src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
 author krauss Mon Feb 21 23:14:36 2011 +0100 (2011-02-21) changeset 41811 7e338ccabff0 parent 41810 588c95c4b53e child 41812 d46c2908a838 permissions -rw-r--r--
strengthened polymul.induct
```     1 (*  Title:      HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
```
```     2     Author:     Amine Chaieb
```
```     3 *)
```
```     4
```
```     5 header {* Implementation and verification of multivariate polynomials *}
```
```     6
```
```     7 theory Reflected_Multivariate_Polynomial
```
```     8 imports Complex_Main "~~/src/HOL/Library/Abstract_Rat" Polynomial_List
```
```     9 begin
```
```    10
```
```    11   (* Implementation *)
```
```    12
```
```    13 subsection{* Datatype of polynomial expressions *}
```
```    14
```
```    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
```
```    17
```
```    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)"
```
```    20
```
```    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"
```
```    31
```
```    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)"
```
```    41
```
```    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))"
```
```    52
```
```    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"
```
```    63
```
```    64 subsection{* Degrees and heads and coefficients *}
```
```    65
```
```    66 fun degree:: "poly \<Rightarrow> nat"
```
```    67 where
```
```    68   "degree (CN c 0 p) = 1 + degree p"
```
```    69 | "degree p = 0"
```
```    70
```
```    71 fun head:: "poly \<Rightarrow> poly"
```
```    72 where
```
```    73   "head (CN c 0 p) = head p"
```
```    74 | "head p = p"
```
```    75
```
```    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)"
```
```    81
```
```    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)"
```
```    86
```
```    87 fun coefficients:: "poly \<Rightarrow> poly list"
```
```    88 where
```
```    89   "coefficients (CN c 0 p) = c#(coefficients p)"
```
```    90 | "coefficients p = [p]"
```
```    91
```
```    92 fun isconstant:: "poly \<Rightarrow> bool"
```
```    93 where
```
```    94   "isconstant (CN c 0 p) = False"
```
```    95 | "isconstant p = True"
```
```    96
```
```    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"
```
```   101
```
```   102 fun headconst:: "poly \<Rightarrow> Num"
```
```   103 where
```
```   104   "headconst (CN c n p) = headconst p"
```
```   105 | "headconst (C n) = n"
```
```   106
```
```   107 subsection{* Operations for normalization *}
```
```   108 consts
```
```   109   polyadd :: "poly\<times>poly \<Rightarrow> poly"
```
```   110   polysub :: "poly\<times>poly \<Rightarrow> poly"
```
```   111   polymul :: "poly\<times>poly \<Rightarrow> poly"
```
```   112
```
```   113 abbreviation poly_add :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
```
```   114   where "a +\<^sub>p b \<equiv> polyadd (a,b)"
```
```   115 abbreviation poly_mul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
```
```   116   where "a *\<^sub>p b \<equiv> polymul (a,b)"
```
```   117 abbreviation poly_sub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
```
```   118   where "a -\<^sub>p b \<equiv> polysub (a,b)"
```
```   119
```
```   120 recdef polyadd "measure (\<lambda> (a,b). polysize a + polysize b)"
```
```   121   "polyadd (C c, C c') = C (c+\<^sub>Nc')"
```
```   122   "polyadd (C c, CN c' n' p') = CN (polyadd (C c, c')) n' p'"
```
```   123   "polyadd (CN c n p, C c') = CN (polyadd (c, C c')) n p"
```
```   124   "polyadd (CN c n p, CN c' n' p') =
```
```   125     (if n < n' then CN (polyadd(c,CN c' n' p')) n p
```
```   126      else if n'<n then CN (polyadd(CN c n p, c')) n' p'
```
```   127      else (let cc' = polyadd (c,c') ;
```
```   128                pp' = polyadd (p,p')
```
```   129            in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
```
```   130   "polyadd (a, b) = Add a b"
```
```   131 (hints recdef_simp add: Let_def measure_def split_def inv_image_def recdef_cong del: if_cong)
```
```   132
```
```   133 fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
```
```   134 where
```
```   135   "polyneg (C c) = C (~\<^sub>N c)"
```
```   136 | "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
```
```   137 | "polyneg a = Neg a"
```
```   138
```
```   139 defs polysub_def[code]: "polysub \<equiv> \<lambda> (p,q). polyadd (p,polyneg q)"
```
```   140
```
```   141 recdef polymul "measure (\<lambda>(a,b). size a + size b)"
```
```   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"
```
```   153 (hints recdef_cong del: if_cong)
```
```   154
```
```   155 fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
```
```   156 where
```
```   157   "polypow 0 = (\<lambda>p. 1\<^sub>p)"
```
```   158 | "polypow n = (\<lambda>p. let q = polypow (n div 2) p ; d = polymul(q,q) in
```
```   159                     if even n then d else polymul(p,d))"
```
```   160
```
```   161 abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
```
```   162   where "a ^\<^sub>p k \<equiv> polypow k a"
```
```   163
```
```   164 function polynate :: "poly \<Rightarrow> poly"
```
```   165 where
```
```   166   "polynate (Bound n) = CN 0\<^sub>p n 1\<^sub>p"
```
```   167 | "polynate (Add p q) = (polynate p +\<^sub>p polynate q)"
```
```   168 | "polynate (Sub p q) = (polynate p -\<^sub>p polynate q)"
```
```   169 | "polynate (Mul p q) = (polynate p *\<^sub>p polynate q)"
```
```   170 | "polynate (Neg p) = (~\<^sub>p (polynate p))"
```
```   171 | "polynate (Pw p n) = ((polynate p) ^\<^sub>p n)"
```
```   172 | "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
```
```   173 | "polynate (C c) = C (normNum c)"
```
```   174 by pat_completeness auto
```
```   175 termination by (relation "measure polysize") auto
```
```   176
```
```   177 fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly" where
```
```   178   "poly_cmul y (C x) = C (y *\<^sub>N x)"
```
```   179 | "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
```
```   180 | "poly_cmul y p = C y *\<^sub>p p"
```
```   181
```
```   182 definition monic :: "poly \<Rightarrow> (poly \<times> bool)" where
```
```   183   "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
```
```   185 subsection{* Pseudo-division *}
```
```   186
```
```   187 definition shift1 :: "poly \<Rightarrow> poly" where
```
```   188   "shift1 p \<equiv> CN 0\<^sub>p 0 p"
```
```   189
```
```   190 abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)" where
```
```   191   "funpow \<equiv> compow"
```
```   192
```
```   193 partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
```
```   194   where
```
```   195   "polydivide_aux a n p k s =
```
```   196   (if s = 0\<^sub>p then (k,s)
```
```   197   else (let b = head s; m = degree s in
```
```   198   (if m < n then (k,s) else
```
```   199   (let p'= funpow (m - n) shift1 p in
```
```   200   (if a = b then polydivide_aux a n p k (s -\<^sub>p p')
```
```   201   else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))))))"
```
```   202
```
```   203 definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)" where
```
```   204   "polydivide s p \<equiv> polydivide_aux (head p) (degree p) p 0 s"
```
```   205
```
```   206 fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly" where
```
```   207   "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
```
```   208 | "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
```
```   209
```
```   210 fun poly_deriv :: "poly \<Rightarrow> poly" where
```
```   211   "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
```
```   212 | "poly_deriv p = 0\<^sub>p"
```
```   213
```
```   214   (* Verification *)
```
```   215 lemma nth_pos2[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
```
```   216 using Nat.gr0_conv_Suc
```
```   217 by clarsimp
```
```   218
```
```   219 subsection{* Semantics of the polynomial representation *}
```
```   220
```
```   221 primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0, field_inverse_zero, power}" where
```
```   222   "Ipoly bs (C c) = INum c"
```
```   223 | "Ipoly bs (Bound n) = bs!n"
```
```   224 | "Ipoly bs (Neg a) = - Ipoly bs a"
```
```   225 | "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
```
```   226 | "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
```
```   227 | "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
```
```   228 | "Ipoly bs (Pw t n) = (Ipoly bs t) ^ n"
```
```   229 | "Ipoly bs (CN c n p) = (Ipoly bs c) + (bs!n)*(Ipoly bs p)"
```
```   230
```
```   231 abbreviation
```
```   232   Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0, field_inverse_zero, power}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
```
```   233   where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"
```
```   234
```
```   235 lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i"
```
```   236   by (simp add: INum_def)
```
```   237 lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"
```
```   238   by (simp  add: INum_def)
```
```   239
```
```   240 lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
```
```   241
```
```   242 subsection {* Normal form and normalization *}
```
```   243
```
```   244 fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
```
```   245 where
```
```   246   "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
```
```   247 | "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))"
```
```   248 | "isnpolyh p = (\<lambda>k. False)"
```
```   249
```
```   250 lemma isnpolyh_mono: "\<lbrakk>n' \<le> n ; isnpolyh p n\<rbrakk> \<Longrightarrow> isnpolyh p n'"
```
```   251 by (induct p rule: isnpolyh.induct, auto)
```
```   252
```
```   253 definition isnpoly :: "poly \<Rightarrow> bool" where
```
```   254   "isnpoly p \<equiv> isnpolyh p 0"
```
```   255
```
```   256 text{* polyadd preserves normal forms *}
```
```   257
```
```   258 lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk>
```
```   259       \<Longrightarrow> isnpolyh (polyadd(p,q)) (min n0 n1)"
```
```   260 proof(induct p q arbitrary: n0 n1 rule: polyadd.induct)
```
```   261   case (2 a b c' n' p' n0 n1)
```
```   262   from prems have  th1: "isnpolyh (C (a,b)) (Suc n')" by simp
```
```   263   from prems(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
```
```   264   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
```
```   265   with prems(1)[OF th1 th2] have th3:"isnpolyh (C (a,b) +\<^sub>p c') (Suc n')" by simp
