src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
 author krauss Mon Feb 21 23:14:36 2011 +0100 (2011-02-21) changeset 41810 588c95c4b53e parent 41808 9f436d00248f child 41811 7e338ccabff0 permissions -rw-r--r--
dropped stupid name
```     1 (*  Title:      HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
```
```     2     Author:     Amine Chaieb
```
```     3 *)
```
```     4
```
```     5 header {* Implementation and verification of multivariate polynomials *}
```
```     6
```
```     7 theory Reflected_Multivariate_Polynomial
```
```     8 imports Complex_Main "~~/src/HOL/Library/Abstract_Rat" Polynomial_List
```
```     9 begin
```
```    10
```
```    11   (* Implementation *)
```
```    12
```
```    13 subsection{* Datatype of polynomial expressions *}
```
```    14
```
```    15 datatype poly = C Num| Bound nat| Add poly poly|Sub poly poly
```
```    16   | Mul poly poly| Neg poly| Pw poly nat| CN poly nat poly
```
```    17
```
```    18 abbreviation poly_0 :: "poly" ("0\<^sub>p") where "0\<^sub>p \<equiv> C (0\<^sub>N)"
```
```    19 abbreviation poly_p :: "int \<Rightarrow> poly" ("_\<^sub>p") where "i\<^sub>p \<equiv> C (i\<^sub>N)"
```
```    20
```
```    21 subsection{* Boundedness, substitution and all that *}
```
```    22 primrec polysize:: "poly \<Rightarrow> nat" where
```
```    23   "polysize (C c) = 1"
```
```    24 | "polysize (Bound n) = 1"
```
```    25 | "polysize (Neg p) = 1 + polysize p"
```
```    26 | "polysize (Add p q) = 1 + polysize p + polysize q"
```
```    27 | "polysize (Sub p q) = 1 + polysize p + polysize q"
```
```    28 | "polysize (Mul p q) = 1 + polysize p + polysize q"
```
```    29 | "polysize (Pw p n) = 1 + polysize p"
```
```    30 | "polysize (CN c n p) = 4 + polysize c + polysize p"
```
```    31
```
```    32 primrec polybound0:: "poly \<Rightarrow> bool" (* a poly is INDEPENDENT of Bound 0 *) where
```
```    33   "polybound0 (C c) = True"
```
```    34 | "polybound0 (Bound n) = (n>0)"
```
```    35 | "polybound0 (Neg a) = polybound0 a"
```
```    36 | "polybound0 (Add a b) = (polybound0 a \<and> polybound0 b)"
```
```    37 | "polybound0 (Sub a b) = (polybound0 a \<and> polybound0 b)"
```
```    38 | "polybound0 (Mul a b) = (polybound0 a \<and> polybound0 b)"
```
```    39 | "polybound0 (Pw p n) = (polybound0 p)"
```
```    40 | "polybound0 (CN c n p) = (n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p)"
```
```    41
```
```    42 primrec polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" (* substitute a poly into a poly for Bound 0 *) where
```
```    43   "polysubst0 t (C c) = (C c)"
```
```    44 | "polysubst0 t (Bound n) = (if n=0 then t else Bound n)"
```
```    45 | "polysubst0 t (Neg a) = Neg (polysubst0 t a)"
```
```    46 | "polysubst0 t (Add a b) = Add (polysubst0 t a) (polysubst0 t b)"
```
```    47 | "polysubst0 t (Sub a b) = Sub (polysubst0 t a) (polysubst0 t b)"
```
```    48 | "polysubst0 t (Mul a b) = Mul (polysubst0 t a) (polysubst0 t b)"
```
```    49 | "polysubst0 t (Pw p n) = Pw (polysubst0 t p) n"
```
```    50 | "polysubst0 t (CN c n p) = (if n=0 then Add (polysubst0 t c) (Mul t (polysubst0 t p))
```
```    51                              else CN (polysubst0 t c) n (polysubst0 t p))"
```
```    52
```
```    53 fun decrpoly:: "poly \<Rightarrow> poly"
```
```    54 where
```
```    55   "decrpoly (Bound n) = Bound (n - 1)"
```
```    56 | "decrpoly (Neg a) = Neg (decrpoly a)"
```
```    57 | "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"
```
```    58 | "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
```
```    59 | "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
```
```    60 | "decrpoly (Pw p n) = Pw (decrpoly p) n"
```
```    61 | "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
```
```    62 | "decrpoly a = a"
```
```    63
```
```    64 subsection{* Degrees and heads and coefficients *}
```
```    65
```
```    66 fun degree:: "poly \<Rightarrow> nat"
```
```    67 where
```
```    68   "degree (CN c 0 p) = 1 + degree p"
```
```    69 | "degree p = 0"
```
```    70
```
```    71 fun head:: "poly \<Rightarrow> poly"
```
```    72 where
```
```    73   "head (CN c 0 p) = head p"
```
```    74 | "head p = p"
```
```    75
```
```    76 (* More general notions of degree and head *)
```
```    77 fun degreen:: "poly \<Rightarrow> nat \<Rightarrow> nat"
```
```    78 where
```
```    79   "degreen (CN c n p) = (\<lambda>m. if n=m then 1 + degreen p n else 0)"
```
```    80  |"degreen p = (\<lambda>m. 0)"
```
```    81
```
```    82 fun headn:: "poly \<Rightarrow> nat \<Rightarrow> poly"
```
```    83 where
```
```    84   "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
```
```    85 | "headn p = (\<lambda>m. p)"
```
```    86
```
```    87 fun coefficients:: "poly \<Rightarrow> poly list"
```
```    88 where
```
```    89   "coefficients (CN c 0 p) = c#(coefficients p)"
```
```    90 | "coefficients p = [p]"
```
```    91
```
```    92 fun isconstant:: "poly \<Rightarrow> bool"
```
```    93 where
```
```    94   "isconstant (CN c 0 p) = False"
```
```    95 | "isconstant p = True"
```
```    96
```
```    97 fun behead:: "poly \<Rightarrow> poly"
```
```    98 where
```
```    99   "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
```
```   100 | "behead p = 0\<^sub>p"
```
```   101
```
```   102 fun headconst:: "poly \<Rightarrow> Num"
```
```   103 where
```
```   104   "headconst (CN c n p) = headconst p"
```
```   105 | "headconst (C n) = n"
```
```   106
```
```   107 subsection{* Operations for normalization *}
```
```   108 consts
```
```   109   polyadd :: "poly\<times>poly \<Rightarrow> poly"
```
```   110   polysub :: "poly\<times>poly \<Rightarrow> poly"
```
```   111   polymul :: "poly\<times>poly \<Rightarrow> poly"
```
```   112
```
```   113 abbreviation poly_add :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
```
```   114   where "a +\<^sub>p b \<equiv> polyadd (a,b)"
```
```   115 abbreviation poly_mul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
```
```   116   where "a *\<^sub>p b \<equiv> polymul (a,b)"
```
```   117 abbreviation poly_sub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
```
```   118   where "a -\<^sub>p b \<equiv> polysub (a,b)"
```
```   119
```
```   120 recdef polyadd "measure (\<lambda> (a,b). polysize a + polysize b)"
```
```   121   "polyadd (C c, C c') = C (c+\<^sub>Nc')"
```
```   122   "polyadd (C c, CN c' n' p') = CN (polyadd (C c, c')) n' p'"
```
```   123   "polyadd (CN c n p, C c') = CN (polyadd (c, C c')) n p"
```
```   124   "polyadd (CN c n p, CN c' n' p') =
```
```   125     (if n < n' then CN (polyadd(c,CN c' n' p')) n p
```
```   126      else if n'<n then CN (polyadd(CN c n p, c')) n' p'
```
```   127      else (let cc' = polyadd (c,c') ;
```
```   128                pp' = polyadd (p,p')
```
```   129            in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
```
```   130   "polyadd (a, b) = Add a b"
```
```   131 (hints recdef_simp add: Let_def measure_def split_def inv_image_def recdef_cong del: if_cong)
```
```   132
```
```   133 fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
```
```   134 where
```
```   135   "polyneg (C c) = C (~\<^sub>N c)"
```
```   136 | "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
```
```   137 | "polyneg a = Neg a"
```
```   138
```
```   139 defs polysub_def[code]: "polysub \<equiv> \<lambda> (p,q). polyadd (p,polyneg q)"
```
```   140
```
```   141 recdef polymul "measure (\<lambda>(a,b). size a + size b)"
```
```   142   "polymul(C c, C c') = C (c*\<^sub>Nc')"
```
```   143   "polymul(C c, CN c' n' p') =
```
```   144       (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul(C c,c')) n' (polymul(C c, p')))"
```
```   145   "polymul(CN c n p, C c') =
```
```   146       (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul(c,C c')) n (polymul(p, C c')))"
```
```   147   "polymul(CN c n p, CN c' n' p') =
```
```   148   (if n<n' then CN (polymul(c,CN c' n' p')) n (polymul(p,CN c' n' p'))
```
```   149   else if n' < n
```
```   150   then CN (polymul(CN c n p,c')) n' (polymul(CN c n p,p'))
```
```   151   else polyadd(polymul(CN c n p, c'),CN 0\<^sub>p n' (polymul(CN c n p, p'))))"
```
```   152   "polymul (a,b) = Mul a b"
```
```   153
```
```   154 fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
```
```   155 where
```
```   156   "polypow 0 = (\<lambda>p. 1\<^sub>p)"
```
```   157 | "polypow n = (\<lambda>p. let q = polypow (n div 2) p ; d = polymul(q,q) in
```
```   158                     if even n then d else polymul(p,d))"
```
```   159
```
```   160 abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
```
```   161   where "a ^\<^sub>p k \<equiv> polypow k a"
```
```   162
```
```   163 function polynate :: "poly \<Rightarrow> poly"
```
```   164 where
```
```   165   "polynate (Bound n) = CN 0\<^sub>p n 1\<^sub>p"
```
```   166 | "polynate (Add p q) = (polynate p +\<^sub>p polynate q)"
```
```   167 | "polynate (Sub p q) = (polynate p -\<^sub>p polynate q)"
```
```   168 | "polynate (Mul p q) = (polynate p *\<^sub>p polynate q)"
```
```   169 | "polynate (Neg p) = (~\<^sub>p (polynate p))"
```
```   170 | "polynate (Pw p n) = ((polynate p) ^\<^sub>p n)"
```
```   171 | "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
```
```   172 | "polynate (C c) = C (normNum c)"
```
```   173 by pat_completeness auto
```
```   174 termination by (relation "measure polysize") auto
```
```   175
```
```   176 fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly" where
```
```   177   "poly_cmul y (C x) = C (y *\<^sub>N x)"
```
```   178 | "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
```
```   179 | "poly_cmul y p = C y *\<^sub>p p"
```
```   180
```
```   181 definition monic :: "poly \<Rightarrow> (poly \<times> bool)" where
```
```   182   "monic p \<equiv> (let h = headconst p in if h = 0\<^sub>N then (p,False) else ((C (Ninv h)) *\<^sub>p p, 0>\<^sub>N h))"
```
```   183
```
```   184 subsection{* Pseudo-division *}
```
```   185
```
```   186 definition shift1 :: "poly \<Rightarrow> poly" where
```
```   187   "shift1 p \<equiv> CN 0\<^sub>p 0 p"
```
```   188
```
```   189 abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)" where
```
```   190   "funpow \<equiv> compow"
```
```   191
```
```   192 partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
```
```   193   where
```
```   194   "polydivide_aux a n p k s =
```
```   195   (if s = 0\<^sub>p then (k,s)
```
```   196   else (let b = head s; m = degree s in
```
```   197   (if m < n then (k,s) else
```
```   198   (let p'= funpow (m - n) shift1 p in
```
```   199   (if a = b then polydivide_aux a n p k (s -\<^sub>p p')
```
```   200   else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))))))"
```
```   201
```
```   202 definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)" where
```
```   203   "polydivide s p \<equiv> polydivide_aux (head p) (degree p) p 0 s"
```
```   204
```
```   205 fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly" where
```
```   206   "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
```
```   207 | "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
```
```   208
```
```   209 fun poly_deriv :: "poly \<Rightarrow> poly" where
```
```   210   "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
```
```   211 | "poly_deriv p = 0\<^sub>p"
```
```   212
```
```   213   (* Verification *)
```
```   214 lemma nth_pos2[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
```
```   215 using Nat.gr0_conv_Suc
```
```   216 by clarsimp
```
```   217
```
```   218 subsection{* Semantics of the polynomial representation *}
```
```   219
```
```   220 primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0, field_inverse_zero, power}" where
```
```   221   "Ipoly bs (C c) = INum c"
