1 (* ========================================================================= *)
2 (* Theory of integers. *)
4 (* The integers are carved out of the real numbers; hence all the *)
5 (* universal theorems can be derived trivially from the real analog. *)
7 (* John Harrison, University of Cambridge Computer Laboratory *)
9 (* (c) Copyright, University of Cambridge 1998 *)
10 (* (c) Copyright, John Harrison 1998-2007 *)
11 (* ========================================================================= *)
15 (* ------------------------------------------------------------------------- *)
16 (* Representing predicate. The "is_int" variant is useful for backwards *)
17 (* compatibility with former definition of "is_int" constant, now removed. *)
18 (* ------------------------------------------------------------------------- *)
20 let integer = new_definition
21 `integer(x) <=> ?n. abs(x) = &n`;;
24 (`integer(x) <=> ?n. x = &n \/ x = -- &n`,
25 REWRITE_TAC[integer] THEN AP_TERM_TAC THEN ABS_TAC THEN REAL_ARITH_TAC);;
27 (* ------------------------------------------------------------------------- *)
28 (* Type of integers. *)
29 (* ------------------------------------------------------------------------- *)
31 let int_tybij = new_type_definition "int" ("int_of_real","real_of_int")
32 (prove(`?x. integer x`,
34 REWRITE_TAC[is_int; REAL_OF_NUM_EQ; EXISTS_OR_THM; GSYM EXISTS_REFL]));;
36 let int_abstr,int_rep =
37 SPEC_ALL(CONJUNCT1 int_tybij),SPEC_ALL(CONJUNCT2 int_tybij);;
39 let dest_int_rep = prove
40 (`!i. ?n. (real_of_int i = &n) \/ (real_of_int i = --(&n))`,
41 REWRITE_TAC[GSYM is_int; int_rep; int_abstr]);;
43 (* ------------------------------------------------------------------------- *)
44 (* We want the following too. *)
45 (* ------------------------------------------------------------------------- *)
48 (`!x y. (x = y) <=> (real_of_int x = real_of_int y)`,
49 REPEAT GEN_TAC THEN EQ_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
50 POP_ASSUM(MP_TAC o AP_TERM `int_of_real`) THEN
51 REWRITE_TAC[int_abstr]);;
53 (* ------------------------------------------------------------------------- *)
54 (* Set up interface map. *)
55 (* ------------------------------------------------------------------------- *)
57 do_list overload_interface
58 ["+",`int_add:int->int->int`; "-",`int_sub:int->int->int`;
59 "*",`int_mul:int->int->int`; "<",`int_lt:int->int->bool`;
60 "<=",`int_le:int->int->bool`; ">",`int_gt:int->int->bool`;
61 ">=",`int_ge:int->int->bool`; "--",`int_neg:int->int`;
62 "pow",`int_pow:int->num->int`; "abs",`int_abs:int->int`;
63 "max",`int_max:int->int->int`; "min",`int_min:int->int->int`;
64 "&",`int_of_num:num->int`];;
66 let prioritize_int() = prioritize_overload(mk_type("int",[]));;
68 (* ------------------------------------------------------------------------- *)
69 (* Definitions and closure derivations of all operations but "inv" and "/". *)
70 (* ------------------------------------------------------------------------- *)
72 let int_le = new_definition
73 `x <= y <=> (real_of_int x) <= (real_of_int y)`;;
75 let int_lt = new_definition
76 `x < y <=> (real_of_int x) < (real_of_int y)`;;
78 let int_ge = new_definition
79 `x >= y <=> (real_of_int x) >= (real_of_int y)`;;
81 let int_gt = new_definition
82 `x > y <=> (real_of_int x) > (real_of_int y)`;;
84 let int_of_num = new_definition
85 `&n = int_of_real(real_of_num n)`;;
87 let int_of_num_th = prove
88 (`!n. real_of_int(int_of_num n) = real_of_num n`,
89 REWRITE_TAC[int_of_num; GSYM int_rep; is_int] THEN
90 REWRITE_TAC[REAL_OF_NUM_EQ; EXISTS_OR_THM; GSYM EXISTS_REFL]);;
92 let int_neg = new_definition
93 `--i = int_of_real(--(real_of_int i))`;;
95 let int_neg_th = prove
96 (`!x. real_of_int(int_neg x) = --(real_of_int x)`,
97 REWRITE_TAC[int_neg; GSYM int_rep; is_int] THEN
98 GEN_TAC THEN STRIP_ASSUME_TAC(SPEC `x:int` dest_int_rep) THEN
99 ASM_REWRITE_TAC[REAL_NEG_NEG; EXISTS_OR_THM; REAL_EQ_NEG2;
100 REAL_OF_NUM_EQ; GSYM EXISTS_REFL]);;
102 let int_add = new_definition
103 `x + y = int_of_real((real_of_int x) + (real_of_int y))`;;
105 let int_add_th = prove
106 (`!x y. real_of_int(x + y) = (real_of_int x) + (real_of_int y)`,
107 REWRITE_TAC[int_add; GSYM int_rep; is_int] THEN REPEAT GEN_TAC THEN
108 X_CHOOSE_THEN `m:num` DISJ_CASES_TAC (SPEC `x:int` dest_int_rep) THEN
109 X_CHOOSE_THEN `n:num` DISJ_CASES_TAC (SPEC `y:int` dest_int_rep) THEN
110 ASM_REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_EQ; EXISTS_OR_THM] THEN
111 REWRITE_TAC[GSYM EXISTS_REFL] THEN
112 DISJ_CASES_THEN MP_TAC (SPECL [`m:num`; `n:num`] LE_CASES) THEN
113 REWRITE_TAC[LE_EXISTS] THEN DISCH_THEN(X_CHOOSE_THEN `d:num` SUBST1_TAC) THEN
114 REWRITE_TAC[GSYM REAL_OF_NUM_ADD; OR_EXISTS_THM; REAL_NEG_ADD] THEN
115 TRY(EXISTS_TAC `d:num` THEN REAL_ARITH_TAC) THEN
116 REWRITE_TAC[EXISTS_OR_THM; GSYM REAL_NEG_ADD; REAL_EQ_NEG2;
117 REAL_OF_NUM_ADD; REAL_OF_NUM_EQ; GSYM EXISTS_REFL]);;
119 let int_sub = new_definition
120 `x - y = int_of_real(real_of_int x - real_of_int y)`;;
122 let int_sub_th = prove
123 (`!x y. real_of_int(x - y) = (real_of_int x) - (real_of_int y)`,
124 REWRITE_TAC[int_sub; real_sub; GSYM int_neg_th; GSYM int_add_th] THEN
125 REWRITE_TAC[int_abstr]);;
127 let int_mul = new_definition
128 `x * y = int_of_real ((real_of_int x) * (real_of_int y))`;;
130 let int_mul_th = prove
131 (`!x y. real_of_int(x * y) = (real_of_int x) * (real_of_int y)`,
132 REPEAT GEN_TAC THEN REWRITE_TAC[int_mul; GSYM int_rep; is_int] THEN
133 X_CHOOSE_THEN `m:num` DISJ_CASES_TAC (SPEC `x:int` dest_int_rep) THEN
134 X_CHOOSE_THEN `n:num` DISJ_CASES_TAC (SPEC `y:int` dest_int_rep) THEN
135 ASM_REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_EQ; EXISTS_OR_THM] THEN
136 REWRITE_TAC[REAL_MUL_LNEG; REAL_MUL_RNEG; REAL_NEG_NEG; REAL_OF_NUM_MUL] THEN
137 REWRITE_TAC[REAL_EQ_NEG2; REAL_OF_NUM_EQ; GSYM EXISTS_REFL]);;
139 let int_abs = new_definition
140 `abs x = int_of_real(abs(real_of_int x))`;;
142 let int_abs_th = prove
143 (`!x. real_of_int(abs x) = abs(real_of_int x)`,
144 GEN_TAC THEN REWRITE_TAC[int_abs; real_abs] THEN COND_CASES_TAC THEN
145 REWRITE_TAC[GSYM int_neg; int_neg_th; int_abstr]);;
147 let int_sgn = new_definition
148 `int_sgn x = int_of_real(real_sgn(real_of_int x))`;;
150 let int_sgn_th = prove
151 (`!x. real_of_int(int_sgn x) = real_sgn(real_of_int x)`,
152 GEN_TAC THEN REWRITE_TAC[int_sgn; real_sgn; GSYM int_rep] THEN
153 REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN
156 let int_max = new_definition
157 `int_max x y = int_of_real(max (real_of_int x) (real_of_int y))`;;
159 let int_max_th = prove
160 (`!x y. real_of_int(max x y) = max (real_of_int x) (real_of_int y)`,
161 REPEAT GEN_TAC THEN REWRITE_TAC[int_max; real_max] THEN
162 COND_CASES_TAC THEN REWRITE_TAC[int_abstr]);;
164 let int_min = new_definition
165 `int_min x y = int_of_real(min (real_of_int x) (real_of_int y))`;;
167 let int_min_th = prove
168 (`!x y. real_of_int(min x y) = min (real_of_int x) (real_of_int y)`,
169 REPEAT GEN_TAC THEN REWRITE_TAC[int_min; real_min] THEN
170 COND_CASES_TAC THEN REWRITE_TAC[int_abstr]);;
172 let int_pow = new_definition
173 `x pow n = int_of_real((real_of_int x) pow n)`;;
175 let int_pow_th = prove
176 (`!x n. real_of_int(x pow n) = (real_of_int x) pow n`,
177 GEN_TAC THEN REWRITE_TAC[int_pow] THEN INDUCT_TAC THEN
178 REWRITE_TAC[real_pow] THENL
179 [REWRITE_TAC[GSYM int_of_num; int_of_num_th];
180 POP_ASSUM(SUBST1_TAC o SYM) THEN
181 ASM_REWRITE_TAC[GSYM int_mul; int_mul_th]]);;
183 (* ------------------------------------------------------------------------- *)
184 (* A couple of theorems peculiar to the integers. *)
185 (* ------------------------------------------------------------------------- *)
