author | paulson |
Fri, 14 Aug 1998 18:37:28 +0200 | |
changeset 5321 | f8848433d240 |
parent 5147 | 825877190618 |
child 5529 | 4a54acae6a15 |
permissions | -rw-r--r-- |
1461 | 1 |
(* Title: ZF/List.ML |
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ID: $Id$ |
1461 | 3 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1993 University of Cambridge |
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||
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Datatype definition of Lists |
|
7 |
*) |
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||
516 | 9 |
open List; |
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|
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(*** Aspects of the datatype definition ***) |
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|
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Addsimps list.case_eqns; |
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||
15 |
||
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(*An elimination rule, for type-checking*) |
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val ConsE = list.mk_cases list.con_defs "Cons(a,l) : list(A)"; |
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|
19 |
(*Proving freeness results*) |
|
516 | 20 |
val Cons_iff = list.mk_free "Cons(a,l)=Cons(a',l') <-> a=a' & l=l'"; |
21 |
val Nil_Cons_iff = list.mk_free "~ Nil=Cons(a,l)"; |
|
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|
23 |
(*Perform induction on l, then prove the major premise using prems. *) |
|
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fun list_ind_tac a prems i = |
|
516 | 25 |
EVERY [res_inst_tac [("x",a)] list.induct i, |
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rename_last_tac a ["1"] (i+2), |
27 |
ares_tac prems i]; |
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|
5067 | 29 |
Goal "list(A) = {0} + (A * list(A))"; |
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30 |
let open list; val rew = rewrite_rule con_defs in |
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by (blast_tac (claset() addSIs (map rew intrs) addEs [rew elim]) 1) |
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32 |
end; |
760 | 33 |
qed "list_unfold"; |
435 | 34 |
|
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(** Lemmas to justify using "list" in other recursive type definitions **) |
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||
5137 | 37 |
Goalw list.defs "A<=B ==> list(A) <= list(B)"; |
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by (rtac lfp_mono 1); |
516 | 39 |
by (REPEAT (rtac list.bnd_mono 1)); |
0 | 40 |
by (REPEAT (ares_tac (univ_mono::basic_monos) 1)); |
760 | 41 |
qed "list_mono"; |
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|
43 |
(*There is a similar proof by list induction.*) |
|
5067 | 44 |
Goalw (list.defs@list.con_defs) "list(univ(A)) <= univ(A)"; |
0 | 45 |
by (rtac lfp_lowerbound 1); |
46 |
by (rtac (A_subset_univ RS univ_mono) 2); |
|
4091 | 47 |
by (blast_tac (claset() addSIs [zero_in_univ, Inl_in_univ, Inr_in_univ, |
1461 | 48 |
Pair_in_univ]) 1); |
760 | 49 |
qed "list_univ"; |
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|
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(*These two theorems justify datatypes involving list(nat), list(A), ...*) |
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bind_thm ("list_subset_univ", ([list_mono, list_univ] MRS subset_trans)); |
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|
5137 | 54 |
Goal "[| l: list(A); A <= univ(B) |] ==> l: univ(B)"; |
