src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
 author huffman Sun Apr 01 16:09:58 2012 +0200 (2012-04-01) changeset 47255 30a1692557b0 parent 46991 196f2d9406c4 child 49962 a8cc904a6820 permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
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   (* Implementation *)
13 subsection{* Datatype of polynomial expressions *}
15 datatype poly = C Num| Bound nat| Add poly poly|Sub poly poly
16   | Mul poly poly| Neg poly| Pw poly nat| CN poly nat poly
18 abbreviation poly_0 :: "poly" ("0\<^sub>p") where "0\<^sub>p \<equiv> C (0\<^sub>N)"
19 abbreviation poly_p :: "int \<Rightarrow> poly" ("_\<^sub>p") where "i\<^sub>p \<equiv> C (i\<^sub>N)"
21 subsection{* Boundedness, substitution and all that *}
22 primrec polysize:: "poly \<Rightarrow> nat" where
23   "polysize (C c) = 1"
24 | "polysize (Bound n) = 1"
25 | "polysize (Neg p) = 1 + polysize p"
26 | "polysize (Add p q) = 1 + polysize p + polysize q"
27 | "polysize (Sub p q) = 1 + polysize p + polysize q"
28 | "polysize (Mul p q) = 1 + polysize p + polysize q"
29 | "polysize (Pw p n) = 1 + polysize p"
30 | "polysize (CN c n p) = 4 + polysize c + polysize p"
32 primrec polybound0:: "poly \<Rightarrow> bool" (* a poly is INDEPENDENT of Bound 0 *) where
33   "polybound0 (C c) = True"
34 | "polybound0 (Bound n) = (n>0)"
35 | "polybound0 (Neg a) = polybound0 a"
36 | "polybound0 (Add a b) = (polybound0 a \<and> polybound0 b)"
37 | "polybound0 (Sub a b) = (polybound0 a \<and> polybound0 b)"
38 | "polybound0 (Mul a b) = (polybound0 a \<and> polybound0 b)"
39 | "polybound0 (Pw p n) = (polybound0 p)"
40 | "polybound0 (CN c n p) = (n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p)"
42 primrec polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" (* substitute a poly into a poly for Bound 0 *) where
43   "polysubst0 t (C c) = (C c)"
44 | "polysubst0 t (Bound n) = (if n=0 then t else Bound n)"
45 | "polysubst0 t (Neg a) = Neg (polysubst0 t a)"
46 | "polysubst0 t (Add a b) = Add (polysubst0 t a) (polysubst0 t b)"
47 | "polysubst0 t (Sub a b) = Sub (polysubst0 t a) (polysubst0 t b)"
48 | "polysubst0 t (Mul a b) = Mul (polysubst0 t a) (polysubst0 t b)"
49 | "polysubst0 t (Pw p n) = Pw (polysubst0 t p) n"
50 | "polysubst0 t (CN c n p) = (if n=0 then Add (polysubst0 t c) (Mul t (polysubst0 t p))
51                              else CN (polysubst0 t c) n (polysubst0 t p))"
53 fun decrpoly:: "poly \<Rightarrow> poly"
54 where
55   "decrpoly (Bound n) = Bound (n - 1)"
56 | "decrpoly (Neg a) = Neg (decrpoly a)"
57 | "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"
58 | "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
59 | "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
60 | "decrpoly (Pw p n) = Pw (decrpoly p) n"
61 | "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
62 | "decrpoly a = a"
64 subsection{* Degrees and heads and coefficients *}
66 fun degree:: "poly \<Rightarrow> nat"
67 where
68   "degree (CN c 0 p) = 1 + degree p"
69 | "degree p = 0"
71 fun head:: "poly \<Rightarrow> poly"
72 where
73   "head (CN c 0 p) = head p"
74 | "head p = p"
76 (* More general notions of degree and head *)
77 fun degreen:: "poly \<Rightarrow> nat \<Rightarrow> nat"
78 where
79   "degreen (CN c n p) = (\<lambda>m. if n=m then 1 + degreen p n else 0)"
80  |"degreen p = (\<lambda>m. 0)"
82 fun headn:: "poly \<Rightarrow> nat \<Rightarrow> poly"
83 where
84   "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
85 | "headn p = (\<lambda>m. p)"
87 fun coefficients:: "poly \<Rightarrow> poly list"
88 where
89   "coefficients (CN c 0 p) = c#(coefficients p)"
90 | "coefficients p = [p]"
92 fun isconstant:: "poly \<Rightarrow> bool"
93 where
94   "isconstant (CN c 0 p) = False"
95 | "isconstant p = True"
97 fun behead:: "poly \<Rightarrow> poly"
98 where
99   "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
100 | "behead p = 0\<^sub>p"
102 fun headconst:: "poly \<Rightarrow> Num"
103 where
104   "headconst (CN c n p) = headconst p"
105 | "headconst (C n) = n"
107 subsection{* Operations for normalization *}
110 declare if_cong[fundef_cong del]
111 declare let_cong[fundef_cong del]
113 fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
114 where
115   "polyadd (C c) (C c') = C (c+\<^sub>Nc')"
116 |  "polyadd (C c) (CN c' n' p') = CN (polyadd (C c) c') n' p'"
117 | "polyadd (CN c n p) (C c') = CN (polyadd c (C c')) n p"
118 | "polyadd (CN c n p) (CN c' n' p') =
119     (if n < n' then CN (polyadd c (CN c' n' p')) n p
120      else if n'<n then CN (polyadd (CN c n p) c') n' p'
121      else (let cc' = polyadd c c' ;
122                pp' = polyadd p p'
123            in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
124 | "polyadd a b = Add a b"
127 fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
128 where
129   "polyneg (C c) = C (~\<^sub>N c)"
130 | "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
131 | "polyneg a = Neg a"
133 definition polysub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
134 where
135   "p -\<^sub>p q = polyadd p (polyneg q)"
137 fun polymul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
138 where
139   "polymul (C c) (C c') = C (c*\<^sub>Nc')"
140 | "polymul (C c) (CN c' n' p') =
141       (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul (C c) c') n' (polymul (C c) p'))"
142 | "polymul (CN c n p) (C c') =
143       (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul c (C c')) n (polymul p (C c')))"
144 | "polymul (CN c n p) (CN c' n' p') =
145   (if n<n' then CN (polymul c (CN c' n' p')) n (polymul p (CN c' n' p'))
146   else if n' < n
147   then CN (polymul (CN c n p) c') n' (polymul (CN c n p) p')
148   else polyadd (polymul (CN c n p) c') (CN 0\<^sub>p n' (polymul (CN c n p) p')))"
149 | "polymul a b = Mul a b"
151 declare if_cong[fundef_cong]
152 declare let_cong[fundef_cong]
154 fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
155 where
156   "polypow 0 = (\<lambda>p. 1\<^sub>p)"
157 | "polypow n = (\<lambda>p. let q = polypow (n div 2) p ; d = polymul q q in
158                     if even n then d else polymul p d)"
160 abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
161   where "a ^\<^sub>p k \<equiv> polypow k a"
163 function polynate :: "poly \<Rightarrow> poly"
164 where
165   "polynate (Bound n) = CN 0\<^sub>p n 1\<^sub>p"
166 | "polynate (Add p q) = (polynate p +\<^sub>p polynate q)"
167 | "polynate (Sub p q) = (polynate p -\<^sub>p polynate q)"
168 | "polynate (Mul p q) = (polynate p *\<^sub>p polynate q)"
169 | "polynate (Neg p) = (~\<^sub>p (polynate p))"
170 | "polynate (Pw p n) = ((polynate p) ^\<^sub>p n)"
171 | "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
172 | "polynate (C c) = C (normNum c)"
173 by pat_completeness auto
174 termination by (relation "measure polysize") auto
176 fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly" where
177   "poly_cmul y (C x) = C (y *\<^sub>N x)"
178 | "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
179 | "poly_cmul y p = C y *\<^sub>p p"
181 definition monic :: "poly \<Rightarrow> (poly \<times> bool)" where
182   "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))"
184 subsection{* Pseudo-division *}
186 definition shift1 :: "poly \<Rightarrow> poly" where
187   "shift1 p \<equiv> CN 0\<^sub>p 0 p"
189 abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)" where
190   "funpow \<equiv> compow"
192 partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
193   where
194   "polydivide_aux a n p k s =
195   (if s = 0\<^sub>p then (k,s)
196   else (let b = head s; m = degree s in
197   (if m < n then (k,s) else
198   (let p'= funpow (m - n) shift1 p in
199   (if a = b then polydivide_aux a n p k (s -\<^sub>p p')
200   else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))))))"
202 definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)" where
203   "polydivide s p \<equiv> polydivide_aux (head p) (degree p) p 0 s"
205 fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly" where
206   "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
207 | "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
209 fun poly_deriv :: "poly \<Rightarrow> poly" where
210   "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
211 | "poly_deriv p = 0\<^sub>p"
213 subsection{* Semantics of the polynomial representation *}
215 primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0, field_inverse_zero, power}" where
216   "Ipoly bs (C c) = INum c"
217 | "Ipoly bs (Bound n) = bs!n"
218 | "Ipoly bs (Neg a) = - Ipoly bs a"
219 | "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
220 | "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
221 | "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
222 | "Ipoly bs (Pw t n) = (Ipoly bs t) ^ n"
223 | "Ipoly bs (CN c n p) = (Ipoly bs c) + (bs!n)*(Ipoly bs p)"
