(* Title: ZF/bt.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
For bt.thy. Binary trees
*)
open BT_Fn;
(** bt_rec -- by Vset recursion **)
goalw BT.thy BT.con_defs "rank(l) < rank(Br(a,l,r))";
by (simp_tac rank_ss 1);
val rank_Br1 = result();
goalw BT.thy BT.con_defs "rank(r) < rank(Br(a,l,r))";
by (simp_tac rank_ss 1);
val rank_Br2 = result();
goal BT_Fn.thy "bt_rec(Lf,c,h) = c";
by (rtac (bt_rec_def RS def_Vrec RS trans) 1);
by (simp_tac (ZF_ss addsimps BT.case_eqns) 1);
val bt_rec_Lf = result();
goal BT_Fn.thy
"bt_rec(Br(a,l,r), c, h) = h(a, l, r, bt_rec(l,c,h), bt_rec(r,c,h))";
by (rtac (bt_rec_def RS def_Vrec RS trans) 1);
by (simp_tac (rank_ss addsimps (BT.case_eqns @ [rank_Br1, rank_Br2])) 1);
val bt_rec_Br = result();
(*Type checking -- proved by induction, as usual*)
val prems = goal BT_Fn.thy
"[| t: bt(A); \
\ c: C(Lf); \
\ !!x y z r s. [| x:A; y:bt(A); z:bt(A); r:C(y); s:C(z) |] ==> \
\ h(x,y,z,r,s): C(Br(x,y,z)) \
\ |] ==> bt_rec(t,c,h) : C(t)";
by (bt_ind_tac "t" prems 1);
by (ALLGOALS (asm_simp_tac (ZF_ss addsimps
(prems@[bt_rec_Lf,bt_rec_Br]))));
val bt_rec_type = result();
(** Versions for use with definitions **)
val [rew] = goal BT_Fn.thy "[| !!t. j(t)==bt_rec(t, c, h) |] ==> j(Lf) = c";
by (rewtac rew);
by (rtac bt_rec_Lf 1);
val def_bt_rec_Lf = result();
val [rew] = goal BT_Fn.thy
"[| !!t. j(t)==bt_rec(t, c, h) |] ==> j(Br(a,l,r)) = h(a,l,r,j(l),j(r))";
by (rewtac rew);
by (rtac bt_rec_Br 1);
val def_bt_rec_Br = result();
fun bt_recs def = map standard ([def] RL [def_bt_rec_Lf, def_bt_rec_Br]);
(** n_nodes **)
val [n_nodes_Lf,n_nodes_Br] = bt_recs n_nodes_def;
val prems = goalw BT_Fn.thy [n_nodes_def]
"xs: bt(A) ==> n_nodes(xs) : nat";
by (REPEAT (ares_tac (prems @ [bt_rec_type, nat_0I, nat_succI, add_type]) 1));
val n_nodes_type = result();
(** n_leaves **)
val [n_leaves_Lf,n_leaves_Br] = bt_recs n_leaves_def;
val prems = goalw BT_Fn.thy [n_leaves_def]
"xs: bt(A) ==> n_leaves(xs) : nat";
by (REPEAT (ares_tac (prems @ [bt_rec_type, nat_0I, nat_succI, add_type]) 1));
val n_leaves_type = result();
(** bt_reflect **)
val [bt_reflect_Lf, bt_reflect_Br] = bt_recs bt_reflect_def;
val prems = goalw BT_Fn.thy [bt_reflect_def]
"xs: bt(A) ==> bt_reflect(xs) : bt(A)";
by (REPEAT (ares_tac (prems @ [bt_rec_type, LfI, BrI]) 1));
val bt_reflect_type = result();
(** BT_Fn simplification **)
val bt_typechecks =
[LfI, BrI, bt_rec_type, n_nodes_type, n_leaves_type, bt_reflect_type];
val bt_ss = arith_ss
addsimps BT.case_eqns
addsimps bt_typechecks
addsimps [bt_rec_Lf, bt_rec_Br,
n_nodes_Lf, n_nodes_Br,
n_leaves_Lf, n_leaves_Br,
bt_reflect_Lf, bt_reflect_Br];
(*** theorems about n_leaves ***)
val prems = goal BT_Fn.thy
"t: bt(A) ==> n_leaves(bt_reflect(t)) = n_leaves(t)";
by (bt_ind_tac "t" prems 1);
by (ALLGOALS (asm_simp_tac bt_ss));
by (REPEAT (ares_tac [add_commute, n_leaves_type] 1));
val n_leaves_reflect = result();
val prems = goal BT_Fn.thy
"t: bt(A) ==> n_leaves(t) = succ(n_nodes(t))";
by (bt_ind_tac "t" prems 1);
by (ALLGOALS (asm_simp_tac (bt_ss addsimps [add_succ_right])));
val n_leaves_nodes = result();
(*** theorems about bt_reflect ***)
val prems = goal BT_Fn.thy
"t: bt(A) ==> bt_reflect(bt_reflect(t))=t";
by (bt_ind_tac "t" prems 1);
by (ALLGOALS (asm_simp_tac bt_ss));
val bt_reflect_bt_reflect_ident = result();