src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
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recdef -> fun; curried
```     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
```
```   109
```
```   110 consts
```
```   111   polysub :: "poly\<times>poly \<Rightarrow> poly"
```
```   112   polymul :: "poly\<times>poly \<Rightarrow> poly"
```
```   113
```
```   114 abbreviation poly_mul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
```
```   115   where "a *\<^sub>p b \<equiv> polymul (a,b)"
```
```   116 abbreviation poly_sub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
```
```   117   where "a -\<^sub>p b \<equiv> polysub (a,b)"
```
```   118
```
```   119 declare if_cong[fundef_cong del]
```
```   120 declare let_cong[fundef_cong del]
```
```   121
```
```   122 fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
```
```   123 where
```
```   124   "polyadd (C c) (C c') = C (c+\<^sub>Nc')"
```
```   125 |  "polyadd (C c) (CN c' n' p') = CN (polyadd (C c) c') n' p'"
```
```   126 | "polyadd (CN c n p) (C c') = CN (polyadd c (C c')) n p"
```
```   127 | "polyadd (CN c n p) (CN c' n' p') =
```
```   128     (if n < n' then CN (polyadd c (CN c' n' p')) n p
```
```   129      else if n'<n then CN (polyadd (CN c n p) c') n' p'
```
```   130      else (let cc' = polyadd c c' ;
```
```   131                pp' = polyadd p p'
```
```   132            in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
```
```   133 | "polyadd a b = Add a b"
```
```   134
```
```   135
```
```   136 fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
```
```   137 where
```
```   138   "polyneg (C c) = C (~\<^sub>N c)"
```
```   139 | "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
```
```   140 | "polyneg a = Neg a"
```
```   141
```
```   142 defs polysub_def[code]: "polysub \<equiv> \<lambda> (p,q). polyadd p (polyneg q)"
```
```   143
```
```   144 recdef polymul "measure (\<lambda>(a,b). size a + size b)"
```
```   145   "polymul(C c, C c') = C (c*\<^sub>Nc')"
```
```   146   "polymul(C c, CN c' n' p') =
```
```   147       (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul(C c,c')) n' (polymul(C c, p')))"
```
```   148   "polymul(CN c n p, C c') =
```
```   149       (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul(c,C c')) n (polymul(p, C c')))"
```
```   150   "polymul(CN c n p, CN c' n' p') =
```
```   151   (if n<n' then CN (polymul(c,CN c' n' p')) n (polymul(p,CN c' n' p'))
```
```   152   else if n' < n
```
```   153   then CN (polymul(CN c n p,c')) n' (polymul(CN c n p,p'))
```
```   154   else polyadd (polymul(CN c n p, c')) (CN 0\<^sub>p n' (polymul(CN c n p, p'))))"
```
```   155   "polymul (a,b) = Mul a b"
```
```   156 (hints recdef_cong del: if_cong)
```
```   157
```
```   158 declare if_cong[fundef_cong]
```
```   159 declare let_cong[fundef_cong]
```
```   160
```
```   161 fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
```
```   162 where
```
```   163   "polypow 0 = (\<lambda>p. 1\<^sub>p)"
```
```   164 | "polypow n = (\<lambda>p. let q = polypow (n div 2) p ; d = polymul(q,q) in
```
```   165                     if even n then d else polymul(p,d))"
```
```   166
```
```   167 abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
```
```   168   where "a ^\<^sub>p k \<equiv> polypow k a"
```
```   169
```
```   170 function polynate :: "poly \<Rightarrow> poly"
```
```   171 where
```
```   172   "polynate (Bound n) = CN 0\<^sub>p n 1\<^sub>p"
```
```   173 | "polynate (Add p q) = (polynate p +\<^sub>p polynate q)"
```
```   174 | "polynate (Sub p q) = (polynate p -\<^sub>p polynate q)"
```
```   175 | "polynate (Mul p q) = (polynate p *\<^sub>p polynate q)"
```
```   176 | "polynate (Neg p) = (~\<^sub>p (polynate p))"
```
```   177 | "polynate (Pw p n) = ((polynate p) ^\<^sub>p n)"
```
```   178 | "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
```
```   179 | "polynate (C c) = C (normNum c)"
```
```   180 by pat_completeness auto
```
```   181 termination by (relation "measure polysize") auto
```
```   182
```
```   183 fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly" where
```
```   184   "poly_cmul y (C x) = C (y *\<^sub>N x)"
```
```   185 | "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
```
```   186 | "poly_cmul y p = C y *\<^sub>p p"
```
```   187
```
```   188 definition monic :: "poly \<Rightarrow> (poly \<times> bool)" where
```
```   189   "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))"
```
```   190
```
```   191 subsection{* Pseudo-division *}
```
```   192
```
```   193 definition shift1 :: "poly \<Rightarrow> poly" where
```
```   194   "shift1 p \<equiv> CN 0\<^sub>p 0 p"
```
```   195
```
```   196 abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)" where
```
```   197   "funpow \<equiv> compow"
```
```   198
```
```   199 partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
```
```   200   where
```
```   201   "polydivide_aux a n p k s =
```
```   202   (if s = 0\<^sub>p then (k,s)
```
```   203   else (let b = head s; m = degree s in
```
```   204   (if m < n then (k,s) else
```
```   205   (let p'= funpow (m - n) shift1 p in
```
```   206   (if a = b then polydivide_aux a n p k (s -\<^sub>p p')
```
```   207   else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))))))"
```
```   208
```
```   209 definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)" where
```
```   210   "polydivide s p \<equiv> polydivide_aux (head p) (degree p) p 0 s"
```
```   211
```
```   212 fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly" where
```
```   213   "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
```
```   214 | "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
```
```   215
```
```   216 fun poly_deriv :: "poly \<Rightarrow> poly" where
```
```   217   "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
```
```   218 | "poly_deriv p = 0\<^sub>p"
```
```   219
```
```   220   (* Verification *)
```
```   221 lemma nth_pos2[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
```
```   222 using Nat.gr0_conv_Suc
```
```   223 by clarsimp
```
```   224
```
```   225 subsection{* Semantics of the polynomial representation *}
```
```   226
```
```   227 primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0, field_inverse_zero, power}" where
```
```   228   "Ipoly bs (C c) = INum c"
```
```   229 | "Ipoly bs (Bound n) = bs!n"
```
```   230 | "Ipoly bs (Neg a) = - Ipoly bs a"
```
```   231 | "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
```
```   232 | "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
```
```   233 | "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
```
```   234 | "Ipoly bs (Pw t n) = (Ipoly bs t) ^ n"
```
```   235 | "Ipoly bs (CN c n p) = (Ipoly bs c) + (bs!n)*(Ipoly bs p)"
```
```   236
```
```   237 abbreviation
```
```   238   Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0, field_inverse_zero, power}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
```
```   239   where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"
```
```   240
```
```   241 lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i"
```
```   242   by (simp add: INum_def)
```
```   243 lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"
```
```   244   by (simp  add: INum_def)
```
```   245
```
```   246 lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
```
```   247
```
```   248 subsection {* Normal form and normalization *}
```
```   249
```
```   250 fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
```
```   251 where
```
```   252   "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
```
```   253 | "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))"
```
```   254 | "isnpolyh p = (\<lambda>k. False)"
```
```   255
```
```   256 lemma isnpolyh_mono: "\<lbrakk>n' \<le> n ; isnpolyh p n\<rbrakk> \<Longrightarrow> isnpolyh p n'"
```
```   257 by (induct p rule: isnpolyh.induct, auto)
```
```   258
```
```   259 definition isnpoly :: "poly \<Rightarrow> bool" where
```
```   260   "isnpoly p \<equiv> isnpolyh p 0"
```
```   261
```
```   262 text{* polyadd preserves normal forms *}
```
```   263
```
```   264 lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk>
```
```   265       \<Longrightarrow> isnpolyh (polyadd p q) (min n0 n1)"
```
```   266 proof(induct p q arbitrary: n0 n1 rule: polyadd.induct)
```
```   267   case (2 ab c' n' p' n0 n1)
```
```   268   from prems have  th1: "isnpolyh (C ab) (Suc n')" by simp
```
```   269   from prems(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
```
```   270   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
```
```   271   with prems(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^sub>p c') (Suc n')" by simp
```
```   272   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
```
```   273   thus ?case using prems th3 by simp
```
```   274 next
```
```   275   case (3 c' n' p' ab n1 n0)
```
```   276   from prems have  th1: "isnpolyh (C ab) (Suc n')" by simp
```
```   277   from prems(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
```
```   278   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
```
```   279   with prems(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (Suc n')" by simp
```
```   280   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
```
```   281   thus ?case using prems th3 by simp
```
```   282 next
```
```   283   case (4 c n p c' n' p' n0 n1)
```
```   284   hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
```
```   285   from prems have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all
```
```   286   from prems have ngen0: "n \<ge> n0" by simp
```
```   287   from prems have n'gen1: "n' \<ge> n1" by simp
```
```   288   have "n < n' \<or> n' < n \<or> n = n'" by auto
```
```   289   moreover {assume eq: "n = n'"
```
```   290     with "4.hyps"(3)[OF nc nc']
```
```   291     have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
```
```   292     hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
```
```   293       using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
```
```   294     from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
```
```   295     have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
```
```   296     from minle npp' ncc'n01 prems ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}
```
```   297   moreover {assume lt: "n < n'"
```
```   298     have "min n0 n1 \<le> n0" by simp
```
```   299     with prems have th1:"min n0 n1 \<le> n" by auto
```
```   300     from prems have th21: "isnpolyh c (Suc n)" by simp
```
