1 (* Title: HOL/Integ/Int.ML
3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory
4 Copyright 1998 University of Cambridge
6 Type "int" is a linear order
8 And many further lemmas
12 Goal "int 0 = (0::int)";
13 by (simp_tac (simpset() addsimps [Zero_int_def]) 1);
17 by (simp_tac (simpset() addsimps [One_int_def]) 1);
20 Goal "int (Suc 0) = 1";
21 by (simp_tac (simpset() addsimps [One_int_def, One_nat_def]) 1);
24 Goalw [zdiff_def,zless_def] "neg x = (x < 0)";
28 Goalw [zle_def] "(~neg x) = (0 <= x)";
29 by (simp_tac (simpset() addsimps [neg_eq_less_0]) 1);
30 qed "not_neg_eq_ge_0";
32 (** Needed to simplify inequalities when Numeral1 can get simplified to 1 **)
35 by (simp_tac (simpset() addsimps [One_int_def, neg_eq_less_0]) 1);
39 by (simp_tac (simpset() addsimps [One_int_def, neg_eq_less_0]) 1);
43 by (simp_tac (simpset() addsimps [iszero_def]) 1);
47 by (simp_tac (simpset() addsimps [Zero_int_def, One_int_def, One_nat_def,
52 by (simp_tac (simpset() addsimps [Zero_int_def, One_int_def, One_nat_def]) 1);
55 Goal "0 \\<noteq> (1::int)";
56 by (simp_tac (simpset() addsimps [Zero_int_def, One_int_def, One_nat_def]) 1);
59 Addsimps [int_0, int_1, int_0_neq_1];
62 (*** Abel_Cancel simproc on the integers ***)
64 (* Lemmas needed for the simprocs *)
66 (*Deletion of other terms in the formula, seeking the -x at the front of z*)
67 Goal "((x::int) + (y + z) = y + u) = ((x + z) = u)";
68 by (stac zadd_left_commute 1);
69 by (rtac zadd_left_cancel 1);
72 (*A further rule to deal with the case that
73 everything gets cancelled on the right.*)
74 Goal "((x::int) + (y + z) = y) = (x = -z)";
75 by (stac zadd_left_commute 1);
76 by (res_inst_tac [("t", "y")] (zadd_0_right RS subst) 1
77 THEN stac zadd_left_cancel 1);
78 by (simp_tac (simpset() addsimps [eq_zdiff_eq RS sym]) 1);
79 qed "zadd_cancel_end";
82 structure Int_Cancel_Data =
85 val eq_reflection = eq_reflection
87 val sg_ref = Sign.self_ref (Theory.sign_of (the_context ()))
89 val zero = Const ("0", HOLogic.intT)
90 val restrict_to_left = restrict_to_left
91 val add_cancel_21 = zadd_cancel_21
92 val add_cancel_end = zadd_cancel_end
93 val add_left_cancel = zadd_left_cancel
94 val add_assoc = zadd_assoc
95 val add_commute = zadd_commute
96 val add_left_commute = zadd_left_commute
98 val add_0_right = zadd_0_right
100 val eq_diff_eq = eq_zdiff_eq
101 val eqI_rules = [zless_eqI, zeq_eqI, zle_eqI]
103 #1 (HOLogic.dest_bin "op =" HOLogic.boolT
104 (HOLogic.dest_Trueprop (concl_of th)))
106 val diff_def = zdiff_def
107 val minus_add_distrib = zminus_zadd_distrib
108 val minus_minus = zminus_zminus
109 val minus_0 = zminus_0
110 val add_inverses = [zadd_zminus_inverse, zadd_zminus_inverse2]
111 val cancel_simps = [zadd_zminus_cancel, zminus_zadd_cancel]
114 structure Int_Cancel = Abel_Cancel (Int_Cancel_Data);
116 Addsimprocs [Int_Cancel.sum_conv, Int_Cancel.rel_conv];
122 Goal "- (z - y) = y - (z::int)";
124 qed "zminus_zdiff_eq";
125 Addsimps [zminus_zdiff_eq];
127 Goal "(w<z) = neg(w-z)";
128 by (simp_tac (simpset() addsimps [zless_def]) 1);
