src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
 author krauss Mon Feb 14 15:27:23 2011 +0100 (2011-02-14) changeset 41763 8ce56536fda7 parent 41413 64cd30d6b0b8 child 41807 ab5d2d81f9fb child 41808 9f436d00248f permissions -rw-r--r--
strengthened induction rule;
clarified some proofs
```     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 consts
```
```    54   decrpoly:: "poly \<Rightarrow> poly"
```
```    55 recdef decrpoly "measure polysize"
```
```    56   "decrpoly (Bound n) = Bound (n - 1)"
```
```    57   "decrpoly (Neg a) = Neg (decrpoly a)"
```
```    58   "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"
```
```    59   "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
```
```    60   "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
```
```    61   "decrpoly (Pw p n) = Pw (decrpoly p) n"
```
```    62   "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
```
```    63   "decrpoly a = a"
```
```    64
```
```    65 subsection{* Degrees and heads and coefficients *}
```
```    66
```
```    67 consts degree:: "poly \<Rightarrow> nat"
```
```    68 recdef degree "measure size"
```
```    69   "degree (CN c 0 p) = 1 + degree p"
```
```    70   "degree p = 0"
```
```    71 consts head:: "poly \<Rightarrow> poly"
```
```    72
```
```    73 recdef head "measure size"
```
```    74   "head (CN c 0 p) = head p"
```
```    75   "head p = p"
```
```    76   (* More general notions of degree and head *)
```
```    77 consts degreen:: "poly \<Rightarrow> nat \<Rightarrow> nat"
```
```    78 recdef degreen "measure size"
```
```    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 consts headn:: "poly \<Rightarrow> nat \<Rightarrow> poly"
```
```    83 recdef headn "measure size"
```
```    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 consts coefficients:: "poly \<Rightarrow> poly list"
```
```    88 recdef coefficients "measure size"
```
```    89   "coefficients (CN c 0 p) = c#(coefficients p)"
```
```    90   "coefficients p = [p]"
```
```    91
```
```    92 consts isconstant:: "poly \<Rightarrow> bool"
```
```    93 recdef isconstant "measure size"
```
```    94   "isconstant (CN c 0 p) = False"
```
```    95   "isconstant p = True"
```
```    96
```
```    97 consts behead:: "poly \<Rightarrow> poly"
```
```    98 recdef behead "measure size"
```
```    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 consts headconst:: "poly \<Rightarrow> Num"
```
```   103 recdef headconst "measure size"
```
```   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   polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
```
```   111   polysub :: "poly\<times>poly \<Rightarrow> poly"
```
```   112   polymul :: "poly\<times>poly \<Rightarrow> poly"
```
```   113   polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
```
```   114 abbreviation poly_add :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
```
```   115   where "a +\<^sub>p b \<equiv> polyadd (a,b)"
```
```   116 abbreviation poly_mul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
```
```   117   where "a *\<^sub>p b \<equiv> polymul (a,b)"
```
```   118 abbreviation poly_sub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
```
```   119   where "a -\<^sub>p b \<equiv> polysub (a,b)"
```
```   120 abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
```
```   121   where "a ^\<^sub>p k \<equiv> polypow k a"
```
```   122
```
```   123 recdef polyadd "measure (\<lambda> (a,b). polysize a + polysize b)"
```
```   124   "polyadd (C c, C c') = C (c+\<^sub>Nc')"
```
```   125   "polyadd (C c, CN c' n' p') = CN (polyadd (C c, c')) n' p'"
```
```   126   "polyadd (CN c n p, C c') = CN (polyadd (c, C c')) n p"
```
```   127 stupid:  "polyadd (CN c n p, CN c' n' p') =
```
```   128     (if n < n' then CN (polyadd(c,CN c' n' p')) n p
```
```   129      else if n'<n then CN (polyadd(CN c n p, c')) n' p'
```
```   130      else (let cc' = polyadd (c,c') ;
```
```   131                pp' = polyadd (p,p')
```
```   132            in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
```
```   133   "polyadd (a, b) = Add a b"
```
```   134 (hints recdef_simp add: Let_def measure_def split_def inv_image_def recdef_cong del: if_cong)
```
```   135
```
```   136 recdef polyneg "measure size"
```
```   137   "polyneg (C c) = C (~\<^sub>N c)"
```
```   138   "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
```
```   139   "polyneg a = Neg a"
```
```   140
```
```   141 defs polysub_def[code]: "polysub \<equiv> \<lambda> (p,q). polyadd (p,polyneg q)"
```
```   142
```
```   143 recdef polymul "measure (\<lambda>(a,b). size a + size b)"
```
```   144   "polymul(C c, C c') = C (c*\<^sub>Nc')"
```
```   145   "polymul(C c, CN c' n' p') =
```
```   146       (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul(C c,c')) n' (polymul(C c, p')))"
```
```   147   "polymul(CN c n p, C c') =
```
```   148       (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul(c,C c')) n (polymul(p, C c')))"
```
```   149   "polymul(CN c n p, CN c' n' p') =
```
```   150   (if n<n' then CN (polymul(c,CN c' n' p')) n (polymul(p,CN c' n' p'))
```
```   151   else if n' < n
```
```   152   then CN (polymul(CN c n p,c')) n' (polymul(CN c n p,p'))
```
```   153   else polyadd(polymul(CN c n p, c'),CN 0\<^sub>p n' (polymul(CN c n p, p'))))"
```
```   154   "polymul (a,b) = Mul a b"
```
```   155 recdef polypow "measure id"
```
```   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 consts polynate :: "poly \<Rightarrow> poly"
```
```   161 recdef polynate "measure polysize"
```
```   162   "polynate (Bound n) = CN 0\<^sub>p n 1\<^sub>p"
```
```   163   "polynate (Add p q) = (polynate p +\<^sub>p polynate q)"
```
```   164   "polynate (Sub p q) = (polynate p -\<^sub>p polynate q)"
```
```   165   "polynate (Mul p q) = (polynate p *\<^sub>p polynate q)"
```
```   166   "polynate (Neg p) = (~\<^sub>p (polynate p))"
```
```   167   "polynate (Pw p n) = ((polynate p) ^\<^sub>p n)"
```
```   168   "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
```
```   169   "polynate (C c) = C (normNum c)"
```
```   170
```
```   171 fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly" where
```
```   172   "poly_cmul y (C x) = C (y *\<^sub>N x)"
```
```   173 | "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
```
```   174 | "poly_cmul y p = C y *\<^sub>p p"
```
```   175
```
```   176 definition monic :: "poly \<Rightarrow> (poly \<times> bool)" where
```
```   177   "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))"
```
```   178
```
```   179 subsection{* Pseudo-division *}
```
```   180
```
```   181 definition shift1 :: "poly \<Rightarrow> poly" where
```
```   182   "shift1 p \<equiv> CN 0\<^sub>p 0 p"
```
```   183
```
```   184 abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)" where
```
```   185   "funpow \<equiv> compow"
```
```   186
```
```   187 partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
```
```   188   where
```
```   189   "polydivide_aux a n p k s =
```
```   190   (if s = 0\<^sub>p then (k,s)
```
```   191   else (let b = head s; m = degree s in
```
```   192   (if m < n then (k,s) else
```
```   193   (let p'= funpow (m - n) shift1 p in
```
```   194   (if a = b then polydivide_aux a n p k (s -\<^sub>p p')
```
```   195   else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))))))"
```
```   196
```
```   197 definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)" where
```
```   198   "polydivide s p \<equiv> polydivide_aux (head p) (degree p) p 0 s"
```
```   199
```
```   200 fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly" where
```
```   201   "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
```
```   202 | "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
```
```   203
```
```   204 fun poly_deriv :: "poly \<Rightarrow> poly" where
```
```   205   "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
```
```   206 | "poly_deriv p = 0\<^sub>p"
```
```   207
```
```   208   (* Verification *)
```
```   209 lemma nth_pos2[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
```
```   210 using Nat.gr0_conv_Suc
```
```   211 by clarsimp
```
```   212
```
```   213 subsection{* Semantics of the polynomial representation *}
```
```   214
```
```   215 primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0, field_inverse_zero, power}" where
```
```   216   "Ipoly bs (C c) = INum c"
```
```   217 | "Ipoly bs (Bound n) = bs!n"
```
```   218 | "Ipoly bs (Neg a) = - Ipoly bs a"
```
```   219 | "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
```
```   220 | "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
```
```   221 | "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
```
```   222 | "Ipoly bs (Pw t n) = (Ipoly bs t) ^ n"
```
```   223 | "Ipoly bs (CN c n p) = (Ipoly bs c) + (bs!n)*(Ipoly bs p)"
```
```   224
```
```   225 abbreviation
```
```   226   Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0, field_inverse_zero, power}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
```
```   227   where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"
```
```   228
```
```   229 lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i"
```
```   230   by (simp add: INum_def)
```
```   231 lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"
```
```   232   by (simp  add: INum_def)
```
```   233
```
```   234 lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
```
```   235
```
```   236 subsection {* Normal form and normalization *}
```
```   237
```
```   238 consts isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
```
```   239 recdef isnpolyh "measure size"
```
```   240   "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
```
```   241   "isnpolyh (CN c n p) = (\<lambda>k. n\<ge> k \<and> (isnpolyh c (Suc n)) \<and> (isnpolyh p n) \<and> (p \<noteq> 0\<^sub>p))"
```
```   242   "isnpolyh p = (\<lambda>k. False)"
```
```   243
```
```   244 lemma isnpolyh_mono: "\<lbrakk>n' \<le> n ; isnpolyh p n\<rbrakk> \<Longrightarrow> isnpolyh p n'"
```
```   245 by (induct p rule: isnpolyh.induct, auto)
```
```   246
```
```   247 definition isnpoly :: "poly \<Rightarrow> bool" where
```
```   248   "isnpoly p \<equiv> isnpolyh p 0"
```
```   249
```
```   250 text{* polyadd preserves normal forms *}
```
```   251
```
```   252 lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk>
```
```   253       \<Longrightarrow> isnpolyh (polyadd(p,q)) (min n0 n1)"
```
```   254 proof(induct p q arbitrary: n0 n1 rule: polyadd.induct)
```
```   255   case (2 a b c' n' p' n0 n1)
```
```   256   from prems have  th1: "isnpolyh (C (a,b)) (Suc n')" by simp
```
```   257   from prems(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
```
```   258   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
```
```   259   with prems(1)[OF th1 th2] have th3:"isnpolyh (C (a,b) +\<^sub>p c') (Suc n')" by simp
```
```   260   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
```
```   261   thus ?case using prems th3 by simp
```
```   262 next
```
```   263   case (3 c' n' p' a b n1 n0)
