# File Nat.ML

    (*  Title:      HOL/Nat.ML
        ID:         $Id: Nat.ML,v 1.80 2005/09/06 14:59:48 wenzelm Exp $
    *)

    (** legacy ML bindings **)

    structure nat =
    struct
      val distinct = thms "nat.distinct";
      val inject = thms "nat.inject";
      val exhaust = thm "nat.exhaust";
      val cases = thms "nat.cases";
      val split = thm "nat.split";
      val split_asm = thm "nat.split_asm";
      val induct = thm "nat.induct";
      val recs = thms "nat.recs";
      val simps = thms "nat.simps";
    end;

    val [nat_rec_0, nat_rec_Suc] = thms "nat.recs";
    bind_thm ("nat_rec_0", nat_rec_0);
    bind_thm ("nat_rec_Suc", nat_rec_Suc);

    val [nat_case_0, nat_case_Suc] = thms "nat.cases";
    bind_thm ("nat_case_0", nat_case_0);
    bind_thm ("nat_case_Suc", nat_case_Suc);

    val leD = thm "leD";  (*Now defined in Orderings.thy*)
    val leI = thm "leI";

    val Least_Suc = thm "Least_Suc";
    val Least_Suc2 = thm "Least_Suc2";
    val One_nat_def = thm "One_nat_def";
    val Suc_Suc_eq = thm "Suc_Suc_eq";
    val Suc_def = thm "Suc_def";
    val Suc_diff_diff = thm "Suc_diff_diff";
    val Suc_diff_le = thm "Suc_diff_le";
    val Suc_inject = thm "Suc_inject";
    val Suc_leD = thm "Suc_leD";
    val Suc_leI = thm "Suc_leI";
    val Suc_le_D = thm "Suc_le_D";
    val Suc_le_eq = thm "Suc_le_eq";
    val Suc_le_lessD = thm "Suc_le_lessD";
    val Suc_le_mono = thm "Suc_le_mono";
    val Suc_lessD = thm "Suc_lessD";
    val Suc_lessE = thm "Suc_lessE";
    val Suc_lessI = thm "Suc_lessI";
    val Suc_less_SucD = thm "Suc_less_SucD";
    val Suc_less_eq = thm "Suc_less_eq";
    val Suc_mono = thm "Suc_mono";
    val Suc_mult_cancel1 = thm "Suc_mult_cancel1";
    val Suc_mult_le_cancel1 = thm "Suc_mult_le_cancel1";
    val Suc_mult_less_cancel1 = thm "Suc_mult_less_cancel1";
    val Suc_n_not_le_n = thm "Suc_n_not_le_n";
    val Suc_n_not_n = thm "Suc_n_not_n";
    val Suc_neq_Zero = thm "Suc_neq_Zero";
    val Suc_not_Zero = thm "Suc_not_Zero";
    val Suc_pred = thm "Suc_pred";
    val Zero_nat_def = thm "Zero_nat_def";
    val Zero_neq_Suc = thm "Zero_neq_Suc";
    val Zero_not_Suc = thm "Zero_not_Suc";
    val add_0 = thm "add_0";
    val add_0_right = thm "add_0_right";
    val add_Suc = thm "add_Suc";
    val add_Suc_right = thm "add_Suc_right";
    val add_ac = thms "add_ac";
    val add_assoc = thm "add_assoc";
    val add_commute = thm "add_commute";
    val add_diff_inverse = thm "add_diff_inverse";
    val add_eq_self_zero = thm "add_eq_self_zero";
    val add_gr_0 = thm "add_gr_0";
    val add_is_0 = thm "add_is_0";
    val add_is_1 = thm "add_is_1";
    val add_leD1 = thm "add_leD1";
    val add_leD2 = thm "add_leD2";
    val add_leE = thm "add_leE";
    val add_le_mono = thm "add_le_mono";
    val add_le_mono1 = thm "add_le_mono1";
    val nat_add_left_cancel = thm "nat_add_left_cancel";
    val nat_add_left_cancel_le = thm "nat_add_left_cancel_le";
    val nat_add_left_cancel_less = thm "nat_add_left_cancel_less";
    val add_left_commute = thm "add_left_commute";
    val add_lessD1 = thm "add_lessD1";
    val add_less_mono = thm "add_less_mono";
    val add_less_mono1 = thm "add_less_mono1";
    val add_mult_distrib = thm "add_mult_distrib";
