src/HOL/Arith.ML
 author paulson Tue Sep 01 10:09:11 1998 +0200 (1998-09-01) changeset 5409 e97558ee8e76 parent 5356 6ef114ba5b55 child 5414 8a458866637c permissions -rw-r--r--
Two new subtraction lemmas
```     1 (*  Title:      HOL/Arith.ML
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1998  University of Cambridge
```
```     5
```
```     6 Proofs about elementary arithmetic: addition, multiplication, etc.
```
```     7 Some from the Hoare example from Norbert Galm
```
```     8 *)
```
```     9
```
```    10 (*** Basic rewrite rules for the arithmetic operators ***)
```
```    11
```
```    12
```
```    13 (** Difference **)
```
```    14
```
```    15 qed_goal "diff_0_eq_0" thy
```
```    16     "0 - n = 0"
```
```    17  (fn _ => [induct_tac "n" 1,  ALLGOALS Asm_simp_tac]);
```
```    18
```
```    19 (*Must simplify BEFORE the induction!!  (Else we get a critical pair)
```
```    20   Suc(m) - Suc(n)   rewrites to   pred(Suc(m) - n)  *)
```
```    21 qed_goal "diff_Suc_Suc" thy
```
```    22     "Suc(m) - Suc(n) = m - n"
```
```    23  (fn _ =>
```
```    24   [Simp_tac 1, induct_tac "n" 1, ALLGOALS Asm_simp_tac]);
```
```    25
```
```    26 Addsimps [diff_0_eq_0, diff_Suc_Suc];
```
```    27
```
```    28 (* Could be (and is, below) generalized in various ways;
```
```    29    However, none of the generalizations are currently in the simpset,
```
```    30    and I dread to think what happens if I put them in *)
```
```    31 Goal "0 < n ==> Suc(n-1) = n";
```
```    32 by (asm_simp_tac (simpset() addsplits [nat.split]) 1);
```
```    33 qed "Suc_pred";
```
```    34 Addsimps [Suc_pred];
```
```    35
```
```    36 Delsimps [diff_Suc];
```
```    37
```
```    38
```
```    39 (**** Inductive properties of the operators ****)
```
```    40
```
```    41 (*** Addition ***)
```
```    42
```
```    43 qed_goal "add_0_right" thy "m + 0 = m"
```
```    44  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```    45
```
```    46 qed_goal "add_Suc_right" thy "m + Suc(n) = Suc(m+n)"
```
```    47  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```    48
```
```    49 Addsimps [add_0_right,add_Suc_right];
```
```    50
```
```    51 (*Associative law for addition*)
```
```    52 qed_goal "add_assoc" thy "(m + n) + k = m + ((n + k)::nat)"
```
```    53  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```    54
```
```    55 (*Commutative law for addition*)
```
```    56 qed_goal "add_commute" thy "m + n = n + (m::nat)"
```
```    57  (fn _ =>  [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```    58
```
```    59 qed_goal "add_left_commute" thy "x+(y+z)=y+((x+z)::nat)"
```
```    60  (fn _ => [rtac (add_commute RS trans) 1, rtac (add_assoc RS trans) 1,
```
```    61            rtac (add_commute RS arg_cong) 1]);
```
```    62
```
```    63 (*Addition is an AC-operator*)
```
```    64 val add_ac = [add_assoc, add_commute, add_left_commute];
```
```    65
```
```    66 Goal "!!k::nat. (k + m = k + n) = (m=n)";
```
```    67 by (induct_tac "k" 1);
```
```    68 by (Simp_tac 1);
```
```    69 by (Asm_simp_tac 1);
```
```    70 qed "add_left_cancel";
```
```    71
```
```    72 Goal "!!k::nat. (m + k = n + k) = (m=n)";
```
```    73 by (induct_tac "k" 1);
```
```    74 by (Simp_tac 1);
```
```    75 by (Asm_simp_tac 1);
```
```    76 qed "add_right_cancel";
```
```    77
```
```    78 Goal "!!k::nat. (k + m <= k + n) = (m<=n)";
```
```    79 by (induct_tac "k" 1);
```
```    80 by (Simp_tac 1);
```
```    81 by (Asm_simp_tac 1);
```
```    82 qed "add_left_cancel_le";
```
```    83
```
```    84 Goal "!!k::nat. (k + m < k + n) = (m<n)";
```
```    85 by (induct_tac "k" 1);
```
```    86 by (Simp_tac 1);
```
```    87 by (Asm_simp_tac 1);
```
```    88 qed "add_left_cancel_less";
```
```    89
```
```    90 Addsimps [add_left_cancel, add_right_cancel,
```
