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src/HOL/List.ML

author | paulson |

Thu, 01 Jun 2000 13:28:00 +0200 | |

changeset 9014 | 4382883421ec |

parent 9003 | 3747ec2aeb86 |

child 9108 | 9fff97d29837 |

permissions | -rw-r--r-- |

simplified the proof of nth_upt

(* Title: HOL/List ID: $Id$ Author: Tobias Nipkow Copyright 1994 TU Muenchen List lemmas *) Goal "!x. xs ~= x#xs"; by (induct_tac "xs" 1); by Auto_tac; qed_spec_mp "not_Cons_self"; bind_thm("not_Cons_self2",not_Cons_self RS not_sym); Addsimps [not_Cons_self,not_Cons_self2]; Goal "(xs ~= []) = (? y ys. xs = y#ys)"; by (induct_tac "xs" 1); by Auto_tac; qed "neq_Nil_conv"; (* Induction over the length of a list: *) val [prem] = Goal "(!!xs. (!ys. length ys < length xs --> P ys) ==> P xs) ==> P(xs)"; by (rtac measure_induct 1 THEN etac prem 1); qed "length_induct"; (** "lists": the list-forming operator over sets **) Goalw lists.defs "A<=B ==> lists A <= lists B"; by (rtac lfp_mono 1); by (REPEAT (ares_tac basic_monos 1)); qed "lists_mono"; val listsE = lists.mk_cases "x#l : lists A"; AddSEs [listsE]; AddSIs lists.intrs; Goal "l: lists A ==> l: lists B --> l: lists (A Int B)"; by (etac lists.induct 1); by (ALLGOALS Blast_tac); qed_spec_mp "lists_IntI"; Goal "lists (A Int B) = lists A Int lists B"; by (rtac (mono_Int RS equalityI) 1); by (simp_tac (simpset() addsimps [mono_def, lists_mono]) 1); by (blast_tac (claset() addSIs [lists_IntI]) 1); qed "lists_Int_eq"; Addsimps [lists_Int_eq]; (** Case analysis **) section "Case analysis"; val prems = Goal "[| P([]); !!x xs. P(x#xs) |] ==> P(xs)"; by (induct_tac "xs" 1); by (REPEAT(resolve_tac prems 1)); qed "list_cases"; Goal "(xs=[] --> P([])) & (!y ys. xs=y#ys --> P(y#ys)) --> P(xs)"; by (induct_tac "xs" 1); by (Blast_tac 1); by (Blast_tac 1); bind_thm("list_eq_cases", impI RSN (2,allI RSN (2,allI RSN (2,impI RS (conjI RS (result() RS mp)))))); (** length **) (* needs to come before "@" because of thm append_eq_append_conv *) section "length"; Goal "length(xs@ys) = length(xs)+length(ys)"; by (induct_tac "xs" 1); by Auto_tac; qed"length_append"; Addsimps [length_append]; Goal "length (map f xs) = length xs"; by (induct_tac "xs" 1); by Auto_tac; qed "length_map"; Addsimps [length_map]; Goal "length(rev xs) = length(xs)"; by (induct_tac "xs" 1); by Auto_tac; qed "length_rev"; Addsimps [length_rev]; Goal "length(tl xs) = (length xs) - 1"; by (case_tac "xs" 1); by Auto_tac; qed "length_tl"; Addsimps [length_tl]; Goal "(length xs = 0) = (xs = [])"; by (induct_tac "xs" 1); by Auto_tac; qed "length_0_conv"; AddIffs [length_0_conv]; Goal "(0 = length xs) = (xs = [])"; by (induct_tac "xs" 1); by Auto_tac; qed "zero_length_conv"; AddIffs [zero_length_conv]; Goal "(0 < length xs) = (xs ~= [])"; by (induct_tac "xs" 1); by Auto_tac; qed "length_greater_0_conv"; AddIffs [length_greater_0_conv]; Goal "(length xs = Suc n) = (? y ys. xs = y#ys & length ys = n)"; by (induct_tac "xs" 1); by Auto_tac; qed "length_Suc_conv"; (** @ - append **) section "@ - append"; Goal "(xs@ys)@zs = xs@(ys@zs)"; by (induct_tac "xs" 1); by Auto_tac; qed "append_assoc"; Addsimps [append_assoc]; Goal "xs @ [] = xs"; by (induct_tac "xs" 1); by Auto_tac; qed "append_Nil2"; Addsimps [append_Nil2]; Goal "(xs@ys = []) = (xs=[] & ys=[])"; by (induct_tac "xs" 1); by Auto_tac; qed "append_is_Nil_conv"; AddIffs [append_is_Nil_conv]; Goal "([] = xs@ys) = (xs=[] & ys=[])"; by (induct_tac "xs" 1); by Auto_tac; qed "Nil_is_append_conv"; AddIffs [Nil_is_append_conv]; Goal "(xs @ ys = xs) = (ys=[])"; by (induct_tac "xs" 1); by Auto_tac; qed "append_self_conv"; Goal "(xs = xs @ ys) = (ys=[])"; by (induct_tac "xs" 1); by Auto_tac; qed "self_append_conv"; AddIffs [append_self_conv,self_append_conv]; Goal "!ys. length xs = length ys | length us = length vs \ \ --> (xs@us = ys@vs) = (xs=ys & us=vs)"; by (induct_tac "xs" 1); by (rtac allI 1); by (case_tac "ys" 1); by (Asm_simp_tac 1); by (Force_tac 1); by (rtac allI 1); by (case_tac "ys" 1); by (Force_tac 1); by (Asm_simp_tac 1); qed_spec_mp "append_eq_append_conv"; Addsimps [append_eq_append_conv]; Goal "(xs @ ys = xs @ zs) = (ys=zs)"; by (Simp_tac 1); qed "same_append_eq"; Goal "(xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; by (Simp_tac 1); qed "append1_eq_conv"; Goal "(ys @ xs = zs @ xs) = (ys=zs)"; by (Simp_tac 1); qed "append_same_eq"; AddIffs [same_append_eq, append1_eq_conv, append_same_eq]; Goal "(xs @ ys = ys) = (xs=[])"; by (cut_inst_tac [("zs","[]")] append_same_eq 1); by Auto_tac; qed "append_self_conv2"; Goal "(ys = xs @ ys) = (xs=[])"; by (simp_tac (simpset() addsimps [simplify (simpset()) (read_instantiate[("ys","[]")]append_same_eq)]) 1); by (Blast_tac 1); qed "self_append_conv2"; AddIffs [append_self_conv2,self_append_conv2]; Goal "xs ~= [] --> hd xs # tl xs = xs"; by (induct_tac "xs" 1); by Auto_tac; qed_spec_mp "hd_Cons_tl"; Addsimps [hd_Cons_tl]; Goal "hd(xs@ys) = (if xs=[] then hd ys else hd xs)"; by (induct_tac "xs" 1); by Auto_tac; qed "hd_append"; Goal "xs ~= [] ==> hd(xs @ ys) = hd xs"; by (asm_simp_tac (simpset() addsimps [hd_append] addsplits [list.split]) 1); qed "hd_append2"; Addsimps [hd_append2]; Goal "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)"; by (simp_tac (simpset() addsplits [list.split]) 1); qed "tl_append"; Goal "xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys"; by (asm_simp_tac (simpset() addsimps [tl_append] addsplits [list.split]) 1); qed "tl_append2"; Addsimps [tl_append2]; (* trivial rules for solving @-equations automatically *) Goal "xs = ys ==> xs = [] @ ys"; by (Asm_simp_tac 1); qed "eq_Nil_appendI"; Goal "[| x#xs1 = ys; xs = xs1 @ zs |] ==> x#xs = ys@zs"; by (dtac sym 1); by (Asm_simp_tac 1); qed "Cons_eq_appendI"; Goal "[| xs@xs1 = zs; ys = xs1 @ us |] ==> xs@ys = zs@us"; by (dtac sym 1); by (Asm_simp_tac 1); qed "append_eq_appendI"; (*** Simplification procedure for all list equalities. Currently only tries to rearranges @ to see if - both lists end in a singleton list, - or both lists end in the same list. ***) local val list_eq_pattern = Thm.read_cterm (Theory.sign_of List.thy) ("(xs::'a list) = ys",HOLogic.boolT); fun last (cons as Const("List.list.Cons",_) $ _ $ xs) = (case xs of Const("List.list.Nil",_) => cons | _ => last xs) | last (Const("List.op @",_) $ _ $ ys) = last ys | last t = t; fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true | list1 _ = false; fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) = (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs) | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys | butlast xs = Const("List.list.Nil",fastype_of xs); val rearr_tac = simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons]); fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) = let val lastl = last lhs and lastr = last rhs fun rearr conv = let val lhs1 = butlast lhs and rhs1 = butlast rhs val Type(_,listT::_) = eqT val appT = [listT,listT] ---> listT val app = Const("List.op @",appT) val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr) val ct = cterm_of sg (HOLogic.mk_Trueprop(HOLogic.mk_eq(F,F2))) val thm = prove_goalw_cterm [] ct (K [rearr_tac 1]) handle ERROR => error("The error(s) above occurred while trying to prove " ^ string_of_cterm ct) in Some((conv RS (thm RS trans)) RS eq_reflection) end in if list1 lastl andalso list1 lastr then rearr append1_eq_conv else if lastl aconv lastr then rearr append_same_eq else None end; in val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq; end; Addsimprocs [list_eq_simproc]; (** map **) section "map"; Goal "(!x. x : set xs --> f x = g x) --> map f xs = map g xs"; by (induct_tac "xs" 1); by Auto_tac; bind_thm("map_ext", impI RS (allI RS (result() RS mp))); Goal "map (%x. x) = (%xs. xs)"; by (rtac ext 1); by (induct_tac "xs" 1); by Auto_tac; qed "map_ident"; Addsimps[map_ident]; Goal "map f (xs@ys) = map f xs @ map f ys"; by (induct_tac "xs" 1); by Auto_tac; qed "map_append"; Addsimps[map_append]; Goalw [o_def] "map (f o g) xs = map f (map g xs)"; by (induct_tac "xs" 1); by Auto_tac; qed "map_compose"; Addsimps[map_compose]; Goal "rev(map f xs) = map f (rev xs)"; by (induct_tac "xs" 1); by Auto_tac; qed "rev_map"; (* a congruence rule for map: *) Goal "xs=ys ==> (!x. x : set ys --> f x = g x) --> map f xs = map g ys"; by (hyp_subst_tac 1); by (induct_tac "ys" 1); by Auto_tac; bind_thm("map_cong", impI RSN (2,allI RSN (2, result() RS mp))); Goal "(map f xs = []) = (xs = [])"; by (case_tac "xs" 1); by Auto_tac; qed "map_is_Nil_conv"; AddIffs [map_is_Nil_conv]; Goal "([] = map f xs) = (xs = [])"; by (case_tac "xs" 1); by Auto_tac; qed "Nil_is_map_conv"; AddIffs [Nil_is_map_conv]; Goal "(map f xs = y#ys) = (? x xs'. xs = x#xs' & f x = y & map f xs' = ys)"; by (case_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "map_eq_Cons"; Goal "!xs. map f xs = map f ys --> (!x y. f x = f y --> x=y) --> xs=ys"; by (induct_tac "ys" 1); by (Asm_simp_tac 1); by (fast_tac (claset() addss (simpset() addsimps [map_eq_Cons])) 1); qed_spec_mp "map_injective"; Goal "inj f ==> inj (map f)"; by (blast_tac (claset() addDs [map_injective,injD] addIs [injI]) 1); qed "inj_mapI"; Goalw [inj_on_def] "inj (map f) ==> inj f"; by (Clarify_tac 1); by (eres_inst_tac [("x","[x]")] ballE 1); by (eres_inst_tac [("x","[y]")] ballE 1); by (Asm_full_simp_tac 1); by (Blast_tac 1); by (Blast_tac 1); qed "inj_mapD"; Goal "inj (map f) = inj f"; by (blast_tac (claset() addDs [inj_mapD] addIs [inj_mapI]) 1); qed "inj_map"; (** rev **) section "rev"; Goal "rev(xs@ys) = rev(ys) @ rev(xs)"; by (induct_tac "xs" 1); by Auto_tac; qed "rev_append"; Addsimps[rev_append]; Goal "rev(rev l) = l"; by (induct_tac "l" 1); by Auto_tac; qed "rev_rev_ident"; Addsimps[rev_rev_ident]; Goal "(rev xs = []) = (xs = [])"; by (induct_tac "xs" 1); by Auto_tac; qed "rev_is_Nil_conv"; AddIffs [rev_is_Nil_conv]; Goal "([] = rev xs) = (xs = [])"; by (induct_tac "xs" 1); by Auto_tac; qed "Nil_is_rev_conv"; AddIffs [Nil_is_rev_conv]; Goal "!ys. (rev xs = rev ys) = (xs = ys)"; by (induct_tac "xs" 1); by (Force_tac 1); by (rtac allI 1); by (case_tac "ys" 1); by (Asm_simp_tac 1); by (Force_tac 1); qed_spec_mp "rev_is_rev_conv"; AddIffs [rev_is_rev_conv]; val prems = Goal "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P xs"; by (stac (rev_rev_ident RS sym) 1); by (res_inst_tac [("list", "rev xs")] list.induct 1); by (ALLGOALS Simp_tac); by (resolve_tac prems 1); by (eresolve_tac prems 1); qed "rev_induct"; fun rev_induct_tac xs = res_inst_tac [("xs",xs)] rev_induct; Goal "(xs = [] --> P) --> (!ys y. xs = ys@[y] --> P) --> P"; by (res_inst_tac [("xs","xs")] rev_induct 1); by Auto_tac; bind_thm ("rev_exhaust", impI RSN (2,allI RSN (2,allI RSN (2,impI RS (result() RS mp RS mp))))); (** set **) section "set"; Goal "finite (set xs)"; by (induct_tac "xs" 1); by Auto_tac; qed "finite_set"; AddIffs [finite_set]; Goal "set (xs@ys) = (set xs Un set ys)"; by (induct_tac "xs" 1); by Auto_tac; qed "set_append"; Addsimps[set_append]; Goal "set l <= set (x#l)"; by Auto_tac; qed "set_subset_Cons"; Goal "(set xs = {}) = (xs = [])"; by (induct_tac "xs" 1); by Auto_tac; qed "set_empty"; Addsimps [set_empty]; Goal "set(rev xs) = set(xs)"; by (induct_tac "xs" 1); by Auto_tac; qed "set_rev"; Addsimps [set_rev]; Goal "set(map f xs) = f``(set xs)"; by (induct_tac "xs" 1); by Auto_tac; qed "set_map"; Addsimps [set_map]; Goal "set(filter P xs) = {x. x : set xs & P x}"; by (induct_tac "xs" 1); by Auto_tac; qed "set_filter"; Addsimps [set_filter]; Goal "set[i..j(] = {k. i <= k & k < j}"; by (induct_tac "j" 1); by Auto_tac; by (arith_tac 1); qed "set_upt"; Addsimps [set_upt]; Goal "(x : set xs) = (? ys zs. xs = ys@x#zs)"; by (induct_tac "xs" 1); by (Simp_tac 1); by (Asm_simp_tac 1); by (rtac iffI 1); by (blast_tac (claset() addIs [eq_Nil_appendI,Cons_eq_appendI]) 1); by (REPEAT(etac exE 1)); by (case_tac "ys" 1); by Auto_tac; qed "in_set_conv_decomp"; (* eliminate `lists' in favour of `set' *) Goal "(xs : lists A) = (!x : set xs. x : A)"; by (induct_tac "xs" 1); by Auto_tac; qed "in_lists_conv_set"; bind_thm("in_listsD",in_lists_conv_set RS iffD1); AddSDs [in_listsD]; bind_thm("in_listsI",in_lists_conv_set RS iffD2); AddSIs [in_listsI]; (** mem **) section "mem"; Goal "(x mem xs) = (x: set xs)"; by (induct_tac "xs" 1); by Auto_tac; qed "set_mem_eq"; (** list_all **) section "list_all"; Goal "list_all P xs = (!x:set xs. P x)"; by (induct_tac "xs" 1); by Auto_tac; qed "list_all_conv"; Goal "list_all P (xs@ys) = (list_all P xs & list_all P ys)"; by (induct_tac "xs" 1); by Auto_tac; qed "list_all_append"; Addsimps [list_all_append]; (** filter **) section "filter"; Goal "filter P (xs@ys) = filter P xs @ filter P ys"; by (induct_tac "xs" 1); by Auto_tac; qed "filter_append"; Addsimps [filter_append]; Goal "filter (%x. True) xs = xs"; by (induct_tac "xs" 1); by Auto_tac; qed "filter_True"; Addsimps [filter_True]; Goal "filter (%x. False) xs = []"; by (induct_tac "xs" 1); by Auto_tac; qed "filter_False"; Addsimps [filter_False]; Goal "length (filter P xs) <= length xs"; by (induct_tac "xs" 1); by Auto_tac; by (asm_simp_tac (simpset() addsimps [le_SucI]) 1); qed "length_filter"; Addsimps[length_filter]; Goal "set (filter P xs) <= set xs"; by Auto_tac; qed "filter_is_subset"; Addsimps [filter_is_subset]; section "concat"; Goal "concat(xs@ys) = concat(xs)@concat(ys)"; by (induct_tac "xs" 1); by Auto_tac; qed"concat_append"; Addsimps [concat_append]; Goal "(concat xss = []) = (!xs:set xss. xs=[])"; by (induct_tac "xss" 1); by Auto_tac; qed "concat_eq_Nil_conv"; AddIffs [concat_eq_Nil_conv]; Goal "([] = concat