225 abbreviation
226   Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0, field_inverse_zero, power}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
227   where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"
229 lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i"
230   by (simp add: INum_def)
231 lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"
232   by (simp  add: INum_def)
234 lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
236 subsection {* Normal form and normalization *}
238 fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
239 where
240   "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
241 | "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))"
242 | "isnpolyh p = (\<lambda>k. False)"
244 lemma isnpolyh_mono: "\<lbrakk>n' \<le> n ; isnpolyh p n\<rbrakk> \<Longrightarrow> isnpolyh p n'"
245 by (induct p rule: isnpolyh.induct, auto)
247 definition isnpoly :: "poly \<Rightarrow> bool" where
248   "isnpoly p \<equiv> isnpolyh p 0"
250 text{* polyadd preserves normal forms *}
252 lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk>
253       \<Longrightarrow> isnpolyh (polyadd p q) (min n0 n1)"
254 proof(induct p q arbitrary: n0 n1 rule: polyadd.induct)
255   case (2 ab c' n' p' n0 n1)
256   from 2 have  th1: "isnpolyh (C ab) (Suc n')" by simp
257   from 2(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
258   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
259   with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^sub>p c') (Suc n')" by simp
260   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
261   thus ?case using 2 th3 by simp
262 next
263   case (3 c' n' p' ab n1 n0)
264   from 3 have  th1: "isnpolyh (C ab) (Suc n')" by simp
265   from 3(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
266   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
267   with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (Suc n')" by simp
268   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
269   thus ?case using 3 th3 by simp
270 next
271   case (4 c n p c' n' p' n0 n1)
272   hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
273   from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all
274   from 4 have ngen0: "n \<ge> n0" by simp
275   from 4 have n'gen1: "n' \<ge> n1" by simp
276   have "n < n' \<or> n' < n \<or> n = n'" by auto
277   moreover {assume eq: "n = n'"
278     with "4.hyps"(3)[OF nc nc']
279     have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
280     hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
281       using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
282     from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
283     have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
284     from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}
285   moreover {assume lt: "n < n'"
286     have "min n0 n1 \<le> n0" by simp
287     with 4 lt have th1:"min n0 n1 \<le> n" by auto
288     from 4 have th21: "isnpolyh c (Suc n)" by simp
289     from 4 have th22: "isnpolyh (CN c' n' p') n'" by simp
290     from lt have th23: "min (Suc n) n' = Suc n" by arith
291     from "4.hyps"(1)[OF th21 th22]
292     have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)" using th23 by simp
293     with 4 lt th1 have ?case by simp }
294   moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
295     have "min n0 n1 \<le> n1"  by simp
296     with 4 gt have th1:"min n0 n1 \<le> n'" by auto
297     from 4 have th21: "isnpolyh c' (Suc n')" by simp_all
298     from 4 have th22: "isnpolyh (CN c n p) n" by simp
299     from gt have th23: "min n (Suc n') = Suc n'" by arith
300     from "4.hyps"(2)[OF th22 th21]
301     have "isnpolyh (polyadd (CN c n p) c') (Suc n')" using th23 by simp
302     with 4 gt th1 have ?case by simp}
303       ultimately show ?case by blast
304 qed auto
306 lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
307 by (induct p q rule: polyadd.induct, auto simp add: Let_def field_simps right_distrib[symmetric] simp del: right_distrib)
309 lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd p q)"
310   using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
312 text{* The degree of addition and other general lemmas needed for the normal form of polymul *}
315   "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow>
316   degreen (polyadd p q) m = max (degreen p m) (degreen q m)"
317 proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
318   case (4 c n p c' n' p' m n0 n1)
319   have "n' = n \<or> n < n' \<or> n' < n" by arith
320   thus ?case
321   proof (elim disjE)
322     assume [simp]: "n' = n"
323     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
324     show ?thesis by (auto simp: Let_def)
325   next
326     assume "n < n'"
327     with 4 show ?thesis by auto
328   next
329     assume "n' < n"
330     with 4 show ?thesis by auto
331   qed
332 qed auto
334 lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
335   by (induct p arbitrary: n rule: headn.induct, auto)
336 lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
337   by (induct p arbitrary: n rule: degree.induct, auto)
338 lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
339   by (induct p arbitrary: n rule: degreen.induct, auto)
341 lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
342   by (induct p arbitrary: n rule: degree.induct, auto)
344 lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
345   using degree_isnpolyh_Suc by auto
346 lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
347   using degreen_0 by auto
351   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1"
352   shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
353   using np nq m
354 proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
355   case (2 c c' n' p' n0 n1) thus ?case  by (cases n', simp_all)
356 next
357   case (3 c n p c' n0 n1) thus ?case by (cases n, auto)
358 next
359   case (4 c n p c' n' p' n0 n1 m)
360   have "n' = n \<or> n < n' \<or> n' < n" by arith
361   thus ?case
362   proof (elim disjE)
363     assume [simp]: "n' = n"
364     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
365     show ?thesis by (auto simp: Let_def)
366   qed simp_all
367 qed auto
369 lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd p q = C c\<rbrakk>
370   \<Longrightarrow> degreen p m = degreen q m"
371 proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
372   case (4 c n p c' n' p' m n0 n1 x)
373   {assume nn': "n' < n" hence ?case using 4 by simp}
374   moreover
375   {assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
376     moreover {assume "n < n'" with 4 have ?case by simp }
377     moreover {assume eq: "n = n'" hence ?case using 4
378         apply (cases "p +\<^sub>p p' = 0\<^sub>p")
379         apply (auto simp add: Let_def)
380         by blast
381       }
382     ultimately have ?case by blast}
383   ultimately show ?case by blast
384 qed simp_all
386 lemma polymul_properties:
387   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
388   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> min n0 n1"
389   shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
390   and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)"
391   and "degreen (p *\<^sub>p q) m = (if (p = 0\<^sub>p \<or> q = 0\<^sub>p) then 0
392                              else degreen p m + degreen q m)"
393   using np nq m
394 proof(induct p q arbitrary: n0 n1 m rule: polymul.induct)
395   case (2 c c' n' p')
396   { case (1 n0 n1)
397     with "2.hyps"(4-6)[of n' n' n']
398       and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']
399     show ?case by (auto simp add: min_def)
400   next
401     case (2 n0 n1) thus ?case by auto
402   next
403     case (3 n0 n1) thus ?case  using "2.hyps" by auto }
404 next
405   case (3 c n p c')
406   { case (1 n0 n1)
407     with "3.hyps"(4-6)[of n n n]
408       "3.hyps"(1-3)[of "Suc n" "Suc n" n]
409     show ?case by (auto simp add: min_def)
410   next
411     case (2 n0 n1) thus ?case by auto
412   next
413     case (3 n0 n1) thus ?case  using "3.hyps" by auto }
414 next
415   case (4 c n p c' n' p')
416   let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
417     {
418       case (1 n0 n1)
419       hence cnp: "isnpolyh ?cnp n" and cnp': "isnpolyh ?cnp' n'"
420         and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)"
421         and np': "isnpolyh p' n'" and nc': "isnpolyh c' (Suc n')"
422         and nn0: "n \<ge> n0" and nn1:"n' \<ge> n1"
423         by simp_all
424       { assume "n < n'"
425         with "4.hyps"(4-5)[OF np cnp', of n]
426           "4.hyps"(1)[OF nc cnp', of n] nn0 cnp
427         have ?case by (simp add: min_def)
428       } moreover {
429         assume "n' < n"
430         with "4.hyps"(16-17)[OF cnp np', of "n'"]