435 | 55 |
by (REPEAT (ares_tac [list_subset_univ RS subsetD] 1)); |
760 | 56 |
qed "list_into_univ"; |
435 | 57 |
|
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val major::prems = Goal |
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"[| l: list(A); \ |
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\ c: C(Nil); \ |
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\ !!x y. [| x: A; y: list(A) |] ==> h(x,y): C(Cons(x,y)) \ |
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62 |
\ |] ==> list_case(c,h,l) : C(l)"; |
516 | 63 |
by (rtac (major RS list.induct) 1); |
4091 | 64 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps (list.case_eqns @ prems)))); |
760 | 65 |
qed "list_case_type"; |
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|
67 |
||
68 |
(** For recursion **) |
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||
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Goalw list.con_defs "rank(a) < rank(Cons(a,l))"; |
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71 |
by (simp_tac rank_ss 1); |
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qed "rank_Cons1"; |
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|
5067 | 74 |
Goalw list.con_defs "rank(l) < rank(Cons(a,l))"; |
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by (simp_tac rank_ss 1); |
760 | 76 |
qed "rank_Cons2"; |
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516 | 78 |
|
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(*** List functions ***) |
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||
81 |
(** hd and tl **) |
|
82 |
||
5067 | 83 |
Goalw [hd_def] "hd(Cons(a,l)) = a"; |
516 | 84 |
by (resolve_tac list.case_eqns 1); |
760 | 85 |
qed "hd_Cons"; |
516 | 86 |
|
5067 | 87 |
Goalw [tl_def] "tl(Nil) = Nil"; |
516 | 88 |
by (resolve_tac list.case_eqns 1); |
760 | 89 |
qed "tl_Nil"; |
516 | 90 |
|
5067 | 91 |
Goalw [tl_def] "tl(Cons(a,l)) = l"; |
516 | 92 |
by (resolve_tac list.case_eqns 1); |
760 | 93 |
qed "tl_Cons"; |
516 | 94 |
|
2469 | 95 |
Addsimps [hd_Cons, tl_Nil, tl_Cons]; |
96 |
||
5137 | 97 |
Goal "l: list(A) ==> tl(l) : list(A)"; |
516 | 98 |
by (etac list.elim 1); |
4091 | 99 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps list.intrs))); |
760 | 100 |
qed "tl_type"; |
516 | 101 |
|
102 |
(** drop **) |
|
103 |
||
5067 | 104 |
Goalw [drop_def] "drop(0, l) = l"; |
516 | 105 |
by (rtac rec_0 1); |
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qed "drop_0"; |
516 | 107 |
|
5137 | 108 |
Goalw [drop_def] "i:nat ==> drop(i, Nil) = Nil"; |
516 | 109 |
by (etac nat_induct 1); |
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by (ALLGOALS Asm_simp_tac); |
760 | 111 |
qed "drop_Nil"; |
516 | 112 |
|
5067 | 113 |
Goalw [drop_def] |
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"i:nat ==> drop(succ(i), Cons(a,l)) = drop(i,l)"; |
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by (rtac sym 1); |
516 | 116 |
by (etac nat_induct 1); |
2469 | 117 |
by (Simp_tac 1); |
118 |
by (Asm_simp_tac 1); |
|
760 | 119 |
qed "drop_succ_Cons"; |
516 | 120 |
|
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Addsimps [drop_0, drop_Nil, drop_succ_Cons]; |
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||
5067 | 123 |
Goalw [drop_def] |
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"[| i:nat; l: list(A) |] ==> drop(i,l) : list(A)"; |
516 | 125 |
by (etac nat_induct 1); |
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by (ALLGOALS (asm_simp_tac (simpset() addsimps [tl_type]))); |
760 | 127 |
qed "drop_type"; |
