Cleaned up and simplified etc.
authornipkow
Mon May 18 17:31:49 1998 +0200 (1998-05-18)
changeset 49351694e2daef8f
parent 4934 683eae4b5d0f
child 4936 e67949e15255
Cleaned up and simplified etc.
snoc_induct/exhaust -> rev_induct_exhaust.
src/HOL/List.ML
     1.1 --- a/src/HOL/List.ML	Fri May 15 11:35:56 1998 +0200
     1.2 +++ b/src/HOL/List.ML	Mon May 18 17:31:49 1998 +0200
     1.3 @@ -6,21 +6,21 @@
     1.4  List lemmas
     1.5  *)
     1.6  
     1.7 -goal thy "!x. xs ~= x#xs";
     1.8 +Goal "!x. xs ~= x#xs";
     1.9  by (induct_tac "xs" 1);
    1.10  by (ALLGOALS Asm_simp_tac);
    1.11  qed_spec_mp "not_Cons_self";
    1.12  bind_thm("not_Cons_self2",not_Cons_self RS not_sym);
    1.13  Addsimps [not_Cons_self,not_Cons_self2];
    1.14  
    1.15 -goal thy "(xs ~= []) = (? y ys. xs = y#ys)";
    1.16 +Goal "(xs ~= []) = (? y ys. xs = y#ys)";
    1.17  by (induct_tac "xs" 1);
    1.18  by (Simp_tac 1);
    1.19  by (Asm_simp_tac 1);
    1.20  qed "neq_Nil_conv";
    1.21  
    1.22  (* Induction over the length of a list: *)
    1.23 -val [prem] = goal thy
    1.24 +val [prem] = Goal
    1.25    "(!!xs. (!ys. length ys < length xs --> P ys) ==> P xs) ==> P(xs)";
    1.26  by(rtac measure_induct 1 THEN etac prem 1);
    1.27  qed "length_induct";
    1.28 @@ -28,7 +28,7 @@
    1.29  
    1.30  (** "lists": the list-forming operator over sets **)
    1.31  
    1.32 -goalw thy lists.defs "!!A B. A<=B ==> lists A <= lists B";
    1.33 +Goalw lists.defs "!!A B. A<=B ==> lists A <= lists B";
    1.34  by (rtac lfp_mono 1);
    1.35  by (REPEAT (ares_tac basic_monos 1));
    1.36  qed "lists_mono";
    1.37 @@ -37,12 +37,12 @@
    1.38  AddSEs [listsE];
    1.39  AddSIs lists.intrs;
    1.40  
    1.41 -goal thy "!!l. l: lists A ==> l: lists B --> l: lists (A Int B)";
    1.42 +Goal "!!l. l: lists A ==> l: lists B --> l: lists (A Int B)";
    1.43  by (etac lists.induct 1);
    1.44  by (ALLGOALS Blast_tac);
    1.45  qed_spec_mp "lists_IntI";
    1.46  
    1.47 -goal thy "lists (A Int B) = lists A Int lists B";
    1.48 +Goal "lists (A Int B) = lists A Int lists B";
    1.49  by (rtac (mono_Int RS equalityI) 1);
    1.50  by (simp_tac (simpset() addsimps [mono_def, lists_mono]) 1);
    1.51  by (blast_tac (claset() addSIs [lists_IntI]) 1);
    1.52 @@ -53,12 +53,12 @@
    1.53  (**  Case analysis **)
    1.54  section "Case analysis";
    1.55  
    1.56 -val prems = goal thy "[| P([]); !!x xs. P(x#xs) |] ==> P(xs)";
    1.57 +val prems = Goal "[| P([]); !!x xs. P(x#xs) |] ==> P(xs)";
    1.58  by (induct_tac "xs" 1);
    1.59  by (REPEAT(resolve_tac prems 1));
    1.60  qed "list_cases";
    1.61  
    1.62 -goal thy  "(xs=[] --> P([])) & (!y ys. xs=y#ys --> P(y#ys)) --> P(xs)";
    1.63 +Goal "(xs=[] --> P([])) & (!y ys. xs=y#ys --> P(y#ys)) --> P(xs)";
    1.64  by (induct_tac "xs" 1);
    1.65  by (Blast_tac 1);
    1.66  by (Blast_tac 1);
    1.67 @@ -70,43 +70,43 @@
    1.68  
    1.69  section "length";
    1.70  
    1.71 -goal thy "length(xs@ys) = length(xs)+length(ys)";
    1.72 +Goal "length(xs@ys) = length(xs)+length(ys)";
    1.73  by (induct_tac "xs" 1);
    1.74  by (ALLGOALS Asm_simp_tac);
    1.75  qed"length_append";
    1.76  Addsimps [length_append];
    1.77  
    1.78 -goal thy "length (map f l) = length l";
    1.79 +Goal "length (map f l) = length l";
    1.80  by (induct_tac "l" 1);
    1.81  by (ALLGOALS Simp_tac);
    1.82  qed "length_map";
    1.83  Addsimps [length_map];
    1.84  
    1.85 -goal thy "length(rev xs) = length(xs)";
    1.86 +Goal "length(rev xs) = length(xs)";
    1.87  by (induct_tac "xs" 1);
    1.88  by (ALLGOALS Asm_simp_tac);
    1.89  qed "length_rev";
    1.90  Addsimps [length_rev];
    1.91  
    1.92 -goal List.thy "!!xs. xs ~= [] ==> length(tl xs) = (length xs) - 1";
    1.93 +Goal "!!xs. xs ~= [] ==> length(tl xs) = (length xs) - 1";
    1.94  by (exhaust_tac "xs" 1);
    1.95  by (ALLGOALS Asm_full_simp_tac);
    1.96  qed "length_tl";
    1.97  Addsimps [length_tl];
    1.98  
    1.99 -goal thy "(length xs = 0) = (xs = [])";
   1.100 +Goal "(length xs = 0) = (xs = [])";
   1.101  by (induct_tac "xs" 1);
   1.102  by (ALLGOALS Asm_simp_tac);
   1.103  qed "length_0_conv";
   1.104  AddIffs [length_0_conv];
   1.105  