431           "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
432         have ?case
433           by (cases "Suc n' = n", simp_all add: min_def)
434       } moreover {
435         assume "n' = n"
436         with "4.hyps"(16-17)[OF cnp np', of n]
437           "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0
438         have ?case
439           apply (auto intro!: polyadd_normh)
440           apply (simp_all add: min_def isnpolyh_mono[OF nn0])
441           done
442       }
443       ultimately show ?case by arith
444     next
445       fix n0 n1 m
446       assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1"
447       and m: "m \<le> min n0 n1"
448       let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
449       let ?d1 = "degreen ?cnp m"
450       let ?d2 = "degreen ?cnp' m"
451       let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
452       have "n'<n \<or> n < n' \<or> n' = n" by auto
453       moreover
454       {assume "n' < n \<or> n < n'"
455         with "4.hyps"(3,6,18) np np' m
456         have ?eq by auto }
457       moreover
458       {assume nn': "n' = n" hence nn:"\<not> n' < n \<and> \<not> n < n'" by arith
459         from "4.hyps"(16,18)[of n n' n]
460           "4.hyps"(13,14)[of n "Suc n'" n]
461           np np' nn'
462         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
463           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
464           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
465           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
466         {assume mn: "m = n"
467           from "4.hyps"(17,18)[OF norm(1,4), of n]
468             "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
469           have degs:  "degreen (?cnp *\<^sub>p c') n =
470             (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"
471             "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n" by (simp_all add: min_def)
472           from degs norm
473           have th1: "degreen(?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" by simp
474           hence neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
475             by simp
476           have nmin: "n \<le> min n n" by (simp add: min_def)
477           from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
478           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
479           from "4.hyps"(16-18)[OF norm(1,4), of n]
480             "4.hyps"(13-15)[OF norm(1,2), of n]
481             mn norm m nn' deg
482           have ?eq by simp}
483         moreover
484         {assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
485           from nn' m np have max1: "m \<le> max n n"  by simp
486           hence min1: "m \<le> min n n" by simp
487           hence min2: "m \<le> min n (Suc n)" by simp
488           from "4.hyps"(16-18)[OF norm(1,4) min1]
489             "4.hyps"(13-15)[OF norm(1,2) min2]
490             degreen_polyadd[OF norm(3,6) max1]
492           have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m
493             \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
494             using mn nn' np np' by simp
495           with "4.hyps"(16-18)[OF norm(1,4) min1]
496             "4.hyps"(13-15)[OF norm(1,2) min2]
497             degreen_0[OF norm(3) mn']
498           have ?eq using nn' mn np np' by clarsimp}
499         ultimately have ?eq by blast}
500       ultimately show ?eq by blast}
501     { case (2 n0 n1)
502       hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1"
503         and m: "m \<le> min n0 n1" by simp_all
504       hence mn: "m \<le> n" by simp
505       let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
506       {assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
507         hence nn: "\<not>n' < n \<and> \<not> n<n'" by simp
508         from "4.hyps"(16-18) [of n n n]
509           "4.hyps"(13-15)[of n "Suc n" n]
510           np np' C(2) mn
511         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
512           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
513           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
514           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
515           "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
516             "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
517           by (simp_all add: min_def)
519           from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
520           have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
521             using norm by simp
522         from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"]  degneq
523         have "False" by simp }
524       thus ?case using "4.hyps" by clarsimp}
525 qed auto
527 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"
528 by(induct p q rule: polymul.induct, auto simp add: field_simps)
530 lemma polymul_normh:
531     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
532   shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
533   using polymul_properties(1)  by blast
534 lemma polymul_eq0_iff:
535   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
536   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) "
537   using polymul_properties(2)  by blast
538 lemma polymul_degreen:
539   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
540   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow> 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)"
541   using polymul_properties(3) by blast
542 lemma polymul_norm:
543   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
544   shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul p q)"
545   using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
547 lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
548   by (induct p arbitrary: n0 rule: headconst.induct, auto)
550 lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
551   by (induct p arbitrary: n0, auto)
553 lemma monic_eqI: assumes np: "isnpolyh p n0"
554   shows "INum (headconst p) * Ipoly bs (fst (monic p)) = (Ipoly bs p ::'a::{field_char_0, field_inverse_zero, power})"
555   unfolding monic_def Let_def
556 proof(cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
557   let ?h = "headconst p"
558   assume pz: "p \<noteq> 0\<^sub>p"
559   {assume hz: "INum ?h = (0::'a)"
560     from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
561     from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
562     with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
563   thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast
564 qed
567 text{* polyneg is a negation and preserves normal forms *}
569 lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
570 by (induct p rule: polyneg.induct, auto)
572 lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"
573   by (induct p arbitrary: n rule: polyneg.induct, auto simp add: Nneg_def)
574 lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
575   by (induct p arbitrary: n0 rule: polyneg.induct, auto)
576 lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "
577 by (induct p rule: polyneg.induct, auto simp add: polyneg0)
579 lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
580   using isnpoly_def polyneg_normh by simp
583 text{* polysub is a substraction and preserves normal forms *}
585 lemma polysub[simp]: "Ipoly bs (polysub p q) = (Ipoly bs p) - (Ipoly bs q)"
586 by (simp add: polysub_def polyneg polyadd)
587 lemma polysub_normh: "\<And> n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
588 by (simp add: polysub_def polyneg_normh polyadd_normh)
590 lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub p q)"
591   using polyadd_norm polyneg_norm by (simp add: polysub_def)
592 lemma polysub_same_0[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
593   shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p"
594 unfolding polysub_def split_def fst_conv snd_conv
595 by (induct p arbitrary: n0,auto simp add: Let_def Nsub0[simplified Nsub_def])
597 lemma polysub_0:
598   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
599   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"
600   unfolding polysub_def split_def fst_conv snd_conv
601   by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
602   (auto simp: Nsub0[simplified Nsub_def] Let_def)
604 text{* polypow is a power function and preserves normal forms *}
606 lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{field_char_0, field_inverse_zero})) ^ n"
607 proof(induct n rule: polypow.induct)
608   case 1 thus ?case by simp
609 next
610   case (2 n)
611   let ?q = "polypow ((Suc n) div 2) p"
612   let ?d = "polymul ?q ?q"
613   have "odd (Suc n) \<or> even (Suc n)" by simp
614   moreover
615   {assume odd: "odd (Suc n)"
616     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" by arith