516 | 128 |
|
129 |
(** list_rec -- by Vset recursion **) |
|
130 |
||
5067 | 131 |
Goal "list_rec(Nil,c,h) = c"; |
516 | 132 |
by (rtac (list_rec_def RS def_Vrec RS trans) 1); |
4091 | 133 |
by (simp_tac (simpset() addsimps list.case_eqns) 1); |
760 | 134 |
qed "list_rec_Nil"; |
516 | 135 |
|
5067 | 136 |
Goal "list_rec(Cons(a,l), c, h) = h(a, l, list_rec(l,c,h))"; |
516 | 137 |
by (rtac (list_rec_def RS def_Vrec RS trans) 1); |
138 |
by (simp_tac (rank_ss addsimps (rank_Cons2::list.case_eqns)) 1); |
|
760 | 139 |
qed "list_rec_Cons"; |
516 | 140 |
|
2469 | 141 |
Addsimps [list_rec_Nil, list_rec_Cons]; |
142 |
||
143 |
||
516 | 144 |
(*Type checking -- proved by induction, as usual*) |
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val prems = Goal |
516 | 146 |
"[| l: list(A); \ |
147 |
\ c: C(Nil); \ |
|
148 |
\ !!x y r. [| x:A; y: list(A); r: C(y) |] ==> h(x,y,r): C(Cons(x,y)) \ |
|
149 |
\ |] ==> list_rec(l,c,h) : C(l)"; |
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150 |
by (list_ind_tac "l" prems 1); |
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by (ALLGOALS (asm_simp_tac (simpset() addsimps prems))); |
760 | 152 |
qed "list_rec_type"; |
516 | 153 |
|
154 |
(** Versions for use with definitions **) |
|
155 |
||
5321 | 156 |
val [rew] = Goal "[| !!l. j(l)==list_rec(l, c, h) |] ==> j(Nil) = c"; |
516 | 157 |
by (rewtac rew); |
158 |
by (rtac list_rec_Nil 1); |
|
760 | 159 |
qed "def_list_rec_Nil"; |
516 | 160 |
|
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val [rew] = Goal "[| !!l. j(l)==list_rec(l, c, h) |] ==> j(Cons(a,l)) = h(a,l,j(l))"; |
516 | 162 |
by (rewtac rew); |
163 |
by (rtac list_rec_Cons 1); |
|
760 | 164 |
qed "def_list_rec_Cons"; |
516 | 165 |
|
166 |
fun list_recs def = map standard |
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([def] RL [def_list_rec_Nil, def_list_rec_Cons]); |
516 | 168 |
|
169 |
(** map **) |
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170 |
||
171 |
val [map_Nil,map_Cons] = list_recs map_def; |
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Addsimps [map_Nil, map_Cons]; |
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|
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val prems = Goalw [map_def] |
516 | 175 |
"[| l: list(A); !!x. x: A ==> h(x): B |] ==> map(h,l) : list(B)"; |
176 |
by (REPEAT (ares_tac (prems @ list.intrs @ [list_rec_type]) 1)); |
|
760 | 177 |
qed "map_type"; |
516 | 178 |
|
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Goal "l: list(A) ==> map(h,l) : list({h(u). u:A})"; |
180 |
by (etac map_type 1); |
|
516 | 181 |
by (etac RepFunI 1); |
760 | 182 |
qed "map_type2"; |
516 | 183 |
|
184 |
(** length **) |
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185 |
||
186 |
val [length_Nil,length_Cons] = list_recs length_def; |
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Addsimps [length_Nil,length_Cons]; |
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Goalw [length_def] |
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"l: list(A) ==> length(l) : nat"; |
516 | 191 |
by (REPEAT (ares_tac [list_rec_type, nat_0I, nat_succI] 1)); |
760 | 192 |
qed "length_type"; |
516 | 193 |
|
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(** app **) |
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195 |
||
196 |
val [app_Nil,app_Cons] = list_recs app_def; |
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Addsimps [app_Nil, app_Cons]; |
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|
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Goalw [app_def] |
