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author | nipkow |

Mon, 18 May 1998 17:31:49 +0200 | |

changeset 4935 | 1694e2daef8f |

parent 4934 | 683eae4b5d0f |

child 4936 | e67949e15255 |

Cleaned up and simplified etc.
snoc_induct/exhaust -> rev_induct_exhaust.

src/HOL/List.ML | file | annotate | diff | comparison | revisions |

--- a/src/HOL/List.ML Fri May 15 11:35:56 1998 +0200 +++ b/src/HOL/List.ML Mon May 18 17:31:49 1998 +0200 @@ -6,21 +6,21 @@ List lemmas *) -goal thy "!x. xs ~= x#xs"; +Goal "!x. xs ~= x#xs"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_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 thy "(xs ~= []) = (? y ys. xs = y#ys)"; +Goal "(xs ~= []) = (? y ys. xs = y#ys)"; by (induct_tac "xs" 1); by (Simp_tac 1); by (Asm_simp_tac 1); qed "neq_Nil_conv"; (* Induction over the length of a list: *) -val [prem] = goal thy +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"; @@ -28,7 +28,7 @@ (** "lists": the list-forming operator over sets **) -goalw thy lists.defs "!!A B. A<=B ==> lists A <= lists B"; +Goalw lists.defs "!!A B. A<=B ==> lists A <= lists B"; by (rtac lfp_mono 1); by (REPEAT (ares_tac basic_monos 1)); qed "lists_mono"; @@ -37,12 +37,12 @@ AddSEs [listsE]; AddSIs lists.intrs; -goal thy "!!l. l: lists A ==> l: lists B --> l: lists (A Int B)"; +Goal "!!l. 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 thy "lists (A Int B) = lists A Int lists B"; +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); @@ -53,12 +53,12 @@ (** Case analysis **) section "Case analysis"; -val prems = goal thy "[| P([]); !!x xs. P(x#xs) |] ==> P(xs)"; +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 thy "(xs=[] --> P([])) & (!y ys. xs=y#ys --> P(y#ys)) --> P(xs)"; +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); @@ -70,43 +70,43 @@ section "length"; -goal thy "length(xs@ys) = length(xs)+length(ys)"; +Goal "length(xs@ys) = length(xs)+length(ys)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed"length_append"; Addsimps [length_append]; -goal thy "length (map f l) = length l"; +Goal "length (map f l) = length l"; by (induct_tac "l" 1); by (ALLGOALS Simp_tac); qed "length_map"; Addsimps [length_map]; -goal thy "length(rev xs) = length(xs)"; +Goal "length(rev xs) = length(xs)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "length_rev"; Addsimps [length_rev]; -goal List.thy "!!xs. xs ~= [] ==> length(tl xs) = (length xs) - 1"; +Goal "!!xs. xs ~= [] ==> length(tl xs) = (length xs) - 1"; by (exhaust_tac "xs" 1); by (ALLGOALS Asm_full_simp_tac); qed "length_tl"; Addsimps [length_tl]; -goal thy "(length xs = 0) = (xs = [])"; +Goal "(length xs = 0) = (xs = [])"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "length_0_conv"; AddIffs [length_0_conv]; -goal thy "(0 = length xs) = (xs = [])"; +Goal "(0 = length xs) = (xs = [])"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "zero_length_conv"; AddIffs [zero_length_conv]; -goal thy "(0 < length xs) = (xs ~= [])"; +Goal "(0 < length xs) = (xs ~= [])"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "length_greater_0_conv"; @@ -116,44 +116,44 @@ section "@ - append"; -goal thy "(xs@ys)@zs = xs@(ys@zs)"; +Goal "(xs@ys)@zs = xs@(ys@zs)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "append_assoc"; Addsimps [append_assoc]; -goal thy "xs @ [] = xs"; +Goal "xs @ [] = xs"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "append_Nil2"; Addsimps [append_Nil2]; -goal thy "(xs@ys = []) = (xs=[] & ys=[])"; +Goal "(xs@ys = []) = (xs=[] & ys=[])"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "append_is_Nil_conv"; AddIffs [append_is_Nil_conv]; -goal thy "([] = xs@ys) = (xs=[] & ys=[])"; +Goal "([] = xs@ys) = (xs=[] & ys=[])"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); by (Blast_tac 1); qed "Nil_is_append_conv"; AddIffs [Nil_is_append_conv]; -goal thy "(xs @ ys = xs) = (ys=[])"; +Goal "(xs @ ys = xs) = (ys=[])"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "append_self_conv"; -goal thy "(xs = xs @ ys) = (ys=[])"; +Goal "(xs = xs @ ys) = (ys=[])"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); by (Blast_tac 1); qed "self_append_conv"; AddIffs [append_self_conv,self_append_conv]; -goal thy "!ys. length xs = length ys | length us = length vs \ +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); @@ -169,15 +169,15 @@ qed_spec_mp "append_eq_append_conv"; Addsimps [append_eq_append_conv]; -goal thy "(xs @ ys = xs @ zs) = (ys=zs)"; +Goal "(xs @ ys = xs @ zs) = (ys=zs)"; by (Simp_tac 1); qed "same_append_eq"; -goal thy "(xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; +Goal "(xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; by (Simp_tac 1); qed "append1_eq_conv"; -goal thy "(ys @ xs = zs @ xs) = (ys=zs)"; +Goal "(ys @ xs = zs @ xs) = (ys=zs)"; by (Simp_tac 1); qed "append_same_eq"; @@ -186,115 +186,82 @@ AddSDs [same_append_eq RS iffD1, append1_eq_conv RS iffD1, append_same_eq RS iffD1]; -goal thy "(xs @ ys = ys) = (xs=[])"; +Goal "(xs @ ys = ys) = (xs=[])"; by(cut_inst_tac [("zs","[]")] append_same_eq 1); by(Asm_full_simp_tac 1); qed "append_self_conv2"; -goal thy "(ys = xs @ ys) = (xs=[])"; +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 thy "xs ~= [] --> hd xs # tl xs = xs"; +Goal "xs ~= [] --> hd xs # tl xs = xs"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed_spec_mp "hd_Cons_tl"; Addsimps [hd_Cons_tl]; -goal thy "hd(xs@ys) = (if xs=[] then hd ys else hd xs)"; +Goal "hd(xs@ys) = (if xs=[] then hd ys else hd xs)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "hd_append"; -goal thy "!!xs. xs ~= [] ==> hd(xs @ ys) = hd xs"; +Goal "!!xs. xs ~= [] ==> hd(xs @ ys) = hd xs"; by (asm_simp_tac (simpset() addsimps [hd_append] addsplits [split_list_case]) 1); qed "hd_append2"; Addsimps [hd_append2]; -goal thy "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)"; +Goal "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)"; by (simp_tac (simpset() addsplits [split_list_case]) 1); qed "tl_append"; -goal thy "!!xs. xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys"; +Goal "!!xs. xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys"; by (asm_simp_tac (simpset() addsimps [tl_append] addsplits [split_list_case]) 1); qed "tl_append2"; Addsimps [tl_append2]; -(** Snoc exhaustion and induction **) -section "Snoc exhaustion and induction"; - -goal thy "xs ~= [] --> (? ys y. xs = ys@[y])"; -by(induct_tac "xs" 1); -by(Simp_tac 1); -by(exhaust_tac "list" 1); - by(Asm_simp_tac 1); - by(res_inst_tac [("x","[]")] exI 1); - by(Simp_tac 1); -by(Asm_full_simp_tac 1); -by(Clarify_tac 1); -by(res_inst_tac [("x","a#ys")] exI 1); -by(Asm_simp_tac 1); -val lemma = result(); - -goal thy "(xs = [] --> P) --> (!ys y. xs = ys@[y] --> P) --> P"; -by(cut_facts_tac [lemma] 1); -by(Blast_tac 1); -bind_thm ("snoc_exhaust", - impI RSN (2,allI RSN (2,allI RSN (2,impI RS (result() RS mp RS mp))))); - -val prems = goal thy "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P xs"; -by(res_inst_tac [("xs","xs")] length_induct 1); -by(res_inst_tac [("xs","xs")] snoc_exhaust 1); - by(Clarify_tac 1); - brs prems 1; -by(Clarify_tac 1); -brs prems 1; -auto(); -qed "snoc_induct"; - - (** map **) section "map"; -goal thy +Goal "(!x. x : set xs --> f x = g x) --> map f xs = map g xs"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); bind_thm("map_ext", impI RS (allI RS (result() RS mp))); -goal thy "map (%x. x) = (%xs. xs)"; +Goal "map (%x. x) = (%xs. xs)"; by (rtac ext 1); by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "map_ident"; Addsimps[map_ident]; -goal thy "map f (xs@ys) = map f xs @ map f ys"; +Goal "map f (xs@ys) = map f xs @ map f ys"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "map_append"; Addsimps[map_append]; -goalw thy [o_def] "map (f o g) xs = map f (map g xs)"; +Goalw [o_def] "map (f o g) xs = map f (map g xs)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "map_compose"; Addsimps[map_compose]; -goal thy "rev(map f xs) = map f (rev xs)"; +Goal "rev(map f xs) = map f (rev xs)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "rev_map"; (* a congruence rule for map: *) -goal thy +Goal "(xs=ys) --> (!x. x : set ys --> f x = g x) --> map f xs = map g ys"; by (rtac impI 1); by (hyp_subst_tac 1); @@ -303,13 +270,13 @@ val lemma = result(); bind_thm("map_cong",impI RSN (2,allI RSN (2,lemma RS mp RS mp))); -goal List.thy "(map f xs = []) = (xs = [])"; +Goal "(map f xs = []) = (xs = [])"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "map_is_Nil_conv"; AddIffs [map_is_Nil_conv]; -goal List.thy "([] = map f xs) = (xs = [])"; +Goal "([] = map f xs) = (xs = [])"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "Nil_is_map_conv"; @@ -320,42 +287,56 @@ section "rev"; -goal thy "rev(xs@ys) = rev(ys) @ rev(xs)"; +Goal "rev(xs@ys) = rev(ys) @ rev(xs)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "rev_append"; Addsimps[rev_append]; -goal thy "rev(rev l) = l"; +Goal "rev(rev l) = l"; by (induct_tac "l" 1); by (ALLGOALS Asm_simp_tac); qed "rev_rev_ident"; Addsimps[rev_rev_ident]; -goal thy "(rev xs = []) = (xs = [])"; +Goal "(rev xs = []) = (xs = [])"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "rev_is_Nil_conv"; AddIffs [rev_is_Nil_conv]; -goal thy "([] = rev xs) = (xs = [])"; +Goal "([] = rev xs) = (xs = [])"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "Nil_is_rev_conv"; AddIffs [Nil_is_rev_conv]; +val prems = Goal "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P xs"; +by(stac (rev_rev_ident RS sym) 1); +br(read_instantiate [("P","%xs. ?P(rev xs)")]list.induct)1; +by(ALLGOALS Simp_tac); +brs prems 1; +bes prems 1; +qed "rev_induct"; + +Goal "(xs = [] --> P) --> (!ys y. xs = ys@[y] --> P) --> P"; +by(res_inst_tac [("xs","xs")] rev_induct 1); +by(ALLGOALS Asm_simp_tac); +bind_thm ("rev_exhaust", + impI RSN (2,allI RSN (2,allI RSN (2,impI RS (result() RS mp RS mp))))); + (** mem **) section "mem"; -goal thy "x mem (xs@ys) = (x mem xs | x mem ys)"; +Goal "x mem (xs@ys) = (x mem xs | x mem ys)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "mem_append"; Addsimps[mem_append]; -goal thy "x mem [x:xs. P(x)] = (x mem xs & P(x))"; +Goal "x mem [x:xs. P(x)] = (x mem xs & P(x))"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "mem_filter"; @@ -365,48 +346,48 @@ section "set"; -goal thy "set (xs@ys) = (set xs Un set ys)"; +Goal "set (xs@ys) = (set xs Un set ys)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "set_append"; Addsimps[set_append]; -goal thy "(x mem xs) = (x: set xs)"; +Goal "(x mem xs) = (x: set xs)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); by (Blast_tac 1); qed "set_mem_eq"; -goal thy "set l <= set (x#l)"; +Goal "set l <= set (x#l)"; by (Simp_tac 1); by (Blast_tac 1); qed "set_subset_Cons"; -goal thy "(set xs = {}) = (xs = [])"; +Goal "(set xs = {}) = (xs = [])"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "set_empty"; Addsimps [set_empty]; -goal thy "set(rev xs) = set(xs)"; +Goal "set(rev xs) = set(xs)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "set_rev"; Addsimps [set_rev]; -goal thy "set(map f xs) = f``(set xs)"; +Goal "set(map f xs) = f``(set xs)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "set_map"; Addsimps [set_map]; -goal thy "set(map f xs) = f``(set xs)"; +Goal "set(map f xs) = f``(set xs)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "set_map"; Addsimps [set_map]; -goal thy "(x : set(filter P xs)) = (x : set xs & P x)"; +Goal "(x : set(filter P xs)) = (x : set xs & P x)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); by(Blast_tac 1); @@ -418,19 +399,19 @@ section "list_all"; -goal thy "list_all (%x. True) xs = True"; +Goal "list_all (%x. True) xs = True"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "list_all_True"; Addsimps [list_all_True]; -goal thy "list_all p (xs@ys) = (list_all p xs & list_all p ys)"; +Goal "list_all p (xs@ys) = (list_all p xs & list_all p ys)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "list_all_append"; Addsimps [list_all_append]; -goal thy "list_all P xs = (!x. x mem xs --> P(x))"; +Goal "list_all P xs = (!x. x mem xs --> P(x))"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); by (Blast_tac 1); @@ -441,25 +422,25 @@ section "filter"; -goal thy "filter P (xs@ys) = filter P xs @ filter P ys"; +Goal "filter P (xs@ys) = filter P xs @ filter P ys"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "filter_append"; Addsimps [filter_append]; -goal thy "filter (%x. True) xs = xs"; +Goal "filter (%x. True) xs = xs"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "filter_True"; Addsimps [filter_True]; -goal thy "filter (%x. False) xs = []"; +Goal "filter (%x. False) xs = []"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "filter_False"; Addsimps [filter_False]; -goal thy "length (filter P xs) <= length xs"; +Goal "length (filter P xs) <= length xs"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "length_filter"; @@ -469,41 +450,41 @@ section "concat"; -goal thy "concat(xs@ys) = concat(xs)@concat(ys)"; +Goal "concat(xs@ys) = concat(xs)@concat(ys)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed"concat_append"; Addsimps [concat_append]; -goal thy "(concat xss = []) = (!xs:set xss. xs=[])"; +Goal "(concat xss = []) = (!xs:set xss. xs=[])"; by (induct_tac "xss" 1); by (ALLGOALS Asm_simp_tac); qed "concat_eq_Nil_conv"; AddIffs [concat_eq_Nil_conv]; -goal thy "([] = concat xss) = (!xs:set xss. xs=[])"; +Goal "([] = concat xss) = (!xs:set xss. xs=[])"; by (induct_tac "xss" 1); by (ALLGOALS Asm_simp_tac); qed "Nil_eq_concat_conv"; AddIffs [Nil_eq_concat_conv]; -goal thy "set(concat xs) = Union(set `` set xs)"; +Goal "set(concat xs) = Union(set `` set