617     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)" by (simp add: Let_def)
618     also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"
619       using "2.hyps" by simp
620     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
621       apply (simp only: power_add power_one_right) by simp
622     also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat)))))"
623       by (simp only: th)
624     finally have ?case
625     using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
626   moreover
627   {assume even: "even (Suc n)"
628     have th: "(Suc (Suc (0\<Colon>nat))) * (Suc n div Suc (Suc (0\<Colon>nat))) = Suc n div 2 + Suc n div 2" by arith
629     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
630     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
631       using "2.hyps" apply (simp only: power_add) by simp
632     finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}
633   ultimately show ?case by blast
634 qed
636 lemma polypow_normh:
637     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
638   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
639 proof (induct k arbitrary: n rule: polypow.induct)
640   case (2 k n)
641   let ?q = "polypow (Suc k div 2) p"
642   let ?d = "polymul ?q ?q"
643   from 2 have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
644   from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
645   from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n" by simp
646   from dn on show ?case by (simp add: Let_def)
647 qed auto
649 lemma polypow_norm:
650   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
651   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
652   by (simp add: polypow_normh isnpoly_def)
654 text{* Finally the whole normalization *}
656 lemma polynate[simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0, field_inverse_zero})"
657 by (induct p rule:polynate.induct, auto)
659 lemma polynate_norm[simp]:
660   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
661   shows "isnpoly (polynate p)"
662   by (induct p rule: polynate.induct, simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm) (simp_all add: isnpoly_def)
664 text{* shift1 *}
667 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
668 by (simp add: shift1_def polymul)
670 lemma shift1_isnpoly:
671   assumes pn: "isnpoly p" and pnz: "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p) "
672   using pn pnz by (simp add: shift1_def isnpoly_def )
674 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
675   by (simp add: shift1_def)
676 lemma funpow_shift1_isnpoly:
677   "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
678   by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
680 lemma funpow_isnpolyh:
681   assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n "and np: "isnpolyh p n"
682   shows "isnpolyh (funpow k f p) n"
683   using f np by (induct k arbitrary: p, auto)
685 lemma funpow_shift1: "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (Mul (Pw (Bound 0) n) p)"
686   by (induct n arbitrary: p, simp_all add: shift1_isnpoly shift1 power_Suc )
688 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
689   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
691 lemma funpow_shift1_1:
692   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (funpow n shift1 1\<^sub>p *\<^sub>p p)"
693   by (simp add: funpow_shift1)
695 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
696   by (induct p rule: poly_cmul.induct) (auto simp add: field_simps)
699   assumes np: "isnpolyh p n"
700   shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = (Ipoly bs p :: 'a :: {field_char_0, field_inverse_zero})"
701   using np
702 proof (induct p arbitrary: n rule: behead.induct)
703   case (1 c p n) hence pn: "isnpolyh p n" by simp
704   from 1(1)[OF pn]
705   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
706   then show ?case using "1.hyps" apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
707     by (simp_all add: th[symmetric] field_simps power_Suc)
708 qed (auto simp add: Let_def)
711   assumes np: "isnpolyh p n" shows "isnpolyh (behead p) n"
712   using np by (induct p rule: behead.induct, auto simp add: Let_def isnpolyh_mono)
714 subsection{* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
715 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
716 proof(induct p arbitrary: n rule: poly.induct, auto)
717   case (goal1 c n p n')
718   hence "n = Suc (n - 1)" by simp
719   hence "isnpolyh p (Suc (n - 1))"  using `isnpolyh p n` by simp
720   with goal1(2) show ?case by simp
721 qed
723 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
724 by (induct p arbitrary: n0 rule: isconstant.induct, auto simp add: isnpolyh_polybound0)
726 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" by (induct p, auto)
728 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
729   apply (induct p arbitrary: n0, auto)
730   apply (atomize)
731   apply (erule_tac x = "Suc nat" in allE)
732   apply auto
733   done
735 lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
736  by (induct p  arbitrary: n0 rule: head.induct, auto intro: isnpolyh_polybound0)
738 lemma polybound0_I:
739   assumes nb: "polybound0 a"
740   shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
741 using nb
742 by (induct a rule: poly.induct) auto
743 lemma polysubst0_I:
744   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
745   by (induct t) simp_all
747 lemma polysubst0_I':
748   assumes nb: "polybound0 a"
749   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t"
750   by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
752 lemma decrpoly: assumes nb: "polybound0 t"
753   shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
754   using nb by (induct t rule: decrpoly.induct, simp_all)
756 lemma polysubst0_polybound0: assumes nb: "polybound0 t"
757   shows "polybound0 (polysubst0 t a)"
758 using nb by (induct a rule: poly.induct, auto)
760 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
761   by (induct p arbitrary: n rule: degree.induct, auto simp add: isnpolyh_polybound0)
763 primrec maxindex :: "poly \<Rightarrow> nat" where
764   "maxindex (Bound n) = n + 1"
765 | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
766 | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
767 | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
768 | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
769 | "maxindex (Neg p) = maxindex p"
770 | "maxindex (Pw p n) = maxindex p"
771 | "maxindex (C x) = 0"
773 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool" where
774   "wf_bs bs p = (length bs \<ge> maxindex p)"
776 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
777 proof(induct p rule: coefficients.induct)
778   case (1 c p)
779   show ?case
780   proof
781     fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"
782     hence "x = c \<or> x \<in> set (coefficients p)" by simp
783     moreover
784     {assume "x = c" hence "wf_bs bs x" using "1.prems"  unfolding wf_bs_def by simp}
785     moreover
786     {assume H: "x \<in> set (coefficients p)"
787       from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp
788       with "1.hyps" H have "wf_bs bs x" by blast }
789     ultimately  show "wf_bs bs x" by blast
790   qed
791 qed simp_all
793 lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
794 by (induct p rule: coefficients.induct, auto)
796 lemma wf_bs_I: "wf_bs bs p ==> Ipoly (bs@bs') p = Ipoly bs p"
797   unfolding wf_bs_def by (induct p, auto simp add: nth_append)
799 lemma take_maxindex_wf: assumes wf: "wf_bs bs p"
800   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
801 proof-
802   let ?ip = "maxindex p"
803   let ?tbs = "take ?ip bs"
804   from wf have "length ?tbs = ?ip" unfolding wf_bs_def by simp
805   hence wf': "wf_bs ?tbs p" unfolding wf_bs_def by  simp
806   have eq: "bs = ?tbs @ (drop ?ip bs)" by simp
807   from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis using eq by simp
808 qed
810 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
811   by (induct p, auto)
813 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
814   unfolding wf_bs_def by simp
816 lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
817   unfolding wf_bs_def by simp
821 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
822 by(induct p rule: coefficients.induct, auto simp add: wf_bs_def)
823 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
824   by (induct p rule: coefficients.induct, simp_all)
827 lemma coefficients_head: "last (coefficients p) = head p"
828   by (induct p rule: coefficients.induct, auto)
830 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
831   unfolding wf_bs_def by (induct p rule: decrpoly.induct, auto)
833 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
