author | paulson |
Fri, 18 Feb 2000 15:35:29 +0100 | |
changeset 8255 | 38f96394c099 |
parent 8252 | af44242c5e7a |
child 8352 | 0fda5ba36934 |
permissions | -rw-r--r-- |
1465 | 1 |
(* Title: HOL/Arith.ML |
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ID: $Id$ |
1465 | 3 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1998 University of Cambridge |
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|
6 |
Proofs about elementary arithmetic: addition, multiplication, etc. |
|
3234 | 7 |
Some from the Hoare example from Norbert Galm |
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*) |
9 |
||
10 |
(*** Basic rewrite rules for the arithmetic operators ***) |
|
11 |
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12 |
|
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(** Difference **) |
14 |
||
7007 | 15 |
Goal "0 - n = 0"; |
16 |
by (induct_tac "n" 1); |
|
17 |
by (ALLGOALS Asm_simp_tac); |
|
18 |
qed "diff_0_eq_0"; |
|
923 | 19 |
|
5429 | 20 |
(*Must simplify BEFORE the induction! (Else we get a critical pair) |
923 | 21 |
Suc(m) - Suc(n) rewrites to pred(Suc(m) - n) *) |
7007 | 22 |
Goal "Suc(m) - Suc(n) = m - n"; |
23 |
by (Simp_tac 1); |
|
24 |
by (induct_tac "n" 1); |
|
25 |
by (ALLGOALS Asm_simp_tac); |
|
26 |
qed "diff_Suc_Suc"; |
|
923 | 27 |
|
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28 |
Addsimps [diff_0_eq_0, diff_Suc_Suc]; |
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|
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(* Could be (and is, below) generalized in various ways; |
31 |
However, none of the generalizations are currently in the simpset, |
|
32 |
and I dread to think what happens if I put them in *) |
|
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33 |
Goal "0 < n ==> Suc(n-1) = n"; |
5183 | 34 |
by (asm_simp_tac (simpset() addsplits [nat.split]) 1); |
4360 | 35 |
qed "Suc_pred"; |
36 |
Addsimps [Suc_pred]; |
|
37 |
||
38 |
Delsimps [diff_Suc]; |
|
39 |
||
923 | 40 |
|
41 |
(**** Inductive properties of the operators ****) |
|
42 |
||
43 |
(*** Addition ***) |
|
44 |
||
7007 | 45 |
Goal "m + 0 = m"; |
46 |
by (induct_tac "m" 1); |
|
47 |
by (ALLGOALS Asm_simp_tac); |
|
48 |
qed "add_0_right"; |
|
923 | 49 |
|
7007 | 50 |
Goal "m + Suc(n) = Suc(m+n)"; |
51 |
by (induct_tac "m" 1); |
|
52 |
by (ALLGOALS Asm_simp_tac); |
|
53 |
qed "add_Suc_right"; |
|
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55 |
Addsimps [add_0_right,add_Suc_right]; |
923 | 56 |
|
7007 | 57 |
|
923 | 58 |
(*Associative law for addition*) |
7007 | 59 |
Goal "(m + n) + k = m + ((n + k)::nat)"; |
60 |
by (induct_tac "m" 1); |
|
61 |
by (ALLGOALS Asm_simp_tac); |
|
62 |
qed "add_assoc"; |
|
923 | 63 |
|
64 |
(*Commutative law for addition*) |
|
7007 | 65 |
Goal "m + n = n + (m::nat)"; |
66 |
by (induct_tac "m" 1); |
|
67 |
by (ALLGOALS Asm_simp_tac); |
|
68 |
qed "add_commute"; |
|
923 | 69 |
|
7007 | 70 |
Goal "x+(y+z)=y+((x+z)::nat)"; |
71 |
by (rtac (add_commute RS trans) 1); |
|
72 |
by (rtac (add_assoc RS trans) 1); |
|
73 |
by (rtac (add_commute RS arg_cong) 1); |
|
74 |
qed "add_left_commute"; |
|
923 | 75 |
|
76 |
(*Addition is an AC-operator*) |
|
7428 | 77 |
bind_thms ("add_ac", [add_assoc, add_commute, add_left_commute]); |
923 | 78 |
|
5429 | 79 |
Goal "(k + m = k + n) = (m=(n::nat))"; |
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by (induct_tac "k" 1); |
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81 |
by (Simp_tac 1); |
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82 |
by (Asm_simp_tac 1); |
923 | 83 |
qed "add_left_cancel"; |
84 |
||
5429 | 85 |
Goal "(m + k = n + k) = (m=(n::nat))"; |
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by (induct_tac "k" 1); |
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|
87 |
by (Simp_tac 1); |
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88 |
by (Asm_simp_tac 1); |
923 | 89 |
qed "add_right_cancel"; |
90 |
||
5429 | 91 |
Goal "(k + m <= k + n) = (m<=(n::nat))"; |
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by (induct_tac "k" 1); |
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93 |
by (Simp_tac 1); |
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|
94 |
by (Asm_simp_tac 1); |
923 | 95 |
qed "add_left_cancel_le"; |
96 |
||
5429 | 97 |
Goal "(k + m < k + n) = (m<(n::nat))"; |
3339 | 98 |
by (induct_tac "k" 1); |
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|
99 |
by (Simp_tac 1); |
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|
100 |
by (Asm_simp_tac 1); |
923 | 101 |
qed "add_left_cancel_less"; |
102 |
||
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|
103 |
Addsimps [add_left_cancel, add_right_cancel, |
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|
104 |
add_left_cancel_le, add_left_cancel_less]; |
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|
105 |
|
3339 | 106 |
(** Reasoning about m+0=0, etc. **) |
107 |
||
5069 | 108 |
Goal "(m+n = 0) = (m=0 & n=0)"; |
5598 | 109 |
by (exhaust_tac "m" 1); |
110 |
by (Auto_tac); |
|
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|
111 |
qed "add_is_0"; |
4360 | 112 |
AddIffs [add_is_0]; |
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|
113 |
|
5598 | 114 |
Goal "(0 = m+n) = (m=0 & n=0)"; |
115 |
by (exhaust_tac "m" 1); |
|
116 |
by (Auto_tac); |
|
117 |
qed "zero_is_add"; |
|
118 |
AddIffs [zero_is_add]; |
|
119 |
||
120 |
Goal "(m+n=1) = (m=1 & n=0 | m=0 & n=1)"; |
|
6301 | 121 |
by (exhaust_tac "m" 1); |
122 |
by (Auto_tac); |
|
5598 | 123 |
qed "add_is_1"; |
124 |
||
125 |
Goal "(1=m+n) = (m=1 & n=0 | m=0 & n=1)"; |
|
6301 | 126 |
by (exhaust_tac "m" 1); |
127 |
by (Auto_tac); |
|
5598 | 128 |
qed "one_is_add"; |
129 |
||
5069 | 130 |
Goal "(0<m+n) = (0<m | 0<n)"; |
4423 | 131 |
by (simp_tac (simpset() delsimps [neq0_conv] addsimps [neq0_conv RS sym]) 1); |
4360 | 132 |
qed "add_gr_0"; |
133 |
AddIffs [add_gr_0]; |
|
134 |
||
5429 | 135 |
(* Could be generalized, eg to "k<n ==> m+(n-(Suc k)) = (m+n)-(Suc k)" *) |
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|
136 |
Goal "0<n ==> m + (n-1) = (m+n)-1"; |
4360 | 137 |
by (exhaust_tac "m" 1); |
6075 | 138 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_Suc, Suc_n_not_n] |
5183 | 139 |
addsplits [nat.split]))); |
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|
140 |
qed "add_pred"; |
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|
141 |
Addsimps [add_pred]; |
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|
142 |
|
5429 | 143 |
Goal "m + n = m ==> n = 0"; |
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by (dtac (add_0_right RS ssubst) 1); |
145 |
by (asm_full_simp_tac (simpset() addsimps [add_assoc] |
|
146 |
delsimps [add_0_right]) 1); |
|
147 |
qed "add_eq_self_zero"; |
|
148 |
||
1626 | 149 |
|
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(**** Additional theorems about "less than" ****) |
151 |
||
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(*Deleted less_natE; instead use less_eq_Suc_add RS exE*) |
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|
153 |
Goal "m<n --> (? k. n=Suc(m+k))"; |
3339 | 154 |
by (induct_tac "n" 1); |
5604 | 155 |
by (ALLGOALS (simp_tac (simpset() addsimps [order_le_less]))); |
