12224
|
1 |
(* Title: Poly.ML
|
|
2 |
Author: Jacques D. Fleuriot
|
|
3 |
Copyright: 2000 University of Edinburgh
|
|
4 |
Description: Properties of real polynomials following
|
|
5 |
John Harrison's HOL-Light implementation.
|
|
6 |
Some early theorems by Lucas Dixon
|
|
7 |
*)
|
|
8 |
|
|
9 |
Goal "p +++ [] = p";
|
|
10 |
by (induct_tac "p" 1);
|
|
11 |
by Auto_tac;
|
|
12 |
qed "padd_Nil2";
|
|
13 |
Addsimps [padd_Nil2];
|
|
14 |
|
|
15 |
Goal "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)";
|
|
16 |
by Auto_tac;
|
|
17 |
qed "padd_Cons_Cons";
|
|
18 |
|
|
19 |
Goal "-- [] = []";
|
|
20 |
by (rewtac poly_minus_def);
|
|
21 |
by (Auto_tac);
|
|
22 |
qed "pminus_Nil";
|
|
23 |
Addsimps [pminus_Nil];
|
|
24 |
|
|
25 |
Goal "[h1] *** p1 = h1 %* p1";
|
|
26 |
by (Simp_tac 1);
|
|
27 |
qed "pmult_singleton";
|
|
28 |
|
|
29 |
Goal "1 %* t = t";
|
|
30 |
by (induct_tac "t" 1);
|
|
31 |
by Auto_tac;
|
|
32 |
qed "poly_ident_mult";
|
|
33 |
Addsimps [poly_ident_mult];
|
|
34 |
|
|
35 |
Goal "[a] +++ ((0)#t) = (a#t)";
|
|
36 |
by (Simp_tac 1);
|
|
37 |
qed "poly_simple_add_Cons";
|
|
38 |
Addsimps [poly_simple_add_Cons];
|
|
39 |
|
|
40 |
(*-------------------------------------------------------------------------*)
|
|
41 |
(* Handy general properties *)
|
|
42 |
(*-------------------------------------------------------------------------*)
|
|
43 |
|
|
44 |
Goal "b +++ a = a +++ b";
|
|
45 |
by (subgoal_tac "ALL a. b +++ a = a +++ b" 1);
|
|
46 |
by (induct_tac "b" 2);
|
|
47 |
by Auto_tac;
|
|
48 |
by (rtac (padd_Cons RS ssubst) 1);
|
|
49 |
by (case_tac "aa" 1);
|
|
50 |
by Auto_tac;
|
|
51 |
qed "padd_commut";
|
|
52 |
|
|
53 |
Goal "(a +++ b) +++ c = a +++ (b +++ c)";
|
|
54 |
by (subgoal_tac "ALL b c. (a +++ b) +++ c = a +++ (b +++ c)" 1);
|
|
55 |
by (Asm_simp_tac 1);
|
|
56 |
by (induct_tac "a" 1);
|
|
57 |
by (Step_tac 2);
|
|
58 |
by (case_tac "b" 2);
|
|
59 |
by (Asm_simp_tac 2);
|
|
60 |
by (Asm_simp_tac 2);
|
|
61 |
by (Asm_simp_tac 1);
|
|
62 |
qed "padd_assoc";
|
|
63 |
|
|
64 |
Goal "a %* ( p +++ q ) = (a %* p +++ a %* q)";
|
|
65 |
by (subgoal_tac "ALL q. a %* ( p +++ q ) = (a %* p +++ a %* q) " 1);
|
|
66 |
by (induct_tac "p" 2);
|
|
67 |
by (Simp_tac 2);
|
|
68 |
by (rtac allI 2 );
|
|
69 |
by (case_tac "q" 2);
|
|
70 |
by (Asm_simp_tac 2);
|
|
71 |
by (asm_simp_tac (simpset() addsimps [real_add_mult_distrib2] ) 2);
|
|
72 |
by (Asm_simp_tac 1);
|
|
73 |
qed "poly_cmult_distr";
|
|
74 |
|
|
75 |
Goal "[0, 1] *** t = ((0)#t)";
|
|
76 |
by (induct_tac "t" 1);
|
|
77 |
by (Simp_tac 1);
|
|
78 |
by (simp_tac (simpset() addsimps [poly_ident_mult, padd_commut]) 1);
|
|
79 |
by (case_tac "list" 1);
|
|
80 |
by (Asm_simp_tac 1);
|
|
81 |
by (asm_full_simp_tac (simpset() addsimps [poly_ident_mult, padd_commut]) 1);
|
|
82 |
qed "pmult_by_x";
|
|
83 |
Addsimps [pmult_by_x];
|
|
84 |
|
|
85 |
|
|
86 |
(*-------------------------------------------------------------------------*)
|
|
87 |
(* properties of evaluation of polynomials. *)
|
|
88 |
(*-------------------------------------------------------------------------*)
|
|
89 |
|
|
90 |
Goal "poly (p1 +++ p2) x = poly p1 x + poly p2 x";
|
|
91 |
by (subgoal_tac "ALL p2. poly (p1 +++ p2) x = poly( p1 ) x + poly( p2 ) x" 1);
|
|
92 |
by (induct_tac "p1" 2);
|
|
93 |
by Auto_tac;
|
|
94 |
by (case_tac "p2" 1);
|
|
95 |
by (auto_tac (claset(),simpset() addsimps [real_add_mult_distrib2]));
|
|
96 |
qed "poly_add";
|
|
97 |
|
|
98 |
Goal "poly (c %* p) x = c * poly p x";
|
|
99 |
by (induct_tac "p" 1);
|
|
100 |
by (case_tac "x=0" 2);
|
|
101 |
by (auto_tac (claset(),simpset() addsimps [real_add_mult_distrib2]
|
|
102 |
@ real_mult_ac));
|
|
103 |
qed "poly_cmult";
|
|
104 |
|
|
105 |
Goalw [poly_minus_def] "poly (-- p) x = - (poly p x)";
|
|
106 |
by (auto_tac (claset(),simpset() addsimps [poly_cmult]));
|
|
107 |
qed "poly_minus";
|
|
108 |
|
|
109 |
Goal "poly (p1 *** p2) x = poly p1 x * poly p2 x";
|
|
110 |
by (subgoal_tac "ALL p2. poly (p1 *** p2) x = poly p1 x * poly p2 x" 1);
|
|
111 |
by (Asm_simp_tac 1);
|
|
112 |
by (induct_tac "p1" 1);
|
|
113 |
by (auto_tac (claset(),simpset() addsimps [poly_cmult]));
|
|
114 |
by (case_tac "list" 1);
|
|
115 |
by (auto_tac (claset(),simpset() addsimps [poly_cmult,poly_add,
|
|
116 |
real_add_mult_distrib,real_add_mult_distrib2] @ real_mult_ac));
|
|
117 |
qed "poly_mult";
|
|
118 |
|
|
119 |
Goal "poly (p %^ n) x = (poly p x) ^ n";
|
|
120 |
by (induct_tac "n" 1);
|
|
121 |
by (auto_tac (claset(),simpset() addsimps [poly_cmult, poly_mult]));
|
|
122 |
qed "poly_exp";
|
|
123 |
|
|
124 |
(*-------------------------------------------------------------------------*)
|
|
125 |
(* More Polynomial Evaluation Lemmas *)
|
|
126 |
(*-------------------------------------------------------------------------*)
|
|
127 |
|
|
128 |
Goal "poly (a +++ []) x = poly a x";
|
|
129 |
by (Simp_tac 1);
|
|
130 |
qed "poly_add_rzero";
|
|
131 |
Addsimps [poly_add_rzero];
|
|
132 |
|
|
133 |
Goal "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x";
|
|
134 |
by (simp_tac (simpset() addsimps [poly_mult,real_mult_assoc]) 1);
|
|
135 |
qed "poly_mult_assoc";
|
|
136 |
|
|
137 |
Goal "poly (p *** []) x = 0";
|
|
138 |
by (induct_tac "p" 1);
|
|
139 |
by Auto_tac;
|
|
140 |
qed "poly_mult_Nil2";
|
|
141 |
Addsimps [poly_mult_Nil2];
|
|
142 |
|
|
143 |
Goal "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d ) x";
|
|
144 |
by (induct_tac "n" 1);
|
|
145 |
by (auto_tac (claset(), simpset() addsimps [poly_mult,real_mult_assoc]));
|
|
146 |
qed "poly_exp_add";
|
|
147 |
|
|
148 |
(*-------------------------------------------------------------------------*)
|
|
149 |
(* The derivative *)
|
|
150 |
(*-------------------------------------------------------------------------*)
|
|
151 |
|
|
152 |
Goalw [pderiv_def] "pderiv [] = []";
|
|
153 |
by (Simp_tac 1);
|
|
154 |
qed "pderiv_Nil";
|
|
155 |
Addsimps [pderiv_Nil];
|
|
156 |
|
|
157 |
Goalw [pderiv_def] "pderiv [c] = []";
|
|
158 |
by (Simp_tac 1);
|
|
159 |
qed "pderiv_singleton";
|
|
160 |
Addsimps [pderiv_singleton];
|
|
161 |
|
|
162 |
Goalw [pderiv_def] "pderiv (h#t) = pderiv_aux 1 t";
|
|
163 |
by (Simp_tac 1);
|
|
164 |
qed "pderiv_Cons";
|
|
165 |
|
|
166 |
Goal "DERIV f x :> D ==> DERIV (%x. (f x) * c) x :> D * c";
|
|
167 |
by (auto_tac (claset() addIs [DERIV_cmult,real_mult_commute RS subst],
|
|
168 |
simpset() addsimps [real_mult_commute]));
|
|
169 |
qed "DERIV_cmult2";
|
|
170 |
|
|
171 |
Goal "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)";
|
|
172 |
by (rtac lemma_DERIV_subst 1 THEN rtac DERIV_pow 1);
|
|
173 |
by (Simp_tac 1);
|
|
174 |
qed "DERIV_pow2";
|
|
175 |
Addsimps [DERIV_pow2,DERIV_pow];
|
|
176 |
|
|
177 |
Goal "ALL n. DERIV (%x. (x ^ (Suc n) * poly p x)) x :> \
|
|
178 |
\ x ^ n * poly (pderiv_aux (Suc n) p) x ";
|
|
179 |
by (induct_tac "p" 1);
|
|
180 |
by (auto_tac (claset() addSIs [DERIV_add,DERIV_cmult2],simpset() addsimps
