(* Title: HOL/List.thy ID: $Id: List.thy,v 1.147 2005/09/29 15:02:57 paulson Exp $ Author: Tobias Nipkow *) header {* The datatype of finite lists *} theory List imports PreList begin datatype 'a list = Nil ("[]") | Cons 'a "'a list" (infixr "#" 65) subsection{*Basic list processing functions*} 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_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" map :: "('a=>'b) => ('a list => 'b list)" nth :: "'a list => nat => 'a" (infixl "!" 100) list_update :: "'a list => nat => 'a => 'a list" take:: "nat => 'a list => 'a list" drop:: "nat => 'a list => 'a list" takeWhile :: "('a => bool) => 'a list => 'a list" dropWhile :: "('a => bool) => 'a list => 'a list" rev :: "'a list => 'a list" zip :: "'a list => 'b list => ('a * 'b) list" upt :: "nat => nat => nat list" ("(1[_..</_'])") remdups :: "'a list => 'a list" remove1 :: "'a => 'a list => 'a list" null:: "'a list => bool" "distinct":: "'a list => bool" replicate :: "nat => 'a => 'a list" rotate1 :: "'a list => 'a list" rotate :: "nat => 'a list => 'a list" sublist :: "'a list => nat set => 'a list" (* For efficiency *) mem :: "'a => 'a list => bool" (infixl 55) list_inter :: "'a list => 'a list => 'a list" list_ex :: "('a => bool) => 'a list => bool" list_all:: "('a => bool) => ('a list => bool)" itrev :: "'a list => 'a list => 'a list" filtermap :: "('a => 'b option) => 'a list => 'b list" map_filter :: "('a => 'b) => ('a => bool) => 'a list => 'b list" nonterminals lupdbinds lupdbind syntax -- {* list Enumeration *} "@list" :: "args => 'a list" ("[(_)]") -- {* Special syntax for filter *} "@filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_:_./ _])") -- {* list update *} "_lupdbind":: "['a, 'a] => lupdbind" ("(2_ :=/ _)") "" :: "lupdbind => lupdbinds" ("_") "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds" ("_,/ _") "_LUpdate" :: "['a, lupdbinds] => 'a" ("_/[(_)]" [900,0] 900) upto:: "nat => nat => nat list" ("(1[_../_])") translations "[x, xs]" == "x#[xs]" "[x]" == "x#[]" "[x:xs . P]"== "filter (%x. P) xs" "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs" "xs[i:=x]" == "list_update xs i x" "[i..j]" == "[i..<(Suc j)]" syntax (xsymbols) "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_∈_ ./ _])") syntax (HTML output) "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_∈_ ./ _])") 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" 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 "hd(x#xs) = x" primrec "tl([]) = []" "tl(x#xs) = xs" primrec "null([]) = True" "null(x#xs) = False" primrec "last(x#xs) = (if xs=[] then x else last xs)" primrec "butlast []= []" "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)" primrec "set [] = {}" "set (x#xs) = insert x (set xs)" primrec "map f [] = []" "map f (x#xs) = f(x)#map f xs" primrec append_Nil:"[]@ys = ys" append_Cons: "(x#xs)@ys = x#(xs@ys)" primrec "rev([]) = []" "rev(x#xs) = rev(xs) @ [x]" primrec "filter P [] = []" "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)" primrec foldl_Nil:"foldl f a [] = a" foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs" primrec "foldr f [] a = a" "foldr f (x#xs) a = f x (foldr f xs a)" primrec "concat([]) = []" "concat(x#xs) = x @ concat(xs)" primrec drop_Nil:"drop n [] = []" drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)" -- {*Warning: simpset does not contain this definition, but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} primrec take_Nil:"take n [] = []" take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)" -- {*Warning: simpset does not contain this definition, but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} primrec nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)" -- {*Warning: simpset does not contain this definition, but separate theorems for @{text "n = 0"} and @{text "n = Suc k"} *} primrec "[][i:=v] = []" "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])" primrec "takeWhile P [] = []" "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])" primrec "dropWhile P [] = []" "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)" primrec "zip xs [] = []" zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)" -- {*Warning: simpset does not contain this definition, but separate theorems for @{text "xs = []"} and @{text "xs = z # zs"} *} primrec upt_0: "[i..<0] = []" upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])" primrec "distinct [] = True" "distinct (x#xs) = (x ~: set xs ∧ distinct xs)" primrec "remdups [] = []" "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)" primrec "remove1 x [] = []" "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)" primrec replicate_0: "replicate 0 x = []" replicate_Suc: "replicate (Suc n) x = x # replicate n x" defs rotate1_def: "rotate1 xs == (case xs of [] => [] | x#xs => xs @ [x])" rotate_def: "rotate n == rotate1 ^ n" list_all2_def: "list_all2 P xs ys == length xs = length ys ∧ (∀(x, y) ∈ set (zip xs ys). P x y)" sublist_def: "sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..<size xs]))" primrec "x mem [] = False" "x mem (y#ys) = (if y=x then True else x mem ys)" primrec "list_inter [] bs = []" "list_inter (a#as) bs = (if a ∈ set bs then a#(list_inter as bs) else list_inter as bs)" primrec "list_all P [] = True" "list_all P (x#xs) = (P(x) ∧ list_all P xs)" primrec "list_ex P [] = False" "list_ex P (x#xs) = (P x ∨ list_ex P xs)" primrec "filtermap f [] = []" "filtermap f (x#xs) = (case f x of None => filtermap f xs | Some y => y # (filtermap f xs))" primrec "map_filter f P [] = []" "map_filter f P (x#xs) = (if P x then f x # map_filter f P xs else map_filter f P xs)" primrec "itrev [] ys = ys" "itrev (x#xs) ys = itrev xs (x#ys)" lemma not_Cons_self [simp]: "xs ≠ x # xs" by (induct xs) auto lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric] lemma neq_Nil_conv: "(xs ≠ []) = (∃y ys. xs = y # ys)" by (induct xs) auto lemma length_induct: "(!!xs. ∀ys. length ys < length xs --> P ys ==> P xs) ==> P xs" by (rule measure_induct [of length]) iprover subsubsection {* @{text length} *} text {* Needs to come before @{text "@"} because of theorem @{text append_eq_append_conv}. *} lemma length_append [simp]: "length (xs @ ys) = length xs + length ys" by (induct xs) auto lemma length_map [simp]: "length (map f xs) = length xs" by (induct xs) auto lemma length_rev [simp]: "length (rev xs) = length xs" by (induct xs) auto lemma length_tl [simp]: "length (tl xs) = length xs - 1" by (cases xs) auto lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])" by (induct xs) auto lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs ≠ [])" by (induct xs) auto lemma length_Suc_conv: "(length xs = Suc n) = (∃y ys. xs = y # ys ∧ length ys = n)" by (induct xs) auto lemma Suc_length_conv: "(Suc n = length xs) = (∃y ys. xs = y # ys ∧ length ys = n)" apply (induct xs, simp, simp) apply blast done lemma impossible_Cons [rule_format]: "length xs <= length ys --> xs = x # ys = False" apply (induct xs, auto) done lemma list_induct2[consumes 1]: "!!ys. [| length xs = length ys; P [] []; !!x xs y ys. [| length xs = length ys; P xs ys |] ==> P (x#xs) (y#ys) |] ==> P xs ys" apply(induct xs) apply simp apply(case_tac ys) apply simp apply(simp) done subsubsection {* @{text "@"} -- append *} lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)" by (induct xs) auto lemma append_Nil2 [simp]: "xs @ [] = xs" by (induct xs) auto lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] ∧ ys = [])" by (induct xs) auto lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] ∧ ys = [])" by (induct xs) auto lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])" by (induct xs) auto lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])" by (induct xs) auto lemma append_eq_append_conv [simp]: "!!ys. length xs = length ys ∨ length us = length vs ==> (xs@us = ys@vs) = (xs=ys ∧ us=vs)" apply (induct xs) apply (case_tac ys, simp, force) apply (case_tac ys, force, simp) done lemma append_eq_append_conv2: "!!ys zs ts. (xs @ ys = zs @ ts) = (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)" apply (induct xs) apply fastsimp apply(case_tac zs) apply simp apply fastsimp done lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)" by simp lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys ∧ x = y)" by simp lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)" by simp lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])" using append_same_eq [of _ _ "[]"] by auto lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])" using append_same_eq [of "[]"] by auto lemma hd_Cons_tl [simp]: "xs ≠ [] ==> hd xs # tl xs = xs" by (induct xs) auto lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)" by (induct xs) auto lemma hd_append2 [simp]: "xs ≠ [] ==> hd (xs @ ys) = hd xs" by (simp add: hd_append split: list.split) lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)" by (simp split: list.split) lemma tl_append2 [simp]: "xs ≠ [] ==> tl (xs @ ys) = tl xs @ ys" by (simp add: tl_append split: list.split) lemma Cons_eq_append_conv: "x#xs = ys@zs = (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))" by(cases ys) auto lemma append_eq_Cons_conv: "(ys@zs = x#xs) = (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))" by(cases ys) auto text {* Trivial rules for solving @{text "@"}-equations automatically. *} lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys" by simp lemma Cons_eq_appendI: "[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs" by (drule sym) simp lemma append_eq_appendI: "[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us" by (drule sym) simp text {* Simplification procedure for all list equalities. Currently only tries to rearrange @{text "@"} to see if - both lists end in a singleton list, - or both lists end in the same list. *} ML_setup {* local val append_assoc = thm "append_assoc"; val append_Nil = thm "append_Nil"; val append_Cons = thm "append_Cons"; val append1_eq_conv = thm "append1_eq_conv"; val append_same_eq = thm "append_same_eq"; 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; fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true | list1 _ = false; fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) = (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs) | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys | butlast xs = Const("List.list.Nil",fastype_of xs); val rearr_ss = HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]; fun list_eq sg ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) = let val lastl = last lhs and lastr = last rhs; fun rearr conv = let val lhs1 = butlast lhs and rhs1 = butlast rhs; val Type(_,listT::_) = eqT val appT = [listT,listT] ---> listT val app = Const("List.op @",appT) val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr) val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2)); val thm = Tactic.prove sg [] [] eq (K (simp_tac (Simplifier.inherit_bounds ss rearr_ss) 1)); 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 val list_eq_simproc = Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq; end; Addsimprocs [list_eq_simproc]; *} subsubsection {* @{text map} *} lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs" by (induct xs) simp_all lemma map_ident [simp]: "map (λx. x) = (λxs. xs)" by (rule ext, induct_tac xs) auto lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys" by (induct xs) auto lemma map_compose: "map (f o g) xs = map f (map g xs)" by (induct xs) (auto simp add: o_def) lemma rev_map: "rev (map f xs) = map f (rev xs)" by (induct xs) auto lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)" by (induct xs) auto lemma map_cong [recdef_cong]: "xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys" -- {* a congruence rule for @{text map} *} by simp lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])" by (cases xs) auto lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])" by (cases xs) auto lemma map_eq_Cons_conv[iff]: "(map f xs = y#ys) = (∃z zs. xs = z#zs ∧ f z = y ∧ map f zs = ys)" by (cases xs) auto lemma Cons_eq_map_conv[iff]: "(x#xs = map f ys) = (∃z zs. ys = z#zs ∧ x = f z ∧ xs = map f zs)" by (cases ys) auto lemma ex_map_conv: "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)" by(induct ys, auto) lemma map_eq_imp_length_eq: "!!xs. map f xs = map f ys ==> length xs = length ys" apply (induct ys) apply simp apply(simp (no_asm_use)) apply clarify apply(simp (no_asm_use)) apply fast done lemma map_inj_on: "[| map f xs = map f ys; inj_on f (set xs Un set ys) |] ==> xs = ys" apply(frule map_eq_imp_length_eq) apply(rotate_tac -1) apply(induct rule:list_induct2) apply simp apply(simp) apply (blast intro:sym) done lemma inj_on_map_eq_map: "inj_on f (set xs Un set ys) ==> (map f xs = map f ys) = (xs = ys)" by(blast dest:map_inj_on) lemma map_injective: "!!xs. map f xs = map f ys ==> inj f ==> xs = ys" by (induct ys) (auto dest!:injD) lemma inj_map_eq_map[simp]: "inj f ==> (map f xs = map f ys) = (xs = ys)" by(blast dest:map_injective) lemma inj_mapI: "inj f ==> inj (map f)" by (iprover dest: map_injective injD intro: inj_onI) lemma inj_mapD: "inj (map f) ==> inj f" apply (unfold inj_on_def, clarify) apply (erule_tac x = "[x]" in ballE) apply (erule_tac x = "[y]" in ballE, simp, blast) apply blast done lemma inj_map[iff]: "inj (map f) = inj f" by (blast dest: inj_mapD intro: inj_mapI) lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) ==> inj_on (map f) A" apply(rule inj_onI) apply(erule map_inj_on) apply(blast intro:inj_onI dest:inj_onD) done lemma map_idI: "(!!x. x ∈ set xs ==> f x = x) ==> map f xs = xs" by (induct xs, auto) lemma map_fun_upd [simp]: "y ∉ set xs ==> map (f(y:=v)) xs = map f xs" by (induct xs) auto lemma map_fst_zip[simp]: "length xs = length ys ==> map fst (zip xs ys) = xs" by (induct rule:list_induct2, simp_all) lemma map_snd_zip[simp]: "length xs = length ys ==> map snd (zip xs ys) = ys" by (induct rule:list_induct2, simp_all) subsubsection {* @{text rev} *} lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs" by (induct xs) auto lemma rev_rev_ident [simp]: "rev (rev xs) = xs" by (induct xs) auto lemma rev_swap: "(rev xs = ys) = (xs = rev ys)" by auto lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])" by (induct xs) auto lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])" by (induct xs) auto lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])" by (cases xs) auto lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])" by (cases xs) auto lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)" apply (induct xs, force) apply (case_tac ys, simp, force) done lemma inj_on_rev[iff]: "inj_on rev A" by(simp add:inj_on_def) lemma rev_induct [case_names Nil snoc]: "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs" apply(simplesubst rev_rev_ident[symmetric]) apply(rule_tac list = "rev xs" in list.induct, simp_all) done ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility" lemma rev_exhaust [case_names Nil snoc]: "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P" by (induct xs rule: rev_induct) auto lemmas rev_cases = rev_exhaust subsubsection {* @{text set} *} lemma finite_set [iff]: "finite (set xs)" by (induct xs) auto lemma set_append [simp]: "set (xs @ ys) = (set xs ∪ set ys)" by (induct xs) auto lemma hd_in_set: "l = x#xs ==> x∈set l" by (case_tac l, auto) lemma set_subset_Cons: "set xs ⊆ set (x # xs)" by auto lemma set_ConsD: "y ∈ set (x # xs) ==> y=x ∨ y ∈ set xs" by auto lemma set_empty [iff]: "(set xs = {}) = (xs = [])" by (induct xs) auto lemma set_empty2[iff]: "({} = set xs) = (xs = [])" by(induct xs) auto lemma set_rev [simp]: "set (rev xs) = set xs" by (induct xs) auto lemma set_map [simp]: "set (map f xs) = f`(set xs)" by (induct xs) auto lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs ∧ P x}" by (induct xs) auto lemma set_upt [simp]: "set[i..<j] = {k. i ≤ k ∧ k < j}" apply (induct j, simp_all) apply (erule ssubst, auto) done lemma in_set_conv_decomp: "(x : set xs) = (∃ys zs. xs = ys @ x # zs)" proof (induct xs) case Nil show ?case by simp case (Cons a xs) show ?case proof assume "x ∈ set (a # xs)" with prems show "∃ys zs. a # xs = ys @ x # zs" by (simp, blast intro: Cons_eq_appendI) next assume "∃ys zs. a # xs = ys @ x # zs" then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast show "x ∈ set (a # xs)" by (cases ys, auto simp add: eq) qed qed lemma finite_list: "finite A ==> EX l. set l = A" apply (erule finite_induct, auto) apply (rule_tac x="x#l" in exI, auto) done lemma card_length: "card (set xs) ≤ length xs" by (induct xs) (auto simp add: card_insert_if) subsubsection {* @{text filter} *} lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys" by (induct xs) auto lemma rev_filter: "rev (filter P xs) = filter P (rev xs)" by (induct xs) simp_all lemma filter_filter [simp]: "filter P (filter Q xs) = filter (λx. Q x ∧ P x) xs" by (induct xs) auto lemma length_filter_le [simp]: "length (filter P xs) ≤ length xs" by (induct xs) (auto simp add: le_SucI) lemma filter_True [simp]: "∀x ∈ set xs. P x ==> filter P xs = xs" by (induct xs) auto lemma filter_False [simp]: "∀x ∈ set xs. ¬ P x ==> filter P xs = []" by (induct xs) auto lemma filter_empty_conv: "(filter P xs = []) = (∀x∈set xs. ¬ P x)" by (induct xs) simp_all lemma filter_id_conv: "(filter P xs = xs) = (∀x∈set xs. P x)" apply (induct xs) apply auto apply(cut_tac P=P and xs=xs in length_filter_le) apply simp done lemma filter_map: "filter P (map f xs) = map f (filter (P o f) xs)" by (induct xs) simp_all lemma length_filter_map[simp]: "length (filter P (map f xs)) = length(filter (P o f) xs)" by (simp add:filter_map) lemma filter_is_subset [simp]: "set (filter P xs) ≤ set xs" by auto lemma length_filter_less: "[| x : set xs; ~ P x |] ==> length(filter P xs) < length xs" proof (induct xs) case Nil thus ?case by simp next case (Cons x xs) thus ?case apply (auto split:split_if_asm) using length_filter_le[of P xs] apply arith done qed lemma length_filter_conv_card: "length(filter p xs) = card{i. i < length xs & p(xs!i)}" proof (induct xs) case Nil thus ?case by simp next case (Cons x xs) let ?S = "{i. i < length xs & p(xs!i)}" have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite) show ?case (is "?l = card ?S'") proof (cases) assume "p x" hence eq: "?S' = insert 0 (Suc ` ?S)" by(auto simp add: nth_Cons image_def split:nat.split elim:lessE) have "length (filter p (x # xs)) = Suc(card ?S)" using Cons by simp also have "… = Suc(card(Suc ` ?S))" using fin by (simp add: card_image inj_Suc) also have "… = card ?S'" using eq fin by (simp add:card_insert_if) (simp add:image_def) finally show ?thesis . next assume "¬ p x" hence eq: "?S' = Suc ` ?S" by(auto simp add: nth_Cons image_def split:nat.split elim:lessE) have "length (filter p (x # xs)) = card ?S" using Cons by simp also have "… = card(Suc ` ?S)" using fin by (simp add: card_image inj_Suc) also have "… = card ?S'" using eq fin by (simp add:card_insert_if) finally show ?thesis . qed qed lemma Cons_eq_filterD: "x#xs = filter P ys ==> ∃us vs. ys = us @ x # vs ∧ (∀u∈set us. ¬ P u) ∧ P x ∧ xs = filter P vs" (concl is "∃us vs. ?P ys us vs") proof(induct ys) case Nil thus ?case by simp next case (Cons y ys) show ?case (is "∃x. ?Q x") proof cases assume Py: "P y" show ?thesis proof cases assume xy: "x = y" show ?thesis proof from Py xy Cons(2) show "?Q []" by simp qed next assume "x ≠ y" with Py Cons(2) show ?thesis by simp qed next assume Py: "¬ P y" with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp show ?thesis (is "? us. ?Q us") proof show "?Q (y#us)" using 1 by simp qed qed qed lemma filter_eq_ConsD: "filter P ys = x#xs ==> ∃us vs. ys = us @ x # vs ∧ (∀u∈set us. ¬ P u) ∧ P x ∧ xs = filter P vs" by(rule Cons_eq_filterD) simp lemma filter_eq_Cons_iff: "(filter P ys = x#xs) = (∃us vs. ys = us @ x # vs ∧ (∀u∈set us. ¬ P u) ∧ P x ∧ xs = filter P vs)" by(auto dest:filter_eq_ConsD) lemma Cons_eq_filter_iff: "(x#xs = filter P ys) = (∃us vs. ys = us @ x # vs ∧ (∀u∈set us. ¬ P u) ∧ P x ∧ xs = filter P vs)" by(auto dest:Cons_eq_filterD) lemma filter_cong: "xs = ys ==> (!!x. x ∈ set ys ==> P x = Q x) ==> filter P xs = filter Q ys" apply simp apply(erule thin_rl) by (induct ys) simp_all subsubsection {* @{text concat} *} lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys" by (induct xs) auto lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (∀xs ∈ set xss. xs = [])" by (induct xss) auto lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (∀xs ∈ set xss. xs = [])" by (induct xss) 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 filter_concat: "filter p (concat xs) = concat (map (filter p) xs)" by (induct xs) auto lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))" by (induct xs) auto subsubsection {* @{text nth} *} lemma nth_Cons_0 [simp]: "(x # xs)!0 = x" by auto lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n" by auto declare nth.simps [simp del] lemma nth_append: "!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))" apply (induct "xs", simp) apply (case_tac n, auto) done lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x" by (induct "xs") auto lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n" by (induct "xs") auto lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)" apply (induct xs, simp) apply (case_tac n, auto) done lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}" apply (induct xs, simp, simp) apply safe apply (rule_tac x = 0 in exI, simp) apply (rule_tac x = "Suc i" in exI, simp) apply (case_tac i, simp) apply (rename_tac j) apply (rule_tac x = j in exI, simp) done lemma in_set_conv_nth: "(x ∈ set xs) = (∃i < length xs. xs!i = x)" by(auto simp:set_conv_nth) lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)" by (auto simp add: set_conv_nth) lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs" by (auto simp add: set_conv_nth) lemma all_nth_imp_all_set: "[| !i < length xs. P(xs!i); x : set xs|] ==> P x" by (auto simp add: set_conv_nth) lemma all_set_conv_all_nth: "(∀x ∈ set xs. P x) = (∀i. i < length xs --> P (xs ! i))" by (auto simp add: set_conv_nth) subsubsection {* @{text list_update} *} 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==> (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_neq [simp]: "!!i j. i ≠ 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) (auto split: nat.split) lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs" apply (induct xs, simp) apply(simp split:nat.splits) done lemma list_update_beyond[simp]: "!!i. length xs ≤ i ==> xs[i:=x] = xs" apply (induct xs) apply simp apply (case_tac i) apply simp_all done lemma list_update_same_conv: "!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)" by (induct xs) (auto split: nat.split) lemma list_update_append1: "!!i. i < size xs ==> (xs @ ys)[i:=x] = xs[i:=x] @ ys" apply (induct xs, simp) apply(simp split:nat.split) done lemma list_update_append: "!!n. (xs @ ys) [n:= x] = (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))" by (induct xs) (auto split:nat.splits) lemma list_update_length [simp]: "(xs @ x # ys)[length xs := y] = (xs @ y # ys)" by (induct xs, auto) lemma update_zip: "!!