834   apply (rule exI[where x="replicate (n - length xs) z"])
835   by simp
836 lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
837 by (cases p, auto) (case_tac "nat", simp_all)
839 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
840   unfolding wf_bs_def
841   apply (induct p q rule: polyadd.induct)
842   apply (auto simp add: Let_def)
843   done
845 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
846   unfolding wf_bs_def
847   apply (induct p q arbitrary: bs rule: polymul.induct)
849   apply clarsimp
850   apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
851   apply auto
852   done
854 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
855   unfolding wf_bs_def by (induct p rule: polyneg.induct, auto)
857 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
858   unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
860 subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}
862 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
863 definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)"
864 definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"
866 lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0"
867 proof (induct p arbitrary: n0 rule: coefficients.induct)
868   case (1 c p n0)
869   have cp: "isnpolyh (CN c 0 p) n0" by fact
870   hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
871     by (auto simp add: isnpolyh_mono[where n'=0])
872   from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp
873 qed auto
875 lemma coefficients_isconst:
876   "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
877   by (induct p arbitrary: n rule: coefficients.induct,
878     auto simp add: isnpolyh_Suc_const)
880 lemma polypoly_polypoly':
881   assumes np: "isnpolyh p n0"
882   shows "polypoly (x#bs) p = polypoly' bs p"
883 proof-
884   let ?cf = "set (coefficients p)"
885   from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
886   {fix q assume q: "q \<in> ?cf"
887     from q cn_norm have th: "isnpolyh q n0" by blast
888     from coefficients_isconst[OF np] q have "isconstant q" by blast
889     with isconstant_polybound0[OF th] have "polybound0 q" by blast}
890   hence "\<forall>q \<in> ?cf. polybound0 q" ..
891   hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"
892     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
893     by auto
895   thus ?thesis unfolding polypoly_def polypoly'_def by simp
896 qed
898 lemma polypoly_poly:
899   assumes np: "isnpolyh p n0" shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
900   using np
901 by (induct p arbitrary: n0 bs rule: coefficients.induct, auto simp add: polypoly_def)
903 lemma polypoly'_poly:
904   assumes np: "isnpolyh p n0" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
905   using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
908 lemma polypoly_poly_polybound0:
909   assumes np: "isnpolyh p n0" and nb: "polybound0 p"
910   shows "polypoly bs p = [Ipoly bs p]"
911   using np nb unfolding polypoly_def
912   by (cases p, auto, case_tac nat, auto)
914 lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
915   by (induct p rule: head.induct, auto)
917 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
918   by (cases p,auto)
921   by (induct p rule: head.induct, simp_all)
923 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
926 lemma isnpolyh_zero_iff:
927   assumes nq: "isnpolyh p n0" 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})"
928   shows "p = 0\<^sub>p"
929 using nq eq
930 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
931   case less
932   note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
933   {assume nz: "maxindex p = 0"
934     then obtain c where "p = C c" using np by (cases p, auto)
935     with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}
936   moreover
937   {assume nz: "maxindex p \<noteq> 0"
938     let ?h = "head p"
939     let ?hd = "decrpoly ?h"
940     let ?ihd = "maxindex ?hd"
941     from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h"
942       by simp_all
943     hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast
945     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
946     have mihn: "maxindex ?h \<le> maxindex p" by auto
947     with decr_maxindex[OF h(2)] nz  have ihd_lt_n: "?ihd < maxindex p" by auto
948     {fix bs:: "'a list"  assume bs: "wf_bs bs ?hd"
949       let ?ts = "take ?ihd bs"
950       let ?rs = "drop ?ihd bs"
951       have ts: "wf_bs ?ts ?hd" using bs unfolding wf_bs_def by simp
952       have bs_ts_eq: "?ts@ ?rs = bs" by simp
953       from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x#?ts) ?h" by simp
954       from ihd_lt_n have "ALL x. length (x#?ts) \<le> maxindex p" by simp
955       with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = maxindex p" by blast
956       hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" unfolding wf_bs_def by simp
957       with zp have "\<forall> x. Ipoly ((x#?ts) @ xs) p = 0" by blast
958       hence "\<forall> x. Ipoly (x#(?ts @ xs)) p = 0" by simp
959       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
960       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"  by simp
961       hence "poly (polypoly' (?ts @ xs) p) = poly []" by (auto intro: ext)
962       hence "\<forall> c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
963         using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
964       with coefficients_head[of p, symmetric]
965       have th0: "Ipoly (?ts @ xs) ?hd = 0" by simp
966       from bs have wf'': "wf_bs ?ts ?hd" unfolding wf_bs_def by simp
967       with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0" by simp
968       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by simp }
969     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast
971     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast
972     hence "?h = 0\<^sub>p" by simp
973     with head_nz[OF np] have "p = 0\<^sub>p" by simp}
974   ultimately show "p = 0\<^sub>p" by blast
975 qed
977 lemma isnpolyh_unique:
978   assumes np:"isnpolyh p n0" and nq: "isnpolyh q n1"
979   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"
980 proof(auto)
981   assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a)= \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
982   hence "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" by simp
983   hence "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
984     using wf_bs_polysub[where p=p and q=q] by auto
985   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq]
986   show "p = q" by blast
987 qed
990 text{* consequences of unicity on the algorithms for polynomial normalization *}
992 lemma polyadd_commute:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
993   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" shows "p +\<^sub>p q = q +\<^sub>p p"
994   using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]] by simp
996 lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp
997 lemma one_normh: "isnpolyh 1\<^sub>p n" by simp
999   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1000   and np: "isnpolyh p n0" shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>p p = p"
1001   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
1002     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
1004 lemma polymul_1[simp]:
1005     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1006   and np: "isnpolyh p n0" shows "p *\<^sub>p 1\<^sub>p = p" and "1\<^sub>p *\<^sub>p p = p"
1007   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
1008     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
1009 lemma polymul_0[simp]:
1010   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1011   and np: "isnpolyh p n0" shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p" and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
1012   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
1013     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
1015 lemma polymul_commute:
1016     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1017   and np:"isnpolyh p n0" and nq: "isnpolyh q n1"
1018   shows "p *\<^sub>p q = q *\<^sub>p p"
1019 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}"] by simp
1021 declare polyneg_polyneg[simp]
1023 lemma isnpolyh_polynate_id[simp]:
1024   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1025   and np:"isnpolyh p n0" shows "polynate p = p"
1026   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}"] by simp
1028 lemma polynate_idempotent[simp]:
1029     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1030   shows "polynate (polynate p) = polynate p"
1031   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
1033 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
1034   unfolding poly_nate_def polypoly'_def ..