```
```   222 | "Ipoly bs (Bound n) = bs!n"
```
```   223 | "Ipoly bs (Neg a) = - Ipoly bs a"
```
```   224 | "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
```
```   225 | "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
```
```   226 | "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
```
```   227 | "Ipoly bs (Pw t n) = (Ipoly bs t) ^ n"
```
```   228 | "Ipoly bs (CN c n p) = (Ipoly bs c) + (bs!n)*(Ipoly bs p)"
```
```   229
```
```   230 abbreviation
```
```   231   Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0, field_inverse_zero, power}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
```
```   232   where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"
```
```   233
```
```   234 lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i"
```
```   235   by (simp add: INum_def)
```
```   236 lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"
```
```   237   by (simp  add: INum_def)
```
```   238
```
```   239 lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
```
```   240
```
```   241 subsection {* Normal form and normalization *}
```
```   242
```
```   243 fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
```
```   244 where
```
```   245   "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
```
```   246 | "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))"
```
```   247 | "isnpolyh p = (\<lambda>k. False)"
```
```   248
```
```   249 lemma isnpolyh_mono: "\<lbrakk>n' \<le> n ; isnpolyh p n\<rbrakk> \<Longrightarrow> isnpolyh p n'"
```
```   250 by (induct p rule: isnpolyh.induct, auto)
```
```   251
```
```   252 definition isnpoly :: "poly \<Rightarrow> bool" where
```
```   253   "isnpoly p \<equiv> isnpolyh p 0"
```
```   254
```
```   255 text{* polyadd preserves normal forms *}
```
```   256
```
```   257 lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk>
```
```   258       \<Longrightarrow> isnpolyh (polyadd(p,q)) (min n0 n1)"
```
```   259 proof(induct p q arbitrary: n0 n1 rule: polyadd.induct)
```
```   260   case (2 a b c' n' p' n0 n1)
```
```   261   from prems have  th1: "isnpolyh (C (a,b)) (Suc n')" by simp
```
```   262   from prems(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
```
```   263   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
```
```   264   with prems(1)[OF th1 th2] have th3:"isnpolyh (C (a,b) +\<^sub>p c') (Suc n')" by simp
```
```   265   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
```
```   266   thus ?case using prems th3 by simp
```
```   267 next
```
```   268   case (3 c' n' p' a b n1 n0)
```
```   269   from prems have  th1: "isnpolyh (C (a,b)) (Suc n')" by simp
```
```   270   from prems(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
```
```   271   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
```
```   272   with prems(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C (a,b)) (Suc n')" by simp
```
```   273   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
```
```   274   thus ?case using prems th3 by simp
```
```   275 next
```
```   276   case (4 c n p c' n' p' n0 n1)
```
```   277   hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
```
```   278   from prems have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all
```
```   279   from prems have ngen0: "n \<ge> n0" by simp
```
```   280   from prems have n'gen1: "n' \<ge> n1" by simp
```
```   281   have "n < n' \<or> n' < n \<or> n = n'" by auto
```
```   282   moreover {assume eq: "n = n'"
```
```   283     with prems(2)[OF nc nc']
```
```   284     have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
```
```   285     hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
```
```   286       using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
```
```   287     from eq prems(1)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
```
```   288     have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
```
```   289     from minle npp' ncc'n01 prems ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}
```
```   290   moreover {assume lt: "n < n'"
```
```   291     have "min n0 n1 \<le> n0" by simp
```
```   292     with prems have th1:"min n0 n1 \<le> n" by auto
```
```   293     from prems have th21: "isnpolyh c (Suc n)" by simp
```
```   294     from prems have th22: "isnpolyh (CN c' n' p') n'" by simp
```
```   295     from lt have th23: "min (Suc n) n' = Suc n" by arith
```
```   296     from prems(4)[OF th21 th22]
```
```   297     have "isnpolyh (polyadd (c, CN c' n' p')) (Suc n)" using th23 by simp
```
```   298     with prems th1 have ?case by simp }
```
```   299   moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
```
```   300     have "min n0 n1 \<le> n1"  by simp
```
```   301     with prems have th1:"min n0 n1 \<le> n'" by auto
```
```   302     from prems have th21: "isnpolyh c' (Suc n')" by simp_all
```
```   303     from prems have th22: "isnpolyh (CN c n p) n" by simp
```
```   304     from gt have th23: "min n (Suc n') = Suc n'" by arith
```
```   305     from prems(3)[OF th22 th21]
```
```   306     have "isnpolyh (polyadd (CN c n p,c')) (Suc n')" using th23 by simp
```
```   307     with prems th1 have ?case by simp}
```
```   308       ultimately show ?case by blast
```
```   309 qed auto
```
```   310
```
```   311 lemma polyadd[simp]: "Ipoly bs (polyadd (p,q)) = (Ipoly bs p) + (Ipoly bs q)"
```
```   312 by (induct p q rule: polyadd.induct, auto simp add: Let_def field_simps right_distrib[symmetric] simp del: right_distrib)
```
```   313
```
```   314 lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd(p,q))"
```
```   315   using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   316
```
```   317 text{* The degree of addition and other general lemmas needed for the normal form of polymul *}
```
```   318
```
```   319 lemma polyadd_different_degreen:
```
```   320   "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow>
```
```   321   degreen (polyadd(p,q)) m = max (degreen p m) (degreen q m)"
```
```   322 proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
```
```   323   case (4 c n p c' n' p' m n0 n1)
```
```   324   have "n' = n \<or> n < n' \<or> n' < n" by arith
```
```   325   thus ?case
```
```   326   proof (elim disjE)
```
```   327     assume [simp]: "n' = n"
```
```   328     from 4(1)[of n n m] 4(2)[of "Suc n" "Suc n" m] 4(5-7)
```
```   329     show ?thesis by (auto simp: Let_def)
```
```   330   next
```
```   331     assume "n < n'"
```
```   332     with 4 show ?thesis by auto
```
```   333   next
```
```   334     assume "n' < n"
```
```   335     with 4 show ?thesis by auto
```
```   336   qed
```
```   337 qed auto
```
```   338
```
```   339 lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
```
```   340   by (induct p arbitrary: n rule: headn.induct, auto)
```
```   341 lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
```
```   342   by (induct p arbitrary: n rule: degree.induct, auto)
```
```   343 lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
```
```   344   by (induct p arbitrary: n rule: degreen.induct, auto)
```
```   345
```
```   346 lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
```
```   347   by (induct p arbitrary: n rule: degree.induct, auto)
```
```   348
```
```   349 lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
```
```   350   using degree_isnpolyh_Suc by auto
```
```   351 lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
```
```   352   using degreen_0 by auto
```
```   353
```
```   354
```
```   355 lemma degreen_polyadd:
```
```   356   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1"
```
```   357   shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
```
```   358   using np nq m
```
```   359 proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
```
```   360   case (2 c c' n' p' n0 n1) thus ?case  by (cases n', simp_all)
```
```   361 next
```
```   362   case (3 c n p c' n0 n1) thus ?case by (cases n, auto)
```
```   363 next
```
```   364   case (4 c n p c' n' p' n0 n1 m)
```
```   365   have "n' = n \<or> n < n' \<or> n' < n" by arith
```
```   366   thus ?case
```
```   367   proof (elim disjE)
```
```   368     assume [simp]: "n' = n"
```
```   369     from 4(1)[of n n m] 4(2)[of "Suc n" "Suc n" m] 4(5-7)
```
```   370     show ?thesis by (auto simp: Let_def)
```
```   371   qed simp_all
```
```   372 qed auto
```
```   373
```
```   374 lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk>
```
```   375   \<Longrightarrow> degreen p m = degreen q m"
```
```   376 proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
```
```   377   case (4 c n p c' n' p' m n0 n1 x)
```
```   378   {assume nn': "n' < n" hence ?case using prems by simp}
```
```   379   moreover
```
```   380   {assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
```
```   381     moreover {assume "n < n'" with prems have ?case by simp }
```
```   382     moreover {assume eq: "n = n'" hence ?case using prems
```
```   383         apply (cases "p +\<^sub>p p' = 0\<^sub>p")
```
```   384         apply (auto simp add: Let_def)
```
```   385         by blast
```
```   386       }
```
```   387     ultimately have ?case by blast}
```
```   388   ultimately show ?case by blast
```
```   389 qed simp_all
```
```   390
```
```   391 lemma polymul_properties:
```
```   392   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   393   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> min n0 n1"
```
```   394   shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   395   and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)"
```
```   396   and "degreen (p *\<^sub>p q) m = (if (p = 0\<^sub>p \<or> q = 0\<^sub>p) then 0
```
```   397                              else degreen p m + degreen q m)"
```
```   398   using np nq m
```
```   399 proof(induct p q arbitrary: n0 n1 m rule: polymul.induct)
```
```   400   case (2 a b c' n' p')
```
```   401   let ?c = "(a,b)"
```
```   402   { case (1 n0 n1)
```
```   403     hence n: "isnpolyh (C ?c) n'" "isnpolyh c' (Suc n')" "isnpolyh p' n'" "isnormNum ?c"
```
```   404       "isnpolyh (CN c' n' p') n1"
```
```   405       by simp_all
```
```   406     {assume "?c = 0\<^sub>N" hence ?case by auto}
```
```   407       moreover {assume cnz: "?c \<noteq> 0\<^sub>N"
```
```   408         from "2.hyps"(1)[rule_format,where xb="n'",  OF cnz n(1) n(3)]
```
```   409           "2.hyps"(2)[rule_format, where x="Suc n'"
```
```   410           and xa="Suc n'" and xb = "n'", OF cnz ] cnz n have ?case
```
```   411           by (auto simp add: min_def)}
```
```   412       ultimately show ?case by blast
```
```   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     hence n: "isnpolyh (C ?c') n" "isnpolyh c (Suc n)" "isnpolyh p n" "isnormNum ?c'"
```
```   422       "isnpolyh (CN c n p) n0"
```
```   423       by simp_all
```
```   424     {assume "?c' = 0\<^sub>N" hence ?case by auto}
```
```   425       moreover {assume cnz: "?c' \<noteq> 0\<^sub>N"
```
```   426         from "3.hyps"(1)[rule_format,where xb="n",  OF cnz n(3) n(1)]
```
```   427           "3.hyps"(2)[rule_format, where x="Suc n"
```
```   428           and xa="Suc n" and xb = "n", OF cnz ] cnz n have ?case
```
```   429           by (auto simp add: min_def)}
```
```   430       ultimately show ?case by blast
```
```   431   next
```
```   432     case (2 n0 n1) thus ?case apply auto done
```
```   433   next
```
```   434     case (3 n0 n1) thus ?case  using "3.hyps" by auto }
```
```   435 next
```
```   436   case (4 c n p c' n' p')
```
```   437   let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
```
```   438     {fix n0 n1
```
```   439       assume "isnpolyh ?cnp n0" and "isnpolyh ?cnp' n1"
```
```   440       hence cnp: "isnpolyh ?cnp n" and cnp': "isnpolyh ?cnp' n'"
```
```   441         and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)"