   1.106 -goal thy "(0 = length xs) = (xs = [])";
   1.107 +Goal "(0 = length xs) = (xs = [])";
   1.108  by (induct_tac "xs" 1);
   1.109  by (ALLGOALS Asm_simp_tac);
   1.110  qed "zero_length_conv";
   1.111  AddIffs [zero_length_conv];
   1.112  
   1.113 -goal thy "(0 < length xs) = (xs ~= [])";
   1.114 +Goal "(0 < length xs) = (xs ~= [])";
   1.115  by (induct_tac "xs" 1);
   1.116  by (ALLGOALS Asm_simp_tac);
   1.117  qed "length_greater_0_conv";
   1.118 @@ -116,44 +116,44 @@
   1.119  
   1.120  section "@ - append";
   1.121  
   1.122 -goal thy "(xs@ys)@zs = xs@(ys@zs)";
   1.123 +Goal "(xs@ys)@zs = xs@(ys@zs)";
   1.124  by (induct_tac "xs" 1);
   1.125  by (ALLGOALS Asm_simp_tac);
   1.126  qed "append_assoc";
   1.127  Addsimps [append_assoc];
   1.128  
   1.129 -goal thy "xs @ [] = xs";
   1.130 +Goal "xs @ [] = xs";
   1.131  by (induct_tac "xs" 1);
   1.132  by (ALLGOALS Asm_simp_tac);
   1.133  qed "append_Nil2";
   1.134  Addsimps [append_Nil2];
   1.135  
   1.136 -goal thy "(xs@ys = []) = (xs=[] & ys=[])";
   1.137 +Goal "(xs@ys = []) = (xs=[] & ys=[])";
   1.138  by (induct_tac "xs" 1);
   1.139  by (ALLGOALS Asm_simp_tac);
   1.140  qed "append_is_Nil_conv";
   1.141  AddIffs [append_is_Nil_conv];
   1.142  
   1.143 -goal thy "([] = xs@ys) = (xs=[] & ys=[])";
   1.144 +Goal "([] = xs@ys) = (xs=[] & ys=[])";
   1.145  by (induct_tac "xs" 1);
   1.146  by (ALLGOALS Asm_simp_tac);
   1.147  by (Blast_tac 1);
   1.148  qed "Nil_is_append_conv";
   1.149  AddIffs [Nil_is_append_conv];
   1.150  
   1.151 -goal thy "(xs @ ys = xs) = (ys=[])";
   1.152 +Goal "(xs @ ys = xs) = (ys=[])";
   1.153  by (induct_tac "xs" 1);
   1.154  by (ALLGOALS Asm_simp_tac);
   1.155  qed "append_self_conv";
   1.156  
   1.157 -goal thy "(xs = xs @ ys) = (ys=[])";
   1.158 +Goal "(xs = xs @ ys) = (ys=[])";
   1.159  by (induct_tac "xs" 1);
   1.160  by (ALLGOALS Asm_simp_tac);
   1.161  by (Blast_tac 1);
   1.162  qed "self_append_conv";
   1.163  AddIffs [append_self_conv,self_append_conv];
   1.164  
   1.165 -goal thy "!ys. length xs = length ys | length us = length vs \
   1.166 +Goal "!ys. length xs = length ys | length us = length vs \
   1.167  \              --> (xs@us = ys@vs) = (xs=ys & us=vs)";
   1.168  by (induct_tac "xs" 1);
   1.169   by (rtac allI 1);
   1.170 @@ -169,15 +169,15 @@
   1.171  qed_spec_mp "append_eq_append_conv";
   1.172  Addsimps [append_eq_append_conv];
   1.173  
   1.174 -goal thy "(xs @ ys = xs @ zs) = (ys=zs)";
   1.175 +Goal "(xs @ ys = xs @ zs) = (ys=zs)";
   1.176  by (Simp_tac 1);
   1.177  qed "same_append_eq";
   1.178  
   1.179 -goal thy "(xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; 
   1.180 +Goal "(xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; 
   1.181  by (Simp_tac 1);
   1.182  qed "append1_eq_conv";
   1.183  
   1.184 -goal thy "(ys @ xs = zs @ xs) = (ys=zs)";
   1.185 +Goal "(ys @ xs = zs @ xs) = (ys=zs)";
   1.186  by (Simp_tac 1);
   1.187  qed "append_same_eq";
   1.188  
   1.189 @@ -186,115 +186,82 @@
   1.190  AddSDs
   1.191   [same_append_eq RS iffD1, append1_eq_conv RS iffD1, append_same_eq RS iffD1];
   1.192  
   1.193 -goal thy "(xs @ ys = ys) = (xs=[])";
   1.194 +Goal "(xs @ ys = ys) = (xs=[])";
   1.195  by(cut_inst_tac [("zs","[]")] append_same_eq 1);
   1.196  by(Asm_full_simp_tac 1);
   1.197  qed "append_self_conv2";
   1.198  
   1.199 -goal thy "(ys = xs @ ys) = (xs=[])";
   1.200 +Goal "(ys = xs @ ys) = (xs=[])";
   1.201  by(simp_tac (simpset() addsimps
   1.202       [simplify (simpset()) (read_instantiate[("ys","[]")]append_same_eq)]) 1);
   1.203  by(Blast_tac 1);
   1.204  qed "self_append_conv2";
   1.205  AddIffs [append_self_conv2,self_append_conv2];
   1.206  
   1.207 -goal thy "xs ~= [] --> hd xs # tl xs = xs";
   1.208 +Goal "xs ~= [] --> hd xs # tl xs = xs";
   1.209  by (induct_tac "xs" 1);
   1.210  by (ALLGOALS Asm_simp_tac);
   1.211  qed_spec_mp "hd_Cons_tl";
   1.212  Addsimps [hd_Cons_tl];
   1.213  
   1.214 -goal thy "hd(xs@ys) = (if xs=[] then hd ys else hd xs)";
   1.215 +Goal "hd(xs@ys) = (if xs=[] then hd ys else hd xs)";
   1.216  by (induct_tac "xs" 1);
   1.217  by (ALLGOALS Asm_simp_tac);
   1.218  qed "hd_append";
   1.219  
   1.220 -goal thy "!!xs. xs ~= [] ==> hd(xs @ ys) = hd xs";
