src/HOL/Nat.ML
author berghofe
Fri Jul 24 13:34:59 1998 +0200 (1998-07-24)
changeset 5188 633ec5f6c155
parent 5069 3ea049f7979d
child 5316 7a8975451a89
permissions -rw-r--r--
Declaration of type 'nat' as a datatype (this allows usage of
exhaust_tac and induct_tac on type 'nat'). Moved some proofs
using natE from NatDef.ML to Nat.ML.
     1 (*  Title:      HOL/Nat.ML
     2     ID:         $Id$
     3     Author:     Tobias Nipkow
     4     Copyright   1997 TU Muenchen
     5 *)
     6 
     7 (** conversion rules for nat_rec **)
     8 
     9 val [nat_rec_0, nat_rec_Suc] = nat.recs;
    10 
    11 (*These 2 rules ease the use of primitive recursion.  NOTE USE OF == *)
    12 val prems = goal thy
    13     "[| !!n. f(n) == nat_rec c h n |] ==> f(0) = c";
    14 by (simp_tac (simpset() addsimps prems) 1);
    15 qed "def_nat_rec_0";
    16 
    17 val prems = goal thy
    18     "[| !!n. f(n) == nat_rec c h n |] ==> f(Suc(n)) = h n (f n)";
    19 by (simp_tac (simpset() addsimps prems) 1);
    20 qed "def_nat_rec_Suc";
    21 
    22 val [nat_case_0, nat_case_Suc] = nat.cases;
    23 
    24 Goal "n ~= 0 ==> EX m. n = Suc m";
    25 by (exhaust_tac "n" 1);
    26 by (REPEAT (Blast_tac 1));
    27 qed "not0_implies_Suc";
    28 
    29 val prems = goal thy "m<n ==> n ~= 0";
    30 by (exhaust_tac "n" 1);
    31 by (cut_facts_tac prems 1);
    32 by (ALLGOALS Asm_full_simp_tac);
    33 qed "gr_implies_not0";
    34 
    35 Goal "(n ~= 0) = (0 < n)";
    36 by (exhaust_tac "n" 1);
    37 by (Blast_tac 1);
    38 by (Blast_tac 1);
    39 qed "neq0_conv";
    40 AddIffs [neq0_conv];
    41 
    42 (*This theorem is useful with blast_tac: (n=0 ==> False) ==> 0<n *)
    43 bind_thm ("gr0I", [neq0_conv, notI] MRS iffD1);
    44 
    45 Goal "(~(0 < n)) = (n=0)";
    46 by (rtac iffI 1);
    47  by (etac swap 1);
    48  by (ALLGOALS Asm_full_simp_tac);
    49 qed "not_gr0";
    50 Addsimps [not_gr0];
    51 
    52 Goal "m<n ==> 0 < n";
    53 by (dtac gr_implies_not0 1);
    54 by (Asm_full_simp_tac 1);
    55 qed "gr_implies_gr0";
    56 Addsimps [gr_implies_gr0];
    57 
    58 qed_goalw "Least_Suc" thy [Least_nat_def]
    59  "!!P. [| ? n. P(Suc n); ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
    60  (fn _ => [
    61         rtac select_equality 1,
    62         fold_goals_tac [Least_nat_def],
    63         safe_tac (claset() addSEs [LeastI]),
    64         rename_tac "j" 1,
    65         exhaust_tac "j" 1,
    66         Blast_tac 1,
    67         blast_tac (claset() addDs [Suc_less_SucD, not_less_Least]) 1,
    68         rename_tac "k n" 1,
    69         exhaust_tac "k" 1,
    70         Blast_tac 1,
    71         hyp_subst_tac 1,
    72         rewtac Least_nat_def,
    73         rtac (select_equality RS arg_cong RS sym) 1,
    74         Safe_tac,
    75         dtac Suc_mono 1,
    76         Blast_tac 1,
    77         cut_facts_tac [less_linear] 1,
    78         Safe_tac,
    79         atac 2,
    80         Blast_tac 2,
    81         dtac Suc_mono 1,
    82         Blast_tac 1]);
    83 
    84 qed_goal "nat_induct2" thy 
    85 "[| P 0; P 1; !!k. P k ==> P (Suc (Suc k)) |] ==> P n" (fn prems => [
    86         cut_facts_tac prems 1,
    87         rtac less_induct 1,
    88         exhaust_tac "n" 1,
    89          hyp_subst_tac 1,
    90          atac 1,
    91         hyp_subst_tac 1,
    92         exhaust_tac "nat" 1,
    93          hyp_subst_tac 1,
    94          atac 1,
    95         hyp_subst_tac 1,
    96         resolve_tac prems 1,
    97         dtac spec 1,
    98         etac mp 1,
    99         rtac (lessI RS less_trans) 1,
   100         rtac (lessI RS Suc_mono) 1]);
   101 
   102 Goal "min 0 n = 0";
   103 by (rtac min_leastL 1);
   104 by (trans_tac 1);
   105 qed "min_0L";
   106 
   107 Goal "min n 0 = 0";
   108 by (rtac min_leastR 1);
   109 by (trans_tac 1);
   110 qed "min_0R";
   111 
   112 Goalw [min_def] "min (Suc m) (Suc n) = Suc(min m n)";
   113 by (Simp_tac 1);
   114 qed "min_Suc_Suc";
   115 
   116 Addsimps [min_0L,min_0R,min_Suc_Suc];