4089 | 156 |
by (blast_tac (claset() addSEs [less_SucE] |
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addSIs [add_0_right RS sym, add_Suc_right RS sym]) 1); |
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|
158 |
qed_spec_mp "less_eq_Suc_add"; |
923 | 159 |
|
5069 | 160 |
Goal "n <= ((m + n)::nat)"; |
3339 | 161 |
by (induct_tac "m" 1); |
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|
162 |
by (ALLGOALS Simp_tac); |
5983 | 163 |
by (etac le_SucI 1); |
923 | 164 |
qed "le_add2"; |
165 |
||
5069 | 166 |
Goal "n <= ((n + m)::nat)"; |
4089 | 167 |
by (simp_tac (simpset() addsimps add_ac) 1); |
923 | 168 |
by (rtac le_add2 1); |
169 |
qed "le_add1"; |
|
170 |
||
171 |
bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans))); |
|
172 |
bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans))); |
|
173 |
||
5429 | 174 |
Goal "(m<n) = (? k. n=Suc(m+k))"; |
175 |
by (blast_tac (claset() addSIs [less_add_Suc1, less_eq_Suc_add]) 1); |
|
176 |
qed "less_iff_Suc_add"; |
|
177 |
||
178 |
||
923 | 179 |
(*"i <= j ==> i <= j+m"*) |
180 |
bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans)); |
|
181 |
||
182 |
(*"i <= j ==> i <= m+j"*) |
|
183 |
bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans)); |
|
184 |
||
185 |
(*"i < j ==> i < j+m"*) |
|
186 |
bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans)); |
|
187 |
||
188 |
(*"i < j ==> i < m+j"*) |
|
189 |
bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans)); |
|
190 |
||
5654 | 191 |
Goal "i+j < (k::nat) --> i<k"; |
3339 | 192 |
by (induct_tac "j" 1); |
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|
193 |
by (ALLGOALS Asm_simp_tac); |
6301 | 194 |
by (blast_tac (claset() addDs [Suc_lessD]) 1); |
5654 | 195 |
qed_spec_mp "add_lessD1"; |
1152 | 196 |
|
5429 | 197 |
Goal "~ (i+j < (i::nat))"; |
3457 | 198 |
by (rtac notI 1); |
199 |
by (etac (add_lessD1 RS less_irrefl) 1); |
|
3234 | 200 |
qed "not_add_less1"; |
201 |
||
5429 | 202 |
Goal "~ (j+i < (i::nat))"; |
4089 | 203 |
by (simp_tac (simpset() addsimps [add_commute, not_add_less1]) 1); |
3234 | 204 |
qed "not_add_less2"; |
205 |
AddIffs [not_add_less1, not_add_less2]; |
|
206 |
||
5069 | 207 |
Goal "m+k<=n --> m<=(n::nat)"; |
3339 | 208 |
by (induct_tac "k" 1); |
5497 | 209 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps le_simps))); |
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210 |
qed_spec_mp "add_leD1"; |
923 | 211 |
|
5429 | 212 |
Goal "m+k<=n ==> k<=(n::nat)"; |
4089 | 213 |
by (full_simp_tac (simpset() addsimps [add_commute]) 1); |
2498 | 214 |
by (etac add_leD1 1); |
215 |
qed_spec_mp "add_leD2"; |
|
216 |
||
5429 | 217 |
Goal "m+k<=n ==> m<=n & k<=(n::nat)"; |
4089 | 218 |
by (blast_tac (claset() addDs [add_leD1, add_leD2]) 1); |
2498 | 219 |
bind_thm ("add_leE", result() RS conjE); |
220 |
||
5429 | 221 |
(*needs !!k for add_ac to work*) |
222 |
Goal "!!k:: nat. [| k<l; m+l = k+n |] ==> m<n"; |
|
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|
223 |
by (force_tac (claset(), |
5497 | 224 |
simpset() delsimps [add_Suc_right] |
5537 | 225 |
addsimps [less_iff_Suc_add, |
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|
226 |
add_Suc_right RS sym] @ add_ac) 1); |
923 | 227 |
qed "less_add_eq_less"; |
228 |
||
229 |
||
1713 | 230 |
(*** Monotonicity of Addition ***) |
923 | 231 |
|
232 |
(*strict, in 1st argument*) |
|
5429 | 233 |
Goal "i < j ==> i + k < j + (k::nat)"; |
3339 | 234 |
by (induct_tac "k" 1); |
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|
235 |
by (ALLGOALS Asm_simp_tac); |
923 | 236 |
qed "add_less_mono1"; |
237 |
||
238 |
(*strict, in both arguments*) |
|
5429 | 239 |
Goal "[|i < j; k < l|] ==> i + k < j + (l::nat)"; |
923 | 240 |
by (rtac (add_less_mono1 RS less_trans) 1); |
1198 | 241 |
by (REPEAT (assume_tac 1)); |
3339 | 242 |
by (induct_tac "j" 1); |
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|
243 |
by (ALLGOALS Asm_simp_tac); |
923 | 244 |
qed "add_less_mono"; |
245 |
||
246 |
(*A [clumsy] way of lifting < monotonicity to <= monotonicity *) |
|
5316 | 247 |
val [lt_mono,le] = Goal |
1465 | 248 |
"[| !!i j::nat. i<j ==> f(i) < f(j); \ |
249 |
\ i <= j \ |
|
923 | 250 |
\ |] ==> f(i) <= (f(j)::nat)"; |
251 |
by (cut_facts_tac [le] 1); |
|
5604 | 252 |
by (asm_full_simp_tac (simpset() addsimps [order_le_less]) 1); |
4089 | 253 |
by (blast_tac (claset() addSIs [lt_mono]) 1); |
923 | 254 |
qed "less_mono_imp_le_mono"; |
255 |
||
256 |
(*non-strict, in 1st argument*) |
|
5429 | 257 |
Goal "i<=j ==> i + k <= j + (k::nat)"; |
3842 | 258 |
by (res_inst_tac [("f", "%j. j+k")] less_mono_imp_le_mono 1); |
1552 | 259 |
by (etac add_less_mono1 1); |
923 | 260 |
by (assume_tac 1); |
261 |
qed "add_le_mono1"; |
|
262 |
||
263 |
(*non-strict, in both arguments*) |
|
5429 | 264 |
Goal "[|i<=j; k<=l |] ==> i + k <= j + (l::nat)"; |
923 | 265 |
by (etac (add_le_mono1 RS le_trans) 1); |
4089 | 266 |
by (simp_tac (simpset() addsimps [add_commute]) 1); |
923 | 267 |
qed "add_le_mono"; |
1713 | 268 |
|
3234 | 269 |
|
270 |
(*** Multiplication ***) |
|
271 |
||
272 |
(*right annihilation in product*) |
|
7007 | 273 |
Goal "m * 0 = 0"; |
274 |
by (induct_tac "m" 1); |
|
275 |
by (ALLGOALS Asm_simp_tac); |
|
276 |
qed "mult_0_right"; |
|
3234 | 277 |
|
3293 | 278 |
(*right successor law for multiplication*) |
7007 | 279 |
Goal "m * Suc(n) = m + (m * n)"; |
280 |
by (induct_tac "m" 1); |
|
281 |
by (ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))); |
|
282 |
qed "mult_Suc_right"; |
|
3234 | 283 |
|
3293 | 284 |
Addsimps [mult_0_right, mult_Suc_right]; |
3234 | 285 |
|
5069 | 286 |
Goal "1 * n = n"; |
3234 | 287 |
by (Asm_simp_tac 1); |
288 |
qed "mult_1"; |
|
289 |
||
5069 | 290 |
Goal "n * 1 = n"; |
3234 | 291 |
by (Asm_simp_tac 1); |
292 |
qed "mult_1_right"; |
|
293 |
||
294 |
(*Commutative law for multiplication*) |
|
7007 | 295 |
Goal "m * n = n * (m::nat)"; |
296 |
by (induct_tac "m" 1); |
|
297 |
by (ALLGOALS Asm_simp_tac); |
|
298 |
qed "mult_commute"; |
|
3234 | 299 |
|
300 |
(*addition distributes over multiplication*) |
|
7007 | 301 |
Goal "(m + n)*k = (m*k) + ((n*k)::nat)"; |
302 |
by (induct_tac "m" 1); |
|
303 |
by (ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))); |
|
304 |
qed "add_mult_distrib"; |
|
3234 | 305 |
|
7007 | 306 |
Goal "k*(m + n) = (k*m) + ((k*n)::nat)"; |
307 |
by (induct_tac "m" 1); |
|
308 |
by (ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))); |
|
309 |
qed "add_mult_distrib2"; |
|
3234 | 310 |
|
311 |
(*Associative law for multiplication*) |
|
7007 | 312 |
Goal "(m * n) * k = m * ((n * k)::nat)"; |
313 |
by (induct_tac "m" 1); |
|
314 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_mult_distrib]))); |
|
315 |
qed "mult_assoc"; |
|
3234 | 316 |
|
7007 | 317 |
Goal "x*(y*z) = y*((x*z)::nat)"; |
318 |
by (rtac trans 1); |
|
319 |
by (rtac mult_commute 1); |
|
320 |
by (rtac trans 1); |
|
321 |
by (rtac mult_assoc 1); |
|
322 |
by (rtac (mult_commute RS arg_cong) 1); |
|
323 |
qed "mult_left_commute"; |
|
3234 | 324 |
|
7428 | 325 |
bind_thms ("mult_ac", [mult_assoc,mult_commute,mult_left_commute]); |
3234 | 326 |
|
5069 | 327 |
Goal "(m*n = 0) = (m=0 | n=0)"; |
3339 | 328 |
by (induct_tac "m" 1); |
