src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
 author wenzelm Sun Sep 13 20:20:16 2015 +0200 (2015-09-13) changeset 61166 5976fe402824 parent 60698 29e8bdc41f90 child 61586 5197a2ecb658 permissions -rw-r--r--
renamed method "goals" to "goal_cases" to emphasize its meaning;
```     1 (*  Title:      HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
```
```     2     Author:     Amine Chaieb
```
```     3 *)
```
```     4
```
```     5 section \<open>Implementation and verification of multivariate polynomials\<close>
```
```     6
```
```     7 theory Reflected_Multivariate_Polynomial
```
```     8 imports Complex_Main Rat_Pair Polynomial_List
```
```     9 begin
```
```    10
```
```    11 subsection \<open>Datatype of polynomial expressions\<close>
```
```    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\<open>Boundedness, substitution and all that\<close>
```
```    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" -- \<open>a poly is INDEPENDENT of Bound 0\<close>
```
```    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" -- \<open>substitute a poly into a poly for Bound 0\<close>
```
```    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 \<open>Degrees and heads and coefficients\<close>
```
```    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 text \<open>More general notions of degree and head.\<close>
```
```    82
```
```    83 fun degreen :: "poly \<Rightarrow> nat \<Rightarrow> nat"
```
```    84 where
```
```    85   "degreen (CN c n p) = (\<lambda>m. if n = m then 1 + degreen p n else 0)"
```
```    86 | "degreen p = (\<lambda>m. 0)"
```
```    87
```
```    88 fun headn :: "poly \<Rightarrow> nat \<Rightarrow> poly"
```
```    89 where
```
```    90   "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
```
```    91 | "headn p = (\<lambda>m. p)"
```
```    92
```
```    93 fun coefficients :: "poly \<Rightarrow> poly list"
```
```    94 where
```
```    95   "coefficients (CN c 0 p) = c # coefficients p"
```
```    96 | "coefficients p = [p]"
```
```    97
```
```    98 fun isconstant :: "poly \<Rightarrow> bool"
```
```    99 where
```
```   100   "isconstant (CN c 0 p) = False"
```
```   101 | "isconstant p = True"
```
```   102
```
```   103 fun behead :: "poly \<Rightarrow> poly"
```
```   104 where
```
```   105   "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
```
```   106 | "behead p = 0\<^sub>p"
```
```   107
```
```   108 fun headconst :: "poly \<Rightarrow> Num"
```
```   109 where
```
```   110   "headconst (CN c n p) = headconst p"
```
```   111 | "headconst (C n) = n"
```
```   112
```
```   113
```
```   114 subsection \<open>Operations for normalization\<close>
```
```   115
```
```   116 declare if_cong[fundef_cong del]
```
```   117 declare let_cong[fundef_cong del]
```
```   118
```
```   119 fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
```
```   120 where
```
```   121   "polyadd (C c) (C c') = C (c +\<^sub>N c')"
```
```   122 | "polyadd (C c) (CN c' n' p') = CN (polyadd (C c) c') n' p'"
```
```   123 | "polyadd (CN c n p) (C c') = CN (polyadd c (C c')) n p"
```
```   124 | "polyadd (CN c n p) (CN c' n' p') =
```
```   125     (if n < n' then CN (polyadd c (CN c' n' p')) n p
```
```   126      else if n' < n then CN (polyadd (CN c n p) c') n' p'
```
```   127      else
```
```   128       let
```
```   129         cc' = polyadd c c';
```
```   130         pp' = polyadd p p'
```
```   131       in if pp' = 0\<^sub>p then cc' else CN cc' n pp')"
```
```   132 | "polyadd a b = Add a b"
```
```   133
```
```   134
```
```   135 fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
```
```   136 where
```
```   137   "polyneg (C c) = C (~\<^sub>N c)"
```
```   138 | "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
```
```   139 | "polyneg a = Neg a"
```
```   140
```
```   141 definition polysub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
```
```   142   where "p -\<^sub>p q = polyadd p (polyneg q)"
```
```   143
```
```   144 fun polymul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
```
```   145 where
```
```   146   "polymul (C c) (C c') = C (c *\<^sub>N c')"
```
```   147 | "polymul (C c) (CN c' n' p') =
```
```   148     (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul (C c) c') n' (polymul (C c) p'))"
```
```   149 | "polymul (CN c n p) (C c') =
```
```   150     (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul c (C c')) n (polymul p (C c')))"
```
```   151 | "polymul (CN c n p) (CN c' n' p') =
```
```   152     (if n < n' then CN (polymul c (CN c' n' p')) n (polymul p (CN c' n' p'))
```
```   153      else if n' < n then CN (polymul (CN c n p) c') n' (polymul (CN c n p) p')
```
```   154      else polyadd (polymul (CN c n p) c') (CN 0\<^sub>p n' (polymul (CN c n p) p')))"
```
```   155 | "polymul a b = Mul a b"
```
```   156
```
```   157 declare if_cong[fundef_cong]
```
```   158 declare let_cong[fundef_cong]
```
```   159
```
```   160 fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
```
```   161 where
```
```   162   "polypow 0 = (\<lambda>p. (1)\<^sub>p)"
```
```   163 | "polypow n =
```
```   164     (\<lambda>p.