187 let INT_IMAGE = prove
188 (`!x. (?n. x = &n) \/ (?n. x = --(&n))`,
190 X_CHOOSE_THEN `n:num` DISJ_CASES_TAC (SPEC `x:int` dest_int_rep) THEN
191 POP_ASSUM(MP_TAC o AP_TERM `int_of_real`) THEN REWRITE_TAC[int_abstr] THEN
192 DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[int_of_num; int_neg] THENL
193 [DISJ1_TAC; DISJ2_TAC] THEN
194 EXISTS_TAC `n:num` THEN REWRITE_TAC[int_abstr] THEN
195 REWRITE_TAC[GSYM int_of_num; int_of_num_th]);;
197 let INT_LT_DISCRETE = prove
198 (`!x y. x < y <=> (x + &1) <= y`,
200 REWRITE_TAC[int_le; int_lt; int_add_th] THEN
201 DISJ_CASES_THEN(X_CHOOSE_THEN `m:num` SUBST1_TAC )
202 (SPEC `x:int` INT_IMAGE) THEN
203 DISJ_CASES_THEN(X_CHOOSE_THEN `n:num` SUBST1_TAC )
204 (SPEC `y:int` INT_IMAGE) THEN
205 REWRITE_TAC[int_neg_th; int_of_num_th] THEN
206 REWRITE_TAC[REAL_LE_NEG2; REAL_LT_NEG2] THEN
207 REWRITE_TAC[REAL_LE_LNEG; REAL_LT_LNEG; REAL_LE_RNEG; REAL_LT_RNEG] THEN
208 REWRITE_TAC[GSYM REAL_ADD_ASSOC] THEN
209 ONCE_REWRITE_TAC[REAL_ADD_SYM] THEN
210 REWRITE_TAC[GSYM real_sub; REAL_LE_SUB_RADD] THEN
211 REWRITE_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_LT; REAL_OF_NUM_ADD] THEN
212 REWRITE_TAC[GSYM ADD1; ONCE_REWRITE_RULE[ADD_SYM] (GSYM ADD1)] THEN
213 REWRITE_TAC[SYM(REWRITE_CONV[ARITH_SUC] `SUC 0`)] THEN
214 REWRITE_TAC[ADD_CLAUSES; LE_SUC_LT; LT_SUC_LE]);;
216 let INT_GT_DISCRETE = prove
217 (`!x y. x > y <=> x >= (y + &1)`,
218 REWRITE_TAC[int_gt; int_ge; real_ge; real_gt; GSYM int_le; GSYM int_lt] THEN
219 MATCH_ACCEPT_TAC INT_LT_DISCRETE);;
221 (* ------------------------------------------------------------------------- *)
222 (* Conversions of integer constants to and from OCaml numbers. *)
223 (* ------------------------------------------------------------------------- *)
227 Comb(Const("int_of_num",_),n) -> is_numeral n
228 | Comb(Const("int_neg",_),Comb(Const("int_of_num",_),n)) ->
229 is_numeral n & not(dest_numeral n = num_0)
232 let dest_intconst tm =
234 Comb(Const("int_of_num",_),n) -> dest_numeral n
235 | Comb(Const("int_neg",_),Comb(Const("int_of_num",_),n)) ->
236 let nn = dest_numeral n in
237 if nn <>/ num_0 then minus_num(dest_numeral n)
238 else failwith "dest_intconst"
239 | _ -> failwith "dest_intconst";;
242 let cast_tm = `int_of_num` and neg_tm = `int_neg` in
243 let mk_numconst n = mk_comb(cast_tm,mk_numeral n) in
244 fun x -> if x </ num_0 then mk_comb(neg_tm,mk_numconst(minus_num x))
247 (* ------------------------------------------------------------------------- *)
248 (* A simple procedure to lift most universal real theorems to integers. *)
249 (* For a more complete procedure, give required term to INT_ARITH (below). *)
250 (* ------------------------------------------------------------------------- *)
252 let INT_OF_REAL_THM =
253 let dest = `real_of_int`
254 and real_ty = `:real`
257 (`real_of_int(if b then x else y) =
258 if b then real_of_int x else real_of_int y`,
259 COND_CASES_TAC THEN REWRITE_TAC[]) in
260 let thlist = map GSYM
261 [int_eq; int_le; int_lt; int_ge; int_gt;
262 int_of_num_th; int_neg_th; int_add_th; int_mul_th; int_sgn_th;
263 int_sub_th; int_abs_th; int_max_th; int_min_th; int_pow_th; cond_th] in
264 let REW_RULE = GEN_REWRITE_RULE DEPTH_CONV thlist in
265 let int_tm_of_real_var v =
266 let s,ty = dest_var v in
267 if ty = real_ty then mk_comb(dest,mk_var(s,int_ty)) else v in
268 let int_of_real_var v =
269 let s,ty = dest_var v in
270 if ty = real_ty then mk_var(s,int_ty) else v in
271 let INT_OF_REAL_THM1 th =
272 let newavs = subtract (frees (concl th)) (freesl (hyp th)) in
273 let avs,bod = strip_forall(concl th) in
274 let allavs = newavs@avs in
275 let avs' = map int_tm_of_real_var allavs in
276 let avs'' = map int_of_real_var avs in
277 GENL avs'' (REW_RULE(SPECL avs' (GENL newavs th))) in
278 let rec INT_OF_REAL_THM th =
279 if is_conj(concl th) then CONJ (INT_OF_REAL_THM (CONJUNCT1 th))
280 (INT_OF_REAL_THM (CONJUNCT2 th))
281 else INT_OF_REAL_THM1 th in
284 (* ------------------------------------------------------------------------- *)
285 (* Collect together all the theorems derived automatically. *)
286 (* ------------------------------------------------------------------------- *)
288 let INT_ABS_0 = INT_OF_REAL_THM REAL_ABS_0;;
289 let INT_ABS_1 = INT_OF_REAL_THM REAL_ABS_1;;
290 let INT_ABS_ABS = INT_OF_REAL_THM REAL_ABS_ABS;;
291 let INT_ABS_BETWEEN = INT_OF_REAL_THM REAL_ABS_BETWEEN;;
292 let INT_ABS_BETWEEN1 = INT_OF_REAL_THM REAL_ABS_BETWEEN1;;
293 let INT_ABS_BETWEEN2 = INT_OF_REAL_THM REAL_ABS_BETWEEN2;;
294 let INT_ABS_BOUND = INT_OF_REAL_THM REAL_ABS_BOUND;;
295 let INT_ABS_CASES = INT_OF_REAL_THM REAL_ABS_CASES;;
296 let INT_ABS_CIRCLE = INT_OF_REAL_THM REAL_ABS_CIRCLE;;
297 let INT_ABS_LE = INT_OF_REAL_THM REAL_ABS_LE;;
298 let INT_ABS_MUL = INT_OF_REAL_THM REAL_ABS_MUL;;
299 let INT_ABS_NEG = INT_OF_REAL_THM REAL_ABS_NEG;;
300 let INT_ABS_NUM = INT_OF_REAL_THM REAL_ABS_NUM;;
301 let INT_ABS_NZ = INT_OF_REAL_THM REAL_ABS_NZ;;
302 let INT_ABS_POS = INT_OF_REAL_THM REAL_ABS_POS;;
303 let INT_ABS_POW = INT_OF_REAL_THM REAL_ABS_POW;;
304 let INT_ABS_REFL = INT_OF_REAL_THM REAL_ABS_REFL;;
305 let INT_ABS_SGN = INT_OF_REAL_THM REAL_ABS_SGN;;
306 let INT_ABS_SIGN = INT_OF_REAL_THM REAL_ABS_SIGN;;
307 let INT_ABS_SIGN2 = INT_OF_REAL_THM REAL_ABS_SIGN2;;
308 let INT_ABS_STILLNZ = INT_OF_REAL_THM REAL_ABS_STILLNZ;;
309 let INT_ABS_SUB = INT_OF_REAL_THM REAL_ABS_SUB;;
310 let INT_ABS_SUB_ABS = INT_OF_REAL_THM REAL_ABS_SUB_ABS;;
311 let INT_ABS_TRIANGLE = INT_OF_REAL_THM REAL_ABS_TRIANGLE;;
312 let INT_ABS_ZERO = INT_OF_REAL_THM REAL_ABS_ZERO;;
313 let INT_ADD2_SUB2 = INT_OF_REAL_THM REAL_ADD2_SUB2;;
314 let INT_ADD_AC = INT_OF_REAL_THM REAL_ADD_AC;;
315 let INT_ADD_ASSOC = INT_OF_REAL_THM REAL_ADD_ASSOC;;
316 let INT_ADD_LDISTRIB = INT_OF_REAL_THM REAL_ADD_LDISTRIB;;
317 let INT_ADD_LID = INT_OF_REAL_THM REAL_ADD_LID;;
318 let INT_ADD_LINV = INT_OF_REAL_THM REAL_ADD_LINV;;
319 let INT_ADD_RDISTRIB = INT_OF_REAL_THM REAL_ADD_RDISTRIB;;
320 let INT_ADD_RID = INT_OF_REAL_THM REAL_ADD_RID;;
321 let INT_ADD_RINV = INT_OF_REAL_THM REAL_ADD_RINV;;
322 let INT_ADD_SUB = INT_OF_REAL_THM REAL_ADD_SUB;;
323 let INT_ADD_SUB2 = INT_OF_REAL_THM REAL_ADD_SUB2;;
324 let INT_ADD_SYM = INT_OF_REAL_THM REAL_ADD_SYM;;
325 let INT_BOUNDS_LE = INT_OF_REAL_THM REAL_BOUNDS_LE;;
326 let INT_BOUNDS_LT = INT_OF_REAL_THM REAL_BOUNDS_LT;;
327 let INT_DIFFSQ = INT_OF_REAL_THM REAL_DIFFSQ;;
328 let INT_ENTIRE = INT_OF_REAL_THM REAL_ENTIRE;;
329 let INT_EQ_ADD_LCANCEL = INT_OF_REAL_THM REAL_EQ_ADD_LCANCEL;;
330 let INT_EQ_ADD_LCANCEL_0 = INT_OF_REAL_THM REAL_EQ_ADD_LCANCEL_0;;
331 let INT_EQ_ADD_RCANCEL = INT_OF_REAL_THM REAL_EQ_ADD_RCANCEL;;
332 let INT_EQ_ADD_RCANCEL_0 = INT_OF_REAL_THM REAL_EQ_ADD_RCANCEL_0;;
333 let INT_EQ_IMP_LE = INT_OF_REAL_THM REAL_EQ_IMP_LE;;
334 let INT_EQ_MUL_LCANCEL = INT_OF_REAL_THM REAL_EQ_MUL_LCANCEL;;
335 let INT_EQ_MUL_RCANCEL = INT_OF_REAL_THM REAL_EQ_MUL_RCANCEL;;
336 let INT_EQ_NEG2 = INT_OF_REAL_THM REAL_EQ_NEG2;;
337 let INT_EQ_SGN_ABS = INT_OF_REAL_THM REAL_EQ_SGN_ABS;;
338 let INT_EQ_SQUARE_ABS = INT_OF_REAL_THM REAL_EQ_SQUARE_ABS;;
339 let INT_EQ_SUB_LADD = INT_OF_REAL_THM REAL_EQ_SUB_LADD;;
340 let INT_EQ_SUB_RADD = INT_OF_REAL_THM REAL_EQ_SUB_RADD;;
341 let INT_LET_ADD = INT_OF_REAL_THM REAL_LET_ADD;;
342 let INT_LET_ADD2 = INT_OF_REAL_THM REAL_LET_ADD2;;
343 let INT_LET_ANTISYM = INT_OF_REAL_THM REAL_LET_ANTISYM;;
344 let INT_LET_TOTAL = INT_OF_REAL_THM REAL_LET_TOTAL;;
345 let INT_LET_TRANS = INT_OF_REAL_THM REAL_LET_TRANS;;
346 let INT_LE_01 = INT_OF_REAL_THM REAL_LE_01;;
347 let INT_LE_ADD = INT_OF_REAL_THM REAL_LE_ADD;;
348 let INT_LE_ADD2 = INT_OF_REAL_THM REAL_LE_ADD2;;
349 let INT_LE_ADDL = INT_OF_REAL_THM REAL_LE_ADDL;;
