixr "#" 65)
+datatype 'a list =
+ Nil ("[]")
+ | Cons 'a "'a list" (infixr "#" 65)
consts
- "@" :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr 65)
- filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list"
- concat :: "'a list list \<Rightarrow> 'a list"
- foldl :: "('b \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"
- foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b"
- hd :: "'a list \<Rightarrow> 'a"
- tl :: "'a list \<Rightarrow> 'a list"
- last :: "'a list \<Rightarrow> 'a"
- butlast :: "'a list \<Rightarrow> 'a list"
- set :: "'a list \<Rightarrow> 'a set"
- list_all :: "('a \<Rightarrow> bool) \<Rightarrow> ('a list \<Rightarrow> bool)"
- list_all2 :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool"
- map :: "('a\<Rightarrow>'b) \<Rightarrow> ('a list \<Rightarrow> 'b list)"
- mem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool" (infixl 55)
- nth :: "'a list \<Rightarrow> nat \<Rightarrow> 'a" (infixl "!" 100)
- list_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a list"
- take :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
- drop :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
- takeWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list"
- dropWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list"
- rev :: "'a list \<Rightarrow> 'a list"
- zip :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a * 'b) list"
- upt :: "nat \<Rightarrow> nat \<Rightarrow> nat list" ("(1[_../_'(])")
- remdups :: "'a list \<Rightarrow> 'a list"
- null :: "'a list \<Rightarrow> bool"
- "distinct" :: "'a list \<Rightarrow> bool"
- replicate :: "nat \<Rightarrow> 'a \<Rightarrow> '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
syntax
- (* list Enumeration *)
- "@list" :: "args \<Rightarrow> 'a list" ("[(_)]")
+ -- {* list Enumeration *}
+ "@list" :: "args => 'a list" ("[(_)]")
- (* Special syntax for filter *)
- "@filter" :: "[pttrn, 'a list, bool] \<Rightarrow> 'a list" ("(1[_:_./ _])")
+ -- {* Special syntax for filter *}
+ "@filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_:_./ _])")
- (* list update *)
- "_lupdbind" :: "['a, 'a] \<Rightarrow> lupdbind" ("(2_ :=/ _)")
- "" :: "lupdbind \<Rightarrow> lupdbinds" ("_")
- "_lupdbinds" :: "[lupdbind, lupdbinds] \<Rightarrow> lupdbinds" ("_,/ _")
- "_LUpdate" :: "['a, lupdbinds] \<Rightarrow> '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 \<Rightarrow> nat \<Rightarrow> nat list" ("(1[_../_])")
+ upto :: "nat => nat => nat list" ("(1[_../_])")
translations
"[x, xs]" == "x#[xs]"
@@ -69,31 +71,22 @@
syntax (xsymbols)
- "@filter" :: "[pttrn, 'a list, bool] \<Rightarrow> 'a list" ("(1[_\<in>_ ./ _])")
-
-
-consts
- lists :: "'a set \<Rightarrow> 'a list set"
-
-inductive "lists A"
-intros
-Nil: "[]: lists A"
-Cons: "\<lbrakk> a: A; l: lists A \<rbrakk> \<Longrightarrow> a#l : lists A"
+ "@filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_\<in>_ ./ _])")
-(*Function "size" is overloaded for all datatypes. Users may refer to the
- list version as "length".*)
-syntax length :: "'a list \<Rightarrow> nat"
-translations "length" => "size:: _ list \<Rightarrow> nat"
+text {*
+ 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"
-(* translating size::list -> length *)
-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
+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
*}
primrec
@@ -117,7 +110,7 @@
"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) & list_all P xs)"
+ 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"
@@ -141,22 +134,23 @@
"concat(x#xs) = x @ concat(xs)"
primrec
drop_Nil: "drop n [] = []"
- drop_Cons: "drop n (x#xs) = (case n of 0 \<Rightarrow> x#xs | Suc(m) \<Rightarrow> drop m xs)"
- (* Warning: simpset does not contain this definition but separate theorems
- for n=0 / n=Suc k*)
+ 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 \<Rightarrow> [] | Suc(m) \<Rightarrow> x # take m xs)"
- (* Warning: simpset does not contain this definition but separate theorems
- for n=0 / n=Suc k*)
-primrec
- nth_Cons: "(x#xs)!n = (case n of 0 \<Rightarrow> x | (Suc k) \<Rightarrow> xs!k)"
- (* Warning: simpset does not contain this definition but separate theorems