```
```   165       let
```
```   166         q = polypow (n div 2) p;
```
```   167         d = polymul q q
```
```   168       in if even n then d else polymul p d)"
```
```   169
```
```   170 abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
```
```   171   where "a ^\<^sub>p k \<equiv> polypow k a"
```
```   172
```
```   173 function polynate :: "poly \<Rightarrow> poly"
```
```   174 where
```
```   175   "polynate (Bound n) = CN 0\<^sub>p n (1)\<^sub>p"
```
```   176 | "polynate (Add p q) = polynate p +\<^sub>p polynate q"
```
```   177 | "polynate (Sub p q) = polynate p -\<^sub>p polynate q"
```
```   178 | "polynate (Mul p q) = polynate p *\<^sub>p polynate q"
```
```   179 | "polynate (Neg p) = ~\<^sub>p (polynate p)"
```
```   180 | "polynate (Pw p n) = polynate p ^\<^sub>p n"
```
```   181 | "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
```
```   182 | "polynate (C c) = C (normNum c)"
```
```   183   by pat_completeness auto
```
```   184 termination by (relation "measure polysize") auto
```
```   185
```
```   186 fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly"
```
```   187 where
```
```   188   "poly_cmul y (C x) = C (y *\<^sub>N x)"
```
```   189 | "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
```
```   190 | "poly_cmul y p = C y *\<^sub>p p"
```
```   191
```
```   192 definition monic :: "poly \<Rightarrow> poly \<times> bool"
```
```   193 where
```
```   194   "monic p =
```
```   195     (let h = headconst p
```
```   196      in if h = 0\<^sub>N then (p, False) else (C (Ninv h) *\<^sub>p p, 0>\<^sub>N h))"
```
```   197
```
```   198
```
```   199 subsection \<open>Pseudo-division\<close>
```
```   200
```
```   201 definition shift1 :: "poly \<Rightarrow> poly"
```
```   202   where "shift1 p = CN 0\<^sub>p 0 p"
```
```   203
```
```   204 abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   205   where "funpow \<equiv> compow"
```
```   206
```
```   207 partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
```
```   208 where
```
```   209   "polydivide_aux a n p k s =
```
```   210     (if s = 0\<^sub>p then (k, s)
```
```   211      else
```
```   212       let
```
```   213         b = head s;
```
```   214         m = degree s
```
```   215       in
```
```   216         if m < n then (k,s)
```
```   217         else
```
```   218           let p' = funpow (m - n) shift1 p
```
```   219           in
```
```   220             if a = b then polydivide_aux a n p k (s -\<^sub>p p')
```
```   221             else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))"
```
```   222
```
```   223 definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
```
```   224   where "polydivide s p = polydivide_aux (head p) (degree p) p 0 s"
```
```   225
```
```   226 fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly"
```
```   227 where
```
```   228   "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
```
```   229 | "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
```
```   230
```
```   231 fun poly_deriv :: "poly \<Rightarrow> poly"
```
```   232 where
```
```   233   "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
```
```   234 | "poly_deriv p = 0\<^sub>p"
```
```   235
```
```   236
```
```   237 subsection \<open>Semantics of the polynomial representation\<close>
```
```   238
```
```   239 primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0,field,power}"
```
```   240 where
```
```   241   "Ipoly bs (C c) = INum c"
```
```   242 | "Ipoly bs (Bound n) = bs!n"
```
```   243 | "Ipoly bs (Neg a) = - Ipoly bs a"
```
```   244 | "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
```
```   245 | "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
```
```   246 | "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
```
```   247 | "Ipoly bs (Pw t n) = Ipoly bs t ^ n"
```
```   248 | "Ipoly bs (CN c n p) = Ipoly bs c + (bs!n) * Ipoly bs p"
```
```   249
```
```   250 abbreviation Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0,field,power}"  ("\<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 \<open>Normal form and normalization\<close>
```
```   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 \<open>polyadd preserves normal forms\<close>
```
```   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   consider (eq) "n = n'" | (lt) "n < n'" | (gt) "n > n'"
```
```   318     by arith
```
```   319   then show ?case
```
```   320   proof cases
```
```   321     case eq
```
```   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' show ?thesis
```
```   333       by (simp add: Let_def)
```
```   334   next
```
```   335     case lt
```
```   336     have "min n0 n1 \<le> n0"
```
```   337       by simp
```
```   338     with 4 lt have th1:"min n0 n1 \<le> n"
```
```   339       by auto
```
```   340     from 4 have th21: "isnpolyh c (Suc n)"
```
```   341       by simp
```
```   342     from 4 have th22: "isnpolyh (CN c' n' p') n'"
```
```   343       by simp
```
```   344     from lt have th23: "min (Suc n) n' = Suc n"
```
```   345       by arith
```
```   346     from "4.hyps"(1)[OF th21 th22] have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)"
```
```   347       using th23 by simp
```
```   348     with 4 lt th1 show ?thesis
```
```   349       by simp
```
```   350   next
```
```   351     case gt
```
```   352     then have gt': "n' < n \<and> \<not> n < n'"
```
```   353       by simp
```
```   354     have "min n0 n1 \<le> n1"
```
```   355       by simp
```
```   356     with 4 gt have th1: "min n0 n1 \<le> n'"
```
```   357       by auto
```
```   358     from 4 have th21: "isnpolyh c' (Suc n')"
```
```   359       by simp_all
```
```   360     from 4 have th22: "isnpolyh (CN c n p) n"
```
```   361       by simp
```
```   362     from gt have th23: "min n (Suc n') = Suc n'"
```
```   363       by arith
```