```    91           add_left_cancel_le, add_left_cancel_less];
```
```    92
```
```    93 (** Reasoning about m+0=0, etc. **)
```
```    94
```
```    95 Goal "(m+n = 0) = (m=0 & n=0)";
```
```    96 by (induct_tac "m" 1);
```
```    97 by (ALLGOALS Asm_simp_tac);
```
```    98 qed "add_is_0";
```
```    99 AddIffs [add_is_0];
```
```   100
```
```   101 Goal "(0<m+n) = (0<m | 0<n)";
```
```   102 by (simp_tac (simpset() delsimps [neq0_conv] addsimps [neq0_conv RS sym]) 1);
```
```   103 qed "add_gr_0";
```
```   104 AddIffs [add_gr_0];
```
```   105
```
```   106 (* FIXME: really needed?? *)
```
```   107 Goal "((m+n)-1 = 0) = (m=0 & n-1 = 0 | m-1 = 0 & n=0)";
```
```   108 by (exhaust_tac "m" 1);
```
```   109 by (ALLGOALS (fast_tac (claset() addss (simpset()))));
```
```   110 qed "pred_add_is_0";
```
```   111 Addsimps [pred_add_is_0];
```
```   112
```
```   113 (* Could be generalized, eg to "!!n. k<n ==> m+(n-(Suc k)) = (m+n)-(Suc k)" *)
```
```   114 Goal "0<n ==> m + (n-1) = (m+n)-1";
```
```   115 by (exhaust_tac "m" 1);
```
```   116 by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_Suc]
```
```   117                                       addsplits [nat.split])));
```
```   118 qed "add_pred";
```
```   119 Addsimps [add_pred];
```
```   120
```
```   121 Goal "!!m::nat. m + n = m ==> n = 0";
```
```   122 by (dtac (add_0_right RS ssubst) 1);
```
```   123 by (asm_full_simp_tac (simpset() addsimps [add_assoc]
```
```   124                                  delsimps [add_0_right]) 1);
```
```   125 qed "add_eq_self_zero";
```
```   126
```
```   127
```
```   128 (**** Additional theorems about "less than" ****)
```
```   129
```
```   130 (*Deleted less_natE; instead use less_eq_Suc_add RS exE*)
```
```   131 Goal "m<n --> (? k. n=Suc(m+k))";
```
```   132 by (induct_tac "n" 1);
```
```   133 by (ALLGOALS (simp_tac (simpset() addsimps [less_Suc_eq])));
```
```   134 by (blast_tac (claset() addSEs [less_SucE]
```
```   135                        addSIs [add_0_right RS sym, add_Suc_right RS sym]) 1);
```
```   136 qed_spec_mp "less_eq_Suc_add";
```
```   137
```
```   138 Goal "n <= ((m + n)::nat)";
```
```   139 by (induct_tac "m" 1);
```
```   140 by (ALLGOALS Simp_tac);
```
```   141 by (etac le_trans 1);
```
```   142 by (rtac (lessI RS less_imp_le) 1);
```
```   143 qed "le_add2";
```
```   144
```
```   145 Goal "n <= ((n + m)::nat)";
```
```   146 by (simp_tac (simpset() addsimps add_ac) 1);
```
```   147 by (rtac le_add2 1);
```
```   148 qed "le_add1";
```
```   149
```
```   150 bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans)));
```
```   151 bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans)));
```
```   152
```
```   153 (*"i <= j ==> i <= j+m"*)
```
```   154 bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans));
```
```   155
```
```   156 (*"i <= j ==> i <= m+j"*)
```
```   157 bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans));
```
```   158
```
```   159 (*"i < j ==> i < j+m"*)
```
```   160 bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans));
```
```   161
```
```   162 (*"i < j ==> i < m+j"*)
```
```   163 bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans));
```
```   164
```
```   165 Goal "i+j < (k::nat) ==> i<k";
```
```   166 by (etac rev_mp 1);
```
```   167 by (induct_tac "j" 1);
```
```   168 by (ALLGOALS Asm_simp_tac);
```
```   169 by (blast_tac (claset() addDs [Suc_lessD]) 1);
```
```   170 qed "add_lessD1";
```
```   171
```
```   172 Goal "!!i::nat. ~ (i+j < i)";
```
```   173 by (rtac notI 1);
```
```   174 by (etac (add_lessD1 RS less_irrefl) 1);
```
```   175 qed "not_add_less1";
```
```   176
```
```   177 Goal "!!i::nat. ~ (j+i < i)";
```
```   178 by (simp_tac (simpset() addsimps [add_commute, not_add_less1]) 1);
```
```   179 qed "not_add_less2";
```
```   180 AddIffs [not_add_less1, not_add_less2];
```
```   181
```