```
```   264   from prems have  th1: "isnpolyh (C (a,b)) (Suc n')" by simp
```
```   265   from prems(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
```
```   266   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
```
```   267   with prems(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C (a,b)) (Suc n')" by simp
```
```   268   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
```
```   269   thus ?case using prems th3 by simp
```
```   270 next
```
```   271   case (4 c n p c' n' p' n0 n1)
```
```   272   hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
```
```   273   from prems have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all
```
```   274   from prems have ngen0: "n \<ge> n0" by simp
```
```   275   from prems have n'gen1: "n' \<ge> n1" by simp
```
```   276   have "n < n' \<or> n' < n \<or> n = n'" by auto
```
```   277   moreover {assume eq: "n = n'"
```
```   278     with prems(2)[OF nc nc']
```
```   279     have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
```
```   280     hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
```
```   281       using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
```
```   282     from eq prems(1)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
```
```   283     have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
```
```   284     from minle npp' ncc'n01 prems ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}
```
```   285   moreover {assume lt: "n < n'"
```
```   286     have "min n0 n1 \<le> n0" by simp
```
```   287     with prems have th1:"min n0 n1 \<le> n" by auto
```
```   288     from prems have th21: "isnpolyh c (Suc n)" by simp
```
```   289     from prems have th22: "isnpolyh (CN c' n' p') n'" by simp
```
```   290     from lt have th23: "min (Suc n) n' = Suc n" by arith
```
```   291     from prems(4)[OF th21 th22]
```
```   292     have "isnpolyh (polyadd (c, CN c' n' p')) (Suc n)" using th23 by simp
```
```   293     with prems th1 have ?case by simp }
```
```   294   moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
```
```   295     have "min n0 n1 \<le> n1"  by simp
```
```   296     with prems have th1:"min n0 n1 \<le> n'" by auto
```
```   297     from prems have th21: "isnpolyh c' (Suc n')" by simp_all
```
```   298     from prems have th22: "isnpolyh (CN c n p) n" by simp
```
```   299     from gt have th23: "min n (Suc n') = Suc n'" by arith
```
```   300     from prems(3)[OF th22 th21]
```
```   301     have "isnpolyh (polyadd (CN c n p,c')) (Suc n')" using th23 by simp
```
```   302     with prems th1 have ?case by simp}
```
```   303       ultimately show ?case by blast
```
```   304 qed auto
```
```   305
```
```   306 lemma polyadd[simp]: "Ipoly bs (polyadd (p,q)) = (Ipoly bs p) + (Ipoly bs q)"
```
```   307 by (induct p q rule: polyadd.induct, auto simp add: Let_def field_simps right_distrib[symmetric] simp del: right_distrib)
```
```   308
```
```   309 lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd(p,q))"
```
```   310   using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   311
```
```   312 text{* The degree of addition and other general lemmas needed for the normal form of polymul *}
```
```   313
```
```   314 lemma polyadd_different_degreen:
```
```   315   "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow>
```
```   316   degreen (polyadd(p,q)) m = max (degreen p m) (degreen q m)"
```
```   317 proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
```
```   318   case (4 c n p c' n' p' m n0 n1)
```
```   319   have "n' = n \<or> n < n' \<or> n' < n" by arith
```
```   320   thus ?case
```
```   321   proof (elim disjE)
```
```   322     assume [simp]: "n' = n"
```
```   323     from 4(1)[of n n m] 4(2)[of "Suc n" "Suc n" m] 4(5-7)
```
```   324     show ?thesis by (auto simp: Let_def)
```
```   325   next
```
```   326     assume "n < n'"
```
```   327     with 4 show ?thesis by auto
```
```   328   next
```
```   329     assume "n' < n"
```
```   330     with 4 show ?thesis by auto
```
```   331   qed
```
```   332 qed auto
```
```   333
```
```   334 lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
```
```   335   by (induct p arbitrary: n rule: headn.induct, auto)
```
```   336 lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
```
```   337   by (induct p arbitrary: n rule: degree.induct, auto)
```
```   338 lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
```
```   339   by (induct p arbitrary: n rule: degreen.induct, auto)
```
```   340
```
```   341 lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
```
```   342   by (induct p arbitrary: n rule: degree.induct, auto)
```
```   343
```
```   344 lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
```
```   345   using degree_isnpolyh_Suc by auto
```
```   346 lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
```
```   347   using degreen_0 by auto
```
```   348
```
```   349
```
```   350 lemma degreen_polyadd:
```
```   351   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1"
```
```   352   shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
```
```   353   using np nq m
```
```   354 proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
```
```   355   case (2 c c' n' p' n0 n1) thus ?case  by (cases n', simp_all)
```
```   356 next
```
```   357   case (3 c n p c' n0 n1) thus ?case by (cases n, auto)
```
```   358 next
```
```   359   case (4 c n p c' n' p' n0 n1 m)
```
```   360   have "n' = n \<or> n < n' \<or> n' < n" by arith
```
```   361   thus ?case
```
```   362   proof (elim disjE)
```
```   363     assume [simp]: "n' = n"
```
```   364     from 4(1)[of n n m] 4(2)[of "Suc n" "Suc n" m] 4(5-7)
```
```   365     show ?thesis by (auto simp: Let_def)
```
```   366   qed simp_all
```
```   367 qed auto
```
```   368
```
```   369 lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk>
```
```   370   \<Longrightarrow> degreen p m = degreen q m"
```
```   371 proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
```
```   372   case (4 c n p c' n' p' m n0 n1 x)
```
```   373   {assume nn': "n' < n" hence ?case using prems by simp}
```
```   374   moreover
```
```   375   {assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
```
```   376     moreover {assume "n < n'" with prems have ?case by simp }
```
```   377     moreover {assume eq: "n = n'" hence ?case using prems
```
```   378         apply (cases "p +\<^sub>p p' = 0\<^sub>p")
```
```   379         apply (auto simp add: Let_def)
```
```   380         by blast
```
```   381       }
```
```   382     ultimately have ?case by blast}
```
```   383   ultimately show ?case by blast
```
```   384 qed simp_all
```
```   385
```
```   386 lemma polymul_properties:
```
```   387   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   388   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> min n0 n1"
```
```   389   shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   390   and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)"
```
```   391   and "degreen (p *\<^sub>p q) m = (if (p = 0\<^sub>p \<or> q = 0\<^sub>p) then 0
```
```   392                              else degreen p m + degreen q m)"
```
```   393   using np nq m
```
```   394 proof(induct p q arbitrary: n0 n1 m rule: polymul.induct)
```
```   395   case (2 a b c' n' p')
```
```   396   let ?c = "(a,b)"
```
```   397   { case (1 n0 n1)
```
```   398     hence n: "isnpolyh (C ?c) n'" "isnpolyh c' (Suc n')" "isnpolyh p' n'" "isnormNum ?c"
```
```   399       "isnpolyh (CN c' n' p') n1"
```
```   400       by simp_all
```
```   401     {assume "?c = 0\<^sub>N" hence ?case by auto}
```
```   402       moreover {assume cnz: "?c \<noteq> 0\<^sub>N"
```
```   403         from "2.hyps"(1)[rule_format,where xb="n'",  OF cnz n(1) n(3)]
```
```   404           "2.hyps"(2)[rule_format, where x="Suc n'"
```
```   405           and xa="Suc n'" and xb = "n'", OF cnz ] cnz n have ?case
```
```   406           by (auto simp add: min_def)}
```
```   407       ultimately show ?case by blast
```
```   408   next
```
```   409     case (2 n0 n1) thus ?case by auto
```
```   410   next
```
```   411     case (3 n0 n1) thus ?case  using "2.hyps" by auto }
```
```   412 next
```
```   413   case (3 c n p a b){
```
```   414     let ?c' = "(a,b)"
```
```   415     case (1 n0 n1)
```
```   416     hence n: "isnpolyh (C ?c') n" "isnpolyh c (Suc n)" "isnpolyh p n" "isnormNum ?c'"
```
```   417       "isnpolyh (CN c n p) n0"
```
```   418       by simp_all
```
```   419     {assume "?c' = 0\<^sub>N" hence ?case by auto}
```
```   420       moreover {assume cnz: "?c' \<noteq> 0\<^sub>N"
```
```   421         from "3.hyps"(1)[rule_format,where xb="n",  OF cnz n(3) n(1)]
```
```   422           "3.hyps"(2)[rule_format, where x="Suc n"
```
```   423           and xa="Suc n" and xb = "n", OF cnz ] cnz n have ?case
```
```   424           by (auto simp add: min_def)}
```
```   425       ultimately show ?case by blast
```
```   426   next
```
```   427     case (2 n0 n1) thus ?case apply auto done
```
```   428   next
```
```   429     case (3 n0 n1) thus ?case  using "3.hyps" by auto }
```
```   430 next
```
```   431   case (4 c n p c' n' p')
```
```   432   let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
```
```   433     {fix n0 n1
```
```   434       assume "isnpolyh ?cnp n0" and "isnpolyh ?cnp' n1"
```
```   435       hence cnp: "isnpolyh ?cnp n" and cnp': "isnpolyh ?cnp' n'"
```
```   436         and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)"
```
```   437         and np': "isnpolyh p' n'" and nc': "isnpolyh c' (Suc n')"
```
```   438         and nn0: "n \<ge> n0" and nn1:"n' \<ge> n1"
```
```   439         by simp_all
```
```   440       have "n < n' \<or> n' < n \<or> n' = n" by auto
```
```   441       moreover
```
```   442       {assume nn': "n < n'"
```
```   443         with "4.hyps"(5)[rule_format, OF nn' np cnp', where xb ="n"]
```
```   444           "4.hyps"(6)[rule_format, OF nn' nc cnp', where xb="n"] nn' nn0 nn1 cnp
```
```   445         have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
```
```   446           by (simp add: min_def) }
```
```   447       moreover
```
```   448
```
```   449       {assume nn': "n > n'" hence stupid: "n' < n \<and> \<not> n < n'" by arith
```
```   450         with "4.hyps"(3)[rule_format, OF stupid cnp np', where xb="n'"]
```
```   451           "4.hyps"(4)[rule_format, OF stupid cnp nc', where xb="Suc n'"]
```
```   452           nn' nn0 nn1 cnp'
```
```   453         have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
```
```   454           by (cases "Suc n' = n", simp_all add: min_def)}
```
```   455       moreover
```
```   456       {assume nn': "n' = n" hence stupid: "\<not> n' < n \<and> \<not> n < n'" by arith
```
```   457         from "4.hyps"(1)[rule_format, OF stupid cnp np', where xb="n"]
```
```   458           "4.hyps"(2)[rule_format, OF stupid cnp nc', where xb="n"] nn' cnp cnp' nn1
```
```   459
```
```   460         have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
