src/HOL/Nat.ML
author clasohm
Tue Jan 30 15:24:36 1996 +0100 (1996-01-30)
changeset 1465 5d7a7e439cec
parent 1327 6c29cfab679c
child 1475 7f5a4cd08209
permissions -rw-r--r--
expanded tabs
clasohm@1465
     1
(*  Title:      HOL/nat
clasohm@923
     2
    ID:         $Id$
clasohm@1465
     3
    Author:     Tobias Nipkow, Cambridge University Computer Laboratory
clasohm@923
     4
    Copyright   1991  University of Cambridge
clasohm@923
     5
clasohm@923
     6
For nat.thy.  Type nat is defined as a set (Nat) over the type ind.
clasohm@923
     7
*)
clasohm@923
     8
clasohm@923
     9
open Nat;
clasohm@923
    10
clasohm@923
    11
goal Nat.thy "mono(%X. {Zero_Rep} Un (Suc_Rep``X))";
clasohm@923
    12
by (REPEAT (ares_tac [monoI, subset_refl, image_mono, Un_mono] 1));
clasohm@923
    13
qed "Nat_fun_mono";
clasohm@923
    14
clasohm@923
    15
val Nat_unfold = Nat_fun_mono RS (Nat_def RS def_lfp_Tarski);
clasohm@923
    16
clasohm@923
    17
(* Zero is a natural number -- this also justifies the type definition*)
clasohm@923
    18
goal Nat.thy "Zero_Rep: Nat";
clasohm@923
    19
by (rtac (Nat_unfold RS ssubst) 1);
clasohm@923
    20
by (rtac (singletonI RS UnI1) 1);
clasohm@923
    21
qed "Zero_RepI";
clasohm@923
    22
clasohm@923
    23
val prems = goal Nat.thy "i: Nat ==> Suc_Rep(i) : Nat";
clasohm@923
    24
by (rtac (Nat_unfold RS ssubst) 1);
clasohm@923
    25
by (rtac (imageI RS UnI2) 1);
clasohm@923
    26
by (resolve_tac prems 1);
clasohm@923
    27
qed "Suc_RepI";
clasohm@923
    28
clasohm@923
    29
(*** Induction ***)
clasohm@923
    30
clasohm@923
    31
val major::prems = goal Nat.thy
clasohm@923
    32
    "[| i: Nat;  P(Zero_Rep);   \
clasohm@923
    33
\       !!j. [| j: Nat; P(j) |] ==> P(Suc_Rep(j)) |]  ==> P(i)";
clasohm@923
    34
by (rtac ([Nat_def, Nat_fun_mono, major] MRS def_induct) 1);
clasohm@923
    35
by (fast_tac (set_cs addIs prems) 1);
clasohm@923
    36
qed "Nat_induct";
clasohm@923
    37
clasohm@923
    38
val prems = goalw Nat.thy [Zero_def,Suc_def]
clasohm@923
    39
    "[| P(0);   \
clasohm@923
    40
\       !!k. P(k) ==> P(Suc(k)) |]  ==> P(n)";
clasohm@923
    41
by (rtac (Rep_Nat_inverse RS subst) 1);   (*types force good instantiation*)
clasohm@923
    42
by (rtac (Rep_Nat RS Nat_induct) 1);
clasohm@923
    43
by (REPEAT (ares_tac prems 1
clasohm@923
    44
     ORELSE eresolve_tac [Abs_Nat_inverse RS subst] 1));
clasohm@923
    45
qed "nat_induct";
clasohm@923
    46
clasohm@923
    47
(*Perform induction on n. *)
clasohm@923
    48
fun nat_ind_tac a i = 
clasohm@923
    49
    EVERY [res_inst_tac [("n",a)] nat_induct i,
clasohm@1465
    50
           rename_last_tac a ["1"] (i+1)];
clasohm@923
    51
clasohm@923
    52
(*A special form of induction for reasoning about m<n and m-n*)
clasohm@923
    53
val prems = goal Nat.thy
