src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
 author blanchet Thu Sep 11 18:54:36 2014 +0200 (2014-09-11) changeset 58306 117ba6cbe414 parent 58259 52c35a59bbf5 child 58310 91ea607a34d8 permissions -rw-r--r--
renamed 'rep_datatype' to 'old_rep_datatype' (HOL)
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_new 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
78   "head (CN c 0 p) = head p"
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
109   "headconst (CN c n p) = headconst p"
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')"
131 | "polyadd a b = Add a b"
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
212         b = head s;
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"
254   by (simp add: INum_def)
256 lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"
257   by (simp  add: INum_def)
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
333       by (simp add: Let_def)
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)
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
544         apply (auto intro!: polyadd_normh)
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"
589           by (simp_all add: min_def)
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"
595           by (simp add: min_def)
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]
618           degreen_polyadd[OF norm(3,6) max1]
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"
657         by (simp_all add: min_def)
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
696 lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
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
704 proof (cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
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"
744   by (simp add: polysub_def)
746 lemma polysub_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
747   by (simp add: polysub_def polyneg_normh polyadd_normh)
749 lemma polysub_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polysub p q)"
750   using polyadd_norm polyneg_norm by (simp add: polysub_def)
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)"
785       by (simp add: Let_def)
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
793     using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp
794   }
795   moreover
796   {
797     assume even: "even (Suc n)"
798     have th: "(Suc (Suc 0)) * (Suc n div Suc (Suc 0)) = Suc n div 2 + Suc n div 2"
799       by arith
800     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d"
801       by (simp add: Let_def)
802     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
803       using "2.hyps" by (simp only: power_add) simp
804     finally have ?case using even_nat_div_two_times_two[OF even]
805       by (simp only: th)
806   }
807   ultimately show ?case by blast
808 qed
810 lemma polypow_normh:
811   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
812   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
813 proof (induct k arbitrary: n rule: polypow.induct)
814   case 1
815   then show ?case by auto
816 next
817   case (2 k n)
818   let ?q = "polypow (Suc k div 2) p"
819   let ?d = "polymul ?q ?q"
820   from 2 have th1: "isnpolyh ?q n" and th2: "isnpolyh p n"
821     by blast+
822   from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n"
823     by simp
824   from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n"
825     by simp
826   from dn on show ?case
827     by (simp add: Let_def)
828 qed
830 lemma polypow_norm:
831   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
832   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
833   by (simp add: polypow_normh isnpoly_def)
835 text{* Finally the whole normalization *}
837 lemma polynate [simp]:
838   "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0,field_inverse_zero})"
839   by (induct p rule:polynate.induct) auto
841 lemma polynate_norm[simp]:
842   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
843   shows "isnpoly (polynate p)"
844   by (induct p rule: polynate.induct)
845      (simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm,
846       simp_all add: isnpoly_def)
848 text{* shift1 *}
851 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
852   by (simp add: shift1_def)
854 lemma shift1_isnpoly:
855   assumes "isnpoly p"
856     and "p \<noteq> 0\<^sub>p"
857   shows "isnpoly (shift1 p) "
858   using assms by (simp add: shift1_def isnpoly_def)
860 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
861   by (simp add: shift1_def)
863 lemma funpow_shift1_isnpoly: "isnpoly p \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpoly (funpow n shift1 p)"
864   by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
866 lemma funpow_isnpolyh:
867   assumes "\<And>p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n"
868     and "isnpolyh p n"
869   shows "isnpolyh (funpow k f p) n"
870   using assms by (induct k arbitrary: p) auto
872 lemma funpow_shift1:
873   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field_inverse_zero}) =
874     Ipoly bs (Mul (Pw (Bound 0) n) p)"
875   by (induct n arbitrary: p) (simp_all add: shift1_isnpoly shift1)
877 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
878   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
880 lemma funpow_shift1_1:
881   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field_inverse_zero}) =
882     Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)"
883   by (simp add: funpow_shift1)
885 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
886   by (induct p rule: poly_cmul.induct) (auto simp add: field_simps)
889   assumes "isnpolyh p n"
890   shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) =
891     (Ipoly bs p :: 'a :: {field_char_0,field_inverse_zero})"
892   using assms
893 proof (induct p arbitrary: n rule: behead.induct)
894   case (1 c p n)
895   then have pn: "isnpolyh p n" by simp
896   from 1(1)[OF pn]
897   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
898   then show ?case using "1.hyps"
899     apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
900     apply (simp_all add: th[symmetric] field_simps)
901     done
902 qed (auto simp add: Let_def)
905   assumes "isnpolyh p n"
906   shows "isnpolyh (behead p) n"
907   using assms by (induct p rule: behead.induct) (auto simp add: Let_def isnpolyh_mono)
910 subsection {* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
912 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
913 proof (induct p arbitrary: n rule: poly.induct, auto)
914   case (goal1 c n p n')
915   then have "n = Suc (n - 1)"
916     by simp
917   then have "isnpolyh p (Suc (n - 1))"
918     using `isnpolyh p n` by simp
919   with goal1(2) show ?case
920     by simp
921 qed
923 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
924   by (induct p arbitrary: n0 rule: isconstant.induct) (auto simp add: isnpolyh_polybound0)
926 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
927   by (induct p) auto
929 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
