src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
 author wenzelm Mon Jul 15 11:29:19 2013 +0200 (2013-07-15) changeset 52658 1e7896c7f781 parent 50282 fe4d4bb9f4c2 child 52803 bcaa5bbf7e6b permissions -rw-r--r--
tuned specifications and proofs;
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 "~~/src/HOL/Library/Abstract_Rat" 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) = True"
36 | "polybound0 (Bound n) = (n>0)"
37 | "polybound0 (Neg a) = polybound0 a"
38 | "polybound0 (Add a b) = (polybound0 a \<and> polybound0 b)"
39 | "polybound0 (Sub a b) = (polybound0 a \<and> polybound0 b)"
40 | "polybound0 (Mul a b) = (polybound0 a \<and> polybound0 b)"
41 | "polybound0 (Pw p n) = (polybound0 p)"
42 | "polybound0 (CN c n p) = (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) = (if n=0 then Add (polysubst0 t c) (Mul t (polysubst0 t p))
54                              else CN (polysubst0 t c) n (polysubst0 t p))"
56 fun decrpoly:: "poly \<Rightarrow> poly"
57 where
58   "decrpoly (Bound n) = Bound (n - 1)"
59 | "decrpoly (Neg a) = Neg (decrpoly a)"
60 | "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"
61 | "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
62 | "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
63 | "decrpoly (Pw p n) = Pw (decrpoly p) n"
64 | "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
65 | "decrpoly a = a"
68 subsection{* Degrees and heads and coefficients *}
70 fun degree:: "poly \<Rightarrow> nat"
71 where
72   "degree (CN c 0 p) = 1 + degree p"
73 | "degree p = 0"
75 fun head:: "poly \<Rightarrow> poly"
76 where
78 | "head p = p"
80 (* More general notions of degree and head *)
81 fun degreen:: "poly \<Rightarrow> nat \<Rightarrow> nat"
82 where
83   "degreen (CN c n p) = (\<lambda>m. if n=m then 1 + degreen p n else 0)"
84  |"degreen p = (\<lambda>m. 0)"
86 fun headn:: "poly \<Rightarrow> nat \<Rightarrow> poly"
87 where
88   "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
89 | "headn p = (\<lambda>m. p)"
91 fun coefficients:: "poly \<Rightarrow> poly list"
92 where
93   "coefficients (CN c 0 p) = c#(coefficients p)"
94 | "coefficients p = [p]"
96 fun isconstant:: "poly \<Rightarrow> bool"
97 where
98   "isconstant (CN c 0 p) = False"
99 | "isconstant p = True"
101 fun behead:: "poly \<Rightarrow> poly"
102 where
103   "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
104 | "behead p = 0\<^sub>p"
106 fun headconst:: "poly \<Rightarrow> Num"
107 where
109 | "headconst (C n) = n"
112 subsection{* Operations for normalization *}
114 declare if_cong[fundef_cong del]
115 declare let_cong[fundef_cong del]
117 fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
118 where
119   "polyadd (C c) (C c') = C (c+\<^sub>Nc')"
120 |  "polyadd (C c) (CN c' n' p') = CN (polyadd (C c) c') n' p'"
121 | "polyadd (CN c n p) (C c') = CN (polyadd c (C c')) n p"
122 | "polyadd (CN c n p) (CN c' n' p') =
123     (if n < n' then CN (polyadd c (CN c' n' p')) n p
124      else if n'<n then CN (polyadd (CN c n p) c') n' p'
125      else (let cc' = polyadd c c' ;
126                pp' = polyadd p p'
127            in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
131 fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
132 where
133   "polyneg (C c) = C (~\<^sub>N c)"
134 | "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
135 | "polyneg a = Neg a"
137 definition polysub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
138   where "p -\<^sub>p q = polyadd p (polyneg q)"
140 fun polymul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
141 where
142   "polymul (C c) (C c') = C (c*\<^sub>Nc')"
143 | "polymul (C c) (CN c' n' p') =
144       (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul (C c) c') n' (polymul (C c) p'))"
145 | "polymul (CN c n p) (C c') =
146       (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul c (C c')) n (polymul p (C c')))"
147 | "polymul (CN c n p) (CN c' n' p') =
148   (if n<n' then CN (polymul c (CN c' n' p')) n (polymul p (CN c' n' p'))
149   else if n' < n
150   then CN (polymul (CN c n p) c') n' (polymul (CN c n p) p')
151   else polyadd (polymul (CN c n p) c') (CN 0\<^sub>p n' (polymul (CN c n p) p')))"
152 | "polymul a b = Mul a b"
154 declare if_cong[fundef_cong]
155 declare let_cong[fundef_cong]
157 fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
158 where
159   "polypow 0 = (\<lambda>p. (1)\<^sub>p)"
160 | "polypow n = (\<lambda>p. let q = polypow (n div 2) p ; d = polymul q q in
161                     if even n then d else polymul p d)"
163 abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
164   where "a ^\<^sub>p k \<equiv> polypow k a"
166 function polynate :: "poly \<Rightarrow> poly"
167 where
168   "polynate (Bound n) = CN 0\<^sub>p n (1)\<^sub>p"
169 | "polynate (Add p q) = (polynate p +\<^sub>p polynate q)"
170 | "polynate (Sub p q) = (polynate p -\<^sub>p polynate q)"
171 | "polynate (Mul p q) = (polynate p *\<^sub>p polynate q)"
172 | "polynate (Neg p) = (~\<^sub>p (polynate p))"
173 | "polynate (Pw p n) = ((polynate p) ^\<^sub>p n)"
174 | "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
175 | "polynate (C c) = C (normNum c)"
176 by pat_completeness auto
177 termination by (relation "measure polysize") auto
179 fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly"
180 where
181   "poly_cmul y (C x) = C (y *\<^sub>N x)"
182 | "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
183 | "poly_cmul y p = C y *\<^sub>p p"
185 definition monic :: "poly \<Rightarrow> (poly \<times> bool)" where
186   "monic p \<equiv> (let h = headconst p in if h = 0\<^sub>N then (p,False) else ((C (Ninv h)) *\<^sub>p p, 0>\<^sub>N h))"
189 subsection{* Pseudo-division *}
191 definition shift1 :: "poly \<Rightarrow> poly"
192   where "shift1 p \<equiv> CN 0\<^sub>p 0 p"
194 abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
195   where "funpow \<equiv> compow"
197 partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
198 where
199   "polydivide_aux a n p k s =
200     (if s = 0\<^sub>p then (k,s)
201     else (let b = head s; m = degree s in
202     (if m < n then (k,s) else
203     (let p'= funpow (m - n) shift1 p in
204     (if a = b then polydivide_aux a n p k (s -\<^sub>p p')
205     else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))))))"
207 definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)"
208   where "polydivide s p \<equiv> polydivide_aux (head p) (degree p) p 0 s"
210 fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly"
211 where
212   "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
213 | "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
215 fun poly_deriv :: "poly \<Rightarrow> poly"
216 where
217   "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
218 | "poly_deriv p = 0\<^sub>p"
221 subsection{* Semantics of the polynomial representation *}
223 primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0, field_inverse_zero, power}" where
224   "Ipoly bs (C c) = INum c"
225 | "Ipoly bs (Bound n) = bs!n"
226 | "Ipoly bs (Neg a) = - Ipoly bs a"
227 | "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
228 | "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
229 | "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
230 | "Ipoly bs (Pw t n) = (Ipoly bs t) ^ n"
231 | "Ipoly bs (CN c n p) = (Ipoly bs c) + (bs!n)*(Ipoly bs p)"
233 abbreviation
234   Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0, field_inverse_zero, power}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
235   where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"
237 lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i"
239 lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"
242 lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
245 subsection {* Normal form and normalization *}
247 fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
248 where
249   "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
250 | "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))"
251 | "isnpolyh p = (\<lambda>k. False)"
253 lemma isnpolyh_mono: "\<lbrakk>n' \<le> n ; isnpolyh p n\<rbrakk> \<Longrightarrow> isnpolyh p n'"
254   by (induct p rule: isnpolyh.induct) auto
256 definition isnpoly :: "poly \<Rightarrow> bool"
257   where "isnpoly p \<equiv> isnpolyh p 0"
259 text{* polyadd preserves normal forms *}
261 lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk>
262       \<Longrightarrow> isnpolyh (polyadd p q) (min n0 n1)"
263 proof(induct p q arbitrary: n0 n1 rule: polyadd.induct)
264   case (2 ab c' n' p' n0 n1)
265   from 2 have  th1: "isnpolyh (C ab) (Suc n')" by simp
266   from 2(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
267   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
268   with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^sub>p c') (Suc n')" by simp