```   182 Goal "m+k<=n --> m<=(n::nat)";
```
```   183 by (induct_tac "k" 1);
```
```   184 by (ALLGOALS Asm_simp_tac);
```
```   185 by (blast_tac (claset() addDs [Suc_leD]) 1);
```
```   186 qed_spec_mp "add_leD1";
```
```   187
```
```   188 Goal "!!n::nat. m+k<=n ==> k<=n";
```
```   189 by (full_simp_tac (simpset() addsimps [add_commute]) 1);
```
```   190 by (etac add_leD1 1);
```
```   191 qed_spec_mp "add_leD2";
```
```   192
```
```   193 Goal "!!n::nat. m+k<=n ==> m<=n & k<=n";
```
```   194 by (blast_tac (claset() addDs [add_leD1, add_leD2]) 1);
```
```   195 bind_thm ("add_leE", result() RS conjE);
```
```   196
```
```   197 Goal "!!k l::nat. [| k<l; m+l = k+n |] ==> m<n";
```
```   198 by (safe_tac (claset() addSDs [less_eq_Suc_add]));
```
```   199 by (asm_full_simp_tac
```
```   200     (simpset() delsimps [add_Suc_right]
```
```   201                 addsimps ([add_Suc_right RS sym, add_left_cancel] @add_ac)) 1);
```
```   202 by (etac subst 1);
```
```   203 by (simp_tac (simpset() addsimps [less_add_Suc1]) 1);
```
```   204 qed "less_add_eq_less";
```
```   205
```
```   206
```
```   207 (*** Monotonicity of Addition ***)
```
```   208
```
```   209 (*strict, in 1st argument*)
```
```   210 Goal "!!i j k::nat. i < j ==> i + k < j + k";
```
```   211 by (induct_tac "k" 1);
```
```   212 by (ALLGOALS Asm_simp_tac);
```
```   213 qed "add_less_mono1";
```
```   214
```
```   215 (*strict, in both arguments*)
```
```   216 Goal "!!i j k::nat. [|i < j; k < l|] ==> i + k < j + l";
```
```   217 by (rtac (add_less_mono1 RS less_trans) 1);
```
```   218 by (REPEAT (assume_tac 1));
```
```   219 by (induct_tac "j" 1);
```
```   220 by (ALLGOALS Asm_simp_tac);
```
```   221 qed "add_less_mono";
```
```   222
```
```   223 (*A [clumsy] way of lifting < monotonicity to <= monotonicity *)
```
```   224 val [lt_mono,le] = Goal
```
```   225      "[| !!i j::nat. i<j ==> f(i) < f(j);       \
```
```   226 \        i <= j                                 \
```
```   227 \     |] ==> f(i) <= (f(j)::nat)";
```
```   228 by (cut_facts_tac [le] 1);
```
```   229 by (asm_full_simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
```
```   230 by (blast_tac (claset() addSIs [lt_mono]) 1);
```
```   231 qed "less_mono_imp_le_mono";
```
```   232
```
```   233 (*non-strict, in 1st argument*)
```
```   234 Goal "!!i j k::nat. i<=j ==> i + k <= j + k";
```
```   235 by (res_inst_tac [("f", "%j. j+k")] less_mono_imp_le_mono 1);
```
```   236 by (etac add_less_mono1 1);
```
```   237 by (assume_tac 1);
```
```   238 qed "add_le_mono1";
```
```   239
```
```   240 (*non-strict, in both arguments*)
```
```   241 Goal "!!k l::nat. [|i<=j;  k<=l |] ==> i + k <= j + l";
```
```   242 by (etac (add_le_mono1 RS le_trans) 1);
```
```   243 by (simp_tac (simpset() addsimps [add_commute]) 1);
```
```   244 (*j moves to the end because it is free while k, l are bound*)
```
```   245 by (etac add_le_mono1 1);
```
```   246 qed "add_le_mono";
```
```   247
```
```   248
```
```   249 (*** Multiplication ***)
```
```   250
```
```   251 (*right annihilation in product*)
```
```   252 qed_goal "mult_0_right" thy "m * 0 = 0"
```
```   253  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```   254
```
```   255 (*right successor law for multiplication*)
```
```   256 qed_goal "mult_Suc_right" thy  "m * Suc(n) = m + (m * n)"
```
```   257  (fn _ => [induct_tac "m" 1,
```
```   258            ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]);
```
```   259
```
```   260 Addsimps [mult_0_right, mult_Suc_right];
```
```   261
```
```   262 Goal "1 * n = n";
```
```   263 by (Asm_simp_tac 1);
```
```   264 qed "mult_1";
```
```   265
```
```   266 Goal "n * 1 = n";
```
```   267 by (Asm_simp_tac 1);
```
```   268 qed "mult_1_right";
```
```   269
```
```   270 (*Commutative law for multiplication*)
```
