src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
 author hoelzl Fri Oct 24 15:07:51 2014 +0200 (2014-10-24) changeset 58776 95e58e04e534 parent 58710 7216a10d69ba child 58834 773b378d9313 permissions -rw-r--r--
use NO_MATCH-simproc for distribution rules in field_simps, otherwise field_simps on '(a / (c + d)) * (e + f)' can be non-terminating
1 (*  Title:      HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
2     Author:     Amine Chaieb
3 *)
5 header {* Implementation and verification of multivariate polynomials *}
7 theory Reflected_Multivariate_Polynomial
8 imports Complex_Main Rat_Pair Polynomial_List
9 begin
11 subsection{* Datatype of polynomial expressions *}
13 datatype poly = C Num | Bound nat | Add poly poly | Sub poly poly
14   | Mul poly poly| Neg poly| Pw poly nat| CN poly nat poly
16 abbreviation poly_0 :: "poly" ("0\<^sub>p") where "0\<^sub>p \<equiv> C (0\<^sub>N)"
17 abbreviation poly_p :: "int \<Rightarrow> poly" ("'((_)')\<^sub>p") where "(i)\<^sub>p \<equiv> C (i)\<^sub>N"
20 subsection{* Boundedness, substitution and all that *}
22 primrec polysize:: "poly \<Rightarrow> nat"
23 where
24   "polysize (C c) = 1"
25 | "polysize (Bound n) = 1"
26 | "polysize (Neg p) = 1 + polysize p"
27 | "polysize (Add p q) = 1 + polysize p + polysize q"
28 | "polysize (Sub p q) = 1 + polysize p + polysize q"
29 | "polysize (Mul p q) = 1 + polysize p + polysize q"
30 | "polysize (Pw p n) = 1 + polysize p"
31 | "polysize (CN c n p) = 4 + polysize c + polysize p"
33 primrec polybound0:: "poly \<Rightarrow> bool" -- {* a poly is INDEPENDENT of Bound 0 *}
34 where
35   "polybound0 (C c) \<longleftrightarrow> True"
36 | "polybound0 (Bound n) \<longleftrightarrow> n > 0"
37 | "polybound0 (Neg a) \<longleftrightarrow> polybound0 a"
38 | "polybound0 (Add a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
39 | "polybound0 (Sub a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
40 | "polybound0 (Mul a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
41 | "polybound0 (Pw p n) \<longleftrightarrow> polybound0 p"
42 | "polybound0 (CN c n p) \<longleftrightarrow> n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p"
44 primrec polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" -- {* substitute a poly into a poly for Bound 0 *}
45 where
46   "polysubst0 t (C c) = C c"
47 | "polysubst0 t (Bound n) = (if n = 0 then t else Bound n)"
48 | "polysubst0 t (Neg a) = Neg (polysubst0 t a)"
49 | "polysubst0 t (Add a b) = Add (polysubst0 t a) (polysubst0 t b)"
50 | "polysubst0 t (Sub a b) = Sub (polysubst0 t a) (polysubst0 t b)"
51 | "polysubst0 t (Mul a b) = Mul (polysubst0 t a) (polysubst0 t b)"
52 | "polysubst0 t (Pw p n) = Pw (polysubst0 t p) n"
53 | "polysubst0 t (CN c n p) =
54     (if n = 0 then Add (polysubst0 t c) (Mul t (polysubst0 t p))
55      else CN (polysubst0 t c) n (polysubst0 t p))"
57 fun decrpoly:: "poly \<Rightarrow> poly"
58 where
59   "decrpoly (Bound n) = Bound (n - 1)"
60 | "decrpoly (Neg a) = Neg (decrpoly a)"
61 | "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"
62 | "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
63 | "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
64 | "decrpoly (Pw p n) = Pw (decrpoly p) n"
65 | "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
66 | "decrpoly a = a"
69 subsection{* Degrees and heads and coefficients *}
71 fun degree :: "poly \<Rightarrow> nat"
72 where
73   "degree (CN c 0 p) = 1 + degree p"
74 | "degree p = 0"
76 fun head :: "poly \<Rightarrow> poly"
77 where
79 | "head p = p"
81 (* More general notions of degree and head *)
82 fun degreen :: "poly \<Rightarrow> nat \<Rightarrow> nat"
83 where
84   "degreen (CN c n p) = (\<lambda>m. if n = m then 1 + degreen p n else 0)"
85 | "degreen p = (\<lambda>m. 0)"
87 fun headn :: "poly \<Rightarrow> nat \<Rightarrow> poly"
88 where
89   "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
90 | "headn p = (\<lambda>m. p)"
92 fun coefficients :: "poly \<Rightarrow> poly list"
93 where
94   "coefficients (CN c 0 p) = c # coefficients p"
95 | "coefficients p = [p]"
97 fun isconstant :: "poly \<Rightarrow> bool"
98 where
99   "isconstant (CN c 0 p) = False"
100 | "isconstant p = True"
102 fun behead :: "poly \<Rightarrow> poly"
103 where
104   "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
105 | "behead p = 0\<^sub>p"
107 fun headconst :: "poly \<Rightarrow> Num"
108 where
110 | "headconst (C n) = n"
113 subsection{* Operations for normalization *}
115 declare if_cong[fundef_cong del]
116 declare let_cong[fundef_cong del]
118 fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
119 where
120   "polyadd (C c) (C c') = C (c +\<^sub>N c')"
121 | "polyadd (C c) (CN c' n' p') = CN (polyadd (C c) c') n' p'"
122 | "polyadd (CN c n p) (C c') = CN (polyadd c (C c')) n p"
123 | "polyadd (CN c n p) (CN c' n' p') =
124     (if n < n' then CN (polyadd c (CN c' n' p')) n p
125      else if n' < n then CN (polyadd (CN c n p) c') n' p'
126      else
127       let
128         cc' = polyadd c c';
129         pp' = polyadd p p'
130       in if pp' = 0\<^sub>p then cc' else CN cc' n pp')"
134 fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
135 where
136   "polyneg (C c) = C (~\<^sub>N c)"
137 | "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
138 | "polyneg a = Neg a"
140 definition polysub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
141   where "p -\<^sub>p q = polyadd p (polyneg q)"
143 fun polymul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
144 where
145   "polymul (C c) (C c') = C (c *\<^sub>N c')"
146 | "polymul (C c) (CN c' n' p') =
147     (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul (C c) c') n' (polymul (C c) p'))"
148 | "polymul (CN c n p) (C c') =
149     (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul c (C c')) n (polymul p (C c')))"
150 | "polymul (CN c n p) (CN c' n' p') =
151     (if n < n' then CN (polymul c (CN c' n' p')) n (polymul p (CN c' n' p'))
152      else if n' < n then CN (polymul (CN c n p) c') n' (polymul (CN c n p) p')
153      else polyadd (polymul (CN c n p) c') (CN 0\<^sub>p n' (polymul (CN c n p) p')))"
154 | "polymul a b = Mul a b"
156 declare if_cong[fundef_cong]
157 declare let_cong[fundef_cong]
159 fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
160 where
161   "polypow 0 = (\<lambda>p. (1)\<^sub>p)"
162 | "polypow n =
163     (\<lambda>p.