|
|
181 |
[pderiv_def,real_add_mult_distrib2,real_mult_assoc RS sym] delsimps
|
|
182 |
[realpow_Suc]));
|
|
183 |
by (rtac (real_mult_commute RS subst) 1);
|
|
184 |
by (simp_tac (simpset() delsimps [realpow_Suc]) 1);
|
|
185 |
by (asm_full_simp_tac (simpset() addsimps [real_mult_commute,realpow_Suc RS sym]
|
|
186 |
delsimps [realpow_Suc]) 1);
|
|
187 |
qed "lemma_DERIV_poly1";
|
|
188 |
|
|
189 |
Goal "DERIV (%x. (x ^ (Suc n) * poly p x)) x :> \
|
|
190 |
\ x ^ n * poly (pderiv_aux (Suc n) p) x ";
|
|
191 |
by (simp_tac (simpset() addsimps [lemma_DERIV_poly1] delsimps [realpow_Suc]) 1);
|
|
192 |
qed "lemma_DERIV_poly";
|
|
193 |
|
|
194 |
Goal "DERIV f x :> D ==> DERIV (%x. a + f x) x :> D";
|
|
195 |
by (rtac lemma_DERIV_subst 1 THEN rtac DERIV_add 1);
|
|
196 |
by Auto_tac;
|
|
197 |
qed "DERIV_add_const";
|
|
198 |
|
|
199 |
Goal "DERIV (%x. poly p x) x :> poly (pderiv p) x";
|
|
200 |
by (induct_tac "p" 1);
|
|
201 |
by (auto_tac (claset(),simpset() addsimps [pderiv_Cons]));
|
|
202 |
by (rtac DERIV_add_const 1);
|
|
203 |
by (rtac lemma_DERIV_subst 1);
|
|
204 |
by (rtac (full_simplify (simpset())
|
|
205 |
(read_instantiate [("n","0")] lemma_DERIV_poly)) 1);
|
|
206 |
by (simp_tac (simpset() addsimps [CLAIM "1 = 1"]) 1);
|
|
207 |
qed "poly_DERIV";
|
|
208 |
Addsimps [poly_DERIV];
|
|
209 |
|
|
210 |
|
|
211 |
(*-------------------------------------------------------------------------*)
|
|
212 |
(* Consequences of the derivative theorem above *)
|
|
213 |
(*-------------------------------------------------------------------------*)
|
|
214 |
|
|
215 |
Goalw [differentiable_def] "(%x. poly p x) differentiable x";
|
|
216 |
by (blast_tac (claset() addIs [poly_DERIV]) 1);
|
|
217 |
qed "poly_differentiable";
|
|
218 |
Addsimps [poly_differentiable];
|
|
219 |
|
|
220 |
Goal "isCont (%x. poly p x) x";
|
|
221 |
by (rtac (poly_DERIV RS DERIV_isCont) 1);
|
|
222 |
qed "poly_isCont";
|
|
223 |
Addsimps [poly_isCont];
|
|
224 |
|
|
225 |
Goal "[| a < b; poly p a < 0; 0 < poly p b |] \
|
|
226 |
\ ==> EX x. a < x & x < b & (poly p x = 0)";
|
|
227 |
by (cut_inst_tac [("f","%x. poly p x"),("a","a"),("b","b"),("y","0")]
|
|
228 |
IVT_objl 1);
|
|
229 |
by (auto_tac (claset(),simpset() addsimps [real_le_less]));
|
|
230 |
qed "poly_IVT_pos";
|
|
231 |
|
|
232 |
Goal "[| a < b; 0 < poly p a; poly p b < 0 |] \
|
|
233 |
\ ==> EX x. a < x & x < b & (poly p x = 0)";
|
|
234 |
by (blast_tac (claset() addIs [full_simplify (simpset()
|
|
235 |
addsimps [poly_minus, rename_numerals real_minus_zero_less_iff2])
|
|
236 |
(read_instantiate [("p","-- p")] poly_IVT_pos)]) 1);
|
|
237 |
qed "poly_IVT_neg";
|
|
238 |
|
|
239 |
Goal "a < b ==> \
|
|
240 |
\ EX x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)";
|
|
241 |
by (dres_inst_tac [("f","poly p")] MVT 1);
|
|
242 |
by Auto_tac;
|
|
243 |
by (res_inst_tac [("x","z")] exI 1);
|
|
244 |
by (auto_tac (claset() addDs [ARITH_PROVE
|
|
245 |
"[| a < z; z < b |] ==> (b - (a::real)) ~= 0"],simpset()
|
|
246 |
addsimps [real_mult_left_cancel,poly_DERIV RS DERIV_unique]));
|
|
247 |
qed "poly_MVT";
|
|
248 |
|
|
249 |
|
|
250 |
(*-------------------------------------------------------------------------*)
|
|
251 |
(* Lemmas for Derivatives *)
|
|
252 |
(*-------------------------------------------------------------------------*)
|
|
253 |
|
|
254 |
Goal "ALL p2 n. poly (pderiv_aux n (p1 +++ p2)) x = \
|
|
255 |
\ poly (pderiv_aux n p1 +++ pderiv_aux n p2) x";
|
|
256 |
by (induct_tac "p1" 1);
|
|
257 |
by (Step_tac 2);
|
|
258 |
by (case_tac "p2" 2);
|
|
259 |
by (auto_tac (claset(),simpset() addsimps [real_add_mult_distrib2]));
|
|
260 |
qed "lemma_poly_pderiv_aux_add";
|
|
261 |
|
|
262 |
Goal "poly (pderiv_aux n (p1 +++ p2)) x = \
|
|
263 |
\ poly (pderiv_aux n p1 +++ pderiv_aux n p2) x";
|
|
264 |
by (simp_tac (simpset() addsimps [lemma_poly_pderiv_aux_add]) 1);
|
|
265 |
qed "poly_pderiv_aux_add";
|
|
266 |
|
|
267 |
Goal "ALL n. poly (pderiv_aux n (c %* p) ) x = poly (c %* pderiv_aux n p) x";
|
|
268 |
by (induct_tac "p" 1);
|
|
269 |
by (auto_tac (claset(),simpset() addsimps [poly_cmult] @ real_mult_ac));
|
|
270 |
qed "lemma_poly_pderiv_aux_cmult";
|
|
271 |
|
|
272 |
Goal "poly (pderiv_aux n (c %* p) ) x = poly (c %* pderiv_aux n p) x";
|
|
273 |
by (simp_tac (simpset() addsimps [lemma_poly_pderiv_aux_cmult]) 1);
|
|
274 |
qed "poly_pderiv_aux_cmult";
|
|
275 |
|
|
276 |
Goalw [poly_minus_def]
|
|
277 |
"poly (pderiv_aux n (-- p)) x = poly (-- pderiv_aux n p) x";
|
|
278 |
by (simp_tac (simpset() addsimps [poly_pderiv_aux_cmult]) 1);
|
|
279 |
qed "poly_pderiv_aux_minus";
|
|
280 |
|
|
281 |
Goal "ALL n. poly (pderiv_aux (Suc n) p) x = poly ((pderiv_aux n p) +++ p) x";
|
|
282 |
by (induct_tac "p" 1);
|
|
283 |
by (auto_tac (claset(),simpset() addsimps [real_of_nat_Suc,
|
|
284 |
real_add_mult_distrib]));
|
|
285 |
qed "lemma_poly_pderiv_aux_mult1";
|
|
286 |
|
|
287 |
Goal "poly (pderiv_aux (Suc n) p) x = poly ((pderiv_aux n p) +++ p) x";
|
|
288 |
by (simp_tac (simpset() addsimps [lemma_poly_pderiv_aux_mult1]) 1);
|
|
289 |
qed "lemma_poly_pderiv_aux_mult";
|
|
290 |
|
|
291 |
Goal "ALL q. poly (pderiv (p +++ q)) x = poly (pderiv p +++ pderiv q) x";
|
|
292 |
by (induct_tac "p" 1);
|
|
293 |
by (Step_tac 2);
|
|
294 |
by (case_tac "q" 2);
|
|
295 |
by (auto_tac (claset(),simpset() addsimps [poly_pderiv_aux_add,poly_add,
|
|
296 |
pderiv_def]));
|
|
297 |
qed "lemma_poly_pderiv_add";
|
|
298 |
|
|
299 |
Goal "poly (pderiv (p +++ q)) x = poly (pderiv p +++ pderiv q) x";
|
|
300 |
by (simp_tac (simpset() addsimps [lemma_poly_pderiv_add]) 1);
|
|
301 |
qed "poly_pderiv_add";
|
|
302 |
|
|
303 |
Goal "poly (pderiv (c %* p)) x = poly (c %* (pderiv p)) x";
|
|
304 |
by (induct_tac "p" 1);
|
|
305 |
by (auto_tac (claset(),simpset() addsimps [poly_pderiv_aux_cmult,poly_cmult,
|
|
306 |
pderiv_def]));
|
|
307 |
qed "poly_pderiv_cmult";
|
|
308 |
|
|
309 |
Goalw [poly_minus_def] "poly (pderiv (--p)) x = poly (--(pderiv p)) x";
|
|
310 |
by (simp_tac (simpset() addsimps [poly_pderiv_cmult]) 1);
|
|
311 |
qed "poly_pderiv_minus";
|
|
312 |
|
|
313 |
Goalw [pderiv_def]
|
|
314 |
"poly (pderiv (h#t)) x = poly ((0 # (pderiv t)) +++ t) x";
|
|
315 |
by (induct_tac "t" 1);
|
|
316 |
by (auto_tac (claset(),simpset() addsimps [poly_add,
|
|
317 |
lemma_poly_pderiv_aux_mult]));
|
|
318 |
qed "lemma_poly_mult_pderiv";
|
|
319 |
|
|
320 |
Goal "ALL q. poly (pderiv (p *** q)) x = \
|
|
321 |
\ poly (p *** (pderiv q) +++ q *** (pderiv p)) x";
|
|
322 |
by (induct_tac "p" 1);
|
|
323 |
by (auto_tac (claset(),simpset() addsimps [poly_add,poly_cmult,
|
|
324 |
poly_pderiv_cmult,poly_pderiv_add,poly_mult]));
|
|
325 |
by (rtac (lemma_poly_mult_pderiv RS ssubst) 1);
|
|
326 |
by (rtac (lemma_poly_mult_pderiv RS ssubst) 1);
|
|
327 |
by (rtac (poly_add RS ssubst) 1);
|
|
328 |
by (rtac (poly_add RS ssubst) 1);
|
|
329 |
by (asm_simp_tac (simpset() addsimps [poly_mult,real_add_mult_distrib2]