329 |
by (induct_tac "n" 2); |
|
3293 | 330 |
by (ALLGOALS Asm_simp_tac); |
331 |
qed "mult_is_0"; |
|
7622 | 332 |
|
333 |
Goal "(0 = m*n) = (0=m | 0=n)"; |
|
334 |
by (cut_facts_tac [mult_is_0] 1); |
|
335 |
by (full_simp_tac (simpset() addsimps [eq_commute]) 1); |
|
336 |
qed "zero_is_mult"; |
|
337 |
||
338 |
Addsimps [mult_is_0, zero_is_mult]; |
|
3293 | 339 |
|
5429 | 340 |
Goal "m <= m*(m::nat)"; |
4158 | 341 |
by (induct_tac "m" 1); |
342 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_assoc RS sym]))); |
|
343 |
by (etac (le_add2 RSN (2,le_trans)) 1); |
|
344 |
qed "le_square"; |
|
345 |
||
3234 | 346 |
|
347 |
(*** Difference ***) |
|
348 |
||
7007 | 349 |
Goal "m - m = 0"; |
350 |
by (induct_tac "m" 1); |
|
351 |
by (ALLGOALS Asm_simp_tac); |
|
352 |
qed "diff_self_eq_0"; |
|
3234 | 353 |
|
354 |
Addsimps [diff_self_eq_0]; |
|
355 |
||
356 |
(*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *) |
|
5069 | 357 |
Goal "~ m<n --> n+(m-n) = (m::nat)"; |
3234 | 358 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
3352 | 359 |
by (ALLGOALS Asm_simp_tac); |
3381
2bac33ec2b0d
New theorems le_add_diff_inverse, le_add_diff_inverse2
paulson
parents:
3366
diff
changeset
|
360 |
qed_spec_mp "add_diff_inverse"; |
2bac33ec2b0d
New theorems le_add_diff_inverse, le_add_diff_inverse2
paulson
parents:
3366
diff
changeset
|
361 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
362 |
Goal "n<=m ==> n+(m-n) = (m::nat)"; |
4089 | 363 |
by (asm_simp_tac (simpset() addsimps [add_diff_inverse, not_less_iff_le]) 1); |
3381
2bac33ec2b0d
New theorems le_add_diff_inverse, le_add_diff_inverse2
paulson
parents:
3366
diff
changeset
|
364 |
qed "le_add_diff_inverse"; |
3234 | 365 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
366 |
Goal "n<=m ==> (m-n)+n = (m::nat)"; |
4089 | 367 |
by (asm_simp_tac (simpset() addsimps [le_add_diff_inverse, add_commute]) 1); |
3381
2bac33ec2b0d
New theorems le_add_diff_inverse, le_add_diff_inverse2
paulson
parents:
3366
diff
changeset
|
368 |
qed "le_add_diff_inverse2"; |
2bac33ec2b0d
New theorems le_add_diff_inverse, le_add_diff_inverse2
paulson
parents:
3366
diff
changeset
|
369 |
|
2bac33ec2b0d
New theorems le_add_diff_inverse, le_add_diff_inverse2
paulson
parents:
3366
diff
changeset
|
370 |
Addsimps [le_add_diff_inverse, le_add_diff_inverse2]; |
3234 | 371 |
|
372 |
||
373 |
(*** More results about difference ***) |
|
374 |
||
5414
8a458866637c
changed Suc_diff_n to Suc_diff_le, with premise n <= m instead of n < Suc(m)
paulson
parents:
5409
diff
changeset
|
375 |
Goal "n <= m ==> Suc(m)-n = Suc(m-n)"; |
5316 | 376 |
by (etac rev_mp 1); |
3352 | 377 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
378 |
by (ALLGOALS Asm_simp_tac); |
|
5414
8a458866637c
changed Suc_diff_n to Suc_diff_le, with premise n <= m instead of n < Suc(m)
paulson
parents:
5409
diff
changeset
|
379 |
qed "Suc_diff_le"; |
3352 | 380 |
|
5069 | 381 |
Goal "m - n < Suc(m)"; |
3234 | 382 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
383 |
by (etac less_SucE 3); |
|
4089 | 384 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq]))); |
3234 | 385 |
qed "diff_less_Suc"; |
386 |
||
5429 | 387 |
Goal "m - n <= (m::nat)"; |
3234 | 388 |
by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1); |
6075 | 389 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [le_SucI]))); |
3234 | 390 |
qed "diff_le_self"; |
3903
1b29151a1009
New simprule diff_le_self, requiring a new proof of diff_diff_cancel
paulson
parents:
3896
diff
changeset
|
391 |
Addsimps [diff_le_self]; |
3234 | 392 |
|
4732 | 393 |
(* j<k ==> j-n < k *) |
394 |
bind_thm ("less_imp_diff_less", diff_le_self RS le_less_trans); |
|
395 |
||
5069 | 396 |
Goal "!!i::nat. i-j-k = i - (j+k)"; |
3352 | 397 |
by (res_inst_tac [("m","i"),("n","j")] diff_induct 1); |
398 |
by (ALLGOALS Asm_simp_tac); |
|
399 |
qed "diff_diff_left"; |
|
400 |
||
5069 | 401 |
Goal "(Suc m - n) - Suc k = m - n - k"; |
4423 | 402 |
by (simp_tac (simpset() addsimps [diff_diff_left]) 1); |
4736 | 403 |
qed "Suc_diff_diff"; |
404 |
Addsimps [Suc_diff_diff]; |
|
4360 | 405 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
406 |
Goal "0<n ==> n - Suc i < n"; |
5183 | 407 |
by (exhaust_tac "n" 1); |
4732 | 408 |
by Safe_tac; |
5497 | 409 |
by (asm_simp_tac (simpset() addsimps le_simps) 1); |
4732 | 410 |
qed "diff_Suc_less"; |
411 |
Addsimps [diff_Suc_less]; |
|
412 |
||
3396 | 413 |
(*This and the next few suggested by Florian Kammueller*) |
5069 | 414 |
Goal "!!i::nat. i-j-k = i-k-j"; |
4089 | 415 |
by (simp_tac (simpset() addsimps [diff_diff_left, add_commute]) 1); |
3352 | 416 |
qed "diff_commute"; |
417 |
||
5429 | 418 |
Goal "k<=j --> j<=i --> i - (j - k) = i - j + (k::nat)"; |
3352 | 419 |
by (res_inst_tac [("m","i"),("n","j")] diff_induct 1); |
420 |
by (ALLGOALS Asm_simp_tac); |
|
5414
8a458866637c
changed Suc_diff_n to Suc_diff_le, with premise n <= m instead of n < Suc(m)
paulson
parents:
5409
diff
changeset
|
421 |
by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_Suc_eq]) 1); |
3352 | 422 |
qed_spec_mp "diff_diff_right"; |
423 |
||
5429 | 424 |
Goal "k <= (j::nat) --> (i + j) - k = i + (j - k)"; |
3352 | 425 |
by (res_inst_tac [("m","j"),("n","k")] diff_induct 1); |
426 |
by (ALLGOALS Asm_simp_tac); |
|
427 |
qed_spec_mp "diff_add_assoc"; |
|
428 |
||
5429 | 429 |
Goal "k <= (j::nat) --> (j + i) - k = i + (j - k)"; |
4732 | 430 |
by (asm_simp_tac (simpset() addsimps [add_commute, diff_add_assoc]) 1); |
431 |
qed_spec_mp "diff_add_assoc2"; |
|
432 |
||
5429 | 433 |
Goal "(n+m) - n = (m::nat)"; |
3339 | 434 |
by (induct_tac "n" 1); |
3234 | 435 |
by (ALLGOALS Asm_simp_tac); |
436 |
qed "diff_add_inverse"; |
|
437 |
Addsimps [diff_add_inverse]; |
|
438 |
||
5429 | 439 |
Goal "(m+n) - n = (m::nat)"; |
4089 | 440 |
by (simp_tac (simpset() addsimps [diff_add_assoc]) 1); |
3234 | 441 |
qed "diff_add_inverse2"; |
442 |
Addsimps [diff_add_inverse2]; |
|
443 |
||
5429 | 444 |
Goal "i <= (j::nat) ==> (j-i=k) = (j=k+i)"; |
3724 | 445 |
by Safe_tac; |
3381
2bac33ec2b0d
New theorems le_add_diff_inverse, le_add_diff_inverse2
paulson
parents:
3366
diff
changeset
|
446 |
by (ALLGOALS Asm_simp_tac); |
3366 | 447 |
qed "le_imp_diff_is_add"; |
448 |
||
5356 | 449 |
Goal "(m-n = 0) = (m <= n)"; |
3234 | 450 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
5497 | 451 |
by (ALLGOALS Asm_simp_tac); |
5356 | 452 |
qed "diff_is_0_eq"; |
7059 | 453 |
|
454 |
Goal "(0 = m-n) = (m <= n)"; |
|
455 |
by (stac (diff_is_0_eq RS sym) 1); |
|
456 |
by (rtac eq_sym_conv 1); |
|
457 |
qed "zero_is_diff_eq"; |
|
458 |
Addsimps [diff_is_0_eq, zero_is_diff_eq]; |
|
3234 | 459 |
|
5333
ea33e66dcedd
Some new theorems. zero_less_diff replaces less_imp_diff_positive
paulson
parents:
5329
diff
changeset
|
460 |
Goal "(0<n-m) = (m<n)"; |
3234 | 461 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
3352 | 462 |
by (ALLGOALS Asm_simp_tac); |
5333
ea33e66dcedd
Some new theorems. zero_less_diff replaces less_imp_diff_positive
paulson
parents:
5329
diff
changeset
|
463 |
qed "zero_less_diff"; |
ea33e66dcedd
Some new theorems. zero_less_diff replaces less_imp_diff_positive
paulson
parents:
5329
diff
changeset
|
464 |
Addsimps [zero_less_diff]; |