xs)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed"set_concat"; Addsimps [set_concat]; -goal thy "map f (concat xs) = concat (map (map f) xs)"; +Goal "map f (concat xs) = concat (map (map f) xs)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "map_concat"; -goal thy "filter p (concat xs) = concat (map (filter p) xs)"; +Goal "filter p (concat xs) = concat (map (filter p) xs)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed"filter_concat"; -goal thy "rev(concat xs) = concat (map rev (rev xs))"; +Goal "rev(concat xs) = concat (map rev (rev xs))"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "rev_concat"; @@ -512,7 +493,7 @@ section "nth"; -goal thy +Goal "!xs. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"; by (nat_ind_tac "n" 1); by (Asm_simp_tac 1); @@ -521,7 +502,7 @@ by (ALLGOALS Asm_simp_tac); qed_spec_mp "nth_append"; -goal thy "!n. n < length xs --> (map f xs)!n = f(xs!n)"; +Goal "!n. n < length xs --> (map f xs)!n = f(xs!n)"; by (induct_tac "xs" 1); (* case [] *) by (Asm_full_simp_tac 1); @@ -532,7 +513,7 @@ qed_spec_mp "nth_map"; Addsimps [nth_map]; -goal thy "!n. n < length xs --> list_all P xs --> P(xs!n)"; +Goal "!n. n < length xs --> list_all P xs --> P(xs!n)"; by (induct_tac "xs" 1); (* case [] *) by (Simp_tac 1); @@ -542,7 +523,7 @@ by (ALLGOALS Asm_full_simp_tac); qed_spec_mp "list_all_nth"; -goal thy "!n. n < length xs --> xs!n mem xs"; +Goal "!n. n < length xs --> xs!n mem xs"; by (induct_tac "xs" 1); (* case [] *) by (Simp_tac 1); @@ -556,83 +537,43 @@ qed_spec_mp "nth_mem"; Addsimps [nth_mem]; -(** More case analysis and induction **) -section "More case analysis and induction"; - -goal thy "xs ~= [] --> (? ys y. xs = ys@[y])"; -by(res_inst_tac [("xs","xs")] length_induct 1); -by(Clarify_tac 1); -bd (neq_Nil_conv RS iffD1) 1; -by(Clarify_tac 1); -by(rename_tac "ys" 1); -by(case_tac "ys = []" 1); - by(res_inst_tac [("x","[]")] exI 1); - by(Asm_full_simp_tac 1); -by(eres_inst_tac [("x","ys")] allE 1); -by(Asm_full_simp_tac 1); -by(REPEAT(etac exE 1)); -by(rename_tac "zs z" 1); -by(hyp_subst_tac 1); -by(res_inst_tac [("x","y#zs")] exI 1); -by(Simp_tac 1); -qed_spec_mp "neq_Nil_snocD"; - -val prems = goal thy - "[| xs=[] ==> P []; !!ys y. xs=ys@[y] ==> P(ys@[y]) |] ==> P xs"; -by(case_tac "xs = []" 1); - by(Asm_simp_tac 1); - bes prems 1; -bd neq_Nil_snocD 1; -by(REPEAT(etac exE 1)); -by(Asm_simp_tac 1); -bes prems 1; -qed "snoc_eq_cases"; - -val prems = goal thy - "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P(xs)"; -by(res_inst_tac [("xs","xs")] length_induct 1); -by(res_inst_tac [("xs","xs")] snoc_eq_cases 1); - brs prems 1; -by(fast_tac (claset() addIs prems addss simpset()) 1); -qed "snoc_induct"; - (** last & butlast **) -goal thy "last(xs@[x]) = x"; +Goal "last(xs@[x]) = x"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "last_snoc"; Addsimps [last_snoc]; -goal thy "butlast(xs@[x]) = xs"; +Goal "butlast(xs@[x]) = xs"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "butlast_snoc"; Addsimps [butlast_snoc]; -goal thy "length(butlast xs) = length xs - 1"; -by (res_inst_tac [("xs","xs")] snoc_induct 1); +Goal "length(butlast xs) = length xs - 1"; +by (res_inst_tac [("xs","xs")] rev_induct 1); by (ALLGOALS Asm_simp_tac); qed "length_butlast"; Addsimps [length_butlast]; -goal thy +Goal "!