- for n=0 / n=Suc k*)
+ 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
- " [][i:=v] = []"
- "(x#xs)[i:=v] = (case i of 0 \<Rightarrow> v # xs
- | Suc j \<Rightarrow> x # xs[j:=v])"
+ nth_Cons: "(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
+ -- {* Warning: simpset does not contain this definition *}
+ -- {* but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
+primrec
+ "[][i:=v] = []"
+ "(x#xs)[i:=v] =
+ (case i of 0 => v # xs
+ | Suc j => x # xs[j:=v])"
primrec
"takeWhile P [] = []"
"takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
@@ -165,16 +159,15 @@
"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 [] \<Rightarrow> [] | z#zs \<Rightarrow> (z,y)#zip zs ys)"
- (* Warning: simpset does not contain this definition but separate theorems
- for xs=[] / xs=z#zs *)
+ zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
+ -- {* Warning: simpset does not contain this definition *}
+ -- {* but separate theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
primrec
upt_0: "[i..0(] = []"
upt_Suc: "[i..(Suc j)(] = (if i <= j then [i..j(] @ [j] else [])"
primrec
"distinct [] = True"
- "distinct (x#xs) = (x ~: set xs & distinct xs)"
+ "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)"
@@ -183,180 +176,190 @@
replicate_Suc: "replicate (Suc n) x = x # replicate n x"
defs
list_all2_def:
- "list_all2 P xs ys == length xs = length ys & (!(x,y):set(zip xs ys). P x y)"
+ "list_all2 P xs ys == length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)"
-(** Lexicographic orderings on lists **)
+subsection {* Lexicographic orderings on lists *}
consts
- lexn :: "('a * 'a)set \<Rightarrow> nat \<Rightarrow> ('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 & 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 * 'a)set \<Rightarrow> ('a list * 'a list)set"
- "lex r == UN n. lexn r n"
+ lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
+ "lex r == \<Union>n. lexn r n"
- lexico :: "('a * 'a)set \<Rightarrow> ('a list * '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] \<Rightarrow> '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]: "\<And>x. xs ~= x#xs"
-by(induct_tac "xs", auto)
+lemma not_Cons_self [simp]: "xs \<noteq> x # xs"
+ by (induct xs) auto
-lemmas not_Cons_self2[simp] = not_Cons_self[THEN not_sym]
+lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
-lemma neq_Nil_conv: "(xs ~= []) = (? y ys. xs = y#ys)"
-by(induct_tac "xs", auto)
+lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
+ by (induct xs) auto
-(* Induction over the length of a list: *)
-(* "(!!xs. (!ys. length ys < length xs --> P ys) ==> P xs) ==> P(xs)" *)
-lemmas length_induct = measure_induct[of length]
+lemma length_induct:
+ "(!!xs. \<forall>ys. length ys < length xs --> P ys ==> P xs) ==> P xs"
+ by (rule measure_induct [of length]) rules
-(** "lists": the list-forming operator over sets **)
+subsection {* @{text lists}: the list-forming operator over sets *}
-lemma lists_mono: "A<=B ==> lists A <= lists B"
-apply(unfold lists.defs)
-apply(blast intro!:lfp_mono)
-done
+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"
-inductive_cases listsE[elim!]: "x#l : lists A"
-declare lists.intros[intro!]
+inductive_cases listsE [elim!]: "x#l : lists A"
-lemma lists_IntI[rule_format]:
- "l: lists A ==> l: lists B --> l: lists (A Int B)"
-apply(erule lists.induct)
-apply blast+
-done
+lemma lists_mono: "A \<subseteq> B ==> lists A \<subseteq> lists B"
+ by (unfold lists.defs) (blast intro!: lfp_mono)
-lemma lists_Int_eq[simp]: "lists (A Int B) = lists A Int lists B"
-apply(rule mono_Int[THEN equalityI])
-apply(simp add:mono_def lists_mono)
-apply(blast intro!: lists_IntI)
-done
+lemma lists_IntI [rule_format]:
+ "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
-lemma append_in_lists_conv[iff]:
- "(xs@ys : lists A) = (xs : lists A & ys : lists A)"
-by(induct_tac "xs", auto)
+lemma append_in_lists_conv [iff]:
+ "(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"
+ by (induct xs) auto
+
+
+subsection {* @{text length} *}
-(** length **)
-(* needs to come before "@" because of thm append_eq_append_conv *)
+text {*
+ Needs to come before @{text "@"} because of theorem @{text
+ append_eq_append_conv}.