131 Goal "(w=z) = iszero(w-z)";
132 by (simp_tac (simpset() addsimps [iszero_def, zdiff_eq_eq]) 1);
135 Goal "(w<=z) = (~ neg(z-w))";
136 by (simp_tac (simpset() addsimps [zle_def, zless_def]) 1);
137 qed "zle_eq_not_neg";
139 (** Inequality reasoning **)
141 Goal "(w < z + (1::int)) = (w<z | w=z)";
142 by (auto_tac (claset(),
143 simpset() addsimps [zless_iff_Suc_zadd, int_Suc,
144 gr0_conv_Suc, zero_reorient]));
145 by (res_inst_tac [("x","Suc n")] exI 1);
146 by (simp_tac (simpset() addsimps [int_Suc]) 1);
149 Goal "(w + (1::int) <= z) = (w<z)";
150 by (asm_full_simp_tac (simpset() addsimps [zle_def, zless_add1_eq]) 1);
151 by (auto_tac (claset() addIs [zle_anti_sym],
152 simpset() addsimps [order_less_imp_le, symmetric zle_def]));
155 Goal "((1::int) + w <= z) = (w<z)";
156 by (stac zadd_commute 1);
157 by (rtac add1_zle_eq 1);
158 qed "add1_left_zle_eq";
161 (*** Monotonicity results ***)
163 Goal "(v+z < w+z) = (v < (w::int))";
165 qed "zadd_right_cancel_zless";
167 Goal "(z+v < z+w) = (v < (w::int))";
169 qed "zadd_left_cancel_zless";
171 Addsimps [zadd_right_cancel_zless, zadd_left_cancel_zless];
173 Goal "(v+z <= w+z) = (v <= (w::int))";
175 qed "zadd_right_cancel_zle";
177 Goal "(z+v <= z+w) = (v <= (w::int))";
179 qed "zadd_left_cancel_zle";
181 Addsimps [zadd_right_cancel_zle, zadd_left_cancel_zle];
183 (*"v<=w ==> v+z <= w+z"*)
184 bind_thm ("zadd_zless_mono1", zadd_right_cancel_zless RS iffD2);
186 (*"v<=w ==> z+v <= z+w"*)
187 bind_thm ("zadd_zless_mono2", zadd_left_cancel_zless RS iffD2);
189 (*"v<=w ==> v+z <= w+z"*)
190 bind_thm ("zadd_zle_mono1", zadd_right_cancel_zle RS iffD2);
192 (*"v<=w ==> z+v <= z+w"*)
193 bind_thm ("zadd_zle_mono2", zadd_left_cancel_zle RS iffD2);
195 Goal "[| w'<=w; z'<=z |] ==> w' + z' <= w + (z::int)";
196 by (etac (zadd_zle_mono1 RS zle_trans) 1);
200 Goal "[| w'<w; z'<=z |] ==> w' + z' < w + (z::int)";
201 by (etac (zadd_zless_mono1 RS order_less_le_trans) 1);
203 qed "zadd_zless_mono";
206 (*** Comparison laws ***)
208 Goal "(- x < - y) = (y < (x::int))";
209 by (simp_tac (simpset() addsimps [zless_def, zdiff_def] @ zadd_ac) 1);
210 qed "zminus_zless_zminus";
211 Addsimps [zminus_zless_zminus];
213 Goal "(- x <= - y) = (y <= (x::int))";
214 by (simp_tac (simpset() addsimps [zle_def]) 1);
215 qed "zminus_zle_zminus";
216 Addsimps [zminus_zle_zminus];
218 (** The next several equations can make the simplifier loop! **)
220 Goal "(x < - y) = (y < - (x::int))";
221 by (simp_tac (simpset() addsimps [zless_def, zdiff_def] @ zadd_ac) 1);
224 Goal "(- x < y) = (- y < (x::int))";
225 by (simp_tac (simpset() addsimps [zless_def, zdiff_def] @ zadd_ac) 1);
228 Goal "(x <= - y) = (y <= - (x::int))";
229 by (simp_tac (simpset() addsimps [zle_def, zminus_zless]) 1);
232 Goal "(- x <= y) = (- y <= (x::int))";
233 by (simp_tac (simpset() addsimps [zle_def, zless_zminus]) 1);
236 Goal "(x = - y) = (y = - (x::int))";
238 qed "equation_zminus";
240 Goal "(- x = y) = (- (y::int) = x)";
242 qed "zminus_equation";
245 (** Instances of the equations above, for zero **)
