src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
 author haftmann Sat Jul 05 11:01:53 2014 +0200 (2014-07-05) changeset 57514 bdc2c6b40bf2 parent 56207 52d5067ca06a child 58249 180f1b3508ed permissions -rw-r--r--
prefer ac_simps collections over separate name bindings for add and mult
```     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 Rat_Pair Polynomial_List
```
```     9 begin
```
```    10
```
```    11 subsection{* Datatype of polynomial expressions *}
```
```    12
```
```    13 datatype poly = C Num | Bound nat | Add poly poly | Sub poly poly
```
```    14   | Mul poly poly| Neg poly| Pw poly nat| CN poly nat poly
```
```    15
```
```    16 abbreviation poly_0 :: "poly" ("0\<^sub>p") where "0\<^sub>p \<equiv> C (0\<^sub>N)"
```
```    17 abbreviation poly_p :: "int \<Rightarrow> poly" ("'((_)')\<^sub>p") where "(i)\<^sub>p \<equiv> C (i)\<^sub>N"
```
```    18
```
```    19
```
```    20 subsection{* Boundedness, substitution and all that *}
```
```    21
```
```    22 primrec polysize:: "poly \<Rightarrow> nat"
```
```    23 where
```
```    24   "polysize (C c) = 1"
```
```    25 | "polysize (Bound n) = 1"
```
```    26 | "polysize (Neg p) = 1 + polysize p"
```
```    27 | "polysize (Add p q) = 1 + polysize p + polysize q"
```
```    28 | "polysize (Sub p q) = 1 + polysize p + polysize q"
```
```    29 | "polysize (Mul p q) = 1 + polysize p + polysize q"
```
```    30 | "polysize (Pw p n) = 1 + polysize p"
```
```    31 | "polysize (CN c n p) = 4 + polysize c + polysize p"
```
```    32
```
```    33 primrec polybound0:: "poly \<Rightarrow> bool" -- {* a poly is INDEPENDENT of Bound 0 *}
```
```    34 where
```
```    35   "polybound0 (C c) \<longleftrightarrow> True"
```
```    36 | "polybound0 (Bound n) \<longleftrightarrow> n > 0"
```
```    37 | "polybound0 (Neg a) \<longleftrightarrow> polybound0 a"
```
```    38 | "polybound0 (Add a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
```
```    39 | "polybound0 (Sub a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
```
```    40 | "polybound0 (Mul a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
```
```    41 | "polybound0 (Pw p n) \<longleftrightarrow> polybound0 p"
```
```    42 | "polybound0 (CN c n p) \<longleftrightarrow> n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p"
```
```    43
```
```    44 primrec polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" -- {* substitute a poly into a poly for Bound 0 *}
```
```    45 where
```
```    46   "polysubst0 t (C c) = C c"
```
```    47 | "polysubst0 t (Bound n) = (if n = 0 then t else Bound n)"
```
```    48 | "polysubst0 t (Neg a) = Neg (polysubst0 t a)"
```
```    49 | "polysubst0 t (Add a b) = Add (polysubst0 t a) (polysubst0 t b)"
```
```    50 | "polysubst0 t (Sub a b) = Sub (polysubst0 t a) (polysubst0 t b)"
```
```    51 | "polysubst0 t (Mul a b) = Mul (polysubst0 t a) (polysubst0 t b)"
```
```    52 | "polysubst0 t (Pw p n) = Pw (polysubst0 t p) n"
```
```    53 | "polysubst0 t (CN c n p) =
```
```    54     (if n = 0 then Add (polysubst0 t c) (Mul t (polysubst0 t p))
```
```    55      else CN (polysubst0 t c) n (polysubst0 t p))"
```
```    56
```
```    57 fun decrpoly:: "poly \<Rightarrow> poly"
```
```    58 where
```
```    59   "decrpoly (Bound n) = Bound (n - 1)"
```
```    60 | "decrpoly (Neg a) = Neg (decrpoly a)"
```
```    61 | "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"
```
```    62 | "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
```
```    63 | "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
```
```    64 | "decrpoly (Pw p n) = Pw (decrpoly p) n"
```
```    65 | "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
```
```    66 | "decrpoly a = a"
```
```    67
```
```    68
```
```    69 subsection{* Degrees and heads and coefficients *}
```
```    70
```
```    71 fun degree :: "poly \<Rightarrow> nat"
```
```    72 where
```
```    73   "degree (CN c 0 p) = 1 + degree p"
```
```    74 | "degree p = 0"
```
```    75
```
```    76 fun head :: "poly \<Rightarrow> poly"
```
```    77 where
```
```    78   "head (CN c 0 p) = head p"
```
```    79 | "head p = p"
```
```    80
```
```    81 (* More general notions of degree and head *)
```
```    82 fun degreen :: "poly \<Rightarrow> nat \<Rightarrow> nat"
```
```    83 where
```
```    84   "degreen (CN c n p) = (\<lambda>m. if n = m then 1 + degreen p n else 0)"
```
```    85 | "degreen p = (\<lambda>m. 0)"
```
```    86
```
```    87 fun headn :: "poly \<Rightarrow> nat \<Rightarrow> poly"
```
```    88 where
```
```    89   "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
```
```    90 | "headn p = (\<lambda>m. p)"
```
```    91
```
```    92 fun coefficients :: "poly \<Rightarrow> poly list"
```
```    93 where
```
```    94   "coefficients (CN c 0 p) = c # coefficients p"
```
```    95 | "coefficients p = [p]"
```
```    96
```
```    97 fun isconstant :: "poly \<Rightarrow> bool"
```
```    98 where
```
```    99   "isconstant (CN c 0 p) = False"
```
```   100 | "isconstant p = True"
```
```   101
```
```   102 fun behead :: "poly \<Rightarrow> poly"
```
```   103 where
```
```   104   "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
```
```   105 | "behead p = 0\<^sub>p"
```
```   106
```
```   107 fun headconst :: "poly \<Rightarrow> Num"
```
```   108 where
```
```   109   "headconst (CN c n p) = headconst p"
```
```   110 | "headconst (C n) = n"
```
```   111
```
```   112
```
```   113 subsection{* Operations for normalization *}
```
```   114
```
```   115 declare if_cong[fundef_cong del]
```
```   116 declare let_cong[fundef_cong del]
```
```   117
```
```   118 fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
```
```   119 where
```
```   120   "polyadd (C c) (C c') = C (c +\<^sub>N c')"
```
```   121 | "polyadd (C c) (CN c' n' p') = CN (polyadd (C c) c') n' p'"
```
```   122 | "polyadd (CN c n p) (C c') = CN (polyadd c (C c')) n p"
```
```   123 | "polyadd (CN c n p) (CN c' n' p') =
```
```   124     (if n < n' then CN (polyadd c (CN c' n' p')) n p
```
```   125      else if n' < n then CN (polyadd (CN c n p) c') n' p'
```
```   126      else
```
```   127       let
```
```   128         cc' = polyadd c c';
```
```   129         pp' = polyadd p p'
```
```   130       in if pp' = 0\<^sub>p then cc' else CN cc' n pp')"
```
```   131 | "polyadd a b = Add a b"
```
```   132
```
```   133
```
```   134 fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
```
```   135 where
```
```   136   "polyneg (C c) = C (~\<^sub>N c)"
```
```   137 | "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
```
```   138 | "polyneg a = Neg a"
```
```   139
```
```   140 definition polysub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
```
```   141   where "p -\<^sub>p q = polyadd p (polyneg q)"
```
```   142
```
```   143 fun polymul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
```
```   144 where
```
```   145   "polymul (C c) (C c') = C (c *\<^sub>N c')"
```
```   146 | "polymul (C c) (CN c' n' p') =
```
```   147     (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul (C c) c') n' (polymul (C c) p'))"
```
```   148 | "polymul (CN c n p) (C c') =
```
```   149     (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul c (C c')) n (polymul p (C c')))"
```
```   150 | "polymul (CN c n p) (CN c' n' p') =
```
```   151     (if n < n' then CN (polymul c (CN c' n' p')) n (polymul p (CN c' n' p'))
```
```   152      else if n' < n 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
```
```   156 declare if_cong[fundef_cong]
```
```   157 declare let_cong[fundef_cong]
```
```   158
```
```   159 fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
```
```   160 where
```
```   161   "polypow 0 = (\<lambda>p. (1)\<^sub>p)"
```
```   162 | "polypow n =
```
```   163     (\<lambda>p.