164       let
165         q = polypow (n div 2) p;
166         d = polymul q q
167       in if even n then d else polymul p d)"
169 abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
170   where "a ^\<^sub>p k \<equiv> polypow k a"
172 function polynate :: "poly \<Rightarrow> poly"
173 where
174   "polynate (Bound n) = CN 0\<^sub>p n (1)\<^sub>p"
175 | "polynate (Add p q) = polynate p +\<^sub>p polynate q"
176 | "polynate (Sub p q) = polynate p -\<^sub>p polynate q"
177 | "polynate (Mul p q) = polynate p *\<^sub>p polynate q"
178 | "polynate (Neg p) = ~\<^sub>p (polynate p)"
179 | "polynate (Pw p n) = polynate p ^\<^sub>p n"
180 | "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
181 | "polynate (C c) = C (normNum c)"
182 by pat_completeness auto
183 termination by (relation "measure polysize") auto
185 fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly"
186 where
187   "poly_cmul y (C x) = C (y *\<^sub>N x)"
188 | "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
189 | "poly_cmul y p = C y *\<^sub>p p"
191 definition monic :: "poly \<Rightarrow> poly \<times> bool"
192 where
193   "monic p =
194     (let h = headconst p
195      in if h = 0\<^sub>N then (p, False) else (C (Ninv h) *\<^sub>p p, 0>\<^sub>N h))"
198 subsection {* Pseudo-division *}
200 definition shift1 :: "poly \<Rightarrow> poly"
201   where "shift1 p = CN 0\<^sub>p 0 p"
203 abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
204   where "funpow \<equiv> compow"
206 partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
207 where
208   "polydivide_aux a n p k s =
209     (if s = 0\<^sub>p then (k, s)
210      else
211       let
213         m = degree s
214       in
215         if m < n then (k,s)
216         else
217           let p' = funpow (m - n) shift1 p
218           in
219             if a = b then polydivide_aux a n p k (s -\<^sub>p p')
220             else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))"
222 definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
223   where "polydivide s p = polydivide_aux (head p) (degree p) p 0 s"
225 fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly"
226 where
227   "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
228 | "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
230 fun poly_deriv :: "poly \<Rightarrow> poly"
231 where
232   "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
233 | "poly_deriv p = 0\<^sub>p"
236 subsection{* Semantics of the polynomial representation *}
238 primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0,field_inverse_zero,power}"
239 where
240   "Ipoly bs (C c) = INum c"
241 | "Ipoly bs (Bound n) = bs!n"
242 | "Ipoly bs (Neg a) = - Ipoly bs a"
243 | "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
244 | "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
245 | "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
246 | "Ipoly bs (Pw t n) = Ipoly bs t ^ n"
247 | "Ipoly bs (CN c n p) = Ipoly bs c + (bs!n) * Ipoly bs p"
249 abbreviation Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0,field_inverse_zero,power}"
250     ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
251   where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"
253 lemma Ipoly_CInt: "Ipoly bs (C (i, 1)) = of_int i"
256 lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"
259 lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
262 subsection {* Normal form and normalization *}
264 fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
265 where
266   "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
267 | "isnpolyh (CN c n p) = (\<lambda>k. n \<ge> k \<and> isnpolyh c (Suc n) \<and> isnpolyh p n \<and> p \<noteq> 0\<^sub>p)"
268 | "isnpolyh p = (\<lambda>k. False)"
270 lemma isnpolyh_mono: "n' \<le> n \<Longrightarrow> isnpolyh p n \<Longrightarrow> isnpolyh p n'"
271   by (induct p rule: isnpolyh.induct) auto
273 definition isnpoly :: "poly \<Rightarrow> bool"
274   where "isnpoly p = isnpolyh p 0"
276 text{* polyadd preserves normal forms *}
278 lemma polyadd_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polyadd p q) (min n0 n1)"
279 proof (induct p q arbitrary: n0 n1 rule: polyadd.induct)
280   case (2 ab c' n' p' n0 n1)
281   from 2 have  th1: "isnpolyh (C ab) (Suc n')"
282     by simp
283   from 2(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1"
284     by simp_all
285   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')"
286     by simp
287   with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^sub>p c') (Suc n')"
288     by simp
289   from nplen1 have n01len1: "min n0 n1 \<le> n'"
290     by simp
291   then show ?case using 2 th3
292     by simp
293 next
294   case (3 c' n' p' ab n1 n0)
295   from 3 have  th1: "isnpolyh (C ab) (Suc n')"
296     by simp
297   from 3(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1"
298     by simp_all
299   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')"
300     by simp
301   with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (Suc n')"
302     by simp
303   from nplen1 have n01len1: "min n0 n1 \<le> n'"
304     by simp
305   then show ?case using 3 th3
306     by simp
307 next
308   case (4 c n p c' n' p' n0 n1)
309   then have nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n"
310     by simp_all
311   from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'"
312     by simp_all
313   from 4 have ngen0: "n \<ge> n0"
314     by simp
315   from 4 have n'gen1: "n' \<ge> n1"
316     by simp
317   have "n < n' \<or> n' < n \<or> n = n'"
318     by auto
319   moreover
320   {
321     assume eq: "n = n'"
322     with "4.hyps"(3)[OF nc nc']
323     have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)"
324       by auto
325     then have ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
326       using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1
327       by auto
328     from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n"
329       by simp
330     have minle: "min n0 n1 \<le> n'"
331       using ngen0 n'gen1 eq by simp
332     from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' have ?case
334   }
335   moreover
336   {
337     assume lt: "n < n'"
338     have "min n0 n1 \<le> n0"
339       by simp
340     with 4 lt have th1:"min n0 n1 \<le> n"
341       by auto
342     from 4 have th21: "isnpolyh c (Suc n)"
343       by simp
344     from 4 have th22: "isnpolyh (CN c' n' p') n'"
345       by simp
346     from lt have th23: "min (Suc n) n' = Suc n"
347       by arith
348     from "4.hyps"(1)[OF th21 th22] have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)"
349       using th23 by simp
350     with 4 lt th1 have ?case
351       by simp
352   }
353   moreover
354   {
355     assume gt: "n' < n"
356     then have gt': "n' < n \<and> \<not> n < n'"
357       by simp
358     have "min n0 n1 \<le> n1"
359       by simp
360     with 4 gt have th1: "min n0 n1 \<le> n'"
361       by auto
362     from 4 have th21: "isnpolyh c' (Suc n')"
363       by simp_all
364     from 4 have th22: "isnpolyh (CN c n p) n"
365       by simp
366     from gt have th23: "min n (Suc n') = Suc n'"
367       by arith
368     from "4.hyps"(2)[OF th22 th21] have "isnpolyh (polyadd (CN c n p) c') (Suc n')"
369       using th23 by simp
370     with 4 gt th1 have ?case
371       by simp
372   }
373   ultimately show ?case by blast
374 qed auto
376 lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
377   by (induct p q rule: polyadd.induct)
378      (auto simp add: Let_def field_simps distrib_left[symmetric] simp del: distrib_left_NO_MATCH)
380 lemma polyadd_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polyadd p q)"
381   using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
383 text{* The degree of addition and other general lemmas needed for the normal form of polymul *}
386   assumes "isnpolyh p n0"
387     and "isnpolyh q n1"
388     and "degreen p m \<noteq> degreen q m"
389     and "m \<le> min n0 n1"
390   shows "degreen (polyadd p q) m = max (degreen p m) (degreen q m)"
391   using assms
392 proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
393   case (4 c n p c' n' p' m n0 n1)
394   have "n' = n \<or> n < n' \<or> n' < n" by arith
395   then show ?case
396   proof (elim disjE)
397     assume [simp]: "n' = n"
398     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
399     show ?thesis by (auto simp: Let_def)
400   next
401     assume "n < n'"
402     with 4 show ?thesis by auto
403   next
404     assume "n' < n"
405     with 4 show ?thesis by auto
406   qed
407 qed auto
409 lemma headnz[simp]: "isnpolyh p n \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
410   by (induct p arbitrary: n rule: headn.induct) auto
412 lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
413   by (induct p arbitrary: n rule: degree.induct) auto
415 lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
416   by (induct p arbitrary: n rule: degreen.induct) auto
418 lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
419   by (induct p arbitrary: n rule: degree.induct) auto
421 lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
422   using degree_isnpolyh_Suc by auto
424 lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
425   using degreen_0 by auto
429   assumes np: "isnpolyh p n0"
430     and nq: "isnpolyh q n1"
431     and m: "m \<le> max n0 n1"
432   shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
433   using np nq m
434 proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
435   case (2 c c' n' p' n0 n1)
436   then show ?case
437     by (cases n') simp_all
438 next
439   case (3 c n p c' n0 n1)
440   then show ?case
441     by (cases n) auto
442 next
443   case (4 c n p c' n' p' n0 n1 m)
444   have "n' = n \<or> n < n' \<or> n' < n" by arith
445   then show ?case
446   proof (elim disjE)
447     assume [simp]: "n' = n"
448     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
449     show ?thesis by (auto simp: Let_def)
450   qed simp_all
451 qed auto
454   assumes "isnpolyh p n0"
455     and "isnpolyh q n1"
456     and "polyadd p q = C c"
457   shows "degreen p m = degreen q m"
458   using assms
459 proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
460   case (4 c n p c' n' p' m n0 n1 x)
461   {
462     assume nn': "n' < n"
463     then have ?case using 4 by simp
464   }
465   moreover
466   {
467     assume nn': "\<not> n' < n"
468     then have "n < n' \<or> n = n'" by arith
469     moreover { assume "n < n'" with 4 have ?case by simp }
470     moreover
471     {
472       assume eq: "n = n'"
473       then have ?case using 4
474         apply (cases "p +\<^sub>p p' = 0\<^sub>p")
475         apply (auto simp add: Let_def)
476         done
477     }
478     ultimately have ?case by blast
479   }
480   ultimately show ?case by blast
481 qed simp_all
483 lemma polymul_properties:
484   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
485     and np: "isnpolyh p n0"
486     and nq: "isnpolyh q n1"
487     and m: "m \<le> min n0 n1"
488   shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
489     and "p *\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p \<or> q = 0\<^sub>p"
490     and "degreen (p *\<^sub>p q) m = (if p = 0\<^sub>p \<or> q = 0\<^sub>p then 0 else degreen p m + degreen q m)"
491   using np nq m
492 proof (induct p q arbitrary: n0 n1 m rule: polymul.induct)