```   301     from prems have th22: "isnpolyh (CN c' n' p') n'" by simp
```
```   302     from lt have th23: "min (Suc n) n' = Suc n" by arith
```
```   303     from "4.hyps"(1)[OF th21 th22]
```
```   304     have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)" using th23 by simp
```
```   305     with prems th1 have ?case by simp }
```
```   306   moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
```
```   307     have "min n0 n1 \<le> n1"  by simp
```
```   308     with prems have th1:"min n0 n1 \<le> n'" by auto
```
```   309     from prems have th21: "isnpolyh c' (Suc n')" by simp_all
```
```   310     from prems have th22: "isnpolyh (CN c n p) n" by simp
```
```   311     from gt have th23: "min n (Suc n') = Suc n'" by arith
```
```   312     from "4.hyps"(2)[OF th22 th21]
```
```   313     have "isnpolyh (polyadd (CN c n p) c') (Suc n')" using th23 by simp
```
```   314     with prems th1 have ?case by simp}
```
```   315       ultimately show ?case by blast
```
```   316 qed auto
```
```   317
```
```   318 lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
```
```   319 by (induct p q rule: polyadd.induct, auto simp add: Let_def field_simps right_distrib[symmetric] simp del: right_distrib)
```
```   320
```
```   321 lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd p q)"
```
```   322   using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   323
```
```   324 text{* The degree of addition and other general lemmas needed for the normal form of polymul *}
```
```   325
```
```   326 lemma polyadd_different_degreen:
```
```   327   "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow>
```
```   328   degreen (polyadd p q) m = max (degreen p m) (degreen q m)"
```
```   329 proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
```
```   330   case (4 c n p c' n' p' m n0 n1)
```
```   331   have "n' = n \<or> n < n' \<or> n' < n" by arith
```
```   332   thus ?case
```
```   333   proof (elim disjE)
```
```   334     assume [simp]: "n' = n"
```
```   335     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
```
```   336     show ?thesis by (auto simp: Let_def)
```
```   337   next
```
```   338     assume "n < n'"
```
```   339     with 4 show ?thesis by auto
```
```   340   next
```
```   341     assume "n' < n"
```
```   342     with 4 show ?thesis by auto
```
```   343   qed
```
```   344 qed auto
```
```   345
```
```   346 lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
```
```   347   by (induct p arbitrary: n rule: headn.induct, auto)
```
```   348 lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
```
```   349   by (induct p arbitrary: n rule: degree.induct, auto)
```
```   350 lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
```
```   351   by (induct p arbitrary: n rule: degreen.induct, auto)
```
```   352
```
```   353 lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
```
```   354   by (induct p arbitrary: n rule: degree.induct, auto)
```
```   355
```
```   356 lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
```
```   357   using degree_isnpolyh_Suc by auto
```
```   358 lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
```
```   359   using degreen_0 by auto
```
```   360
```
```   361
```
```   362 lemma degreen_polyadd:
```
```   363   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1"
```
```   364   shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
```
```   365   using np nq m
```
```   366 proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
```
```   367   case (2 c c' n' p' n0 n1) thus ?case  by (cases n', simp_all)
```
```   368 next
```
```   369   case (3 c n p c' n0 n1) thus ?case by (cases n, auto)
```
```   370 next
```
```   371   case (4 c n p c' n' p' n0 n1 m)
```
```   372   have "n' = n \<or> n < n' \<or> n' < n" by arith
```
```   373   thus ?case
```
```   374   proof (elim disjE)
```
```   375     assume [simp]: "n' = n"
```
```   376     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
```
```   377     show ?thesis by (auto simp: Let_def)
```
```   378   qed simp_all
```
```   379 qed auto
```
```   380
```
```   381 lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd p q = C c\<rbrakk>
```
```   382   \<Longrightarrow> degreen p m = degreen q m"
```
```   383 proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
```
```   384   case (4 c n p c' n' p' m n0 n1 x)
```
```   385   {assume nn': "n' < n" hence ?case using prems by simp}
```
```   386   moreover
```
```   387   {assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
```
```   388     moreover {assume "n < n'" with prems have ?case by simp }
```
```   389     moreover {assume eq: "n = n'" hence ?case using prems
```
```   390         apply (cases "p +\<^sub>p p' = 0\<^sub>p")
```
```   391         apply (auto simp add: Let_def)
```
```   392         by blast
```
```   393       }
```
```   394     ultimately have ?case by blast}
```
```   395   ultimately show ?case by blast
```
```   396 qed simp_all
```
```   397
```
```   398 lemma polymul_properties:
```
```   399   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   400   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> min n0 n1"
```
```   401   shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   402   and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)"
```
```   403   and "degreen (p *\<^sub>p q) m = (if (p = 0\<^sub>p \<or> q = 0\<^sub>p) then 0
```
```   404                              else degreen p m + degreen q m)"
```
```   405   using np nq m
```
```   406 proof(induct p q arbitrary: n0 n1 m rule: polymul.induct)
```
```   407   case (2 a b c' n' p')
```
```   408   let ?c = "(a,b)"
```
```   409   { case (1 n0 n1)
```
```   410     with "2.hyps"(1-3)[of n' n' n']
```
```   411       and "2.hyps"(4-6)[of "Suc n'" "Suc n'" n']
```
```   412     show ?case by (auto simp add: min_def)
```
```   413   next
```
```   414     case (2 n0 n1) thus ?case by auto
```
```   415   next
```
```   416     case (3 n0 n1) thus ?case  using "2.hyps" by auto }
```
```   417 next
```
```   418   case (3 c n p a b)
```
```   419   let ?c' = "(a,b)"
```
```   420   { case (1 n0 n1)
```
```   421     with "3.hyps"(1-3)[of n n n]
```
```   422       "3.hyps"(4-6)[of "Suc n" "Suc n" n]
```
```   423     show ?case by (auto simp add: min_def)
```
```   424   next
```
```   425     case (2 n0 n1) thus ?case by auto
```
```   426   next
```
```   427     case (3 n0 n1) thus ?case  using "3.hyps" by auto }
```
```   428 next
```
```   429   case (4 c n p c' n' p')
```
```   430   let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
```
```   431     {
```
```   432       case (1 n0 n1)
```
```   433       hence cnp: "isnpolyh ?cnp n" and cnp': "isnpolyh ?cnp' n'"
```
```   434         and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)"
```
```   435         and np': "isnpolyh p' n'" and nc': "isnpolyh c' (Suc n')"
```
```   436         and nn0: "n \<ge> n0" and nn1:"n' \<ge> n1"
```
```   437         by simp_all
```
```   438       { assume "n < n'"
```
```   439         with "4.hyps"(13-14)[OF np cnp', of n]
```
```   440           "4.hyps"(16)[OF nc cnp', of n] nn0 cnp
```
```   441         have ?case by (simp 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 "Suc n'"] nn1 cnp'
```
```   446         have ?case
```
```   447           by (cases "Suc n' = n", simp_all add: min_def)
```
```   448       } moreover {
```
```   449         assume "n' = n"
```
```   450         with "4.hyps"(1-2)[OF cnp np', of n]
```
```   451           "4.hyps"(4)[OF cnp nc', of n] cnp cnp' nn1 nn0
```
```   452         have ?case
```
```   453           apply (auto intro!: polyadd_normh)
```
```   454           apply (simp_all add: min_def isnpolyh_mono[OF nn0])
```
```   455           done
```
```   456       }
```
```   457       ultimately show ?case by arith
```
```   458     next
```
```   459       fix n0 n1 m
```
```   460       assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1"
```
```   461       and m: "m \<le> min n0 n1"
```
```   462       let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
```
```   463       let ?d1 = "degreen ?cnp m"
```
```   464       let ?d2 = "degreen ?cnp' m"
```
```   465       let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
```
```   466       have "n'<n \<or> n < n' \<or> n' = n" by auto
```
```   467       moreover
```
```   468       {assume "n' < n \<or> n < n'"
```
```   469         with "4.hyps"(3,15,18) np np' m
```
```   470         have ?eq by auto }
```
```   471       moreover
```
```   472       {assume nn': "n' = n" hence nn:"\<not> n' < n \<and> \<not> n < n'" by arith
```
```   473         from "4.hyps"(1,3)[of n n' n]
```
```   474           "4.hyps"(4,5)[of n "Suc n'" n]
```
```   475           np np' nn'
```
```   476         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
```
```   477           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   478           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
```
```   479           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
```
```   480         {assume mn: "m = n"
```
```   481           from "4.hyps"(2,3)[OF norm(1,4), of n]
```
```   482             "4.hyps"(4,6)[OF norm(1,2), of n] norm nn' mn
```
```   483           have degs:  "degreen (?cnp *\<^sub>p c') n =
```
```   484             (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"
```
```   485             "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n" by (simp_all add: min_def)
```
```   486           from degs norm
```
```   487           have th1: "degreen(?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" by simp
```
```   488           hence neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   489             by simp
```
```   490           have nmin: "n \<le> min n n" by (simp add: min_def)
```
```   491           from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
```
```   492           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
```
```   493           from "4.hyps"(1-3)[OF norm(1,4), of n]
```
```   494             "4.hyps"(4-6)[OF norm(1,2), of n]
```
```   495             mn norm m nn' deg
```
```   496           have ?eq by simp}
```
```   497         moreover
```
```   498         {assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
```
```   499           from nn' m np have max1: "m \<le> max n n"  by simp
```
```   500           hence min1: "m \<le> min n n" by simp
```
```   501           hence min2: "m \<le> min n (Suc n)" by simp
```
```   502           from "4.hyps"(1-3)[OF norm(1,4) min1]
```
