src/ZF/ListFn.ML

Initial revision

(*  Title: 	ZF/list-fn.ML
    ID:         $Id$
    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1992  University of Cambridge

For list-fn.thy.  Lists in Zermelo-Fraenkel Set Theory 
*)

open ListFn;


(** list_rec -- by Vset recursion **)

(*Used to verify list_rec*)
val list_rec_ss = ZF_ss 
      addcongs (mk_typed_congs ListFn.thy [("h", "[i,i,i]=>i")])
      addrews List.case_eqns;

goal ListFn.thy "list_rec(Nil,c,h) = c";
by (rtac (list_rec_def RS def_Vrec RS trans) 1);
by (SIMP_TAC list_rec_ss 1);
val list_rec_Nil = result();

goal ListFn.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 (list_rec_ss addrews [Vset_rankI, rank_Cons2]) 1);
val list_rec_Cons = result();

(*Type checking -- proved by induction, as usual*)
val prems = goal ListFn.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 addrews (prems@[list_rec_Nil,list_rec_Cons]))));
val list_rec_type = result();

(** Versions for use with definitions **)

val [rew] = goal ListFn.thy
    "[| !!l. j(l)==list_rec(l, c, h) |] ==> j(Nil) = c";
by (rewtac rew);
by (rtac list_rec_Nil 1);
val def_list_rec_Nil = result();

val [rew] = goal ListFn.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);
val def_list_rec_Cons = result();

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 ListFn.thy [map_def] 
    "[| l: list(A);  !!x. x: A ==> h(x): B |] ==> map(h,l) : list(B)";
by (REPEAT (ares_tac (prems@[list_rec_type, NilI, ConsI]) 1));
val map_type = result();

val [major] = goal ListFn.thy "l: list(A) ==> map(h,l) : list({h(u). u:A})";
by (rtac (major RS map_type) 1);
by (etac RepFunI 1);
val map_type2 = result();

(** length **)

val [length_Nil,length_Cons] = list_recs length_def;

val prems = goalw ListFn.thy [length_def] 
    "l: list(A) ==> length(l) : nat";
by (REPEAT (ares_tac (prems @ [list_rec_type, nat_0I, nat_succI]) 1));
val length_type = result();

(** app **)

val [app_Nil,app_Cons] = list_recs app_def;

val prems = goalw ListFn.thy [app_def] 
    "[| xs: list(A);  ys: list(A) |] ==> xs@ys : list(A)";
by (REPEAT (ares_tac (prems @ [list_rec_type, ConsI]) 1));
val app_type = result();

(** rev **)

val [rev_Nil,rev_Cons] = list_recs rev_def;

val prems = goalw ListFn.thy [rev_def] 
    "xs: list(A) ==> rev(xs) : list(A)";
by (REPEAT (ares_tac (prems @ [list_rec_type, NilI, ConsI, app_type]) 1));
val rev_type = result();


(** flat **)

val [flat_Nil,flat_Cons] = list_recs flat_def;

val prems = goalw ListFn.thy [flat_def] 
    "ls: list(list(A)) ==> flat(ls) : list(A)";
by (REPEAT (ares_tac (prems @ [list_rec_type, NilI, ConsI, app_type]) 1));
val flat_type = result();


(** list_add **)

val [list_add_Nil,list_add_Cons] = list_recs list_add_def;

val prems = goalw ListFn.thy [list_add_def] 
    "xs: list(nat) ==> list_add(xs) : nat";
by (REPEAT (ares_tac (prems @ [list_rec_type, nat_0I, add_type]) 1));
val list_add_type = result();

(** ListFn simplification **)

val list_typechecks =
      [NilI, ConsI, list_rec_type,
       map_type, map_type2, app_type, length_type, rev_type, flat_type,
       list_add_type];

val list_congs = 
    List.congs @ 
    mk_congs ListFn.thy
        ["list_rec","map","op @","length","rev","flat","list_add"];

val list_ss = arith_ss 
    addcongs list_congs
    addrews List.case_eqns
    addrews list_typechecks
    addrews [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];