350 let INT_LE_ADDR = INT_OF_REAL_THM REAL_LE_ADDR;;
351 let INT_LE_ANTISYM = INT_OF_REAL_THM REAL_LE_ANTISYM;;
352 let INT_LE_DOUBLE = INT_OF_REAL_THM REAL_LE_DOUBLE;;
353 let INT_LE_LADD = INT_OF_REAL_THM REAL_LE_LADD;;
354 let INT_LE_LADD_IMP = INT_OF_REAL_THM REAL_LE_LADD_IMP;;
355 let INT_LE_LMUL = INT_OF_REAL_THM REAL_LE_LMUL;;
356 let INT_LE_LNEG = INT_OF_REAL_THM REAL_LE_LNEG;;
357 let INT_LE_LT = INT_OF_REAL_THM REAL_LE_LT;;
358 let INT_LE_MAX = INT_OF_REAL_THM REAL_LE_MAX;;
359 let INT_LE_MIN = INT_OF_REAL_THM REAL_LE_MIN;;
360 let INT_LE_MUL = INT_OF_REAL_THM REAL_LE_MUL;;
361 let INT_LE_MUL_EQ = INT_OF_REAL_THM REAL_LE_MUL_EQ;;
362 let INT_LE_NEG = INT_OF_REAL_THM REAL_LE_NEG;;
363 let INT_LE_NEG2 = INT_OF_REAL_THM REAL_LE_NEG2;;
364 let INT_LE_NEGL = INT_OF_REAL_THM REAL_LE_NEGL;;
365 let INT_LE_NEGR = INT_OF_REAL_THM REAL_LE_NEGR;;
366 let INT_LE_NEGTOTAL = INT_OF_REAL_THM REAL_LE_NEGTOTAL;;
367 let INT_LE_POW2 = INT_OF_REAL_THM REAL_LE_POW2;;
368 let INT_LE_RADD = INT_OF_REAL_THM REAL_LE_RADD;;
369 let INT_LE_REFL = INT_OF_REAL_THM REAL_LE_REFL;;
370 let INT_LE_RMUL = INT_OF_REAL_THM REAL_LE_RMUL;;
371 let INT_LE_RNEG = INT_OF_REAL_THM REAL_LE_RNEG;;
372 let INT_LE_SQUARE = INT_OF_REAL_THM REAL_LE_SQUARE;;
373 let INT_LE_SQUARE_ABS = INT_OF_REAL_THM REAL_LE_SQUARE_ABS;;
374 let INT_LE_SUB_LADD = INT_OF_REAL_THM REAL_LE_SUB_LADD;;
375 let INT_LE_SUB_RADD = INT_OF_REAL_THM REAL_LE_SUB_RADD;;
376 let INT_LE_TOTAL = INT_OF_REAL_THM REAL_LE_TOTAL;;
377 let INT_LE_TRANS = INT_OF_REAL_THM REAL_LE_TRANS;;
378 let INT_LNEG_UNIQ = INT_OF_REAL_THM REAL_LNEG_UNIQ;;
379 let INT_LTE_ADD = INT_OF_REAL_THM REAL_LTE_ADD;;
380 let INT_LTE_ADD2 = INT_OF_REAL_THM REAL_LTE_ADD2;;
381 let INT_LTE_ANTISYM = INT_OF_REAL_THM REAL_LTE_ANTISYM;;
382 let INT_LTE_TOTAL = INT_OF_REAL_THM REAL_LTE_TOTAL;;
383 let INT_LTE_TRANS = INT_OF_REAL_THM REAL_LTE_TRANS;;
384 let INT_LT_01 = INT_OF_REAL_THM REAL_LT_01;;
385 let INT_LT_ADD = INT_OF_REAL_THM REAL_LT_ADD;;
386 let INT_LT_ADD1 = INT_OF_REAL_THM REAL_LT_ADD1;;
387 let INT_LT_ADD2 = INT_OF_REAL_THM REAL_LT_ADD2;;
388 let INT_LT_ADDL = INT_OF_REAL_THM REAL_LT_ADDL;;
389 let INT_LT_ADDNEG = INT_OF_REAL_THM REAL_LT_ADDNEG;;
390 let INT_LT_ADDNEG2 = INT_OF_REAL_THM REAL_LT_ADDNEG2;;
391 let INT_LT_ADDR = INT_OF_REAL_THM REAL_LT_ADDR;;
392 let INT_LT_ADD_SUB = INT_OF_REAL_THM REAL_LT_ADD_SUB;;
393 let INT_LT_ANTISYM = INT_OF_REAL_THM REAL_LT_ANTISYM;;
394 let INT_LT_GT = INT_OF_REAL_THM REAL_LT_GT;;
395 let INT_LT_IMP_LE = INT_OF_REAL_THM REAL_LT_IMP_LE;;
396 let INT_LT_IMP_NE = INT_OF_REAL_THM REAL_LT_IMP_NE;;
397 let INT_LT_LADD = INT_OF_REAL_THM REAL_LT_LADD;;
398 let INT_LT_LE = INT_OF_REAL_THM REAL_LT_LE;;
399 let INT_LT_LMUL_EQ = INT_OF_REAL_THM REAL_LT_LMUL_EQ;;
400 let INT_LT_MAX = INT_OF_REAL_THM REAL_LT_MAX;;
401 let INT_LT_MIN = INT_OF_REAL_THM REAL_LT_MIN;;
402 let INT_LT_MUL = INT_OF_REAL_THM REAL_LT_MUL;;
403 let INT_LT_MUL_EQ = INT_OF_REAL_THM REAL_LT_MUL_EQ;;
404 let INT_LT_NEG = INT_OF_REAL_THM REAL_LT_NEG;;
405 let INT_LT_NEG2 = INT_OF_REAL_THM REAL_LT_NEG2;;
406 let INT_LT_NEGTOTAL = INT_OF_REAL_THM REAL_LT_NEGTOTAL;;
407 let INT_LT_POW2 = INT_OF_REAL_THM REAL_LT_POW2;;
408 let INT_LT_RADD = INT_OF_REAL_THM REAL_LT_RADD;;
409 let INT_LT_REFL = INT_OF_REAL_THM REAL_LT_REFL;;
410 let INT_LT_RMUL_EQ = INT_OF_REAL_THM REAL_LT_RMUL_EQ;;
411 let INT_LT_SQUARE_ABS = INT_OF_REAL_THM REAL_LT_SQUARE_ABS;;
412 let INT_LT_SUB_LADD = INT_OF_REAL_THM REAL_LT_SUB_LADD;;
413 let INT_LT_SUB_RADD = INT_OF_REAL_THM REAL_LT_SUB_RADD;;
414 let INT_LT_TOTAL = INT_OF_REAL_THM REAL_LT_TOTAL;;
415 let INT_LT_TRANS = INT_OF_REAL_THM REAL_LT_TRANS;;
416 let INT_MAX_ACI = INT_OF_REAL_THM REAL_MAX_ACI;;
417 let INT_MAX_ASSOC = INT_OF_REAL_THM REAL_MAX_ASSOC;;
418 let INT_MAX_LE = INT_OF_REAL_THM REAL_MAX_LE;;
419 let INT_MAX_LT = INT_OF_REAL_THM REAL_MAX_LT;;
420 let INT_MAX_MAX = INT_OF_REAL_THM REAL_MAX_MAX;;
421 let INT_MAX_MIN = INT_OF_REAL_THM REAL_MAX_MIN;;
422 let INT_MAX_SYM = INT_OF_REAL_THM REAL_MAX_SYM;;
423 let INT_MIN_ACI = INT_OF_REAL_THM REAL_MIN_ACI;;
424 let INT_MIN_ASSOC = INT_OF_REAL_THM REAL_MIN_ASSOC;;
425 let INT_MIN_LE = INT_OF_REAL_THM REAL_MIN_LE;;
426 let INT_MIN_LT = INT_OF_REAL_THM REAL_MIN_LT;;
427 let INT_MIN_MAX = INT_OF_REAL_THM REAL_MIN_MAX;;
428 let INT_MIN_MIN = INT_OF_REAL_THM REAL_MIN_MIN;;
429 let INT_MIN_SYM = INT_OF_REAL_THM REAL_MIN_SYM;;
430 let INT_MUL_AC = INT_OF_REAL_THM REAL_MUL_AC;;
431 let INT_MUL_ASSOC = INT_OF_REAL_THM REAL_MUL_ASSOC;;
432 let INT_MUL_LID = INT_OF_REAL_THM REAL_MUL_LID;;
433 let INT_MUL_LNEG = INT_OF_REAL_THM REAL_MUL_LNEG;;
434 let INT_MUL_LZERO = INT_OF_REAL_THM REAL_MUL_LZERO;;
435 let INT_MUL_POS_LE = INT_OF_REAL_THM REAL_MUL_POS_LE;;
436 let INT_MUL_POS_LT = INT_OF_REAL_THM REAL_MUL_POS_LT;;
437 let INT_MUL_RID = INT_OF_REAL_THM REAL_MUL_RID;;
438 let INT_MUL_RNEG = INT_OF_REAL_THM REAL_MUL_RNEG;;
439 let INT_MUL_RZERO = INT_OF_REAL_THM REAL_MUL_RZERO;;
440 let INT_MUL_SYM = INT_OF_REAL_THM REAL_MUL_SYM;;
441 let INT_NEGNEG = INT_OF_REAL_THM REAL_NEGNEG;;
442 let INT_NEG_0 = INT_OF_REAL_THM REAL_NEG_0;;
443 let INT_NEG_ADD = INT_OF_REAL_THM REAL_NEG_ADD;;
444 let INT_NEG_EQ = INT_OF_REAL_THM REAL_NEG_EQ;;
445 let INT_NEG_EQ_0 = INT_OF_REAL_THM REAL_NEG_EQ_0;;
446 let INT_NEG_GE0 = INT_OF_REAL_THM REAL_NEG_GE0;;
447 let INT_NEG_GT0 = INT_OF_REAL_THM REAL_NEG_GT0;;
448 let INT_NEG_LE0 = INT_OF_REAL_THM REAL_NEG_LE0;;
449 let INT_NEG_LMUL = INT_OF_REAL_THM REAL_NEG_LMUL;;
450 let INT_NEG_LT0 = INT_OF_REAL_THM REAL_NEG_LT0;;
451 let INT_NEG_MINUS1 = INT_OF_REAL_THM REAL_NEG_MINUS1;;
452 let INT_NEG_MUL2 = INT_OF_REAL_THM REAL_NEG_MUL2;;
453 let INT_NEG_NEG = INT_OF_REAL_THM REAL_NEG_NEG;;
454 let INT_NEG_RMUL = INT_OF_REAL_THM REAL_NEG_RMUL;;
455 let INT_NEG_SUB = INT_OF_REAL_THM REAL_NEG_SUB;;
456 let INT_NOT_EQ = INT_OF_REAL_THM REAL_NOT_EQ;;
457 let INT_NOT_LE = INT_OF_REAL_THM REAL_NOT_LE;;
458 let INT_NOT_LT = INT_OF_REAL_THM REAL_NOT_LT;;
459 let INT_OF_NUM_ADD = INT_OF_REAL_THM REAL_OF_NUM_ADD;;
460 let INT_OF_NUM_EQ = INT_OF_REAL_THM REAL_OF_NUM_EQ;;
461 let INT_OF_NUM_GE = INT_OF_REAL_THM REAL_OF_NUM_GE;;
462 let INT_OF_NUM_GT = INT_OF_REAL_THM REAL_OF_NUM_GT;;
463 let INT_OF_NUM_LE = INT_OF_REAL_THM REAL_OF_NUM_LE;;
464 let INT_OF_NUM_LT = INT_OF_REAL_THM REAL_OF_NUM_LT;;
465 let INT_OF_NUM_MAX = INT_OF_REAL_THM REAL_OF_NUM_MAX;;
466 let INT_OF_NUM_MIN = INT_OF_REAL_THM REAL_OF_NUM_MIN;;
467 let INT_OF_NUM_MUL = INT_OF_REAL_THM REAL_OF_NUM_MUL;;
468 let INT_OF_NUM_POW = INT_OF_REAL_THM REAL_OF_NUM_POW;;
469 let INT_OF_NUM_SUB = INT_OF_REAL_THM REAL_OF_NUM_SUB;;
470 let INT_OF_NUM_SUC = INT_OF_REAL_THM REAL_OF_NUM_SUC;;
471 let INT_POS = INT_OF_REAL_THM REAL_POS;;
472 let INT_POS_NZ = INT_OF_REAL_THM REAL_POS_NZ;;
473 let INT_POW2_ABS = INT_OF_REAL_THM REAL_POW2_ABS;;
474 let INT_POW_1 = INT_OF_REAL_THM REAL_POW_1;;
475 let INT_POW_1_LE = INT_OF_REAL_THM REAL_POW_1_LE;;
476 let INT_POW_1_LT = INT_OF_REAL_THM REAL_POW_1_LT;;
477 let INT_POW_2 = INT_OF_REAL_THM REAL_POW_2;;
478 let INT_POW_ADD = INT_OF_REAL_THM REAL_POW_ADD;;
479 let INT_POW_EQ = INT_OF_REAL_THM REAL_POW_EQ;;
480 let INT_POW_EQ_0 = INT_OF_REAL_THM REAL_POW_EQ_0;;
481 let INT_POW_EQ_ABS = INT_OF_REAL_THM REAL_POW_EQ_ABS;;
482 let INT_POW_LE = INT_OF_REAL_THM REAL_POW_LE;;
483 let INT_POW_LE2 = INT_OF_REAL_THM REAL_POW_LE2;;
484 let INT_POW_LE2_ODD = INT_OF_REAL_THM REAL_POW_LE2_ODD;;
485 let INT_POW_LE2_REV = INT_OF_REAL_THM REAL_POW_LE2_REV;;
486 let INT_POW_LE_1 = INT_OF_REAL_THM REAL_POW_LE_1;;
487 let INT_POW_LT = INT_OF_REAL_THM REAL_POW_LT;;
488 let INT_POW_LT2 = INT_OF_REAL_THM REAL_POW_LT2;;
489 let INT_POW_LT2_REV = INT_OF_REAL_THM REAL_POW_LT2_REV;;
