# File Nat.ML

    (*  Title:      FOLP/ex/nat.ML
        ID:         $Id: Nat.ML,v 1.4 2005/09/18 12:25:49 wenzelm Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
        Copyright   1992  University of Cambridge
    *)

    Goal "?p : ~ (Suc(k) = k)";
    by (res_inst_tac [("n","k")] induct 1);
    by (rtac notI 1);
    by (etac Suc_neq_0 1);
    by (rtac notI 1);
    by (etac notE 1);
    by (etac Suc_inject 1);
    val Suc_n_not_n = result();

    Goal "?p : (k+m)+n = k+(m+n)";
    prths ([induct] RL [topthm()]);  (*prints all 14 next states!*)
    by (rtac induct 1);
    back();
    back();
    back();
    back();
    back();
    back();

    Goalw [add_def] "?p : 0+n = n";
    by (rtac rec_0 1);
    val add_0 = result();

    Goalw [add_def] "?p : Suc(m)+n = Suc(m+n)";
    by (rtac rec_Suc 1);
    val add_Suc = result();

    (*
    val nat_congs = mk_congs (the_context ()) ["Suc", "op +"];
    prths nat_congs;
    *)
    val prems = goal (the_context ()) "p: x=y ==> ?p : Suc(x) = Suc(y)";
    by (resolve_tac (prems RL [subst]) 1);
    by (rtac refl 1);
    val Suc_cong = result();

    val prems = goal (the_context ()) "[| p : a=x;  q: b=y |] ==> ?p : a+b=x+y";
    by (resolve_tac (prems RL [subst]) 1 THEN resolve_tac (prems RL [subst]) 1 THEN
        rtac refl 1);
    val Plus_cong = result();

    val nat_congs = [Suc_cong,Plus_cong];

    val add_ss = FOLP_ss  addcongs nat_congs
                         addrews  [add_0, add_Suc];

    Goal "?p : (k+m)+n = k+(m+n)";
    by (res_inst_tac [("n","k")] induct 1);
    by (SIMP_TAC add_ss 1);
    by (ASM_SIMP_TAC add_ss 1);
    val add_assoc = result();

    Goal "?p : m+0 = m";
    by (res_inst_tac [("n","m")] induct 1);
    by (SIMP_TAC add_ss 1);
    by (ASM_SIMP_TAC add_ss 1);
    val add_0_right = result();

    Goal "?p : m+Suc(n) = Suc(m+n)";
    by (res_inst_tac [("n","m")] induct 1);
    by (ALLGOALS (ASM_SIMP_TAC add_ss));
    val add_Suc_right = result();

    (*mk_typed_congs appears not to work with FOLP's version of subst*)