269   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
270   thus ?case using 2 th3 by simp
271 next
272   case (3 c' n' p' ab n1 n0)
273   from 3 have  th1: "isnpolyh (C ab) (Suc n')" by simp
274   from 3(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
275   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
276   with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (Suc n')" by simp
277   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
278   thus ?case using 3 th3 by simp
279 next
280   case (4 c n p c' n' p' n0 n1)
281   hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
282   from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all
283   from 4 have ngen0: "n \<ge> n0" by simp
284   from 4 have n'gen1: "n' \<ge> n1" by simp
285   have "n < n' \<or> n' < n \<or> n = n'" by auto
286   moreover {assume eq: "n = n'"
287     with "4.hyps"(3)[OF nc nc']
288     have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
289     hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
290       using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
291     from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
292     have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
293     from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}
294   moreover {assume lt: "n < n'"
295     have "min n0 n1 \<le> n0" by simp
296     with 4 lt have th1:"min n0 n1 \<le> n" by auto
297     from 4 have th21: "isnpolyh c (Suc n)" by simp
298     from 4 have th22: "isnpolyh (CN c' n' p') n'" by simp
299     from lt have th23: "min (Suc n) n' = Suc n" by arith
300     from "4.hyps"(1)[OF th21 th22]
301     have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)" using th23 by simp
302     with 4 lt th1 have ?case by simp }
303   moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
304     have "min n0 n1 \<le> n1"  by simp
305     with 4 gt have th1:"min n0 n1 \<le> n'" by auto
306     from 4 have th21: "isnpolyh c' (Suc n')" by simp_all
307     from 4 have th22: "isnpolyh (CN c n p) n" by simp
308     from gt have th23: "min n (Suc n') = Suc n'" by arith
309     from "4.hyps"(2)[OF th22 th21]
310     have "isnpolyh (polyadd (CN c n p) c') (Suc n')" using th23 by simp
311     with 4 gt th1 have ?case by simp}
312       ultimately show ?case by blast
313 qed auto
315 lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
316   by (induct p q rule: polyadd.induct)
317     (auto simp add: Let_def field_simps distrib_left[symmetric] simp del: distrib_left)
319 lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd p q)"
320   using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
322 text{* The degree of addition and other general lemmas needed for the normal form of polymul *}
325   "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow>
326   degreen (polyadd p q) m = max (degreen p m) (degreen q m)"
327 proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
328   case (4 c n p c' n' p' m n0 n1)
329   have "n' = n \<or> n < n' \<or> n' < n" by arith
330   thus ?case
331   proof (elim disjE)
332     assume [simp]: "n' = n"
333     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
334     show ?thesis by (auto simp: Let_def)
335   next
336     assume "n < n'"
337     with 4 show ?thesis by auto
338   next
339     assume "n' < n"
340     with 4 show ?thesis by auto
341   qed
342 qed auto
344 lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
345   by (induct p arbitrary: n rule: headn.induct) auto
346 lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
347   by (induct p arbitrary: n rule: degree.induct) auto
348 lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
349   by (induct p arbitrary: n rule: degreen.induct) auto
351 lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
352   by (induct p arbitrary: n rule: degree.induct) auto
354 lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
355   using degree_isnpolyh_Suc by auto
356 lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
357   using degreen_0 by auto
361   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1"
362   shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
363   using np nq m
364 proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
365   case (2 c c' n' p' n0 n1) thus ?case  by (cases n') simp_all
366 next
367   case (3 c n p c' n0 n1) thus ?case by (cases n) auto
368 next
369   case (4 c n p c' n' p' n0 n1 m)
370   have "n' = n \<or> n < n' \<or> n' < n" by arith
371   thus ?case
372   proof (elim disjE)
373     assume [simp]: "n' = n"
374     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
375     show ?thesis by (auto simp: Let_def)
376   qed simp_all
377 qed auto
379 lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd p q = C c\<rbrakk>
380   \<Longrightarrow> degreen p m = degreen q m"
381 proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
382   case (4 c n p c' n' p' m n0 n1 x)
383   {assume nn': "n' < n" hence ?case using 4 by simp}
384   moreover
385   {assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
386     moreover {assume "n < n'" with 4 have ?case by simp }
387     moreover {assume eq: "n = n'" hence ?case using 4
388         apply (cases "p +\<^sub>p p' = 0\<^sub>p")
389         apply (auto simp add: Let_def)
390         apply blast
391         done
392       }
393     ultimately have ?case by blast}
394   ultimately show ?case by blast
395 qed simp_all
397 lemma polymul_properties:
398   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
399     and np: "isnpolyh p n0"
400     and nq: "isnpolyh q n1"
401     and m: "m \<le> min n0 n1"
402   shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
403     and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)"
404     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)"
405   using np nq m
406 proof (induct p q arbitrary: n0 n1 m rule: polymul.induct)
407   case (2 c c' n' p')
408   { case (1 n0 n1)
409     with "2.hyps"(4-6)[of n' n' n']
410       and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']
411     show ?case by (auto simp add: min_def)
412   next
413     case (2 n0 n1) thus ?case by auto
414   next
415     case (3 n0 n1) thus ?case  using "2.hyps" by auto }
416 next
417   case (3 c n p c')
418   { case (1 n0 n1)
419     with "3.hyps"(4-6)[of n n n]
420       "3.hyps"(1-3)[of "Suc n" "Suc n" n]
421     show ?case by (auto simp add: min_def)
422   next
423     case (2 n0 n1) thus ?case by auto
424   next
425     case (3 n0 n1) thus ?case  using "3.hyps" by auto }
426 next
427   case (4 c n p c' n' p')
428   let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
429     {
430       case (1 n0 n1)
431       hence cnp: "isnpolyh ?cnp n" and cnp': "isnpolyh ?cnp' n'"
432         and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)"
433         and np': "isnpolyh p' n'" and nc': "isnpolyh c' (Suc n')"
434         and nn0: "n \<ge> n0" and nn1:"n' \<ge> n1"
435         by simp_all
436       { assume "n < n'"
437         with "4.hyps"(4-5)[OF np cnp', of n]
438           "4.hyps"(1)[OF nc cnp', of n] nn0 cnp
439         have ?case by (simp add: min_def)
440       } moreover {
441         assume "n' < n"
442         with "4.hyps"(16-17)[OF cnp np', of "n'"]
443           "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
444         have ?case
445           by (cases "Suc n' = n") (simp_all add: min_def)
446       } moreover {
447         assume "n' = n"
448         with "4.hyps"(16-17)[OF cnp np', of n]
449           "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0
450         have ?case
452           apply (simp_all add: min_def isnpolyh_mono[OF nn0])
453           done
454       }
455       ultimately show ?case by arith
456     next
457       fix n0 n1 m
458       assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1"
459       and m: "m \<le> min n0 n1"
460       let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
461       let ?d1 = "degreen ?cnp m"
462       let ?d2 = "degreen ?cnp' m"
463       let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
464       have "n'<n \<or> n < n' \<or> n' = n" by auto
465       moreover
466       {assume "n' < n \<or> n < n'"
467         with "4.hyps"(3,6,18) np np' m
468         have ?eq by auto }
469       moreover
470       {assume nn': "n' = n" hence nn:"\<not> n' < n \<and> \<not> n < n'" by arith
471         from "4.hyps"(16,18)[of n n' n]
472           "4.hyps"(13,14)[of n "Suc n'" n]
473           np np' nn'
474         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
475           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
476           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
477           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
478         {assume mn: "m = n"
479           from "4.hyps"(17,18)[OF norm(1,4), of n]