493   case (2 c c' n' p')
494   {
495     case (1 n0 n1)
496     with "2.hyps"(4-6)[of n' n' n'] and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']
497     show ?case by (auto simp add: min_def)
498   next
499     case (2 n0 n1)
500     then show ?case by auto
501   next
502     case (3 n0 n1)
503     then show ?case  using "2.hyps" by auto
504   }
505 next
506   case (3 c n p c')
507   {
508     case (1 n0 n1)
509     with "3.hyps"(4-6)[of n n n] and "3.hyps"(1-3)[of "Suc n" "Suc n" n]
510     show ?case by (auto simp add: min_def)
511   next
512     case (2 n0 n1)
513     then show ?case by auto
514   next
515     case (3 n0 n1)
516     then show ?case  using "3.hyps" by auto
517   }
518 next
519   case (4 c n p c' n' p')
520   let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
521   {
522     case (1 n0 n1)
523     then have cnp: "isnpolyh ?cnp n"
524       and cnp': "isnpolyh ?cnp' n'"
525       and np: "isnpolyh p n"
526       and nc: "isnpolyh c (Suc n)"
527       and np': "isnpolyh p' n'"
528       and nc': "isnpolyh c' (Suc n')"
529       and nn0: "n \<ge> n0"
530       and nn1: "n' \<ge> n1"
531       by simp_all
532     {
533       assume "n < n'"
534       with "4.hyps"(4-5)[OF np cnp', of n] and "4.hyps"(1)[OF nc cnp', of n] nn0 cnp
535       have ?case by (simp add: min_def)
536     } moreover {
537       assume "n' < n"
538       with "4.hyps"(16-17)[OF cnp np', of "n'"] and "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
539       have ?case by (cases "Suc n' = n") (simp_all add: min_def)
540     } moreover {
541       assume "n' = n"
542       with "4.hyps"(16-17)[OF cnp np', of n] and "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0
543       have ?case
545         apply (simp_all add: min_def isnpolyh_mono[OF nn0])
546         done
547     }
548     ultimately show ?case by arith
549   next
550     fix n0 n1 m
551     assume np: "isnpolyh ?cnp n0"
552     assume np':"isnpolyh ?cnp' n1"
553     assume m: "m \<le> min n0 n1"
554     let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
555     let ?d1 = "degreen ?cnp m"
556     let ?d2 = "degreen ?cnp' m"
557     let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
558     have "n' < n \<or> n < n' \<or> n' = n" by auto
559     moreover
560     {
561       assume "n' < n \<or> n < n'"
562       with "4.hyps"(3,6,18) np np' m have ?eq
563         by auto
564     }
565     moreover
566     {
567       assume nn': "n' = n"
568       then have nn: "\<not> n' < n \<and> \<not> n < n'" by arith
569       from "4.hyps"(16,18)[of n n' n]
570         "4.hyps"(13,14)[of n "Suc n'" n]
571         np np' nn'
572       have norm:
573         "isnpolyh ?cnp n"
574         "isnpolyh c' (Suc n)"
575         "isnpolyh (?cnp *\<^sub>p c') n"
576         "isnpolyh p' n"
577         "isnpolyh (?cnp *\<^sub>p p') n"
578         "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
579         "?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p"
580         "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
581         by (auto simp add: min_def)
582       {
583         assume mn: "m = n"
584         from "4.hyps"(17,18)[OF norm(1,4), of n]
585           "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
586         have degs:
587           "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else ?d1 + degreen c' n)"
588           "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n"
590         from degs norm have th1: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
591           by simp
592         then have neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
593           by simp
594         have nmin: "n \<le> min n n"
596         from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
597         have deg: "degreen (CN c n p *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n =
598             degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
599           by simp
600         from "4.hyps"(16-18)[OF norm(1,4), of n]
601           "4.hyps"(13-15)[OF norm(1,2), of n]
602           mn norm m nn' deg
603         have ?eq by simp
604       }
605       moreover
606       {
607         assume mn: "m \<noteq> n"
608         then have mn': "m < n"
609           using m np by auto
610         from nn' m np have max1: "m \<le> max n n"
611           by simp
612         then have min1: "m \<le> min n n"
613           by simp
614         then have min2: "m \<le> min n (Suc n)"
615           by simp
616         from "4.hyps"(16-18)[OF norm(1,4) min1]
617           "4.hyps"(13-15)[OF norm(1,2) min2]
619         have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m \<le>
620             max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
621           using mn nn' np np' by simp
622         with "4.hyps"(16-18)[OF norm(1,4) min1]
623           "4.hyps"(13-15)[OF norm(1,2) min2]
624           degreen_0[OF norm(3) mn']
625         have ?eq using nn' mn np np' by clarsimp
626       }
627       ultimately have ?eq by blast
628     }
629     ultimately show ?eq by blast
630   }
631   {
632     case (2 n0 n1)
633     then have np: "isnpolyh ?cnp n0"
634       and np': "isnpolyh ?cnp' n1"
635       and m: "m \<le> min n0 n1"
636       by simp_all
637     then have mn: "m \<le> n" by simp
638     let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
639     {
640       assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
641       then have nn: "\<not> n' < n \<and> \<not> n < n'"
642         by simp
643       from "4.hyps"(16-18) [of n n n]
644         "4.hyps"(13-15)[of n "Suc n" n]
645         np np' C(2) mn
646       have norm:
647         "isnpolyh ?cnp n"
648         "isnpolyh c' (Suc n)"
649         "isnpolyh (?cnp *\<^sub>p c') n"
650         "isnpolyh p' n"
651         "isnpolyh (?cnp *\<^sub>p p') n"
652         "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
653         "?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p"
654         "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
655         "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
656         "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
658       from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
659         by simp
660       have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
661         using norm by simp
662       from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"] degneq
663       have False by simp
664     }
665     then show ?case using "4.hyps" by clarsimp
666   }
667 qed auto
669 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = Ipoly bs p * Ipoly bs q"
670   by (induct p q rule: polymul.induct) (auto simp add: field_simps)
672 lemma polymul_normh:
673   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
674   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
675   using polymul_properties(1) by blast
677 lemma polymul_eq0_iff:
678   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
679   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p *\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p \<or> q = 0\<^sub>p"
680   using polymul_properties(2) by blast
682 lemma polymul_degreen:
683   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
684   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> m \<le> min n0 n1 \<Longrightarrow>
685     degreen (p *\<^sub>p q) m = (if p = 0\<^sub>p \<or> q = 0\<^sub>p then 0 else degreen p m + degreen q m)"
686   by (fact polymul_properties(3))
688 lemma polymul_norm:
689   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
690   shows "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polymul p q)"
691   using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
693 lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
694   by (induct p arbitrary: n0 rule: headconst.induct) auto
697   by (induct p arbitrary: n0) auto
699 lemma monic_eqI:
700   assumes np: "isnpolyh p n0"
701   shows "INum (headconst p) * Ipoly bs (fst (monic p)) =
702     (Ipoly bs p ::'a::{field_char_0,field_inverse_zero, power})"
703   unfolding monic_def Let_def
705   let ?h = "headconst p"
706   assume pz: "p \<noteq> 0\<^sub>p"
707   {
708     assume hz: "INum ?h = (0::'a)"
709     from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N"
710       by simp_all
711     from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N"
712       by simp
713     with headconst_zero[OF np] have "p = 0\<^sub>p"
714       by blast
715     with pz have False
716       by blast
717   }
718   then show "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
719     by blast
720 qed
723 text{* polyneg is a negation and preserves normal forms *}
725 lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
726   by (induct p rule: polyneg.induct) auto
728 lemma polyneg0: "isnpolyh p n \<Longrightarrow> (~\<^sub>p p) = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
729   by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: Nneg_def)
731 lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
732   by (induct p arbitrary: n0 rule: polyneg.induct) auto
734 lemma polyneg_normh: "isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n"
735   by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: polyneg0)
737 lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
738   using isnpoly_def polyneg_normh by simp
741 text{* polysub is a substraction and preserves normal forms *}
743 lemma polysub[simp]: "Ipoly bs (polysub p q) = Ipoly bs p - Ipoly bs q"
746 lemma polysub_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
749 lemma polysub_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polysub p q)"
752 lemma polysub_same_0[simp]:
753   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
754   shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p"
755   unfolding polysub_def split_def fst_conv snd_conv
756   by (induct p arbitrary: n0) (auto simp add: Let_def Nsub0[simplified Nsub_def])
758 lemma polysub_0:
759   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
760   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p -\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = q"
761   unfolding polysub_def split_def fst_conv snd_conv
762   by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
763     (auto simp: Nsub0[simplified Nsub_def] Let_def)
765 text{* polypow is a power function and preserves normal forms *}
767 lemma polypow[simp]:
768   "Ipoly bs (polypow n p) = (Ipoly bs p :: 'a::{field_char_0,field_inverse_zero}) ^ n"
769 proof (induct n rule: polypow.induct)
770   case 1
771   then show ?case
772     by simp
773 next
774   case (2 n)
775   let ?q = "polypow ((Suc n) div 2) p"
776   let ?d = "polymul ?q ?q"
777   have "odd (Suc n) \<or> even (Suc n)"
778     by simp
779   moreover
780   {
781     assume odd: "odd (Suc n)"
782     have th: "(Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0)))) = Suc n div 2 + Suc n div 2 + 1"
783       by arith
784     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)"
786     also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2) * (Ipoly bs p)^(Suc n div 2)"
787       using "2.hyps" by simp
788     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
789       by (simp only: power_add power_one_right) simp
790     also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0))))"