   1.221 +Goal "!!xs. xs ~= [] ==> hd(xs @ ys) = hd xs";
   1.222  by (asm_simp_tac (simpset() addsimps [hd_append]
   1.223                             addsplits [split_list_case]) 1);
   1.224  qed "hd_append2";
   1.225  Addsimps [hd_append2];
   1.226  
   1.227 -goal thy "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)";
   1.228 +Goal "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)";
   1.229  by (simp_tac (simpset() addsplits [split_list_case]) 1);
   1.230  qed "tl_append";
   1.231  
   1.232 -goal thy "!!xs. xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys";
   1.233 +Goal "!!xs. xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys";
   1.234  by (asm_simp_tac (simpset() addsimps [tl_append]
   1.235                             addsplits [split_list_case]) 1);
   1.236  qed "tl_append2";
   1.237  Addsimps [tl_append2];
   1.238  
   1.239  
   1.240 -(** Snoc exhaustion and induction **)
   1.241 -section "Snoc exhaustion and induction";
   1.242 -
   1.243 -goal thy "xs ~= [] --> (? ys y. xs = ys@[y])";
   1.244 -by(induct_tac "xs" 1);
   1.245 -by(Simp_tac 1);
   1.246 -by(exhaust_tac "list" 1);
   1.247 - by(Asm_simp_tac 1);
   1.248 - by(res_inst_tac [("x","[]")] exI 1);
   1.249 - by(Simp_tac 1);
   1.250 -by(Asm_full_simp_tac 1);
   1.251 -by(Clarify_tac 1);
   1.252 -by(res_inst_tac [("x","a#ys")] exI 1);
   1.253 -by(Asm_simp_tac 1);
   1.254 -val lemma = result();
   1.255 -
   1.256 -goal thy  "(xs = [] --> P) -->  (!ys y. xs = ys@[y] --> P) --> P";
   1.257 -by(cut_facts_tac [lemma] 1);
   1.258 -by(Blast_tac 1);
   1.259 -bind_thm ("snoc_exhaust",
   1.260 -  impI RSN (2,allI RSN (2,allI RSN (2,impI RS (result() RS mp RS mp)))));
   1.261 -
   1.262 -val prems = goal thy "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P xs";
   1.263 -by(res_inst_tac [("xs","xs")] length_induct 1);
   1.264 -by(res_inst_tac [("xs","xs")] snoc_exhaust 1);
   1.265 - by(Clarify_tac 1);
   1.266 - brs prems 1;
   1.267 -by(Clarify_tac 1);
   1.268 -brs prems 1;
   1.269 -auto();
   1.270 -qed "snoc_induct";
   1.271 -
   1.272 -
   1.273  (** map **)
   1.274  
   1.275  section "map";
   1.276  
   1.277 -goal thy
   1.278 +Goal
   1.279    "(!x. x : set xs --> f x = g x) --> map f xs = map g xs";
   1.280  by (induct_tac "xs" 1);
   1.281  by (ALLGOALS Asm_simp_tac);
   1.282  bind_thm("map_ext", impI RS (allI RS (result() RS mp)));
   1.283  
   1.284 -goal thy "map (%x. x) = (%xs. xs)";
   1.285 +Goal "map (%x. x) = (%xs. xs)";
   1.286  by (rtac ext 1);
   1.287  by (induct_tac "xs" 1);
   1.288  by (ALLGOALS Asm_simp_tac);
   1.289  qed "map_ident";
   1.290  Addsimps[map_ident];
   1.291  
   1.292 -goal thy "map f (xs@ys) = map f xs @ map f ys";
   1.293 +Goal "map f (xs@ys) = map f xs @ map f ys";
   1.294  by (induct_tac "xs" 1);
   1.295  by (ALLGOALS Asm_simp_tac);
   1.296  qed "map_append";
   1.297  Addsimps[map_append];
   1.298  
   1.299 -goalw thy [o_def] "map (f o g) xs = map f (map g xs)";
   1.300 +Goalw [o_def] "map (f o g) xs = map f (map g xs)";
   1.301  by (induct_tac "xs" 1);
   1.302  by (ALLGOALS Asm_simp_tac);
   1.303  qed "map_compose";
   1.304  Addsimps[map_compose];
   1.305  
   1.306 -goal thy "rev(map f xs) = map f (rev xs)";
   1.307 +Goal "rev(map f xs) = map f (rev xs)";
   1.308  by (induct_tac "xs" 1);
   1.309  by (ALLGOALS Asm_simp_tac);
   1.310  qed "rev_map";
   1.311  
   1.312  (* a congruence rule for map: *)
   1.313 -goal thy
   1.314 +Goal
   1.315   "(xs=ys) --> (!x. x : set ys --> f x = g x) --> map f xs = map g ys";
   1.316  by (rtac impI 1);
   1.317  by (hyp_subst_tac 1);
   1.318 @@ -303,13 +270,13 @@
   1.319  val lemma = result();
   1.320  bind_thm("map_cong",impI RSN (2,allI RSN (2,lemma RS mp RS mp)));
   1.321  
   1.322 -goal List.thy "(map f xs = []) = (xs = [])";
   1.323 +Goal "(map f xs = []) = (xs = [])";
   1.324  by (induct_tac "xs" 1);
   1.325  by (ALLGOALS Asm_simp_tac);
   1.326  qed "map_is_Nil_conv";
   1.327  AddIffs [map_is_Nil_conv];
   1.328  
   1.329 -goal List.thy "([] = map f xs) = (xs = [])";
   1.330 +Goal "([] = map f xs) = (xs = [])";
   1.331  by (induct_tac "xs" 1);
   1.332  by (ALLGOALS Asm_simp_tac);
   1.333  qed "Nil_is_map_conv";