490 let INT_POW_LT_1 = INT_OF_REAL_THM REAL_POW_LT_1;;
491 let INT_POW_MONO = INT_OF_REAL_THM REAL_POW_MONO;;
492 let INT_POW_MONO_LT = INT_OF_REAL_THM REAL_POW_MONO_LT;;
493 let INT_POW_MUL = INT_OF_REAL_THM REAL_POW_MUL;;
494 let INT_POW_NEG = INT_OF_REAL_THM REAL_POW_NEG;;
495 let INT_POW_NZ = INT_OF_REAL_THM REAL_POW_NZ;;
496 let INT_POW_ONE = INT_OF_REAL_THM REAL_POW_ONE;;
497 let INT_POW_POW = INT_OF_REAL_THM REAL_POW_POW;;
498 let INT_POW_ZERO = INT_OF_REAL_THM REAL_POW_ZERO;;
499 let INT_RNEG_UNIQ = INT_OF_REAL_THM REAL_RNEG_UNIQ;;
500 let INT_SGN = INT_OF_REAL_THM real_sgn;;
501 let INT_SGN_0 = INT_OF_REAL_THM REAL_SGN_0;;
502 let INT_SGN_ABS = INT_OF_REAL_THM REAL_SGN_ABS;;
503 let INT_SGN_CASES = INT_OF_REAL_THM REAL_SGN_CASES;;
504 let INT_SGN_EQ = INT_OF_REAL_THM REAL_SGN_EQ;;
505 let INT_SGN_INEQS = INT_OF_REAL_THM REAL_SGN_INEQS;;
506 let INT_SGN_MUL = INT_OF_REAL_THM REAL_SGN_MUL;;
507 let INT_SGN_NEG = INT_OF_REAL_THM REAL_SGN_NEG;;
508 let INT_SGN_POW = INT_OF_REAL_THM REAL_SGN_POW;;
509 let INT_SGN_POW_2 = INT_OF_REAL_THM REAL_SGN_POW_2;;
510 let INT_SGN_INT_SGN = INT_OF_REAL_THM REAL_SGN_REAL_SGN;;
511 let INT_SOS_EQ_0 = INT_OF_REAL_THM REAL_SOS_EQ_0;;
512 let INT_SUB_0 = INT_OF_REAL_THM REAL_SUB_0;;
513 let INT_SUB_ABS = INT_OF_REAL_THM REAL_SUB_ABS;;
514 let INT_SUB_ADD = INT_OF_REAL_THM REAL_SUB_ADD;;
515 let INT_SUB_ADD2 = INT_OF_REAL_THM REAL_SUB_ADD2;;
516 let INT_SUB_LDISTRIB = INT_OF_REAL_THM REAL_SUB_LDISTRIB;;
517 let INT_SUB_LE = INT_OF_REAL_THM REAL_SUB_LE;;
518 let INT_SUB_LNEG = INT_OF_REAL_THM REAL_SUB_LNEG;;
519 let INT_SUB_LT = INT_OF_REAL_THM REAL_SUB_LT;;
520 let INT_SUB_LZERO = INT_OF_REAL_THM REAL_SUB_LZERO;;
521 let INT_SUB_NEG2 = INT_OF_REAL_THM REAL_SUB_NEG2;;
522 let INT_SUB_RDISTRIB = INT_OF_REAL_THM REAL_SUB_RDISTRIB;;
523 let INT_SUB_REFL = INT_OF_REAL_THM REAL_SUB_REFL;;
524 let INT_SUB_RNEG = INT_OF_REAL_THM REAL_SUB_RNEG;;
525 let INT_SUB_RZERO = INT_OF_REAL_THM REAL_SUB_RZERO;;
526 let INT_SUB_SUB = INT_OF_REAL_THM REAL_SUB_SUB;;
527 let INT_SUB_SUB2 = INT_OF_REAL_THM REAL_SUB_SUB2;;
528 let INT_SUB_TRIANGLE = INT_OF_REAL_THM REAL_SUB_TRIANGLE;;
530 (* ------------------------------------------------------------------------- *)
531 (* More useful "image" theorems. *)
532 (* ------------------------------------------------------------------------- *)
534 let INT_FORALL_POS = prove
535 (`!P. (!n. P(&n)) <=> (!i:int. &0 <= i ==> P(i))`,
536 GEN_TAC THEN EQ_TAC THEN DISCH_TAC THEN GEN_TAC THENL
537 [DISJ_CASES_THEN (CHOOSE_THEN SUBST1_TAC) (SPEC `i:int` INT_IMAGE) THEN
538 ASM_REWRITE_TAC[INT_LE_RNEG; INT_ADD_LID; INT_OF_NUM_LE; LE] THEN
539 DISCH_THEN SUBST1_TAC THEN ASM_REWRITE_TAC[INT_NEG_0];
540 FIRST_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[INT_OF_NUM_LE; LE_0]]);;
542 let INT_EXISTS_POS = prove
543 (`!P. (?n. P(&n)) <=> (?i:int. &0 <= i /\ P(i))`,
544 GEN_TAC THEN GEN_REWRITE_TAC I [TAUT `(p <=> q) <=> (~p <=> ~q)`] THEN
545 REWRITE_TAC[NOT_EXISTS_THM; INT_FORALL_POS] THEN MESON_TAC[]);;
547 let INT_FORALL_ABS = prove
548 (`!P. (!n. P(&n)) <=> (!x:int. P(abs x))`,
549 REWRITE_TAC[INT_FORALL_POS] THEN MESON_TAC[INT_ABS_POS; INT_ABS_REFL]);;
551 let INT_EXISTS_ABS = prove
552 (`!P. (?n. P(&n)) <=> (?x:int. P(abs x))`,
553 GEN_TAC THEN GEN_REWRITE_TAC I [TAUT `(p <=> q) <=> (~p <=> ~q)`] THEN
554 REWRITE_TAC[NOT_EXISTS_THM; INT_FORALL_ABS] THEN MESON_TAC[]);;
556 (* ------------------------------------------------------------------------- *)
557 (* Sometimes handy in number-theoretic applications. *)
558 (* ------------------------------------------------------------------------- *)
560 let INT_ABS_MUL_1 = prove
561 (`!x y. (abs(x * y) = &1) <=> (abs(x) = &1) /\ (abs(y) = &1)`,
562 REPEAT GEN_TAC THEN REWRITE_TAC[INT_ABS_MUL] THEN
563 MP_TAC(SPEC `y:int` INT_ABS_POS) THEN SPEC_TAC(`abs(y)`,`b:int`) THEN
564 MP_TAC(SPEC `x:int` INT_ABS_POS) THEN SPEC_TAC(`abs(x)`,`a:int`) THEN
565 REWRITE_TAC[GSYM INT_FORALL_POS; INT_OF_NUM_MUL; INT_OF_NUM_EQ; MULT_EQ_1]);;
568 (`(?x. &0 <= x /\ P x) <=>
569 (?x. &0 <= x /\ P x /\ !y. &0 <= y /\ P y ==> x <= y)`,
570 ONCE_REWRITE_TAC[MESON[] `(?x. P x /\ Q x) <=> ~(!x. P x ==> ~Q x)`] THEN
571 REWRITE_TAC[IMP_CONJ; GSYM INT_FORALL_POS; INT_OF_NUM_LE] THEN
572 REWRITE_TAC[NOT_FORALL_THM] THEN GEN_REWRITE_TAC LAND_CONV [num_WOP] THEN
573 REWRITE_TAC[GSYM NOT_LE; CONTRAPOS_THM]);;
575 (* ------------------------------------------------------------------------- *)
576 (* A few "pseudo definitions". *)
577 (* ------------------------------------------------------------------------- *)
581 (!n. x pow (SUC n) = x * x pow n)`,
582 REWRITE_TAC(map INT_OF_REAL_THM (CONJUNCTS real_pow)));;
585 (`!x. abs(x) = if &0 <= x then x else --x`,
586 GEN_TAC THEN MP_TAC(INT_OF_REAL_THM(SPEC `x:real` real_abs)) THEN
587 COND_CASES_TAC THEN REWRITE_TAC[int_eq]);;
590 (`!x y. x >= y <=> y <= x`,
591 REWRITE_TAC[int_ge; int_le; real_ge]);;
594 (`!x y. x > y <=> y < x`,
595 REWRITE_TAC[int_gt; int_lt; real_gt]);;
598 (`!x y. x < y <=> ~(y <= x)`,
599 REWRITE_TAC[int_lt; int_le; real_lt]);;
601 (* ------------------------------------------------------------------------- *)
602 (* Now a decision procedure for the integers. *)
603 (* ------------------------------------------------------------------------- *)
608 (`(~(x <= y) <=> y + &1 <= x) /\
609 (~(x < y) <=> y <= x) /\
610 (~(x = y) <=> x + &1 <= y \/ y + &1 <= x) /\
611 (x < y <=> x + &1 <= y)`,
612 REWRITE_TAC[INT_NOT_LE; INT_NOT_LT; INT_NOT_EQ; INT_LT_DISCRETE]) in
613 GEN_REWRITE_CONV I [pth]
614 and bub_CONV = GEN_REWRITE_CONV TOP_SWEEP_CONV
615 [int_eq; int_le; int_lt; int_ge; int_gt;
616 int_of_num_th; int_neg_th; int_add_th; int_mul_th;
617 int_sub_th; int_pow_th; int_abs_th; int_max_th; int_min_th] in
618 let base_CONV = TRY_CONV atom_CONV THENC bub_CONV in
619 let NNF_NORM_CONV = GEN_NNF_CONV false
620 (base_CONV,fun t -> base_CONV t,base_CONV(mk_neg t)) in
622 TOP_DEPTH_CONV BETA_CONV THENC
624 GEN_REWRITE_CONV DEPTH_CONV [INT_GT; INT_GE] THENC
625 NNF_CONV THENC DEPTH_BINOP_CONV `(\/)` CONDS_ELIM_CONV THENC
628 and not_tm = `(~)` in
629 let pth = TAUT(mk_eq(mk_neg(mk_neg p_tm),p_tm)) in
631 let th0 = INST [tm,p_tm] pth
632 and th1 = init_CONV (mk_neg tm) in
633 let th2 = REAL_ARITH(mk_neg(rand(concl th1))) in
634 EQ_MP th0 (EQ_MP (AP_TERM not_tm (SYM th1)) th2);;
636 let INT_ARITH_TAC = CONV_TAC(EQT_INTRO o INT_ARITH);;
638 let ASM_INT_ARITH_TAC =
639 REPEAT(FIRST_X_ASSUM(MP_TAC o check (not o is_forall o concl))) THEN
642 (* ------------------------------------------------------------------------- *)
643 (* Some pseudo-definitions. *)
644 (* ------------------------------------------------------------------------- *)
646 let INT_SUB = INT_ARITH `!x y. x - y = x + --y`;;
648 let INT_MAX = INT_ARITH `!x y. max x y = if x <= y then y else x`;;
650 let INT_MIN = INT_ARITH `!x y. min x y = if x <= y then x else y`;;
652 (* ------------------------------------------------------------------------- *)
653 (* Additional useful lemmas. *)
654 (* ------------------------------------------------------------------------- *)
656 let INT_OF_NUM_EXISTS = prove
657 (`!x:int. (?n. x = &n) <=> &0 <= x`,
658 GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN ASM_SIMP_TAC[INT_POS] THEN
659 MP_TAC(ISPEC `x:int` INT_IMAGE) THEN
660 REWRITE_TAC[OR_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN
663 let INT_LE_DISCRETE = INT_ARITH `!x y:int. x <= y <=> x < y + &1`;;
665 (* ------------------------------------------------------------------------- *)
666 (* Archimedian property for the integers. *)
667 (* ------------------------------------------------------------------------- *)
670 (`!x d. ~(d = &0) ==> ?c. x < c * d`,