|
|
330 |
@ real_add_ac @ real_mult_ac) 1);
|
|
331 |
qed "poly_pderiv_mult";
|
|
332 |
|
|
333 |
Goal "poly (pderiv (p %^ (Suc n))) x = \
|
|
334 |
\ poly ((real (Suc n)) %* (p %^ n) *** pderiv p ) x";
|
|
335 |
by (induct_tac "n" 1);
|
|
336 |
by (auto_tac (claset(),simpset() addsimps [poly_add,poly_pderiv_cmult,
|
|
337 |
poly_cmult,poly_pderiv_mult,real_of_nat_zero,poly_mult,
|
|
338 |
real_of_nat_Suc,real_add_mult_distrib2,real_add_mult_distrib]
|
|
339 |
@ real_mult_ac));
|
|
340 |
qed "poly_pderiv_exp";
|
|
341 |
|
|
342 |
Goal "poly (pderiv ([-a, 1] %^ (Suc n))) x = \
|
|
343 |
\ poly (real (Suc n) %* ([-a, 1] %^ n)) x";
|
|
344 |
by (simp_tac (simpset() addsimps [poly_pderiv_exp,poly_mult]
|
|
345 |
delsimps [pexp_Suc]) 1);
|
|
346 |
by (simp_tac (simpset() addsimps [poly_cmult,pderiv_def]) 1);
|
|
347 |
qed "poly_pderiv_exp_prime";
|
|
348 |
|
|
349 |
(* ----------------------------------------------------------------------- *)
|
|
350 |
(* Key property that f(a) = 0 ==> (x - a) divides p(x). *)
|
|
351 |
(* ----------------------------------------------------------------------- *)
|
|
352 |
|
|
353 |
Goal "ALL h. EX q r. h#t = [r] +++ [-a, 1] *** q";
|
|
354 |
by (induct_tac "t" 1);
|
|
355 |
by (Step_tac 1);
|
|
356 |
by (res_inst_tac [("x","[]")] exI 1);
|
|
357 |
by (res_inst_tac [("x","h")] exI 1);
|
|
358 |
by (Simp_tac 1);
|
|
359 |
by (dres_inst_tac [("x","aa")] spec 1);
|
|
360 |
by (Step_tac 1);
|
|
361 |
by (res_inst_tac [("x","r#q")] exI 1);
|
|
362 |
by (res_inst_tac [("x","a*r + h")] exI 1);
|
|
363 |
by (case_tac "q" 1);
|
|
364 |
by (auto_tac (claset(),simpset() addsimps [real_minus_mult_eq1 RS sym]));
|
|
365 |
qed "lemma_poly_linear_rem";
|
|
366 |
|
|
367 |
Goal "EX q r. h#t = [r] +++ [-a, 1] *** q";
|
|
368 |
by (cut_inst_tac [("t","t"),("a","a")] lemma_poly_linear_rem 1);
|
|
369 |
by Auto_tac;
|
|
370 |
qed "poly_linear_rem";
|
|
371 |
|
|
372 |
|
|
373 |
Goal "(poly p a = 0) = ((p = []) | (EX q. p = [-a, 1] *** q))";
|
|
374 |
by (auto_tac (claset(),simpset() addsimps [poly_add,poly_cmult,
|
|
375 |
real_add_mult_distrib2]));
|
|
376 |
by (case_tac "p" 1);
|
|
377 |
by (cut_inst_tac [("h","aa"),("t","list"),("a","a")] poly_linear_rem 2);
|
|
378 |
by (Step_tac 2);
|
|
379 |
by (case_tac "q" 1);
|
|
380 |
by Auto_tac;
|
|
381 |
by (dres_inst_tac [("x","[]")] spec 1);
|
|
382 |
by (Asm_full_simp_tac 1);
|
|
383 |
by (auto_tac (claset(),simpset() addsimps [poly_add,poly_cmult,
|
|
384 |
real_add_assoc]));
|
|
385 |
by (dres_inst_tac [("x","aa#lista")] spec 1);
|
|
386 |
by Auto_tac;
|
|
387 |
qed "poly_linear_divides";
|
|
388 |
|
|
389 |
Goal "ALL h k a. length (k %* p +++ (h # (a %* p))) = Suc (length p)";
|
|
390 |
by (induct_tac "p" 1);
|
|
391 |
by Auto_tac;
|
|
392 |
qed "lemma_poly_length_mult";
|
|
393 |
Addsimps [lemma_poly_length_mult];
|
|
394 |
|
|
395 |
Goal "ALL h k. length (k %* p +++ (h # p)) = Suc (length p)";
|
|
396 |
by (induct_tac "p" 1);
|
|
397 |
by Auto_tac;
|
|
398 |
qed "lemma_poly_length_mult2";
|
|
399 |
Addsimps [lemma_poly_length_mult2];
|
|
400 |
|
|
401 |
Goal "length([-a ,1] *** q) = Suc (length q)";
|
|
402 |
by Auto_tac;
|
|
403 |
qed "poly_length_mult";
|
|
404 |
Addsimps [poly_length_mult];
|
|
405 |
|
|
406 |
|
|
407 |
(*-------------------------------------------------------------------------*)
|
|
408 |
(* Polynomial length *)
|
|
409 |
(*-------------------------------------------------------------------------*)
|
|
410 |
|
|
411 |
Goal "length (a %* p) = length p";
|
|
412 |
by (induct_tac "p" 1);
|
|
413 |
by Auto_tac;
|
|
414 |
qed "poly_cmult_length";
|
|
415 |
Addsimps [poly_cmult_length];
|
|
416 |
|
|
417 |
Goal "length (p1 +++ p2) = (if (length( p1 ) < length( p2 )) \
|
|
418 |
\ then (length( p2 )) else (length( p1) ))";
|
|
419 |
by (subgoal_tac "ALL p2. length (p1 +++ p2) = (if (length( p1 ) < \
|
|
420 |
\ length( p2 )) then (length( p2 )) else (length( p1) ))" 1);
|
|
421 |
by (induct_tac "p1" 2);
|
|
422 |
by (Simp_tac 2);
|
|
423 |
by (Simp_tac 2);
|
|
424 |
by (Step_tac 2);
|
|
425 |
by (Asm_full_simp_tac 2);
|
|
426 |
by (arith_tac 2);
|
|
427 |
by (Asm_full_simp_tac 2);
|
|
428 |
by (arith_tac 2);
|
|
429 |
by (induct_tac "p2" 1);
|
|
430 |
by (Asm_full_simp_tac 1);
|
|
431 |
by (Asm_full_simp_tac 1);
|
|
432 |
qed "poly_add_length";
|
|
433 |
|
|
434 |
Goal "length([a,b] *** p) = Suc (length p)";
|
|
435 |
by (asm_full_simp_tac (simpset() addsimps [poly_cmult_length,
|
|
436 |
poly_add_length]) 1);
|
|
437 |
qed "poly_root_mult_length";
|
|
438 |
Addsimps [poly_root_mult_length];
|
|
439 |
|
|
440 |
Goal "(poly (p *** q) x ~= poly [] x) = \
|
|
441 |
\ (poly p x ~= poly [] x & poly q x ~= poly [] x)";
|
|
442 |
by (auto_tac (claset(),simpset() addsimps [poly_mult,rename_numerals
|
|
443 |
real_mult_not_zero]));
|
|
444 |
qed "poly_mult_not_eq_poly_Nil";
|
|
445 |
Addsimps [poly_mult_not_eq_poly_Nil];
|
|
446 |
|
|
447 |
Goal "(poly (p *** q) x = 0) = (poly p x = 0 | poly q x = 0)";
|
|
448 |
by (auto_tac (claset() addDs [CLAIM "x * y = 0 ==> x = 0 | y = (0::real)"],
|
|
449 |
simpset() addsimps [poly_mult]));
|
|
450 |
qed "poly_mult_eq_zero_disj";
|
|
451 |
|
|
452 |
(*-------------------------------------------------------------------------*)
|
|
453 |
(* Normalisation Properties *)
|
|
454 |
(*-------------------------------------------------------------------------*)
|
|
455 |
|
|
456 |
Goal "(pnormalize p = []) --> (poly p x = 0)";
|
|
457 |
by (induct_tac "p" 1);
|
|
458 |
by Auto_tac;
|
|
459 |
qed "poly_normalized_nil";
|
|
460 |
|
|
461 |
(*-------------------------------------------------------------------------*)
|
|
462 |
(* A nontrivial polynomial of degree n has no more than n roots *)
|
|
463 |
(*-------------------------------------------------------------------------*)
|
|
464 |
|
|
465 |
Goal
|
|
466 |
"ALL p x. (poly p x ~= poly [] x & length p = n \
|
|
467 |
\ --> (EX i. ALL x. (poly p x = (0::real)) --> (EX m. (m <= n & x = i m))))";
|
|
468 |
by (induct_tac "n" 1);
|
|
469 |
by (Step_tac 1);
|
|
470 |
by (rtac ccontr 1);
|
|
471 |
by (subgoal_tac "EX a. poly p a = 0" 1 THEN Step_tac 1);
|
|
472 |
by (dtac (poly_linear_divides RS iffD1) 1);
|
|
473 |
by (Step_tac 1);
|
|
474 |
by (dres_inst_tac [("x","q")] spec 1);
|
|
475 |
by (dres_inst_tac [("x","x")] spec 1);
|
|
476 |
by (asm_full_simp_tac (simpset() delsimps [poly_Nil,pmult_Cons]) 1);
|
|
477 |
by (etac exE 1);
|
|
478 |
by (dres_inst_tac [("x","%m. if m = Suc n then a else i m")] spec 1);
|
|
479 |
by (Step_tac 1);
|
|
480 |
by (dtac (poly_mult_eq_zero_disj RS iffD1) 1);
|
|
481 |
by (Step_tac 1);
|
|
482 |
by (dres_inst_tac [("x","Suc(length q)")] spec 1);
|
|
483 |
by (Asm_full_simp_tac 1);
|
|
484 |
by (dres_inst_tac [("x","xa")] spec 1 THEN Step_tac 1);
|
|
485 |
by (dres_inst_tac [("x","m")] spec 1);
|
|
486 |
by (Asm_full_simp_tac 1);
|
|
487 |
by (Blast_tac 1);
|
|
488 |
qed_spec_mp "poly_roots_index_lemma";