```
```   442         and np': "isnpolyh p' n'" and nc': "isnpolyh c' (Suc n')"
```
```   443         and nn0: "n \<ge> n0" and nn1:"n' \<ge> n1"
```
```   444         by simp_all
```
```   445       have "n < n' \<or> n' < n \<or> n' = n" by auto
```
```   446       moreover
```
```   447       {assume nn': "n < n'"
```
```   448         with "4.hyps"(5)[rule_format, OF nn' np cnp', where xb ="n"]
```
```   449           "4.hyps"(6)[rule_format, OF nn' nc cnp', where xb="n"] nn' nn0 nn1 cnp
```
```   450         have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
```
```   451           by (simp add: min_def) }
```
```   452       moreover
```
```   453
```
```   454       {assume nn': "n > n'" hence stupid: "n' < n \<and> \<not> n < n'" by arith
```
```   455         with "4.hyps"(3)[rule_format, OF stupid cnp np', where xb="n'"]
```
```   456           "4.hyps"(4)[rule_format, OF stupid cnp nc', where xb="Suc n'"]
```
```   457           nn' nn0 nn1 cnp'
```
```   458         have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
```
```   459           by (cases "Suc n' = n", simp_all add: min_def)}
```
```   460       moreover
```
```   461       {assume nn': "n' = n" hence stupid: "\<not> n' < n \<and> \<not> n < n'" by arith
```
```   462         from "4.hyps"(1)[rule_format, OF stupid cnp np', where xb="n"]
```
```   463           "4.hyps"(2)[rule_format, OF stupid cnp nc', where xb="n"] nn' cnp cnp' nn1
```
```   464
```
```   465         have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
```
```   466           by simp (rule polyadd_normh,simp_all add: min_def isnpolyh_mono[OF nn0]) }
```
```   467       ultimately show "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)" by blast }
```
```   468     note th = this
```
```   469     {fix n0 n1 m
```
```   470       assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1"
```
```   471       and m: "m \<le> min n0 n1"
```
```   472       let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
```
```   473       let ?d1 = "degreen ?cnp m"
```
```   474       let ?d2 = "degreen ?cnp' m"
```
```   475       let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
```
```   476       have "n'<n \<or> n < n' \<or> n' = n" by auto
```
```   477       moreover
```
```   478       {assume "n' < n \<or> n < n'"
```
```   479         with "4.hyps" np np' m
```
```   480         have ?eq apply (cases "n' < n", simp_all)
```
```   481         apply (erule allE[where x="n"],erule allE[where x="n"],auto)
```
```   482         done }
```
```   483       moreover
```
```   484       {assume nn': "n' = n"  hence nn:"\<not> n' < n \<and> \<not> n < n'" by arith
```
```   485         from "4.hyps"(1)[rule_format, OF nn, where x="n" and xa ="n'" and xb="n"]
```
```   486           "4.hyps"(2)[rule_format, OF nn, where x="n" and xa ="Suc n'" and xb="n"]
```
```   487           np np' nn'
```
```   488         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
```
```   489           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   490           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
```
```   491           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
```
```   492         {assume mn: "m = n"
```
```   493           from "4.hyps"(1)[rule_format, OF nn norm(1,4), where xb="n"]
```
```   494             "4.hyps"(2)[rule_format, OF nn norm(1,2), where xb="n"] norm nn' mn
```
```   495           have degs:  "degreen (?cnp *\<^sub>p c') n =
```
```   496             (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"
```
```   497             "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n" by (simp_all add: min_def)
```
```   498           from degs norm
```
```   499           have th1: "degreen(?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" by simp
```
```   500           hence neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   501             by simp
```
```   502           have nmin: "n \<le> min n n" by (simp add: min_def)
```
```   503           from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
```
```   504           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
```
```   505           from "4.hyps"(1)[rule_format, OF nn norm(1,4), where xb="n"]
```
```   506             "4.hyps"(2)[rule_format, OF nn norm(1,2), where xb="n"]
```
```   507             mn norm m nn' deg
```
```   508           have ?eq by simp}
```
```   509         moreover
```
```   510         {assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
```
```   511           from nn' m np have max1: "m \<le> max n n"  by simp
```
```   512           hence min1: "m \<le> min n n" by simp
```
```   513           hence min2: "m \<le> min n (Suc n)" by simp
```
```   514           {assume "c' = 0\<^sub>p"
```
```   515             from `c' = 0\<^sub>p` have ?eq
```
```   516               using "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
```
```   517             "4.hyps"(2)[rule_format, OF nn norm(1,2) min2] mn nn'
```
```   518               apply simp
```
```   519               done}
```
```   520           moreover
```
```   521           {assume cnz: "c' \<noteq> 0\<^sub>p"
```
```   522             from "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
```
```   523               "4.hyps"(2)[rule_format, OF nn norm(1,2) min2]
```
```   524               degreen_polyadd[OF norm(3,6) max1]
```
```   525
```
```   526             have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m
```
```   527               \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
```
```   528               using mn nn' cnz np np' by simp
```
```   529             with "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
```
```   530               "4.hyps"(2)[rule_format, OF nn norm(1,2) min2]
```
```   531               degreen_0[OF norm(3) mn'] have ?eq using nn' mn cnz np np' by clarsimp}
```
```   532           ultimately have ?eq by blast }
```
```   533         ultimately have ?eq by blast}
```
```   534       ultimately show ?eq by blast}
```
```   535     note degth = this
```
```   536     { case (2 n0 n1)
```
```   537       hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1"
```
```   538         and m: "m \<le> min n0 n1" by simp_all
```
```   539       hence mn: "m \<le> n" by simp
```
```   540       let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
```
```   541       {assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
```
```   542         hence nn: "\<not>n' < n \<and> \<not> n<n'" by simp
```
```   543         from "4.hyps"(1) [rule_format, OF nn, where x="n" and xa = "n" and xb="n"]
```
```   544           "4.hyps"(2) [rule_format, OF nn, where x="n" and xa = "Suc n" and xb="n"]
```
```   545           np np' C(2) mn
```
```   546         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
```
```   547           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   548           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
```
```   549           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
```
```   550           "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
```
```   551             "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
```
```   552           by (simp_all add: min_def)
```
```   553
```
```   554           from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```   555           have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   556             using norm by simp
```
```   557         from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"]  degneq
```
```   558         have "False" by simp }
```
```   559       thus ?case using "4.hyps" by clarsimp}
```
```   560 qed auto
```
```   561
```
```   562 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"
```
```   563 by(induct p q rule: polymul.induct, auto simp add: field_simps)
```
```   564
```
```   565 lemma polymul_normh:
```
```   566     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   567   shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   568   using polymul_properties(1)  by blast
```
```   569 lemma polymul_eq0_iff:
```
```   570   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   571   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) "
```
```   572   using polymul_properties(2)  by blast
```
```   573 lemma polymul_degreen:
```
```   574   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   575   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)"
```
```   576   using polymul_properties(3) by blast
```
```   577 lemma polymul_norm:
```
```   578   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   579   shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul (p,q))"
```
```   580   using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   581
```
```   582 lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
```
```   583   by (induct p arbitrary: n0 rule: headconst.induct, auto)
```
```   584
```
```   585 lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
```
```   586   by (induct p arbitrary: n0, auto)
```
```   587
```
```   588 lemma monic_eqI: assumes np: "isnpolyh p n0"
```
```   589   shows "INum (headconst p) * Ipoly bs (fst (monic p)) = (Ipoly bs p ::'a::{field_char_0, field_inverse_zero, power})"
```
```   590   unfolding monic_def Let_def
```
```   591 proof(cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
```
```   592   let ?h = "headconst p"
```
```   593   assume pz: "p \<noteq> 0\<^sub>p"
```
```   594   {assume hz: "INum ?h = (0::'a)"
```
```   595     from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
```
```   596     from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
```
```   597     with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
```
```   598   thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast
```
```   599 qed
```
```   600
```
```   601
```
```   602 text{* polyneg is a negation and preserves normal forms *}
```
```   603
```
```   604 lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
```
```   605 by (induct p rule: polyneg.induct, auto)
```
```   606
```
```   607 lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   608   by (induct p arbitrary: n rule: polyneg.induct, auto simp add: Nneg_def)
```
```   609 lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
```
```   610   by (induct p arbitrary: n0 rule: polyneg.induct, auto)
```
```   611 lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "
```
```   612 by (induct p rule: polyneg.induct, auto simp add: polyneg0)
```
```   613
```
```   614 lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
```
```   615   using isnpoly_def polyneg_normh by simp
```
```   616
```
```   617
```
```   618 text{* polysub is a substraction and preserves normal forms *}
```
```   619
```
```   620 lemma polysub[simp]: "Ipoly bs (polysub (p,q)) = (Ipoly bs p) - (Ipoly bs q)"
```
```   621 by (simp add: polysub_def polyneg polyadd)
```
```   622 lemma polysub_normh: "\<And> n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub(p,q)) (min n0 n1)"
```
```   623 by (simp add: polysub_def polyneg_normh polyadd_normh)
```
```   624
```
```   625 lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub(p,q))"
```
```   626   using polyadd_norm polyneg_norm by (simp add: polysub_def)
```
```   627 lemma polysub_same_0[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   628   shows "isnpolyh p n0 \<Longrightarrow> polysub (p, p) = 0\<^sub>p"
```
```   629 unfolding polysub_def split_def fst_conv snd_conv
```
```   630 by (induct p arbitrary: n0,auto simp add: Let_def Nsub0[simplified Nsub_def])
```
```   631
```
```   632 lemma polysub_0:
```
```   633   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   634   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"