```
```   266   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
```
```   267   thus ?case using prems th3 by simp
```
```   268 next
```
```   269   case (3 c' n' p' a b n1 n0)
```
```   270   from prems have  th1: "isnpolyh (C (a,b)) (Suc n')" by simp
```
```   271   from prems(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
```
```   272   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
```
```   273   with prems(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C (a,b)) (Suc n')" by simp
```
```   274   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
```
```   275   thus ?case using prems th3 by simp
```
```   276 next
```
```   277   case (4 c n p c' n' p' n0 n1)
```
```   278   hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
```
```   279   from prems have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all
```
```   280   from prems have ngen0: "n \<ge> n0" by simp
```
```   281   from prems have n'gen1: "n' \<ge> n1" by simp
```
```   282   have "n < n' \<or> n' < n \<or> n = n'" by auto
```
```   283   moreover {assume eq: "n = n'"
```
```   284     with prems(2)[OF nc nc']
```
```   285     have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
```
```   286     hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
```
```   287       using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
```
```   288     from eq prems(1)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
```
```   289     have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
```
```   290     from minle npp' ncc'n01 prems ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}
```
```   291   moreover {assume lt: "n < n'"
```
```   292     have "min n0 n1 \<le> n0" by simp
```
```   293     with prems have th1:"min n0 n1 \<le> n" by auto
```
```   294     from prems have th21: "isnpolyh c (Suc n)" by simp
```
```   295     from prems have th22: "isnpolyh (CN c' n' p') n'" by simp
```
```   296     from lt have th23: "min (Suc n) n' = Suc n" by arith
```
```   297     from prems(4)[OF th21 th22]
```
```   298     have "isnpolyh (polyadd (c, CN c' n' p')) (Suc n)" using th23 by simp
```
```   299     with prems th1 have ?case by simp }
```
```   300   moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
```
```   301     have "min n0 n1 \<le> n1"  by simp
```
```   302     with prems have th1:"min n0 n1 \<le> n'" by auto
```
```   303     from prems have th21: "isnpolyh c' (Suc n')" by simp_all
```
```   304     from prems have th22: "isnpolyh (CN c n p) n" by simp
```
```   305     from gt have th23: "min n (Suc n') = Suc n'" by arith
```
```   306     from prems(3)[OF th22 th21]
```
```   307     have "isnpolyh (polyadd (CN c n p,c')) (Suc n')" using th23 by simp
```
```   308     with prems th1 have ?case by simp}
```
```   309       ultimately show ?case by blast
```
```   310 qed auto
```
```   311
```
```   312 lemma polyadd[simp]: "Ipoly bs (polyadd (p,q)) = (Ipoly bs p) + (Ipoly bs q)"
```
```   313 by (induct p q rule: polyadd.induct, auto simp add: Let_def field_simps right_distrib[symmetric] simp del: right_distrib)
```
```   314
```
```   315 lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd(p,q))"
```
```   316   using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   317
```
```   318 text{* The degree of addition and other general lemmas needed for the normal form of polymul *}
```
```   319
```
```   320 lemma polyadd_different_degreen:
```
```   321   "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow>
```
```   322   degreen (polyadd(p,q)) m = max (degreen p m) (degreen q m)"
```
```   323 proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
```
```   324   case (4 c n p c' n' p' m n0 n1)
```
```   325   have "n' = n \<or> n < n' \<or> n' < n" by arith
```
```   326   thus ?case
```
```   327   proof (elim disjE)
```
```   328     assume [simp]: "n' = n"
```
```   329     from 4(1)[of n n m] 4(2)[of "Suc n" "Suc n" m] 4(5-7)
```
```   330     show ?thesis by (auto simp: Let_def)
```
```   331   next
```
```   332     assume "n < n'"
```
```   333     with 4 show ?thesis by auto
```
```   334   next
```
```   335     assume "n' < n"
```
```   336     with 4 show ?thesis by auto
```
```   337   qed
```
```   338 qed auto
```
```   339
```
```   340 lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
```
```   341   by (induct p arbitrary: n rule: headn.induct, auto)
```
```   342 lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
```
```   343   by (induct p arbitrary: n rule: degree.induct, auto)
```
```   344 lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
```
```   345   by (induct p arbitrary: n rule: degreen.induct, auto)
```
```   346
```
```   347 lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
```
```   348   by (induct p arbitrary: n rule: degree.induct, auto)
```
```   349
```
```   350 lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
```
```   351   using degree_isnpolyh_Suc by auto
```
```   352 lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
```
```   353   using degreen_0 by auto
```
```   354
```
```   355
```
```   356 lemma degreen_polyadd:
```
```   357   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1"
```
```   358   shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
```
```   359   using np nq m
```
```   360 proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
```
```   361   case (2 c c' n' p' n0 n1) thus ?case  by (cases n', simp_all)
```
```   362 next
```
```   363   case (3 c n p c' n0 n1) thus ?case by (cases n, auto)
```
```   364 next
```
```   365   case (4 c n p c' n' p' n0 n1 m)
```
```   366   have "n' = n \<or> n < n' \<or> n' < n" by arith
```
```   367   thus ?case
```
```   368   proof (elim disjE)
```
```   369     assume [simp]: "n' = n"
```
```   370     from 4(1)[of n n m] 4(2)[of "Suc n" "Suc n" m] 4(5-7)
```
```   371     show ?thesis by (auto simp: Let_def)
```
```   372   qed simp_all
```
```   373 qed auto
```
```   374
```
```   375 lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk>
```
```   376   \<Longrightarrow> degreen p m = degreen q m"
```
```   377 proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
```
```   378   case (4 c n p c' n' p' m n0 n1 x)
```
```   379   {assume nn': "n' < n" hence ?case using prems by simp}
```
```   380   moreover
```
```   381   {assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
```
```   382     moreover {assume "n < n'" with prems have ?case by simp }
```
```   383     moreover {assume eq: "n = n'" hence ?case using prems
```
```   384         apply (cases "p +\<^sub>p p' = 0\<^sub>p")
```
```   385         apply (auto simp add: Let_def)
```
```   386         by blast
```
```   387       }
```
```   388     ultimately have ?case by blast}
```
```   389   ultimately show ?case by blast
```
```   390 qed simp_all
```
```   391
```
```   392 lemma polymul_properties:
```
```   393   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   394   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> min n0 n1"
```
```   395   shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   396   and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)"
```
```   397   and "degreen (p *\<^sub>p q) m = (if (p = 0\<^sub>p \<or> q = 0\<^sub>p) then 0
```
```   398                              else degreen p m + degreen q m)"
```
```   399   using np nq m
```
```   400 proof(induct p q arbitrary: n0 n1 m rule: polymul.induct)
```
```   401   case (2 a b c' n' p')
```
```   402   let ?c = "(a,b)"
```
```   403   { case (1 n0 n1)
```
```   404     with "2.hyps"(1-3)[of n' n' n']
```
```   405       and "2.hyps"(4-6)[of "Suc n'" "Suc n'" n']
```
```   406     show ?case by (auto simp add: min_def)
```
```   407   next
```
```   408     case (2 n0 n1) thus ?case by auto
```
```   409   next
```
```   410     case (3 n0 n1) thus ?case  using "2.hyps" by auto }
```
```   411 next
```
```   412   case (3 c n p a b)
```
```   413   let ?c' = "(a,b)"
```
```   414   { case (1 n0 n1)
```
```   415     with "3.hyps"(1-3)[of n n n]
```
```   416       "3.hyps"(4-6)[of "Suc n" "Suc n" n]
```
```   417     show ?case by (auto simp add: min_def)
```
```   418   next
```
```   419     case (2 n0 n1) thus ?case by auto
```
```   420   next
```
```   421     case (3 n0 n1) thus ?case  using "3.hyps" by auto }
```
```   422 next
```
```   423   case (4 c n p c' n' p')
```
```   424   let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
```
```   425     {
```
```   426       case (1 n0 n1)
```
```   427       hence cnp: "isnpolyh ?cnp n" and cnp': "isnpolyh ?cnp' n'"
```
```   428         and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)"
```
```   429         and np': "isnpolyh p' n'" and nc': "isnpolyh c' (Suc n')"
```
```   430         and nn0: "n \<ge> n0" and nn1:"n' \<ge> n1"
```
```   431         by simp_all
```
```   432       { assume "n < n'"
```
```   433         with "4.hyps"(13-14)[OF np cnp', of n]
```
```   434           "4.hyps"(16)[OF nc cnp', of n] nn0 cnp
```
```   435         have ?case by (simp add: min_def)
```
```   436       } moreover {
```
```   437         assume "n' < n"
```
```   438         with "4.hyps"(1-2)[OF cnp np', of "n'"]
```
```   439           "4.hyps"(4)[OF cnp nc', of "Suc n'"] nn1 cnp'
```
```   440         have ?case
```
```   441           by (cases "Suc n' = n", simp_all add: min_def)
```
```   442       } moreover {
```
```   443         assume "n' = n"
```
```   444         with "4.hyps"(1-2)[OF cnp np', of n]
```
```   445           "4.hyps"(4)[OF cnp nc', of n] cnp cnp' nn1 nn0
```
```   446         have ?case
```
```   447           apply (auto intro!: polyadd_normh)
```
```   448           apply (simp_all add: min_def isnpolyh_mono[OF nn0])
```
```   449           done
```
```   450       }
```
```   451       ultimately show ?case by arith
```
```   452     next
```
```   453       fix n0 n1 m
```
```   454       assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1"
```
```   455       and m: "m \<le> min n0 n1"
```
```   456       let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
```
```   457       let ?d1 = "degreen ?cnp m"
```
```   458       let ?d2 = "degreen ?cnp' m"
```
```   459       let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
```
```   460       have "n'<n \<or> n < n' \<or> n' = n" by auto
```
```   461       moreover
```
```   462       {assume "n' < n \<or> n < n'"