clasohm@923
    54
    "[| !!x. P x 0;  \
clasohm@923
    55
\       !!y. P 0 (Suc y);  \
clasohm@923
    56
\       !!x y. [| P x y |] ==> P (Suc x) (Suc y)  \
clasohm@923
    57
\    |] ==> P m n";
clasohm@923
    58
by (res_inst_tac [("x","m")] spec 1);
clasohm@923
    59
by (nat_ind_tac "n" 1);
clasohm@923
    60
by (rtac allI 2);
clasohm@923
    61
by (nat_ind_tac "x" 2);
clasohm@923
    62
by (REPEAT (ares_tac (prems@[allI]) 1 ORELSE etac spec 1));
clasohm@923
    63
qed "diff_induct";
clasohm@923
    64
clasohm@923
    65
(*Case analysis on the natural numbers*)
clasohm@923
    66
val prems = goal Nat.thy 
clasohm@923
    67
    "[| n=0 ==> P;  !!x. n = Suc(x) ==> P |] ==> P";
clasohm@923
    68
by (subgoal_tac "n=0 | (EX x. n = Suc(x))" 1);
clasohm@923
    69
by (fast_tac (HOL_cs addSEs prems) 1);
clasohm@923
    70
by (nat_ind_tac "n" 1);
clasohm@923
    71
by (rtac (refl RS disjI1) 1);
clasohm@923
    72
by (fast_tac HOL_cs 1);
clasohm@923
    73
qed "natE";
clasohm@923
    74
clasohm@923
    75
(*** Isomorphisms: Abs_Nat and Rep_Nat ***)
clasohm@923
    76
clasohm@923
    77
(*We can't take these properties as axioms, or take Abs_Nat==Inv(Rep_Nat),
clasohm@923
    78
  since we assume the isomorphism equations will one day be given by Isabelle*)
clasohm@923
    79
clasohm@923
    80
goal Nat.thy "inj(Rep_Nat)";
clasohm@923
    81
by (rtac inj_inverseI 1);
clasohm@923
    82
by (rtac Rep_Nat_inverse 1);
clasohm@923
    83
qed "inj_Rep_Nat";
clasohm@923
    84
clasohm@923
    85
goal Nat.thy "inj_onto Abs_Nat Nat";
clasohm@923
    86
by (rtac inj_onto_inverseI 1);
clasohm@923
    87
by (etac Abs_Nat_inverse 1);
clasohm@923
    88
qed "inj_onto_Abs_Nat";
clasohm@923
    89
clasohm@923
    90
(*** Distinctness of constructors ***)
clasohm@923
    91
clasohm@923
    92
goalw Nat.thy [Zero_def,Suc_def] "Suc(m) ~= 0";
clasohm@923
    93
by (rtac (inj_onto_Abs_Nat RS inj_onto_contraD) 1);
clasohm@923
    94
by (rtac Suc_Rep_not_Zero_Rep 1);
clasohm@923
    95
by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI, Zero_RepI] 1));
clasohm@923
    96
qed "Suc_not_Zero";
clasohm@923
    97
clasohm@923
    98
bind_thm ("Zero_not_Suc", (Suc_not_Zero RS not_sym));
clasohm@923
    99
nipkow@1301
   100
Addsimps [Suc_not_Zero,Zero_not_Suc];
nipkow@1301
   101
clasohm@923
   102
bind_thm ("Suc_neq_Zero", (Suc_not_Zero RS notE));
clasohm@923
   103
val Zero_neq_Suc = sym RS Suc_neq_Zero;
clasohm@923
   104
clasohm@923
   105
(** Injectiveness of Suc **)
clasohm@923
   106
clasohm@923
   107
goalw Nat.thy [Suc_def] "inj(Suc)";
clasohm@923
   108
by (rtac injI 1);
clasohm@923
   109
by (dtac (inj_onto_Abs_Nat RS inj_ontoD) 1);
clasohm@923
   110
by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI] 1));
clasohm@923
   111
by (dtac (inj_Suc_Rep RS injD) 1);
clasohm@923