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) lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A" by (blast dest!: set_update_subset_insert [THEN subsetD]) lemma set_update_memI: "!!n. n < length xs ==> x ∈ set (xs[n := x])" by (induct xs) (auto split:nat.splits) subsubsection {* @{text last} and @{text butlast} *} lemma last_snoc [simp]: "last (xs @ [x]) = x" by (induct xs) auto lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs" by (induct xs) auto lemma last_ConsL: "xs = [] ==> last(x#xs) = x" by(simp add:last.simps) lemma last_ConsR: "xs ≠ [] ==> last(x#xs) = last xs" by(simp add:last.simps) lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)" by (induct xs) (auto) lemma last_appendL[simp]: "ys = [] ==> last(xs @ ys) = last xs" by(simp add:last_append) lemma last_appendR[simp]: "ys ≠ [] ==> last(xs @ ys) = last ys" by(simp add:last_append) 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 lemma append_butlast_last_id [simp]: "xs ≠ [] ==> butlast xs @ [last xs] = xs" by (induct xs) auto lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs" by (induct xs) (auto split: split_if_asm) lemma in_set_butlast_appendI: "x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))" by (auto dest: in_set_butlastD simp add: butlast_append) lemma last_drop[simp]: "!!n. n < length xs ==> last (drop n xs) = last xs" apply (induct xs) apply simp apply (auto split:nat.split) done lemma last_conv_nth: "xs≠[] ==> last xs = xs!(length xs - 1)" by(induct xs)(auto simp:neq_Nil_conv) subsubsection {* @{text take} and @{text drop} *} lemma take_0 [simp]: "take 0 xs = []" by (induct xs) auto lemma drop_0 [simp]: "drop 0 xs = xs" by (induct xs) auto lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs" by simp lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs" by simp declare take_Cons [simp del] and drop_Cons [simp del] lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)" by(clarsimp simp add:neq_Nil_conv) lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)" by(cases xs, simp_all) lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)" by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split) lemma nth_via_drop: "!!n. drop n xs = y#ys ==> xs!n = y" apply (induct xs, simp) apply(simp add:drop_Cons nth_Cons split:nat.splits) done lemma take_Suc_conv_app_nth: "!!i. i < length xs ==> take (Suc i) xs = take i xs @ [xs!i]" apply (induct xs, simp) apply (case_tac i, auto) done lemma drop_Suc_conv_tl: "!!i. i < length xs ==> (xs!i) # (drop (Suc i) xs) = drop i xs" apply (induct xs, simp) apply (case_tac i, auto) done 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 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 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 take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs" apply (induct m, auto) apply (case_tac xs, auto) apply (case_tac n, auto) done lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs" apply (induct m, auto) apply (case_tac xs, auto) done lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)" apply (induct m, auto) apply (case_tac xs, auto) done lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)" apply(induct xs) apply simp apply(simp add: take_Cons drop_Cons split:nat.split) done lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs" apply (induct n, auto) apply (case_tac xs, auto) done lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 ∨ xs = [])" apply(induct xs) apply simp apply(simp add:take_Cons split:nat.split) done lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)" apply(induct xs) apply simp apply(simp add:drop_Cons split:nat.split) done lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)" apply (induct n, auto) apply (case_tac xs, auto) done lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)" apply (induct n, auto) apply (case_tac xs, auto) done lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)" apply (induct xs, auto) apply (case_tac i, auto) done lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)" apply (induct xs, auto) apply (case_tac i, auto) done lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i" apply (induct xs, auto) apply (case_tac n, blast) apply (case_tac i, auto) done lemma nth_drop [simp]: "!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)" apply (induct n, auto) apply (case_tac xs, auto) done lemma set_take_subset: "!!n. set(take n xs) ⊆ set xs" by(induct xs)(auto simp:take_Cons split:nat.split) lemma set_drop_subset: "!!n. set(drop n xs) ⊆ set xs" by(induct xs)(auto simp:drop_Cons split:nat.split) lemma in_set_takeD: "x : set(take n xs) ==> x : set xs" using set_take_subset by fast lemma in_set_dropD: "x : set(drop n xs) ==> x : set xs" using set_drop_subset by fast lemma append_eq_conv_conj: "!!zs. (xs @ ys = zs) = (xs = take (length xs) zs ∧ ys = drop (length xs) zs)" apply (induct xs, simp, clarsimp) apply (case_tac zs, auto) done lemma take_add [rule_format]: "∀i. i+j ≤ length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)" apply (induct xs, auto) apply (case_tac i, simp_all) done lemma append_eq_append_conv_if: "!! ys1. (xs1 @ xs2 = ys1 @ ys2) = (if size xs1 ≤ size ys1 then xs1 = take (size xs1) ys1 ∧ xs2 = drop (size xs1) ys1 @ ys2 else take (size ys1) xs1 = ys1 ∧ drop (size ys1) xs1 @ xs2 = ys2)" apply(induct xs1) apply simp apply(case_tac ys1) apply simp_all done lemma take_hd_drop: "!!n. n < length xs ==> take n xs @ [hd (drop n xs)] = take (n+1) xs" apply(induct xs) apply simp apply(simp add:drop_Cons split:nat.split) done lemma id_take_nth_drop: "i < length xs ==> xs = take i xs @ xs!i # drop (Suc i) xs" proof - assume si: "i < length xs" hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto moreover from si have "take (Suc i) xs = take i xs @ [xs!i]" apply (rule_tac take_Suc_conv_app_nth) by arith ultimately show ?thesis by auto qed lemma upd_conv_take_nth_drop: "i < length xs ==> xs[i:=a] = take i xs @ a # drop (Suc i) xs" proof - assume i: "i < length xs" have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]" by(rule arg_cong[OF id_take_nth_drop[OF i]]) also have "… = take i xs @ a # drop (Suc i) xs" using i by (simp add: list_update_append) finally show ?thesis . qed subsubsection {* @{text takeWhile} and @{text dropWhile} *} lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs" by (induct xs) auto lemma takeWhile_append1 [simp]: "[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs" by (induct xs) auto lemma takeWhile_append2 [simp]: "(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys" by (induct xs) auto lemma takeWhile_tail: "¬ P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs" by (induct xs) auto lemma dropWhile_append1 [simp]: "[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys" by (induct xs) auto lemma dropWhile_append2 [simp]: "(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys" by (induct xs) auto lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs ∧ P x" by (induct xs) (auto split: split_if_asm) lemma takeWhile_eq_all_conv[simp]: "(takeWhile P xs = xs) = (∀x ∈ set xs. P x)" by(induct xs, auto) lemma dropWhile_eq_Nil_conv[simp]: "(dropWhile P xs = []) = (∀x ∈ set xs. P x)" by(induct xs, auto) lemma dropWhile_eq_Cons_conv: "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & ¬ P y)" by(induct xs, auto) text{* The following two lemmmas could be generalized to an arbitrary property. *} lemma takeWhile_neq_rev: "[|distinct xs; x ∈ set xs|] ==> takeWhile (λy. y ≠ x) (rev xs) = rev (tl (dropWhile (λy. y ≠ x) xs))" by(induct xs) (auto simp: takeWhile_tail[where l="[]"]) lemma dropWhile_neq_rev: "[|distinct xs; x ∈ set xs|] ==> dropWhile (λy. y ≠ x) (rev xs) = x # rev (takeWhile (λy. y ≠ x) xs)" apply(induct xs) apply simp apply auto apply(subst dropWhile_append2) apply auto done subsubsection {* @{text zip} *} lemma zip_Nil [simp]: "zip [] ys = []" by (induct ys) auto lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys" by simp declare zip_Cons [simp del] lemma zip_Cons1: "zip (x#xs) ys = (case ys of [] => [] | y#ys => (x,y)#zip xs ys)" by(auto split:list.split) lemma length_zip [simp]: "!!xs. length (zip xs ys) = min (length xs) (length ys)" apply (induct ys, simp) apply (case_tac xs, 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, simp) apply (case_tac xs, 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, simp) apply (case_tac ys, 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_rev: "length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)" by (induct rule:list_induct2, simp_all) lemma nth_zip [simp]: "!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)" apply (induct ys, simp) apply (case_tac xs) apply (simp_all add: nth.simps split: nat.split) done lemma set_zip: "set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}" by (simp add: set_conv_nth cong: rev_conj_cong) lemma zip_update: "length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]" by (rule sym, simp add: update_zip) lemma zip_replicate [simp]: "!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)" apply (induct i, auto) apply (case_tac j, auto) done subsubsection {* @{text list_all2} *} lemma list_all2_lengthD [intro?]: "list_all2 P xs ys ==> length xs = length ys" by (simp add: list_all2_def) lemma list_all2_Nil [iff,code]: "list_all2 P [] ys = (ys = [])" by (simp add: list_all2_def) lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])" by (simp add: list_all2_def) lemma list_all2_Cons [iff,code]: "list_all2 P (x # xs) (y # ys) = (P x y ∧ 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 (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 (cases xs) auto lemma list_all2_rev [iff]: "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys" by (simp add: list_all2_def zip_rev cong: conj_cong) lemma list_all2_rev1: "list_all2 P (rev xs) ys = list_all2 P xs (rev ys)" by (subst list_all2_rev [symmetric]) simp lemma list_all2_append1: "list_all2 P (xs @ ys) zs = (EX us vs. zs = us @ vs ∧ 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, 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, clarify) apply (simp add: ball_Un) done lemma list_all2_append: "length xs = length ys ==> list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys ∧ list_all2 P us vs)" by (induct rule:list_induct2, simp_all) lemma list_all2_appendI [intro?, trans]: "[| list_all2 P a b; list_all2 P c d |] ==> list_all2 P (a@c) (b@d)" by (simp add: list_all2_append list_all2_lengthD) lemma list_all2_conv_all_nth: "list_all2 P xs ys = (length xs = length ys ∧ (∀i < length xs. P (xs!i) (ys!i)))" by (force simp add: list_all2_def set_zip) lemma list_all2_trans: assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c" shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs" (is "!!bs cs. PROP ?Q as bs cs") proof (induct as) fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs" show "!!cs. PROP ?Q (x # xs) bs cs" proof (induct bs) fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs" show "PROP ?Q (x # xs) (y # ys) cs" by (induct cs) (auto intro: tr I1 I2) qed simp qed simp lemma list_all2_all_nthI [intro?]: "length a = length b ==> (!!n. n < length a ==> P (a!n) (b!n)) ==> list_all2 P a b" by (simp add: list_all2_conv_all_nth) lemma list_all2I: "∀x ∈ set (zip a b). split P x ==> length a = length b ==> list_all2 P a b" by (simp add: list_all2_def) lemma list_all2_nthD: "[| list_all2 P xs ys; p < size xs |] ==> P (xs!p) (ys!p)" by (simp add: list_all2_conv_all_nth) lemma list_all2_nthD2: "[|list_all2 P xs ys; p < size ys|] ==> P (xs!p) (ys!p)" by (frule list_all2_lengthD) (auto intro: list_all2_nthD) lemma list_all2_map1: "list_all2 P (map f as) bs = list_all2 (λx y. P (f x) y) as bs" by (simp add: list_all2_conv_all_nth) lemma list_all2_map2: "list_all2 P as (map f bs) = list_all2 (λx y. P x (f y)) as bs" by (auto simp add: list_all2_conv_all_nth) lemma list_all2_refl [intro?]: "(!!x. P x x) ==> list_all2 P xs xs" by (simp add: list_all2_conv_all_nth) lemma list_all2_update_cong: "[| i<size xs; list_all2 P xs ys; P x y |] ==> list_all2 P (xs[i:=x]) (ys[i:=y])" by (simp add: list_all2_conv_all_nth nth_list_update) lemma list_all2_update_cong2: "[|list_all2 P xs ys; P x y; i < length ys|] ==> list_all2 P (xs[i:=x]) (ys[i:=y])" by (simp add: list_all2_lengthD list_all2_update_cong) lemma list_all2_takeI [simp,intro?]: "!!n ys. list_all2 P xs ys ==> list_all2 P (take n xs) (take n ys)" apply (induct xs) apply simp apply (clarsimp simp add: list_all2_Cons1) apply (case_tac n) apply auto done lemma list_all2_dropI [simp,intro?]: "!!n bs. list_all2 P as bs ==> list_all2 P (drop n as) (drop n bs)" apply (induct as, simp) apply (clarsimp simp add: list_all2_Cons1) apply (case_tac n, simp, simp) done lemma list_all2_mono [intro?]: "!!y. list_all2 P x y ==> (!!x y. P x y ==> Q x y) ==> list_all2 Q x y" apply (induct x, simp) apply (case_tac y, auto) done subsubsection {* @{text foldl} and @{text foldr} *} lemma foldl_append [simp]: "!