```
```   463         with "4.hyps"(3,15,18) np np' m
```
```   464         have ?eq by auto }
```
```   465       moreover
```
```   466       {assume nn': "n' = n" hence nn:"\<not> n' < n \<and> \<not> n < n'" by arith
```
```   467         from "4.hyps"(1,3)[of n n' n]
```
```   468           "4.hyps"(4,5)[of n "Suc n'" n]
```
```   469           np np' nn'
```
```   470         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
```
```   471           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   472           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
```
```   473           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
```
```   474         {assume mn: "m = n"
```
```   475           from "4.hyps"(2,3)[OF norm(1,4), of n]
```
```   476             "4.hyps"(4,6)[OF norm(1,2), of n] norm nn' mn
```
```   477           have degs:  "degreen (?cnp *\<^sub>p c') n =
```
```   478             (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"
```
```   479             "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n" by (simp_all add: min_def)
```
```   480           from degs norm
```
```   481           have th1: "degreen(?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" by simp
```
```   482           hence neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   483             by simp
```
```   484           have nmin: "n \<le> min n n" by (simp add: min_def)
```
```   485           from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
```
```   486           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
```
```   487           from "4.hyps"(1-3)[OF norm(1,4), of n]
```
```   488             "4.hyps"(4-6)[OF norm(1,2), of n]
```
```   489             mn norm m nn' deg
```
```   490           have ?eq by simp}
```
```   491         moreover
```
```   492         {assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
```
```   493           from nn' m np have max1: "m \<le> max n n"  by simp
```
```   494           hence min1: "m \<le> min n n" by simp
```
```   495           hence min2: "m \<le> min n (Suc n)" by simp
```
```   496           from "4.hyps"(1-3)[OF norm(1,4) min1]
```
```   497             "4.hyps"(4-6)[OF norm(1,2) min2]
```
```   498             degreen_polyadd[OF norm(3,6) max1]
```
```   499
```
```   500           have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m
```
```   501             \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
```
```   502             using mn nn' np np' by simp
```
```   503           with "4.hyps"(1-3)[OF norm(1,4) min1]
```
```   504             "4.hyps"(4-6)[OF norm(1,2) min2]
```
```   505             degreen_0[OF norm(3) mn']
```
```   506           have ?eq using nn' mn np np' by clarsimp}
```
```   507         ultimately have ?eq by blast}
```
```   508       ultimately show ?eq by blast}
```
```   509     { case (2 n0 n1)
```
```   510       hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1"
```
```   511         and m: "m \<le> min n0 n1" by simp_all
```
```   512       hence mn: "m \<le> n" by simp
```
```   513       let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
```
```   514       {assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
```
```   515         hence nn: "\<not>n' < n \<and> \<not> n<n'" by simp
```
```   516         from "4.hyps"(1-3) [of n n n]
```
```   517           "4.hyps"(4-6)[of n "Suc n" n]
```
```   518           np np' C(2) mn
```
```   519         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
```
```   520           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   521           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
```
```   522           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
```
```   523           "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
```
```   524             "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
```
```   525           by (simp_all add: min_def)
```
```   526
```
```   527           from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```   528           have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   529             using norm by simp
```
```   530         from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"]  degneq
```
```   531         have "False" by simp }
```
```   532       thus ?case using "4.hyps" by clarsimp}
```
```   533 qed auto
```
```   534
```
```   535 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"
```
```   536 by(induct p q rule: polymul.induct, auto simp add: field_simps)
```
```   537
```
```   538 lemma polymul_normh:
```
```   539     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   540   shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   541   using polymul_properties(1)  by blast
```
```   542 lemma polymul_eq0_iff:
```
```   543   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   544   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) "
```
```   545   using polymul_properties(2)  by blast
```
```   546 lemma polymul_degreen:
```
```   547   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   548   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)"
```
```   549   using polymul_properties(3) by blast
```
```   550 lemma polymul_norm:
```
```   551   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   552   shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul (p,q))"
```
```   553   using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   554
```
```   555 lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
```
```   556   by (induct p arbitrary: n0 rule: headconst.induct, auto)
```
```   557
```
```   558 lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
```
```   559   by (induct p arbitrary: n0, auto)
```
```   560
```
```   561 lemma monic_eqI: assumes np: "isnpolyh p n0"
```
```   562   shows "INum (headconst p) * Ipoly bs (fst (monic p)) = (Ipoly bs p ::'a::{field_char_0, field_inverse_zero, power})"
```
```   563   unfolding monic_def Let_def
```
```   564 proof(cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
```
```   565   let ?h = "headconst p"
```
```   566   assume pz: "p \<noteq> 0\<^sub>p"
```
```   567   {assume hz: "INum ?h = (0::'a)"
```
```   568     from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
```
```   569     from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
```
```   570     with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
```
```   571   thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast
```
```   572 qed
```
```   573
```
```   574
```
```   575 text{* polyneg is a negation and preserves normal forms *}
```
```   576
```
```   577 lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
```
```   578 by (induct p rule: polyneg.induct, auto)
```
```   579
```
```   580 lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   581   by (induct p arbitrary: n rule: polyneg.induct, auto simp add: Nneg_def)
```
```   582 lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
```
```   583   by (induct p arbitrary: n0 rule: polyneg.induct, auto)
```
```   584 lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "
```
```   585 by (induct p rule: polyneg.induct, auto simp add: polyneg0)
```
```   586
```
```   587 lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
```
```   588   using isnpoly_def polyneg_normh by simp
```
```   589
```
```   590
```
```   591 text{* polysub is a substraction and preserves normal forms *}
```
```   592
```
```   593 lemma polysub[simp]: "Ipoly bs (polysub (p,q)) = (Ipoly bs p) - (Ipoly bs q)"
```
```   594 by (simp add: polysub_def polyneg polyadd)
```
```   595 lemma polysub_normh: "\<And> n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub(p,q)) (min n0 n1)"
```
```   596 by (simp add: polysub_def polyneg_normh polyadd_normh)
```
```   597
```
```   598 lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub(p,q))"
```
```   599   using polyadd_norm polyneg_norm by (simp add: polysub_def)
```
```   600 lemma polysub_same_0[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   601   shows "isnpolyh p n0 \<Longrightarrow> polysub (p, p) = 0\<^sub>p"
```
```   602 unfolding polysub_def split_def fst_conv snd_conv
```
```   603 by (induct p arbitrary: n0,auto simp add: Let_def Nsub0[simplified Nsub_def])
```
```   604
```
```   605 lemma polysub_0:
```
```   606   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   607   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"
```
```   608   unfolding polysub_def split_def fst_conv snd_conv
```
```   609   by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
```
```   610   (auto simp: Nsub0[simplified Nsub_def] Let_def)
```
```   611
```
```   612 text{* polypow is a power function and preserves normal forms *}
```
```   613
```
```   614 lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{field_char_0, field_inverse_zero})) ^ n"
```
```   615 proof(induct n rule: polypow.induct)
```
```   616   case 1 thus ?case by simp
```
```   617 next
```
```   618   case (2 n)
```
```   619   let ?q = "polypow ((Suc n) div 2) p"
```
```   620   let ?d = "polymul(?q,?q)"
```
```   621   have "odd (Suc n) \<or> even (Suc n)" by simp
```
```   622   moreover
```
```   623   {assume odd: "odd (Suc n)"
```
```   624     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
```
```   625     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul(p, ?d))" by (simp add: Let_def)
```
```   626     also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"
```
```   627       using "2.hyps" by simp
```
```   628     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
```
```   629       apply (simp only: power_add power_one_right) by simp
```
```   630     also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat)))))"
```
```   631       by (simp only: th)
```
```   632     finally have ?case
```
```   633     using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
```
```   634   moreover
```
```   635   {assume even: "even (Suc n)"
```
```   636     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
```
```   637     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
```
```   638     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
```
```   639       using "2.hyps" apply (simp only: power_add) by simp
```
```   640     finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}
```
```   641   ultimately show ?case by blast
```