xss) = (!xs:set xss. xs=[])"; by (induct_tac "xss" 1); by Auto_tac; qed "Nil_eq_concat_conv"; AddIffs [Nil_eq_concat_conv]; Goal "set(concat xs) = Union(set `` set xs)"; by (induct_tac "xs" 1); by Auto_tac; qed"set_concat"; Addsimps [set_concat]; Goal "map f (concat xs) = concat (map (map f) xs)"; by (induct_tac "xs" 1); by Auto_tac; qed "map_concat"; Goal "filter p (concat xs) = concat (map (filter p) xs)"; by (induct_tac "xs" 1); by Auto_tac; qed"filter_concat"; Goal "rev(concat xs) = concat (map rev (rev xs))"; by (induct_tac "xs" 1); by Auto_tac; qed "rev_concat"; (** nth **) section "nth"; Goal "(x#xs)!0 = x"; by Auto_tac; qed "nth_Cons_0"; Addsimps [nth_Cons_0]; Goal "(x#xs)!(Suc n) = xs!n"; by Auto_tac; qed "nth_Cons_Suc"; Addsimps [nth_Cons_Suc]; Delsimps (thms "nth.simps"); Goal "!n. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"; by (induct_tac "xs" 1); by (Asm_simp_tac 1); by (rtac allI 1); by (case_tac "n" 1); by Auto_tac; qed_spec_mp "nth_append"; Goal "!n. n < length xs --> (map f xs)!n = f(xs!n)"; by (induct_tac "xs" 1); by (Asm_full_simp_tac 1); by (rtac allI 1); by (induct_tac "n" 1); by Auto_tac; qed_spec_mp "nth_map"; Addsimps [nth_map]; Goal "set xs = {xs!i |i. i < length xs}"; by (induct_tac "xs" 1); by (Simp_tac 1); by (Asm_simp_tac 1); by Safe_tac; by (res_inst_tac [("x","0")] exI 1); by (Simp_tac 1); by (res_inst_tac [("x","Suc i")] exI 1); by (Asm_simp_tac 1); by (case_tac "i" 1); by (Asm_full_simp_tac 1); by (rename_tac "j" 1); by (res_inst_tac [("x","j")] exI 1); by (Asm_simp_tac 1); qed "set_conv_nth"; Goal "n < length xs ==> Ball (set xs) P --> P(xs!n)"; by (simp_tac (simpset() addsimps [set_conv_nth]) 1); by (Blast_tac 1); qed_spec_mp "list_ball_nth"; Goal "n < length xs ==> xs!n : set xs"; by (simp_tac (simpset() addsimps [set_conv_nth]) 1); by (Blast_tac 1); qed_spec_mp "nth_mem"; Addsimps [nth_mem]; Goal "(!i. i < length xs --> P(xs!i)) --> (!x : set xs. P x)"; by (simp_tac (simpset() addsimps [set_conv_nth]) 1); by (Blast_tac 1); qed_spec_mp "all_nth_imp_all_set"; Goal "(!x : set xs. P x) = (!i. i<length xs --> P (xs ! i))"; by (simp_tac (simpset() addsimps [set_conv_nth]) 1); by (Blast_tac 1); qed_spec_mp "all_set_conv_all_nth"; (** list update **) section "list update"; Goal "!i. length(xs[i:=x]) = length xs"; by (induct_tac "xs" 1); by (Simp_tac 1); by (asm_full_simp_tac (simpset() addsplits [nat.split]) 1); qed_spec_mp "length_list_update"; Addsimps [length_list_update]; Goal "!i j. i < length xs --> (xs[i:=x])!j = (if i=j then x else xs!j)"; by (induct_tac "xs" 1); by (Simp_tac 1); by (auto_tac (claset(), simpset() addsimps [nth_Cons] addsplits [nat.split])); qed_spec_mp "nth_list_update"; Goal "i < length xs ==> (xs[i:=x])!i = x"; by (asm_simp_tac (simpset() addsimps [nth_list_update]) 1); qed "nth_list_update_eq"; Addsimps [nth_list_update_eq]; Goal "!i j. i ~= j --> xs[i:=x]!j = xs!j"; by (induct_tac "xs" 1); by (Simp_tac 1); by (auto_tac (claset(), simpset() addsimps [nth_Cons] addsplits [nat.split])); qed_spec_mp "nth_list_update_neq"; Addsimps [nth_list_update_neq]; Goal "!i. i < size xs --> xs[i:=x, i:=y] = xs[i:=y]"; by (induct_tac "xs" 1); by (Simp_tac 1); by (asm_simp_tac (simpset() addsplits [nat.split]) 1); qed_spec_mp "list_update_overwrite"; Addsimps [list_update_overwrite]; Goal "!i < length xs. (xs[i := x] = xs) = (xs!i = x)"; by (induct_tac "xs" 1); by (Simp_tac 1); by (simp_tac (simpset() addsplits [nat.split]) 1); by (Blast_tac 1); qed_spec_mp "list_update_same_conv"; Goal "!i xy xs. length xs = length ys --> \ \ (zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"; by (induct_tac "ys" 1); by Auto_tac; by (case_tac "xs" 1); by (auto_tac (claset(), simpset() addsplits [nat.split])); qed_spec_mp "update_zip"; Goal "!i. set(xs[i:=x]) <= insert x (set xs)"; by (induct_tac "xs" 1); by (asm_full_simp_tac (simpset() addsimps []) 1); by (asm_full_simp_tac (simpset() addsplits [nat.split]) 1); by (Fast_tac 1); qed_spec_mp "set_update_subset_insert"; Goal "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"; by(fast_tac (claset() addSDs [set_update_subset_insert RS subsetD]) 1); qed "set_update_subsetI"; (** last & butlast **) section "last / butlast"; Goal "last(xs@[x]) = x"; by (induct_tac "xs" 1); by Auto_tac; qed "last_snoc"; Addsimps [last_snoc]; Goal "butlast(xs@[x]) = xs"; by (induct_tac "xs" 1); by Auto_tac; qed "butlast_snoc"; Addsimps [butlast_snoc]; Goal "length(butlast xs) = length xs - 1"; by (res_inst_tac [("xs","xs")] rev_induct 1); by Auto_tac; qed "length_butlast"; Addsimps [length_butlast]; Goal "!