3234 | 465 |
|
5333
ea33e66dcedd
Some new theorems. zero_less_diff replaces less_imp_diff_positive
paulson
parents:
5329
diff
changeset
|
466 |
Goal "i < j ==> ? k. 0<k & i+k = j"; |
5078 | 467 |
by (res_inst_tac [("x","j - i")] exI 1); |
5333
ea33e66dcedd
Some new theorems. zero_less_diff replaces less_imp_diff_positive
paulson
parents:
5329
diff
changeset
|
468 |
by (asm_simp_tac (simpset() addsimps [add_diff_inverse, less_not_sym]) 1); |
5078 | 469 |
qed "less_imp_add_positive"; |
470 |
||
5069 | 471 |
Goal "P(k) --> (!n. P(Suc(n))--> P(n)) --> P(k-i)"; |
3234 | 472 |
by (res_inst_tac [("m","k"),("n","i")] diff_induct 1); |
3718 | 473 |
by (ALLGOALS (Clarify_tac THEN' Simp_tac THEN' TRY o Blast_tac)); |
3234 | 474 |
qed "zero_induct_lemma"; |
475 |
||
5316 | 476 |
val prems = Goal "[| P(k); !!n. P(Suc(n)) ==> P(n) |] ==> P(0)"; |
3234 | 477 |
by (rtac (diff_self_eq_0 RS subst) 1); |
478 |
by (rtac (zero_induct_lemma RS mp RS mp) 1); |
|
479 |
by (REPEAT (ares_tac ([impI,allI]@prems) 1)); |
|
480 |
qed "zero_induct"; |
|
481 |
||
5429 | 482 |
Goal "(k+m) - (k+n) = m - (n::nat)"; |
3339 | 483 |
by (induct_tac "k" 1); |
3234 | 484 |
by (ALLGOALS Asm_simp_tac); |
485 |
qed "diff_cancel"; |
|
486 |
Addsimps [diff_cancel]; |
|
487 |
||
5429 | 488 |
Goal "(m+k) - (n+k) = m - (n::nat)"; |
3234 | 489 |
val add_commute_k = read_instantiate [("n","k")] add_commute; |
5537 | 490 |
by (asm_simp_tac (simpset() addsimps [add_commute_k]) 1); |
3234 | 491 |
qed "diff_cancel2"; |
492 |
Addsimps [diff_cancel2]; |
|
493 |
||
5429 | 494 |
Goal "n - (n+m) = 0"; |
3339 | 495 |
by (induct_tac "n" 1); |
3234 | 496 |
by (ALLGOALS Asm_simp_tac); |
497 |
qed "diff_add_0"; |
|
498 |
Addsimps [diff_add_0]; |
|
499 |
||
5409 | 500 |
|
3234 | 501 |
(** Difference distributes over multiplication **) |
502 |
||
5069 | 503 |
Goal "!!m::nat. (m - n) * k = (m * k) - (n * k)"; |
3234 | 504 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
505 |
by (ALLGOALS Asm_simp_tac); |
|
506 |
qed "diff_mult_distrib" ; |
|
507 |
||
5069 | 508 |
Goal "!!m::nat. k * (m - n) = (k * m) - (k * n)"; |
3234 | 509 |
val mult_commute_k = read_instantiate [("m","k")] mult_commute; |
4089 | 510 |
by (simp_tac (simpset() addsimps [diff_mult_distrib, mult_commute_k]) 1); |
3234 | 511 |
qed "diff_mult_distrib2" ; |
512 |
(*NOT added as rewrites, since sometimes they are used from right-to-left*) |
|
513 |
||
514 |
||
1713 | 515 |
(*** Monotonicity of Multiplication ***) |
516 |
||
5429 | 517 |
Goal "i <= (j::nat) ==> i*k<=j*k"; |
3339 | 518 |
by (induct_tac "k" 1); |
4089 | 519 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_le_mono]))); |
1713 | 520 |
qed "mult_le_mono1"; |
521 |
||
6987 | 522 |
Goal "i <= (j::nat) ==> k*i <= k*j"; |
523 |
by (dtac mult_le_mono1 1); |
|
524 |
by (asm_simp_tac (simpset() addsimps [mult_commute]) 1); |
|
525 |
qed "mult_le_mono2"; |
|
526 |
||
527 |
(* <= monotonicity, BOTH arguments*) |
|
5429 | 528 |
Goal "[| i <= (j::nat); k <= l |] ==> i*k <= j*l"; |
2007 | 529 |
by (etac (mult_le_mono1 RS le_trans) 1); |
6987 | 530 |
by (etac mult_le_mono2 1); |
1713 | 531 |
qed "mult_le_mono"; |
532 |
||
533 |
(*strict, in 1st argument; proof is by induction on k>0*) |
|
5429 | 534 |
Goal "[| i<j; 0<k |] ==> k*i < k*j"; |
5078 | 535 |
by (eres_inst_tac [("m1","0")] (less_eq_Suc_add RS exE) 1); |
1713 | 536 |
by (Asm_simp_tac 1); |
3339 | 537 |
by (induct_tac "x" 1); |
4089 | 538 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_less_mono]))); |
1713 | 539 |
qed "mult_less_mono2"; |
540 |
||
5429 | 541 |
Goal "[| i<j; 0<k |] ==> i*k < j*k"; |
3457 | 542 |
by (dtac mult_less_mono2 1); |
4089 | 543 |
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [mult_commute]))); |
3234 | 544 |
qed "mult_less_mono1"; |
545 |
||
5069 | 546 |
Goal "(0 < m*n) = (0<m & 0<n)"; |
3339 | 547 |
by (induct_tac "m" 1); |
548 |
by (induct_tac "n" 2); |
|
1713 | 549 |
by (ALLGOALS Asm_simp_tac); |
550 |
qed "zero_less_mult_iff"; |
|
4356 | 551 |
Addsimps [zero_less_mult_iff]; |
1713 | 552 |
|
5069 | 553 |
Goal "(m*n = 1) = (m=1 & n=1)"; |
3339 | 554 |
by (induct_tac "m" 1); |
1795 | 555 |
by (Simp_tac 1); |
3339 | 556 |
by (induct_tac "n" 1); |
1795 | 557 |
by (Simp_tac 1); |
4089 | 558 |
by (fast_tac (claset() addss simpset()) 1); |
1795 | 559 |
qed "mult_eq_1_iff"; |
4356 | 560 |
Addsimps [mult_eq_1_iff]; |
1795 | 561 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
562 |
Goal "0<k ==> (m*k < n*k) = (m<n)"; |
4089 | 563 |
by (safe_tac (claset() addSIs [mult_less_mono1])); |
3234 | 564 |
by (cut_facts_tac [less_linear] 1); |
4389 | 565 |
by (blast_tac (claset() addIs [mult_less_mono1] addEs [less_asym]) 1); |
3234 | 566 |
qed "mult_less_cancel2"; |
567 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
568 |
Goal "0<k ==> (k*m < k*n) = (m<n)"; |
3457 | 569 |
by (dtac mult_less_cancel2 1); |
4089 | 570 |
by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1); |
3234 | 571 |
qed "mult_less_cancel1"; |
572 |
Addsimps [mult_less_cancel1, mult_less_cancel2]; |
|
573 |
||
6864 | 574 |
Goal "0<k ==> (m*k <= n*k) = (m<=n)"; |
575 |
by (asm_full_simp_tac (simpset() addsimps [linorder_not_less RS sym]) 1); |
|
576 |
qed "mult_le_cancel2"; |
|
577 |
||
578 |
Goal "0<k ==> (k*m <= k*n) = (m<=n)"; |
|
579 |
by (asm_full_simp_tac (simpset() addsimps [linorder_not_less RS sym]) 1); |
|
580 |
qed "mult_le_cancel1"; |
|
581 |
Addsimps [mult_le_cancel1, mult_le_cancel2]; |
|
582 |
||
5069 | 583 |
Goal "(Suc k * m < Suc k * n) = (m < n)"; |
4423 | 584 |
by (rtac mult_less_cancel1 1); |
4297
5defc2105cc8
added Suc_mult_less_cancel1, Suc_mult_le_cancel1, Suc_mult_cancel1;
wenzelm
parents:
4158
diff
changeset
|
585 |
by (Simp_tac 1); |
5defc2105cc8
added Suc_mult_less_cancel1, Suc_mult_le_cancel1, Suc_mult_cancel1;
wenzelm
parents:
4158
diff
changeset
|
586 |
qed "Suc_mult_less_cancel1"; |
5defc2105cc8
added Suc_mult_less_cancel1, Suc_mult_le_cancel1, Suc_mult_cancel1;
wenzelm
parents:
4158
diff
changeset
|
587 |
|
5069 | 588 |
Goalw [le_def] "(Suc k * m <= Suc k * n) = (m <= n)"; |
4297
5defc2105cc8
added Suc_mult_less_cancel1, Suc_mult_le_cancel1, Suc_mult_cancel1;
wenzelm
parents:
4158
diff
changeset
|
589 |
by (simp_tac (simpset_of HOL.thy) 1); |
4423 | 590 |
by (rtac Suc_mult_less_cancel1 1); |
4297
5defc2105cc8
added Suc_mult_less_cancel1, Suc_mult_le_cancel1, Suc_mult_cancel1;
wenzelm
parents:
4158
diff
changeset
|
591 |
qed "Suc_mult_le_cancel1"; |
5defc2105cc8
added Suc_mult_less_cancel1, Suc_mult_le_cancel1, Suc_mult_cancel1;
wenzelm
parents:
4158
diff
changeset
|
592 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
593 |
Goal "0<k ==> (m*k = n*k) = (m=n)"; |
3234 | 594 |
by (cut_facts_tac [less_linear] 1); |
3724 | 595 |
by Safe_tac; |
3457 | 596 |
by (assume_tac 2); |
3234 | 597 |
by (ALLGOALS (dtac mult_less_mono1 THEN' assume_tac)); |
598 |
by (ALLGOALS Asm_full_simp_tac); |
|
599 |
qed "mult_cancel2"; |
|
600 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
601 |
Goal "0<k ==> (k*m = k*n) = (m=n)"; |
3457 | 602 |
by (dtac mult_cancel2 1); |
4089 | 603 |
by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1); |
3234 | 604 |
qed "mult_cancel1"; |
605 |
Addsimps [mult_cancel1, mult_cancel2]; |