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed_spec_mp "butlast_append"; -goal thy "x:set(butlast xs) --> x:set xs"; +Goal "x:set(butlast xs) --> x:set xs"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed_spec_mp "in_set_butlastD"; -goal thy "!!xs. x:set(butlast xs) ==> x:set(butlast(xs@ys))"; +Goal "!!xs. x:set(butlast xs) ==> x:set(butlast(xs@ys))"; by (asm_simp_tac (simpset() addsimps [butlast_append]) 1); by (blast_tac (claset() addDs [in_set_butlastD]) 1); qed "in_set_butlast_appendI1"; -goal thy "!!xs. x:set(butlast ys) ==> x:set(butlast(xs@ys))"; +Goal "!!xs. x:set(butlast ys) ==> x:set(butlast(xs@ys))"; by (asm_simp_tac (simpset() addsimps [butlast_append]) 1); by (Clarify_tac 1); by (Full_simp_tac 1); @@ -641,28 +582,28 @@ (** take & drop **) section "take & drop"; -goal thy "take 0 xs = []"; +Goal "take 0 xs = []"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "take_0"; -goal thy "drop 0 xs = xs"; +Goal "drop 0 xs = xs"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); qed "drop_0"; -goal thy "take (Suc n) (x#xs) = x # take n xs"; +Goal "take (Suc n) (x#xs) = x # take n xs"; by (Simp_tac 1); qed "take_Suc_Cons"; -goal thy "drop (Suc n) (x#xs) = drop n xs"; +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 thy "!xs. length(take n xs) = min (length xs) n"; +Goal "!xs. length(take n xs) = min (length xs) n"; by (nat_ind_tac "n" 1); by (ALLGOALS Asm_simp_tac); by (rtac allI 1); @@ -671,7 +612,7 @@ qed_spec_mp "length_take"; Addsimps [length_take]; -goal thy "!xs. length(drop n xs) = (length xs - n)"; +Goal "!xs. length(drop n xs) = (length xs - n)"; by (nat_ind_tac "n" 1); by (ALLGOALS Asm_simp_tac); by (rtac allI 1); @@ -680,7 +621,7 @@ qed_spec_mp "length_drop"; Addsimps [length_drop]; -goal thy "!xs. length xs <= n --> take n xs = xs"; +Goal "!xs. length xs <= n --> take n xs = xs"; by (nat_ind_tac "n" 1); by (ALLGOALS Asm_simp_tac); by (rtac allI 1); @@ -688,7 +629,7 @@ by (ALLGOALS Asm_simp_tac); qed_spec_mp "take_all"; -goal thy "!xs. length xs <= n --> drop n xs = []"; +Goal "!xs. length xs <= n --> drop n xs = []"; by (nat_ind_tac "n" 1); by (ALLGOALS Asm_simp_tac); by (rtac allI 1); @@ -696,7 +637,7 @@ by (ALLGOALS Asm_simp_tac); qed_spec_mp "drop_all"; -goal thy +Goal "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"; by (nat_ind_tac "n" 1); by (ALLGOALS Asm_simp_tac); @@ -706,7 +647,7 @@ qed_spec_mp "take_append"; Addsimps [take_append]; -goal thy "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; +Goal "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; by (nat_ind_tac "n" 1); by (ALLGOALS Asm_simp_tac); by (rtac allI 1); @@ -715,7 +656,7 @@ qed_spec_mp "drop_append"; Addsimps [drop_append]; -goal thy "!xs n. take n (take m xs) = take (min n m) xs"; +Goal "!xs n. take n (take m xs) = take (min n m) xs"; by (nat_ind_tac "m" 1); by (ALLGOALS Asm_simp_tac); by (rtac allI 1); @@ -726,7 +667,7 @@ by (ALLGOALS Asm_simp_tac); qed_spec_mp "take_take"; -goal thy "!xs. drop n (drop m xs) = drop (n + m) xs"; +Goal "!xs. drop n (drop m xs) = drop (n + m) xs"; by (nat_ind_tac "m" 1); by (ALLGOALS Asm_simp_tac); by (rtac allI 1); @@ -734,7 +675,7 @@ by (ALLGOALS Asm_simp_tac); qed_spec_mp "drop_drop"; -goal thy "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; +Goal "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; by (nat_ind_tac "m" 1); by (ALLGOALS Asm_simp_tac); by (rtac allI 1); @@ -742,7 +683,7 @@ by (ALLGOALS Asm_simp_tac); qed_spec_mp "take_drop"; -goal thy "!xs. take n (map f xs) = map f (take n xs)"; +Goal "!xs. take