```   642 qed
```
```   643
```
```   644 lemma polypow_normh:
```
```   645     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   646   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
```
```   647 proof (induct k arbitrary: n rule: polypow.induct)
```
```   648   case (2 k n)
```
```   649   let ?q = "polypow (Suc k div 2) p"
```
```   650   let ?d = "polymul (?q,?q)"
```
```   651   from prems have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
```
```   652   from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
```
```   653   from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul(p,?d)) n" by simp
```
```   654   from dn on show ?case by (simp add: Let_def)
```
```   655 qed auto
```
```   656
```
```   657 lemma polypow_norm:
```
```   658   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   659   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
```
```   660   by (simp add: polypow_normh isnpoly_def)
```
```   661
```
```   662 text{* Finally the whole normalization *}
```
```   663
```
```   664 lemma polynate[simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0, field_inverse_zero})"
```
```   665 by (induct p rule:polynate.induct, auto)
```
```   666
```
```   667 lemma polynate_norm[simp]:
```
```   668   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   669   shows "isnpoly (polynate p)"
```
```   670   by (induct p rule: polynate.induct, simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm) (simp_all add: isnpoly_def)
```
```   671
```
```   672 text{* shift1 *}
```
```   673
```
```   674
```
```   675 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
```
```   676 by (simp add: shift1_def polymul)
```
```   677
```
```   678 lemma shift1_isnpoly:
```
```   679   assumes pn: "isnpoly p" and pnz: "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p) "
```
```   680   using pn pnz by (simp add: shift1_def isnpoly_def )
```
```   681
```
```   682 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
```
```   683   by (simp add: shift1_def)
```
```   684 lemma funpow_shift1_isnpoly:
```
```   685   "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
```
```   686   by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
```
```   687
```
```   688 lemma funpow_isnpolyh:
```
```   689   assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n "and np: "isnpolyh p n"
```
```   690   shows "isnpolyh (funpow k f p) n"
```
```   691   using f np by (induct k arbitrary: p, auto)
```
```   692
```
```   693 lemma funpow_shift1: "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (Mul (Pw (Bound 0) n) p)"
```
```   694   by (induct n arbitrary: p, simp_all add: shift1_isnpoly shift1 power_Suc )
```
```   695
```
```   696 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
```
```   697   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
```
```   698
```
```   699 lemma funpow_shift1_1:
```
```   700   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (funpow n shift1 1\<^sub>p *\<^sub>p p)"
```
```   701   by (simp add: funpow_shift1)
```
```   702
```
```   703 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
```
```   704 by (induct p  arbitrary: n0 rule: poly_cmul.induct, auto simp add: field_simps)
```
```   705
```
```   706 lemma behead:
```
```   707   assumes np: "isnpolyh p n"
```
```   708   shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = (Ipoly bs p :: 'a :: {field_char_0, field_inverse_zero})"
```
```   709   using np
```
```   710 proof (induct p arbitrary: n rule: behead.induct)
```
```   711   case (1 c p n) hence pn: "isnpolyh p n" by simp
```
```   712   from prems(2)[OF pn]
```
```   713   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
```
```   714   then show ?case using "1.hyps" apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
```
```   715     by (simp_all add: th[symmetric] field_simps power_Suc)
```
```   716 qed (auto simp add: Let_def)
```
```   717
```
```   718 lemma behead_isnpolyh:
```
```   719   assumes np: "isnpolyh p n" shows "isnpolyh (behead p) n"
```
```   720   using np by (induct p rule: behead.induct, auto simp add: Let_def isnpolyh_mono)
```
```   721
```
```   722 subsection{* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
```
```   723 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
```
```   724 proof(induct p arbitrary: n rule: poly.induct, auto)
```
```   725   case (goal1 c n p n')
```
```   726   hence "n = Suc (n - 1)" by simp
```
```   727   hence "isnpolyh p (Suc (n - 1))"  using `isnpolyh p n` by simp
```
```   728   with prems(2) show ?case by simp
```
```   729 qed
```
```   730
```
```   731 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
```
```   732 by (induct p arbitrary: n0 rule: isconstant.induct, auto simp add: isnpolyh_polybound0)
```
```   733
```
```   734 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" by (induct p, auto)
```
```   735
```
```   736 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
```
```   737   apply (induct p arbitrary: n0, auto)
```
```   738   apply (atomize)
```
```   739   apply (erule_tac x = "Suc nat" in allE)
```
```   740   apply auto
```
```   741   done
```
```   742
```
```   743 lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
```
```   744  by (induct p  arbitrary: n0 rule: head.induct, auto intro: isnpolyh_polybound0)
```
```   745
```
```   746 lemma polybound0_I:
```
```   747   assumes nb: "polybound0 a"
```
```   748   shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
```
```   749 using nb
```
```   750 by (induct a rule: poly.induct) auto
```
```   751 lemma polysubst0_I:
```
```   752   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
```
```   753   by (induct t) simp_all
```
```   754
```
```   755 lemma polysubst0_I':
```
```   756   assumes nb: "polybound0 a"
```
```   757   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t"
```
```   758   by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
```
```   759
```
```   760 lemma decrpoly: assumes nb: "polybound0 t"
```
```   761   shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
```
```   762   using nb by (induct t rule: decrpoly.induct, simp_all)
```
```   763
```
```   764 lemma polysubst0_polybound0: assumes nb: "polybound0 t"
```
```   765   shows "polybound0 (polysubst0 t a)"
```
```   766 using nb by (induct a rule: poly.induct, auto)
```
```   767
```
```   768 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
```
```   769   by (induct p arbitrary: n rule: degree.induct, auto simp add: isnpolyh_polybound0)
```
```   770
```
```   771 primrec maxindex :: "poly \<Rightarrow> nat" where
```
```   772   "maxindex (Bound n) = n + 1"
```
```   773 | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
```
```   774 | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
```
```   775 | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
```
```   776 | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
```
```   777 | "maxindex (Neg p) = maxindex p"
```
```   778 | "maxindex (Pw p n) = maxindex p"
```
```   779 | "maxindex (C x) = 0"
```
```   780
```
```   781 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool" where
```
```   782   "wf_bs bs p = (length bs \<ge> maxindex p)"
```
```   783
```
```   784 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
```
```   785 proof(induct p rule: coefficients.induct)
```
```   786   case (1 c p)
```
```   787   show ?case
```
```   788   proof
```
```   789     fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"
```
```   790     hence "x = c \<or> x \<in> set (coefficients p)" by simp
```
```   791     moreover
```
```   792     {assume "x = c" hence "wf_bs bs x" using "1.prems"  unfolding wf_bs_def by simp}
```
```   793     moreover
```
```   794     {assume H: "x \<in> set (coefficients p)"
```
```   795       from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp
```
```   796       with "1.hyps" H have "wf_bs bs x" by blast }
```
```   797     ultimately  show "wf_bs bs x" by blast
```
```   798   qed
```
```   799 qed simp_all
```
```   800
```
```   801 lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
```
```   802 by (induct p rule: coefficients.induct, auto)
```
```   803
```
```   804 lemma wf_bs_I: "wf_bs bs p ==> Ipoly (bs@bs') p = Ipoly bs p"
```
```   805   unfolding wf_bs_def by (induct p, auto simp add: nth_append)
```
```   806
```
```   807 lemma take_maxindex_wf: assumes wf: "wf_bs bs p"
```
```   808   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
```
```   809 proof-
```
```   810   let ?ip = "maxindex p"
```
```   811   let ?tbs = "take ?ip bs"
```
```   812   from wf have "length ?tbs = ?ip" unfolding wf_bs_def by simp
```
```   813   hence wf': "wf_bs ?tbs p" unfolding wf_bs_def by  simp
```
```   814   have eq: "bs = ?tbs @ (drop ?ip bs)" by simp
```
```   815   from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis using eq by simp
```
```   816 qed
```
```   817
```
```   818 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
```
```   819   by (induct p, auto)
```
```   820
```
```   821 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
```
```   822   unfolding wf_bs_def by simp
```
```   823
```
```   824 lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
```
```   825   unfolding wf_bs_def by simp
```
```   826
```
```   827
```
```   828
```
```   829 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
```
```   830 by(induct p rule: coefficients.induct, auto simp add: wf_bs_def)
```
```   831 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
```
```   832   by (induct p rule: coefficients.induct, simp_all)
```
```   833
```
```   834
```
```   835 lemma coefficients_head: "last (coefficients p) = head p"
```
```   836   by (induct p rule: coefficients.induct, auto)
```
```   837
```
```   838 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
```
```   839   unfolding wf_bs_def by (induct p rule: decrpoly.induct, auto)
```
```   840
```
```   841 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
```
```   842   apply (rule exI[where x="replicate (n - length xs) z"])
```
```   843   by simp
```
```   844 lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
```
```   845 by (cases p, auto) (case_tac "nat", simp_all)
```
```   846
```
```   847 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
```
```   848   unfolding wf_bs_def
```
```   849   apply (induct p q rule: polyadd.induct)
```
