(* Title: HOL/List.thy Author: Tobias Nipkow *) section ‹The datatype of finite lists› theory List imports Sledgehammer Code_Numeral Lifting_Set begin datatype (set: 'a) list = Nil ("[]") | Cons (hd: 'a) (tl: "'a list") (infixr "#" 65) for map: map rel: list_all2 pred: list_all where "tl [] = []" datatype_compat list lemma [case_names Nil Cons, cases type: list]: ― ‹for backward compatibility -- names of variables differ› "(y = [] ⟹ P) ⟹ (⋀a list. y = a # list ⟹ P) ⟹ P" by (rule list.exhaust) lemma [case_names Nil Cons, induct type: list]: ― ‹for backward compatibility -- names of variables differ› "P [] ⟹ (⋀a list. P list ⟹ P (a # list)) ⟹ P list" by (rule list.induct) text ‹Compatibility:› setup ‹Sign.mandatory_path "list"› lemmas inducts = list.induct lemmas recs = list.rec lemmas cases = list.case setup ‹Sign.parent_path› lemmas set_simps = list.set (* legacy *) syntax ― ‹list Enumeration› "_list" :: "args => 'a list" ("[(_)]") translations "[x, xs]" == "x#[xs]" "[x]" == "x#[]" subsection ‹Basic list processing functions› primrec (nonexhaustive) last :: "'a list ⇒ 'a" where "last (x # xs) = (if xs = [] then x else last xs)" primrec butlast :: "'a list ⇒ 'a list" where "butlast [] = []" | "butlast (x # xs) = (if xs = [] then [] else x # butlast xs)" lemma set_rec: "set xs = rec_list {} (λx _. insert x) xs" by (induct xs) auto definition coset :: "'a list ⇒ 'a set" where [simp]: "coset xs = - set xs" primrec append :: "'a list ⇒ 'a list ⇒ 'a list" (infixr "@" 65) where append_Nil: "[] @ ys = ys" | append_Cons: "(x#xs) @ ys = x # xs @ ys" primrec rev :: "'a list ⇒ 'a list" where "rev [] = []" | "rev (x # xs) = rev xs @ [x]" primrec filter:: "('a ⇒ bool) ⇒ 'a list ⇒ 'a list" where "filter P [] = []" | "filter P (x # xs) = (if P x then x # filter P xs else filter P xs)" text ‹Special input syntax for filter:› syntax (ASCII) "_filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_<-_./ _])") syntax "_filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_←_ ./ _])") translations "[x<-xs . P]" ⇀ "CONST filter (λx. P) xs" primrec fold :: "('a ⇒ 'b ⇒ 'b) ⇒ 'a list ⇒ 'b ⇒ 'b" where fold_Nil: "fold f [] = id" | fold_Cons: "fold f (x # xs) = fold f xs ∘ f x" primrec foldr :: "('a ⇒ 'b ⇒ 'b) ⇒ 'a list ⇒ 'b ⇒ 'b" where foldr_Nil: "foldr f [] = id" | foldr_Cons: "foldr f (x # xs) = f x ∘ foldr f xs" primrec foldl :: "('b ⇒ 'a ⇒ 'b) ⇒ 'b ⇒ 'a list ⇒ 'b" where foldl_Nil: "foldl f a [] = a" | foldl_Cons: "foldl f a (x # xs) = foldl f (f a x) xs" primrec concat:: "'a list list ⇒ 'a list" where "concat [] = []" | "concat (x # xs) = x @ concat xs" primrec drop:: "nat ⇒ 'a list ⇒ 'a list" where 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 ‹n = 0› and ‹n = Suc k›› primrec take:: "nat ⇒ 'a list ⇒ 'a list" where 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 ‹n = 0› and ‹n = Suc k›› primrec (nonexhaustive) nth :: "'a list => nat => 'a" (infixl "!" 100) where 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 ‹n = 0› and ‹n = Suc k›› primrec list_update :: "'a list ⇒ nat ⇒ 'a ⇒ 'a list" where "list_update [] i v = []" | "list_update (x # xs) i v = (case i of 0 ⇒ v # xs | Suc j ⇒ x # list_update xs j v)" nonterminal lupdbinds and lupdbind syntax "_lupdbind":: "['a, 'a] => lupdbind" ("(2_ :=/ _)") "" :: "lupdbind => lupdbinds" ("_") "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds" ("_,/ _") "_LUpdate" :: "['a, lupdbinds] => 'a" ("_/[(_)]" [900,0] 900) translations "_LUpdate xs (_lupdbinds b bs)" == "_LUpdate (_LUpdate xs b) bs" "xs[i:=x]" == "CONST list_update xs i x" primrec takeWhile :: "('a ⇒ bool) ⇒ 'a list ⇒ 'a list" where "takeWhile P [] = []" | "takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])" primrec dropWhile :: "('a ⇒ bool) ⇒ 'a list ⇒ 'a list" where "dropWhile P [] = []" | "dropWhile P (x # xs) = (if P x then dropWhile P xs else x # xs)" primrec zip :: "'a list ⇒ 'b list ⇒ ('a × 'b) list" where "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 ‹xs = []› and ‹xs = z # zs›› abbreviation map2 :: "('a ⇒ 'b ⇒ 'c) ⇒ 'a list ⇒ 'b list ⇒ 'c list" where "map2 f xs ys ≡ map (λ(x,y). f x y) (zip xs ys)" primrec product :: "'a list ⇒ 'b list ⇒ ('a × 'b) list" where "product [] _ = []" | "product (x#xs) ys = map (Pair x) ys @ product xs ys" hide_const (open) product primrec product_lists :: "'a list list ⇒ 'a list list" where "product_lists [] = [[]]" | "product_lists (xs # xss) = concat (map (λx. map (Cons x) (product_lists xss)) xs)" primrec upt :: "nat ⇒ nat ⇒ nat list" ("(1[_..</_'])") where upt_0: "[i..<0] = []" | upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])" definition insert :: "'a ⇒ 'a list ⇒ 'a list" where "insert x xs = (if x ∈ set xs then xs else x # xs)" definition union :: "'a list ⇒ 'a list ⇒ 'a list" where "union = fold insert" hide_const (open) insert union hide_fact (open) insert_def union_def primrec find :: "('a ⇒ bool) ⇒ 'a list ⇒ 'a option" where "find _ [] = None" | "find P (x#xs) = (if P x then Some x else find P xs)" text ‹In the context of multisets, ‹count_list› is equivalent to @{term "count ∘ mset"} and it it advisable to use the latter.› primrec count_list :: "'a list ⇒ 'a ⇒ nat" where "count_list [] y = 0" | "count_list (x#xs) y = (if x=y then count_list xs y + 1 else count_list xs y)" definition "extract" :: "('a ⇒ bool) ⇒ 'a list ⇒ ('a list * 'a * 'a list) option" where "extract P xs = (case dropWhile (Not ∘ P) xs of [] ⇒ None | y#ys ⇒ Some(takeWhile (Not ∘ P) xs, y, ys))" hide_const (open) "extract" primrec those :: "'a option list ⇒ 'a list option" where "those [] = Some []" | "those (x # xs) = (case x of None ⇒ None | Some y ⇒ map_option (Cons y) (those xs))" primrec remove1 :: "'a ⇒ 'a list ⇒ 'a list" where "remove1 x [] = []" | "remove1 x (y # xs) = (if x = y then xs else y # remove1 x xs)" primrec removeAll :: "'a ⇒ 'a list ⇒ 'a list" where "removeAll x [] = []" | "removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)" primrec distinct :: "'a list ⇒ bool" where "distinct [] ⟷ True" | "distinct (x # xs) ⟷ x ∉ set xs ∧ distinct xs" primrec remdups :: "'a list ⇒ 'a list" where "remdups [] = []" | "remdups (x # xs) = (if x ∈ set xs then remdups xs else x # remdups xs)" fun remdups_adj :: "'a list ⇒ 'a list" where "remdups_adj [] = []" | "remdups_adj [x] = [x]" | "remdups_adj (x # y # xs) = (if x = y then remdups_adj (x # xs) else x # remdups_adj (y # xs))" primrec replicate :: "nat ⇒ 'a ⇒ 'a list" where replicate_0: "replicate 0 x = []" | replicate_Suc: "replicate (Suc n) x = x # replicate n x" text ‹ Function ‹size› is overloaded for all datatypes. Users may refer to the list version as ‹length›.› abbreviation length :: "'a list ⇒ nat" where "length ≡ size" definition enumerate :: "nat ⇒ 'a list ⇒ (nat × 'a) list" where enumerate_eq_zip: "enumerate n xs = zip [n..<n + length xs] xs" primrec rotate1 :: "'a list ⇒ 'a list" where "rotate1 [] = []" | "rotate1 (x # xs) = xs @ [x]" definition rotate :: "nat ⇒ 'a list ⇒ 'a list" where "rotate n = rotate1 ^^ n" definition nths :: "'a list => nat set => 'a list" where "nths xs A = map fst (filter (λp. snd p ∈ A) (zip xs [0..<size xs]))" primrec subseqs :: "'a list ⇒ 'a list list" where "subseqs [] = [[]]" | "subseqs (x#xs) = (let xss = subseqs xs in map (Cons x) xss @ xss)" primrec n_lists :: "nat ⇒ 'a list ⇒ 'a list list" where "n_lists 0 xs = [[]]" | "n_lists (Suc n) xs = concat (map (λys. map (λy. y # ys) xs) (n_lists n xs))" hide_const (open) n_lists fun splice :: "'a list ⇒ 'a list ⇒ 'a list" where "splice [] ys = ys" | "splice xs [] = xs" | "splice (x#xs) (y#ys) = x # y # splice xs ys" function shuffle where "shuffle [] ys = {ys}" | "shuffle xs [] = {xs}" | "shuffle (x # xs) (y # ys) = (#) x ` shuffle xs (y # ys) ∪ (#) y ` shuffle (x # xs) ys" by pat_completeness simp_all termination by lexicographic_order text‹Use only if you cannot use @{const Min} instead:› fun min_list :: "'a::ord list ⇒ 'a" where "min_list (x # xs) = (case xs of [] ⇒ x | _ ⇒ min x (min_list xs))" text‹Returns first minimum:› fun arg_min_list :: "('a ⇒ ('b::linorder)) ⇒ 'a list ⇒ 'a" where "arg_min_list f [x] = x" | "arg_min_list f (x#y#zs) = (let m = arg_min_list f (y#zs) in if f x ≤ f m then x else m)" text‹ \begin{figure}[htbp] \fbox{ \begin{tabular}{l} @{lemma "[a,b]@[c,d] = [a,b,c,d]" by simp}\\ @{lemma "length [a,b,c] = 3" by simp}\\ @{lemma "set [a,b,c] = {a,b,c}" by simp}\\ @{lemma "map f [a,b,c] = [f a, f b, f c]" by simp}\\ @{lemma "rev [a,b,c] = [c,b,a]" by simp}\\ @{lemma "hd [a,b,c,d] = a" by simp}\\ @{lemma "tl [a,b,c,d] = [b,c,d]" by simp}\\ @{lemma "last [a,b,c,d] = d" by simp}\\ @{lemma "butlast [a,b,c,d] = [a,b,c]" by simp}\\ @{lemma[source] "filter (λn::nat. n<2) [0,2,1] = [0,1]" by simp}\\ @{lemma "concat [[a,b],[c,d,e],[],[f]] = [a,b,c,d,e,f]" by simp}\\ @{lemma "fold f [a,b,c] x = f c (f b (f a x))" by simp}\\ @{lemma "foldr f [a,b,c] x = f a (f b (f c x))" by simp}\\ @{lemma "foldl f x [a,b,c] = f (f (f x a) b) c" by simp}\\ @{lemma "zip [a,b,c] [x,y,z] = [(a,x),(b,y),(c,z)]" by simp}\\ @{lemma "zip [a,b] [x,y,z] = [(a,x),(b,y)]" by simp}\\ @{lemma "enumerate 3 [a,b,c] = [(3,a),(4,b),(5,c)]" by normalization}\\ @{lemma "List.product [a,b] [c,d] = [(a, c), (a, d), (b, c), (b, d)]" by simp}\\ @{lemma "product_lists [[a,b], [c], [d,e]] = [[a,c,d], [a,c,e], [b,c,d], [b,c,e]]" by simp}\\ @{lemma "splice [a,b,c] [x,y,z] = [a,x,b,y,c,z]" by simp}\\ @{lemma "splice [a,b,c,d] [x,y] = [a,x,b,y,c,d]" by simp}\\ @{lemma "shuffle [a,b] [c,d] = {[a,b,c,d],[a,c,b,d],[a,c,d,b],[c,a,b,d],[c,a,d,b],[c,d,a,b]}" by (simp add: insert_commute)}\\ @{lemma "take 2 [a,b,c,d] = [a,b]" by simp}\\ @{lemma "take 6 [a,b,c,d] = [a,b,c,d]" by simp}\\ @{lemma "drop 2 [a,b,c,d] = [c,d]" by simp}\\ @{lemma "drop 6 [a,b,c,d] = []" by simp}\\ @{lemma "takeWhile (%n::nat. n<3) [1,2,3,0] = [1,2]" by simp}\\ @{lemma "dropWhile (%n::nat. n<3) [1,2,3,0] = [3,0]" by simp}\\ @{lemma "distinct [2,0,1::nat]" by simp}\\ @{lemma "remdups [2,0,2,1::nat,2] = [0,1,2]" by simp}\\ @{lemma "remdups_adj [2,2,3,1,1::nat,2,1] = [2,3,1,2,1]" by simp}\\ @{lemma "List.insert 2 [0::nat,1,2] = [0,1,2]" by (simp add: List.insert_def)}\\ @{lemma "List.insert 3 [0::nat,1,2] = [3,0,1,2]" by (simp add: List.insert_def)}\\ @{lemma "List.union [2,3,4] [0::int,1,2] = [4,3,0,1,2]" by (simp add: List.insert_def List.union_def)}\\ @{lemma "List.find (%i::int. i>0) [0,0] = None" by simp}\\ @{lemma "List.find (%i::int. i>0) [0,1,0,2] = Some 1" by simp}\\ @{lemma "count_list [0,1,0,2::int] 0 = 2" by (simp)}\\ @{lemma "List.extract (%i::int. i>0) [0,0] = None" by(simp add: extract_def)}\\ @{lemma "List.extract (%i::int. i>0) [0,1,0,2] = Some([0], 1, [0,2])" by(simp add: extract_def)}\\ @{lemma "remove1 2 [2,0,2,1::nat,2] = [0,2,1,2]" by simp}\\ @{lemma "removeAll 2 [2,0,2,1::nat,2] = [0,1]" by simp}\\ @{lemma "nth [a,b,c,d] 2 = c" by simp}\\ @{lemma "[a,b,c,d][2 := x] = [a,b,x,d]" by simp}\\ @{lemma "nths [a,b,c,d,e] {0,2,3} = [a,c,d]" by (simp add:nths_def)}\\ @{lemma "subseqs [a,b] = [[a, b], [a], [b], []]" by simp}\\ @{lemma "List.n_lists 2 [a,b,c] = [[a, a], [b, a], [c, a], [a, b], [b, b], [c, b], [a, c], [b, c], [c, c]]" by (simp add: eval_nat_numeral)}\\ @{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by simp}\\ @{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate_def eval_nat_numeral)}\\ @{lemma "replicate 4 a = [a,a,a,a]" by (simp add:eval_nat_numeral)}\\ @{lemma "[2..<5] = [2,3,4]" by (simp add:eval_nat_numeral)}\\ @{lemma "min_list [3,1,-2::int] = -2" by (simp)}\\ @{lemma "arg_min_list (λi. i*i) [3,-1,1,-2::int] = -1" by (simp)} \end{tabular}} \caption{Characteristic examples} \label{fig:Characteristic} \end{figure} Figure~\ref{fig:Characteristic} shows characteristic examples that should give an intuitive understanding of the above functions. › text‹The following simple sort functions are intended for proofs, not for efficient implementations.› text ‹A sorted predicate w.r.t. a relation:› fun sorted_wrt :: "('a ⇒ 'a ⇒ bool) ⇒ 'a list ⇒ bool" where "sorted_wrt P [] = True" | "sorted_wrt P (x # ys) = ((∀y ∈ set ys. P x y) ∧ sorted_wrt P ys)" (* FIXME: define sorted in terms of sorted_wrt *) text ‹A class-based sorted predicate:› context linorder begin fun sorted :: "'a list ⇒ bool" where "sorted [] = True" | "sorted (x # ys) = ((∀y ∈ set ys. x ≤ y) ∧ sorted ys)" lemma sorted_sorted_wrt: "sorted = sorted_wrt (≤)" proof (rule ext) fix xs show "sorted xs = sorted_wrt (≤) xs" by(induction xs rule: sorted.induct) auto qed primrec insort_key :: "('b ⇒ 'a) ⇒ 'b ⇒ 'b list ⇒ 'b list" where "insort_key f x [] = [x]" | "insort_key f x (y#ys) = (if f x ≤ f y then (x#y#ys) else y#(insort_key f x ys))" definition sort_key :: "('b ⇒ 'a) ⇒ 'b list ⇒ 'b list" where "sort_key f xs = foldr (insort_key f) xs []" definition insort_insert_key :: "('b ⇒ 'a) ⇒ 'b ⇒ 'b list ⇒ 'b list" where "insort_insert_key f x xs = (if f x ∈ f ` set xs then xs else insort_key f x xs)" abbreviation "sort ≡ sort_key (λx. x)" abbreviation "insort ≡ insort_key (λx. x)" abbreviation "insort_insert ≡ insort_insert_key (λx. x)" definition stable_sort_key :: "(('b ⇒ 'a) ⇒ 'b list ⇒ 'b list) ⇒ bool" where "stable_sort_key sk = (∀f xs k. filter (λy. f y = k) (sk f xs) = filter (λy. f y = k) xs)" end subsubsection ‹List comprehension› text‹Input syntax for Haskell-like list comprehension notation. Typical example: ‹[(x,y). x ← xs, y ← ys, x ≠ y]›, the list of all pairs of distinct elements from ‹xs› and ‹ys›. The syntax is as in Haskell, except that ‹|› becomes a dot (like in Isabelle's set comprehension): ‹[e. x ← xs, …]› rather than \verb![e| x <- xs, ...]!. The qualifiers after the dot are \begin{description} \item[generators] ‹p ← xs›, where ‹p› is a pattern and ‹xs› an expression of list type, or \item[guards] ‹b›, where ‹b› is a boolean expression. %\item[local bindings] @ {text"let x = e"}. \end{description} Just like in Haskell, list comprehension is just a shorthand. To avoid misunderstandings, the translation into desugared form is not reversed upon output. Note that the translation of ‹[e. x ← xs]› is optmized to @{term"map (%x. e) xs"}. It is easy to write short list comprehensions which stand for complex expressions. During proofs, they may become unreadable (and mangled). In such cases it can be advisable to introduce separate definitions for the list comprehensions in question.› nonterminal lc_qual and lc_quals syntax "_listcompr" :: "'a ⇒ lc_qual ⇒ lc_quals ⇒ 'a list" ("[_ . __") "_lc_gen" :: "'a ⇒ 'a list ⇒ lc_qual" ("_ ← _") "_lc_test" :: "bool ⇒ lc_qual" ("_") (*"_lc_let" :: "letbinds => lc_qual" ("let _")*) "_lc_end" :: "lc_quals" ("]") "_lc_quals" :: "lc_qual ⇒ lc_quals ⇒ lc_quals" (", __") syntax (ASCII) "_lc_gen" :: "'a ⇒ 'a list ⇒ lc_qual" ("_ <- _") parse_translation ‹ let val NilC = Syntax.const @{const_syntax Nil}; val ConsC = Syntax.const @{const_syntax Cons}; val mapC = Syntax.const @{const_syntax map}; val concatC = Syntax.const @{const_syntax concat}; val IfC = Syntax.const @{const_syntax If}; val dummyC = Syntax.const @{const_syntax Pure.dummy_pattern} fun single x = ConsC $ x $ NilC; fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *) let (* FIXME proper name context!? *) val x = Free (singleton (Name.variant_list (fold Term.add_free_names [p, e] [])) "x", dummyT); val e = if opti then single e else e; val case1 = Syntax.const @{syntax_const "_case1"} $ p $ e; val case2 = Syntax.const @{syntax_const "_case1"} $ dummyC $ NilC; val cs = Syntax.const @{syntax_const "_case2"} $ case1 $ case2; in Syntax_Trans.abs_tr [x, Case_Translation.case_tr false ctxt [x, cs]] end; fun pair_pat_tr (x as Free _) e = Syntax_Trans.abs_tr [x, e] | pair_pat_tr (_ $ p1 $ p2) e = Syntax.const @{const_syntax case_prod} $ pair_pat_tr p1 (pair_pat_tr p2 e) | pair_pat_tr dummy e = Syntax_Trans.abs_tr [Syntax.const "_idtdummy", e] fun pair_pat ctxt (Const (@{const_syntax "Pair"},_) $ s $ t) = pair_pat ctxt s andalso pair_pat ctxt t | pair_pat ctxt (Free (s,_)) = let val thy = Proof_Context.theory_of ctxt; val s' = Proof_Context.intern_const ctxt s; in not (Sign.declared_const thy s') end | pair_pat _ t = (t = dummyC); fun abs_tr ctxt p e opti = let val p = Term_Position.strip_positions p in if pair_pat ctxt p then (pair_pat_tr p e, true) else (pat_tr ctxt p e opti, false) end fun lc_tr ctxt [e, Const (@{syntax_const "_lc_test"}, _) $ b, qs] = let val res = (case qs of Const (@{syntax_const "_lc_end"}, _) => single e | Const (@{syntax_const "_lc_quals"}, _) $ q $ qs => lc_tr ctxt [e, q, qs]); in IfC $ b $ res $ NilC end | lc_tr ctxt [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es, Const(@{syntax_const "_lc_end"}, _)] = (case abs_tr ctxt p e true of (f, true) => mapC $ f $ es | (f, false) => concatC $ (mapC $ f $ es)) | lc_tr ctxt [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es, Const (@{syntax_const "_lc_quals"}, _) $ q $ qs] = let val e' = lc_tr ctxt [e, q, qs]; in concatC $ (mapC $ (fst (abs_tr ctxt p e' false)) $ es) end; in [(@{syntax_const "_listcompr"}, lc_tr)] end › ML_val ‹ let val read = Syntax.read_term @{context} o Syntax.implode_input; fun check s1 s2 = read s1 aconv read s2 orelse error ("Check failed: " ^ quote (Input.source_content s1) ^ Position.here_list [Input.pos_of s1, Input.pos_of s2]); in check ‹[(x,y,z). b]› ‹if b then [(x, y, z)] else []›; check ‹[(x,y,z). (x,_,y)←xs]› ‹map (λ(x,_,y). (x, y, z)) xs›; check ‹[e x y. (x,_)←xs, y←ys]› ‹concat (map (λ(x,_). map (λy. e x y) ys) xs)›; check ‹[(x,y,z). x<a, x>b]› ‹if x < a then if b < x then [(x, y, z)] else [] else []›; check ‹[(x,y,z). x←xs, x>b]› ‹concat (map (λx. if b < x then [(x, y, z)] else []) xs)›; check ‹[(x,y,z). x<a, x←xs]› ‹if x < a then map (λx. (x, y, z)) xs else []›; check ‹[(x,y). Cons True x ← xs]› ‹concat (map (λxa. case xa of [] ⇒ [] | True # x ⇒ [(x, y)] | False # x ⇒ []) xs)›; check ‹[(x,y,z). Cons x [] ← xs]› ‹concat (map (λxa. case xa of [] ⇒ [] | [x] ⇒ [(x, y, z)] | x # aa # lista ⇒ []) xs)›; check ‹[(x,y,z). x<a, x>b, x=d]› ‹if x < a then if b < x then if x = d then [(x, y, z)] else [] else [] else []›; check ‹[(x,y,z). x<a, x>b, y←ys]› ‹if x < a then if b < x then map (λy. (x, y, z)) ys else [] else []›; check ‹[(x,y,z). x<a, (_,x)←xs,y>b]› ‹if x < a then concat (map (λ(_,x). if b < y then [(x, y, z)] else []) xs) else []›; check ‹[(x,y,z). x<a, x←xs, y←ys]› ‹if x < a then concat (map (λx. map (λy. (x, y, z)) ys) xs) else []›; check ‹[(x,y,z). x←xs, x>b, y<a]› ‹concat (map (λx. if b < x then if y < a then [(x, y, z)] else [] else []) xs)›; check ‹[(x,y,z). x←xs, x>b, y←ys]› ‹concat (map (λx. if b < x then map (λy. (x, y, z)) ys else []) xs)›; check ‹[(x,y,z). x←xs, (y,_)←ys,y>x]› ‹concat (map (λx. concat (map (λ(y,_). if x < y then [(x, y, z)] else []) ys)) xs)›; check ‹[(x,y,z). x←xs, y←ys,z←zs]› ‹concat (map (λx. concat (map (λy. map (λz. (x, y, z)) zs) ys)) xs)› end; › ML ‹ (* Simproc for rewriting list comprehensions applied to List.set to set comprehension. *) signature LIST_TO_SET_COMPREHENSION = sig val simproc : Proof.context -> cterm -> thm option end structure List_to_Set_Comprehension : LIST_TO_SET_COMPREHENSION = struct (* conversion *) fun all_exists_conv cv ctxt ct = (case Thm.term_of ct of Const (@{const_name Ex}, _) $ Abs _ => Conv.arg_conv (Conv.abs_conv (all_exists_conv cv o #2) ctxt) ct | _ => cv ctxt ct) fun all_but_last_exists_conv cv ctxt ct = (case Thm.term_of ct of Const (@{const_name Ex}, _) $ Abs (_, _, Const (@{const_name Ex}, _) $ _) => Conv.arg_conv (Conv.abs_conv (all_but_last_exists_conv cv o #2) ctxt) ct | _ => cv ctxt ct) fun Collect_conv cv ctxt ct = (case Thm.term_of ct of Const (@{const_name Collect}, _) $ Abs _ => Conv.arg_conv (Conv.abs_conv cv ctxt) ct | _ => raise CTERM ("Collect_conv", [ct])) fun rewr_conv' th = Conv.rewr_conv (mk_meta_eq th) fun conjunct_assoc_conv ct = Conv.try_conv (rewr_conv' @{thm conj_assoc} then_conv HOLogic.conj_conv Conv.all_conv conjunct_assoc_conv) ct fun right_hand_set_comprehension_conv conv ctxt = HOLogic.Trueprop_conv (HOLogic.eq_conv Conv.all_conv (Collect_conv (all_exists_conv conv o #2) ctxt)) (* term abstraction of list comprehension patterns *) datatype termlets = If | Case of typ * int local val set_Nil_I = @{lemma "set [] = {x. False}" by (simp add: empty_def [symmetric])} val set_singleton = @{lemma "set [a] = {x. x = a}" by simp} val inst_Collect_mem_eq = @{lemma "set A = {x. x ∈ set A}" by simp} val del_refl_eq = @{lemma "(t = t ∧ P) ≡ P" by simp} fun mk_set T = Const (@{const_name set}, HOLogic.listT T --> HOLogic.mk_setT T) fun dest_set (Const (@{const_name set}, _) $ xs) = xs fun dest_singleton_list (Const (@{const_name Cons}, _) $ t $ (Const (@{const_name Nil}, _))) = t | dest_singleton_list t = raise TERM ("dest_singleton_list", [t]) (*We check that one case returns a singleton list and all other cases return [], and return the index of the one singleton list case.*) fun possible_index_of_singleton_case cases = let fun check (i, case_t) s = (case strip_abs_body case_t of (Const (@{const_name Nil}, _)) => s | _ => (case s of SOME NONE => SOME (SOME i) | _ => NONE)) in fold_index check cases (SOME NONE) |> the_default NONE end (*returns condition continuing term option*) fun dest_if (Const (@{const_name If}, _) $ cond $ then_t $ Const (@{const_name Nil}, _)) = SOME (cond, then_t) | dest_if _ = NONE (*returns (case_expr type index chosen_case constr_name) option*) fun dest_case ctxt case_term = let val (case_const, args) = strip_comb case_term in (case try dest_Const case_const of SOME (c, T) => (case Ctr_Sugar.ctr_sugar_of_case ctxt c of SOME {ctrs, ...} => (case possible_index_of_singleton_case (fst (split_last args)) of SOME i => let val constr_names = map (fst o dest_Const) ctrs val (Ts, _) = strip_type T val T' = List.last Ts in SOME (List.last args, T', i, nth args i, nth constr_names i) end | NONE => NONE) | NONE => NONE) | NONE => NONE) end fun tac ctxt [] = resolve_tac ctxt [set_singleton] 1 ORELSE resolve_tac ctxt [inst_Collect_mem_eq] 1 | tac ctxt (If :: cont) = Splitter.split_tac ctxt @{thms if_split} 1 THEN resolve_tac ctxt @{thms conjI} 1 THEN resolve_tac ctxt @{thms impI} 1 THEN Subgoal.FOCUS (fn {prems, context = ctxt', ...} => CONVERSION (right_hand_set_comprehension_conv (K (HOLogic.conj_conv (Conv.rewr_conv (List.last prems RS @{thm Eq_TrueI})) Conv.all_conv then_conv rewr_conv' @{lemma "(True ∧ P) = P" by simp})) ctxt') 1) ctxt 1 THEN tac ctxt cont THEN resolve_tac ctxt @{thms impI} 1 THEN Subgoal.FOCUS (fn {prems, context = ctxt', ...} => CONVERSION (right_hand_set_comprehension_conv (K (HOLogic.conj_conv (Conv.rewr_conv (List.last prems RS @{thm Eq_FalseI})) Conv.all_conv then_conv rewr_conv' @{lemma "(False ∧ P) = False" by simp})) ctxt') 1) ctxt 1 THEN resolve_tac ctxt [set_Nil_I] 1 | tac ctxt (Case (T, i) :: cont) = let val SOME {injects, distincts, case_thms, split, ...} = Ctr_Sugar.ctr_sugar_of ctxt (fst (dest_Type T)) in (* do case distinction *) Splitter.split_tac ctxt [split] 1 THEN EVERY (map_index (fn (i', _) => (if i' < length case_thms - 1 then resolve_tac ctxt @{thms conjI} 1 else all_tac) THEN REPEAT_DETERM (resolve_tac ctxt @{thms allI} 1) THEN resolve_tac ctxt @{thms impI} 1 THEN (if i' = i then (* continue recursively *) Subgoal.FOCUS (fn {prems, context = ctxt', ...} => CONVERSION (Thm.eta_conversion then_conv right_hand_set_comprehension_conv (K ((HOLogic.conj_conv (HOLogic.eq_conv Conv.all_conv (rewr_conv' (List.last prems)) then_conv (Conv.try_conv (Conv.rewrs_conv (map mk_meta_eq injects)))) Conv.all_conv) then_conv (Conv.try_conv (Conv.rewr_conv del_refl_eq)) then_conv conjunct_assoc_conv)) ctxt' then_conv (HOLogic.Trueprop_conv (HOLogic.eq_conv Conv.all_conv (Collect_conv (fn (_, ctxt'') => Conv.repeat_conv (all_but_last_exists_conv (K (rewr_conv' @{lemma "(∃x. x = t ∧ P x) = P t" by simp})) ctxt'')) ctxt')))) 1) ctxt 1 THEN tac ctxt cont else Subgoal.FOCUS (fn {prems, context = ctxt', ...} => CONVERSION (right_hand_set_comprehension_conv (K (HOLogic.conj_conv ((HOLogic.eq_conv Conv.all_conv (rewr_conv' (List.last prems))) then_conv (Conv.rewrs_conv (map (fn th => th RS @{thm Eq_FalseI}) distincts))) Conv.all_conv then_conv (rewr_conv' @{lemma "(False ∧ P) = False" by simp}))) ctxt' then_conv HOLogic.Trueprop_conv (HOLogic.eq_conv Conv.all_conv (Collect_conv (fn (_, ctxt'') => Conv.repeat_conv (Conv.bottom_conv (K (rewr_conv' @{lemma "(∃x. P) = P" by simp})) ctxt'')) ctxt'))) 1) ctxt 1 THEN resolve_tac ctxt [set_Nil_I] 1)) case_thms) end in fun simproc ctxt redex = let fun make_inner_eqs bound_vs Tis eqs t = (case dest_case ctxt t of SOME (x, T, i, cont, constr_name) => let val (vs, body) = strip_abs (Envir.eta_long (map snd bound_vs) cont) val x' = incr_boundvars (length vs) x val eqs' = map (incr_boundvars (length vs)) eqs val constr_t = list_comb (Const (constr_name, map snd vs ---> T), map Bound (((length vs) - 1) downto 0)) val constr_eq = Const (@{const_name HOL.eq}, T --> T --> @{typ bool}) $ constr_t $ x' in make_inner_eqs (rev vs @ bound_vs) (Case (T, i) :: Tis) (constr_eq :: eqs') body end | NONE => (case dest_if t of SOME (condition, cont) => make_inner_eqs bound_vs (If :: Tis) (condition :: eqs) cont | NONE => if null eqs then NONE (*no rewriting, nothing to be done*) else let val Type (@{type_name list}, [rT]) = fastype_of1 (map snd bound_vs, t) val pat_eq = (case try dest_singleton_list t of SOME t' => Const (@{const_name HOL.eq}, rT --> rT --> @{typ bool}) $ Bound (length bound_vs) $ t' | NONE => Const (@{const_name Set.member}, rT --> HOLogic.mk_setT rT --> @{typ bool}) $ Bound (length bound_vs) $ (mk_set rT $ t)) val reverse_bounds = curry subst_bounds ((map Bound ((length bound_vs - 1) downto 0)) @ [Bound (length bound_vs)]) val eqs' = map reverse_bounds eqs val pat_eq' = reverse_bounds pat_eq val inner_t = fold (fn (_, T) => fn t => HOLogic.exists_const T $ absdummy T t) (rev bound_vs) (fold (curry HOLogic.mk_conj) eqs' pat_eq') val lhs = Thm.term_of redex val rhs = HOLogic.mk_Collect ("x", rT, inner_t) val rewrite_rule_t = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)) in SOME ((Goal.prove ctxt [] [] rewrite_rule_t (fn {context = ctxt', ...} => tac ctxt' (rev Tis))) RS @{thm eq_reflection}) end)) in make_inner_eqs [] [] [] (dest_set (Thm.term_of redex)) end end end › simproc_setup list_to_set_comprehension ("set xs") = ‹K List_to_Set_Comprehension.simproc› code_datatype set coset hide_const (open) coset subsubsection ‹@{const Nil} and @{const Cons}› lemma not_Cons_self [simp]: "xs ≠ x # xs" by (induct xs) auto lemma not_Cons_self2 [simp]: "x # xs ≠ xs" by (rule not_Cons_self [symmetric]) lemma neq_Nil_conv: "(xs ≠ []) = (∃y ys. xs = y # ys)" by (induct xs) auto lemma tl_Nil: "tl xs = [] ⟷ xs = [] ∨ (∃x. xs = [x])" by (cases xs) auto lemma Nil_tl: "[] = tl xs ⟷ xs = [] ∨ (∃x. xs = [x])" by (cases xs) auto lemma length_induct: "(⋀xs. ∀ys. length ys < length xs ⟶ P ys ⟹ P xs) ⟹ P xs" by (fact measure_induct) lemma induct_list012: "⟦P []; ⋀x. P [x]; ⋀x y zs. P (y # zs) ⟹ P (x # y # zs)⟧ ⟹ P xs" by induction_schema (pat_completeness, lexicographic_order) lemma list_nonempty_induct [consumes 1, case_names single cons]: "⟦ xs ≠ []; ⋀x. P [x]; ⋀x xs. xs ≠ [] ⟹ P xs ⟹ P (x # xs)⟧ ⟹ P xs" by(induction xs rule: induct_list012) auto lemma inj_split_Cons: "inj_on (λ(xs, n). n#xs) X" by (auto intro!: inj_onI) lemma inj_on_Cons1 [simp]: "inj_on ((#) x) A" by(simp add: inj_on_def) subsubsection ‹@{const length}› text ‹ Needs to come before ‹@› because of theorem ‹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_pos_if_in_set: "x ∈ set xs ⟹ length xs > 0" by 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: "length xs <= length ys ==> xs = x # ys = False" by (induct xs) auto lemma list_induct2 [consumes 1, case_names Nil Cons]: "length xs = length ys ⟹ P [] [] ⟹ (⋀x xs y ys. length xs = length ys ⟹ P xs ys ⟹ P (x#xs) (y#ys)) ⟹ P xs ys" proof (induct xs arbitrary: ys) case Nil then show ?case by simp next case (Cons x xs ys) then show ?case by (cases ys) simp_all qed lemma list_induct3 [consumes 2, case_names Nil Cons]: "length xs = length ys ⟹ length ys = length zs ⟹ P [] [] [] ⟹ (⋀x xs y ys z zs. length xs = length ys ⟹ length ys = length zs ⟹ P xs ys zs ⟹ P (x#xs) (y#ys) (z#zs)) ⟹ P xs ys zs" proof (induct xs arbitrary: ys zs) case Nil then show ?case by simp next case (Cons x xs ys zs) then show ?case by (cases ys, simp_all) (cases zs, simp_all) qed lemma list_induct4 [consumes 3, case_names Nil Cons]: "length xs = length ys ⟹ length ys = length zs ⟹ length zs = length ws ⟹ P [] [] [] [] ⟹ (⋀x xs y ys z zs w ws. length xs = length ys ⟹ length ys = length zs ⟹ length zs = length ws ⟹ P xs ys zs ws ⟹ P (x#xs) (y#ys) (z#zs) (w#ws)) ⟹ P xs ys zs ws" proof (induct xs arbitrary: ys zs ws) case Nil then show ?case by simp next case (Cons x xs ys zs ws) then show ?case by ((cases ys, simp_all), (cases zs,simp_all)) (cases ws, simp_all) qed lemma list_induct2': "⟦ P [] []; ⋀x xs. P (x#xs) []; ⋀y ys. P [] (y#ys); ⋀x xs y ys. P xs ys ⟹ P (x#xs) (y#ys) ⟧ ⟹ P xs ys" by (induct xs arbitrary: ys) (case_tac x, auto)+ lemma list_all2_iff: "list_all2 P xs ys ⟷ length xs = length ys ∧ (∀(x, y) ∈ set (zip xs ys). P x y)" by (induct xs ys rule: list_induct2') auto lemma neq_if_length_neq: "length xs ≠ length ys ⟹ (xs = ys) == False" by (rule Eq_FalseI) auto simproc_setup list_neq ("(xs::'a list) = ys") = ‹ (* Reduces xs=ys to False if xs and ys cannot be of the same length. This is the case if the atomic sublists of one are a submultiset of those of the other list and there are fewer Cons's in one than the other. *) let fun len (Const(@{const_name Nil},_)) acc = acc | len (Const(@{const_name Cons},_) $ _ $ xs) (ts,n) = len xs (ts,n+1) | len (Const(@{const_name append},_) $ xs $ ys) acc = len xs (len ys acc) | len (Const(@{const_name rev},_) $ xs) acc = len xs acc | len (Const(@{const_name map},_) $ _ $ xs) acc = len xs acc | len t (ts,n) = (t::ts,n); val ss = simpset_of @{context}; fun list_neq ctxt ct = let val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct; val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0); fun prove_neq() = let val Type(_,listT::_) = eqT; val size = HOLogic.size_const listT; val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs); val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len); val thm = Goal.prove ctxt [] [] neq_len (K (simp_tac (put_simpset ss ctxt) 1)); in SOME (thm RS @{thm neq_if_length_neq}) end in if m < n andalso submultiset (aconv) (ls,rs) orelse n < m andalso submultiset (aconv) (rs,ls) then prove_neq() else NONE end; in K list_neq end; › subsubsection ‹‹@› -- append› global_interpretation append: monoid append Nil proof fix xs ys zs :: "'a list" show "(xs @ ys) @ zs = xs @ (ys @ zs)" by (induct xs) simp_all show "xs @ [] = xs" by (induct xs) simp_all qed simp lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)" by (fact append.assoc) lemma append_Nil2: "xs @ [] = xs" by (fact append.right_neutral) 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]: "length xs = length ys ∨ length us = length vs ==> (xs@us = ys@vs) = (xs=ys ∧ us=vs)" apply (induct xs arbitrary: ys) apply (case_tac ys, simp, force) apply (case_tac ys, force, simp) done lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) = (∃us. xs = zs @ us ∧ us @ ys = ts ∨ xs @ us = zs ∧ ys = us @ ts)" apply (induct xs arbitrary: ys zs ts) apply fastforce apply(case_tac zs) apply simp apply fastforce done lemma same_append_eq [iff, induct_simp]: "(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, induct_simp]: "(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: "xs ≠ [] ==> hd xs # tl xs = xs" by (fact list.collapse) 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 ∨ (∃ys'. x#ys' = ys ∧ xs = ys'@zs))" by(cases ys) auto lemma append_eq_Cons_conv: "(ys@zs = x#xs) = (ys = [] ∧ zs = x#xs ∨ (∃ys'. ys = x#ys' ∧ ys'@zs = xs))" by(cases ys) auto lemma longest_common_prefix: "∃ps xs' ys'. xs = ps @ xs' ∧ ys = ps @ ys' ∧ (xs' = [] ∨ ys' = [] ∨ hd xs' ≠ hd ys')" by (induct xs ys rule: list_induct2') (blast, blast, blast, metis (no_types, hide_lams) append_Cons append_Nil list.sel(1)) text ‹Trivial rules for solving ‹@›-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 ‹@› to see if - both lists end in a singleton list, - or both lists end in the same list. › simproc_setup list_eq ("(xs::'a list) = ys") = ‹ let fun last (cons as Const (@{const_name Cons}, _) $ _ $ xs) = (case xs of Const (@{const_name Nil}, _) => cons | _ => last xs) | last (Const(@{const_name append},_) $ _ $ ys) = last ys | last t = t; fun list1 (Const(@{const_name Cons},_) $ _ $ Const(@{const_name Nil},_)) = true | list1 _ = false; fun butlast ((cons as Const(@{const_name Cons},_) $ x) $ xs) = (case xs of Const (@{const_name Nil}, _) => xs | _ => cons $ butlast xs) | butlast ((app as Const (@{const_name append}, _) $ xs) $ ys) = app $ butlast ys | butlast xs = Const(@{const_name Nil}, fastype_of xs); val rearr_ss = simpset_of (put_simpset HOL_basic_ss @{context} addsimps [@{thm append_assoc}, @{thm append_Nil}, @{thm append_Cons}]); fun list_eq ctxt (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(@{const_name append},appT) val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr) val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2)); val thm = Goal.prove ctxt [] [] eq (K (simp_tac (put_simpset rearr_ss ctxt) 1)); in SOME ((conv RS (thm RS trans)) RS eq_reflection) end; in if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv} else if lastl aconv lastr then rearr @{thm append_same_eq} else NONE end; in fn _ => fn ctxt => fn ct => list_eq ctxt (Thm.term_of ct) end; › subsubsection ‹@{const map}› lemma hd_map: "xs ≠ [] ⟹ hd (map f xs) = f (hd xs)" by (cases xs) simp_all lemma map_tl: "map f (tl xs) = tl (map f xs)" by (cases xs) simp_all 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_map [simp]: "map f (map g xs) = map (f ∘ g) xs" by (induct xs) auto lemma map_comp_map[simp]: "((map f) ∘ (map g)) = map(f ∘ g)" by (rule ext) simp 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 [fundef_cong]: "xs = ys ⟹ (⋀x. x ∈ set ys ⟹ f x = g x) ⟹ map f xs = map g ys" 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: "(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: "(x#xs = map f ys) = (∃z zs. ys = z#zs ∧ x = f z ∧ xs = map f zs)" by (cases ys) auto lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1] lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1] declare map_eq_Cons_D [dest!] Cons_eq_map_D [dest!] lemma ex_map_conv: "(∃xs. ys = map f xs) = (∀y ∈ set ys. ∃x. y = f x)" by(induct ys, auto simp add: Cons_eq_map_conv) lemma map_eq_imp_length_eq: assumes "map f xs = map g ys" shows "length xs = length ys" using assms proof (induct ys arbitrary: xs) case Nil then show ?case by simp next case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto from Cons xs have "map f zs = map g ys" by simp with Cons have "length zs = length ys" by blast with xs show ?case by simp qed 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: "map f xs = map f ys ==> inj f ==> xs = ys" by (induct ys arbitrary: xs) (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_def) apply clarify apply (erule_tac x = "[x]" in allE) apply (erule_tac x = "[y]" in allE) apply auto 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 (⋃(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) lemma map_fst_zip_take: "map fst (zip xs ys) = take (min (length xs) (length ys)) xs" by (induct xs ys rule: list_induct2') simp_all lemma map_snd_zip_take: "map snd (zip xs ys) = take (min (length xs) (length ys)) ys" by (induct xs ys rule: list_induct2') simp_all lemma map2_map_map: "map2 h (map f xs) (map g xs) = map (λx. h (f x) (g x)) xs" by (induction xs) (auto) functor map: map by (simp_all add: id_def) declare map.id [simp] subsubsection ‹@{const 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]: "(rev xs = rev ys) = (xs = ys)" apply (induct xs arbitrary: ys, 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 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 lemma rev_nonempty_induct [consumes 1, case_names single snoc]: assumes "xs ≠ []" and single: "⋀x. P [x]" and snoc': "⋀x xs. xs ≠ [] ⟹ P xs ⟹ P (xs@[x])" shows "P xs" using ‹xs ≠ []› proof (induct xs rule: rev_induct) case (snoc x xs) then show ?case proof (cases xs) case Nil thus ?thesis by (simp add: single) next case Cons with snoc show ?thesis by (fastforce intro!: snoc') qed qed simp lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])" by(rule rev_cases[of xs]) auto subsubsection ‹@{const set}› declare list.set[code_post] ― ‹pretty output› 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[simp]: "xs ≠ [] ⟹ hd xs ∈ set xs" by(cases xs) 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] = {i..