```   364     from "4.hyps"(2)[OF th22 th21] have "isnpolyh (polyadd (CN c n p) c') (Suc n')"
```
```   365       using th23 by simp
```
```   366     with 4 gt th1 show ?thesis
```
```   367       by simp
```
```   368   qed
```
```   369 qed auto
```
```   370
```
```   371 lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
```
```   372   by (induct p q rule: polyadd.induct)
```
```   373      (auto simp add: Let_def field_simps distrib_left[symmetric] simp del: distrib_left_NO_MATCH)
```
```   374
```
```   375 lemma polyadd_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polyadd p q)"
```
```   376   using polyadd_normh[of p 0 q 0] isnpoly_def by simp
```
```   377
```
```   378 text \<open>The degree of addition and other general lemmas needed for the normal form of polymul.\<close>
```
```   379
```
```   380 lemma polyadd_different_degreen:
```
```   381   assumes "isnpolyh p n0"
```
```   382     and "isnpolyh q n1"
```
```   383     and "degreen p m \<noteq> degreen q m"
```
```   384     and "m \<le> min n0 n1"
```
```   385   shows "degreen (polyadd p q) m = max (degreen p m) (degreen q m)"
```
```   386   using assms
```
```   387 proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
```
```   388   case (4 c n p c' n' p' m n0 n1)
```
```   389   show ?case
```
```   390   proof (cases "n = n'")
```
```   391     case True
```
```   392     with 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
```
```   393     show ?thesis by (auto simp: Let_def)
```
```   394   next
```
```   395     case False
```
```   396     with 4 show ?thesis by auto
```
```   397   qed
```
```   398 qed auto
```
```   399
```
```   400 lemma headnz[simp]: "isnpolyh p n \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
```
```   401   by (induct p arbitrary: n rule: headn.induct) auto
```
```   402
```
```   403 lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
```
```   404   by (induct p arbitrary: n rule: degree.induct) auto
```
```   405
```
```   406 lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
```
```   407   by (induct p arbitrary: n rule: degreen.induct) auto
```
```   408
```
```   409 lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
```
```   410   by (induct p arbitrary: n rule: degree.induct) auto
```
```   411
```
```   412 lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
```
```   413   using degree_isnpolyh_Suc by auto
```
```   414
```
```   415 lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
```
```   416   using degreen_0 by auto
```
```   417
```
```   418
```
```   419 lemma degreen_polyadd:
```
```   420   assumes np: "isnpolyh p n0"
```
```   421     and nq: "isnpolyh q n1"
```
```   422     and m: "m \<le> max n0 n1"
```
```   423   shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
```
```   424   using np nq m
```
```   425 proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
```
```   426   case (2 c c' n' p' n0 n1)
```
```   427   then show ?case
```
```   428     by (cases n') simp_all
```
```   429 next
```
```   430   case (3 c n p c' n0 n1)
```
```   431   then show ?case
```
```   432     by (cases n) auto
```
```   433 next
```
```   434   case (4 c n p c' n' p' n0 n1 m)
```
```   435   show ?case
```
```   436   proof (cases "n = n'")
```
```   437     case True
```
```   438     with 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
```
```   439     show ?thesis by (auto simp: Let_def)
```
```   440   next
```
```   441     case False
```
```   442     then show ?thesis by simp
```
```   443   qed
```
```   444 qed auto
```
```   445
```
```   446 lemma polyadd_eq_const_degreen:
```
```   447   assumes "isnpolyh p n0"
```
```   448     and "isnpolyh q n1"
```
```   449     and "polyadd p q = C c"
```
```   450   shows "degreen p m = degreen q m"
```
```   451   using assms
```
```   452 proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
```
```   453   case (4 c n p c' n' p' m n0 n1 x)
```
```   454   consider "n = n'" | "n > n' \<or> n < n'" by arith
```
```   455   then show ?case
```
```   456   proof cases
```
```   457     case 1
```
```   458     with 4 show ?thesis
```
```   459       by (cases "p +\<^sub>p p' = 0\<^sub>p") (auto simp add: Let_def)
```
```   460   next
```
```   461     case 2
```
```   462     with 4 show ?thesis by auto
```
```   463   qed
```
```   464 qed simp_all
```
```   465
```
```   466 lemma polymul_properties:
```
```   467   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```   468     and np: "isnpolyh p n0"
```
```   469     and nq: "isnpolyh q n1"
```
```   470     and m: "m \<le> min n0 n1"
```
```   471   shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   472     and "p *\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p \<or> q = 0\<^sub>p"
```
```   473     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)"
```
```   474   using np nq m
```
```   475 proof (induct p q arbitrary: n0 n1 m rule: polymul.induct)
```
```   476   case (2 c c' n' p')
```
```   477   {
```
```   478     case (1 n0 n1)
```
```   479     with "2.hyps"(4-6)[of n' n' n'] and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']
```
```   480     show ?case by (auto simp add: min_def)
```
```   481   next
```
```   482     case (2 n0 n1)
```
```   483     then show ?case by auto
```
```   484   next
```
```   485     case (3 n0 n1)
```
```   486     then show ?case using "2.hyps" by auto
```
```   487   }
```
```   488 next
```
```   489   case (3 c n p c')
```
```   490   {
```
```   491     case (1 n0 n1)
```
```   492     with "3.hyps"(4-6)[of n n n] and "3.hyps"(1-3)[of "Suc n" "Suc n" n]
```
```   493     show ?case by (auto simp add: min_def)
```
```   494   next
```
```   495     case (2 n0 n1)
```
```   496     then show ?case by auto
```
```   497   next
```
```   498     case (3 n0 n1)
```
```   499     then show ?case  using "3.hyps" by auto
```