247 (*instantiate a variable to zero and simplify*)
248 fun zero_instance v th = simplify (simpset()) (inst v "0" th);
250 Addsimps [zero_instance "x" zless_zminus,
251 zero_instance "y" zminus_zless,
252 zero_instance "x" zle_zminus,
253 zero_instance "y" zminus_zle,
254 zero_instance "x" equation_zminus,
255 zero_instance "y" zminus_equation];
258 Goal "- (int (Suc n)) < 0";
259 by (simp_tac (simpset() addsimps [zless_def]) 1);
260 qed "negative_zless_0";
262 Goal "- (int (Suc n)) < int m";
263 by (rtac (negative_zless_0 RS order_less_le_trans) 1);
265 qed "negative_zless";
266 AddIffs [negative_zless];
269 by (simp_tac (simpset() addsimps [zminus_zle]) 1);
270 qed "negative_zle_0";
272 Goal "- int n <= int m";
273 by (simp_tac (simpset() addsimps [zless_def, zle_def, zdiff_def, zadd_int]) 1);
275 AddIffs [negative_zle];
277 Goal "~(0 <= - (int (Suc n)))";
278 by (stac zle_zminus 1);
280 qed "not_zle_0_negative";
281 Addsimps [not_zle_0_negative];
283 Goal "(int n <= - int m) = (n = 0 & m = 0)";
286 by (dtac (zle_zminus RS iffD1) 2);
287 by (ALLGOALS (dtac (negative_zle_0 RSN(2,zle_trans))));
288 by (ALLGOALS Asm_full_simp_tac);
291 Goal "~(int n < - int m)";
292 by (simp_tac (simpset() addsimps [symmetric zle_def]) 1);
293 qed "not_int_zless_negative";
295 Goal "(- int n = int m) = (n = 0 & m = 0)";
297 by (rtac (int_zle_neg RS iffD1) 1);
299 by (ALLGOALS Asm_simp_tac);
300 qed "negative_eq_positive";
302 Addsimps [negative_eq_positive, not_int_zless_negative];
305 Goal "(w <= z) = (EX n. z = w + int n)";
306 by (auto_tac (claset() addIs [inst "x" "0::nat" exI]
307 addSIs [not_sym RS not0_implies_Suc],
308 simpset() addsimps [zless_iff_Suc_zadd, int_le_less]));
311 Goal "abs (int m) = int m";
312 by (simp_tac (simpset() addsimps [zabs_def]) 1);
314 Addsimps [abs_int_eq];
317 (**** nat: magnitide of an integer, as a natural number ****)
319 Goalw [nat_def] "nat(int n) = n";
324 Goalw [nat_def] "nat(- (int n)) = 0";
325 by (auto_tac (claset(),
326 simpset() addsimps [neg_eq_less_0, zero_reorient, zminus_zless]));
327 qed "nat_zminus_int";
328 Addsimps [nat_zminus_int];
330 Goalw [Zero_int_def] "nat 0 = 0";
335 Goal "~ neg z ==> int (nat z) = z";
336 by (dtac (not_neg_eq_ge_0 RS iffD1) 1);
337 by (dtac zle_imp_zless_or_eq 1);
338 by (auto_tac (claset(), simpset() addsimps [zless_iff_Suc_zadd]));
341 Goal "neg x ==> EX n. x = - (int (Suc n))";
342 by (auto_tac (claset(),
343 simpset() addsimps [neg_eq_less_0, zless_iff_Suc_zadd,
344 zdiff_eq_eq RS sym, zdiff_def]));
347 Goalw [nat_def] "neg z ==> nat z = 0";
351 Goal "(m < nat z) = (int m < z)";
352 by (case_tac "neg z" 1);
353 by (etac (not_neg_nat RS subst) 2);
354 by (auto_tac (claset(), simpset() addsimps [neg_nat]));
355 by (auto_tac (claset() addDs [order_less_trans],
356 simpset() addsimps [neg_eq_less_0]));
357 qed "zless_nat_eq_int_zless";
359 Goal "0 <= z ==> int (nat z) = z";
360 by (asm_full_simp_tac
361 (simpset() addsimps [neg_eq_less_0, zle_def, not_neg_nat]) 1);
364 Goal "z <= 0 ==> nat z = 0";
365 by (auto_tac (claset(),