930   apply (induct p arbitrary: n0)
931   apply auto
932   apply atomize
933   apply (rename_tac nat a b, erule_tac x = "Suc nat" in allE)
934   apply auto
935   done
937 lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
938   by (induct p  arbitrary: n0 rule: head.induct) (auto intro: isnpolyh_polybound0)
940 lemma polybound0_I:
941   assumes "polybound0 a"
942   shows "Ipoly (b # bs) a = Ipoly (b' # bs) a"
943   using assms by (induct a rule: poly.induct) auto
945 lemma polysubst0_I: "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b # bs) a) # bs) t"
946   by (induct t) simp_all
948 lemma polysubst0_I':
949   assumes "polybound0 a"
950   shows "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b' # bs) a) # bs) t"
951   by (induct t) (simp_all add: polybound0_I[OF assms, where b="b" and b'="b'"])
953 lemma decrpoly:
954   assumes "polybound0 t"
955   shows "Ipoly (x # bs) t = Ipoly bs (decrpoly t)"
956   using assms by (induct t rule: decrpoly.induct) simp_all
958 lemma polysubst0_polybound0:
959   assumes "polybound0 t"
960   shows "polybound0 (polysubst0 t a)"
961   using assms by (induct a rule: poly.induct) auto
963 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
964   by (induct p arbitrary: n rule: degree.induct) (auto simp add: isnpolyh_polybound0)
966 primrec maxindex :: "poly \<Rightarrow> nat"
967 where
968   "maxindex (Bound n) = n + 1"
969 | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
970 | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
971 | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
972 | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
973 | "maxindex (Neg p) = maxindex p"
974 | "maxindex (Pw p n) = maxindex p"
975 | "maxindex (C x) = 0"
977 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool"
978   where "wf_bs bs p \<longleftrightarrow> length bs \<ge> maxindex p"
980 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall>c \<in> set (coefficients p). wf_bs bs c"
981 proof (induct p rule: coefficients.induct)
982   case (1 c p)
983   show ?case
984   proof
985     fix x
986     assume xc: "x \<in> set (coefficients (CN c 0 p))"
987     then have "x = c \<or> x \<in> set (coefficients p)"
988       by simp
989     moreover
990     {
991       assume "x = c"
992       then have "wf_bs bs x"
993         using "1.prems" unfolding wf_bs_def by simp
994     }
995     moreover
996     {
997       assume H: "x \<in> set (coefficients p)"
998       from "1.prems" have "wf_bs bs p"
999         unfolding wf_bs_def by simp
1000       with "1.hyps" H have "wf_bs bs x"
1001         by blast
1002     }
1003     ultimately show "wf_bs bs x"
1004       by blast
1005   qed
1006 qed simp_all
1008 lemma maxindex_coefficients: "\<forall>c \<in> set (coefficients p). maxindex c \<le> maxindex p"
1009   by (induct p rule: coefficients.induct) auto
1011 lemma wf_bs_I: "wf_bs bs p \<Longrightarrow> Ipoly (bs @ bs') p = Ipoly bs p"
1012   unfolding wf_bs_def by (induct p) (auto simp add: nth_append)
1014 lemma take_maxindex_wf:
1015   assumes wf: "wf_bs bs p"
1016   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
1017 proof -
1018   let ?ip = "maxindex p"
1019   let ?tbs = "take ?ip bs"
1020   from wf have "length ?tbs = ?ip"
1021     unfolding wf_bs_def by simp
1022   then have wf': "wf_bs ?tbs p"
1023     unfolding wf_bs_def by  simp
1024   have eq: "bs = ?tbs @ drop ?ip bs"
1025     by simp
1026   from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis
1027     using eq by simp
1028 qed
1030 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
1031   by (induct p) auto
1033 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
1034   unfolding wf_bs_def by simp
1036 lemma wf_bs_insensitive': "wf_bs (x # bs) p = wf_bs (y # bs) p"
1037   unfolding wf_bs_def by simp
1039 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x # bs) p"
1040   by (induct p rule: coefficients.induct) (auto simp add: wf_bs_def)
1042 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
1043   by (induct p rule: coefficients.induct) simp_all
1045 lemma coefficients_head: "last (coefficients p) = head p"
1046   by (induct p rule: coefficients.induct) auto
1048 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x # bs) p"
1049   unfolding wf_bs_def by (induct p rule: decrpoly.induct) auto
1051 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists>ys. length (xs @ ys) = n"
1052   apply (rule exI[where x="replicate (n - length xs) z" for z])
1053   apply simp
1054   done
1056 lemma isnpolyh_Suc_const: "isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
1057   apply (cases p)
1058   apply auto
1059   apply (rename_tac nat a, case_tac "nat")
1060   apply simp_all
1061   done
1063 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
1064   unfolding wf_bs_def by (induct p q rule: polyadd.induct) (auto simp add: Let_def)
1066 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
1067   unfolding wf_bs_def
1068   apply (induct p q arbitrary: bs rule: polymul.induct)
1070   apply clarsimp
1071   apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
1072   apply auto
1073   done
1075 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
1076   unfolding wf_bs_def by (induct p rule: polyneg.induct) auto
1078 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
1079   unfolding polysub_def split_def fst_conv snd_conv
1080   using wf_bs_polyadd wf_bs_polyneg by blast
1083 subsection {* Canonicity of polynomial representation, see lemma isnpolyh_unique *}
1085 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
1086 definition "polypoly' bs p = map (Ipoly bs \<circ> decrpoly) (coefficients p)"
1087 definition "poly_nate bs p = map (Ipoly bs \<circ> decrpoly) (coefficients (polynate p))"
1089 lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall>q \<in> set (coefficients p). isnpolyh q n0"
1090 proof (induct p arbitrary: n0 rule: coefficients.induct)
1091   case (1 c p n0)
1092   have cp: "isnpolyh (CN c 0 p) n0"
1093     by fact
1094   then have norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
1095     by (auto simp add: isnpolyh_mono[where n'=0])
1096   from "1.hyps"[OF norm(2)] norm(1) norm(4) show ?case
1097     by simp
1098 qed auto
1100 lemma coefficients_isconst: "isnpolyh p n \<Longrightarrow> \<forall>q \<in> set (coefficients p). isconstant q"
1101   by (induct p arbitrary: n rule: coefficients.induct) (auto simp add: isnpolyh_Suc_const)
1103 lemma polypoly_polypoly':
1104   assumes np: "isnpolyh p n0"
1105   shows "polypoly (x # bs) p = polypoly' bs p"
1106 proof -
1107   let ?cf = "set (coefficients p)"
1108   from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
1109   {
1110     fix q
1111     assume q: "q \<in> ?cf"
1112     from q cn_norm have th: "isnpolyh q n0"
1113       by blast
1114     from coefficients_isconst[OF np] q have "isconstant q"
1115       by blast
1116     with isconstant_polybound0[OF th] have "polybound0 q"
1117       by blast
1118   }
1119   then have "\<forall>q \<in> ?cf. polybound0 q" ..