1035 lemma poly_nate_poly: shows "poly (poly_nate bs p) = (\<lambda>x:: 'a ::{field_char_0, field_inverse_zero}. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
1036   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
1037   unfolding poly_nate_polypoly' by (auto intro: ext)
1039 subsection{* heads, degrees and all that *}
1040 lemma degree_eq_degreen0: "degree p = degreen p 0"
1041   by (induct p rule: degree.induct, simp_all)
1043 lemma degree_polyneg: assumes n: "isnpolyh p n"
1044   shows "degree (polyneg p) = degree p"
1045   using n
1046   by (induct p arbitrary: n rule: polyneg.induct, simp_all) (case_tac na, auto)
1049   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
1050   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
1051 using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
1054 lemma degree_polysub: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
1055   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
1056 proof-
1057   from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp
1058   from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])
1059 qed
1062   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1063   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q"
1064   and d: "degree p = degree q"
1065   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
1066 unfolding polysub_def split_def fst_conv snd_conv
1067 using np nq h d
1068 proof(induct p q rule:polyadd.induct)
1069   case (1 c c') thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def])
1070 next
1071   case (2 c c' n' p')
1072   from 2 have "degree (C c) = degree (CN c' n' p')" by simp
1073   hence nz:"n' > 0" by (cases n', auto)
1074   hence "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
1075   with 2 show ?case by simp
1076 next
1077   case (3 c n p c')
1078   hence "degree (C c') = degree (CN c n p)" by simp
1079   hence nz:"n > 0" by (cases n, auto)
1080   hence "head (CN c n p) = CN c n p" by (cases n, auto)
1081   with 3 show ?case by simp
1082 next
1083   case (4 c n p c' n' p')
1084   hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1"
1085     "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+
1086   hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all
1087   hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
1088     using H(1-2) degree_polyneg by auto
1089   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"  by simp+
1090   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"  by simp
1091   from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto
1092   have "n = n' \<or> n < n' \<or> n > n'" by arith
1093   moreover
1094   {assume nn': "n = n'"
1095     have "n = 0 \<or> n >0" by arith
1096     moreover {assume nz: "n = 0" hence ?case using 4 nn' by (auto simp add: Let_def degcmc')}
1097     moreover {assume nz: "n > 0"
1098       with nn' H(3) have  cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
1099       hence ?case 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] using nn' 4 by (simp add: Let_def)}
1100     ultimately have ?case by blast}
1101   moreover
1102   {assume nn': "n < n'" hence n'p: "n' > 0" by simp
1103     hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n', simp_all)
1104     have degcnp': "degree (CN c' n' p') = 0" and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')" using 4 nn' by (cases n', simp_all)
1105     hence "n > 0" by (cases n, simp_all)
1106     hence headcnp: "head (CN c n p) = CN c n p" by (cases n, auto)
1107     from H(3) headcnp headcnp' nn' have ?case by auto}
1108   moreover
1109   {assume nn': "n > n'"  hence np: "n > 0" by simp
1110     hence headcnp:"head (CN c n p) = CN c n p"  by (cases n, simp_all)
1111     from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
1112     from np have degcnp: "degree (CN c n p) = 0" by (cases n, simp_all)
1113     with degcnpeq have "n' > 0" by (cases n', simp_all)
1114     hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
1115     from H(3) headcnp headcnp' nn' have ?case by auto}
1116   ultimately show ?case  by blast
1117 qed auto
1119 lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
1120 by (induct p arbitrary: n0 rule: head.induct, simp_all add: shift1_def)
1122 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
1123 proof(induct k arbitrary: n0 p)
1124   case (Suc k n0 p) hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
1125   with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
1127   thus ?case by (simp add: funpow_swap1)
1128 qed auto
1130 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
1131   by (simp add: shift1_def)
1132 lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
1133   by (induct k arbitrary: p) (auto simp add: shift1_degree)
1135 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
1136   by (induct n arbitrary: p) (simp_all add: funpow.simps)
1138 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
1139   by (induct p arbitrary: n rule: degree.induct, auto)
1140 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
1141   by (induct p arbitrary: n rule: degreen.induct, auto)
1142 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
1143   by (induct p arbitrary: n rule: degree.induct, auto)
1145   by (induct p rule: head.induct, auto)
1148   "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd p q = C c\<rbrakk> \<Longrightarrow> degree p = degree q"
1149   using polyadd_eq_const_degreen degree_eq_degreen0 by simp
1151 lemma polyadd_head: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
1152   and deg: "degree p \<noteq> degree q"
1153   shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
1154 using np nq deg
1155 apply(induct p q arbitrary: n0 n1 rule: polyadd.induct,simp_all)
1156 apply (case_tac n', simp, simp)
1157 apply (case_tac n, simp, simp)
1158 apply (case_tac n, case_tac n', simp add: Let_def)
1159 apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p")
1161 apply (metis head_nz)
1162 apply (metis head_nz)
1163 apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
1164 by (metis degree.simps(9) gr0_conv_Suc nat_less_le order_le_less_trans)
1167    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1168   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"
1169 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
1170   case (2 c c' n' p' n0 n1)
1171   hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"  by (simp_all add: head_isnpolyh)
1172   thus ?case using 2 by (cases n', auto)
1173 next
1174   case (3 c n p c' n0 n1)
1175   hence "isnpolyh (head (CN c n p)) n0" "isnormNum c'"  by (simp_all add: head_isnpolyh)
1176   thus ?case using 3 by (cases n, auto)
1177 next
1178   case (4 c n p c' n' p' n0 n1)
1179   hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
1180     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
1181     by simp_all
1182   have "n < n' \<or> n' < n \<or> n = n'" by arith
1183   moreover
1184   {assume nn': "n < n'" hence ?case
1185       using norm
1186     "4.hyps"(2)[OF norm(1,6)]
1187     "4.hyps"(1)[OF norm(2,6)] by (simp, cases n, simp,cases n', simp_all)}
1188   moreover {assume nn': "n'< n"
1189     hence ?case using norm "4.hyps"(6) [OF norm(5,3)]
1190       "4.hyps"(5)[OF norm(5,4)]
1191       by (simp,cases n',simp,cases n,auto)}
1192   moreover {assume nn': "n' = n"
1193     from nn' polymul_normh[OF norm(5,4)]
1194     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
1195     from nn' polymul_normh[OF norm(5,3)] norm
1196     have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
1197     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
1198     have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
1199     from polyadd_normh[OF ncnpc' ncnpp0']
1200     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"
1201       by (simp add: min_def)
1202     {assume np: "n > 0"
1203       with nn' head_isnpolyh_Suc'[OF np nth]
1204         head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
1205       have ?case by simp}
1206     moreover
1207     {moreover assume nz: "n = 0"
1208       from polymul_degreen[OF norm(5,4), where m="0"]
1209         polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
1210       norm(5,6) degree_npolyhCN[OF norm(6)]
1211     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
1212     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
1213     from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
1214     have ?case   using norm "4.hyps"(6)[OF norm(5,3)]
1215         "4.hyps"(5)[OF norm(5,4)] nn' nz by simp }
1216     ultimately have ?case by (cases n) auto}
1217   ultimately show ?case by blast
1218 qed simp_all
1220 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
1221   by(induct p rule: degree.induct, auto)
1223 lemma degree_head[simp]: "degree (head p) = 0"
1224   by (induct p rule: head.induct, auto)
1226 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
1227   by (cases n, simp_all)
1228 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
1229   by (cases n, simp_all)
1231 lemma polyadd_different_degree: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degree p \<noteq> degree q\<rbrakk> \<Longrightarrow> degree (polyadd p q) = max (degree p) (degree q)"
1232   using polyadd_different_degreen degree_eq_degreen0 by simp
1234 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
1235   by (induct p arbitrary: n0 rule: polyneg.induct, auto)
1237 lemma degree_polymul:
1238   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1239   and np: "isnpolyh p n0" and nq: "isnpolyh q n1"
1240   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
1241   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
1243 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
1244   by (induct p arbitrary: n rule: degree.induct, auto)
1246 lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"
1247   by (induct p arbitrary: n rule: degree.induct, auto)
1249 subsection {* Correctness of polynomial pseudo division *}
1251 lemma polydivide_aux_properties:
1252   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1253   and np: "isnpolyh p n0" and ns: "isnpolyh s n1"
1254   and ap: "head p = a" and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
1255   shows "(polydivide_aux a n p k s = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p)
1256           \<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)))"
1257   using ns
1258 proof(induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
1259   case less
1260   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
1261   let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p)
1262     \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
1263   let ?b = "head s"
1264   let ?p' = "funpow (degree s - n) shift1 p"
1265   let ?xdn = "funpow (degree s - n) shift1 1\<^sub>p"
1266   let ?akk' = "a ^\<^sub>p (k' - k)"
1267   note ns = `isnpolyh s n1`
1268   from np have np0: "isnpolyh p 0"
1269     using isnpolyh_mono[where n="n0" and n'="0" and p="p"]  by simp
1270   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
1271   have headp': "head ?p' = head p" using funpow_shift1_head[OF np pnz] by simp
1272   from funpow_shift1_isnpoly[where p="1\<^sub>p"] have nxdn: "isnpolyh ?xdn 0" by (simp add: isnpoly_def)
1273   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
1274   have nakk':"isnpolyh ?akk' 0" by blast
1275   {assume sz: "s = 0\<^sub>p"
1276    hence ?ths using np polydivide_aux.simps apply clarsimp by (rule exI[where x="0\<^sub>p"], simp) }
1277   moreover
1278   {assume sz: "s \<noteq> 0\<^sub>p"
1279     {assume dn: "degree s < n"
1280       hence "?ths" using ns ndp np polydivide_aux.simps by auto (rule exI[where x="0\<^sub>p"],simp) }
1281     moreover
1282     {assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith
1283       have degsp': "degree s = degree ?p'"
1284         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp
1285       {assume ba: "?b = a"
1287         have nr: "isnpolyh (s -\<^sub>p ?p') 0" using polysub_normh[OF ns np'] by simp
1288         from degree_polysub_samehead[OF ns np' headsp' degsp']
1289         have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
1290         moreover
1291         {assume deglt:"degree (s -\<^sub>p ?p') < degree s"
1292           from polydivide_aux.simps sz dn' ba
1293           have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
1294             by (simp add: Let_def)
1295           {assume h1: "polydivide_aux a n p k s = (k', r)"
1296             from less(1)[OF deglt nr, of k k' r]
1297               trans[OF eq[symmetric] h1]
1298             have kk': "k \<le> k'" and nr:"\<exists>nr. isnpolyh r nr" and dr: "degree r = 0 \<or> degree r < degree p"
1299               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
1300             from q1 obtain q n1 where nq: "isnpolyh q n1"
1301               and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"  by blast
1302             from nr obtain nr where nr': "isnpolyh r nr" by blast
1303             from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0" by simp
1304             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
1305             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
1306             from polyadd_normh[OF polymul_normh[OF np
1307               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
1308             have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0" by simp
1309             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')) =
1310               Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
1311             hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
1312               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
1313               by (simp add: field_simps)
1314             hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1315               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p *\<^sub>p p)
1316               + Ipoly bs p * Ipoly bs q + Ipoly bs r"
1317               by (auto simp only: funpow_shift1_1)
1318             hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1319               Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p)
1320               + Ipoly bs q) + Ipoly bs r" by (simp add: field_simps)
1321             hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1322               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
1323             with isnpolyh_unique[OF nakks' nqr']
1324             have "a ^\<^sub>p (k' - k) *\<^sub>p s =
1325               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
1326             hence ?qths using nq'
1327               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 1\<^sub>p) +\<^sub>p q" in exI)
1328               apply (rule_tac x="0" in exI) by simp
1329             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"
1330               by blast } hence ?ths by blast }
1331         moreover
1332         {assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
1333           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}"]
1334           have " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs s = Ipoly bs ?p'" by simp
1335           hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)" using np nxdn apply simp
1336             by (simp only: funpow_shift1_1) simp
1337           hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]] by blast
1338           {assume h1: "polydivide_aux a n p k s = (k',r)"
1339             from polydivide_aux.simps sz dn' ba
1340             have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
1341               by (simp add: Let_def)
1342             also have "\<dots> = (k,0\<^sub>p)" using polydivide_aux.simps spz by simp
1343             finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
1344             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
1345               polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
1346               apply auto
1347               apply (rule exI[where x="?xdn"])
1348               apply (auto simp add: polymul_commute[of p])
1349               done} }
1350         ultimately have ?ths by blast }
1351       moreover
1352       {assume ba: "?b \<noteq> a"
1353         from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
1354           polymul_normh[OF head_isnpolyh[OF ns] np']]
1355         have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by(simp add: min_def)
1356         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
1357           using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
1358             polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
1359             funpow_shift1_nz[OF pnz] by simp_all
1361           polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
1362         have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
1365           by (simp add: ap)
1366         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
1367           head_nz[OF np] pnz sz ap[symmetric]
1368           funpow_shift1_nz[OF pnz, where n="degree s - n"]
1369           polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"]  head_nz[OF ns]
1370           ndp dn
1371         have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p') "
1372           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
1373         {assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
1374           from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns] polymul_normh[OF head_isnpolyh[OF ns]np']]
1375           ap have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by simp
1376           {assume h1:"polydivide_aux a n p k s = (k', r)"
1377             from h1 polydivide_aux.simps sz dn' ba
1378             have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
1379               by (simp add: Let_def)
1380             with less(1)[OF dth nasbp', of "Suc k" k' r]
1381             obtain q nq nr where kk': "Suc k \<le> k'" and nr: "isnpolyh r nr" and nq: "isnpolyh q nq"
1382               and dr: "degree r = 0 \<or> degree r < degree p"
1383               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
1384             from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith
1385             {fix bs:: "'a::{field_char_0, field_inverse_zero} list"
1387             from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
1388             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)" by simp
1389             hence "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s = Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
1390               by (simp add: field_simps power_Suc)
1391             hence "Ipoly bs a ^ (k' - k)  * Ipoly bs s = Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
1392               by (simp add:kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