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys" by (induct xs) auto lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)" by (induct xs) auto lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)" by (induct xs) auto lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a" by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"]) text {* Note: @{text "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 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 subsubsection {* @{text upto} *} lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])" -- {* simp does not terminate! *} by (induct j) auto lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []" by (subst upt_rec) simp lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 ∨ j <= i)" by(induct j)simp_all lemma upt_eq_Cons_conv: "!!x xs. ([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)" apply(induct j) apply simp apply(clarsimp simp add: append_eq_Cons_conv) apply arith done lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]" -- {* Only needed if @{text upt_Suc} is deleted from the simpset. *} by simp lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]" apply(rule trans) apply(subst upt_rec) prefer 2 apply (rule refl, 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 lemma length_upt [simp]: "length [i..<j] = j - i" by (induct j) (auto simp add: Suc_diff_le) lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k" apply (induct j) apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split) done lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..<n] = [i..<i+m]" apply (induct m, simp) apply (subst upt_rec) apply (rule sym) apply (subst upt_rec) apply (simp del: upt.simps) done lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]" apply(induct j) apply auto apply arith done lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..n]" 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 lemma nth_take_lemma: "!!xs ys. k <= length xs ==> k <= length ys ==> (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys" apply (atomize, induct k) apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify) txt {* Both lists must be non-empty *} apply (case_tac xs, simp) apply (case_tac ys, clarify) apply (simp (no_asm_use)) apply clarify txt {* prenexing's needed, not miniscoping *} apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps) apply blast done lemma nth_equalityI: "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys" apply (frule nth_take_lemma [OF le_refl eq_imp_le]) apply (simp_all add: take_all) done (* needs nth_equalityI *) lemma list_all2_antisym: "[| (!!x y. [|P x y; Q y x|] ==> x = y); list_all2 P xs ys; list_all2 Q ys xs |] ==> xs = ys" apply (simp add: list_all2_conv_all_nth) apply (rule nth_equalityI, blast, simp) done lemma take_equalityI: "(∀i. take i xs = take i ys) ==> xs = ys" -- {* The famous take-lemma. *} apply (drule_tac x = "max (length xs) (length ys)" in spec) apply (simp add: le_max_iff_disj take_all) done lemma take_Cons': "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)" by (cases n) simp_all lemma drop_Cons': "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)" by (cases n) simp_all lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))" by (cases n) simp_all lemmas [simp] = take_Cons'[of "number_of v",standard] drop_Cons'[of "number_of v",standard] nth_Cons'[of _ _ "number_of v",standard] subsubsection {* @{text "distinct"} and @{text remdups} *} lemma distinct_append [simp]: "distinct (xs @ ys) = (distinct xs ∧ distinct ys ∧ set xs ∩ set ys = {})" by (induct xs) auto lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs" by(induct xs) auto lemma set_remdups [simp]: "set (remdups xs) = set xs" by (induct xs) (auto simp add: insert_absorb) lemma distinct_remdups [iff]: "distinct (remdups xs)" by (induct xs) auto lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])" by (induct x, auto) lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])" by (induct x, auto) lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs" by (induct xs) auto lemma length_remdups_eq[iff]: "(length (remdups xs) = length xs) = (remdups xs = xs)" apply(induct xs) apply auto apply(subgoal_tac "length (remdups xs) <= length xs") apply arith apply(rule length_remdups_leq) done lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)" by (induct xs) auto lemma distinct_map_filterI: "distinct(map f xs) ==> distinct(map f (filter P xs))" apply(induct xs) apply simp apply force done lemma distinct_upt[simp]: "distinct[i..<j]" by (induct j) auto lemma distinct_take[simp]: "!!i. distinct xs ==> distinct (take i xs)" apply(induct xs) apply simp apply (case_tac i) apply simp_all apply(blast dest:in_set_takeD) done lemma distinct_drop[simp]: "!!i. distinct xs ==> distinct (drop i xs)" apply(induct xs) apply simp apply (case_tac i) apply simp_all done lemma distinct_list_update: assumes d: "distinct xs" and a: "a ∉ set xs - {xs!i}" shows "distinct (xs[i:=a])" proof (cases "i < length xs") case True with a have "a ∉ set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}" apply (drule_tac id_take_nth_drop) by simp with d True show ?thesis apply (simp add: upd_conv_take_nth_drop) apply (drule subst [OF id_take_nth_drop]) apply assumption apply simp apply (cases "a = xs!i") apply simp by blast next case False with d show ?thesis by auto qed text {* It is best to avoid this indexed version of distinct, but sometimes it is useful. *} lemma distinct_conv_nth: "distinct xs = (∀i < size xs. ∀j < size xs. i ≠ j --> xs!i ≠ xs!j)" apply (induct xs, simp, simp) apply (rule iffI, clarsimp) apply (case_tac i) apply (case_tac j, simp) apply (simp add: set_conv_nth) apply (case_tac j) apply (clarsimp simp add: set_conv_nth, simp) apply (rule conjI) apply (clarsimp simp add: set_conv_nth) apply (erule_tac x = 0 in allE, simp) apply (erule_tac x = "Suc i" in allE, simp, clarsimp) apply (erule_tac x = "Suc i" in allE, simp) apply (erule_tac x = "Suc j" in allE, simp) done lemma distinct_card: "distinct xs ==> card (set xs) = size xs" by (induct xs) auto lemma card_distinct: "card (set xs) = size xs ==> distinct xs" proof (induct xs) case Nil thus ?case by simp next case (Cons x xs) show ?case proof (cases "x ∈ set xs") case False with Cons show ?thesis by simp next case True with Cons.prems have "card (set xs) = Suc (length xs)" by (simp add: card_insert_if split: split_if_asm) moreover have "card (set xs) ≤ length xs" by (rule card_length) ultimately have False by simp thus ?thesis .. qed qed lemma inj_on_setI: "distinct(map f xs) ==> inj_on f (set xs)" apply(induct xs) apply simp apply fastsimp done lemma inj_on_set_conv: "distinct xs ==> inj_on f (set xs) = distinct(map f xs)" apply(induct xs) apply simp apply fastsimp done subsubsection {* @{text remove1} *} lemma set_remove1_subset: "set(remove1 x xs) <= set xs" apply(induct xs) apply simp apply simp apply blast done lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}" apply(induct xs) apply simp apply simp apply blast done lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)" apply(insert set_remove1_subset) apply fast done lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)" by (induct xs) simp_all subsubsection {* @{text replicate} *} lemma length_replicate [simp]: "length (replicate n x) = n" by (induct n) auto lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)" by (induct n) auto lemma replicate_app_Cons_same: "(replicate n x) @ (x # xs) = x # replicate n x @ xs" by (induct n) auto lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x" apply (induct n, simp) apply (simp add: replicate_app_Cons_same) done lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x" by (induct n) auto text{* Courtesy of Matthias Daum: *} lemma append_replicate_commute: "replicate n x @ replicate k x = replicate k x @ replicate n x" apply (simp add: replicate_add [THEN sym]) apply (simp add: add_commute) done lemma hd_replicate [simp]: "n ≠ 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 last_replicate [simp]: "n ≠ 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, simp) apply (simp add: nth_Cons split: nat.split) done text{* Courtesy of Matthias Daum (2 lemmas): *} lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x" apply (case_tac "k ≤ i") apply (simp add: min_def) apply (drule not_leE) apply (simp add: min_def) apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x") apply simp apply (simp add: replicate_add [symmetric]) done lemma drop_replicate[simp]: "!!i. drop i (replicate k x) = replicate (k-i) x" apply (induct k) apply simp apply clarsimp apply (case_tac i) apply simp apply clarsimp done lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}" by (induct n) auto lemma set_replicate [simp]: "n ≠ 0 ==> set (replicate n x) = {x}" by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc) lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})" by auto lemma in_set_replicateD: "x : set (replicate n y) ==> x = y" by (simp add: set_replicate_conv_if split: split_if_asm) subsubsection{*@{text rotate1} and @{text rotate}*} lemma rotate_simps[simp]: "rotate1 [] = [] ∧ rotate1 (x#xs) = xs @ [x]" by(simp add:rotate1_def) lemma rotate0[simp]: "rotate 0 = id" by(simp add:rotate_def) lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)" by(simp add:rotate_def) lemma rotate_add: "rotate (m+n) = rotate m o rotate n" by(simp add:rotate_def funpow_add) lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs" by(simp add:rotate_add) lemma rotate1_length01[simp]: "length xs <= 1 ==> rotate1 xs = xs" by(cases xs) simp_all lemma rotate_length01[simp]: "length xs <= 1 ==> rotate n xs = xs" apply(induct n) apply simp apply (simp add:rotate_def) done lemma rotate1_hd_tl: "xs ≠ [] ==> rotate1 xs = tl xs @ [hd xs]" by(simp add:rotate1_def split:list.split) lemma rotate_drop_take: "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs" apply(induct n) apply simp apply(simp add:rotate_def) apply(cases "xs = []") apply (simp) apply(case_tac "n mod length xs = 0") apply(simp add:mod_Suc) apply(simp add: rotate1_hd_tl drop_Suc take_Suc) apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric] take_hd_drop linorder_not_le) done lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs" by(simp add:rotate_drop_take) lemma rotate_id[simp]: "n mod length xs = 0 ==> rotate n xs = xs" by(simp add:rotate_drop_take) lemma length_rotate1[simp]: "length(rotate1 xs) = length xs" by(simp add:rotate1_def split:list.split) lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs" by (induct n) (simp_all add:rotate_def) lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs" by(simp add:rotate1_def split:list.split) blast lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs" by (induct n) (simp_all add:rotate_def) lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)" by(simp add:rotate_drop_take take_map drop_map) lemma set_rotate1[simp]: "set(rotate1 xs) = set xs" by(simp add:rotate1_def split:list.split) lemma set_rotate[simp]: "set(rotate n xs) = set xs" by (induct n) (simp_all add:rotate_def) lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])" by(simp add:rotate1_def split:list.split) lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])" by (induct n) (simp_all add:rotate_def) lemma rotate_rev: "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)" apply(simp add:rotate_drop_take rev_drop rev_take) apply(cases "length xs = 0") apply simp apply(cases "n mod length xs = 0") apply simp apply(simp add:rotate_drop_take rev_drop rev_take) done subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *} lemma sublist_empty [simp]: "sublist xs {} = []" by (auto simp add: sublist_def) lemma sublist_nil [simp]: "sublist [] A = []" by (auto simp add: sublist_def) lemma length_sublist: "length(sublist xs I) = card{i. i < length xs ∧ i : I}" by(simp add: sublist_def length_filter_conv_card cong:conj_cong) lemma sublist_shift_lemma_Suc: "!