```
```   164       let
```
```   165         q = polypow (n div 2) p;
```
```   166         d = polymul q q
```
```   167       in if even n then d else polymul p d)"
```
```   168
```
```   169 abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
```
```   170   where "a ^\<^sub>p k \<equiv> polypow k a"
```
```   171
```
```   172 function polynate :: "poly \<Rightarrow> poly"
```
```   173 where
```
```   174   "polynate (Bound n) = CN 0\<^sub>p n (1)\<^sub>p"
```
```   175 | "polynate (Add p q) = polynate p +\<^sub>p polynate q"
```
```   176 | "polynate (Sub p q) = polynate p -\<^sub>p polynate q"
```
```   177 | "polynate (Mul p q) = polynate p *\<^sub>p polynate q"
```
```   178 | "polynate (Neg p) = ~\<^sub>p (polynate p)"
```
```   179 | "polynate (Pw p n) = polynate p ^\<^sub>p n"
```
```   180 | "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
```
```   181 | "polynate (C c) = C (normNum c)"
```
```   182 by pat_completeness auto
```
```   183 termination by (relation "measure polysize") auto
```
```   184
```
```   185 fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly"
```
```   186 where
```
```   187   "poly_cmul y (C x) = C (y *\<^sub>N x)"
```
```   188 | "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
```
```   189 | "poly_cmul y p = C y *\<^sub>p p"
```
```   190
```
```   191 definition monic :: "poly \<Rightarrow> poly \<times> bool"
```
```   192 where
```
```   193   "monic p =
```
```   194     (let h = headconst p
```
```   195      in if h = 0\<^sub>N then (p, False) else (C (Ninv h) *\<^sub>p p, 0>\<^sub>N h))"
```
```   196
```
```   197
```
```   198 subsection {* Pseudo-division *}
```
```   199
```
```   200 definition shift1 :: "poly \<Rightarrow> poly"
```
```   201   where "shift1 p = CN 0\<^sub>p 0 p"
```
```   202
```
```   203 abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   204   where "funpow \<equiv> compow"
```
```   205
```
```   206 partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
```
```   207 where
```
```   208   "polydivide_aux a n p k s =
```
```   209     (if s = 0\<^sub>p then (k, s)
```
```   210      else
```
```   211       let
```
```   212         b = head s;
```
```   213         m = degree s
```
```   214       in
```
```   215         if m < n then (k,s)
```
```   216         else
```
```   217           let p' = funpow (m - n) shift1 p
```
```   218           in
```
```   219             if a = b then polydivide_aux a n p k (s -\<^sub>p p')
```
```   220             else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))"
```
```   221
```
```   222 definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
```
```   223   where "polydivide s p = polydivide_aux (head p) (degree p) p 0 s"
```
```   224
```
```   225 fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly"
```
```   226 where
```
```   227   "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
```
```   228 | "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
```
```   229
```
```   230 fun poly_deriv :: "poly \<Rightarrow> poly"
```
```   231 where
```
```   232   "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
```
```   233 | "poly_deriv p = 0\<^sub>p"
```
```   234
```
```   235
```
```   236 subsection{* Semantics of the polynomial representation *}
```
```   237
```
```   238 primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0,field_inverse_zero,power}"
```
```   239 where
```
```   240   "Ipoly bs (C c) = INum c"
```
```   241 | "Ipoly bs (Bound n) = bs!n"
```
```   242 | "Ipoly bs (Neg a) = - Ipoly bs a"
```
```   243 | "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
```
```   244 | "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
```
```   245 | "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
```
```   246 | "Ipoly bs (Pw t n) = Ipoly bs t ^ n"
```
```   247 | "Ipoly bs (CN c n p) = Ipoly bs c + (bs!n) * Ipoly bs p"
```
```   248
```
```   249 abbreviation Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0,field_inverse_zero,power}"
```
```   250     ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
```
```   251   where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"
```
```   252
```
```   253 lemma Ipoly_CInt: "Ipoly bs (C (i, 1)) = of_int i"
```
```   254   by (simp add: INum_def)
```
```   255
```
```   256 lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"
```
```   257   by (simp  add: INum_def)
```
```   258
```
```   259 lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
```
```   260
```
```   261
```
```   262 subsection {* Normal form and normalization *}
```
```   263
```
```   264 fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
```
```   265 where
```
```   266   "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
```
```   267 | "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)"
```
```   268 | "isnpolyh p = (\<lambda>k. False)"
```
```   269
```
```   270 lemma isnpolyh_mono: "n' \<le> n \<Longrightarrow> isnpolyh p n \<Longrightarrow> isnpolyh p n'"
```
```   271   by (induct p rule: isnpolyh.induct) auto
```
```   272
```
```   273 definition isnpoly :: "poly \<Rightarrow> bool"
```
```   274   where "isnpoly p = isnpolyh p 0"
```
```   275
```
```   276 text{* polyadd preserves normal forms *}
```
```   277
```
```   278 lemma polyadd_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polyadd p q) (min n0 n1)"
```
```   279 proof (induct p q arbitrary: n0 n1 rule: polyadd.induct)
```
```   280   case (2 ab c' n' p' n0 n1)
```
```   281   from 2 have  th1: "isnpolyh (C ab) (Suc n')"
```
```   282     by simp
```
```   283   from 2(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1"
```
```   284     by simp_all
```
```   285   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')"
```
```   286     by simp
```
```   287   with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^sub>p c') (Suc n')"
```
```   288     by simp
```
```   289   from nplen1 have n01len1: "min n0 n1 \<le> n'"
```
```   290     by simp
```
```   291   then show ?case using 2 th3
```
```   292     by simp
```
```   293 next
```
```   294   case (3 c' n' p' ab n1 n0)
```
```   295   from 3 have  th1: "isnpolyh (C ab) (Suc n')"
```
```   296     by simp
```
```   297   from 3(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1"
```
```   298     by simp_all
```
```   299   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')"
```
```   300     by simp
```
```   301   with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (Suc n')"
```
```   302     by simp
```
```   303   from nplen1 have n01len1: "min n0 n1 \<le> n'"
```
```   304     by simp
```
```   305   then show ?case using 3 th3
```
```   306     by simp
```
```   307 next
```
```   308   case (4 c n p c' n' p' n0 n1)
```
```   309   then have nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n"
```
```   310     by simp_all
```
```   311   from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'"
```
```   312     by simp_all
```
```   313   from 4 have ngen0: "n \<ge> n0"
```
```   314     by simp
```
```   315   from 4 have n'gen1: "n' \<ge> n1"
```
```   316     by simp
```
```   317   have "n < n' \<or> n' < n \<or> n = n'"
```
```   318     by auto
```
```   319   moreover
```
```   320   {
```
```   321     assume eq: "n = n'"
```
```   322     with "4.hyps"(3)[OF nc nc']
```
```   323     have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)"
```
```   324       by auto
```
```   325     then have ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
```
```   326       using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1
```
```   327       by auto
```
```   328     from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n"
```
```   329       by simp
```
```   330     have minle: "min n0 n1 \<le> n'"
```
```   331       using ngen0 n'gen1 eq by simp
```
```   332     from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' have ?case
```
```   333       by (simp add: Let_def)
```
```   334   }
```
```   335   moreover
```
```   336   {
```
```   337     assume lt: "n < n'"
```
```   338     have "min n0 n1 \<le> n0"
```
```   339       by simp
```
```   340     with 4 lt have th1:"min n0 n1 \<le> n"
```
```   341       by auto
```
```   342     from 4 have th21: "isnpolyh c (Suc n)"
```
```   343       by simp
```
```   344     from 4 have th22: "isnpolyh (CN c' n' p') n'"
```
```   345       by simp
```
```   346     from lt have th23: "min (Suc n) n' = Suc n"
```
```   347       by arith
```
```   348     from "4.hyps"(1)[OF th21 th22] have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)"
```
```   349       using th23 by simp
```
```   350     with 4 lt th1 have ?case
```
```   351       by simp
```
```   352   }
```
```   353   moreover
```
```   354   {
```
```   355     assume gt: "n' < n"
```
```   356     then have gt': "n' < n \<and> \<not> n < n'"
```
```   357       by simp
```
```   358     have "min n0 n1 \<le> n1"
```
```   359       by simp
```
```   360     with 4 gt have th1: "min n0 n1 \<le> n'"
```
```   361       by auto
```
```   362     from 4 have th21: "isnpolyh c' (Suc n')"
```
```   363       by simp_all
```
```   364     from 4 have th22: "isnpolyh (CN c n p) n"
```
```   365       by simp
```
```   366     from gt have th23: "min n (Suc n') = Suc n'"
```
```   367       by arith
```
```   368     from "4.hyps"(2)[OF th22 th21] have "isnpolyh (polyadd (CN c n p) c') (Suc n')"
```
```   369       using th23 by simp
```
```   370     with 4 gt th1 have ?case
```
```   371       by simp
```
```   372   }
```
```   373   ultimately show ?case by blast
```
```   374 qed auto
```
```   375
```
```   376 lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
```
```   377   by (induct p q rule: polyadd.induct)
```
```   378     (auto simp add: Let_def field_simps distrib_left[symmetric] simp del: distrib_left)
```
```   379
```
```   380 lemma polyadd_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polyadd p q)"
```
```   381   using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   382
```
```   383 text{* The degree of addition and other general lemmas needed for the normal form of polymul *}
```
```   384
```
```   385 lemma polyadd_different_degreen:
```
```   386   assumes "isnpolyh p n0"
```
```   387     and "isnpolyh q n1"
```
```   388     and "degreen p m \<noteq> degreen q m"
```
```   389     and "m \<le> min n0 n1"
```
```   390   shows "degreen (polyadd p q) m = max (degreen p m) (degreen q m)"
```
```   391   using assms
```
```   392 proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
```
```   393   case (4 c n p c' n' p' m n0 n1)
```
```   394   have "n' = n \<or> n < n' \<or> n' < n" by arith
```
```   395   then show ?case
```
```   396   proof (elim disjE)
```
```   397     assume [simp]: "n' = n"
```
```   398     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
```
```   399     show ?thesis by (auto simp: Let_def)
```
```   400   next
```
```   401     assume "n < n'"
```
```   402     with 4 show ?thesis by auto
```
```   403   next
```
```   404     assume "n' < n"
```
```   405     with 4 show ?thesis by auto
```
```   406   qed
```
```   407 qed auto
```
```   408
```
```   409 lemma headnz[simp]: "isnpolyh p n \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
```
```   410   by (induct p arbitrary: n rule: headn.induct) auto
```
```   411
```
```   412 lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
```
```   413   by (induct p arbitrary: n rule: degree.induct) auto
```
```   414
```
```   415 lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
```
```   416   by (induct p arbitrary: n rule: degreen.induct) auto
```
```   417
```
```   418 lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
```
```   419   by (induct p arbitrary: n rule: degree.induct) auto
```
```   420
```
```   421 lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
```
```   422   using degree_isnpolyh_Suc by auto
```
```   423
```
```   424 lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
```
```   425   using degreen_0 by auto
```
```   426
```
```   427
```
```   428 lemma degreen_polyadd:
```
```   429   assumes np: "isnpolyh p n0"
```
```   430     and nq: "isnpolyh q n1"
```
```   431     and m: "m \<le> max n0 n1"
```
```   432   shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
```
```   433   using np nq m
```
```   434 proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
```
```   435   case (2 c c' n' p' n0 n1)
```
```   436   then show ?case
```
```   437     by (cases n') simp_all
```
```   438 next
```
```   439   case (3 c n p c' n0 n1)
```
```   440   then show ?case
```
```   441     by (cases n) auto
```
```   442 next
```
```   443   case (4 c n p c' n' p' n0 n1 m)
```
```   444   have "n' = n \<or> n < n' \<or> n' < n" by arith
```
```   445   then show ?case
```
```   446   proof (elim disjE)
```
```   447     assume [simp]: "n' = n"
```
```   448     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
```
```   449     show ?thesis by (auto simp: Let_def)
```
```   450   qed simp_all
```
```   451 qed auto
```
```   452
```
```   453 lemma polyadd_eq_const_degreen:
```
```   454   assumes "isnpolyh p n0"
```
```   455     and "isnpolyh q n1"
```
```   456     and "polyadd p q = C c"
```
```   457   shows "degreen p m = degreen q m"
```
```   458   using assms
```
```   459 proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
```
```   460   case (4 c n p c' n' p' m n0 n1 x)
```
```   461   {
```
```   462     assume nn': "n' < n"
```
```   463     then have ?case using 4 by simp
```
```   464   }
```
```   465   moreover
```
```   466   {
```
```   467     assume nn': "\<not> n' < n"
```
```   468     then have "n < n' \<or> n = n'" by arith
```
```   469     moreover { assume "n < n'" with 4 have ?case by simp }
```
```   470     moreover
```
```   471     {
```
```   472       assume eq: "n = n'"
```
```   473       then have ?case using 4
```
```   474         apply (cases "p +\<^sub>p p' = 0\<^sub>p")
```
```   475         apply (auto simp add: Let_def)
```
```   476         done
```
```   477     }
```
```   478     ultimately have ?case by blast
```
```   479   }
```
```   480   ultimately show ?case by blast
```
```   481 qed simp_all
```
```   482
```
```   483 lemma polymul_properties:
```
```   484   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   485     and np: "isnpolyh p n0"
```
```   486     and nq: "isnpolyh q n1"
```
```   487     and m: "m \<le> min n0 n1"
```
```   488   shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   489     and "p *\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p \<or> q = 0\<^sub>p"
```
```   490     and "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)"
```
```   491   using np nq m
```
```   492 proof (induct p q arbitrary: n0 n1 m rule: polymul.induct)
```
```   493   case (2 c c' n' p')
```
```   494   {
```
```   495     case (1 n0 n1)
```