```   500   }
```
```   501 next
```
```   502   case (4 c n p c' n' p')
```
```   503   let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
```
```   504   {
```
```   505     case (1 n0 n1)
```
```   506     then have cnp: "isnpolyh ?cnp n"
```
```   507       and cnp': "isnpolyh ?cnp' n'"
```
```   508       and np: "isnpolyh p n"
```
```   509       and nc: "isnpolyh c (Suc n)"
```
```   510       and np': "isnpolyh p' n'"
```
```   511       and nc': "isnpolyh c' (Suc n')"
```
```   512       and nn0: "n \<ge> n0"
```
```   513       and nn1: "n' \<ge> n1"
```
```   514       by simp_all
```
```   515     {
```
```   516       assume "n < n'"
```
```   517       with "4.hyps"(4-5)[OF np cnp', of n] and "4.hyps"(1)[OF nc cnp', of n] nn0 cnp
```
```   518       have ?case by (simp add: min_def)
```
```   519     } moreover {
```
```   520       assume "n' < n"
```
```   521       with "4.hyps"(16-17)[OF cnp np', of "n'"] and "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
```
```   522       have ?case by (cases "Suc n' = n") (simp_all add: min_def)
```
```   523     } moreover {
```
```   524       assume "n' = n"
```
```   525       with "4.hyps"(16-17)[OF cnp np', of n] and "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0
```
```   526       have ?case
```
```   527         apply (auto intro!: polyadd_normh)
```
```   528         apply (simp_all add: min_def isnpolyh_mono[OF nn0])
```
```   529         done
```
```   530     }
```
```   531     ultimately show ?case by arith
```
```   532   next
```
```   533     fix n0 n1 m
```
```   534     assume np: "isnpolyh ?cnp n0"
```
```   535     assume np':"isnpolyh ?cnp' n1"
```
```   536     assume m: "m \<le> min n0 n1"
```
```   537     let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
```
```   538     let ?d1 = "degreen ?cnp m"
```
```   539     let ?d2 = "degreen ?cnp' m"
```
```   540     let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
```
```   541     have "n' < n \<or> n < n' \<or> n' = n" by auto
```
```   542     moreover
```
```   543     {
```
```   544       assume "n' < n \<or> n < n'"
```
```   545       with "4.hyps"(3,6,18) np np' m have ?eq
```
```   546         by auto
```
```   547     }
```
```   548     moreover
```
```   549     {
```
```   550       assume nn': "n' = n"
```
```   551       then have nn: "\<not> n' < n \<and> \<not> n < n'" by arith
```
```   552       from "4.hyps"(16,18)[of n n' n]
```
```   553         "4.hyps"(13,14)[of n "Suc n'" n]
```
```   554         np np' nn'
```
```   555       have norm:
```
```   556         "isnpolyh ?cnp n"
```
```   557         "isnpolyh c' (Suc n)"
```
```   558         "isnpolyh (?cnp *\<^sub>p c') n"
```
```   559         "isnpolyh p' n"
```
```   560         "isnpolyh (?cnp *\<^sub>p p') n"
```
```   561         "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   562         "?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p"
```
```   563         "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
```
```   564         by (auto simp add: min_def)
```
```   565       {
```
```   566         assume mn: "m = n"
```
```   567         from "4.hyps"(17,18)[OF norm(1,4), of n]
```
```   568           "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
```
```   569         have degs:
```
```   570           "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else ?d1 + degreen c' n)"
```
```   571           "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n"
```
```   572           by (simp_all add: min_def)
```
```   573         from degs norm have th1: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   574           by simp
```
```   575         then have neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   576           by simp
```
```   577         have nmin: "n \<le> min n n"
```
```   578           by (simp add: min_def)
```
```   579         from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
```
```   580         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 =
```
```   581             degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   582           by simp
```
```   583         from "4.hyps"(16-18)[OF norm(1,4), of n]
```
```   584           "4.hyps"(13-15)[OF norm(1,2), of n]
```
```   585           mn norm m nn' deg
```
```   586         have ?eq by simp
```
```   587       }
```
```   588       moreover
```
```   589       {
```
```   590         assume mn: "m \<noteq> n"
```
```   591         then have mn': "m < n"
```
```   592           using m np by auto
```
```   593         from nn' m np have max1: "m \<le> max n n"
```
```   594           by simp
```
```   595         then have min1: "m \<le> min n n"
```
```   596           by simp
```
```   597         then have min2: "m \<le> min n (Suc n)"
```
```   598           by simp
```
```   599         from "4.hyps"(16-18)[OF norm(1,4) min1]
```
```   600           "4.hyps"(13-15)[OF norm(1,2) min2]
```
```   601           degreen_polyadd[OF norm(3,6) max1]
```
```   602         have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m \<le>
```
```   603             max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
```
```   604           using mn nn' np np' by simp
```
```   605         with "4.hyps"(16-18)[OF norm(1,4) min1]
```
```   606           "4.hyps"(13-15)[OF norm(1,2) min2]
```
```   607           degreen_0[OF norm(3) mn']
```
```   608         have ?eq using nn' mn np np' by clarsimp
```
```   609       }
```
```   610       ultimately have ?eq by blast
```
```   611     }
```
```   612     ultimately show ?eq by blast
```
```   613   }
```
```   614   {
```
```   615     case (2 n0 n1)
```
```   616     then have np: "isnpolyh ?cnp n0"
```
```   617       and np': "isnpolyh ?cnp' n1"
```
```   618       and m: "m \<le> min n0 n1"
```
```   619       by simp_all
```