+*}
-section "length"
-
-lemma length_append[simp]: "length(xs@ys) = length(xs)+length(ys)"
-by(induct_tac "xs", auto)
+lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
+ by (induct xs) auto
-lemma length_map[simp]: "length (map f xs) = length xs"
-by(induct_tac "xs", auto)
+lemma length_map [simp]: "length (map f xs) = length xs"
+ by (induct xs) auto
-lemma length_rev[simp]: "length(rev xs) = length(xs)"
-by(induct_tac "xs", auto)
+lemma length_rev [simp]: "length (rev xs) = length xs"
+ by (induct xs) auto
-lemma length_tl[simp]: "length(tl xs) = (length xs) - 1"
-by(case_tac "xs", auto)
+lemma length_tl [simp]: "length (tl xs) = length xs - 1"
+ by (cases xs) auto
-lemma length_0_conv[iff]: "(length xs = 0) = (xs = [])"
-by(induct_tac "xs", auto)
+lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
+ by (induct xs) auto
-lemma length_greater_0_conv[iff]: "(0 < length xs) = (xs ~= [])"
-by(induct_tac xs, auto)
+lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
+ by (induct xs) auto
lemma length_Suc_conv:
- "(length xs = Suc n) = (? y ys. xs = y#ys & length ys = n)"
-by(induct_tac "xs", auto)
+ "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
+ by (induct xs) auto
+
-(** @ - append **)
+subsection {* @{text "@"} -- append *}
-section "@ - append"
+lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
+ by (induct xs) auto
-lemma append_assoc[simp]: "(xs@ys)@zs = xs@(ys@zs)"
-by(induct_tac "xs", auto)
+lemma append_Nil2 [simp]: "xs @ [] = xs"
+ by (induct xs) auto
-lemma append_Nil2[simp]: "xs @ [] = xs"
-by(induct_tac "xs", auto)
+lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
+ by (induct xs) auto
-lemma append_is_Nil_conv[iff]: "(xs@ys = []) = (xs=[] & ys=[])"
-by(induct_tac "xs", auto)
+lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
+ by (induct xs) auto
-lemma Nil_is_append_conv[iff]: "([] = xs@ys) = (xs=[] & ys=[])"
-by(induct_tac "xs", auto)
+lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
+ by (induct xs) auto
-lemma append_self_conv[iff]: "(xs @ ys = xs) = (ys=[])"
-by(induct_tac "xs", auto)
+lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
+ by (induct xs) auto
-lemma self_append_conv[iff]: "(xs = xs @ ys) = (ys=[])"
-by(induct_tac "xs", auto)
-
-lemma append_eq_append_conv[rule_format,simp]:
- "!ys. length xs = length ys | length us = length vs
- --> (xs@us = ys@vs) = (xs=ys & us=vs)"
-apply(induct_tac "xs")
- apply(rule allI)
- apply(case_tac "ys")
+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
- apply force
-apply(rule allI)
-apply(case_tac "ys")
- apply force
-apply simp
-done
+ done
+
+lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
+ by simp
+
+lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
+ by simp
-lemma same_append_eq[iff]: "(xs @ ys = xs @ zs) = (ys=zs)"
-by simp
-
-lemma append1_eq_conv[iff]: "(xs @ [x] = ys @ [y]) = (xs = ys & x = y)"
-by simp
+lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
+ by simp
-lemma append_same_eq[iff]: "(ys @ xs = zs @ xs) = (ys=zs)"
-by simp
+lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
+ using append_same_eq [of _ _ "[]"] by auto
-lemma append_self_conv2[iff]: "(xs @ ys = ys) = (xs=[])"
-by(insert append_same_eq[of _ _ "[]"], auto)
+lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
+ using append_same_eq [of "[]"] by auto
-lemma self_append_conv2[iff]: "(ys = xs @ ys) = (xs=[])"
-by(auto simp add: append_same_eq[of "[]", simplified])
+lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
+ by (induct xs) auto
-lemma hd_Cons_tl[rule_format,simp]: "xs ~= [] --> hd xs # tl xs = xs"
-by(induct_tac "xs", auto)
+lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
+ by (induct xs) auto
-lemma hd_append: "hd(xs@ys) = (if xs=[] then hd ys else hd xs)"
-by(induct_tac "xs", auto)
+lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
+ by (simp add: hd_append split: list.split)
-lemma hd_append2[simp]: "xs ~= [] ==> hd(xs @ ys) = hd xs"
-by(simp add: hd_append split: list.split)
+lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
+ by (simp split: list.split)
-lemma tl_append: "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)"
-by(simp split: list.split)
+lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
+ by (simp add: tl_append split: list.split)
-lemma tl_append2[simp]: "xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys"
-by(simp add: tl_append split: list.split)
-(* trivial rules for solving @-equations automatically *)
+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)
+lemma Cons_eq_appendI:
+ "[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
+ by (drule sym) simp
-lemma append_eq_appendI: "[| xs@xs1 = zs; ys = xs1 @ us |] ==> xs@ys = zs@us"
-by(drule sym, simp)
+lemma append_eq_appendI:
+ "[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
+ by (drule sym) simp
-(***
-Simplification procedure for all list equalities.
-Currently only tries to rearrange @ to see if
-- both lists end in a singleton list,
-- or both lists end in the same list.
-***)
-ML_setup{*
+text {*
+ Simplification procedure for all list equalities.
+ Currently only tries to rearrange @{text "@"} to see if
+ - both lists end in a singleton list,
+ - or both lists end in the same list.