480             "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
481           have degs:  "degreen (?cnp *\<^sub>p c') n =
482             (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"
483             "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n" by (simp_all add: min_def)
484           from degs norm
485           have th1: "degreen(?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" by simp
486           hence neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
487             by simp
488           have nmin: "n \<le> min n n" by (simp add: min_def)
489           from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
490           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 = degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
491           from "4.hyps"(16-18)[OF norm(1,4), of n]
492             "4.hyps"(13-15)[OF norm(1,2), of n]
493             mn norm m nn' deg
494           have ?eq by simp}
495         moreover
496         {assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
497           from nn' m np have max1: "m \<le> max n n"  by simp
498           hence min1: "m \<le> min n n" by simp
499           hence min2: "m \<le> min n (Suc n)" by simp
500           from "4.hyps"(16-18)[OF norm(1,4) min1]
501             "4.hyps"(13-15)[OF norm(1,2) min2]
504           have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m
505             \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
506             using mn nn' np np' by simp
507           with "4.hyps"(16-18)[OF norm(1,4) min1]
508             "4.hyps"(13-15)[OF norm(1,2) min2]
509             degreen_0[OF norm(3) mn']
510           have ?eq using nn' mn np np' by clarsimp}
511         ultimately have ?eq by blast}
512       ultimately show ?eq by blast}
513     { case (2 n0 n1)
514       hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1"
515         and m: "m \<le> min n0 n1" by simp_all
516       hence mn: "m \<le> n" by simp
517       let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
518       {assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
519         hence nn: "\<not>n' < n \<and> \<not> n<n'" by simp
520         from "4.hyps"(16-18) [of n n n]
521           "4.hyps"(13-15)[of n "Suc n" n]
522           np np' C(2) mn
523         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
524           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
525           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
526           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
527           "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
528             "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
531           from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
532           have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
533             using norm by simp
534         from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"]  degneq
535         have "False" by simp }
536       thus ?case using "4.hyps" by clarsimp}
537 qed auto
539 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"
540   by (induct p q rule: polymul.induct) (auto simp add: field_simps)
542 lemma polymul_normh:
543   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
544   shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
545   using polymul_properties(1)  by blast
547 lemma polymul_eq0_iff:
548   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
549   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p) "
550   using polymul_properties(2)  by blast
552 lemma polymul_degreen:  (* FIXME duplicate? *)
553   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
554   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow>
555     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)"
556   using polymul_properties(3) by blast
558 lemma polymul_norm:
559   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
560   shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul p q)"
561   using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
563 lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
564   by (induct p arbitrary: n0 rule: headconst.induct) auto
567   by (induct p arbitrary: n0) auto
569 lemma monic_eqI:
570   assumes np: "isnpolyh p n0"
571   shows "INum (headconst p) * Ipoly bs (fst (monic p)) =
572     (Ipoly bs p ::'a::{field_char_0, field_inverse_zero, power})"
573   unfolding monic_def Let_def
575   let ?h = "headconst p"
576   assume pz: "p \<noteq> 0\<^sub>p"
577   {assume hz: "INum ?h = (0::'a)"
578     from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
579     from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
580     with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
581   thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast
582 qed
585 text{* polyneg is a negation and preserves normal forms *}
587 lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
588   by (induct p rule: polyneg.induct) auto
590 lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"
591   by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: Nneg_def)
592 lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
593   by (induct p arbitrary: n0 rule: polyneg.induct) auto
594 lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "
595   by (induct p rule: polyneg.induct) (auto simp add: polyneg0)
597 lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
598   using isnpoly_def polyneg_normh by simp
601 text{* polysub is a substraction and preserves normal forms *}
603 lemma polysub[simp]: "Ipoly bs (polysub p q) = (Ipoly bs p) - (Ipoly bs q)"
605 lemma polysub_normh: "\<And> n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
608 lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub p q)"
610 lemma polysub_same_0[simp]:
611   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
612   shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p"
613   unfolding polysub_def split_def fst_conv snd_conv
614   by (induct p arbitrary: n0) (auto simp add: Let_def Nsub0[simplified Nsub_def])
616 lemma polysub_0:
617   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
618   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"
619   unfolding polysub_def split_def fst_conv snd_conv
620   by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
621     (auto simp: Nsub0[simplified Nsub_def] Let_def)
623 text{* polypow is a power function and preserves normal forms *}
625 lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{field_char_0, field_inverse_zero})) ^ n"
626 proof (induct n rule: polypow.induct)
627   case 1
628   thus ?case by simp
629 next
630   case (2 n)
631   let ?q = "polypow ((Suc n) div 2) p"
632   let ?d = "polymul ?q ?q"
633   have "odd (Suc n) \<or> even (Suc n)" by simp
634   moreover
635   {assume odd: "odd (Suc n)"
636     have th: "(Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat))))) = Suc n div 2 + Suc n div 2 + 1"
637       by arith
638     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)" by (simp add: Let_def)
639     also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"
640       using "2.hyps" by simp
641     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
642       by (simp only: power_add power_one_right) simp
643     also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat)))))"
644       by (simp only: th)
645     finally have ?case
646     using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
647   moreover
648   {assume even: "even (Suc n)"
649     have th: "(Suc (Suc (0\<Colon>nat))) * (Suc n div Suc (Suc (0\<Colon>nat))) = Suc n div 2 + Suc n div 2"
650       by arith
651     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
652     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
653       using "2.hyps" apply (simp only: power_add) by simp
654     finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}
655   ultimately show ?case by blast
656 qed
658 lemma polypow_normh:
659   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
660   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
661 proof (induct k arbitrary: n rule: polypow.induct)
662   case (2 k n)
663   let ?q = "polypow (Suc k div 2) p"
664   let ?d = "polymul ?q ?q"
665   from 2 have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
666   from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
667   from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n" by simp
668   from dn on show ?case by (simp add: Let_def)
669 qed auto
671 lemma polypow_norm:
672   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
673   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
674   by (simp add: polypow_normh isnpoly_def)
676 text{* Finally the whole normalization *}
678 lemma polynate [simp]:
679   "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0, field_inverse_zero})"