791       by (simp only: th)
792     finally have ?case unfolding numeral_2_eq_2 [symmetric]
793     using odd_two_times_div_two_Suc [OF odd] by simp
794   }
795   moreover
796   {
797     assume even: "even (Suc n)"
798     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d"
800     also have "\<dots> = (Ipoly bs p) ^ (2 * (Suc n div 2))"
801       using "2.hyps" by (simp only: mult_2 power_add) simp
802     finally have ?case using even_two_times_div_two [OF even]
803       by simp
804   }
805   ultimately show ?case by blast
806 qed
808 lemma polypow_normh:
809   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
810   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
811 proof (induct k arbitrary: n rule: polypow.induct)
812   case 1
813   then show ?case by auto
814 next
815   case (2 k n)
816   let ?q = "polypow (Suc k div 2) p"
817   let ?d = "polymul ?q ?q"
818   from 2 have th1: "isnpolyh ?q n" and th2: "isnpolyh p n"
819     by blast+
820   from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n"
821     by simp
822   from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n"
823     by simp
824   from dn on show ?case by (simp, unfold Let_def) auto
826 qed
828 lemma polypow_norm:
829   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
830   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
831   by (simp add: polypow_normh isnpoly_def)
833 text{* Finally the whole normalization *}
835 lemma polynate [simp]:
836   "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0,field_inverse_zero})"
837   by (induct p rule:polynate.induct) auto
839 lemma polynate_norm[simp]:
840   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
841   shows "isnpoly (polynate p)"
842   by (induct p rule: polynate.induct)
846 text{* shift1 *}
849 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
852 lemma shift1_isnpoly:
853   assumes "isnpoly p"
854     and "p \<noteq> 0\<^sub>p"
855   shows "isnpoly (shift1 p) "
856   using assms by (simp add: shift1_def isnpoly_def)
858 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
861 lemma funpow_shift1_isnpoly: "isnpoly p \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpoly (funpow n shift1 p)"
862   by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
864 lemma funpow_isnpolyh:
865   assumes "\<And>p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n"
866     and "isnpolyh p n"
867   shows "isnpolyh (funpow k f p) n"
868   using assms by (induct k arbitrary: p) auto
870 lemma funpow_shift1:
871   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field_inverse_zero}) =
872     Ipoly bs (Mul (Pw (Bound 0) n) p)"
873   by (induct n arbitrary: p) (simp_all add: shift1_isnpoly shift1)
875 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
876   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
878 lemma funpow_shift1_1:
879   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field_inverse_zero}) =
880     Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)"
883 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
884   by (induct p rule: poly_cmul.induct) (auto simp add: field_simps)
887   assumes "isnpolyh p n"
889     (Ipoly bs p :: 'a :: {field_char_0,field_inverse_zero})"
890   using assms
891 proof (induct p arbitrary: n rule: behead.induct)
892   case (1 c p n)
893   then have pn: "isnpolyh p n" by simp
894   from 1(1)[OF pn]
895   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
896   then show ?case using "1.hyps"
898     apply (simp_all add: th[symmetric] field_simps)
899     done
900 qed (auto simp add: Let_def)
903   assumes "isnpolyh p n"
904   shows "isnpolyh (behead p) n"
905   using assms by (induct p rule: behead.induct) (auto simp add: Let_def isnpolyh_mono)
908 subsection {* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
910 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
911 proof (induct p arbitrary: n rule: poly.induct, auto)
912   case (goal1 c n p n')
913   then have "n = Suc (n - 1)"
914     by simp
915   then have "isnpolyh p (Suc (n - 1))"
916     using `isnpolyh p n` by simp
917   with goal1(2) show ?case
918     by simp
919 qed
921 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
922   by (induct p arbitrary: n0 rule: isconstant.induct) (auto simp add: isnpolyh_polybound0)
924 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
925   by (induct p) auto
927 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
928   apply (induct p arbitrary: n0)
929   apply auto
930   apply atomize
931   apply (rename_tac nat a b, erule_tac x = "Suc nat" in allE)
932   apply auto
933   done
936   by (induct p  arbitrary: n0 rule: head.induct) (auto intro: isnpolyh_polybound0)
938 lemma polybound0_I:
939   assumes "polybound0 a"
940   shows "Ipoly (b # bs) a = Ipoly (b' # bs) a"
941   using assms by (induct a rule: poly.induct) auto
943 lemma polysubst0_I: "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b # bs) a) # bs) t"
944   by (induct t) simp_all
946 lemma polysubst0_I':
947   assumes "polybound0 a"
948   shows "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b' # bs) a) # bs) t"
949   by (induct t) (simp_all add: polybound0_I[OF assms, where b="b" and b'="b'"])
951 lemma decrpoly:
952   assumes "polybound0 t"
953   shows "Ipoly (x # bs) t = Ipoly bs (decrpoly t)"
954   using assms by (induct t rule: decrpoly.induct) simp_all
956 lemma polysubst0_polybound0:
957   assumes "polybound0 t"
958   shows "polybound0 (polysubst0 t a)"
959   using assms by (induct a rule: poly.induct) auto
961 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
962   by (induct p arbitrary: n rule: degree.induct) (auto simp add: isnpolyh_polybound0)
964 primrec maxindex :: "poly \<Rightarrow> nat"
965 where
966   "maxindex (Bound n) = n + 1"
967 | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
968 | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
969 | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
970 | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
971 | "maxindex (Neg p) = maxindex p"
972 | "maxindex (Pw p n) = maxindex p"
973 | "maxindex (C x) = 0"
975 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool"
976   where "wf_bs bs p \<longleftrightarrow> length bs \<ge> maxindex p"
978 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall>c \<in> set (coefficients p). wf_bs bs c"
979 proof (induct p rule: coefficients.induct)
980   case (1 c p)
981   show ?case
982   proof
983     fix x
984     assume xc: "x \<in> set (coefficients (CN c 0 p))"
985     then have "x = c \<or> x \<in> set (coefficients p)"
986       by simp
987     moreover
988     {
989       assume "x = c"
990       then have "wf_bs bs x"
991         using "1.prems" unfolding wf_bs_def by simp
992     }
993     moreover
994     {
995       assume H: "x \<in> set (coefficients p)"
996       from "1.prems" have "wf_bs bs p"
997         unfolding wf_bs_def by simp
998       with "1.hyps" H have "wf_bs bs x"
999         by blast
1000     }
1001     ultimately show "wf_bs bs x"
1002       by blast
1003   qed
1004 qed simp_all
1006 lemma maxindex_coefficients: "\<forall>c \<in> set (coefficients p). maxindex c \<le> maxindex p"
1007   by (induct p rule: coefficients.induct) auto
1009 lemma wf_bs_I: "wf_bs bs p \<Longrightarrow> Ipoly (bs @ bs') p = Ipoly bs p"
1010   unfolding wf_bs_def by (induct p) (auto simp add: nth_append)
1012 lemma take_maxindex_wf:
1013   assumes wf: "wf_bs bs p"
1014   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
1015 proof -
1016   let ?ip = "maxindex p"
1017   let ?tbs = "take ?ip bs"
1018   from wf have "length ?tbs = ?ip"
1019     unfolding wf_bs_def by simp
1020   then have wf': "wf_bs ?tbs p"
1021     unfolding wf_bs_def by  simp
1022   have eq: "bs = ?tbs @ drop ?ip bs"
1023     by simp
1024   from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis
1025     using eq by simp
1026 qed
1028 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
1029   by (induct p) auto
1031 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
1032   unfolding wf_bs_def by simp
1034 lemma wf_bs_insensitive': "wf_bs (x # bs) p = wf_bs (y # bs) p"
1035   unfolding wf_bs_def by simp
1037 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x # bs) p"
1038   by (induct p rule: coefficients.induct) (auto simp add: wf_bs_def)
1040 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
1041   by (induct p rule: coefficients.induct) simp_all
1044   by (induct p rule: coefficients.induct) auto
1046 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x # bs) p"
1047   unfolding wf_bs_def by (induct p rule: decrpoly.induct) auto
1049 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists>ys. length (xs @ ys) = n"
1050   apply (rule exI[where x="replicate (n - length xs) z" for z])
1051   apply simp
1052   done
1054 lemma isnpolyh_Suc_const: "isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
1055   apply (cases p)
1056   apply auto
1057   apply (rename_tac nat a, case_tac "nat")
1058   apply simp_all
1059   done
1061 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
1062   unfolding wf_bs_def by (induct p q rule: polyadd.induct) (auto simp add: Let_def)
1064 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
1065   unfolding wf_bs_def
1066   apply (induct p q arbitrary: bs rule: polymul.induct)
1068   apply clarsimp
1069   apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
1070   apply auto
1071   done
1073 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
1074   unfolding wf_bs_def by (induct p rule: polyneg.induct) auto
1076 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
1077   unfolding polysub_def split_def fst_conv snd_conv
1078   using wf_bs_polyadd wf_bs_polyneg by blast
1081 subsection {* Canonicity of polynomial representation, see lemma isnpolyh_unique *}
1083 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
1084 definition "polypoly' bs p = map (Ipoly bs \<circ> decrpoly) (coefficients p)"
1085 definition "poly_nate bs p = map (Ipoly bs \<circ> decrpoly) (coefficients (polynate p))"
1087 lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall>q \<in> set (coefficients p). isnpolyh q n0"
1088 proof (induct p arbitrary: n0 rule: coefficients.induct)
1089   case (1 c p n0)
1090   have cp: "isnpolyh (CN c 0 p) n0"
1091     by fact
1092   then have norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
1093     by (auto simp add: isnpolyh_mono[where n'=0])
1094   from "1.hyps"[OF norm(2)] norm(1) norm(4) show ?case
1095     by simp
1096 qed auto
1098 lemma coefficients_isconst: "isnpolyh p n \<Longrightarrow> \<forall>q \<in> set (coefficients p). isconstant q"
1099   by (induct p arbitrary: n rule: coefficients.induct) (auto simp add: isnpolyh_Suc_const)
1101 lemma polypoly_polypoly':
1102   assumes np: "isnpolyh p n0"
1103   shows "polypoly (x # bs) p = polypoly' bs p"
1104 proof -
1105   let ?cf = "set (coefficients p)"
1106   from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
1107   {
1108     fix q
1109     assume q: "q \<in> ?cf"
1110     from q cn_norm have th: "isnpolyh q n0"
1111       by blast
1112     from coefficients_isconst[OF np] q have "isconstant q"
1113       by blast
1114     with isconstant_polybound0[OF th] have "polybound0 q"
1115       by blast
1116   }
1117   then have "\<forall>q \<in> ?cf. polybound0 q" ..