+*}
+
+ML_setup {*
local
val append_assoc = thm "append_assoc";
@@ -415,967 +418,947 @@
*}
-(** map **)
+subsection {* @{text map} *}
-section "map"
+lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
+ by (induct xs) simp_all
-lemma map_ext: "(\<And>x. x : set xs \<longrightarrow> f x = g x) \<Longrightarrow> map f xs = map g xs"
-by (induct xs, simp_all)
+lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
+ by (rule ext, induct_tac xs) auto
-lemma map_ident[simp]: "map (%x. x) = (%xs. xs)"
-by(rule ext, induct_tac "xs", auto)
+lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
+ by (induct xs) auto
-lemma map_append[simp]: "map f (xs@ys) = map f xs @ map f ys"
-by(induct_tac "xs", auto)
+lemma map_compose: "map (f o g) xs = map f (map g xs)"
+ by (induct xs) (auto simp add: o_def)
-lemma map_compose(*[simp]*): "map (f o g) xs = map f (map g xs)"
-by(unfold o_def, induct_tac "xs", auto)
+lemma rev_map: "rev (map f xs) = map f (rev xs)"
+ by (induct xs) auto
-lemma rev_map: "rev(map f xs) = map f (rev xs)"
-by(induct_tac xs, auto)
-
-(* a congruence rule for map: *)
lemma map_cong:
- "xs=ys ==> (!!x. x : set ys \<Longrightarrow> f x = g x) \<Longrightarrow> map f xs = map g ys"
-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(case_tac xs, auto)
+lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
+ by (cases xs) auto
-lemma Nil_is_map_conv[iff]: "([] = map f xs) = (xs = [])"
-by(case_tac xs, auto)
+lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
+ by (cases xs) auto
lemma map_eq_Cons:
- "(map f xs = y#ys) = (? x xs'. xs = x#xs' & f x = y & map f xs' = ys)"
-by(case_tac 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:
- "\<And>xs. map f xs = map f ys \<Longrightarrow> (!x y. f x = f y --> x=y) \<Longrightarrow> xs=ys"
-by(induct "ys", simp, fastsimp 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(blast 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)
-(** rev **)
-section "rev"
+subsection {* @{text rev} *}
-lemma rev_append[simp]: "rev(xs@ys) = rev(ys) @ rev(xs)"
-by(induct_tac xs, auto)
+lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
+ by (induct xs) auto
-lemma rev_rev_ident[simp]: "rev(rev xs) = xs"
-by(induct_tac xs, auto)
+lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
+ by (induct xs) auto
-lemma rev_is_Nil_conv[iff]: "(rev xs = []) = (xs = [])"
-by(induct_tac xs, auto)
+lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
+ by (induct xs) auto
-lemma Nil_is_rev_conv[iff]: "([] = rev xs) = (xs = [])"
-by(induct_tac xs, auto)
+lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
+ 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
+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
-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
+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
-(* Instead of (rev_induct_tac xs) use (induct_tac xs rule: rev_induct) *)
+ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *} -- "compatibility"
-lemma rev_exhaust: "(xs = [] \<Longrightarrow> P) \<Longrightarrow> (!!ys y. xs = ys@[y] \<Longrightarrow> P) \<Longrightarrow> 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
-(** set **)
+subsection {* @{text set} *}
-section "set"
+lemma finite_set [iff]: "finite (set xs)"
+ by (induct xs) auto
-lemma finite_set[iff]: "finite (set xs)"
-by(induct_tac xs, auto)
+lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
+ by (induct xs) auto
-lemma set_append[simp]: "set (xs@ys) = (set xs Un set ys)"
-by(induct_tac xs, auto)
+lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
+ by auto
-lemma set_subset_Cons: "set xs \<subseteq> set (x#xs)"
-by auto
+lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
+ by (induct xs) auto
-lemma set_empty[iff]: "(set xs = {}) = (xs = [])"
-by(induct_tac xs, auto)
+lemma set_rev [simp]: "set (rev xs) = set xs"
+ by (induct xs) auto
-lemma set_rev[simp]: "set(rev xs) = set(xs)"
-by(induct_tac xs, auto)
+lemma set_map [simp]: "set (map f xs) = f`(set xs)"
+ by (induct xs) auto
-lemma set_map[simp]: "set(map f xs) = f`(set xs)"
-by(induct_tac xs, auto)
+lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
+ by (induct xs) auto
-lemma set_filter[simp]: "set(filter P xs) = {x. x : set xs & P x}"
-by(induct_tac xs, auto)
-
-lemma set_upt[simp]: "set[i..j(] = {k. i <= k & k < j}"
-apply(induct_tac j)
- apply simp_all
-apply(erule ssubst)
-apply auto
-apply arith
-done
+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
-lemma in_set_conv_decomp: "(x : set xs) = (? ys zs. xs = ys@x#zs)"
-apply(induct_tac "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_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
+
+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
+
+lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"
+ 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])
-(* eliminate `lists' in favour of `set' *)
-
-lemma in_lists_conv_set: "(xs : lists A) = (!x : set xs. x : A)"
-by(induct_tac xs, auto)
-
-lemmas in_listsD[dest!] = in_lists_conv_set[THEN iffD1]
-lemmas in_listsI[intro!] = in_lists_conv_set[THEN iffD2]
-
-
-(** mem **)
-
-section "mem"
+subsection {* @{text mem} *}
lemma set_mem_eq: "(x mem xs) = (x : set xs)"
-by(induct_tac xs, auto)
+ by (induct xs) auto
-(** list_all **)
-
-section "list_all"
+subsection {* @{text list_all} *}
-lemma list_all_conv: "list_all P xs = (!x:set xs. P x)"
-by(induct_tac xs, auto)
+lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)"
+ by (induct xs) auto
-lemma list_all_append[simp]:
- "list_all P (xs@ys) = (list_all P xs & list_all P ys)"
-by(induct_tac 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
-(** filter **)
-
-section "filter"