680   by (induct p rule:polynate.induct) auto
682 lemma polynate_norm[simp]:
683   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
684   shows "isnpoly (polynate p)"
685   by (induct p rule: polynate.induct)
689 text{* shift1 *}
692 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
695 lemma shift1_isnpoly:
696   assumes pn: "isnpoly p"
697     and pnz: "p \<noteq> 0\<^sub>p"
698   shows "isnpoly (shift1 p) "
699   using pn pnz by (simp add: shift1_def isnpoly_def)
701 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
703 lemma funpow_shift1_isnpoly:
704   "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
705   by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
707 lemma funpow_isnpolyh:
708   assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n"
709     and np: "isnpolyh p n"
710   shows "isnpolyh (funpow k f p) n"
711   using f np by (induct k arbitrary: p) auto
713 lemma funpow_shift1:
714   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) =
715     Ipoly bs (Mul (Pw (Bound 0) n) p)"
716   by (induct n arbitrary: p) (simp_all add: shift1_isnpoly shift1)
718 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
719   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
721 lemma funpow_shift1_1:
722   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) =
723     Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)"
726 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
727   by (induct p rule: poly_cmul.induct) (auto simp add: field_simps)
730   assumes np: "isnpolyh p n"
732     (Ipoly bs p :: 'a :: {field_char_0, field_inverse_zero})"
733   using np
734 proof (induct p arbitrary: n rule: behead.induct)
735   case (1 c p n) hence pn: "isnpolyh p n" by simp
736   from 1(1)[OF pn]
737   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
738   then show ?case using "1.hyps"
740     apply (simp_all add: th[symmetric] field_simps)
741     done
742 qed (auto simp add: Let_def)
745   assumes np: "isnpolyh p n"
746   shows "isnpolyh (behead p) n"
747   using np by (induct p rule: behead.induct) (auto simp add: Let_def isnpolyh_mono)
750 subsection{* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
752 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
753 proof (induct p arbitrary: n rule: poly.induct, auto)
754   case (goal1 c n p n')
755   hence "n = Suc (n - 1)" by simp
756   hence "isnpolyh p (Suc (n - 1))"  using `isnpolyh p n` by simp
757   with goal1(2) show ?case by simp
758 qed
760 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
761   by (induct p arbitrary: n0 rule: isconstant.induct) (auto simp add: isnpolyh_polybound0)
763 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
764   by (induct p) auto
766 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
767   apply (induct p arbitrary: n0)
768   apply auto
769   apply (atomize)
770   apply (erule_tac x = "Suc nat" in allE)
771   apply auto
772   done
775   by (induct p  arbitrary: n0 rule: head.induct) (auto intro: isnpolyh_polybound0)
777 lemma polybound0_I:
778   assumes nb: "polybound0 a"
779   shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
780   using nb
781   by (induct a rule: poly.induct) auto
783 lemma polysubst0_I: "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
784   by (induct t) simp_all
786 lemma polysubst0_I':
787   assumes nb: "polybound0 a"
788   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t"
789   by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
791 lemma decrpoly:
792   assumes nb: "polybound0 t"
793   shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
794   using nb by (induct t rule: decrpoly.induct) simp_all
796 lemma polysubst0_polybound0:
797   assumes nb: "polybound0 t"
798   shows "polybound0 (polysubst0 t a)"
799   using nb by (induct a rule: poly.induct) auto
801 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
802   by (induct p arbitrary: n rule: degree.induct) (auto simp add: isnpolyh_polybound0)
804 primrec maxindex :: "poly \<Rightarrow> nat" where
805   "maxindex (Bound n) = n + 1"
806 | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
807 | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
808 | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
809 | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
810 | "maxindex (Neg p) = maxindex p"
811 | "maxindex (Pw p n) = maxindex p"
812 | "maxindex (C x) = 0"
814 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool"
815   where "wf_bs bs p = (length bs \<ge> maxindex p)"
817 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
818 proof (induct p rule: coefficients.induct)
819   case (1 c p)
820   show ?case
821   proof
822     fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"
823     hence "x = c \<or> x \<in> set (coefficients p)" by simp
824     moreover
825     {assume "x = c" hence "wf_bs bs x" using "1.prems"  unfolding wf_bs_def by simp}
826     moreover
827     {assume H: "x \<in> set (coefficients p)"
828       from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp
829       with "1.hyps" H have "wf_bs bs x" by blast }
830     ultimately  show "wf_bs bs x" by blast
831   qed
832 qed simp_all
834 lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
835   by (induct p rule: coefficients.induct) auto
837 lemma wf_bs_I: "wf_bs bs p ==> Ipoly (bs@bs') p = Ipoly bs p"
838   unfolding wf_bs_def by (induct p) (auto simp add: nth_append)
840 lemma take_maxindex_wf:
841   assumes wf: "wf_bs bs p"
842   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
843 proof-
844   let ?ip = "maxindex p"
845   let ?tbs = "take ?ip bs"
846   from wf have "length ?tbs = ?ip" unfolding wf_bs_def by simp
847   hence wf': "wf_bs ?tbs p" unfolding wf_bs_def by  simp
848   have eq: "bs = ?tbs @ (drop ?ip bs)" by simp
849   from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis using eq by simp
850 qed
852 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
853   by (induct p) auto
855 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
856   unfolding wf_bs_def by simp
858 lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
859   unfolding wf_bs_def by simp
863 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
864   by (induct p rule: coefficients.induct) (auto simp add: wf_bs_def)
865 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
866   by (induct p rule: coefficients.induct) simp_all
870   by (induct p rule: coefficients.induct) auto
872 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
873   unfolding wf_bs_def by (induct p rule: decrpoly.induct) auto
875 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
876   apply (rule exI[where x="replicate (n - length xs) z"])
877   apply simp
878   done
880 lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
881   apply (cases p)
882   apply auto
883   apply (case_tac "nat")
884   apply simp_all
885   done
887 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
888   unfolding wf_bs_def
889   apply (induct p q rule: polyadd.induct)
890   apply (auto simp add: Let_def)
891   done
893 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
894   unfolding wf_bs_def
895   apply (induct p q arbitrary: bs rule: polymul.induct)
897   apply clarsimp
898   apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
899   apply auto
900   done
902 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
903   unfolding wf_bs_def by (induct p rule: polyneg.induct) auto
905 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
906   unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
909 subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}
911 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
912 definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)"
913 definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"
915 lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0"
916 proof (induct p arbitrary: n0 rule: coefficients.induct)
917   case (1 c p n0)
918   have cp: "isnpolyh (CN c 0 p) n0" by fact
919   hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
920     by (auto simp add: isnpolyh_mono[where n'=0])
921   from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp
922 qed auto
924 lemma coefficients_isconst:
925   "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
926   by (induct p arbitrary: n rule: coefficients.induct)
929 lemma polypoly_polypoly':
930   assumes np: "isnpolyh p n0"
931   shows "polypoly (x#bs) p = polypoly' bs p"
932 proof-
933   let ?cf = "set (coefficients p)"
934   from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
935   {fix q assume q: "q \<in> ?cf"
936     from q cn_norm have th: "isnpolyh q n0" by blast
937     from coefficients_isconst[OF np] q have "isconstant q" by blast
938     with isconstant_polybound0[OF th] have "polybound0 q" by blast}
939   hence "\<forall>q \<in> ?cf. polybound0 q" ..