671 SUBGOAL_THEN `!x. &0 <= x ==> ?n. x <= &n` ASSUME_TAC THENL
672 [REWRITE_TAC[GSYM INT_FORALL_POS; INT_OF_NUM_LE] THEN MESON_TAC[LE_REFL];
674 SUBGOAL_THEN `!x. ?n. x <= &n` ASSUME_TAC THENL
675 [ASM_MESON_TAC[INT_LE_TOTAL]; ALL_TAC] THEN
676 SUBGOAL_THEN `!x d. &0 < d ==> ?c. x < c * d` ASSUME_TAC THENL
677 [REPEAT GEN_TAC THEN REWRITE_TAC[INT_LT_DISCRETE; INT_ADD_LID] THEN
678 ASM_MESON_TAC[INT_POS; INT_LE_LMUL; INT_ARITH
679 `x + &1 <= &n /\ &n * &1 <= &n * d ==> x + &1 <= &n * d`];
681 SUBGOAL_THEN `!x d. ~(d = &0) ==> ?c. x < c * d` ASSUME_TAC THENL
682 [ASM_MESON_TAC[INT_ARITH `--x * y = x * --y`;
683 INT_ARITH `~(d = &0) ==> &0 < d \/ &0 < --d`];
685 ASM_MESON_TAC[INT_ARITH `--x * y = x * --y`;
686 INT_ARITH `~(d = &0) ==> &0 < d \/ &0 < --d`]);;
688 (* ------------------------------------------------------------------------- *)
689 (* Definitions of ("Euclidean") integer division and remainder. *)
690 (* ------------------------------------------------------------------------- *)
692 let INT_DIVMOD_EXIST_0 = prove
693 (`!m n:int. ?q r. if n = &0 then q = &0 /\ r = m
694 else &0 <= r /\ r < abs(n) /\ m = q * n + r`,
695 REPEAT GEN_TAC THEN ASM_CASES_TAC `n = &0` THEN
696 ASM_REWRITE_TAC[RIGHT_EXISTS_AND_THM; EXISTS_REFL] THEN
697 GEN_REWRITE_TAC I [SWAP_EXISTS_THM] THEN
698 SUBGOAL_THEN `?r. &0 <= r /\ ?q:int. m = n * q + r` MP_TAC THENL
699 [FIRST_ASSUM(MP_TAC o SPEC `--m:int` o MATCH_MP INT_ARCH) THEN
700 DISCH_THEN(X_CHOOSE_TAC `s:int`) THEN
701 EXISTS_TAC `m + s * n:int` THEN CONJ_TAC THENL
702 [ASM_INT_ARITH_TAC; EXISTS_TAC `--s:int` THEN INT_ARITH_TAC];
703 GEN_REWRITE_TAC LAND_CONV [INT_WOP] THEN
704 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:int` THEN
705 REWRITE_TAC[LEFT_AND_EXISTS_THM; RIGHT_AND_EXISTS_THM] THEN
706 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `q:int` THEN STRIP_TAC THEN
707 ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPEC `r - abs n`) THEN
708 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
709 DISCH_THEN(MP_TAC o SPEC `if &0 <= n then q + &1 else q - &1`) THEN
710 ASM_INT_ARITH_TAC]);;
712 parse_as_infix("div",(22,"left"));;
713 parse_as_infix("rem",(22,"left"));;
715 let INT_DIVISION_0 = new_specification ["div"; "rem"]
716 (REWRITE_RULE[SKOLEM_THM] INT_DIVMOD_EXIST_0);;
718 let INT_DIVISION = prove
720 ==> m = m div n * n + m rem n /\ &0 <= m rem n /\ m rem n < abs n`,
721 MESON_TAC[INT_DIVISION_0]);;
723 (* ------------------------------------------------------------------------- *)
724 (* Arithmetic operations on integers. Essentially a clone of stuff for reals *)
725 (* in the file "calc_int.ml", except for div and rem, which are more like N. *)
726 (* ------------------------------------------------------------------------- *)
728 let INT_LE_CONV,INT_LT_CONV,INT_GE_CONV,INT_GT_CONV,INT_EQ_CONV =
730 TAUT `(F /\ F <=> F) /\ (F /\ T <=> F) /\
731 (T /\ F <=> F) /\ (T /\ T <=> T)` in
732 let nth = TAUT `(~T <=> F) /\ (~F <=> T)` in
733 let NUM2_EQ_CONV = BINOP_CONV NUM_EQ_CONV THENC GEN_REWRITE_CONV I [tth] in
735 RAND_CONV NUM2_EQ_CONV THENC
736 GEN_REWRITE_CONV I [nth] in
737 let [pth_le1; pth_le2a; pth_le2b; pth_le3] = (CONJUNCTS o prove)
738 (`(--(&m) <= &n <=> T) /\
739 (&m <= &n <=> m <= n) /\
740 (--(&m) <= --(&n) <=> n <= m) /\
741 (&m <= --(&n) <=> (m = 0) /\ (n = 0))`,
742 REWRITE_TAC[INT_LE_NEG2] THEN
743 REWRITE_TAC[INT_LE_LNEG; INT_LE_RNEG] THEN
744 REWRITE_TAC[INT_OF_NUM_ADD; INT_OF_NUM_LE; LE_0] THEN
745 REWRITE_TAC[LE; ADD_EQ_0]) in
746 let INT_LE_CONV = FIRST_CONV
747 [GEN_REWRITE_CONV I [pth_le1];
748 GEN_REWRITE_CONV I [pth_le2a; pth_le2b] THENC NUM_LE_CONV;
749 GEN_REWRITE_CONV I [pth_le3] THENC NUM2_EQ_CONV] in
750 let [pth_lt1; pth_lt2a; pth_lt2b; pth_lt3] = (CONJUNCTS o prove)
751 (`(&m < --(&n) <=> F) /\
752 (&m < &n <=> m < n) /\
753 (--(&m) < --(&n) <=> n < m) /\
754 (--(&m) < &n <=> ~((m = 0) /\ (n = 0)))`,
755 REWRITE_TAC[pth_le1; pth_le2a; pth_le2b; pth_le3;
756 GSYM NOT_LE; INT_LT] THEN
758 let INT_LT_CONV = FIRST_CONV
759 [GEN_REWRITE_CONV I [pth_lt1];
760 GEN_REWRITE_CONV I [pth_lt2a; pth_lt2b] THENC NUM_LT_CONV;
761 GEN_REWRITE_CONV I [pth_lt3] THENC NUM2_NE_CONV] in
762 let [pth_ge1; pth_ge2a; pth_ge2b; pth_ge3] = (CONJUNCTS o prove)
763 (`(&m >= --(&n) <=> T) /\
764 (&m >= &n <=> n <= m) /\
765 (--(&m) >= --(&n) <=> m <= n) /\
766 (--(&m) >= &n <=> (m = 0) /\ (n = 0))`,
767 REWRITE_TAC[pth_le1; pth_le2a; pth_le2b; pth_le3; INT_GE] THEN
769 let INT_GE_CONV = FIRST_CONV
770 [GEN_REWRITE_CONV I [pth_ge1];
771 GEN_REWRITE_CONV I [pth_ge2a; pth_ge2b] THENC NUM_LE_CONV;
772 GEN_REWRITE_CONV I [pth_ge3] THENC NUM2_EQ_CONV] in
773 let [pth_gt1; pth_gt2a; pth_gt2b; pth_gt3] = (CONJUNCTS o prove)
774 (`(--(&m) > &n <=> F) /\
775 (&m > &n <=> n < m) /\
776 (--(&m) > --(&n) <=> m < n) /\
777 (&m > --(&n) <=> ~((m = 0) /\ (n = 0)))`,
778 REWRITE_TAC[pth_lt1; pth_lt2a; pth_lt2b; pth_lt3; INT_GT] THEN
780 let INT_GT_CONV = FIRST_CONV
781 [GEN_REWRITE_CONV I [pth_gt1];
782 GEN_REWRITE_CONV I [pth_gt2a; pth_gt2b] THENC NUM_LT_CONV;
783 GEN_REWRITE_CONV I [pth_gt3] THENC NUM2_NE_CONV] in
784 let [pth_eq1a; pth_eq1b; pth_eq2a; pth_eq2b] = (CONJUNCTS o prove)
785 (`((&m = &n) <=> (m = n)) /\
786 ((--(&m) = --(&n)) <=> (m = n)) /\
787 ((--(&m) = &n) <=> (m = 0) /\ (n = 0)) /\
788 ((&m = --(&n)) <=> (m = 0) /\ (n = 0))`,
789 REWRITE_TAC[GSYM INT_LE_ANTISYM; GSYM LE_ANTISYM] THEN
790 REWRITE_TAC[pth_le1; pth_le2a; pth_le2b; pth_le3; LE; LE_0] THEN
792 let INT_EQ_CONV = FIRST_CONV
793 [GEN_REWRITE_CONV I [pth_eq1a; pth_eq1b] THENC NUM_EQ_CONV;
794 GEN_REWRITE_CONV I [pth_eq2a; pth_eq2b] THENC NUM2_EQ_CONV] in
795 INT_LE_CONV,INT_LT_CONV,
796 INT_GE_CONV,INT_GT_CONV,INT_EQ_CONV;;
802 REWRITE_TAC[INT_NEG_NEG; INT_NEG_0]) in
803 GEN_REWRITE_CONV I [pth];;
808 (&0 * --(&x) = &0) /\
811 REWRITE_TAC[INT_MUL_LZERO; INT_MUL_RZERO])
812 and pth1,pth2 = (CONJ_PAIR o prove)
813 (`((&m * &n = &(m * n)) /\
814 (--(&m) * --(&n) = &(m * n))) /\
815 ((--(&m) * &n = --(&(m * n))) /\
816 (&m * --(&n) = --(&(m * n))))`,
817 REWRITE_TAC[INT_MUL_LNEG; INT_MUL_RNEG; INT_NEG_NEG] THEN
818 REWRITE_TAC[INT_OF_NUM_MUL]) in
820 [GEN_REWRITE_CONV I [pth0];
821 GEN_REWRITE_CONV I [pth1] THENC RAND_CONV NUM_MULT_CONV;
822 GEN_REWRITE_CONV I [pth2] THENC RAND_CONV(RAND_CONV NUM_MULT_CONV)];;
825 let neg_tm = `(--)` in
827 let add_tm = `(+)` in
828 let dest = dest_binop `(+)` in
829 let m_tm = `m:num` and n_tm = `n:num` in
831 (`(--(&m) + &m = &0) /\
833 REWRITE_TAC[INT_ADD_LINV; INT_ADD_RINV]) in
834 let [pth1; pth2; pth3; pth4; pth5; pth6] = (CONJUNCTS o prove)
835 (`(--(&m) + --(&n) = --(&(m + n))) /\
836 (--(&m) + &(m + n) = &n) /\
837 (--(&(m + n)) + &m = --(&n)) /\
838 (&(m + n) + --(&m) = &n) /\
839 (&m + --(&(m + n)) = --(&n)) /\
840 (&m + &n = &(m + n))`,
841 REWRITE_TAC[GSYM INT_OF_NUM_ADD; INT_NEG_ADD] THEN
842 REWRITE_TAC[INT_ADD_ASSOC; INT_ADD_LINV; INT_ADD_LID] THEN
843 REWRITE_TAC[INT_ADD_RINV; INT_ADD_LID] THEN
844 ONCE_REWRITE_TAC[INT_ADD_SYM] THEN
845 REWRITE_TAC[INT_ADD_ASSOC; INT_ADD_LINV; INT_ADD_LID] THEN
846 REWRITE_TAC[INT_ADD_RINV; INT_ADD_LID]) in
847 GEN_REWRITE_CONV I [pth0] ORELSEC
849 try let l,r = dest tm in
850 if rator l = neg_tm then
851 if rator r = neg_tm then
852 let th1 = INST [rand(rand l),m_tm; rand(rand r),n_tm] pth1 in
853 let tm1 = rand(rand(rand(concl th1))) in
854 let th2 = AP_TERM neg_tm (AP_TERM amp_tm (NUM_ADD_CONV tm1)) in
857 let m = rand(rand l) and n = rand r in
858 let m' = dest_numeral m and n' = dest_numeral n in
860 let p = mk_numeral (n' -/ m') in
861 let th1 = INST [m,m_tm; p,n_tm] pth2 in
862 let th2 = NUM_ADD_CONV (rand(rand(lhand(concl th1)))) in
863 let th3 = AP_TERM (rator tm) (AP_TERM amp_tm (SYM th2)) in
866 let p = mk_numeral (m' -/ n') in