+subsection {* @{text filter} *}
-lemma filter_append[simp]: "filter P (xs@ys) = filter P xs @ filter P ys"
-by(induct_tac xs, auto)
+lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
+ by (induct xs) auto
-lemma filter_filter[simp]: "filter P (filter Q xs) = filter (%x. Q x & P x) xs"
-by(induct_tac xs, auto)
+lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
+ by (induct xs) auto
-lemma filter_True[simp]: "!x : set xs. P x \<Longrightarrow> filter P xs = xs"
-by(induct xs, auto)
+lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
+ by (induct xs) auto
-lemma filter_False[simp]: "!x : set xs. ~P x \<Longrightarrow> filter P xs = []"
-by(induct xs, auto)
+lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
+ by (induct xs) auto
-lemma length_filter[simp]: "length (filter P xs) <= length xs"
-by(induct xs, auto simp add: le_SucI)
+lemma length_filter [simp]: "length (filter P xs) \<le> length xs"
+ by (induct xs) (auto simp add: le_SucI)
-lemma filter_is_subset[simp]: "set (filter P xs) <= set xs"
-by auto
+lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
+ by auto
-section "concat"
+subsection {* @{text concat} *}
-lemma concat_append[simp]: "concat(xs@ys) = concat(xs)@concat(ys)"
-by(induct xs, auto)
+lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
+ by (induct xs) auto
-lemma concat_eq_Nil_conv[iff]: "(concat xss = []) = (!xs:set xss. xs=[])"
-by(induct xss, auto)
+lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
+ by (induct xss) auto
-lemma Nil_eq_concat_conv[iff]: "([] = concat xss) = (!xs:set xss. xs=[])"
-by(induct xss, auto)
+lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
+ by (induct xss) auto
-lemma set_concat[simp]: "set(concat xs) = Union(set ` set xs)"
-by(induct xs, auto)
+lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
+ by (induct xs) auto
-lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
-by(induct xs, auto)
+lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
+ by (induct xs) auto
-lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
-by(induct xs, auto)
+lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
+ by (induct xs) auto
-lemma rev_concat: "rev(concat xs) = concat (map rev (rev xs))"
-by(induct xs, auto)
+lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
+ by (induct xs) auto
-(** nth **)
-section "nth"
+subsection {* @{text nth} *}
-lemma nth_Cons_0[simp]: "(x#xs)!0 = x"
-by auto
+lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
+ by auto
-lemma nth_Cons_Suc[simp]: "(x#xs)!(Suc n) = xs!n"
-by auto
+lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
+ by auto
-declare nth.simps[simp del]
+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 \<Longrightarrow> (map f xs)!n = f(xs!n)"
-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
-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)
+lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
+ apply (induct_tac xs)
+ apply simp
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 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: "\<lbrakk> n < length xs; !x : set xs. P x \<rbrakk> \<Longrightarrow> P(xs!n)"
-by(simp add:set_conv_nth, blast)
+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(simp add:set_conv_nth, blast)
+lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
+ by (auto simp add: set_conv_nth)
lemma all_nth_imp_all_set:
- "\<lbrakk> !i < length xs. P(xs!i); x : set xs \<rbrakk> \<Longrightarrow> P x"
-by(simp add:set_conv_nth, blast)
+ "[| !i < length xs. P(xs!i); x : set xs |] ==> P x"
+ by (auto simp add: set_conv_nth)
lemma all_set_conv_all_nth:
- "(!x : set xs. P x) = (!i. i<length xs --> P (xs ! i))"
-by(simp add:set_conv_nth, blast)
+ "(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
+ by (auto simp add: set_conv_nth)
-(** list update **)
+subsection {* @{text list_update} *}
-section "list update"
-
-lemma length_list_update[simp]: "!!i. length(xs[i:=x]) = length xs"
-by(induct xs, simp, simp split:nat.split)
+lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
+ by (induct xs) (auto split: nat.split)
lemma nth_list_update:
- "!!i j. i < length xs \<Longrightarrow> (xs[i:=x])!j = (if i=j then x else xs!j)"
-by(induct xs, simp, 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)
+lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
+ by (simp add: nth_list_update)
-lemma nth_list_update_neq[simp]: "!!i j. i ~= j \<Longrightarrow> xs[i:=x]!j = xs!j"
-by(induct xs, simp, auto simp add:nth_Cons split:nat.split)
+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)
-lemma list_update_overwrite[simp]:
- "!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
-by(induct xs, simp, simp 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)
lemma list_update_same_conv:
- "!!i. i < length xs \<Longrightarrow> (xs[i := x] = xs) = (xs!i = x)"
-by(induct xs, simp, simp split:nat.split, blast)
+ "!!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 \<Longrightarrow>
+ "!!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)
+ 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, simp, simp split:nat.split, fast)
+ by (induct xs) (auto split: nat.split)
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
-by(fast dest!:set_update_subset_insert[THEN subsetD])
+ by (blast dest!: set_update_subset_insert [THEN subsetD])
-(** last & butlast **)
+subsection {* @{text last} and @{text butlast} *}
-section "last / butlast"
+lemma last_snoc [simp]: "last (xs @ [x]) = x"
+ by (induct xs) auto
-lemma last_snoc[simp]: "last(xs@[x]) = x"
-by(induct xs, auto)
+lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
+ by (induct xs) auto
-lemma butlast_snoc[simp]:"butlast(xs@[x]) = xs"
-by(induct xs, auto)