```
```   461           by simp (rule polyadd_normh,simp_all add: min_def isnpolyh_mono[OF nn0]) }
```
```   462       ultimately show "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)" by blast }
```
```   463     note th = this
```
```   464     {fix n0 n1 m
```
```   465       assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1"
```
```   466       and m: "m \<le> min n0 n1"
```
```   467       let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
```
```   468       let ?d1 = "degreen ?cnp m"
```
```   469       let ?d2 = "degreen ?cnp' m"
```
```   470       let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
```
```   471       have "n'<n \<or> n < n' \<or> n' = n" by auto
```
```   472       moreover
```
```   473       {assume "n' < n \<or> n < n'"
```
```   474         with "4.hyps" np np' m
```
```   475         have ?eq apply (cases "n' < n", simp_all)
```
```   476         apply (erule allE[where x="n"],erule allE[where x="n"],auto)
```
```   477         done }
```
```   478       moreover
```
```   479       {assume nn': "n' = n"  hence nn:"\<not> n' < n \<and> \<not> n < n'" by arith
```
```   480         from "4.hyps"(1)[rule_format, OF nn, where x="n" and xa ="n'" and xb="n"]
```
```   481           "4.hyps"(2)[rule_format, OF nn, where x="n" and xa ="Suc n'" and xb="n"]
```
```   482           np np' nn'
```
```   483         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
```
```   484           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   485           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
```
```   486           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
```
```   487         {assume mn: "m = n"
```
```   488           from "4.hyps"(1)[rule_format, OF nn norm(1,4), where xb="n"]
```
```   489             "4.hyps"(2)[rule_format, OF nn norm(1,2), where xb="n"] norm nn' mn
```
```   490           have degs:  "degreen (?cnp *\<^sub>p c') n =
```
```   491             (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"
```
```   492             "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n" by (simp_all add: min_def)
```
```   493           from degs norm
```
```   494           have th1: "degreen(?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" by simp
```
```   495           hence neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   496             by simp
```
```   497           have nmin: "n \<le> min n n" by (simp add: min_def)
```
```   498           from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
```
```   499           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
```
```   500           from "4.hyps"(1)[rule_format, OF nn norm(1,4), where xb="n"]
```
```   501             "4.hyps"(2)[rule_format, OF nn norm(1,2), where xb="n"]
```
```   502             mn norm m nn' deg
```
```   503           have ?eq by simp}
```
```   504         moreover
```
```   505         {assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
```
```   506           from nn' m np have max1: "m \<le> max n n"  by simp
```
```   507           hence min1: "m \<le> min n n" by simp
```
```   508           hence min2: "m \<le> min n (Suc n)" by simp
```
```   509           {assume "c' = 0\<^sub>p"
```
```   510             from `c' = 0\<^sub>p` have ?eq
```
```   511               using "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
```
```   512             "4.hyps"(2)[rule_format, OF nn norm(1,2) min2] mn nn'
```
```   513               apply simp
```
```   514               done}
```
```   515           moreover
```
```   516           {assume cnz: "c' \<noteq> 0\<^sub>p"
```
```   517             from "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
```
```   518               "4.hyps"(2)[rule_format, OF nn norm(1,2) min2]
```
```   519               degreen_polyadd[OF norm(3,6) max1]
```
```   520
```
```   521             have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m
```
```   522               \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
```
```   523               using mn nn' cnz np np' by simp
```
```   524             with "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
```
```   525               "4.hyps"(2)[rule_format, OF nn norm(1,2) min2]
```
```   526               degreen_0[OF norm(3) mn'] have ?eq using nn' mn cnz np np' by clarsimp}
```
```   527           ultimately have ?eq by blast }
```
```   528         ultimately have ?eq by blast}
```
```   529       ultimately show ?eq by blast}
```
```   530     note degth = this
```
```   531     { case (2 n0 n1)
```
```   532       hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1"
```
```   533         and m: "m \<le> min n0 n1" by simp_all
```
```   534       hence mn: "m \<le> n" by simp
```
```   535       let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
```
```   536       {assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
```
```   537         hence nn: "\<not>n' < n \<and> \<not> n<n'" by simp
```
```   538         from "4.hyps"(1) [rule_format, OF nn, where x="n" and xa = "n" and xb="n"]
```
```   539           "4.hyps"(2) [rule_format, OF nn, where x="n" and xa = "Suc n" and xb="n"]
```
```   540           np np' C(2) mn
```
```   541         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
```
```   542           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   543           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
```
```   544           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
```
```   545           "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
```
```   546             "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
```
```   547           by (simp_all add: min_def)
```
```   548
```
```   549           from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```   550           have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   551             using norm by simp
```
```   552         from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"]  degneq
```
```   553         have "False" by simp }
```
```   554       thus ?case using "4.hyps" by clarsimp}
```
```   555 qed auto
```
```   556
```
```   557 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"
```
```   558 by(induct p q rule: polymul.induct, auto simp add: field_simps)
```
```   559
```
```   560 lemma polymul_normh:
```
```   561     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   562   shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   563   using polymul_properties(1)  by blast
```
```   564 lemma polymul_eq0_iff:
```
```   565   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   566   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) "
```
```   567   using polymul_properties(2)  by blast
```
```   568 lemma polymul_degreen:
```
```   569   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   570   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)"
```
```   571   using polymul_properties(3) by blast
```
```   572 lemma polymul_norm:
```
```   573   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   574   shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul (p,q))"
```
```   575   using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   576
```
```   577 lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
```
```   578   by (induct p arbitrary: n0 rule: headconst.induct, auto)
```
```   579
```
```   580 lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
```
```   581   by (induct p arbitrary: n0, auto)
```
```   582
```
```   583 lemma monic_eqI: assumes np: "isnpolyh p n0"
```
```   584   shows "INum (headconst p) * Ipoly bs (fst (monic p)) = (Ipoly bs p ::'a::{field_char_0, field_inverse_zero, power})"
```
```   585   unfolding monic_def Let_def
```
```   586 proof(cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
```
```   587   let ?h = "headconst p"
```
```   588   assume pz: "p \<noteq> 0\<^sub>p"
```
```   589   {assume hz: "INum ?h = (0::'a)"
```
```   590     from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
```
```   591     from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
```
```   592     with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
```
```   593   thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast
```
```   594 qed
```
```   595
```
```   596
```
```   597 text{* polyneg is a negation and preserves normal forms *}
```
```   598
```
```   599 lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
```
```   600 by (induct p rule: polyneg.induct, auto)
```
```   601
```
```   602 lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   603   by (induct p arbitrary: n rule: polyneg.induct, auto simp add: Nneg_def)
```
```   604 lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
```
```   605   by (induct p arbitrary: n0 rule: polyneg.induct, auto)
```
```   606 lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "
```
```   607 by (induct p rule: polyneg.induct, auto simp add: polyneg0)
```
```   608
```
```   609 lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
```
```   610   using isnpoly_def polyneg_normh by simp
```
```   611
```
```   612
```
```   613 text{* polysub is a substraction and preserves normal forms *}
```
```   614
```
```   615 lemma polysub[simp]: "Ipoly bs (polysub (p,q)) = (Ipoly bs p) - (Ipoly bs q)"
```
```   616 by (simp add: polysub_def polyneg polyadd)
```
```   617 lemma polysub_normh: "\<And> n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub(p,q)) (min n0 n1)"
```
```   618 by (simp add: polysub_def polyneg_normh polyadd_normh)
```
```   619
```
```   620 lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub(p,q))"
```
```   621   using polyadd_norm polyneg_norm by (simp add: polysub_def)
```
```   622 lemma polysub_same_0[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   623   shows "isnpolyh p n0 \<Longrightarrow> polysub (p, p) = 0\<^sub>p"
```
```   624 unfolding polysub_def split_def fst_conv snd_conv
```
```   625 by (induct p arbitrary: n0,auto simp add: Let_def Nsub0[simplified Nsub_def])
```
```   626
```
```   627 lemma polysub_0:
```
```   628   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   629   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"
```
```   630   unfolding polysub_def split_def fst_conv snd_conv
```
```   631   by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
```
```   632   (auto simp: Nsub0[simplified Nsub_def] Let_def)
```
```   633
```
```   634 text{* polypow is a power function and preserves normal forms *}
```
```   635
```
```   636 lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{field_char_0, field_inverse_zero})) ^ n"
```
```   637 proof(induct n rule: polypow.induct)
```
```   638   case 1 thus ?case by simp
```
```   639 next
```
```   640   case (2 n)
```
```   641   let ?q = "polypow ((Suc n) div 2) p"
```
```   642   let ?d = "polymul(?q,?q)"
```
```   643   have "odd (Suc n) \<or> even (Suc n)" by simp
```
```   644   moreover
```