```   620     then have mn: "m \<le> n" by simp
```
```   621     let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
```
```   622     {
```
```   623       assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
```
```   624       then have nn: "\<not> n' < n \<and> \<not> n < n'"
```
```   625         by simp
```
```   626       from "4.hyps"(16-18) [of n n n]
```
```   627         "4.hyps"(13-15)[of n "Suc n" n]
```
```   628         np np' C(2) mn
```
```   629       have norm:
```
```   630         "isnpolyh ?cnp n"
```
```   631         "isnpolyh c' (Suc n)"
```
```   632         "isnpolyh (?cnp *\<^sub>p c') n"
```
```   633         "isnpolyh p' n"
```
```   634         "isnpolyh (?cnp *\<^sub>p p') n"
```
```   635         "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   636         "?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p"
```
```   637         "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
```
```   638         "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
```
```   639         "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
```
```   640         by (simp_all add: min_def)
```
```   641       from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   642         by simp
```
```   643       have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   644         using norm by simp
```
```   645       from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"] degneq
```
```   646       have False by simp
```
```   647     }
```
```   648     then show ?case using "4.hyps" by clarsimp
```
```   649   }
```
```   650 qed auto
```
```   651
```
```   652 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = Ipoly bs p * Ipoly bs q"
```
```   653   by (induct p q rule: polymul.induct) (auto simp add: field_simps)
```
```   654
```
```   655 lemma polymul_normh:
```
```   656   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```   657   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   658   using polymul_properties(1) by blast
```
```   659
```
```   660 lemma polymul_eq0_iff:
```
```   661   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```   662   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"
```
```   663   using polymul_properties(2) by blast
```
```   664
```
```   665 lemma polymul_degreen:
```
```   666   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```   667   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> m \<le> min n0 n1 \<Longrightarrow>
```
```   668     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)"
```
```   669   by (fact polymul_properties(3))
```
```   670
```
```   671 lemma polymul_norm:
```
```   672   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```   673   shows "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polymul p q)"
```
```   674   using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   675
```
```   676 lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
```
```   677   by (induct p arbitrary: n0 rule: headconst.induct) auto
```
```   678
```
```   679 lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
```
```   680   by (induct p arbitrary: n0) auto
```
```   681
```
```   682 lemma monic_eqI:
```
```   683   assumes np: "isnpolyh p n0"
```
```   684   shows "INum (headconst p) * Ipoly bs (fst (monic p)) =
```
```   685     (Ipoly bs p ::'a::{field_char_0,field, power})"
```
```   686   unfolding monic_def Let_def
```
```   687 proof (cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
```
```   688   let ?h = "headconst p"
```
```   689   assume pz: "p \<noteq> 0\<^sub>p"
```
```   690   {
```
```   691     assume hz: "INum ?h = (0::'a)"
```
```   692     from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N"
```
```   693       by simp_all
```
```   694     from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N"
```
```   695       by simp
```
```   696     with headconst_zero[OF np] have "p = 0\<^sub>p"
```
```   697       by blast
```
```   698     with pz have False
```
```   699       by blast
```
```   700   }
```
```   701   then show "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
```
```   702     by blast
```
```   703 qed
```
```   704
```
```   705
```
```   706 text \<open>polyneg is a negation and preserves normal forms\<close>
```
```   707
```
```   708 lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
```
```   709   by (induct p rule: polyneg.induct) auto
```
```   710
```
```   711 lemma polyneg0: "isnpolyh p n \<Longrightarrow> (~\<^sub>p p) = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
```
```   712   by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: Nneg_def)
```
```   713
```
```   714 lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
```
```   715   by (induct p arbitrary: n0 rule: polyneg.induct) auto
```
```   716
```
```   717 lemma polyneg_normh: "isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n"
```
```   718   by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: polyneg0)
```
```   719
```
```   720 lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
```
```   721   using isnpoly_def polyneg_normh by simp
```
```   722
```
```   723
```
```   724 text \<open>polysub is a substraction and preserves normal forms\<close>
```
```   725
```
```   726 lemma polysub[simp]: "Ipoly bs (polysub p q) = Ipoly bs p - Ipoly bs q"
```
```   727   by (simp add: polysub_def)
```
```   728
```
```   729 lemma polysub_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
```
```   730   by (simp add: polysub_def polyneg_normh polyadd_normh)
```
```   731
```
```   732 lemma polysub_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polysub p q)"
```
```   733   using polyadd_norm polyneg_norm by (simp add: polysub_def)
```
```   734
```
```   735 lemma polysub_same_0[simp]:
```