867 let th1 = INST [n,m_tm; p,n_tm] pth3 in
868 let th2 = NUM_ADD_CONV (rand(rand(lhand(lhand(concl th1))))) in
869 let th3 = AP_TERM neg_tm (AP_TERM amp_tm (SYM th2)) in
870 let th4 = AP_THM (AP_TERM add_tm th3) (rand tm) in
873 if rator r = neg_tm then
874 let m = rand l and n = rand(rand r) in
875 let m' = dest_numeral m and n' = dest_numeral n in
877 let p = mk_numeral (m' -/ n') in
878 let th1 = INST [n,m_tm; p,n_tm] pth4 in
879 let th2 = NUM_ADD_CONV (rand(lhand(lhand(concl th1)))) in
880 let th3 = AP_TERM add_tm (AP_TERM amp_tm (SYM th2)) in
881 let th4 = AP_THM th3 (rand tm) in
884 let p = mk_numeral (n' -/ m') in
885 let th1 = INST [m,m_tm; p,n_tm] pth5 in
886 let th2 = NUM_ADD_CONV (rand(rand(rand(lhand(concl th1))))) in
887 let th3 = AP_TERM neg_tm (AP_TERM amp_tm (SYM th2)) in
888 let th4 = AP_TERM (rator tm) th3 in
891 let th1 = INST [rand l,m_tm; rand r,n_tm] pth6 in
892 let tm1 = rand(rand(concl th1)) in
893 let th2 = AP_TERM amp_tm (NUM_ADD_CONV tm1) in
895 with Failure _ -> failwith "INT_ADD_CONV");;
898 GEN_REWRITE_CONV I [INT_SUB] THENC
899 TRY_CONV(RAND_CONV INT_NEG_CONV) THENC
903 let pth1,pth2 = (CONJ_PAIR o prove)
904 (`(&x pow n = &(x EXP n)) /\
905 ((--(&x)) pow n = if EVEN n then &(x EXP n) else --(&(x EXP n)))`,
906 REWRITE_TAC[INT_OF_NUM_POW; INT_POW_NEG]) in
908 (`((if T then x:int else y) = x) /\ ((if F then x:int else y) = y)`,
910 let neg_tm = `(--)` in
911 (GEN_REWRITE_CONV I [pth1] THENC RAND_CONV NUM_EXP_CONV) ORELSEC
912 (GEN_REWRITE_CONV I [pth2] THENC
913 RATOR_CONV(RATOR_CONV(RAND_CONV NUM_EVEN_CONV)) THENC
914 GEN_REWRITE_CONV I [tth] THENC
915 (fun tm -> if rator tm = neg_tm then RAND_CONV(RAND_CONV NUM_EXP_CONV) tm
916 else RAND_CONV NUM_EXP_CONV tm));;
920 (`(abs(--(&x)) = &x) /\
922 REWRITE_TAC[INT_ABS_NEG; INT_ABS_NUM]) in
923 GEN_REWRITE_CONV I [pth];;
926 REWR_CONV INT_MAX THENC
927 RATOR_CONV(RATOR_CONV(RAND_CONV INT_LE_CONV)) THENC
928 GEN_REWRITE_CONV I [COND_CLAUSES];;
931 REWR_CONV INT_MIN THENC
932 RATOR_CONV(RATOR_CONV(RAND_CONV INT_LE_CONV)) THENC
933 GEN_REWRITE_CONV I [COND_CLAUSES];;
935 (* ------------------------------------------------------------------------- *)
936 (* Instantiate the normalizer. *)
937 (* ------------------------------------------------------------------------- *)
941 (`(!x y z. x + (y + z) = (x + y) + z) /\
942 (!x y. x + y = y + x) /\
944 (!x y z. x * (y * z) = (x * y) * z) /\
945 (!x y. x * y = y * x) /\
948 (!x y z. x * (y + z) = x * y + x * z) /\
949 (!x. x pow 0 = &1) /\
950 (!x n. x pow (SUC n) = x * x pow n)`,
951 REWRITE_TAC[INT_POW] THEN INT_ARITH_TAC)
953 (`(!x. --x = --(&1) * x) /\
954 (!x y. x - y = x + --(&1) * y)`,
956 and is_semiring_constant = is_intconst
957 and SEMIRING_ADD_CONV = INT_ADD_CONV
958 and SEMIRING_MUL_CONV = INT_MUL_CONV
959 and SEMIRING_POW_CONV = INT_POW_CONV in
960 let _,_,_,_,_,INT_POLY_CONV =
961 SEMIRING_NORMALIZERS_CONV sth rth
962 (is_semiring_constant,
963 SEMIRING_ADD_CONV,SEMIRING_MUL_CONV,SEMIRING_POW_CONV)
967 (* ------------------------------------------------------------------------- *)
968 (* Instantiate the ring and ideal procedures. *)
969 (* ------------------------------------------------------------------------- *)
971 let INT_RING,int_ideal_cofactors =
972 let INT_INTEGRAL = prove
973 (`(!x. &0 * x = &0) /\
974 (!x y z. (x + y = x + z) <=> (y = z)) /\
975 (!w x y z. (w * y + x * z = w * z + x * y) <=> (w = x) \/ (y = z))`,
976 REWRITE_TAC[MULT_CLAUSES; EQ_ADD_LCANCEL] THEN
977 REWRITE_TAC[GSYM INT_OF_NUM_EQ;
978 GSYM INT_OF_NUM_ADD; GSYM INT_OF_NUM_MUL] THEN
979 ONCE_REWRITE_TAC[GSYM INT_SUB_0] THEN
980 REWRITE_TAC[GSYM INT_ENTIRE] THEN INT_ARITH_TAC)
981 and int_ty = `:int` in
984 (dest_intconst,mk_intconst,INT_EQ_CONV,
985 `(--):int->int`,`(+):int->int->int`,`(-):int->int->int`,
986 genvar bool_ty,`(*):int->int->int`,genvar bool_ty,
987 `(pow):int->num->int`,
988 INT_INTEGRAL,TRUTH,INT_POLY_CONV) in
990 (fun tms tm -> if forall (fun t -> type_of t = int_ty) (tm::tms)
993 "int_ideal_cofactors: not all terms have type :int");;
995 (* ------------------------------------------------------------------------- *)
996 (* Arithmetic operations also on div and rem, hence the whole lot. *)
997 (* ------------------------------------------------------------------------- *)
999 let INT_DIVMOD_UNIQ = prove
1000 (`!m n q r:int. m = q * n + r /\ &0 <= r /\ r < abs n
1001 ==> m div n = q /\ m rem n = r`,
1002 REPEAT GEN_TAC THEN STRIP_TAC THEN
1003 SUBGOAL_THEN `~(n = &0)` MP_TAC THENL [ASM_INT_ARITH_TAC; ALL_TAC] THEN
1004 DISCH_THEN(STRIP_ASSUME_TAC o SPEC `m:int` o MATCH_MP INT_DIVISION) THEN
1005 ASM_CASES_TAC `m div n = q` THENL
1006 [REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC INT_RING; ALL_TAC] THEN
1007 SUBGOAL_THEN `abs(m rem n - r) < abs n` MP_TAC THENL
1008 [ASM_INT_ARITH_TAC; MATCH_MP_TAC(TAUT `~p ==> p ==> q`)] THEN
1009 MATCH_MP_TAC(INT_ARITH
1010 `&1 * abs n <= abs(q - m div n) * abs n /\
1011 abs(m rem n - r) = abs((q - m div n) * n)
1012 ==> ~(abs(m rem n - r) < abs n)`) THEN
1014 [MATCH_MP_TAC INT_LE_RMUL THEN ASM_INT_ARITH_TAC;
1015 AP_TERM_TAC THEN REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC INT_RING]);;
1017 let INT_DIV_CONV,INT_REM_CONV =
1019 (`q * n + r = m ==> &0 <= r ==> r < abs n ==> m div n = q /\ m rem n = r`,
1020 MESON_TAC[INT_DIVMOD_UNIQ])
1021 and m = `m:int` and n = `n:int` and q = `q:int` and r = `r:int`
1022 and dtm = `(div)` and mtm = `(rem)` in
1024 let r = mod_num x y in
1025 if r </ Int 0 then r +/ abs_num y else r in
1026 let equo_num x y = quo_num (x -/ emod_num x y) y in
1027 let INT_DIVMOD_CONV x y =
1028 let k = equo_num x y
1029 and l = emod_num x y in
1030 let th0 = INST [mk_intconst x,m; mk_intconst y,n;
1031 mk_intconst k,q; mk_intconst l,r] pth in
1032 let tm0 = lhand(lhand(concl th0)) in
1033 let th1 = (LAND_CONV INT_MUL_CONV THENC INT_ADD_CONV) tm0 in
1034 let th2 = MP th0 th1 in
1035 let tm2 = lhand(concl th2) in
1036 let th3 = MP th2 (EQT_ELIM(INT_LE_CONV tm2)) in
1037 let tm3 = lhand(concl th3) in
1038 MP th3 (EQT_ELIM((RAND_CONV INT_ABS_CONV THENC INT_LT_CONV) tm3)) in
1039 (fun tm -> try let l,r = dest_binop dtm tm in
1040 CONJUNCT1(INT_DIVMOD_CONV (dest_intconst l) (dest_intconst r))
1041 with Failure _ -> failwith "INT_DIV_CONV"),
1042 (fun tm -> try let l,r = dest_binop mtm tm in
1043 CONJUNCT2(INT_DIVMOD_CONV (dest_intconst l) (dest_intconst r))
1044 with Failure _ -> failwith "INT_MOD_CONV");;
1047 let gconv_net = itlist (uncurry net_of_conv)
1048 [`x <= y`,INT_LE_CONV;
1049 `x < y`,INT_LT_CONV;
1050 `x >= y`,INT_GE_CONV;
1051 `x > y`,INT_GT_CONV;
1052 `x:int = y`,INT_EQ_CONV;
1053 `--x`,CHANGED_CONV INT_NEG_CONV;
1054 `abs(x)`,INT_ABS_CONV;
1055 `x + y`,INT_ADD_CONV;
1056 `x - y`,INT_SUB_CONV;
1057 `x * y`,INT_MUL_CONV;
1058 `x div y`,INT_DIV_CONV;
1059 `x rem y`,INT_REM_CONV;
1060 `x pow n`,INT_POW_CONV;
1061 `max x y`,INT_MAX_CONV;
1062 `min x y`,INT_MIN_CONV]
1064 REWRITES_CONV gconv_net;;
1066 let INT_REDUCE_CONV = DEPTH_CONV INT_RED_CONV;;
1068 (* ------------------------------------------------------------------------- *)
1069 (* Set up overloading so we can use same symbols for N, Z and even R. *)
1070 (* ------------------------------------------------------------------------- *)
1072 make_overloadable "divides" `:A->A->bool`;;
1073 make_overloadable "mod" `:A->A->A->bool`;;
1074 make_overloadable "coprime" `:A#A->bool`;;
1075 make_overloadable "gcd" `:A#A->A`;;
1077 (* ------------------------------------------------------------------------- *)
1078 (* The general notion of congruence: just syntax for equivalence relation. *)
1079 (* ------------------------------------------------------------------------- *)