|
|
489 |
bind_thm ("poly_roots_index_lemma2",conjI RS poly_roots_index_lemma);
|
|
490 |
|
|
491 |
Goal "poly p x ~= poly [] x ==> \
|
|
492 |
\ EX i. ALL x. (poly p x = 0) --> (EX n. n <= length p & x = i n)";
|
|
493 |
by (blast_tac (claset() addIs [poly_roots_index_lemma2]) 1);
|
|
494 |
qed "poly_roots_index_length";
|
|
495 |
|
|
496 |
Goal "poly p x ~= poly [] x ==> \
|
|
497 |
\ EX N i. ALL x. (poly p x = 0) --> (EX n. (n::nat) < N & x = i n)";
|
|
498 |
by (dtac poly_roots_index_length 1 THEN Step_tac 1);
|
|
499 |
by (res_inst_tac [("x","Suc (length p)")] exI 1);
|
|
500 |
by (res_inst_tac [("x","i")] exI 1);
|
|
501 |
by (auto_tac (claset(),simpset() addsimps
|
|
502 |
[ARITH_PROVE "(m < Suc n) = (m <= n)"]));
|
|
503 |
qed "poly_roots_finite_lemma";
|
|
504 |
|
|
505 |
(* annoying proof *)
|
|
506 |
Goal "ALL P. (ALL x. P x --> (EX n. (n::nat) < N & x = (j::nat=>real) n)) \
|
|
507 |
\ --> (EX a. ALL x. P x --> x < a)";
|
|
508 |
by (induct_tac "N" 1);
|
|
509 |
by (Asm_full_simp_tac 1);
|
|
510 |
by (Step_tac 1);
|
|
511 |
by (dres_inst_tac [("x","%z. P z & (z ~= (j::nat=>real) n)")] spec 1);
|
|
512 |
by Auto_tac;
|
|
513 |
by (dres_inst_tac [("x","x")] spec 1);
|
|
514 |
by (Step_tac 1);
|
|
515 |
by (res_inst_tac [("x","na")] exI 1);
|
|
516 |
by (auto_tac (claset() addDs [ARITH_PROVE "na < Suc n ==> na = n | na < n"],
|
|
517 |
simpset()));
|
|
518 |
by (res_inst_tac [("x","abs a + abs(j n) + 1")] exI 1);
|
|
519 |
by (Step_tac 1);
|
|
520 |
by (dres_inst_tac [("x","x")] spec 1);
|
|
521 |
by (Step_tac 1);
|
|
522 |
by (dres_inst_tac [("x","j na")] spec 1);
|
|
523 |
by (Step_tac 1);
|
|
524 |
by (ALLGOALS(arith_tac));
|
|
525 |
qed_spec_mp "real_finite_lemma";
|
|
526 |
|
|
527 |
Goal "(poly p ~= poly []) = \
|
|
528 |
\ (EX N j. ALL x. poly p x = 0 --> (EX n. (n::nat) < N & x = j n))";
|
|
529 |
by (Step_tac 1);
|
|
530 |
by (etac swap 1 THEN rtac ext 1);
|
|
531 |
by (rtac ccontr 1);
|
|
532 |
by (clarify_tac (claset() addSDs [poly_roots_finite_lemma]) 1);
|
|
533 |
by (clarify_tac (claset() addSDs [real_finite_lemma]) 1);
|
|
534 |
by (dres_inst_tac [("x","a")] fun_cong 1);
|
|
535 |
by Auto_tac;
|
|
536 |
qed "poly_roots_finite";
|
|
537 |
|
|
538 |
(*-------------------------------------------------------------------------*)
|
|
539 |
(* Entirety and Cancellation for polynomials *)
|
|
540 |
(*-------------------------------------------------------------------------*)
|
|
541 |
|
|
542 |
Goal "[| poly p ~= poly [] ; poly q ~= poly [] |] \
|
|
543 |
\ ==> poly (p *** q) ~= poly []";
|
|
544 |
by (auto_tac (claset(),simpset() addsimps [poly_roots_finite]));
|
|
545 |
by (res_inst_tac [("x","N + Na")] exI 1);
|
|
546 |
by (res_inst_tac [("x","%n. if n < N then j n else ja (n - N)")] exI 1);
|
|
547 |
by (auto_tac (claset(),simpset() addsimps [poly_mult_eq_zero_disj]));
|
|
548 |
by (flexflex_tac THEN rotate_tac 1 1);
|
|
549 |
by (dtac spec 1 THEN Auto_tac);
|
|
550 |
qed "poly_entire_lemma";
|
|
551 |
|
|
552 |
Goal "(poly (p *** q) = poly []) = ((poly p = poly []) | (poly q = poly []))";
|
|
553 |
by (auto_tac (claset() addIs [ext] addDs [fun_cong],simpset()
|
|
554 |
addsimps [poly_entire_lemma,poly_mult]));
|
|
555 |
by (blast_tac (claset() addIs [ccontr] addDs [poly_entire_lemma,
|
|
556 |
poly_mult RS subst]) 1);
|
|
557 |
qed "poly_entire";
|
|
558 |
|
|
559 |
Goal "(poly (p *** q) ~= poly []) = ((poly p ~= poly []) & (poly q ~= poly []))";
|
|
560 |
by (asm_full_simp_tac (simpset() addsimps [poly_entire]) 1);
|
|
561 |
qed "poly_entire_neg";
|
|
562 |
|
|
563 |
Goal " (f = g) = (ALL x. f x = g x)";
|
|
564 |
by (auto_tac (claset() addSIs [ext],simpset()));
|
|
565 |
qed "fun_eq";
|
|
566 |
|
|
567 |
Goal "(poly (p +++ -- q) = poly []) = (poly p = poly q)";
|
|
568 |
by (auto_tac (claset(),simpset() addsimps [poly_add,poly_minus_def,
|
|
569 |
fun_eq,poly_cmult,ARITH_PROVE "(p + -q = 0) = (p = (q::real))"]));
|
|
570 |
qed "poly_add_minus_zero_iff";
|
|
571 |
|
|
572 |
Goal "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))";
|
|
573 |
by (auto_tac (claset(),simpset() addsimps [poly_add,poly_minus_def,
|
|
574 |
fun_eq,poly_mult,poly_cmult,real_add_mult_distrib2]));
|
|
575 |
qed "poly_add_minus_mult_eq";
|
|
576 |
|
|
577 |
Goal "(poly (p *** q) = poly (p *** r)) = (poly p = poly [] | poly q = poly r)";
|
|
578 |
by (res_inst_tac [("p1","p *** q")] (poly_add_minus_zero_iff RS subst) 1);
|
|
579 |
by (auto_tac (claset() addIs [ext], simpset() addsimps [poly_add_minus_mult_eq,
|
|
580 |
poly_entire,poly_add_minus_zero_iff]));
|
|
581 |
qed "poly_mult_left_cancel";
|
|
582 |
|
|
583 |
Goal "(x * y = 0) = (x = (0::real) | y = 0)";
|
|
584 |
by (auto_tac (claset() addDs [CLAIM "x * y = 0 ==> x = 0 | y = (0::real)"],
|
|
585 |
simpset()));
|
|
586 |
qed "real_mult_zero_disj_iff";
|
|
587 |
|
|
588 |
Goal "(poly (p %^ n) = poly []) = (poly p = poly [] & n ~= 0)";
|
|
589 |
by (simp_tac (simpset() addsimps [fun_eq]) 1);
|
|
590 |
by (rtac (CLAIM "((ALL x. P x) & Q) = (ALL x. P x & Q)" RS ssubst) 1);
|
|
591 |
by (rtac (CLAIM "f = g ==> (ALL x. f x) = (ALL x. g x)") 1);
|
|
592 |
by (rtac ext 1);
|
|
593 |
by (induct_tac "n" 1);
|
|
594 |
by (auto_tac (claset(),simpset() addsimps [poly_mult,
|
|
595 |
real_mult_zero_disj_iff]));
|
|
596 |
qed "poly_exp_eq_zero";
|
|
597 |
Addsimps [poly_exp_eq_zero];
|
|
598 |
|
|
599 |
Goal "poly [a,1] ~= poly []";
|
|
600 |
by (simp_tac (simpset() addsimps [fun_eq]) 1);
|
|
601 |
by (res_inst_tac [("x","1 - a")] exI 1);
|
|
602 |
by (Simp_tac 1);
|
|
603 |
qed "poly_prime_eq_zero";
|
|
604 |
Addsimps [poly_prime_eq_zero];
|
|
605 |
|
|
606 |
Goal "(poly ([a, 1] %^ n) ~= poly [])";
|
|
607 |
by Auto_tac;
|
|
608 |
qed "poly_exp_prime_eq_zero";
|
|
609 |
Addsimps [poly_exp_prime_eq_zero];
|
|
610 |
|
|
611 |
(*-------------------------------------------------------------------------*)
|
|
612 |
(* A more constructive notion of polynomials being trivial *)
|
|
613 |
(*-------------------------------------------------------------------------*)
|
|
614 |
|
|
615 |
Goal "poly (h # t) = poly [] ==> h = 0 & poly t = poly []";
|
|
616 |
by (asm_full_simp_tac (simpset() addsimps [fun_eq]) 1);
|
|
617 |
by (case_tac "h = 0" 1);
|
|
618 |
by (dres_inst_tac [("x","0")] spec 2);
|
|
619 |
by (rtac conjI 1);
|
|
620 |
by (rtac ((simplify (simpset()) (read_instantiate [("g","poly []")] fun_eq))
|
|
621 |
RS iffD1) 2 THEN rtac ccontr 2);
|
|
622 |
by (auto_tac (claset(),simpset() addsimps [poly_roots_finite,
|
|
623 |
real_mult_zero_disj_iff]));
|
|
624 |
by (dtac real_finite_lemma 1 THEN Step_tac 1);
|
|
625 |
by (REPEAT(dres_inst_tac [("x","abs a + 1")] spec 1));
|
|
626 |
by (arith_tac 1);
|
|
627 |
qed "poly_zero_lemma";
|
|
628 |
|
|
629 |
Goal "(poly p = poly []) = list_all (%c. c = 0) p";
|
|
630 |
by (induct_tac "p" 1);
|
|
631 |
by (Asm_full_simp_tac 1);
|
|
632 |
by (rtac iffI 1);
|
|
633 |
by (dtac poly_zero_lemma 1);
|
|