940   hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"
941     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
942     by auto
944   thus ?thesis unfolding polypoly_def polypoly'_def by simp
945 qed
947 lemma polypoly_poly:
948   assumes np: "isnpolyh p n0"
949   shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
950   using np
951   by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def)
953 lemma polypoly'_poly:
954   assumes np: "isnpolyh p n0"
955   shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
956   using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
959 lemma polypoly_poly_polybound0:
960   assumes np: "isnpolyh p n0" and nb: "polybound0 p"
961   shows "polypoly bs p = [Ipoly bs p]"
962   using np nb unfolding polypoly_def
963   apply (cases p)
964   apply auto
965   apply (case_tac nat)
966   apply auto
967   done
970   by (induct p rule: head.induct) auto
972 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
973   by (cases p) auto
976   by (induct p rule: head.induct) simp_all
978 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
981 lemma isnpolyh_zero_iff:
982   assumes nq: "isnpolyh p n0"
983     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})"
984   shows "p = 0\<^sub>p"
985   using nq eq
986 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
987   case less
988   note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
989   {assume nz: "maxindex p = 0"
990     then obtain c where "p = C c" using np by (cases p) auto
991     with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}
992   moreover
993   {assume nz: "maxindex p \<noteq> 0"
994     let ?h = "head p"
995     let ?hd = "decrpoly ?h"
996     let ?ihd = "maxindex ?hd"
997     from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h"
998       by simp_all
999     hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast
1001     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
1002     have mihn: "maxindex ?h \<le> maxindex p" by auto
1003     with decr_maxindex[OF h(2)] nz  have ihd_lt_n: "?ihd < maxindex p" by auto
1004     {fix bs:: "'a list"  assume bs: "wf_bs bs ?hd"
1005       let ?ts = "take ?ihd bs"
1006       let ?rs = "drop ?ihd bs"
1007       have ts: "wf_bs ?ts ?hd" using bs unfolding wf_bs_def by simp
1008       have bs_ts_eq: "?ts@ ?rs = bs" by simp
1009       from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x#?ts) ?h" by simp
1010       from ihd_lt_n have "ALL x. length (x#?ts) \<le> maxindex p" by simp
1011       with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = maxindex p" by blast
1012       hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" unfolding wf_bs_def by simp
1013       with zp have "\<forall> x. Ipoly ((x#?ts) @ xs) p = 0" by blast
1014       hence "\<forall> x. Ipoly (x#(?ts @ xs)) p = 0" by simp
1015       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
1016       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"  by simp
1017       hence "poly (polypoly' (?ts @ xs) p) = poly []" by auto
1018       hence "\<forall> c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
1019         using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
1021       have th0: "Ipoly (?ts @ xs) ?hd = 0" by simp
1022       from bs have wf'': "wf_bs ?ts ?hd" unfolding wf_bs_def by simp
1023       with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0" by simp
1024       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by simp }
1025     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast
1027     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast
1028     hence "?h = 0\<^sub>p" by simp
1029     with head_nz[OF np] have "p = 0\<^sub>p" by simp}
1030   ultimately show "p = 0\<^sub>p" by blast
1031 qed
1033 lemma isnpolyh_unique:
1034   assumes np:"isnpolyh p n0"
1035     and nq: "isnpolyh q n1"
1036   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"
1037 proof(auto)
1038   assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a)= \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
1039   hence "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" by simp
1040   hence "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
1041     using wf_bs_polysub[where p=p and q=q] by auto
1042   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq]
1043   show "p = q" by blast
1044 qed
1047 text{* consequences of unicity on the algorithms for polynomial normalization *}
1050   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1051     and np: "isnpolyh p n0"
1052     and nq: "isnpolyh q n1"
1053   shows "p +\<^sub>p q = q +\<^sub>p p"
1056 lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp
1057 lemma one_normh: "isnpolyh (1)\<^sub>p n" by simp
1060   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1061     and np: "isnpolyh p n0"
1062   shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>p p = p"
1063   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
1064     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
1066 lemma polymul_1[simp]:
1067   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1068     and np: "isnpolyh p n0"
1069   shows "p *\<^sub>p (1)\<^sub>p = p" and "(1)\<^sub>p *\<^sub>p p = p"
1070   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
1071     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
1073 lemma polymul_0[simp]:
1074   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1075     and np: "isnpolyh p n0"
1076   shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p" and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
1077   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
1078     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
1080 lemma polymul_commute:
1081   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1082     and np:"isnpolyh p n0"
1083     and nq: "isnpolyh q n1"
1084   shows "p *\<^sub>p q = q *\<^sub>p p"
1085   using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np], where ?'a = "'a\<Colon>{field_char_0, field_inverse_zero, power}"]
1086   by simp
1088 declare polyneg_polyneg [simp]
1090 lemma isnpolyh_polynate_id [simp]:
1091   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1092     and np:"isnpolyh p n0"
1093   shows "polynate p = p"
1094   using isnpolyh_unique[where ?'a= "'a::{field_char_0, field_inverse_zero}", OF polynate_norm[of p, unfolded isnpoly_def] np] polynate[where ?'a = "'a::{field_char_0, field_inverse_zero}"]
1095   by simp
1097 lemma polynate_idempotent[simp]:
1098   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1099   shows "polynate (polynate p) = polynate p"
1100   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
1102 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
1103   unfolding poly_nate_def polypoly'_def ..