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)"; by (induct_tac "xs" 1); by Auto_tac; qed_spec_mp "butlast_append"; Goal "xs ~= [] --> butlast xs @ [last xs] = xs"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed_spec_mp "append_butlast_last_id"; Addsimps [append_butlast_last_id]; Goal "x:set(butlast xs) --> x:set xs"; by (induct_tac "xs" 1); by Auto_tac; qed_spec_mp "in_set_butlastD"; Goal "x:set(butlast xs) | x:set(butlast ys) ==> x:set(butlast(xs@ys))"; by (auto_tac (claset() addDs [in_set_butlastD], simpset() addsimps [butlast_append])); qed "in_set_butlast_appendI"; (** take & drop **) section "take & drop"; Goal "take 0 xs = []"; by (induct_tac "xs" 1); by Auto_tac; qed "take_0"; Goal "drop 0 xs = xs"; by (induct_tac "xs" 1); by Auto_tac; qed "drop_0"; Goal "take (Suc n) (x#xs) = x # take n xs"; by (Simp_tac 1); qed "take_Suc_Cons"; Goal "drop (Suc n) (x#xs) = drop n xs"; by (Simp_tac 1); qed "drop_Suc_Cons"; Delsimps [take_Cons,drop_Cons]; Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons]; Goal "!xs. length(take n xs) = min (length xs) n"; by (induct_tac "n" 1); by Auto_tac; by (case_tac "xs" 1); by Auto_tac; qed_spec_mp "length_take"; Addsimps [length_take]; Goal "!xs. length(drop n xs) = (length xs - n)"; by (induct_tac "n" 1); by Auto_tac; by (case_tac "xs" 1); by Auto_tac; qed_spec_mp "length_drop"; Addsimps [length_drop]; Goal "!xs. length xs <= n --> take n xs = xs"; by (induct_tac "n" 1); by Auto_tac; by (case_tac "xs" 1); by Auto_tac; qed_spec_mp "take_all"; Addsimps [take_all]; Goal "!xs. length xs <= n --> drop n xs = []"; by (induct_tac "n" 1); by Auto_tac; by (case_tac "xs" 1); by Auto_tac; qed_spec_mp "drop_all"; Addsimps [drop_all]; Goal "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"; by (induct_tac "n" 1); by Auto_tac; by (case_tac "xs" 1); by Auto_tac; qed_spec_mp "take_append"; Addsimps [take_append]; Goal "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; by (induct_tac "n" 1); by Auto_tac; by (case_tac "xs" 1); by Auto_tac; qed_spec_mp "drop_append"; Addsimps [drop_append]; Goal "!xs n. take n (take m xs) = take (min n m) xs"; by (induct_tac "m" 1); by Auto_tac; by (case_tac "xs" 1); by Auto_tac; by (case_tac "na" 1); by Auto_tac; qed_spec_mp "take_take"; Addsimps [take_take]; Goal "!xs. drop n (drop m xs) = drop (n + m) xs"; by (induct_tac "m" 1); by Auto_tac; by (case_tac "xs" 1); by Auto_tac; qed_spec_mp "drop_drop"; Addsimps [drop_drop]; Goal "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; by (induct_tac "m" 1); by Auto_tac; by (case_tac "xs" 1); by Auto_tac; qed_spec_mp "take_drop"; Goal "!xs. take n xs @ drop n xs = xs"; by (induct_tac "n" 1); by Auto_tac; by (case_tac "xs" 1); by Auto_tac; qed_spec_mp "append_take_drop_id"; Addsimps [append_take_drop_id]; Goal "!xs. take n (map f xs) = map f (take n xs)"; by (induct_tac "n" 1); by Auto_tac; by (case_tac "xs" 1); by Auto_tac; qed_spec_mp "take_map"; Goal "!xs. drop n (map f xs) = map f (drop n xs)"; by (induct_tac "n" 1); by Auto_tac; by (case_tac "xs" 1); by Auto_tac; qed_spec_mp "drop_map"; Goal "!n i. i < n --> (take n xs)!i = xs!i"; by (induct_tac "xs" 1); by Auto_tac; by (case_tac "n" 1); by (Blast_tac 1); by (case_tac "i" 1); by Auto_tac; qed_spec_mp "nth_take"; Addsimps [nth_take]; Goal "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)"; by (induct_tac "n" 1); by Auto_tac; by (case_tac "xs" 1); by Auto_tac; qed_spec_mp "nth_drop"; Addsimps [nth_drop]; Goal "!zs. (xs@ys = zs) = (xs = take (length xs) zs & ys = drop (length xs) zs)"; by (induct_tac "xs" 1); by (Simp_tac 1); by (Asm_full_simp_tac 1); by (Clarify_tac 1); by (case_tac "zs" 1); by (Auto_tac); qed_spec_mp "append_eq_conv_conj"; (** takeWhile & dropWhile **) section "takeWhile & dropWhile"; Goal "takeWhile P xs @ dropWhile P xs = xs"; by (induct_tac "xs" 1); by Auto_tac; qed "takeWhile_dropWhile_id"; Addsimps [takeWhile_dropWhile_id]; Goal "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs"; by (induct_tac "xs" 1); by Auto_tac; bind_thm("takeWhile_append1", conjI RS (result() RS mp)); Addsimps [takeWhile_append1]; Goal "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys"; by (induct_tac "xs" 1); by Auto_tac; bind_thm("takeWhile_append2", ballI RS (result() RS mp)); Addsimps [takeWhile_append2]; Goal "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"; by (induct_tac "xs" 1); by Auto_tac; bind_thm("dropWhile_append1", conjI RS (result() RS mp)); Addsimps [dropWhile_append1]; Goal "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys"; by (induct_tac "xs" 1); by Auto_tac; bind_thm("dropWhile_append2", ballI RS (result() RS mp)); Addsimps [dropWhile_append2]; Goal "x:set(takeWhile P xs) --> x:set xs & P x"; by (induct_tac "xs" 1); by Auto_tac; qed_spec_mp"set_take_whileD"; (** zip **) section "zip"; Goal "zip [] ys = []"; by (induct_tac "ys" 1); by Auto_tac; qed "zip_Nil"; Addsimps [zip_Nil]; Goal "zip (x#xs) (y#ys) = (x,y)#zip xs ys"; by (Simp_tac 1); qed "zip_Cons_Cons"; Addsimps [zip_Cons_Cons]; Delsimps(tl (thms"zip.simps")); Goal "!xs. length (zip xs ys) = min (length xs) (length ys)"; by (induct_tac "ys" 1); by (Simp_tac 1); by (Clarify_tac 1); by (case_tac "xs" 1); by (Auto_tac); qed_spec_mp "length_zip"; Addsimps [length_zip]; Goal "!xs. zip (xs@ys) zs = \ \ zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"; by (induct_tac "zs" 1); by (Simp_tac 1); by (Clarify_tac 1); by (case_tac "xs" 1); by (Asm_simp_tac 1); by (Asm_simp_tac 1); qed_spec_mp "zip_append1"; Goal "!ys. zip xs (ys@zs) = \ \ zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"; by (induct_tac "xs" 1); by (Simp_tac 1); by (Clarify_tac 1); by (case_tac "ys" 1); by (Asm_simp_tac 1); by (Asm_simp_tac 1); qed_spec_mp "zip_append2"; Goal "[| length xs = length us; length ys = length vs |] ==> \ \ zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"; by (asm_simp_tac (simpset() addsimps [zip_append1]) 1); qed_spec_mp "zip_append"; Addsimps [zip_append]; Goal "!xs. length xs = length ys --> zip (rev xs) (rev ys) = rev (zip xs ys)"; by (induct_tac "ys" 1); by (Asm_full_simp_tac 1); by (Asm_full_simp_tac 1); by (Clarify_tac 1); by (case_tac "xs" 1); by (Auto_tac); qed_spec_mp "zip_rev"; Goal "!i xs. i < length xs --> i < length ys --> (zip xs ys)!i = (xs!i, ys!i)"; by (induct_tac "ys" 1); by (Simp_tac 1); by (Clarify_tac 1); by (case_tac "xs" 1); by (Auto_tac); by (asm_full_simp_tac (simpset() addsimps (thms"nth.simps") addsplits [nat.split]) 1); qed_spec_mp "nth_zip"; Addsimps [nth_zip]; Goal "set(zip xs ys) = {(xs!i,ys!i) |i. i < min (length xs) (length ys)}"; by (simp_tac (simpset() addsimps [set_conv_nth]addcongs [rev_conj_cong]) 1); qed_spec_mp "set_zip"; Goal "length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"; by (rtac sym 1); by (asm_simp_tac (simpset() addsimps [update_zip]) 1); qed_spec_mp "zip_update"; Goal "!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"; by (induct_tac "i" 1); by (Auto_tac); by (case_tac "j" 1); by (Auto_tac); qed "zip_replicate"; Addsimps [zip_replicate]; (** list_all2 **) section "list_all2"; Goalw [list_all2_def] "list_all2 P xs ys ==> length xs = length ys"; by (Asm_simp_tac 1); qed "list_all2_lengthD"; Goalw [list_all2_def] "list_all2 P [] ys = (ys=[])"; by (Simp_tac 1); qed "list_all2_Nil"; AddIffs [list_all2_Nil]; Goalw [list_all2_def] "list_all2 P xs [] = (xs=[])"; by (Simp_tac 1); qed "list_all2_Nil2"; AddIffs [list_all2_Nil2]; Goalw [list_all2_def] "list_all2 P (x#xs) (y#ys) = (P x y & list_all2 P xs ys)"; by (Auto_tac); qed "list_all2_Cons"; AddIffs[list_all2_Cons]; Goalw [list_all2_def] "list_all2 P (x#xs) ys = (? z zs. ys = z#zs & P x z & list_all2 P xs zs)"; by (case_tac "ys" 1); by (Auto_tac); qed "list_all2_Cons1"; Goalw [list_all2_def] "list_all2 P xs (y#ys) = (? z zs. xs = z#zs & P z y & list_all2 P zs ys)"; by (case_tac "xs" 1); by (Auto_tac); qed "list_all2_Cons2"; Goalw [list_all2_def] "list_all2 P (xs@ys) zs = \ \ (EX us vs. zs = us@vs & length us = length xs & length vs = length ys & \ \ list_all2 P xs us & list_all2 P ys vs)"; by (simp_tac (simpset() addsimps [zip_append1]) 1); by (rtac iffI 1); by (res_inst_tac [("x","take (length xs) zs")] exI 1); by (res_inst_tac [("x","drop (length xs) zs")] exI 1); by (asm_full_simp_tac (simpset() addsimps [min_def,eq_sym_conv]) 1); by (Clarify_tac 1); by (asm_full_simp_tac (simpset() addsimps [ball_Un]) 1); qed "list_all2_append1"; Goalw [list_all2_def] "list_all2 P xs (ys@zs) = \ \ (EX us vs. xs = us@vs & length us = length ys & length vs = length zs & \ \ list_all2 P us ys & list_all2 P vs zs)"; by (simp_tac (simpset() addsimps [zip_append2]) 1); by (rtac iffI 1); by (res_inst_tac [("x","take (length ys) xs")] exI 1); by (res_inst_tac [("x","drop (length ys) xs")] exI 1); by (asm_full_simp_tac (simpset() addsimps [min_def,eq_sym_conv]) 1); by (Clarify_tac 1); by (asm_full_simp_tac (simpset() addsimps [ball_Un]) 1); qed "list_all2_append2"; Goalw [list_all2_def] "list_all2 P xs ys = \ \ (length xs = length ys & (!i<length xs. P (xs!i) (ys!i)))"; by (force_tac (claset(), simpset() addsimps [set_zip]) 1); qed "list_all2_conv_all_nth"; (** foldl **) section "foldl"; Goal "!