1118   then have "\<forall>q \<in> ?cf. Ipoly (x # bs) q = Ipoly bs (decrpoly q)"
1119     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
1120     by auto
1121   then show ?thesis
1122     unfolding polypoly_def polypoly'_def by simp
1123 qed
1125 lemma polypoly_poly:
1126   assumes "isnpolyh p n0"
1127   shows "Ipoly (x # bs) p = poly (polypoly (x # bs) p) x"
1128   using assms
1129   by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def)
1131 lemma polypoly'_poly:
1132   assumes "isnpolyh p n0"
1133   shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
1134   using polypoly_poly[OF assms, simplified polypoly_polypoly'[OF assms]] .
1137 lemma polypoly_poly_polybound0:
1138   assumes "isnpolyh p n0"
1139     and "polybound0 p"
1140   shows "polypoly bs p = [Ipoly bs p]"
1141   using assms
1142   unfolding polypoly_def
1143   apply (cases p)
1144   apply auto
1145   apply (rename_tac nat a, case_tac nat)
1146   apply auto
1147   done
1150   by (induct p rule: head.induct) auto
1152 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> headn p m = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
1153   by (cases p) auto
1156   by (induct p rule: head.induct) simp_all
1158 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> head p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
1161 lemma isnpolyh_zero_iff:
1162   assumes nq: "isnpolyh p n0"
1163     and eq :"\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a::{field_char_0,field_inverse_zero, power})"
1164   shows "p = 0\<^sub>p"
1165   using nq eq
1166 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
1167   case less
1168   note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
1169   {
1170     assume nz: "maxindex p = 0"
1171     then obtain c where "p = C c"
1172       using np by (cases p) auto
1173     with zp np have "p = 0\<^sub>p"
1174       unfolding wf_bs_def by simp
1175   }
1176   moreover
1177   {
1178     assume nz: "maxindex p \<noteq> 0"
1179     let ?h = "head p"
1180     let ?hd = "decrpoly ?h"
1181     let ?ihd = "maxindex ?hd"
1183     have h: "isnpolyh ?h n0" "polybound0 ?h"
1184       by simp_all
1185     then have nhd: "isnpolyh ?hd (n0 - 1)"
1186       using decrpoly_normh by blast
1188     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
1189     have mihn: "maxindex ?h \<le> maxindex p"
1190       by auto
1191     with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < maxindex p"
1192       by auto
1193     {
1194       fix bs :: "'a list"
1195       assume bs: "wf_bs bs ?hd"
1196       let ?ts = "take ?ihd bs"
1197       let ?rs = "drop ?ihd bs"
1198       have ts: "wf_bs ?ts ?hd"
1199         using bs unfolding wf_bs_def by simp
1200       have bs_ts_eq: "?ts @ ?rs = bs"
1201         by simp
1202       from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x # ?ts) ?h"
1203         by simp
1204       from ihd_lt_n have "\<forall>x. length (x # ?ts) \<le> maxindex p"
1205         by simp
1206       with length_le_list_ex obtain xs where xs: "length ((x # ?ts) @ xs) = maxindex p"
1207         by blast
1208       then have "\<forall>x. wf_bs ((x # ?ts) @ xs) p"
1209         unfolding wf_bs_def by simp
1210       with zp have "\<forall>x. Ipoly ((x # ?ts) @ xs) p = 0"
1211         by blast
1212       then have "\<forall>x. Ipoly (x # (?ts @ xs)) p = 0"
1213         by simp
1214       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
1215       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"
1216         by simp
1217       then have "poly (polypoly' (?ts @ xs) p) = poly []"
1218         by auto
1219       then have "\<forall>c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
1220         using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
1222       have th0: "Ipoly (?ts @ xs) ?hd = 0"
1223         by simp
1224       from bs have wf'': "wf_bs ?ts ?hd"
1225         unfolding wf_bs_def by simp
1226       with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0"
1227         by simp
1228       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
1229         by simp
1230     }
1231     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
1232       by blast
1233     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p"
1234       by blast
1235     then have "?h = 0\<^sub>p" by simp
1236     with head_nz[OF np] have "p = 0\<^sub>p" by simp
1237   }
1238   ultimately show "p = 0\<^sub>p"
1239     by blast
1240 qed
1242 lemma isnpolyh_unique:
1243   assumes np: "isnpolyh p n0"
1244     and nq: "isnpolyh q n1"
1245   shows "(\<forall>bs. \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (\<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup> :: 'a::{field_char_0,field_inverse_zero,power})) \<longleftrightarrow> p = q"
1246 proof auto
1247   assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a) = \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
1248   then have "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)"
1249     by simp
1250   then have "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
1251     using wf_bs_polysub[where p=p and q=q] by auto
1252   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq] show "p = q"
1253     by blast
1254 qed
1257 text{* consequences of unicity on the algorithms for polynomial normalization *}
1260   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1261     and np: "isnpolyh p n0"
1262     and nq: "isnpolyh q n1"
1263   shows "p +\<^sub>p q = q +\<^sub>p p"
1265   by simp
1267 lemma zero_normh: "isnpolyh 0\<^sub>p n"
1268   by simp
1270 lemma one_normh: "isnpolyh (1)\<^sub>p n"
1271   by simp
1274   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1275     and np: "isnpolyh p n0"
1276   shows "p +\<^sub>p 0\<^sub>p = p"
1277     and "0\<^sub>p +\<^sub>p p = p"
1278   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
1279     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
1281 lemma polymul_1[simp]:
1282   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1283     and np: "isnpolyh p n0"
1284   shows "p *\<^sub>p (1)\<^sub>p = p"
1285     and "(1)\<^sub>p *\<^sub>p p = p"
1286   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
1287     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
1289 lemma polymul_0[simp]:
1290   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1291     and np: "isnpolyh p n0"
1292   shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p"
1293     and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
1294   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
1295     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
1297 lemma polymul_commute:
1298   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1299     and np: "isnpolyh p n0"
1300     and nq: "isnpolyh q n1"
1301   shows "p *\<^sub>p q = q *\<^sub>p p"
1302   using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np],
1303     where ?'a = "'a::{field_char_0,field_inverse_zero, power}"]
1304   by simp
1306 declare polyneg_polyneg [simp]
1308 lemma isnpolyh_polynate_id [simp]:
1309   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1310     and np: "isnpolyh p n0"
1311   shows "polynate p = p"
1312   using isnpolyh_unique[where ?'a= "'a::{field_char_0,field_inverse_zero}",
1313       OF polynate_norm[of p, unfolded isnpoly_def] np]
1314     polynate[where ?'a = "'a::{field_char_0,field_inverse_zero}"]
1315   by simp
1317 lemma polynate_idempotent[simp]:
1318   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1319   shows "polynate (polynate p) = polynate p"
1320   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
1322 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
1323   unfolding poly_nate_def polypoly'_def ..