366 simpset() addsimps [order_le_less, neg_eq_less_0,
369 Addsimps [nat_0_le, nat_le_0];
371 (*An alternative condition is 0 <= w *)
372 Goal "0 < z ==> (nat w < nat z) = (w < z)";
373 by (stac (zless_int RS sym) 1);
374 by (asm_simp_tac (simpset() addsimps [not_neg_nat, not_neg_eq_ge_0,
376 by (case_tac "neg w" 1);
377 by (asm_simp_tac (simpset() addsimps [not_neg_nat]) 2);
378 by (asm_full_simp_tac (simpset() addsimps [neg_eq_less_0, neg_nat]) 1);
379 by (blast_tac (claset() addIs [order_less_trans]) 1);
380 val lemma = result();
382 Goal "(nat w < nat z) = (0 < z & w < z)";
383 by (case_tac "0 < z" 1);
384 by (auto_tac (claset(), simpset() addsimps [lemma, linorder_not_less]));
385 qed "zless_nat_conj";
388 (* a case theorem distinguishing non-negative and negative int *)
391 "[|!! n. z = int n ==> P; !! n. z = - (int (Suc n)) ==> P |] ==> P";
392 by (case_tac "neg z" 1);
393 by (fast_tac (claset() addSDs [negD] addSEs prems) 1);
394 by (dtac (not_neg_nat RS sym) 1);
395 by (eresolve_tac prems 1);
398 fun int_case_tac x = res_inst_tac [("z",x)] int_cases;
401 (*** Monotonicity of Multiplication ***)
403 Goal "i <= (j::int) ==> i * int k <= j * int k";
404 by (induct_tac "k" 1);
407 (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2, zadd_zle_mono,
409 val lemma = result();
411 Goal "[| i <= j; (0::int) <= k |] ==> i*k <= j*k";
412 by (res_inst_tac [("t", "k")] (not_neg_nat RS subst) 1);
414 by (full_simp_tac (simpset() addsimps [not_neg_eq_ge_0]) 1);
415 qed "zmult_zle_mono1";
417 Goal "[| i <= j; k <= (0::int) |] ==> j*k <= i*k";
418 by (rtac (zminus_zle_zminus RS iffD1) 1);
419 by (asm_simp_tac (simpset() addsimps [zmult_zminus_right RS sym,
420 zmult_zle_mono1, zle_zminus]) 1);
421 qed "zmult_zle_mono1_neg";
423 Goal "[| i <= j; (0::int) <= k |] ==> k*i <= k*j";
424 by (dtac zmult_zle_mono1 1);
425 by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [zmult_commute])));
426 qed "zmult_zle_mono2";
428 Goal "[| i <= j; k <= (0::int) |] ==> k*j <= k*i";
429 by (dtac zmult_zle_mono1_neg 1);
430 by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [zmult_commute])));
431 qed "zmult_zle_mono2_neg";
433 (* <= monotonicity, BOTH arguments*)
434 Goal "[| i <= j; k <= l; (0::int) <= j; (0::int) <= k |] ==> i*k <= j*l";
435 by (etac (zmult_zle_mono1 RS order_trans) 1);
437 by (etac zmult_zle_mono2 1);
439 qed "zmult_zle_mono";
442 (** strict, in 1st argument; proof is by induction on k>0 **)
444 Goal "i<j ==> 0<k --> int k * i < int k * j";
445 by (induct_tac "k" 1);
447 by (case_tac "n=0" 2);
448 by (ALLGOALS (asm_full_simp_tac
449 (simpset() addsimps [zadd_zmult_distrib, zadd_zless_mono,
450 int_Suc0_eq_1, order_le_less])));
451 val lemma = result();
453 Goal "[| i<j; (0::int) < k |] ==> k*i < k*j";
454 by (res_inst_tac [("t", "k")] (not_neg_nat RS subst) 1);
455 by (etac (lemma RS mp) 2);
456 by (asm_simp_tac (simpset() addsimps [not_neg_eq_ge_0,
458 by (forward_tac [conjI RS (zless_nat_conj RS iffD2)] 1);
460 qed "zmult_zless_mono2";
462 Goal "[| i<j; (0::int) < k |] ==> i*k < j*k";