```   503             "4.hyps"(4-6)[OF norm(1,2) min2]
```
```   504             degreen_polyadd[OF norm(3,6) max1]
```
```   505
```
```   506           have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m
```
```   507             \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
```
```   508             using mn nn' np np' by simp
```
```   509           with "4.hyps"(1-3)[OF norm(1,4) min1]
```
```   510             "4.hyps"(4-6)[OF norm(1,2) min2]
```
```   511             degreen_0[OF norm(3) mn']
```
```   512           have ?eq using nn' mn np np' by clarsimp}
```
```   513         ultimately have ?eq by blast}
```
```   514       ultimately show ?eq by blast}
```
```   515     { case (2 n0 n1)
```
```   516       hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1"
```
```   517         and m: "m \<le> min n0 n1" by simp_all
```
```   518       hence mn: "m \<le> n" by simp
```
```   519       let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
```
```   520       {assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
```
```   521         hence nn: "\<not>n' < n \<and> \<not> n<n'" by simp
```
```   522         from "4.hyps"(1-3) [of n n n]
```
```   523           "4.hyps"(4-6)[of n "Suc n" n]
```
```   524           np np' C(2) mn
```
```   525         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
```
```   526           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   527           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
```
```   528           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
```
```   529           "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
```
```   530             "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
```
```   531           by (simp_all add: min_def)
```
```   532
```
```   533           from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```   534           have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   535             using norm by simp
```
```   536         from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"]  degneq
```
```   537         have "False" by simp }
```
```   538       thus ?case using "4.hyps" by clarsimp}
```
```   539 qed auto
```
```   540
```
```   541 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"
```
```   542 by(induct p q rule: polymul.induct, auto simp add: field_simps)
```
```   543
```
```   544 lemma polymul_normh:
```
```   545     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   546   shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   547   using polymul_properties(1)  by blast
```
```   548 lemma polymul_eq0_iff:
```
```   549   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   550   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) "
```
```   551   using polymul_properties(2)  by blast
```
```   552 lemma polymul_degreen:
```
```   553   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   554   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow> 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)"
```
```   555   using polymul_properties(3) by blast
```
```   556 lemma polymul_norm:
```
```   557   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   558   shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul (p,q))"
```
```   559   using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   560
```
```   561 lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
```
```   562   by (induct p arbitrary: n0 rule: headconst.induct, auto)
```
```   563
```
```   564 lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
```
```   565   by (induct p arbitrary: n0, auto)
```
```   566
```
```   567 lemma monic_eqI: assumes np: "isnpolyh p n0"
```
```   568   shows "INum (headconst p) * Ipoly bs (fst (monic p)) = (Ipoly bs p ::'a::{field_char_0, field_inverse_zero, power})"
```
```   569   unfolding monic_def Let_def
```
```   570 proof(cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
```
```   571   let ?h = "headconst p"
```
```   572   assume pz: "p \<noteq> 0\<^sub>p"
```
```   573   {assume hz: "INum ?h = (0::'a)"
```
```   574     from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
```
```   575     from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
```
```   576     with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
```
```   577   thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast
```
```   578 qed
```
```   579
```
```   580
```
```   581 text{* polyneg is a negation and preserves normal forms *}
```
```   582
```
```   583 lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
```
```   584 by (induct p rule: polyneg.induct, auto)
```
```   585
```
```   586 lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   587   by (induct p arbitrary: n rule: polyneg.induct, auto simp add: Nneg_def)
```
```   588 lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
```
```   589   by (induct p arbitrary: n0 rule: polyneg.induct, auto)
```
```   590 lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "
```
```   591 by (induct p rule: polyneg.induct, auto simp add: polyneg0)
```
```   592
```
```   593 lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
```
```   594   using isnpoly_def polyneg_normh by simp
```
```   595
```
```   596
```
```   597 text{* polysub is a substraction and preserves normal forms *}
```
```   598
```
```   599 lemma polysub[simp]: "Ipoly bs (polysub (p,q)) = (Ipoly bs p) - (Ipoly bs q)"
```
```   600 by (simp add: polysub_def polyneg polyadd)
```
```   601 lemma polysub_normh: "\<And> n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub(p,q)) (min n0 n1)"
```
```   602 by (simp add: polysub_def polyneg_normh polyadd_normh)
```
```   603
```
```   604 lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub(p,q))"
```
```   605   using polyadd_norm polyneg_norm by (simp add: polysub_def)
```
```   606 lemma polysub_same_0[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   607   shows "isnpolyh p n0 \<Longrightarrow> polysub (p, p) = 0\<^sub>p"
```
```   608 unfolding polysub_def split_def fst_conv snd_conv
```
```   609 by (induct p arbitrary: n0,auto simp add: Let_def Nsub0[simplified Nsub_def])
```
```   610
```
```   611 lemma polysub_0:
```
```   612   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   613   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"
```
```   614   unfolding polysub_def split_def fst_conv snd_conv
```
```   615   by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
```
```   616   (auto simp: Nsub0[simplified Nsub_def] Let_def)
```
```   617
```
```   618 text{* polypow is a power function and preserves normal forms *}
```
```   619
```
```   620 lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{field_char_0, field_inverse_zero})) ^ n"
```
```   621 proof(induct n rule: polypow.induct)
```
```   622   case 1 thus ?case by simp
```
```   623 next
```
```   624   case (2 n)
```
```   625   let ?q = "polypow ((Suc n) div 2) p"
```
```   626   let ?d = "polymul(?q,?q)"
```
```   627   have "odd (Suc n) \<or> even (Suc n)" by simp
```
```   628   moreover
```
```   629   {assume odd: "odd (Suc n)"
```
```   630     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
```
```   631     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul(p, ?d))" by (simp add: Let_def)
```
```   632     also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"
```
```   633       using "2.hyps" by simp
```
```   634     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
```
```   635       apply (simp only: power_add power_one_right) by simp
```
```   636     also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat)))))"
```
```   637       by (simp only: th)
```
```   638     finally have ?case
```
```   639     using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
```
```   640   moreover
```
```   641   {assume even: "even (Suc n)"
```
```   642     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
```
```   643     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
```
```   644     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
```
```   645       using "2.hyps" apply (simp only: power_add) by simp
```
```   646     finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}
```
```   647   ultimately show ?case by blast
```
```   648 qed
```
```   649
```
```   650 lemma polypow_normh:
```
```   651     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   652   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
```
```   653 proof (induct k arbitrary: n rule: polypow.induct)
```
```   654   case (2 k n)
```
```   655   let ?q = "polypow (Suc k div 2) p"
```
```   656   let ?d = "polymul (?q,?q)"
```
```   657   from prems have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
```
```   658   from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
```
```   659   from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul(p,?d)) n" by simp
```
```   660   from dn on show ?case by (simp add: Let_def)
```
```   661 qed auto
```
```   662
```
```   663 lemma polypow_norm:
```
```   664   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   665   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
```
```   666   by (simp add: polypow_normh isnpoly_def)
```
```   667
```
```   668 text{* Finally the whole normalization *}
```
```   669
```
```   670 lemma polynate[simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0, field_inverse_zero})"
```
```   671 by (induct p rule:polynate.induct, auto)
```
```   672
```
```   673 lemma polynate_norm[simp]:
```
```   674   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   675   shows "isnpoly (polynate p)"
```
```   676   by (induct p rule: polynate.induct, simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm) (simp_all add: isnpoly_def)
```
```   677
```
```   678 text{* shift1 *}
```
```   679
```
```   680
```
```   681 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
```
```   682 by (simp add: shift1_def polymul)
```
```   683
```
```   684 lemma shift1_isnpoly:
```
```   685   assumes pn: "isnpoly p" and pnz: "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p) "
```
```   686   using pn pnz by (simp add: shift1_def isnpoly_def )
```
```   687
```
```   688 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
```
```   689   by (simp add: shift1_def)
```
```   690 lemma funpow_shift1_isnpoly:
```
```   691   "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
```
```   692   by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
```
```   693
```
```   694 lemma funpow_isnpolyh:
```