```   496     with "2.hyps"(4-6)[of n' n' n'] and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']
```
```   497     show ?case by (auto simp add: min_def)
```
```   498   next
```
```   499     case (2 n0 n1)
```
```   500     then show ?case by auto
```
```   501   next
```
```   502     case (3 n0 n1)
```
```   503     then show ?case  using "2.hyps" by auto
```
```   504   }
```
```   505 next
```
```   506   case (3 c n p c')
```
```   507   {
```
```   508     case (1 n0 n1)
```
```   509     with "3.hyps"(4-6)[of n n n] and "3.hyps"(1-3)[of "Suc n" "Suc n" n]
```
```   510     show ?case by (auto simp add: min_def)
```
```   511   next
```
```   512     case (2 n0 n1)
```
```   513     then show ?case by auto
```
```   514   next
```
```   515     case (3 n0 n1)
```
```   516     then show ?case  using "3.hyps" by auto
```
```   517   }
```
```   518 next
```
```   519   case (4 c n p c' n' p')
```
```   520   let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
```
```   521   {
```
```   522     case (1 n0 n1)
```
```   523     then have cnp: "isnpolyh ?cnp n"
```
```   524       and cnp': "isnpolyh ?cnp' n'"
```
```   525       and np: "isnpolyh p n"
```
```   526       and nc: "isnpolyh c (Suc n)"
```
```   527       and np': "isnpolyh p' n'"
```
```   528       and nc': "isnpolyh c' (Suc n')"
```
```   529       and nn0: "n \<ge> n0"
```
```   530       and nn1: "n' \<ge> n1"
```
```   531       by simp_all
```
```   532     {
```
```   533       assume "n < n'"
```
```   534       with "4.hyps"(4-5)[OF np cnp', of n] and "4.hyps"(1)[OF nc cnp', of n] nn0 cnp
```
```   535       have ?case by (simp add: min_def)
```
```   536     } moreover {
```
```   537       assume "n' < n"
```
```   538       with "4.hyps"(16-17)[OF cnp np', of "n'"] and "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
```
```   539       have ?case by (cases "Suc n' = n") (simp_all add: min_def)
```
```   540     } moreover {
```
```   541       assume "n' = n"
```
```   542       with "4.hyps"(16-17)[OF cnp np', of n] and "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0
```
```   543       have ?case
```
```   544         apply (auto intro!: polyadd_normh)
```
```   545         apply (simp_all add: min_def isnpolyh_mono[OF nn0])
```
```   546         done
```
```   547     }
```
```   548     ultimately show ?case by arith
```
```   549   next
```
```   550     fix n0 n1 m
```
```   551     assume np: "isnpolyh ?cnp n0"
```
```   552     assume np':"isnpolyh ?cnp' n1"
```
```   553     assume m: "m \<le> min n0 n1"
```
```   554     let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
```
```   555     let ?d1 = "degreen ?cnp m"
```
```   556     let ?d2 = "degreen ?cnp' m"
```
```   557     let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
```
```   558     have "n' < n \<or> n < n' \<or> n' = n" by auto
```
```   559     moreover
```
```   560     {
```
```   561       assume "n' < n \<or> n < n'"
```
```   562       with "4.hyps"(3,6,18) np np' m have ?eq
```
```   563         by auto
```
```   564     }
```
```   565     moreover
```
```   566     {
```
```   567       assume nn': "n' = n"
```
```   568       then have nn: "\<not> n' < n \<and> \<not> n < n'" by arith
```
```   569       from "4.hyps"(16,18)[of n n' n]
```
```   570         "4.hyps"(13,14)[of n "Suc n'" n]
```
```   571         np np' nn'
```
```   572       have norm:
```
```   573         "isnpolyh ?cnp n"
```
```   574         "isnpolyh c' (Suc n)"
```
```   575         "isnpolyh (?cnp *\<^sub>p c') n"
```
```   576         "isnpolyh p' n"
```
```   577         "isnpolyh (?cnp *\<^sub>p p') n"
```
```   578         "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   579         "?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p"
```
```   580         "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
```
```   581         by (auto simp add: min_def)
```
```   582       {
```
```   583         assume mn: "m = n"
```
```   584         from "4.hyps"(17,18)[OF norm(1,4), of n]
```
```   585           "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
```
```   586         have degs:
```
```   587           "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else ?d1 + degreen c' n)"
```
```   588           "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n"
```
```   589           by (simp_all add: min_def)
```
```   590         from degs norm have th1: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   591           by simp
```
```   592         then have neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   593           by simp
```
```   594         have nmin: "n \<le> min n n"
```
```   595           by (simp add: min_def)
```
```   596         from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
```
```   597         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 =
```
```   598             degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   599           by simp
```
```   600         from "4.hyps"(16-18)[OF norm(1,4), of n]
```
```   601           "4.hyps"(13-15)[OF norm(1,2), of n]
```
```   602           mn norm m nn' deg
```
```   603         have ?eq by simp
```
```   604       }
```
```   605       moreover
```
```   606       {
```
```   607         assume mn: "m \<noteq> n"
```
```   608         then have mn': "m < n"
```
```   609           using m np by auto
```
```   610         from nn' m np have max1: "m \<le> max n n"
```
```   611           by simp
```
```   612         then have min1: "m \<le> min n n"
```
```   613           by simp
```
```   614         then have min2: "m \<le> min n (Suc n)"
```
```   615           by simp
```
```   616         from "4.hyps"(16-18)[OF norm(1,4) min1]
```
```   617           "4.hyps"(13-15)[OF norm(1,2) min2]
```
```   618           degreen_polyadd[OF norm(3,6) max1]
```
```   619         have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m \<le>
```
```   620             max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
```
```   621           using mn nn' np np' by simp
```
```   622         with "4.hyps"(16-18)[OF norm(1,4) min1]
```
```   623           "4.hyps"(13-15)[OF norm(1,2) min2]
```
```   624           degreen_0[OF norm(3) mn']
```
```   625         have ?eq using nn' mn np np' by clarsimp
```
```   626       }
```
```   627       ultimately have ?eq by blast
```
```   628     }
```
```   629     ultimately show ?eq by blast
```
```   630   }
```
```   631   {
```
```   632     case (2 n0 n1)
```
```   633     then have np: "isnpolyh ?cnp n0"
```
```   634       and np': "isnpolyh ?cnp' n1"
```
```   635       and m: "m \<le> min n0 n1"
```
```   636       by simp_all
```
```   637     then have mn: "m \<le> n" by simp
```
```   638     let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
```
```   639     {
```
```   640       assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
```
```   641       then have nn: "\<not> n' < n \<and> \<not> n < n'"
```
```   642         by simp
```
```   643       from "4.hyps"(16-18) [of n n n]
```
```   644         "4.hyps"(13-15)[of n "Suc n" n]
```
```   645         np np' C(2) mn
```
```   646       have norm:
```
```   647         "isnpolyh ?cnp n"
```
```   648         "isnpolyh c' (Suc n)"
```
```   649         "isnpolyh (?cnp *\<^sub>p c') n"
```
```   650         "isnpolyh p' n"
```
```   651         "isnpolyh (?cnp *\<^sub>p p') n"
```
```   652         "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   653         "?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p"
```
```   654         "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
```
```   655         "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
```
```   656         "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
```
```   657         by (simp_all add: min_def)
```
```   658       from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   659         by simp
```
```   660       have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   661         using norm by simp
```
```   662       from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"] degneq
```
```   663       have False by simp
```
```   664     }
```
```   665     then show ?case using "4.hyps" by clarsimp
```
```   666   }
```
```   667 qed auto
```
```   668
```
```   669 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = Ipoly bs p * Ipoly bs q"
```
```   670   by (induct p q rule: polymul.induct) (auto simp add: field_simps)
```
```   671
```
```   672 lemma polymul_normh:
```
```   673   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   674   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   675   using polymul_properties(1) by blast
```
```   676
```
```   677 lemma polymul_eq0_iff:
```
```   678   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   679   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p *\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p \<or> q = 0\<^sub>p"
```
```   680   using polymul_properties(2) by blast
```
```   681
```
```   682 lemma polymul_degreen:
```
```   683   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   684   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> m \<le> min n0 n1 \<Longrightarrow>
```
```   685     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)"
```
```   686   by (fact polymul_properties(3))
```
```   687
```
```   688 lemma polymul_norm:
```
```   689   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   690   shows "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polymul p q)"
```
```   691   using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   692
```
```   693 lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
```
```   694   by (induct p arbitrary: n0 rule: headconst.induct) auto
```
```   695
```
```   696 lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
```
```   697   by (induct p arbitrary: n0) auto
```
```   698
```
```   699 lemma monic_eqI:
```
```   700   assumes np: "isnpolyh p n0"
```
```   701   shows "INum (headconst p) * Ipoly bs (fst (monic p)) =
```
```   702     (Ipoly bs p ::'a::{field_char_0,field_inverse_zero, power})"
```
```   703   unfolding monic_def Let_def
```
```   704 proof (cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
```
```   705   let ?h = "headconst p"
```
```   706   assume pz: "p \<noteq> 0\<^sub>p"
```
```   707   {
```
```   708     assume hz: "INum ?h = (0::'a)"
```
```   709     from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N"
```
```   710       by simp_all
```
```   711     from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N"
```
```   712       by simp
```
```   713     with headconst_zero[OF np] have "p = 0\<^sub>p"
```
```   714       by blast
```
```   715     with pz have False
```
```   716       by blast
```
```   717   }
```
```   718   then show "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
```
```   719     by blast
```
```   720 qed
```
```   721
```
```   722
```
```   723 text{* polyneg is a negation and preserves normal forms *}
```
```   724
```
```   725 lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
```
```   726   by (induct p rule: polyneg.induct) auto
```
```   727
```
```   728 lemma polyneg0: "isnpolyh p n \<Longrightarrow> (~\<^sub>p p) = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
```
```   729   by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: Nneg_def)
```
```   730
```
```   731 lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
```
```   732   by (induct p arbitrary: n0 rule: polyneg.induct) auto
```
```   733
```
```   734 lemma polyneg_normh: "isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n"
```
```   735   by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: polyneg0)
```
```   736
```
```   737 lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
```
```   738   using isnpoly_def polyneg_normh by simp
```
```   739
```
```   740
```
```   741 text{* polysub is a substraction and preserves normal forms *}
```
```   742
```
```   743 lemma polysub[simp]: "Ipoly bs (polysub p q) = Ipoly bs p - Ipoly bs q"
```
```   744   by (simp add: polysub_def)
```
```   745
```
```   746 lemma polysub_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
```
```   747   by (simp add: polysub_def polyneg_normh polyadd_normh)
```
```   748
```
```   749 lemma polysub_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polysub p q)"
```
```   750   using polyadd_norm polyneg_norm by (simp add: polysub_def)
```
```   751
```
```   752 lemma polysub_same_0[simp]:
```
```   753   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   754   shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p"
```
```   755   unfolding polysub_def split_def fst_conv snd_conv
```
```   756   by (induct p arbitrary: n0) (auto simp add: Let_def Nsub0[simplified Nsub_def])
```
```   757
```
```   758 lemma polysub_0:
```
```   759   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   760   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p -\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = q"
```
```   761   unfolding polysub_def split_def fst_conv snd_conv
```
```   762   by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
```
```   763     (auto simp: Nsub0[simplified Nsub_def] Let_def)
```
```   764
```
```   765 text{* polypow is a power function and preserves normal forms *}
```
```   766
```
```   767 lemma polypow[simp]:
```
```   768   "Ipoly bs (polypow n p) = (Ipoly bs p :: 'a::{field_char_0,field_inverse_zero}) ^ n"
```
```   769 proof (induct n rule: polypow.induct)
```
```   770   case 1
```
```   771   then show ?case
```
```   772     by simp
```
```   773 next
```
```   774   case (2 n)
```
```   775   let ?q = "polypow ((Suc n) div 2) p"
```
```   776   let ?d = "polymul ?q ?q"
```
```   777   have "odd (Suc n) \<or> even (Suc n)"
```
```   778     by simp
```
```   779   moreover
```
```   780   {
```
```   781     assume odd: "odd (Suc n)"
```
```   782     have th: "(Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0)))) = Suc n div 2 + Suc n div 2 + 1"
```
```   783       by arith
```
```   784     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)"
```
```   785       by (simp add: Let_def)
```
```   786     also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2) * (Ipoly bs p)^(Suc n div 2)"
```
```   787       using "2.hyps" by simp
```
```   788     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
```
```   789       by (simp only: power_add power_one_right) simp
```
```   790     also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0))))"
```
```   791       by (simp only: th)
```
```   792     finally have ?case
```
```   793     using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp
```
```   794   }
```
```   795   moreover
```
```   796   {
```
```   797     assume even: "even (Suc n)"
```
```   798     have th: "(Suc (Suc 0)) * (Suc n div Suc (Suc 0)) = Suc n div 2 + Suc n div 2"