```   271 qed_goal "mult_commute" thy "m * n = n * (m::nat)"
```
```   272  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```   273
```
```   274 (*addition distributes over multiplication*)
```
```   275 qed_goal "add_mult_distrib" thy "(m + n)*k = (m*k) + ((n*k)::nat)"
```
```   276  (fn _ => [induct_tac "m" 1,
```
```   277            ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]);
```
```   278
```
```   279 qed_goal "add_mult_distrib2" thy "k*(m + n) = (k*m) + ((k*n)::nat)"
```
```   280  (fn _ => [induct_tac "m" 1,
```
```   281            ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]);
```
```   282
```
```   283 (*Associative law for multiplication*)
```
```   284 qed_goal "mult_assoc" thy "(m * n) * k = m * ((n * k)::nat)"
```
```   285   (fn _ => [induct_tac "m" 1,
```
```   286             ALLGOALS (asm_simp_tac (simpset() addsimps [add_mult_distrib]))]);
```
```   287
```
```   288 qed_goal "mult_left_commute" thy "x*(y*z) = y*((x*z)::nat)"
```
```   289  (fn _ => [rtac trans 1, rtac mult_commute 1, rtac trans 1,
```
```   290            rtac mult_assoc 1, rtac (mult_commute RS arg_cong) 1]);
```
```   291
```
```   292 val mult_ac = [mult_assoc,mult_commute,mult_left_commute];
```
```   293
```
```   294 Goal "(m*n = 0) = (m=0 | n=0)";
```
```   295 by (induct_tac "m" 1);
```
```   296 by (induct_tac "n" 2);
```
```   297 by (ALLGOALS Asm_simp_tac);
```
```   298 qed "mult_is_0";
```
```   299 Addsimps [mult_is_0];
```
```   300
```
```   301 Goal "!!m::nat. m <= m*m";
```
```   302 by (induct_tac "m" 1);
```
```   303 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_assoc RS sym])));
```
```   304 by (etac (le_add2 RSN (2,le_trans)) 1);
```
```   305 qed "le_square";
```
```   306
```
```   307
```
```   308 (*** Difference ***)
```
```   309
```
```   310
```
```   311 qed_goal "diff_self_eq_0" thy "m - m = 0"
```
```   312  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```   313 Addsimps [diff_self_eq_0];
```
```   314
```
```   315 (*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
```
```   316 Goal "~ m<n --> n+(m-n) = (m::nat)";
```
```   317 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   318 by (ALLGOALS Asm_simp_tac);
```
```   319 qed_spec_mp "add_diff_inverse";
```
```   320
```
```   321 Goal "n<=m ==> n+(m-n) = (m::nat)";
```
```   322 by (asm_simp_tac (simpset() addsimps [add_diff_inverse, not_less_iff_le]) 1);
```
```   323 qed "le_add_diff_inverse";
```
```   324
```
```   325 Goal "n<=m ==> (m-n)+n = (m::nat)";
```
```   326 by (asm_simp_tac (simpset() addsimps [le_add_diff_inverse, add_commute]) 1);
```
```   327 qed "le_add_diff_inverse2";
```
```   328
```
```   329 Addsimps  [le_add_diff_inverse, le_add_diff_inverse2];
```
```   330
```
```   331
```
```   332 (*** More results about difference ***)
```
```   333
```
```   334 Goal "n < Suc(m) ==> Suc(m)-n = Suc(m-n)";
```
```   335 by (etac rev_mp 1);
```
```   336 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   337 by (ALLGOALS Asm_simp_tac);
```
```   338 qed "Suc_diff_n";
```
```   339
```
```   340 Goal "m - n < Suc(m)";
```
```   341 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   342 by (etac less_SucE 3);
```
```   343 by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq])));
```
```   344 qed "diff_less_Suc";
```
```   345
```
```   346 Goal "!!m::nat. m - n <= m";
```
```   347 by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1);
```
```   348 by (ALLGOALS Asm_simp_tac);
```
```   349 qed "diff_le_self";
```
```   350 Addsimps [diff_le_self];
```
```   351
```
```   352 (* j<k ==> j-n < k *)
```
```   353 bind_thm ("less_imp_diff_less", diff_le_self RS le_less_trans);
```
```   354
```
```   355 Goal "!!i::nat. i-j-k = i - (j+k)";
```
```   356 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
```
```   357 by (ALLGOALS Asm_simp_tac);
```