463 by (dtac zmult_zless_mono2 1);
464 by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [zmult_commute])));
465 qed "zmult_zless_mono1";
467 (* < monotonicity, BOTH arguments*)
468 Goal "[| i < j; k < l; (0::int) < j; (0::int) < k |] ==> i*k < j*l";
469 by (etac (zmult_zless_mono1 RS order_less_trans) 1);
471 by (etac zmult_zless_mono2 1);
473 qed "zmult_zless_mono";
475 Goal "[| i<j; k < (0::int) |] ==> j*k < i*k";
476 by (rtac (zminus_zless_zminus RS iffD1) 1);
477 by (asm_simp_tac (simpset() addsimps [zmult_zminus_right RS sym,
478 zmult_zless_mono1, zless_zminus]) 1);
479 qed "zmult_zless_mono1_neg";
481 Goal "[| i<j; k < (0::int) |] ==> k*j < k*i";
482 by (rtac (zminus_zless_zminus RS iffD1) 1);
483 by (asm_simp_tac (simpset() addsimps [zmult_zminus RS sym,
484 zmult_zless_mono2, zless_zminus]) 1);
485 qed "zmult_zless_mono2_neg";
488 Goal "(m*n = (0::int)) = (m = 0 | n = 0)";
489 by (case_tac "m < (0::int)" 1);
490 by (auto_tac (claset(),
491 simpset() addsimps [linorder_not_less, order_le_less,
494 (force_tac (claset() addDs [zmult_zless_mono1_neg, zmult_zless_mono1],
496 qed "zmult_eq_0_iff";
497 AddIffs [zmult_eq_0_iff];
500 (** Cancellation laws for k*m < k*n and m*k < n*k, also for <= and =,
501 but not (yet?) for k*m < n*k. **)
503 Goal "(m*k < n*k) = (((0::int) < k & m<n) | (k < 0 & n<m))";
504 by (case_tac "k = (0::int)" 1);
505 by (auto_tac (claset(), simpset() addsimps [linorder_neq_iff,
506 zmult_zless_mono1, zmult_zless_mono1_neg]));
507 by (auto_tac (claset(),
508 simpset() addsimps [linorder_not_less,
509 inst "y1" "m*k" (linorder_not_le RS sym),
510 inst "y1" "m" (linorder_not_le RS sym)]));
511 by (ALLGOALS (etac notE));
512 by (auto_tac (claset(), simpset() addsimps [order_less_imp_le, zmult_zle_mono1,
513 zmult_zle_mono1_neg]));
514 qed "zmult_zless_cancel2";
517 Goal "(k*m < k*n) = (((0::int) < k & m<n) | (k < 0 & n<m))";
518 by (simp_tac (simpset() addsimps [inst "z" "k" zmult_commute,
519 zmult_zless_cancel2]) 1);
520 qed "zmult_zless_cancel1";
522 Goal "(m*k <= n*k) = (((0::int) < k --> m<=n) & (k < 0 --> n<=m))";
523 by (simp_tac (simpset() addsimps [linorder_not_less RS sym,
524 zmult_zless_cancel2]) 1);
525 qed "zmult_zle_cancel2";
527 Goal "(k*m <= k*n) = (((0::int) < k --> m<=n) & (k < 0 --> n<=m))";
528 by (simp_tac (simpset() addsimps [linorder_not_less RS sym,
529 zmult_zless_cancel1]) 1);
530 qed "zmult_zle_cancel1";
532 Goal "(m*k = n*k) = (k = (0::int) | m=n)";
533 by (cut_facts_tac [linorder_less_linear] 1);
537 (force_tac (claset() addD2 ("mono_neg", zmult_zless_mono1_neg)
538 addD2 ("mono_pos", zmult_zless_mono1),
539 simpset() addsimps [linorder_neq_iff]) 1));
543 Goal "(k*m = k*n) = (k = (0::int) | m=n)";
544 by (simp_tac (simpset() addsimps [inst "z" "k" zmult_commute,
547 Addsimps [zmult_cancel1, zmult_cancel2];
550 (*Analogous to zadd_int*)
551 Goal "n<=m --> int m - int n = int (m-n)";
552 by (induct_thm_tac diff_induct "m n" 1);
553 by (auto_tac (claset(), simpset() addsimps [int_Suc, symmetric zdiff_def]));
554 qed_spec_mp "zdiff_int";