```   695   assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n "and np: "isnpolyh p n"
```
```   696   shows "isnpolyh (funpow k f p) n"
```
```   697   using f np by (induct k arbitrary: p, auto)
```
```   698
```
```   699 lemma funpow_shift1: "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (Mul (Pw (Bound 0) n) p)"
```
```   700   by (induct n arbitrary: p, simp_all add: shift1_isnpoly shift1 power_Suc )
```
```   701
```
```   702 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
```
```   703   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
```
```   704
```
```   705 lemma funpow_shift1_1:
```
```   706   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (funpow n shift1 1\<^sub>p *\<^sub>p p)"
```
```   707   by (simp add: funpow_shift1)
```
```   708
```
```   709 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
```
```   710 by (induct p  arbitrary: n0 rule: poly_cmul.induct, auto simp add: field_simps)
```
```   711
```
```   712 lemma behead:
```
```   713   assumes np: "isnpolyh p n"
```
```   714   shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = (Ipoly bs p :: 'a :: {field_char_0, field_inverse_zero})"
```
```   715   using np
```
```   716 proof (induct p arbitrary: n rule: behead.induct)
```
```   717   case (1 c p n) hence pn: "isnpolyh p n" by simp
```
```   718   from prems(2)[OF pn]
```
```   719   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
```
```   720   then show ?case using "1.hyps" apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
```
```   721     by (simp_all add: th[symmetric] field_simps power_Suc)
```
```   722 qed (auto simp add: Let_def)
```
```   723
```
```   724 lemma behead_isnpolyh:
```
```   725   assumes np: "isnpolyh p n" shows "isnpolyh (behead p) n"
```
```   726   using np by (induct p rule: behead.induct, auto simp add: Let_def isnpolyh_mono)
```
```   727
```
```   728 subsection{* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
```
```   729 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
```
```   730 proof(induct p arbitrary: n rule: poly.induct, auto)
```
```   731   case (goal1 c n p n')
```
```   732   hence "n = Suc (n - 1)" by simp
```
```   733   hence "isnpolyh p (Suc (n - 1))"  using `isnpolyh p n` by simp
```
```   734   with prems(2) show ?case by simp
```
```   735 qed
```
```   736
```
```   737 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
```
```   738 by (induct p arbitrary: n0 rule: isconstant.induct, auto simp add: isnpolyh_polybound0)
```
```   739
```
```   740 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" by (induct p, auto)
```
```   741
```
```   742 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
```
```   743   apply (induct p arbitrary: n0, auto)
```
```   744   apply (atomize)
```
```   745   apply (erule_tac x = "Suc nat" in allE)
```
```   746   apply auto
```
```   747   done
```
```   748
```
```   749 lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
```
```   750  by (induct p  arbitrary: n0 rule: head.induct, auto intro: isnpolyh_polybound0)
```
```   751
```
```   752 lemma polybound0_I:
```
```   753   assumes nb: "polybound0 a"
```
```   754   shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
```
```   755 using nb
```
```   756 by (induct a rule: poly.induct) auto
```
```   757 lemma polysubst0_I:
```
```   758   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
```
```   759   by (induct t) simp_all
```
```   760
```
```   761 lemma polysubst0_I':
```
```   762   assumes nb: "polybound0 a"
```
```   763   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t"
```
```   764   by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
```
```   765
```
```   766 lemma decrpoly: assumes nb: "polybound0 t"
```
```   767   shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
```
```   768   using nb by (induct t rule: decrpoly.induct, simp_all)
```
```   769
```
```   770 lemma polysubst0_polybound0: assumes nb: "polybound0 t"
```
```   771   shows "polybound0 (polysubst0 t a)"
```
```   772 using nb by (induct a rule: poly.induct, auto)
```
```   773
```
```   774 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
```
```   775   by (induct p arbitrary: n rule: degree.induct, auto simp add: isnpolyh_polybound0)
```
```   776
```
```   777 primrec maxindex :: "poly \<Rightarrow> nat" where
```
```   778   "maxindex (Bound n) = n + 1"
```
```   779 | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
```
```   780 | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
```
```   781 | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
```
```   782 | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
```
```   783 | "maxindex (Neg p) = maxindex p"
```
```   784 | "maxindex (Pw p n) = maxindex p"
```
```   785 | "maxindex (C x) = 0"
```
```   786
```
```   787 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool" where
```
```   788   "wf_bs bs p = (length bs \<ge> maxindex p)"
```
```   789
```
```   790 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
```
```   791 proof(induct p rule: coefficients.induct)
```
```   792   case (1 c p)
```
```   793   show ?case
```
```   794   proof
```
```   795     fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"
```
```   796     hence "x = c \<or> x \<in> set (coefficients p)" by simp
```
```   797     moreover
```
```   798     {assume "x = c" hence "wf_bs bs x" using "1.prems"  unfolding wf_bs_def by simp}
```
```   799     moreover
```
```   800     {assume H: "x \<in> set (coefficients p)"
```
```   801       from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp
```
```   802       with "1.hyps" H have "wf_bs bs x" by blast }
```
```   803     ultimately  show "wf_bs bs x" by blast
```
```   804   qed
```
```   805 qed simp_all
```
```   806
```
```   807 lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
```
```   808 by (induct p rule: coefficients.induct, auto)
```
```   809
```
```   810 lemma wf_bs_I: "wf_bs bs p ==> Ipoly (bs@bs') p = Ipoly bs p"
```
```   811   unfolding wf_bs_def by (induct p, auto simp add: nth_append)
```
```   812
```
```   813 lemma take_maxindex_wf: assumes wf: "wf_bs bs p"
```
```   814   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
```
```   815 proof-
```
```   816   let ?ip = "maxindex p"
```
```   817   let ?tbs = "take ?ip bs"
```
```   818   from wf have "length ?tbs = ?ip" unfolding wf_bs_def by simp
```
```   819   hence wf': "wf_bs ?tbs p" unfolding wf_bs_def by  simp
```
```   820   have eq: "bs = ?tbs @ (drop ?ip bs)" by simp
```
```   821   from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis using eq by simp
```
```   822 qed
```
```   823
```
```   824 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
```
```   825   by (induct p, auto)
```
```   826
```
```   827 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
```
```   828   unfolding wf_bs_def by simp
```
```   829
```
```   830 lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
```
```   831   unfolding wf_bs_def by simp
```
```   832
```
```   833
```
```   834
```
```   835 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
```
```   836 by(induct p rule: coefficients.induct, auto simp add: wf_bs_def)
```
```   837 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
```
```   838   by (induct p rule: coefficients.induct, simp_all)
```
```   839
```
```   840
```
```   841 lemma coefficients_head: "last (coefficients p) = head p"
```
```   842   by (induct p rule: coefficients.induct, auto)
```
```   843
```
```   844 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
```
```   845   unfolding wf_bs_def by (induct p rule: decrpoly.induct, auto)
```
```   846
```
```   847 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
```
```   848   apply (rule exI[where x="replicate (n - length xs) z"])
```
```   849   by simp
```
```   850 lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
```
```   851 by (cases p, auto) (case_tac "nat", simp_all)
```
```   852
```
```   853 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
```
```   854   unfolding wf_bs_def
```
```   855   apply (induct p q rule: polyadd.induct)
```
```   856   apply (auto simp add: Let_def)
```
```   857   done
```
```   858
```
```   859 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
```
```   860   unfolding wf_bs_def
```
```   861   apply (induct p q arbitrary: bs rule: polymul.induct)
```
```   862   apply (simp_all add: wf_bs_polyadd)
```
```   863   apply clarsimp
```
```   864   apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
```
```   865   apply auto
```
```   866   done
```
```   867
```
```   868 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
```
```   869   unfolding wf_bs_def by (induct p rule: polyneg.induct, auto)
```
```   870
```
```   871 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
```
```   872   unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
```
```   873
```
```   874 subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}
```
```   875
```
```   876 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
```
```   877 definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)"
```
```   878 definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"
```
```   879
```
```   880 lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0"
```
```   881 proof (induct p arbitrary: n0 rule: coefficients.induct)
```
```   882   case (1 c p n0)
```
```   883   have cp: "isnpolyh (CN c 0 p) n0" by fact
```
```   884   hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
```
```   885     by (auto simp add: isnpolyh_mono[where n'=0])
```
```   886   from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp
```
```   887 qed auto
```
```   888
```
```   889 lemma coefficients_isconst:
```
```   890   "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
```
```   891   by (induct p arbitrary: n rule: coefficients.induct,
```
```   892     auto simp add: isnpolyh_Suc_const)
```
```   893
```
```   894 lemma polypoly_polypoly':
```
```   895   assumes np: "isnpolyh p n0"
```
```   896   shows "polypoly (x#bs) p = polypoly' bs p"
```
```   897 proof-
```
```   898   let ?cf = "set (coefficients p)"
```
```   899   from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
```
```   900   {fix q assume q: "q \<in> ?cf"
```
```   901     from q cn_norm have th: "isnpolyh q n0" by blast
```
```   902     from coefficients_isconst[OF np] q have "isconstant q" by blast
```
```   903     with isconstant_polybound0[OF th] have "polybound0 q" by blast}
```
```   904   hence "\<forall>q \<in> ?cf. polybound0 q" ..