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) = map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))" apply(induct xs) apply simp apply (case_tac "is") apply simp apply simp done lemma sublist_shift_lemma: "map fst [p:zip xs [i..<i + length xs] . snd p : A] = map fst [p:zip xs [0..<length xs] . snd p + i : A]" by (induct xs rule: rev_induct) (simp_all add: add_commute) lemma sublist_append: "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}" apply (unfold sublist_def) apply (induct l' rule: rev_induct, simp) apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma) apply (simp add: add_commute) done lemma sublist_Cons: "sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}" apply (induct l rule: rev_induct) apply (simp add: sublist_def) apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append) done lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs ∧ i ∈ I}" apply(induct xs) apply simp apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE) apply(erule lessE) apply auto apply(erule lessE) apply auto done lemma set_sublist_subset: "set(sublist xs I) ⊆ set xs" by(auto simp add:set_sublist) lemma notin_set_sublistI[simp]: "x ∉ set xs ==> x ∉ set(sublist xs I)" by(auto simp add:set_sublist) lemma in_set_sublistD: "x ∈ set(sublist xs I) ==> x ∈ set xs" by(auto simp add:set_sublist) lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])" by (simp add: sublist_Cons) lemma distinct_sublistI[simp]: "!!I. distinct xs ==> distinct(sublist xs I)" apply(induct xs) apply simp apply(auto simp add:sublist_Cons) done lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l" apply (induct l rule: rev_induct, simp) apply (simp split: nat_diff_split add: sublist_append) done lemma filter_in_sublist: "!!s. distinct xs ==> filter (%x. x ∈ set(sublist xs s)) xs = sublist xs s" proof (induct xs) case Nil thus ?case by simp next case (Cons a xs) moreover hence "!x. x: set xs --> x ≠ a" by auto ultimately show ?case by(simp add: sublist_Cons cong:filter_cong) qed subsubsection{*Sets of Lists*} subsubsection {* @{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" inductive_cases listsE [elim!]: "x#l : lists A" lemma lists_mono [mono]: "A ⊆ B ==> lists A ⊆ lists B" by (unfold lists.defs) (blast intro!: lfp_mono) lemma lists_IntI: assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l by induct blast+ lemma lists_Int_eq [simp]: "lists (A ∩ B) = lists A ∩ lists B" proof (rule mono_Int [THEN equalityI]) show "mono lists" by (simp add: mono_def lists_mono) show "lists A ∩ lists B ⊆ lists (A ∩ B)" by (blast intro: lists_IntI) qed lemma append_in_lists_conv [iff]: "(xs @ ys : lists A) = (xs : lists A ∧ ys : lists A)" by (induct xs) auto lemma in_lists_conv_set: "(xs : lists A) = (∀x ∈ set xs. x : A)" -- {* eliminate @{text lists} in favour of @{text set} *} by (induct xs) auto lemma in_listsD [dest!]: "xs ∈ lists A ==> ∀x∈set xs. x ∈ A" by (rule in_lists_conv_set [THEN iffD1]) lemma in_listsI [intro!]: "∀x∈set xs. x ∈ A ==> xs ∈ lists A" by (rule in_lists_conv_set [THEN iffD2]) lemma lists_UNIV [simp]: "lists UNIV = UNIV" by auto subsubsection {* For efficiency *} text{* Only use @{text mem} for generating executable code. Otherwise use @{prop"x : set xs"} instead --- it is much easier to reason about. The same is true for @{const list_all} and @{const list_ex}: write @{text"∀x∈set xs"} and @{text"∃x∈set xs"} instead because the HOL quantifiers are aleady known to the automatic provers. In fact, the declarations in the Code subsection make sure that @{text"∈"}, @{text"∀x∈set xs"} and @{text"∃x∈set xs"} are implemented efficiently. The functions @{const itrev}, @{const filtermap} and @{const map_filter} are just there to generate efficient code. Do not use them for modelling and proving. *} lemma mem_iff: "(x mem xs) = (x : set xs)" by (induct xs) auto lemma list_inter_conv: "set(list_inter xs ys) = set xs ∩ set ys" by (induct xs) auto lemma list_all_iff: "list_all P xs = (∀x ∈ 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 xs) auto lemma list_all_rev [simp]: "list_all P (rev xs) = list_all P xs" by (simp add: list_all_iff) lemma list_ex_iff: "list_ex P xs = (∃x ∈ set xs. P x)" by (induct xs) simp_all lemma itrev[simp]: "ALL ys. itrev xs ys = rev xs @ ys" by (induct xs) simp_all lemma filtermap_conv: "filtermap f xs = map (%x. the(f x)) (filter (%x. f x ≠ None) xs)" by (induct xs) auto lemma map_filter_conv[simp]: "map_filter f P xs = map f (filter P xs)" by (induct xs) auto subsubsection {* Code generation *} text{* Defaults for generating efficient code for some standard functions. *} lemmas in_set_code[code unfold] = mem_iff[symmetric, THEN eq_reflection] lemma rev_code[code unfold]: "rev xs == itrev xs []" by simp lemma distinct_Cons_mem[code]: "distinct (x#xs) = (~(x mem xs) ∧ distinct xs)" by (simp add:mem_iff) lemma remdups_Cons_mem[code]: "remdups (x#xs) = (if x mem xs then remdups xs else x # remdups xs)" by (simp add:mem_iff) lemma list_inter_Cons_mem[code]: "list_inter (a#as) bs = (if a mem bs then a#(list_inter as bs) else list_inter as bs)" by(simp add:mem_iff) text{* For implementing bounded quantifiers over lists by @{const list_ex}/@{const list_all}: *} lemmas list_bex_code[code unfold] = list_ex_iff[symmetric, THEN eq_reflection] lemmas list_ball_code[code unfold] = list_all_iff[symmetric, THEN eq_reflection] subsubsection{* Inductive definition for membership *} consts ListMem :: "('a × 'a list)set" inductive ListMem intros elem: "(x,x#xs) ∈ ListMem" insert: "(x,xs) ∈ ListMem ==> (x,y#xs) ∈ ListMem" lemma ListMem_iff: "((x,xs) ∈ ListMem) = (x ∈ set xs)" apply (rule iffI) apply (induct set: ListMem) apply auto apply (induct xs) apply (auto intro: ListMem.intros) done subsubsection{*Lists as Cartesian products*} text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from @{term A} and tail drawn from @{term Xs}.*} constdefs set_Cons :: "'a set => 'a list set => 'a list set" "set_Cons A XS == {z. ∃x xs. z = x#xs & x ∈ A & xs ∈ XS}" lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A" by (auto simp add: set_Cons_def) text{*Yields the set of lists, all of the same length as the argument and with elements drawn from the corresponding element of the argument.*} consts listset :: "'a set list => 'a list set" primrec "listset [] = {[]}" "listset(A#As) = set_Cons A (listset As)" subsection{*Relations on Lists*} subsubsection {* Length Lexicographic Ordering *} text{*These orderings preserve well-foundedness: shorter lists precede longer lists. These ordering are not used in dictionaries.*} consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set" --{*The lexicographic ordering for lists of the specified length*} primrec "lexn r 0 = {}" "lexn r (Suc n) = (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int {(xs,ys). length xs = Suc n ∧ length ys = Suc n}" constdefs lex :: "('a × 'a) set => ('a list × 'a list) set" "lex r == \<Union>n. lexn r n" --{*Holds only between lists of the same length*} lenlex :: "('a × 'a) set => ('a list × 'a list) set" "lenlex r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))" --{*Compares lists by their length and then lexicographically*} lemma wf_lexn: "wf r ==> wf (lexn r n)" apply (induct n, simp, simp) apply(rule wf_subset) prefer 2 apply (rule Int_lower1) apply(rule wf_prod_fun_image) prefer 2 apply (rule inj_onI, auto) done lemma lexn_length: "!!xs ys. (xs, ys) : lexn r n ==> length xs = n ∧ 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, clarify) apply (rename_tac m n) apply (subgoal_tac "m ≠ 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 n, simp, blast) apply (simp add: image_Collect lex_prod_def, safe, blast) apply (rule_tac x = "ab # xys" in exI, simp) apply (case_tac xys, simp_all, blast) done lemma lex_conv: "lex r = {(xs,ys). length xs = length ys ∧ (∃xys x y xs' ys'. xs = xys @ x # xs' ∧ ys = xys @ y # ys' ∧ (x, y):r)}" by (force simp add: lex_def lexn_conv) lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)" by (unfold lenlex_def) blast lemma lenlex_conv: "lenlex r = {(xs,ys). length xs < length ys | length xs = length ys ∧ (xs, ys) : lex r}" by (simp add: lenlex_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 Nil2_notin_lex [iff]: "(xs, []) ∉ 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, clarify) apply (case_tac xys, simp, simp) apply blast done subsubsection {* Lexicographic Ordering *} text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b". This ordering does \emph{not} preserve well-foundedness. Author: N. Voelker, March 2005. *} constdefs lexord :: "('a * 'a)set => ('a list * 'a list) set" "lexord r == {(x,y). ∃ a v. y = x @ a # v ∨ (∃ u a b v w. (a,b) ∈ r ∧ x = u @ (a # v) ∧ y = u @ (b # w))}" lemma lexord_Nil_left[simp]: "([],y) ∈ lexord r = (∃ a x. y = a # x)" by (unfold lexord_def, induct_tac y, auto) lemma lexord_Nil_right[simp]: "(x,[]) ∉ lexord r" by (unfold lexord_def, induct_tac x, auto) lemma lexord_cons_cons[simp]: "((a # x, b # y) ∈ lexord r) = ((a,b)∈ r | (a = b & (x,y)∈ lexord r))" apply (unfold lexord_def, safe, simp_all) apply (case_tac u, simp, simp) apply (case_tac u, simp, clarsimp, blast, blast, clarsimp) apply (erule_tac x="b # u" in allE) by force lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons lemma lexord_append_rightI: "∃ b z. y = b # z ==> (x, x @ y) ∈ lexord r" by (induct_tac x, auto) lemma lexord_append_left_rightI: "(a,b) ∈ r ==> (u @ a # x, u @ b # y) ∈ lexord r" by (induct_tac u, auto) lemma lexord_append_leftI: " (u,v) ∈ lexord r ==> (x @ u, x @ v) ∈ lexord r" by (induct x, auto) lemma lexord_append_leftD: "[| (x @ u, x @ v) ∈ lexord r; (! a. (a,a) ∉ r) |] ==> (u,v) ∈ lexord r" by (erule rev_mp, induct_tac x, auto) lemma lexord_take_index_conv: "((x,y) : lexord r) = ((length x < length y ∧ take (length x) y = x) ∨ (∃i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) ∈ r))" apply (unfold lexord_def Let_def, clarsimp) apply (rule_tac f = "(% a b. a ∨ b)" in arg_cong2) apply auto apply (rule_tac x="hd (drop (length x) y)" in exI) apply (rule_tac x="tl (drop (length x) y)" in exI) apply (erule subst, simp add: min_def) apply (rule_tac x ="length u" in exI, simp) apply (rule_tac x ="take i x" in exI) apply (rule_tac x ="x ! i" in exI) apply (rule_tac x ="y ! i" in exI, safe) apply (rule_tac x="drop (Suc i) x" in exI) apply (drule sym, simp add: drop_Suc_conv_tl) apply (rule_tac x="drop (Suc i) y" in exI) by (simp add: drop_Suc_conv_tl) -- {* lexord is extension of partial ordering List.lex *} lemma lexord_lex: " (x,y) ∈ lex r = ((x,y) ∈ lexord r ∧ length x = length y)" apply (rule_tac x = y in spec) apply (induct_tac x, clarsimp) by (clarify, case_tac x, simp, force) lemma lexord_irreflexive: "(! x. (x,x) ∉ r) ==> (y,y) ∉ lexord r" by (induct y, auto) lemma lexord_trans: "[| (x, y) ∈ lexord r; (y, z) ∈ lexord r; trans r |] ==> (x, z) ∈ lexord