```   736   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```   737   shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p"
```
```   738   unfolding polysub_def split_def fst_conv snd_conv
```
```   739   by (induct p arbitrary: n0) (auto simp add: Let_def Nsub0[simplified Nsub_def])
```
```   740
```
```   741 lemma polysub_0:
```
```   742   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```   743   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p -\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = q"
```
```   744   unfolding polysub_def split_def fst_conv snd_conv
```
```   745   by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
```
```   746     (auto simp: Nsub0[simplified Nsub_def] Let_def)
```
```   747
```
```   748 text \<open>polypow is a power function and preserves normal forms\<close>
```
```   749
```
```   750 lemma polypow[simp]:
```
```   751   "Ipoly bs (polypow n p) = (Ipoly bs p :: 'a::{field_char_0,field}) ^ n"
```
```   752 proof (induct n rule: polypow.induct)
```
```   753   case 1
```
```   754   then show ?case
```
```   755     by simp
```
```   756 next
```
```   757   case (2 n)
```
```   758   let ?q = "polypow ((Suc n) div 2) p"
```
```   759   let ?d = "polymul ?q ?q"
```
```   760   have "odd (Suc n) \<or> even (Suc n)"
```
```   761     by simp
```
```   762   moreover
```
```   763   {
```
```   764     assume odd: "odd (Suc n)"
```
```   765     have th: "(Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0)))) = Suc n div 2 + Suc n div 2 + 1"
```
```   766       by arith
```
```   767     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)"
```
```   768       by (simp add: Let_def)
```
```   769     also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2) * (Ipoly bs p)^(Suc n div 2)"
```
```   770       using "2.hyps" by simp
```
```   771     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
```
```   772       by (simp only: power_add power_one_right) simp
```
```   773     also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0))))"
```
```   774       by (simp only: th)
```
```   775     finally have ?case unfolding numeral_2_eq_2 [symmetric]
```
```   776     using odd_two_times_div_two_nat [OF odd] by simp
```
```   777   }
```
```   778   moreover
```
```   779   {
```
```   780     assume even: "even (Suc n)"
```
```   781     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d"
```
```   782       by (simp add: Let_def)
```
```   783     also have "\<dots> = (Ipoly bs p) ^ (2 * (Suc n div 2))"
```
```   784       using "2.hyps" by (simp only: mult_2 power_add) simp
```
```   785     finally have ?case using even_two_times_div_two [OF even]
```
```   786       by simp
```
```   787   }
```
```   788   ultimately show ?case by blast
```
```   789 qed
```
```   790
```
```   791 lemma polypow_normh:
```
```   792   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```   793   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
```
```   794 proof (induct k arbitrary: n rule: polypow.induct)
```
```   795   case 1
```
```   796   then show ?case by auto
```
```   797 next
```
```   798   case (2 k n)
```
```   799   let ?q = "polypow (Suc k div 2) p"
```
```   800   let ?d = "polymul ?q ?q"
```
```   801   from 2 have th1: "isnpolyh ?q n" and th2: "isnpolyh p n"
```
```   802     by blast+
```
```   803   from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n"
```
```   804     by simp
```
```   805   from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n"
```
```   806     by simp
```
```   807   from dn on show ?case by (simp, unfold Let_def) auto
```
```   808
```
```   809 qed
```
```   810
```
```   811 lemma polypow_norm:
```
```   812   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```   813   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
```
```   814   by (simp add: polypow_normh isnpoly_def)
```
```   815
```
```   816 text \<open>Finally the whole normalization\<close>
```
```   817
```
```   818 lemma polynate [simp]:
```
```   819   "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0,field})"
```
```   820   by (induct p rule:polynate.induct) auto
```
```   821
```
```   822 lemma polynate_norm[simp]:
```
```   823   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```   824   shows "isnpoly (polynate p)"
```
```   825   by (induct p rule: polynate.induct)
```
```   826      (simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm,
```
```   827       simp_all add: isnpoly_def)
```
```   828
```
```   829 text \<open>shift1\<close>
```
```   830
```
```   831
```
```   832 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
```
```   833   by (simp add: shift1_def)
```
```   834
```
```   835 lemma shift1_isnpoly:
```
```   836   assumes "isnpoly p"
```
```   837     and "p \<noteq> 0\<^sub>p"
```
```   838   shows "isnpoly (shift1 p) "
```
```   839   using assms by (simp add: shift1_def isnpoly_def)
```
```   840
```
```   841 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
```
```   842   by (simp add: shift1_def)
```
```   843
```
```   844 lemma funpow_shift1_isnpoly: "isnpoly p \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpoly (funpow n shift1 p)"
```
```   845   by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
```
```   846
```
```   847 lemma funpow_isnpolyh:
```
```   848   assumes "\<And>p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n"
```
```   849     and "isnpolyh p n"
```
```   850   shows "isnpolyh (funpow k f p) n"
```
```   851   using assms by (induct k arbitrary: p) auto
```
```   852
```
```   853 lemma funpow_shift1:
```
```   854   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field}) =
```
```   855     Ipoly bs (Mul (Pw (Bound 0) n) p)"
```
```   856   by (induct n arbitrary: p) (simp_all add: shift1_isnpoly shift1)
```
```   857
```
```   858 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
```