```
```   905   hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"
```
```   906     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
```
```   907     by auto
```
```   908
```
```   909   thus ?thesis unfolding polypoly_def polypoly'_def by simp
```
```   910 qed
```
```   911
```
```   912 lemma polypoly_poly:
```
```   913   assumes np: "isnpolyh p n0" shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
```
```   914   using np
```
```   915 by (induct p arbitrary: n0 bs rule: coefficients.induct, auto simp add: polypoly_def)
```
```   916
```
```   917 lemma polypoly'_poly:
```
```   918   assumes np: "isnpolyh p n0" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
```
```   919   using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
```
```   920
```
```   921
```
```   922 lemma polypoly_poly_polybound0:
```
```   923   assumes np: "isnpolyh p n0" and nb: "polybound0 p"
```
```   924   shows "polypoly bs p = [Ipoly bs p]"
```
```   925   using np nb unfolding polypoly_def
```
```   926   by (cases p, auto, case_tac nat, auto)
```
```   927
```
```   928 lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
```
```   929   by (induct p rule: head.induct, auto)
```
```   930
```
```   931 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   932   by (cases p,auto)
```
```   933
```
```   934 lemma head_eq_headn0: "head p = headn p 0"
```
```   935   by (induct p rule: head.induct, simp_all)
```
```   936
```
```   937 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   938   by (simp add: head_eq_headn0)
```
```   939
```
```   940 lemma isnpolyh_zero_iff:
```
```   941   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})"
```
```   942   shows "p = 0\<^sub>p"
```
```   943 using nq eq
```
```   944 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
```
```   945   case less
```
```   946   note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
```
```   947   {assume nz: "maxindex p = 0"
```
```   948     then obtain c where "p = C c" using np by (cases p, auto)
```
```   949     with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}
```
```   950   moreover
```
```   951   {assume nz: "maxindex p \<noteq> 0"
```
```   952     let ?h = "head p"
```
```   953     let ?hd = "decrpoly ?h"
```
```   954     let ?ihd = "maxindex ?hd"
```
```   955     from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h"
```
```   956       by simp_all
```
```   957     hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast
```
```   958
```
```   959     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
```
```   960     have mihn: "maxindex ?h \<le> maxindex p" by auto
```
```   961     with decr_maxindex[OF h(2)] nz  have ihd_lt_n: "?ihd < maxindex p" by auto
```
```   962     {fix bs:: "'a list"  assume bs: "wf_bs bs ?hd"
```
```   963       let ?ts = "take ?ihd bs"
```
```   964       let ?rs = "drop ?ihd bs"
```
```   965       have ts: "wf_bs ?ts ?hd" using bs unfolding wf_bs_def by simp
```
```   966       have bs_ts_eq: "?ts@ ?rs = bs" by simp
```
```   967       from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x#?ts) ?h" by simp
```
```   968       from ihd_lt_n have "ALL x. length (x#?ts) \<le> maxindex p" by simp
```
```   969       with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = maxindex p" by blast
```
```   970       hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" unfolding wf_bs_def by simp
```
```   971       with zp have "\<forall> x. Ipoly ((x#?ts) @ xs) p = 0" by blast
```
```   972       hence "\<forall> x. Ipoly (x#(?ts @ xs)) p = 0" by simp
```
```   973       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
```
```   974       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"  by simp
```
```   975       hence "poly (polypoly' (?ts @ xs) p) = poly []" by (auto intro: ext)
```
```   976       hence "\<forall> c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
```
```   977         using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
```
```   978       with coefficients_head[of p, symmetric]
```
```   979       have th0: "Ipoly (?ts @ xs) ?hd = 0" by simp
```
```   980       from bs have wf'': "wf_bs ?ts ?hd" unfolding wf_bs_def by simp
```
```   981       with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0" by simp
```
```   982       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by simp }
```
```   983     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast
```
```   984
```
```   985     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast
```
```   986     hence "?h = 0\<^sub>p" by simp
```
```   987     with head_nz[OF np] have "p = 0\<^sub>p" by simp}
```
```   988   ultimately show "p = 0\<^sub>p" by blast
```
```   989 qed
```
```   990
```
```   991 lemma isnpolyh_unique:
```
```   992   assumes np:"isnpolyh p n0" and nq: "isnpolyh q n1"
```
```   993   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"
```
```   994 proof(auto)
```
```   995   assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a)= \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
```
```   996   hence "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" by simp
```
```   997   hence "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
```
```   998     using wf_bs_polysub[where p=p and q=q] by auto
```
```   999   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq]
```
```  1000   show "p = q" by blast
```
```  1001 qed
```
```  1002
```
```  1003
```
```  1004 text{* consequences of unicity on the algorithms for polynomial normalization *}
```
```  1005
```
```  1006 lemma polyadd_commute:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1007   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" shows "p +\<^sub>p q = q +\<^sub>p p"
```
```  1008   using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]] by simp
```
```  1009
```
```  1010 lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp
```
```  1011 lemma one_normh: "isnpolyh 1\<^sub>p n" by simp
```
```  1012 lemma polyadd_0[simp]:
```
```  1013   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1014   and np: "isnpolyh p n0" shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>p p = p"
```
```  1015   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
```
```  1016     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
```
```  1017
```
```  1018 lemma polymul_1[simp]:
```
```  1019     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1020   and np: "isnpolyh p n0" shows "p *\<^sub>p 1\<^sub>p = p" and "1\<^sub>p *\<^sub>p p = p"
```
```  1021   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
```
```  1022     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
```
```  1023 lemma polymul_0[simp]:
```
```  1024   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1025   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"
```
```  1026   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
```
```  1027     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
```
```  1028
```
```  1029 lemma polymul_commute:
```
```  1030     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1031   and np:"isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1032   shows "p *\<^sub>p q = q *\<^sub>p p"
```
```  1033 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
```
```  1034
```
```  1035 declare polyneg_polyneg[simp]
```
```  1036
```
```  1037 lemma isnpolyh_polynate_id[simp]:
```
```  1038   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1039   and np:"isnpolyh p n0" shows "polynate p = p"
```
```  1040   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
```
```  1041
```
```  1042 lemma polynate_idempotent[simp]:
```
```  1043     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1044   shows "polynate (polynate p) = polynate p"
```
```  1045   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
```
```  1046
```
```  1047 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
```
```  1048   unfolding poly_nate_def polypoly'_def ..