```
```   635   unfolding polysub_def split_def fst_conv snd_conv
```
```   636   by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
```
```   637   (auto simp: Nsub0[simplified Nsub_def] Let_def)
```
```   638
```
```   639 text{* polypow is a power function and preserves normal forms *}
```
```   640
```
```   641 lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{field_char_0, field_inverse_zero})) ^ n"
```
```   642 proof(induct n rule: polypow.induct)
```
```   643   case 1 thus ?case by simp
```
```   644 next
```
```   645   case (2 n)
```
```   646   let ?q = "polypow ((Suc n) div 2) p"
```
```   647   let ?d = "polymul(?q,?q)"
```
```   648   have "odd (Suc n) \<or> even (Suc n)" by simp
```
```   649   moreover
```
```   650   {assume odd: "odd (Suc n)"
```
```   651     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
```
```   652     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul(p, ?d))" by (simp add: Let_def)
```
```   653     also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"
```
```   654       using "2.hyps" by simp
```
```   655     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
```
```   656       apply (simp only: power_add power_one_right) by simp
```
```   657     also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat)))))"
```
```   658       by (simp only: th)
```
```   659     finally have ?case
```
```   660     using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
```
```   661   moreover
```
```   662   {assume even: "even (Suc n)"
```
```   663     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
```
```   664     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
```
```   665     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
```
```   666       using "2.hyps" apply (simp only: power_add) by simp
```
```   667     finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}
```
```   668   ultimately show ?case by blast
```
```   669 qed
```
```   670
```
```   671 lemma polypow_normh:
```
```   672     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   673   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
```
```   674 proof (induct k arbitrary: n rule: polypow.induct)
```
```   675   case (2 k n)
```
```   676   let ?q = "polypow (Suc k div 2) p"
```
```   677   let ?d = "polymul (?q,?q)"
```
```   678   from prems have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
```
```   679   from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
```
```   680   from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul(p,?d)) n" by simp
```
```   681   from dn on show ?case by (simp add: Let_def)
```
```   682 qed auto
```
```   683
```
```   684 lemma polypow_norm:
```
```   685   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   686   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
```
```   687   by (simp add: polypow_normh isnpoly_def)
```
```   688
```
```   689 text{* Finally the whole normalization *}
```
```   690
```
```   691 lemma polynate[simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0, field_inverse_zero})"
```
```   692 by (induct p rule:polynate.induct, auto)
```
```   693
```
```   694 lemma polynate_norm[simp]:
```
```   695   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   696   shows "isnpoly (polynate p)"
```
```   697   by (induct p rule: polynate.induct, simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm) (simp_all add: isnpoly_def)
```
```   698
```
```   699 text{* shift1 *}
```
```   700
```
```   701
```
```   702 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
```
```   703 by (simp add: shift1_def polymul)
```
```   704
```
```   705 lemma shift1_isnpoly:
```
```   706   assumes pn: "isnpoly p" and pnz: "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p) "
```
```   707   using pn pnz by (simp add: shift1_def isnpoly_def )
```
```   708
```
```   709 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
```
```   710   by (simp add: shift1_def)
```
```   711 lemma funpow_shift1_isnpoly:
```
```   712   "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
```
```   713   by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
```
```   714
```
```   715 lemma funpow_isnpolyh:
```
```   716   assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n "and np: "isnpolyh p n"
```
```   717   shows "isnpolyh (funpow k f p) n"
```
```   718   using f np by (induct k arbitrary: p, auto)
```
```   719
```
```   720 lemma funpow_shift1: "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (Mul (Pw (Bound 0) n) p)"
```
```   721   by (induct n arbitrary: p, simp_all add: shift1_isnpoly shift1 power_Suc )
```
```   722
```
```   723 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
```
```   724   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
```
```   725
```
```   726 lemma funpow_shift1_1:
```
```   727   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (funpow n shift1 1\<^sub>p *\<^sub>p p)"
```
```   728   by (simp add: funpow_shift1)
```
```   729
```
```   730 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
```
```   731 by (induct p  arbitrary: n0 rule: poly_cmul.induct, auto simp add: field_simps)
```
```   732
```
```   733 lemma behead:
```
```   734   assumes np: "isnpolyh p n"
```
```   735   shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = (Ipoly bs p :: 'a :: {field_char_0, field_inverse_zero})"
```
```   736   using np
```
```   737 proof (induct p arbitrary: n rule: behead.induct)
```
```   738   case (1 c p n) hence pn: "isnpolyh p n" by simp
```
```   739   from prems(2)[OF pn]
```
```   740   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
```
```   741   then show ?case using "1.hyps" apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
```
```   742     by (simp_all add: th[symmetric] field_simps power_Suc)
```
```   743 qed (auto simp add: Let_def)
```
```   744
```
```   745 lemma behead_isnpolyh:
```
```   746   assumes np: "isnpolyh p n" shows "isnpolyh (behead p) n"
```
```   747   using np by (induct p rule: behead.induct, auto simp add: Let_def isnpolyh_mono)
```
```   748
```
```   749 subsection{* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
```
```   750 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
```
```   751 proof(induct p arbitrary: n rule: poly.induct, auto)
```
```   752   case (goal1 c n p n')
```
```   753   hence "n = Suc (n - 1)" by simp
```
```   754   hence "isnpolyh p (Suc (n - 1))"  using `isnpolyh p n` by simp
```
```   755   with prems(2) show ?case by simp
```
```   756 qed
```
```   757
```
```   758 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
```
```   759 by (induct p arbitrary: n0 rule: isconstant.induct, auto simp add: isnpolyh_polybound0)
```
```   760
```
```   761 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" by (induct p, auto)
```
```   762
```
```   763 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
```
```   764   apply (induct p arbitrary: n0, auto)
```
```   765   apply (atomize)
```
```   766   apply (erule_tac x = "Suc nat" in allE)
```
```   767   apply auto
```
```   768   done
```
```   769
```
```   770 lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
```
```   771  by (induct p  arbitrary: n0 rule: head.induct, auto intro: isnpolyh_polybound0)
```
```   772
```
```   773 lemma polybound0_I:
```
```   774   assumes nb: "polybound0 a"
```
```   775   shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
```
```   776 using nb
```
```   777 by (induct a rule: poly.induct) auto
```
```   778 lemma polysubst0_I:
```
```   779   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
```
```   780   by (induct t) simp_all
```
```   781
```
```   782 lemma polysubst0_I':
```
```   783   assumes nb: "polybound0 a"
```
```   784   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t"
```
```   785   by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
```
```   786
```
```   787 lemma decrpoly: assumes nb: "polybound0 t"
```
```   788   shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
```
```   789   using nb by (induct t rule: decrpoly.induct, simp_all)
```
```   790
```
```   791 lemma polysubst0_polybound0: assumes nb: "polybound0 t"
```
```   792   shows "polybound0 (polysubst0 t a)"
```
```   793 using nb by (induct a rule: poly.induct, auto)
```
```   794
```
```   795 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
```
```   796   by (induct p arbitrary: n rule: degree.induct, auto simp add: isnpolyh_polybound0)
```
```   797
```
```   798 primrec maxindex :: "poly \<Rightarrow> nat" where
```
```   799   "maxindex (Bound n) = n + 1"
```
```   800 | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
```
```   801 | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
```
```   802 | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
```
```   803 | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
```
```   804 | "maxindex (Neg p) = maxindex p"
```
```   805 | "maxindex (Pw p n) = maxindex p"
```
```   806 | "maxindex (C x) = 0"
```
```   807
```
```   808 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool" where
```
```   809   "wf_bs bs p = (length bs \<ge> maxindex p)"
```
```   810
```
```   811 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
```
```   812 proof(induct p rule: coefficients.induct)
```
```   813   case (1 c p)
```
```   814   show ?case
```
```   815   proof
```
```   816     fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"
```
```   817     hence "x = c \<or> x \<in> set (coefficients p)" by simp
```
```   818     moreover
```
```   819     {assume "x = c" hence "wf_bs bs x" using "1.prems"  unfolding wf_bs_def by simp}
```
```   820     moreover
```
```   821     {assume H: "x \<in> set (coefficients p)"
```
```   822       from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp
```
```   823       with "1.hyps" H have "wf_bs bs x" by blast }
```
```   824     ultimately  show "wf_bs bs x" by blast
```
```   825   qed
```
```   826 qed simp_all
```
```   827
```
```   828 lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
```
```   829 by (induct p rule: coefficients.induct, auto)
```
```   830
```
```   831 lemma wf_bs_I: "wf_bs bs p ==> Ipoly (bs@bs') p = Ipoly bs p"
```
```   832   unfolding wf_bs_def by (induct p, auto simp add: nth_append)
```
```   833
```
```   834 lemma take_maxindex_wf: assumes wf: "wf_bs bs p"
```
```   835   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
```
```   836 proof-
```
```   837   let ?ip = "maxindex p"
```
```   838   let ?tbs = "take ?ip bs"
```
```   839   from wf have "length ?tbs = ?ip" unfolding wf_bs_def by simp
```
```   840   hence wf': "wf_bs ?tbs p" unfolding wf_bs_def by  simp
```
```   841   have eq: "bs = ?tbs @ (drop ?ip bs)" by simp
```
```   842   from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis using eq by simp
```
```   843 qed
```
```   844
```
```   845 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
```
```   846   by (induct p, auto)
```
```   847
```
```   848 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
```
```   849   unfolding wf_bs_def by simp
```
```   850
```
```   851 lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
```
```   852   unfolding wf_bs_def by simp
```
```   853
```
```   854
```
```   855
```
```   856 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
```
```   857 by(induct p rule: coefficients.induct, auto simp add: wf_bs_def)
```
```   858 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
```
```   859   by (induct p rule: coefficients.induct, simp_all)
```
```   860
```
```   861
```
```   862 lemma coefficients_head: "last (coefficients p) = head p"
```
```   863   by (induct p rule: coefficients.induct, auto)
```
```   864
