New unified treatment of sequent calculi by Sara Kalvala
combines the old LK and Modal with the new ILL (Int. Linear Logic)
(* Title: ZF/List.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Datatype definition of Lists
*)
open List;
(*** Aspects of the datatype definition ***)
(*An elimination rule, for type-checking*)
val ConsE = list.mk_cases list.con_defs "Cons(a,l) : list(A)";
(*Proving freeness results*)
val Cons_iff = list.mk_free "Cons(a,l)=Cons(a',l') <-> a=a' & l=l'";
val Nil_Cons_iff = list.mk_free "~ Nil=Cons(a,l)";
(*Perform induction on l, then prove the major premise using prems. *)
fun list_ind_tac a prems i =
EVERY [res_inst_tac [("x",a)] list.induct i,
rename_last_tac a ["1"] (i+2),
ares_tac prems i];
goal List.thy "list(A) = {0} + (A * list(A))";
let open list; val rew = rewrite_rule con_defs in
by (fast_tac (sum_cs addSIs (equalityI :: map rew intrs)
addEs [rew elim]) 1)
end;
qed "list_unfold";
(** Lemmas to justify using "list" in other recursive type definitions **)
goalw List.thy list.defs "!!A B. A<=B ==> list(A) <= list(B)";
by (rtac lfp_mono 1);
by (REPEAT (rtac list.bnd_mono 1));
by (REPEAT (ares_tac (univ_mono::basic_monos) 1));
qed "list_mono";
(*There is a similar proof by list induction.*)
goalw List.thy (list.defs@list.con_defs) "list(univ(A)) <= univ(A)";
by (rtac lfp_lowerbound 1);
by (rtac (A_subset_univ RS univ_mono) 2);
by (fast_tac (ZF_cs addSIs [zero_in_univ, Inl_in_univ, Inr_in_univ,
Pair_in_univ]) 1);
qed "list_univ";
(*These two theorems justify datatypes involving list(nat), list(A), ...*)
bind_thm ("list_subset_univ", ([list_mono, list_univ] MRS subset_trans));
goal List.thy "!!l A B. [| l: list(A); A <= univ(B) |] ==> l: univ(B)";
by (REPEAT (ares_tac [list_subset_univ RS subsetD] 1));
qed "list_into_univ";
val major::prems = goal List.thy
"[| l: list(A); \
\ c: C(Nil); \
\ !!x y. [| x: A; y: list(A) |] ==> h(x,y): C(Cons(x,y)) \
\ |] ==> list_case(c,h,l) : C(l)";
by (rtac (major RS list.induct) 1);
by (ALLGOALS (asm_simp_tac (ZF_ss addsimps (list.case_eqns @ prems))));
qed "list_case_type";
(** For recursion **)
goalw List.thy list.con_defs "rank(a) < rank(Cons(a,l))";
by (simp_tac rank_ss 1);
qed "rank_Cons1";
goalw List.thy list.con_defs "rank(l) < rank(Cons(a,l))";
by (simp_tac rank_ss 1);
qed "rank_Cons2";
(*** List functions ***)
(** hd and tl **)
goalw List.thy [hd_def] "hd(Cons(a,l)) = a";
by (resolve_tac list.case_eqns 1);
qed "hd_Cons";
goalw List.thy [tl_def] "tl(Nil) = Nil";
by (resolve_tac list.case_eqns 1);
qed "tl_Nil";
goalw List.thy [tl_def] "tl(Cons(a,l)) = l";
by (resolve_tac list.case_eqns 1);
qed "tl_Cons";
goal List.thy "!!l. l: list(A) ==> tl(l) : list(A)";
by (etac list.elim 1);
by (ALLGOALS (asm_simp_tac (ZF_ss addsimps (list.intrs @ [tl_Nil,tl_Cons]))));
qed "tl_type";
(** drop **)
goalw List.thy [drop_def] "drop(0, l) = l";
by (rtac rec_0 1);
qed "drop_0";
goalw List.thy [drop_def] "!!i. i:nat ==> drop(i, Nil) = Nil";
by (etac nat_induct 1);
by (ALLGOALS (asm_simp_tac (nat_ss addsimps [tl_Nil])));
qed "drop_Nil";
goalw List.thy [drop_def]
"!!i. i:nat ==> drop(succ(i), Cons(a,l)) = drop(i,l)";
by (etac nat_induct 1);
by (simp_tac (nat_ss addsimps [tl_Cons]) 1);
by (stac rec_succ 1);
by (stac rec_succ 1);
by (asm_simp_tac ZF_ss 1);
qed "drop_succ_Cons";