```   358 qed "diff_diff_left";
```
```   359
```
```   360 Goal "(Suc m - n) - Suc k = m - n - k";
```
```   361 by (simp_tac (simpset() addsimps [diff_diff_left]) 1);
```
```   362 qed "Suc_diff_diff";
```
```   363 Addsimps [Suc_diff_diff];
```
```   364
```
```   365 Goal "0<n ==> n - Suc i < n";
```
```   366 by (exhaust_tac "n" 1);
```
```   367 by Safe_tac;
```
```   368 by (asm_simp_tac (simpset() addsimps [le_eq_less_Suc RS sym]) 1);
```
```   369 qed "diff_Suc_less";
```
```   370 Addsimps [diff_Suc_less];
```
```   371
```
```   372 Goal "i<n ==> n - Suc i < n - i";
```
```   373 by (exhaust_tac "n" 1);
```
```   374 by Safe_tac;
```
```   375 by (asm_simp_tac (simpset() addsimps [Suc_diff_n]) 1);
```
```   376 qed "diff_Suc_less_diff";
```
```   377
```
```   378 Goal "m - n <= Suc m - n";
```
```   379 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   380 by (ALLGOALS Asm_simp_tac);
```
```   381 qed "diff_le_Suc_diff";
```
```   382
```
```   383 (*This and the next few suggested by Florian Kammueller*)
```
```   384 Goal "!!i::nat. i-j-k = i-k-j";
```
```   385 by (simp_tac (simpset() addsimps [diff_diff_left, add_commute]) 1);
```
```   386 qed "diff_commute";
```
```   387
```
```   388 Goal "!!i j k:: nat. k<=j --> j<=i --> i - (j - k) = i - j + k";
```
```   389 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
```
```   390 by (ALLGOALS Asm_simp_tac);
```
```   391 by (asm_simp_tac
```
```   392     (simpset() addsimps [Suc_diff_n, le_imp_less_Suc, le_Suc_eq]) 1);
```
```   393 qed_spec_mp "diff_diff_right";
```
```   394
```
```   395 Goal "!!i j k:: nat. k<=j --> (i + j) - k = i + (j - k)";
```
```   396 by (res_inst_tac [("m","j"),("n","k")] diff_induct 1);
```
```   397 by (ALLGOALS Asm_simp_tac);
```
```   398 qed_spec_mp "diff_add_assoc";
```
```   399
```
```   400 Goal "!!i j k:: nat. k<=j --> (j + i) - k = i + (j - k)";
```
```   401 by (asm_simp_tac (simpset() addsimps [add_commute, diff_add_assoc]) 1);
```
```   402 qed_spec_mp "diff_add_assoc2";
```
```   403
```
```   404 Goal "!!n::nat. (n+m) - n = m";
```
```   405 by (induct_tac "n" 1);
```
```   406 by (ALLGOALS Asm_simp_tac);
```
```   407 qed "diff_add_inverse";
```
```   408 Addsimps [diff_add_inverse];
```
```   409
```
```   410 Goal "!!n::nat.(m+n) - n = m";
```
```   411 by (simp_tac (simpset() addsimps [diff_add_assoc]) 1);
```
```   412 qed "diff_add_inverse2";
```
```   413 Addsimps [diff_add_inverse2];
```
```   414
```
```   415 Goal "!!i j k::nat. i<=j ==> (j-i=k) = (j=k+i)";
```
```   416 by Safe_tac;
```
```   417 by (ALLGOALS Asm_simp_tac);
```
```   418 qed "le_imp_diff_is_add";
```
```   419
```
```   420 Goal "(m-n = 0) = (m <= n)";
```
```   421 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   422 by (ALLGOALS (asm_simp_tac (simpset() addsimps [le_eq_less_Suc])));
```
```   423 qed "diff_is_0_eq";
```
```   424 Addsimps [diff_is_0_eq RS iffD2];
```
```   425
```
```   426 Goal "m-n = 0  -->  n-m = 0  -->  m=n";
```
```   427 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   428 by (REPEAT(Simp_tac 1 THEN TRY(atac 1)));
```
```   429 qed_spec_mp "diffs0_imp_equal";
```
```   430
```
```   431 Goal "(0<n-m) = (m<n)";
```
```   432 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   433 by (ALLGOALS Asm_simp_tac);
```
```   434 qed "zero_less_diff";
```
```   435 Addsimps [zero_less_diff];
```
```   436
```
```   437 Goal "i < j  ==> ? k. 0<k & i+k = j";
```
```   438 by (res_inst_tac [("x","j - i")] exI 1);
```
```   439 by (asm_simp_tac (simpset() addsimps [add_diff_inverse, less_not_sym]) 1);
```
```   440 qed "less_imp_add_positive";
```
```   441
```
```   442 Goal "Suc(m)-n = (if m<n then 0 else Suc(m-n))";
```
```   443 by (simp_tac (simpset() addsimps [leI, Suc_le_eq,
```
```   444 				  le_imp_less_Suc RS Suc_diff_n]) 1);
```
```   445 qed "if_Suc_diff_n";
```