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"[| xs: list(A); ys: list(A) |] ==> xs@ys : list(A)"; |
516 | 201 |
by (REPEAT (ares_tac [list_rec_type, list.Cons_I] 1)); |
760 | 202 |
qed "app_type"; |
516 | 203 |
|
204 |
(** rev **) |
|
205 |
||
206 |
val [rev_Nil,rev_Cons] = list_recs rev_def; |
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Addsimps [rev_Nil,rev_Cons] ; |
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|
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Goalw [rev_def] |
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210 |
"xs: list(A) ==> rev(xs) : list(A)"; |
516 | 211 |
by (REPEAT (ares_tac (list.intrs @ [list_rec_type, app_type]) 1)); |
760 | 212 |
qed "rev_type"; |
516 | 213 |
|
214 |
||
215 |
(** flat **) |
|
216 |
||
217 |
val [flat_Nil,flat_Cons] = list_recs flat_def; |
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Addsimps [flat_Nil,flat_Cons]; |
516 | 219 |
|
5067 | 220 |
Goalw [flat_def] |
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221 |
"ls: list(list(A)) ==> flat(ls) : list(A)"; |
516 | 222 |
by (REPEAT (ares_tac (list.intrs @ [list_rec_type, app_type]) 1)); |
760 | 223 |
qed "flat_type"; |
516 | 224 |
|
225 |
||
1926 | 226 |
(** set_of_list **) |
227 |
||
228 |
val [set_of_list_Nil,set_of_list_Cons] = list_recs set_of_list_def; |
|
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Addsimps [set_of_list_Nil,set_of_list_Cons]; |
1926 | 230 |
|
5067 | 231 |
Goalw [set_of_list_def] |
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"l: list(A) ==> set_of_list(l) : Pow(A)"; |
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by (etac list_rec_type 1); |
3016 | 234 |
by (ALLGOALS (Blast_tac)); |
1926 | 235 |
qed "set_of_list_type"; |
236 |
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Goal "xs: list(A) ==> \ |
1926 | 238 |
\ set_of_list (xs@ys) = set_of_list(xs) Un set_of_list(ys)"; |
239 |
by (etac list.induct 1); |
|
4091 | 240 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [Un_cons]))); |
1926 | 241 |
qed "set_of_list_append"; |
242 |
||
243 |
||
516 | 244 |
(** list_add **) |
245 |
||
246 |
val [list_add_Nil,list_add_Cons] = list_recs list_add_def; |
|
2469 | 247 |
Addsimps [list_add_Nil,list_add_Cons]; |
516 | 248 |
|
5067 | 249 |
Goalw [list_add_def] |
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250 |
"xs: list(nat) ==> list_add(xs) : nat"; |
516 | 251 |
by (REPEAT (ares_tac [list_rec_type, nat_0I, add_type] 1)); |
760 | 252 |
qed "list_add_type"; |
516 | 253 |
|
254 |
val list_typechecks = |
|
255 |
list.intrs @ |
|
256 |
[list_rec_type, map_type, map_type2, app_type, length_type, |
|
257 |
rev_type, flat_type, list_add_type]; |
|
258 |
||
4091 | 259 |
simpset_ref() := simpset() setSolver (type_auto_tac list_typechecks); |
516 | 260 |
|
261 |
||
262 |
(*** theorems about map ***) |
|
263 |
||
5321 | 264 |
Goal "l: list(A) ==> map(%u. u, l) = l"; |
265 |
by (list_ind_tac "l" [] 1); |
|
3016 | 266 |
by (ALLGOALS Asm_simp_tac); |
760 | 267 |
qed "map_ident"; |
516 | 268 |
|
5321 | 269 |
Goal "l: list(A) ==> map(h, map(j,l)) = map(%u. h(j(u)), l)"; |
270 |
by (list_ind_tac "l" [] 1); |
|
3016 | 271 |
by (ALLGOALS Asm_simp_tac); |
760 | 272 |
qed "map_compose"; |
516 | 273 |
|
5321 | 274 |
Goal "xs: list(A) ==> map(h, xs@ys) = map(h,xs) @ map(h,ys)"; |
275 |
by (list_ind_tac "xs" [] 1); |
|
3016 | 276 |
by (ALLGOALS Asm_simp_tac); |
760 | 277 |
qed "map_app_distrib"; |
516 | 278 |
|
5321 | 279 |
Goal "ls: list(list(A)) ==> map(h, flat(ls)) = flat(map(map(h),ls))"; |
280 |