```   645   {assume odd: "odd (Suc n)"
```
```   646     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
```
```   647     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul(p, ?d))" by (simp add: Let_def)
```
```   648     also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"
```
```   649       using "2.hyps" by simp
```
```   650     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
```
```   651       apply (simp only: power_add power_one_right) by simp
```
```   652     also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat)))))"
```
```   653       by (simp only: th)
```
```   654     finally have ?case
```
```   655     using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
```
```   656   moreover
```
```   657   {assume even: "even (Suc n)"
```
```   658     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
```
```   659     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
```
```   660     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
```
```   661       using "2.hyps" apply (simp only: power_add) by simp
```
```   662     finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}
```
```   663   ultimately show ?case by blast
```
```   664 qed
```
```   665
```
```   666 lemma polypow_normh:
```
```   667     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   668   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
```
```   669 proof (induct k arbitrary: n rule: polypow.induct)
```
```   670   case (2 k n)
```
```   671   let ?q = "polypow (Suc k div 2) p"
```
```   672   let ?d = "polymul (?q,?q)"
```
```   673   from prems have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
```
```   674   from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
```
```   675   from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul(p,?d)) n" by simp
```
```   676   from dn on show ?case by (simp add: Let_def)
```
```   677 qed auto
```
```   678
```
```   679 lemma polypow_norm:
```
```   680   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   681   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
```
```   682   by (simp add: polypow_normh isnpoly_def)
```
```   683
```
```   684 text{* Finally the whole normalization *}
```
```   685
```
```   686 lemma polynate[simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0, field_inverse_zero})"
```
```   687 by (induct p rule:polynate.induct, auto)
```
```   688
```
```   689 lemma polynate_norm[simp]:
```
```   690   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   691   shows "isnpoly (polynate p)"
```
```   692   by (induct p rule: polynate.induct, simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm) (simp_all add: isnpoly_def)
```
```   693
```
```   694 text{* shift1 *}
```
```   695
```
```   696
```
```   697 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
```
```   698 by (simp add: shift1_def polymul)
```
```   699
```
```   700 lemma shift1_isnpoly:
```
```   701   assumes pn: "isnpoly p" and pnz: "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p) "
```
```   702   using pn pnz by (simp add: shift1_def isnpoly_def )
```
```   703
```
```   704 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
```
```   705   by (simp add: shift1_def)
```
```   706 lemma funpow_shift1_isnpoly:
```
```   707   "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
```
```   708   by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
```
```   709
```
```   710 lemma funpow_isnpolyh:
```
```   711   assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n "and np: "isnpolyh p n"
```
```   712   shows "isnpolyh (funpow k f p) n"
```
```   713   using f np by (induct k arbitrary: p, auto)
```
```   714
```
```   715 lemma funpow_shift1: "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (Mul (Pw (Bound 0) n) p)"
```
```   716   by (induct n arbitrary: p, simp_all add: shift1_isnpoly shift1 power_Suc )
```
```   717
```
```   718 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
```
```   719   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
```
```   720
```
```   721 lemma funpow_shift1_1:
```
```   722   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (funpow n shift1 1\<^sub>p *\<^sub>p p)"
```
```   723   by (simp add: funpow_shift1)
```
```   724
```
```   725 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
```
```   726 by (induct p  arbitrary: n0 rule: poly_cmul.induct, auto simp add: field_simps)
```
```   727
```
```   728 lemma behead:
```
```   729   assumes np: "isnpolyh p n"
```
```   730   shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = (Ipoly bs p :: 'a :: {field_char_0, field_inverse_zero})"
```
```   731   using np
```
```   732 proof (induct p arbitrary: n rule: behead.induct)
```
```   733   case (1 c p n) hence pn: "isnpolyh p n" by simp
```
```   734   from prems(2)[OF pn]
```
```   735   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
```
```   736   then show ?case using "1.hyps" apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
```
```   737     by (simp_all add: th[symmetric] field_simps power_Suc)
```
```   738 qed (auto simp add: Let_def)
```
```   739
```
```   740 lemma behead_isnpolyh:
```
```   741   assumes np: "isnpolyh p n" shows "isnpolyh (behead p) n"
```
```   742   using np by (induct p rule: behead.induct, auto simp add: Let_def isnpolyh_mono)
```
```   743
```
```   744 subsection{* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
```
```   745 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
```
```   746 proof(induct p arbitrary: n rule: poly.induct, auto)
```
```   747   case (goal1 c n p n')
```
```   748   hence "n = Suc (n - 1)" by simp
```
```   749   hence "isnpolyh p (Suc (n - 1))"  using `isnpolyh p n` by simp
```
```   750   with prems(2) show ?case by simp
```
```   751 qed
```
```   752
```
```   753 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
```
```   754 by (induct p arbitrary: n0 rule: isconstant.induct, auto simp add: isnpolyh_polybound0)
```
```   755
```
```   756 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" by (induct p, auto)
```
```   757
```
```   758 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
```
```   759   apply (induct p arbitrary: n0, auto)
```
```   760   apply (atomize)
```
```   761   apply (erule_tac x = "Suc nat" in allE)
```
```   762   apply auto
```
```   763   done
```
```   764
```
```   765 lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
```
```   766  by (induct p  arbitrary: n0 rule: head.induct, auto intro: isnpolyh_polybound0)
```
```   767
```
```   768 lemma polybound0_I:
```
```   769   assumes nb: "polybound0 a"
```
```   770   shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
```
```   771 using nb
```
```   772 by (induct a rule: poly.induct) auto
```
```   773 lemma polysubst0_I:
```
```   774   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
```
```   775   by (induct t) simp_all
```
```   776
```
```   777 lemma polysubst0_I':
```
```   778   assumes nb: "polybound0 a"
```
```   779   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t"
```
```   780   by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
```
```   781
```
```   782 lemma decrpoly: assumes nb: "polybound0 t"
```
```   783   shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
```
```   784   using nb by (induct t rule: decrpoly.induct, simp_all)
```
```   785
```
```   786 lemma polysubst0_polybound0: assumes nb: "polybound0 t"
```
```   787   shows "polybound0 (polysubst0 t a)"
```
```   788 using nb by (induct a rule: poly.induct, auto)
```
```   789
```
```   790 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
```
```   791   by (induct p arbitrary: n rule: degree.induct, auto simp add: isnpolyh_polybound0)
```
```   792
```
```   793 primrec maxindex :: "poly \<Rightarrow> nat" where
```
```   794   "maxindex (Bound n) = n + 1"
```
```   795 | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
```
```   796 | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
```
```   797 | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
```
```   798 | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
```
```   799 | "maxindex (Neg p) = maxindex p"
```
```   800 | "maxindex (Pw p n) = maxindex p"
```
```   801 | "maxindex (C x) = 0"
```
```   802
```
```   803 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool" where
```
```   804   "wf_bs bs p = (length bs \<ge> maxindex p)"
```
```   805
```
```   806 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
```
```   807 proof(induct p rule: coefficients.induct)
```
```   808   case (1 c p)
```
```   809   show ?case
```
```   810   proof
```
```   811     fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"
```
```   812     hence "x = c \<or> x \<in> set (coefficients p)" by simp
```
```   813     moreover
```
```   814     {assume "x = c" hence "wf_bs bs x" using "1.prems"  unfolding wf_bs_def by simp}
```
```   815     moreover
```
```   816     {assume H: "x \<in> set (coefficients p)"
```
```   817       from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp
```
```   818       with "1.hyps" H have "wf_bs bs x" by blast }
```
```   819     ultimately  show "wf_bs bs x" by blast
```
```   820   qed
```
```   821 qed simp_all
```
```   822
```
```   823 lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
```
```   824 by (induct p rule: coefficients.induct, auto)
```
```   825
```
```   826 lemma wf_bs_I: "wf_bs bs p ==> Ipoly (bs@bs') p = Ipoly bs p"
```
```   827   unfolding wf_bs_def by (induct p, auto simp add: nth_append)
```
```   828
```
```   829 lemma take_maxindex_wf: assumes wf: "wf_bs bs p"
```
```   830   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
```
```   831 proof-
```
```   832   let ?ip = "maxindex p"
```
```   833   let ?tbs = "take ?ip bs"
```
```   834   from wf have "length ?tbs = ?ip" unfolding wf_bs_def by simp
```
```   835   hence wf': "wf_bs ?tbs p" unfolding wf_bs_def by  simp
```
```   836   have eq: "bs = ?tbs @ (drop ?ip bs)" by simp
```
```   837   from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis using eq by simp
```
```   838 qed
```
```   839
```
```   840 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
```
```   841   by (induct p, auto)
```
```   842
```
```   843 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
```
```   844   unfolding wf_bs_def by simp
```
```   845
```
```   846 lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
```
```   847   unfolding wf_bs_def by simp
```
```   848
```
```   849
```
```   850
```
```   851 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
```
```   852 by(induct p rule: coefficients.induct, auto simp add: wf_bs_def)
```
```   853 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
```
```   854   by (induct p rule: coefficients.induct, simp_all)
```
```   855
```
```   856
```
```   857 lemma coefficients_head: "last (coefficients p) = head p"
```
```   858   by (induct p rule: coefficients.induct, auto)
```
```   859
```
```   860 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