r" apply (erule rev_mp)+ apply (rule_tac x = x in spec) apply (rule_tac x = z in spec) apply ( induct_tac y, simp, clarify) apply (case_tac xa, erule ssubst) apply (erule allE, erule allE) -- {* avoid simp recursion *} apply (case_tac x, simp, simp) apply (case_tac x, erule allE, erule allE, simp) apply (erule_tac x = listb in allE) apply (erule_tac x = lista in allE, simp) apply (unfold trans_def) by blast lemma lexord_transI: "trans r ==> trans (lexord r)" by (rule transI, drule lexord_trans, blast) lemma lexord_linear: "(! a b. (a,b)∈ r | a = b | (b,a) ∈ r) ==> (x,y) : lexord r | x = y | (y,x) : lexord r" apply (rule_tac x = y in spec) apply (induct_tac x, rule allI) apply (case_tac x, simp, simp) apply (rule allI, case_tac x, simp, simp) by blast subsubsection{*Lifting a Relation on List Elements to the Lists*} consts listrel :: "('a * 'a)set => ('a list * 'a list)set" inductive "listrel(r)" intros Nil: "([],[]) ∈ listrel r" Cons: "[| (x,y) ∈ r; (xs,ys) ∈ listrel r |] ==> (x#xs, y#ys) ∈ listrel r" inductive_cases listrel_Nil1 [elim!]: "([],xs) ∈ listrel r" inductive_cases listrel_Nil2 [elim!]: "(xs,[]) ∈ listrel r" inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) ∈ listrel r" inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) ∈ listrel r" lemma listrel_mono: "r ⊆ s ==> listrel r ⊆ listrel s" apply clarify apply (erule listrel.induct) apply (blast intro: listrel.intros)+ done lemma listrel_subset: "r ⊆ A × A ==> listrel r ⊆ lists A × lists A" apply clarify apply (erule listrel.induct, auto) done lemma listrel_refl: "refl A r ==> refl (lists A) (listrel r)" apply (simp add: refl_def listrel_subset Ball_def) apply (rule allI) apply (induct_tac x) apply (auto intro: listrel.intros) done lemma listrel_sym: "sym r ==> sym (listrel r)" apply (auto simp add: sym_def) apply (erule listrel.induct) apply (blast intro: listrel.intros)+ done lemma listrel_trans: "trans r ==> trans (listrel r)" apply (simp add: trans_def) apply (intro allI) apply (rule impI) apply (erule listrel.induct) apply (blast intro: listrel.intros)+ done theorem equiv_listrel: "equiv A r ==> equiv (lists A) (listrel r)" by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}" by (blast intro: listrel.intros) lemma listrel_Cons: "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})"; by (auto simp add: set_Cons_def intro: listrel.intros) subsection{*Miscellany*} subsubsection {* Characters and strings *} datatype nibble = Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7 | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF datatype char = Char nibble nibble -- "Note: canonical order of character encoding coincides with standard term ordering" types string = "char list" syntax "_Char" :: "xstr => char" ("CHR _") "_String" :: "xstr => string" ("_") parse_ast_translation {* let val constants = Syntax.Appl o map Syntax.Constant; fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10)); fun mk_char c = if Symbol.is_ascii c andalso Symbol.is_printable c then constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)] else error ("Printable ASCII character expected: " ^ quote c); fun mk_string [] = Syntax.Constant "Nil" | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs]; fun char_ast_tr [Syntax.Variable xstr] = (case Syntax.explode_xstr xstr of [c] => mk_char c | _ => error ("Single character expected: " ^ xstr)) | char_ast_tr asts = raise AST ("char_ast_tr", asts); fun string_ast_tr [Syntax.Variable xstr] = (case Syntax.explode_xstr xstr of [] => constants [Syntax.constrainC, "Nil", "string"] | cs => mk_string cs) | string_ast_tr asts = raise AST ("string_tr", asts); in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end; *} ML {* fun int_of_nibble h = if "0" <= h andalso h <= "9" then ord h - ord "0" else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10 else raise Match; fun nibble_of_int i = if i <= 9 then chr (ord "0" + i) else chr (ord "A" + i - 10); *} print_ast_translation {* let fun dest_nib (Syntax.Constant c) = (case explode c of ["N", "i", "b", "b", "l", "e", h] => int_of_nibble h | _ => raise Match) | dest_nib _ = raise Match; fun dest_chr c1 c2 = let val c = chr (dest_nib c1 * 16 + dest_nib c2) in if Symbol.is_printable c then c else raise Match end; fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2 | dest_char _ = raise Match; fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)]; fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]] | char_ast_tr' _ = raise Match; fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String", xstr (map dest_char (Syntax.unfold_ast "_args" args))] | list_ast_tr' ts = raise Match; in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end; *} subsubsection {* Code generator setup *} ML {* local fun list_codegen thy defs gr dep thyname b t = let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy defs dep thyname false) (gr, HOLogic.dest_list t) in SOME (gr', Pretty.list "[" "]" ps) end handle TERM _ => NONE; fun dest_nibble (Const (s, _)) = int_of_nibble (unprefix "List.nibble.Nibble" s) | dest_nibble _ = raise Match; fun char_codegen thy defs gr dep thyname b (Const ("List.char.Char", _) $ c1 $ c2) = (let val c = chr (dest_nibble c1 * 16 + dest_nibble c2) in if Symbol.is_printable c then SOME (gr, Pretty.quote (Pretty.str c)) else NONE end handle Fail _ => NONE | Match => NONE) | char_codegen thy defs gr dep thyname b _ = NONE; in val list_codegen_setup = [Codegen.add_codegen "list_codegen" list_codegen, Codegen.add_codegen "char_codegen" char_codegen]; end; *} types_code "list" ("_ list") attach (term_of) {* val term_of_list = HOLogic.mk_list; *} attach (test) {* fun gen_list' aG i j = frequency [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] () and gen_list aG i = gen_list' aG i i; *} "char" ("string") attach (term_of) {* val nibbleT = Type ("List.nibble", []); fun term_of_char c = Const ("List.char.Char", nibbleT --> nibbleT --> Type ("List.char", [])) $ Const ("List.nibble.Nibble" ^ nibble_of_int (ord c div 16), nibbleT) $ Const ("List.nibble.Nibble" ^ nibble_of_int (ord c mod 16), nibbleT); *} attach (test) {* fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z"))); *} consts_code "Cons" ("(_ ::/ _)") setup list_codegen_setup end
lemma not_Cons_self:
xs ≠ x # xs
lemmas not_Cons_self2:
x1 # t ≠ t
lemmas not_Cons_self2:
x1 # t ≠ t
lemma neq_Nil_conv:
(xs ≠ []) = (∃y ys. xs = y # ys)
lemma length_induct:
(!!xs. ∀ys. length ys < length xs --> P ys ==> P xs) ==> P xs
lemma length_append:
length (xs @ ys) = length xs + length ys
lemma length_map:
length (map f xs) = length xs
lemma length_rev:
length (rev xs) = length xs
lemma length_tl:
length (tl xs) = length xs - 1
lemma length_0_conv:
(length xs = 0) = (xs = [])
lemma length_greater_0_conv:
(0 < length xs) = (xs ≠ [])
lemma length_Suc_conv:
(length xs = Suc n) = (∃y ys. xs = y # ys ∧ length ys = n)
lemma Suc_length_conv:
(Suc n = length xs) = (∃y ys. xs = y # ys ∧ length ys = n)
lemma impossible_Cons:
length xs ≤ length ys ==> (xs = x # ys) = False
lemma list_induct2:
[| length xs = length ys; P [] []; !!x xs y ys. [| length xs = length ys; P xs ys |] ==> P (x # xs) (y # ys) |] ==> P xs ys
lemma append_assoc:
(xs @ ys) @ zs = xs @ ys @ zs
lemma append_Nil2:
xs @ [] = xs
lemma append_is_Nil_conv:
(xs @ ys = []) = (xs = [] ∧ ys = [])
lemma Nil_is_append_conv:
([] = xs @ ys) = (xs = [] ∧ ys = [])
lemma append_self_conv:
(xs @ ys = xs) = (ys = [])
lemma self_append_conv:
(xs = xs @ ys) = (ys = [])
lemma append_eq_append_conv:
length xs = length ys ∨ length us = length vs ==> (xs @ us = ys @ vs) = (xs = ys ∧ us = vs)
lemma append_eq_append_conv2:
(xs @ ys = zs @ ts) = (∃us. xs = zs @ us ∧ us @ ys = ts ∨ xs @ us = zs ∧ ys = us @ ts)
lemma same_append_eq:
(xs @ ys = xs @ zs) = (ys = zs)
lemma append1_eq_conv:
(xs @ [x] = ys @ [y]) = (xs = ys ∧ x = y)
lemma append_same_eq:
(ys @ xs = zs @ xs) = (ys = zs)
lemma append_self_conv2:
(xs @ ys = ys) = (xs = [])
lemma self_append_conv2:
(ys = xs @ ys) = (xs = [])
lemma hd_Cons_tl:
xs ≠ [] ==> hd xs # tl xs = xs
lemma hd_append:
hd (xs @ ys) = (if xs = [] then hd ys else hd xs)
lemma hd_append2:
xs ≠ [] ==> hd (xs @ ys) = hd xs
lemma tl_append:
tl (xs @ ys) = (case xs of [] => tl ys | z # zs => zs @ ys)
lemma tl_append2:
xs ≠ [] ==> tl (xs @ ys) = tl xs @ ys
lemma Cons_eq_append_conv:
(x # xs = ys @ zs) = (ys = [] ∧ x # xs = zs ∨ (∃ys'. x # ys' = ys ∧ xs = ys' @ zs))
lemma append_eq_Cons_conv:
(ys @ zs = x # xs) = (ys = [] ∧ zs = x # xs ∨ (∃ys'. ys = x # ys' ∧ ys' @ zs = xs))
lemma eq_Nil_appendI:
xs = ys ==> xs = [] @ ys
lemma Cons_eq_appendI:
[| x # xs1.0 = ys; xs = xs1.0 @ zs |] ==> x # xs = ys @ zs
lemma append_eq_appendI:
[| xs @ xs1.0 = zs; ys = xs1.0 @ us |] ==> xs @ ys = zs @ us
lemma map_ext:
(!!x. x ∈ set xs --> f x = g x) ==> map f xs = map g xs
lemma map_ident:
map (%x. x) = (%xs. xs)
lemma map_append:
map f (xs @ ys) = map f xs @ map f ys
lemma map_compose:
map (f o g) xs = map f (map g xs)
lemma rev_map:
rev (map f xs) = map f (rev xs)
lemma map_eq_conv:
(map f xs = map g xs) = (∀x∈set xs. f x = g x)
lemma map_cong:
[| xs = ys; !!x. x ∈ set ys ==> f x = g x |] ==> map f xs = map g ys
lemma map_is_Nil_conv:
(map f xs = []) = (xs = [])
lemma Nil_is_map_conv:
([] = map f xs) = (xs = [])
lemma map_eq_Cons_conv:
(map f xs = y # ys) = (∃z zs. xs = z # zs ∧ f z = y ∧ map f zs = ys)
lemma Cons_eq_map_conv:
(x # xs = map f ys) = (∃z zs. ys = z # zs ∧ x = f z ∧ xs = map f zs)
lemma ex_map_conv:
(∃xs. ys = map f xs) = (∀y∈set ys. ∃x. y = f x)
lemma map_eq_imp_length_eq:
map f xs = map f ys ==> length xs = length ys
lemma map_inj_on:
[| map f xs = map f ys; inj_on f (set xs ∪ set ys) |] ==> xs = ys
lemma inj_on_map_eq_map:
inj_on f (set xs ∪ set ys) ==> (map f xs = map f ys) = (xs = ys)
lemma map_injective:
[| map f xs = map f ys; inj f |] ==> xs = ys
lemma inj_map_eq_map:
inj f ==> (map f xs = map f ys) = (xs = ys)
lemma inj_mapI:
inj f ==> inj (map f)
lemma inj_mapD:
inj (map f) ==> inj f
lemma inj_map:
inj (map f) = inj f
lemma inj_on_mapI:
inj_on f (Union (set ` A)) ==> inj_on (map f) A
lemma map_idI:
(!!x. x ∈ set xs ==> f x = x) ==> map f xs = xs
lemma map_fun_upd:
y ∉ set xs ==> map (f(y := v)) xs = map f xs
lemma map_fst_zip:
length xs = length ys ==> map fst (zip xs ys) = xs
lemma map_snd_zip:
length xs = length ys ==> map snd (zip xs ys) = ys
lemma rev_append:
rev (xs @ ys) = rev ys @ rev xs
lemma rev_rev_ident:
rev (rev xs) = xs
lemma rev_swap:
(rev xs = ys) = (xs = rev ys)
lemma rev_is_Nil_conv:
(rev xs = []) = (xs = [])
lemma Nil_is_rev_conv:
([] = rev xs) = (xs = [])
lemma rev_singleton_conv:
(rev xs = [x]) = (xs = [x])
lemma singleton_rev_conv:
([x] = rev xs) = (xs = [x])
lemma rev_is_rev_conv:
(rev xs = rev ys) = (xs = ys)
lemma inj_on_rev:
inj_on rev A
lemma rev_induct:
[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs
lemma rev_exhaust:
[| xs = [] ==> P; !!ys y. xs = ys @ [y] ==> P |] ==> P
lemmas rev_cases:
[| xs = [] ==> P; !!ys y. xs = ys @ [y] ==> P |] ==> P
lemmas rev_cases:
[| xs = [] ==> P; !!ys y. xs = ys @ [y] ==> P |] ==> P
lemma finite_set:
finite (set xs)
lemma set_append:
set (xs @ ys) = set xs ∪ set ys
lemma hd_in_set:
l = x # xs ==> x ∈ set l
lemma set_subset_Cons:
set xs ⊆ set (x # xs)
lemma set_ConsD:
y ∈ set (x # xs) ==> y = x ∨ y ∈ set xs
lemma set_empty:
(set xs = {}) = (xs = [])
lemma set_empty2:
({} = set xs) = (xs = [])
lemma set_rev:
set (rev xs) = set xs
lemma set_map:
set (map f xs) = f ` set xs
lemma set_filter:
set (filter P xs) = {x : set xs. P x}
lemma set_upt:
set [i..<j] = {k. i ≤ k ∧ k < j}
lemma in_set_conv_decomp:
(x ∈ set xs) = (∃ys zs. xs = ys @ x # zs)
lemma finite_list:
finite A ==> ∃l. set l = A
lemma card_length:
card (set xs) ≤ length xs
lemma filter_append:
filter P (xs @ ys) = filter P xs @ filter P ys