1120   then have "\<forall>q \<in> ?cf. Ipoly (x # bs) q = Ipoly bs (decrpoly q)"
1121     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
1122     by auto
1123   then show ?thesis
1124     unfolding polypoly_def polypoly'_def by simp
1125 qed
1127 lemma polypoly_poly:
1128   assumes "isnpolyh p n0"
1129   shows "Ipoly (x # bs) p = poly (polypoly (x # bs) p) x"
1130   using assms
1131   by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def)
1133 lemma polypoly'_poly:
1134   assumes "isnpolyh p n0"
1135   shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
1136   using polypoly_poly[OF assms, simplified polypoly_polypoly'[OF assms]] .
1139 lemma polypoly_poly_polybound0:
1140   assumes "isnpolyh p n0"
1141     and "polybound0 p"
1142   shows "polypoly bs p = [Ipoly bs p]"
1143   using assms
1144   unfolding polypoly_def
1145   apply (cases p)
1146   apply auto
1147   apply (rename_tac nat a, case_tac nat)
1148   apply auto
1149   done
1151 lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
1152   by (induct p rule: head.induct) auto
1154 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> headn p m = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
1155   by (cases p) auto
1158   by (induct p rule: head.induct) simp_all
1160 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> head p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
1163 lemma isnpolyh_zero_iff:
1164   assumes nq: "isnpolyh p n0"
1165     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})"
1166   shows "p = 0\<^sub>p"
1167   using nq eq
1168 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
1169   case less
1170   note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
1171   {
1172     assume nz: "maxindex p = 0"
1173     then obtain c where "p = C c"
1174       using np by (cases p) auto
1175     with zp np have "p = 0\<^sub>p"
1176       unfolding wf_bs_def by simp
1177   }
1178   moreover
1179   {
1180     assume nz: "maxindex p \<noteq> 0"
1181     let ?h = "head p"
1182     let ?hd = "decrpoly ?h"
1183     let ?ihd = "maxindex ?hd"
1185     have h: "isnpolyh ?h n0" "polybound0 ?h"
1186       by simp_all
1187     then have nhd: "isnpolyh ?hd (n0 - 1)"
1188       using decrpoly_normh by blast
1190     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
1191     have mihn: "maxindex ?h \<le> maxindex p"
1192       by auto
1193     with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < maxindex p"
1194       by auto
1195     {
1196       fix bs :: "'a list"
1197       assume bs: "wf_bs bs ?hd"
1198       let ?ts = "take ?ihd bs"
1199       let ?rs = "drop ?ihd bs"
1200       have ts: "wf_bs ?ts ?hd"
1201         using bs unfolding wf_bs_def by simp
1202       have bs_ts_eq: "?ts @ ?rs = bs"
1203         by simp
1204       from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x # ?ts) ?h"
1205         by simp
1206       from ihd_lt_n have "\<forall>x. length (x # ?ts) \<le> maxindex p"
1207         by simp
1208       with length_le_list_ex obtain xs where xs: "length ((x # ?ts) @ xs) = maxindex p"
1209         by blast
1210       then have "\<forall>x. wf_bs ((x # ?ts) @ xs) p"
1211         unfolding wf_bs_def by simp
1212       with zp have "\<forall>x. Ipoly ((x # ?ts) @ xs) p = 0"
1213         by blast
1214       then have "\<forall>x. Ipoly (x # (?ts @ xs)) p = 0"
1215         by simp
1216       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
1217       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"
1218         by simp
1219       then have "poly (polypoly' (?ts @ xs) p) = poly []"
1220         by auto
1221       then have "\<forall>c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
1222         using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
1223       with coefficients_head[of p, symmetric]
1224       have th0: "Ipoly (?ts @ xs) ?hd = 0"
1225         by simp
1226       from bs have wf'': "wf_bs ?ts ?hd"
1227         unfolding wf_bs_def by simp
1228       with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0"
1229         by simp
1230       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
1231         by simp
1232     }
1233     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
1234       by blast
1235     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p"
1236       by blast
1237     then have "?h = 0\<^sub>p" by simp
1238     with head_nz[OF np] have "p = 0\<^sub>p" by simp
1239   }
1240   ultimately show "p = 0\<^sub>p"
1241     by blast
1242 qed
1244 lemma isnpolyh_unique:
1245   assumes np: "isnpolyh p n0"
1246     and nq: "isnpolyh q n1"
1247   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"
1248 proof auto
1249   assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a) = \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
1250   then have "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)"
1251     by simp
1252   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)"
1253     using wf_bs_polysub[where p=p and q=q] by auto
1254   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq] show "p = q"
1255     by blast
1256 qed
1259 text{* consequences of unicity on the algorithms for polynomial normalization *}
1262   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1263     and np: "isnpolyh p n0"
1264     and nq: "isnpolyh q n1"
1265   shows "p +\<^sub>p q = q +\<^sub>p p"
1266   using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]]
1267   by simp
1269 lemma zero_normh: "isnpolyh 0\<^sub>p n"
1270   by simp
1272 lemma one_normh: "isnpolyh (1)\<^sub>p n"
1273   by simp
1276   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1277     and np: "isnpolyh p n0"
1278   shows "p +\<^sub>p 0\<^sub>p = p"
1279     and "0\<^sub>p +\<^sub>p p = p"
1280   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
1281     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
1283 lemma polymul_1[simp]:
1284   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1285     and np: "isnpolyh p n0"
1286   shows "p *\<^sub>p (1)\<^sub>p = p"
1287     and "(1)\<^sub>p *\<^sub>p p = p"