```
```   861   unfolding wf_bs_def by (induct p rule: decrpoly.induct, auto)
```
```   862
```
```   863 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
```
```   864   apply (rule exI[where x="replicate (n - length xs) z"])
```
```   865   by simp
```
```   866 lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
```
```   867 by (cases p, auto) (case_tac "nat", simp_all)
```
```   868
```
```   869 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
```
```   870   unfolding wf_bs_def
```
```   871   apply (induct p q rule: polyadd.induct)
```
```   872   apply (auto simp add: Let_def)
```
```   873   done
```
```   874
```
```   875 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
```
```   876
```
```   877  unfolding wf_bs_def
```
```   878   apply (induct p q arbitrary: bs rule: polymul.induct)
```
```   879   apply (simp_all add: wf_bs_polyadd)
```
```   880   apply clarsimp
```
```   881   apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
```
```   882   apply auto
```
```   883   done
```
```   884
```
```   885 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
```
```   886   unfolding wf_bs_def by (induct p rule: polyneg.induct, auto)
```
```   887
```
```   888 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
```
```   889   unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
```
```   890
```
```   891 subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}
```
```   892
```
```   893 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
```
```   894 definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)"
```
```   895 definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"
```
```   896
```
```   897 lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0"
```
```   898 proof (induct p arbitrary: n0 rule: coefficients.induct)
```
```   899   case (1 c p n0)
```
```   900   have cp: "isnpolyh (CN c 0 p) n0" by fact
```
```   901   hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
```
```   902     by (auto simp add: isnpolyh_mono[where n'=0])
```
```   903   from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp
```
```   904 qed auto
```
```   905
```
```   906 lemma coefficients_isconst:
```
```   907   "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
```
```   908   by (induct p arbitrary: n rule: coefficients.induct,
```
```   909     auto simp add: isnpolyh_Suc_const)
```
```   910
```
```   911 lemma polypoly_polypoly':
```
```   912   assumes np: "isnpolyh p n0"
```
```   913   shows "polypoly (x#bs) p = polypoly' bs p"
```
```   914 proof-
```
```   915   let ?cf = "set (coefficients p)"
```
```   916   from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
```
```   917   {fix q assume q: "q \<in> ?cf"
```
```   918     from q cn_norm have th: "isnpolyh q n0" by blast
```
```   919     from coefficients_isconst[OF np] q have "isconstant q" by blast
```
```   920     with isconstant_polybound0[OF th] have "polybound0 q" by blast}
```
```   921   hence "\<forall>q \<in> ?cf. polybound0 q" ..
```
```   922   hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"
```
```   923     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
```
```   924     by auto
```
```   925
```
```   926   thus ?thesis unfolding polypoly_def polypoly'_def by simp
```
```   927 qed
```
```   928
```
```   929 lemma polypoly_poly:
```
```   930   assumes np: "isnpolyh p n0" shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
```
```   931   using np
```
```   932 by (induct p arbitrary: n0 bs rule: coefficients.induct, auto simp add: polypoly_def)
```
```   933
```
```   934 lemma polypoly'_poly:
```
```   935   assumes np: "isnpolyh p n0" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
```
```   936   using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
```
```   937
```
```   938
```
```   939 lemma polypoly_poly_polybound0:
```
```   940   assumes np: "isnpolyh p n0" and nb: "polybound0 p"
```
```   941   shows "polypoly bs p = [Ipoly bs p]"
```
```   942   using np nb unfolding polypoly_def
```
```   943   by (cases p, auto, case_tac nat, auto)
```
```   944
```
```   945 lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
```
```   946   by (induct p rule: head.induct, auto)
```
```   947
```
```   948 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   949   by (cases p,auto)
```
```   950
```
```   951 lemma head_eq_headn0: "head p = headn p 0"
```
```   952   by (induct p rule: head.induct, simp_all)
```
```   953
```
```   954 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   955   by (simp add: head_eq_headn0)
```
```   956
```
```   957 lemma isnpolyh_zero_iff:
```
```   958   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})"
```
```   959   shows "p = 0\<^sub>p"
```
```   960 using nq eq
```
```   961 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
```
```   962   case less
```
```   963   note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
```
```   964   {assume nz: "maxindex p = 0"
```
```   965     then obtain c where "p = C c" using np by (cases p, auto)
```
```   966     with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}
```
```   967   moreover
```
```   968   {assume nz: "maxindex p \<noteq> 0"
```
```   969     let ?h = "head p"
```
```   970     let ?hd = "decrpoly ?h"
```
```   971     let ?ihd = "maxindex ?hd"
```
```   972     from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h"
```
```   973       by simp_all
```
```   974     hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast
```
```   975
```
```   976     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
```
```   977     have mihn: "maxindex ?h \<le> maxindex p" by auto
```
```   978     with decr_maxindex[OF h(2)] nz  have ihd_lt_n: "?ihd < maxindex p" by auto
```
```   979     {fix bs:: "'a list"  assume bs: "wf_bs bs ?hd"
```
```   980       let ?ts = "take ?ihd bs"
```
```   981       let ?rs = "drop ?ihd bs"
```
```   982       have ts: "wf_bs ?ts ?hd" using bs unfolding wf_bs_def by simp
```
```   983       have bs_ts_eq: "?ts@ ?rs = bs" by simp
```
```   984       from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x#?ts) ?h" by simp
```
```   985       from ihd_lt_n have "ALL x. length (x#?ts) \<le> maxindex p" by simp
```
```   986       with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = maxindex p" by blast
```
```   987       hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" unfolding wf_bs_def by simp
```
```   988       with zp have "\<forall> x. Ipoly ((x#?ts) @ xs) p = 0" by blast
```
```   989       hence "\<forall> x. Ipoly (x#(?ts @ xs)) p = 0" by simp
```
```   990       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
```
```   991       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"  by simp
```
```   992       hence "poly (polypoly' (?ts @ xs) p) = poly []" by (auto intro: ext)
```
```   993       hence "\<forall> c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
```
```   994         using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
```
```   995       with coefficients_head[of p, symmetric]
```
```   996       have th0: "Ipoly (?ts @ xs) ?hd = 0" by simp
```
```   997       from bs have wf'': "wf_bs ?ts ?hd" unfolding wf_bs_def by simp
```
```   998       with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0" by simp
```
```   999       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by simp }
```
```  1000     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast
```
```  1001
```
```  1002     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast
```
```  1003     hence "?h = 0\<^sub>p" by simp
```
```  1004     with head_nz[OF np] have "p = 0\<^sub>p" by simp}
```
```  1005   ultimately show "p = 0\<^sub>p" by blast
```
```  1006 qed
```
```  1007
```
```  1008 lemma isnpolyh_unique:
```
```  1009   assumes np:"isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1010   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"
```
```  1011 proof(auto)
```
```  1012   assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a)= \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
```
```  1013   hence "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" by simp
```
```  1014   hence "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
```
```  1015     using wf_bs_polysub[where p=p and q=q] by auto
```
```  1016   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq]
```
```  1017   show "p = q" by blast
```
```  1018 qed
```
```  1019
```
```  1020
```
```  1021 text{* consequences of unicity on the algorithms for polynomial normalization *}
```
```  1022
```
```  1023 lemma polyadd_commute:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1024   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" shows "p +\<^sub>p q = q +\<^sub>p p"
```
```  1025   using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]] by simp
```
```  1026
```
```  1027 lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp
```
```  1028 lemma one_normh: "isnpolyh 1\<^sub>p n" by simp
```
```  1029 lemma polyadd_0[simp]:
```
```  1030   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1031   and np: "isnpolyh p n0" shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>p p = p"
```
```  1032   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
```
```  1033     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
```
```  1034
```
```  1035 lemma polymul_1[simp]:
```
```  1036     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1037   and np: "isnpolyh p n0" shows "p *\<^sub>p 1\<^sub>p = p" and "1\<^sub>p *\<^sub>p p = p"
```
```  1038   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
```
```  1039     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
```
```  1040 lemma polymul_0[simp]:
```
```  1041   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1042   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"
```
```  1043   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
```
```  1044     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
```
```  1045
```
```  1046 lemma polymul_commute:
```
```  1047     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1048   and np:"isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1049   shows "p *\<^sub>p q = q *\<^sub>p p"
```
```  1050 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
```
```  1051
```
```  1052 declare polyneg_polyneg[simp]
```
```  1053
```
```  1054 lemma isnpolyh_polynate_id[simp]:
```
```  1055   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1056   and np:"isnpolyh p n0" shows "polynate p = p"
```
```  1057   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
```
```  1058
```
```  1059 lemma polynate_idempotent[simp]:
```
```  1060     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1061   shows "polynate (polynate p) = polynate p"
```
```  1062   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
```
```  1063
```
```  1064 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
```
```  1065   unfolding poly_nate_def polypoly'_def ..