```   859   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
```
```   860
```
```   861 lemma funpow_shift1_1:
```
```   862   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field}) =
```
```   863     Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)"
```
```   864   by (simp add: funpow_shift1)
```
```   865
```
```   866 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
```
```   867   by (induct p rule: poly_cmul.induct) (auto simp add: field_simps)
```
```   868
```
```   869 lemma behead:
```
```   870   assumes "isnpolyh p n"
```
```   871   shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) =
```
```   872     (Ipoly bs p :: 'a :: {field_char_0,field})"
```
```   873   using assms
```
```   874 proof (induct p arbitrary: n rule: behead.induct)
```
```   875   case (1 c p n)
```
```   876   then have pn: "isnpolyh p n" by simp
```
```   877   from 1(1)[OF pn]
```
```   878   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
```
```   879   then show ?case using "1.hyps"
```
```   880     apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
```
```   881     apply (simp_all add: th[symmetric] field_simps)
```
```   882     done
```
```   883 qed (auto simp add: Let_def)
```
```   884
```
```   885 lemma behead_isnpolyh:
```
```   886   assumes "isnpolyh p n"
```
```   887   shows "isnpolyh (behead p) n"
```
```   888   using assms by (induct p rule: behead.induct) (auto simp add: Let_def isnpolyh_mono)
```
```   889
```
```   890
```
```   891 subsection \<open>Miscellaneous lemmas about indexes, decrementation, substitution  etc ...\<close>
```
```   892
```
```   893 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
```
```   894 proof (induct p arbitrary: n rule: poly.induct, auto, goal_cases)
```
```   895   case prems: (1 c n p n')
```
```   896   then have "n = Suc (n - 1)"
```
```   897     by simp
```
```   898   then have "isnpolyh p (Suc (n - 1))"
```
```   899     using \<open>isnpolyh p n\<close> by simp
```
```   900   with prems(2) show ?case
```
```   901     by simp
```
```   902 qed
```
```   903
```
```   904 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
```
```   905   by (induct p arbitrary: n0 rule: isconstant.induct) (auto simp add: isnpolyh_polybound0)
```
```   906
```
```   907 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
```
```   908   by (induct p) auto
```
```   909
```
```   910 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
```
```   911   apply (induct p arbitrary: n0)
```
```   912   apply auto
```
```   913   apply atomize
```
```   914   apply (rename_tac nat a b, erule_tac x = "Suc nat" in allE)
```
```   915   apply auto
```
```   916   done
```
```   917
```
```   918 lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
```
```   919   by (induct p  arbitrary: n0 rule: head.induct) (auto intro: isnpolyh_polybound0)
```
```   920
```
```   921 lemma polybound0_I:
```
```   922   assumes "polybound0 a"
```
```   923   shows "Ipoly (b # bs) a = Ipoly (b' # bs) a"
```
```   924   using assms by (induct a rule: poly.induct) auto
```
```   925
```
```   926 lemma polysubst0_I: "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b # bs) a) # bs) t"
```
```   927   by (induct t) simp_all
```
```   928
```
```   929 lemma polysubst0_I':
```
```   930   assumes "polybound0 a"
```
```   931   shows "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b' # bs) a) # bs) t"
```
```   932   by (induct t) (simp_all add: polybound0_I[OF assms, where b="b" and b'="b'"])
```
```   933
```
```   934 lemma decrpoly:
```
```   935   assumes "polybound0 t"
```
```   936   shows "Ipoly (x # bs) t = Ipoly bs (decrpoly t)"
```
```   937   using assms by (induct t rule: decrpoly.induct) simp_all
```
```   938
```
```   939 lemma polysubst0_polybound0:
```
```   940   assumes "polybound0 t"
```
```   941   shows "polybound0 (polysubst0 t a)"
```
```   942   using assms by (induct a rule: poly.induct) auto
```
```   943
```
```   944 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
```
```   945   by (induct p arbitrary: n rule: degree.induct) (auto simp add: isnpolyh_polybound0)
```
```   946
```
```   947 primrec maxindex :: "poly \<Rightarrow> nat"
```
```   948 where
```
```   949   "maxindex (Bound n) = n + 1"
```
```   950 | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
```
```   951 | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
```
```   952 | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
```
```   953 | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
```
```   954 | "maxindex (Neg p) = maxindex p"
```
```   955 | "maxindex (Pw p n) = maxindex p"
```
```   956 | "maxindex (C x) = 0"
```
```   957
```
```   958 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool"
```
```   959   where "wf_bs bs p \<longleftrightarrow> length bs \<ge> maxindex p"
```
```   960
```
```   961 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall>c \<in> set (coefficients p). wf_bs bs c"
```
```   962 proof (induct p rule: coefficients.induct)
```
```   963   case (1 c p)
```
```   964   show ?case
```
```   965   proof
```
```   966     fix x
```
```   967     assume xc: "x \<in> set (coefficients (CN c 0 p))"
```
```   968     then have "x = c \<or> x \<in> set (coefficients p)"
```
```   969       by simp
```
```   970     moreover
```
```   971     {
```
```   972       assume "x = c"
```
```   973       then have "wf_bs bs x"
```
```   974         using "1.prems" unfolding wf_bs_def by simp
```
```   975     }
```
```   976     moreover
```
```   977     {
```
```   978       assume H: "x \<in> set (coefficients p)"
```
```   979       from "1.prems" have "wf_bs bs p"
```
```   980         unfolding wf_bs_def by simp
```
```   981       with "1.hyps" H have "wf_bs bs x"
```
```   982         by blast
```
```   983     }
```
```   984     ultimately show "wf_bs bs x"
```