1288   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
1289     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
1291 lemma polymul_0[simp]:
1292   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1293     and np: "isnpolyh p n0"
1294   shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p"
1295     and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
1296   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
1297     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
1299 lemma polymul_commute:
1300   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1301     and np: "isnpolyh p n0"
1302     and nq: "isnpolyh q n1"
1303   shows "p *\<^sub>p q = q *\<^sub>p p"
1304   using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np],
1305     where ?'a = "'a::{field_char_0,field_inverse_zero, power}"]
1306   by simp
1308 declare polyneg_polyneg [simp]
1310 lemma isnpolyh_polynate_id [simp]:
1311   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1312     and np: "isnpolyh p n0"
1313   shows "polynate p = p"
1314   using isnpolyh_unique[where ?'a= "'a::{field_char_0,field_inverse_zero}",
1315       OF polynate_norm[of p, unfolded isnpoly_def] np]
1316     polynate[where ?'a = "'a::{field_char_0,field_inverse_zero}"]
1317   by simp
1319 lemma polynate_idempotent[simp]:
1320   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1321   shows "polynate (polynate p) = polynate p"
1322   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
1324 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
1325   unfolding poly_nate_def polypoly'_def ..
1327 lemma poly_nate_poly:
1328   "poly (poly_nate bs p) = (\<lambda>x:: 'a ::{field_char_0,field_inverse_zero}. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
1329   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
1330   unfolding poly_nate_polypoly' by auto
1333 subsection{* heads, degrees and all that *}
1335 lemma degree_eq_degreen0: "degree p = degreen p 0"
1336   by (induct p rule: degree.induct) simp_all
1338 lemma degree_polyneg:
1339   assumes "isnpolyh p n"
1340   shows "degree (polyneg p) = degree p"
1341   apply (induct p rule: polyneg.induct)
1342   using assms
1343   apply simp_all
1344   apply (case_tac na)
1345   apply auto
1346   done
1349   assumes np: "isnpolyh p n0"
1350     and nq: "isnpolyh q n1"
1351   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
1352   using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
1355 lemma degree_polysub:
1356   assumes np: "isnpolyh p n0"
1357     and nq: "isnpolyh q n1"
1358   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
1359 proof-
1360   from nq have nq': "isnpolyh (~\<^sub>p q) n1"
1361     using polyneg_normh by simp
1362   from degree_polyadd[OF np nq'] show ?thesis
1363     by (simp add: polysub_def degree_polyneg[OF nq])
1364 qed
1367   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1368     and np: "isnpolyh p n0"
1369     and nq: "isnpolyh q n1"
1370     and h: "head p = head q"
1371     and d: "degree p = degree q"
1372   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
1373   unfolding polysub_def split_def fst_conv snd_conv
1374   using np nq h d
1375 proof (induct p q rule: polyadd.induct)
1376   case (1 c c')
1377   then show ?case
1378     by (simp add: Nsub_def Nsub0[simplified Nsub_def])
1379 next
1380   case (2 c c' n' p')
1381   from 2 have "degree (C c) = degree (CN c' n' p')"
1382     by simp
1383   then have nz: "n' > 0"
1384     by (cases n') auto
1385   then have "head (CN c' n' p') = CN c' n' p'"
1386     by (cases n') auto
1387   with 2 show ?case
1388     by simp
1389 next
1390   case (3 c n p c')
1391   then have "degree (C c') = degree (CN c n p)"
1392     by simp
1393   then have nz: "n > 0"
1394     by (cases n) auto
1395   then have "head (CN c n p) = CN c n p"
1396     by (cases n) auto
1397   with 3 show ?case by simp
1398 next
1399   case (4 c n p c' n' p')
1400   then have H:
1401     "isnpolyh (CN c n p) n0"
1402     "isnpolyh (CN c' n' p') n1"
1403     "head (CN c n p) = head (CN c' n' p')"
1404     "degree (CN c n p) = degree (CN c' n' p')"
1405     by simp_all
1406   then have degc: "degree c = 0" and degc': "degree c' = 0"
1407     by simp_all
1408   then have degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
1409     using H(1-2) degree_polyneg by auto
1410   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"
1411     by simp_all
1412   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc'
1413   have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"
1414     by simp
1415   from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'"
1416     by auto
1417   have "n = n' \<or> n < n' \<or> n > n'"
1418     by arith
1419   moreover
1420   {
1421     assume nn': "n = n'"
1422     have "n = 0 \<or> n > 0" by arith
1423     moreover
1424     {
1425       assume nz: "n = 0"
1426       then have ?case using 4 nn'
1427         by (auto simp add: Let_def degcmc')
1428     }
1429     moreover
1430     {
1431       assume nz: "n > 0"
1432       with nn' H(3) have  cc': "c = c'" and pp': "p = p'"
1433         by (cases n, auto)+
1434       then have ?case
1435         using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv]
1436         using polysub_same_0[OF c'nh, simplified polysub_def]
1437         using nn' 4 by (simp add: Let_def)
1438     }
1439     ultimately have ?case by blast
1440   }
1441   moreover
1442   {
1443     assume nn': "n < n'"
1444     then have n'p: "n' > 0"
1445       by simp
1446     then have headcnp':"head (CN c' n' p') = CN c' n' p'"
1447       by (cases n') simp_all
1448     have degcnp': "degree (CN c' n' p') = 0"
1449       and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
1450       using 4 nn' by (cases n', simp_all)
1451     then have "n > 0"
1452       by (cases n) simp_all
1453     then have headcnp: "head (CN c n p) = CN c n p"
1454       by (cases n) auto
1455     from H(3) headcnp headcnp' nn' have ?case
1456       by auto
1457   }
1458   moreover
1459   {
1460     assume nn': "n > n'"
1461     then have np: "n > 0" by simp