   1.334 @@ -320,42 +287,56 @@
   1.335  
   1.336  section "rev";
   1.337  
   1.338 -goal thy "rev(xs@ys) = rev(ys) @ rev(xs)";
   1.339 +Goal "rev(xs@ys) = rev(ys) @ rev(xs)";
   1.340  by (induct_tac "xs" 1);
   1.341  by (ALLGOALS Asm_simp_tac);
   1.342  qed "rev_append";
   1.343  Addsimps[rev_append];
   1.344  
   1.345 -goal thy "rev(rev l) = l";
   1.346 +Goal "rev(rev l) = l";
   1.347  by (induct_tac "l" 1);
   1.348  by (ALLGOALS Asm_simp_tac);
   1.349  qed "rev_rev_ident";
   1.350  Addsimps[rev_rev_ident];
   1.351  
   1.352 -goal thy "(rev xs = []) = (xs = [])";
   1.353 +Goal "(rev xs = []) = (xs = [])";
   1.354  by (induct_tac "xs" 1);
   1.355  by (ALLGOALS Asm_simp_tac);
   1.356  qed "rev_is_Nil_conv";
   1.357  AddIffs [rev_is_Nil_conv];
   1.358  
   1.359 -goal thy "([] = rev xs) = (xs = [])";
   1.360 +Goal "([] = rev xs) = (xs = [])";
   1.361  by (induct_tac "xs" 1);
   1.362  by (ALLGOALS Asm_simp_tac);
   1.363  qed "Nil_is_rev_conv";
   1.364  AddIffs [Nil_is_rev_conv];
   1.365  
   1.366 +val prems = Goal "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P xs";
   1.367 +by(stac (rev_rev_ident RS sym) 1);
   1.368 +br(read_instantiate [("P","%xs. ?P(rev xs)")]list.induct)1;
   1.369 +by(ALLGOALS Simp_tac);
   1.370 +brs prems 1;
   1.371 +bes prems 1;
   1.372 +qed "rev_induct";
   1.373 +
   1.374 +Goal  "(xs = [] --> P) -->  (!ys y. xs = ys@[y] --> P) --> P";
   1.375 +by(res_inst_tac [("xs","xs")] rev_induct 1);
   1.376 +by(ALLGOALS Asm_simp_tac);
   1.377 +bind_thm ("rev_exhaust",
   1.378 +  impI RSN (2,allI RSN (2,allI RSN (2,impI RS (result() RS mp RS mp)))));
   1.379 +
   1.380  
   1.381  (** mem **)
   1.382  
   1.383  section "mem";
   1.384  
   1.385 -goal thy "x mem (xs@ys) = (x mem xs | x mem ys)";
   1.386 +Goal "x mem (xs@ys) = (x mem xs | x mem ys)";
   1.387  by (induct_tac "xs" 1);
   1.388  by (ALLGOALS Asm_simp_tac);
   1.389  qed "mem_append";
   1.390  Addsimps[mem_append];
   1.391  
   1.392 -goal thy "x mem [x:xs. P(x)] = (x mem xs & P(x))";
   1.393 +Goal "x mem [x:xs. P(x)] = (x mem xs & P(x))";
   1.394  by (induct_tac "xs" 1);
   1.395  by (ALLGOALS Asm_simp_tac);
   1.396  qed "mem_filter";
   1.397 @@ -365,48 +346,48 @@
   1.398  
   1.399  section "set";
   1.400  
   1.401 -goal thy "set (xs@ys) = (set xs Un set ys)";
   1.402 +Goal "set (xs@ys) = (set xs Un set ys)";
   1.403  by (induct_tac "xs" 1);
   1.404  by (ALLGOALS Asm_simp_tac);
   1.405  qed "set_append";
   1.406  Addsimps[set_append];
   1.407  
   1.408 -goal thy "(x mem xs) = (x: set xs)";
   1.409 +Goal "(x mem xs) = (x: set xs)";
   1.410  by (induct_tac "xs" 1);
   1.411  by (ALLGOALS Asm_simp_tac);
   1.412  by (Blast_tac 1);
   1.413  qed "set_mem_eq";
   1.414  
   1.415 -goal thy "set l <= set (x#l)";
   1.416 +Goal "set l <= set (x#l)";
   1.417  by (Simp_tac 1);
   1.418  by (Blast_tac 1);
   1.419  qed "set_subset_Cons";
   1.420  
   1.421 -goal thy "(set xs = {}) = (xs = [])";
   1.422 +Goal "(set xs = {}) = (xs = [])";
   1.423  by (induct_tac "xs" 1);
   1.424  by (ALLGOALS Asm_simp_tac);
   1.425  qed "set_empty";
   1.426  Addsimps [set_empty];
   1.427  
   1.428 -goal thy "set(rev xs) = set(xs)";
   1.429 +Goal "set(rev xs) = set(xs)";
   1.430  by (induct_tac "xs" 1);
   1.431  by (ALLGOALS Asm_simp_tac);
   1.432  qed "set_rev";
   1.433  Addsimps [set_rev];
   1.434  
   1.435 -goal thy "set(map f xs) = f``(set xs)";
   1.436 +Goal "set(map f xs) = f``(set xs)";
   1.437  by (induct_tac "xs" 1);
   1.438  by (ALLGOALS Asm_simp_tac);
   1.439  qed "set_map";
   1.440  Addsimps [set_map];
   1.441  
   1.442 -goal thy "set(map f xs) = f``(set xs)";
   1.443 +Goal "set(map f xs) = f``(set xs)";
   1.444  by (induct_tac "xs" 1);
   1.445  by (ALLGOALS Asm_simp_tac);
   1.446  qed "set_map";
   1.447  Addsimps [set_map];
   1.448  
   1.449 -goal thy "(x : set(filter P xs)) = (x : set xs & P x)";
   1.450 +Goal "(x : set(filter P xs)) = (x : set xs & P x)";
   1.451  by (induct_tac "xs" 1);
   1.452  by (ALLGOALS Asm_simp_tac);
   1.453  by(Blast_tac 1);
   1.454 @@ -418,19 +399,19 @@
   1.455  
   1.456  section "list_all";
   1.457  
   1.458 -goal thy "list_all (%x. True) xs = True";
   1.459 +Goal "list_all (%x. True) xs = True";