|
606 |
||
5069 | 607 |
Goal "(Suc k * m = Suc k * n) = (m = n)"; |
4423 | 608 |
by (rtac mult_cancel1 1); |
4297
5defc2105cc8
added Suc_mult_less_cancel1, Suc_mult_le_cancel1, Suc_mult_cancel1;
wenzelm
parents:
4158
diff
changeset
|
609 |
by (Simp_tac 1); |
5defc2105cc8
added Suc_mult_less_cancel1, Suc_mult_le_cancel1, Suc_mult_cancel1;
wenzelm
parents:
4158
diff
changeset
|
610 |
qed "Suc_mult_cancel1"; |
5defc2105cc8
added Suc_mult_less_cancel1, Suc_mult_le_cancel1, Suc_mult_cancel1;
wenzelm
parents:
4158
diff
changeset
|
611 |
|
3234 | 612 |
|
1795 | 613 |
(** Lemma for gcd **) |
614 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
615 |
Goal "m = m*n ==> n=1 | m=0"; |
1795 | 616 |
by (dtac sym 1); |
617 |
by (rtac disjCI 1); |
|
618 |
by (rtac nat_less_cases 1 THEN assume_tac 2); |
|
4089 | 619 |
by (fast_tac (claset() addSEs [less_SucE] addss simpset()) 1); |
4356 | 620 |
by (best_tac (claset() addDs [mult_less_mono2] addss simpset()) 1); |
1795 | 621 |
qed "mult_eq_self_implies_10"; |
622 |
||
623 |
||
5983 | 624 |
|
625 |
||
626 |
(*---------------------------------------------------------------------------*) |
|
627 |
(* Various arithmetic proof procedures *) |
|
628 |
(*---------------------------------------------------------------------------*) |
|
629 |
||
630 |
(*---------------------------------------------------------------------------*) |
|
631 |
(* 1. Cancellation of common terms *) |
|
632 |
(*---------------------------------------------------------------------------*) |
|
633 |
||
634 |
(* Title: HOL/arith_data.ML |
|
635 |
ID: $Id$ |
|
636 |
Author: Markus Wenzel and Stefan Berghofer, TU Muenchen |
|
637 |
||
638 |
Setup various arithmetic proof procedures. |
|
639 |
*) |
|
640 |
||
641 |
signature ARITH_DATA = |
|
642 |
sig |
|
6055 | 643 |
val nat_cancel_sums_add: simproc list |
5983 | 644 |
val nat_cancel_sums: simproc list |
645 |
val nat_cancel_factor: simproc list |
|
646 |
val nat_cancel: simproc list |
|
647 |
end; |
|
648 |
||
649 |
structure ArithData: ARITH_DATA = |
|
650 |
struct |
|
651 |
||
652 |
||
653 |
(** abstract syntax of structure nat: 0, Suc, + **) |
|
654 |
||
655 |
(* mk_sum, mk_norm_sum *) |
|
656 |
||
657 |
val one = HOLogic.mk_nat 1; |
|
658 |
val mk_plus = HOLogic.mk_binop "op +"; |
|
659 |
||
660 |
fun mk_sum [] = HOLogic.zero |
|
661 |
| mk_sum [t] = t |
|
662 |
| mk_sum (t :: ts) = mk_plus (t, mk_sum ts); |
|
663 |
||
664 |
(*normal form of sums: Suc (... (Suc (a + (b + ...))))*) |
|
665 |
fun mk_norm_sum ts = |
|
666 |
let val (ones, sums) = partition (equal one) ts in |
|
667 |
funpow (length ones) HOLogic.mk_Suc (mk_sum sums) |
|
668 |
end; |
|
669 |
||
670 |
||
671 |
(* dest_sum *) |
|
672 |
||
673 |
val dest_plus = HOLogic.dest_bin "op +" HOLogic.natT; |
|
674 |
||
675 |
fun dest_sum tm = |
|
676 |
if HOLogic.is_zero tm then [] |
|
677 |
else |
|
678 |
(case try HOLogic.dest_Suc tm of |
|
679 |
Some t => one :: dest_sum t |
|
680 |
| None => |
|
681 |
(case try dest_plus tm of |
|
682 |
Some (t, u) => dest_sum t @ dest_sum u |
|
683 |
| None => [tm])); |
|
684 |
||
685 |
||
686 |
(** generic proof tools **) |
|
687 |
||
688 |
(* prove conversions *) |
|
689 |
||
690 |
val mk_eqv = HOLogic.mk_Trueprop o HOLogic.mk_eq; |
|
691 |
||
692 |
fun prove_conv expand_tac norm_tac sg (t, u) = |
|
693 |
mk_meta_eq (prove_goalw_cterm_nocheck [] (cterm_of sg (mk_eqv (t, u))) |
|
694 |
(K [expand_tac, norm_tac])) |
|
695 |
handle ERROR => error ("The error(s) above occurred while trying to prove " ^ |
|
696 |
(string_of_cterm (cterm_of sg (mk_eqv (t, u))))); |
|
697 |
||
698 |
val subst_equals = prove_goal HOL.thy "[| t = s; u = t |] ==> u = s" |
|
699 |
(fn prems => [cut_facts_tac prems 1, SIMPSET' asm_simp_tac 1]); |
|
700 |
||
701 |
||
702 |
(* rewriting *) |
|
703 |
||
704 |
fun simp_all rules = ALLGOALS (simp_tac (HOL_ss addsimps rules)); |
|
705 |
||
706 |
val add_rules = [add_Suc, add_Suc_right, add_0, add_0_right]; |
|
707 |
val mult_rules = [mult_Suc, mult_Suc_right, mult_0, mult_0_right]; |
|
708 |
||
709 |
||
710 |
||
711 |
(** cancel common summands **) |
|
712 |
||
713 |
structure Sum = |
|
714 |
struct |
|
715 |
val mk_sum = mk_norm_sum; |
|
716 |
val dest_sum = dest_sum; |
|
717 |
val prove_conv = prove_conv; |
|
718 |
val norm_tac = simp_all add_rules THEN simp_all add_ac; |
|
719 |
end; |
|
720 |
||
721 |
fun gen_uncancel_tac rule ct = |
|
722 |
rtac (instantiate' [] [None, Some ct] (rule RS subst_equals)) 1; |
|
723 |
||
724 |
||
725 |
(* nat eq *) |
|
726 |
||
727 |
structure EqCancelSums = CancelSumsFun |
|
728 |
(struct |
|
729 |
open Sum; |
|
730 |
val mk_bal = HOLogic.mk_eq; |
|
731 |
val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT; |
|
732 |
val uncancel_tac = gen_uncancel_tac add_left_cancel; |
|
733 |
end); |
|
734 |
||
735 |
||
736 |
(* nat less *) |
|
737 |
||
738 |
structure LessCancelSums = CancelSumsFun |
|
739 |
(struct |
|
740 |
open Sum; |
|
741 |
val mk_bal = HOLogic.mk_binrel "op <"; |
|
742 |
val dest_bal = HOLogic.dest_bin "op <" HOLogic.natT; |
|
743 |
val uncancel_tac = gen_uncancel_tac add_left_cancel_less; |
|
744 |
end); |
|
745 |
||
746 |
||
747 |
(* nat le *) |
|
748 |
||
749 |
structure LeCancelSums = CancelSumsFun |
|
750 |
(struct |
|
751 |
open Sum; |
|
752 |
val mk_bal = HOLogic.mk_binrel "op <="; |
|
753 |
val dest_bal = HOLogic.dest_bin "op <=" HOLogic.natT; |
|
754 |
val uncancel_tac = gen_uncancel_tac add_left_cancel_le; |
|
755 |
end); |
|
756 |
||
757 |
||
758 |
(* nat diff *) |
|
759 |
||
760 |
structure DiffCancelSums = CancelSumsFun |
|
761 |
(struct |
|
762 |
open Sum; |
|
763 |
val mk_bal = HOLogic.mk_binop "op -"; |
|
764 |
val dest_bal = HOLogic.dest_bin "op -" HOLogic.natT; |
|
765 |
val uncancel_tac = gen_uncancel_tac diff_cancel; |
|
766 |
end); |
|
767 |
||
768 |
||
769 |
||
770 |
(** cancel common factor **) |
|
771 |
||
772 |
structure Factor = |
|
773 |
struct |
|
774 |
val mk_sum = mk_norm_sum; |
|
775 |
val dest_sum = dest_sum; |
|
776 |
val prove_conv = prove_conv; |
|
777 |
val norm_tac = simp_all (add_rules @ mult_rules) THEN simp_all add_ac; |
|
778 |
end; |
|
779 |
||
6394 | 780 |
fun mk_cnat n = cterm_of (Theory.sign_of Nat.thy) (HOLogic.mk_nat n); |
5983 | 781 |
|
782 |
fun gen_multiply_tac rule k = |
|
783 |
if k > 0 then |
|
784 |
rtac (instantiate' [] [None, Some (mk_cnat (k - 1))] (rule RS subst_equals)) 1 |
|
785 |
else no_tac; |
|
786 |
||
787 |
||
788 |
(* nat eq *) |
|
789 |
||
790 |
structure EqCancelFactor = CancelFactorFun |
|
791 |
(struct |
|
792 |
open Factor; |
|
793 |
val mk_bal = HOLogic.mk_eq; |
|
794 |
val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT; |
|
795 |
val multiply_tac = gen_multiply_tac Suc_mult_cancel1; |
|
796 |
end); |
|
797 |
||
798 |
||
799 |
(* nat less *) |
|
800 |
||
801 |
structure LessCancelFactor = CancelFactorFun |
|
802 |
(struct |
|
803 |
open Factor; |
|
804 |
val mk_bal = HOLogic.mk_binrel "op <"; |
|
805 |
val dest_bal = HOLogic.dest_bin "op <" HOLogic.natT; |
|
806 |
val multiply_tac = gen_multiply_tac Suc_mult_less_cancel1; |
|
807 |
end); |
|
808 |
||
809 |
||
810 |
(* nat le *) |
|
811 |
||
812 |
structure LeCancelFactor = CancelFactorFun |
|
813 |
(struct |
|
814 |