    val add_mult_distrib2 = thm "add_mult_distrib2";
    val nat_add_right_cancel = thm "nat_add_right_cancel";
    val def_nat_rec_0 = thm "def_nat_rec_0";
    val def_nat_rec_Suc = thm "def_nat_rec_Suc";
    val diff_0 = thm "diff_0";
    val diff_0_eq_0 = thm "diff_0_eq_0";
    val diff_Suc = thm "diff_Suc";
    val diff_Suc_Suc = thm "diff_Suc_Suc";
    val diff_Suc_less = thm "diff_Suc_less";
    val diff_add_0 = thm "diff_add_0";
    val diff_add_assoc = thm "diff_add_assoc";
    val diff_add_assoc2 = thm "diff_add_assoc2";
    val diff_add_inverse = thm "diff_add_inverse";
    val diff_add_inverse2 = thm "diff_add_inverse2";
    val diff_cancel = thm "diff_cancel";
    val diff_cancel2 = thm "diff_cancel2";
    val diff_commute = thm "diff_commute";
    val diff_diff_left = thm "diff_diff_left";
    val diff_induct = thm "diff_induct";
    val diff_is_0_eq = thm "diff_is_0_eq";
    val diff_le_self = thm "diff_le_self";
    val diff_less_Suc = thm "diff_less_Suc";
    val diff_mult_distrib = thm "diff_mult_distrib";
    val diff_mult_distrib2 = thm "diff_mult_distrib2";
    val diff_self_eq_0 = thm "diff_self_eq_0";
    val eq_imp_le = thm "eq_imp_le";
    val gr0I = thm "gr0I";
    val gr0_conv_Suc = thm "gr0_conv_Suc";
    val gr_implies_not0 = thm "gr_implies_not0";
    val inj_Suc = thm "inj_Suc";
    val le0 = thm "le0";
    val le_0_eq = thm "le_0_eq";
    val le_SucE = thm "le_SucE";
    val le_SucI = thm "le_SucI";
    val le_Suc_eq = thm "le_Suc_eq";
    val le_add1 = thm "le_add1";
    val le_add2 = thm "le_add2";
    val le_add_diff_inverse = thm "le_add_diff_inverse";
    val le_add_diff_inverse2 = thm "le_add_diff_inverse2";
    val le_anti_sym = thm "le_anti_sym";
    val le_def = thm "le_def";
    val le_eq_less_or_eq = thm "le_eq_less_or_eq";
    val le_imp_diff_is_add = thm "le_imp_diff_is_add";
    val le_imp_less_Suc = thm "le_imp_less_Suc";
    val le_imp_less_or_eq = thm "le_imp_less_or_eq";
    val le_less_trans = thm "le_less_trans";
    val le_neq_implies_less = thm "le_neq_implies_less";
    val le_refl = thm "le_refl";
    val le_simps = thms "le_simps";
    val le_trans = thm "le_trans";
    val lessE = thm "lessE";
    val lessI = thm "lessI";
    val less_Suc0 = thm "less_Suc0";
    val less_SucE = thm "less_SucE";
    val less_SucI = thm "less_SucI";
    val less_Suc_eq = thm "less_Suc_eq";
    val less_Suc_eq_0_disj = thm "less_Suc_eq_0_disj";
    val less_Suc_eq_le = thm "less_Suc_eq_le";
    val less_add_Suc1 = thm "less_add_Suc1";
    val less_add_Suc2 = thm "less_add_Suc2";
    val less_add_eq_less = thm "less_add_eq_less";
    val less_asym = thm "less_asym";
    val less_def = thm "less_def";
    val less_eq = thm "less_eq";
    val less_iff_Suc_add = thm "less_iff_Suc_add";
    val less_imp_Suc_add = thm "less_imp_Suc_add";
    val less_imp_add_positive = thm "less_imp_add_positive";
    val less_imp_diff_less = thm "less_imp_diff_less";
    val less_imp_le = thm "less_imp_le";
    val less_irrefl = thm "less_irrefl";
    val less_le_trans = thm "less_le_trans";
    val less_linear = thm "less_linear";
    val less_mono_imp_le_mono = thm "less_mono_imp_le_mono";
    val less_not_refl = thm "less_not_refl";
    val less_not_refl2 = thm "less_not_refl2";