1081 parse_as_infix("==",(10,"right"));;
1083 let cong = new_definition
1084 `(x == y) (rel:A->A->bool) <=> rel x y`;;
1086 (* ------------------------------------------------------------------------- *)
1087 (* Get real moduli defined and out of the way first. *)
1088 (* ------------------------------------------------------------------------- *)
1090 let real_mod = new_definition
1091 `real_mod n (x:real) y = ?q. integer q /\ x - y = q * n`;;
1093 overload_interface ("mod",`real_mod`);;
1095 (* ------------------------------------------------------------------------- *)
1096 (* Integer divisibility. *)
1097 (* ------------------------------------------------------------------------- *)
1099 parse_as_infix("divides",(12,"right"));;
1100 overload_interface("divides",`int_divides:int->int->bool`);;
1102 let int_divides = new_definition
1103 `a divides b <=> ?x. b = a * x`;;
1105 (* ------------------------------------------------------------------------- *)
1106 (* Integer congruences. *)
1107 (* ------------------------------------------------------------------------- *)
1109 parse_as_prefix "mod";;
1110 overload_interface ("mod",`int_mod:int->int->int->bool`);;
1112 let int_mod = new_definition
1113 `(mod n) x y = n divides (x - y)`;;
1115 let int_congruent = prove
1116 (`!x y n. (x == y) (mod n) <=> ?d. x - y = n * d`,
1117 REWRITE_TAC[int_mod; cong; int_divides]);;
1119 (* ------------------------------------------------------------------------- *)
1120 (* Integer coprimality. *)
1121 (* ------------------------------------------------------------------------- *)
1123 overload_interface("coprime",`int_coprime:int#int->bool`);;
1125 let int_coprime = new_definition
1126 `!a b. coprime(a,b) <=> ?x y. a * x + b * y = &1`;;
1128 (* ------------------------------------------------------------------------- *)
1129 (* A tactic for simple divisibility/congruence/coprimality goals. *)
1130 (* ------------------------------------------------------------------------- *)
1133 let int_ty = `:int` in
1134 let INT_POLYEQ_CONV =
1135 GEN_REWRITE_CONV I [GSYM INT_SUB_0] THENC LAND_CONV INT_POLY_CONV in
1136 let ISOLATE_VARIABLE =
1137 let pth = INT_ARITH `!a x. a = &0 <=> x = x + a` in
1138 let is_defined v t =
1139 let mons = striplist(dest_binary "int_add") t in
1140 mem v mons & forall (fun m -> v = m or not(free_in v m)) mons in
1142 let th = INT_POLYEQ_CONV tm
1143 and th' = (SYM_CONV THENC INT_POLYEQ_CONV) tm in
1145 try find (fun v -> is_defined v (lhand(rand(concl th)))) vars,th'
1147 find (fun v -> is_defined v (lhand(rand(concl th')))) vars,th in
1148 let th2 = TRANS th1 (SPECL [lhs(rand(concl th1)); v] pth) in
1149 CONV_RULE(RAND_CONV(RAND_CONV INT_POLY_CONV)) th2 in
1150 let UNWIND_POLYS_CONV tm =
1151 let vars,bod = strip_exists tm in
1152 let cjs = conjuncts bod in
1153 let th1 = tryfind (ISOLATE_VARIABLE vars) cjs in
1154 let eq = lhand(concl th1) in
1155 let bod' = list_mk_conj(eq::(subtract cjs [eq])) in
1156 let th2 = CONJ_ACI_RULE(mk_eq(bod,bod')) in
1157 let th3 = TRANS th2 (MK_CONJ th1 (REFL(rand(rand(concl th2))))) in
1158 let v = lhs(lhand(rand(concl th3))) in
1159 let vars' = (subtract vars [v]) @ [v] in
1160 let th4 = CONV_RULE(RAND_CONV(REWR_CONV UNWIND_THM2)) (MK_EXISTS v th3) in
1162 DISCH_ALL(itlist SIMPLE_CHOOSE v (itlist SIMPLE_EXISTS v' (ASSUME bod))) in
1163 let th5 = IMP_ANTISYM_RULE (IMP_RULE vars vars') (IMP_RULE vars' vars) in
1164 TRANS th5 (itlist MK_EXISTS (subtract vars [v]) th4) in
1165 let zero_tm = `&0` and one_tm = `&1` in
1166 let isolate_monomials =
1167 let mul_tm = `(int_mul)` and add_tm = `(int_add)`
1168 and neg_tm = `(int_neg)` in
1169 let dest_mul = dest_binop mul_tm
1170 and dest_add = dest_binop add_tm
1171 and mk_mul = mk_binop mul_tm
1172 and mk_add = mk_binop add_tm in
1174 let ps = striplist dest_mul m in
1175 let ps' = subtract ps [v] in
1176 if ps' = [] then one_tm else end_itlist mk_mul ps' in
1177 let find_multipliers v mons =
1178 let mons1 = filter (fun m -> free_in v m) mons in
1179 let mons2 = map (scrub_var v) mons1 in
1180 if mons2 = [] then zero_tm else end_itlist mk_add mons2 in
1183 partition (fun m -> intersect (frees m) vars = [])
1184 (striplist dest_add tm) in
1185 let cofactors = map (fun v -> find_multipliers v vmons) vars
1186 and cnc = if cmons = [] then zero_tm
1187 else mk_comb(neg_tm,end_itlist mk_add cmons) in
1189 let isolate_variables evs ps eq =
1190 let vars = filter (fun v -> vfree_in v eq) evs in
1191 let qs,p = isolate_monomials vars eq in
1192 let rs = filter (fun t -> type_of t = int_ty) (qs @ ps) in
1193 let rs = int_ideal_cofactors rs p in
1194 eq,zip (fst(chop_list(length qs) rs)) vars in
1195 let subst_in_poly i p = rhs(concl(INT_POLY_CONV (vsubst i p))) in
1196 let rec solve_idealism evs ps eqs =
1197 if evs = [] then [] else
1198 let eq,cfs = tryfind (isolate_variables evs ps) eqs in
1199 let evs' = subtract evs (map snd cfs)
1200 and eqs' = map (subst_in_poly cfs) (subtract eqs [eq]) in
1201 cfs @ solve_idealism evs' ps eqs' in
1202 let rec GENVAR_EXISTS_CONV tm =
1203 if not(is_exists tm) then REFL tm else
1204 let ev,bod = dest_exists tm in
1205 let gv = genvar(type_of ev) in
1206 (GEN_ALPHA_CONV gv THENC BINDER_CONV GENVAR_EXISTS_CONV) tm in
1207 let EXISTS_POLY_TAC (asl,w as gl) =
1208 let evs,bod = strip_exists w
1209 and ps = mapfilter (check (fun t -> type_of t = int_ty) o
1210 lhs o concl o snd) asl in
1211 let cfs = solve_idealism evs ps (map lhs (conjuncts bod)) in
1212 (MAP_EVERY EXISTS_TAC(map (fun v -> rev_assocd v cfs zero_tm) evs) THEN
1213 REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC INT_RING) gl in
1214 let SCRUB_NEQ_TAC = MATCH_MP_TAC o MATCH_MP (MESON[]
1215 `~(x = y) ==> x = y \/ p ==> p`) in
1216 REWRITE_TAC[int_coprime; int_congruent; int_divides] THEN
1217 REPEAT(STRIP_TAC ORELSE EQ_TAC) THEN
1218 REWRITE_TAC[LEFT_AND_EXISTS_THM; RIGHT_AND_EXISTS_THM;
1219 LEFT_OR_EXISTS_THM; RIGHT_OR_EXISTS_THM] THEN
1220 CONV_TAC(REPEATC UNWIND_POLYS_CONV) THEN
1221 REPEAT(FIRST_X_ASSUM SCRUB_NEQ_TAC) THEN
1222 REWRITE_TAC[LEFT_AND_EXISTS_THM; RIGHT_AND_EXISTS_THM;
1223 LEFT_OR_EXISTS_THM; RIGHT_OR_EXISTS_THM] THEN
1224 REPEAT(FIRST_X_ASSUM(MP_TAC o SYM)) THEN
1225 CONV_TAC(ONCE_DEPTH_CONV INT_POLYEQ_CONV) THEN
1226 REWRITE_TAC[GSYM INT_ENTIRE;
1227 TAUT `a \/ (b /\ c) <=> (a \/ b) /\ (a \/ c)`] THEN
1228 POP_ASSUM_LIST(K ALL_TAC) THEN
1229 REPEAT DISCH_TAC THEN CONV_TAC GENVAR_EXISTS_CONV THEN
1230 CONV_TAC(ONCE_DEPTH_CONV INT_POLYEQ_CONV) THEN EXISTS_POLY_TAC;;
1232 let INTEGER_RULE tm = prove(tm,INTEGER_TAC);;
1234 (* ------------------------------------------------------------------------- *)
1235 (* Existence of integer gcd, and the Bezout identity. *)
1236 (* ------------------------------------------------------------------------- *)
1238 let WF_INT_MEASURE = prove
1239 (`!P m. (!x. &0 <= m(x)) /\ (!x. (!y. m(y) < m(x) ==> P(y)) ==> P(x))
1241 REPEAT STRIP_TAC THEN SUBGOAL_THEN `!n x:A. m(x) = &n ==> P(x)` MP_TAC THENL
1242 [MATCH_MP_TAC num_WF; ALL_TAC] THEN
1243 REWRITE_TAC[GSYM INT_OF_NUM_LT; INT_FORALL_POS] THEN ASM_MESON_TAC[]);;
1245 let WF_INT_MEASURE_2 = prove
1246 (`!P m. (!x y. &0 <= m x y) /\
1247 (!x y. (!x' y'. m x' y' < m x y ==> P x' y') ==> P x y)
1248 ==> !x:A y:B. P x y`,
1249 REWRITE_TAC[FORALL_UNCURRY; GSYM FORALL_PAIR_THM; WF_INT_MEASURE]);;
1251 let INT_GCD_EXISTS = prove
1252 (`!a b. ?d. d divides a /\ d divides b /\ ?x y. d = a * x + b * y`,
1253 let INT_GCD_EXISTS_CASES = INT_ARITH
1254 `(a = &0 \/ b = &0) \/
1255 abs(a - b) + abs b < abs a + abs b \/ abs(a + b) + abs b < abs a + abs b \/
1256 abs a + abs(b - a) < abs a + abs b \/ abs a + abs(b + a) < abs a + abs b` in
1257 MATCH_MP_TAC WF_INT_MEASURE_2 THEN EXISTS_TAC `\x y. abs(x) + abs(y)` THEN