```
```  1066 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>)"
```
```  1067   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
```
```  1068   unfolding poly_nate_polypoly' by (auto intro: ext)
```
```  1069
```
```  1070 subsection{* heads, degrees and all that *}
```
```  1071 lemma degree_eq_degreen0: "degree p = degreen p 0"
```
```  1072   by (induct p rule: degree.induct, simp_all)
```
```  1073
```
```  1074 lemma degree_polyneg: assumes n: "isnpolyh p n"
```
```  1075   shows "degree (polyneg p) = degree p"
```
```  1076   using n
```
```  1077   by (induct p arbitrary: n rule: polyneg.induct, simp_all) (case_tac na, auto)
```
```  1078
```
```  1079 lemma degree_polyadd:
```
```  1080   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1081   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1082 using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
```
```  1083
```
```  1084
```
```  1085 lemma degree_polysub: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1086   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1087 proof-
```
```  1088   from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp
```
```  1089   from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])
```
```  1090 qed
```
```  1091
```
```  1092 lemma degree_polysub_samehead:
```
```  1093   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1094   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q"
```
```  1095   and d: "degree p = degree q"
```
```  1096   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
```
```  1097 unfolding polysub_def split_def fst_conv snd_conv
```
```  1098 using np nq h d
```
```  1099 proof(induct p q rule:polyadd.induct)
```
```  1100   case (1 a b a' b') thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def])
```
```  1101 next
```
```  1102   case (2 a b c' n' p')
```
```  1103   let ?c = "(a,b)"
```
```  1104   from prems have "degree (C ?c) = degree (CN c' n' p')" by simp
```
```  1105   hence nz:"n' > 0" by (cases n', auto)
```
```  1106   hence "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
```
```  1107   with prems show ?case by simp
```
```  1108 next
```
```  1109   case (3 c n p a' b')
```
```  1110   let ?c' = "(a',b')"
```
```  1111   from prems have "degree (C ?c') = degree (CN c n p)" by simp
```
```  1112   hence nz:"n > 0" by (cases n, auto)
```
```  1113   hence "head (CN c n p) = CN c n p" by (cases n, auto)
```
```  1114   with prems show ?case by simp
```
```  1115 next
```
```  1116   case (4 c n p c' n' p')
```
```  1117   hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1"
```
```  1118     "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+
```
```  1119   hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all
```
```  1120   hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
```
```  1121     using H(1-2) degree_polyneg by auto
```
```  1122   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"  by simp+
```
```  1123   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"  by simp
```
```  1124   from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto
```
```  1125   have "n = n' \<or> n < n' \<or> n > n'" by arith
```
```  1126   moreover
```
```  1127   {assume nn': "n = n'"
```
```  1128     have "n = 0 \<or> n >0" by arith
```
```  1129     moreover {assume nz: "n = 0" hence ?case using prems by (auto simp add: Let_def degcmc')}
```
```  1130     moreover {assume nz: "n > 0"
```
```  1131       with nn' H(3) have  cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
```
```  1132       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)}
```
```  1133     ultimately have ?case by blast}
```
```  1134   moreover
```
```  1135   {assume nn': "n < n'" hence n'p: "n' > 0" by simp
```
```  1136     hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n', simp_all)
```
```  1137     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)
```
```  1138     hence "n > 0" by (cases n, simp_all)
```
```  1139     hence headcnp: "head (CN c n p) = CN c n p" by (cases n, auto)
```
```  1140     from H(3) headcnp headcnp' nn' have ?case by auto}
```
```  1141   moreover
```
```  1142   {assume nn': "n > n'"  hence np: "n > 0" by simp
```
```  1143     hence headcnp:"head (CN c n p) = CN c n p"  by (cases n, simp_all)
```
```  1144     from prems have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
```
```  1145     from np have degcnp: "degree (CN c n p) = 0" by (cases n, simp_all)
```
```  1146     with degcnpeq have "n' > 0" by (cases n', simp_all)
```
```  1147     hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
```
```  1148     from H(3) headcnp headcnp' nn' have ?case by auto}
```
```  1149   ultimately show ?case  by blast
```
```  1150 qed auto
```
```  1151
```
```  1152 lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
```
```  1153 by (induct p arbitrary: n0 rule: head.induct, simp_all add: shift1_def)
```
```  1154
```
```  1155 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
```
```  1156 proof(induct k arbitrary: n0 p)
```
```  1157   case (Suc k n0 p) hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
```
```  1158   with prems have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
```
```  1159     and "head (shift1 p) = head p" by (simp_all add: shift1_head)
```
```  1160   thus ?case by (simp add: funpow_swap1)
```
```  1161 qed auto
```
```  1162
```
```  1163 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
```
```  1164   by (simp add: shift1_def)
```
```  1165 lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
```
```  1166   by (induct k arbitrary: p, auto simp add: shift1_degree)
```
```  1167
```
```  1168 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
```
```  1169   by (induct n arbitrary: p, simp_all add: funpow_def)
```
```  1170
```
```  1171 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
```
```  1172   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1173 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
```
```  1174   by (induct p arbitrary: n rule: degreen.induct, auto)
```
```  1175 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
```
```  1176   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1177 lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
```
```  1178   by (induct p rule: head.induct, auto)
```
```  1179
```
```  1180 lemma polyadd_eq_const_degree:
```
```  1181   "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk> \<Longrightarrow> degree p = degree q"
```
```  1182   using polyadd_eq_const_degreen degree_eq_degreen0 by simp
```
```  1183
```
```  1184 lemma polyadd_head: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1185   and deg: "degree p \<noteq> degree q"
```
```  1186   shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
```
```  1187 using np nq deg
```
```  1188 apply(induct p q arbitrary: n0 n1 rule: polyadd.induct,simp_all)
```
```  1189 apply (case_tac n', simp, simp)
```
```  1190 apply (case_tac n, simp, simp)
```
```  1191 apply (case_tac n, case_tac n', simp add: Let_def)
```
```  1192 apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p")
```
```  1193 apply (auto simp add: polyadd_eq_const_degree)
```
```  1194 apply (metis head_nz)
```
```  1195 apply (metis head_nz)
```
```  1196 apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
```
```  1197 by (metis degree.simps(9) gr0_conv_Suc nat_less_le order_le_less_trans)
```
```  1198
```
```  1199 lemma polymul_head_polyeq:
```
```  1200    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1201   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"
```
```  1202 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
```
```  1203   case (2 a b c' n' p' n0 n1)
```
```  1204   hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum (a,b)"  by (simp_all add: head_isnpolyh)
```
```  1205   thus ?case using prems by (cases n', auto)
```
```  1206 next
```
```  1207   case (3 c n p a' b' n0 n1)
```
```  1208   hence "isnpolyh (head (CN c n p)) n0" "isnormNum (a',b')"  by (simp_all add: head_isnpolyh)
```
```  1209   thus ?case using prems by (cases n, auto)
```
```  1210 next
```
```  1211   case (4 c n p c' n' p' n0 n1)
```
```  1212   hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
```
```  1213     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
```
```  1214     by simp_all
```
```  1215   have "n < n' \<or> n' < n \<or> n = n'" by arith
```
```  1216   moreover
```
```  1217   {assume nn': "n < n'" hence ?case
```
```  1218       thm prems
```
```  1219       using norm
```
```  1220     prems(6)[rule_format, OF nn' norm(1,6)]
```
```  1221     prems(7)[rule_format, OF nn' norm(2,6)] by (simp, cases n, simp,cases n', simp_all)}
```
```  1222   moreover {assume nn': "n'< n"
```
```  1223     hence stupid: "n' < n \<and> \<not> n < n'" by simp
```
```  1224     hence ?case using norm prems(4) [rule_format, OF stupid norm(5,3)]
```
```  1225       prems(5)[rule_format, OF stupid norm(5,4)]
```
```  1226       by (simp,cases n',simp,cases n,auto)}
```
```  1227   moreover {assume nn': "n' = n"
```
```  1228     hence stupid: "\<not> n' < n \<and> \<not> n < n'" by simp
```
```  1229     from nn' polymul_normh[OF norm(5,4)]
```
```  1230     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
```
```  1231     from nn' polymul_normh[OF norm(5,3)] norm
```
```  1232     have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
```
```  1233     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
```
```  1234     have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```  1235     from polyadd_normh[OF ncnpc' ncnpp0']
```
```  1236     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"
```
```  1237       by (simp add: min_def)
```
```  1238     {assume np: "n > 0"
```
```  1239       with nn' head_isnpolyh_Suc'[OF np nth]
```
```  1240         head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
```
```  1241       have ?case by simp}
```
```  1242     moreover
```
```  1243     {moreover assume nz: "n = 0"
```
```  1244       from polymul_degreen[OF norm(5,4), where m="0"]
```
```  1245         polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
```
```  1246       norm(5,6) degree_npolyhCN[OF norm(6)]
```
```  1247     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
```
```  1248     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
```
```  1249     from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
```
```  1250     have ?case   using norm prems(2)[rule_format, OF stupid norm(5,3)]
```
```  1251         prems(3)[rule_format, OF stupid norm(5,4)] nn' nz by simp }
```
```  1252     ultimately have ?case by (cases n) auto}
```
```  1253   ultimately show ?case by blast
```
```  1254 qed simp_all
```
```  1255
```
```  1256 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
```
```  1257   by(induct p rule: degree.induct, auto)
```
```  1258
```
```  1259 lemma degree_head[simp]: "degree (head p) = 0"
```
```  1260   by (induct p rule: head.induct, auto)
```
```  1261
```
```  1262 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1+ degree p"
```
```  1263   by (cases n, simp_all)
```
```  1264 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
```
```  1265   by (cases n, simp_all)
```
```  1266
```
```  1267 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)"
```
```  1268   using polyadd_different_degreen degree_eq_degreen0 by simp
```
```  1269
```
```  1270 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
```
```  1271   by (induct p arbitrary: n0 rule: polyneg.induct, auto)
```
```  1272
```
```  1273 lemma degree_polymul:
```
```  1274   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1275   and np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1276   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
```
```  1277   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
```
```  1278
```
```  1279 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
```
```  1280   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1281
```
```  1282 lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"
```
```  1283   by (induct p arbitrary: n rule: degree.induct, auto)