    val less_not_refl3 = thm "less_not_refl3";
    val less_not_sym = thm "less_not_sym";
    val less_one = thm "less_one";
    val less_or_eq_imp_le = thm "less_or_eq_imp_le";
    val less_trans = thm "less_trans";
    val less_trans_Suc = thm "less_trans_Suc";
    val less_zeroE = thm "less_zeroE";
    val max_0L = thm "max_0L";
    val max_0R = thm "max_0R";
    val max_Suc_Suc = thm "max_Suc_Suc";
    val min_0L = thm "min_0L";
    val min_0R = thm "min_0R";
    val min_Suc_Suc = thm "min_Suc_Suc";
    val mult_0 = thm "mult_0";
    val mult_0_right = thm "mult_0_right";
    val mult_1 = thm "mult_1";
    val mult_1_right = thm "mult_1_right";
    val mult_Suc = thm "mult_Suc";
    val mult_Suc_right = thm "mult_Suc_right";
    val mult_ac = thms "mult_ac";
    val mult_assoc = thm "mult_assoc";
    val mult_cancel1 = thm "mult_cancel1";
    val mult_cancel2 = thm "mult_cancel2";
    val mult_commute = thm "mult_commute";
    val mult_eq_1_iff = thm "mult_eq_1_iff";
    val mult_eq_self_implies_10 = thm "mult_eq_self_implies_10";
    val mult_is_0 = thm "mult_is_0";
    val mult_le_cancel1 = thm "mult_le_cancel1";
    val mult_le_cancel2 = thm "mult_le_cancel2";
    val mult_le_mono = thm "mult_le_mono";
    val mult_le_mono1 = thm "mult_le_mono1";
    val mult_le_mono2 = thm "mult_le_mono2";
    val mult_left_commute = thm "mult_left_commute";
    val mult_less_cancel1 = thm "mult_less_cancel1";
    val mult_less_cancel2 = thm "mult_less_cancel2";
    val mult_less_mono1 = thm "mult_less_mono1";
    val mult_less_mono2 = thm "mult_less_mono2";
    val n_not_Suc_n = thm "n_not_Suc_n";
    val nat_distrib = thms "nat_distrib";
    val nat_induct = thm "nat_induct";
    val nat_induct2 = thm "nat_induct2";
    val nat_le_linear = thm "nat_le_linear";
    val nat_less_cases = thm "nat_less_cases";
    val nat_less_induct = thm "nat_less_induct";
    val nat_less_le = thm "nat_less_le";
    val nat_neq_iff = thm "nat_neq_iff";
    val nat_not_singleton = thm "nat_not_singleton";
    val neq0_conv = thm "neq0_conv";
    val not0_implies_Suc = thm "not0_implies_Suc";
    val not_add_less1 = thm "not_add_less1";
    val not_add_less2 = thm "not_add_less2";
    val not_gr0 = thm "not_gr0";
    val not_leE = thm "not_leE";
    val not_less0 = thm "not_less0";
    val not_less_eq = thm "not_less_eq";
    val not_less_less_Suc_eq = thm "not_less_less_Suc_eq";
    val not_less_simps = thms "not_less_simps";
    val one_eq_mult_iff = thm "one_eq_mult_iff";
    val one_is_add = thm "one_is_add";
    val one_le_mult_iff = thm "one_le_mult_iff";
    val one_reorient = thm "one_reorient";
    val pred_nat_def = thm "pred_nat_def";
    val trans_le_add1 = thm "trans_le_add1";
    val trans_le_add2 = thm "trans_le_add2";
    val trans_less_add1 = thm "trans_less_add1";
    val trans_less_add2 = thm "trans_less_add2";
    val wf_less = thm "wf_less";
    val wf_pred_nat = thm "wf_pred_nat";
    val zero_induct = thm "zero_induct";
    val zero_induct_lemma = thm "zero_induct_lemma";
    val zero_less_Suc = thm "zero_less_Suc";
    val zero_less_diff = thm "zero_less_diff";
    val zero_less_mult_iff = thm "zero_less_mult_iff";
    val zero_reorient = thm "zero_reorient";
    val linorder_neqE_nat = thm "linorder_neqE_nat";