-
-lemma length_butlast[simp]: "length(butlast xs) = length xs - 1"
-by(induct xs rule:rev_induct, auto)
+lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
+ 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 ~= [] --> butlast xs @ [last xs] = xs"
-by(induct xs, auto)
+lemma append_butlast_last_id [simp]:
+ "xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
+ by (induct xs) auto
-lemma in_set_butlastD: "x:set(butlast xs) ==> x:set xs"
-by(induct xs, auto split:split_if_asm)
+lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
+ by (induct xs) (auto split: split_if_asm)
lemma in_set_butlast_appendI:
- "x:set(butlast xs) | x:set(butlast ys) ==> x:set(butlast(xs@ys))"
-by(auto dest:in_set_butlastD simp add:butlast_append)
+ "x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
+ by (auto dest: in_set_butlastD simp add: butlast_append)
-(** take & drop **)
-section "take & drop"
+
+subsection {* @{text take} and @{text drop} *}
-lemma take_0[simp]: "take 0 xs = []"
-by(induct xs, auto)
+lemma take_0 [simp]: "take 0 xs = []"
+ by (induct xs) auto
-lemma drop_0[simp]: "drop 0 xs = xs"
-by(induct xs, auto)
+lemma drop_0 [simp]: "drop 0 xs = xs"
+ by (induct xs) auto
-lemma take_Suc_Cons[simp]: "take (Suc n) (x#xs) = x # take n xs"
-by simp
+lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
+ by simp
-lemma drop_Suc_Cons[simp]: "drop (Suc n) (x#xs) = drop n xs"
-by simp
+lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
+ by simp
-declare take_Cons[simp del] drop_Cons[simp del]
+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)
+lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
+ 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)
+lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
+ 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)
+lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
+ 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)
+lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
+ 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)
+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)
-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)
+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)
-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
+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
-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
+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
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
+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
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
+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
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
+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
-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
+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
lemma append_eq_conv_conj:
- "!!zs. (xs@ys = zs) = (xs = take (length xs) zs & 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
+
-(** takeWhile & dropWhile **)
+subsection {* @{text takeWhile} and @{text dropWhile} *}
-section "takeWhile & dropWhile"
+lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
+ by (induct xs) auto
-lemma takeWhile_dropWhile_id[simp]: "takeWhile P xs @ dropWhile P xs = xs"
-by(induct xs, auto)
+lemma takeWhile_append1 [simp]:
+ "[| x:set xs; ~P(x) |] ==> takeWhile P (xs @ ys) = takeWhile P xs"
+ by (induct xs) auto
-lemma takeWhile_append1[simp]:
- "\<lbrakk> x:set xs; ~P(x) \<rbrakk> \<Longrightarrow> takeWhile P (xs @ ys) = takeWhile P xs"
-by(induct xs, auto)
+lemma takeWhile_append2 [simp]:
+ "(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
+ by (induct xs) auto
-lemma takeWhile_append2[simp]:
- "(!!x. x : set xs \<Longrightarrow> P(x)) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
-by(induct xs, auto)
-
-lemma takeWhile_tail: "~P(x) ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
-by(induct xs, auto)
+lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
+ by (induct xs) auto
-lemma dropWhile_append1[simp]:
- "\<lbrakk> x : set xs; ~P(x) \<rbrakk> \<Longrightarrow> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
-by(induct xs, auto)
+lemma dropWhile_append1 [simp]:
+ "[| x : set xs; ~P(x) |] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
+ by (induct xs) auto
-lemma dropWhile_append2[simp]:
- "(!!x. x:set xs \<Longrightarrow> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
-by(induct xs, auto)
+lemma dropWhile_append2 [simp]:
+ "(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
+ by (induct xs) auto
-lemma set_take_whileD: "x:set(takeWhile P xs) ==> x:set xs & P x"
-by(induct xs, auto split:split_if_asm)
+lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
+ by (induct xs) (auto split: split_if_asm)
-(** zip **)
-section "zip"
+subsection {* @{text zip} *}
-lemma zip_Nil[simp]: "zip [] ys = []"
-by(induct ys, auto)
+lemma zip_Nil [simp]: "zip [] ys = []"
+ by (induct ys) auto
-lemma zip_Cons_Cons[simp]: "zip (x#xs) (y#ys) = (x,y)#zip xs ys"
-by simp
+lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
+ by simp
-declare zip_Cons[simp del]
+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
+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
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)
+lemma zip_append [simp]:
+ "[| length xs = length us; length ys = length vs |] ==>
+ zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
+ by (simp add: zip_append1)
lemma zip_rev:
- "!!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. \<lbrakk> i < length xs; i < length ys \<rbrakk> \<Longrightarrow> (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 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