by (list_ind_tac "ls" [] 1); |
|
4091 | 281 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [map_app_distrib]))); |
760 | 282 |
qed "map_flat"; |
516 | 283 |
|
5321 | 284 |
Goal "l: list(A) ==> \ |
516 | 285 |
\ list_rec(map(h,l), c, d) = \ |
286 |
\ list_rec(l, c, %x xs r. d(h(x), map(h,xs), r))"; |
|
5321 | 287 |
by (list_ind_tac "l" [] 1); |
3016 | 288 |
by (ALLGOALS Asm_simp_tac); |
760 | 289 |
qed "list_rec_map"; |
516 | 290 |
|
291 |
(** theorems about list(Collect(A,P)) -- used in ex/term.ML **) |
|
292 |
||
293 |
(* c : list(Collect(B,P)) ==> c : list(B) *) |
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294 |
bind_thm ("list_CollectD", (Collect_subset RS list_mono RS subsetD)); |
516 | 295 |
|
5321 | 296 |
Goal "l: list({x:A. h(x)=j(x)}) ==> map(h,l) = map(j,l)"; |
297 |
by (list_ind_tac "l" [] 1); |
|
3016 | 298 |
by (ALLGOALS Asm_simp_tac); |
760 | 299 |
qed "map_list_Collect"; |
516 | 300 |
|
301 |
(*** theorems about length ***) |
|
302 |
||
5321 | 303 |
Goal "xs: list(A) ==> length(map(h,xs)) = length(xs)"; |
304 |
by (list_ind_tac "xs" [] 1); |
|
3016 | 305 |
by (ALLGOALS Asm_simp_tac); |
760 | 306 |
qed "length_map"; |
516 | 307 |
|
5321 | 308 |
Goal "xs: list(A) ==> length(xs@ys) = length(xs) #+ length(ys)"; |
309 |
by (list_ind_tac "xs" [] 1); |
|
3016 | 310 |
by (ALLGOALS Asm_simp_tac); |
760 | 311 |
qed "length_app"; |
516 | 312 |
|
313 |
(* [| m: nat; n: nat |] ==> m #+ succ(n) = succ(n) #+ m |
|
314 |
Used for rewriting below*) |
|
315 |
val add_commute_succ = nat_succI RSN (2,add_commute); |
|
316 |
||
5321 | 317 |
Goal "xs: list(A) ==> length(rev(xs)) = length(xs)"; |
318 |
by (list_ind_tac "xs" [] 1); |
|
4091 | 319 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [length_app, add_commute_succ]))); |
760 | 320 |
qed "length_rev"; |
516 | 321 |
|
5321 | 322 |
Goal "ls: list(list(A)) ==> length(flat(ls)) = list_add(map(length,ls))"; |
323 |
by (list_ind_tac "ls" [] 1); |
|
4091 | 324 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [length_app]))); |
760 | 325 |
qed "length_flat"; |
516 | 326 |
|
327 |
(** Length and drop **) |
|
328 |
||
329 |
(*Lemma for the inductive step of drop_length*) |
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|
330 |
Goal "xs: list(A) ==> \ |
516 | 331 |
\ ALL x. EX z zs. drop(length(xs), Cons(x,xs)) = Cons(z,zs)"; |
332 |
by (etac list.induct 1); |
|
2469 | 333 |
by (ALLGOALS Asm_simp_tac); |
3016 | 334 |
by (Blast_tac 1); |
760 | 335 |
qed "drop_length_Cons_lemma"; |
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|
336 |
bind_thm ("drop_length_Cons", (drop_length_Cons_lemma RS spec)); |
516 | 337 |
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|
338 |
Goal "l: list(A) ==> ALL i: length(l). EX z zs. drop(i,l) = Cons(z,zs)"; |
516 | 339 |
by (etac list.induct 1); |
2469 | 340 |
by (ALLGOALS Asm_simp_tac); |
516 | 341 |
by (rtac conjI 1); |
342 |
by (etac drop_length_Cons 1); |
|
343 |
by (rtac ballI 1); |
|
344 |
by (rtac natE 1); |
|
345 |
by (etac ([asm_rl, length_type, Ord_nat] MRS Ord_trans) 1); |
|
346 |
by (assume_tac 1); |
|
3016 | 347 |
by (ALLGOALS Asm_simp_tac); |
4091 | 348 |
by (ALLGOALS (blast_tac (claset() addIs [succ_in_naturalD, length_type]))); |
760 | 349 |
qed "drop_length_lemma"; |
782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
760
diff
changeset
|
350 |
bind_thm ("drop_length", (drop_length_lemma RS bspec)); |
516 | 351 |
|
352 |
||
353 |
(*** theorems about app ***) |
|
354 |
||
5321 | 355 |
Goal "xs: list(A) ==> xs@Nil=xs"; |
356 |
by (etac list.induct 1); |
|
3016 | 357 |