```
```   799       by arith
```
```   800     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d"
```
```   801       by (simp add: Let_def)
```
```   802     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
```
```   803       using "2.hyps" by (simp only: power_add) simp
```
```   804     finally have ?case using even_nat_div_two_times_two[OF even]
```
```   805       by (simp only: th)
```
```   806   }
```
```   807   ultimately show ?case by blast
```
```   808 qed
```
```   809
```
```   810 lemma polypow_normh:
```
```   811   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   812   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
```
```   813 proof (induct k arbitrary: n rule: polypow.induct)
```
```   814   case 1
```
```   815   then show ?case by auto
```
```   816 next
```
```   817   case (2 k n)
```
```   818   let ?q = "polypow (Suc k div 2) p"
```
```   819   let ?d = "polymul ?q ?q"
```
```   820   from 2 have th1: "isnpolyh ?q n" and th2: "isnpolyh p n"
```
```   821     by blast+
```
```   822   from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n"
```
```   823     by simp
```
```   824   from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n"
```
```   825     by simp
```
```   826   from dn on show ?case
```
```   827     by (simp add: Let_def)
```
```   828 qed
```
```   829
```
```   830 lemma polypow_norm:
```
```   831   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   832   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
```
```   833   by (simp add: polypow_normh isnpoly_def)
```
```   834
```
```   835 text{* Finally the whole normalization *}
```
```   836
```
```   837 lemma polynate [simp]:
```
```   838   "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0,field_inverse_zero})"
```
```   839   by (induct p rule:polynate.induct) auto
```
```   840
```
```   841 lemma polynate_norm[simp]:
```
```   842   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   843   shows "isnpoly (polynate p)"
```
```   844   by (induct p rule: polynate.induct)
```
```   845      (simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm,
```
```   846       simp_all add: isnpoly_def)
```
```   847
```
```   848 text{* shift1 *}
```
```   849
```
```   850
```
```   851 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
```
```   852   by (simp add: shift1_def)
```
```   853
```
```   854 lemma shift1_isnpoly:
```
```   855   assumes "isnpoly p"
```
```   856     and "p \<noteq> 0\<^sub>p"
```
```   857   shows "isnpoly (shift1 p) "
```
```   858   using assms by (simp add: shift1_def isnpoly_def)
```
```   859
```
```   860 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
```
```   861   by (simp add: shift1_def)
```
```   862
```
```   863 lemma funpow_shift1_isnpoly: "isnpoly p \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpoly (funpow n shift1 p)"
```
```   864   by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
```
```   865
```
```   866 lemma funpow_isnpolyh:
```
```   867   assumes "\<And>p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n"
```
```   868     and "isnpolyh p n"
```
```   869   shows "isnpolyh (funpow k f p) n"
```
```   870   using assms by (induct k arbitrary: p) auto
```
```   871
```
```   872 lemma funpow_shift1:
```
```   873   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field_inverse_zero}) =
```
```   874     Ipoly bs (Mul (Pw (Bound 0) n) p)"
```
```   875   by (induct n arbitrary: p) (simp_all add: shift1_isnpoly shift1)
```
```   876
```
```   877 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
```
```   878   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
```
```   879
```
```   880 lemma funpow_shift1_1:
```
```   881   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field_inverse_zero}) =
```
```   882     Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)"
```
```   883   by (simp add: funpow_shift1)
```
```   884
```
```   885 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
```
```   886   by (induct p rule: poly_cmul.induct) (auto simp add: field_simps)
```
```   887
```
```   888 lemma behead:
```
```   889   assumes "isnpolyh p n"
```
```   890   shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) =
```
```   891     (Ipoly bs p :: 'a :: {field_char_0,field_inverse_zero})"
```
```   892   using assms
```
```   893 proof (induct p arbitrary: n rule: behead.induct)
```
```   894   case (1 c p n)
```
```   895   then have pn: "isnpolyh p n" by simp
```
```   896   from 1(1)[OF pn]
```
```   897   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
```
```   898   then show ?case using "1.hyps"
```
```   899     apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
```
```   900     apply (simp_all add: th[symmetric] field_simps)
```
```   901     done
```
```   902 qed (auto simp add: Let_def)
```
```   903
```
```   904 lemma behead_isnpolyh:
```
```   905   assumes "isnpolyh p n"
```
```   906   shows "isnpolyh (behead p) n"
```
```   907   using assms by (induct p rule: behead.induct) (auto simp add: Let_def isnpolyh_mono)
```
```   908
```
```   909
```
```   910 subsection {* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
```
```   911
```
```   912 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
```
```   913 proof (induct p arbitrary: n rule: poly.induct, auto)
```
```   914   case (goal1 c n p n')
```
```   915   then have "n = Suc (n - 1)"
```
```   916     by simp
```
```   917   then have "isnpolyh p (Suc (n - 1))"
```
```   918     using `isnpolyh p n` by simp
```
```   919   with goal1(2) show ?case
```
```   920     by simp
```
```   921 qed
```
```   922
```
```   923 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
```
```   924   by (induct p arbitrary: n0 rule: isconstant.induct) (auto simp add: isnpolyh_polybound0)
```
```   925
```
```   926 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
```
```   927   by (induct p) auto
```
```   928
```
```   929 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
```
```   930   apply (induct p arbitrary: n0)
```
```   931   apply auto
```
```   932   apply atomize
```
```   933   apply (erule_tac x = "Suc nat" in allE)
```
```   934   apply auto
```
```   935   done
```
```   936
```
```   937 lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
```
```   938   by (induct p  arbitrary: n0 rule: head.induct) (auto intro: isnpolyh_polybound0)
```
```   939
```
```   940 lemma polybound0_I:
```
```   941   assumes "polybound0 a"
```
```   942   shows "Ipoly (b # bs) a = Ipoly (b' # bs) a"
```
```   943   using assms by (induct a rule: poly.induct) auto
```
```   944
```
```   945 lemma polysubst0_I: "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b # bs) a) # bs) t"
```
```   946   by (induct t) simp_all
```
```   947
```
```   948 lemma polysubst0_I':
```
```   949   assumes "polybound0 a"
```
```   950   shows "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b' # bs) a) # bs) t"
```
```   951   by (induct t) (simp_all add: polybound0_I[OF assms, where b="b" and b'="b'"])
```
```   952
```
```   953 lemma decrpoly:
```
```   954   assumes "polybound0 t"
```
```   955   shows "Ipoly (x # bs) t = Ipoly bs (decrpoly t)"
```
```   956   using assms by (induct t rule: decrpoly.induct) simp_all
```
```   957
```
```   958 lemma polysubst0_polybound0:
```
```   959   assumes "polybound0 t"
```
```   960   shows "polybound0 (polysubst0 t a)"
```
```   961   using assms by (induct a rule: poly.induct) auto
```
```   962
```
```   963 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
```
```   964   by (induct p arbitrary: n rule: degree.induct) (auto simp add: isnpolyh_polybound0)
```
```   965
```
```   966 primrec maxindex :: "poly \<Rightarrow> nat"
```
```   967 where
```
```   968   "maxindex (Bound n) = n + 1"
```
```   969 | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
```
```   970 | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
```
```   971 | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
```
```   972 | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
```
```   973 | "maxindex (Neg p) = maxindex p"
```
```   974 | "maxindex (Pw p n) = maxindex p"
```
```   975 | "maxindex (C x) = 0"
```
```   976
```
```   977 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool"
```
```   978   where "wf_bs bs p \<longleftrightarrow> length bs \<ge> maxindex p"
```
```   979
```
```   980 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall>c \<in> set (coefficients p). wf_bs bs c"
```
```   981 proof (induct p rule: coefficients.induct)
```
```   982   case (1 c p)
```
```   983   show ?case
```
```   984   proof
```
```   985     fix x
```
```   986     assume xc: "x \<in> set (coefficients (CN c 0 p))"
```
```   987     then have "x = c \<or> x \<in> set (coefficients p)"
```
```   988       by simp
```
```   989     moreover
```
```   990     {
```
```   991       assume "x = c"
```
```   992       then have "wf_bs bs x"
```
```   993         using "1.prems" unfolding wf_bs_def by simp
```
```   994     }
```
```   995     moreover
```
```   996     {
```
```   997       assume H: "x \<in> set (coefficients p)"
```
```   998       from "1.prems" have "wf_bs bs p"
```
```   999         unfolding wf_bs_def by simp
```
```  1000       with "1.hyps" H have "wf_bs bs x"
```
```  1001         by blast
```
```  1002     }
```
```  1003     ultimately show "wf_bs bs x"
```
```  1004       by blast
```
```  1005   qed
```
```  1006 qed simp_all
```
```  1007
```
```  1008 lemma maxindex_coefficients: "\<forall>c \<in> set (coefficients p). maxindex c \<le> maxindex p"
```
```  1009   by (induct p rule: coefficients.induct) auto
```
```  1010
```
```  1011 lemma wf_bs_I: "wf_bs bs p \<Longrightarrow> Ipoly (bs @ bs') p = Ipoly bs p"
```
```  1012   unfolding wf_bs_def by (induct p) (auto simp add: nth_append)
```
```  1013
```
```  1014 lemma take_maxindex_wf:
```
```  1015   assumes wf: "wf_bs bs p"
```
```  1016   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
```
```  1017 proof -
```
```  1018   let ?ip = "maxindex p"
```
```  1019   let ?tbs = "take ?ip bs"
```
```  1020   from wf have "length ?tbs = ?ip"
```
```  1021     unfolding wf_bs_def by simp
```
```  1022   then have wf': "wf_bs ?tbs p"
```
```  1023     unfolding wf_bs_def by  simp
```
```  1024   have eq: "bs = ?tbs @ drop ?ip bs"
```
```  1025     by simp
```
```  1026   from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis
```
```  1027     using eq by simp
```
```  1028 qed
```
```  1029
```
```  1030 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
```
```  1031   by (induct p) auto
```
```  1032
```
```  1033 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
```
```  1034   unfolding wf_bs_def by simp
```
```  1035
```
```  1036 lemma wf_bs_insensitive': "wf_bs (x # bs) p = wf_bs (y # bs) p"
```
```  1037   unfolding wf_bs_def by simp
```
```  1038
```
```  1039 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x # bs) p"
```
```  1040   by (induct p rule: coefficients.induct) (auto simp add: wf_bs_def)
```
```  1041
```
```  1042 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
```
```  1043   by (induct p rule: coefficients.induct) simp_all
```
```  1044
```
```  1045 lemma coefficients_head: "last (coefficients p) = head p"
```
```  1046   by (induct p rule: coefficients.induct) auto
```
```  1047
```
```  1048 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x # bs) p"
```
```  1049   unfolding wf_bs_def by (induct p rule: decrpoly.induct) auto
```
```  1050
```
```  1051 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists>ys. length (xs @ ys) = n"
```
```  1052   apply (rule exI[where x="replicate (n - length xs) z" for z])
```
```  1053   apply simp
```
```  1054   done
```
```  1055
```
```  1056 lemma isnpolyh_Suc_const: "isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
```
```  1057   apply (cases p)
```
```  1058   apply auto
```
```  1059   apply (case_tac "nat")
```
```  1060   apply simp_all
```
```  1061   done
```
```  1062
```
```  1063 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
```
```  1064   unfolding wf_bs_def by (induct p q rule: polyadd.induct) (auto simp add: Let_def)
```
```  1065
```
```  1066 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
```
```  1067   unfolding wf_bs_def
```
```  1068   apply (induct p q arbitrary: bs rule: polymul.induct)
```
```  1069   apply (simp_all add: wf_bs_polyadd)
```
```  1070   apply clarsimp
```
```  1071   apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
```
```  1072   apply auto
```
```  1073   done
```
```  1074
```
```  1075 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
```
```  1076   unfolding wf_bs_def by (induct p rule: polyneg.induct) auto
```
```  1077
```
```  1078 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
```
```  1079   unfolding polysub_def split_def fst_conv snd_conv
```
```  1080   using wf_bs_polyadd wf_bs_polyneg by blast
```
```  1081
```
```  1082
```
```  1083 subsection {* Canonicity of polynomial representation, see lemma isnpolyh_unique *}
```
```  1084
```
```  1085 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
```
```  1086 definition "polypoly' bs p = map (Ipoly bs \<circ> decrpoly) (coefficients p)"
```
```  1087 definition "poly_nate bs p = map (Ipoly bs \<circ> decrpoly) (coefficients (polynate p))"
```
```  1088
```
```  1089 lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall>q \<in> set (coefficients p). isnpolyh q n0"
```
```  1090 proof (induct p arbitrary: n0 rule: coefficients.induct)
```
```  1091   case (1 c p n0)
```
```  1092   have cp: "isnpolyh (CN c 0 p) n0"
```
```  1093     by fact
```
```  1094   then have norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
```
```  1095     by (auto simp add: isnpolyh_mono[where n'=0])
```
```  1096   from "1.hyps"[OF norm(2)] norm(1) norm(4) show ?case
```
```  1097     by simp
```
```  1098 qed auto
```
```  1099
```
```  1100 lemma coefficients_isconst: "isnpolyh p n \<Longrightarrow> \<forall>q \<in> set (coefficients p). isconstant q"
```
```  1101   by (induct p arbitrary: n rule: coefficients.induct) (auto simp add: isnpolyh_Suc_const)
```
```  1102
```
```  1103 lemma polypoly_polypoly':
```
```  1104   assumes np: "isnpolyh p n0"
```
```  1105   shows "polypoly (x # bs) p = polypoly' bs p"
```
```  1106 proof -
```
```  1107   let ?cf = "set (coefficients p)"
```
```  1108   from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
```
```  1109   {
```
```  1110     fix q
```
```  1111     assume q: "q \<in> ?cf"
```
```  1112     from q cn_norm have th: "isnpolyh q n0"
```
```  1113       by blast
```
```  1114     from coefficients_isconst[OF np] q have "isconstant q"
```
```  1115       by blast
```
```  1116     with isconstant_polybound0[OF th] have "polybound0 q"
```
```  1117       by blast
```
```  1118   }
```
```  1119   then have "\<forall>q \<in> ?cf. polybound0 q" ..