1105 lemma poly_nate_poly:
1106   "poly (poly_nate bs p) = (\<lambda>x:: 'a ::{field_char_0, field_inverse_zero}. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
1107   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
1108   unfolding poly_nate_polypoly' by auto
1111 subsection{* heads, degrees and all that *}
1113 lemma degree_eq_degreen0: "degree p = degreen p 0"
1114   by (induct p rule: degree.induct) simp_all
1116 lemma degree_polyneg:
1117   assumes n: "isnpolyh p n"
1118   shows "degree (polyneg p) = degree p"
1119   apply (induct p arbitrary: n rule: polyneg.induct)
1120   using n apply simp_all
1121   apply (case_tac na)
1122   apply auto
1123   done
1126   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
1127   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
1128   using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
1131 lemma degree_polysub:
1132   assumes np: "isnpolyh p n0"
1133     and nq: "isnpolyh q n1"
1134   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
1135 proof-
1136   from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp
1137   from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])
1138 qed
1141   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1142     and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q"
1143     and d: "degree p = degree q"
1144   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
1145   unfolding polysub_def split_def fst_conv snd_conv
1146   using np nq h d
1147 proof (induct p q rule: polyadd.induct)
1148   case (1 c c')
1149   thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def])
1150 next
1151   case (2 c c' n' p')
1152   from 2 have "degree (C c) = degree (CN c' n' p')" by simp
1153   hence nz:"n' > 0" by (cases n') auto
1154   hence "head (CN c' n' p') = CN c' n' p'" by (cases n') auto
1155   with 2 show ?case by simp
1156 next
1157   case (3 c n p c')
1158   hence "degree (C c') = degree (CN c n p)" by simp
1159   hence nz:"n > 0" by (cases n) auto
1160   hence "head (CN c n p) = CN c n p" by (cases n) auto
1161   with 3 show ?case by simp
1162 next
1163   case (4 c n p c' n' p')
1164   hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1"
1165     "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+
1166   hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all
1167   hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
1168     using H(1-2) degree_polyneg by auto
1169   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"  by simp+
1170   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"  by simp
1171   from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto
1172   have "n = n' \<or> n < n' \<or> n > n'" by arith
1173   moreover
1174   {assume nn': "n = n'"
1175     have "n = 0 \<or> n >0" by arith
1176     moreover {assume nz: "n = 0" hence ?case using 4 nn' by (auto simp add: Let_def degcmc')}
1177     moreover {assume nz: "n > 0"
1178       with nn' H(3) have  cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
1179       hence ?case
1180         using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv] polysub_same_0[OF c'nh, simplified polysub_def]
1181         using nn' 4 by (simp add: Let_def)}
1182     ultimately have ?case by blast}
1183   moreover
1184   {assume nn': "n < n'" hence n'p: "n' > 0" by simp
1185     hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n') simp_all
1186     have degcnp': "degree (CN c' n' p') = 0" and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
1187       using 4 nn' by (cases n', simp_all)
1188     hence "n > 0" by (cases n) simp_all
1189     hence headcnp: "head (CN c n p) = CN c n p" by (cases n) auto
1191   moreover
1192   {assume nn': "n > n'"  hence np: "n > 0" by simp
1193     hence headcnp:"head (CN c n p) = CN c n p"  by (cases n) simp_all
1194     from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
1195     from np have degcnp: "degree (CN c n p) = 0" by (cases n) simp_all
1196     with degcnpeq have "n' > 0" by (cases n') simp_all
1197     hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n') auto
1199   ultimately show ?case  by blast
1200 qed auto
1205 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
1206 proof (induct k arbitrary: n0 p)
1207   case 0
1208   thus ?case by auto
1209 next
1210   case (Suc k n0 p)
1211   hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
1212   with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
1214   thus ?case by (simp add: funpow_swap1)
1215 qed
1217 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
1219 lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
1220   by (induct k arbitrary: p) (auto simp add: shift1_degree)
1222 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
1223   by (induct n arbitrary: p) simp_all
1225 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
1226   by (induct p arbitrary: n rule: degree.induct) auto
1227 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
1228   by (induct p arbitrary: n rule: degreen.induct) auto
1229 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
1230   by (induct p arbitrary: n rule: degree.induct) auto
1232   by (induct p rule: head.induct) auto
1235   "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> polyadd p q = C c \<Longrightarrow> degree p = degree q"
1236   using polyadd_eq_const_degreen degree_eq_degreen0 by simp
1239   assumes np: "isnpolyh p n0"
1240     and nq: "isnpolyh q n1"
1241     and deg: "degree p \<noteq> degree q"
1242   shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
1243   using np nq deg
1244   apply (induct p q arbitrary: n0 n1 rule: polyadd.induct)
1245   using np
1246   apply simp_all
1247   apply (case_tac n', simp, simp)
1248   apply (case_tac n, simp, simp)
1249   apply (case_tac n, case_tac n', simp add: Let_def)
1250   apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p")
1254   apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
1255   apply (metis degree.simps(9) gr0_conv_Suc nat_less_le order_le_less_trans)
1256   done
1259   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1260   shows "\<lbrakk>isnpolyh p n0; isnpolyh q n1 ; p \<noteq> 0\<^sub>p ; q \<noteq> 0\<^sub>p \<rbrakk> \<Longrightarrow> head (p *\<^sub>p q) = head p *\<^sub>p head q"
1261 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
1262   case (2 c c' n' p' n0 n1)
1264   thus ?case using 2 by (cases n') auto
1265 next
1266   case (3 c n p c' n0 n1)
1268   thus ?case using 3 by (cases n) auto
1269 next
1270   case (4 c n p c' n' p' n0 n1)
1271   hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
1272     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
1273     by simp_all
1274   have "n < n' \<or> n' < n \<or> n = n'" by arith
1275   moreover
1276   {assume nn': "n < n'" hence ?case
1277       using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)]
1278       apply simp
1279       apply (cases n)
1280       apply simp
1281       apply (cases n')
1282       apply simp_all
1283       done }
1284   moreover {assume nn': "n'< n"
1285     hence ?case
1286       using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)]
1287       apply simp
1288       apply (cases n')
1289       apply simp
1290       apply (cases n)
1291       apply auto
1292       done }
1293   moreover {assume nn': "n' = n"
1294     from nn' polymul_normh[OF norm(5,4)]
1295     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
1296     from nn' polymul_normh[OF norm(5,3)] norm
1297     have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
1298     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
1299     have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
1301     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"
1303     {assume np: "n > 0"
1304       with nn' head_isnpolyh_Suc'[OF np nth]
1306       have ?case by simp}
1307     moreover
1308     {moreover assume nz: "n = 0"
1309       from polymul_degreen[OF norm(5,4), where m="0"]
1310         polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
1311       norm(5,6) degree_npolyhCN[OF norm(6)]
1312     have dth:"degree (CN c 0 p *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))" by simp
1313     hence dth':"degree (CN c 0 p *\<^sub>p c') \<noteq> degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))" by simp
1315     have ?case   using norm "4.hyps"(6)[OF norm(5,3)]
1316         "4.hyps"(5)[OF norm(5,4)] nn' nz by simp }
1317     ultimately have ?case by (cases n) auto}
1318   ultimately show ?case by blast
1319 qed simp_all
1321 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
1322   by (induct p rule: degree.induct) auto
1325   by (induct p rule: head.induct) auto
1327 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
1328   by (cases n) simp_all
1329 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
1330   by (cases n) simp_all
1333   "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degree p \<noteq> degree q\<rbrakk> \<Longrightarrow>
1334     degree (polyadd p q) = max (degree p) (degree q)"
1335   using polyadd_different_degreen degree_eq_degreen0 by simp
1337 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
1338   by (induct p arbitrary: n0 rule: polyneg.induct) auto
1340 lemma degree_polymul:
1341   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1342     and np: "isnpolyh p n0"