1462     then have headcnp:"head (CN c n p) = CN c n p"
1463       by (cases n) simp_all
1464     from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)"
1465       by simp
1466     from np have degcnp: "degree (CN c n p) = 0"
1467       by (cases n) simp_all
1468     with degcnpeq have "n' > 0"
1469       by (cases n') simp_all
1470     then have headcnp': "head (CN c' n' p') = CN c' n' p'"
1471       by (cases n') auto
1472     from H(3) headcnp headcnp' nn' have ?case by auto
1473   }
1474   ultimately show ?case by blast
1475 qed auto
1477 lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
1478   by (induct p arbitrary: n0 rule: head.induct) (simp_all add: shift1_def)
1480 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
1481 proof (induct k arbitrary: n0 p)
1482   case 0
1483   then show ?case
1484     by auto
1485 next
1486   case (Suc k n0 p)
1487   then have "isnpolyh (shift1 p) 0"
1488     by (simp add: shift1_isnpolyh)
1489   with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
1490     and "head (shift1 p) = head p"
1492   then show ?case
1493     by (simp add: funpow_swap1)
1494 qed
1496 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
1497   by (simp add: shift1_def)
1499 lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
1500   by (induct k arbitrary: p) (auto simp add: shift1_degree)
1502 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
1503   by (induct n arbitrary: p) simp_all
1505 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
1506   by (induct p arbitrary: n rule: degree.induct) auto
1507 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
1508   by (induct p arbitrary: n rule: degreen.induct) auto
1509 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
1510   by (induct p arbitrary: n rule: degree.induct) auto
1512   by (induct p rule: head.induct) auto
1515   "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> polyadd p q = C c \<Longrightarrow> degree p = degree q"
1516   using polyadd_eq_const_degreen degree_eq_degreen0 by simp
1519   assumes np: "isnpolyh p n0"
1520     and nq: "isnpolyh q n1"
1521     and deg: "degree p \<noteq> degree q"
1522   shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
1523   using np nq deg
1524   apply (induct p q arbitrary: n0 n1 rule: polyadd.induct)
1525   apply simp_all
1526   apply (case_tac n', simp, simp)
1527   apply (case_tac n, simp, simp)
1528   apply (case_tac n, case_tac n', simp add: Let_def)
1530   apply (metis head_nz)
1531   apply (metis head_nz)
1532   apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
1533   done
1536   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1537   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"
1538 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
1539   case (2 c c' n' p' n0 n1)
1540   then have "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"
1542   then show ?case
1543     using 2 by (cases n') auto
1544 next
1545   case (3 c n p c' n0 n1)
1546   then have "isnpolyh (head (CN c n p)) n0" "isnormNum c'"
1548   then show ?case
1549     using 3 by (cases n) auto
1550 next
1551   case (4 c n p c' n' p' n0 n1)
1552   then have norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
1553     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
1554     by simp_all
1555   have "n < n' \<or> n' < n \<or> n = n'" by arith
1556   moreover
1557   {
1558     assume nn': "n < n'"
1559     then have ?case
1560       using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)]
1561       apply simp
1562       apply (cases n)
1563       apply simp
1564       apply (cases n')
1565       apply simp_all
1566       done
1567   }
1568   moreover {
1569     assume nn': "n'< n"
1570     then have ?case
1571       using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)]
1572       apply simp
1573       apply (cases n')
1574       apply simp
1575       apply (cases n)
1576       apply auto
1577       done
1578   }
1579   moreover
1580   {
1581     assume nn': "n' = n"
1582     from nn' polymul_normh[OF norm(5,4)]
1583     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
1584     from nn' polymul_normh[OF norm(5,3)] norm
1585     have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
1586     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
1587     have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
1588     from polyadd_normh[OF ncnpc' ncnpp0']
1589     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"
1590       by (simp add: min_def)
1591     {
1592       assume np: "n > 0"
1593       with nn' head_isnpolyh_Suc'[OF np nth]
1594         head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
1595       have ?case by simp
1596     }
1597     moreover
1598     {
1599       assume nz: "n = 0"
1600       from polymul_degreen[OF norm(5,4), where m="0"]
1601         polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
1602       norm(5,6) degree_npolyhCN[OF norm(6)]
1603     have dth: "degree (CN c 0 p *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))"
1604       by simp
1605     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'))"
1606       by simp
1607     from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
1608     have ?case
1609       using norm "4.hyps"(6)[OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)] nn' nz
1610       by simp
1611     }
1612     ultimately have ?case
1613       by (cases n) auto
1614   }
1615   ultimately show ?case by blast
1616 qed simp_all
1618 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
1619   by (induct p rule: degree.induct) auto
1621 lemma degree_head[simp]: "degree (head p) = 0"
1622   by (induct p rule: head.induct) auto
1624 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
1625   by (cases n) simp_all
1627 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
1628   by (cases n) simp_all
1631   "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> degree p \<noteq> degree q \<Longrightarrow>
1632     degree (polyadd p q) = max (degree p) (degree q)"
1633   using polyadd_different_degreen degree_eq_degreen0 by simp