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
+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
-(** list_all2 **)
-section "list_all2"
+
+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)
+lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])"
+ by (simp add: list_all2_def)
-lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs=[])"
-by(simp add:list_all2_def)
+lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])"
+ by (simp add: list_all2_def)
-lemma list_all2_Cons[iff]:
- "list_all2 P (x#xs) (y#ys) = (P x y & list_all2 P xs ys)"
-by(auto 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)
lemma list_all2_Cons1:
- "list_all2 P (x#xs) ys = (? z zs. ys = z#zs & P x z & list_all2 P xs zs)"
-by(case_tac 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) = (? z zs. xs = z#zs & P z y & list_all2 P zs ys)"
-by(case_tac 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)
+lemma list_all2_rev [iff]:
+ "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
+ by (simp add: list_all2_def zip_rev cong: conj_cong)
lemma list_all2_append1:
- "list_all2 P (xs@ys) zs =
- (EX us vs. zs = us@vs & length us = length xs & length vs = length ys &
- list_all2 P xs us & 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 & length us = length ys & length vs = length zs &
- list_all2 P us ys & 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 & (!i<length xs. P (xs!i) (ys!i)))"
-by(force simp add:list_all2_def set_zip)
+ (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]:
- "ALL a b c. P1 a b --> P2 b c --> P3 a c ==>
- ALL 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
+
+text {*
+ Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
+ difficult to use because it requires an additional transitivity step.
+*}
+
+lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
+ 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)
+
+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
-section "foldl"
-
-lemma foldl_append[simp]:
- "!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
-by(induct xs, auto)
+subsection {* @{text upto} *}
-(* Note: `n <= 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)
+lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])"
+ -- {* Does not terminate! *}
+ by (induct j) auto
+
+lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []"
+ by (subst upt_rec) simp
-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)
-
-lemma sum_eq_0_conv[iff]:
- "!!m::nat. (foldl op+ m ns = 0) = (m=0 & (!n : set ns. n=0))"
-by(induct ns, auto)
+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
-(** upto **)
-
-(* Does not terminate! *)
-lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])"
-by(induct j, auto)
-
-lemma upt_conv_Nil[simp]: "j<=i ==> [i..j(] = []"
-by(subst upt_rec, 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
-(*Only needed if upt_Suc is deleted from the simpset*)
-lemma upt_Suc_append: "i<=j ==> [i..(Suc j)(] = [i..j(]@[j]"
-by simp
+lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]"
+ -- {* LOOPS as a simprule, since @{text "j <= j"}. *}
+ by (induct k) auto
-lemma 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
-
-(*LOOPS as a simprule, since j<=j*)
-lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]"
-by(induct_tac "k", auto)
+lemma length_upt [simp]: "length [i..j(] = j - i"
+ by (induct j) (auto simp add: Suc_diff_le)
-lemma length_upt[simp]: "length [i..j(] = j-i"
-by(induct_tac j, simp, simp add: Suc_diff_le)
-
-lemma nth_upt[simp]: "i+k < j ==> [i..j(] ! k = i+k"
-apply(induct j)
-apply(auto simp add: less_Suc_eq nth_append split:nat_diff_split)
-done
+lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k"
+ apply (induct j)
+ apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
+ done
-lemma 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
+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
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)"
-thm diff_induct
-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_tac k)
-apply(simp_all add: less_Suc_eq_0_disj all_conj_distrib)
-apply clarify
-(*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
-(*prenexing's needed, not miniscoping*)
-apply(simp (no_asm_use) add: all_simps[symmetric] del: all_simps)
-apply blast
-(*prenexing's needed, not miniscoping*)
-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
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
+
+
+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
+
+lemma set_remdups [simp]: "set (remdups xs) = set xs"
+ by (induct xs) (auto simp add: insert_absorb)
+
+lemma distinct_remdups [iff]: "distinct (remdups xs)"
+ by (induct xs) auto
+
+lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
+ by (induct xs) auto
-(*The famous take-lemma*)
-lemma take_equalityI: "(ALL i. take i xs = take i ys) ==> xs = ys"
-apply(drule_tac x = "max (length xs) (length ys)" in spec)
-apply(simp add: le_max_iff_disj take_all)
-done
+text {*
+ 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 & remdups **)
-section "distinct & remdups"
-
-lemma distinct_append[simp]:
- "distinct(xs@ys) = (distinct xs & distinct ys & set xs Int set ys = {})"
-by(induct xs, auto)
-
-lemma set_remdups[simp]: "set(remdups xs) = set xs"
-by(induct xs, simp, simp add:insert_absorb)
-
-lemma distinct_remdups[iff]: "distinct(remdups xs)"
-by(induct xs, auto)
-
-lemma distinct_filter[simp]: "distinct xs ==> distinct (filter P xs)"
-by(induct xs, auto)
+subsection {* @{text replicate} *}
-(* 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 \<longrightarrow> 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
+lemma length_replicate [simp]: "length (replicate n x) = n"