   1.460  by (induct_tac "xs" 1);
   1.461  by (ALLGOALS Asm_simp_tac);
   1.462  qed "list_all_True";
   1.463  Addsimps [list_all_True];
   1.464  
   1.465 -goal thy "list_all p (xs@ys) = (list_all p xs & list_all p ys)";
   1.466 +Goal "list_all p (xs@ys) = (list_all p xs & list_all p ys)";
   1.467  by (induct_tac "xs" 1);
   1.468  by (ALLGOALS Asm_simp_tac);
   1.469  qed "list_all_append";
   1.470  Addsimps [list_all_append];
   1.471  
   1.472 -goal thy "list_all P xs = (!x. x mem xs --> P(x))";
   1.473 +Goal "list_all P xs = (!x. x mem xs --> P(x))";
   1.474  by (induct_tac "xs" 1);
   1.475  by (ALLGOALS Asm_simp_tac);
   1.476  by (Blast_tac 1);
   1.477 @@ -441,25 +422,25 @@
   1.478  
   1.479  section "filter";
   1.480  
   1.481 -goal thy "filter P (xs@ys) = filter P xs @ filter P ys";
   1.482 +Goal "filter P (xs@ys) = filter P xs @ filter P ys";
   1.483  by (induct_tac "xs" 1);
   1.484  by (ALLGOALS Asm_simp_tac);
   1.485  qed "filter_append";
   1.486  Addsimps [filter_append];
   1.487  
   1.488 -goal thy "filter (%x. True) xs = xs";
   1.489 +Goal "filter (%x. True) xs = xs";
   1.490  by (induct_tac "xs" 1);
   1.491  by (ALLGOALS Asm_simp_tac);
   1.492  qed "filter_True";
   1.493  Addsimps [filter_True];
   1.494  
   1.495 -goal thy "filter (%x. False) xs = []";
   1.496 +Goal "filter (%x. False) xs = []";
   1.497  by (induct_tac "xs" 1);
   1.498  by (ALLGOALS Asm_simp_tac);
   1.499  qed "filter_False";
   1.500  Addsimps [filter_False];
   1.501  
   1.502 -goal thy "length (filter P xs) <= length xs";
   1.503 +Goal "length (filter P xs) <= length xs";
   1.504  by (induct_tac "xs" 1);
   1.505  by (ALLGOALS Asm_simp_tac);
   1.506  qed "length_filter";
   1.507 @@ -469,41 +450,41 @@
   1.508  
   1.509  section "concat";
   1.510  
   1.511 -goal thy  "concat(xs@ys) = concat(xs)@concat(ys)";
   1.512 +Goal  "concat(xs@ys) = concat(xs)@concat(ys)";
   1.513  by (induct_tac "xs" 1);
   1.514  by (ALLGOALS Asm_simp_tac);
   1.515  qed"concat_append";
   1.516  Addsimps [concat_append];
   1.517  
   1.518 -goal thy "(concat xss = []) = (!xs:set xss. xs=[])";
   1.519 +Goal "(concat xss = []) = (!xs:set xss. xs=[])";
   1.520  by (induct_tac "xss" 1);
   1.521  by (ALLGOALS Asm_simp_tac);
   1.522  qed "concat_eq_Nil_conv";
   1.523  AddIffs [concat_eq_Nil_conv];
   1.524  
   1.525 -goal thy "([] = concat xss) = (!xs:set xss. xs=[])";
   1.526 +Goal "([] = concat xss) = (!xs:set xss. xs=[])";
   1.527  by (induct_tac "xss" 1);
   1.528  by (ALLGOALS Asm_simp_tac);
   1.529  qed "Nil_eq_concat_conv";
   1.530  AddIffs [Nil_eq_concat_conv];
   1.531  
   1.532 -goal thy  "set(concat xs) = Union(set `` set xs)";
   1.533 +Goal  "set(concat xs) = Union(set `` set xs)";
   1.534  by (induct_tac "xs" 1);
   1.535  by (ALLGOALS Asm_simp_tac);
   1.536  qed"set_concat";
   1.537  Addsimps [set_concat];
   1.538  
   1.539 -goal thy "map f (concat xs) = concat (map (map f) xs)"; 
   1.540 +Goal "map f (concat xs) = concat (map (map f) xs)"; 
   1.541  by (induct_tac "xs" 1);
   1.542  by (ALLGOALS Asm_simp_tac);
   1.543  qed "map_concat";
   1.544  
   1.545 -goal thy "filter p (concat xs) = concat (map (filter p) xs)"; 
   1.546 +Goal "filter p (concat xs) = concat (map (filter p) xs)"; 
   1.547  by (induct_tac "xs" 1);
   1.548  by (ALLGOALS Asm_simp_tac);
   1.549  qed"filter_concat"; 
   1.550  
   1.551 -goal thy "rev(concat xs) = concat (map rev (rev xs))";
   1.552 +Goal "rev(concat xs) = concat (map rev (rev xs))";
   1.553  by (induct_tac "xs" 1);
   1.554  by (ALLGOALS Asm_simp_tac);
   1.555  qed "rev_concat";
   1.556 @@ -512,7 +493,7 @@
   1.557  
   1.558  section "nth";
   1.559  
   1.560 -goal thy
   1.561 +Goal
   1.562    "!xs. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))";
   1.563  by (nat_ind_tac "n" 1);
   1.564   by (Asm_simp_tac 1);
   1.565 @@ -521,7 +502,7 @@
   1.566    by (ALLGOALS Asm_simp_tac);
   1.567  qed_spec_mp "nth_append";
   1.568  
   1.569 -goal thy "!n. n < length xs --> (map f xs)!n = f(xs!n)";
   1.570 +Goal "!n. n < length xs --> (map f xs)!n = f(xs!n)";
   1.571  by (induct_tac "xs" 1);
   1.572  (* case [] *)
   1.573  by (Asm_full_simp_tac 1);
   1.574 @@ -532,7 +513,7 @@
   1.575  qed_spec_mp "nth_map";