goalw List.thy [drop_def]
"!!i l. [| i:nat; l: list(A) |] ==> drop(i,l) : list(A)";
by (etac nat_induct 1);
by (ALLGOALS (asm_simp_tac (nat_ss addsimps [tl_type])));
qed "drop_type";
(** list_rec -- by Vset recursion **)
goal List.thy "list_rec(Nil,c,h) = c";
by (rtac (list_rec_def RS def_Vrec RS trans) 1);
by (simp_tac (ZF_ss addsimps list.case_eqns) 1);
qed "list_rec_Nil";
goal List.thy "list_rec(Cons(a,l), c, h) = h(a, l, list_rec(l,c,h))";
by (rtac (list_rec_def RS def_Vrec RS trans) 1);
by (simp_tac (rank_ss addsimps (rank_Cons2::list.case_eqns)) 1);
qed "list_rec_Cons";
(*Type checking -- proved by induction, as usual*)
val prems = goal List.thy
"[| l: list(A); \
\ c: C(Nil); \
\ !!x y r. [| x:A; y: list(A); r: C(y) |] ==> h(x,y,r): C(Cons(x,y)) \
\ |] ==> list_rec(l,c,h) : C(l)";
by (list_ind_tac "l" prems 1);
by (ALLGOALS (asm_simp_tac
(ZF_ss addsimps (prems@[list_rec_Nil,list_rec_Cons]))));
qed "list_rec_type";
(** Versions for use with definitions **)
val [rew] = goal List.thy
"[| !!l. j(l)==list_rec(l, c, h) |] ==> j(Nil) = c";
by (rewtac rew);
by (rtac list_rec_Nil 1);
qed "def_list_rec_Nil";
val [rew] = goal List.thy
"[| !!l. j(l)==list_rec(l, c, h) |] ==> j(Cons(a,l)) = h(a,l,j(l))";
by (rewtac rew);
by (rtac list_rec_Cons 1);
qed "def_list_rec_Cons";
fun list_recs def = map standard
([def] RL [def_list_rec_Nil, def_list_rec_Cons]);
(** map **)
val [map_Nil,map_Cons] = list_recs map_def;
val prems = goalw List.thy [map_def]
"[| l: list(A); !!x. x: A ==> h(x): B |] ==> map(h,l) : list(B)";
by (REPEAT (ares_tac (prems @ list.intrs @ [list_rec_type]) 1));
qed "map_type";
val [major] = goal List.thy "l: list(A) ==> map(h,l) : list({h(u). u:A})";
by (rtac (major RS map_type) 1);
by (etac RepFunI 1);
qed "map_type2";
(** length **)
val [length_Nil,length_Cons] = list_recs length_def;
goalw List.thy [length_def]
"!!l. l: list(A) ==> length(l) : nat";
by (REPEAT (ares_tac [list_rec_type, nat_0I, nat_succI] 1));
qed "length_type";
(** app **)
val [app_Nil,app_Cons] = list_recs app_def;
goalw List.thy [app_def]
"!!xs ys. [| xs: list(A); ys: list(A) |] ==> xs@ys : list(A)";
by (REPEAT (ares_tac [list_rec_type, list.Cons_I] 1));
qed "app_type";
(** rev **)
val [rev_Nil,rev_Cons] = list_recs rev_def;
goalw List.thy [rev_def]
"!!xs. xs: list(A) ==> rev(xs) : list(A)";
by (REPEAT (ares_tac (list.intrs @ [list_rec_type, app_type]) 1));
qed "rev_type";
(** flat **)
val [flat_Nil,flat_Cons] = list_recs flat_def;
goalw List.thy [flat_def]
"!!ls. ls: list(list(A)) ==> flat(ls) : list(A)";
by (REPEAT (ares_tac (list.intrs @ [list_rec_type, app_type]) 1));
qed "flat_type";
(** set_of_list **)
val [set_of_list_Nil,set_of_list_Cons] = list_recs set_of_list_def;
goalw List.thy [set_of_list_def]
"!!l. l: list(A) ==> set_of_list(l) : Pow(A)";
by (etac list_rec_type 1);
by (ALLGOALS (fast_tac ZF_cs));
qed "set_of_list_type";
goal List.thy
"!!l. xs: list(A) ==> \
\ set_of_list (xs@ys) = set_of_list(xs) Un set_of_list(ys)";
by (etac list.induct 1);
by (ALLGOALS (asm_simp_tac (ZF_ss addsimps [set_of_list_Nil,set_of_list_Cons,
app_Nil,app_Cons,Un_cons])));
qed "set_of_list_append";
(** list_add **)
val [list_add_Nil,list_add_Cons] = list_recs list_add_def;
goalw List.thy [list_add_def]
"!!xs. xs: list(nat) ==> list_add(xs) : nat";