lemma rev_filter:
rev (filter P xs) = filter P (rev xs)
lemma filter_filter:
filter P (filter Q xs) = [x∈xs . Q x ∧ P x]
lemma length_filter_le:
length (filter P xs) ≤ length xs
lemma filter_True:
∀x∈set xs. P x ==> filter P xs = xs
lemma filter_False:
∀x∈set xs. ¬ P x ==> filter P xs = []
lemma filter_empty_conv:
(filter P xs = []) = (∀x∈set xs. ¬ P x)
lemma filter_id_conv:
(filter P xs = xs) = (∀x∈set xs. P x)
lemma filter_map:
filter P (map f xs) = map f (filter (P o f) xs)
lemma length_filter_map:
length (filter P (map f xs)) = length (filter (P o f) xs)
lemma filter_is_subset:
set (filter P xs) ⊆ set xs
lemma length_filter_less:
[| x ∈ set xs; ¬ P x |] ==> length (filter P xs) < length xs
lemma length_filter_conv_card:
length (filter p xs) = card {i. i < length xs ∧ p (xs ! i)}
lemma Cons_eq_filterD:
x # xs = filter P ys ==> ∃us vs. ys = us @ x # vs ∧ (∀u∈set us. ¬ P u) ∧ P x ∧ xs = filter P vs
lemma filter_eq_ConsD:
filter P ys = x # xs ==> ∃us vs. ys = us @ x # vs ∧ (∀u∈set us. ¬ P u) ∧ P x ∧ xs = filter P vs
lemma filter_eq_Cons_iff:
(filter P ys = x # xs) = (∃us vs. ys = us @ x # vs ∧ (∀u∈set us. ¬ P u) ∧ P x ∧ xs = filter P vs)
lemma Cons_eq_filter_iff:
(x # xs = filter P ys) = (∃us vs. ys = us @ x # vs ∧ (∀u∈set us. ¬ P u) ∧ P x ∧ xs = filter P vs)
lemma filter_cong:
[| xs = ys; !!x. x ∈ set ys ==> P x = Q x |] ==> filter P xs = filter Q ys
lemma concat_append:
concat (xs @ ys) = concat xs @ concat ys
lemma concat_eq_Nil_conv:
(concat xss = []) = (∀xs∈set xss. xs = [])
lemma Nil_eq_concat_conv:
([] = concat xss) = (∀xs∈set xss. xs = [])
lemma set_concat:
set (concat xs) = Union (set ` set xs)
lemma map_concat:
map f (concat xs) = concat (map (map f) xs)
lemma filter_concat:
filter p (concat xs) = concat (map (filter p) xs)
lemma rev_concat:
rev (concat xs) = concat (map rev (rev xs))
lemma nth_Cons_0:
(x # xs) ! 0 = x
lemma nth_Cons_Suc:
(x # xs) ! Suc n = xs ! n
lemma nth_append:
(xs @ ys) ! n = (if n < length xs then xs ! n else ys ! (n - length xs))
lemma nth_append_length:
(xs @ x # ys) ! length xs = x
lemma nth_append_length_plus:
(xs @ ys) ! (length xs + n) = ys ! n
lemma nth_map:
n < length xs ==> map f xs ! n = f (xs ! n)
lemma set_conv_nth:
set xs = {xs ! i |i. i < length xs}
lemma in_set_conv_nth:
(x ∈ set xs) = (∃i<length xs. xs ! i = x)
lemma list_ball_nth:
[| n < length xs; ∀x∈set xs. P x |] ==> P (xs ! n)
lemma nth_mem:
n < length xs ==> xs ! n ∈ set xs
lemma all_nth_imp_all_set:
[| ∀i<length xs. P (xs ! i); x ∈ set xs |] ==> P x
lemma all_set_conv_all_nth:
(∀x∈set xs. P x) = (∀i<length xs. P (xs ! i))
lemma length_list_update:
length (xs[i := x]) = length xs
lemma nth_list_update:
i < length xs ==> xs[i := x] ! j = (if i = j then x else xs ! j)
lemma nth_list_update_eq:
i < length xs ==> xs[i := x] ! i = x
lemma nth_list_update_neq:
i ≠ j ==> xs[i := x] ! j = xs ! j
lemma list_update_overwrite:
i < length xs ==> xs[i := x, i := y] = xs[i := y]
lemma list_update_id:
i < length xs ==> xs[i := xs ! i] = xs
lemma list_update_beyond:
length xs ≤ i ==> xs[i := x] = xs
lemma list_update_same_conv:
i < length xs ==> (xs[i := x] = xs) = (xs ! i = x)
lemma list_update_append1:
i < length xs ==> (xs @ ys)[i := x] = xs[i := x] @ ys
lemma list_update_append:
(xs @ ys)[n := x] = (if n < length xs then xs[n := x] @ ys else xs @ ys[n - length xs := x])
lemma list_update_length:
(xs @ x # ys)[length xs := y] = xs @ y # ys
lemma update_zip:
length xs = length ys ==> zip xs ys[i := xy] = zip (xs[i := fst xy]) (ys[i := snd xy])
lemma set_update_subset_insert:
set (xs[i := x]) ⊆ insert x (set xs)
lemma set_update_subsetI:
[| set xs ⊆ A; x ∈ A |] ==> set (xs[i := x]) ⊆ A
lemma set_update_memI:
n < length xs ==> x ∈ set (xs[n := x])
lemma last_snoc:
last (xs @ [x]) = x
lemma butlast_snoc:
butlast (xs @ [x]) = xs
lemma last_ConsL:
xs = [] ==> last (x # xs) = x
lemma last_ConsR:
xs ≠ [] ==> last (x # xs) = last xs
lemma last_append:
last (xs @ ys) = (if ys = [] then last xs else last ys)
lemma last_appendL:
ys = [] ==> last (xs @ ys) = last xs
lemma last_appendR:
ys ≠ [] ==> last (xs @ ys) = last ys
lemma length_butlast:
length (butlast xs) = length xs - 1
lemma butlast_append:
butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)
lemma append_butlast_last_id:
xs ≠ [] ==> butlast xs @ [last xs] = xs
lemma in_set_butlastD:
x ∈ set (butlast xs) ==> x ∈ set xs
lemma in_set_butlast_appendI:
x ∈ set (butlast xs) ∨ x ∈ set (butlast ys) ==> x ∈ set (butlast (xs @ ys))
lemma last_drop:
n < length xs ==> last (drop n xs) = last xs
lemma last_conv_nth:
xs ≠ [] ==> last xs = xs ! (length xs - 1)
lemma take_0:
take 0 xs = []
lemma drop_0:
drop 0 xs = xs
lemma take_Suc_Cons:
take (Suc n) (x # xs) = x # take n xs
lemma drop_Suc_Cons:
drop (Suc n) (x # xs) = drop n xs
lemma take_Suc:
xs ≠ [] ==> take (Suc n) xs = hd xs # take n (tl xs)
lemma drop_Suc:
drop (Suc n) xs = drop n (tl xs)
lemma drop_tl:
drop n (tl xs) = tl (drop n xs)
lemma nth_via_drop:
drop n xs = y # ys ==> xs ! n = y
lemma take_Suc_conv_app_nth:
i < length xs ==> take (Suc i) xs = take i xs @ [xs ! i]
lemma drop_Suc_conv_tl:
i < length xs ==> xs ! i # drop (Suc i) xs = drop i xs
lemma length_take:
length (take n xs) = min (length xs) n
lemma length_drop:
length (drop n xs) = length xs - n
lemma take_all:
length xs ≤ n ==> take n xs = xs
lemma drop_all:
length xs ≤ n ==> drop n xs = []
lemma take_append:
take n (xs @ ys) = take n xs @ take (n - length xs) ys
lemma drop_append:
drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys
lemma take_take:
take n (take m xs) = take (min n m) xs
lemma drop_drop:
drop n (drop m xs) = drop (n + m) xs
lemma take_drop:
take n (drop m xs) = drop m (take (n + m) xs)
lemma drop_take:
drop n (take m xs) = take (m - n) (drop n xs)
lemma append_take_drop_id:
take n xs @ drop n xs = xs
lemma take_eq_Nil:
(take n xs = []) = (n = 0 ∨ xs = [])
lemma drop_eq_Nil:
(drop n xs = []) = (length xs ≤ n)
lemma take_map:
take n (map f xs) = map f (take n xs)
lemma drop_map:
drop n (map f xs) = map f (drop n xs)
lemma rev_take:
rev (take i xs) = drop (length xs - i) (rev xs)
lemma rev_drop:
rev (drop i xs) = take (length xs - i) (rev xs)
lemma nth_take:
i < n ==> take n xs ! i = xs ! i
lemma nth_drop:
n + i ≤ length xs ==> drop n xs ! i = xs ! (n + i)
lemma set_take_subset:
set (take n xs) ⊆ set xs
lemma set_drop_subset:
set (drop n xs) ⊆ set xs
lemma in_set_takeD:
x ∈ set (take n xs) ==> x ∈ set xs
lemma in_set_dropD:
x ∈ set (drop n xs) ==> x ∈ set xs
lemma append_eq_conv_conj:
(xs @ ys = zs) = (xs = take (length xs) zs ∧ ys = drop (length xs) zs)
lemma take_add:
i + j ≤ length xs ==> take (i + j) xs = take i xs @ take j (drop i xs)
lemma append_eq_append_conv_if:
(xs1 @ xs2 = ys1 @ ys2) = (if length xs1 ≤ length ys1 then xs1 = take (length xs1) ys1 ∧ xs2 = drop (length xs1) ys1 @ ys2 else take (length ys1) xs1 = ys1 ∧ drop (length ys1) xs1 @ xs2 = ys2)
lemma take_hd_drop:
n < length xs ==> take n xs @ [hd (drop n xs)] = take (n + 1) xs
lemma id_take_nth_drop:
i < length xs ==> xs = take i xs @ xs ! i # drop (Suc i) xs
lemma upd_conv_take_nth_drop:
i < length xs ==> xs[i := a] = take i xs @ a # drop (Suc i) xs
lemma takeWhile_dropWhile_id:
takeWhile P xs @ dropWhile P xs = xs
lemma takeWhile_append1:
[| x ∈ set xs; ¬ P x |] ==> takeWhile P (xs @ ys) = takeWhile P xs
lemma takeWhile_append2:
(!!x. x ∈ set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys
lemma takeWhile_tail:
¬ P x ==> takeWhile P (xs @ x # l) = takeWhile P xs
lemma dropWhile_append1:
[| x ∈ set xs; ¬ P x |] ==> dropWhile P (xs @ ys) = dropWhile P xs @ ys
lemma dropWhile_append2:
(!!x. x ∈ set xs ==> P x) ==> dropWhile P (xs @ ys) = dropWhile P ys
lemma set_take_whileD:
x ∈ set (takeWhile P xs) ==> x ∈ set xs ∧ P x
lemma takeWhile_eq_all_conv:
(takeWhile P xs = xs) = (∀x∈set xs. P x)
lemma dropWhile_eq_Nil_conv:
(dropWhile P xs = []) = (∀x∈set xs. P x)
lemma dropWhile_eq_Cons_conv:
(dropWhile P xs = y # ys) = (xs = takeWhile P xs @ y # ys ∧ ¬ P y)
lemma takeWhile_neq_rev:
[| distinct xs; x ∈ set xs |] ==> takeWhile (%y. y ≠ x) (rev xs) = rev (tl (dropWhile (%y. y ≠ x) xs))
lemma dropWhile_neq_rev:
[| distinct xs; x ∈ set xs |] ==> dropWhile (%y. y ≠ x) (rev xs) = x # rev (takeWhile (%y. y ≠ x) xs)
lemma zip_Nil:
zip [] ys = []
lemma zip_Cons_Cons:
zip (x # xs) (y # ys) = (x, y) # zip xs ys
lemma zip_Cons1:
zip (x # xs) ys = (case ys of [] => [] | y # ys => (x, y) # zip xs ys)
lemma length_zip:
length (zip xs ys) = min (length xs) (length ys)
lemma zip_append1:
zip (xs @ ys) zs = zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)
lemma zip_append2:
zip xs (ys @ zs) = zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs
lemma zip_append:
[| length xs = length us; length ys = length vs |] ==> zip (xs @ ys) (us @ vs) = zip xs us @ zip ys vs
lemma zip_rev:
length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)
lemma nth_zip:
[| i < length xs; i < length ys |] ==> zip xs ys ! i = (xs ! i, ys ! i)
lemma set_zip:
set (zip xs ys) = {(xs ! i, ys ! i) |i. i < min (length xs) (length ys)}
lemma zip_update:
length xs = length ys ==> zip (xs[i := x]) (ys[i := y]) = zip xs ys[i := (x, y)]
lemma zip_replicate:
zip (replicate i x) (replicate j y) = replicate (min i j) (x, y)
lemma list_all2_lengthD:
list_all2 P xs ys ==> length xs = length ys
lemma list_all2_Nil:
list_all2 P [] ys = (ys = [])
lemma list_all2_Nil2:
list_all2 P xs [] = (xs = [])
lemma list_all2_Cons:
list_all2 P (x # xs) (y # ys) = (P x y ∧ list_all2 P xs ys)
lemma list_all2_Cons1:
list_all2 P (x # xs) ys = (∃z zs. ys = z # zs ∧ P x z ∧ list_all2 P xs zs)
lemma list_all2_Cons2:
list_all2 P xs (y # ys) = (∃z zs. xs = z # zs ∧ P z y ∧ list_all2 P zs ys)
lemma list_all2_rev:
list_all2 P (rev xs) (rev ys) = list_all2 P xs ys
lemma list_all2_rev1:
list_all2 P (rev xs) ys = list_all2 P xs (rev ys)
lemma list_all2_append1:
list_all2 P (xs @ ys) zs = (∃us vs. zs = us @ vs ∧ length us = length xs ∧ length vs = length ys ∧ list_all2 P xs us ∧ list_all2 P ys vs)
lemma list_all2_append2:
list_all2 P xs (ys @ zs) = (∃us vs. xs = us @ vs ∧ length us = length ys ∧ length vs = length zs ∧ list_all2 P us ys ∧ list_all2 P vs zs)
lemma list_all2_append:
length xs = length ys ==> list_all2 P (xs @ us) (ys @ vs) = (list_all2 P xs ys ∧ list_all2 P us vs)
lemma list_all2_appendI:
[| list_all2 P a b; list_all2 P c d |] ==> list_all2 P (a @ c) (b @ d)
lemma list_all2_conv_all_nth:
list_all2 P xs ys = (length xs = length ys ∧ (∀i<length xs. P (xs ! i) (ys ! i)))
lemma list_all2_trans:
[| !!a b c. [| P1.0 a b; P2.0 b c |] ==> P3.0 a c; list_all2 P1.0 as bs; list_all2 P2.0 bs cs |] ==> list_all2 P3.0 as cs
lemma list_all2_all_nthI:
[| length a = length b; !!n. n < length a ==> P (a ! n) (b ! n) |] ==> list_all2 P a b
lemma list_all2I:
[| ∀x∈set (zip a b). split P x; length a = length b |] ==> list_all2 P a b
lemma list_all2_nthD:
[| list_all2 P xs ys; p < length xs |] ==> P (xs ! p) (ys ! p)
lemma list_all2_nthD2:
[| list_all2 P xs ys; p < length ys |] ==> P (xs ! p) (ys ! p)
lemma list_all2_map1:
list_all2 P (map f as) bs = list_all2 (%x. P (f x)) as bs
lemma list_all2_map2:
list_all2 P as (map f bs) = list_all2 (%x y. P x (f y)) as bs
lemma list_all2_refl:
(!!x. P x x) ==> list_all2 P xs xs
lemma list_all2_update_cong:
[| i < length xs; list_all2 P xs ys; P x y |] ==> list_all2 P (xs[i := x]) (ys[i := y])
lemma list_all2_update_cong2:
[| list_all2 P xs ys; P x y; i < length ys |] ==> list_all2 P (xs[i := x]) (ys[i := y])
lemma list_all2_takeI:
list_all2 P xs ys ==> list_all2 P (take n xs) (take n ys)
lemma list_all2_dropI:
list_all2 P as bs ==> list_all2 P (drop n as) (drop n bs)
lemma list_all2_mono:
[| list_all2 P x y; !!x y. P x y ==> Q x y |] ==> list_all2 Q x y
lemma foldl_append:
foldl f a (xs @ ys) = foldl f (foldl f a xs) ys