```   850   apply (auto simp add: Let_def)
```
```   851   done
```
```   852
```
```   853 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
```
```   854   unfolding wf_bs_def
```
```   855   apply (induct p q arbitrary: bs rule: polymul.induct)
```
```   856   apply (simp_all add: wf_bs_polyadd)
```
```   857   apply clarsimp
```
```   858   apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
```
```   859   apply auto
```
```   860   done
```
```   861
```
```   862 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
```
```   863   unfolding wf_bs_def by (induct p rule: polyneg.induct, auto)
```
```   864
```
```   865 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
```
```   866   unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
```
```   867
```
```   868 subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}
```
```   869
```
```   870 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
```
```   871 definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)"
```
```   872 definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"
```
```   873
```
```   874 lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0"
```
```   875 proof (induct p arbitrary: n0 rule: coefficients.induct)
```
```   876   case (1 c p n0)
```
```   877   have cp: "isnpolyh (CN c 0 p) n0" by fact
```
```   878   hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
```
```   879     by (auto simp add: isnpolyh_mono[where n'=0])
```
```   880   from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp
```
```   881 qed auto
```
```   882
```
```   883 lemma coefficients_isconst:
```
```   884   "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
```
```   885   by (induct p arbitrary: n rule: coefficients.induct,
```
```   886     auto simp add: isnpolyh_Suc_const)
```
```   887
```
```   888 lemma polypoly_polypoly':
```
```   889   assumes np: "isnpolyh p n0"
```
```   890   shows "polypoly (x#bs) p = polypoly' bs p"
```
```   891 proof-
```
```   892   let ?cf = "set (coefficients p)"
```
```   893   from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
```
```   894   {fix q assume q: "q \<in> ?cf"
```
```   895     from q cn_norm have th: "isnpolyh q n0" by blast
```
```   896     from coefficients_isconst[OF np] q have "isconstant q" by blast
```
```   897     with isconstant_polybound0[OF th] have "polybound0 q" by blast}
```
```   898   hence "\<forall>q \<in> ?cf. polybound0 q" ..
```
```   899   hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"
```
```   900     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
```
```   901     by auto
```
```   902
```
```   903   thus ?thesis unfolding polypoly_def polypoly'_def by simp
```
```   904 qed
```
```   905
```
```   906 lemma polypoly_poly:
```
```   907   assumes np: "isnpolyh p n0" shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
```
```   908   using np
```
```   909 by (induct p arbitrary: n0 bs rule: coefficients.induct, auto simp add: polypoly_def)
```
```   910
```
```   911 lemma polypoly'_poly:
```
```   912   assumes np: "isnpolyh p n0" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
```
```   913   using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
```
```   914
```
```   915
```
```   916 lemma polypoly_poly_polybound0:
```
```   917   assumes np: "isnpolyh p n0" and nb: "polybound0 p"
```
```   918   shows "polypoly bs p = [Ipoly bs p]"
```
```   919   using np nb unfolding polypoly_def
```
```   920   by (cases p, auto, case_tac nat, auto)
```
```   921
```
```   922 lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
```
```   923   by (induct p rule: head.induct, auto)
```
```   924
```
```   925 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   926   by (cases p,auto)
```
```   927
```
```   928 lemma head_eq_headn0: "head p = headn p 0"
```
```   929   by (induct p rule: head.induct, simp_all)
```
```   930
```
```   931 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   932   by (simp add: head_eq_headn0)
```
```   933
```
```   934 lemma isnpolyh_zero_iff:
```
```   935   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})"
```
```   936   shows "p = 0\<^sub>p"
```
```   937 using nq eq
```
```   938 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
```
```   939   case less
```
```   940   note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
```
```   941   {assume nz: "maxindex p = 0"
```
```   942     then obtain c where "p = C c" using np by (cases p, auto)
```
```   943     with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}
```
```   944   moreover
```
```   945   {assume nz: "maxindex p \<noteq> 0"
```
```   946     let ?h = "head p"
```
```   947     let ?hd = "decrpoly ?h"
```
```   948     let ?ihd = "maxindex ?hd"
```
```   949     from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h"
```
```   950       by simp_all
```
```   951     hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast
```
```   952
```
```   953     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
```
```   954     have mihn: "maxindex ?h \<le> maxindex p" by auto
```
```   955     with decr_maxindex[OF h(2)] nz  have ihd_lt_n: "?ihd < maxindex p" by auto
```
```   956     {fix bs:: "'a list"  assume bs: "wf_bs bs ?hd"
```
```   957       let ?ts = "take ?ihd bs"
```
```   958       let ?rs = "drop ?ihd bs"
```
```   959       have ts: "wf_bs ?ts ?hd" using bs unfolding wf_bs_def by simp
```
```   960       have bs_ts_eq: "?ts@ ?rs = bs" by simp
```
```   961       from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x#?ts) ?h" by simp
```
```   962       from ihd_lt_n have "ALL x. length (x#?ts) \<le> maxindex p" by simp
```
```   963       with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = maxindex p" by blast
```
```   964       hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" unfolding wf_bs_def by simp
```
```   965       with zp have "\<forall> x. Ipoly ((x#?ts) @ xs) p = 0" by blast
```
```   966       hence "\<forall> x. Ipoly (x#(?ts @ xs)) p = 0" by simp
```
```   967       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
```
```   968       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"  by simp
```
```   969       hence "poly (polypoly' (?ts @ xs) p) = poly []" by (auto intro: ext)
```
```   970       hence "\<forall> c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
```
```   971         using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
```
```   972       with coefficients_head[of p, symmetric]
```
```   973       have th0: "Ipoly (?ts @ xs) ?hd = 0" by simp
```
```   974       from bs have wf'': "wf_bs ?ts ?hd" unfolding wf_bs_def by simp
```
```   975       with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0" by simp
```
```   976       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by simp }
```
```   977     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast
```
```   978
```
```   979     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast
```
```   980     hence "?h = 0\<^sub>p" by simp
```
```   981     with head_nz[OF np] have "p = 0\<^sub>p" by simp}
```
```   982   ultimately show "p = 0\<^sub>p" by blast
```
```   983 qed
```
```   984
```
```   985 lemma isnpolyh_unique:
```
```   986   assumes np:"isnpolyh p n0" and nq: "isnpolyh q n1"
```
```   987   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"
```
```   988 proof(auto)
```
```   989   assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a)= \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
```
```   990   hence "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" by simp
```
```   991   hence "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
```
```   992     using wf_bs_polysub[where p=p and q=q] by auto
```
```   993   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq]
```
```   994   show "p = q" by blast
```
```   995 qed
```
```   996
```
```   997
```
```   998 text{* consequences of unicity on the algorithms for polynomial normalization *}
```
```   999
```
```  1000 lemma polyadd_commute:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1001   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" shows "p +\<^sub>p q = q +\<^sub>p p"
```
```  1002   using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]] by simp
```
```  1003
```
```  1004 lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp
```
```  1005 lemma one_normh: "isnpolyh 1\<^sub>p n" by simp
```
```  1006 lemma polyadd_0[simp]:
```
```  1007   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1008   and np: "isnpolyh p n0" shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>p p = p"
```
```  1009   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
```
```  1010     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
```
```  1011
```
```  1012 lemma polymul_1[simp]:
```
```  1013     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1014   and np: "isnpolyh p n0" shows "p *\<^sub>p 1\<^sub>p = p" and "1\<^sub>p *\<^sub>p p = p"
```
```  1015   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
```
```  1016     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
```
```  1017 lemma polymul_0[simp]:
```
```  1018   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1019   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"
```
```  1020   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
```
```  1021     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
```
```  1022
```
```  1023 lemma polymul_commute:
```
```  1024     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1025   and np:"isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1026   shows "p *\<^sub>p q = q *\<^sub>p p"
```
```  1027 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
```
```  1028
```
```  1029 declare polyneg_polyneg[simp]
```
```  1030
```
```  1031 lemma isnpolyh_polynate_id[simp]:
```
```  1032   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1033   and np:"isnpolyh p n0" shows "polynate p = p"
```
```  1034   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
```
```  1035
```
```  1036 lemma polynate_idempotent[simp]:
```
```  1037     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1038   shows "polynate (polynate p) = polynate p"
```
```  1039   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
```
```  1040
```
```  1041 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
```
```  1042   unfolding poly_nate_def polypoly'_def ..