by (REPEAT (ares_tac [list_rec_type, nat_0I, add_type] 1));
qed "list_add_type";
(** List simplification **)
val list_typechecks =
list.intrs @
[list_rec_type, map_type, map_type2, app_type, length_type,
rev_type, flat_type, list_add_type];
val list_ss = arith_ss
addsimps list.case_eqns
addsimps [list_rec_Nil, list_rec_Cons,
map_Nil, map_Cons, app_Nil, app_Cons,
length_Nil, length_Cons, rev_Nil, rev_Cons,
flat_Nil, flat_Cons, list_add_Nil, list_add_Cons]
setsolver (type_auto_tac list_typechecks);
(*** theorems about map ***)
val prems = goal List.thy
"l: list(A) ==> map(%u.u, l) = l";
by (list_ind_tac "l" prems 1);
by (ALLGOALS (asm_simp_tac list_ss));
qed "map_ident";
val prems = goal List.thy
"l: list(A) ==> map(h, map(j,l)) = map(%u.h(j(u)), l)";
by (list_ind_tac "l" prems 1);
by (ALLGOALS (asm_simp_tac list_ss));
qed "map_compose";
val prems = goal List.thy
"xs: list(A) ==> map(h, xs@ys) = map(h,xs) @ map(h,ys)";
by (list_ind_tac "xs" prems 1);
by (ALLGOALS (asm_simp_tac list_ss));
qed "map_app_distrib";
val prems = goal List.thy
"ls: list(list(A)) ==> map(h, flat(ls)) = flat(map(map(h),ls))";
by (list_ind_tac "ls" prems 1);
by (ALLGOALS (asm_simp_tac (list_ss addsimps [map_app_distrib])));
qed "map_flat";
val prems = goal List.thy
"l: list(A) ==> \
\ list_rec(map(h,l), c, d) = \
\ list_rec(l, c, %x xs r. d(h(x), map(h,xs), r))";
by (list_ind_tac "l" prems 1);
by (ALLGOALS (asm_simp_tac list_ss));
qed "list_rec_map";
(** theorems about list(Collect(A,P)) -- used in ex/term.ML **)
(* c : list(Collect(B,P)) ==> c : list(B) *)
bind_thm ("list_CollectD", (Collect_subset RS list_mono RS subsetD));
val prems = goal List.thy
"l: list({x:A. h(x)=j(x)}) ==> map(h,l) = map(j,l)";
by (list_ind_tac "l" prems 1);
by (ALLGOALS (asm_simp_tac list_ss));
qed "map_list_Collect";
(*** theorems about length ***)
val prems = goal List.thy
"xs: list(A) ==> length(map(h,xs)) = length(xs)";
by (list_ind_tac "xs" prems 1);
by (ALLGOALS (asm_simp_tac list_ss));
qed "length_map";
val prems = goal List.thy
"xs: list(A) ==> length(xs@ys) = length(xs) #+ length(ys)";
by (list_ind_tac "xs" prems 1);
by (ALLGOALS (asm_simp_tac list_ss));
qed "length_app";
(* [| m: nat; n: nat |] ==> m #+ succ(n) = succ(n) #+ m
Used for rewriting below*)
val add_commute_succ = nat_succI RSN (2,add_commute);
val prems = goal List.thy
"xs: list(A) ==> length(rev(xs)) = length(xs)";
by (list_ind_tac "xs" prems 1);
by (ALLGOALS (asm_simp_tac (list_ss addsimps [length_app, add_commute_succ])));
qed "length_rev";
val prems = goal List.thy
"ls: list(list(A)) ==> length(flat(ls)) = list_add(map(length,ls))";
by (list_ind_tac "ls" prems 1);
by (ALLGOALS (asm_simp_tac (list_ss addsimps [length_app])));
qed "length_flat";
(** Length and drop **)
(*Lemma for the inductive step of drop_length*)
goal List.thy
"!!xs. xs: list(A) ==> \
\ ALL x. EX z zs. drop(length(xs), Cons(x,xs)) = Cons(z,zs)";
by (etac list.induct 1);
by (ALLGOALS (asm_simp_tac (list_ss addsimps [drop_0,drop_succ_Cons])));
by (fast_tac ZF_cs 1);
qed "drop_length_Cons_lemma";
bind_thm ("drop_length_Cons", (drop_length_Cons_lemma RS spec));
goal List.thy
"!!l. l: list(A) ==> ALL i: length(l). EX z zs. drop(i,l) = Cons(z,zs)";
by (etac list.induct 1);
by (ALLGOALS (asm_simp_tac (list_ss addsimps bquant_simps)));
by (rtac conjI 1);