```
```  1120   then have "\<forall>q \<in> ?cf. Ipoly (x # bs) q = Ipoly bs (decrpoly q)"
```
```  1121     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
```
```  1122     by auto
```
```  1123   then show ?thesis
```
```  1124     unfolding polypoly_def polypoly'_def by simp
```
```  1125 qed
```
```  1126
```
```  1127 lemma polypoly_poly:
```
```  1128   assumes "isnpolyh p n0"
```
```  1129   shows "Ipoly (x # bs) p = poly (polypoly (x # bs) p) x"
```
```  1130   using assms
```
```  1131   by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def)
```
```  1132
```
```  1133 lemma polypoly'_poly:
```
```  1134   assumes "isnpolyh p n0"
```
```  1135   shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
```
```  1136   using polypoly_poly[OF assms, simplified polypoly_polypoly'[OF assms]] .
```
```  1137
```
```  1138
```
```  1139 lemma polypoly_poly_polybound0:
```
```  1140   assumes "isnpolyh p n0"
```
```  1141     and "polybound0 p"
```
```  1142   shows "polypoly bs p = [Ipoly bs p]"
```
```  1143   using assms
```
```  1144   unfolding polypoly_def
```
```  1145   apply (cases p)
```
```  1146   apply auto
```
```  1147   apply (case_tac nat)
```
```  1148   apply auto
```
```  1149   done
```
```  1150
```
```  1151 lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
```
```  1152   by (induct p rule: head.induct) auto
```
```  1153
```
```  1154 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> headn p m = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
```
```  1155   by (cases p) auto
```
```  1156
```
```  1157 lemma head_eq_headn0: "head p = headn p 0"
```
```  1158   by (induct p rule: head.induct) simp_all
```
```  1159
```
```  1160 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> head p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
```
```  1161   by (simp add: head_eq_headn0)
```
```  1162
```
```  1163 lemma isnpolyh_zero_iff:
```
```  1164   assumes nq: "isnpolyh p n0"
```
```  1165     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})"
```
```  1166   shows "p = 0\<^sub>p"
```
```  1167   using nq eq
```
```  1168 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
```
```  1169   case less
```
```  1170   note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
```
```  1171   {
```
```  1172     assume nz: "maxindex p = 0"
```
```  1173     then obtain c where "p = C c"
```
```  1174       using np by (cases p) auto
```
```  1175     with zp np have "p = 0\<^sub>p"
```
```  1176       unfolding wf_bs_def by simp
```
```  1177   }
```
```  1178   moreover
```
```  1179   {
```
```  1180     assume nz: "maxindex p \<noteq> 0"
```
```  1181     let ?h = "head p"
```
```  1182     let ?hd = "decrpoly ?h"
```
```  1183     let ?ihd = "maxindex ?hd"
```
```  1184     from head_isnpolyh[OF np] head_polybound0[OF np]
```
```  1185     have h: "isnpolyh ?h n0" "polybound0 ?h"
```
```  1186       by simp_all
```
```  1187     then have nhd: "isnpolyh ?hd (n0 - 1)"
```
```  1188       using decrpoly_normh by blast
```
```  1189
```
```  1190     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
```
```  1191     have mihn: "maxindex ?h \<le> maxindex p"
```
```  1192       by auto
```
```  1193     with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < maxindex p"
```
```  1194       by auto
```
```  1195     {
```
```  1196       fix bs :: "'a list"
```
```  1197       assume bs: "wf_bs bs ?hd"
```
```  1198       let ?ts = "take ?ihd bs"
```
```  1199       let ?rs = "drop ?ihd bs"
```
```  1200       have ts: "wf_bs ?ts ?hd"
```
```  1201         using bs unfolding wf_bs_def by simp
```
```  1202       have bs_ts_eq: "?ts @ ?rs = bs"
```
```  1203         by simp
```
```  1204       from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x # ?ts) ?h"
```
```  1205         by simp
```
```  1206       from ihd_lt_n have "\<forall>x. length (x # ?ts) \<le> maxindex p"
```
```  1207         by simp
```
```  1208       with length_le_list_ex obtain xs where xs: "length ((x # ?ts) @ xs) = maxindex p"
```
```  1209         by blast
```
```  1210       then have "\<forall>x. wf_bs ((x # ?ts) @ xs) p"
```
```  1211         unfolding wf_bs_def by simp
```
```  1212       with zp have "\<forall>x. Ipoly ((x # ?ts) @ xs) p = 0"
```
```  1213         by blast
```
```  1214       then have "\<forall>x. Ipoly (x # (?ts @ xs)) p = 0"
```
```  1215         by simp
```
```  1216       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
```
```  1217       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"
```
```  1218         by simp
```
```  1219       then have "poly (polypoly' (?ts @ xs) p) = poly []"
```
```  1220         by auto
```
```  1221       then have "\<forall>c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
```
```  1222         using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
```
```  1223       with coefficients_head[of p, symmetric]
```
```  1224       have th0: "Ipoly (?ts @ xs) ?hd = 0"
```
```  1225         by simp
```
```  1226       from bs have wf'': "wf_bs ?ts ?hd"
```
```  1227         unfolding wf_bs_def by simp
```
```  1228       with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0"
```
```  1229         by simp
```
```  1230       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
```
```  1231         by simp
```
```  1232     }
```
```  1233     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
```
```  1234       by blast
```
```  1235     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p"
```
```  1236       by blast
```
```  1237     then have "?h = 0\<^sub>p" by simp
```
```  1238     with head_nz[OF np] have "p = 0\<^sub>p" by simp
```
```  1239   }
```
```  1240   ultimately show "p = 0\<^sub>p"
```
```  1241     by blast
```
```  1242 qed
```
```  1243
```
```  1244 lemma isnpolyh_unique:
```
```  1245   assumes np: "isnpolyh p n0"
```
```  1246     and nq: "isnpolyh q n1"
```
```  1247   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"
```
```  1248 proof auto
```
```  1249   assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a) = \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
```
```  1250   then have "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)"
```
```  1251     by simp
```
```  1252   then have "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
```
```  1253     using wf_bs_polysub[where p=p and q=q] by auto
```
```  1254   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq] show "p = q"
```
```  1255     by blast
```
```  1256 qed
```
```  1257
```
```  1258
```
```  1259 text{* consequences of unicity on the algorithms for polynomial normalization *}
```
```  1260
```
```  1261 lemma polyadd_commute:
```
```  1262   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1263     and np: "isnpolyh p n0"
```
```  1264     and nq: "isnpolyh q n1"
```
```  1265   shows "p +\<^sub>p q = q +\<^sub>p p"
```
```  1266   using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]]
```
```  1267   by simp
```
```  1268
```
```  1269 lemma zero_normh: "isnpolyh 0\<^sub>p n"
```
```  1270   by simp
```
```  1271
```
```  1272 lemma one_normh: "isnpolyh (1)\<^sub>p n"
```
```  1273   by simp
```
```  1274
```
```  1275 lemma polyadd_0[simp]:
```
```  1276   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1277     and np: "isnpolyh p n0"
```
```  1278   shows "p +\<^sub>p 0\<^sub>p = p"
```
```  1279     and "0\<^sub>p +\<^sub>p p = p"
```
```  1280   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
```
```  1281     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
```
```  1282
```
```  1283 lemma polymul_1[simp]:
```
```  1284   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1285     and np: "isnpolyh p n0"
```
```  1286   shows "p *\<^sub>p (1)\<^sub>p = p"
```
```  1287     and "(1)\<^sub>p *\<^sub>p p = p"
```
```  1288   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
```
```  1289     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
```
```  1290
```
```  1291 lemma polymul_0[simp]:
```
```  1292   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1293     and np: "isnpolyh p n0"
```
```  1294   shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p"
```
```  1295     and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
```
```  1296   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
```
```  1297     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
```
```  1298
```
```  1299 lemma polymul_commute:
```
```  1300   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1301     and np: "isnpolyh p n0"
```
```  1302     and nq: "isnpolyh q n1"
```
```  1303   shows "p *\<^sub>p q = q *\<^sub>p p"
```
```  1304   using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np],
```
```  1305     where ?'a = "'a::{field_char_0,field_inverse_zero, power}"]
```
```  1306   by simp
```
```  1307
```
```  1308 declare polyneg_polyneg [simp]
```
```  1309
```
```  1310 lemma isnpolyh_polynate_id [simp]:
```
```  1311   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1312     and np: "isnpolyh p n0"
```
```  1313   shows "polynate p = p"
```
```  1314   using isnpolyh_unique[where ?'a= "'a::{field_char_0,field_inverse_zero}",
```
```  1315       OF polynate_norm[of p, unfolded isnpoly_def] np]
```
```  1316     polynate[where ?'a = "'a::{field_char_0,field_inverse_zero}"]
```
```  1317   by simp
```
```  1318
```
```  1319 lemma polynate_idempotent[simp]:
```
```  1320   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1321   shows "polynate (polynate p) = polynate p"
```
```  1322   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
```
```  1323
```
```  1324 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
```
```  1325   unfolding poly_nate_def polypoly'_def ..