   112
by (etac (inj_Rep_Nat RS injD) 1);
clasohm@923
   113
qed "inj_Suc";
clasohm@923
   114
clasohm@1264
   115
val Suc_inject = inj_Suc RS injD;
clasohm@923
   116
clasohm@923
   117
goal Nat.thy "(Suc(m)=Suc(n)) = (m=n)";
clasohm@923
   118
by (EVERY1 [rtac iffI, etac Suc_inject, etac arg_cong]); 
clasohm@923
   119
qed "Suc_Suc_eq";
clasohm@923
   120
clasohm@923
   121
goal Nat.thy "n ~= Suc(n)";
clasohm@923
   122
by (nat_ind_tac "n" 1);
nipkow@1301
   123
by (ALLGOALS(asm_simp_tac (!simpset addsimps [Suc_Suc_eq])));
clasohm@923
   124
qed "n_not_Suc_n";
clasohm@923
   125
clasohm@923
   126
val Suc_n_not_n = n_not_Suc_n RS not_sym;
clasohm@923
   127
clasohm@923
   128
(*** nat_case -- the selection operator for nat ***)
clasohm@923
   129
clasohm@923
   130
goalw Nat.thy [nat_case_def] "nat_case a f 0 = a";
clasohm@923
   131
by (fast_tac (set_cs addIs [select_equality] addEs [Zero_neq_Suc]) 1);
clasohm@923
   132
qed "nat_case_0";
clasohm@923
   133
clasohm@923
   134
goalw Nat.thy [nat_case_def] "nat_case a f (Suc k) = f(k)";
clasohm@923
   135
by (fast_tac (set_cs addIs [select_equality] 
clasohm@1465
   136
                       addEs [make_elim Suc_inject, Suc_neq_Zero]) 1);
clasohm@923
   137
qed "nat_case_Suc";
clasohm@923
   138
clasohm@923
   139
(** Introduction rules for 'pred_nat' **)
clasohm@923
   140
clasohm@972
   141
goalw Nat.thy [pred_nat_def] "(n, Suc(n)) : pred_nat";
clasohm@923
   142
by (fast_tac set_cs 1);
clasohm@923
   143
qed "pred_natI";
clasohm@923
   144
clasohm@923
   145
val major::prems = goalw Nat.thy [pred_nat_def]
clasohm@972
   146
    "[| p : pred_nat;  !!x n. [| p = (n, Suc(n)) |] ==> R \
clasohm@923
   147
\    |] ==> R";
clasohm@923
   148
by (rtac (major RS CollectE) 1);
clasohm@923
   149
by (REPEAT (eresolve_tac ([asm_rl,exE]@prems) 1));
clasohm@923
   150
qed "pred_natE";
clasohm@923
   151
clasohm@923
   152
goalw Nat.thy [wf_def] "wf(pred_nat)";
clasohm@923
   153
by (strip_tac 1);
clasohm@923
   154
by (nat_ind_tac "x" 1);
clasohm@923
   155
by (fast_tac (HOL_cs addSEs [mp, pred_natE, Pair_inject, 
clasohm@1465
   156
                             make_elim Suc_inject]) 2);
clasohm@923
   157
by (fast_tac (HOL_cs addSEs [mp, pred_natE, Pair_inject, Zero_neq_Suc]) 1);
clasohm@923
   158
qed "wf_pred_nat";
clasohm@923
   159
clasohm@923
   160
clasohm@923
   161
(*** nat_rec -- by wf recursion on pred_nat ***)
clasohm@923
   162
clasohm@923
   163
bind_thm ("nat_rec_unfold", (wf_pred_nat RS (nat_rec_def RS def_wfrec)));
clasohm@923
   164
clasohm@923
   165
(** conversion rules **)
clasohm@923
   166
clasohm@923
   167
goal Nat.thy "nat_rec 0 c h = c";
clasohm@923
   168
by (rtac (nat_rec_unfold RS trans) 1);
clasohm@1264
   169
by (simp_tac (!simpset addsimps [nat_case_0]) 1);
clasohm@923
   170
qed "nat_rec_0";