+ by (induct n) auto
-(** replicate **)
-section "replicate"
-
-lemma length_replicate[simp]: "length(replicate n x) = n"
-by(induct n, auto)
-
-lemma map_replicate[simp]: "map f (replicate n x) = replicate n (f x)"
-by(induct n, auto)
+lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
+ by (induct n) auto
lemma replicate_app_Cons_same:
- "(replicate n x) @ (x#xs) = x # replicate n x @ xs"
-by(induct n, auto)
+ "(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
+lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
+ 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)
+lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
+ by (induct n) auto
-lemma hd_replicate[simp]: "n ~= 0 ==> hd(replicate n x) = x"
-by(induct n, auto)
+lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
+ by (induct n) auto
-lemma tl_replicate[simp]: "n ~= 0 ==> tl(replicate n x) = replicate (n - 1) x"
-by(induct n, auto)
+lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
+ by (induct n) auto
-lemma last_replicate[rule_format,simp]:
- "n ~= 0 --> last(replicate n x) = x"
-by(induct_tac n, auto)
+lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
+ by (atomize (full), induct n) auto
-lemma nth_replicate[simp]: "!!i. i<n ==> (replicate n x)!i = x"
-apply(induct n)
- apply simp
-apply(simp add: nth_Cons split:nat.split)
-done
+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
-lemma set_replicate_Suc: "set(replicate (Suc n) x) = {x}"
-by(induct n, auto)
+lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
+ by (induct n) auto
-lemma set_replicate[simp]: "n ~= 0 ==> set(replicate n x) = {x}"
-by(fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
+lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
+ by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
-lemma set_replicate_conv_if: "set(replicate n x) = (if n=0 then {} else {x})"
-by auto
+lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
+ by auto
-lemma in_set_replicateD: "x : set(replicate n y) ==> x=y"
-by(simp add: set_replicate_conv_if split:split_if_asm)
+lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
+ by (simp add: set_replicate_conv_if split: split_if_asm)
-(*** Lexcicographic orderings on lists ***)
-section"Lexcicographic orderings on lists"
+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
+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
lemma lexn_length:
- "!!xs ys. (xs,ys) : lexn r n ==> length xs = n & 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 ~= n")
- prefer 2 apply blast
-apply(blast dest: lexn_length not_sym)
-done
-
+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
lemma lexn_conv:
- "lexn r n =
- {(xs,ys). length xs = n & length ys = n &
- (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}"
-apply(induct_tac n)
- apply simp
- apply blast
-apply(simp add: image_Collect lex_prod_def)
-apply auto
+ "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
- 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
+ done
lemma lex_conv:
- "lex r =
- {(xs,ys). length xs = length ys &
- (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (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)
+lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)"
+ by (unfold lexico_def) blast
lemma lexico_conv:
-"lexico r = {(xs,ys). length xs < length ys |
- length xs = length ys & (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) ~: lex r"
-by(simp add:lex_conv)
+lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
+ by (simp add: lex_conv)
-lemma Nil2_notin_lex[iff]: "(xs,[]) ~: lex r"
-by(simp add:lex_conv)
+lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
+ by (simp add:lex_conv)
-lemma Cons_in_lex[iff]:
- "((x#xs,y#ys) : lex r) =
- ((x,y) : r & length xs = length ys | x=y & (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
+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
-(*** sublist (a generalization of nth to sets) ***)
+subsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
-lemma sublist_empty[simp]: "sublist xs {} = []"
-by(auto simp add:sublist_def)
+lemma sublist_empty [simp]: "sublist xs {} = []"
+ by (auto simp add: sublist_def)
-lemma sublist_nil[simp]: "sublist [] A = []"
-by(auto simp add:sublist_def)
+lemma sublist_nil [simp]: "sublist [] A = []"
+ 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]"
-apply(induct_tac xs rule: rev_induct)
- apply simp
-apply(simp add:add_commute)
-done
+ "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_tac 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_tac 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)
+lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
+ by (simp add: sublist_Cons)
-lemma sublist_upt_eq_take[simp]: "sublist l {..n(} = take n l"
-apply(induct_tac l rule: rev_induct)
- apply simp
-apply(simp split:nat_diff_split add:sublist_append)
-done
+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
-lemma take_Cons': "take n (x#xs) = (if n=0 then [] else x # take (n - 1) xs)"
-by(case_tac n, simp_all)
-
-lemma drop_Cons': "drop n (x#xs) = (if n=0 then x#xs else drop (n - 1) xs)"
-by(case_tac n, simp_all)
+lemma take_Cons':
+ "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
+ by (cases n) simp_all
-lemma nth_Cons': "(x#xs)!n = (if n=0 then x else xs!(n - 1))"
-by(case_tac 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
-lemmas [simp] = take_Cons'[of "number_of v",standard]
- drop_Cons'[of "number_of v",standard]
- nth_Cons'[of "number_of v",standard]
+lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
+ by (cases n) simp_all
+
+lemmas [of "number_of v", standard, simp] =
+ take_Cons' drop_Cons' nth_Cons'
end