open Factor; |
|
815 |
val mk_bal = HOLogic.mk_binrel "op <="; |
|
816 |
val dest_bal = HOLogic.dest_bin "op <=" HOLogic.natT; |
|
817 |
val multiply_tac = gen_multiply_tac Suc_mult_le_cancel1; |
|
818 |
end); |
|
819 |
||
820 |
||
821 |
||
822 |
(** prepare nat_cancel simprocs **) |
|
823 |
||
8100 | 824 |
fun prep_pat s = Thm.read_cterm (Theory.sign_of Arith.thy) (s, HOLogic.termT); |
5983 | 825 |
val prep_pats = map prep_pat; |
826 |
||
827 |
fun prep_simproc (name, pats, proc) = Simplifier.mk_simproc name pats proc; |
|
828 |
||
829 |
val eq_pats = prep_pats ["(l::nat) + m = n", "(l::nat) = m + n", "Suc m = n", "m = Suc n"]; |
|
830 |
val less_pats = prep_pats ["(l::nat) + m < n", "(l::nat) < m + n", "Suc m < n", "m < Suc n"]; |
|
831 |
val le_pats = prep_pats ["(l::nat) + m <= n", "(l::nat) <= m + n", "Suc m <= n", "m <= Suc n"]; |
|
832 |
val diff_pats = prep_pats ["((l::nat) + m) - n", "(l::nat) - (m + n)", "Suc m - n", "m - Suc n"]; |
|
833 |
||
6055 | 834 |
val nat_cancel_sums_add = map prep_simproc |
5983 | 835 |
[("nateq_cancel_sums", eq_pats, EqCancelSums.proc), |
836 |
("natless_cancel_sums", less_pats, LessCancelSums.proc), |
|
6055 | 837 |
("natle_cancel_sums", le_pats, LeCancelSums.proc)]; |
838 |
||
839 |
val nat_cancel_sums = nat_cancel_sums_add @ |
|
840 |
[prep_simproc("natdiff_cancel_sums", diff_pats, DiffCancelSums.proc)]; |
|
5983 | 841 |
|
842 |
val nat_cancel_factor = map prep_simproc |
|
843 |
[("nateq_cancel_factor", eq_pats, EqCancelFactor.proc), |
|
844 |
("natless_cancel_factor", less_pats, LessCancelFactor.proc), |
|
845 |
("natle_cancel_factor", le_pats, LeCancelFactor.proc)]; |
|
846 |
||
847 |
val nat_cancel = nat_cancel_factor @ nat_cancel_sums; |
|
848 |
||
849 |
||
850 |
end; |
|
851 |
||
852 |
open ArithData; |
|
853 |
||
854 |
Addsimprocs nat_cancel; |
|
855 |
||
856 |
(*---------------------------------------------------------------------------*) |
|
857 |
(* 2. Linear arithmetic *) |
|
858 |
(*---------------------------------------------------------------------------*) |
|
859 |
||
6101 | 860 |
(* Parameters data for general linear arithmetic functor *) |
861 |
||
862 |
structure LA_Logic: LIN_ARITH_LOGIC = |
|
5983 | 863 |
struct |
864 |
val ccontr = ccontr; |
|
865 |
val conjI = conjI; |
|
6101 | 866 |
val neqE = linorder_neqE; |
5983 | 867 |
val notI = notI; |
868 |
val sym = sym; |
|
6109 | 869 |
val not_lessD = linorder_not_less RS iffD1; |
6128 | 870 |
val not_leD = linorder_not_le RS iffD1; |
5983 | 871 |
|
6128 | 872 |
|
6968 | 873 |
fun mk_Eq thm = (thm RS Eq_FalseI) handle THM _ => (thm RS Eq_TrueI); |
6128 | 874 |
|
6073 | 875 |
val mk_Trueprop = HOLogic.mk_Trueprop; |
876 |
||
6079 | 877 |
fun neg_prop(TP$(Const("Not",_)$t)) = TP$t |
878 |
| neg_prop(TP$t) = TP $ (Const("Not",HOLogic.boolT-->HOLogic.boolT)$t); |
|
6073 | 879 |
|
6101 | 880 |
fun is_False thm = |
881 |
let val _ $ t = #prop(rep_thm thm) |
|
882 |
in t = Const("False",HOLogic.boolT) end; |
|
883 |
||
6128 | 884 |
fun is_nat(t) = fastype_of1 t = HOLogic.natT; |
885 |
||
886 |
fun mk_nat_thm sg t = |
|
887 |
let val ct = cterm_of sg t and cn = cterm_of sg (Var(("n",0),HOLogic.natT)) |
|
888 |
in instantiate ([],[(cn,ct)]) le0 end; |
|
889 |
||
6101 | 890 |
end; |
891 |
||
7582 | 892 |
signature LIN_ARITH_DATA2 = |
893 |
sig |
|
894 |
include LIN_ARITH_DATA |
|
895 |
val discrete: (string * bool)list ref |
|
896 |
end; |
|
5983 | 897 |
|
7582 | 898 |
structure LA_Data_Ref: LIN_ARITH_DATA2 = |
899 |
struct |
|
900 |
val add_mono_thms = ref ([]:thm list); |
|
901 |
val lessD = ref ([]:thm list); |
|
902 |
val ss_ref = ref HOL_basic_ss; |
|
903 |
val discrete = ref ([]:(string*bool)list); |
|
5983 | 904 |
|
7548
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
nipkow
parents:
7428
diff
changeset
|
905 |
(* Decomposition of terms *) |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
7428
diff
changeset
|
906 |
|
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
7428
diff
changeset
|
907 |
fun nT (Type("fun",[N,_])) = N = HOLogic.natT |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
7428
diff
changeset
|
908 |
| nT _ = false; |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
7428
diff
changeset
|
909 |
|
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
nipkow
parents:
7428
diff
changeset
|
910 |
fun add_atom(t,m,(p,i)) = (case assoc(p,t) of None => ((t,m)::p,i) |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
7428
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changeset
|
911 |
| Some n => (overwrite(p,(t,n+m:int)), i)); |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
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changeset
|
912 |
|
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
7428
diff
changeset
|
913 |
(* Turn term into list of summand * multiplicity plus a constant *) |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
7428
diff
changeset
|
914 |
fun poly(Const("op +",_) $ s $ t, m, pi) = poly(s,m,poly(t,m,pi)) |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
7428
diff
changeset
|
915 |
| poly(all as Const("op -",T) $ s $ t, m, pi) = |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
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changeset
|
916 |
if nT T then add_atom(all,m,pi) |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
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changeset
|
917 |
else poly(s,m,poly(t,~1*m,pi)) |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
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changeset
|
918 |
| poly(Const("uminus",_) $ t, m, pi) = poly(t,~1*m,pi) |
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ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
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changeset
|
919 |
| poly(Const("0",_), _, pi) = pi |
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ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
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changeset
|
920 |
| poly(Const("Suc",_)$t, m, (p,i)) = poly(t, m, (p,i+m)) |
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ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
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changeset
|
921 |
| poly(all as Const("op *",_) $ (Const("Numeral.number_of",_)$c) $ t, m, pi)= |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
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changeset
|
922 |
(poly(t,m*HOLogic.dest_binum c,pi) |
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ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
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changeset
|
923 |
handle TERM _ => add_atom(all,m,pi)) |
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ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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changeset
|
924 |
| poly(all as Const("op *",_) $ t $ (Const("Numeral.number_of",_)$c), m, pi)= |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
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changeset
|
925 |
(poly(t,m*HOLogic.dest_binum c,pi) |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
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changeset