1258 REWRITE_TAC[] THEN REPEAT STRIP_TAC THENL [INT_ARITH_TAC; ALL_TAC] THEN
1259 DISJ_CASES_THEN MP_TAC INT_GCD_EXISTS_CASES THENL
1260 [STRIP_TAC THEN ASM_REWRITE_TAC[INTEGER_RULE `d divides &0`] THEN
1261 REWRITE_TAC[INT_MUL_LZERO; INT_ADD_LID; INT_ADD_RID] THEN
1262 MESON_TAC[INTEGER_RULE `d divides d`; INT_MUL_RID];
1263 DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN (ANTE_RES_THEN MP_TAC)) THEN
1264 MATCH_MP_TAC MONO_EXISTS THEN INTEGER_TAC]);;
1266 let INT_GCD_EXISTS_POS = prove
1267 (`!a b. ?d. &0 <= d /\ d divides a /\ d divides b /\ ?x y. d = a * x + b * y`,
1269 X_CHOOSE_TAC `d:int` (SPECL [`a:int`; `b:int`] INT_GCD_EXISTS) THEN
1270 DISJ_CASES_TAC(SPEC `d:int` INT_LE_NEGTOTAL) THEN
1271 ASM_MESON_TAC[INTEGER_RULE `(--d) divides x <=> d divides x`;
1272 INT_ARITH `a * --x + b * --y = --(a * x + b * y)`]);;
1274 (* ------------------------------------------------------------------------- *)
1275 (* Hence define (positive) gcd function; add elimination to INTEGER_TAC. *)
1276 (* ------------------------------------------------------------------------- *)
1278 overload_interface("gcd",`int_gcd:int#int->int`);;
1280 let int_gcd = new_specification ["int_gcd"]
1281 (REWRITE_RULE[EXISTS_UNCURRY; SKOLEM_THM] INT_GCD_EXISTS_POS);;
1285 let gcd_tm = `gcd` in
1287 let l,r = dest_comb tm in
1288 if l = gcd_tm then dest_pair r else failwith "dest_gcd" in
1291 let gts = find_terms (can dest_gcd) w in
1293 (fun tm -> let a,b = dest_gcd tm in SPECL [a;b] int_gcd) gts in
1294 MAP_EVERY MP_TAC ths THEN
1295 MAP_EVERY SPEC_TAC (zip gts (map (genvar o type_of) gts))) in
1296 REPEAT(GEN_TAC ORELSE CONJ_TAC) THEN GCD_ELIM_TAC THEN INTEGER_TAC;;
1298 let INTEGER_RULE tm = prove(tm,INTEGER_TAC);;
1300 (* ------------------------------------------------------------------------- *)
1301 (* Mapping from nonnegative integers back to natural numbers. *)
1302 (* ------------------------------------------------------------------------- *)
1304 let num_of_int = new_definition
1305 `num_of_int x = @n. &n = x`;;
1307 let NUM_OF_INT_OF_NUM = prove
1308 (`!n. num_of_int(&n) = n`,
1309 REWRITE_TAC[num_of_int; INT_OF_NUM_EQ; SELECT_UNIQUE]);;
1311 let INT_OF_NUM_OF_INT = prove
1312 (`!x. &0 <= x ==> &(num_of_int x) = x`,
1313 REWRITE_TAC[GSYM INT_FORALL_POS; num_of_int] THEN
1314 GEN_TAC THEN CONV_TAC SELECT_CONV THEN MESON_TAC[]);;
1316 let NUM_OF_INT = prove
1317 (`!x. &0 <= x <=> (&(num_of_int x) = x)`,
1318 MESON_TAC[INT_OF_NUM_OF_INT; INT_POS]);;
1320 (* ------------------------------------------------------------------------- *)
1321 (* Now define similar notions over the natural numbers. *)
1322 (* ------------------------------------------------------------------------- *)
1324 overload_interface("divides",`num_divides:num->num->bool`);;
1325 overload_interface ("mod",`num_mod:num->num->num->bool`);;
1326 overload_interface("coprime",`num_coprime:num#num->bool`);;
1327 overload_interface("gcd",`num_gcd:num#num->num`);;
1329 let num_divides = new_definition
1330 `a divides b <=> &a divides &b`;;
1332 let num_mod = new_definition
1333 `(mod n) x y <=> (mod &n) (&x) (&y)`;;
1335 let num_congruent = prove
1336 (`!x y n. (x == y) (mod n) <=> (&x == &y) (mod &n)`,
1337 REWRITE_TAC[cong; num_mod]);;
1339 let num_coprime = new_definition
1340 `coprime(a,b) <=> coprime(&a,&b)`;;
1342 let num_gcd = new_definition
1343 `gcd(a,b) = num_of_int(gcd(&a,&b))`;;
1345 (* ------------------------------------------------------------------------- *)
1346 (* Map an assertion over N to an integer equivalent. *)
1347 (* To make this work nicely, all variables of type num should be quantified. *)
1348 (* ------------------------------------------------------------------------- *)
1350 let NUM_TO_INT_CONV =
1351 let pth_relativize = prove
1352 (`((!n. P(&n)) <=> (!i. ~(&0 <= i) \/ P i)) /\
1353 ((?n. P(&n)) <=> (?i. &0 <= i /\ P i))`,
1354 REWRITE_TAC[INT_EXISTS_POS; INT_FORALL_POS] THEN MESON_TAC[]) in
1355 let relation_conv = (GEN_REWRITE_CONV TOP_SWEEP_CONV o map GSYM)
1356 [INT_OF_NUM_EQ; INT_OF_NUM_LE; INT_OF_NUM_LT; INT_OF_NUM_GE; INT_OF_NUM_GT;
1357 INT_OF_NUM_SUC; INT_OF_NUM_ADD; INT_OF_NUM_MUL; INT_OF_NUM_POW]
1358 and quantifier_conv = GEN_REWRITE_CONV DEPTH_CONV [pth_relativize] in
1359 NUM_SIMPLIFY_CONV THENC relation_conv THENC quantifier_conv;;
1361 (* ------------------------------------------------------------------------- *)
1362 (* Linear decision procedure for the naturals at last! *)
1363 (* ------------------------------------------------------------------------- *)
1367 NUM_SIMPLIFY_CONV THENC
1368 GEN_REWRITE_CONV DEPTH_CONV [ADD1] THENC
1369 PROP_ATOM_CONV (BINOP_CONV NUM_NORMALIZE_CONV) THENC
1371 (GEN_REWRITE_CONV TOP_SWEEP_CONV o map GSYM)
1372 [INT_OF_NUM_EQ; INT_OF_NUM_LE; INT_OF_NUM_LT; INT_OF_NUM_GE;
1373 INT_OF_NUM_GT; INT_OF_NUM_ADD; SPEC `NUMERAL k` INT_OF_NUM_MUL;
1374 INT_OF_NUM_MAX; INT_OF_NUM_MIN]
1377 Comb(Const("int_of_num",_),n) when not(is_numeral n) -> true
1380 let th1 = init_conv tm in
1381 let tm1 = rand(concl th1) in
1382 let avs,bod = strip_forall tm1 in
1383 let nim = setify(find_terms is_numimage bod) in
1384 let gvs = map (genvar o type_of) nim in
1385 let pths = map (fun v -> SPEC (rand v) INT_POS) nim in
1386 let ibod = itlist (curry mk_imp o concl) pths bod in
1387 let gbod = subst (zip gvs nim) ibod in
1388 let th2 = INST (zip nim gvs) (INT_ARITH gbod) in
1389 let th3 = GENL avs (rev_itlist (C MP) pths th2) in
1390 EQ_MP (SYM th1) th3;;
1392 let ARITH_TAC = CONV_TAC(EQT_INTRO o ARITH_RULE);;
1395 REPEAT(FIRST_X_ASSUM(MP_TAC o check (not o is_forall o concl))) THEN
1398 (* ------------------------------------------------------------------------- *)
1399 (* Also a similar divisibility procedure for natural numbers. *)
1400 (* ------------------------------------------------------------------------- *)
1403 (`!a b. &(gcd(a,b)) = gcd(&a,&b)`,
1404 REWRITE_TAC[num_gcd; GSYM NUM_OF_INT; int_gcd]);;
1407 let pth_relativize = prove
1408 (`((!n. P(&n)) <=> (!i. &0 <= i ==> P i)) /\
1409 ((?n. P(&n)) <=> (?i. &0 <= i /\ P i))`,
1410 GEN_REWRITE_TAC RAND_CONV [TAUT `(a <=> b) <=> (~a <=> ~b)`] THEN
1411 REWRITE_TAC[NOT_EXISTS_THM; INT_FORALL_POS] THEN MESON_TAC[]) in
1413 GEN_REWRITE_CONV TOP_SWEEP_CONV
1414 (num_divides::num_congruent::num_coprime::NUM_GCD::(map GSYM
1415 [INT_OF_NUM_EQ; INT_OF_NUM_LE; INT_OF_NUM_LT; INT_OF_NUM_GE; INT_OF_NUM_GT;
1416 INT_OF_NUM_SUC; INT_OF_NUM_ADD; INT_OF_NUM_MUL; INT_OF_NUM_POW]))
1417 and quantifier_conv = GEN_REWRITE_CONV DEPTH_CONV [pth_relativize] in
1418 W(fun (_,w) -> MAP_EVERY (fun v -> SPEC_TAC(v,v)) (frees w)) THEN
1419 CONV_TAC(relation_conv THENC quantifier_conv) THEN
1420 REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN REPEAT GEN_TAC THEN
1423 let NUMBER_RULE tm = prove(tm,NUMBER_TAC);;
1426 (`a divides b <=> ?x. b = a * x`,
1427 EQ_TAC THENL [REWRITE_TAC[num_divides; int_divides]; NUMBER_TAC] THEN
1428 DISCH_THEN(X_CHOOSE_TAC `x:int`) THEN EXISTS_TAC `num_of_int(abs x)` THEN
1429 SIMP_TAC[GSYM INT_OF_NUM_EQ;
1430 INT_ARITH `&m:int = &n <=> abs(&m :int) = abs(&n)`] THEN
1431 ASM_REWRITE_TAC[GSYM INT_OF_NUM_MUL; INT_ABS_MUL] THEN
1432 SIMP_TAC[INT_OF_NUM_OF_INT; INT_ABS_POS; INT_ABS_ABS]);;
1434 let DIVIDES_LE = prove
1435 (`!m n. m divides n ==> m <= n \/ n = 0`,
1436 SUBGOAL_THEN `!m n. m <= m * n \/ m * n = 0`
1437 (fun th -> MESON_TAC[divides; th]) THEN
1438 REWRITE_TAC[LE_MULT_LCANCEL; MULT_EQ_0; ARITH_RULE
1439 `m <= m * n <=> m * 1 <= m * n`] THEN
1442 (* ------------------------------------------------------------------------- *)
1443 (* Make sure we give priority to N. *)
1444 (* ------------------------------------------------------------------------- *)