634 |
by Auto_tac;
|
|
635 |
qed "poly_zero";
|
|
636 |
|
|
637 |
Addsimps [real_mult_zero_disj_iff];
|
|
638 |
Goal "ALL n. (list_all (%c. c = 0) (pderiv_aux (Suc n) p) = \
|
|
639 |
\ list_all (%c. c = 0) p)";
|
|
640 |
by (induct_tac "p" 1);
|
|
641 |
by Auto_tac;
|
|
642 |
qed_spec_mp "pderiv_aux_iszero";
|
|
643 |
Addsimps [pderiv_aux_iszero];
|
|
644 |
|
|
645 |
Goal "(number_of n :: nat) ~= 0 \
|
|
646 |
\ ==> (list_all (%c. c = 0) (pderiv_aux (number_of n) p) = \
|
|
647 |
\ list_all (%c. c = 0) p)";
|
|
648 |
by (res_inst_tac [("n1","number_of n"),("m1","0")] (less_imp_Suc_add RS exE) 1);
|
|
649 |
by (Force_tac 1);
|
|
650 |
by (res_inst_tac [("n1","0 + x")] (pderiv_aux_iszero RS subst) 1);
|
|
651 |
by (asm_simp_tac (simpset() delsimps [pderiv_aux_iszero]) 1);
|
|
652 |
qed "pderiv_aux_iszero_num";
|
|
653 |
|
|
654 |
Goal "poly (pderiv p) = poly [] --> (EX h. poly p = poly [h])";
|
|
655 |
by (asm_full_simp_tac (simpset() addsimps [poly_zero]) 1);
|
|
656 |
by (induct_tac "p" 1);
|
|
657 |
by (Force_tac 1);
|
|
658 |
by (asm_full_simp_tac (simpset() addsimps [pderiv_Cons,
|
|
659 |
pderiv_aux_iszero_num] delsimps [poly_Cons]) 1);
|
|
660 |
by (auto_tac (claset(),simpset() addsimps [poly_zero RS sym]));
|
|
661 |
qed_spec_mp "pderiv_iszero";
|
|
662 |
|
|
663 |
Goal "poly p = poly [] --> (poly (pderiv p) = poly [])";
|
|
664 |
by (asm_full_simp_tac (simpset() addsimps [poly_zero]) 1);
|
|
665 |
by (induct_tac "p" 1);
|
|
666 |
by (Force_tac 1);
|
|
667 |
by (asm_full_simp_tac (simpset() addsimps [pderiv_Cons,
|
|
668 |
pderiv_aux_iszero_num] delsimps [poly_Cons]) 1);
|
|
669 |
qed "pderiv_zero_obj";
|
|
670 |
|
|
671 |
Goal "poly p = poly [] ==> (poly (pderiv p) = poly [])";
|
|
672 |
by (blast_tac (claset() addEs [pderiv_zero_obj RS impE]) 1);
|
|
673 |
qed "pderiv_zero";
|
|
674 |
Addsimps [pderiv_zero];
|
|
675 |
|
|
676 |
Goal "poly p = poly q ==> (poly (pderiv p) = poly (pderiv q))";
|
|
677 |
by (cut_inst_tac [("p","p +++ --q")] pderiv_zero_obj 1);
|
|
678 |
by (auto_tac (claset() addIs [ ARITH_PROVE "x + - y = 0 ==> x = (y::real)"],
|
|
679 |
simpset() addsimps [fun_eq,poly_add,poly_minus,poly_pderiv_add,
|
|
680 |
poly_pderiv_minus] delsimps [pderiv_zero]));
|
|
681 |
qed "poly_pderiv_welldef";
|
|
682 |
|
|
683 |
(* ------------------------------------------------------------------------- *)
|
|
684 |
(* Basics of divisibility. *)
|
|
685 |
(* ------------------------------------------------------------------------- *)
|
|
686 |
|
|
687 |
Goal "([a, 1] divides (p *** q)) = ([a, 1] divides p | [a, 1] divides q)";
|
|
688 |
by (auto_tac (claset(),simpset() addsimps [divides_def,fun_eq,poly_mult,
|
|
689 |
poly_add,poly_cmult,real_add_mult_distrib RS sym]));
|
|
690 |
by (dres_inst_tac [("x","-a")] spec 1);
|
|
691 |
by (auto_tac (claset(),simpset() addsimps [poly_linear_divides,poly_add,
|
|
692 |
poly_cmult,real_add_mult_distrib RS sym]));
|
|
693 |
by (res_inst_tac [("x","qa *** q")] exI 1);
|
|
694 |
by (res_inst_tac [("x","p *** qa")] exI 2);
|
|
695 |
by (auto_tac (claset(),simpset() addsimps [poly_add,poly_mult,
|
|
696 |
poly_cmult] @ real_mult_ac));
|
|
697 |
qed "poly_primes";
|
|
698 |
|
|
699 |
Goalw [divides_def] "p divides p";
|
|
700 |
by (res_inst_tac [("x","[1]")] exI 1);
|
|
701 |
by (auto_tac (claset(),simpset() addsimps [poly_mult,fun_eq]));
|
|
702 |
qed "poly_divides_refl";
|
|
703 |
Addsimps [poly_divides_refl];
|
|
704 |
|
|
705 |
Goalw [divides_def] "[| p divides q; q divides r |] ==> p divides r";
|
|
706 |
by (Step_tac 1);
|
|
707 |
by (res_inst_tac [("x","qa *** qaa")] exI 1);
|
|
708 |
by (auto_tac (claset(),simpset() addsimps [poly_mult,fun_eq,
|
|
709 |
real_mult_assoc]));
|
|
710 |
qed "poly_divides_trans";
|
|
711 |
|
|
712 |
Goal "(m::nat) <= n = (EX d. n = m + d)";
|
|
713 |
by (auto_tac (claset(),simpset() addsimps [le_eq_less_or_eq,
|
|
714 |
less_iff_Suc_add]));
|
|
715 |
qed "le_iff_add";
|
|
716 |
|
|
717 |
Goal "m <= n ==> (p %^ m) divides (p %^ n)";
|
|
718 |
by (auto_tac (claset(),simpset() addsimps [le_iff_add]));
|
|
719 |
by (induct_tac "d" 1);
|
|
720 |
by (rtac poly_divides_trans 2);
|
|
721 |
by (auto_tac (claset(),simpset() addsimps [divides_def]));
|
|
722 |
by (res_inst_tac [("x","p")] exI 1);
|
|
723 |
by (auto_tac (claset(),simpset() addsimps [poly_mult,fun_eq]
|
|
724 |
@ real_mult_ac));
|
|
725 |
qed "poly_divides_exp";
|
|
726 |
|
|
727 |
Goal "[| (p %^ n) divides q; m <= n |] ==> (p %^ m) divides q";
|
|
728 |
by (blast_tac (claset() addIs [poly_divides_exp,poly_divides_trans]) 1);
|
|
729 |
qed "poly_exp_divides";
|
|
730 |
|
|
731 |
Goalw [divides_def]
|
|
732 |
"[| p divides q; p divides r |] ==> p divides (q +++ r)";
|
|
733 |
by Auto_tac;
|
|
734 |
by (res_inst_tac [("x","qa +++ qaa")] exI 1);
|
|
735 |
by (auto_tac (claset(),simpset() addsimps [poly_add,fun_eq,poly_mult,
|
|
736 |
real_add_mult_distrib2]));
|
|
737 |
qed "poly_divides_add";
|
|
738 |
|
|
739 |
Goalw [divides_def]
|
|
740 |
"[| p divides q; p divides (q +++ r) |] ==> p divides r";
|
|
741 |
by Auto_tac;
|
|
742 |
by (res_inst_tac [("x","qaa +++ -- qa")] exI 1);
|
|
743 |
by (auto_tac (claset(),simpset() addsimps [poly_add,fun_eq,poly_mult,
|
|
744 |
poly_minus,real_add_mult_distrib2,real_minus_mult_eq2 RS sym,
|
|
745 |
ARITH_PROVE "(x = y + -z) = (z + x = (y::real))"]));
|
|
746 |
qed "poly_divides_diff";
|
|
747 |
|
|
748 |
Goal "[| p divides r; p divides (q +++ r) |] ==> p divides q";
|
|
749 |
by (etac poly_divides_diff 1);
|
|
750 |
by (auto_tac (claset(),simpset() addsimps [poly_add,fun_eq,poly_mult,
|
|
751 |
divides_def] @ real_add_ac));
|
|
752 |
qed "poly_divides_diff2";
|
|
753 |
|
|
754 |
Goalw [divides_def] "poly p = poly [] ==> q divides p";
|
|
755 |
by (auto_tac (claset(),simpset() addsimps [fun_eq,poly_mult]));
|
|
756 |
qed "poly_divides_zero";
|
|
757 |
|
|
758 |
Goalw [divides_def] "q divides []";
|
|
759 |
by (res_inst_tac [("x","[]")] exI 1);
|
|
760 |
by (auto_tac (claset(),simpset() addsimps [fun_eq]));
|
|
761 |
qed "poly_divides_zero2";
|
|
762 |
Addsimps [poly_divides_zero2];
|
|
763 |
|
|
764 |
(* ------------------------------------------------------------------------- *)
|
|
765 |
(* At last, we can consider the order of a root. *)
|
|
766 |
(* ------------------------------------------------------------------------- *)
|
|
767 |
|
|
768 |
(* FIXME: Tidy up *)
|
|
769 |
Goal "[| length p = d; poly p ~= poly [] |] \
|
|
770 |
\ ==> EX n. ([-a, 1] %^ n) divides p & \
|
|
771 |
\ ~(([-a, 1] %^ (Suc n)) divides p)";
|
|
772 |
by (subgoal_tac "ALL p. length p = d & poly p ~= poly [] \
|
|
773 |
\ --> (EX n q. p = mulexp n [-a, 1] q & poly q a ~= 0)" 1);
|
|
774 |
by (induct_tac "d" 2);
|
|
775 |
by (asm_full_simp_tac (simpset() addsimps [fun_eq]) 2);
|
|
776 |
by (Step_tac 2);
|
|
777 |
by (case_tac "poly pa a = 0" 2);
|
|
778 |
by (dtac (poly_linear_divides RS iffD1) 2);
|
|
779 |
by (Step_tac 2);
|
|
780 |