by (ALLGOALS Asm_simp_tac); |
760 | 358 |
qed "app_right_Nil"; |
516 | 359 |
|
5321 | 360 |
Goal "xs: list(A) ==> (xs@ys)@zs = xs@(ys@zs)"; |
361 |
by (list_ind_tac "xs" [] 1); |
|
3016 | 362 |
by (ALLGOALS Asm_simp_tac); |
760 | 363 |
qed "app_assoc"; |
516 | 364 |
|
5321 | 365 |
Goal "ls: list(list(A)) ==> flat(ls@ms) = flat(ls)@flat(ms)"; |
366 |
by (list_ind_tac "ls" [] 1); |
|
4091 | 367 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [app_assoc]))); |
760 | 368 |
qed "flat_app_distrib"; |
516 | 369 |
|
370 |
(*** theorems about rev ***) |
|
371 |
||
5321 | 372 |
Goal "l: list(A) ==> rev(map(h,l)) = map(h,rev(l))"; |
373 |
by (list_ind_tac "l" [] 1); |
|
4091 | 374 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [map_app_distrib]))); |
760 | 375 |
qed "rev_map_distrib"; |
516 | 376 |
|
377 |
(*Simplifier needs the premises as assumptions because rewriting will not |
|
378 |
instantiate the variable ?A in the rules' typing conditions; note that |
|
379 |
rev_type does not instantiate ?A. Only the premises do. |
|
380 |
*) |
|
5147
825877190618
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5137
diff
changeset
|
381 |
Goal "[| xs: list(A); ys: list(A) |] ==> rev(xs@ys) = rev(ys)@rev(xs)"; |
516 | 382 |
by (etac list.induct 1); |
4091 | 383 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [app_right_Nil,app_assoc]))); |
760 | 384 |
qed "rev_app_distrib"; |
516 | 385 |
|
5321 | 386 |
Goal "l: list(A) ==> rev(rev(l))=l"; |
387 |
by (list_ind_tac "l" [] 1); |
|
4091 | 388 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [rev_app_distrib]))); |
760 | 389 |
qed "rev_rev_ident"; |
516 | 390 |
|
5321 | 391 |
Goal "ls: list(list(A)) ==> rev(flat(ls)) = flat(map(rev,rev(ls)))"; |
392 |
by (list_ind_tac "ls" [] 1); |
|
4091 | 393 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps |
516 | 394 |
[map_app_distrib, flat_app_distrib, rev_app_distrib, app_right_Nil]))); |
760 | 395 |
qed "rev_flat"; |
516 | 396 |
|
397 |
||
398 |
(*** theorems about list_add ***) |
|
399 |
||
5321 | 400 |
Goal "[| xs: list(nat); ys: list(nat) |] ==> \ |
516 | 401 |
\ list_add(xs@ys) = list_add(ys) #+ list_add(xs)"; |
5321 | 402 |
by (list_ind_tac "xs" [] 1); |
516 | 403 |
by (ALLGOALS |
4091 | 404 |
(asm_simp_tac (simpset() addsimps [add_0_right, add_assoc RS sym]))); |
516 | 405 |
by (rtac (add_commute RS subst_context) 1); |
406 |
by (REPEAT (ares_tac [refl, list_add_type] 1)); |
|
760 | 407 |
qed "list_add_app"; |
516 | 408 |
|
5321 | 409 |
Goal "l: list(nat) ==> list_add(rev(l)) = list_add(l)"; |
410 |
by (list_ind_tac "l" [] 1); |
|
516 | 411 |
by (ALLGOALS |
4091 | 412 |
(asm_simp_tac (simpset() addsimps [list_add_app, add_0_right]))); |
760 | 413 |
qed "list_add_rev"; |
516 | 414 |
|
5321 | 415 |
Goal "ls: list(list(nat)) ==> list_add(flat(ls)) = list_add(map(list_add,ls))"; |
416 |
by (list_ind_tac "ls" [] 1); |
|
4091 | 417 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [list_add_app]))); |
516 | 418 |
by (REPEAT (ares_tac [refl, list_add_type, map_type, add_commute] 1)); |
760 | 419 |
qed "list_add_flat"; |
516 | 420 |
|
421 |
(** New induction rule **) |
|
422 |
||
5321 | 423 |
val major::prems = Goal |
516 | 424 |
"[| l: list(A); \ |
425 |
\ P(Nil); \ |
|
426 |
\ !!x y. [| x: A; y: list(A); P(y) |] ==> P(y @ [x]) \ |
|
427 |
\ |] ==> P(l)"; |
|
428 |
by (rtac (major RS rev_rev_ident RS subst) 1); |
|
429 |
by (rtac (major RS rev_type RS list.induct) 1); |
|
4091 | 430 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps prems))); |
760 | 431 |
qed "list_append_induct"; |
516 | 432 |