1325 lemma poly_nate_poly:
1326   "poly (poly_nate bs p) = (\<lambda>x:: 'a ::{field_char_0,field_inverse_zero}. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
1327   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
1328   unfolding poly_nate_polypoly' by auto
1331 subsection{* heads, degrees and all that *}
1333 lemma degree_eq_degreen0: "degree p = degreen p 0"
1334   by (induct p rule: degree.induct) simp_all
1336 lemma degree_polyneg:
1337   assumes "isnpolyh p n"
1338   shows "degree (polyneg p) = degree p"
1339   apply (induct p rule: polyneg.induct)
1340   using assms
1341   apply simp_all
1342   apply (case_tac na)
1343   apply auto
1344   done
1347   assumes np: "isnpolyh p n0"
1348     and nq: "isnpolyh q n1"
1349   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
1350   using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
1353 lemma degree_polysub:
1354   assumes np: "isnpolyh p n0"
1355     and nq: "isnpolyh q n1"
1356   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
1357 proof-
1358   from nq have nq': "isnpolyh (~\<^sub>p q) n1"
1359     using polyneg_normh by simp
1360   from degree_polyadd[OF np nq'] show ?thesis
1361     by (simp add: polysub_def degree_polyneg[OF nq])
1362 qed
1365   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1366     and np: "isnpolyh p n0"
1367     and nq: "isnpolyh q n1"
1369     and d: "degree p = degree q"
1370   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
1371   unfolding polysub_def split_def fst_conv snd_conv
1372   using np nq h d
1373 proof (induct p q rule: polyadd.induct)
1374   case (1 c c')
1375   then show ?case
1376     by (simp add: Nsub_def Nsub0[simplified Nsub_def])
1377 next
1378   case (2 c c' n' p')
1379   from 2 have "degree (C c) = degree (CN c' n' p')"
1380     by simp
1381   then have nz: "n' > 0"
1382     by (cases n') auto
1383   then have "head (CN c' n' p') = CN c' n' p'"
1384     by (cases n') auto
1385   with 2 show ?case
1386     by simp
1387 next
1388   case (3 c n p c')
1389   then have "degree (C c') = degree (CN c n p)"
1390     by simp
1391   then have nz: "n > 0"
1392     by (cases n) auto
1393   then have "head (CN c n p) = CN c n p"
1394     by (cases n) auto
1395   with 3 show ?case by simp
1396 next
1397   case (4 c n p c' n' p')
1398   then have H:
1399     "isnpolyh (CN c n p) n0"
1400     "isnpolyh (CN c' n' p') n1"
1401     "head (CN c n p) = head (CN c' n' p')"
1402     "degree (CN c n p) = degree (CN c' n' p')"
1403     by simp_all
1404   then have degc: "degree c = 0" and degc': "degree c' = 0"
1405     by simp_all
1406   then have degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
1407     using H(1-2) degree_polyneg by auto
1408   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"
1409     by simp_all
1410   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc'
1411   have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"
1412     by simp
1413   from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'"
1414     by auto
1415   have "n = n' \<or> n < n' \<or> n > n'"
1416     by arith
1417   moreover
1418   {
1419     assume nn': "n = n'"
1420     have "n = 0 \<or> n > 0" by arith
1421     moreover
1422     {
1423       assume nz: "n = 0"
1424       then have ?case using 4 nn'
1425         by (auto simp add: Let_def degcmc')
1426     }
1427     moreover
1428     {
1429       assume nz: "n > 0"
1430       with nn' H(3) have  cc': "c = c'" and pp': "p = p'"
1431         by (cases n, auto)+
1432       then have ?case
1433         using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv]
1434         using polysub_same_0[OF c'nh, simplified polysub_def]
1435         using nn' 4 by (simp add: Let_def)
1436     }
1437     ultimately have ?case by blast
1438   }
1439   moreover
1440   {
1441     assume nn': "n < n'"
1442     then have n'p: "n' > 0"
1443       by simp
1444     then have headcnp':"head (CN c' n' p') = CN c' n' p'"
1445       by (cases n') simp_all
1446     have degcnp': "degree (CN c' n' p') = 0"
1447       and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
1448       using 4 nn' by (cases n', simp_all)
1449     then have "n > 0"
1450       by (cases n) simp_all
1451     then have headcnp: "head (CN c n p) = CN c n p"
1452       by (cases n) auto
1454       by auto
1455   }
1456   moreover
1457   {
1458     assume nn': "n > n'"
1459     then have np: "n > 0" by simp
1460     then have headcnp:"head (CN c n p) = CN c n p"
1461       by (cases n) simp_all
1462     from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)"
1463       by simp
1464     from np have degcnp: "degree (CN c n p) = 0"
1465       by (cases n) simp_all
1466     with degcnpeq have "n' > 0"
1467       by (cases n') simp_all
1468     then have headcnp': "head (CN c' n' p') = CN c' n' p'"
1469       by (cases n') auto
1471   }
1472   ultimately show ?case by blast
1473 qed auto
1478 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
1479 proof (induct k arbitrary: n0 p)
1480   case 0
1481   then show ?case
1482     by auto
1483 next
1484   case (Suc k n0 p)
1485   then have "isnpolyh (shift1 p) 0"
1487   with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
1490   then show ?case
1492 qed
1494 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
1497 lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
1498   by (induct k arbitrary: p) (auto simp add: shift1_degree)
1500 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
1501   by (induct n arbitrary: p) simp_all
1503 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
1504   by (induct p arbitrary: n rule: degree.induct) auto
1505 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
1506   by (induct p arbitrary: n rule: degreen.induct) auto
1507 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
1508   by (induct p arbitrary: n rule: degree.induct) auto
1510   by (induct p rule: head.induct) auto
1513   "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> polyadd p q = C c \<Longrightarrow> degree p = degree q"
1514   using polyadd_eq_const_degreen degree_eq_degreen0 by simp
1517   assumes np: "isnpolyh p n0"
1518     and nq: "isnpolyh q n1"
1519     and deg: "degree p \<noteq> degree q"
1520   shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
1521   using np nq deg
1522   apply (induct p q arbitrary: n0 n1 rule: polyadd.induct)
1523   apply simp_all
1524   apply (case_tac n', simp, simp)
1525   apply (case_tac n, simp, simp)
1526   apply (case_tac n, case_tac n', simp add: Let_def)
1530   apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
1531   done
1534   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1535   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> q \<noteq> 0\<^sub>p \<Longrightarrow> head (p *\<^sub>p q) = head p *\<^sub>p head q"
1536 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
1537   case (2 c c' n' p' n0 n1)
1538   then have "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"
1540   then show ?case
1541     using 2 by (cases n') auto
1542 next
1543   case (3 c n p c' n0 n1)
1544   then have "isnpolyh (head (CN c n p)) n0" "isnormNum c'"
1546   then show ?case
1547     using 3 by (cases n) auto
1548 next
1549   case (4 c n p c' n' p' n0 n1)
1550   then have norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
1551     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
1552     by simp_all
1553   have "n < n' \<or> n' < n \<or> n = n'" by arith
1554   moreover
1555   {
1556     assume nn': "n < n'"
1557     then have ?case
1558       using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)]
1559       apply simp
1560       apply (cases n)
1561       apply simp
1562       apply (cases n')
1563       apply simp_all
1564       done
1565   }
1566   moreover {
1567     assume nn': "n'< n"
1568     then have ?case
1569       using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)]
1570       apply simp
1571       apply (cases n')
1572       apply simp
1573       apply (cases n)
1574       apply auto
1575       done
1576   }
1577   moreover
1578   {
1579     assume nn': "n' = n"
1580     from nn' polymul_normh[OF norm(5,4)]
1581     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
1582     from nn' polymul_normh[OF norm(5,3)] norm
1583     have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
1584     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
1585     have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
1587     have nth: "isnpolyh ((CN c n p *\<^sub>p c') +\<^sub>p (CN 0\<^sub>p n (CN c n p *\<^sub>p p'))) n"
1589     {
1590       assume np: "n > 0"
1591       with nn' head_isnpolyh_Suc'[OF np nth]
1593       have ?case by simp
1594     }
1595     moreover
1596     {
1597       assume nz: "n = 0"
1598       from polymul_degreen[OF norm(5,4), where m="0"]
1599         polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
1600       norm(5,6) degree_npolyhCN[OF norm(6)]
1601     have dth: "degree (CN c 0 p *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))"
1602       by simp
1603     then have dth': "degree (CN c 0 p *\<^sub>p c') \<noteq> degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))"
1604       by simp
1606     have ?case
1607       using norm "4.hyps"(6)[OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)] nn' nz
1608       by simp
1609     }
1610     ultimately have ?case
1611       by (cases n) auto
1612   }
1613   ultimately show ?case by blast
1614 qed simp_all
1616 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
1617   by (induct p rule: degree.induct) auto
1620   by (induct p rule: head.induct) auto
1622 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
1623   by (cases n) simp_all
1625 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
1626   by (cases n) simp_all
1629   "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> degree p \<noteq> degree q \<Longrightarrow>
1630     degree (polyadd p q) = max (degree p) (degree q)"
1631   using polyadd_different_degreen degree_eq_degreen0 by simp
1633 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
1634   by (induct p arbitrary: n0 rule: polyneg.induct) auto
1636 lemma degree_polymul:
1637   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1638     and np: "isnpolyh p n0"
1639     and nq: "isnpolyh q n1"
1640   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
1641   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
1643 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
1644   by (induct p arbitrary: n rule: degree.induct) auto
1647   by (induct p arbitrary: n rule: degree.induct) auto
1650 subsection {* Correctness of polynomial pseudo division *}