1635 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
1636   by (induct p arbitrary: n0 rule: polyneg.induct) auto
1638 lemma degree_polymul:
1639   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1640     and np: "isnpolyh p n0"
1641     and nq: "isnpolyh q n1"
1642   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
1643   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
1645 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
1646   by (induct p arbitrary: n rule: degree.induct) auto
1648 lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head (polyneg p) = polyneg (head p)"
1649   by (induct p arbitrary: n rule: degree.induct) auto
1652 subsection {* Correctness of polynomial pseudo division *}
1654 lemma polydivide_aux_properties:
1655   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1656     and np: "isnpolyh p n0"
1657     and ns: "isnpolyh s n1"
1658     and ap: "head p = a"
1659     and ndp: "degree p = n"
1660     and pnz: "p \<noteq> 0\<^sub>p"
1661   shows "polydivide_aux a n p k s = (k', r) \<longrightarrow> k' \<ge> k \<and> (degree r = 0 \<or> degree r < degree p) \<and>
1662     (\<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)"
1663   using ns
1664 proof (induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
1665   case less
1666   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
1667   let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and>
1668     (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
1669   let ?b = "head s"
1670   let ?p' = "funpow (degree s - n) shift1 p"
1671   let ?xdn = "funpow (degree s - n) shift1 (1)\<^sub>p"
1672   let ?akk' = "a ^\<^sub>p (k' - k)"
1673   note ns = `isnpolyh s n1`
1674   from np have np0: "isnpolyh p 0"
1675     using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by simp
1676   have np': "isnpolyh ?p' 0"
1677     using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def
1678     by simp
1680     using funpow_shift1_head[OF np pnz] by simp
1681   from funpow_shift1_isnpoly[where p="(1)\<^sub>p"] have nxdn: "isnpolyh ?xdn 0"
1682     by (simp add: isnpoly_def)
1683   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
1684   have nakk':"isnpolyh ?akk' 0" by blast
1685   {
1686     assume sz: "s = 0\<^sub>p"
1687     then have ?ths
1688       using np polydivide_aux.simps
1689       apply clarsimp
1690       apply (rule exI[where x="0\<^sub>p"])
1691       apply simp
1692       done
1693   }
1694   moreover
1695   {
1696     assume sz: "s \<noteq> 0\<^sub>p"
1697     {
1698       assume dn: "degree s < n"
1699       then have "?ths"
1700         using ns ndp np polydivide_aux.simps
1701         apply auto
1702         apply (rule exI[where x="0\<^sub>p"])
1703         apply simp
1704         done
1705     }
1706     moreover
1707     {
1708       assume dn': "\<not> degree s < n"
1709       then have dn: "degree s \<ge> n"
1710         by arith
1711       have degsp': "degree s = degree ?p'"
1712         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"]
1713         by simp
1714       {
1715         assume ba: "?b = a"
1717           using ap headp' by simp
1718         have nr: "isnpolyh (s -\<^sub>p ?p') 0"
1719           using polysub_normh[OF ns np'] by simp
1720         from degree_polysub_samehead[OF ns np' headsp' degsp']
1721         have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p"
1722           by simp
1723         moreover
1724         {
1725           assume deglt:"degree (s -\<^sub>p ?p') < degree s"
1726           from polydivide_aux.simps sz dn' ba
1727           have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
1728             by (simp add: Let_def)
1729           {
1730             assume h1: "polydivide_aux a n p k s = (k', r)"
1731             from less(1)[OF deglt nr, of k k' r] trans[OF eq[symmetric] h1]
1732             have kk': "k \<le> k'"
1733               and nr: "\<exists>nr. isnpolyh r nr"
1734               and dr: "degree r = 0 \<or> degree r < degree p"
1735               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"
1736               by auto
1737             from q1 obtain q n1 where nq: "isnpolyh q n1"
1738               and asp: "a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"
1739               by blast
1740             from nr obtain nr where nr': "isnpolyh r nr"
1741               by blast
1742             from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0"
1743               by simp
1744             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
1745             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
1746             from polyadd_normh[OF polymul_normh[OF np
1747               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
1748             have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0"
1749               by simp
1750             from asp have "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
1751               Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)"
1752               by simp
1753             then have "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
1754               Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
1755               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
1756               by (simp add: field_simps)
1757             then have "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
1758               Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1759               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p *\<^sub>p p) +
1760               Ipoly bs p * Ipoly bs q + Ipoly bs r"
1761               by (auto simp only: funpow_shift1_1)
1762             then have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list.
1763               Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1764               Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p) +
1765               Ipoly bs q) + Ipoly bs r"
1766               by (simp add: field_simps)
1767             then have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list.