by (dres_inst_tac [("x","q")] spec 2);
|
|
781 |
by (dtac (poly_entire_neg RS iffD1) 2);
|
|
782 |
by (Step_tac 2);
|
|
783 |
by (Force_tac 2 THEN Blast_tac 2);
|
|
784 |
by (res_inst_tac [("x","Suc na")] exI 2);
|
|
785 |
by (res_inst_tac [("x","qa")] exI 2);
|
|
786 |
by (asm_full_simp_tac (simpset() delsimps [pmult_Cons]) 2);
|
|
787 |
by (res_inst_tac [("x","0")] exI 2);
|
|
788 |
by (Force_tac 2);
|
|
789 |
by (dres_inst_tac [("x","p")] spec 1 THEN Step_tac 1);
|
|
790 |
by (res_inst_tac [("x","n")] exI 1 THEN Step_tac 1);
|
|
791 |
by (rewtac divides_def);
|
|
792 |
by (res_inst_tac [("x","q")] exI 1);
|
|
793 |
by (induct_tac "n" 1);
|
|
794 |
by (Simp_tac 1);
|
|
795 |
by (asm_simp_tac (simpset() addsimps [poly_add,poly_cmult,poly_mult,
|
|
796 |
real_add_mult_distrib2] @ real_mult_ac) 1);
|
|
797 |
by (Step_tac 1);
|
|
798 |
by (rotate_tac 2 1);
|
|
799 |
by (rtac swap 1 THEN assume_tac 2);
|
|
800 |
by (induct_tac "n" 1);
|
|
801 |
by (asm_full_simp_tac (simpset() delsimps [pmult_Cons,pexp_Suc]) 1);
|
|
802 |
by (eres_inst_tac [("Pa","poly q a = 0")] swap 1);
|
|
803 |
by (asm_full_simp_tac (simpset() addsimps [poly_add,poly_cmult,
|
|
804 |
real_minus_mult_eq1 RS sym]) 1);
|
|
805 |
by (rtac (pexp_Suc RS ssubst) 1);
|
|
806 |
by (rtac ccontr 1);
|
|
807 |
by (asm_full_simp_tac (simpset() addsimps [poly_mult_left_cancel,
|
|
808 |
poly_mult_assoc] delsimps [pmult_Cons,pexp_Suc]) 1);
|
|
809 |
qed "poly_order_exists";
|
|
810 |
|
|
811 |
Goalw [divides_def] "[1] divides p";
|
|
812 |
by Auto_tac;
|
|
813 |
qed "poly_one_divides";
|
|
814 |
Addsimps [poly_one_divides];
|
|
815 |
|
|
816 |
Goal "poly p ~= poly [] \
|
|
817 |
\ ==> EX! n. ([-a, 1] %^ n) divides p & \
|
|
818 |
\ ~(([-a, 1] %^ (Suc n)) divides p)";
|
|
819 |
by (auto_tac (claset() addIs [poly_order_exists],
|
|
820 |
simpset() addsimps [less_linear] delsimps [pmult_Cons,pexp_Suc]));
|
|
821 |
by (cut_inst_tac [("m","y"),("n","n")] less_linear 1);
|
|
822 |
by (dres_inst_tac [("m","n")] poly_exp_divides 1);
|
|
823 |
by (auto_tac (claset() addDs [ARITH_PROVE "n < m ==> Suc n <= m" RSN
|
|
824 |
(2,poly_exp_divides)],simpset() delsimps [pmult_Cons,pexp_Suc]));
|
|
825 |
qed "poly_order";
|
|
826 |
|
|
827 |
(* ------------------------------------------------------------------------- *)
|
|
828 |
(* Order *)
|
|
829 |
(* ------------------------------------------------------------------------- *)
|
|
830 |
|
|
831 |
Goal "[| n = (@n. P n); EX! n. P n |] ==> P n";
|
|
832 |
by (blast_tac (claset() addIs [someI2]) 1);
|
|
833 |
qed "some1_equalityD";
|
|
834 |
|
|
835 |
Goalw [order_def]
|
|
836 |
"(([-a, 1] %^ n) divides p & \
|
|
837 |
\ ~(([-a, 1] %^ (Suc n)) divides p)) = \
|
|
838 |
\ ((n = order a p) & ~(poly p = poly []))";
|
|
839 |
by (rtac iffI 1);
|
|
840 |
by (blast_tac (claset() addDs [poly_divides_zero]
|
|
841 |
addSIs [some1_equality RS sym, poly_order]) 1);
|
|
842 |
by (blast_tac (claset() addSIs [poly_order RSN (2,some1_equalityD)]) 1);
|
|
843 |
qed "order";
|
|
844 |
|
|
845 |
Goal "[| poly p ~= poly [] |] \
|
|
846 |
\ ==> ([-a, 1] %^ (order a p)) divides p & \
|
|
847 |
\ ~(([-a, 1] %^ (Suc(order a p))) divides p)";
|
|
848 |
by (asm_full_simp_tac (simpset() addsimps [order] delsimps [pexp_Suc]) 1);
|
|
849 |
qed "order2";
|
|
850 |
|
|
851 |
Goal "[| poly p ~= poly []; ([-a, 1] %^ n) divides p; \
|
|
852 |
\ ~(([-a, 1] %^ (Suc n)) divides p) \
|
|
853 |
\ |] ==> (n = order a p)";
|
|
854 |
by (cut_inst_tac [("p","p"),("a","a"),("n","n")] order 1);
|
|
855 |
by Auto_tac;
|
|
856 |
qed "order_unique";
|
|
857 |
|
|
858 |
Goal "(poly p ~= poly [] & ([-a, 1] %^ n) divides p & \
|
|
859 |
\ ~(([-a, 1] %^ (Suc n)) divides p)) \
|
|
860 |
\ ==> (n = order a p)";
|
|
861 |
by (blast_tac (claset() addIs [order_unique]) 1);
|
|
862 |
qed "order_unique_lemma";
|
|
863 |
|
|
864 |
Goal "poly p = poly q ==> order a p = order a q";
|
|
865 |
by (auto_tac (claset(),simpset() addsimps [fun_eq,divides_def,poly_mult,
|
|
866 |
order_def]));
|
|
867 |
qed "order_poly";
|
|
868 |
|
|
869 |
Goal "p %^ (Suc 0) = p";
|
|
870 |
by (induct_tac "p" 1);
|
|
871 |
by (auto_tac (claset(),simpset() addsimps [numeral_1_eq_1]));
|
|
872 |
qed "pexp_one";
|
|
873 |
Addsimps [pexp_one];
|
|
874 |
|
|
875 |
Goal "ALL p a. 0 < n & [- a, 1] %^ n divides p & \
|
|
876 |
\ ~ [- a, 1] %^ (Suc n) divides p \
|
|
877 |
\ --> poly p a = 0";
|
|
878 |
by (induct_tac "n" 1);
|
|
879 |
by (Blast_tac 1);
|
|
880 |
by (auto_tac (claset(),simpset() addsimps [divides_def,poly_mult]
|
|
881 |
delsimps [pmult_Cons]));
|
|
882 |
qed_spec_mp "lemma_order_root";
|
|
883 |
|
|
884 |
Goal "(poly p a = 0) = ((poly p = poly []) | order a p ~= 0)";
|
|
885 |
by (case_tac "poly p = poly []" 1);
|
|
886 |
by Auto_tac;
|
|
887 |
by (asm_full_simp_tac (simpset() addsimps [poly_linear_divides]
|
|
888 |
delsimps [pmult_Cons]) 1);
|
|
889 |
by (Step_tac 1);
|
|
890 |
by (ALLGOALS(dres_inst_tac [("a","a")] order2));
|
|
891 |
by (rtac ccontr 1);
|
|
892 |
by (asm_full_simp_tac (simpset() addsimps [divides_def,poly_mult,fun_eq]
|
|
893 |
delsimps [pmult_Cons]) 1);
|
|
894 |
by (Blast_tac 1);
|
|
895 |
by (blast_tac (claset() addIs [lemma_order_root]) 1);
|
|
896 |
qed "order_root";
|
|
897 |
|
|
898 |
Goal "(([-a, 1] %^ n) divides p) = ((poly p = poly []) | n <= order a p)";
|
|
899 |
by (case_tac "poly p = poly []" 1);
|
|
900 |
by Auto_tac;
|
|
901 |
by (asm_full_simp_tac (simpset() addsimps [divides_def,fun_eq,poly_mult]) 1);
|
|
902 |
by (res_inst_tac [("x","[]")] exI 1);
|
|
903 |
by (TRYALL(dres_inst_tac [("a","a")] order2));
|
|
904 |
by (auto_tac (claset() addIs [poly_exp_divides],simpset()
|
|
905 |
delsimps [pexp_Suc]));
|
|
906 |
qed "order_divides";
|
|
907 |
|
|
908 |
Goalw [divides_def]
|
|
909 |
"poly p ~= poly [] \
|
|
910 |
\ ==> EX q. (poly p = poly (([-a, 1] %^ (order a p)) *** q)) & \
|
|
911 |
\ ~([-a, 1] divides q)";
|
|
912 |
by (dres_inst_tac [("a","a")] order2 1);
|
|
913 |
by (asm_full_simp_tac (simpset() addsimps [divides_def]
|
|
914 |
delsimps [pexp_Suc,pmult_Cons]) 1);
|
|
915 |
by (Step_tac 1);
|
|
916 |
by (res_inst_tac [("x","q")] exI 1);
|
|
917 |
by (Step_tac 1);
|
|
918 |
by (dres_inst_tac [("x","qa")] spec 1);
|
|
919 |
by (auto_tac (claset(),simpset() addsimps [poly_mult,fun_eq,poly_exp]
|
|
920 |
@ real_mult_ac delsimps [pmult_Cons]));
|
|
921 |
qed "order_decomp";
|
|
922 |
|
|
923 |
(* ------------------------------------------------------------------------- *)
|
|
924 |
(* Important composition properties of orders. *)
|
|
925 |
(* ------------------------------------------------------------------------- *)
|
|
926 |
|
|
927 |
Goal "poly (p *** q) ~= poly [] \
|
|
928 |
\ ==> order a (p *** q) = order a p + order a q";
|
|
929 |
by (cut_inst_tac [("a","a"),("p","p***q"),("n","order a p + order a q")]
|
|
930 |
order 1);
|
|
931 |
by (auto_tac (claset(),simpset() addsimps [poly_entire] delsimps [pmult_Cons]));
|
|
932 |