clasohm@923
   171
clasohm@923
   172
goal Nat.thy "nat_rec (Suc n) c h = h n (nat_rec n c h)";
clasohm@923
   173
by (rtac (nat_rec_unfold RS trans) 1);
clasohm@1264
   174
by (simp_tac (!simpset addsimps [nat_case_Suc, pred_natI, cut_apply]) 1);
clasohm@923
   175
qed "nat_rec_Suc";
clasohm@923
   176
clasohm@923
   177
(*These 2 rules ease the use of primitive recursion.  NOTE USE OF == *)
clasohm@923
   178
val [rew] = goal Nat.thy
clasohm@923
   179
    "[| !!n. f(n) == nat_rec n c h |] ==> f(0) = c";
clasohm@923
   180
by (rewtac rew);
clasohm@923
   181
by (rtac nat_rec_0 1);
clasohm@923
   182
qed "def_nat_rec_0";
clasohm@923
   183
clasohm@923
   184
val [rew] = goal Nat.thy
clasohm@923
   185
    "[| !!n. f(n) == nat_rec n c h |] ==> f(Suc(n)) = h n (f n)";
clasohm@923
   186
by (rewtac rew);
clasohm@923
   187
by (rtac nat_rec_Suc 1);
clasohm@923
   188
qed "def_nat_rec_Suc";
clasohm@923
   189
clasohm@923
   190
fun nat_recs def =
clasohm@923
   191
      [standard (def RS def_nat_rec_0),
clasohm@923
   192
       standard (def RS def_nat_rec_Suc)];
clasohm@923
   193
clasohm@923
   194
clasohm@923
   195
(*** Basic properties of "less than" ***)
clasohm@923
   196
clasohm@923
   197
(** Introduction properties **)
clasohm@923
   198
clasohm@923
   199
val prems = goalw Nat.thy [less_def] "[| i<j;  j<k |] ==> i<(k::nat)";
clasohm@923
   200
by (rtac (trans_trancl RS transD) 1);
clasohm@923
   201
by (resolve_tac prems 1);
clasohm@923
   202
by (resolve_tac prems 1);
clasohm@923
   203
qed "less_trans";
clasohm@923
   204
clasohm@923
   205
goalw Nat.thy [less_def] "n < Suc(n)";
clasohm@923
   206
by (rtac (pred_natI RS r_into_trancl) 1);
clasohm@923
   207
qed "lessI";
nipkow@1301
   208
Addsimps [lessI];
clasohm@923
   209
clasohm@972
   210
(* i(j ==> i(Suc(j) *)
clasohm@923
   211
val less_SucI = lessI RSN (2, less_trans);
clasohm@923
   212
clasohm@923
   213
goal Nat.thy "0 < Suc(n)";
clasohm@923
   214
by (nat_ind_tac "n" 1);
clasohm@923
   215
by (rtac lessI 1);
clasohm@923
   216
by (etac less_trans 1);
clasohm@923
   217
by (rtac lessI 1);
clasohm@923
   218
qed "zero_less_Suc";
nipkow@1301
   219
Addsimps [zero_less_Suc];
clasohm@923
   220
clasohm@923
   221
(** Elimination properties **)
clasohm@923
   222
clasohm@923
   223
val prems = goalw Nat.thy [less_def] "n<m ==> ~ m<(n::nat)";
clasohm@923
   224
by(fast_tac (HOL_cs addIs ([wf_pred_nat, wf_trancl RS wf_asym]@prems))1);
clasohm@923
   225
qed "less_not_sym";
clasohm@923
   226
clasohm@972
   227
(* [| n(m; m(n |] ==> R *)
clasohm@923
   228
bind_thm ("less_asym", (less_not_sym RS notE));
clasohm@923
   229
clasohm@923
   230
goalw Nat.thy [less_def] "~ n<(n::nat)";
clasohm@923
   231
by (rtac notI 1);
clasohm@923
   232
by (etac (wf_pred_nat RS wf_trancl RS wf_anti_refl) 1);
clasohm@923
   233