(*** theorems about map ***)

val prems = goal ListFn.thy
    "l: list(A) ==> map(%u.u, l) = l";
by (list_ind_tac "l" prems 1);
by (ALLGOALS (ASM_SIMP_TAC list_ss));
val map_ident = result();

val prems = goal ListFn.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));
val map_compose = result();

val prems = goal ListFn.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));
val map_app_distrib = result();

val prems = goal ListFn.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 addrews [map_app_distrib])));
val map_flat = result();

val prems = goal ListFn.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 addcongs (mk_typed_congs ListFn.thy [("d", "[i,i,i]=>i")]))));
val list_rec_map = result();

(** theorems about list(Collect(A,P)) -- used in ex/term.ML **)

(* c : list(Collect(B,P)) ==> c : list(B) *)
val list_CollectD = standard (Collect_subset RS list_mono RS subsetD);

val prems = goal ListFn.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));
val map_list_Collect = result();

(*** theorems about length ***)

val prems = goal ListFn.thy
    "xs: list(A) ==> length(map(h,xs)) = length(xs)";
by (list_ind_tac "xs" prems 1);
by (ALLGOALS (ASM_SIMP_TAC list_ss));
val length_map = result();

val prems = goal ListFn.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));
val length_app = result();

(* [| 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 ListFn.thy
    "xs: list(A) ==> length(rev(xs)) = length(xs)";
by (list_ind_tac "xs" prems 1);
by (ALLGOALS (ASM_SIMP_TAC (list_ss addrews [length_app, add_commute_succ])));
val length_rev = result();

val prems = goal ListFn.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 addrews [length_app])));
val length_flat = result();

(*** theorems about app ***)

val [major] = goal ListFn.thy "xs: list(A) ==> xs@Nil=xs";
by (rtac (major RS List.induct) 1);
by (ALLGOALS (ASM_SIMP_TAC list_ss));
val app_right_Nil = result();

val prems = goal ListFn.thy "xs: list(A) ==> (xs@ys)@zs = xs@(ys@zs)";
by (list_ind_tac "xs" prems 1);
by (ALLGOALS (ASM_SIMP_TAC list_ss));
val app_assoc = result();

val prems = goal ListFn.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 addrews [app_assoc])));
val flat_app_distrib = result();

(*** theorems about rev ***)

val prems = goal ListFn.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 addrews [map_app_distrib])));
val rev_map_distrib = result();

(*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.
*)
val prems = goal ListFn.thy
    "[| xs: list(A);  ys: list(A) |] ==> rev(xs@ys) = rev(ys)@rev(xs)";
by (cut_facts_tac prems 1);
by (etac List.induct 1);
by (ALLGOALS (ASM_SIMP_TAC (list_ss addrews [app_right_Nil,app_assoc])));
val rev_app_distrib = result();

val prems = goal ListFn.thy "l: list(A) ==> rev(rev(l))=l";
by (list_ind_tac "l" prems 1);
by (ALLGOALS (ASM_SIMP_TAC (list_ss addrews [rev_app_distrib])));
val rev_rev_ident = result();

val prems = goal ListFn.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 addrews 
       [map_app_distrib, flat_app_distrib, rev_app_distrib, app_right_Nil])));
val rev_flat = result();


(*** theorems about list_add ***)

val prems = goal ListFn.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 addrews [add_0_right, add_assoc RS sym])));
by (resolve_tac arith_congs 1);
by (REPEAT (ares_tac [refl, list_add_type, add_commute] 1));
val list_add_app = result();

val prems = goal ListFn.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 addrews [list_add_app, add_0_right])));
val list_add_rev = result();

val prems = goal ListFn.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 addrews [list_add_app])));
by (REPEAT (ares_tac [refl, list_add_type, map_type, add_commute] 1));
val list_add_flat = result();

(** New induction rule **)

val major::prems = goal ListFn.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 addrews prems)));
val list_append_induct = result();