1768               Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1769               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)"
1770               by simp
1771             with isnpolyh_unique[OF nakks' nqr']
1772             have "a ^\<^sub>p (k' - k) *\<^sub>p s =
1773               p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r"
1774               by blast
1775             then have ?qths using nq'
1776               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q" in exI)
1777               apply (rule_tac x="0" in exI)
1778               apply simp
1779               done
1780             with kk' nr dr have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and>
1781               (\<exists>nr. isnpolyh r nr) \<and> ?qths"
1782               by blast
1783           }
1784           then have ?ths by blast
1785         }
1786         moreover
1787         {
1788           assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
1789           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}"]
1790           have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list. Ipoly bs s = Ipoly bs ?p'"
1791             by simp
1792           then have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list. Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)"
1793             using np nxdn
1794             apply simp
1795             apply (simp only: funpow_shift1_1)
1796             apply simp
1797             done
1798           then have sp': "s = ?xdn *\<^sub>p p"
1799             using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]]
1800             by blast
1801           {
1802             assume h1: "polydivide_aux a n p k s = (k', r)"
1803             from polydivide_aux.simps sz dn' ba
1804             have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
1805               by (simp add: Let_def)
1806             also have "\<dots> = (k,0\<^sub>p)"
1807               using polydivide_aux.simps spz by simp
1808             finally have "(k', r) = (k, 0\<^sub>p)"
1809               using h1 by simp
1810             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
1811               polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
1812               apply auto
1813               apply (rule exI[where x="?xdn"])
1814               apply (auto simp add: polymul_commute[of p])
1815               done
1816           }
1817         }
1818         ultimately have ?ths by blast
1819       }
1820       moreover
1821       {
1822         assume ba: "?b \<noteq> a"
1823         from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
1824           polymul_normh[OF head_isnpolyh[OF ns] np']]
1825         have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
1826           by (simp add: min_def)
1827         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
1828           using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
1829             polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
1830             funpow_shift1_nz[OF pnz]
1831           by simp_all
1833           polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
1834         have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
1837           by (simp add: ap)
1838         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
1839           head_nz[OF np] pnz sz ap[symmetric]
1840           funpow_shift1_nz[OF pnz, where n="degree s - n"]
1841           polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"] head_nz[OF ns]
1842           ndp dn
1843         have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p')"
1844           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
1845         {
1846           assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
1847           from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns]
1848             polymul_normh[OF head_isnpolyh[OF ns]np']] ap
1849           have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
1850             by simp
1851           {
1852             assume h1:"polydivide_aux a n p k s = (k', r)"
1853             from h1 polydivide_aux.simps sz dn' ba
1854             have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
1855               by (simp add: Let_def)
1856             with less(1)[OF dth nasbp', of "Suc k" k' r]
1857             obtain q nq nr where kk': "Suc k \<le> k'"
1858               and nr: "isnpolyh r nr"
1859               and nq: "isnpolyh q nq"
1860               and dr: "degree r = 0 \<or> degree r < degree p"
1861               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"
1862               by auto
1863             from kk' have kk'': "Suc (k' - Suc k) = k' - k"
1864               by arith
1865             {
1866               fix bs :: "'a::{field_char_0,field_inverse_zero} list"
1867               from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
1868               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)"
1869                 by simp
1870               then have "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s =
1871                 Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
1872                 by (simp add: field_simps)
1873               then have "Ipoly bs a ^ (k' - k)  * Ipoly bs s =
1874                 Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
1875                 by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
1876               then have "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1877                 Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
1878                 by (simp add: field_simps)
1879             }
1880             then have ieq:"\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
1881                 Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1882                 Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)"
1883               by auto
1884             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
1885             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]]
1886             have nqw: "isnpolyh ?q 0"
1887               by simp
1888             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]]
1889             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"
1890               by blast
1891             from dr kk' nr h1 asth nqw have ?ths
1892               apply simp
1893               apply (rule conjI)
1894               apply (rule exI[where x="nr"], simp)
1895               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
1896               apply (rule exI[where x="0"], simp)
1897               done
1898           }
1899           then have ?ths by blast
1900         }
1901         moreover
1902         {
1903           assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
1904           {
1905             fix bs :: "'a::{field_char_0,field_inverse_zero} list"
1906             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
1907             have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'"
1908               by simp
1909             then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
1910               by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
1911             then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))"
1912               by simp
1913           }
1914           then have hth: "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
1915             Ipoly bs (a *\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
1916           from hth have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
1917             using isnpolyh_unique[where ?'a = "'a::{field_char_0,field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
1918                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
1919               simplified ap]
1920             by simp
1921           {
1922             assume h1: "polydivide_aux a n p k s = (k', r)"
1923             from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
1924             have "(k', r) = (Suc k, 0\<^sub>p)"
1925               by (simp add: Let_def)
1926             with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
1927               polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
1928             have ?ths
1929               apply (clarsimp simp add: Let_def)
1930               apply (rule exI[where x="?b *\<^sub>p ?xdn"])
1931               apply simp
1932               apply (rule exI[where x="0"], simp)
1933               done
1934           }
1935           then have ?ths by blast
1936         }
1937         ultimately have ?ths
1938           using degree_polysub_samehead[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns] polymul_normh[OF head_isnpolyh[OF ns] np'] hdth degth] polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
1939             head_nz[OF np] pnz sz ap[symmetric]
1940           by (auto simp add: degree_eq_degreen0[symmetric])
1941       }
1942       ultimately have ?ths by blast
1943     }
1944     ultimately have ?ths by blast
1945   }
1946   ultimately show ?ths by blast
1947 qed
1949 lemma polydivide_properties:
1950   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
1951     and np: "isnpolyh p n0"
1952     and ns: "isnpolyh s n1"
1953     and pnz: "p \<noteq> 0\<^sub>p"
1954   shows "\<exists>k r. polydivide s p = (k, r) \<and>
1955     (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) \<and>
1956     (\<exists>q n1. isnpolyh q n1 \<and> polypow k (head p) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
1957 proof -
1958   have trv: "head p = head p" "degree p = degree p"
1959     by simp_all
1960   from polydivide_def[where s="s" and p="p"] have ex: "\<exists> k r. polydivide s p = (k,r)"
1961     by auto
1962   then obtain k r where kr: "polydivide s p = (k,r)"
1963     by blast
1964   from trans[OF polydivide_def[where s="s"and p="p", symmetric] kr]
1965     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
1966   have "(degree r = 0 \<or> degree r < degree p) \<and>
1967     (\<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)"
1968     by blast
1969   with kr show ?thesis
1970     apply -
1971     apply (rule exI[where x="k"])
1972     apply (rule exI[where x="r"])
1973     apply simp
1974     done
1975 qed
1978 subsection {* More about polypoly and pnormal etc *}
1980 definition "isnonconstant p \<longleftrightarrow> \<not> isconstant p"
1982 lemma isnonconstant_pnormal_iff:
1983   assumes "isnonconstant p"
1984   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
1985 proof
1986   let ?p = "polypoly bs p"
1987   assume H: "pnormal ?p"
1988   have csz: "coefficients p \<noteq> []"
1989     using assms by (cases p) auto
1990   from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"] pnormal_last_nonzero[OF H]
1991   show "Ipoly bs (head p) \<noteq> 0"
1992     by (simp add: polypoly_def)
1993 next
1994   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
1995   let ?p = "polypoly bs p"
1996   have csz: "coefficients p \<noteq> []"
1997     using assms by (cases p) auto
1998   then have pz: "?p \<noteq> []"
1999     by (simp add: polypoly_def)
2000   then have lg: "length ?p > 0"
2001     by simp
2002   from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
2003   have lz: "last ?p \<noteq> 0"
2004     by (simp add: polypoly_def)
2005   from pnormal_last_length[OF lg lz] show "pnormal ?p" .