```
```  1043 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>)"
```
```  1044   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
```
```  1045   unfolding poly_nate_polypoly' by (auto intro: ext)
```
```  1046
```
```  1047 subsection{* heads, degrees and all that *}
```
```  1048 lemma degree_eq_degreen0: "degree p = degreen p 0"
```
```  1049   by (induct p rule: degree.induct, simp_all)
```
```  1050
```
```  1051 lemma degree_polyneg: assumes n: "isnpolyh p n"
```
```  1052   shows "degree (polyneg p) = degree p"
```
```  1053   using n
```
```  1054   by (induct p arbitrary: n rule: polyneg.induct, simp_all) (case_tac na, auto)
```
```  1055
```
```  1056 lemma degree_polyadd:
```
```  1057   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1058   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1059 using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
```
```  1060
```
```  1061
```
```  1062 lemma degree_polysub: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1063   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1064 proof-
```
```  1065   from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp
```
```  1066   from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])
```
```  1067 qed
```
```  1068
```
```  1069 lemma degree_polysub_samehead:
```
```  1070   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1071   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q"
```
```  1072   and d: "degree p = degree q"
```
```  1073   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
```
```  1074 unfolding polysub_def split_def fst_conv snd_conv
```
```  1075 using np nq h d
```
```  1076 proof(induct p q rule:polyadd.induct)
```
```  1077   case (1 a b a' b') thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def])
```
```  1078 next
```
```  1079   case (2 a b c' n' p')
```
```  1080   let ?c = "(a,b)"
```
```  1081   from prems have "degree (C ?c) = degree (CN c' n' p')" by simp
```
```  1082   hence nz:"n' > 0" by (cases n', auto)
```
```  1083   hence "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
```
```  1084   with prems show ?case by simp
```
```  1085 next
```
```  1086   case (3 c n p a' b')
```
```  1087   let ?c' = "(a',b')"
```
```  1088   from prems have "degree (C ?c') = degree (CN c n p)" by simp
```
```  1089   hence nz:"n > 0" by (cases n, auto)
```
```  1090   hence "head (CN c n p) = CN c n p" by (cases n, auto)
```
```  1091   with prems show ?case by simp
```
```  1092 next
```
```  1093   case (4 c n p c' n' p')
```
```  1094   hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1"
```
```  1095     "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+
```
```  1096   hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all
```
```  1097   hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
```
```  1098     using H(1-2) degree_polyneg by auto
```
```  1099   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"  by simp+
```
```  1100   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"  by simp
```
```  1101   from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto
```
```  1102   have "n = n' \<or> n < n' \<or> n > n'" by arith
```
```  1103   moreover
```
```  1104   {assume nn': "n = n'"
```
```  1105     have "n = 0 \<or> n >0" by arith
```
```  1106     moreover {assume nz: "n = 0" hence ?case using prems by (auto simp add: Let_def degcmc')}
```
```  1107     moreover {assume nz: "n > 0"
```
```  1108       with nn' H(3) have  cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
```
```  1109       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 split_def fst_conv snd_conv] using nn' prems by (simp add: Let_def)}
```
```  1110     ultimately have ?case by blast}
```
```  1111   moreover
```
```  1112   {assume nn': "n < n'" hence n'p: "n' > 0" by simp
```
```  1113     hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n', simp_all)
```
```  1114     have degcnp': "degree (CN c' n' p') = 0" and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')" using prems by (cases n', simp_all)
```
```  1115     hence "n > 0" by (cases n, simp_all)
```
```  1116     hence headcnp: "head (CN c n p) = CN c n p" by (cases n, auto)
```
```  1117     from H(3) headcnp headcnp' nn' have ?case by auto}
```
```  1118   moreover
```
```  1119   {assume nn': "n > n'"  hence np: "n > 0" by simp
```
```  1120     hence headcnp:"head (CN c n p) = CN c n p"  by (cases n, simp_all)
```
```  1121     from prems have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
```
```  1122     from np have degcnp: "degree (CN c n p) = 0" by (cases n, simp_all)
```
```  1123     with degcnpeq have "n' > 0" by (cases n', simp_all)
```
```  1124     hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
```
```  1125     from H(3) headcnp headcnp' nn' have ?case by auto}
```
```  1126   ultimately show ?case  by blast
```
```  1127 qed auto
```
```  1128
```
```  1129 lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
```
```  1130 by (induct p arbitrary: n0 rule: head.induct, simp_all add: shift1_def)
```
```  1131
```
```  1132 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
```
```  1133 proof(induct k arbitrary: n0 p)
```
```  1134   case (Suc k n0 p) hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
```
```  1135   with prems have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
```
```  1136     and "head (shift1 p) = head p" by (simp_all add: shift1_head)
```
```  1137   thus ?case by (simp add: funpow_swap1)
```
```  1138 qed auto
```
```  1139
```
```  1140 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
```
```  1141   by (simp add: shift1_def)
```
```  1142 lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
```
```  1143   by (induct k arbitrary: p, auto simp add: shift1_degree)
```
```  1144
```
```  1145 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
```
```  1146   by (induct n arbitrary: p, simp_all add: funpow_def)
```
```  1147
```
```  1148 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
```
```  1149   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1150 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
```
```  1151   by (induct p arbitrary: n rule: degreen.induct, auto)
```
```  1152 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
```
```  1153   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1154 lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
```
```  1155   by (induct p rule: head.induct, auto)
```
```  1156
```
```  1157 lemma polyadd_eq_const_degree:
```
```  1158   "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk> \<Longrightarrow> degree p = degree q"
```
```  1159   using polyadd_eq_const_degreen degree_eq_degreen0 by simp
```
```  1160
```
```  1161 lemma polyadd_head: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1162   and deg: "degree p \<noteq> degree q"
```
```  1163   shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
```
```  1164 using np nq deg
```
```  1165 apply(induct p q arbitrary: n0 n1 rule: polyadd.induct,simp_all)
```
```  1166 apply (case_tac n', simp, simp)
```
```  1167 apply (case_tac n, simp, simp)
```
```  1168 apply (case_tac n, case_tac n', simp add: Let_def)
```
```  1169 apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p")
```
```  1170 apply (auto simp add: polyadd_eq_const_degree)
```
```  1171 apply (metis head_nz)
```
```  1172 apply (metis head_nz)
```
```  1173 apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
```
```  1174 by (metis degree.simps(9) gr0_conv_Suc nat_less_le order_le_less_trans)
```
```  1175
```
```  1176 lemma polymul_head_polyeq:
```
```  1177    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1178   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"
```
```  1179 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
```
```  1180   case (2 a b c' n' p' n0 n1)
```
```  1181   hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum (a,b)"  by (simp_all add: head_isnpolyh)
```
```  1182   thus ?case using prems by (cases n', auto)
```
```  1183 next
```
```  1184   case (3 c n p a' b' n0 n1)
```
```  1185   hence "isnpolyh (head (CN c n p)) n0" "isnormNum (a',b')"  by (simp_all add: head_isnpolyh)
```
```  1186   thus ?case using prems by (cases n, auto)
```
```  1187 next
```
```  1188   case (4 c n p c' n' p' n0 n1)
```
```  1189   hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
```
```  1190     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
```
```  1191     by simp_all
```
```  1192   have "n < n' \<or> n' < n \<or> n = n'" by arith
```
```  1193   moreover
```
```  1194   {assume nn': "n < n'" hence ?case
```
```  1195       using norm
```
```  1196     "4.hyps"(5)[OF norm(1,6)]
```
```  1197     "4.hyps"(6)[OF norm(2,6)] by (simp, cases n, simp,cases n', simp_all)}
```
```  1198   moreover {assume nn': "n'< n"
```
```  1199     hence ?case using norm "4.hyps"(1) [OF norm(5,3)]
```
```  1200       "4.hyps"(2)[OF norm(5,4)]
```
```  1201       by (simp,cases n',simp,cases n,auto)}
```
```  1202   moreover {assume nn': "n' = n"
```
```  1203     from nn' polymul_normh[OF norm(5,4)]
```
```  1204     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
```
```  1205     from nn' polymul_normh[OF norm(5,3)] norm
```
```  1206     have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
```
```  1207     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
```
```  1208     have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```  1209     from polyadd_normh[OF ncnpc' ncnpp0']
```
```  1210     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"
```
```  1211       by (simp add: min_def)
```
```  1212     {assume np: "n > 0"
```
```  1213       with nn' head_isnpolyh_Suc'[OF np nth]
```
```  1214         head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
```
```  1215       have ?case by simp}
```
```  1216     moreover
```
```  1217     {moreover assume nz: "n = 0"
```
```  1218       from polymul_degreen[OF norm(5,4), where m="0"]
```
```  1219         polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
```
```  1220       norm(5,6) degree_npolyhCN[OF norm(6)]
```
```  1221     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
```
```  1222     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
```
```  1223     from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
```
```  1224     have ?case   using norm "4.hyps"(1)[OF norm(5,3)]
```
```  1225         "4.hyps"(2)[OF norm(5,4)] nn' nz by simp }
```
```  1226     ultimately have ?case by (cases n) auto}
```
```  1227   ultimately show ?case by blast
```
```  1228 qed simp_all
```
```  1229
```
```  1230 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
```
```  1231   by(induct p rule: degree.induct, auto)
```
```  1232
```
```  1233 lemma degree_head[simp]: "degree (head p) = 0"
```
```  1234   by (induct p rule: head.induct, auto)
```
```  1235
```
```  1236 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1+ degree p"
```
```  1237   by (cases n, simp_all)
```
```  1238 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
```
```  1239   by (cases n, simp_all)
```
```  1240
```
```  1241 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)"
```
```  1242   using polyadd_different_degreen degree_eq_degreen0 by simp
```
```  1243
```
```  1244 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
```
```  1245   by (induct p arbitrary: n0 rule: polyneg.induct, auto)
```
```  1246
```
```  1247 lemma degree_polymul:
```
```  1248   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1249   and np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1250   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
```
```  1251   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
```
```  1252
```
```  1253 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
```
```  1254   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1255
```
```  1256 lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"
```
```  1257   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1258