by (etac drop_length_Cons 1);
by (rtac ballI 1);
by (rtac natE 1);
by (etac ([asm_rl, length_type, Ord_nat] MRS Ord_trans) 1);
by (assume_tac 1);
by (asm_simp_tac (list_ss addsimps [drop_0]) 1);
by (fast_tac ZF_cs 1);
by (asm_simp_tac (list_ss addsimps [drop_succ_Cons]) 1);
by (dtac bspec 1);
by (fast_tac ZF_cs 2);
by (fast_tac (ZF_cs addEs [succ_in_naturalD,length_type]) 1);
qed "drop_length_lemma";
bind_thm ("drop_length", (drop_length_lemma RS bspec));
(*** theorems about app ***)
val [major] = goal List.thy "xs: list(A) ==> xs@Nil=xs";
by (rtac (major RS list.induct) 1);
by (ALLGOALS (asm_simp_tac list_ss));
qed "app_right_Nil";
val prems = goal List.thy "xs: list(A) ==> (xs@ys)@zs = xs@(ys@zs)";
by (list_ind_tac "xs" prems 1);
by (ALLGOALS (asm_simp_tac list_ss));
qed "app_assoc";
val prems = goal List.thy
"ls: list(list(A)) ==> flat(ls@ms) = flat(ls)@flat(ms)";
by (list_ind_tac "ls" prems 1);
by (ALLGOALS (asm_simp_tac (list_ss addsimps [app_assoc])));
qed "flat_app_distrib";
(*** theorems about rev ***)
val prems = goal List.thy "l: list(A) ==> rev(map(h,l)) = map(h,rev(l))";
by (list_ind_tac "l" prems 1);
by (ALLGOALS (asm_simp_tac (list_ss addsimps [map_app_distrib])));
qed "rev_map_distrib";
(*Simplifier needs the premises as assumptions because rewriting will not
instantiate the variable ?A in the rules' typing conditions; note that
rev_type does not instantiate ?A. Only the premises do.
*)
goal List.thy
"!!xs. [| xs: list(A); ys: list(A) |] ==> rev(xs@ys) = rev(ys)@rev(xs)";
by (etac list.induct 1);
by (ALLGOALS (asm_simp_tac (list_ss addsimps [app_right_Nil,app_assoc])));
qed "rev_app_distrib";
val prems = goal List.thy "l: list(A) ==> rev(rev(l))=l";
by (list_ind_tac "l" prems 1);
by (ALLGOALS (asm_simp_tac (list_ss addsimps [rev_app_distrib])));
qed "rev_rev_ident";
val prems = goal List.thy
"ls: list(list(A)) ==> rev(flat(ls)) = flat(map(rev,rev(ls)))";
by (list_ind_tac "ls" prems 1);
by (ALLGOALS (asm_simp_tac (list_ss addsimps
[map_app_distrib, flat_app_distrib, rev_app_distrib, app_right_Nil])));
qed "rev_flat";
(*** theorems about list_add ***)
val prems = goal List.thy
"[| xs: list(nat); ys: list(nat) |] ==> \
\ list_add(xs@ys) = list_add(ys) #+ list_add(xs)";
by (cut_facts_tac prems 1);
by (list_ind_tac "xs" prems 1);
by (ALLGOALS
(asm_simp_tac (list_ss addsimps [add_0_right, add_assoc RS sym])));
by (rtac (add_commute RS subst_context) 1);
by (REPEAT (ares_tac [refl, list_add_type] 1));
qed "list_add_app";
val prems = goal List.thy
"l: list(nat) ==> list_add(rev(l)) = list_add(l)";
by (list_ind_tac "l" prems 1);
by (ALLGOALS
(asm_simp_tac (list_ss addsimps [list_add_app, add_0_right])));
qed "list_add_rev";
val prems = goal List.thy
"ls: list(list(nat)) ==> list_add(flat(ls)) = list_add(map(list_add,ls))";
by (list_ind_tac "ls" prems 1);
by (ALLGOALS (asm_simp_tac (list_ss addsimps [list_add_app])));
by (REPEAT (ares_tac [refl, list_add_type, map_type, add_commute] 1));
qed "list_add_flat";
(** New induction rule **)
val major::prems = goal List.thy
"[| l: list(A); \
\ P(Nil); \
\ !!x y. [| x: A; y: list(A); P(y) |] ==> P(y @ [x]) \
\ |] ==> P(l)";
by (rtac (major RS rev_rev_ident RS subst) 1);
by (rtac (major RS rev_type RS list.induct) 1);
by (ALLGOALS (asm_simp_tac (list_ss addsimps prems)));
qed "list_append_induct";