   1.576  Addsimps [nth_map];
   1.577  
   1.578 -goal thy "!n. n < length xs --> list_all P xs --> P(xs!n)";
   1.579 +Goal "!n. n < length xs --> list_all P xs --> P(xs!n)";
   1.580  by (induct_tac "xs" 1);
   1.581  (* case [] *)
   1.582  by (Simp_tac 1);
   1.583 @@ -542,7 +523,7 @@
   1.584  by (ALLGOALS Asm_full_simp_tac);
   1.585  qed_spec_mp "list_all_nth";
   1.586  
   1.587 -goal thy "!n. n < length xs --> xs!n mem xs";
   1.588 +Goal "!n. n < length xs --> xs!n mem xs";
   1.589  by (induct_tac "xs" 1);
   1.590  (* case [] *)
   1.591  by (Simp_tac 1);
   1.592 @@ -556,83 +537,43 @@
   1.593  qed_spec_mp "nth_mem";
   1.594  Addsimps [nth_mem];
   1.595  
   1.596 -(**  More case analysis and induction **)
   1.597 -section "More case analysis and induction";
   1.598 -
   1.599 -goal thy "xs ~= [] --> (? ys y. xs = ys@[y])";
   1.600 -by(res_inst_tac [("xs","xs")] length_induct 1);
   1.601 -by(Clarify_tac 1);
   1.602 -bd (neq_Nil_conv RS iffD1) 1;
   1.603 -by(Clarify_tac 1);
   1.604 -by(rename_tac "ys" 1);
   1.605 -by(case_tac "ys = []" 1);
   1.606 - by(res_inst_tac [("x","[]")] exI 1);
   1.607 - by(Asm_full_simp_tac 1);
   1.608 -by(eres_inst_tac [("x","ys")] allE 1);
   1.609 -by(Asm_full_simp_tac 1);
   1.610 -by(REPEAT(etac exE 1));
   1.611 -by(rename_tac "zs z" 1);
   1.612 -by(hyp_subst_tac 1);
   1.613 -by(res_inst_tac [("x","y#zs")] exI 1);
   1.614 -by(Simp_tac 1);
   1.615 -qed_spec_mp "neq_Nil_snocD";
   1.616 -
   1.617 -val prems = goal thy
   1.618 -  "[| xs=[] ==> P []; !!ys y. xs=ys@[y] ==> P(ys@[y]) |] ==> P xs";
   1.619 -by(case_tac "xs = []" 1);
   1.620 - by(Asm_simp_tac 1);
   1.621 - bes prems 1;
   1.622 -bd neq_Nil_snocD 1;
   1.623 -by(REPEAT(etac exE 1));
   1.624 -by(Asm_simp_tac 1);
   1.625 -bes prems 1;
   1.626 -qed "snoc_eq_cases";
   1.627 -
   1.628 -val prems = goal thy
   1.629 -  "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P(xs)";
   1.630 -by(res_inst_tac [("xs","xs")] length_induct 1);
   1.631 -by(res_inst_tac [("xs","xs")] snoc_eq_cases 1);
   1.632 - brs prems 1;
   1.633 -by(fast_tac (claset() addIs prems addss simpset()) 1);
   1.634 -qed "snoc_induct";
   1.635 -
   1.636  (** last & butlast **)
   1.637  
   1.638 -goal thy "last(xs@[x]) = x";
   1.639 +Goal "last(xs@[x]) = x";
   1.640  by (induct_tac "xs" 1);
   1.641  by (ALLGOALS Asm_simp_tac);
   1.642  qed "last_snoc";
   1.643  Addsimps [last_snoc];
   1.644  
   1.645 -goal thy "butlast(xs@[x]) = xs";
   1.646 +Goal "butlast(xs@[x]) = xs";
   1.647  by (induct_tac "xs" 1);
   1.648  by (ALLGOALS Asm_simp_tac);
   1.649  qed "butlast_snoc";
   1.650  Addsimps [butlast_snoc];
   1.651  
   1.652 -goal thy "length(butlast xs) = length xs - 1";
   1.653 -by (res_inst_tac [("xs","xs")] snoc_induct 1);
   1.654 +Goal "length(butlast xs) = length xs - 1";
   1.655 +by (res_inst_tac [("xs","xs")] rev_induct 1);
   1.656  by (ALLGOALS Asm_simp_tac);
   1.657  qed "length_butlast";
   1.658  Addsimps [length_butlast];
   1.659  
   1.660 -goal thy
   1.661 +Goal
   1.662    "!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)";
   1.663  by (induct_tac "xs" 1);
   1.664  by (ALLGOALS Asm_simp_tac);
   1.665  qed_spec_mp "butlast_append";
   1.666  
   1.667 -goal thy "x:set(butlast xs) --> x:set xs";
   1.668 +Goal "x:set(butlast xs) --> x:set xs";
   1.669  by (induct_tac "xs" 1);
   1.670  by (ALLGOALS Asm_simp_tac);
   1.671  qed_spec_mp "in_set_butlastD";
   1.672  
   1.673 -goal thy "!!xs. x:set(butlast xs) ==> x:set(butlast(xs@ys))";
   1.674 +Goal "!!xs. x:set(butlast xs) ==> x:set(butlast(xs@ys))";
   1.675  by (asm_simp_tac (simpset() addsimps [butlast_append]) 1);
   1.676  by (blast_tac (claset() addDs [in_set_butlastD]) 1);
   1.677  qed "in_set_butlast_appendI1";
   1.678  
   1.679 -goal thy "!!xs. x:set(butlast ys) ==> x:set(butlast(xs@ys))";
   1.680 +Goal "!!xs. x:set(butlast ys) ==> x:set(butlast(xs@ys))";
   1.681  by (asm_simp_tac (simpset() addsimps [butlast_append]) 1);
   1.682  by (Clarify_tac 1);
   1.683  by (Full_simp_tac 1);
   1.684 @@ -641,28 +582,28 @@
   1.685  (** take  & drop **)
   1.686  section "take & drop";
   1.687  
   1.688 -goal thy "take 0 xs = []";
   1.689 +Goal "take 0 xs = []";
   1.690  by (induct_tac "xs" 1);