by (REPEAT(dres_inst_tac [("a","a")] order2 1));
|
|
933 |
by (Step_tac 1);
|
|
934 |
by (asm_full_simp_tac (simpset() addsimps [divides_def,fun_eq,poly_exp_add,
|
|
935 |
poly_mult] delsimps [pmult_Cons]) 1);
|
|
936 |
by (Step_tac 1);
|
|
937 |
by (res_inst_tac [("x","qa *** qaa")] exI 1);
|
|
938 |
by (asm_full_simp_tac (simpset() addsimps [poly_mult] @ real_mult_ac
|
|
939 |
delsimps [pmult_Cons]) 1);
|
|
940 |
by (REPEAT(dres_inst_tac [("a","a")] order_decomp 1));
|
|
941 |
by (Step_tac 1);
|
|
942 |
by (subgoal_tac "[-a,1] divides (qa *** qaa)" 1);
|
|
943 |
by (asm_full_simp_tac (simpset() addsimps [poly_primes]
|
|
944 |
delsimps [pmult_Cons]) 1);
|
|
945 |
by (auto_tac (claset(),simpset() addsimps [divides_def]
|
|
946 |
delsimps [pmult_Cons]));
|
|
947 |
by (res_inst_tac [("x","qb")] exI 1);
|
|
948 |
by (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) = \
|
|
949 |
\ poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))" 1);
|
|
950 |
by (dtac (poly_mult_left_cancel RS iffD1) 1);
|
|
951 |
by (Force_tac 1);
|
|
952 |
by (subgoal_tac "poly ([-a, 1] %^ (order a q) *** \
|
|
953 |
\ ([-a, 1] %^ (order a p) *** (qa *** qaa))) = \
|
|
954 |
\ poly ([-a, 1] %^ (order a q) *** \
|
|
955 |
\ ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb)))" 1);
|
|
956 |
by (dtac (poly_mult_left_cancel RS iffD1) 1);
|
|
957 |
by (Force_tac 1);
|
|
958 |
by (asm_full_simp_tac (simpset() addsimps [fun_eq,poly_exp_add,poly_mult]
|
|
959 |
@ real_mult_ac delsimps [pmult_Cons]) 1);
|
|
960 |
qed "order_mult";
|
|
961 |
|
|
962 |
(* FIXME: too too long! *)
|
|
963 |
Goal "ALL p q a. 0 < n & \
|
|
964 |
\ poly (pderiv p) ~= poly [] & \
|
|
965 |
\ poly p = poly ([- a, 1] %^ n *** q) & ~ [- a, 1] divides q \
|
|
966 |
\ --> n = Suc (order a (pderiv p))";
|
|
967 |
by (induct_tac "n" 1);
|
|
968 |
by (Step_tac 1);
|
|
969 |
by (rtac order_unique_lemma 1 THEN rtac conjI 1);
|
|
970 |
by (assume_tac 1);
|
|
971 |
by (subgoal_tac "ALL r. r divides (pderiv p) = \
|
|
972 |
\ r divides (pderiv ([-a, 1] %^ Suc n *** q))" 1);
|
|
973 |
by (dtac poly_pderiv_welldef 2);
|
|
974 |
by (asm_full_simp_tac (simpset() addsimps [divides_def] delsimps [pmult_Cons,
|
|
975 |
pexp_Suc]) 2);
|
|
976 |
by (asm_full_simp_tac (simpset() delsimps [pmult_Cons,pexp_Suc]) 1);
|
|
977 |
by (rtac conjI 1);
|
|
978 |
by (asm_full_simp_tac (simpset() addsimps [divides_def,fun_eq]
|
|
979 |
delsimps [pmult_Cons,pexp_Suc]) 1);
|
|
980 |
by (res_inst_tac
|
|
981 |
[("x","[-a, 1] *** (pderiv q) +++ real (Suc n) %* q")] exI 1);
|
|
982 |
by (asm_full_simp_tac (simpset() addsimps [poly_pderiv_mult,
|
|
983 |
poly_pderiv_exp_prime,poly_add,poly_mult,poly_cmult,
|
|
984 |
real_add_mult_distrib2] @ real_mult_ac
|
|
985 |
delsimps [pmult_Cons,pexp_Suc]) 1);
|
|
986 |
by (asm_full_simp_tac (simpset() addsimps [poly_mult,real_add_mult_distrib2,
|
|
987 |
real_add_mult_distrib] @ real_mult_ac delsimps [pmult_Cons]) 1);
|
|
988 |
by (thin_tac "ALL r. \
|
|
989 |
\ r divides pderiv p = \
|
|
990 |
\ r divides pderiv ([- a, 1] %^ Suc n *** q)" 1);
|
|
991 |
by (rewtac divides_def);
|
|
992 |
by (simp_tac (simpset() addsimps [poly_pderiv_mult,
|
|
993 |
poly_pderiv_exp_prime,fun_eq,poly_add,poly_mult]
|
|
994 |
delsimps [pmult_Cons,pexp_Suc]) 1);
|
|
995 |
by (rtac swap 1 THEN assume_tac 1);
|
|
996 |
by (rotate_tac 3 1 THEN etac swap 1);
|
|
997 |
by (asm_full_simp_tac (simpset() delsimps [pmult_Cons,pexp_Suc]) 1);
|
|
998 |
by (Step_tac 1);
|
|
999 |
by (res_inst_tac [("x","inverse(real (Suc n)) %* (qa +++ --(pderiv q))")]
|
|
1000 |
exI 1);
|
|
1001 |
by (subgoal_tac "poly ([-a, 1] %^ n *** q) = \
|
|
1002 |
\ poly ([-a, 1] %^ n *** ([-a, 1] *** (inverse (real (Suc n)) %* \
|
|
1003 |
\ (qa +++ -- (pderiv q)))))" 1);
|
|
1004 |
by (dtac (poly_mult_left_cancel RS iffD1) 1);
|
|
1005 |
by (Asm_full_simp_tac 1);
|
|
1006 |
by (asm_full_simp_tac (simpset() addsimps [fun_eq,poly_mult,poly_add,poly_cmult,
|
|
1007 |
poly_minus] delsimps [pmult_Cons]) 1);
|
|
1008 |
by (Step_tac 1);
|
|
1009 |
by (res_inst_tac [("c1","real (Suc n)")] (real_mult_left_cancel
|
|
1010 |
RS iffD1) 1);
|
|
1011 |
by (Simp_tac 1);
|
|
1012 |
by (rtac ((CLAIM_SIMP
|
|
1013 |
"a * (b * (c * (d * e))) = e * (b * (c * (d * (a::real))))"
|
|
1014 |
real_mult_ac) RS ssubst) 1);
|
|
1015 |
by (rotate_tac 2 1);
|
|
1016 |
by (dres_inst_tac [("x","xa")] spec 1);
|
|
1017 |
by (asm_full_simp_tac (simpset() addsimps [real_add_mult_distrib,
|
|
1018 |
real_minus_mult_eq1 RS sym] @ real_mult_ac
|
|
1019 |
delsimps [pmult_Cons]) 1);
|
|
1020 |
qed_spec_mp "lemma_order_pderiv";
|
|
1021 |
|
|
1022 |
Goal "[| poly (pderiv p) ~= poly []; order a p ~= 0 |] \
|
|
1023 |
\ ==> (order a p = Suc (order a (pderiv p)))";
|
|
1024 |
by (case_tac "poly p = poly []" 1);
|
|
1025 |
by (auto_tac (claset() addDs [pderiv_zero],simpset()));
|
|
1026 |
by (dres_inst_tac [("a","a"),("p","p")] order_decomp 1);
|
|
1027 |
by (blast_tac (claset() addIs [lemma_order_pderiv]) 1);
|
|
1028 |
qed "order_pderiv";
|
|
1029 |
|
|
1030 |
(* ------------------------------------------------------------------------- *)
|
|
1031 |
(* Now justify the standard squarefree decomposition, i.e. f / gcd(f,f'). *)
|
|
1032 |
(* `a la Harrison *)
|
|
1033 |
(* ------------------------------------------------------------------------- *)
|
|
1034 |
|
|
1035 |
Goal "[| poly (pderiv p) ~= poly []; \
|
|
1036 |
\ poly p = poly (q *** d); \
|
|
1037 |
\ poly (pderiv p) = poly (e *** d); \
|
|
1038 |
\ poly d = poly (r *** p +++ s *** pderiv p) \
|
|
1039 |
\ |] ==> order a q = (if order a p = 0 then 0 else 1)";
|
|
1040 |
by (subgoal_tac "order a p = order a q + order a d" 1);
|
|
1041 |
by (res_inst_tac [("s","order a (q *** d)")] trans 2);
|
|
1042 |
by (blast_tac (claset() addIs [order_poly]) 2);
|
|
1043 |
by (rtac order_mult 2);
|
|
1044 |
by (rtac notI 2 THEN Asm_full_simp_tac 2);
|
|
1045 |
by (dres_inst_tac [("p","p")] pderiv_zero 2);
|
|
1046 |
by (Asm_full_simp_tac 2);
|
|
1047 |
by (case_tac "order a p = 0" 1);
|
|
1048 |
by (Asm_full_simp_tac 1);
|
|
1049 |
by (subgoal_tac "order a (pderiv p) = order a e + order a d" 1);
|
|
1050 |
by (res_inst_tac [("s","order a (e *** d)")] trans 2);
|
|
1051 |
by (blast_tac (claset() addIs [order_poly]) 2);
|
|
1052 |
by (rtac order_mult 2);
|
|
1053 |
by (rtac notI 2 THEN Asm_full_simp_tac 2);
|
|
1054 |
by (case_tac "poly p = poly []" 1);
|
|
1055 |
by (dres_inst_tac [("p","p")] pderiv_zero 1);
|
|
1056 |
by (Asm_full_simp_tac 1);
|
|
1057 |
by (dtac order_pderiv 1 THEN assume_tac 1);
|
|
1058 |
by (subgoal_tac "order a (pderiv p) <= order a d" 1);
|
|
1059 |
by (subgoal_tac "([-a, 1] %^ (order a (pderiv p))) divides d" 2);
|
|
1060 |
by (asm_full_simp_tac (simpset() addsimps [poly_entire,order_divides]) 2);
|
|
1061 |
by (subgoal_tac "([-a, 1] %^ (order a (pderiv p))) divides p & \
|
|
1062 |
\ ([-a, 1] %^ (order a (pderiv p))) divides (pderiv p)" 2);
|
|
1063 |
by (asm_simp_tac (simpset() addsimps [order_divides]) 3);