```
```  1049 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>)"
```
```  1050   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
```
```  1051   unfolding poly_nate_polypoly' by (auto intro: ext)
```
```  1052
```
```  1053 subsection{* heads, degrees and all that *}
```
```  1054 lemma degree_eq_degreen0: "degree p = degreen p 0"
```
```  1055   by (induct p rule: degree.induct, simp_all)
```
```  1056
```
```  1057 lemma degree_polyneg: assumes n: "isnpolyh p n"
```
```  1058   shows "degree (polyneg p) = degree p"
```
```  1059   using n
```
```  1060   by (induct p arbitrary: n rule: polyneg.induct, simp_all) (case_tac na, auto)
```
```  1061
```
```  1062 lemma degree_polyadd:
```
```  1063   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1064   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1065 using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
```
```  1066
```
```  1067
```
```  1068 lemma degree_polysub: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1069   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1070 proof-
```
```  1071   from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp
```
```  1072   from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])
```
```  1073 qed
```
```  1074
```
```  1075 lemma degree_polysub_samehead:
```
```  1076   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1077   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q"
```
```  1078   and d: "degree p = degree q"
```
```  1079   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
```
```  1080 unfolding polysub_def split_def fst_conv snd_conv
```
```  1081 using np nq h d
```
```  1082 proof(induct p q rule:polyadd.induct)
```
```  1083   case (1 c c') thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def])
```
```  1084 next
```
```  1085   case (2 c c' n' p')
```
```  1086   from prems have "degree (C c) = degree (CN c' n' p')" by simp
```
```  1087   hence nz:"n' > 0" by (cases n', auto)
```
```  1088   hence "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
```
```  1089   with prems show ?case by simp
```
```  1090 next
```
```  1091   case (3 c n p c')
```
```  1092   from prems have "degree (C c') = degree (CN c n p)" by simp
```
```  1093   hence nz:"n > 0" by (cases n, auto)
```
```  1094   hence "head (CN c n p) = CN c n p" by (cases n, auto)
```
```  1095   with prems show ?case by simp
```
```  1096 next
```
```  1097   case (4 c n p c' n' p')
```
```  1098   hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1"
```
```  1099     "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+
```
```  1100   hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all
```
```  1101   hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
```
```  1102     using H(1-2) degree_polyneg by auto
```
```  1103   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"  by simp+
```
```  1104   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"  by simp
```
```  1105   from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto
```
```  1106   have "n = n' \<or> n < n' \<or> n > n'" by arith
```
```  1107   moreover
```
```  1108   {assume nn': "n = n'"
```
```  1109     have "n = 0 \<or> n >0" by arith
```
```  1110     moreover {assume nz: "n = 0" hence ?case using prems by (auto simp add: Let_def degcmc')}
```
```  1111     moreover {assume nz: "n > 0"
```
```  1112       with nn' H(3) have  cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
```
```  1113       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)}
```
```  1114     ultimately have ?case by blast}
```
```  1115   moreover
```
```  1116   {assume nn': "n < n'" hence n'p: "n' > 0" by simp
```
```  1117     hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n', simp_all)
```
```  1118     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)
```
```  1119     hence "n > 0" by (cases n, simp_all)
```
```  1120     hence headcnp: "head (CN c n p) = CN c n p" by (cases n, auto)
```
```  1121     from H(3) headcnp headcnp' nn' have ?case by auto}
```
```  1122   moreover
```
```  1123   {assume nn': "n > n'"  hence np: "n > 0" by simp
```
```  1124     hence headcnp:"head (CN c n p) = CN c n p"  by (cases n, simp_all)
```
```  1125     from prems have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
```
```  1126     from np have degcnp: "degree (CN c n p) = 0" by (cases n, simp_all)
```
```  1127     with degcnpeq have "n' > 0" by (cases n', simp_all)
```
```  1128     hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
```
```  1129     from H(3) headcnp headcnp' nn' have ?case by auto}
```
```  1130   ultimately show ?case  by blast
```
```  1131 qed auto
```
```  1132
```
```  1133 lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
```
```  1134 by (induct p arbitrary: n0 rule: head.induct, simp_all add: shift1_def)
```
```  1135
```
```  1136 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
```
```  1137 proof(induct k arbitrary: n0 p)
```
```  1138   case (Suc k n0 p) hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
```
```  1139   with prems have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
```
```  1140     and "head (shift1 p) = head p" by (simp_all add: shift1_head)
```
```  1141   thus ?case by (simp add: funpow_swap1)
```
```  1142 qed auto
```
```  1143
```
```  1144 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
```
```  1145   by (simp add: shift1_def)
```
```  1146 lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
```
```  1147   by (induct k arbitrary: p, auto simp add: shift1_degree)
```
```  1148
```
```  1149 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
```
```  1150   by (induct n arbitrary: p, simp_all add: funpow_def)
```
```  1151
```
```  1152 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
```
```  1153   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1154 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
```
```  1155   by (induct p arbitrary: n rule: degreen.induct, auto)
```
```  1156 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
```
```  1157   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1158 lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
```
```  1159   by (induct p rule: head.induct, auto)
```
```  1160
```
```  1161 lemma polyadd_eq_const_degree:
```
```  1162   "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd p q = C c\<rbrakk> \<Longrightarrow> degree p = degree q"
```
```  1163   using polyadd_eq_const_degreen degree_eq_degreen0 by simp
```
```  1164
```
```  1165 lemma polyadd_head: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1166   and deg: "degree p \<noteq> degree q"
```
```  1167   shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
```
```  1168 using np nq deg
```
```  1169 apply(induct p q arbitrary: n0 n1 rule: polyadd.induct,simp_all)
```
```  1170 apply (case_tac n', simp, simp)
```
```  1171 apply (case_tac n, simp, simp)
```
```  1172 apply (case_tac n, case_tac n', simp add: Let_def)
```
```  1173 apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p")
```
```  1174 apply (auto simp add: polyadd_eq_const_degree)
```
```  1175 apply (metis head_nz)
```
```  1176 apply (metis head_nz)
```
```  1177 apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
```
```  1178 by (metis degree.simps(9) gr0_conv_Suc nat_less_le order_le_less_trans)
```
```  1179
```
```  1180 lemma polymul_head_polyeq:
```
```  1181    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1182   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"
```
```  1183 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
```
```  1184   case (2 a b c' n' p' n0 n1)
```
```  1185   hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum (a,b)"  by (simp_all add: head_isnpolyh)
```
```  1186   thus ?case using prems by (cases n', auto)
```
```  1187 next
```
```  1188   case (3 c n p a' b' n0 n1)
```
```  1189   hence "isnpolyh (head (CN c n p)) n0" "isnormNum (a',b')"  by (simp_all add: head_isnpolyh)
```
```  1190   thus ?case using prems by (cases n, auto)
```
```  1191 next
```
```  1192   case (4 c n p c' n' p' n0 n1)
```
```  1193   hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
```
```  1194     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
```
```  1195     by simp_all
```
```  1196   have "n < n' \<or> n' < n \<or> n = n'" by arith
```
```  1197   moreover
```
```  1198   {assume nn': "n < n'" hence ?case
```
```  1199       using norm
```
```  1200     "4.hyps"(5)[OF norm(1,6)]
```
```  1201     "4.hyps"(6)[OF norm(2,6)] by (simp, cases n, simp,cases n', simp_all)}
```
```  1202   moreover {assume nn': "n'< n"
```
```  1203     hence ?case using norm "4.hyps"(1) [OF norm(5,3)]
```
```  1204       "4.hyps"(2)[OF norm(5,4)]
```
```  1205       by (simp,cases n',simp,cases n,auto)}
```
```  1206   moreover {assume nn': "n' = n"
```
```  1207     from nn' polymul_normh[OF norm(5,4)]
```
```  1208     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
```
```  1209     from nn' polymul_normh[OF norm(5,3)] norm
```
```  1210     have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
```
```  1211     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
```
```  1212     have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```  1213     from polyadd_normh[OF ncnpc' ncnpp0']
```
```  1214     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"
```
```  1215       by (simp add: min_def)
```
```  1216     {assume np: "n > 0"
```
```  1217       with nn' head_isnpolyh_Suc'[OF np nth]
```
```  1218         head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
```
```  1219       have ?case by simp}
```
```  1220     moreover
```
```  1221     {moreover assume nz: "n = 0"
```
```  1222       from polymul_degreen[OF norm(5,4), where m="0"]
```
```  1223         polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
```
```  1224       norm(5,6) degree_npolyhCN[OF norm(6)]
```
```  1225     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
```
```  1226     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
```
```  1227     from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
```
```  1228     have ?case   using norm "4.hyps"(1)[OF norm(5,3)]
```
```  1229         "4.hyps"(2)[OF norm(5,4)] nn' nz by simp }
```
```  1230     ultimately have ?case by (cases n) auto}
```
```  1231   ultimately show ?case by blast
```
```  1232 qed simp_all
```
```  1233
```
```  1234 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
```
```  1235   by(induct p rule: degree.induct, auto)
```
```  1236
```
```  1237 lemma degree_head[simp]: "degree (head p) = 0"
```
```  1238   by (induct p rule: head.induct, auto)
```
```  1239
```
```  1240 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
```
```  1241   by (cases n, simp_all)
```
```  1242 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
```
```  1243   by (cases n, simp_all)
```
```  1244
```
```  1245 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)"
```
```  1246   using polyadd_different_degreen degree_eq_degreen0 by simp
```
```  1247
```
```  1248 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
```
```  1249   by (induct p arbitrary: n0 rule: polyneg.induct, auto)
```
```  1250
```
```  1251 lemma degree_polymul:
```
```  1252   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1253   and np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1254   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
```
```  1255   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
```
```  1256
```
```  1257 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
```
```  1258   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1259
```
```  1260 lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"
```
```  1261   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1262
```
```  1263 subsection {* Correctness of polynomial pseudo division *}
```
```  1264
```
```  1265 lemma polydivide_aux_properties:
```
```  1266   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1267   and np: "isnpolyh p n0" and ns: "isnpolyh s n1"
```
```  1268   and ap: "head p = a" and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
```
```  1269   shows "(polydivide_aux a n p k s = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1270           \<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)))"
```
```  1271   using ns
```
```  1272 proof(induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
```
```  1273   case less
```
```  1274   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
```
```  1275   let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1276     \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
```
```  1277   let ?b = "head s"
```
```  1278   let ?p' = "funpow (degree s - n) shift1 p"
```
```  1279   let ?xdn = "funpow (degree s - n) shift1 1\<^sub>p"
```
```  1280   let ?akk' = "a ^\<^sub>p (k' - k)"
```
```  1281   note ns = `isnpolyh s n1`
```
```  1282   from np have np0: "isnpolyh p 0"
```
```  1283     using isnpolyh_mono[where n="n0" and n'="0" and p="p"]  by simp
```
```  1284   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
```
```  1285   have headp': "head ?p' = head p" using funpow_shift1_head[OF np pnz] by simp
```
```  1286   from funpow_shift1_isnpoly[where p="1\<^sub>p"] have nxdn: "isnpolyh ?xdn 0" by (simp add: isnpoly_def)
```
```  1287   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
```
```  1288   have nakk':"isnpolyh ?akk' 0" by blast
```
```  1289   {assume sz: "s = 0\<^sub>p"
```
```  1290    hence ?ths using np polydivide_aux.simps apply clarsimp by (rule exI[where x="0\<^sub>p"], simp) }
```
```  1291   moreover
```
```  1292   {assume sz: "s \<noteq> 0\<^sub>p"
```
```  1293     {assume dn: "degree s < n"
```
```  1294       hence "?ths" using ns ndp np polydivide_aux.simps by auto (rule exI[where x="0\<^sub>p"],simp) }
```
```  1295     moreover
```
```  1296     {assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith
```
```  1297       have degsp': "degree s = degree ?p'"
```
```  1298         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp
```
```  1299       {assume ba: "?b = a"
```
```  1300         hence headsp': "head s = head ?p'" using ap headp' by simp
```
```  1301         have nr: "isnpolyh (s -\<^sub>p ?p') 0" using polysub_normh[OF ns np'] by simp
```
```  1302         from degree_polysub_samehead[OF ns np' headsp' degsp']
```
```  1303         have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
```
```  1304         moreover
```
```  1305         {assume deglt:"degree (s -\<^sub>p ?p') < degree s"
```
```  1306           from polydivide_aux.simps sz dn' ba
```
```  1307           have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
```
```  1308             by (simp add: Let_def)
```
```  1309           {assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1310             from less(1)[OF deglt nr, of k k' r]
```
```  1311               trans[OF eq[symmetric] h1]
```
```  1312             have kk': "k \<le> k'" and nr:"\<exists>nr. isnpolyh r nr" and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1313               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
```
```  1314             from q1 obtain q n1 where nq: "isnpolyh q n1"
```
```  1315               and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"  by blast
```
```  1316             from nr obtain nr where nr': "isnpolyh r nr" by blast
```
```  1317             from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0" by simp
```
```  1318             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
```
```  1319             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
```
```  1320             from polyadd_normh[OF polymul_normh[OF np
```
```  1321               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
```
```  1322             have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0" by simp
```
```  1323             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')) =
```
```  1324               Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
```
```  1325             hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
```
```  1326               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1327               by (simp add: field_simps)
```
```  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 (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p *\<^sub>p p)
```
```  1330               + Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1331               by (auto simp only: funpow_shift1_1)
```
```  1332             hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1333               Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p)
```
```  1334               + Ipoly bs q) + Ipoly bs r" by (simp add: field_simps)
```
```  1335             hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1336               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
```
```  1337             with isnpolyh_unique[OF nakks' nqr']
```
```  1338             have "a ^\<^sub>p (k' - k) *\<^sub>p s =
```
```  1339               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
```
```  1340             hence ?qths using nq'
```
```  1341               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 1\<^sub>p) +\<^sub>p q" in exI)
```
```  1342               apply (rule_tac x="0" in exI) by simp
```
```  1343             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"
```
```  1344               by blast } hence ?ths by blast }
```
```  1345         moreover
```
```  1346         {assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
```
```  1347           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}"]
```
```  1348           have " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs s = Ipoly bs ?p'" by simp
```
```  1349           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
```
```  1350             by (simp only: funpow_shift1_1) simp
```
```  1351           hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]] by blast
```
```  1352           {assume h1: "polydivide_aux a n p k s = (k',r)"
```
```  1353             from polydivide_aux.simps sz dn' ba
```
```  1354             have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
```
```  1355               by (simp add: Let_def)
```
```  1356             also have "\<dots> = (k,0\<^sub>p)" using polydivide_aux.simps spz by simp
```
```  1357             finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
```
```  1358             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
```
```  1359               polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
```
```  1360               apply auto
```
```  1361               apply (rule exI[where x="?xdn"])
```
```  1362               apply (auto simp add: polymul_commute[of p])
```
```  1363               done} }
```
```  1364         ultimately have ?ths by blast }
```
```  1365       moreover
```
```  1366       {assume ba: "?b \<noteq> a"
```
```  1367         from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1368           polymul_normh[OF head_isnpolyh[OF ns] np']]
```
```  1369         have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by(simp add: min_def)
```
```  1370         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
```
```  1371           using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1372             polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
```
```  1373             funpow_shift1_nz[OF pnz] by simp_all
```
```  1374         from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
```
```  1375           polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1376         have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
```
```  1377           using head_head[OF ns] funpow_shift1_head[OF np pnz]
```
```  1378             polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
```
```  1379           by (simp add: ap)
```
```  1380         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
```
```  1381           head_nz[OF np] pnz sz ap[symmetric]
```
```  1382           funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1383           polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"]  head_nz[OF ns]
```
```  1384           ndp dn
```
```  1385         have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p') "
```
```  1386           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
```
```  1387         {assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
```
```  1388           from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns] polymul_normh[OF head_isnpolyh[OF ns]np']]
```
```  1389           ap have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by simp
```
```  1390           {assume h1:"polydivide_aux a n p k s = (k', r)"
```
```  1391             from h1 polydivide_aux.simps sz dn' ba
```
```  1392             have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
```
```  1393               by (simp add: Let_def)
```
```  1394             with less(1)[OF dth nasbp', of "Suc k" k' r]
```
```  1395             obtain q nq nr where kk': "Suc k \<le> k'" and nr: "isnpolyh r nr" and nq: "isnpolyh q nq"
```
```  1396               and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1397               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
```
```  1398             from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith
```
```  1399             {fix bs:: "'a::{field_char_0, field_inverse_zero} list"
```
```  1400
```
```  1401             from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
```
```  1402             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
```
```  1403             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"
```
```  1404               by (simp add: field_simps power_Suc)
```
```  1405             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"
```
```  1406               by (simp add:kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1407             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"
```
```  1408               by (simp add: field_simps)}
```
```  1409             hence ieq:"\<forall>(bs :: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1410               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
```
```  1411             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
```
```  1412             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]]
```
```  1413             have nqw: "isnpolyh ?q 0" by simp
```
```  1414             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]]
```
```  1415             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
```
```  1416             from dr kk' nr h1 asth nqw have ?ths apply simp
```
```  1417               apply (rule conjI)
```
```  1418               apply (rule exI[where x="nr"], simp)
```
```  1419               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
```
```  1420               apply (rule exI[where x="0"], simp)
```
```  1421               done}
```
```  1422           hence ?ths by blast }
```
```  1423         moreover
```
```  1424         {assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
```
```  1425           {fix bs :: "'a::{field_char_0, field_inverse_zero} list"
```
```  1426             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
```
```  1427           have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'" by simp
```
```  1428           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
```
```  1429             by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1430           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" by simp
```
```  1431         }
```
```  1432         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))" ..