```
```  1284
```
```  1285 subsection {* Correctness of polynomial pseudo division *}
```
```  1286
```
```  1287 lemma polydivide_aux_properties:
```
```  1288   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1289   and np: "isnpolyh p n0" and ns: "isnpolyh s n1"
```
```  1290   and ap: "head p = a" and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
```
```  1291   shows "(polydivide_aux a n p k s = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1292           \<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)))"
```
```  1293   using ns
```
```  1294 proof(induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
```
```  1295   case less
```
```  1296   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
```
```  1297   let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1298     \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
```
```  1299   let ?b = "head s"
```
```  1300   let ?p' = "funpow (degree s - n) shift1 p"
```
```  1301   let ?xdn = "funpow (degree s - n) shift1 1\<^sub>p"
```
```  1302   let ?akk' = "a ^\<^sub>p (k' - k)"
```
```  1303   note ns = `isnpolyh s n1`
```
```  1304   from np have np0: "isnpolyh p 0"
```
```  1305     using isnpolyh_mono[where n="n0" and n'="0" and p="p"]  by simp
```
```  1306   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
```
```  1307   have headp': "head ?p' = head p" using funpow_shift1_head[OF np pnz] by simp
```
```  1308   from funpow_shift1_isnpoly[where p="1\<^sub>p"] have nxdn: "isnpolyh ?xdn 0" by (simp add: isnpoly_def)
```
```  1309   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
```
```  1310   have nakk':"isnpolyh ?akk' 0" by blast
```
```  1311   {assume sz: "s = 0\<^sub>p"
```
```  1312    hence ?ths using np polydivide_aux.simps apply clarsimp by (rule exI[where x="0\<^sub>p"], simp) }
```
```  1313   moreover
```
```  1314   {assume sz: "s \<noteq> 0\<^sub>p"
```
```  1315     {assume dn: "degree s < n"
```
```  1316       hence "?ths" using ns ndp np polydivide_aux.simps by auto (rule exI[where x="0\<^sub>p"],simp) }
```
```  1317     moreover
```
```  1318     {assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith
```
```  1319       have degsp': "degree s = degree ?p'"
```
```  1320         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp
```
```  1321       {assume ba: "?b = a"
```
```  1322         hence headsp': "head s = head ?p'" using ap headp' by simp
```
```  1323         have nr: "isnpolyh (s -\<^sub>p ?p') 0" using polysub_normh[OF ns np'] by simp
```
```  1324         from degree_polysub_samehead[OF ns np' headsp' degsp']
```
```  1325         have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
```
```  1326         moreover
```
```  1327         {assume deglt:"degree (s -\<^sub>p ?p') < degree s"
```
```  1328           from polydivide_aux.simps sz dn' ba
```
```  1329           have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
```
```  1330             by (simp add: Let_def)
```
```  1331           {assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1332             from less(1)[OF deglt nr, of k k' r]
```
```  1333               trans[OF eq[symmetric] h1]
```
```  1334             have kk': "k \<le> k'" and nr:"\<exists>nr. isnpolyh r nr" and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1335               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
```
```  1336             from q1 obtain q n1 where nq: "isnpolyh q n1"
```
```  1337               and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"  by blast
```
```  1338             from nr obtain nr where nr': "isnpolyh r nr" by blast
```
```  1339             from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0" by simp
```
```  1340             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
```
```  1341             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
```
```  1342             from polyadd_normh[OF polymul_normh[OF np
```
```  1343               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
```
```  1344             have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0" by simp
```
```  1345             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')) =
```
```  1346               Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
```
```  1347             hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
```
```  1348               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1349               by (simp add: field_simps)
```
```  1350             hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1351               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p *\<^sub>p p)
```
```  1352               + Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1353               by (auto simp only: funpow_shift1_1)
```
```  1354             hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1355               Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p)
```
```  1356               + Ipoly bs q) + Ipoly bs r" by (simp add: field_simps)
```
```  1357             hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1358               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
```
```  1359             with isnpolyh_unique[OF nakks' nqr']
```
```  1360             have "a ^\<^sub>p (k' - k) *\<^sub>p s =
```
```  1361               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
```
```  1362             hence ?qths using nq'
```
```  1363               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 1\<^sub>p) +\<^sub>p q" in exI)
```
```  1364               apply (rule_tac x="0" in exI) by simp
```
```  1365             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"
```
```  1366               by blast } hence ?ths by blast }
```
```  1367         moreover
```
```  1368         {assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
```
```  1369           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}"]
```
```  1370           have " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs s = Ipoly bs ?p'" by simp
```
```  1371           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
```
```  1372             by (simp only: funpow_shift1_1) simp
```
```  1373           hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]] by blast
```
```  1374           {assume h1: "polydivide_aux a n p k s = (k',r)"
```
```  1375             from polydivide_aux.simps sz dn' ba
```
```  1376             have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
```
```  1377               by (simp add: Let_def)
```
```  1378             also have "\<dots> = (k,0\<^sub>p)" using polydivide_aux.simps spz by simp
```
```  1379             finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
```
```  1380             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
```
```  1381               polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
```
```  1382               apply auto
```
```  1383               apply (rule exI[where x="?xdn"])
```
```  1384               apply (auto simp add: polymul_commute[of p])
```
```  1385               done} }
```
```  1386         ultimately have ?ths by blast }
```
```  1387       moreover
```
```  1388       {assume ba: "?b \<noteq> a"
```
```  1389         from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1390           polymul_normh[OF head_isnpolyh[OF ns] np']]
```
```  1391         have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by(simp add: min_def)
```
```  1392         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
```
```  1393           using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1394             polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
```
```  1395             funpow_shift1_nz[OF pnz] by simp_all
```
```  1396         from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
```
```  1397           polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1398         have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
```
```  1399           using head_head[OF ns] funpow_shift1_head[OF np pnz]
```
```  1400             polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
```
```  1401           by (simp add: ap)
```
```  1402         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
```
```  1403           head_nz[OF np] pnz sz ap[symmetric]
```
```  1404           funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1405           polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"]  head_nz[OF ns]
```
```  1406           ndp dn
```
```  1407         have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p') "
```
```  1408           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
```
```  1409         {assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
```
```  1410           from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns] polymul_normh[OF head_isnpolyh[OF ns]np']]
```
```  1411           ap have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by simp
```
```  1412           {assume h1:"polydivide_aux a n p k s = (k', r)"
```
```  1413             from h1 polydivide_aux.simps sz dn' ba
```
```  1414             have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
```
```  1415               by (simp add: Let_def)
```
```  1416             with less(1)[OF dth nasbp', of "Suc k" k' r]
```
```  1417             obtain q nq nr where kk': "Suc k \<le> k'" and nr: "isnpolyh r nr" and nq: "isnpolyh q nq"
```
```  1418               and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1419               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
```
```  1420             from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith
```
```  1421             {fix bs:: "'a::{field_char_0, field_inverse_zero} list"
```
```  1422
```
```  1423             from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
```
```  1424             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
```
```  1425             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"
```
```  1426               by (simp add: field_simps power_Suc)
```
```  1427             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"
```
```  1428               by (simp add:kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1429             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"
```
```  1430               by (simp add: field_simps)}
```
```  1431             hence ieq:"\<forall>(bs :: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1432               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
```
```  1433             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
```
```  1434             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]]
```
```  1435             have nqw: "isnpolyh ?q 0" by simp
```
```  1436             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]]
```
```  1437             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
```
```  1438             from dr kk' nr h1 asth nqw have ?ths apply simp
```
```  1439               apply (rule conjI)
```
```  1440               apply (rule exI[where x="nr"], simp)
```
```  1441               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
```
```  1442               apply (rule exI[where x="0"], simp)
```
```  1443               done}
```
```  1444           hence ?ths by blast }
```
```  1445         moreover
```
```  1446         {assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
```
```  1447           {fix bs :: "'a::{field_char_0, field_inverse_zero} list"
```
```  1448             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
```
```  1449           have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'" by simp
```
```  1450           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
```
```  1451             by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1452           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" by simp
```
```  1453         }
```
```  1454         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))" ..