qed "less_not_refl";
clasohm@923
   234
clasohm@972
   235
(* n(n ==> R *)
clasohm@923
   236
bind_thm ("less_anti_refl", (less_not_refl RS notE));
clasohm@923
   237
clasohm@923
   238
goal Nat.thy "!!m. n<m ==> m ~= (n::nat)";
clasohm@923
   239
by(fast_tac (HOL_cs addEs [less_anti_refl]) 1);
clasohm@923
   240
qed "less_not_refl2";
clasohm@923
   241
clasohm@923
   242
clasohm@923
   243
val major::prems = goalw Nat.thy [less_def]
clasohm@923
   244
    "[| i<k;  k=Suc(i) ==> P;  !!j. [| i<j;  k=Suc(j) |] ==> P \
clasohm@923
   245
\    |] ==> P";
clasohm@923
   246
by (rtac (major RS tranclE) 1);
nipkow@1024
   247
by (REPEAT_FIRST (bound_hyp_subst_tac ORELSE'
clasohm@1465
   248
                  eresolve_tac (prems@[pred_natE, Pair_inject])));
nipkow@1024
   249
by (rtac refl 1);
clasohm@923
   250
qed "lessE";
clasohm@923
   251
clasohm@923
   252
goal Nat.thy "~ n<0";
clasohm@923
   253
by (rtac notI 1);
clasohm@923
   254
by (etac lessE 1);
clasohm@923
   255
by (etac Zero_neq_Suc 1);
clasohm@923
   256
by (etac Zero_neq_Suc 1);
clasohm@923
   257
qed "not_less0";
nipkow@1301
   258
Addsimps [not_less0];
clasohm@923
   259
clasohm@923
   260
(* n<0 ==> R *)
clasohm@923
   261
bind_thm ("less_zeroE", (not_less0 RS notE));
clasohm@923
   262
clasohm@923
   263
val [major,less,eq] = goal Nat.thy
clasohm@923
   264
    "[| m < Suc(n);  m<n ==> P;  m=n ==> P |] ==> P";
clasohm@923
   265
by (rtac (major RS lessE) 1);
clasohm@923
   266
by (rtac eq 1);
clasohm@923
   267
by (fast_tac (HOL_cs addSDs [Suc_inject]) 1);
clasohm@923
   268
by (rtac less 1);
clasohm@923
   269
by (fast_tac (HOL_cs addSDs [Suc_inject]) 1);
clasohm@923
   270
qed "less_SucE";
clasohm@923
   271
clasohm@923
   272
goal Nat.thy "(m < Suc(n)) = (m < n | m = n)";
clasohm@923
   273
by (fast_tac (HOL_cs addSIs [lessI]
clasohm@1465
   274
                     addEs  [less_trans, less_SucE]) 1);
clasohm@923
   275
qed "less_Suc_eq";
clasohm@923
   276
nipkow@1301
   277
val prems = goal Nat.thy "m<n ==> n ~= 0";
nipkow@1301
   278
by(res_inst_tac [("n","n")] natE 1);
nipkow@1301
   279
by(cut_facts_tac prems 1);
nipkow@1301
   280
by(ALLGOALS Asm_full_simp_tac);
nipkow@1301
   281
qed "gr_implies_not0";
nipkow@1301
   282
Addsimps [gr_implies_not0];
clasohm@923
   283
clasohm@923
   284
(** Inductive (?) properties **)
clasohm@923
   285
clasohm@923
   286
val [prem] = goal Nat.thy "Suc(m) < n ==> m<n";
clasohm@923
   287
by (rtac (prem RS rev_mp) 1);
clasohm@923
   288
by (nat_ind_tac "n" 1);
clasohm@923
   289
by (rtac impI 1);
clasohm@923
   290
by (etac less_zeroE 1);
clasohm@923
   291
by (fast_tac (HOL_cs addSIs [lessI RS less_SucI]
clasohm@1465
   292
                     addSDs [Suc_inject]
clasohm@1465
   293
                     addEs  [less_trans, lessE]) 1);
clasohm@923
   294
qed "Suc_lessD";
