(* Title: ZF/List.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Datatype definition of Lists
*)
(*** 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)";
Goal "list(A) = {0} + (A * list(A))";
let open list; val rew = rewrite_rule con_defs in
by (blast_tac (claset() addSIs (map rew intrs) addEs [rew elim]) 1)
end;
qed "list_unfold";
(** Lemmas to justify using "list" in other recursive type definitions **)
Goalw list.defs "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.defs@list.con_defs) "list(univ(A)) <= univ(A)";
by (rtac lfp_lowerbound 1);
by (rtac (A_subset_univ RS univ_mono) 2);
by (blast_tac (claset() 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 "[| 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
"[| 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 (simpset() addsimps prems)));
qed "list_case_type";
(*** List functions ***)
Goal "l: list(A) ==> tl(l) : list(A)";
by (exhaust_tac "l" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps list.intrs)));
qed "tl_type";
(** drop **)
Goal "i:nat ==> drop(i, Nil) = Nil";
by (induct_tac "i" 1);
by (ALLGOALS Asm_simp_tac);
qed "drop_Nil";
Goal "i:nat ==> drop(succ(i), Cons(a,l)) = drop(i,l)";
by (rtac sym 1);
by (induct_tac "i" 1);
by (Simp_tac 1);
by (Asm_simp_tac 1);
qed "drop_succ_Cons";
Addsimps [drop_Nil, drop_succ_Cons];
Goal "[| i:nat; l: list(A) |] ==> drop(i,l) : list(A)";
by (induct_tac "i" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [tl_type])));
qed "drop_type";
Delsimps [drop_SUCC];
(** Type checking -- proved by induction, as usual **)
val prems = Goal
"[| 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(c,h,l) : C(l)";
by (cut_facts_tac prems 1);
by (induct_tac "l" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps prems)));
qed "list_rec_type";
(** map **)
val prems = Goalw [get_def thy "map_list"]
"[| 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";
Goal "l: list(A) ==> map(h,l) : list({h(u). u:A})";
by (etac map_type 1);
by (etac RepFunI 1);
qed "map_type2";
(** length **)
Goalw [get_def thy "length_list"]
"l: list(A) ==> length(l) : nat";
by (REPEAT (ares_tac [list_rec_type, nat_0I, nat_succI] 1));
qed "length_type";
(** app **)
Goalw [get_def thy "op @_list"]
"[| 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 **)
Goalw [get_def thy "rev_list"]
"xs: list(A) ==> rev(xs) : list(A)";
by (REPEAT (ares_tac (list.intrs @ [list_rec_type, app_type]) 1));
qed "rev_type";
(** flat **)
Goalw [get_def thy "flat_list"]
"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 **)
Goalw [get_def thy "set_of_list_list"]
"l: list(A) ==> set_of_list(l) : Pow(A)";
by (etac list_rec_type 1);
by (ALLGOALS (Blast_tac));
qed "set_of_list_type";
Goal "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 (simpset() addsimps [Un_cons])));
qed "set_of_list_append";
(** list_add **)
Goalw [get_def thy "list_add_list"]
"xs: list(nat) ==> list_add(xs) : nat";
by (REPEAT (ares_tac [list_rec_type, nat_0I, add_type] 1));
qed "list_add_type";
val list_typechecks =
list.intrs @
[list_rec_type, map_type, map_type2, app_type, length_type,
rev_type, flat_type, list_add_type];
Addsimps list_typechecks;
simpset_ref() := simpset() setSolver (type_auto_tac list_typechecks);
(*** theorems about map ***)
Goal "l: list(A) ==> map(%u. u, l) = l";
by (induct_tac "l" 1);
by (ALLGOALS Asm_simp_tac);
qed "map_ident";
Addsimps [map_ident];
Goal "l: list(A) ==> map(h, map(j,l)) = map(%u. h(j(u)), l)";
by (induct_tac "l" 1);
by (ALLGOALS Asm_simp_tac);
qed "map_compose";
Goal "xs: list(A) ==> map(h, xs@ys) = map(h,xs) @ map(h,ys)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "map_app_distrib";