```
```  1433           from hth
```
```  1434           have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
```
```  1435             using isnpolyh_unique[where ?'a = "'a::{field_char_0, field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
```
```  1436                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
```
```  1437               simplified ap] by simp
```
```  1438           {assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1439           from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
```
```  1440           have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
```
```  1441           with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
```
```  1442             polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
```
```  1443           have ?ths apply (clarsimp simp add: Let_def)
```
```  1444             apply (rule exI[where x="?b *\<^sub>p ?xdn"]) apply simp
```
```  1445             apply (rule exI[where x="0"], simp)
```
```  1446             done}
```
```  1447         hence ?ths by blast}
```
```  1448         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"]
```
```  1449           head_nz[OF np] pnz sz ap[symmetric]
```
```  1450           by (simp add: degree_eq_degreen0[symmetric]) blast }
```
```  1451       ultimately have ?ths by blast
```
```  1452     }
```
```  1453     ultimately have ?ths by blast}
```
```  1454   ultimately show ?ths by blast
```
```  1455 qed
```
```  1456
```
```  1457 lemma polydivide_properties:
```
```  1458   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1459   and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
```
```  1460   shows "(\<exists> k r. polydivide s p = (k,r) \<and> (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1461   \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
```
```  1462 proof-
```
```  1463   have trv: "head p = head p" "degree p = degree p" by simp_all
```
```  1464   from polydivide_def[where s="s" and p="p"]
```
```  1465   have ex: "\<exists> k r. polydivide s p = (k,r)" by auto
```
```  1466   then obtain k r where kr: "polydivide s p = (k,r)" by blast
```
```  1467   from trans[OF meta_eq_to_obj_eq[OF polydivide_def[where s="s"and p="p"], symmetric] kr]
```
```  1468     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
```
```  1469   have "(degree r = 0 \<or> degree r < degree p) \<and>
```
```  1470    (\<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
```
```  1471   with kr show ?thesis
```
```  1472     apply -
```
```  1473     apply (rule exI[where x="k"])
```
```  1474     apply (rule exI[where x="r"])
```
```  1475     apply simp
```
```  1476     done
```
```  1477 qed
```
```  1478
```
```  1479 subsection{* More about polypoly and pnormal etc *}
```
```  1480
```
```  1481 definition "isnonconstant p = (\<not> isconstant p)"
```
```  1482
```
```  1483 lemma isnonconstant_pnormal_iff: assumes nc: "isnonconstant p"
```
```  1484   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1485 proof
```
```  1486   let ?p = "polypoly bs p"
```
```  1487   assume H: "pnormal ?p"
```
```  1488   have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
```
```  1489
```
```  1490   from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1491     pnormal_last_nonzero[OF H]
```
```  1492   show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)
```
```  1493 next
```
```  1494   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1495   let ?p = "polypoly bs p"
```
```  1496   have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
```
```  1497   hence pz: "?p \<noteq> []" by (simp add: polypoly_def)
```
```  1498   hence lg: "length ?p > 0" by simp
```
```  1499   from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1500   have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)
```
```  1501   from pnormal_last_length[OF lg lz] show "pnormal ?p" .
```
```  1502 qed
```
```  1503
```
```  1504 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
```
```  1505   unfolding isnonconstant_def
```
```  1506   apply (cases p, simp_all)
```
```  1507   apply (case_tac nat, auto)
```
```  1508   done
```
```  1509 lemma isnonconstant_nonconstant: assumes inc: "isnonconstant p"
```
```  1510   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1511 proof
```
```  1512   let ?p = "polypoly bs p"
```
```  1513   assume nc: "nonconstant ?p"
```
```  1514   from isnonconstant_pnormal_iff[OF inc, of bs] nc
```
```  1515   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast
```
```  1516 next
```
```  1517   let ?p = "polypoly bs p"
```
```  1518   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1519   from isnonconstant_pnormal_iff[OF inc, of bs] h
```
```  1520   have pn: "pnormal ?p" by blast
```
```  1521   {fix x assume H: "?p = [x]"
```
```  1522     from H have "length (coefficients p) = 1" unfolding polypoly_def by auto
```
```  1523     with isnonconstant_coefficients_length[OF inc] have False by arith}
```
```  1524   thus "nonconstant ?p" using pn unfolding nonconstant_def by blast
```
```  1525 qed
```
```  1526
```
```  1527 lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
```
```  1528   unfolding pnormal_def
```
```  1529  apply (induct p)
```
```  1530  apply (simp_all, case_tac "p=[]", simp_all)
```
```  1531  done
```
```  1532
```
```  1533 lemma degree_degree: assumes inc: "isnonconstant p"
```
```  1534   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1535 proof
```
```  1536   let  ?p = "polypoly bs p"
```
```  1537   assume H: "degree p = Polynomial_List.degree ?p"
```
```  1538   from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
```
```  1539     unfolding polypoly_def by auto
```
```  1540   from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
```
```  1541   have lg:"length (pnormalize ?p) = length ?p"
```
```  1542     unfolding Polynomial_List.degree_def polypoly_def by simp
```
```  1543   hence "pnormal ?p" using pnormal_length[OF pz] by blast
```
```  1544   with isnonconstant_pnormal_iff[OF inc]
```
```  1545   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast
```
```  1546 next
```
```  1547   let  ?p = "polypoly bs p"
```
```  1548   assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1549   with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast
```
```  1550   with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
```
```  1551   show "degree p = Polynomial_List.degree ?p"
```
```  1552     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
```
```  1553 qed
```
```  1554
```
```  1555 section{* Swaps ; Division by a certain variable *}
```
```  1556 primrec swap:: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly" where
```
```  1557   "swap n m (C x) = C x"
```
```  1558 | "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"
```
```  1559 | "swap n m (Neg t) = Neg (swap n m t)"
```
```  1560 | "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
```
```  1561 | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
```
```  1562 | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
```
```  1563 | "swap n m (Pw t k) = Pw (swap n m t) k"
```
```  1564 | "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)
```
```  1565   (swap n m p)"
```
```  1566
```
```  1567 lemma swap: assumes nbs: "n < length bs" and mbs: "m < length bs"
```
```  1568   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  1569 proof (induct t)
```
```  1570   case (Bound k) thus ?case using nbs mbs by simp
```
```  1571 next
```
```  1572   case (CN c k p) thus ?case using nbs mbs by simp
```
```  1573 qed simp_all
```
```  1574 lemma swap_swap_id[simp]: "swap n m (swap m n t) = t"
```
```  1575   by (induct t,simp_all)
```
```  1576
```
```  1577 lemma swap_commute: "swap n m p = swap m n p" by (induct p, simp_all)
```
```  1578
```
```  1579 lemma swap_same_id[simp]: "swap n n t = t"
```
```  1580   by (induct t, simp_all)
```
```  1581
```
```  1582 definition "swapnorm n m t = polynate (swap n m t)"
```
```  1583
```
```  1584 lemma swapnorm: assumes nbs: "n < length bs" and mbs: "m < length bs"
```
```  1585   shows "((Ipoly bs (swapnorm n m t) :: 'a\<Colon>{field_char_0, field_inverse_zero})) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  1586   using swap[OF prems] swapnorm_def by simp
```
```  1587
```
```  1588 lemma swapnorm_isnpoly[simp]:
```
```  1589     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1590   shows "isnpoly (swapnorm n m p)"
```
```  1591   unfolding swapnorm_def by simp
```
```  1592
```
```  1593 definition "polydivideby n s p =
```
```  1594     (let ss = swapnorm 0 n s ; sp = swapnorm 0 n p ; h = head sp; (k,r) = polydivide ss sp
```
```  1595      in (k,swapnorm 0 n h,swapnorm 0 n r))"
```
```  1596
```
```  1597 lemma swap_nz [simp]: " (swap n m p = 0\<^sub>p) = (p = 0\<^sub>p)" by (induct p, simp_all)
```
```  1598
```
```  1599 fun isweaknpoly :: "poly \<Rightarrow> bool"
```
```  1600 where
```
```  1601   "isweaknpoly (C c) = True"
```
```  1602 | "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
```
```  1603 | "isweaknpoly p = False"
```
```  1604
```
```  1605 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
```
```  1606   by (induct p arbitrary: n0, auto)
```
```  1607
```
```  1608 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
```
```  1609   by (induct p, auto)
```
```  1610
```
`  1611 end`