```
```   865 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
```
```   866   unfolding wf_bs_def by (induct p rule: decrpoly.induct, auto)
```
```   867
```
```   868 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
```
```   869   apply (rule exI[where x="replicate (n - length xs) z"])
```
```   870   by simp
```
```   871 lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
```
```   872 by (cases p, auto) (case_tac "nat", simp_all)
```
```   873
```
```   874 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
```
```   875   unfolding wf_bs_def
```
```   876   apply (induct p q rule: polyadd.induct)
```
```   877   apply (auto simp add: Let_def)
```
```   878   done
```
```   879
```
```   880 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
```
```   881
```
```   882  unfolding wf_bs_def
```
```   883   apply (induct p q arbitrary: bs rule: polymul.induct)
```
```   884   apply (simp_all add: wf_bs_polyadd)
```
```   885   apply clarsimp
```
```   886   apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
```
```   887   apply auto
```
```   888   done
```
```   889
```
```   890 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
```
```   891   unfolding wf_bs_def by (induct p rule: polyneg.induct, auto)
```
```   892
```
```   893 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
```
```   894   unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
```
```   895
```
```   896 subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}
```
```   897
```
```   898 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
```
```   899 definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)"
```
```   900 definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"
```
```   901
```
```   902 lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0"
```
```   903 proof (induct p arbitrary: n0 rule: coefficients.induct)
```
```   904   case (1 c p n0)
```
```   905   have cp: "isnpolyh (CN c 0 p) n0" by fact
```
```   906   hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
```
```   907     by (auto simp add: isnpolyh_mono[where n'=0])
```
```   908   from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp
```
```   909 qed auto
```
```   910
```
```   911 lemma coefficients_isconst:
```
```   912   "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
```
```   913   by (induct p arbitrary: n rule: coefficients.induct,
```
```   914     auto simp add: isnpolyh_Suc_const)
```
```   915
```
```   916 lemma polypoly_polypoly':
```
```   917   assumes np: "isnpolyh p n0"
```
```   918   shows "polypoly (x#bs) p = polypoly' bs p"
```
```   919 proof-
```
```   920   let ?cf = "set (coefficients p)"
```
```   921   from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
```
```   922   {fix q assume q: "q \<in> ?cf"
```
```   923     from q cn_norm have th: "isnpolyh q n0" by blast
```
```   924     from coefficients_isconst[OF np] q have "isconstant q" by blast
```
```   925     with isconstant_polybound0[OF th] have "polybound0 q" by blast}
```
```   926   hence "\<forall>q \<in> ?cf. polybound0 q" ..
```
```   927   hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"
```
```   928     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
```
```   929     by auto
```
```   930
```
```   931   thus ?thesis unfolding polypoly_def polypoly'_def by simp
```
```   932 qed
```
```   933
```
```   934 lemma polypoly_poly:
```
```   935   assumes np: "isnpolyh p n0" shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
```
```   936   using np
```
```   937 by (induct p arbitrary: n0 bs rule: coefficients.induct, auto simp add: polypoly_def)
```
```   938
```
```   939 lemma polypoly'_poly:
```
```   940   assumes np: "isnpolyh p n0" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
```
```   941   using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
```
```   942
```
```   943
```
```   944 lemma polypoly_poly_polybound0:
```
```   945   assumes np: "isnpolyh p n0" and nb: "polybound0 p"
```
```   946   shows "polypoly bs p = [Ipoly bs p]"
```
```   947   using np nb unfolding polypoly_def
```
```   948   by (cases p, auto, case_tac nat, auto)
```
```   949
```
```   950 lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
```
```   951   by (induct p rule: head.induct, auto)
```
```   952
```
```   953 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   954   by (cases p,auto)
```
```   955
```
```   956 lemma head_eq_headn0: "head p = headn p 0"
```
```   957   by (induct p rule: head.induct, simp_all)
```
```   958
```
```   959 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   960   by (simp add: head_eq_headn0)
```
```   961
```
```   962 lemma isnpolyh_zero_iff:
```
```   963   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})"
```
```   964   shows "p = 0\<^sub>p"
```
```   965 using nq eq
```
```   966 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
```
```   967   case less
```
```   968   note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
```
```   969   {assume nz: "maxindex p = 0"
```
```   970     then obtain c where "p = C c" using np by (cases p, auto)
```
```   971     with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}
```
```   972   moreover
```
```   973   {assume nz: "maxindex p \<noteq> 0"
```
```   974     let ?h = "head p"
```
```   975     let ?hd = "decrpoly ?h"
```
```   976     let ?ihd = "maxindex ?hd"
```
```   977     from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h"
```
```   978       by simp_all
```
```   979     hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast
```
```   980
```
```   981     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
```
```   982     have mihn: "maxindex ?h \<le> maxindex p" by auto
```
```   983     with decr_maxindex[OF h(2)] nz  have ihd_lt_n: "?ihd < maxindex p" by auto
```
```   984     {fix bs:: "'a list"  assume bs: "wf_bs bs ?hd"
```
```   985       let ?ts = "take ?ihd bs"
```
```   986       let ?rs = "drop ?ihd bs"
```
```   987       have ts: "wf_bs ?ts ?hd" using bs unfolding wf_bs_def by simp
```
```   988       have bs_ts_eq: "?ts@ ?rs = bs" by simp
```
```   989       from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x#?ts) ?h" by simp
```
```   990       from ihd_lt_n have "ALL x. length (x#?ts) \<le> maxindex p" by simp
```
```   991       with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = maxindex p" by blast
```
```   992       hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" unfolding wf_bs_def by simp
```
```   993       with zp have "\<forall> x. Ipoly ((x#?ts) @ xs) p = 0" by blast
```
```   994       hence "\<forall> x. Ipoly (x#(?ts @ xs)) p = 0" by simp
```
```   995       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
```
```   996       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"  by simp
```
```   997       hence "poly (polypoly' (?ts @ xs) p) = poly []" by (auto intro: ext)
```
```   998       hence "\<forall> c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
```
```   999         using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
```
```  1000       with coefficients_head[of p, symmetric]
```
```  1001       have th0: "Ipoly (?ts @ xs) ?hd = 0" by simp
```
```  1002       from bs have wf'': "wf_bs ?ts ?hd" unfolding wf_bs_def by simp
```
```  1003       with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0" by simp
```
```  1004       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by simp }
```
```  1005     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast
```
```  1006
```
```  1007     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast
```
```  1008     hence "?h = 0\<^sub>p" by simp
```
```  1009     with head_nz[OF np] have "p = 0\<^sub>p" by simp}
```
```  1010   ultimately show "p = 0\<^sub>p" by blast
```
```  1011 qed
```
```  1012
```
```  1013 lemma isnpolyh_unique:
```
```  1014   assumes np:"isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1015   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"
```
```  1016 proof(auto)
```
```  1017   assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a)= \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
```
```  1018   hence "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" by simp
```
```  1019   hence "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
```
```  1020     using wf_bs_polysub[where p=p and q=q] by auto
```
```  1021   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq]
```
```  1022   show "p = q" by blast
```
```  1023 qed
```
```  1024
```
```  1025
```
```  1026 text{* consequences of unicity on the algorithms for polynomial normalization *}
```
```  1027
```
```  1028 lemma polyadd_commute:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1029   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" shows "p +\<^sub>p q = q +\<^sub>p p"
```
```  1030   using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]] by simp
```
```  1031
```
```  1032 lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp
```
```  1033 lemma one_normh: "isnpolyh 1\<^sub>p n" by simp
```
```  1034 lemma polyadd_0[simp]:
```
```  1035   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1036   and np: "isnpolyh p n0" shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>p p = p"
```
```  1037   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
```
```  1038     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
```
```  1039
```
```  1040 lemma polymul_1[simp]:
```
```  1041     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1042   and np: "isnpolyh p n0" shows "p *\<^sub>p 1\<^sub>p = p" and "1\<^sub>p *\<^sub>p p = p"
```
```  1043   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
```
```  1044     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
```
```  1045 lemma polymul_0[simp]:
```
```  1046   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1047   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"
```
```  1048   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
```
```  1049     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
```
```  1050
```
```  1051 lemma polymul_commute:
```
```  1052     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1053   and np:"isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1054   shows "p *\<^sub>p q = q *\<^sub>p p"
```
```  1055 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
```
```  1056
```
```  1057 declare polyneg_polyneg[simp]
```
```  1058
```
```  1059 lemma isnpolyh_polynate_id[simp]:
```
```  1060   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1061   and np:"isnpolyh p n0" shows "polynate p = p"
```
```  1062   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
```
```  1063
```
```  1064 lemma polynate_idempotent[simp]:
```
```  1065     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1066   shows "polynate (polynate p) = polynate p"
```
```  1067   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
```
```  1068
```
```  1069 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
```
```  1070   unfolding poly_nate_def polypoly'_def ..
```
```  1071 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>)"
```
```  1072   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
```
```  1073   unfolding poly_nate_polypoly' by (auto intro: ext)
```
```  1074
```
```  1075 subsection{* heads, degrees and all that *}
```
```  1076 lemma degree_eq_degreen0: "degree p = degreen p 0"
```
```  1077   by (induct p rule: degree.induct, simp_all)
```
```  1078
```
```  1079 lemma degree_polyneg: assumes n: "isnpolyh p n"
```
```  1080   shows "degree (polyneg p) = degree p"
```
```  1081   using n
```
```  1082   by (induct p arbitrary: n rule: polyneg.induct, simp_all) (case_tac na, auto)
```
```  1083
```
```  1084 lemma degree_polyadd:
```
```  1085   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1086   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1087 using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
```
```  1088
```
```  1089
```
```  1090 lemma degree_polysub: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1091   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1092 proof-
```
```  1093   from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp
```
```  1094   from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])
```
```  1095 qed
```
```  1096
```
```  1097 lemma degree_polysub_samehead:
```
```  1098   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1099   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q"
```
```  1100   and d: "degree p = degree q"
```
```  1101   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
```
```  1102 unfolding polysub_def split_def fst_conv snd_conv
```
```  1103 using np nq h d
```
```  1104 proof(induct p q rule:polyadd.induct)
```
```  1105   case (1 a b a' b') thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def])
```
```  1106 next
```
```  1107   case (2 a b c' n' p')
```
```  1108   let ?c = "(a,b)"
```
```  1109   from prems have "degree (C ?c) = degree (CN c' n' p')" by simp
```
```  1110   hence nz:"n' > 0" by (cases n', auto)
```
```  1111   hence "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
```
```  1112   with prems show ?case by simp
```
```  1113 next
```
```  1114   case (3 c n p a' b')
```
```  1115   let ?c' = "(a',b')"
```
```  1116   from prems have "degree (C ?c') = degree (CN c n p)" by simp
```
```  1117   hence nz:"n > 0" by (cases n, auto)
```
```  1118   hence "head (CN c n p) = CN c n p" by (cases n, auto)
```
```  1119   with prems show ?case by simp
```
```  1120 next
```
```  1121   case (4 c n p c' n' p')
```
```  1122   hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1"
```
```  1123     "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+
```
```  1124   hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all
```
```  1125   hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
```
```  1126     using H(1-2) degree_polyneg by auto
```
```  1127   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"  by simp+
```
```  1128   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"  by simp
```
```  1129   from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto
```
```  1130   have "n = n' \<or> n < n' \<or> n > n'" by arith
```
```  1131   moreover
```
```  1132   {assume nn': "n = n'"
```
```  1133     have "n = 0 \<or> n >0" by arith
```
```  1134     moreover {assume nz: "n = 0" hence ?case using prems by (auto simp add: Let_def degcmc')}
```
```  1135     moreover {assume nz: "n > 0"
```
```  1136       with nn' H(3) have  cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
```
```  1137       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)}
```
```  1138     ultimately have ?case by blast}
```
```  1139   moreover
```
```  1140   {assume nn': "n < n'" hence n'p: "n' > 0" by simp
```
```  1141     hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n', simp_all)
```
```  1142     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)
```
```  1143     hence "n > 0" by (cases n, simp_all)
```
```  1144     hence headcnp: "head (CN c n p) = CN c n p" by (cases n, auto)
```
```  1145     from H(3) headcnp headcnp' nn' have ?case by auto}
```
```  1146   moreover
```
```  1147   {assume nn': "n > n'"  hence np: "n > 0" by simp
```
```  1148     hence headcnp:"head (CN c n p) = CN c n p"  by (cases n, simp_all)
```
```  1149     from prems have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
```
```  1150     from np have degcnp: "degree (CN c n p) = 0" by (cases n, simp_all)
```
```  1151     with degcnpeq have "n' > 0" by (cases n', simp_all)
```
```  1152     hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
```
```  1153     from H(3) headcnp headcnp' nn' have ?case by auto}
```
```  1154   ultimately show ?case  by blast
```
```  1155 qed auto
```
```  1156
```
```  1157 lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
```
```  1158 by (induct p arbitrary: n0 rule: head.induct, simp_all add: shift1_def)
```
```  1159
```
```  1160 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
```
```  1161 proof(induct k arbitrary: n0 p)
```
```  1162   case (Suc k n0 p) hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
```
```  1163   with prems have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
```
```  1164     and "head (shift1 p) = head p" by (simp_all add: shift1_head)
```
```  1165   thus ?case by (simp add: funpow_swap1)
```
```  1166 qed auto
```
```  1167
```
```  1168 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
```
```  1169   by (simp add: shift1_def)
```
```  1170 lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
```
```  1171   by (induct k arbitrary: p, auto simp add: shift1_degree)
```
```  1172
```
```  1173 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
```
```  1174   by (induct n arbitrary: p, simp_all add: funpow_def)
```
```  1175
```
```  1176 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
```
```  1177   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1178 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
```
```  1179   by (induct p arbitrary: n rule: degreen.induct, auto)
```
```  1180 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
```
```  1181   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1182 lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
```
```  1183   by (induct p rule: head.induct, auto)
```
```  1184
```
```  1185 lemma polyadd_eq_const_degree:
```
```  1186   "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk> \<Longrightarrow> degree p = degree q"
```
```  1187   using polyadd_eq_const_degreen degree_eq_degreen0 by simp
```
```  1188
```
```  1189 lemma polyadd_head: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1190   and deg: "degree p \<noteq> degree q"
```
```  1191   shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
```
```  1192 using np nq deg
```
```  1193 apply(induct p q arbitrary: n0 n1 rule: polyadd.induct,simp_all)
```
```  1194 apply (case_tac n', simp, simp)
```
```  1195 apply (case_tac n, simp, simp)
```
```  1196 apply (case_tac n, case_tac n', simp add: Let_def)
```
```  1197 apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p")
```
```  1198 apply (auto simp add: polyadd_eq_const_degree)
```
```  1199 apply (metis head_nz)
```
```  1200 apply (metis head_nz)
```
```  1201 apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
```
```  1202 by (metis degree.simps(9) gr0_conv_Suc nat_less_le order_le_less_trans)
```
```  1203
```
```  1204 lemma polymul_head_polyeq:
```
```  1205    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1206   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"
```
```  1207 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
```
```  1208   case (2 a b c' n' p' n0 n1)
```
```  1209   hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum (a,b)"  by (simp_all add: head_isnpolyh)
```
```  1210   thus ?case using prems by (cases n', auto)
```
```  1211 next
```
```  1212   case (3 c n p a' b' n0 n1)
```
```  1213   hence "isnpolyh (head (CN c n p)) n0" "isnormNum (a',b')"  by (simp_all add: head_isnpolyh)
```
```  1214   thus ?case using prems by (cases n, auto)
```
```  1215 next
```
```  1216   case (4 c n p c' n' p' n0 n1)
```
```  1217   hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
```
```  1218     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
```
```  1219     by simp_all
```
```  1220   have "n < n' \<or> n' < n \<or> n = n'" by arith
```
```  1221   moreover
```
```  1222   {assume nn': "n < n'" hence ?case
```
```  1223       thm prems
```
```  1224       using norm
```
```  1225     prems(6)[rule_format, OF nn' norm(1,6)]
```
```  1226     prems(7)[rule_format, OF nn' norm(2,6)] by (simp, cases n, simp,cases n', simp_all)}
```
```  1227   moreover {assume nn': "n'< n"
```
```  1228     hence stupid: "n' < n \<and> \<not> n < n'" by simp
```
```  1229     hence ?case using norm prems(4) [rule_format, OF stupid norm(5,3)]
```
```  1230       prems(5)[rule_format, OF stupid norm(5,4)]
```
```  1231       by (simp,cases n',simp,cases n,auto)}
```
```  1232   moreover {assume nn': "n' = n"
```
```  1233     hence stupid: "\<not> n' < n \<and> \<not> n < n'" by simp
```
```  1234     from nn' polymul_normh[OF norm(5,4)]
```
```  1235     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
```
```  1236     from nn' polymul_normh[OF norm(5,3)] norm
```
```  1237     have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
```
```  1238     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
```
```  1239     have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```  1240     from polyadd_normh[OF ncnpc' ncnpp0']
```
```  1241     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"
```
```  1242       by (simp add: min_def)
```
```  1243     {assume np: "n > 0"
```
```  1244       with nn' head_isnpolyh_Suc'[OF np nth]
```
```  1245         head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
```
```  1246       have ?case by simp}
```
```  1247     moreover
```
```  1248     {moreover assume nz: "n = 0"
```
```  1249       from polymul_degreen[OF norm(5,4), where m="0"]
```
```  1250         polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
```
```  1251       norm(5,6) degree_npolyhCN[OF norm(6)]
```
```  1252     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
```
```  1253     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
```
```  1254     from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
```
```  1255     have ?case   using norm prems(2)[rule_format, OF stupid norm(5,3)]
```
```  1256         prems(3)[rule_format, OF stupid norm(5,4)] nn' nz by simp }
```
```  1257     ultimately have ?case by (cases n) auto}
```
```  1258   ultimately show ?case by blast
```
```  1259 qed simp_all
```
```  1260
```
```  1261 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
```
```  1262   by(induct p rule: degree.induct, auto)
```
```  1263
```
```  1264 lemma degree_head[simp]: "degree (head p) = 0"
```
```  1265   by (induct p rule: head.induct, auto)
```
```  1266
```
```  1267 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1+ degree p"
```
```  1268   by (cases n, simp_all)
```
```  1269 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
```
```  1270   by (cases n, simp_all)
```
```  1271
```
```  1272 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)"
```
```  1273   using polyadd_different_degreen degree_eq_degreen0 by simp
```
```  1274
```
```  1275 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
```
```  1276   by (induct p arbitrary: n0 rule: polyneg.induct, auto)
```
```  1277
```
```  1278 lemma degree_polymul:
```
```  1279   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1280   and np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1281   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
```
```  1282   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
```
```  1283
```
```  1284 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
```
```  1285   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1286
```
```  1287 lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"
```
```  1288   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1289
```
```  1290 subsection {* Correctness of polynomial pseudo division *}
```
```  1291
```
```  1292 lemma polydivide_aux_properties:
```
```  1293   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1294   and np: "isnpolyh p n0" and ns: "isnpolyh s n1"
```
```  1295   and ap: "head p = a" and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