```   446
```
```   447 Goal "Suc(m)-n <= Suc(m-n)";
```
```   448 by (simp_tac (simpset() addsimps [if_Suc_diff_n]) 1);
```
```   449 qed "diff_Suc_le_Suc_diff";
```
```   450
```
```   451 Goal "P(k) --> (!n. P(Suc(n))--> P(n)) --> P(k-i)";
```
```   452 by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);
```
```   453 by (ALLGOALS (Clarify_tac THEN' Simp_tac THEN' TRY o Blast_tac));
```
```   454 qed "zero_induct_lemma";
```
```   455
```
```   456 val prems = Goal "[| P(k);  !!n. P(Suc(n)) ==> P(n) |] ==> P(0)";
```
```   457 by (rtac (diff_self_eq_0 RS subst) 1);
```
```   458 by (rtac (zero_induct_lemma RS mp RS mp) 1);
```
```   459 by (REPEAT (ares_tac ([impI,allI]@prems) 1));
```
```   460 qed "zero_induct";
```
```   461
```
```   462 Goal "!!k::nat. (k+m) - (k+n) = m - n";
```
```   463 by (induct_tac "k" 1);
```
```   464 by (ALLGOALS Asm_simp_tac);
```
```   465 qed "diff_cancel";
```
```   466 Addsimps [diff_cancel];
```
```   467
```
```   468 Goal "!!m::nat. (m+k) - (n+k) = m - n";
```
```   469 val add_commute_k = read_instantiate [("n","k")] add_commute;
```
```   470 by (asm_simp_tac (simpset() addsimps ([add_commute_k])) 1);
```
```   471 qed "diff_cancel2";
```
```   472 Addsimps [diff_cancel2];
```
```   473
```
```   474 (*From Clemens Ballarin*)
```
```   475 Goal "!!n::nat. [| k<=n; n<=m |] ==> (m-k) - (n-k) = m-n";
```
```   476 by (subgoal_tac "k<=n --> n<=m --> (m-k) - (n-k) = m-n" 1);
```
```   477 by (Asm_full_simp_tac 1);
```
```   478 by (induct_tac "k" 1);
```
```   479 by (Simp_tac 1);
```
```   480 (* Induction step *)
```
```   481 by (subgoal_tac "Suc na <= m --> n <= m --> Suc na <= n --> \
```
```   482 \                Suc (m - Suc na) - Suc (n - Suc na) = m-n" 1);
```
```   483 by (Asm_full_simp_tac 1);
```
```   484 by (blast_tac (claset() addIs [le_trans]) 1);
```
```   485 by (auto_tac (claset() addIs [Suc_leD], simpset() delsimps [diff_Suc_Suc]));
```
```   486 by (asm_full_simp_tac (simpset() delsimps [Suc_less_eq]
```
```   487 		       addsimps [le_imp_less_Suc RS Suc_diff_n RS sym]) 1);
```
```   488 qed "diff_right_cancel";
```
```   489
```
```   490 Goal "!!n::nat. n - (n+m) = 0";
```
```   491 by (induct_tac "n" 1);
```
```   492 by (ALLGOALS Asm_simp_tac);
```
```   493 qed "diff_add_0";
```
```   494 Addsimps [diff_add_0];
```
```   495
```
```   496
```
```   497 (** Difference distributes over multiplication **)
```
```   498
```
```   499 Goal "!!m::nat. (m - n) * k = (m * k) - (n * k)";
```
```   500 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   501 by (ALLGOALS Asm_simp_tac);
```
```   502 qed "diff_mult_distrib" ;
```
```   503
```
```   504 Goal "!!m::nat. k * (m - n) = (k * m) - (k * n)";
```
```   505 val mult_commute_k = read_instantiate [("m","k")] mult_commute;
```
```   506 by (simp_tac (simpset() addsimps [diff_mult_distrib, mult_commute_k]) 1);
```
```   507 qed "diff_mult_distrib2" ;
```
```   508 (*NOT added as rewrites, since sometimes they are used from right-to-left*)
```
```   509
```
```   510
```
```   511 (*** Monotonicity of Multiplication ***)
```
```   512
```
```   513 Goal "!!i::nat. i<=j ==> i*k<=j*k";
```
```   514 by (induct_tac "k" 1);
```
```   515 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_le_mono])));
```
```   516 qed "mult_le_mono1";
```
```   517
```
```   518 (*<=monotonicity, BOTH arguments*)
```
```   519 Goal "!!i::nat. [| i<=j; k<=l |] ==> i*k<=j*l";
```
```   520 by (etac (mult_le_mono1 RS le_trans) 1);
```
```   521 by (rtac le_trans 1);
```
```   522 by (stac mult_commute 2);
```
```   523 by (etac mult_le_mono1 2);
```
```   524 by (simp_tac (simpset() addsimps [mult_commute]) 1);
```
```   525 qed "mult_le_mono";
```
```   526
```
```   527 (*strict, in 1st argument; proof is by induction on k>0*)
```
```   528 Goal "!!i::nat. [| i<j; 0<k |] ==> k*i < k*j";
```