clasohm@923
   295
clasohm@923
   296
val [major,minor] = goal Nat.thy 
clasohm@923
   297
    "[| Suc(i)<k;  !!j. [| i<j;  k=Suc(j) |] ==> P \
clasohm@923
   298
\    |] ==> P";
clasohm@923
   299
by (rtac (major RS lessE) 1);
clasohm@923
   300
by (etac (lessI RS minor) 1);
clasohm@923
   301
by (etac (Suc_lessD RS minor) 1);
clasohm@923
   302
by (assume_tac 1);
clasohm@923
   303
qed "Suc_lessE";
clasohm@923
   304
clasohm@923
   305
val [major] = goal Nat.thy "Suc(m) < Suc(n) ==> m<n";
clasohm@923
   306
by (rtac (major RS lessE) 1);
clasohm@923
   307
by (REPEAT (rtac lessI 1
clasohm@923
   308
     ORELSE eresolve_tac [make_elim Suc_inject, ssubst, Suc_lessD] 1));
clasohm@923
   309
qed "Suc_less_SucD";
clasohm@923
   310
clasohm@923
   311
val prems = goal Nat.thy "m<n ==> Suc(m) < Suc(n)";
clasohm@923
   312
by (subgoal_tac "m<n --> Suc(m) < Suc(n)" 1);
clasohm@923
   313
by (fast_tac (HOL_cs addIs prems) 1);
clasohm@923
   314
by (nat_ind_tac "n" 1);
clasohm@923
   315
by (rtac impI 1);
clasohm@923
   316
by (etac less_zeroE 1);
clasohm@923
   317
by (fast_tac (HOL_cs addSIs [lessI]
clasohm@1465
   318
                     addSDs [Suc_inject]
clasohm@1465
   319
                     addEs  [less_trans, lessE]) 1);
clasohm@923
   320
qed "Suc_mono";
clasohm@923
   321
clasohm@923
   322
goal Nat.thy "(Suc(m) < Suc(n)) = (m<n)";
clasohm@923
   323
by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]);
clasohm@923
   324
qed "Suc_less_eq";
nipkow@1301
   325
Addsimps [Suc_less_eq];
clasohm@923
   326
clasohm@923
   327
goal Nat.thy "~(Suc(n) < n)";
clasohm@923
   328
by(fast_tac (HOL_cs addEs [Suc_lessD RS less_anti_refl]) 1);
clasohm@923
   329
qed "not_Suc_n_less_n";
nipkow@1301
   330
Addsimps [not_Suc_n_less_n];
nipkow@1301
   331
nipkow@1301
   332
goal Nat.thy "!!i. i<j ==> j<k --> Suc i < k";
nipkow@1301
   333
by(nat_ind_tac "k" 1);
nipkow@1301
   334
by(ALLGOALS (asm_simp_tac (!simpset addsimps [less_Suc_eq])));
nipkow@1301
   335
by(fast_tac (HOL_cs addDs [Suc_lessD]) 1);
nipkow@1301
   336
bind_thm("less_trans_Suc",result() RS mp);
clasohm@923
   337
clasohm@923
   338
(*"Less than" is a linear ordering*)
clasohm@923
   339
goal Nat.thy "m<n | m=n | n<(m::nat)";
clasohm@923
   340
by (nat_ind_tac "m" 1);
clasohm@923
   341
by (nat_ind_tac "n" 1);
clasohm@923
   342
by (rtac (refl RS disjI1 RS disjI2) 1);
clasohm@923
   343
by (rtac (zero_less_Suc RS disjI1) 1);
clasohm@923
   344
by (fast_tac (HOL_cs addIs [lessI, Suc_mono, less_SucI] addEs [lessE]) 1);
clasohm@923
   345
qed "less_linear";
clasohm@923
   346
clasohm@923
   347
(*Can be used with less_Suc_eq to get n=m | n<m *)
clasohm@923
   348
goal Nat.thy "(~ m < n) = (n < Suc(m))";
clasohm@923
   349
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
nipkow@1301
   350
by(ALLGOALS Asm_simp_tac);