Goal "ls: list(list(A)) ==> map(h, flat(ls)) = flat(map(map(h),ls))";
by (induct_tac "ls" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [map_app_distrib])));
qed "map_flat";
Goal "l: list(A) ==> \
\ list_rec(c, d, map(h,l)) = \
\ list_rec(c, %x xs r. d(h(x), map(h,xs), r), l)";
by (induct_tac "l" 1);
by (ALLGOALS Asm_simp_tac);
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);
Goal "l: list({x:A. h(x)=j(x)}) ==> map(h,l) = map(j,l)";
by (induct_tac "l" 1);
by (ALLGOALS Asm_simp_tac);
qed "map_list_Collect";
(*** theorems about length ***)
Goal "xs: list(A) ==> length(map(h,xs)) = length(xs)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "length_map";
Goal "xs: list(A) ==> length(xs@ys) = length(xs) #+ length(ys)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "length_app";
Goal "xs: list(A) ==> length(rev(xs)) = length(xs)";
by (induct_tac "xs" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [length_app])));
qed "length_rev";
Goal "ls: list(list(A)) ==> length(flat(ls)) = list_add(map(length,ls))";
by (induct_tac "ls" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [length_app])));
qed "length_flat";
(** Length and drop **)
(*Lemma for the inductive step of drop_length*)
Goal "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);
by (Blast_tac 1);
qed_spec_mp "drop_length_Cons";
Goal "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);
by Safe_tac;
by (etac drop_length_Cons 1);
by (rtac natE 1);
by (etac ([asm_rl, length_type, Ord_nat] MRS Ord_trans) 1);
by (assume_tac 1);
by (ALLGOALS Asm_simp_tac);
by (ALLGOALS (blast_tac (claset() addIs [succ_in_naturalD, length_type])));
qed_spec_mp "drop_length";
(*** theorems about app ***)
Goal "xs: list(A) ==> xs@Nil=xs";
by (etac list.induct 1);
by (ALLGOALS Asm_simp_tac);
qed "app_right_Nil";
Addsimps [app_right_Nil];
Goal "xs: list(A) ==> (xs@ys)@zs = xs@(ys@zs)";
by (induct_tac "xs" 1);
by (ALLGOALS Asm_simp_tac);
qed "app_assoc";
Goal "ls: list(list(A)) ==> flat(ls@ms) = flat(ls)@flat(ms)";
by (induct_tac "ls" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [app_assoc])));
qed "flat_app_distrib";
(*** theorems about rev ***)
Goal "l: list(A) ==> rev(map(h,l)) = map(h,rev(l))";
by (induct_tac "l" 1);
by (ALLGOALS (asm_simp_tac (simpset() 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 "[| xs: list(A); ys: list(A) |] ==> rev(xs@ys) = rev(ys)@rev(xs)";
by (etac list.induct 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [app_assoc])));
qed "rev_app_distrib";
Goal "l: list(A) ==> rev(rev(l))=l";
by (induct_tac "l" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [rev_app_distrib])));
qed "rev_rev_ident";
Addsimps [rev_rev_ident];
Goal "ls: list(list(A)) ==> rev(flat(ls)) = flat(map(rev,rev(ls)))";
by (induct_tac "ls" 1);
by (ALLGOALS
(asm_simp_tac (simpset() addsimps
[map_app_distrib, flat_app_distrib, rev_app_distrib])));
qed "rev_flat";
(*** theorems about list_add ***)
Goal "[| xs: list(nat); ys: list(nat) |] ==> \
\ list_add(xs@ys) = list_add(ys) #+ list_add(xs)";
by (induct_tac "xs" 1);
by (ALLGOALS
(asm_simp_tac (simpset() 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";
Goal "l: list(nat) ==> list_add(rev(l)) = list_add(l)";
by (induct_tac "l" 1);
by (ALLGOALS
(asm_simp_tac (simpset() addsimps [list_add_app, add_0_right])));
qed "list_add_rev";
Goal "ls: list(list(nat)) ==> list_add(flat(ls)) = list_add(map(list_add,ls))";
by (induct_tac "ls" 1);
by (ALLGOALS (asm_simp_tac (simpset() 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
"[| 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 (simpset() addsimps prems)));
qed "list_append_induct";