1652 lemma polydivide_aux_properties:
1653   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1654     and np: "isnpolyh p n0"
1655     and ns: "isnpolyh s n1"
1656     and ap: "head p = a"
1657     and ndp: "degree p = n"
1658     and pnz: "p \<noteq> 0\<^sub>p"
1659   shows "polydivide_aux a n p k s = (k', r) \<longrightarrow> k' \<ge> k \<and> (degree r = 0 \<or> degree r < degree p) \<and>
1660     (\<exists>nr. isnpolyh r nr) \<and> (\<exists>q n1. isnpolyh q n1 \<and> (polypow (k' - k) a) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
1661   using ns
1662 proof (induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
1663   case less
1664   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
1665   let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and>
1666     (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
1667   let ?b = "head s"
1668   let ?p' = "funpow (degree s - n) shift1 p"
1669   let ?xdn = "funpow (degree s - n) shift1 (1)\<^sub>p"
1670   let ?akk' = "a ^\<^sub>p (k' - k)"
1671   note ns = `isnpolyh s n1`
1672   from np have np0: "isnpolyh p 0"
1673     using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by simp
1674   have np': "isnpolyh ?p' 0"
1675     using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def
1676     by simp
1678     using funpow_shift1_head[OF np pnz] by simp
1679   from funpow_shift1_isnpoly[where p="(1)\<^sub>p"] have nxdn: "isnpolyh ?xdn 0"
1681   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
1682   have nakk':"isnpolyh ?akk' 0" by blast
1683   {
1684     assume sz: "s = 0\<^sub>p"
1685     then have ?ths
1686       using np polydivide_aux.simps
1687       apply clarsimp
1688       apply (rule exI[where x="0\<^sub>p"])
1689       apply simp
1690       done
1691   }
1692   moreover
1693   {
1694     assume sz: "s \<noteq> 0\<^sub>p"
1695     {
1696       assume dn: "degree s < n"
1697       then have "?ths"
1698         using ns ndp np polydivide_aux.simps
1699         apply auto
1700         apply (rule exI[where x="0\<^sub>p"])
1701         apply simp
1702         done
1703     }
1704     moreover
1705     {
1706       assume dn': "\<not> degree s < n"
1707       then have dn: "degree s \<ge> n"
1708         by arith
1709       have degsp': "degree s = degree ?p'"
1710         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"]
1711         by simp
1712       {
1713         assume ba: "?b = a"
1715           using ap headp' by simp
1716         have nr: "isnpolyh (s -\<^sub>p ?p') 0"
1717           using polysub_normh[OF ns np'] by simp
1719         have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p"
1720           by simp
1721         moreover
1722         {
1723           assume deglt:"degree (s -\<^sub>p ?p') < degree s"
1724           from polydivide_aux.simps sz dn' ba
1725           have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
1727           {
1728             assume h1: "polydivide_aux a n p k s = (k', r)"
1729             from less(1)[OF deglt nr, of k k' r] trans[OF eq[symmetric] h1]
1730             have kk': "k \<le> k'"
1731               and nr: "\<exists>nr. isnpolyh r nr"
1732               and dr: "degree r = 0 \<or> degree r < degree p"
1733               and q1: "\<exists>q nq. isnpolyh q nq \<and> a ^\<^sub>p k' - k *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"
1734               by auto
1735             from q1 obtain q n1 where nq: "isnpolyh q n1"
1736               and asp: "a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"
1737               by blast
1738             from nr obtain nr where nr': "isnpolyh r nr"
1739               by blast
1740             from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0"
1741               by simp
1742             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
1743             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
1745               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
1746             have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0"
1747               by simp
1748             from asp have "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
1749               Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)"
1750               by simp
1751             then have "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
1752               Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
1753               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
1755             then have "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
1756               Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1757               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p *\<^sub>p p) +
1758               Ipoly bs p * Ipoly bs q + Ipoly bs r"
1759               by (auto simp only: funpow_shift1_1)
1760             then have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list.
1761               Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1762               Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p) +
1763               Ipoly bs q) + Ipoly bs r"
1765             then have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list.
1766               Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1767               Ipoly bs (p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r)"
1768               by simp
1769             with isnpolyh_unique[OF nakks' nqr']
1770             have "a ^\<^sub>p (k' - k) *\<^sub>p s =
1771               p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r"
1772               by blast
1773             then have ?qths using nq'
1774               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q" in exI)
1775               apply (rule_tac x="0" in exI)
1776               apply simp
1777               done
1778             with kk' nr dr have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and>
1779               (\<exists>nr. isnpolyh r nr) \<and> ?qths"
1780               by blast
1781           }
1782           then have ?ths by blast
1783         }
1784         moreover
1785         {
1786           assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
1787           from spz isnpolyh_unique[OF polysub_normh[OF ns np'], where q="0\<^sub>p", symmetric, where ?'a = "'a::{field_char_0,field_inverse_zero}"]
1788           have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list. Ipoly bs s = Ipoly bs ?p'"
1789             by simp
1790           then have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list. Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)"
1791             using np nxdn
1792             apply simp
1793             apply (simp only: funpow_shift1_1)
1794             apply simp
1795             done
1796           then have sp': "s = ?xdn *\<^sub>p p"
1797             using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]]
1798             by blast
1799           {
1800             assume h1: "polydivide_aux a n p k s = (k', r)"
1801             from polydivide_aux.simps sz dn' ba
1802             have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
1804             also have "\<dots> = (k,0\<^sub>p)"
1805               using polydivide_aux.simps spz by simp
1806             finally have "(k', r) = (k, 0\<^sub>p)"
1807               using h1 by simp
1808             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
1809               polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
1810               apply auto
1811               apply (rule exI[where x="?xdn"])
1812               apply (auto simp add: polymul_commute[of p])
1813               done
1814           }
1815         }
1816         ultimately have ?ths by blast
1817       }
1818       moreover
1819       {
1820         assume ba: "?b \<noteq> a"
1821         from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
1823         have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
1825         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
1826           using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
1828             funpow_shift1_nz[OF pnz]
1829           by simp_all
1832         have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
1836         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
1837           head_nz[OF np] pnz sz ap[symmetric]
1838           funpow_shift1_nz[OF pnz, where n="degree s - n"]
1840           ndp dn
1841         have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p')"
1842           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
1843         {
1844           assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
1845           from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns]
1847           have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
1848             by simp
1849           {
1850             assume h1:"polydivide_aux a n p k s = (k', r)"
1851             from h1 polydivide_aux.simps sz dn' ba
1852             have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
1854             with less(1)[OF dth nasbp', of "Suc k" k' r]
1855             obtain q nq nr where kk': "Suc k \<le> k'"
1856               and nr: "isnpolyh r nr"
1857               and nq: "isnpolyh q nq"
1858               and dr: "degree r = 0 \<or> degree r < degree p"
1859               and qr: "a ^\<^sub>p (k' - Suc k) *\<^sub>p ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = p *\<^sub>p q +\<^sub>p r"
1860               by auto
1861             from kk' have kk'': "Suc (k' - Suc k) = k' - k"
1862               by arith
1863             {
1864               fix bs :: "'a::{field_char_0,field_inverse_zero} list"
1865               from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
1866               have "Ipoly bs (a ^\<^sub>p (k' - Suc k) *\<^sub>p ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p'))) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)"
1867                 by simp
1868               then have "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s =
1869                 Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
1871               then have "Ipoly bs a ^ (k' - k)  * Ipoly bs s =
1872                 Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
1873                 by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
1874               then have "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1875                 Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
1877             }