lemma foldr_append:
foldr f (xs @ ys) a = foldr f xs (foldr f ys a)
lemma foldr_foldl:
foldr f xs a = foldl (%x y. f y x) a (rev xs)
lemma foldl_foldr:
foldl f a xs = foldr (%x y. f y x) (rev xs) a
lemma start_le_sum:
m ≤ n ==> m ≤ foldl op + n ns
lemma elem_le_sum:
n ∈ set ns ==> n ≤ foldl op + 0 ns
lemma sum_eq_0_conv:
(foldl op + m ns = 0) = (m = 0 ∧ (∀n∈set ns. n = 0))
lemma upt_rec:
[i..<j] = (if i < j then i # [Suc i..<j] else [])
lemma upt_conv_Nil:
j ≤ i ==> [i..<j] = []
lemma upt_eq_Nil_conv:
([i..<j] = []) = (j = 0 ∨ j ≤ i)
lemma upt_eq_Cons_conv:
([i..<j] = x # xs) = (i < j ∧ i = x ∧ [i + 1..<j] = xs)
lemma upt_Suc_append:
i ≤ j ==> [i..j] = [i..<j] @ [j]
lemma upt_conv_Cons:
i < j ==> [i..<j] = i # [Suc i..<j]
lemma upt_add_eq_append:
i ≤ j ==> [i..<j + k] = [i..<j] @ [j..<j + k]
lemma length_upt:
length [i..<j] = j - i
lemma nth_upt:
i + k < j ==> [i..<j] ! k = i + k
lemma take_upt:
i + m ≤ n ==> take m [i..<n] = [i..<i + m]
lemma drop_upt:
drop m [i..<j] = [i + m..<j]
lemma map_Suc_upt:
map Suc [m..<n] = [Suc m..n]
lemma nth_map_upt:
i < n - m ==> map f [m..<n] ! i = f (m + i)
lemma nth_take_lemma:
[| k ≤ length xs; k ≤ length ys; !!i. i < k --> xs ! i = ys ! i |] ==> take k xs = take k ys
lemma nth_equalityI:
[| length xs = length ys; ∀i<length xs. xs ! i = ys ! i |] ==> xs = ys
lemma list_all2_antisym:
[| !!x y. [| P x y; Q y x |] ==> x = y; list_all2 P xs ys; list_all2 Q ys xs |] ==> xs = ys
lemma take_equalityI:
∀i. take i xs = take i ys ==> xs = ys
lemma take_Cons':
take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)
lemma drop_Cons':
drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)
lemma nth_Cons':
(x # xs) ! n = (if n = 0 then x else xs ! (n - 1))
lemmas
take (number_of v) (x # xs) = (if number_of v = 0 then [] else x # take (number_of v - 1) xs)
drop (number_of v) (x # xs) = (if number_of v = 0 then x # xs else drop (number_of v - 1) xs)
(x # xs) ! number_of v = (if number_of v = 0 then x else xs ! (number_of v - 1))
lemmas
take (number_of v) (x # xs) = (if number_of v = 0 then [] else x # take (number_of v - 1) xs)
drop (number_of v) (x # xs) = (if number_of v = 0 then x # xs else drop (number_of v - 1) xs)
(x # xs) ! number_of v = (if number_of v = 0 then x else xs ! (number_of v - 1))
lemma distinct_append:
distinct (xs @ ys) = (distinct xs ∧ distinct ys ∧ set xs ∩ set ys = {})
lemma distinct_rev:
distinct (rev xs) = distinct xs
lemma set_remdups:
set (remdups xs) = set xs
lemma distinct_remdups:
distinct (remdups xs)
lemma remdups_eq_nil_iff:
(remdups x = []) = (x = [])
lemma remdups_eq_nil_right_iff:
([] = remdups x) = (x = [])
lemma length_remdups_leq:
length (remdups xs) ≤ length xs
lemma length_remdups_eq:
(length (remdups xs) = length xs) = (remdups xs = xs)
lemma distinct_filter:
distinct xs ==> distinct (filter P xs)
lemma distinct_map_filterI:
distinct (map f xs) ==> distinct (map f (filter P xs))
lemma distinct_upt:
distinct [i..<j]
lemma distinct_take:
distinct xs ==> distinct (take i xs)
lemma distinct_drop:
distinct xs ==> distinct (drop i xs)
lemma distinct_list_update:
[| distinct xs; a ∉ set xs - {xs ! i} |] ==> distinct (xs[i := a])
lemma distinct_conv_nth:
distinct xs = (∀i<length xs. ∀j<length xs. i ≠ j --> xs ! i ≠ xs ! j)
lemma distinct_card:
distinct xs ==> card (set xs) = length xs
lemma card_distinct:
card (set xs) = length xs ==> distinct xs
lemma inj_on_setI:
distinct (map f xs) ==> inj_on f (set xs)
lemma inj_on_set_conv:
distinct xs ==> inj_on f (set xs) = distinct (map f xs)
lemma set_remove1_subset:
set (remove1 x xs) ⊆ set xs
lemma set_remove1_eq:
distinct xs ==> set (remove1 x xs) = set xs - {x}
lemma notin_set_remove1:
x ∉ set xs ==> x ∉ set (remove1 y xs)
lemma distinct_remove1:
distinct xs ==> distinct (remove1 x xs)
lemma length_replicate:
length (replicate n x) = n
lemma map_replicate:
map f (replicate n x) = replicate n (f x)
lemma replicate_app_Cons_same:
replicate n x @ x # xs = x # replicate n x @ xs
lemma rev_replicate:
rev (replicate n x) = replicate n x
lemma replicate_add:
replicate (n + m) x = replicate n x @ replicate m x
lemma append_replicate_commute:
replicate n x @ replicate k x = replicate k x @ replicate n x
lemma hd_replicate:
n ≠ 0 ==> hd (replicate n x) = x
lemma tl_replicate:
n ≠ 0 ==> tl (replicate n x) = replicate (n - 1) x
lemma last_replicate:
n ≠ 0 ==> last (replicate n x) = x
lemma nth_replicate:
i < n ==> replicate n x ! i = x
lemma take_replicate:
take i (replicate k x) = replicate (min i k) x
lemma drop_replicate:
drop i (replicate k x) = replicate (k - i) x
lemma set_replicate_Suc:
set (replicate (Suc n) x) = {x}
lemma set_replicate:
n ≠ 0 ==> set (replicate n x) = {x}
lemma set_replicate_conv_if:
set (replicate n x) = (if n = 0 then {} else {x})
lemma in_set_replicateD:
x ∈ set (replicate n y) ==> x = y
lemma rotate_simps:
rotate1 [] = [] ∧ rotate1 (x # xs) = xs @ [x]
lemma rotate0:
rotate 0 = id
lemma rotate_Suc:
rotate (Suc n) xs = rotate1 (rotate n xs)
lemma rotate_add:
rotate (m + n) = rotate m o rotate n
lemma rotate_rotate:
rotate m (rotate n xs) = rotate (m + n) xs
lemma rotate1_length01:
length xs ≤ 1 ==> rotate1 xs = xs
lemma rotate_length01:
length xs ≤ 1 ==> rotate n xs = xs
lemma rotate1_hd_tl:
xs ≠ [] ==> rotate1 xs = tl xs @ [hd xs]
lemma rotate_drop_take:
rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs
lemma rotate_conv_mod:
rotate n xs = rotate (n mod length xs) xs
lemma rotate_id:
n mod length xs = 0 ==> rotate n xs = xs
lemma length_rotate1:
length (rotate1 xs) = length xs
lemma length_rotate:
length (rotate n xs) = length xs
lemma distinct1_rotate:
distinct (rotate1 xs) = distinct xs
lemma distinct_rotate:
distinct (rotate n xs) = distinct xs
lemma rotate_map:
rotate n (map f xs) = map f (rotate n xs)
lemma set_rotate1:
set (rotate1 xs) = set xs
lemma set_rotate:
set (rotate n xs) = set xs
lemma rotate1_is_Nil_conv:
(rotate1 xs = []) = (xs = [])
lemma rotate_is_Nil_conv:
(rotate n xs = []) = (xs = [])
lemma rotate_rev:
rotate n (rev xs) = rev (rotate (length xs - n mod length xs) xs)
lemma sublist_empty:
sublist xs {} = []
lemma sublist_nil:
sublist [] A = []
lemma length_sublist:
length (sublist xs I) = card {i. i < length xs ∧ i ∈ I}
lemma sublist_shift_lemma_Suc:
map fst [p∈zip xs is . P (Suc (snd p))] = map fst [p∈zip xs (map Suc is) . P (snd p)]
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]
lemma sublist_append:
sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l ∈ A}
lemma sublist_Cons:
sublist (x # l) A = (if 0 ∈ A then [x] else []) @ sublist l {j. Suc j ∈ A}
lemma set_sublist:
set (sublist xs I) = {xs ! i |i. i < length xs ∧ i ∈ I}
lemma set_sublist_subset:
set (sublist xs I) ⊆ set xs
lemma notin_set_sublistI:
x ∉ set xs ==> x ∉ set (sublist xs I)
lemma in_set_sublistD:
x ∈ set (sublist xs I) ==> x ∈ set xs
lemma sublist_singleton:
sublist [x] A = (if 0 ∈ A then [x] else [])
lemma distinct_sublistI:
distinct xs ==> distinct (sublist xs I)
lemma sublist_upt_eq_take:
sublist l {..<n} = take n l
lemma filter_in_sublist:
distinct xs ==> [x∈xs . x ∈ set (sublist xs s)] = sublist xs s
lemmas listsE:
[| x # l ∈ lists A; [| x ∈ A; l ∈ lists A |] ==> P |] ==> P
lemma lists_mono:
A ⊆ B ==> lists A ⊆ lists B
lemma lists_IntI:
[| l ∈ lists A; l ∈ lists B |] ==> l ∈ lists (A ∩ B)
lemma lists_Int_eq:
lists (A ∩ B) = lists A ∩ lists B
lemma append_in_lists_conv:
(xs @ ys ∈ lists A) = (xs ∈ lists A ∧ ys ∈ lists A)
lemma in_lists_conv_set:
(xs ∈ lists A) = (∀x∈set xs. x ∈ A)
lemma in_listsD:
xs ∈ lists A ==> ∀x∈set xs. x ∈ A
lemma in_listsI:
∀x∈set xs. x ∈ A ==> xs ∈ lists A
lemma lists_UNIV:
lists UNIV = UNIV
lemma mem_iff:
x mem xs = (x ∈ set xs)
lemma list_inter_conv:
set (list_inter xs ys) = set xs ∩ set ys
lemma list_all_iff:
list_all P xs = (∀x∈set xs. P x)
lemma list_all_append:
list_all P (xs @ ys) = (list_all P xs ∧ list_all P ys)
lemma list_all_rev:
list_all P (rev xs) = list_all P xs
lemma list_ex_iff:
list_ex P xs = (∃x∈set xs. P x)
lemma itrev:
∀ys. itrev xs ys = rev xs @ ys
lemma filtermap_conv:
filtermap f xs = map (%x. the (f x)) [x∈xs . f x ≠ None]
lemma map_filter_conv:
map_filter f P xs = map f (filter P xs)
lemmas in_set_code:
x2 ∈ set xs2 == x2 mem xs2
lemmas in_set_code:
x2 ∈ set xs2 == x2 mem xs2
lemma rev_code:
rev xs == itrev xs []
lemma distinct_Cons_mem:
distinct (x # xs) = (¬ x mem xs ∧ distinct xs)
lemma remdups_Cons_mem:
remdups (x # xs) = (if x mem xs then remdups xs else x # remdups xs)
lemma list_inter_Cons_mem:
list_inter (a # as) bs = (if a mem bs then a # list_inter as bs else list_inter as bs)
lemmas list_bex_code:
∃x∈set xs2. P2 x == list_ex P2 xs2
lemmas list_bex_code:
∃x∈set xs2. P2 x == list_ex P2 xs2
lemmas list_ball_code:
∀x∈set xs2. P2 x == list_all P2 xs2
lemmas list_ball_code:
∀x∈set xs2. P2 x == list_all P2 xs2
lemma ListMem_iff:
((x, xs) ∈ ListMem) = (x ∈ set xs)
lemma set_Cons_sing_Nil:
set_Cons A {[]} = (%x. [x]) ` A
lemma wf_lexn:
wf r ==> wf (lexn r n)
lemma lexn_length:
(xs, ys) ∈ lexn r n ==> length xs = n ∧ length ys = n
lemma wf_lex:
wf r ==> wf (lex r)
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)}
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)}
lemma wf_lenlex:
wf r ==> wf (lenlex r)
lemma lenlex_conv:
lenlex r = {(xs, ys). length xs < length ys ∨ length xs = length ys ∧ (xs, ys) ∈ lex r}
lemma Nil_notin_lex:
([], ys) ∉ lex r
lemma Nil2_notin_lex:
(xs, []) ∉ lex r
lemma Cons_in_lex:
((x # xs, y # ys) ∈ lex r) = ((x, y) ∈ r ∧ length xs = length ys ∨ x = y ∧ (xs, ys) ∈ lex r)
lemma lexord_Nil_left:
(([], y) ∈ lexord r) = (∃a x. y = a # x)
lemma lexord_Nil_right:
(x, []) ∉ lexord r
lemma lexord_cons_cons:
((a # x, b # y) ∈ lexord r) = ((a, b) ∈ r ∨ a = b ∧ (x, y) ∈ lexord r)
lemmas lexord_simps:
(([], y) ∈ lexord r) = (∃a x. y = a # x)
(x, []) ∉ lexord r
((a # x, b # y) ∈ lexord r) = ((a, b) ∈ r ∨ a = b ∧ (x, y) ∈ lexord r)
lemmas lexord_simps:
(([], y) ∈ lexord r) = (∃a x. y = a # x)
(x, []) ∉ lexord r
((a # x, b # y) ∈ lexord r) = ((a, b) ∈ r ∨ a = b ∧ (x, y) ∈ lexord r)
lemma lexord_append_rightI:
∃b z. y = b # z ==> (x, x @ y) ∈ lexord r
lemma lexord_append_left_rightI:
(a, b) ∈ r ==> (u @ a # x, u @ b # y) ∈ lexord r
lemma lexord_append_leftI:
(u, v) ∈ lexord r ==> (x @ u, x @ v) ∈ lexord r
lemma lexord_append_leftD:
[| (x @ u, x @ v) ∈ lexord r; ∀a. (a, a) ∉ r |] ==> (u, v) ∈ lexord r
lemma lexord_take_index_conv:
((x, y) ∈ lexord r) = (length x < length y ∧ take (length x) y = x ∨ (∃i<min (length x) (length y). take i x = take i y ∧ (x ! i, y ! i) ∈ r))
lemma lexord_lex:
((x, y) ∈ lex r) = ((x, y) ∈ lexord r ∧ length x = length y)
lemma lexord_irreflexive:
∀x. (x, x) ∉ r ==> (y, y) ∉ lexord r
lemma lexord_trans:
[| (x, y) ∈ lexord r; (y, z) ∈ lexord r; trans r |] ==> (x, z) ∈ lexord r
lemma lexord_transI:
trans r ==> trans (lexord r)
lemma lexord_linear:
∀a b. (a, b) ∈ r ∨ a = b ∨ (b, a) ∈ r ==> (x, y) ∈ lexord r ∨ x = y ∨ (y, x) ∈ lexord r
lemmas listrel_Nil1:
[| ([], xs) ∈ listrel r; xs = [] ==> P |] ==> P
lemmas listrel_Nil2:
[| (xs, []) ∈ listrel r; xs = [] ==> P |] ==> P
lemmas listrel_Cons1:
[| (y # ys, xs) ∈ listrel r; !!y ys. [| (y, y) ∈ r; (ys, ys) ∈ listrel r; xs = y # ys |] ==> P |] ==> P
lemmas listrel_Cons2:
[| (xs, y # ys) ∈ listrel r; !!x xs. [| (x, y) ∈ r; (xs, ys) ∈ listrel r; xs = x # xs |] ==> P |] ==> P
lemma listrel_mono:
r ⊆ s ==> listrel r ⊆ listrel s
lemma listrel_subset:
r ⊆ A × A ==> listrel r ⊆ lists A × lists A
lemma listrel_refl:
refl A r ==> refl (lists A) (listrel r)
lemma listrel_sym:
sym r ==> sym (listrel r)
lemma listrel_trans:
trans r ==> trans (listrel r)
theorem equiv_listrel:
equiv A r ==> equiv (lists A) (listrel r)
lemma listrel_Nil:
listrel r `` {[]} = {[]}
lemma listrel_Cons:
listrel r `` {x # xs} = set_Cons (r `` {x}) (listrel r `` {xs})