2006 qed
2008 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
2009   unfolding isnonconstant_def
2010   apply (cases p)
2011   apply simp_all
2012   apply (rename_tac nat a, case_tac nat)
2013   apply auto
2014   done
2016 lemma isnonconstant_nonconstant:
2017   assumes "isnonconstant p"
2018   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
2019 proof
2020   let ?p = "polypoly bs p"
2021   assume nc: "nonconstant ?p"
2022   from isnonconstant_pnormal_iff[OF assms, of bs] nc
2023   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
2024     unfolding nonconstant_def by blast
2025 next
2026   let ?p = "polypoly bs p"
2027   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
2028   from isnonconstant_pnormal_iff[OF assms, of bs] h
2029   have pn: "pnormal ?p"
2030     by blast
2031   {
2032     fix x
2033     assume H: "?p = [x]"
2034     from H have "length (coefficients p) = 1"
2035       unfolding polypoly_def by auto
2036     with isnonconstant_coefficients_length[OF assms]
2037     have False by arith
2038   }
2039   then show "nonconstant ?p"
2040     using pn unfolding nonconstant_def by blast
2041 qed
2043 lemma pnormal_length: "p \<noteq> [] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
2044   apply (induct p)
2045   apply (simp_all add: pnormal_def)
2046   apply (case_tac "p = []")
2047   apply simp_all
2048   done
2050 lemma degree_degree:
2051   assumes "isnonconstant p"
2052   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
2053 proof
2054   let ?p = "polypoly bs p"
2055   assume H: "degree p = Polynomial_List.degree ?p"
2056   from isnonconstant_coefficients_length[OF assms] have pz: "?p \<noteq> []"
2057     unfolding polypoly_def by auto
2058   from H degree_coefficients[of p] isnonconstant_coefficients_length[OF assms]
2059   have lg: "length (pnormalize ?p) = length ?p"
2060     unfolding Polynomial_List.degree_def polypoly_def by simp
2061   then have "pnormal ?p"
2062     using pnormal_length[OF pz] by blast
2063   with isnonconstant_pnormal_iff[OF assms] show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
2064     by blast
2065 next
2066   let ?p = "polypoly bs p"
2067   assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
2068   with isnonconstant_pnormal_iff[OF assms] have "pnormal ?p"
2069     by blast
2070   with degree_coefficients[of p] isnonconstant_coefficients_length[OF assms]
2071   show "degree p = Polynomial_List.degree ?p"
2072     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
2073 qed
2076 section {* Swaps ; Division by a certain variable *}
2078 primrec swap :: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly"
2079 where
2080   "swap n m (C x) = C x"
2081 | "swap n m (Bound k) = Bound (if k = n then m else if k = m then n else k)"
2082 | "swap n m (Neg t) = Neg (swap n m t)"
2083 | "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
2084 | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
2085 | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
2086 | "swap n m (Pw t k) = Pw (swap n m t) k"
2087 | "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)"
2089 lemma swap:
2090   assumes "n < length bs"
2091     and "m < length bs"
2092   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
2093 proof (induct t)
2094   case (Bound k)
2095   then show ?case
2096     using assms by simp
2097 next
2098   case (CN c k p)
2099   then show ?case
2100     using assms by simp
2101 qed simp_all
2103 lemma swap_swap_id [simp]: "swap n m (swap m n t) = t"
2104   by (induct t) simp_all
2106 lemma swap_commute: "swap n m p = swap m n p"
2107   by (induct p) simp_all
2109 lemma swap_same_id[simp]: "swap n n t = t"
2110   by (induct t) simp_all
2112 definition "swapnorm n m t = polynate (swap n m t)"
2114 lemma swapnorm:
2115   assumes nbs: "n < length bs"
2116     and mbs: "m < length bs"
2117   shows "((Ipoly bs (swapnorm n m t) :: 'a::{field_char_0,field_inverse_zero})) =
2118     Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
2119   using swap[OF assms] swapnorm_def by simp
2121 lemma swapnorm_isnpoly [simp]:
2122   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
2123   shows "isnpoly (swapnorm n m p)"
2124   unfolding swapnorm_def by simp
2126 definition "polydivideby n s p =
2127   (let
2128     ss = swapnorm 0 n s;
2129     sp = swapnorm 0 n p;
2130     h = head sp;
2131     (k, r) = polydivide ss sp
2132    in (k, swapnorm 0 n h, swapnorm 0 n r))"
2134 lemma swap_nz [simp]: "swap n m p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
2135   by (induct p) simp_all
2137 fun isweaknpoly :: "poly \<Rightarrow> bool"
2138 where
2139   "isweaknpoly (C c) = True"
2140 | "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
2141 | "isweaknpoly p = False"
2143 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
2144   by (induct p arbitrary: n0) auto
2146 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
2147   by (induct p) auto
2149 end