n (map f xs) = map f (take n xs)"; by (nat_ind_tac "n" 1); by (ALLGOALS Asm_simp_tac); by (rtac allI 1); @@ -750,7 +691,7 @@ by (ALLGOALS Asm_simp_tac); qed_spec_mp "take_map"; -goal thy "!xs. drop n (map f xs) = map f (drop n xs)"; +Goal "!xs. drop n (map f xs) = map f (drop n xs)"; by (nat_ind_tac "n" 1); by (ALLGOALS Asm_simp_tac); by (rtac allI 1); @@ -758,7 +699,7 @@ by (ALLGOALS Asm_simp_tac); qed_spec_mp "drop_map"; -goal thy "!n i. i < n --> (take n xs)!i = xs!i"; +Goal "!n i. i < n --> (take n xs)!i = xs!i"; by (induct_tac "xs" 1); by (ALLGOALS Asm_simp_tac); by (Clarify_tac 1); @@ -769,7 +710,7 @@ qed_spec_mp "nth_take"; Addsimps [nth_take]; -goal thy "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)"; +Goal "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)"; by (nat_ind_tac "n" 1); by (ALLGOALS Asm_simp_tac); by (rtac allI 1); @@ -782,42 +723,39 @@ section "takeWhile & dropWhile"; -goal thy "takeWhile P xs @ dropWhile P xs = xs"; +Goal "takeWhile P xs @ dropWhile P xs = xs"; by (induct_tac "xs" 1); by (ALLGOALS Asm_full_simp_tac); qed "takeWhile_dropWhile_id"; Addsimps [takeWhile_dropWhile_id]; -goal thy "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs"; +Goal "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs"; by (induct_tac "xs" 1); by (ALLGOALS Asm_full_simp_tac); by (Blast_tac 1); bind_thm("takeWhile_append1", conjI RS (result() RS mp)); Addsimps [takeWhile_append1]; -goal thy - "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys"; +Goal "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys"; by (induct_tac "xs" 1); by (ALLGOALS Asm_full_simp_tac); bind_thm("takeWhile_append2", ballI RS (result() RS mp)); Addsimps [takeWhile_append2]; -goal thy - "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"; +Goal "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"; by (induct_tac "xs" 1); by (ALLGOALS Asm_full_simp_tac); by (Blast_tac 1); bind_thm("dropWhile_append1", conjI RS (result() RS mp)); Addsimps [dropWhile_append1]; -goal thy - "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys"; +Goal "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys"; by (induct_tac "xs" 1); by (ALLGOALS Asm_full_simp_tac); bind_thm("dropWhile_append2", ballI RS (result() RS mp)); Addsimps [dropWhile_append2]; -goal thy "x:set(takeWhile P xs) --> x:set xs & P x"; +Goal "x:set(takeWhile P xs) --> x:set xs & P x"; by (induct_tac "xs" 1); by (ALLGOALS Asm_full_simp_tac); qed_spec_mp"set_take_whileD"; @@ -829,19 +767,19 @@ (** nodups & remdups **) section "nodups & remdups"; -goal thy "set(remdups xs) = set xs"; +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 thy "nodups(remdups xs)"; +Goal "nodups(remdups xs)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_full_simp_tac); qed "nodups_remdups"; -goal thy "nodups xs --> nodups (filter P xs)"; +Goal "nodups xs --> nodups (filter P xs)"; by (induct_tac "xs" 1); by (ALLGOALS Asm_full_simp_tac); qed_spec_mp "nodups_filter"; @@ -849,12 +787,12 @@ (** replicate **) section "replicate"; -goal thy "set(replicate (Suc n) x) = {x}"; +Goal "set(replicate (Suc n) x) = {x}"; by (induct_tac "n" 1); by (ALLGOALS Asm_full_simp_tac); val lemma = result(); -goal thy "!!n. n ~= 0 ==> set(replicate n x) = {x}"; +Goal "!!n. n ~= 0 ==> set(replicate n x) = {x}"; by (fast_tac (claset() addSDs [not0_implies_Suc] addSIs [lemma]) 1); qed "set_replicate"; Addsimps [set_replicate];