<j}" by (induct j) auto lemma split_list: "x ∈ set xs ⟹ ∃ys zs. xs = ys @ x # zs" proof (induct xs) case Nil thus ?case by simp next case Cons thus ?case by (auto intro: Cons_eq_appendI) qed lemma in_set_conv_decomp: "x ∈ set xs ⟷ (∃ys zs. xs = ys @ x # zs)" by (auto elim: split_list) lemma split_list_first: "x ∈ set xs ⟹ ∃ys zs. xs = ys @ x # zs ∧ x ∉ set ys" proof (induct xs) case Nil thus ?case by simp next case (Cons a xs) show ?case proof cases assume "x = a" thus ?case using Cons by fastforce next assume "x ≠ a" thus ?case using Cons by(fastforce intro!: Cons_eq_appendI) qed qed lemma in_set_conv_decomp_first: "(x ∈ set xs) = (∃ys zs. xs = ys @ x # zs ∧ x ∉ set ys)" by (auto dest!: split_list_first) lemma split_list_last: "x ∈ set xs ⟹ ∃ys zs. xs = ys @ x # zs ∧ x ∉ set zs" proof (induct xs rule: rev_induct) case Nil thus ?case by simp next case (snoc a xs) show ?case proof cases assume "x = a" thus ?case using snoc by (auto intro!: exI) next assume "x ≠ a" thus ?case using snoc by fastforce qed qed lemma in_set_conv_decomp_last: "(x ∈ set xs) = (∃ys zs. xs = ys @ x # zs ∧ x ∉ set zs)" by (auto dest!: split_list_last) lemma split_list_prop: "∃x ∈ set xs. P x ⟹ ∃ys x zs. xs = ys @ x # zs ∧ P x" proof (induct xs) case Nil thus ?case by simp next case Cons thus ?case by(simp add:Bex_def)(metis append_Cons append.simps(1)) qed lemma split_list_propE: assumes "∃x ∈ set xs. P x" obtains ys x zs where "xs = ys @ x # zs" and "P x" using split_list_prop [OF assms] by blast lemma split_list_first_prop: "∃x ∈ set xs. P x ⟹ ∃ys x zs. xs = ys@x#zs ∧ P x ∧ (∀y ∈ set ys. ¬ P y)" proof (induct xs) case Nil thus ?case by simp next case (Cons x xs) show ?case proof cases assume "P x" hence "x # xs = [] @ x # xs ∧ P x ∧ (∀y∈set []. ¬ P y)" by simp thus ?thesis by fast next assume "¬ P x" hence "∃x∈set xs. P x" using Cons(2) by simp thus ?thesis using ‹¬ P x› Cons(1) by (metis append_Cons set_ConsD) qed qed lemma split_list_first_propE: assumes "∃x ∈ set xs. P x" obtains ys x zs where "xs = ys @ x # zs" and "P x" and "∀y ∈ set ys. ¬ P y" using split_list_first_prop [OF assms] by blast lemma split_list_first_prop_iff: "(∃x ∈ set xs. P x) ⟷ (∃ys x zs. xs = ys@x#zs ∧ P x ∧ (∀y ∈ set ys. ¬ P y))" by (rule, erule split_list_first_prop) auto lemma split_list_last_prop: "∃x ∈ set xs. P x ⟹ ∃ys x zs. xs = ys@x#zs ∧ P x ∧ (∀z ∈ set zs. ¬ P z)" proof(induct xs rule:rev_induct) case Nil thus ?case by simp next case (snoc x xs) show ?case proof cases assume "P x" thus ?thesis by (auto intro!: exI) next assume "¬ P x" hence "∃x∈set xs. P x" using snoc(2) by simp thus ?thesis using ‹¬ P x› snoc(1) by fastforce qed qed lemma split_list_last_propE: assumes "∃x ∈ set xs. P x" obtains ys x zs where "xs = ys @ x # zs" and "P x" and "∀z ∈ set zs. ¬ P z" using split_list_last_prop [OF assms] by blast lemma split_list_last_prop_iff: "(∃x ∈ set xs. P x) ⟷ (∃ys x zs. xs = ys@x#zs ∧ P x ∧ (∀z ∈ set zs. ¬ P z))" by rule (erule split_list_last_prop, auto) lemma finite_list: "finite A ⟹ ∃xs. set xs = A" by (erule finite_induct) (auto simp add: list.set(2)[symmetric] simp del: list.set(2)) lemma card_length: "card (set xs) ≤ length xs" by (induct xs) (auto simp add: card_insert_if) lemma set_minus_filter_out: "set xs - {y} = set (filter (λx. ¬ (x = y)) xs)" by (induct xs) auto lemma append_Cons_eq_iff: "⟦ x ∉ set xs; x ∉ set ys ⟧ ⟹ xs @ x # ys = xs' @ x # ys' ⟷ (xs = xs' ∧ ys = ys')" by(auto simp: append_eq_Cons_conv Cons_eq_append_conv append_eq_append_conv2) subsubsection ‹@{const 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 sum_length_filter_compl: "length(filter P xs) + length(filter (λx. ¬P x) xs) = length xs" by(induct xs) simp_all 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 ∘ f) xs)" by (induct xs) simp_all lemma length_filter_map[simp]: "length (filter P (map f xs)) = length(filter (P ∘ 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:if_split_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: image_def split:nat.split dest:gr0_implies_Suc) have "length (filter p (x # xs)) = Suc(card ?S)" using Cons ‹p x› by simp also have "… = Suc(card(Suc ` ?S))" using fin by (simp add: card_image) 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: image_def split:nat.split elim:lessE) have "length (filter p (x # xs)) = card ?S" using Cons ‹¬ p x› by simp also have "… = card(Suc ` ?S)" using fin by (simp add: card_image) 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" (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 "x = y" with Py Cons.prems have "?Q []" by simp then show ?thesis .. next assume "x ≠ y" with Py Cons.prems show ?thesis by simp qed next assume "¬ P y" with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastforce then have "?Q (y#us)" by simp then show ?thesis .. 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 inj_on_filter_key_eq: assumes "inj_on f (insert y (set xs))" shows "filter (λx. f y = f x) xs = filter (HOL.eq y) xs" using assms by (induct xs) auto lemma filter_cong[fundef_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 ‹List partitioning› primrec partition :: "('a ⇒ bool) ⇒'a list ⇒ 'a list × 'a list" where "partition P [] = ([], [])" | "partition P (x # xs) = (let (yes, no) = partition P xs in if P x then (x # yes, no) else (yes, x # no))" lemma partition_filter1: "fst (partition P xs) = filter P xs" by (induct xs) (auto simp add: Let_def split_def) lemma partition_filter2: "snd (partition P xs) = filter (Not ∘ P) xs" by (induct xs) (auto simp add: Let_def split_def) lemma partition_P: assumes "partition P xs = (yes, no)" shows "(∀p ∈ set yes. P p) ∧ (∀p ∈ set no. ¬ P p)" proof - from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)" by simp_all then show ?thesis by (simp_all add: partition_filter1 partition_filter2) qed lemma partition_set: assumes "partition P xs = (yes, no)" shows "set yes ∪ set no = set xs" proof - from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)" by simp_all then show ?thesis by (auto simp add: partition_filter1 partition_filter2) qed lemma partition_filter_conv[simp]: "partition f xs = (filter f xs,filter (Not ∘ f) xs)" unfolding partition_filter2[symmetric] unfolding partition_filter1[symmetric] by simp declare partition.simps[simp del] subsubsection ‹@{const concat}› lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys" by (induct xs) auto lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (∀xs ∈ set xss. xs = [])" by (induct xss) auto lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (∀xs ∈ set xss. xs = [])" by (induct xss) auto lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)" by (induct xs) auto lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs" by (induct xs) auto lemma map_concat: "map f (concat xs) = concat (map (map f) xs)" by (induct xs) auto lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)" by (induct xs) auto lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))" by (induct xs) auto lemma concat_eq_concat_iff: "∀(x, y) ∈ set (zip xs ys). length x = length y ==> length xs = length ys ==> (concat xs = concat ys) = (xs = ys)" proof (induct xs arbitrary: ys) case (Cons x xs ys) thus ?case by (cases ys) auto qed (auto) lemma concat_injective: "concat xs = concat ys ==> length xs = length ys ==> ∀(x, y) ∈ set (zip xs ys). length x = length y ==> xs = ys" by (simp add: concat_eq_concat_iff) subsubsection ‹@{const nth}› lemma nth_Cons_0 [simp, code]: "(x # xs)!0 = x" by auto lemma nth_Cons_Suc [simp, code]: "(x # xs)!(Suc n) = xs!n" by auto declare nth.simps [simp del] lemma nth_Cons_pos[simp]: "0 < n ⟹ (x#xs) ! n = xs ! (n - 1)" by(auto simp: Nat.gr0_conv_Suc) lemma nth_append: "(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))" apply (induct xs arbitrary: n, 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 < length xs ==> (map f xs)!n = f(xs!n)" apply (induct xs arbitrary: n, simp) apply (case_tac n, auto) done lemma nth_tl: "n < length (tl xs) ⟹ tl xs ! n = xs ! Suc n" by (induction xs) auto lemma hd_conv_nth: "xs ≠ [] ⟹ hd xs = xs!0" by(cases xs) simp_all lemma list_eq_iff_nth_eq: "(xs = ys) = (length xs = length ys ∧ (∀i<length xs. xs!i = ys!i))" apply(induct xs arbitrary: ys) apply force apply(case_tac ys) apply simp apply(simp add:nth_Cons split:nat.split)apply blast done lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}" apply (induct xs, simp, simp) apply safe apply (metis nat.case(1) nth.simps zero_less_Suc) apply (metis less_Suc_eq_0_disj nth_Cons_Suc) apply (case_tac i, simp) apply (metis diff_Suc_Suc nat.case(2) nth.simps zero_less_diff) done lemma in_set_conv_nth: "(x ∈ set xs) = (∃i < length xs. xs!i = x)" by(auto simp:set_conv_nth) lemma nth_equal_first_eq: assumes "x ∉ set xs" assumes "n ≤ length xs" shows "(x # xs) ! n = x ⟷ n = 0" (is "?lhs ⟷ ?rhs") proof assume ?lhs show ?rhs proof (rule ccontr) assume "n ≠ 0" then have "n > 0" by simp with ‹?lhs› have "xs ! (n - 1) = x" by simp moreover from ‹n > 0› ‹n ≤ length xs› have "n - 1 < length xs" by simp ultimately have "∃i<length xs. xs ! i = x" by auto with ‹x ∉ set xs› in_set_conv_nth [of x xs] show False by simp qed next assume ?rhs then show ?lhs by simp qed lemma nth_non_equal_first_eq: assumes "x ≠ y" shows "(x # xs) ! n = y ⟷ xs ! (n - 1) = y ∧ n > 0" (is "?lhs ⟷ ?rhs") proof assume "?lhs" with assms have "n > 0" by (cases n) simp_all with ‹?lhs› show ?rhs by simp next assume "?rhs" then show "?lhs" by simp qed 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) lemma rev_nth: "n < size xs ⟹ rev xs ! n = xs ! (length xs - Suc n)" proof (induct xs arbitrary: n) case Nil thus ?case by simp next case (Cons x xs) hence n: "n < Suc (length xs)" by simp moreover { assume "n < length xs" with n obtain n' where n': "length xs - n = Suc n'" by (cases "length xs - n", auto) moreover from n' have "length xs - Suc n = n'" by simp ultimately have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp } ultimately show ?case by (clarsimp simp add: Cons nth_append) qed lemma Skolem_list_nth: "(∀i<k. ∃x. P i x) = (∃xs. size xs = k ∧ (∀i<k. P i (xs!i)))" (is "_ = (∃xs. ?P k xs)") proof(induct k) case 0 show ?case by simp next case (Suc k) show ?case (is "?L = ?R" is "_ = (∃xs. ?P' xs)") proof assume "?R" thus "?L" using Suc by auto next assume "?L" with Suc obtain x xs where "?P k xs ∧ P k x" by (metis less_Suc_eq) hence "?P'(xs@[x])" by(simp add:nth_append less_Suc_eq) thus "?R" .. qed qed subsubsection ‹@{const list_update}› lemma length_list_update [simp]: "length(xs[i:=x]) = length xs" by (induct xs arbitrary: i) (auto split: nat.split) lemma nth_list_update: "i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)" by (induct xs arbitrary: i j) (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 ==> xs[i:=x]!j = xs!j" by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split) lemma list_update_id[simp]: "xs[i := xs!i] = xs" by (induct xs arbitrary: i) (simp_all split:nat.splits) lemma list_update_beyond[simp]: "length xs ≤ i ⟹ xs[i:=x] = xs" apply (induct xs arbitrary: i) apply simp apply (case_tac i) apply simp_all done lemma list_update_nonempty[simp]: "xs[k:=x] = [] ⟷ xs=[]" by (simp only: length_0_conv[symmetric] length_list_update) lemma list_update_same_conv: "i < length xs ==> (xs[i := x] = xs) = (xs!i = x)" by (induct xs arbitrary: i) (auto split: nat.split) lemma list_update_append1: "i < size xs ⟹ (xs @ ys)[i:=x] = xs[i:=x] @ ys" by (induct xs arbitrary: i)(auto split:nat.split) lemma list_update_append: "(xs @ ys) [n:= x] = (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))" by (induct xs arbitrary: n) (auto split:nat.splits) lemma list_update_length [simp]: "(xs @ x # ys)[length xs := y] = (xs @ y # ys)" by (induct xs, auto) lemma map_update: "map f (xs[k:= y]) = (map f xs)[k := f y]" by(induct xs arbitrary: k)(auto split:nat.splits) lemma rev_update: "k < length xs ⟹ rev (xs[k:= y]) = (rev xs)[length xs - k - 1 := y]" by (induct xs arbitrary: k) (auto simp: list_update_append split:nat.splits) lemma update_zip: "(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])" by (induct ys arbitrary: i xy xs) (auto, case_tac xs, auto split: nat.split) lemma set_update_subset_insert: "set(xs[i:=x]) <= insert x (set xs)" by (induct xs arbitrary: i) (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 < length xs ⟹ x ∈ set (xs[n := x])" by (induct xs arbitrary: n) (auto split:nat.splits) lemma list_update_overwrite[simp]: "xs [i := x, i := y] = xs [i := y]" apply (induct xs arbitrary: i) apply simp apply (case_tac i, simp_all) done lemma list_update_swap: "i ≠ i' ⟹ xs [i := x, i' := x'] = xs [i' := x', i := x]" apply (induct xs arbitrary: i i') apply simp apply (case_tac i, case_tac i') apply auto apply (case_tac i') apply auto done lemma list_update_code [code]: "[][i := y] = []" "(x # xs)[0 := y] = y # xs" "(x # xs)[Suc i := y] = x # xs[i := y]" by simp_all subsubsection ‹@{const last} and @{const 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 lemma last_ConsR: "xs ≠ [] ⟹ last(x#xs) = last xs" by simp 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 last_tl: "xs = [] ∨ tl xs ≠ [] ⟹last (tl xs) = last xs" by (induct xs) simp_all lemma butlast_tl: "butlast (tl xs) = tl (butlast xs)" by (induct xs) simp_all lemma hd_rev: "xs ≠ [] ⟹ hd(rev xs) = last xs" by(rule rev_exhaust[of xs]) simp_all lemma last_rev: "xs ≠ [] ⟹ last(rev xs) = hd xs" by(cases xs) simp_all lemma last_in_set[simp]: "as ≠ [] ⟹ last as ∈ set as" by (induct as) auto lemma length_butlast [simp]: "length (butlast xs) = length xs - 1" by (induct xs rule: rev_induct) auto lemma butlast_append: "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)" by (induct xs arbitrary: ys) 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: if_split_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 < length xs ⟹ last (drop n xs) = last xs" by (induct xs arbitrary: n)(auto split:nat.split) lemma nth_butlast: assumes "n < length (butlast xs)" shows "butlast xs ! n = xs ! n" proof (cases xs) case (Cons y ys) moreover from assms have "butlast xs ! n = (butlast xs @ [last xs]) ! n" by (simp add: nth_append) ultimately show ?thesis using append_butlast_last_id by simp qed simp lemma last_conv_nth: "xs≠[] ⟹ last xs = xs!