clasohm@923
   351
qed "not_less_eq";
clasohm@923
   352
clasohm@923
   353
(*Complete induction, aka course-of-values induction*)
clasohm@923
   354
val prems = goalw Nat.thy [less_def]
clasohm@923
   355
    "[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |]  ==>  P(n)";
clasohm@923
   356
by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1);
clasohm@923
   357
by (eresolve_tac prems 1);
clasohm@923
   358
qed "less_induct";
clasohm@923
   359
clasohm@923
   360
clasohm@923
   361
(*** Properties of <= ***)
clasohm@923
   362
clasohm@923
   363
goalw Nat.thy [le_def] "0 <= n";
clasohm@923
   364
by (rtac not_less0 1);
clasohm@923
   365
qed "le0";
clasohm@923
   366
nipkow@1301
   367
goalw Nat.thy [le_def] "~ Suc n <= n";
nipkow@1301
   368
by(Simp_tac 1);
nipkow@1301
   369
qed "Suc_n_not_le_n";
nipkow@1301
   370
nipkow@1301
   371
goalw Nat.thy [le_def] "(i <= 0) = (i = 0)";
nipkow@1301
   372
by(nat_ind_tac "i" 1);
nipkow@1301
   373
by(ALLGOALS Asm_simp_tac);
nipkow@1301
   374
qed "le_0";
nipkow@1301
   375
nipkow@1301
   376
Addsimps [less_not_refl,
nipkow@1301
   377
          less_Suc_eq, le0, le_0,
nipkow@1301
   378
          Suc_Suc_eq, Suc_n_not_le_n,
clasohm@1264
   379
          n_not_Suc_n, Suc_n_not_n,
clasohm@1264
   380
          nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc];
clasohm@923
   381
clasohm@923
   382
(*Prevents simplification of f and g: much faster*)
clasohm@923
   383
qed_goal "nat_case_weak_cong" Nat.thy
clasohm@923
   384
  "m=n ==> nat_case a f m = nat_case a f n"
clasohm@923
   385
  (fn [prem] => [rtac (prem RS arg_cong) 1]);
clasohm@923
   386
clasohm@923
   387
qed_goal "nat_rec_weak_cong" Nat.thy
clasohm@923
   388
  "m=n ==> nat_rec m a f = nat_rec n a f"
clasohm@923
   389
  (fn [prem] => [rtac (prem RS arg_cong) 1]);
clasohm@923
   390
clasohm@923
   391
val prems = goalw Nat.thy [le_def] "~(n<m) ==> m<=(n::nat)";
clasohm@923
   392
by (resolve_tac prems 1);
clasohm@923
   393
qed "leI";
clasohm@923
   394
clasohm@923
   395
val prems = goalw Nat.thy [le_def] "m<=n ==> ~(n<(m::nat))";
clasohm@923
   396
by (resolve_tac prems 1);
clasohm@923
   397
qed "leD";
clasohm@923
   398
clasohm@923
   399
val leE = make_elim leD;
clasohm@923
   400
clasohm@923
   401
goalw Nat.thy [le_def] "!!m. ~ m <= n ==> n<(m::nat)";
clasohm@923
   402
by (fast_tac HOL_cs 1);
clasohm@923
   403
qed "not_leE";
clasohm@923
   404
clasohm@923
   405
goalw Nat.thy [le_def] "!!m. m < n ==> Suc(m) <= n";
clasohm@1264
   406
by(Simp_tac 1);
clasohm@923
   407
by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1);
clasohm@923
   408
qed "lessD";
clasohm@923
   409
clasohm@923
   410
goalw Nat.thy [le_def] "!!m. Suc(m) <= n ==> m <= n";
clasohm@1264
   411
by(Asm_full_simp_tac 1);
clasohm@923
   412
by(fast_tac HOL_cs 1);