```
```  1455           from hth
```
```  1456           have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
```
```  1457             using isnpolyh_unique[where ?'a = "'a::{field_char_0, field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
```
```  1458                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
```
```  1459               simplified ap] by simp
```
```  1460           {assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1461           from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
```
```  1462           have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
```
```  1463           with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
```
```  1464             polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
```
```  1465           have ?ths apply (clarsimp simp add: Let_def)
```
```  1466             apply (rule exI[where x="?b *\<^sub>p ?xdn"]) apply simp
```
```  1467             apply (rule exI[where x="0"], simp)
```
```  1468             done}
```
```  1469         hence ?ths by blast}
```
```  1470         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"]
```
```  1471           head_nz[OF np] pnz sz ap[symmetric]
```
```  1472           by (simp add: degree_eq_degreen0[symmetric]) blast }
```
```  1473       ultimately have ?ths by blast
```
```  1474     }
```
```  1475     ultimately have ?ths by blast}
```
```  1476   ultimately show ?ths by blast
```
```  1477 qed
```
```  1478
```
```  1479 lemma polydivide_properties:
```
```  1480   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1481   and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
```
```  1482   shows "(\<exists> k r. polydivide s p = (k,r) \<and> (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1483   \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
```
```  1484 proof-
```
```  1485   have trv: "head p = head p" "degree p = degree p" by simp_all
```
```  1486   from polydivide_def[where s="s" and p="p"]
```
```  1487   have ex: "\<exists> k r. polydivide s p = (k,r)" by auto
```
```  1488   then obtain k r where kr: "polydivide s p = (k,r)" by blast
```
```  1489   from trans[OF meta_eq_to_obj_eq[OF polydivide_def[where s="s"and p="p"], symmetric] kr]
```
```  1490     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
```
```  1491   have "(degree r = 0 \<or> degree r < degree p) \<and>
```
```  1492    (\<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
```
```  1493   with kr show ?thesis
```
```  1494     apply -
```
```  1495     apply (rule exI[where x="k"])
```
```  1496     apply (rule exI[where x="r"])
```
```  1497     apply simp
```
```  1498     done
```
```  1499 qed
```
```  1500
```
```  1501 subsection{* More about polypoly and pnormal etc *}
```
```  1502
```
```  1503 definition "isnonconstant p = (\<not> isconstant p)"
```
```  1504
```
```  1505 lemma isnonconstant_pnormal_iff: assumes nc: "isnonconstant p"
```
```  1506   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1507 proof
```
```  1508   let ?p = "polypoly bs p"
```
```  1509   assume H: "pnormal ?p"
```
```  1510   have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
```
```  1511
```
```  1512   from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1513     pnormal_last_nonzero[OF H]
```
```  1514   show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)
```
```  1515 next
```
```  1516   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1517   let ?p = "polypoly bs p"
```
```  1518   have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
```
```  1519   hence pz: "?p \<noteq> []" by (simp add: polypoly_def)
```
```  1520   hence lg: "length ?p > 0" by simp
```
```  1521   from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1522   have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)
```
```  1523   from pnormal_last_length[OF lg lz] show "pnormal ?p" .
```
```  1524 qed
```
```  1525
```
```  1526 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
```
```  1527   unfolding isnonconstant_def
```
```  1528   apply (cases p, simp_all)
```
```  1529   apply (case_tac nat, auto)
```
```  1530   done
```
```  1531 lemma isnonconstant_nonconstant: assumes inc: "isnonconstant p"
```
```  1532   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1533 proof
```
```  1534   let ?p = "polypoly bs p"
```
```  1535   assume nc: "nonconstant ?p"
```
```  1536   from isnonconstant_pnormal_iff[OF inc, of bs] nc
```
```  1537   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast
```
```  1538 next
```
```  1539   let ?p = "polypoly bs p"
```
```  1540   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1541   from isnonconstant_pnormal_iff[OF inc, of bs] h
```
```  1542   have pn: "pnormal ?p" by blast
```
```  1543   {fix x assume H: "?p = [x]"
```
```  1544     from H have "length (coefficients p) = 1" unfolding polypoly_def by auto
```
```  1545     with isnonconstant_coefficients_length[OF inc] have False by arith}
```
```  1546   thus "nonconstant ?p" using pn unfolding nonconstant_def by blast
```
```  1547 qed
```
```  1548
```
```  1549 lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
```
```  1550   unfolding pnormal_def
```
```  1551  apply (induct p)
```
```  1552  apply (simp_all, case_tac "p=[]", simp_all)
```
```  1553  done
```
```  1554
```
```  1555 lemma degree_degree: assumes inc: "isnonconstant p"
```
```  1556   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1557 proof
```
```  1558   let  ?p = "polypoly bs p"
```
```  1559   assume H: "degree p = Polynomial_List.degree ?p"
```
```  1560   from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
```
```  1561     unfolding polypoly_def by auto
```
```  1562   from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
```
```  1563   have lg:"length (pnormalize ?p) = length ?p"
```
```  1564     unfolding Polynomial_List.degree_def polypoly_def by simp
```
```  1565   hence "pnormal ?p" using pnormal_length[OF pz] by blast
```
```  1566   with isnonconstant_pnormal_iff[OF inc]
```
```  1567   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast
```
```  1568 next
```
```  1569   let  ?p = "polypoly bs p"
```
```  1570   assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1571   with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast
```
```  1572   with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
```
```  1573   show "degree p = Polynomial_List.degree ?p"
```
```  1574     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
```
```  1575 qed
```
```  1576
```
```  1577 section{* Swaps ; Division by a certain variable *}
```
```  1578 primrec swap:: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly" where
```
```  1579   "swap n m (C x) = C x"
```
```  1580 | "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"
```
```  1581 | "swap n m (Neg t) = Neg (swap n m t)"
```
```  1582 | "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
```
```  1583 | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
```
```  1584 | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
```
```  1585 | "swap n m (Pw t k) = Pw (swap n m t) k"
```
```  1586 | "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)
```
```  1587   (swap n m p)"
```
```  1588
```
```  1589 lemma swap: assumes nbs: "n < length bs" and mbs: "m < length bs"
```
```  1590   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  1591 proof (induct t)
```
```  1592   case (Bound k) thus ?case using nbs mbs by simp
```
```  1593 next
```
```  1594   case (CN c k p) thus ?case using nbs mbs by simp
```
```  1595 qed simp_all
```
```  1596 lemma swap_swap_id[simp]: "swap n m (swap m n t) = t"
```
```  1597   by (induct t,simp_all)
```
```  1598
```
```  1599 lemma swap_commute: "swap n m p = swap m n p" by (induct p, simp_all)
```
```  1600
```
```  1601 lemma swap_same_id[simp]: "swap n n t = t"
```
```  1602   by (induct t, simp_all)
```
```  1603
```
```  1604 definition "swapnorm n m t = polynate (swap n m t)"
```
```  1605
```
```  1606 lemma swapnorm: assumes nbs: "n < length bs" and mbs: "m < length bs"
```
```  1607   shows "((Ipoly bs (swapnorm n m t) :: 'a\<Colon>{field_char_0, field_inverse_zero})) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  1608   using swap[OF prems] swapnorm_def by simp
```
```  1609
```
```  1610 lemma swapnorm_isnpoly[simp]:
```
```  1611     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1612   shows "isnpoly (swapnorm n m p)"
```
```  1613   unfolding swapnorm_def by simp
```
```  1614
```
```  1615 definition "polydivideby n s p =
```
```  1616     (let ss = swapnorm 0 n s ; sp = swapnorm 0 n p ; h = head sp; (k,r) = polydivide ss sp
```
```  1617      in (k,swapnorm 0 n h,swapnorm 0 n r))"
```
```  1618
```
```  1619 lemma swap_nz [simp]: " (swap n m p = 0\<^sub>p) = (p = 0\<^sub>p)" by (induct p, simp_all)
```
```  1620
```
```  1621 consts isweaknpoly :: "poly \<Rightarrow> bool"
```
```  1622 recdef isweaknpoly "measure size"
```
```  1623   "isweaknpoly (C c) = True"
```
```  1624   "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
```
```  1625   "isweaknpoly p = False"
```
```  1626
```
```  1627 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
```
```  1628   by (induct p arbitrary: n0, auto)
```
```  1629
```
```  1630 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
```
```  1631   by (induct p, auto)
```
```  1632
```
`  1633 end`