|
|
1064 |
by (asm_full_simp_tac (simpset() addsimps [divides_def]
|
|
1065 |
delsimps [pexp_Suc,pmult_Cons]) 2);
|
|
1066 |
by (Step_tac 2);
|
|
1067 |
by (res_inst_tac [("x","r *** qa +++ s *** qaa")] exI 2);
|
|
1068 |
by (asm_full_simp_tac (simpset() addsimps [fun_eq,poly_add,poly_mult,
|
|
1069 |
real_add_mult_distrib, real_add_mult_distrib2] @ real_mult_ac
|
|
1070 |
delsimps [pexp_Suc,pmult_Cons]) 2);
|
|
1071 |
by Auto_tac;
|
|
1072 |
qed "poly_squarefree_decomp_order";
|
|
1073 |
|
|
1074 |
|
|
1075 |
Goal "[| poly (pderiv p) ~= poly []; \
|
|
1076 |
\ poly p = poly (q *** d); \
|
|
1077 |
\ poly (pderiv p) = poly (e *** d); \
|
|
1078 |
\ poly d = poly (r *** p +++ s *** pderiv p) \
|
|
1079 |
\ |] ==> ALL a. order a q = (if order a p = 0 then 0 else 1)";
|
|
1080 |
by (blast_tac (claset() addIs [poly_squarefree_decomp_order]) 1);
|
|
1081 |
qed "poly_squarefree_decomp_order2";
|
|
1082 |
|
|
1083 |
Goal "poly p ~= poly [] ==> (poly p a = 0) = (order a p ~= 0)";
|
|
1084 |
by (rtac (order_root RS ssubst) 1);
|
|
1085 |
by Auto_tac;
|
|
1086 |
qed "order_root2";
|
|
1087 |
|
|
1088 |
Goal "[| poly (pderiv p) ~= poly []; order a p ~= 0 |] \
|
|
1089 |
\ ==> (order a (pderiv p) = n) = (order a p = Suc n)";
|
|
1090 |
by (auto_tac (claset() addDs [order_pderiv],simpset()));
|
|
1091 |
qed "order_pderiv2";
|
|
1092 |
|
|
1093 |
Goalw [rsquarefree_def]
|
|
1094 |
"rsquarefree p = (ALL a. ~(poly p a = 0 & poly (pderiv p) a = 0))";
|
|
1095 |
by (case_tac "poly p = poly []" 1);
|
|
1096 |
by (Asm_full_simp_tac 1);
|
|
1097 |
by (Asm_full_simp_tac 1);
|
|
1098 |
by (case_tac "poly (pderiv p) = poly []" 1);
|
|
1099 |
by (Asm_full_simp_tac 1);
|
|
1100 |
by (dtac pderiv_iszero 1 THEN Clarify_tac 1);
|
|
1101 |
by (subgoal_tac "ALL a. order a p = order a [h]" 1);
|
|
1102 |
by (asm_full_simp_tac (simpset() addsimps [fun_eq]) 1);
|
|
1103 |
by (rtac allI 1);
|
|
1104 |
by (cut_inst_tac [("p","[h]"),("a","a")] order_root 1);
|
|
1105 |
by (asm_full_simp_tac (simpset() addsimps [fun_eq]) 1);
|
|
1106 |
by (blast_tac (claset() addIs [order_poly]) 1);
|
|
1107 |
by (auto_tac (claset(),simpset() addsimps [order_root,order_pderiv2]));
|
|
1108 |
by (dtac spec 1 THEN Auto_tac);
|
|
1109 |
qed "rsquarefree_roots";
|
|
1110 |
|
|
1111 |
Goal "[1] *** p = p";
|
|
1112 |
by Auto_tac;
|
|
1113 |
qed "pmult_one";
|
|
1114 |
Addsimps [pmult_one];
|
|
1115 |
|
|
1116 |
Goal "poly [] = poly [0]";
|
|
1117 |
by (simp_tac (simpset() addsimps [fun_eq]) 1);
|
|
1118 |
qed "poly_Nil_zero";
|
|
1119 |
|
|
1120 |
Goalw [rsquarefree_def]
|
|
1121 |
"[| rsquarefree p; poly p a = 0 |] \
|
|
1122 |
\ ==> EX q. (poly p = poly ([-a, 1] *** q)) & poly q a ~= 0";
|
|
1123 |
by (Step_tac 1);
|
|
1124 |
by (forw_inst_tac [("a","a")] order_decomp 1);
|
|
1125 |
by (dres_inst_tac [("x","a")] spec 1);
|
|
1126 |
by (dres_inst_tac [("a1","a")] (order_root2 RS sym) 1);
|
|
1127 |
by (auto_tac (claset(),simpset() delsimps [pmult_Cons]));
|
|
1128 |
by (res_inst_tac [("x","q")] exI 1 THEN Step_tac 1);
|
|
1129 |
by (asm_full_simp_tac (simpset() addsimps [poly_mult,fun_eq]) 1);
|
|
1130 |
by (dres_inst_tac [("p1","q")] (poly_linear_divides RS iffD1) 1);
|
|
1131 |
by (asm_full_simp_tac (simpset() addsimps [divides_def]
|
|
1132 |
delsimps [pmult_Cons]) 1);
|
|
1133 |
by (Step_tac 1);
|
|
1134 |
by (dres_inst_tac [("x","[]")] spec 1);
|
|
1135 |
by (auto_tac (claset(),simpset() addsimps [fun_eq]));
|
|
1136 |
qed "rsquarefree_decomp";
|
|
1137 |
|
|
1138 |
Goal "[| poly (pderiv p) ~= poly []; \
|
|
1139 |
\ poly p = poly (q *** d); \
|
|
1140 |
\ poly (pderiv p) = poly (e *** d); \
|
|
1141 |
\ poly d = poly (r *** p +++ s *** pderiv p) \
|
|
1142 |
\ |] ==> rsquarefree q & (ALL a. (poly q a = 0) = (poly p a = 0))";
|
|
1143 |
by (ftac poly_squarefree_decomp_order2 1);
|
|
1144 |
by (TRYALL(assume_tac));
|
|
1145 |
by (case_tac "poly p = poly []" 1);
|
|
1146 |
by (blast_tac (claset() addDs [pderiv_zero]) 1);
|
|
1147 |
by (simp_tac (simpset() addsimps [rsquarefree_def,
|
|
1148 |
order_root] delsimps [pmult_Cons]) 1);
|
|
1149 |
by (asm_full_simp_tac (simpset() addsimps [poly_entire]
|
|
1150 |
delsimps [pmult_Cons]) 1);
|
|
1151 |
qed "poly_squarefree_decomp";
|
|
1152 |
|
|
1153 |
|
|
1154 |
(* ------------------------------------------------------------------------- *)
|
|
1155 |
(* Normalization of a polynomial. *)
|
|
1156 |
(* ------------------------------------------------------------------------- *)
|
|
1157 |
|
|
1158 |
Goal "poly (pnormalize p) = poly p";
|
|
1159 |
by (induct_tac "p" 1);
|
|
1160 |
by (auto_tac (claset(),simpset() addsimps [fun_eq]));
|
|
1161 |
qed "poly_normalize";
|
|
1162 |
Addsimps [poly_normalize];
|
|
1163 |
|
|
1164 |
|
|
1165 |
(* ------------------------------------------------------------------------- *)
|
|
1166 |
(* The degree of a polynomial. *)
|
|
1167 |
(* ------------------------------------------------------------------------- *)
|
|
1168 |
|
|
1169 |
Goal "list_all (%c. c = 0) p --> pnormalize p = []";
|
|
1170 |
by (induct_tac "p" 1);
|
|
1171 |
by Auto_tac;
|
|
1172 |
qed_spec_mp "lemma_degree_zero";
|
|
1173 |
|
|
1174 |
Goalw [degree_def] "poly p = poly [] ==> degree p = 0";
|
|
1175 |
by (case_tac "pnormalize p = []" 1);
|
|
1176 |
by (auto_tac (claset() addDs [lemma_degree_zero],simpset()
|
|
1177 |
addsimps [poly_zero]));
|
|
1178 |
qed "degree_zero";
|
|
1179 |
|
|
1180 |
(* ------------------------------------------------------------------------- *)
|
|
1181 |
(* Tidier versions of finiteness of roots. *)
|
|
1182 |
(* ------------------------------------------------------------------------- *)
|
|
1183 |
|
|
1184 |
Goal "poly p ~= poly [] ==> finite {x. poly p x = 0}";
|
|
1185 |
by (auto_tac (claset(),simpset() addsimps [poly_roots_finite]));
|
|
1186 |
by (res_inst_tac [("B","{x::real. EX n. (n::nat) < N & (x = j n)}")]
|
|
1187 |
finite_subset 1);
|
|
1188 |
by (induct_tac "N" 2);
|
|
1189 |
by Auto_tac;
|
|
1190 |
by (subgoal_tac "{x::real. EX na. na < Suc n & (x = j na)} = \
|
|
1191 |
\ {(j n)} Un {x. EX na. na < n & (x = j na)}" 1);
|
|
1192 |
by (auto_tac (claset(),simpset() addsimps [ARITH_PROVE
|
|
1193 |
"(na < Suc n) = (na = n | na < n)"]));
|
|
1194 |
qed "poly_roots_finite_set";
|
|
1195 |
|
|
1196 |
(* ------------------------------------------------------------------------- *)
|
|
1197 |
(* bound for polynomial. *)
|
|
1198 |
(* ------------------------------------------------------------------------- *)
|
|
1199 |
|
|
1200 |
Goal "abs(x) <= k --> abs(poly p x) <= poly (map abs p) k";
|
|
1201 |
by (induct_tac "p" 1);
|
|
1202 |
by Auto_tac;
|
|
1203 |
by (res_inst_tac [("j","abs a + abs(x * poly list x)")] real_le_trans 1);
|
|
1204 |
by (rtac abs_triangle_ineq 1);
|
|
1205 |
by (auto_tac (claset() addSIs [real_mult_le_mono],simpset()
|
|
1206 |
addsimps [abs_mult]));
|
|
1207 |
by (arith_tac 1);
|
|
1208 |
qed_spec_mp "poly_mono";
|