1878             then have ieq:"\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
1879                 Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1880                 Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)"
1881               by auto
1882             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
1883             from polyadd_normh[OF nq polymul_normh[OF polymul_normh[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k"] head_isnpolyh[OF ns], simplified ap] nxdn]]
1884             have nqw: "isnpolyh ?q 0"
1885               by simp
1886             from ieq isnpolyh_unique[OF polymul_normh[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - k"] ns, simplified ap] polyadd_normh[OF polymul_normh[OF np nqw] nr]]
1887             have asth: "(a ^\<^sub>p (k' - k) *\<^sub>p s) = p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r"
1888               by blast
1889             from dr kk' nr h1 asth nqw have ?ths
1890               apply simp
1891               apply (rule conjI)
1892               apply (rule exI[where x="nr"], simp)
1893               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
1894               apply (rule exI[where x="0"], simp)
1895               done
1896           }
1897           then have ?ths by blast
1898         }
1899         moreover
1900         {
1901           assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
1902           {
1903             fix bs :: "'a::{field_char_0,field_inverse_zero} list"
1904             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
1905             have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'"
1906               by simp
1907             then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
1908               by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
1909             then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))"
1910               by simp
1911           }
1912           then have hth: "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
1913             Ipoly bs (a *\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
1914           from hth have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
1915             using isnpolyh_unique[where ?'a = "'a::{field_char_0,field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
1916                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
1917               simplified ap]
1918             by simp
1919           {
1920             assume h1: "polydivide_aux a n p k s = (k', r)"
1921             from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
1922             have "(k', r) = (Suc k, 0\<^sub>p)"
1924             with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
1925               polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
1926             have ?ths
1927               apply (clarsimp simp add: Let_def)
1928               apply (rule exI[where x="?b *\<^sub>p ?xdn"])
1929               apply simp
1930               apply (rule exI[where x="0"], simp)
1931               done
1932           }
1933           then have ?ths by blast
1934         }
1935         ultimately have ?ths
1937             head_nz[OF np] pnz sz ap[symmetric]
1938           by (auto simp add: degree_eq_degreen0[symmetric])
1939       }
1940       ultimately have ?ths by blast
1941     }
1942     ultimately have ?ths by blast
1943   }
1944   ultimately show ?ths by blast
1945 qed
1947 lemma polydivide_properties:
1948   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1949     and np: "isnpolyh p n0"
1950     and ns: "isnpolyh s n1"
1951     and pnz: "p \<noteq> 0\<^sub>p"
1952   shows "\<exists>k r. polydivide s p = (k, r) \<and>
1953     (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) \<and>
1954     (\<exists>q n1. isnpolyh q n1 \<and> polypow k (head p) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
1955 proof -
1956   have trv: "head p = head p" "degree p = degree p"
1957     by simp_all
1958   from polydivide_def[where s="s" and p="p"] have ex: "\<exists> k r. polydivide s p = (k,r)"
1959     by auto
1960   then obtain k r where kr: "polydivide s p = (k,r)"
1961     by blast
1962   from trans[OF polydivide_def[where s="s"and p="p", symmetric] kr]
1963     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
1964   have "(degree r = 0 \<or> degree r < degree p) \<and>
1965     (\<exists>nr. isnpolyh r nr) \<and> (\<exists>q n1. isnpolyh q n1 \<and> head p ^\<^sub>p k - 0 *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
1966     by blast
1967   with kr show ?thesis
1968     apply -
1969     apply (rule exI[where x="k"])
1970     apply (rule exI[where x="r"])
1971     apply simp
1972     done
1973 qed
1976 subsection {* More about polypoly and pnormal etc *}
1978 definition "isnonconstant p \<longleftrightarrow> \<not> isconstant p"
1980 lemma isnonconstant_pnormal_iff:
1981   assumes "isnonconstant p"
1982   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
1983 proof
1984   let ?p = "polypoly bs p"
1985   assume H: "pnormal ?p"
1986   have csz: "coefficients p \<noteq> []"
1987     using assms by (cases p) auto
1988   from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"] pnormal_last_nonzero[OF H]
1989   show "Ipoly bs (head p) \<noteq> 0"
1991 next
1992   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
1993   let ?p = "polypoly bs p"
1994   have csz: "coefficients p \<noteq> []"
1995     using assms by (cases p) auto
1996   then have pz: "?p \<noteq> []"
1998   then have lg: "length ?p > 0"
1999     by simp
2000   from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
2001   have lz: "last ?p \<noteq> 0"
2003   from pnormal_last_length[OF lg lz] show "pnormal ?p" .
2004 qed
2006 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
2007   unfolding isnonconstant_def
2008   apply (cases p)
2009   apply simp_all
2010   apply (rename_tac nat a, case_tac nat)
2011   apply auto
2012   done
2014 lemma isnonconstant_nonconstant:
2015   assumes "isnonconstant p"
2016   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
2017 proof
2018   let ?p = "polypoly bs p"
2019   assume nc: "nonconstant ?p"
2020   from isnonconstant_pnormal_iff[OF assms, of bs] nc
2021   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
2022     unfolding nonconstant_def by blast
2023 next
2024   let ?p = "polypoly bs p"
2025   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
2026   from isnonconstant_pnormal_iff[OF assms, of bs] h
2027   have pn: "pnormal ?p"
2028     by blast
2029   {
2030     fix x
2031     assume H: "?p = [x]"
2032     from H have "length (coefficients p) = 1"
2033       unfolding polypoly_def by auto
2034     with isnonconstant_coefficients_length[OF assms]
2035     have False by arith
2036   }
2037   then show "nonconstant ?p"
2038     using pn unfolding nonconstant_def by blast
2039 qed
2041 lemma pnormal_length: "p \<noteq> [] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
2042   apply (induct p)
2044   apply (case_tac "p = []")
2045   apply simp_all
2046   done
2048 lemma degree_degree:
2049   assumes "isnonconstant p"
2050   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
2051 proof
2052   let ?p = "polypoly bs p"
2053   assume H: "degree p = Polynomial_List.degree ?p"
2054   from isnonconstant_coefficients_length[OF assms] have pz: "?p \<noteq> []"
2055     unfolding polypoly_def by auto
2056   from H degree_coefficients[of p] isnonconstant_coefficients_length[OF assms]
2057   have lg: "length (pnormalize ?p) = length ?p"
2058     unfolding Polynomial_List.degree_def polypoly_def by simp
2059   then have "pnormal ?p"
2060     using pnormal_length[OF pz] by blast
2061   with isnonconstant_pnormal_iff[OF assms] show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
2062     by blast
2063 next
2064   let ?p = "polypoly bs p"
2065   assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
2066   with isnonconstant_pnormal_iff[OF assms] have "pnormal ?p"
2067     by blast
2068   with degree_coefficients[of p] isnonconstant_coefficients_length[OF assms]
2069   show "degree p = Polynomial_List.degree ?p"
2070     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
2071 qed
2074 section {* Swaps ; Division by a certain variable *}
2076 primrec swap :: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly"
2077 where
2078   "swap n m (C x) = C x"
2079 | "swap n m (Bound k) = Bound (if k = n then m else if k = m then n else k)"
2080 | "swap n m (Neg t) = Neg (swap n m t)"
2081 | "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
2082 | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
2083 | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
2084 | "swap n m (Pw t k) = Pw (swap n m t) k"
2085 | "swap n m (CN c k p) = CN (swap n m c) (if k = n then m else if k=m then n else k) (swap n m p)"
2087 lemma swap:
2088   assumes "n < length bs"
2089     and "m < length bs"
2090   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
2091 proof (induct t)
2092   case (Bound k)
2093   then show ?case
2094     using assms by simp
2095 next
2096   case (CN c k p)
2097   then show ?case
2098     using assms by simp
2099 qed simp_all
2101 lemma swap_swap_id [simp]: "swap n m (swap m n t) = t"
2102   by (induct t) simp_all
2104 lemma swap_commute: "swap n m p = swap m n p"
2105   by (induct p) simp_all
2107 lemma swap_same_id[simp]: "swap n n t = t"
2108   by (induct t) simp_all
2110 definition "swapnorm n m t = polynate (swap n m t)"
2112 lemma swapnorm:
2113   assumes nbs: "n < length bs"
2114     and mbs: "m < length bs"
2115   shows "((Ipoly bs (swapnorm n m t) :: 'a::{field_char_0,field_inverse_zero})) =
2116     Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
2117   using swap[OF assms] swapnorm_def by simp
2119 lemma swapnorm_isnpoly [simp]:
2120   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
2121   shows "isnpoly (swapnorm n m p)"
2122   unfolding swapnorm_def by simp
2124 definition "polydivideby n s p =
2125   (let
2126     ss = swapnorm 0 n s;
2127     sp = swapnorm 0 n p;
2129     (k, r) = polydivide ss sp
2130    in (k, swapnorm 0 n h, swapnorm 0 n r))"
2132 lemma swap_nz [simp]: "swap n m p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
2133   by (induct p) simp_all
2135 fun isweaknpoly :: "poly \<Rightarrow> bool"
2136 where
2137   "isweaknpoly (C c) = True"
2138 | "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
2139 | "isweaknpoly p = False"
2141 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
2142   by (induct p arbitrary: n0) auto
2144 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
2145   by (induct p) auto
2147 end