```
```  1326
```
```  1327 lemma poly_nate_poly:
```
```  1328   "poly (poly_nate bs p) = (\<lambda>x:: 'a ::{field_char_0,field_inverse_zero}. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
```
```  1329   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
```
```  1330   unfolding poly_nate_polypoly' by auto
```
```  1331
```
```  1332
```
```  1333 subsection{* heads, degrees and all that *}
```
```  1334
```
```  1335 lemma degree_eq_degreen0: "degree p = degreen p 0"
```
```  1336   by (induct p rule: degree.induct) simp_all
```
```  1337
```
```  1338 lemma degree_polyneg:
```
```  1339   assumes "isnpolyh p n"
```
```  1340   shows "degree (polyneg p) = degree p"
```
```  1341   apply (induct p rule: polyneg.induct)
```
```  1342   using assms
```
```  1343   apply simp_all
```
```  1344   apply (case_tac na)
```
```  1345   apply auto
```
```  1346   done
```
```  1347
```
```  1348 lemma degree_polyadd:
```
```  1349   assumes np: "isnpolyh p n0"
```
```  1350     and nq: "isnpolyh q n1"
```
```  1351   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1352   using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
```
```  1353
```
```  1354
```
```  1355 lemma degree_polysub:
```
```  1356   assumes np: "isnpolyh p n0"
```
```  1357     and nq: "isnpolyh q n1"
```
```  1358   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1359 proof-
```
```  1360   from nq have nq': "isnpolyh (~\<^sub>p q) n1"
```
```  1361     using polyneg_normh by simp
```
```  1362   from degree_polyadd[OF np nq'] show ?thesis
```
```  1363     by (simp add: polysub_def degree_polyneg[OF nq])
```
```  1364 qed
```
```  1365
```
```  1366 lemma degree_polysub_samehead:
```
```  1367   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1368     and np: "isnpolyh p n0"
```
```  1369     and nq: "isnpolyh q n1"
```
```  1370     and h: "head p = head q"
```
```  1371     and d: "degree p = degree q"
```
```  1372   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
```
```  1373   unfolding polysub_def split_def fst_conv snd_conv
```
```  1374   using np nq h d
```
```  1375 proof (induct p q rule: polyadd.induct)
```
```  1376   case (1 c c')
```
```  1377   then show ?case
```
```  1378     by (simp add: Nsub_def Nsub0[simplified Nsub_def])
```
```  1379 next
```
```  1380   case (2 c c' n' p')
```
```  1381   from 2 have "degree (C c) = degree (CN c' n' p')"
```
```  1382     by simp
```
```  1383   then have nz: "n' > 0"
```
```  1384     by (cases n') auto
```
```  1385   then have "head (CN c' n' p') = CN c' n' p'"
```
```  1386     by (cases n') auto
```
```  1387   with 2 show ?case
```
```  1388     by simp
```
```  1389 next
```
```  1390   case (3 c n p c')
```
```  1391   then have "degree (C c') = degree (CN c n p)"
```
```  1392     by simp
```
```  1393   then have nz: "n > 0"
```
```  1394     by (cases n) auto
```
```  1395   then have "head (CN c n p) = CN c n p"
```
```  1396     by (cases n) auto
```
```  1397   with 3 show ?case by simp
```
```  1398 next
```
```  1399   case (4 c n p c' n' p')
```
```  1400   then have H:
```
```  1401     "isnpolyh (CN c n p) n0"
```
```  1402     "isnpolyh (CN c' n' p') n1"
```
```  1403     "head (CN c n p) = head (CN c' n' p')"
```
```  1404     "degree (CN c n p) = degree (CN c' n' p')"
```
```  1405     by simp_all
```
```  1406   then have degc: "degree c = 0" and degc': "degree c' = 0"
```
```  1407     by simp_all
```
```  1408   then have degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
```
```  1409     using H(1-2) degree_polyneg by auto
```
```  1410   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"
```
```  1411     by simp_all
```
```  1412   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc'
```
```  1413   have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"
```
```  1414     by simp
```
```  1415   from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'"
```
```  1416     by auto
```
```  1417   have "n = n' \<or> n < n' \<or> n > n'"
```
```  1418     by arith
```
```  1419   moreover
```
```  1420   {
```
```  1421     assume nn': "n = n'"
```
```  1422     have "n = 0 \<or> n > 0" by arith
```
```  1423     moreover
```
```  1424     {
```
```  1425       assume nz: "n = 0"
```
```  1426       then have ?case using 4 nn'
```
```  1427         by (auto simp add: Let_def degcmc')
```
```  1428     }
```
```  1429     moreover
```
```  1430     {
```
```  1431       assume nz: "n > 0"
```
```  1432       with nn' H(3) have  cc': "c = c'" and pp': "p = p'"
```
```  1433         by (cases n, auto)+
```
```  1434       then have ?case
```
```  1435         using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv]
```
```  1436         using polysub_same_0[OF c'nh, simplified polysub_def]
```
```  1437         using nn' 4 by (simp add: Let_def)
```
```  1438     }
```
```  1439     ultimately have ?case by blast
```
```  1440   }
```
```  1441   moreover
```
```  1442   {
```
```  1443     assume nn': "n < n'"
```
```  1444     then have n'p: "n' > 0"
```
```  1445       by simp
```
```  1446     then have headcnp':"head (CN c' n' p') = CN c' n' p'"
```
```  1447       by (cases n') simp_all
```
```  1448     have degcnp': "degree (CN c' n' p') = 0"
```
```  1449       and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
```
```  1450       using 4 nn' by (cases n', simp_all)
```
```  1451     then have "n > 0"
```
```  1452       by (cases n) simp_all
```
```  1453     then have headcnp: "head (CN c n p) = CN c n p"
```
```  1454       by (cases n) auto
```
```  1455     from H(3) headcnp headcnp' nn' have ?case
```
```  1456       by auto
```
```  1457   }
```
```  1458   moreover
```
```  1459   {
```
```  1460     assume nn': "n > n'"
```
```  1461     then have np: "n > 0" by simp
```
```  1462     then have headcnp:"head (CN c n p) = CN c n p"
```
```  1463       by (cases n) simp_all
```
```  1464     from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)"
```
```  1465       by simp
```
```  1466     from np have degcnp: "degree (CN c n p) = 0"
```
```  1467       by (cases n) simp_all
```
```  1468     with degcnpeq have "n' > 0"
```
```  1469       by (cases n') simp_all
```
```  1470     then have headcnp': "head (CN c' n' p') = CN c' n' p'"
```
```  1471       by (cases n') auto
```
```  1472     from H(3) headcnp headcnp' nn' have ?case by auto
```
```  1473   }
```
```  1474   ultimately show ?case by blast
```
```  1475 qed auto
```
```  1476
```
```  1477 lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
```
```  1478   by (induct p arbitrary: n0 rule: head.induct) (simp_all add: shift1_def)
```
```  1479
```
```  1480 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
```
```  1481 proof (induct k arbitrary: n0 p)
```
```  1482   case 0
```
```  1483   then show ?case
```
```  1484     by auto
```
```  1485 next
```
```  1486   case (Suc k n0 p)
```
```  1487   then have "isnpolyh (shift1 p) 0"
```
```  1488     by (simp add: shift1_isnpolyh)
```
```  1489   with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
```
```  1490     and "head (shift1 p) = head p"
```
```  1491     by (simp_all add: shift1_head)
```
```  1492   then show ?case
```
```  1493     by (simp add: funpow_swap1)
```
```  1494 qed
```
```  1495
```
```  1496 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
```
```  1497   by (simp add: shift1_def)
```
```  1498
```
```  1499 lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
```
```  1500   by (induct k arbitrary: p) (auto simp add: shift1_degree)
```
```  1501
```
```  1502 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
```
```  1503   by (induct n arbitrary: p) simp_all
```
```  1504
```
```  1505 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
```
```  1506   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1507 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
```
```  1508   by (induct p arbitrary: n rule: degreen.induct) auto
```
```  1509 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
```
```  1510   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1511 lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
```
```  1512   by (induct p rule: head.induct) auto
```
```  1513
```
```  1514 lemma polyadd_eq_const_degree:
```
```  1515   "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> polyadd p q = C c \<Longrightarrow> degree p = degree q"
```
```  1516   using polyadd_eq_const_degreen degree_eq_degreen0 by simp
```
```  1517
```
```  1518 lemma polyadd_head:
```
```  1519   assumes np: "isnpolyh p n0"
```
```  1520     and nq: "isnpolyh q n1"
```
```  1521     and deg: "degree p \<noteq> degree q"
```
```  1522   shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
```
```  1523   using np nq deg
```
```  1524   apply (induct p q arbitrary: n0 n1 rule: polyadd.induct)
```
```  1525   apply simp_all
```
```  1526   apply (case_tac n', simp, simp)
```
```  1527   apply (case_tac n, simp, simp)
```
```  1528   apply (case_tac n, case_tac n', simp add: Let_def)
```
```  1529   apply (auto simp add: polyadd_eq_const_degree)
```
```  1530   apply (metis head_nz)
```
```  1531   apply (metis head_nz)
```
```  1532   apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
```
```  1533   done
```
```  1534
```
```  1535 lemma polymul_head_polyeq:
```
```  1536   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1537   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> q \<noteq> 0\<^sub>p \<Longrightarrow> head (p *\<^sub>p q) = head p *\<^sub>p head q"
```
```  1538 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
```
```  1539   case (2 c c' n' p' n0 n1)
```
```  1540   then have "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"
```
```  1541     by (simp_all add: head_isnpolyh)
```
```  1542   then show ?case
```
```  1543     using 2 by (cases n') auto
```
```  1544 next
```
```  1545   case (3 c n p c' n0 n1)
```
```  1546   then have "isnpolyh (head (CN c n p)) n0" "isnormNum c'"
```
```  1547     by (simp_all add: head_isnpolyh)
```
```  1548   then show ?case
```
```  1549     using 3 by (cases n) auto
```
```  1550 next
```
```  1551   case (4 c n p c' n' p' n0 n1)
```
```  1552   then have norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
```
```  1553     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
```
```  1554     by simp_all
```
```  1555   have "n < n' \<or> n' < n \<or> n = n'" by arith
```
```  1556   moreover
```
```  1557   {
```
```  1558     assume nn': "n < n'"
```
```  1559     then have ?case
```
```  1560       using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)]
```
```  1561       apply simp
```
```  1562       apply (cases n)
```
```  1563       apply simp
```
```  1564       apply (cases n')
```
```  1565       apply simp_all
```
```  1566       done
```
```  1567   }
```
```  1568   moreover {
```
```  1569     assume nn': "n'< n"
```
```  1570     then have ?case
```
```  1571       using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)]
```
```  1572       apply simp
```
```  1573       apply (cases n')
```
```  1574       apply simp
```
```  1575       apply (cases n)
```
```  1576       apply auto
```
```  1577       done
```
```  1578   }
```
```  1579   moreover
```
```  1580   {
```
```  1581     assume nn': "n' = n"
```
```  1582     from nn' polymul_normh[OF norm(5,4)]
```
```  1583     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
```
```  1584     from nn' polymul_normh[OF norm(5,3)] norm
```
```  1585     have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
```
```  1586     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
```
```  1587     have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```  1588     from polyadd_normh[OF ncnpc' ncnpp0']
```
```  1589     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"
```
```  1590       by (simp add: min_def)
```
```  1591     {
```
```  1592       assume np: "n > 0"
```
```  1593       with nn' head_isnpolyh_Suc'[OF np nth]
```
```  1594         head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
```
```  1595       have ?case by simp
```
```  1596     }
```
```  1597     moreover
```
```  1598     {
```
```  1599       assume nz: "n = 0"
```
```  1600       from polymul_degreen[OF norm(5,4), where m="0"]
```
```  1601         polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
```
```  1602       norm(5,6) degree_npolyhCN[OF norm(6)]
```
```  1603     have dth: "degree (CN c 0 p *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))"
```
```  1604       by simp
```
```  1605     then have dth': "degree (CN c 0 p *\<^sub>p c') \<noteq> degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))"
```
```  1606       by simp
```
```  1607     from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
```
```  1608     have ?case
```
```  1609       using norm "4.hyps"(6)[OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)] nn' nz
```
```  1610       by simp
```
```  1611     }
```
```  1612     ultimately have ?case
```
```  1613       by (cases n) auto
```
```  1614   }
```
```  1615   ultimately show ?case by blast
```
```  1616 qed simp_all
```
```  1617
```
```  1618 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
```
```  1619   by (induct p rule: degree.induct) auto
```
```  1620
```
```  1621 lemma degree_head[simp]: "degree (head p) = 0"
```
```  1622   by (induct p rule: head.induct) auto
```
```  1623
```
```  1624 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
```
```  1625   by (cases n) simp_all
```
```  1626
```
```  1627 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
```
```  1628   by (cases n) simp_all
```
```  1629
```
```  1630 lemma polyadd_different_degree:
```
```  1631   "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> degree p \<noteq> degree q \<Longrightarrow>
```
```  1632     degree (polyadd p q) = max (degree p) (degree q)"
```
```  1633   using polyadd_different_degreen degree_eq_degreen0 by simp
```
```  1634
```
```  1635 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
```
```  1636   by (induct p arbitrary: n0 rule: polyneg.induct) auto
```
```  1637
```
```  1638 lemma degree_polymul:
```
```  1639   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1640     and np: "isnpolyh p n0"
```
```  1641     and nq: "isnpolyh q n1"
```
```  1642   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
```
```  1643   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
```
```  1644
```
```  1645 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
```
```  1646   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1647
```
```  1648 lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head (polyneg p) = polyneg (head p)"
```
```  1649   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1650
```
```  1651
```
```  1652 subsection {* Correctness of polynomial pseudo division *}
```
```  1653
```
```  1654 lemma polydivide_aux_properties:
```
```  1655   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1656     and np: "isnpolyh p n0"
```
```  1657     and ns: "isnpolyh s n1"
```
```  1658     and ap: "head p = a"
```
```  1659     and ndp: "degree p = n"
```
```  1660     and pnz: "p \<noteq> 0\<^sub>p"
```
```  1661   shows "polydivide_aux a n p k s = (k', r) \<longrightarrow> k' \<ge> k \<and> (degree r = 0 \<or> degree r < degree p) \<and>
```
```  1662     (\<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)"
```
```  1663   using ns
```
```  1664 proof (induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
```
```  1665   case less
```
```  1666   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
```
```  1667   let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and>
```
```  1668     (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
```
```  1669   let ?b = "head s"
```
```  1670   let ?p' = "funpow (degree s - n) shift1 p"
```
```  1671   let ?xdn = "funpow (degree s - n) shift1 (1)\<^sub>p"
```
```  1672   let ?akk' = "a ^\<^sub>p (k' - k)"
```
```  1673   note ns = `isnpolyh s n1`
```
```  1674   from np have np0: "isnpolyh p 0"
```
```  1675     using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by simp
```
```  1676   have np': "isnpolyh ?p' 0"
```
```  1677     using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def
```
```  1678     by simp
```
```  1679   have headp': "head ?p' = head p"
```
```  1680     using funpow_shift1_head[OF np pnz] by simp
```
```  1681   from funpow_shift1_isnpoly[where p="(1)\<^sub>p"] have nxdn: "isnpolyh ?xdn 0"
```
```  1682     by (simp add: isnpoly_def)
```
```  1683   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
```
```  1684   have nakk':"isnpolyh ?akk' 0" by blast
```
```  1685   {
```
```  1686     assume sz: "s = 0\<^sub>p"
```
```  1687     then have ?ths
```
```  1688       using np polydivide_aux.simps
```
```  1689       apply clarsimp
```
```  1690       apply (rule exI[where x="0\<^sub>p"])
```
```  1691       apply simp
```
```  1692       done
```
```  1693   }
```
```  1694   moreover
```
```  1695   {
```
```  1696     assume sz: "s \<noteq> 0\<^sub>p"
```
```  1697     {
```
```  1698       assume dn: "degree s < n"
```
```  1699       then have "?ths"
```
```  1700         using ns ndp np polydivide_aux.simps
```
```  1701         apply auto
```
```  1702         apply (rule exI[where x="0\<^sub>p"])
```
```  1703         apply simp
```
```  1704         done
```
```  1705     }
```
```  1706     moreover
```
```  1707     {
```
```  1708       assume dn': "\<not> degree s < n"
```
```  1709       then have dn: "degree s \<ge> n"
```
```  1710         by arith
```
```  1711       have degsp': "degree s = degree ?p'"
```
```  1712         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"]
```
```  1713         by simp
```
```  1714       {
```
```  1715         assume ba: "?b = a"
```
```  1716         then have headsp': "head s = head ?p'"
```
```  1717           using ap headp' by simp
```
```  1718         have nr: "isnpolyh (s -\<^sub>p ?p') 0"
```
```  1719           using polysub_normh[OF ns np'] by simp
```
```  1720         from degree_polysub_samehead[OF ns np' headsp' degsp']
```
```  1721         have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p"
```
```  1722           by simp
```
```  1723         moreover
```
```  1724         {
```
```  1725           assume deglt:"degree (s -\<^sub>p ?p') < degree s"
```
```  1726           from polydivide_aux.simps sz dn' ba
```
```  1727           have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
```
```  1728             by (simp add: Let_def)
```
```  1729           {
```
```  1730             assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1731             from less(1)[OF deglt nr, of k k' r] trans[OF eq[symmetric] h1]
```
```  1732             have kk': "k \<le> k'"
```
```  1733               and nr: "\<exists>nr. isnpolyh r nr"
```
```  1734               and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1735               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"
```
```  1736               by auto
```
```  1737             from q1 obtain q n1 where nq: "isnpolyh q n1"
```
```  1738               and asp: "a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"
```
```  1739               by blast
```
```  1740             from nr obtain nr where nr': "isnpolyh r nr"
```
```  1741               by blast
```
```  1742             from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0"
```
```  1743               by simp
```
```  1744             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
```
```  1745             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
```
```  1746             from polyadd_normh[OF polymul_normh[OF np
```
```  1747               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
```
```  1748             have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0"
```
```  1749               by simp
```
```  1750             from asp have "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
```
```  1751               Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)"
```
```  1752               by simp
```
```  1753             then have "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
```
```  1754               Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
```
```  1755               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1756               by (simp add: field_simps)
```
```  1757             then have "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
```
```  1758               Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1759               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p *\<^sub>p p) +
```
```  1760               Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1761               by (auto simp only: funpow_shift1_1)
```
```  1762             then have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list.
```
```  1763               Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1764               Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p) +
```
```  1765               Ipoly bs q) + Ipoly bs r"
```
```  1766               by (simp add: field_simps)
```
```  1767             then have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list.
```
```  1768               Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1769               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)"
```
```  1770               by simp
```
```  1771             with isnpolyh_unique[OF nakks' nqr']
```
```  1772             have "a ^\<^sub>p (k' - k) *\<^sub>p s =
```
```  1773               p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r"
```
```  1774               by blast
```
```  1775             then have ?qths using nq'
```
```  1776               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q" in exI)
```
```  1777               apply (rule_tac x="0" in exI)
```
```  1778               apply simp
```
```  1779               done
```
```  1780             with kk' nr dr have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and>
```
```  1781               (\<exists>nr. isnpolyh r nr) \<and> ?qths"
```
```  1782               by blast
```
```  1783           }
```
```  1784           then have ?ths by blast
```
```  1785         }
```
```  1786         moreover
```
```  1787         {
```
```  1788           assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
```
```  1789           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}"]
```
```  1790           have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list. Ipoly bs s = Ipoly bs ?p'"
```
```  1791             by simp
```
```  1792           then have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list. Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)"
```
```  1793             using np nxdn
```
```  1794             apply simp
```
```  1795             apply (simp only: funpow_shift1_1)
```
```  1796             apply simp
```
```  1797             done
```
```  1798           then have sp': "s = ?xdn *\<^sub>p p"
```
```  1799             using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]]
```
```  1800             by blast
```
```  1801           {
```
```  1802             assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1803             from polydivide_aux.simps sz dn' ba
```
```  1804             have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
```
```  1805               by (simp add: Let_def)
```
```  1806             also have "\<dots> = (k,0\<^sub>p)"
```
```  1807               using polydivide_aux.simps spz by simp
```
```  1808             finally have "(k', r) = (k, 0\<^sub>p)"
```
```  1809               using h1 by simp
```
```  1810             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
```
```  1811               polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
```
```  1812               apply auto
```
```  1813               apply (rule exI[where x="?xdn"])
```
```  1814               apply (auto simp add: polymul_commute[of p])
```
```  1815               done
```
```  1816           }
```
```  1817         }
```
```  1818         ultimately have ?ths by blast
```
```  1819       }
```
```  1820       moreover
```
```  1821       {
```
```  1822         assume ba: "?b \<noteq> a"
```
```  1823         from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1824           polymul_normh[OF head_isnpolyh[OF ns] np']]
```
```  1825         have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
```
```  1826           by (simp add: min_def)
```
```  1827         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
```
```  1828           using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1829             polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
```
```  1830             funpow_shift1_nz[OF pnz]
```
```  1831           by simp_all
```
```  1832         from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
```
```  1833           polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1834         have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
```
```  1835           using head_head[OF ns] funpow_shift1_head[OF np pnz]
```
```  1836             polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
```
```  1837           by (simp add: ap)
```
```  1838         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
```
```  1839           head_nz[OF np] pnz sz ap[symmetric]
```
```  1840           funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1841           polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"] head_nz[OF ns]
```
```  1842           ndp dn
```
```  1843         have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p')"
```
```  1844           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
```
```  1845         {
```
```  1846           assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
```
```  1847           from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns]
```
```  1848             polymul_normh[OF head_isnpolyh[OF ns]np']] ap
```
```  1849           have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
```
```  1850             by simp
```
```  1851           {
```
```  1852             assume h1:"polydivide_aux a n p k s = (k', r)"
```
```  1853             from h1 polydivide_aux.simps sz dn' ba
```
```  1854             have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
```
```  1855               by (simp add: Let_def)
```
```  1856             with less(1)[OF dth nasbp', of "Suc k" k' r]
```
```  1857             obtain q nq nr where kk': "Suc k \<le> k'"
```
```  1858               and nr: "isnpolyh r nr"
```
```  1859               and nq: "isnpolyh q nq"
```
```  1860               and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1861               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"
```
```  1862               by auto
```
```  1863             from kk' have kk'': "Suc (k' - Suc k) = k' - k"
```
```  1864               by arith
```
```  1865             {
```
```  1866               fix bs :: "'a::{field_char_0,field_inverse_zero} list"
```
```  1867               from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
```
```  1868               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)"
```