```   529 by (eres_inst_tac [("m1","0")] (less_eq_Suc_add RS exE) 1);
```
```   530 by (Asm_simp_tac 1);
```
```   531 by (induct_tac "x" 1);
```
```   532 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_less_mono])));
```
```   533 qed "mult_less_mono2";
```
```   534
```
```   535 Goal "!!i::nat. [| i<j; 0<k |] ==> i*k < j*k";
```
```   536 by (dtac mult_less_mono2 1);
```
```   537 by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [mult_commute])));
```
```   538 qed "mult_less_mono1";
```
```   539
```
```   540 Goal "(0 < m*n) = (0<m & 0<n)";
```
```   541 by (induct_tac "m" 1);
```
```   542 by (induct_tac "n" 2);
```
```   543 by (ALLGOALS Asm_simp_tac);
```
```   544 qed "zero_less_mult_iff";
```
```   545 Addsimps [zero_less_mult_iff];
```
```   546
```
```   547 Goal "(m*n = 1) = (m=1 & n=1)";
```
```   548 by (induct_tac "m" 1);
```
```   549 by (Simp_tac 1);
```
```   550 by (induct_tac "n" 1);
```
```   551 by (Simp_tac 1);
```
```   552 by (fast_tac (claset() addss simpset()) 1);
```
```   553 qed "mult_eq_1_iff";
```
```   554 Addsimps [mult_eq_1_iff];
```
```   555
```
```   556 Goal "0<k ==> (m*k < n*k) = (m<n)";
```
```   557 by (safe_tac (claset() addSIs [mult_less_mono1]));
```
```   558 by (cut_facts_tac [less_linear] 1);
```
```   559 by (blast_tac (claset() addIs [mult_less_mono1] addEs [less_asym]) 1);
```
```   560 qed "mult_less_cancel2";
```
```   561
```
```   562 Goal "0<k ==> (k*m < k*n) = (m<n)";
```
```   563 by (dtac mult_less_cancel2 1);
```
```   564 by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1);
```
```   565 qed "mult_less_cancel1";
```
```   566 Addsimps [mult_less_cancel1, mult_less_cancel2];
```
```   567
```
```   568 Goal "(Suc k * m < Suc k * n) = (m < n)";
```
```   569 by (rtac mult_less_cancel1 1);
```
```   570 by (Simp_tac 1);
```
```   571 qed "Suc_mult_less_cancel1";
```
```   572
```
```   573 Goalw [le_def] "(Suc k * m <= Suc k * n) = (m <= n)";
```
```   574 by (simp_tac (simpset_of HOL.thy) 1);
```
```   575 by (rtac Suc_mult_less_cancel1 1);
```
```   576 qed "Suc_mult_le_cancel1";
```
```   577
```
```   578 Goal "0<k ==> (m*k = n*k) = (m=n)";
```
```   579 by (cut_facts_tac [less_linear] 1);
```
```   580 by Safe_tac;
```
```   581 by (assume_tac 2);
```
```   582 by (ALLGOALS (dtac mult_less_mono1 THEN' assume_tac));
```
```   583 by (ALLGOALS Asm_full_simp_tac);
```
```   584 qed "mult_cancel2";
```
```   585
```
```   586 Goal "0<k ==> (k*m = k*n) = (m=n)";
```
```   587 by (dtac mult_cancel2 1);
```
```   588 by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1);
```
```   589 qed "mult_cancel1";
```
```   590 Addsimps [mult_cancel1, mult_cancel2];
```
```   591
```
```   592 Goal "(Suc k * m = Suc k * n) = (m = n)";
```
```   593 by (rtac mult_cancel1 1);
```
```   594 by (Simp_tac 1);
```
```   595 qed "Suc_mult_cancel1";
```
```   596
```
```   597
```
```   598 (** Lemma for gcd **)
```
```   599
```
```   600 Goal "m = m*n ==> n=1 | m=0";
```
```   601 by (dtac sym 1);
```
```   602 by (rtac disjCI 1);
```
```   603 by (rtac nat_less_cases 1 THEN assume_tac 2);
```
```   604 by (fast_tac (claset() addSEs [less_SucE] addss simpset()) 1);
```
```   605 by (best_tac (claset() addDs [mult_less_mono2] addss simpset()) 1);
```
```   606 qed "mult_eq_self_implies_10";
```
```   607
```
```   608
```
```   609 (*** Subtraction laws -- mostly from Clemens Ballarin ***)
```
```   610
```
```   611 Goal "!! a b c::nat. [| a < b; c <= a |] ==> a-c < b-c";
```
```   612 by (subgoal_tac "c+(a-c) < c+(b-c)" 1);
```
```   613 by (Full_simp_tac 1);
```
```   614 by (subgoal_tac "c <= b" 1);
```
```   615 by (blast_tac (claset() addIs [less_imp_le, le_trans]) 2);
```
```   616 by (Asm_simp_tac 1);
```
```   617 qed "diff_less_mono";
```
```   618
```
```   619 Goal "!! a b c::nat. a+b < c ==> a < c-b";
```
```   620 by (dtac diff_less_mono 1);