```
```  1259 subsection {* Correctness of polynomial pseudo division *}
```
```  1260
```
```  1261 lemma polydivide_aux_properties:
```
```  1262   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1263   and np: "isnpolyh p n0" and ns: "isnpolyh s n1"
```
```  1264   and ap: "head p = a" and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
```
```  1265   shows "(polydivide_aux a n p k s = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1266           \<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)))"
```
```  1267   using ns
```
```  1268 proof(induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
```
```  1269   case less
```
```  1270   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
```
```  1271   let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1272     \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
```
```  1273   let ?b = "head s"
```
```  1274   let ?p' = "funpow (degree s - n) shift1 p"
```
```  1275   let ?xdn = "funpow (degree s - n) shift1 1\<^sub>p"
```
```  1276   let ?akk' = "a ^\<^sub>p (k' - k)"
```
```  1277   note ns = `isnpolyh s n1`
```
```  1278   from np have np0: "isnpolyh p 0"
```
```  1279     using isnpolyh_mono[where n="n0" and n'="0" and p="p"]  by simp
```
```  1280   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
```
```  1281   have headp': "head ?p' = head p" using funpow_shift1_head[OF np pnz] by simp
```
```  1282   from funpow_shift1_isnpoly[where p="1\<^sub>p"] have nxdn: "isnpolyh ?xdn 0" by (simp add: isnpoly_def)
```
```  1283   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
```
```  1284   have nakk':"isnpolyh ?akk' 0" by blast
```
```  1285   {assume sz: "s = 0\<^sub>p"
```
```  1286    hence ?ths using np polydivide_aux.simps apply clarsimp by (rule exI[where x="0\<^sub>p"], simp) }
```
```  1287   moreover
```
```  1288   {assume sz: "s \<noteq> 0\<^sub>p"
```
```  1289     {assume dn: "degree s < n"
```
```  1290       hence "?ths" using ns ndp np polydivide_aux.simps by auto (rule exI[where x="0\<^sub>p"],simp) }
```
```  1291     moreover
```
```  1292     {assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith
```
```  1293       have degsp': "degree s = degree ?p'"
```
```  1294         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp
```
```  1295       {assume ba: "?b = a"
```
```  1296         hence headsp': "head s = head ?p'" using ap headp' by simp
```
```  1297         have nr: "isnpolyh (s -\<^sub>p ?p') 0" using polysub_normh[OF ns np'] by simp
```
```  1298         from degree_polysub_samehead[OF ns np' headsp' degsp']
```
```  1299         have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
```
```  1300         moreover
```
```  1301         {assume deglt:"degree (s -\<^sub>p ?p') < degree s"
```
```  1302           from polydivide_aux.simps sz dn' ba
```
```  1303           have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
```
```  1304             by (simp add: Let_def)
```
```  1305           {assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1306             from less(1)[OF deglt nr, of k k' r]
```
```  1307               trans[OF eq[symmetric] h1]
```
```  1308             have kk': "k \<le> k'" and nr:"\<exists>nr. isnpolyh r nr" and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1309               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
```
```  1310             from q1 obtain q n1 where nq: "isnpolyh q n1"
```
```  1311               and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"  by blast
```
```  1312             from nr obtain nr where nr': "isnpolyh r nr" by blast
```
```  1313             from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0" by simp
```
```  1314             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
```
```  1315             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
```
```  1316             from polyadd_normh[OF polymul_normh[OF np
```
```  1317               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
```
```  1318             have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0" by simp
```
```  1319             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')) =
```
```  1320               Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
```
```  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 (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1323               by (simp add: field_simps)
```
```  1324             hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1325               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p *\<^sub>p p)
```
```  1326               + Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1327               by (auto simp only: funpow_shift1_1)
```
```  1328             hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1329               Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p)
```
```  1330               + Ipoly bs q) + Ipoly bs r" by (simp add: field_simps)
```
```  1331             hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1332               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
```
```  1333             with isnpolyh_unique[OF nakks' nqr']
```
```  1334             have "a ^\<^sub>p (k' - k) *\<^sub>p s =
```
```  1335               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
```
```  1336             hence ?qths using nq'
```
```  1337               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 1\<^sub>p) +\<^sub>p q" in exI)
```
```  1338               apply (rule_tac x="0" in exI) by simp
```
```  1339             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"
```
```  1340               by blast } hence ?ths by blast }
```
```  1341         moreover
```
```  1342         {assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
```
```  1343           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}"]
```
```  1344           have " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs s = Ipoly bs ?p'" by simp
```
```  1345           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
```
```  1346             by (simp only: funpow_shift1_1) simp
```
```  1347           hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]] by blast
```
```  1348           {assume h1: "polydivide_aux a n p k s = (k',r)"
```
```  1349             from polydivide_aux.simps sz dn' ba
```
```  1350             have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
```
```  1351               by (simp add: Let_def)
```
```  1352             also have "\<dots> = (k,0\<^sub>p)" using polydivide_aux.simps spz by simp
```
```  1353             finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
```
```  1354             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
```
```  1355               polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
```
```  1356               apply auto
```
```  1357               apply (rule exI[where x="?xdn"])
```
```  1358               apply (auto simp add: polymul_commute[of p])
```
```  1359               done} }
```
```  1360         ultimately have ?ths by blast }
```
```  1361       moreover
```
```  1362       {assume ba: "?b \<noteq> a"
```
```  1363         from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1364           polymul_normh[OF head_isnpolyh[OF ns] np']]
```
```  1365         have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by(simp add: min_def)
```
```  1366         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
```
```  1367           using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1368             polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
```
```  1369             funpow_shift1_nz[OF pnz] by simp_all
```
```  1370         from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
```
```  1371           polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1372         have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
```
```  1373           using head_head[OF ns] funpow_shift1_head[OF np pnz]
```
```  1374             polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
```
```  1375           by (simp add: ap)
```
```  1376         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
```
```  1377           head_nz[OF np] pnz sz ap[symmetric]
```
```  1378           funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1379           polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"]  head_nz[OF ns]
```
```  1380           ndp dn
```
```  1381         have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p') "
```
```  1382           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
```
```  1383         {assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
```
```  1384           from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns] polymul_normh[OF head_isnpolyh[OF ns]np']]
```
```  1385           ap have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by simp
```
```  1386           {assume h1:"polydivide_aux a n p k s = (k', r)"
```
```  1387             from h1 polydivide_aux.simps sz dn' ba
```
```  1388             have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
```
```  1389               by (simp add: Let_def)
```
```  1390             with less(1)[OF dth nasbp', of "Suc k" k' r]
```
```  1391             obtain q nq nr where kk': "Suc k \<le> k'" and nr: "isnpolyh r nr" and nq: "isnpolyh q nq"
```
```  1392               and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1393               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
```
```  1394             from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith
```
```  1395             {fix bs:: "'a::{field_char_0, field_inverse_zero} list"
```
```  1396
```
```  1397             from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
```
```  1398             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
```
```  1399             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"
```
```  1400               by (simp add: field_simps power_Suc)
```
```  1401             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"
```
```  1402               by (simp add:kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1403             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"
```
```  1404               by (simp add: field_simps)}
```
```  1405             hence ieq:"\<forall>(bs :: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1406               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
```
```  1407             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
```
```  1408             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]]
```
```  1409             have nqw: "isnpolyh ?q 0" by simp
```
```  1410             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]]
```
```  1411             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
```
```  1412             from dr kk' nr h1 asth nqw have ?ths apply simp
```
```  1413               apply (rule conjI)
```
```  1414               apply (rule exI[where x="nr"], simp)
```
```  1415               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
```
```  1416               apply (rule exI[where x="0"], simp)
```
```  1417               done}
```
```  1418           hence ?ths by blast }
```
```  1419         moreover
```
```  1420         {assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
```
```  1421           {fix bs :: "'a::{field_char_0, field_inverse_zero} list"
```
```  1422             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
```
```  1423           have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'" by simp
```
```  1424           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
```
```  1425             by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1426           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" by simp
```
```  1427         }
```
```  1428         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))" ..