(length xs - 1)" by(induct xs)(auto simp:neq_Nil_conv) lemma butlast_conv_take: "butlast xs = take (length xs - 1) xs" by (induction xs rule: induct_list012) simp_all lemma last_list_update: "xs ≠ [] ⟹ last(xs[k:=x]) = (if k = size xs - 1 then x else last xs)" by (auto simp: last_conv_nth) lemma butlast_list_update: "butlast(xs[k:=x]) = (if k = size xs - 1 then butlast xs else (butlast xs)[k:=x])" by(cases xs rule:rev_cases)(auto simp: list_update_append split: nat.splits) lemma last_map: "xs ≠ [] ⟹ last (map f xs) = f (last xs)" by (cases xs rule: rev_cases) simp_all lemma map_butlast: "map f (butlast xs) = butlast (map f xs)" by (induct xs) simp_all lemma snoc_eq_iff_butlast: "xs @ [x] = ys ⟷ (ys ≠ [] ∧ butlast ys = xs ∧ last ys = x)" by fastforce corollary longest_common_suffix: "∃ss xs' ys'. xs = xs' @ ss ∧ ys = ys' @ ss ∧ (xs' = [] ∨ ys' = [] ∨ last xs' ≠ last ys')" using longest_common_prefix[of "rev xs" "rev ys"] unfolding rev_swap rev_append by (metis last_rev rev_is_Nil_conv) subsubsection ‹@{const take} and @{const drop}› lemma take_0: "take 0 xs = []" by (induct xs) auto lemma drop_0: "drop 0 xs = xs" by (induct xs) auto lemma take0[simp]: "take 0 = (λxs. [])" by(rule ext) (rule take_0) lemma drop0[simp]: "drop 0 = (λx. x)" by(rule ext) (rule drop_0) 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 hd_take[simp]: "j > 0 ⟹ hd (take j xs) = hd xs" by (metis gr0_conv_Suc list.sel(1) take.simps(1) take_Suc) lemma take_tl: "take n (tl xs) = tl (take (Suc n) xs)" by (induct xs arbitrary: n) simp_all lemma drop_tl: "drop n (tl xs) = tl(drop n xs)" by(induct xs arbitrary: n, simp_all add:drop_Cons drop_Suc split:nat.split) lemma tl_take: "tl (take n xs) = take (n - 1) (tl xs)" by (cases n, simp, cases xs, auto) lemma tl_drop: "tl (drop n xs) = drop n (tl xs)" by (simp only: drop_tl) lemma nth_via_drop: "drop n xs = y#ys ⟹ xs!n = y" by (induct xs arbitrary: n, simp)(auto simp: drop_Cons nth_Cons split: nat.splits) lemma take_Suc_conv_app_nth: "i < length xs ⟹ take (Suc i) xs = take i xs @ [xs!i]" apply (induct xs arbitrary: i, simp) apply (case_tac i, auto) done lemma Cons_nth_drop_Suc: "i < length xs ⟹ (xs!i) # (drop (Suc i) xs) = drop i xs" apply (induct xs arbitrary: i, simp) apply (case_tac i, auto) done lemma length_take [simp]: "length (take n xs) = min (length xs) n" by (induct n arbitrary: xs) (auto, case_tac xs, auto) lemma length_drop [simp]: "length (drop n xs) = (length xs - n)" by (induct n arbitrary: xs) (auto, case_tac xs, auto) lemma take_all [simp]: "length xs <= n ==> take n xs = xs" by (induct n arbitrary: xs) (auto, case_tac xs, auto) lemma drop_all [simp]: "length xs <= n ==> drop n xs = []" by (induct n arbitrary: xs) (auto, case_tac xs, auto) lemma take_append [simp]: "take n (xs @ ys) = (take n xs @ take (n - length xs) ys)" by (induct n arbitrary: xs) (auto, case_tac xs, auto) lemma drop_append [simp]: "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys" by (induct n arbitrary: xs) (auto, case_tac xs, auto) lemma take_take [simp]: "take n (take m xs) = take (min n m) xs" apply (induct m arbitrary: xs n, auto) apply (case_tac xs, auto) apply (case_tac n, auto) done lemma drop_drop [simp]: "drop n (drop m xs) = drop (n + m) xs" apply (induct m arbitrary: xs, auto) apply (case_tac xs, auto) done lemma take_drop: "take n (drop m xs) = drop m (take (n + m) xs)" apply (induct m arbitrary: xs n, auto) apply (case_tac xs, auto) done lemma drop_take: "drop n (take m xs) = take (m-n) (drop n xs)" by(induct xs arbitrary: m n)(auto simp: take_Cons drop_Cons split: nat.split) lemma append_take_drop_id [simp]: "take n xs @ drop n xs = xs" apply (induct n arbitrary: xs, auto) apply (case_tac xs, auto) done lemma take_eq_Nil[simp]: "(take n xs = []) = (n = 0 ∨ xs = [])" by(induct xs arbitrary: n)(auto simp: take_Cons split:nat.split) lemma drop_eq_Nil[simp]: "(drop n xs = []) = (length xs <= n)" by (induct xs arbitrary: n) (auto simp: drop_Cons split:nat.split) lemma take_map: "take n (map f xs) = map f (take n xs)" apply (induct n arbitrary: xs, auto) apply (case_tac xs, auto) done lemma drop_map: "drop n (map f xs) = map f (drop n xs)" apply (induct n arbitrary: xs, auto) apply (case_tac xs, auto) done lemma rev_take: "rev (take i xs) = drop (length xs - i) (rev xs)" apply (induct xs arbitrary: i, auto) apply (case_tac i, auto) done lemma rev_drop: "rev (drop i xs) = take (length xs - i) (rev xs)" apply (induct xs arbitrary: i, auto) apply (case_tac i, auto) done lemma drop_rev: "drop n (rev xs) = rev (take (length xs - n) xs)" by (cases "length xs < n") (auto simp: rev_take) lemma take_rev: "take n (rev xs) = rev (drop (length xs - n) xs)" by (cases "length xs < n") (auto simp: rev_drop) lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i" apply (induct xs arbitrary: i n, auto) apply (case_tac n, blast) apply (case_tac i, auto) done lemma nth_drop [simp]: "n <= length xs ==> (drop n xs)!i = xs!(n + i)" apply (induct n arbitrary: xs i, auto) apply (case_tac xs, auto) done lemma butlast_take: "n <= length xs ==> butlast (take n xs) = take (n - 1) xs" by (simp add: butlast_conv_take min.absorb1 min.absorb2) lemma butlast_drop: "butlast (drop n xs) = drop n (butlast xs)" by (simp add: butlast_conv_take drop_take ac_simps) lemma take_butlast: "n < length xs ==> take n (butlast xs) = take n xs" by (simp add: butlast_conv_take min.absorb1) lemma drop_butlast: "drop n (butlast xs) = butlast (drop n xs)" by (simp add: butlast_conv_take drop_take ac_simps) lemma hd_drop_conv_nth: "n < length xs ⟹ hd(drop n xs) = xs!n" by(simp add: hd_conv_nth) lemma set_take_subset_set_take: "m <= n ⟹ set(take m xs) <= set(take n xs)" apply (induct xs arbitrary: m n) apply simp apply (case_tac n) apply (auto simp: take_Cons) done lemma set_take_subset: "set(take n xs) ⊆ set xs" by(induct xs arbitrary: n)(auto simp:take_Cons split:nat.split) lemma set_drop_subset: "set(drop n xs) ⊆ set xs" by(induct xs arbitrary: n)(auto simp:drop_Cons split:nat.split) lemma set_drop_subset_set_drop: "m >= n ⟹ set(drop m xs) <= set(drop n xs)" apply(induct xs arbitrary: m n) apply(auto simp:drop_Cons split:nat.split) by (metis set_drop_subset subset_iff) 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: "(xs @ ys = zs) = (xs = take (length xs) zs ∧ ys = drop (length xs) zs)" apply (induct xs arbitrary: zs, simp, clarsimp) apply (case_tac zs, auto) done lemma take_add: "take (i+j) xs = take i xs @ take j (drop i xs)" apply (induct xs arbitrary: i, auto) apply (case_tac i, simp_all) done lemma append_eq_append_conv_if: "(xs⇩_{1}@ xs⇩_{2}= ys⇩_{1}@ ys⇩_{2}) = (if size xs⇩_{1}≤ size ys⇩_{1}then xs⇩_{1}= take (size xs⇩_{1}) ys⇩_{1}∧ xs⇩_{2}= drop (size xs⇩_{1}) ys⇩_{1}@ ys⇩_{2}else take (size ys⇩_{1}) xs⇩_{1}= ys⇩_{1}∧ drop (size ys⇩_{1}) xs⇩_{1}@ xs⇩_{2}= ys⇩_{2})" apply(induct xs⇩_{1}arbitrary: ys⇩_{1}) apply simp apply(case_tac ys⇩_{1}) apply simp_all done lemma take_hd_drop: "n < length xs ⟹ take n xs @ [hd (drop n xs)] = take (Suc n) xs" apply(induct xs arbitrary: n) 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 take_update_cancel[simp]: "n ≤ m ⟹ take n (xs[m := y]) = take n xs" by(simp add: list_eq_iff_nth_eq) lemma drop_update_cancel[simp]: "n < m ⟹ drop m (xs[n := x]) = drop m xs" by(simp add: list_eq_iff_nth_eq) 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 lemma take_update_swap: "take m (xs[n := x]) = (take m xs)[n := x]" apply(cases "n ≥ length xs") apply simp apply(simp add: upd_conv_take_nth_drop take_Cons drop_take min_def diff_Suc split: nat.split) done lemma drop_update_swap: "m ≤ n ⟹ drop m (xs[n := x]) = (drop m xs)[n-m := x]" apply(cases "n ≥ length xs") apply simp apply(simp add: upd_conv_take_nth_drop drop_take) done lemma nth_image: "l ≤ size xs ⟹ nth xs ` {0..<l} = set(take l xs)" by(auto simp: set_conv_nth image_def) (metis Suc_le_eq nth_take order_trans) subsubsection ‹@{const takeWhile} and @{const dropWhile}› lemma length_takeWhile_le: "length (takeWhile P xs) ≤ length xs" by (induct xs) auto lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs" by (induct xs) auto lemma takeWhile_append1 [simp]: "⟦x ∈ set xs; ¬P(x)⟧ ⟹ takeWhile P (xs @ ys) = takeWhile P xs" by (induct xs) auto lemma takeWhile_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 takeWhile_nth: "j < length (takeWhile P xs) ⟹ takeWhile P xs ! j = xs ! j" apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto lemma dropWhile_nth: "j < length (dropWhile P xs) ⟹ dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))" apply (subst (3) takeWhile_dropWhile_id[symmetric]) unfolding nth_append by auto lemma length_dropWhile_le: "length (dropWhile P xs) ≤ length 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 dropWhile_append3: "¬ P y ⟹dropWhile P (xs @ y # ys) = dropWhile P xs @ y # ys" by (induct xs) auto lemma dropWhile_last: "x ∈ set xs ⟹ ¬ P x ⟹ last (dropWhile P xs) = last xs" by (auto simp add: dropWhile_append3 in_set_conv_decomp) lemma set_dropWhileD: "x ∈ set (dropWhile P xs) ⟹ x ∈ set xs" by (induct xs) (auto split: if_split_asm) lemma set_takeWhileD: "x ∈ set (takeWhile P xs) ⟹ x ∈ set xs ∧ P x" by (induct xs) (auto split: if_split_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) lemma distinct_takeWhile[simp]: "distinct xs ==> distinct (takeWhile P xs)" by (induct xs) (auto dest: set_takeWhileD) lemma distinct_dropWhile[simp]: "distinct xs ==> distinct (dropWhile P xs)" by (induct xs) auto lemma takeWhile_map: "takeWhile P (map f xs) = map f (takeWhile (P ∘ f) xs)" by (induct xs) auto lemma dropWhile_map: "dropWhile P (map f xs) = map f (dropWhile (P ∘ f) xs)" by (induct xs) auto lemma takeWhile_eq_take: "takeWhile P xs = take (length (takeWhile P xs)) xs" by (induct xs) auto lemma dropWhile_eq_drop: "dropWhile P xs = drop (length (takeWhile P xs)) xs" by (induct xs) auto lemma hd_dropWhile: "dropWhile P xs ≠ [] ⟹ ¬ P (hd (dropWhile P xs))" by (induct xs) auto lemma takeWhile_eq_filter: assumes "⋀ x. x ∈ set (dropWhile P xs) ⟹ ¬ P x" shows "takeWhile P xs = filter P xs" proof - have A: "filter P xs = filter P (takeWhile P xs @ dropWhile P xs)" by simp have B: "filter P (dropWhile P xs) = []" unfolding filter_empty_conv using assms by blast have "filter P xs = takeWhile P xs" unfolding A filter_append B by (auto simp add: filter_id_conv dest: set_takeWhileD) thus ?thesis .. qed lemma takeWhile_eq_take_P_nth: "⟦ ⋀ i. ⟦ i < n ; i < length xs ⟧ ⟹ P (xs ! i) ; n < length xs ⟹ ¬ P (xs ! n) ⟧ ⟹ takeWhile P xs = take n xs" proof (induct xs arbitrary: n) case Nil thus ?case by simp next case (Cons x xs) show ?case proof (cases n) case 0 with Cons show ?thesis by simp next case [simp]: (Suc n') have "P x" using Cons.prems(1)[of 0] by simp moreover have "takeWhile P xs = take n' xs" proof (rule Cons.hyps) fix i assume "i < n'" "i < length xs" thus "P (xs ! i)" using Cons.prems(1)[of "Suc i"] by simp next assume "n' < length xs" thus "¬ P (xs ! n')" using Cons by auto qed ultimately show ?thesis by simp qed qed lemma nth_length_takeWhile: "length (takeWhile P xs) < length xs ⟹ ¬ P (xs ! length (takeWhile P xs))" by (induct xs) auto lemma length_takeWhile_less_P_nth: assumes all: "⋀ i. i < j ⟹ P (xs ! i)" and "j ≤ length xs" shows "j ≤ length (takeWhile P xs)" proof (rule classical) assume "¬ ?thesis" hence "length (takeWhile P xs) < length xs" using assms by simp thus ?thesis using all ‹¬ ?thesis› nth_length_takeWhile[of P xs] by auto qed 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 lemma takeWhile_not_last: "distinct xs ⟹ takeWhile (λy. y ≠ last xs) xs = butlast xs" by(induction xs rule: induct_list012) auto lemma takeWhile_cong [fundef_cong]: "⟦l = k; ⋀x. x ∈ set l ⟹ P x = Q x⟧ ⟹ takeWhile P l = takeWhile Q k" by (induct k arbitrary: l) (simp_all) lemma dropWhile_cong [fundef_cong]: "⟦l = k; ⋀x. x ∈ set l ⟹ P x = Q x⟧ ⟹ dropWhile P l = dropWhile Q k" by (induct k arbitrary: l, simp_all) lemma takeWhile_idem [simp]: "takeWhile P (takeWhile P xs) = takeWhile P xs" by (induct xs) auto lemma dropWhile_idem [simp]: "dropWhile P (dropWhile P xs) = dropWhile P xs" by (induct xs) auto subsubsection ‹@{const 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 [code]: "zip [] ys = []" "zip xs [] = []" "zip (x # xs) (y # ys) = (x, y) # zip xs ys" by (fact zip_Nil zip.simps(1) zip_Cons_Cons)+ 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]: "length (zip xs ys) = min (length xs) (length ys)" by (induct xs ys rule:list_induct2') auto lemma zip_obtain_same_length: assumes "⋀zs ws n. length zs = length ws ⟹ n = min (length xs) (length ys) ⟹ zs = take n xs ⟹ ws = take n ys ⟹ P (zip zs ws)" shows "P (zip xs ys)" proof - let ?n = "min (length xs) (length ys)" have "P (zip (take ?n xs) (take ?n ys))" by (rule assms) simp_all moreover have "zip xs ys = zip (take ?n xs) (take ?n ys)" proof (induct xs arbitrary: ys) case Nil then show ?case by simp next case (Cons x xs) then show ?case by (cases ys) simp_all qed ultimately show ?thesis by simp qed lemma zip_append1: "zip (xs @ ys) zs = zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)" by (induct xs zs rule:list_induct2') auto lemma zip_append2: "zip xs (ys @ zs) = zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs" by (induct xs ys rule:list_induct2') auto lemma zip_append [simp]: "[| length xs = length us |] ==> 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 zip_map_map: "zip (map f xs) (map g ys) = map (λ (x, y). (f x, g y)) (zip xs ys)" proof (induct xs arbitrary: ys) case (Cons x xs) note Cons_x_xs = Cons.hyps show ?case proof (cases ys) case (Cons y ys') show ?thesis unfolding Cons using Cons_x_xs by simp qed simp qed simp lemma zip_map1: "zip (map f xs) ys = map (λ(x, y). (f x, y)) (zip xs ys)" using zip_map_map[of f xs "λx. x" ys] by simp lemma zip_map2: "zip xs (map f ys) = map (λ(x, y). (x, f y)) (zip xs ys)" using zip_map_map[of "λx. x" xs f ys] by simp lemma map_zip_map: "map f (zip (map g xs) ys) = map (%(x,y). f(g x, y)) (zip xs ys)" by (auto simp: zip_map1) lemma map_zip_map2: "map f (zip xs (map g ys)) = map (%(x,y). f(x, g y)) (zip xs ys)" by (auto simp: zip_map2) text‹Courtesy of Andreas Lochbihler:› lemma zip_same_conv_map: "zip xs xs = map (λx. (x, x)) xs" by(induct xs) auto lemma nth_zip [simp]: "[| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)" apply (induct ys arbitrary: i xs, 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_same: "((a,b) ∈ set (zip xs xs)) = (a ∈ set xs ∧ a = b)" by(induct xs) auto lemma zip_update: "zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]" by(rule sym, simp add: update_zip) lemma zip_replicate [simp]: "zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)" apply (induct i arbitrary: j, auto) apply (case_tac j, auto) done lemma zip_replicate1: "zip (replicate n x) ys = map (Pair x) (take n ys)" by(induction ys arbitrary: n)(case_tac [2] n, simp_all) lemma take_zip: "take n (zip xs ys) = zip (take n xs) (take n ys)" apply (induct n arbitrary: xs ys) apply simp apply (case_tac xs, simp) apply (case_tac ys, simp_all) done lemma drop_zip: "drop n (zip xs ys) = zip (drop n xs) (drop n ys)" apply (induct n arbitrary: xs ys) apply simp apply (case_tac xs, simp) apply (case_tac ys, simp_all) done lemma zip_takeWhile_fst: "zip (takeWhile P xs) ys = takeWhile (P ∘ fst) (zip xs ys)" proof (induct xs arbitrary: ys) case (Cons x xs) thus ?case by (cases ys) auto qed simp lemma zip_takeWhile_snd: "zip xs (takeWhile P ys) = takeWhile (P ∘ snd) (zip xs ys)" proof (induct xs arbitrary: ys) case (Cons x xs) thus ?case by (cases ys) auto qed simp lemma set_zip_leftD: "(x,y)∈ set (zip xs ys) ⟹ x ∈ set xs" by (induct xs ys rule:list_induct2') auto lemma set_zip_rightD: "(x,y)∈ set (zip xs ys) ⟹ y ∈ set ys" by (induct xs ys rule:list_induct2') auto lemma in_set_zipE: "(x,y) ∈ set(zip xs ys) ⟹ (⟦ x ∈ set xs; y ∈ set ys ⟧ ⟹ R) ⟹ R" by(blast dest: set_zip_leftD set_zip_rightD) lemma zip_map_fst_snd: "zip (map fst zs) (map snd zs) = zs" by (induct zs) simp_all lemma zip_eq_conv: "length xs = length ys ⟹ zip xs ys = zs ⟷ map fst zs = xs ∧ map snd zs = ys" by (auto simp add: zip_map_fst_snd) lemma in_set_zip: "p ∈ set (zip xs ys) ⟷ (∃n. xs ! n = fst p ∧ ys ! n = snd p ∧ n < length xs ∧ n < length ys)" by (cases p) (auto simp add: set_zip) lemma in_set_impl_in_set_zip1: assumes "length xs = length ys" assumes "x ∈ set xs" obtains y where "(x, y) ∈ set (zip xs ys)" proof - from assms have "x ∈ set (map fst (zip xs ys))" by simp from this that show ?thesis by fastforce qed lemma in_set_impl_in_set_zip2: assumes "length xs = length ys" assumes "y ∈ set ys" obtains x where "(x, y) ∈ set (zip xs ys)" proof - from assms have "y ∈ set (map snd (zip xs ys))" by simp from this that show ?thesis by fastforce qed lemma pair_list_eqI: assumes "map fst xs = map fst ys" and "map snd xs = map snd ys" shows "xs = ys" proof - from assms(1) have "length xs = length ys" by (rule map_eq_imp_length_eq) from this assms show ?thesis by (induct xs ys rule: list_induct2) (simp_all add: prod_eqI) qed subsubsection ‹@{const list_all2}› lemma list_all2_lengthD [intro?]: "list_all2 P xs ys ==> length xs = length ys" by (simp add: list_all2_iff) lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])" by (simp add: list_all2_iff) lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])" by (simp add: list_all2_iff) 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_iff) 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_induct [consumes 1, case_names Nil Cons, induct set: list_all2]: assumes P: "list_all2 P xs ys" assumes Nil: "R [] []" assumes Cons: "⋀x xs y ys. ⟦P x y; list_all2 P xs ys; R xs ys⟧ ⟹ R (x # xs) (y # ys)" shows "R xs ys" using P by (induct xs arbitrary: ys) (auto simp add: list_all2_Cons1 Nil Cons) lemma list_all2_rev [iff]: "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys" by (simp add: list_all2_iff 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 = (∃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_iff 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) = (∃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_iff 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_iff 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). case_prod P x ⟹ length a = length b ⟹ list_all2 P a b" by (simp add: list_all2_iff) 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: "⟦ list_all2 P xs ys; P x y ⟧ ⟹ list_all2 P (xs[i:=x]) (ys[i:=y])" by (cases "i < length ys") (auto simp add: list_all2_conv_all_nth nth_list_update) lemma list_all2_takeI [simp,intro?]: "list_all2 P xs ys ⟹ list_all2 P (take n xs) (take n ys)" apply (induct xs arbitrary: n ys) apply simp apply (clarsimp simp add: list_all2_Cons1) apply (case_tac n) apply auto done lemma list_all2_dropI [simp,intro?]: "list_all2 P as bs ⟹ list_all2 P (drop n as) (drop n bs)" apply (induct as arbitrary: n bs, simp) apply (clarsimp simp add: list_all2_Cons1) apply (case_tac n, simp, simp) done lemma list_all2_mono [intro?]: "list_all2 P xs ys ⟹ (⋀xs ys. P xs ys ⟹ Q xs ys) ⟹ list_all2 Q xs ys" apply (induct xs arbitrary: ys, simp) apply (case_tac ys, auto) done lemma list_all2_eq: "xs = ys ⟷ list_all2 (=) xs ys" by (induct xs ys rule: list_induct2') auto lemma list_eq_iff_zip_eq: "xs = ys ⟷ length xs = length ys ∧ (∀(x,y) ∈ set (zip xs ys). x = y)" by(auto simp add: set_zip list_all2_eq list_all2_conv_all_nth cong: conj_cong) lemma list_all2_same: "list_all2 P xs xs ⟷ (∀x∈set xs. P x x)" by(auto simp add: list_all2_conv_all_nth set_conv_nth) lemma zip_assoc: "zip xs (zip ys zs) = map (λ((x, y), z). (x, y, z)) (zip (zip xs ys) zs)" by(rule list_all2_all_nthI[where P="(=)", unfolded list.rel_eq]) simp_all lemma zip_commute: "zip xs ys = map (λ(x, y). (y, x)) (zip ys xs)" by(rule list_all2_all_nthI[where P="(=)", unfolded list.rel_eq]) simp_all lemma zip_left_commute: "zip xs (zip ys zs) = map (λ(y, (x, z)). (x, y, z)) (zip ys (zip xs zs))" by(rule list_all2_all_nthI[where P="(=)", unfolded list.rel_eq]) simp_all lemma zip_replicate2: "zip xs (replicate n y) = map (λx. (x, y)) (take n xs)" by(subst zip_commute)(simp add: zip_replicate1) subsubsection ‹@{const List.product} and @{const product_lists}› lemma product_concat_map: "List.product xs ys = concat (map (λx. map (λy. (x,y)) ys) xs)" by(induction xs) (simp)+ lemma set_product[simp]: "set (List.product xs ys) = set xs × set ys" by (induct xs) auto lemma length_product [simp]: "length (List.product xs ys) = length xs * length ys" by (induct xs) simp_all lemma product_nth: assumes "n < length xs * length ys" shows "List.product xs ys ! n = (xs ! (n div length ys), ys ! (n mod length ys))" using assms proof (induct xs arbitrary: n) case Nil then show ?case by simp next case (Cons x xs n) then have "length ys > 0" by auto with Cons show ?case by (auto simp add: nth_append not_less le_mod_geq le_div_geq) qed lemma in_set_product_lists_length: "xs ∈ set (product_lists xss) ⟹ length xs = length xss" by (induct xss arbitrary: xs) auto lemma product_lists_set: "set (product_lists xss) = {xs. list_all2 (λx ys. x ∈ set ys) xs xss}" (is "?L = Collect ?R") proof (intro equalityI subsetI, unfold mem_Collect_eq) fix xs assume "xs ∈ ?L" then have "length xs = length xss" by (rule in_set_product_lists_length) from this ‹xs ∈ ?L› show "?R xs" by (induct xs xss rule: list_induct2) auto next fix xs assume "?R xs" then show "xs ∈ ?L" by induct auto qed subsubsection ‹@{const fold} with natural argument order› lemma fold_simps [code]: ― ‹eta-expanded variant for generated code -- enables tail-recursion optimisation in Scala› "fold f [] s = s" "fold f (x # xs) s = fold f xs (f x s)" by simp_all lemma fold_remove1_split: "⟦ ⋀x y. x ∈ set xs ⟹ y ∈ set xs ⟹ f x ∘ f y = f y ∘ f x; x ∈ set xs ⟧ ⟹ fold f xs = fold f (remove1 x xs) ∘ f x" by (induct xs) (auto simp add: comp_assoc) lemma fold_cong [fundef_cong]: "a = b ⟹ xs = ys ⟹ (⋀x. x ∈ set xs ⟹ f x = g x) ⟹ fold f xs a = fold g ys b" by (induct ys arbitrary: a b xs) simp_all lemma fold_id: "(⋀x. x ∈ set xs ⟹ f x = id) ⟹ fold f xs = id" by (induct xs) simp_all lemma fold_commute: "(⋀x. x ∈ set xs ⟹ h ∘ g x = f x ∘ h) ⟹ h ∘ fold g xs = fold f xs ∘ h" by (induct xs) (simp_all add: fun_eq_iff) lemma fold_commute_apply: assumes "⋀x. x ∈ set xs ⟹ h ∘ g x = f x ∘ h" shows "h (fold g xs s) = fold f xs (h s)" proof - from assms have "h ∘ fold g xs = fold f xs ∘ h" by (rule fold_commute) then show ?thesis by (simp add: fun_eq_iff) qed lemma fold_invariant: "⟦ ⋀x. x ∈ set xs ⟹ Q x; P s; ⋀x s. Q x ⟹ P s ⟹ P (f x s) ⟧ ⟹ P (fold f xs s)" by (induct xs arbitrary: s) simp_all lemma fold_append [simp]: "fold f (xs @ ys) = fold f ys ∘ fold f xs" by (induct xs) simp_all lemma fold_map [code_unfold]: "fold g (map f xs) = fold (g ∘ f) xs" by (induct xs) simp_all lemma fold_filter: "fold f (filter P xs) = fold (λx. if P x then f x else id) xs" by (induct xs) simp_all lemma fold_rev: "(⋀x y. x ∈ set xs ⟹ y ∈ set xs ⟹ f y ∘ f x = f x ∘ f y) ⟹ fold f (rev xs) = fold f xs" by (induct xs) (simp_all add: fold_commute_apply fun_eq_iff) lemma fold_Cons_rev: "fold Cons xs = append (rev xs)" by (induct xs) simp_all lemma rev_conv_fold [code]: "rev xs = fold Cons xs []" by (simp add: fold_Cons_rev) lemma fold_append_concat_rev: "fold append xss = append (concat (rev xss))" by (induct xss) simp_all text ‹@{const Finite_Set.fold} and @{const fold}› lemma (in comp_fun_commute) fold_set_fold_remdups: "Finite_Set.fold f y (set xs) = fold f (remdups xs) y" by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_left_comm insert_absorb) lemma (in comp_fun_idem) fold_set_fold: "Finite_Set.fold f y (set xs) = fold f xs y" by (rule sym, induct xs arbitrary: y) (simp_all add: fold_fun_left_comm) lemma union_set_fold [code]: "set xs ∪ A = fold Set.insert xs A" proof - interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert) show ?thesis by (simp add: union_fold_insert fold_set_fold) qed lemma union_coset_filter [code]: "List.coset xs ∪ A = List.coset (List.filter (λx. x ∉ A) xs)" by auto lemma minus_set_fold [code]: "A - set xs = fold Set.remove xs A" proof - interpret comp_fun_idem Set.remove by (fact comp_fun_idem_remove) show ?thesis by (simp add: minus_fold_remove [of _ A] fold_set_fold) qed lemma minus_coset_filter [code]: "A - List.coset xs = set (List.filter (λx. x ∈ A) xs)" by auto lemma inter_set_filter [code]: "A ∩ set xs = set (List.filter (λx. x ∈ A) xs)" by auto lemma inter_coset_fold [code]: "A ∩ List.coset xs = fold Set.remove xs A" by (simp add: Diff_eq [symmetric] minus_set_fold) lemma (in semilattice_set) set_eq_fold [code]: "F (set (x # xs)) = fold f xs x" proof - interpret comp_fun_idem f by standard (simp_all add: fun_eq_iff left_commute) show ?thesis by (simp add: eq_fold fold_set_fold) qed lemma (in complete_lattice) Inf_set_fold: "Inf (set xs) = fold inf xs top" proof - interpret comp_fun_idem "inf :: 'a ⇒ 'a ⇒ 'a" by (fact comp_fun_idem_inf) show ?thesis by (simp add: Inf_fold_inf fold_set_fold inf_commute) qed declare Inf_set_fold [where 'a = "'a set", code] lemma (in complete_lattice) Sup_set_fold: "Sup (set xs) = fold sup xs bot" proof - interpret comp_fun_idem "sup :: 'a ⇒ 'a ⇒ 'a" by (fact comp_fun_idem_sup) show ?thesis by (simp add: Sup_fold_sup fold_set_fold sup_commute) qed declare Sup_set_fold [where 'a = "'a set", code] lemma (in complete_lattice) INF_set_fold: "INFIMUM (set xs) f = fold (inf ∘ f) xs top" using Inf_set_fold [of "map f xs "] by (simp add: fold_map) declare INF_set_fold [code] lemma (in complete_lattice) SUP_set_fold: "SUPREMUM (set xs) f = fold (sup ∘ f) xs bot" using Sup_set_fold [of "map f xs "] by (simp add: fold_map) declare SUP_set_fold [code] subsubsection ‹Fold variants: @{const foldr} and @{const foldl}› text ‹Correspondence› lemma foldr_conv_fold [code_abbrev]: "foldr f xs = fold f (rev xs)" by (induct xs) simp_all lemma foldl_conv_fold: "foldl f s xs = fold (λx s. f s x) xs s" by (induct xs arbitrary: s) simp_all lemma foldr_conv_foldl: ― ‹The ``Third Duality Theorem'' in Bird \& Wadler:› "foldr f xs a = foldl (λx y. f y x) a (rev xs)" by (simp add: foldr_conv_fold foldl_conv_fold) lemma foldl_conv_foldr: "foldl f a xs = foldr (λx y. f y x) (rev xs) a" by (simp add: foldr_conv_fold foldl_conv_fold) lemma foldr_fold: "(⋀x y. x ∈ set xs ⟹ y ∈ set xs ⟹ f y ∘ f x = f x ∘ f y) ⟹ foldr f xs = fold f xs" unfolding foldr_conv_fold by (rule fold_rev) lemma foldr_cong [fundef_cong]: "a = b ⟹ l = k ⟹ (⋀a x. x ∈ set l ⟹ f x a = g x a) ⟹ foldr f l a = foldr g k b" by (auto simp add: foldr_conv_fold intro!: fold_cong) lemma foldl_cong [fundef_cong]: "a = b ⟹ l = k ⟹ (⋀a x. x ∈ set l ⟹ f a x = g a x) ⟹ foldl f a l = foldl g b k" by (auto simp add: foldl_conv_fold intro!: fold_cong) lemma foldr_append [simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)" by (simp add: foldr_conv_fold) lemma foldl_append [simp]: "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys" by (simp add: foldl_conv_fold) lemma foldr_map [code_unfold]: "foldr g (map f xs) a = foldr (g ∘ f) xs a" by (simp add: foldr_conv_fold fold_map rev_map) lemma foldr_filter: "foldr f (filter P xs) = foldr (λx. if P x then f x else id) xs" by (simp add: foldr_conv_fold rev_filter fold_filter) lemma foldl_map [code_unfold]: "foldl g a (map f xs) = foldl (λa x. g a (f x)) a xs" by (simp add: foldl_conv_fold fold_map comp_def) lemma concat_conv_foldr [code]: "concat xss = foldr append xss []" by (simp add: fold_append_concat_rev foldr_conv_fold) subsubsection ‹@{const upt}› lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])" ― ‹simp does not terminate!› by (induct j) auto lemmas upt_rec_numeral[simp] = upt_rec[of "numeral m" "numeral n"] for m n 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: "([i..<j] = x#xs) = (i < j ∧ i = x ∧ [i+1..<j] = xs)" apply(induct j arbitrary: x xs) 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 ‹upt_Suc› is deleted from the simpset.› by simp lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]" by (simp add: upt_rec) lemma upt_conv_Cons_Cons: ― ‹no precondition› "m # n # ns = [m..<q] ⟷ n # ns = [Suc m..<q]" proof (cases "m < q") case False then show ?thesis by simp next case True then show ?thesis by (simp add: upt_conv_Cons) qed lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]" ― ‹LOOPS as a simprule, since ‹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" by (induct j) (auto simp add: less_Suc_eq nth_append split: nat_diff_split) lemma hd_upt[simp]: "i < j ⟹ hd[i..<j] = i" by(simp add:upt_conv_Cons) lemma tl_upt: "tl [m..<n] = [Suc m..<n]" by (simp add: upt_rec) lemma last_upt[simp]: "i < j ⟹ last[i..<j] = j - 1" by(cases j)(auto simp: upt_Suc_append) lemma take_upt [simp]: "i+m <= n ==> take m [i..<n] = [i..<i+m]" apply (induct m arbitrary: i, 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]" by(induct j) auto lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..<Suc n]" by (induct n) auto lemma map_add_upt: "map (λi. i + n) [0..<m] = [n..<m + n]" by (induct m) simp_all lemma nth_map_upt: "i < n-m ==> (map f [m..<n]) ! i = f(m+i)" apply (induct n m arbitrary: i rule: diff_induct) prefer 3 apply (subst map_Suc_upt[symmetric]) apply (auto simp add: less_diff_conv) done lemma map_decr_upt: "map (λn. n - Suc 0) [Suc m..<Suc n] = [m..<n]" by (induct n) simp_all lemma map_upt_Suc: "map f [0 ..< Suc n] = f 0 # map (λi. f (Suc i)) [0 ..< n]" by (induct n arbitrary: f) auto lemma nth_take_lemma: "k ≤ length xs ⟹ k ≤ length ys ⟹ (⋀i. i < k ⟶ xs!i = ys!i) ⟹ take k xs = take k ys" apply (atomize, induct k arbitrary: xs ys) 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; ∀i < length xs. xs!i = ys!i |] ==> xs = ys" by (frule nth_take_lemma [OF le_refl eq_imp_le]) simp_all lemma map_nth: "map (λi. xs ! i) [0..<length xs] = xs" by (rule nth_equalityI, auto) 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) 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 lemma take_Cons_numeral [simp]: "take (numeral v) (x # xs) = x # take (numeral v - 1) xs" by (simp add: take_Cons') lemma drop_Cons_numeral [simp]: "drop (numeral v) (x # xs) = drop (numeral v - 1) xs" by (simp add: drop_Cons') lemma nth_Cons_numeral [simp]: "(x # xs) ! numeral v = xs ! (numeral v - 1)" by (simp add: nth_Cons') subsubsection ‹‹upto›: interval-list on @{typ int}› function upto :: "int ⇒ int ⇒ int list" ("(1[_../_])") where "upto i j = (if i ≤ j then i # [i+1..j] else [])" by auto termination by(relation "measure(%(i::int,j). nat(j - i + 1))") auto declare upto.simps[simp del] lemmas upto_rec_numeral [simp] = upto.simps[of "numeral m" "numeral n"] upto.simps[of "numeral m" "- numeral n"] upto.simps[of "- numeral m" "numeral n"] upto.simps[of "- numeral m" "- numeral n"] for m n lemma upto_empty[simp]: "j < i ⟹ [i..j] = []" by(simp add: upto.simps) lemma upto_single[simp]: "[i..i] = [i]" by(simp add: upto.simps) lemma upto_Nil[simp]: "[i..j] = [] ⟷ j < i" by (simp add: upto.simps) lemma upto_Nil2[simp]: "[] = [i..j] ⟷ j < i" by (simp add: upto.simps) lemma upto_rec1: "i ≤ j ⟹ [i..j] = i#[i+1..j]" by(simp add: upto.simps) lemma upto_rec2: "i ≤ j ⟹ [i..j] = [i..j - 1]@[j]" proof(induct "nat(j-i)" arbitrary: i j) case 0 thus ?case by(simp add: upto.simps) next case (Suc n) hence "n = nat (j - (i + 1))" "i < j" by linarith+ from this(2) Suc.hyps(1)[OF this(1)] Suc(2,3) upto_rec1 show ?case by simp qed lemma length_upto[simp]: "length [i..j] = nat(j - i + 1)" by(induction i j rule: upto.induct) (auto simp: upto.simps) lemma set_upto[simp]: "set[i..j] = {i..j}" proof(induct i j rule:upto.induct) case (1 i j) from this show ?case unfolding upto.simps[of i j] by auto qed lemma nth_upto: "i + int k ≤ j ⟹ [i..j] ! k = i + int k" apply(induction i j arbitrary: k rule: upto.induct) apply(subst upto_rec1) apply(auto simp add: nth_Cons') done lemma upto_split1: "i ≤ j ⟹ j ≤ k ⟹ [i..k] = [i..j-1] @ [j..k]" proof (induction j rule: int_ge_induct) case base thus ?case by (simp add: upto_rec1) next case step thus ?case using upto_rec1 upto_rec2 by simp qed lemma upto_split2: "i ≤ j ⟹ j ≤ k ⟹ [i..k] = [i..j] @ [j+1..k]" using upto_rec1 upto_rec2 upto_split1 by auto lemma upto_split3: "⟦ i ≤ j; j ≤ k ⟧ ⟹ [i..k] = [i..j-1] @ j # [j+1..k]" using upto_rec1 upto_split1 by auto text‹Tail recursive version for code generation:› definition upto_aux :: "int ⇒ int ⇒ int list ⇒ int list" where "upto_aux i j js = [i..j] @ js" lemma upto_aux_rec [code]: "upto_aux i j js = (if j<i then js else upto_aux i (j - 1) (j#js))" by (simp add: upto_aux_def upto_rec2) lemma upto_code[code]: "[i..j] = upto_aux i j []" by(simp add: upto_aux_def) subsubsection ‹@{const distinct} and @{const remdups} and @{const remdups_adj}› lemma distinct_tl: "distinct xs ⟹ distinct (tl xs)" by (cases xs) simp_all 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 distinct_remdups_id: "distinct xs ==> remdups xs = xs" by (induct xs, auto) lemma remdups_id_iff_distinct [simp]: "remdups xs = xs ⟷ distinct xs" by (metis distinct_remdups distinct_remdups_id) lemma finite_distinct_list: "finite A ⟹ ∃xs. set xs = A ∧ distinct xs" by (metis distinct_remdups finite_list set_remdups) 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 remdups_filter: "remdups(filter P xs) = filter P (remdups xs)" by (induct xs) auto lemma distinct_map: "distinct(map f xs) = (distinct xs ∧ inj_on f (set xs))" by (induct xs) auto lemma distinct_map_filter: "distinct (map f xs) ⟹ distinct (map f (filter P xs))" by (induct xs) auto lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)" by (induct xs) auto lemma distinct_upt[simp]: "distinct[i..<j]" by (induct j) auto lemma distinct_upto[simp]: "distinct[i..j]" apply(induct i j rule:upto.induct) apply(subst upto.simps) apply(simp) done lemma distinct_take[simp]: "distinct xs ⟹ distinct (take i xs)" apply(induct xs arbitrary: i) apply simp apply (case_tac i) apply simp_all apply(blast dest:in_set_takeD) done lemma distinct_drop[simp]: "distinct xs ⟹ distinct (drop i xs)" apply(induct xs arbitrary: i) 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 lemma distinct_concat: "⟦ distinct xs; ⋀ ys. ys ∈ set xs ⟹ distinct ys; ⋀ ys zs. ⟦ ys ∈ set xs ; zs ∈ set xs ; ys ≠ zs ⟧ ⟹ set ys ∩ set zs = {} ⟧ ⟹ distinct (concat xs)" by (induct xs) auto 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 nth_eq_iff_index_eq: "⟦ distinct xs; i < length xs; j < length xs ⟧ ⟹ (xs!i = xs!j) = (i = j)" by(auto simp: distinct_conv_nth) lemma distinct_Ex1: "distinct xs ⟹ x ∈ set xs ⟹ (∃!i. i < length xs ∧ xs ! i = x)" by (auto simp: in_set_conv_nth nth_eq_iff_index_eq) lemma inj_on_nth: "distinct xs ⟹ ∀i ∈ I. i < length xs ⟹ inj_on (nth xs) I" by (rule inj_onI) (simp add: nth_eq_iff_index_eq) lemma bij_betw_nth: assumes "distinct xs" "A = {..