src/FOL/ex/List.ML

updated version

(*  Title: 	FOL/ex/list
    ID:         $Id$
    Author: 	Tobias Nipkow
    Copyright   1991  University of Cambridge

For ex/list.thy.  Examples of simplification and induction on lists
*)

open List;

val prems = goal List.thy "[| P([]);  !!x l. P(x.l) |] ==> All(P)";
by (rtac list_ind 1);
by (REPEAT (resolve_tac (prems@[allI,impI]) 1));
qed "list_exh";

val list_rew_thms = [list_distinct1,list_distinct2,app_nil,app_cons,
		     hd_eq,tl_eq,forall_nil,forall_cons,list_free,
		     len_nil,len_cons,at_0,at_succ];

val list_ss = nat_ss addsimps list_rew_thms;

goal List.thy "~l=[] --> hd(l).tl(l) = l";
by(IND_TAC list_exh (simp_tac list_ss) "l" 1);
result();

goal List.thy "(l1++l2)++l3 = l1++(l2++l3)";
by(IND_TAC list_ind (simp_tac list_ss) "l1" 1);
qed "append_assoc";

goal List.thy "l++[] = l";
by(IND_TAC list_ind (simp_tac list_ss) "l" 1);
qed "app_nil_right";

goal List.thy "l1++l2=[] <-> l1=[] & l2=[]";
by(IND_TAC list_exh (simp_tac list_ss) "l1" 1);
qed "app_eq_nil_iff";

goal List.thy "forall(l++l',P) <-> forall(l,P) & forall(l',P)";
by(IND_TAC list_ind (simp_tac list_ss) "l" 1);
qed "forall_app";

goal List.thy "forall(l,%x.P(x)&Q(x)) <-> forall(l,P) & forall(l,Q)";
by(IND_TAC list_ind (simp_tac list_ss) "l" 1);
by(fast_tac FOL_cs 1);
qed "forall_conj";

goal List.thy "~l=[] --> forall(l,P) <-> P(hd(l)) & forall(tl(l),P)";
by(IND_TAC list_ind (simp_tac list_ss) "l" 1);
qed "forall_ne";

(*** Lists with natural numbers ***)

goal List.thy "len(l1++l2) = len(l1)+len(l2)";
by (IND_TAC list_ind (simp_tac list_ss) "l1" 1);
qed "len_app";

goal List.thy "i<len(l1) --> at(l1++l2,i) = at(l1,i)";
by (IND_TAC list_ind (simp_tac list_ss) "l1" 1);
by (REPEAT (rtac allI 1));
by (rtac impI 1);
by (ALL_IND_TAC nat_exh (asm_simp_tac list_ss) 1);
qed "at_app1";

goal List.thy "at(l1++(x.l2), len(l1)) = x";
by (IND_TAC list_ind (simp_tac list_ss) "l1" 1);
qed "at_app_hd2";