```  1869                 by simp
```
```  1870               then have "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s =
```
```  1871                 Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
```
```  1872                 by (simp add: field_simps)
```
```  1873               then have "Ipoly bs a ^ (k' - k)  * Ipoly bs s =
```
```  1874                 Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
```
```  1875                 by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1876               then have "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1877                 Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
```
```  1878                 by (simp add: field_simps)
```
```  1879             }
```
```  1880             then have ieq:"\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
```
```  1881                 Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1882                 Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)"
```
```  1883               by auto
```
```  1884             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
```
```  1885             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]]
```
```  1886             have nqw: "isnpolyh ?q 0"
```
```  1887               by simp
```
```  1888             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]]
```
```  1889             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"
```
```  1890               by blast
```
```  1891             from dr kk' nr h1 asth nqw have ?ths
```
```  1892               apply simp
```
```  1893               apply (rule conjI)
```
```  1894               apply (rule exI[where x="nr"], simp)
```
```  1895               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
```
```  1896               apply (rule exI[where x="0"], simp)
```
```  1897               done
```
```  1898           }
```
```  1899           then have ?ths by blast
```
```  1900         }
```
```  1901         moreover
```
```  1902         {
```
```  1903           assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
```
```  1904           {
```
```  1905             fix bs :: "'a::{field_char_0,field_inverse_zero} list"
```
```  1906             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
```
```  1907             have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'"
```
```  1908               by simp
```
```  1909             then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
```
```  1910               by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1911             then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))"
```
```  1912               by simp
```
```  1913           }
```
```  1914           then have hth: "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
```
```  1915             Ipoly bs (a *\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
```
```  1916           from hth have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
```
```  1917             using isnpolyh_unique[where ?'a = "'a::{field_char_0,field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
```
```  1918                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
```
```  1919               simplified ap]
```
```  1920             by simp
```
```  1921           {
```
```  1922             assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1923             from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
```
```  1924             have "(k', r) = (Suc k, 0\<^sub>p)"
```
```  1925               by (simp add: Let_def)
```
```  1926             with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
```
```  1927               polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
```
```  1928             have ?ths
```
```  1929               apply (clarsimp simp add: Let_def)
```
```  1930               apply (rule exI[where x="?b *\<^sub>p ?xdn"])
```
```  1931               apply simp
```
```  1932               apply (rule exI[where x="0"], simp)
```
```  1933               done
```
```  1934           }
```
```  1935           then have ?ths by blast
```
```  1936         }
```
```  1937         ultimately have ?ths
```
```  1938           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"]
```
```  1939             head_nz[OF np] pnz sz ap[symmetric]
```
```  1940           by (auto simp add: degree_eq_degreen0[symmetric])
```
```  1941       }
```
```  1942       ultimately have ?ths by blast
```
```  1943     }
```
```  1944     ultimately have ?ths by blast
```
```  1945   }
```
```  1946   ultimately show ?ths by blast
```
```  1947 qed
```
```  1948
```
```  1949 lemma polydivide_properties:
```
```  1950   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1951     and np: "isnpolyh p n0"
```
```  1952     and ns: "isnpolyh s n1"
```
```  1953     and pnz: "p \<noteq> 0\<^sub>p"
```
```  1954   shows "\<exists>k r. polydivide s p = (k, r) \<and>
```
```  1955     (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) \<and>
```
```  1956     (\<exists>q n1. isnpolyh q n1 \<and> polypow k (head p) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
```
```  1957 proof -
```
```  1958   have trv: "head p = head p" "degree p = degree p"
```
```  1959     by simp_all
```
```  1960   from polydivide_def[where s="s" and p="p"] have ex: "\<exists> k r. polydivide s p = (k,r)"
```
```  1961     by auto
```
```  1962   then obtain k r where kr: "polydivide s p = (k,r)"
```
```  1963     by blast
```
```  1964   from trans[OF polydivide_def[where s="s"and p="p", symmetric] kr]
```
```  1965     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
```
```  1966   have "(degree r = 0 \<or> degree r < degree p) \<and>
```
```  1967     (\<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)"
```
```  1968     by blast
```
```  1969   with kr show ?thesis
```
```  1970     apply -
```
```  1971     apply (rule exI[where x="k"])
```
```  1972     apply (rule exI[where x="r"])
```
```  1973     apply simp
```
```  1974     done
```
```  1975 qed
```
```  1976
```
```  1977
```
```  1978 subsection {* More about polypoly and pnormal etc *}
```
```  1979
```
```  1980 definition "isnonconstant p \<longleftrightarrow> \<not> isconstant p"
```
```  1981
```
```  1982 lemma isnonconstant_pnormal_iff:
```
```  1983   assumes "isnonconstant p"
```
```  1984   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1985 proof
```
```  1986   let ?p = "polypoly bs p"
```
```  1987   assume H: "pnormal ?p"
```
```  1988   have csz: "coefficients p \<noteq> []"
```
```  1989     using assms by (cases p) auto
```
```  1990   from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"] pnormal_last_nonzero[OF H]
```
```  1991   show "Ipoly bs (head p) \<noteq> 0"
```
```  1992     by (simp add: polypoly_def)
```
```  1993 next
```
```  1994   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1995   let ?p = "polypoly bs p"
```
```  1996   have csz: "coefficients p \<noteq> []"
```
```  1997     using assms by (cases p) auto
```
```  1998   then have pz: "?p \<noteq> []"
```
```  1999     by (simp add: polypoly_def)
```
```  2000   then have lg: "length ?p > 0"
```
```  2001     by simp
```
```  2002   from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  2003   have lz: "last ?p \<noteq> 0"
```
```  2004     by (simp add: polypoly_def)
```
```  2005   from pnormal_last_length[OF lg lz] show "pnormal ?p" .
```
```  2006 qed
```
```  2007
```
```  2008 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
```
```  2009   unfolding isnonconstant_def
```
```  2010   apply (cases p)
```
```  2011   apply simp_all
```
```  2012   apply (case_tac nat)
```
```  2013   apply auto
```
```  2014   done
```
```  2015
```
```  2016 lemma isnonconstant_nonconstant:
```
```  2017   assumes "isnonconstant p"
```
```  2018   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  2019 proof
```
```  2020   let ?p = "polypoly bs p"
```
```  2021   assume nc: "nonconstant ?p"
```
```  2022   from isnonconstant_pnormal_iff[OF assms, of bs] nc
```
```  2023   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  2024     unfolding nonconstant_def by blast
```
```  2025 next
```
```  2026   let ?p = "polypoly bs p"
```
```  2027   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  2028   from isnonconstant_pnormal_iff[OF assms, of bs] h
```
```  2029   have pn: "pnormal ?p"
```
```  2030     by blast
```
```  2031   {
```
```  2032     fix x
```
```  2033     assume H: "?p = [x]"
```
```  2034     from H have "length (coefficients p) = 1"
```
```  2035       unfolding polypoly_def by auto
```
```  2036     with isnonconstant_coefficients_length[OF assms]
```
```  2037     have False by arith
```
```  2038   }
```
```  2039   then show "nonconstant ?p"
```
```  2040     using pn unfolding nonconstant_def by blast
```
```  2041 qed
```
```  2042
```
```  2043 lemma pnormal_length: "p \<noteq> [] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
```
```  2044   apply (induct p)
```
```  2045   apply (simp_all add: pnormal_def)
```
```  2046   apply (case_tac "p = []")
```
```  2047   apply simp_all
```
```  2048   done
```
```  2049
```
```  2050 lemma degree_degree:
```
```  2051   assumes "isnonconstant p"
```
```  2052   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  2053 proof
```
```  2054   let ?p = "polypoly bs p"
```
```  2055   assume H: "degree p = Polynomial_List.degree ?p"
```
```  2056   from isnonconstant_coefficients_length[OF assms] have pz: "?p \<noteq> []"
```
```  2057     unfolding polypoly_def by auto
```
```  2058   from H degree_coefficients[of p] isnonconstant_coefficients_length[OF assms]
```
```  2059   have lg: "length (pnormalize ?p) = length ?p"
```
```  2060     unfolding Polynomial_List.degree_def polypoly_def by simp
```
```  2061   then have "pnormal ?p"
```
```  2062     using pnormal_length[OF pz] by blast
```
```  2063   with isnonconstant_pnormal_iff[OF assms] show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  2064     by blast
```
```  2065 next
```
```  2066   let ?p = "polypoly bs p"
```
```  2067   assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  2068   with isnonconstant_pnormal_iff[OF assms] have "pnormal ?p"
```
```  2069     by blast
```
```  2070   with degree_coefficients[of p] isnonconstant_coefficients_length[OF assms]
```
```  2071   show "degree p = Polynomial_List.degree ?p"
```
```  2072     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
```
```  2073 qed
```
```  2074
```
```  2075
```
```  2076 section {* Swaps ; Division by a certain variable *}
```
```  2077
```
```  2078 primrec swap :: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly"
```
```  2079 where
```
```  2080   "swap n m (C x) = C x"
```
```  2081 | "swap n m (Bound k) = Bound (if k = n then m else if k = m then n else k)"
```
```  2082 | "swap n m (Neg t) = Neg (swap n m t)"
```
```  2083 | "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
```
```  2084 | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
```
```  2085 | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
```
```  2086 | "swap n m (Pw t k) = Pw (swap n m t) k"
```
```  2087 | "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) (swap n m p)"
```
```  2088
```
```  2089 lemma swap:
```
```  2090   assumes "n < length bs"
```
```  2091     and "m < length bs"
```
```  2092   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  2093 proof (induct t)
```
```  2094   case (Bound k)
```
```  2095   then show ?case
```
```  2096     using assms by simp
```
```  2097 next
```
```  2098   case (CN c k p)
```
```  2099   then show ?case
```
```  2100     using assms by simp
```
```  2101 qed simp_all
```
```  2102
```
```  2103 lemma swap_swap_id [simp]: "swap n m (swap m n t) = t"
```
```  2104   by (induct t) simp_all
```
```  2105
```
```  2106 lemma swap_commute: "swap n m p = swap m n p"
```
```  2107   by (induct p) simp_all
```
```  2108
```
```  2109 lemma swap_same_id[simp]: "swap n n t = t"
```
```  2110   by (induct t) simp_all
```
```  2111
```
```  2112 definition "swapnorm n m t = polynate (swap n m t)"
```
```  2113
```
```  2114 lemma swapnorm:
```
```  2115   assumes nbs: "n < length bs"
```
```  2116     and mbs: "m < length bs"
```
```  2117   shows "((Ipoly bs (swapnorm n m t) :: 'a::{field_char_0,field_inverse_zero})) =
```
```  2118     Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  2119   using swap[OF assms] swapnorm_def by simp
```
```  2120
```
```  2121 lemma swapnorm_isnpoly [simp]:
```
```  2122   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  2123   shows "isnpoly (swapnorm n m p)"
```
```  2124   unfolding swapnorm_def by simp
```
```  2125
```
```  2126 definition "polydivideby n s p =
```
```  2127   (let
```
```  2128     ss = swapnorm 0 n s;
```
```  2129     sp = swapnorm 0 n p;
```
```  2130     h = head sp;
```
```  2131     (k, r) = polydivide ss sp
```
```  2132    in (k, swapnorm 0 n h, swapnorm 0 n r))"
```
```  2133
```
```  2134 lemma swap_nz [simp]: "swap n m p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
```
```  2135   by (induct p) simp_all
```
```  2136
```
```  2137 fun isweaknpoly :: "poly \<Rightarrow> bool"
```
```  2138 where
```
```  2139   "isweaknpoly (C c) = True"
```
```  2140 | "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
```
```  2141 | "isweaknpoly p = False"
```
```  2142
```
```  2143 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
```
```  2144   by (induct p arbitrary: n0) auto
```
```  2145
```
```  2146 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
```
```  2147   by (induct p) auto
```
```  2148
```
`  2149 end`