<length xs}" "B = set xs" shows "bij_betw ((!) xs) A B" using assms unfolding bij_betw_def by (auto intro!: inj_on_nth simp: set_conv_nth) lemma set_update_distinct: "⟦ distinct xs; n < length xs ⟧ ⟹ set(xs[n := x]) = insert x (set xs - {xs!n})" by(auto simp: set_eq_iff in_set_conv_nth nth_list_update nth_eq_iff_index_eq) lemma distinct_swap[simp]: "⟦ i < size xs; j < size xs ⟧ ⟹ distinct(xs[i := xs!j, j := xs!i]) = distinct xs" apply (simp add: distinct_conv_nth nth_list_update) apply safe apply metis+ done lemma set_swap[simp]: "⟦ i < size xs; j < size xs ⟧ ⟹ set(xs[i := xs!j, j := xs!i]) = set xs" by(simp add: set_conv_nth nth_list_update) metis 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: if_split_asm) moreover have "card (set xs) ≤ length xs" by (rule card_length) ultimately have False by simp thus ?thesis .. qed qed lemma distinct_length_filter: "distinct xs ⟹ length (filter P xs) = card ({x. P x} Int set xs)" by (induct xs) (auto) lemma not_distinct_decomp: "¬ distinct ws ⟹ ∃xs ys zs y. ws = xs@[y]@ys@[y]@zs" apply (induct n == "length ws" arbitrary:ws) apply simp apply(case_tac ws) apply simp apply (simp split:if_split_asm) apply (metis Cons_eq_appendI eq_Nil_appendI split_list) done lemma not_distinct_conv_prefix: defines "dec as xs y ys ≡ y ∈ set xs ∧ distinct xs ∧ as = xs @ y # ys" shows "¬distinct as ⟷ (∃xs y ys. dec as xs y ys)" (is "?L = ?R") proof assume "?L" then show "?R" proof (induct "length as" arbitrary: as rule: less_induct) case less obtain xs ys zs y where decomp: "as = (xs @ y # ys) @ y # zs" using not_distinct_decomp[OF less.prems] by auto show ?case proof (cases "distinct (xs @ y # ys)") case True with decomp have "dec as (xs @ y # ys) y zs" by (simp add: dec_def) then show ?thesis by blast next case False with less decomp obtain xs' y' ys' where "dec (xs @ y # ys) xs' y' ys'" by atomize_elim auto with decomp have "dec as xs' y' (ys' @ y # zs)" by (simp add: dec_def) then show ?thesis by blast qed qed qed (auto simp: dec_def) lemma distinct_product: "distinct xs ⟹ distinct ys ⟹ distinct (List.product xs ys)" by (induct xs) (auto intro: inj_onI simp add: distinct_map) lemma distinct_product_lists: assumes "∀xs ∈ set xss. distinct xs" shows "distinct (product_lists xss)" using assms proof (induction xss) case (Cons xs xss) note * = this then show ?case proof (cases "product_lists xss") case Nil then show ?thesis by (induct xs) simp_all next case (Cons ps pss) with * show ?thesis by (auto intro!: inj_onI distinct_concat simp add: distinct_map) qed qed simp lemma length_remdups_concat: "length (remdups (concat xss)) = card (⋃xs∈set xss. set xs)" by (simp add: distinct_card [symmetric]) lemma length_remdups_card_conv: "length(remdups xs) = card(set xs)" proof - have xs: "concat[xs] = xs" by simp from length_remdups_concat[of "[xs]"] show ?thesis unfolding xs by simp qed lemma remdups_remdups: "remdups (remdups xs) = remdups xs" by (induct xs) simp_all lemma distinct_butlast: assumes "distinct xs" shows "distinct (butlast xs)" proof (cases "xs = []") case False from ‹xs ≠ []› obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto with ‹distinct xs› show ?thesis by simp qed (auto) lemma remdups_map_remdups: "remdups (map f (remdups xs)) = remdups (map f xs)" by (induct xs) simp_all lemma distinct_zipI1: assumes "distinct xs" shows "distinct (zip xs ys)" proof (rule zip_obtain_same_length) fix xs' :: "'a list" and ys' :: "'b list" and n assume "length xs' = length ys'" assume "xs' = take n xs" with assms have "distinct xs'" by simp with ‹length xs' = length ys'› show "distinct (zip xs' ys')" by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE) qed lemma distinct_zipI2: assumes "distinct ys" shows "distinct (zip xs ys)" proof (rule zip_obtain_same_length) fix xs' :: "'b list" and ys' :: "'a list" and n assume "length xs' = length ys'" assume "ys' = take n ys" with assms have "distinct ys'" by simp with ‹length xs' = length ys'› show "distinct (zip xs' ys')" by (induct xs' ys' rule: list_induct2) (auto elim: in_set_zipE) qed lemma set_take_disj_set_drop_if_distinct: "distinct vs ⟹ i ≤ j ⟹ set (take i vs) ∩ set (drop j vs) = {}" by (auto simp: in_set_conv_nth distinct_conv_nth) (* The next two lemmas help Sledgehammer. *) lemma distinct_singleton: "distinct [x]" by simp lemma distinct_length_2_or_more: "distinct (a # b # xs) ⟷ (a ≠ b ∧ distinct (a # xs) ∧ distinct (b # xs))" by force lemma remdups_adj_altdef: "(remdups_adj xs = ys) ⟷ (∃f::nat => nat. mono f ∧ f ` {0 ..< size xs} = {0 ..< size ys} ∧ (∀i < size xs. xs!i = ys!(f i)) ∧ (∀i. i + 1 < size xs ⟶ (xs!i = xs!(i+1) ⟷ f i = f(i+1))))" (is "?L ⟷ (∃f. ?p f xs ys)") proof assume ?L then show "∃f. ?p f xs ys" proof (induct xs arbitrary: ys rule: remdups_adj.induct) case (1 ys) thus ?case by (intro exI[of _ id]) (auto simp: mono_def) next case (2 x ys) thus ?case by (intro exI[of _ id]) (auto simp: mono_def) next case (3 x1 x2 xs ys) let ?xs = "x1 # x2 # xs" let ?cond = "x1 = x2" define zs where "zs = remdups_adj (x2 # xs)" from 3(1-2)[of zs] obtain f where p: "?p f (x2 # xs) zs" unfolding zs_def by (cases ?cond) auto then have f0: "f 0 = 0" by (intro mono_image_least[where f=f]) blast+ from p have mono: "mono f" and f_xs_zs: "f ` {0..<length (x2 # xs)} = {0..<length zs}" by auto have ys: "ys = (if x1 = x2 then zs else x1 # zs)" unfolding 3(3)[symmetric] zs_def by auto have zs0: "zs ! 0 = x2" unfolding zs_def by (induct xs) auto have zsne: "zs ≠ []" unfolding zs_def by (induct xs) auto let ?Succ = "if ?cond then id else Suc" let ?x1 = "if ?cond then id else Cons x1" let ?f = "λ i. if i = 0 then 0 else ?Succ (f (i - 1))" have ys: "ys = ?x1 zs" unfolding ys by (cases ?cond, auto) have mono: "mono ?f" using ‹mono f› unfolding mono_def by auto show ?case unfolding ys proof (intro exI[of _ ?f] conjI allI impI) show "mono ?f" by fact next fix i assume i: "i < length ?xs" with p show "?xs ! i = ?x1 zs ! (?f i)" using zs0 by auto next fix i assume i: "i + 1 < length ?xs" with p show "(?xs ! i = ?xs ! (i + 1)) = (?f i = ?f (i + 1))" by (cases i) (auto simp: f0) next have id: "{0 ..< length (?x1 zs)} = insert 0 (?Succ ` {0 ..< length zs})" using zsne by (cases ?cond, auto) { fix i assume "i < Suc (length xs)" hence "Suc i ∈ {0..<Suc (Suc (length xs))} ∩ Collect ((<) 0)" by auto from imageI[OF this, of "λi. ?Succ (f (i - Suc 0))"] have "?Succ (f i) ∈ (λi. ?Succ (f (i - Suc 0))) ` ({0..<Suc (Suc (length xs))} ∩ Collect ((<) 0))" by auto } then show "?f ` {0 ..< length ?xs} = {0 ..< length (?x1 zs)}" unfolding id f_xs_zs[symmetric] by auto qed qed next assume "∃ f. ?p f xs ys" then show ?L proof (induct xs arbitrary: ys rule: remdups_adj.induct) case 1 then show ?case by auto next case (2 x) then obtain f where f_img: "f ` {0 ..< size [x]} = {0 ..< size ys}" and f_nth: "⋀i. i < size [x] ⟹ [x]!i = ys!(f i)" by blast have "length ys = card (f ` {0 ..< size [x]})" using f_img by auto then have *: "length ys = 1" by auto then have "f 0 = 0" using f_img by auto with * show ?case using f_nth by (cases ys) auto next case (3 x1 x2 xs) from "3.prems" obtain f where f_mono: "mono f" and f_img: "f ` {0..<length (x1 # x2 # xs)} = {0..<length ys}" and f_nth: "⋀i. i < length (x1 # x2 # xs) ⟹ (x1 # x2 # xs) ! i = ys ! f i" "⋀i. i + 1 < length (x1 # x2 #xs) ⟹ ((x1 # x2 # xs) ! i = (x1 # x2 # xs) ! (i + 1)) = (f i = f (i + 1))" by blast show ?case proof cases assume "x1 = x2" let ?f' = "f ∘ Suc" have "remdups_adj (x1 # xs) = ys" proof (intro "3.hyps" exI conjI impI allI) show "mono ?f'" using f_mono by (simp add: mono_iff_le_Suc) next have "?f' ` {0 ..< length (x1 # xs)} = f ` {Suc 0 ..< length (x1 # x2 # xs)}" by safe (fastforce, rename_tac x, case_tac x, auto) also have "… = f ` {0 ..< length (x1 # x2 # xs)}" proof - have "f 0 = f (Suc 0)" using ‹x1 = x2› f_nth[of 0] by simp then show ?thesis by safe (fastforce, rename_tac x, case_tac x, auto) qed also have "… = {0 ..< length ys}" by fact finally show "?f' ` {0 ..< length (x1 # xs)} = {0 ..< length ys}" . qed (insert f_nth[of "Suc i" for i], auto simp: ‹x1 = x2›) then show ?thesis using ‹x1 = x2› by simp next assume "x1 ≠ x2" have "2 ≤ length ys" proof - have "2 = card {f 0, f 1}" using ‹x1 ≠ x2› f_nth[of 0] by auto also have "… ≤ card (f ` {0..< length (x1 # x2 # xs)})" by (rule card_mono) auto finally show ?thesis using f_img by simp qed have "f 0 = 0" using f_mono f_img by (rule mono_image_least) simp have "f (Suc 0) = Suc 0" proof (rule ccontr) assume "f (Suc 0) ≠ Suc 0" then have "Suc 0 < f (Suc 0)" using f_nth[of 0] ‹x1 ≠ x2› ‹f 0 = 0› by auto then have "⋀i. Suc 0 < f (Suc i)" using f_mono by (meson Suc_le_mono le0 less_le_trans monoD) then have "⋀i. Suc 0 ≠ f i" using ‹f 0 = 0› by (case_tac i) fastforce+ then have "Suc 0 ∉ f ` {0 ..< length (x1 # x2 # xs)}" by auto then show False using f_img ‹2 ≤ length ys› by auto qed obtain ys' where "ys = x1 # x2 # ys'" using ‹2 ≤ length ys› f_nth[of 0] f_nth[of 1] apply (cases ys) apply (rename_tac [2] ys', case_tac [2] ys') by (auto simp: ‹f 0 = 0› ‹f (Suc 0) = Suc 0›) define f' where "f' x = f (Suc x) - 1" for x { fix i have "Suc 0 ≤ f (Suc 0)" using f_nth[of 0] ‹x1 ≠ x2› ‹f 0 = 0› by auto also have "… ≤ f (Suc i)" using f_mono by (rule monoD) arith finally have "Suc 0 ≤ f (Suc i)" . } note Suc0_le_f_Suc = this { fix i have "f (Suc i) = Suc (f' i)" using Suc0_le_f_Suc[of i] by (auto simp: f'_def) } note f_Suc = this have "remdups_adj (x2 # xs) = (x2 # ys')" proof (intro "3.hyps" exI conjI impI allI) show "mono f'" using Suc0_le_f_Suc f_mono by (auto simp: f'_def mono_iff_le_Suc le_diff_iff) next have "f' ` {0 ..< length (x2 # xs)} = (λx. f x - 1) ` {0 ..< length (x1 # x2 #xs)}" apply safe apply (rename_tac [!] n, case_tac [!] n) apply (auto simp: f'_def ‹f 0 = 0› ‹f (Suc 0) = Suc 0› intro: rev_image_eqI) done also have "… = (λx. x - 1) ` f ` {0 ..< length (x1 # x2 #xs)}" by (auto simp: image_comp) also have "… = (λx. x - 1) ` {0 ..< length ys}" by (simp only: f_img) also have "… = {0 ..< length (x2 # ys')}" using ‹ys = _› by (fastforce intro: rev_image_eqI) finally show "f' ` {0 ..< length (x2 # xs)} = {0 ..< length (x2 # ys')}" . qed (insert f_nth[of "Suc i" for i] ‹x1 ≠ x2›, auto simp add: f_Suc ‹ys = _›) then show ?case using ‹ys = _› ‹x1 ≠ x2› by simp qed qed qed lemma hd_remdups_adj[simp]: "hd (remdups_adj xs) = hd xs" by (induction xs rule: remdups_adj.induct) simp_all lemma remdups_adj_Cons: "remdups_adj (x # xs) = (case remdups_adj xs of [] ⇒ [x] | y # xs ⇒ if x = y then y # xs else x # y # xs)" by (induct xs arbitrary: x) (auto split: list.splits) lemma remdups_adj_append_two: "remdups_adj (xs @ [x,y]) = remdups_adj (xs @ [x]) @ (if x = y then [] else [y])" by (induct xs rule: remdups_adj.induct, simp_all) lemma remdups_adj_adjacent: "Suc i < length (remdups_adj xs) ⟹ remdups_adj xs ! i ≠ remdups_adj xs ! Suc i" proof (induction xs arbitrary: i rule: remdups_adj.induct) case (3 x y xs i) thus ?case by (cases i, cases "x = y") (simp, auto simp: hd_conv_nth[symmetric]) qed simp_all lemma remdups_adj_rev[simp]: "remdups_adj (rev xs) = rev (remdups_adj xs)" by (induct xs rule: remdups_adj.induct, simp_all add: remdups_adj_append_two) lemma remdups_adj_length[simp]: "length (remdups_adj xs) ≤ length xs" by (induct xs rule: remdups_adj.induct, auto) lemma remdups_adj_length_ge1[simp]: "xs ≠ [] ⟹ length (remdups_adj xs) ≥ Suc 0" by (induct xs rule: remdups_adj.induct, simp_all) lemma remdups_adj_Nil_iff[simp]: "remdups_adj xs = [] ⟷ xs = []" by (induct xs rule: remdups_adj.induct, simp_all) lemma remdups_adj_set[simp]: "set (remdups_adj xs) = set xs" by (induct xs rule: remdups_adj.induct, simp_all) lemma remdups_adj_Cons_alt[simp]: "x # tl (remdups_adj (x # xs)) = remdups_adj (x # xs)" by (induct xs rule: remdups_adj.induct, auto) lemma remdups_adj_distinct: "distinct xs ⟹ remdups_adj xs = xs" by (induct xs rule: remdups_adj.induct, simp_all) lemma remdups_adj_append: "remdups_adj (xs⇩_{1}@ x # xs⇩_{2}) = remdups_adj (xs⇩_{1}@ [x]) @ tl (remdups_adj (x # xs⇩_{2}))" by (induct xs⇩_{1}rule: remdups_adj.induct, simp_all) lemma remdups_adj_singleton: "remdups_adj xs = [x] ⟹ xs = replicate (length xs) x" by (induct xs rule: remdups_adj.induct, auto split: if_split_asm) lemma remdups_adj_map_injective: assumes "inj f" shows "remdups_adj (map f xs) = map f (remdups_adj xs)" by (induct xs rule: remdups_adj.induct) (auto simp add: injD[OF assms]) lemma remdups_adj_replicate: "remdups_adj (replicate n x) = (if n = 0 then [] else [x])" by (induction n) (auto simp: remdups_adj_Cons) lemma remdups_upt [simp]: "remdups [m..<n] = [m..<n]" proof (cases "m ≤ n") case False then show ?thesis by simp next case True then obtain q where "n = m + q" by (auto simp add: le_iff_add) moreover have "remdups [m..<m + q] = [m..<m + q]" by (induct q) simp_all ultimately show ?thesis by simp qed subsubsection ‹@{const insert}› lemma in_set_insert [simp]: "x ∈ set xs ⟹ List.insert x xs = xs" by (simp add: List.insert_def) lemma not_in_set_insert [simp]: "x ∉ set xs ⟹ List.insert x xs = x # xs" by (simp add: List.insert_def) lemma insert_Nil [simp]: "List.insert x [] = [x]" by simp lemma set_insert [simp]: "set (List.insert x xs) = insert x (set xs)" by (auto simp add: List.insert_def) lemma distinct_insert [simp]: "distinct (List.insert x xs) = distinct xs" by (simp add: List.insert_def) lemma insert_remdups: "List.insert x (remdups xs) = remdups (List.insert x xs)" by (simp add: List.insert_def) subsubsection ‹@{const List.union}› text‹This is all one should need to know about union:› lemma set_union[simp]: "set (List.union xs ys) = set xs ∪ set ys" unfolding List.union_def by(induct xs arbitrary: ys) simp_all lemma distinct_union[simp]: "distinct(List.union xs ys) = distinct ys" unfolding List.union_def by(induct xs arbitrary: ys) simp_all subsubsection ‹@{const List.find}› lemma find_None_iff: "List.find P xs = None ⟷ ¬ (∃x. x ∈ set xs ∧ P x)" proof (induction xs) case Nil thus ?case by simp next case (Cons x xs) thus ?case by (fastforce split: if_splits) qed lemma find_Some_iff: "List.find P xs = Some x ⟷ (∃i<length xs. P (xs!i) ∧ x = xs!i ∧ (∀j<i. ¬ P (xs!j)))" proof (induction xs) case Nil thus ?case by simp next case (Cons x xs) thus ?case apply(auto simp: nth_Cons' split: if_splits) using diff_Suc_1[unfolded One_nat_def] less_Suc_eq_0_disj by fastforce qed lemma find_cong[fundef_cong]: assumes "xs = ys" and "⋀x. x ∈ set ys ⟹ P x = Q x" shows "List.find P xs = List.find Q ys" proof (cases "List.find P xs") case None thus ?thesis by (metis find_None_iff assms) next case (Some x) hence "List.find Q ys = Some x" using assms by (auto simp add: find_Some_iff) thus ?thesis using Some by auto qed lemma find_dropWhile: "List.find P xs = (case dropWhile (Not ∘ P) xs of [] ⇒ None | x # _ ⇒ Some x)" by (induct xs) simp_all subsubsection ‹@{const count_list}› lemma count_notin[simp]: "x ∉ set xs ⟹ count_list xs x = 0" by (induction xs) auto lemma count_le_length: "count_list xs x ≤ length xs" by (induction xs) auto lemma sum_count_set: "set xs ⊆ X ⟹ finite X ⟹ sum (count_list xs) X = length xs" apply(induction xs arbitrary: X) apply simp apply (simp add: sum.If_cases) by (metis Diff_eq sum.remove) subsubsection ‹@{const List.extract}› lemma extract_None_iff: "List.extract P xs = None ⟷ ¬ (∃ x∈set xs. P x)" by(auto simp: extract_def dropWhile_eq_Cons_conv split: list.splits) (metis in_set_conv_decomp) lemma extract_SomeE: "List.extract P xs = Some (ys, y, zs) ⟹ xs = ys @ y # zs ∧ P y ∧ ¬ (∃ y ∈ set ys. P y)" by(auto simp: extract_def dropWhile_eq_Cons_conv split: list.splits) lemma extract_Some_iff: "List.extract P xs = Some (ys, y, zs) ⟷ xs = ys @ y # zs ∧ P y ∧ ¬ (∃ y ∈ set ys. P y)" by(auto simp: extract_def dropWhile_eq_Cons_conv dest: set_takeWhileD split: list.splits) lemma extract_Nil_code[code]: "List.extract P [] = None" by(simp add: extract_def) lemma extract_Cons_code[code]: "List.extract P (x # xs) = (if P x then Some ([], x, xs) else (case List.extract P xs of None ⇒ None | Some (ys, y, zs) ⇒ Some (x#ys, y, zs)))" by(auto simp add: extract_def comp_def split: list.splits) (metis dropWhile_eq_Nil_conv list.distinct(1)) subsubsection ‹@{const remove1}› lemma remove1_append: "remove1 x (xs @ ys) = (if x ∈ set xs then remove1 x xs @ ys else xs @ remove1 x ys)" by (induct xs) auto lemma remove1_commute: "remove1 x (remove1 y zs) = remove1 y (remove1 x zs)" by (induct zs) auto lemma in_set_remove1[simp]: "a ≠ b ⟹ a ∈ set(remove1 b xs) = (a ∈ set xs)" apply (induct xs) apply auto done 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 length_remove1: "length(remove1 x xs) = (if x ∈ set xs then length xs - 1 else length xs)" by (induct xs) (auto dest!:length_pos_if_in_set) lemma remove1_filter_not[simp]: "¬ P x ⟹ remove1 x (filter P xs) = filter P xs" by(induct xs) auto lemma filter_remove1: "filter Q (remove1 x xs) = remove1 x (filter Q xs)" by (induct xs) auto lemma notin_set_remove1[simp]: "x ∉ set xs ⟹ x ∉ set(remove1 y xs)" by(insert set_remove1_subset) fast lemma distinct_remove1[simp]: "distinct xs ⟹ distinct(remove1 x xs)" by (induct xs) simp_all lemma remove1_remdups: "distinct xs ⟹ remove1 x (remdups xs) = remdups (remove1 x xs)" by (induct xs) simp_all lemma remove1_idem: "x ∉ set xs ⟹ remove1 x xs = xs" by (induct xs) simp_all subsubsection ‹@{const removeAll}› lemma removeAll_filter_not_eq: "removeAll x = filter (λy. x ≠ y)" proof fix xs show "removeAll x xs = filter (λy. x ≠ y) xs" by (induct xs) auto qed lemma removeAll_append[simp]: "removeAll x (xs @ ys) = removeAll x xs @ removeAll x ys" by (induct xs) auto lemma set_removeAll[simp]: "set(removeAll x xs) = set xs - {x}" by (induct xs) auto lemma removeAll_id[simp]: "x ∉ set xs ⟹ removeAll x xs = xs" by (induct xs) auto (* Needs count:: 'a ⇒ 'a list ⇒ nat lemma length_removeAll: "length(removeAll x xs) = length xs - count x xs" *) lemma removeAll_filter_not[simp]: "¬ P x ⟹ removeAll x (filter P xs) = filter P xs" by(induct xs) auto lemma distinct_removeAll: "distinct xs ⟹ distinct (removeAll x xs)" by (simp add: removeAll_filter_not_eq) lemma distinct_remove1_removeAll: "distinct xs ⟹ remove1 x xs = removeAll x xs" by (induct xs) simp_all lemma map_removeAll_inj_on: "inj_on f (insert x (set xs)) ⟹ map f (removeAll x xs) = removeAll (f x) (map f xs)" by (induct xs) (simp_all add:inj_on_def) lemma map_removeAll_inj: "inj f ⟹ map f (removeAll x xs) = removeAll (f x) (map f xs)" by (rule map_removeAll_inj_on, erule subset_inj_on, rule subset_UNIV) lemma length_removeAll_less_eq [simp]: "length (removeAll x xs) ≤ length xs" by (simp add: removeAll_filter_not_eq) lemma length_removeAll_less [termination_simp]: "x ∈ set xs ⟹ length (removeAll x xs) < length xs" by (auto dest: length_filter_less simp add: removeAll_filter_not_eq) subsubsection ‹@{const replicate}› lemma length_replicate [simp]: "length (replicate n x) = n" by (induct n) auto lemma replicate_eqI: assumes "length xs = n" and "⋀y. y ∈ set xs ⟹ y = x" shows "xs = replicate n x" using assms proof (induct xs arbitrary: n) case Nil then show ?case by simp next case (Cons x xs) then show ?case by (cases n) simp_all qed lemma Ex_list_of_length: "∃xs. length xs = n" by (rule exI[of _ "replicate n undefined"]) simp lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)" by (induct n) auto lemma map_replicate_const: "map (λ x. k) lst = replicate (length lst) k" by (induct lst) 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" by (induct n) (auto simp: replicate_app_Cons_same) 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 [symmetric]) apply (simp add: add.commute) done text‹Courtesy of Andreas Lochbihler:› lemma filter_replicate: "filter P (replicate n x) = (if P x then replicate n x else [])" by(induct n) auto lemma hd_replicate [simp]: "n ≠ 0 ==> hd (replicate n x) = x" by (induct n) auto lemma tl_replicate [simp]: "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 < n ==> (replicate n x)!i = x" by (induct n arbitrary: i)(auto simp: nth_Cons split: nat.split) 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_le_imp_less) 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]: "drop i (replicate k x) = replicate (k-i) x" apply (induct k arbitrary: i) 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_replicate[simp]: "(x ∈ set (replicate n y)) = (x = y ∧ n ≠ 0)" by (simp add: set_replicate_conv_if) lemma Ball_set_replicate[simp]: "(∀x ∈ set(replicate n a). P x) = (P a ∨ n=0)" by(simp add: set_replicate_conv_if) lemma Bex_set_replicate[simp]: "(∃x ∈ set(replicate n a). P x) = (P a ∧ n≠0)" by(simp add: set_replicate_conv_if) lemma replicate_append_same: "replicate i x @ [x] = x # replicate i x" by (induct i) simp_all lemma map_replicate_trivial: "map (λi. x) [0..