1343     and nq: "isnpolyh q n1"
1344   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
1345   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
1347 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
1348   by (induct p arbitrary: n rule: degree.induct) auto
1351   by (induct p arbitrary: n rule: degree.induct) auto
1354 subsection {* Correctness of polynomial pseudo division *}
1356 lemma polydivide_aux_properties:
1357   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1358     and np: "isnpolyh p n0"
1359     and ns: "isnpolyh s n1"
1360     and ap: "head p = a"
1361     and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
1362   shows "(polydivide_aux a n p k s = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p)
1363           \<and> (\<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)))"
1364   using ns
1365 proof (induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
1366   case less
1367   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
1368   let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p)
1369     \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
1370   let ?b = "head s"
1371   let ?p' = "funpow (degree s - n) shift1 p"
1372   let ?xdn = "funpow (degree s - n) shift1 (1)\<^sub>p"
1373   let ?akk' = "a ^\<^sub>p (k' - k)"
1374   note ns = `isnpolyh s n1`
1375   from np have np0: "isnpolyh p 0"
1376     using isnpolyh_mono[where n="n0" and n'="0" and p="p"]  by simp
1377   have np': "isnpolyh ?p' 0" using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def by simp
1379   from funpow_shift1_isnpoly[where p="(1)\<^sub>p"] have nxdn: "isnpolyh ?xdn 0" by (simp add: isnpoly_def)
1380   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
1381   have nakk':"isnpolyh ?akk' 0" by blast
1382   { assume sz: "s = 0\<^sub>p"
1383     hence ?ths using np polydivide_aux.simps
1384       apply clarsimp
1385       apply (rule exI[where x="0\<^sub>p"])
1386       apply simp
1387       done }
1388   moreover
1389   {assume sz: "s \<noteq> 0\<^sub>p"
1390     {assume dn: "degree s < n"
1391       hence "?ths" using ns ndp np polydivide_aux.simps
1392         apply auto
1393         apply (rule exI[where x="0\<^sub>p"])
1394         apply simp
1395         done }
1396     moreover
1397     {assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith
1398       have degsp': "degree s = degree ?p'"
1399         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp
1400       {assume ba: "?b = a"
1402         have nr: "isnpolyh (s -\<^sub>p ?p') 0" using polysub_normh[OF ns np'] by simp
1404         have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
1405         moreover
1406         {assume deglt:"degree (s -\<^sub>p ?p') < degree s"
1407           from polydivide_aux.simps sz dn' ba
1408           have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
1410           {assume h1: "polydivide_aux a n p k s = (k', r)"
1411             from less(1)[OF deglt nr, of k k' r]
1412               trans[OF eq[symmetric] h1]
1413             have kk': "k \<le> k'" and nr:"\<exists>nr. isnpolyh r nr" and dr: "degree r = 0 \<or> degree r < degree p"
1414               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)" by auto
1415             from q1 obtain q n1 where nq: "isnpolyh q n1"
1416               and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"  by blast
1417             from nr obtain nr where nr': "isnpolyh r nr" by blast
1418             from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0" by simp
1419             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
1420             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
1422               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
1423             have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0" by simp
1424             from asp have "\<forall> (bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) =
1425               Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
1426             hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
1427               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
1429             hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1430               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p *\<^sub>p p)
1431               + Ipoly bs p * Ipoly bs q + Ipoly bs r"
1432               by (auto simp only: funpow_shift1_1)
1433             hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1434               Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p)
1435               + Ipoly bs q) + Ipoly bs r" by (simp add: field_simps)
1436             hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1437               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)" by simp
1438             with isnpolyh_unique[OF nakks' nqr']
1439             have "a ^\<^sub>p (k' - k) *\<^sub>p s =
1440               p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r" by blast
1441             hence ?qths using nq'
1442               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q" in exI)
1443               apply (rule_tac x="0" in exI) by simp
1444             with kk' nr dr have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
1445               by blast } hence ?ths by blast }
1446         moreover
1447         {assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
1448           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}"]
1449           have " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs s = Ipoly bs ?p'" by simp
1450           hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)"
1451             using np nxdn
1452             apply simp
1453             apply (simp only: funpow_shift1_1)
1454             apply simp
1455             done
1456           hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]]
1457             by blast
1458           {assume h1: "polydivide_aux a n p k s = (k',r)"
1459             from polydivide_aux.simps sz dn' ba
1460             have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
1462             also have "\<dots> = (k,0\<^sub>p)" using polydivide_aux.simps spz by simp
1463             finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
1464             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
1465               polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
1466               apply auto
1467               apply (rule exI[where x="?xdn"])
1468               apply (auto simp add: polymul_commute[of p])
1469               done} }
1470         ultimately have ?ths by blast }
1471       moreover
1472       {assume ba: "?b \<noteq> a"
1473         from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
1475         have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by(simp add: min_def)
1476         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
1477           using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
1479             funpow_shift1_nz[OF pnz] by simp_all
1482         have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
1486         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
1487           head_nz[OF np] pnz sz ap[symmetric]
1488           funpow_shift1_nz[OF pnz, where n="degree s - n"]
1490           ndp dn
1491         have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p') "
1492           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
1493         {assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
1495           ap have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by simp
1496           {assume h1:"polydivide_aux a n p k s = (k', r)"
1497             from h1 polydivide_aux.simps sz dn' ba
1498             have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
1500             with less(1)[OF dth nasbp', of "Suc k" k' r]
1501             obtain q nq nr where kk': "Suc k \<le> k'" and nr: "isnpolyh r nr" and nq: "isnpolyh q nq"
1502               and dr: "degree r = 0 \<or> degree r < degree p"
1503               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" by auto
1504             from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith
1505             {fix bs:: "'a::{field_char_0, field_inverse_zero} list"
1507             from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
1508             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)"
1509               by simp
1510             hence "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s =
1511               Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
1513             hence "Ipoly bs a ^ (k' - k)  * Ipoly bs s =
1514               Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
1515               by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
1516             hence "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1517               Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
1519             hence ieq:"\<forall>(bs :: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1520               Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)" by auto
1521             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