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"; by (induct_tac "xs" 1); by Auto_tac; qed_spec_mp "foldl_append"; Addsimps [foldl_append]; (* Note: `n <= foldl op+ n ns' looks simpler, but is more difficult to use because it requires an additional transitivity step *) Goal "!n::nat. m <= n --> m <= foldl op+ n ns"; by (induct_tac "ns" 1); by Auto_tac; qed_spec_mp "start_le_sum"; Goal "!!n::nat. n : set ns ==> n <= foldl op+ 0 ns"; by (force_tac (claset() addIs [start_le_sum], simpset() addsimps [in_set_conv_decomp]) 1); qed "elem_le_sum"; Goal "!m::nat. (foldl op+ m ns = 0) = (m=0 & (!n : set ns. n=0))"; by (induct_tac "ns" 1); by Auto_tac; qed_spec_mp "sum_eq_0_conv"; AddIffs [sum_eq_0_conv]; (** upto **) (* Does not terminate! *) Goal "[i..j(] = (if i<j then i#[Suc i..j(] else [])"; by (induct_tac "j" 1); by Auto_tac; qed "upt_rec"; Goal "j<=i ==> [i..j(] = []"; by (stac upt_rec 1); by (Asm_simp_tac 1); qed "upt_conv_Nil"; Addsimps [upt_conv_Nil]; (*Only needed if upt_Suc is deleted from the simpset*) Goal "i<=j ==> [i..(Suc j)(] = [i..j(]@[j]"; by (Asm_simp_tac 1); qed "upt_Suc_append"; Goal "i<j ==> [i..j(] = i#[Suc i..j(]"; by (rtac trans 1); by (stac upt_rec 1); by (rtac refl 2); by (Asm_simp_tac 1); qed "upt_conv_Cons"; (*LOOPS as a simprule, since j<=j*) Goal "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]"; by (induct_tac "k" 1); by Auto_tac; qed "upt_add_eq_append"; Goal "length [i..j(] = j-i"; by (induct_tac "j" 1); by (Simp_tac 1); by (asm_simp_tac (simpset() addsimps [Suc_diff_le]) 1); qed "length_upt"; Addsimps [length_upt]; Goal "i+k < j --> [i..j(] ! k = i+k"; by (induct_tac "j" 1); by (asm_simp_tac (simpset() addsimps [less_Suc_eq, nth_append] addsplits [nat_diff_split]) 2); by (Simp_tac 1); qed_spec_mp "nth_upt"; Addsimps [nth_upt]; Goal "!i. i+m <= n --> take m [i..n(] = [i..i+m(]"; by (induct_tac "m" 1); by (Simp_tac 1); by (Clarify_tac 1); by (stac upt_rec 1); by (rtac sym 1); by (stac upt_rec 1); by (asm_simp_tac (simpset() delsimps (thms"upt.simps")) 1); qed_spec_mp "take_upt"; Addsimps [take_upt]; Goal "map Suc [m..n(] = [Suc m..n]"; by (induct_tac "n" 1); by Auto_tac; qed "map_Suc_upt"; Goal "ALL i. i < n-m --> (map f [m..n(]) ! i = f(m+i)"; by (res_inst_tac [("m","n"),("n","m")] diff_induct 1); by (stac (map_Suc_upt RS sym) 3); by (auto_tac (claset(), simpset() addsimps [less_diff_conv, nth_upt])); qed_spec_mp "nth_map_upt"; Goal "ALL xs ys. k <= length xs --> k <= length ys --> \ \ (ALL i. i < k --> xs!i = ys!i) \ \ --> take k xs = take k ys"; by (induct_tac "k" 1); by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq_0_disj, all_conj_distrib]))); by (Clarify_tac 1); (*Both lists must be non-empty*) by (case_tac "xs" 1); by (case_tac "ys" 2); by (ALLGOALS Clarify_tac); (*prenexing's needed, not miniscoping*) by (ALLGOALS (full_simp_tac (simpset() addsimps (all_simps RL [sym]) delsimps (all_simps)))); by (Blast_tac 1); qed_spec_mp "nth_take_lemma"; Goal "[| length xs = length ys; \ \ ALL i. i < length xs --> xs!i = ys!i |] \ \ ==> xs = ys"; by (forward_tac [[le_refl, eq_imp_le] MRS nth_take_lemma] 1); by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [take_all]))); qed_spec_mp "nth_equalityI"; (*The famous take-lemma*) Goal "(ALL i. take i xs = take i ys) ==> xs = ys"; by (dres_inst_tac [("x", "max (length xs) (length ys)")] spec 1); by (full_simp_tac (simpset() addsimps [le_max_iff_disj, take_all]) 1); qed_spec_mp "take_equalityI"; (** nodups & remdups **) section "nodups & remdups"; Goal "set(remdups xs) = set xs"; by (induct_tac "xs" 1); by (Simp_tac 1); by (asm_full_simp_tac (simpset() addsimps [insert_absorb]) 1); qed "set_remdups"; Addsimps [set_remdups]; Goal "nodups(remdups xs)"; by (induct_tac "xs" 1); by Auto_tac; qed "nodups_remdups"; Goal "nodups xs --> nodups (filter P xs)"; by (induct_tac "xs" 1); by Auto_tac; qed_spec_mp "nodups_filter"; (** replicate **) section "replicate"; Goal "length(replicate n x) = n"; by (induct_tac "n" 1); by Auto_tac; qed "length_replicate"; Addsimps [length_replicate]; Goal "map f (replicate n x) = replicate n (f x)"; by (induct_tac "n" 1); by Auto_tac; qed "map_replicate"; Addsimps [map_replicate]; Goal "(replicate n x) @ (x#xs) = x # replicate n x @ xs"; by (induct_tac "n" 1); by Auto_tac; qed "replicate_app_Cons_same"; Goal "rev(replicate n x) = replicate n x"; by (induct_tac "n" 1); by (Simp_tac 1); by (asm_simp_tac (simpset() addsimps [replicate_app_Cons_same]) 1); qed "rev_replicate"; Addsimps [rev_replicate]; Goal "replicate (n+m) x = replicate n x @ replicate m x"; by (induct_tac "n" 1); by Auto_tac; qed "replicate_add"; Goal"n ~= 0 --> hd(replicate n x) = x"; by (induct_tac "n" 1); by Auto_tac; qed_spec_mp "hd_replicate"; Addsimps [hd_replicate]; Goal "n ~= 0 --> tl(replicate n x) = replicate (n-1) x"; by (induct_tac "n" 1); by Auto_tac; qed_spec_mp "tl_replicate"; Addsimps [tl_replicate]; Goal "n ~= 0 --> last(replicate n x) = x"; by (induct_tac "n" 1); by Auto_tac; qed_spec_mp "last_replicate"; Addsimps [last_replicate]; Goal "!i. i<n --> (replicate n x)!i = x"; by (induct_tac "n" 1); by (Simp_tac 1); by (asm_simp_tac (simpset() addsimps [nth_Cons] addsplits [nat.split]) 1); qed_spec_mp "nth_replicate"; Addsimps [nth_replicate]; Goal "set(replicate (Suc n) x) = {x}"; by (induct_tac "n" 1); by Auto_tac; val lemma = result(); Goal "n ~= 0 ==> set(replicate n x) = {x}"; by (fast_tac (claset() addSDs [not0_implies_Suc] addSIs [lemma]) 1); qed "set_replicate"; Addsimps [set_replicate]; Goal "set(replicate n x) = (if n=0 then {} else {x})"; by (Auto_tac); qed "set_replicate_conv_if"; Goal "x : set(replicate n y) --> x=y"; by (asm_simp_tac (simpset() addsimps [set_replicate_conv_if]) 1); qed_spec_mp "in_set_replicateD"; (*** Lexcicographic orderings on lists ***) section"Lexcicographic orderings on lists"; Goal "wf r ==> wf(lexn r n)"; by (induct_tac "n" 1); by (Simp_tac 1); by (Simp_tac 1); by (rtac wf_subset 1); by (rtac Int_lower1 2); by (rtac wf_prod_fun_image 1); by (rtac injI 2); by Auto_tac; qed "wf_lexn"; Goal "!xs ys. (xs,ys) : lexn r n --> length xs = n & length ys = n"; by (induct_tac "n" 1); by Auto_tac; qed_spec_mp "lexn_length"; Goalw [lex_def] "wf r ==> wf(lex r)"; by (rtac wf_UN 1); by (blast_tac (claset() addIs [wf_lexn]) 1); by (Clarify_tac 1); by (rename_tac "m n" 1); by (subgoal_tac "m ~= n" 1); by (Blast_tac 2); by (blast_tac (claset() addDs [lexn_length,not_sym]) 1); qed "wf_lex"; AddSIs [wf_lex]; Goal "lexn r n = \ \ {(xs,ys). length xs = n & length ys = n & \ \ (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}"; by (induct_tac "n" 1); by (Simp_tac 1); by (Blast_tac 1); by (asm_full_simp_tac (simpset() addsimps [lex_prod_def]) 1); by (auto_tac (claset(), simpset())); by (Blast_tac 1); by (rename_tac "a xys x xs' y ys'" 1); by (res_inst_tac [("x","a#xys")] exI 1); by (Simp_tac 1); by (case_tac "xys" 1); by (ALLGOALS (asm_full_simp_tac (simpset()))); by (Blast_tac 1); qed "lexn_conv"; Goalw [lex_def] "lex r = \ \ {(xs,ys). length xs = length ys & \ \ (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}"; by (force_tac (claset(), simpset() addsimps [lexn_conv]) 1); qed "lex_conv"; Goalw [lexico_def] "wf r ==> wf(lexico r)"; by (Blast_tac 1); qed "wf_lexico"; AddSIs [wf_lexico]; Goalw [lexico_def,diag_def,lex_prod_def,measure_def,inv_image_def] "lexico r = {(xs,ys). length xs < length ys | \ \ length xs = length ys & (xs,ys) : lex r}"; by (Simp_tac 1); qed "lexico_conv"; Goal "([],ys) ~: lex r"; by (simp_tac (simpset() addsimps [lex_conv]) 1); qed "Nil_notin_lex"; Goal "(xs,[]) ~: lex r"; by (simp_tac (simpset() addsimps [lex_conv]) 1); qed "Nil2_notin_lex"; AddIffs [Nil_notin_lex,Nil2_notin_lex]; Goal "((x#xs,y#ys) : lex r) = \ \ ((x,y) : r & length xs = length ys | x=y & (xs,ys) : lex r)"; by (simp_tac (simpset() addsimps [lex_conv]) 1); by (rtac iffI 1); by (blast_tac (claset() addIs [Cons_eq_appendI]) 2); by (REPEAT(eresolve_tac [conjE, exE] 1)); by (case_tac "xys" 1); by (Asm_full_simp_tac 1); by (Asm_full_simp_tac 1); by (Blast_tac 1); qed "Cons_in_lex"; AddIffs [Cons_in_lex]; (*** Versions of some theorems above using binary numerals ***) AddIffs (map (rename_numerals thy) [length_0_conv, zero_length_conv, length_greater_0_conv, sum_eq_0_conv]); Goal "take n (x#xs) = (if n = #0 then [] else x # take (n-#1) xs)"; by (case_tac "n" 1); by (ALLGOALS (asm_simp_tac (simpset() addsimps [numeral_0_eq_0, numeral_1_eq_1]))); qed "take_Cons'"; Goal "drop n (x#xs) = (if n = #0 then x#xs else drop (n-#1) xs)"; by (case_tac "n" 1); by (ALLGOALS (asm_simp_tac (simpset() addsimps [numeral_0_eq_0, numeral_1_eq_1]))); qed "drop_Cons'"; Goal "(x#xs)!n = (if n = #0 then x else xs!(n-#1))"; by (case_tac "n" 1); by (ALLGOALS (asm_simp_tac (simpset() addsimps [numeral_0_eq_0, numeral_1_eq_1]))); qed "nth_Cons'"; Addsimps (map (inst "n" "number_of ?v") [take_Cons', drop_Cons', nth_Cons']);