```
```   621 by (rtac le_add2 1);
```
```   622 by (Asm_full_simp_tac 1);
```
```   623 qed "add_less_imp_less_diff";
```
```   624
```
```   625 Goal "n <= m ==> Suc m - n = Suc (m - n)";
```
```   626 by (asm_full_simp_tac (simpset() addsimps [le_imp_less_Suc RS Suc_diff_n]) 1);
```
```   627 qed "Suc_diff_le";
```
```   628
```
```   629 Goal "Suc i <= n ==> Suc (n - Suc i) = n - i";
```
```   630 by (asm_full_simp_tac
```
```   631     (simpset() addsimps [Suc_diff_n RS sym, le_eq_less_Suc]) 1);
```
```   632 qed "Suc_diff_Suc";
```
```   633
```
```   634 Goal "!! i::nat. i <= n ==> n - (n - i) = i";
```
```   635 by (etac rev_mp 1);
```
```   636 by (res_inst_tac [("m","n"),("n","i")] diff_induct 1);
```
```   637 by (ALLGOALS (asm_simp_tac  (simpset() addsimps [Suc_diff_le])));
```
```   638 qed "diff_diff_cancel";
```
```   639 Addsimps [diff_diff_cancel];
```
```   640
```
```   641 Goal "!!k::nat. k <= n ==> m <= n + m - k";
```
```   642 by (etac rev_mp 1);
```
```   643 by (res_inst_tac [("m", "k"), ("n", "n")] diff_induct 1);
```
```   644 by (Simp_tac 1);
```
```   645 by (simp_tac (simpset() addsimps [less_add_Suc2, less_imp_le]) 1);
```
```   646 by (Simp_tac 1);
```
```   647 qed "le_add_diff";
```
```   648
```
```   649 Goal "!!i::nat. 0<k ==> j<i --> j+k-i < k";
```
```   650 by (res_inst_tac [("m","j"),("n","i")] diff_induct 1);
```
```   651 by (ALLGOALS Asm_simp_tac);
```
```   652 qed_spec_mp "add_diff_less";
```
```   653
```
```   654
```
```   655 Goal "m-1 < n ==> m <= n";
```
```   656 by (exhaust_tac "m" 1);
```
```   657 by (auto_tac (claset(), simpset() addsimps [Suc_le_eq]));
```
```   658 qed "pred_less_imp_le";
```
```   659
```
```   660 Goal "j<=i ==> i - j < Suc i - j";
```
```   661 by (REPEAT (etac rev_mp 1));
```
```   662 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
```
```   663 by Auto_tac;
```
```   664 qed "diff_less_Suc_diff";
```
```   665
```
```   666 Goal "i - j <= Suc i - j";
```
```   667 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
```
```   668 by Auto_tac;
```
```   669 qed "diff_le_Suc_diff";
```
```   670 AddIffs [diff_le_Suc_diff];
```
```   671
```
```   672 Goal "n - Suc i <= n - i";
```
```   673 by (case_tac "i<n" 1);
```
```   674 bd diff_Suc_less_diff 1;
```
```   675 by (auto_tac (claset(), simpset() addsimps [leI RS le_imp_less_Suc]));
```
```   676 qed "diff_Suc_le_diff";
```
```   677 AddIffs [diff_Suc_le_diff];
```
```   678
```
```   679 Goal "0 < n ==> (m <= n-1) = (m<n)";
```
```   680 by (exhaust_tac "n" 1);
```
```   681 by Auto_tac;
```
```   682 by (ALLGOALS trans_tac);
```
```   683 qed "le_pred_eq";
```
```   684
```
```   685 Goal "0 < n ==> (m-1 < n) = (m<=n)";
```
```   686 by (exhaust_tac "m" 1);
```
```   687 by (auto_tac (claset(), simpset() addsimps [Suc_le_eq]));
```
```   688 qed "less_pred_eq";
```
```   689
```
```   690
```
```   691
```
```   692 (** (Anti)Monotonicity of subtraction -- by Stefan Merz **)
```
```   693
```
```   694 (* Monotonicity of subtraction in first argument *)
```
```   695 Goal "!!n::nat. m<=n --> (m-l) <= (n-l)";
```
```   696 by (induct_tac "n" 1);
```
```   697 by (Simp_tac 1);
```
```   698 by (simp_tac (simpset() addsimps [le_Suc_eq]) 1);
```
```   699 by (blast_tac (claset() addIs [diff_le_Suc_diff, le_trans]) 1);
```
```   700 qed_spec_mp "diff_le_mono";
```
```   701
```
```   702 Goal "!!n::nat. m<=n ==> (l-n) <= (l-m)";
```
```   703 by (induct_tac "l" 1);
```
```   704 by (Simp_tac 1);
```
```   705 by (case_tac "n <= na" 1);
```
```   706 by (subgoal_tac "m <= na" 1);
```
```   707 by (asm_simp_tac (simpset() addsimps [Suc_diff_le]) 1);
```
```   708 by (fast_tac (claset() addEs [le_trans]) 1);
```
```   709 by (dtac not_leE 1);
```
```   710 by (asm_simp_tac (simpset() addsimps [if_Suc_diff_n]) 1);
```
```   711 qed_spec_mp "diff_le_mono2";
```