```
```  1296   shows "(polydivide_aux a n p k s = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1297           \<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)))"
```
```  1298   using ns
```
```  1299 proof(induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
```
```  1300   case less
```
```  1301   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
```
```  1302   let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1303     \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
```
```  1304   let ?b = "head s"
```
```  1305   let ?p' = "funpow (degree s - n) shift1 p"
```
```  1306   let ?xdn = "funpow (degree s - n) shift1 1\<^sub>p"
```
```  1307   let ?akk' = "a ^\<^sub>p (k' - k)"
```
```  1308   note ns = `isnpolyh s n1`
```
```  1309   from np have np0: "isnpolyh p 0"
```
```  1310     using isnpolyh_mono[where n="n0" and n'="0" and p="p"]  by simp
```
```  1311   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
```
```  1312   have headp': "head ?p' = head p" using funpow_shift1_head[OF np pnz] by simp
```
```  1313   from funpow_shift1_isnpoly[where p="1\<^sub>p"] have nxdn: "isnpolyh ?xdn 0" by (simp add: isnpoly_def)
```
```  1314   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
```
```  1315   have nakk':"isnpolyh ?akk' 0" by blast
```
```  1316   {assume sz: "s = 0\<^sub>p"
```
```  1317    hence ?ths using np polydivide_aux.simps apply clarsimp by (rule exI[where x="0\<^sub>p"], simp) }
```
```  1318   moreover
```
```  1319   {assume sz: "s \<noteq> 0\<^sub>p"
```
```  1320     {assume dn: "degree s < n"
```
```  1321       hence "?ths" using ns ndp np polydivide_aux.simps by auto (rule exI[where x="0\<^sub>p"],simp) }
```
```  1322     moreover
```
```  1323     {assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith
```
```  1324       have degsp': "degree s = degree ?p'"
```
```  1325         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp
```
```  1326       {assume ba: "?b = a"
```
```  1327         hence headsp': "head s = head ?p'" using ap headp' by simp
```
```  1328         have nr: "isnpolyh (s -\<^sub>p ?p') 0" using polysub_normh[OF ns np'] by simp
```
```  1329         from degree_polysub_samehead[OF ns np' headsp' degsp']
```
```  1330         have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
```
```  1331         moreover
```
```  1332         {assume deglt:"degree (s -\<^sub>p ?p') < degree s"
```
```  1333           from polydivide_aux.simps sz dn' ba
```
```  1334           have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
```
```  1335             by (simp add: Let_def)
```
```  1336           {assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1337             from less(1)[OF deglt nr, of k k' r]
```
```  1338               trans[OF eq[symmetric] h1]
```
```  1339             have kk': "k \<le> k'" and nr:"\<exists>nr. isnpolyh r nr" and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1340               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
```
```  1341             from q1 obtain q n1 where nq: "isnpolyh q n1"
```
```  1342               and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"  by blast
```
```  1343             from nr obtain nr where nr': "isnpolyh r nr" by blast
```
```  1344             from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0" by simp
```
```  1345             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
```
```  1346             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
```
```  1347             from polyadd_normh[OF polymul_normh[OF np
```
```  1348               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
```
```  1349             have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0" by simp
```
```  1350             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')) =
```
```  1351               Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
```
```  1352             hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
```
```  1353               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1354               by (simp add: field_simps)
```
```  1355             hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1356               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p *\<^sub>p p)
```
```  1357               + Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1358               by (auto simp only: funpow_shift1_1)
```
```  1359             hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1360               Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p)
```
```  1361               + Ipoly bs q) + Ipoly bs r" by (simp add: field_simps)
```
```  1362             hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1363               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
```
```  1364             with isnpolyh_unique[OF nakks' nqr']
```
```  1365             have "a ^\<^sub>p (k' - k) *\<^sub>p s =
```
```  1366               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
```
```  1367             hence ?qths using nq'
```
```  1368               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 1\<^sub>p) +\<^sub>p q" in exI)
```
```  1369               apply (rule_tac x="0" in exI) by simp
```
```  1370             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"
```
```  1371               by blast } hence ?ths by blast }
```
```  1372         moreover
```
```  1373         {assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
```
```  1374           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}"]
```
```  1375           have " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs s = Ipoly bs ?p'" by simp
```
```  1376           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
```
```  1377             by (simp only: funpow_shift1_1) simp
```
```  1378           hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]] by blast
```
```  1379           {assume h1: "polydivide_aux a n p k s = (k',r)"
```
```  1380             from polydivide_aux.simps sz dn' ba
```
```  1381             have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
```
```  1382               by (simp add: Let_def)
```
```  1383             also have "\<dots> = (k,0\<^sub>p)" using polydivide_aux.simps spz by simp
```
```  1384             finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
```
```  1385             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
```
```  1386               polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
```
```  1387               apply auto
```
```  1388               apply (rule exI[where x="?xdn"])
```
```  1389               apply (auto simp add: polymul_commute[of p])
```
```  1390               done} }
```
```  1391         ultimately have ?ths by blast }
```
```  1392       moreover
```
```  1393       {assume ba: "?b \<noteq> a"
```
```  1394         from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1395           polymul_normh[OF head_isnpolyh[OF ns] np']]
```
```  1396         have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by(simp add: min_def)
```
```  1397         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
```
```  1398           using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1399             polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
```
```  1400             funpow_shift1_nz[OF pnz] by simp_all
```
```  1401         from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
```
```  1402           polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1403         have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
```
```  1404           using head_head[OF ns] funpow_shift1_head[OF np pnz]
```
```  1405             polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
```
```  1406           by (simp add: ap)
```
```  1407         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
```
```  1408           head_nz[OF np] pnz sz ap[symmetric]
```
```  1409           funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1410           polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"]  head_nz[OF ns]
```
```  1411           ndp dn
```
```  1412         have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p') "
```
```  1413           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
```
```  1414         {assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
```
```  1415           from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns] polymul_normh[OF head_isnpolyh[OF ns]np']]
```
```  1416           ap have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by simp
```
```  1417           {assume h1:"polydivide_aux a n p k s = (k', r)"
```
```  1418             from h1 polydivide_aux.simps sz dn' ba
```
```  1419             have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
```
```  1420               by (simp add: Let_def)
```
```  1421             with less(1)[OF dth nasbp', of "Suc k" k' r]
```
```  1422             obtain q nq nr where kk': "Suc k \<le> k'" and nr: "isnpolyh r nr" and nq: "isnpolyh q nq"
```
```  1423               and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1424               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
```
```  1425             from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith
```
```  1426             {fix bs:: "'a::{field_char_0, field_inverse_zero} list"
```
```  1427
```
```  1428             from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
```
```  1429             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
```
```  1430             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"
```
```  1431               by (simp add: field_simps power_Suc)
```
```  1432             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"
```
```  1433               by (simp add:kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1434             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"
```
```  1435               by (simp add: field_simps)}
```
```  1436             hence ieq:"\<forall>(bs :: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1437               Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)" by auto
```
```  1438             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
```
```  1439             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]]
```
```  1440             have nqw: "isnpolyh ?q 0" by simp
```
```  1441             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]]
```
```  1442             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
```
```  1443             from dr kk' nr h1 asth nqw have ?ths apply simp
```
```  1444               apply (rule conjI)
```
```  1445               apply (rule exI[where x="nr"], simp)
```
```  1446               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
```
```  1447               apply (rule exI[where x="0"], simp)
```
```  1448               done}
```
```  1449           hence ?ths by blast }
```
```  1450         moreover
```
```  1451         {assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
```
```  1452           {fix bs :: "'a::{field_char_0, field_inverse_zero} list"
```
```  1453             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
```
```  1454           have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'" by simp
```
```  1455           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
```
```  1456             by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1457           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" by simp
```
```  1458         }
```
```  1459         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))" ..