clasohm@923
   413
qed "Suc_leD";
clasohm@923
   414
nipkow@1327
   415
goalw Nat.thy [le_def] "!!m. m <= n ==> m <= Suc n";
nipkow@1327
   416
by (fast_tac (HOL_cs addDs [Suc_lessD]) 1);
nipkow@1327
   417
qed "le_SucI";
nipkow@1327
   418
Addsimps[le_SucI];
nipkow@1327
   419
clasohm@923
   420
goalw Nat.thy [le_def] "!!m. m < n ==> m <= (n::nat)";
clasohm@923
   421
by (fast_tac (HOL_cs addEs [less_asym]) 1);
clasohm@923
   422
qed "less_imp_le";
clasohm@923
   423
clasohm@923
   424
goalw Nat.thy [le_def] "!!m. m <= n ==> m < n | m=(n::nat)";
clasohm@923
   425
by (cut_facts_tac [less_linear] 1);
clasohm@923
   426
by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1);
clasohm@923
   427
qed "le_imp_less_or_eq";
clasohm@923
   428
clasohm@923
   429
goalw Nat.thy [le_def] "!!m. m<n | m=n ==> m <=(n::nat)";
clasohm@923
   430
by (cut_facts_tac [less_linear] 1);
clasohm@923
   431
by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1);
clasohm@923
   432
by (flexflex_tac);
clasohm@923
   433
qed "less_or_eq_imp_le";
clasohm@923
   434
clasohm@923
   435
goal Nat.thy "(x <= (y::nat)) = (x < y | x=y)";
clasohm@923
   436
by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1));
clasohm@923
   437
qed "le_eq_less_or_eq";
clasohm@923
   438
clasohm@923
   439
goal Nat.thy "n <= (n::nat)";
clasohm@1264
   440
by(simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);
clasohm@923
   441
qed "le_refl";
clasohm@923
   442
clasohm@923
   443
val prems = goal Nat.thy "!!i. [| i <= j; j < k |] ==> i < (k::nat)";
clasohm@923
   444
by (dtac le_imp_less_or_eq 1);
clasohm@923
   445
by (fast_tac (HOL_cs addIs [less_trans]) 1);
clasohm@923
   446
qed "le_less_trans";
clasohm@923
   447
clasohm@923
   448
goal Nat.thy "!!i. [| i < j; j <= k |] ==> i < (k::nat)";
clasohm@923
   449
by (dtac le_imp_less_or_eq 1);
clasohm@923
   450
by (fast_tac (HOL_cs addIs [less_trans]) 1);
clasohm@923
   451
qed "less_le_trans";
clasohm@923
   452
clasohm@923
   453
goal Nat.thy "!!i. [| i <= j; j <= k |] ==> i <= (k::nat)";
clasohm@923
   454
by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq,
clasohm@923
   455
          rtac less_or_eq_imp_le, fast_tac (HOL_cs addIs [less_trans])]);
clasohm@923
   456
qed "le_trans";
clasohm@923
   457
clasohm@923
   458
val prems = goal Nat.thy "!!m. [| m <= n; n <= m |] ==> m = (n::nat)";
clasohm@923
   459
by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq,
clasohm@923
   460
          fast_tac (HOL_cs addEs [less_anti_refl,less_asym])]);
clasohm@923
   461
qed "le_anti_sym";
clasohm@923
   462
clasohm@923
   463
goal Nat.thy "(Suc(n) <= Suc(m)) = (n <= m)";
clasohm@1264
   464
by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);
clasohm@923
   465
qed "Suc_le_mono";
clasohm@923
   466
clasohm@1264
   467
Addsimps [le_refl,Suc_le_mono];