```
```  1429           from hth
```
```  1430           have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
```
```  1431             using isnpolyh_unique[where ?'a = "'a::{field_char_0, field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
```
```  1432                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
```
```  1433               simplified ap] by simp
```
```  1434           {assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1435           from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
```
```  1436           have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
```
```  1437           with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
```
```  1438             polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
```
```  1439           have ?ths apply (clarsimp simp add: Let_def)
```
```  1440             apply (rule exI[where x="?b *\<^sub>p ?xdn"]) apply simp
```
```  1441             apply (rule exI[where x="0"], simp)
```
```  1442             done}
```
```  1443         hence ?ths by blast}
```
```  1444         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"]
```
```  1445           head_nz[OF np] pnz sz ap[symmetric]
```
```  1446           by (simp add: degree_eq_degreen0[symmetric]) blast }
```
```  1447       ultimately have ?ths by blast
```
```  1448     }
```
```  1449     ultimately have ?ths by blast}
```
```  1450   ultimately show ?ths by blast
```
```  1451 qed
```
```  1452
```
```  1453 lemma polydivide_properties:
```
```  1454   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1455   and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
```
```  1456   shows "(\<exists> k r. polydivide s p = (k,r) \<and> (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1457   \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
```
```  1458 proof-
```
```  1459   have trv: "head p = head p" "degree p = degree p" by simp_all
```
```  1460   from polydivide_def[where s="s" and p="p"]
```
```  1461   have ex: "\<exists> k r. polydivide s p = (k,r)" by auto
```
```  1462   then obtain k r where kr: "polydivide s p = (k,r)" by blast
```
```  1463   from trans[OF meta_eq_to_obj_eq[OF polydivide_def[where s="s"and p="p"], symmetric] kr]
```
```  1464     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
```
```  1465   have "(degree r = 0 \<or> degree r < degree p) \<and>
```
```  1466    (\<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
```
```  1467   with kr show ?thesis
```
```  1468     apply -
```
```  1469     apply (rule exI[where x="k"])
```
```  1470     apply (rule exI[where x="r"])
```
```  1471     apply simp
```
```  1472     done
```
```  1473 qed
```
```  1474
```
```  1475 subsection{* More about polypoly and pnormal etc *}
```
```  1476
```
```  1477 definition "isnonconstant p = (\<not> isconstant p)"
```
```  1478
```
```  1479 lemma isnonconstant_pnormal_iff: assumes nc: "isnonconstant p"
```
```  1480   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1481 proof
```
```  1482   let ?p = "polypoly bs p"
```
```  1483   assume H: "pnormal ?p"
```
```  1484   have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
```
```  1485
```
```  1486   from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1487     pnormal_last_nonzero[OF H]
```
```  1488   show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)
```
```  1489 next
```
```  1490   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1491   let ?p = "polypoly bs p"
```
```  1492   have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
```
```  1493   hence pz: "?p \<noteq> []" by (simp add: polypoly_def)
```
```  1494   hence lg: "length ?p > 0" by simp
```
```  1495   from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1496   have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)
```
```  1497   from pnormal_last_length[OF lg lz] show "pnormal ?p" .
```
```  1498 qed
```
```  1499
```
```  1500 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
```
```  1501   unfolding isnonconstant_def
```
```  1502   apply (cases p, simp_all)
```
```  1503   apply (case_tac nat, auto)
```
```  1504   done
```
```  1505 lemma isnonconstant_nonconstant: assumes inc: "isnonconstant p"
```
```  1506   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1507 proof
```
```  1508   let ?p = "polypoly bs p"
```
```  1509   assume nc: "nonconstant ?p"
```
```  1510   from isnonconstant_pnormal_iff[OF inc, of bs] nc
```
```  1511   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast
```
```  1512 next
```
```  1513   let ?p = "polypoly bs p"
```
```  1514   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1515   from isnonconstant_pnormal_iff[OF inc, of bs] h
```
```  1516   have pn: "pnormal ?p" by blast
```
```  1517   {fix x assume H: "?p = [x]"
```
```  1518     from H have "length (coefficients p) = 1" unfolding polypoly_def by auto
```
```  1519     with isnonconstant_coefficients_length[OF inc] have False by arith}
```
```  1520   thus "nonconstant ?p" using pn unfolding nonconstant_def by blast
```
```  1521 qed
```
```  1522
```
```  1523 lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
```
```  1524   unfolding pnormal_def
```
```  1525  apply (induct p)
```
```  1526  apply (simp_all, case_tac "p=[]", simp_all)
```
```  1527  done
```
```  1528
```
```  1529 lemma degree_degree: assumes inc: "isnonconstant p"
```
```  1530   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1531 proof
```
```  1532   let  ?p = "polypoly bs p"
```
```  1533   assume H: "degree p = Polynomial_List.degree ?p"
```
```  1534   from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
```
```  1535     unfolding polypoly_def by auto
```
```  1536   from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
```
```  1537   have lg:"length (pnormalize ?p) = length ?p"
```
```  1538     unfolding Polynomial_List.degree_def polypoly_def by simp
```
```  1539   hence "pnormal ?p" using pnormal_length[OF pz] by blast
```
```  1540   with isnonconstant_pnormal_iff[OF inc]
```
```  1541   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast
```
```  1542 next
```
```  1543   let  ?p = "polypoly bs p"
```
```  1544   assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1545   with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast
```
```  1546   with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
```
```  1547   show "degree p = Polynomial_List.degree ?p"
```
```  1548     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
```
```  1549 qed
```
```  1550
```
```  1551 section{* Swaps ; Division by a certain variable *}
```
```  1552 primrec swap:: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly" where
```
```  1553   "swap n m (C x) = C x"
```
```  1554 | "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"
```
```  1555 | "swap n m (Neg t) = Neg (swap n m t)"
```
```  1556 | "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
```
```  1557 | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
```
```  1558 | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
```
```  1559 | "swap n m (Pw t k) = Pw (swap n m t) k"
```
```  1560 | "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)
```
```  1561   (swap n m p)"
```
```  1562
```
```  1563 lemma swap: assumes nbs: "n < length bs" and mbs: "m < length bs"
```
```  1564   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  1565 proof (induct t)
```
```  1566   case (Bound k) thus ?case using nbs mbs by simp
```
```  1567 next
```
```  1568   case (CN c k p) thus ?case using nbs mbs by simp
```
```  1569 qed simp_all
```
```  1570 lemma swap_swap_id[simp]: "swap n m (swap m n t) = t"
```
```  1571   by (induct t,simp_all)
```
```  1572
```
```  1573 lemma swap_commute: "swap n m p = swap m n p" by (induct p, simp_all)
```
```  1574
```
```  1575 lemma swap_same_id[simp]: "swap n n t = t"
```
```  1576   by (induct t, simp_all)
```
```  1577
```
```  1578 definition "swapnorm n m t = polynate (swap n m t)"
```
```  1579
```
```  1580 lemma swapnorm: assumes nbs: "n < length bs" and mbs: "m < length bs"
```
```  1581   shows "((Ipoly bs (swapnorm n m t) :: 'a\<Colon>{field_char_0, field_inverse_zero})) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  1582   using swap[OF prems] swapnorm_def by simp
```
```  1583
```
```  1584 lemma swapnorm_isnpoly[simp]:
```
```  1585     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1586   shows "isnpoly (swapnorm n m p)"
```
```  1587   unfolding swapnorm_def by simp
```
```  1588
```
```  1589 definition "polydivideby n s p =
```
```  1590     (let ss = swapnorm 0 n s ; sp = swapnorm 0 n p ; h = head sp; (k,r) = polydivide ss sp
```
```  1591      in (k,swapnorm 0 n h,swapnorm 0 n r))"
```
```  1592
```
```  1593 lemma swap_nz [simp]: " (swap n m p = 0\<^sub>p) = (p = 0\<^sub>p)" by (induct p, simp_all)
```
```  1594
```
```  1595 fun isweaknpoly :: "poly \<Rightarrow> bool"
```
```  1596 where
```
```  1597   "isweaknpoly (C c) = True"
```
```  1598 | "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
```
```  1599 | "isweaknpoly p = False"
```
```  1600
```
```  1601 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
```
```  1602   by (induct p arbitrary: n0, auto)
```
```  1603
```
```  1604 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
```
```  1605   by (induct p, auto)
```
```  1606
```
`  1607 end`