<i] = replicate i x" by (induct i) (simp_all add: replicate_append_same) lemma concat_replicate_trivial[simp]: "concat (replicate i []) = []" by (induct i) (auto simp add: map_replicate_const) lemma replicate_empty[simp]: "(replicate n x = []) ⟷ n=0" by (induct n) auto lemma empty_replicate[simp]: "([] = replicate n x) ⟷ n=0" by (induct n) auto lemma replicate_eq_replicate[simp]: "(replicate m x = replicate n y) ⟷ (m=n ∧ (m≠0 ⟶ x=y))" apply(induct m arbitrary: n) apply simp apply(induct_tac n) apply auto done lemma replicate_length_filter: "replicate (length (filter (λy. x = y) xs)) x = filter (λy. x = y) xs" by (induct xs) auto lemma comm_append_are_replicate: "⟦ xs ≠ []; ys ≠ []; xs @ ys = ys @ xs ⟧ ⟹ ∃m n zs. concat (replicate m zs) = xs ∧ concat (replicate n zs) = ys" proof (induction "length (xs @ ys)" arbitrary: xs ys rule: less_induct) case less define xs' ys' where "xs' = (if (length xs ≤ length ys) then xs else ys)" and "ys' = (if (length xs ≤ length ys) then ys else xs)" then have prems': "length xs' ≤ length ys'" "xs' @ ys' = ys' @ xs'" and "xs' ≠ []" and len: "length (xs @ ys) = length (xs' @ ys')" using less by (auto intro: less.hyps) from prems' obtain ws where "ys' = xs' @ ws" by (auto simp: append_eq_append_conv2) have "∃m n zs. concat (replicate m zs) = xs' ∧ concat (replicate n zs) = ys'" proof (cases "ws = []") case True then have "concat (replicate 1 xs') = xs'" and "concat (replicate 1 xs') = ys'" using ‹ys' = xs' @ ws› by auto then show ?thesis by blast next case False from ‹ys' = xs' @ ws› and ‹xs' @ ys' = ys' @ xs'› have "xs' @ ws = ws @ xs'" by simp then have "∃m n zs. concat (replicate m zs) = xs' ∧ concat (replicate n zs) = ws" using False and ‹xs' ≠ []› and ‹ys' = xs' @ ws› and len by (intro less.hyps) auto then obtain m n zs where *: "concat (replicate m zs) = xs'" and "concat (replicate n zs) = ws" by blast then have "concat (replicate (m + n) zs) = ys'" using ‹ys' = xs' @ ws› by (simp add: replicate_add) with * show ?thesis by blast qed then show ?case using xs'_def ys'_def by meson qed lemma comm_append_is_replicate: fixes xs ys :: "'a list" assumes "xs ≠ []" "ys ≠ []" assumes "xs @ ys = ys @ xs" shows "∃n zs. n > 1 ∧ concat (replicate n zs) = xs @ ys" proof - obtain m n zs where "concat (replicate m zs) = xs" and "concat (replicate n zs) = ys" using comm_append_are_replicate[of xs ys, OF assms] by blast then have "m + n > 1" and "concat (replicate (m+n) zs) = xs @ ys" using ‹xs ≠ []› and ‹ys ≠ []› by (auto simp: replicate_add) then show ?thesis by blast qed lemma Cons_replicate_eq: "x # xs = replicate n y ⟷ x = y ∧ n > 0 ∧ xs = replicate (n - 1) x" by (induct n) auto lemma replicate_length_same: "(∀y∈set xs. y = x) ⟹ replicate (length xs) x = xs" by (induct xs) simp_all lemma foldr_replicate [simp]: "foldr f (replicate n x) = f x ^^ n" by (induct n) (simp_all) lemma fold_replicate [simp]: "fold f (replicate n x) = f x ^^ n" by (subst foldr_fold [symmetric]) simp_all subsubsection ‹@{const enumerate}› lemma enumerate_simps [simp, code]: "enumerate n [] = []" "enumerate n (x # xs) = (n, x) # enumerate (Suc n) xs" apply (auto simp add: enumerate_eq_zip not_le) apply (cases "n < n + length xs") apply (auto simp add: upt_conv_Cons) done lemma length_enumerate [simp]: "length (enumerate n xs) = length xs" by (simp add: enumerate_eq_zip) lemma map_fst_enumerate [simp]: "map fst (enumerate n xs) = [n..<n + length xs]" by (simp add: enumerate_eq_zip) lemma map_snd_enumerate [simp]: "map snd (enumerate n xs) = xs" by (simp add: enumerate_eq_zip) lemma in_set_enumerate_eq: "p ∈ set (enumerate n xs) ⟷ n ≤ fst p ∧ fst p < length xs + n ∧ nth xs (fst p - n) = snd p" proof - { fix m assume "n ≤ m" moreover assume "m < length xs + n" ultimately have "[n..<n + length xs] ! (m - n) = m ∧ xs ! (m - n) = xs ! (m - n) ∧ m - n < length xs" by auto then have "∃q. [n..<n + length xs] ! q = m ∧ xs ! q = xs ! (m - n) ∧ q < length xs" .. } then show ?thesis by (cases p) (auto simp add: enumerate_eq_zip in_set_zip) qed lemma nth_enumerate_eq: "m < length xs ⟹ enumerate n xs ! m = (n + m, xs ! m)" by (simp add: enumerate_eq_zip) lemma enumerate_replicate_eq: "enumerate n (replicate m a) = map (λq. (q, a)) [n..<n + m]" by (rule pair_list_eqI) (simp_all add: enumerate_eq_zip comp_def map_replicate_const) lemma enumerate_Suc_eq: "enumerate (Suc n) xs = map (apfst Suc) (enumerate n xs)" by (rule pair_list_eqI) (simp_all add: not_le, simp del: map_map add: map_Suc_upt map_map [symmetric]) lemma distinct_enumerate [simp]: "distinct (enumerate n xs)" by (simp add: enumerate_eq_zip distinct_zipI1) lemma enumerate_append_eq: "enumerate n (xs @ ys) = enumerate n xs @ enumerate (n + length xs) ys" unfolding enumerate_eq_zip apply auto apply (subst zip_append [symmetric]) apply simp apply (subst upt_add_eq_append [symmetric]) apply (simp_all add: ac_simps) done lemma enumerate_map_upt: "enumerate n (map f [n..<m]) = map (λk. (k, f k)) [n..<m]" by (cases "n ≤ m") (simp_all add: zip_map2 zip_same_conv_map enumerate_eq_zip) subsubsection ‹@{const rotate1} and @{const rotate}› 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 ∘ 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_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)" by(simp add:rotate_def funpow_swap1) 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 (cases xs) simp_all 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 (cases xs) simp_all lemma length_rotate[simp]: "length(rotate n xs) = length xs" by (induct n arbitrary: xs) (simp_all add:rotate_def) lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs" by (cases xs) auto 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 (cases xs) auto 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 (cases xs) auto 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)" proof (cases "length xs = 0 ∨ n mod length xs = 0") case False then show ?thesis by(simp add:rotate_drop_take rev_drop rev_take) qed force lemma hd_rotate_conv_nth: "xs ≠ [] ⟹ hd(rotate n xs) = xs!(n mod length xs)" apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth) apply(subgoal_tac "length xs ≠ 0") prefer 2 apply simp using mod_less_divisor[of "length xs" n] by arith lemma rotate_append: "rotate (length l) (l @ q) = q @ l" by (induct l arbitrary: q) (auto simp add: rotate1_rotate_swap) subsubsection ‹@{const nths} --- a generalization of @{const nth} to sets› lemma nths_empty [simp]: "nths xs {} = []" by (auto simp add: nths_def) lemma nths_nil [simp]: "nths [] A = []" by (auto simp add: nths_def) lemma length_nths: "length (nths xs I) = card{i. i < length xs ∧ i ∈ I}" by(simp add: nths_def length_filter_conv_card cong:conj_cong) lemma nths_shift_lemma_Suc: "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 arbitrary: "is") apply simp apply (case_tac "is") apply simp apply simp done lemma nths_shift_lemma: "map fst (filter (λp. snd p ∈ A) (zip xs [i..<i + length xs])) = map fst (filter (λp. snd p + i ∈ A) (zip xs [0..<length xs]))" by (induct xs rule: rev_induct) (simp_all add: add.commute) lemma nths_append: "nths (l @ l') A = nths l A @ nths l' {j. j + length l ∈ A}" apply (unfold nths_def) apply (induct l' rule: rev_induct, simp) apply (simp add: upt_add_eq_append[of 0] nths_shift_lemma) apply (simp add: add.commute) done lemma nths_Cons: "nths (x # l) A = (if 0 ∈ A then [x] else []) @ nths l {j. Suc j ∈ A}" apply (induct l rule: rev_induct) apply (simp add: nths_def) apply (simp del: append_Cons add: append_Cons[symmetric] nths_append) done lemma nths_map: "nths (map f xs) I = map f (nths xs I)" by(induction xs arbitrary: I) (simp_all add: nths_Cons) lemma set_nths: "set(nths xs I) = {xs!i|i. i<size xs ∧ i ∈ I}" apply(induct xs arbitrary: I) apply(auto simp: nths_Cons nth_Cons split:nat.split dest!: gr0_implies_Suc) done lemma set_nths_subset: "set(nths xs I) ⊆ set xs" by(auto simp add:set_nths) lemma notin_set_nthsI[simp]: "x ∉ set xs ⟹ x ∉ set(nths xs I)" by(auto simp add:set_nths) lemma in_set_nthsD: "x ∈ set(nths xs I) ⟹ x ∈ set xs" by(auto simp add:set_nths) lemma nths_singleton [simp]: "nths [x] A = (if 0 ∈ A then [x] else [])" by (simp add: nths_Cons) lemma distinct_nthsI[simp]: "distinct xs ⟹ distinct (nths xs I)" by (induct xs arbitrary: I) (auto simp: nths_Cons) lemma nths_upt_eq_take [simp]: "nths l {..<n} = take n l" by (induct l rule: rev_induct) (simp_all split: nat_diff_split add: nths_append) lemma filter_eq_nths: "filter P xs = nths xs {i. i<length xs ∧ P(xs!i)}" by(induction xs) (auto simp: nths_Cons) lemma filter_in_nths: "distinct xs ⟹ filter (%x. x ∈ set (nths xs s)) xs = nths xs s" proof (induct xs arbitrary: s) case Nil thus ?case by simp next case (Cons a xs) then have "∀x. x ∈ set xs ⟶ x ≠ a" by auto with Cons show ?case by(simp add: nths_Cons cong:filter_cong) qed subsubsection ‹@{const subseqs} and @{const List.n_lists}› lemma length_subseqs: "length (subseqs xs) = 2 ^ length xs" by (induct xs) (simp_all add: Let_def) lemma subseqs_powset: "set ` set (subseqs xs) = Pow (set xs)" proof - have aux: "⋀x A. set ` Cons x ` A = insert x ` set ` A" by (auto simp add: image_def) have "set (map set (subseqs xs)) = Pow (set xs)" by (induct xs) (simp_all add: aux Let_def Pow_insert Un_commute comp_def del: map_map) then show ?thesis by simp qed lemma distinct_set_subseqs: assumes "distinct xs" shows "distinct (map set (subseqs xs))" proof (rule card_distinct) have "finite (set xs)" .. then have "card (Pow (set xs)) = 2 ^ card (set xs)" by (rule card_Pow) with assms distinct_card [of xs] have "card (Pow (set xs)) = 2 ^ length xs" by simp then show "card (set (map set (subseqs xs))) = length (map set (subseqs xs))" by (simp add: subseqs_powset length_subseqs) qed lemma n_lists_Nil [simp]: "List.n_lists n [] = (if n = 0 then [[]] else [])" by (induct n) simp_all lemma length_n_lists_elem: "ys ∈ set (List.n_lists n xs) ⟹ length ys = n" by (induct n arbitrary: ys) auto lemma set_n_lists: "set (List.n_lists n xs) = {ys. length ys = n ∧ set ys ⊆ set xs}" proof (rule set_eqI) fix ys :: "'a list" show "ys ∈ set (List.n_lists n xs) ⟷ ys ∈ {ys. length ys = n ∧ set ys ⊆ set xs}" proof - have "ys ∈ set (List.n_lists n xs) ⟹ length ys = n" by (induct n arbitrary: ys) auto moreover have "⋀x. ys ∈ set (List.n_lists n xs) ⟹ x ∈ set ys ⟹ x ∈ set xs" by (induct n arbitrary: ys) auto moreover have "set ys ⊆ set xs ⟹ ys ∈ set (List.n_lists (length ys) xs)" by (induct ys) auto ultimately show ?thesis by auto qed qed lemma subseqs_refl: "xs ∈ set (subseqs xs)" by (induct xs) (simp_all add: Let_def) lemma subset_subseqs: "X ⊆ set xs ⟹ X ∈ set ` set (subseqs xs)" unfolding subseqs_powset by simp lemma Cons_in_subseqsD: "y # ys ∈ set (subseqs xs) ⟹ ys ∈ set (subseqs xs)" by (induct xs) (auto simp: Let_def) lemma subseqs_distinctD: "⟦ ys ∈ set (subseqs xs); distinct xs ⟧ ⟹ distinct ys" proof (induct xs arbitrary: ys) case (Cons x xs ys) then show ?case by (auto simp: Let_def) (metis Pow_iff contra_subsetD image_eqI subseqs_powset) qed simp subsubsection ‹@{const splice}› lemma splice_Nil2 [simp, code]: "splice xs [] = xs" by (cases xs) simp_all declare splice.simps(1,3)[code] declare splice.simps(2)[simp del] lemma length_splice[simp]: "length(splice xs ys) = length xs + length ys" by (induct xs ys rule: splice.induct) auto subsubsection ‹@{const shuffle}› lemma Nil_in_shuffle[simp]: "[] ∈ shuffle xs ys ⟷ xs = [] ∧ ys = []" by (induct xs ys rule: shuffle.induct) auto lemma shuffleE: "zs ∈ shuffle xs ys ⟹ (zs = xs ⟹ ys = [] ⟹ P) ⟹ (zs = ys ⟹ xs = [] ⟹ P) ⟹ (⋀x xs' z zs'. xs = x # xs' ⟹ zs = z # zs' ⟹ x = z ⟹ zs' ∈ shuffle xs' ys ⟹ P) ⟹ (⋀y ys' z zs'. ys = y # ys' ⟹ zs = z # zs' ⟹ y = z ⟹ zs' ∈ shuffle xs ys' ⟹ P) ⟹ P" by (induct xs ys rule: shuffle.induct) auto lemma Cons_in_shuffle_iff: "z # zs ∈ shuffle xs ys ⟷ (xs ≠ [] ∧ hd xs = z ∧ zs ∈ shuffle (tl xs) ys ∨ ys ≠ [] ∧ hd ys = z ∧ zs ∈ shuffle xs (tl ys))" by (induct xs ys rule: shuffle.induct) auto lemma splice_in_shuffle [simp, intro]: "splice xs ys ∈ shuffle xs ys" by (induction xs ys rule: splice.induct) (simp_all add: Cons_in_shuffle_iff) lemma Nil_in_shuffleI: "xs = [] ⟹ ys = [] ⟹ [] ∈ shuffle xs ys" by simp lemma Cons_in_shuffle_leftI: "zs ∈ shuffle xs ys ⟹ z # zs ∈ shuffle (z # xs) ys" by (cases ys) auto lemma Cons_in_shuffle_rightI: "zs ∈ shuffle xs ys ⟹ z # zs ∈ shuffle xs (z # ys)" by (cases xs) auto lemma finite_shuffle [simp, intro]: "finite (shuffle xs ys)" by (induction xs ys rule: shuffle.induct) simp_all lemma length_shuffle: "zs ∈ shuffle xs ys ⟹ length zs = length xs + length ys" by (induction xs ys arbitrary: zs rule: shuffle.induct) auto lemma set_shuffle: "zs ∈ shuffle xs ys ⟹ set zs = set xs ∪ set ys" by (induction xs ys arbitrary: zs rule: shuffle.induct) auto lemma distinct_disjoint_shuffle: assumes "distinct xs" "distinct ys" "set xs ∩ set ys = {}" "zs ∈ shuffle xs ys" shows "distinct zs" using assms proof (induction xs ys arbitrary: zs rule: shuffle.induct) case (3 x xs y ys) show ?case proof (cases zs) case (Cons z zs') with "3.prems" and "3.IH"[of zs'] show ?thesis by (force dest: set_shuffle) qed simp_all qed simp_all lemma shuffle_commutes: "shuffle xs ys = shuffle ys xs" by (induction xs ys rule: shuffle.induct) (simp_all add: Un_commute) lemma Cons_shuffle_subset1: "(#) x ` shuffle xs ys ⊆ shuffle (x # xs) ys" by (cases ys) auto lemma Cons_shuffle_subset2: "(#) y ` shuffle xs ys ⊆ shuffle xs (y # ys)" by (cases xs) auto lemma filter_shuffle: "filter P ` shuffle xs ys = shuffle (filter P xs) (filter P ys)" proof - have *: "filter P ` (#) x ` A = (if P x then (#) x ` filter P ` A else filter P ` A)" for x A by (auto simp: image_image) show ?thesis by (induction xs ys rule: shuffle.induct) (simp_all split: if_splits add: image_Un * Un_absorb1 Un_absorb2 Cons_shuffle_subset1 Cons_shuffle_subset2) qed lemma filter_shuffle_disjoint1: assumes "set xs ∩ set ys = {}" "zs ∈ shuffle xs ys" shows "filter (λx. x ∈ set xs) zs = xs" (is "filter ?P _ = _") and "filter (λx. x ∉ set xs) zs = ys" (is "filter ?Q _ = _") using assms proof - from assms have "filter ?P zs ∈ filter ?P ` shuffle xs ys" by blast also have "filter ?P ` shuffle xs ys = shuffle (filter ?P xs) (filter ?P ys)" by (rule filter_shuffle) also have "filter ?P xs = xs" by (rule filter_True) simp_all also have "filter ?P ys = []" by (rule filter_False) (insert assms(1), auto) also have "shuffle xs [] = {xs}" by simp finally show "filter ?P zs = xs" by simp next from assms have "filter ?Q zs ∈ filter ?Q ` shuffle xs ys" by blast also have "filter ?Q ` shuffle xs ys = shuffle (filter ?Q xs) (filter ?Q ys)" by (rule filter_shuffle) also have "filter ?Q ys = ys" by (rule filter_True) (insert assms(1), auto) also have "filter ?Q xs = []" by (rule filter_False) (insert assms(1), auto) also have "shuffle [] ys = {ys}" by simp finally show "filter ?Q zs = ys" by simp qed lemma filter_shuffle_disjoint2: assumes "set xs ∩ set ys = {}" "zs ∈ shuffle xs ys" shows "filter (λx. x ∈ set ys) zs = ys" "filter (λx. x ∉ set ys) zs = xs" using