|
926 |
handle TERM _ => add_atom(all,m,pi)) |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
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changeset
|
927 |
| poly(all as Const("Numeral.number_of",_)$t,m,(p,i)) = |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
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diff
changeset
|
928 |
((p,i + m*HOLogic.dest_binum t) |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
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changeset
|
929 |
handle TERM _ => add_atom(all,m,(p,i))) |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
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|
930 |
| poly x = add_atom x; |
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ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
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changeset
|
931 |
|
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ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
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changeset
|
932 |
fun decomp2(rel,lhs,rhs) = |
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ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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changeset
|
933 |
let val (p,i) = poly(lhs,1,([],0)) and (q,j) = poly(rhs,1,([],0)) |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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changeset
|
934 |
in case rel of |
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ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
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parents:
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changeset
|
935 |
"op <" => Some(p,i,"<",q,j) |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
nipkow
parents:
7428
diff
changeset
|
936 |
| "op <=" => Some(p,i,"<=",q,j) |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
nipkow
parents:
7428
diff
changeset
|
937 |
| "op =" => Some(p,i,"=",q,j) |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
nipkow
parents:
7428
diff
changeset
|
938 |
| _ => None |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
nipkow
parents:
7428
diff
changeset
|
939 |
end; |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
nipkow
parents:
7428
diff
changeset
|
940 |
|
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
nipkow
parents:
7428
diff
changeset
|
941 |
fun negate(Some(x,i,rel,y,j,d)) = Some(x,i,"~"^rel,y,j,d) |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
nipkow
parents:
7428
diff
changeset
|
942 |
| negate None = None; |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
nipkow
parents:
7428
diff
changeset
|
943 |
|
7582 | 944 |
fun decomp1 (T,xxx) = |
7548
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
nipkow
parents:
7428
diff
changeset
|
945 |
(case T of |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
nipkow
parents:
7428
diff
changeset
|
946 |
Type("fun",[Type(D,[]),_]) => |
7582 | 947 |
(case assoc(!discrete,D) of |
7548
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
nipkow
parents:
7428
diff
changeset
|
948 |
None => None |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
nipkow
parents:
7428
diff
changeset
|
949 |
| Some d => (case decomp2 xxx of |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
nipkow
parents:
7428
diff
changeset
|
950 |
None => None |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
nipkow
parents:
7428
diff
changeset
|
951 |
| Some(p,i,rel,q,j) => Some(p,i,rel,q,j,d))) |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
nipkow
parents:
7428
diff
changeset
|
952 |
| _ => None); |
9e29a3af64ab
ROOT: Integ/bin_simprocs.ML now loaded in Integ/Bin.ML
nipkow
parents:
7428
diff
changeset
|
953 |
|
7582 | 954 |
fun decomp (_$(Const(rel,T)$lhs$rhs)) = decomp1 (T,(rel,lhs,rhs)) |
955 |
| decomp (_$(Const("Not",_)$(Const(rel,T)$lhs$rhs))) = |
|
956 |
negate(decomp1 (T,(rel,lhs,rhs))) |
|
957 |
| decomp _ = None |
|
6128 | 958 |
end; |
6055 | 959 |
|
7582 | 960 |
let |
961 |
||
962 |
(* reduce contradictory <= to False. |
|
963 |
Most of the work is done by the cancel tactics. |
|
964 |
*) |
|
965 |
val add_rules = [add_0,add_0_right,Zero_not_Suc,Suc_not_Zero,le_0_eq]; |
|
966 |
||
967 |
val add_mono_thms = map (fn s => prove_goal Arith.thy s |
|
968 |
(fn prems => [cut_facts_tac prems 1, |
|
969 |
blast_tac (claset() addIs [add_le_mono]) 1])) |
|
970 |
["(i <= j) & (k <= l) ==> i + k <= j + (l::nat)", |
|
971 |
"(i = j) & (k <= l) ==> i + k <= j + (l::nat)", |
|
972 |
"(i <= j) & (k = l) ==> i + k <= j + (l::nat)", |
|
973 |
"(i = j) & (k = l) ==> i + k = j + (l::nat)" |
|
974 |
]; |
|
975 |
||
976 |
in |
|
977 |
LA_Data_Ref.add_mono_thms := !LA_Data_Ref.add_mono_thms @ add_mono_thms; |
|
978 |
LA_Data_Ref.lessD := !LA_Data_Ref.lessD @ [Suc_leI]; |
|
979 |
LA_Data_Ref.ss_ref := !LA_Data_Ref.ss_ref addsimps add_rules |
|
980 |
addsimprocs nat_cancel_sums_add; |
|
981 |
LA_Data_Ref.discrete := !LA_Data_Ref.discrete @ [("nat",true)] |
|
5983 | 982 |
end; |
983 |
||
6128 | 984 |
structure Fast_Arith = |
985 |
Fast_Lin_Arith(structure LA_Logic=LA_Logic and LA_Data=LA_Data_Ref); |
|
5983 | 986 |
|
6128 | 987 |
val fast_arith_tac = Fast_Arith.lin_arith_tac; |
6073 | 988 |
|
7582 | 989 |
let |
6128 | 990 |
val nat_arith_simproc_pats = |
6394 | 991 |
map (fn s => Thm.read_cterm (Theory.sign_of Arith.thy) (s, HOLogic.boolT)) |
6128 | 992 |
["(m::nat) < n","(m::nat) <= n", "(m::nat) = n"]; |
5983 | 993 |
|
7582 | 994 |
val fast_nat_arith_simproc = mk_simproc |
995 |
"fast_nat_arith" nat_arith_simproc_pats Fast_Arith.lin_arith_prover; |
|
996 |
in |
|
997 |
Addsimprocs [fast_nat_arith_simproc] |
|
998 |
end; |
|
6073 | 999 |
|
1000 |
(* Because of fast_nat_arith_simproc, the arithmetic solver is really only |
|
1001 |
useful to detect inconsistencies among the premises for subgoals which are |
|
1002 |
*not* themselves (in)equalities, because the latter activate |
|
1003 |
fast_nat_arith_simproc anyway. However, it seems cheaper to activate the |
|
1004 |
solver all the time rather than add the additional check. *) |
|
1005 |
||
7570 | 1006 |
simpset_ref () := (simpset() addSolver |
1007 |
(mk_solver "lin. arith." Fast_Arith.cut_lin_arith_tac)); |
|
6055 | 1008 |
|
8252 | 1009 |
(*Elimination of `-' on nat due to John Harrison. *) |
7945
3aca6352f063
got rid of split_diff, which duplicated nat_diff_split, and
paulson
parents:
7622
diff
changeset
|
1010 |
Goal "P(a - b::nat) = (ALL d. (b = a + d --> P 0) & (a = b + d --> P d))"; |
6301 | 1011 |
by (case_tac "a <= b" 1); |
7059 | 1012 |
by Auto_tac; |
6301 | 1013 |
by (eres_inst_tac [("x","b-a")] allE 1); |
7059 | 1014 |
by (asm_simp_tac (simpset() addsimps [diff_is_0_eq RS iffD2]) 1); |
6055 | 1015 |
qed "nat_diff_split"; |
1016 |
||
8252 | 1017 |
(*LCP's version, replacing b=a+d by a<b, which sometimes works better*) |
1018 |
Goal "P(a - b::nat) = (ALL d. (a<b --> P 0) & (a = b + d --> P d))"; |
|
1019 |
by (auto_tac (claset(), simpset() addsplits [nat_diff_split])); |
|
1020 |
qed "nat_diff_split'"; |
|
1021 |
||
7945
3aca6352f063
got rid of split_diff, which duplicated nat_diff_split, and
paulson
parents:
7622
diff