   1.691  by (ALLGOALS Asm_simp_tac);
   1.692  qed "take_0";
   1.693  
   1.694 -goal thy "drop 0 xs = xs";
   1.695 +Goal "drop 0 xs = xs";
   1.696  by (induct_tac "xs" 1);
   1.697  by (ALLGOALS Asm_simp_tac);
   1.698  qed "drop_0";
   1.699  
   1.700 -goal thy "take (Suc n) (x#xs) = x # take n xs";
   1.701 +Goal "take (Suc n) (x#xs) = x # take n xs";
   1.702  by (Simp_tac 1);
   1.703  qed "take_Suc_Cons";
   1.704  
   1.705 -goal thy "drop (Suc n) (x#xs) = drop n xs";
   1.706 +Goal "drop (Suc n) (x#xs) = drop n xs";
   1.707  by (Simp_tac 1);
   1.708  qed "drop_Suc_Cons";
   1.709  
   1.710  Delsimps [take_Cons,drop_Cons];
   1.711  Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons];
   1.712  
   1.713 -goal thy "!xs. length(take n xs) = min (length xs) n";
   1.714 +Goal "!xs. length(take n xs) = min (length xs) n";
   1.715  by (nat_ind_tac "n" 1);
   1.716   by (ALLGOALS Asm_simp_tac);
   1.717  by (rtac allI 1);
   1.718 @@ -671,7 +612,7 @@
   1.719  qed_spec_mp "length_take";
   1.720  Addsimps [length_take];
   1.721  
   1.722 -goal thy "!xs. length(drop n xs) = (length xs - n)";
   1.723 +Goal "!xs. length(drop n xs) = (length xs - n)";
   1.724  by (nat_ind_tac "n" 1);
   1.725   by (ALLGOALS Asm_simp_tac);
   1.726  by (rtac allI 1);
   1.727 @@ -680,7 +621,7 @@
   1.728  qed_spec_mp "length_drop";
   1.729  Addsimps [length_drop];
   1.730  
   1.731 -goal thy "!xs. length xs <= n --> take n xs = xs";
   1.732 +Goal "!xs. length xs <= n --> take n xs = xs";
   1.733  by (nat_ind_tac "n" 1);
   1.734   by (ALLGOALS Asm_simp_tac);
   1.735  by (rtac allI 1);
   1.736 @@ -688,7 +629,7 @@
   1.737   by (ALLGOALS Asm_simp_tac);
   1.738  qed_spec_mp "take_all";
   1.739  
   1.740 -goal thy "!xs. length xs <= n --> drop n xs = []";
   1.741 +Goal "!xs. length xs <= n --> drop n xs = []";
   1.742  by (nat_ind_tac "n" 1);
   1.743   by (ALLGOALS Asm_simp_tac);
   1.744  by (rtac allI 1);
   1.745 @@ -696,7 +637,7 @@
   1.746   by (ALLGOALS Asm_simp_tac);
   1.747  qed_spec_mp "drop_all";
   1.748  
   1.749 -goal thy 
   1.750 +Goal 
   1.751    "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)";
   1.752  by (nat_ind_tac "n" 1);
   1.753   by (ALLGOALS Asm_simp_tac);
   1.754 @@ -706,7 +647,7 @@
   1.755  qed_spec_mp "take_append";
   1.756  Addsimps [take_append];
   1.757  
   1.758 -goal thy "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
   1.759 +Goal "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; 
   1.760  by (nat_ind_tac "n" 1);
   1.761   by (ALLGOALS Asm_simp_tac);
   1.762  by (rtac allI 1);
   1.763 @@ -715,7 +656,7 @@
   1.764  qed_spec_mp "drop_append";
   1.765  Addsimps [drop_append];
   1.766  
   1.767 -goal thy "!xs n. take n (take m xs) = take (min n m) xs"; 
   1.768 +Goal "!xs n. take n (take m xs) = take (min n m) xs"; 
   1.769  by (nat_ind_tac "m" 1);
   1.770   by (ALLGOALS Asm_simp_tac);
   1.771  by (rtac allI 1);
   1.772 @@ -726,7 +667,7 @@
   1.773   by (ALLGOALS Asm_simp_tac);
   1.774  qed_spec_mp "take_take";
   1.775  
   1.776 -goal thy "!xs. drop n (drop m xs) = drop (n + m) xs"; 
   1.777 +Goal "!xs. drop n (drop m xs) = drop (n + m) xs"; 
   1.778  by (nat_ind_tac "m" 1);
   1.779   by (ALLGOALS Asm_simp_tac);
   1.780  by (rtac allI 1);
   1.781 @@ -734,7 +675,7 @@
   1.782   by (ALLGOALS Asm_simp_tac);
   1.783  qed_spec_mp "drop_drop";
   1.784  
   1.785 -goal thy "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
   1.786 +Goal "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; 
   1.787  by (nat_ind_tac "m" 1);
   1.788   by (ALLGOALS Asm_simp_tac);
   1.789  by (rtac allI 1);
   1.790 @@ -742,7 +683,7 @@
   1.791   by (ALLGOALS Asm_simp_tac);
   1.792  qed_spec_mp "take_drop";
   1.793  
   1.794 -goal thy "!xs. take n (map f xs) = map f (take n xs)"; 
   1.795 +Goal "!xs. take n (map f xs) = map f (take n xs)"; 
   1.796  by (nat_ind_tac "n" 1);
   1.797  by (ALLGOALS Asm_simp_tac);
   1.798  by (rtac allI 1);
   1.799 @@ -750,7 +691,7 @@
   1.800  by (ALLGOALS Asm_simp_tac);
   1.801  qed_spec_mp "take_map"; 
   1.802  
   1.803 -goal thy "!xs. drop n (map f xs) = map f (drop n xs)"; 
   1.804 +Goal "!xs. drop n (map f xs) = map f (drop n xs)"; 
   1.805  by (nat_ind_tac "n" 1);