filter_shuffle_disjoint1[of ys xs zs] assms by (simp_all add: shuffle_commutes Int_commute) lemma partition_in_shuffle: "xs ∈ shuffle (filter P xs) (filter (λx. ¬P x) xs)" proof (induction xs) case (Cons x xs) show ?case proof (cases "P x") case True hence "x # xs ∈ (#) x ` shuffle (filter P xs) (filter (λx. ¬P x) xs)" by (intro imageI Cons.IH) also have "… ⊆ shuffle (filter P (x # xs)) (filter (λx. ¬P x) (x # xs))" by (simp add: True Cons_shuffle_subset1) finally show ?thesis . next case False hence "x # xs ∈ (#) x ` shuffle (filter P xs) (filter (λx. ¬P x) xs)" by (intro imageI Cons.IH) also have "… ⊆ shuffle (filter P (x # xs)) (filter (λx. ¬P x) (x # xs))" by (simp add: False Cons_shuffle_subset2) finally show ?thesis . qed qed auto lemma inv_image_partition: assumes "⋀x. x ∈ set xs ⟹ P x" "⋀y. y ∈ set ys ⟹ ¬P y" shows "partition P -` {(xs, ys)} = shuffle xs ys" proof (intro equalityI subsetI) fix zs assume zs: "zs ∈ shuffle xs ys" hence [simp]: "set zs = set xs ∪ set ys" by (rule set_shuffle) from assms have "filter P zs = filter (λx. x ∈ set xs) zs" "filter (λx. ¬P x) zs = filter (λx. x ∈ set ys) zs" by (intro filter_cong refl; force)+ moreover from assms have "set xs ∩ set ys = {}" by auto ultimately show "zs ∈ partition P -` {(xs, ys)}" using zs by (simp add: o_def filter_shuffle_disjoint1 filter_shuffle_disjoint2) next fix zs assume "zs ∈ partition P -` {(xs, ys)}" thus "zs ∈ shuffle xs ys" using partition_in_shuffle[of zs] by (auto simp: o_def) qed subsubsection ‹Transpose› function transpose where "transpose [] = []" | "transpose ([] # xss) = transpose xss" | "transpose ((x#xs) # xss) = (x # [h. (h#t) ← xss]) # transpose (xs # [t. (h#t) ← xss])" by pat_completeness auto lemma transpose_aux_filter_head: "concat (map (case_list [] (λh t. [h])) xss) = map (λxs. hd xs) (filter (λys. ys ≠ []) xss)" by (induct xss) (auto split: list.split) lemma transpose_aux_filter_tail: "concat (map (case_list [] (λh t. [t])) xss) = map (λxs. tl xs) (filter (λys. ys ≠ []) xss)" by (induct xss) (auto split: list.split) lemma transpose_aux_max: "max (Suc (length xs)) (foldr (λxs. max (length xs)) xss 0) = Suc (max (length xs) (foldr (λx. max (length x - Suc 0)) (filter (λys. ys ≠ []) xss) 0))" (is "max _ ?foldB = Suc (max _ ?foldA)") proof (cases "(filter (λys. ys ≠ []) xss) = []") case True hence "foldr (λxs. max (length xs)) xss 0 = 0" proof (induct xss) case (Cons x xs) then have "x = []" by (cases x) auto with Cons show ?case by auto qed simp thus ?thesis using True by simp next case False have foldA: "?foldA = foldr (λx. max (length x)) (filter (λys. ys ≠ []) xss) 0 - 1" by (induct xss) auto have foldB: "?foldB = foldr (λx. max (length x)) (filter (λys. ys ≠ []) xss) 0" by (induct xss) auto have "0 < ?foldB" proof - from False obtain z zs where zs: "(filter (λys. ys ≠ []) xss) = z#zs" by (auto simp: neq_Nil_conv) hence "z ∈ set (filter (λys. ys ≠ []) xss)" by auto hence "z ≠ []" by auto thus ?thesis unfolding foldB zs by (auto simp: max_def intro: less_le_trans) qed thus ?thesis unfolding foldA foldB max_Suc_Suc[symmetric] by simp qed termination transpose by (relation "measure (λxs. foldr (λxs. max (length xs)) xs 0 + length xs)") (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max less_Suc_eq_le) lemma transpose_empty: "(transpose xs = []) ⟷ (∀x ∈ set xs. x = [])" by (induct rule: transpose.induct) simp_all lemma length_transpose: fixes xs :: "'a list list" shows "length (transpose xs) = foldr (λxs. max (length xs)) xs 0" by (induct rule: transpose.induct) (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max max_Suc_Suc[symmetric] simp del: max_Suc_Suc) lemma nth_transpose: fixes xs :: "'a list list" assumes "i < length (transpose xs)" shows "transpose xs ! i = map (λxs. xs ! i) (filter (λys. i < length ys) xs)" using assms proof (induct arbitrary: i rule: transpose.induct) case (3 x xs xss) define XS where "XS = (x # xs) # xss" hence [simp]: "XS ≠ []" by auto thus ?case proof (cases i) case 0 thus ?thesis by (simp add: transpose_aux_filter_head hd_conv_nth) next case (Suc j) have *: "⋀xss. xs # map tl xss = map tl ((x#xs)#xss)" by simp have **: "⋀xss. (x#xs) # filter (λys. ys ≠ []) xss = filter (λys. ys ≠ []) ((x#xs)#xss)" by simp { fix x have "Suc j < length x ⟷ x ≠ [] ∧ j < length x - Suc 0" by (cases x) simp_all } note *** = this have j_less: "j < length (transpose (xs # concat (map (case_list [] (λh t. [t])) xss)))" using "3.prems" by (simp add: transpose_aux_filter_tail length_transpose Suc) show ?thesis unfolding transpose.simps ‹i = Suc j› nth_Cons_Suc "3.hyps"[OF j_less] apply (auto simp: transpose_aux_filter_tail filter_map comp_def length_transpose * ** *** XS_def[symmetric]) apply (rule list.exhaust) by auto qed qed simp_all lemma transpose_map_map: "transpose (map (map f) xs) = map (map f) (transpose xs)" proof (rule nth_equalityI, safe) have [simp]: "length (transpose (map (map f) xs)) = length (transpose xs)" by (simp add: length_transpose foldr_map comp_def) show "length (transpose (map (map f) xs)) = length (map (map f) (transpose xs))" by simp fix i assume "i < length (transpose (map (map f) xs))" thus "transpose (map (map f) xs) ! i = map (map f) (transpose xs) ! i" by (simp add: nth_transpose filter_map comp_def) qed subsubsection ‹@{const min} and @{const arg_min}› lemma min_list_Min: "xs ≠ [] ⟹ min_list xs = Min (set xs)" by (induction xs rule: induct_list012)(auto) lemma f_arg_min_list_f: "xs ≠ [] ⟹ f (arg_min_list f xs) = Min (f ` (set xs))" by(induction f xs rule: arg_min_list.induct) (auto simp: min_def intro!: antisym) lemma arg_min_list_in: "xs ≠ [] ⟹ arg_min_list f xs ∈ set xs" by(induction xs rule: induct_list012) (auto simp: Let_def) subsubsection ‹(In)finiteness› lemma finite_maxlen: "finite (M::'a list set) ⟹ ∃n. ∀s∈M. size s < n" proof (induct rule: finite.induct) case emptyI show ?case by simp next case (insertI M xs) then obtain n where "∀s∈M. length s < n" by blast hence "∀s∈insert xs M. size s < max n (size xs) + 1" by auto thus ?case .. qed lemma lists_length_Suc_eq: "{xs. set xs ⊆ A ∧ length xs = Suc n} = (λ(xs, n). n#xs) ` ({xs. set xs ⊆ A ∧ length xs = n} × A)" by (auto simp: length_Suc_conv) lemma assumes "finite A" shows finite_lists_length_eq: "finite {xs. set xs ⊆ A ∧ length xs = n}" and card_lists_length_eq: "card {xs. set xs ⊆ A ∧ length xs = n} = (card A)^n" using ‹finite A› by (induct n) (auto simp: card_image inj_split_Cons lists_length_Suc_eq cong: conj_cong) lemma finite_lists_length_le: assumes "finite A" shows "finite {xs. set xs ⊆ A ∧ length xs ≤ n}" (is "finite ?S") proof- have "?S = (⋃n∈{0..n}. {xs. set xs ⊆ A ∧ length xs = n})" by auto thus ?thesis by (auto intro!: finite_lists_length_eq[OF ‹finite A›] simp only:) qed lemma card_lists_length_le: assumes "finite A" shows "card {xs. set xs ⊆ A ∧ length xs ≤ n} = (∑i≤n. card A^i)" proof - have "(∑i≤n. card A^i) = card (⋃i≤n. {xs. set xs ⊆ A ∧ length xs = i})" using ‹finite A› by (subst card_UN_disjoint) (auto simp add: card_lists_length_eq finite_lists_length_eq) also have "(⋃i≤n. {xs. set xs ⊆ A ∧ length xs = i}) = {xs. set xs ⊆ A ∧ length xs ≤ n}" by auto finally show ?thesis by simp qed lemma finite_lists_distinct_length_eq [intro]: assumes "finite A" shows "finite {xs. length xs = n ∧ distinct xs ∧ set xs ⊆ A}" (is "finite ?S") proof - have "finite {xs. set xs ⊆ A ∧ length xs = n}" using ‹finite A› by (rule finite_lists_length_eq) moreover have "?S ⊆ {xs. set xs ⊆ A ∧ length xs = n}" by auto ultimately show ?thesis using finite_subset by auto qed lemma card_lists_distinct_length_eq: assumes "finite A" "k ≤ card A" shows "card {xs. length xs = k ∧ distinct xs ∧ set xs ⊆ A} = ∏{card A - k + 1 .. card A}" using assms proof (induct k) case 0 then have "{xs. length xs = 0 ∧ distinct xs ∧ set xs ⊆ A} = {[]}" by auto then show ?case by simp next case (Suc k) let "?k_list" = "λk xs. length xs = k ∧ distinct xs ∧ set xs ⊆ A" have inj_Cons: "⋀A. inj_on (λ(xs, n). n # xs) A" by (rule inj_onI) auto from Suc have "k ≤ card A" by simp moreover note ‹finite A› moreover have "finite {xs. ?k_list k xs}" by (rule finite_subset) (use finite_lists_length_eq[OF ‹finite A›, of k] in auto) moreover have "⋀i j. i ≠ j ⟶ {i} × (A - set i) ∩ {j} × (A - set j) = {}" by auto moreover have "⋀i. i ∈ {xs. ?k_list k xs} ⟹ card (A - set i) = card A - k" by (simp add: card_Diff_subset distinct_card) moreover have "{xs. ?k_list (Suc k) xs} = (λ(xs, n). n#xs) ` ⋃((λxs. {xs} × (A - set xs)) ` {xs. ?k_list k xs})" by (auto simp: length_Suc_conv) moreover have "Suc (card A - Suc k) = card A - k" using Suc.prems by simp then have "(card A - k) * ∏{Suc (card A - k)..card A} = ∏{Suc (card A - Suc k)..card A}" by (subst prod.insert[symmetric]) (simp add: atLeastAtMost_insertL)+ ultimately show ?case by (simp add: card_image inj_Cons card_UN_disjoint Suc.hyps algebra_simps) qed lemma card_lists_distinct_length_eq': assumes "k < card A" shows "card {xs. length xs = k ∧ distinct xs ∧ set xs ⊆ A} = ∏{card A - k + 1 .. card A}" proof - from ‹k < card A› have "finite A" and "k ≤ card A" using card_infinite by force+ from this show ?thesis by (rule card_lists_distinct_length_eq) qed lemma infinite_UNIV_listI: "¬ finite(UNIV::'a list set)" apply (rule notI) apply (drule finite_maxlen) apply clarsimp apply (erule_tac x = "replicate n undefined" in allE) by simp subsection ‹Sorting› subsubsection ‹@{const sorted_wrt}› text ‹Sometimes the second equation in the definition of @{const sorted_wrt} is too aggressive because it relates each list element to \emph{all} its successors. Then this equation should be removed and ‹sorted_wrt2_simps› should be added instead.› lemma sorted_wrt1: "sorted_wrt P [x] = True" by(simp) lemma sorted_wrt2: "transp P ⟹ sorted_wrt P (x # y # zs) = (P x y ∧ sorted_wrt P (y # zs))" apply(induction zs arbitrary: x y) apply(auto dest: transpD) apply (meson transpD) done lemmas sorted_wrt2_simps = sorted_wrt1 sorted_wrt2 lemma sorted_wrt_true [simp]: "sorted_wrt (λ_ _. True) xs" by (induction xs) simp_all lemma sorted_wrt_append: "sorted_wrt P (xs @ ys) ⟷ sorted_wrt P xs ∧ sorted_wrt P ys ∧ (∀x∈set xs. ∀y∈set ys. P x y)" by (induction xs) auto lemma sorted_wrt_map: "sorted_wrt R (map f xs) = sorted_wrt (λx y. R (f x) (f y)) xs" by (induction xs) simp_all lemma sorted_wrt_rev: "sorted_wrt P (rev xs) = sorted_wrt (λx y. P y x) xs" by (induction xs) (auto simp add: sorted_wrt_append) lemma sorted_wrt_mono_rel: "(⋀x y. ⟦ x ∈ set xs; y ∈ set xs; P x y ⟧ ⟹ Q x y) ⟹ sorted_wrt P xs ⟹ sorted_wrt Q xs" by(induction xs)(auto) lemma sorted_wrt01: "length xs ≤ 1 ⟹ sorted_wrt P xs" by(auto simp: le_Suc_eq length_Suc_conv) lemma sorted_wrt_iff_nth_less: "sorted_wrt P xs = (∀i j. i < j ⟶ j < length xs ⟶ P (xs ! i) (xs ! j))" apply(induction xs) apply(auto simp add: in_set_conv_nth Ball_def nth_Cons split: nat.split) done lemma sorted_wrt_nth_less: "⟦ sorted_wrt P xs; i < j; j < length xs ⟧ ⟹ P (xs ! i) (xs ! j)" by(auto simp: sorted_wrt_iff_nth_less) lemma sorted_wrt_upt[simp]: "sorted_wrt (<) [m..<n]" by(induction n) (auto simp: sorted_wrt_append) lemma sorted_wrt_upto[simp]: "sorted_wrt (<) [i..j]" apply(induction i j rule: upto.induct) apply(subst upto.simps) apply(simp) done text ‹Each element is greater or equal to its index:› lemma sorted_wrt_less_idx: "sorted_wrt (<) ns ⟹ i < length ns ⟹ i ≤ ns!i" proof (induction ns arbitrary: i rule: rev_induct) case Nil thus ?case by simp next case snoc thus ?case by (auto simp: nth_append sorted_wrt_append) (metis less_antisym not_less nth_mem) qed subsubsection ‹@{const sorted}› context linorder begin text ‹Sometimes the second equation in the definition of @{const sorted} is too aggressive because it relates each list element to \emph{all} its successors. Then this equation should be removed and ‹sorted2_simps› should be added instead. Executable code is one such use case.› lemma sorted1: "sorted [x] = True" by simp lemma sorted2: "sorted (x # y # zs) = (x ≤ y ∧ sorted (y # zs))" by(induction zs) auto lemmas sorted2_simps = sorted1 sorted2 lemmas [code] = sorted.simps(1) sorted2_simps lemma sorted_append: "sorted (xs@ys) = (sorted xs ∧ sorted ys ∧ (∀x ∈ set xs. ∀y ∈ set ys. x≤y))" by (simp add: sorted_sorted_wrt sorted_wrt_append) lemma sorted_map: "sorted (map f xs) = sorted_wrt (λx y. f x ≤ f y) xs" by (simp add: sorted_sorted_wrt sorted_wrt_map) lemma sorted01: "length xs ≤ 1 ⟹ sorted xs" by (simp add: sorted_sorted_wrt sorted_wrt01) lemma sorted_tl: "sorted xs ⟹ sorted (tl xs)" by (cases xs) (simp_all) lemma sorted_iff_nth_mono_less: "sorted xs = (∀i j. i < j ⟶ j < length xs ⟶ xs ! i ≤ xs ! j)" by (simp add: sorted_sorted_wrt sorted_wrt_iff_nth_less) lemma sorted_iff_nth_mono: "sorted xs = (∀i j. i ≤ j ⟶ j < length xs ⟶ xs ! i ≤ xs ! j)" by (auto simp: sorted_iff_nth_mono_less nat_less_le) lemma sorted_nth_mono: "sorted xs ⟹ i ≤ j ⟹ j < length xs ⟹ xs!i ≤ xs!j" by (auto simp: sorted_iff_nth_mono) lemma sorted_rev_nth_mono: "sorted (rev xs) ⟹ i ≤ j ⟹ j < length xs ⟹ xs!j ≤ xs!i" using sorted_nth_mono[ of "rev xs" "length xs - j - 1" "length xs - i - 1"] rev_nth[of "length xs - i - 1" "xs"] rev_nth[of "length xs - j - 1" "xs"] by auto lemma sorted_map_remove1: "sorted (map f xs) ⟹ sorted (map f (remove1 x xs))" by (induct xs) (auto) lemma sorted_remove1: "sorted xs ⟹ sorted (remove1 a xs)" using sorted_map_remove1 [of "λx. x"] by simp lemma sorted_butlast: assumes "xs ≠ []" and "sorted xs" shows "sorted (butlast xs)" proof - from ‹xs ≠ []› obtain ys y where "xs = ys @ [y]" by (cases xs rule: rev_cases) auto with ‹sorted xs› show ?thesis by (simp add: sorted_append) qed lemma sorted_replicate [simp]: "sorted(replicate n x)" by(induction n) (auto) lemma sorted_remdups[simp]: "sorted xs ⟹ sorted (remdups xs)" by (induct xs) (auto) lemma sorted_remdups_adj[simp]: "sorted xs ⟹ sorted (remdups_adj xs)" by (induct xs rule: remdups_adj.induct, simp_all split: if_split_asm) lemma sorted_nths: "sorted xs ⟹ sorted (nths xs I)" by(induction xs arbitrary: I)(auto simp: nths_Cons) lemma sorted_distinct_set_unique: assumes "sorted xs" "distinct xs" "sorted ys" "distinct ys" "set xs = set ys" shows "xs = ys" proof - from assms have 1: "length xs = length ys" by (auto dest!: distinct_card) from assms show ?thesis proof(induct rule:list_induct2[OF 1]) case 1 show ?case by simp next case 2 thus ?case by simp (metis Diff_insert_absorb antisym insertE insert_iff) qed qed lemma map_sorted_distinct_set_unique: assumes "inj_on f (set xs ∪ set ys)" assumes "sorted (map f xs)" "distinct (map f xs)" "sorted (map f ys)" "distinct (map f ys)" assumes "set xs = set ys" shows "xs = ys" proof - from assms have "map f xs = map f ys" by (simp add: sorted_distinct_set_unique) with ‹inj_on f (set xs ∪ set ys)› show "xs = ys" by (blast intro: map_inj_on) qed lemma assumes "sorted xs" shows sorted_take: "sorted (take n xs)" and sorted_drop: "sorted (drop n xs)" proof - from assms have "sorted (take n xs @ drop n xs)" by simp then show "sorted (take n xs)" and "sorted (drop n xs)" unfolding sorted_append by simp_all qed lemma sorted_dropWhile: "sorted xs ⟹ sorted (dropWhile P xs)" by (auto dest: sorted_drop simp add: dropWhile_eq_drop) lemma sorted_takeWhile: "sorted xs ⟹ sorted (takeWhile P xs)" by (subst takeWhile_eq_take) (auto dest: sorted_take) lemma sorted_filter: "sorted (map f xs) ⟹ sorted (map f (filter P xs))" by (induct xs) simp_all lemma foldr_max_sorted: assumes "sorted (rev xs)" shows "foldr max xs y = (if xs = [] then y else max (xs ! 0) y)" using assms proof (induct xs) case (Cons x xs) then have "sorted (rev xs)" using sorted_append by auto with Cons show ?case by (cases xs) (auto simp add: sorted_append max_def) qed simp lemma filter_equals_takeWhile_sorted_rev: assumes sorted: "sorted (rev (map f xs))" shows "filter (λx. t < f x) xs = takeWhile (λ x. t < f x) xs" (is "filter ?P xs = ?tW") proof (rule takeWhile_eq_filter[symmetric]) let "?dW" = "dropWhile ?P xs" fix x assume "x ∈ set ?dW" then obtain i where i: "i < length ?dW" and nth_i: "x = ?dW ! i" unfolding in_set_conv_nth by auto hence "length ?tW + i < length (?tW @ ?dW)" unfolding length_append by simp hence i': "length (map f ?tW) + i < length (map f xs)" by simp have "(map f ?tW @ map f ?dW) ! (length (map f ?tW) + i) ≤ (map f ?tW @ map f ?dW) ! (length (map f ?tW) + 0)" using sorted_rev_nth_mono[OF sorted _ i', of "length ?tW"] unfolding map_append[symmetric] by simp hence "f x ≤ f (?dW ! 0)" unfolding nth_append_length_plus nth_i using i preorder_class.le_less_trans[OF le0 i] by simp also have "... ≤ t" using hd_dropWhile[of "?P" xs] le0[THEN preorder_class.le_less_trans, OF i] using hd_conv_nth[of "?dW"] by simp finally show "¬ t < f x" by simp qed lemma sorted_map_same: "sorted (map f (filter (λx. f x = g xs) xs))" proof (induct xs arbitrary: g) case Nil then show ?case by simp next case (Cons x xs) then have "sorted (map f (filter (λy. f y = (λxs. f x) xs) xs))" . moreover from Cons have "sorted (map f (filter (λy. f y = (g ∘ Cons x) xs) xs))" . ultimately show ?case by simp_all qed lemma sorted_same: "sorted (filter (λx. x = g xs) xs)" using sorted_map_same [of "λx. x"] by simp end lemma sorted_upt[simp]: "sorted [m..<n]" by(simp add: sorted_sorted_wrt sorted_wrt_mono_rel[OF _ sorted_wrt_upt]) lemma sorted_upto[simp]: "sorted [m..n]" by(simp add: sorted_sorted_wrt sorted_wrt_mono_rel[OF</