changeset
|
1022 |
|
6055 | 1023 |
(* FIXME: K true should be replaced by a sensible test to speed things up |
6157 | 1024 |
in case there are lots of irrelevant terms involved; |
1025 |
elimination of min/max can be optimized: |
|
1026 |
(max m n + k <= r) = (m+k <= r & n+k <= r) |
|
1027 |
(l <= min m n + k) = (l <= m+k & l <= n+k) |
|
6055 | 1028 |
*) |
7582 | 1029 |
val arith_tac_split_thms = ref [nat_diff_split,split_min,split_max]; |
1030 |
fun arith_tac i = |
|
1031 |
refute_tac (K true) (REPEAT o split_tac (!arith_tac_split_thms)) |
|
1032 |
((REPEAT_DETERM o etac linorder_neqE) THEN' fast_arith_tac) i; |
|
6055 | 1033 |
|
7131 | 1034 |
|
1035 |
(* proof method setup *) |
|
1036 |
||
7428 | 1037 |
val arith_method = |
1038 |
Method.METHOD (fn facts => FIRSTGOAL (Method.insert_tac facts THEN' arith_tac)); |
|
7131 | 1039 |
|
1040 |
val arith_setup = |
|
1041 |
[Method.add_methods |
|
1042 |
[("arith", Method.no_args arith_method, "decide linear arithmethic")]]; |
|
1043 |
||
5983 | 1044 |
(*---------------------------------------------------------------------------*) |
1045 |
(* End of proof procedures. Now go and USE them! *) |
|
1046 |
(*---------------------------------------------------------------------------*) |
|
1047 |
||
4736 | 1048 |
(*** Subtraction laws -- mostly from Clemens Ballarin ***) |
3234 | 1049 |
|
5429 | 1050 |
Goal "[| a < (b::nat); c <= a |] ==> a-c < b-c"; |
6301 | 1051 |
by (arith_tac 1); |
3234 | 1052 |
qed "diff_less_mono"; |
1053 |
||
5429 | 1054 |
Goal "a+b < (c::nat) ==> a < c-b"; |
6301 | 1055 |
by (arith_tac 1); |
3234 | 1056 |
qed "add_less_imp_less_diff"; |
1057 |
||
5427 | 1058 |
Goal "(i < j-k) = (i+k < (j::nat))"; |
6301 | 1059 |
by (arith_tac 1); |
5427 | 1060 |
qed "less_diff_conv"; |
1061 |
||
5497 | 1062 |
Goal "(j-k <= (i::nat)) = (j <= i+k)"; |
6301 | 1063 |
by (arith_tac 1); |
5485 | 1064 |
qed "le_diff_conv"; |
1065 |
||
5497 | 1066 |
Goal "k <= j ==> (i <= j-k) = (i+k <= (j::nat))"; |
6301 | 1067 |
by (arith_tac 1); |
5497 | 1068 |
qed "le_diff_conv2"; |
1069 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
1070 |
Goal "Suc i <= n ==> Suc (n - Suc i) = n - i"; |
6301 | 1071 |
by (arith_tac 1); |
3234 | 1072 |
qed "Suc_diff_Suc"; |
1073 |
||
5429 | 1074 |
Goal "i <= (n::nat) ==> n - (n - i) = i"; |
6301 | 1075 |
by (arith_tac 1); |
3234 | 1076 |
qed "diff_diff_cancel"; |
3381
2bac33ec2b0d
New theorems le_add_diff_inverse, le_add_diff_inverse2
paulson
parents:
3366
diff
changeset
|
1077 |
Addsimps [diff_diff_cancel]; |
3234 | 1078 |
|
5429 | 1079 |
Goal "k <= (n::nat) ==> m <= n + m - k"; |
6301 | 1080 |
by (arith_tac 1); |
3234 | 1081 |
qed "le_add_diff"; |
1082 |
||
6055 | 1083 |
Goal "[| 0<k; j<i |] ==> j+k-i < k"; |
6301 | 1084 |
by (arith_tac 1); |
6055 | 1085 |
qed "add_diff_less"; |
3234 | 1086 |
|
5356 | 1087 |
Goal "m-1 < n ==> m <= n"; |
6301 | 1088 |
by (arith_tac 1); |
5356 | 1089 |
qed "pred_less_imp_le"; |
1090 |
||
1091 |
Goal "j<=i ==> i - j < Suc i - j"; |
|
6301 | 1092 |
by (arith_tac 1); |
5356 | 1093 |
qed "diff_less_Suc_diff"; |
1094 |
||
1095 |
Goal "i - j <= Suc i - j"; |
|
6301 | 1096 |
by (arith_tac 1); |
5356 | 1097 |
qed "diff_le_Suc_diff"; |
1098 |
AddIffs [diff_le_Suc_diff]; |
|
1099 |
||
1100 |
Goal "n - Suc i <= n - i"; |
|
6301 | 1101 |
by (arith_tac 1); |
5356 | 1102 |
qed "diff_Suc_le_diff"; |
1103 |
AddIffs [diff_Suc_le_diff]; |
|
1104 |
||
5409 | 1105 |
Goal "0 < n ==> (m <= n-1) = (m<n)"; |
6301 | 1106 |
by (arith_tac 1); |
5409 | 1107 |
qed "le_pred_eq"; |
1108 |
||
1109 |
Goal "0 < n ==> (m-1 < n) = (m<=n)"; |
|
6301 | 1110 |
by (arith_tac 1); |
5409 | 1111 |
qed "less_pred_eq"; |
1112 |
||
7059 | 1113 |
(*Replaces the previous diff_less and le_diff_less, which had the stronger |
1114 |
second premise n<=m*) |
|
1115 |
Goal "[| 0<n; 0<m |] ==> m - n < m"; |
|
6301 | 1116 |
by (arith_tac 1); |
5414
8a458866637c
changed Suc_diff_n to Suc_diff_le, with premise n <= m instead of n < Suc(m)
paulson
parents:
5409
diff
changeset
|
1117 |
qed "diff_less"; |
8a458866637c
changed Suc_diff_n to Suc_diff_le, with premise n <= m instead of n < Suc(m)
paulson
parents:
5409
diff
changeset
|
1118 |
|
4732 | 1119 |
|
7128 | 1120 |
(*** Reducting subtraction to addition ***) |
1121 |
||
1122 |
Goal "n<=(l::nat) --> Suc l - n + m = Suc (l - n + m)"; |
|
7945
3aca6352f063
got rid of split_diff, which duplicated nat_diff_split, and
paulson
parents:
7622
diff
changeset
|
1123 |
by (simp_tac (simpset() addsplits [nat_diff_split]) 1); |
7128 | 1124 |
qed_spec_mp "Suc_diff_add_le"; |
1125 |
||
1126 |
Goal "i<n ==> n - Suc i < n - i"; |
|
7945
3aca6352f063
got rid of split_diff, which duplicated nat_diff_split, and
paulson
parents:
7622
diff
changeset
|
1127 |
by (asm_simp_tac (simpset() addsplits [nat_diff_split]) 1); |
7128 | 1128 |
qed "diff_Suc_less_diff"; |
1129 |
||
1130 |
Goal "Suc(m)-n = (if m<n then 0 else Suc(m-n))"; |
|
7945
3aca6352f063
got rid of split_diff, which duplicated nat_diff_split, and
paulson
parents:
7622
diff
changeset
|
1131 |
by (simp_tac (simpset() addsplits [nat_diff_split]) 1); |
7128 | 1132 |
qed "if_Suc_diff_le"; |
1133 |
||
1134 |
Goal "Suc(m)-n <= Suc(m-n)"; |
|
7945
3aca6352f063
got rid of split_diff, which duplicated nat_diff_split, and
paulson
parents:
7622
diff
changeset
|
1135 |
by (simp_tac (simpset() addsplits [nat_diff_split]) 1); |
7128 | 1136 |
qed "diff_Suc_le_Suc_diff"; |
1137 |
||
1138 |
Goal "[| k<=n; n<=m |] ==> (m-k) - (n-k) = m-(n::nat)"; |
|
7945
3aca6352f063
got rid of split_diff, which duplicated nat_diff_split, and
paulson
parents:
7622
diff
changeset
|
1139 |
by (asm_simp_tac (simpset() addsplits [nat_diff_split]) 1); |
7128 | 1140 |
qed "diff_right_cancel"; |
1141 |
||
1142 |
||
7108 | 1143 |
(** (Anti)Monotonicity of subtraction -- by Stephan Merz **) |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
1144 |
|
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
1145 |
(* Monotonicity of subtraction in first argument *) |
6055 | 1146 |
Goal "m <= (n::nat) ==> (m-l) <= (n-l)"; |
7945
3aca6352f063
got rid of split_diff, which duplicated nat_diff_split, and
paulson
parents:
7622
diff
changeset
|
1147 |
by (asm_simp_tac (simpset() addsplits [nat_diff_split]) 1); |
6055 | 1148 |
qed "diff_le_mono"; |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
1149 |
|
5429 | 1150 |
Goal "m <= (n::nat) ==> (l-n) <= (l-m)"; |
7945
3aca6352f063
got rid of split_diff, which duplicated nat_diff_split, and
paulson
parents:
7622
diff
changeset
|
1151 |
by (asm_simp_tac (simpset() addsplits [nat_diff_split]) 1); |
6055 | 1152 |
qed "diff_le_mono2"; |
5983 | 1153 |
|
6055 | 1154 |
Goal "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"; |
7945
3aca6352f063
got rid of split_diff, which duplicated nat_diff_split, and
paulson
parents:
7622
diff
changeset
|
1155 |
by (asm_simp_tac (simpset() addsplits [nat_diff_split]) 1); |
6055 | 1156 |
qed "diff_less_mono2"; |
5983 | 1157 |
|
6055 | 1158 |
Goal "[| m-n = 0; n-m = 0 |] ==> m=n"; |
7945
3aca6352f063
got rid of split_diff, which duplicated nat_diff_split, and
paulson
parents:
7622
diff
changeset
|
1159 |
by (asm_full_simp_tac (simpset() addsplits [nat_diff_split]) 1); |
6055 | 1160 |
qed "diffs0_imp_equal"; |