   1.806  by (ALLGOALS Asm_simp_tac);
   1.807  by (rtac allI 1);
   1.808 @@ -758,7 +699,7 @@
   1.809  by (ALLGOALS Asm_simp_tac);
   1.810  qed_spec_mp "drop_map";
   1.811  
   1.812 -goal thy "!n i. i < n --> (take n xs)!i = xs!i";
   1.813 +Goal "!n i. i < n --> (take n xs)!i = xs!i";
   1.814  by (induct_tac "xs" 1);
   1.815   by (ALLGOALS Asm_simp_tac);
   1.816  by (Clarify_tac 1);
   1.817 @@ -769,7 +710,7 @@
   1.818  qed_spec_mp "nth_take";
   1.819  Addsimps [nth_take];
   1.820  
   1.821 -goal thy  "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)";
   1.822 +Goal  "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)";
   1.823  by (nat_ind_tac "n" 1);
   1.824   by (ALLGOALS Asm_simp_tac);
   1.825  by (rtac allI 1);
   1.826 @@ -782,42 +723,39 @@
   1.827  
   1.828  section "takeWhile & dropWhile";
   1.829  
   1.830 -goal thy "takeWhile P xs @ dropWhile P xs = xs";
   1.831 +Goal "takeWhile P xs @ dropWhile P xs = xs";
   1.832  by (induct_tac "xs" 1);
   1.833  by (ALLGOALS Asm_full_simp_tac);
   1.834  qed "takeWhile_dropWhile_id";
   1.835  Addsimps [takeWhile_dropWhile_id];
   1.836  
   1.837 -goal thy  "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
   1.838 +Goal  "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs";
   1.839  by (induct_tac "xs" 1);
   1.840  by (ALLGOALS Asm_full_simp_tac);
   1.841  by (Blast_tac 1);
   1.842  bind_thm("takeWhile_append1", conjI RS (result() RS mp));
   1.843  Addsimps [takeWhile_append1];
   1.844  
   1.845 -goal thy
   1.846 -  "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
   1.847 +Goal "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys";
   1.848  by (induct_tac "xs" 1);
   1.849  by (ALLGOALS Asm_full_simp_tac);
   1.850  bind_thm("takeWhile_append2", ballI RS (result() RS mp));
   1.851  Addsimps [takeWhile_append2];
   1.852  
   1.853 -goal thy
   1.854 -  "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
   1.855 +Goal "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys";
   1.856  by (induct_tac "xs" 1);
   1.857  by (ALLGOALS Asm_full_simp_tac);
   1.858  by (Blast_tac 1);
   1.859  bind_thm("dropWhile_append1", conjI RS (result() RS mp));
   1.860  Addsimps [dropWhile_append1];
   1.861  
   1.862 -goal thy
   1.863 -  "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
   1.864 +Goal "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys";
   1.865  by (induct_tac "xs" 1);
   1.866  by (ALLGOALS Asm_full_simp_tac);
   1.867  bind_thm("dropWhile_append2", ballI RS (result() RS mp));
   1.868  Addsimps [dropWhile_append2];
   1.869  
   1.870 -goal thy "x:set(takeWhile P xs) --> x:set xs & P x";
   1.871 +Goal "x:set(takeWhile P xs) --> x:set xs & P x";
   1.872  by (induct_tac "xs" 1);
   1.873  by (ALLGOALS Asm_full_simp_tac);
   1.874  qed_spec_mp"set_take_whileD";
   1.875 @@ -829,19 +767,19 @@
   1.876  (** nodups & remdups **)
   1.877  section "nodups & remdups";
   1.878  
   1.879 -goal thy "set(remdups xs) = set xs";
   1.880 +Goal "set(remdups xs) = set xs";
   1.881  by (induct_tac "xs" 1);
   1.882   by (Simp_tac 1);
   1.883  by (asm_full_simp_tac (simpset() addsimps [insert_absorb]) 1);
   1.884  qed "set_remdups";
   1.885  Addsimps [set_remdups];
   1.886  
   1.887 -goal thy "nodups(remdups xs)";
   1.888 +Goal "nodups(remdups xs)";
   1.889  by (induct_tac "xs" 1);
   1.890  by (ALLGOALS Asm_full_simp_tac);
   1.891  qed "nodups_remdups";
   1.892  
   1.893 -goal thy "nodups xs --> nodups (filter P xs)";
   1.894 +Goal "nodups xs --> nodups (filter P xs)";
   1.895  by (induct_tac "xs" 1);
   1.896  by (ALLGOALS Asm_full_simp_tac);
   1.897  qed_spec_mp "nodups_filter";
   1.898 @@ -849,12 +787,12 @@
   1.899  (** replicate **)
   1.900  section "replicate";
   1.901  
   1.902 -goal thy "set(replicate (Suc n) x) = {x}";
   1.903 +Goal "set(replicate (Suc n) x) = {x}";
   1.904  by (induct_tac "n" 1);
   1.905  by (ALLGOALS Asm_full_simp_tac);
   1.906  val lemma = result();
   1.907  
   1.908 -goal thy "!!n. n ~= 0 ==> set(replicate n x) = {x}";
   1.909 +Goal "!!n. n ~= 0 ==> set(replicate n x) = {x}";
   1.910  by (fast_tac (claset() addSDs [not0_implies_Suc] addSIs [lemma]) 1);
   1.911  qed "set_replicate";
   1.912  Addsimps [set_replicate];