1393             hence "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
1394               by (simp add: field_simps)}
1395             hence ieq:"\<forall>(bs :: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
1396               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
1397             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
1398             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]]
1399             have nqw: "isnpolyh ?q 0" by simp
1400             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]]
1401             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
1402             from dr kk' nr h1 asth nqw have ?ths apply simp
1403               apply (rule conjI)
1404               apply (rule exI[where x="nr"], simp)
1405               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
1406               apply (rule exI[where x="0"], simp)
1407               done}
1408           hence ?ths by blast }
1409         moreover
1410         {assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
1411           {fix bs :: "'a::{field_char_0, field_inverse_zero} list"
1412             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
1413           have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'" by simp
1414           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
1415             by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
1416           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" by simp
1417         }
1418         hence hth: "\<forall> (bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
1419           from hth
1420           have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
1421             using isnpolyh_unique[where ?'a = "'a::{field_char_0, field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
1422                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
1423               simplified ap] by simp
1424           {assume h1: "polydivide_aux a n p k s = (k', r)"
1425           from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
1426           have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
1427           with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
1428             polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
1429           have ?ths apply (clarsimp simp add: Let_def)
1430             apply (rule exI[where x="?b *\<^sub>p ?xdn"]) apply simp
1431             apply (rule exI[where x="0"], simp)
1432             done}
1433         hence ?ths by blast}
1434         ultimately have ?ths 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"]
1435           head_nz[OF np] pnz sz ap[symmetric]
1436           by (simp add: degree_eq_degreen0[symmetric]) blast }
1437       ultimately have ?ths by blast
1438     }
1439     ultimately have ?ths by blast}
1440   ultimately show ?ths by blast
1441 qed
1443 lemma polydivide_properties:
1444   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1445   and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
1446   shows "(\<exists> k r. polydivide s p = (k,r) \<and> (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p)
1447   \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
1448 proof-
1449   have trv: "head p = head p" "degree p = degree p" by simp_all
1450   from polydivide_def[where s="s" and p="p"]
1451   have ex: "\<exists> k r. polydivide s p = (k,r)" by auto
1452   then obtain k r where kr: "polydivide s p = (k,r)" by blast
1453   from trans[OF meta_eq_to_obj_eq[OF polydivide_def[where s="s"and p="p"], symmetric] kr]
1454     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
1455   have "(degree r = 0 \<or> degree r < degree p) \<and>
1456    (\<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
1457   with kr show ?thesis
1458     apply -
1459     apply (rule exI[where x="k"])
1460     apply (rule exI[where x="r"])
1461     apply simp
1462     done
1463 qed
1465 subsection{* More about polypoly and pnormal etc *}
1467 definition "isnonconstant p = (\<not> isconstant p)"
1469 lemma isnonconstant_pnormal_iff: assumes nc: "isnonconstant p"
1470   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
1471 proof
1472   let ?p = "polypoly bs p"
1473   assume H: "pnormal ?p"
1474   have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
1476   from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
1477     pnormal_last_nonzero[OF H]
1478   show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)
1479 next
1480   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
1481   let ?p = "polypoly bs p"
1482   have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
1483   hence pz: "?p \<noteq> []" by (simp add: polypoly_def)
1484   hence lg: "length ?p > 0" by simp
1485   from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
1486   have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)
1487   from pnormal_last_length[OF lg lz] show "pnormal ?p" .
1488 qed
1490 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
1491   unfolding isnonconstant_def
1492   apply (cases p, simp_all)
1493   apply (case_tac nat, auto)
1494   done
1495 lemma isnonconstant_nonconstant: assumes inc: "isnonconstant p"
1496   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
1497 proof
1498   let ?p = "polypoly bs p"
1499   assume nc: "nonconstant ?p"
1500   from isnonconstant_pnormal_iff[OF inc, of bs] nc
1501   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast
1502 next
1503   let ?p = "polypoly bs p"
1504   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
1505   from isnonconstant_pnormal_iff[OF inc, of bs] h
1506   have pn: "pnormal ?p" by blast
1507   {fix x assume H: "?p = [x]"
1508     from H have "length (coefficients p) = 1" unfolding polypoly_def by auto
1509     with isnonconstant_coefficients_length[OF inc] have False by arith}
1510   thus "nonconstant ?p" using pn unfolding nonconstant_def by blast
1511 qed
1513 lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
1514   unfolding pnormal_def
1515  apply (induct p)
1516  apply (simp_all, case_tac "p=[]", simp_all)
1517  done
1519 lemma degree_degree: assumes inc: "isnonconstant p"
1520   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
1521 proof
1522   let  ?p = "polypoly bs p"
1523   assume H: "degree p = Polynomial_List.degree ?p"
1524   from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
1525     unfolding polypoly_def by auto
1526   from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
1527   have lg:"length (pnormalize ?p) = length ?p"
1528     unfolding Polynomial_List.degree_def polypoly_def by simp
1529   hence "pnormal ?p" using pnormal_length[OF pz] by blast
1530   with isnonconstant_pnormal_iff[OF inc]
1531   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast
1532 next
1533   let  ?p = "polypoly bs p"
1534   assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
1535   with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast
1536   with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
1537   show "degree p = Polynomial_List.degree ?p"
1538     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
1539 qed
1541 section{* Swaps ; Division by a certain variable *}
1542 primrec swap:: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly" where
1543   "swap n m (C x) = C x"
1544 | "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"
1545 | "swap n m (Neg t) = Neg (swap n m t)"
1546 | "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
1547 | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
1548 | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
1549 | "swap n m (Pw t k) = Pw (swap n m t) k"
1550 | "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)
1551   (swap n m p)"
1553 lemma swap: assumes nbs: "n < length bs" and mbs: "m < length bs"
1554   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
1555 proof (induct t)
1556   case (Bound k) thus ?case using nbs mbs by simp
1557 next
1558   case (CN c k p) thus ?case using nbs mbs by simp
1559 qed simp_all
1560 lemma swap_swap_id[simp]: "swap n m (swap m n t) = t"
1561   by (induct t,simp_all)
1563 lemma swap_commute: "swap n m p = swap m n p" by (induct p, simp_all)
1565 lemma swap_same_id[simp]: "swap n n t = t"
1566   by (induct t, simp_all)
1568 definition "swapnorm n m t = polynate (swap n m t)"
1570 lemma swapnorm: assumes nbs: "n < length bs" and mbs: "m < length bs"
1571   shows "((Ipoly bs (swapnorm n m t) :: 'a\<Colon>{field_char_0, field_inverse_zero})) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
1572   using swap[OF assms] swapnorm_def by simp
1574 lemma swapnorm_isnpoly[simp]:
1575     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
1576   shows "isnpoly (swapnorm n m p)"
1577   unfolding swapnorm_def by simp
1579 definition "polydivideby n s p =
1580     (let ss = swapnorm 0 n s ; sp = swapnorm 0 n p ; h = head sp; (k,r) = polydivide ss sp
1581      in (k,swapnorm 0 n h,swapnorm 0 n r))"
1583 lemma swap_nz [simp]: " (swap n m p = 0\<^sub>p) = (p = 0\<^sub>p)" by (induct p, simp_all)
1585 fun isweaknpoly :: "poly \<Rightarrow> bool"
1586 where
1587   "isweaknpoly (C c) = True"
1588 | "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
1589 | "isweaknpoly p = False"
1591 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
1592   by (induct p arbitrary: n0, auto)
1594 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
1595   by (induct p, auto)
1597 end