```
```  1460           from hth
```
```  1461           have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
```
```  1462             using isnpolyh_unique[where ?'a = "'a::{field_char_0, field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
```
```  1463                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
```
```  1464               simplified ap] by simp
```
```  1465           {assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1466           from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
```
```  1467           have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
```
```  1468           with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
```
```  1469             polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
```
```  1470           have ?ths apply (clarsimp simp add: Let_def)
```
```  1471             apply (rule exI[where x="?b *\<^sub>p ?xdn"]) apply simp
```
```  1472             apply (rule exI[where x="0"], simp)
```
```  1473             done}
```
```  1474         hence ?ths by blast}
```
```  1475         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"]
```
```  1476           head_nz[OF np] pnz sz ap[symmetric]
```
```  1477           by (simp add: degree_eq_degreen0[symmetric]) blast }
```
```  1478       ultimately have ?ths by blast
```
```  1479     }
```
```  1480     ultimately have ?ths by blast}
```
```  1481   ultimately show ?ths by blast
```
```  1482 qed
```
```  1483
```
```  1484 lemma polydivide_properties:
```
```  1485   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1486   and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
```
```  1487   shows "(\<exists> k r. polydivide s p = (k,r) \<and> (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1488   \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
```
```  1489 proof-
```
```  1490   have trv: "head p = head p" "degree p = degree p" by simp_all
```
```  1491   from polydivide_def[where s="s" and p="p"]
```
```  1492   have ex: "\<exists> k r. polydivide s p = (k,r)" by auto
```
```  1493   then obtain k r where kr: "polydivide s p = (k,r)" by blast
```
```  1494   from trans[OF meta_eq_to_obj_eq[OF polydivide_def[where s="s"and p="p"], symmetric] kr]
```
```  1495     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
```
```  1496   have "(degree r = 0 \<or> degree r < degree p) \<and>
```
```  1497    (\<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
```
```  1498   with kr show ?thesis
```
```  1499     apply -
```
```  1500     apply (rule exI[where x="k"])
```
```  1501     apply (rule exI[where x="r"])
```
```  1502     apply simp
```
```  1503     done
```
```  1504 qed
```
```  1505
```
```  1506 subsection{* More about polypoly and pnormal etc *}
```
```  1507
```
```  1508 definition "isnonconstant p = (\<not> isconstant p)"
```
```  1509
```
```  1510 lemma isnonconstant_pnormal_iff: assumes nc: "isnonconstant p"
```
```  1511   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1512 proof
```
```  1513   let ?p = "polypoly bs p"
```
```  1514   assume H: "pnormal ?p"
```
```  1515   have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
```
```  1516
```
```  1517   from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1518     pnormal_last_nonzero[OF H]
```
```  1519   show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)
```
```  1520 next
```
```  1521   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1522   let ?p = "polypoly bs p"
```
```  1523   have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
```
```  1524   hence pz: "?p \<noteq> []" by (simp add: polypoly_def)
```
```  1525   hence lg: "length ?p > 0" by simp
```
```  1526   from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1527   have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)
```
```  1528   from pnormal_last_length[OF lg lz] show "pnormal ?p" .
```
```  1529 qed
```
```  1530
```
```  1531 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
```
```  1532   unfolding isnonconstant_def
```
```  1533   apply (cases p, simp_all)
```
```  1534   apply (case_tac nat, auto)
```
```  1535   done
```
```  1536 lemma isnonconstant_nonconstant: assumes inc: "isnonconstant p"
```
```  1537   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1538 proof
```
```  1539   let ?p = "polypoly bs p"
```
```  1540   assume nc: "nonconstant ?p"
```
```  1541   from isnonconstant_pnormal_iff[OF inc, of bs] nc
```
```  1542   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast
```
```  1543 next
```
```  1544   let ?p = "polypoly bs p"
```
```  1545   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1546   from isnonconstant_pnormal_iff[OF inc, of bs] h
```
```  1547   have pn: "pnormal ?p" by blast
```
```  1548   {fix x assume H: "?p = [x]"
```
```  1549     from H have "length (coefficients p) = 1" unfolding polypoly_def by auto
```
```  1550     with isnonconstant_coefficients_length[OF inc] have False by arith}
```
```  1551   thus "nonconstant ?p" using pn unfolding nonconstant_def by blast
```
```  1552 qed
```
```  1553
```
```  1554 lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
```
```  1555   unfolding pnormal_def
```
```  1556  apply (induct p)
```
```  1557  apply (simp_all, case_tac "p=[]", simp_all)
```
```  1558  done
```
```  1559
```
```  1560 lemma degree_degree: assumes inc: "isnonconstant p"
```
```  1561   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1562 proof
```
```  1563   let  ?p = "polypoly bs p"
```
```  1564   assume H: "degree p = Polynomial_List.degree ?p"
```
```  1565   from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
```
```  1566     unfolding polypoly_def by auto
```
```  1567   from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
```
```  1568   have lg:"length (pnormalize ?p) = length ?p"
```
```  1569     unfolding Polynomial_List.degree_def polypoly_def by simp
```
```  1570   hence "pnormal ?p" using pnormal_length[OF pz] by blast
```
```  1571   with isnonconstant_pnormal_iff[OF inc]
```
```  1572   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast
```
```  1573 next
```
```  1574   let  ?p = "polypoly bs p"
```
```  1575   assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1576   with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast
```
```  1577   with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
```
```  1578   show "degree p = Polynomial_List.degree ?p"
```
```  1579     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
```
```  1580 qed
```
```  1581
```
```  1582 section{* Swaps ; Division by a certain variable *}
```
```  1583 primrec swap:: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly" where
```
```  1584   "swap n m (C x) = C x"
```
```  1585 | "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"
```
```  1586 | "swap n m (Neg t) = Neg (swap n m t)"
```
```  1587 | "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
```
```  1588 | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
```
```  1589 | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
```
```  1590 | "swap n m (Pw t k) = Pw (swap n m t) k"
```
```  1591 | "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)
```
```  1592   (swap n m p)"
```
```  1593
```
```  1594 lemma swap: assumes nbs: "n < length bs" and mbs: "m < length bs"
```
```  1595   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  1596 proof (induct t)
```
```  1597   case (Bound k) thus ?case using nbs mbs by simp
```
```  1598 next
```
```  1599   case (CN c k p) thus ?case using nbs mbs by simp
```
```  1600 qed simp_all
```
```  1601 lemma swap_swap_id[simp]: "swap n m (swap m n t) = t"
```
```  1602   by (induct t,simp_all)
```
```  1603
```
```  1604 lemma swap_commute: "swap n m p = swap m n p" by (induct p, simp_all)
```
```  1605
```
```  1606 lemma swap_same_id[simp]: "swap n n t = t"
```
```  1607   by (induct t, simp_all)
```
```  1608
```
```  1609 definition "swapnorm n m t = polynate (swap n m t)"
```
```  1610
```
```  1611 lemma swapnorm: assumes nbs: "n < length bs" and mbs: "m < length bs"
```
```  1612   shows "((Ipoly bs (swapnorm n m t) :: 'a\<Colon>{field_char_0, field_inverse_zero})) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  1613   using swap[OF prems] swapnorm_def by simp
```
```  1614
```
```  1615 lemma swapnorm_isnpoly[simp]:
```
```  1616     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1617   shows "isnpoly (swapnorm n m p)"
```
```  1618   unfolding swapnorm_def by simp
```
```  1619
```
```  1620 definition "polydivideby n s p =
```
```  1621     (let ss = swapnorm 0 n s ; sp = swapnorm 0 n p ; h = head sp; (k,r) = polydivide ss sp
```
```  1622      in (k,swapnorm 0 n h,swapnorm 0 n r))"
```
```  1623
```
```  1624 lemma swap_nz [simp]: " (swap n m p = 0\<^sub>p) = (p = 0\<^sub>p)" by (induct p, simp_all)
```
```  1625
```
```  1626 fun isweaknpoly :: "poly \<Rightarrow> bool"
```
```  1627 where
```
```  1628   "isweaknpoly (C c) = True"
```
```  1629 | "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
```
```  1630 | "isweaknpoly p = False"
```
```  1631
```
```  1632 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
```
```  1633   by (induct p arbitrary: n0, auto)
```
```  1634
```
```  1635 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
```
```  1636   by (induct p, auto)
```
```  1637
```
`  1638 end`