1522             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]]
1523             have nqw: "isnpolyh ?q 0" by simp
1524             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]]
1525             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" by blast
1526             from dr kk' nr h1 asth nqw have ?ths apply simp
1527               apply (rule conjI)
1528               apply (rule exI[where x="nr"], simp)
1529               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
1530               apply (rule exI[where x="0"], simp)
1531               done}
1532           hence ?ths by blast }
1533         moreover
1534         {assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
1535           {fix bs :: "'a::{field_char_0, field_inverse_zero} list"
1536             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
1537           have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'" by simp
1538           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
1539             by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
1540           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" by simp
1541         }
1542         hence hth: "\<forall> (bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a*\<^sub>p s) =
1543             Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
1544           from hth
1545           have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
1546             using isnpolyh_unique[where ?'a = "'a::{field_char_0, field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
1547                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
1548               simplified ap] by simp
1549           {assume h1: "polydivide_aux a n p k s = (k', r)"
1550           from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
1551           have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
1552           with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
1553             polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
1554           have ?ths
1555             apply (clarsimp simp add: Let_def)
1556             apply (rule exI[where x="?b *\<^sub>p ?xdn"])
1557             apply simp
1558             apply (rule exI[where x="0"], simp)
1559             done }
1560         hence ?ths by blast }
1561         ultimately have ?ths
1563             head_nz[OF np] pnz sz ap[symmetric]
1564           by (simp add: degree_eq_degreen0[symmetric]) blast }
1565       ultimately have ?ths by blast
1566     }
1567     ultimately have ?ths by blast }
1568   ultimately show ?ths by blast
1569 qed
1571 lemma polydivide_properties:
1572   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1573   and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
1574   shows "(\<exists> k r. polydivide s p = (k,r) \<and> (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p)
1575   \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
1576 proof-
1577   have trv: "head p = head p" "degree p = degree p" by simp_all
1578   from polydivide_def[where s="s" and p="p"]
1579   have ex: "\<exists> k r. polydivide s p = (k,r)" by auto
1580   then obtain k r where kr: "polydivide s p = (k,r)" by blast
1581   from trans[OF meta_eq_to_obj_eq[OF polydivide_def[where s="s"and p="p"], symmetric] kr]
1582     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
1583   have "(degree r = 0 \<or> degree r < degree p) \<and>
1584    (\<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)" by blast
1585   with kr show ?thesis
1586     apply -
1587     apply (rule exI[where x="k"])
1588     apply (rule exI[where x="r"])
1589     apply simp
1590     done
1591 qed
1594 subsection{* More about polypoly and pnormal etc *}
1596 definition "isnonconstant p = (\<not> isconstant p)"
1598 lemma isnonconstant_pnormal_iff:
1599   assumes nc: "isnonconstant p"
1600   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
1601 proof
1602   let ?p = "polypoly bs p"
1603   assume H: "pnormal ?p"
1604   have csz: "coefficients p \<noteq> []" using nc by (cases p) auto
1606   from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
1607     pnormal_last_nonzero[OF H]
1608   show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)
1609 next
1610   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
1611   let ?p = "polypoly bs p"
1612   have csz: "coefficients p \<noteq> []" using nc by (cases p) auto
1613   hence pz: "?p \<noteq> []" by (simp add: polypoly_def)
1614   hence lg: "length ?p > 0" by simp
1615   from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
1616   have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)
1617   from pnormal_last_length[OF lg lz] show "pnormal ?p" .
1618 qed
1620 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
1621   unfolding isnonconstant_def
1622   apply (cases p)
1623   apply simp_all
1624   apply (case_tac nat)
1625   apply auto
1626   done
1628 lemma isnonconstant_nonconstant:
1629   assumes inc: "isnonconstant p"
1630   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
1631 proof
1632   let ?p = "polypoly bs p"
1633   assume nc: "nonconstant ?p"
1634   from isnonconstant_pnormal_iff[OF inc, of bs] nc
1635   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast
1636 next
1637   let ?p = "polypoly bs p"
1638   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
1639   from isnonconstant_pnormal_iff[OF inc, of bs] h
1640   have pn: "pnormal ?p" by blast
1641   {fix x assume H: "?p = [x]"
1642     from H have "length (coefficients p) = 1" unfolding polypoly_def by auto
1643     with isnonconstant_coefficients_length[OF inc] have False by arith}
1644   thus "nonconstant ?p" using pn unfolding nonconstant_def by blast
1645 qed
1647 lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
1648   apply (induct p)
1650   apply (case_tac "p = []")
1651   apply simp_all
1652   done
1654 lemma degree_degree:
1655   assumes inc: "isnonconstant p"
1656   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
1657 proof
1658   let  ?p = "polypoly bs p"
1659   assume H: "degree p = Polynomial_List.degree ?p"
1660   from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
1661     unfolding polypoly_def by auto
1662   from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
1663   have lg:"length (pnormalize ?p) = length ?p"
1664     unfolding Polynomial_List.degree_def polypoly_def by simp
1665   hence "pnormal ?p" using pnormal_length[OF pz] by blast
1666   with isnonconstant_pnormal_iff[OF inc]
1667   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast
1668 next
1669   let  ?p = "polypoly bs p"
1670   assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
1671   with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast
1672   with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
1673   show "degree p = Polynomial_List.degree ?p"
1674     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
1675 qed
1678 section{* Swaps ; Division by a certain variable *}
1680 primrec swap:: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly" where
1681   "swap n m (C x) = C x"
1682 | "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"
1683 | "swap n m (Neg t) = Neg (swap n m t)"
1684 | "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
1685 | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
1686 | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
1687 | "swap n m (Pw t k) = Pw (swap n m t) k"
1688 | "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)
1689   (swap n m p)"
1691 lemma swap:
1692   assumes nbs: "n < length bs"
1693     and mbs: "m < length bs"
1694   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
1695 proof (induct t)
1696   case (Bound k)
1697   thus ?case using nbs mbs by simp
1698 next
1699   case (CN c k p)
1700   thus ?case using nbs mbs by simp
1701 qed simp_all
1703 lemma swap_swap_id [simp]: "swap n m (swap m n t) = t"
1704   by (induct t) simp_all
1706 lemma swap_commute: "swap n m p = swap m n p"
1707   by (induct p) simp_all
1709 lemma swap_same_id[simp]: "swap n n t = t"
1710   by (induct t) simp_all
1712 definition "swapnorm n m t = polynate (swap n m t)"
1714 lemma swapnorm:
1715   assumes nbs: "n < length bs"
1716     and mbs: "m < length bs"
1717   shows "((Ipoly bs (swapnorm n m t) :: 'a\<Colon>{field_char_0, field_inverse_zero})) =
1718     Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
1719   using swap[OF assms] swapnorm_def by simp
1721 lemma swapnorm_isnpoly [simp]:
1722   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1723   shows "isnpoly (swapnorm n m p)"
1724   unfolding swapnorm_def by simp
1726 definition "polydivideby n s p =
1727   (let ss = swapnorm 0 n s ; sp = swapnorm 0 n p ; h = head sp; (k,r) = polydivide ss sp
1728    in (k,swapnorm 0 n h,swapnorm 0 n r))"
1730 lemma swap_nz [simp]: " (swap n m p = 0\<^sub>p) = (p = 0\<^sub>p)"
1731   by (induct p) simp_all
1733 fun isweaknpoly :: "poly \<Rightarrow> bool"
1734 where
1735   "isweaknpoly (C c) = True"
1736 | "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
1737 | "isweaknpoly p = False"
1739 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
1740   by (induct p arbitrary: n0) auto
1742 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
1743   by (induct p) auto
1745 end