```   985       by blast
```
```   986   qed
```
```   987 qed simp_all
```
```   988
```
```   989 lemma maxindex_coefficients: "\<forall>c \<in> set (coefficients p). maxindex c \<le> maxindex p"
```
```   990   by (induct p rule: coefficients.induct) auto
```
```   991
```
```   992 lemma wf_bs_I: "wf_bs bs p \<Longrightarrow> Ipoly (bs @ bs') p = Ipoly bs p"
```
```   993   unfolding wf_bs_def by (induct p) (auto simp add: nth_append)
```
```   994
```
```   995 lemma take_maxindex_wf:
```
```   996   assumes wf: "wf_bs bs p"
```
```   997   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
```
```   998 proof -
```
```   999   let ?ip = "maxindex p"
```
```  1000   let ?tbs = "take ?ip bs"
```
```  1001   from wf have "length ?tbs = ?ip"
```
```  1002     unfolding wf_bs_def by simp
```
```  1003   then have wf': "wf_bs ?tbs p"
```
```  1004     unfolding wf_bs_def by  simp
```
```  1005   have eq: "bs = ?tbs @ drop ?ip bs"
```
```  1006     by simp
```
```  1007   from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis
```
```  1008     using eq by simp
```
```  1009 qed
```
```  1010
```
```  1011 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
```
```  1012   by (induct p) auto
```
```  1013
```
```  1014 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
```
```  1015   unfolding wf_bs_def by simp
```
```  1016
```
```  1017 lemma wf_bs_insensitive': "wf_bs (x # bs) p = wf_bs (y # bs) p"
```
```  1018   unfolding wf_bs_def by simp
```
```  1019
```
```  1020 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x # bs) p"
```
```  1021   by (induct p rule: coefficients.induct) (auto simp add: wf_bs_def)
```
```  1022
```
```  1023 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
```
```  1024   by (induct p rule: coefficients.induct) simp_all
```
```  1025
```
```  1026 lemma coefficients_head: "last (coefficients p) = head p"
```
```  1027   by (induct p rule: coefficients.induct) auto
```
```  1028
```
```  1029 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x # bs) p"
```
```  1030   unfolding wf_bs_def by (induct p rule: decrpoly.induct) auto
```
```  1031
```
```  1032 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists>ys. length (xs @ ys) = n"
```
```  1033   apply (rule exI[where x="replicate (n - length xs) z" for z])
```
```  1034   apply simp
```
```  1035   done
```
```  1036
```
```  1037 lemma isnpolyh_Suc_const: "isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
```
```  1038   apply (cases p)
```
```  1039   apply auto
```
```  1040   apply (rename_tac nat a, case_tac "nat")
```
```  1041   apply simp_all
```
```  1042   done
```
```  1043
```
```  1044 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
```
```  1045   unfolding wf_bs_def by (induct p q rule: polyadd.induct) (auto simp add: Let_def)
```
```  1046
```
```  1047 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
```
```  1048   unfolding wf_bs_def
```
```  1049   apply (induct p q arbitrary: bs rule: polymul.induct)
```
```  1050   apply (simp_all add: wf_bs_polyadd)
```
```  1051   apply clarsimp
```
```  1052   apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
```
```  1053   apply auto
```
```  1054   done
```
```  1055
```
```  1056 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
```
```  1057   unfolding wf_bs_def by (induct p rule: polyneg.induct) auto
```
```  1058
```
```  1059 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
```
```  1060   unfolding polysub_def split_def fst_conv snd_conv
```
```  1061   using wf_bs_polyadd wf_bs_polyneg by blast
```
```  1062
```
```  1063
```
```  1064 subsection \<open>Canonicity of polynomial representation, see lemma isnpolyh_unique\<close>
```
```  1065
```
```  1066 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
```
```  1067 definition "polypoly' bs p = map (Ipoly bs \<circ> decrpoly) (coefficients p)"
```
```  1068 definition "poly_nate bs p = map (Ipoly bs \<circ> decrpoly) (coefficients (polynate p))"
```
```  1069
```
```  1070 lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall>q \<in> set (coefficients p). isnpolyh q n0"
```
```  1071 proof (induct p arbitrary: n0 rule: coefficients.induct)
```
```  1072   case (1 c p n0)
```
```  1073   have cp: "isnpolyh (CN c 0 p) n0"
```
```  1074     by fact
```
```  1075   then have norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
```
```  1076     by (auto simp add: isnpolyh_mono[where n'=0])
```
```  1077   from "1.hyps"[OF norm(2)] norm(1) norm(4) show ?case
```
```  1078     by simp
```
```  1079 qed auto
```
```  1080
```
```  1081 lemma coefficients_isconst: "isnpolyh p n \<Longrightarrow> \<forall>q \<in> set (coefficients p). isconstant q"
```
```  1082   by (induct p arbitrary: n rule: coefficients.induct) (auto simp add: isnpolyh_Suc_const)
```
```  1083
```
```  1084 lemma polypoly_polypoly':
```
```  1085   assumes np: "isnpolyh p n0"
```
```  1086   shows "polypoly (x # bs) p = polypoly' bs p"
```
```  1087 proof -
```
```  1088   let ?cf = "set (coefficients p)"
```
```  1089   from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
```
```  1090   {
```
```  1091     fix q
```
```  1092     assume q: "q \<in> ?cf"
```
```  1093     from q cn_norm have th: "isnpolyh q n0"
```
```  1094       by blast
```
```  1095     from coefficients_isconst[OF np] q have "isconstant q"
```
```  1096       by blast
```
```  1097     with isconstant_polybound0[OF th] have "polybound0 q"
```
```  1098       by blast
```
```  1099   }
```
```  1100   then have "\<forall>q \<in> ?cf. polybound0 q" ..
```
```  1101   then have "\<forall>q \<in> ?cf. Ipoly (x # bs) q = Ipoly bs (decrpoly q)"
```
```  1102     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
```
```  1103     by auto
```
```  1104   then show ?thesis
```
```  1105     unfolding polypoly_def polypoly'_def by simp
```
```  1106 qed
```
```  1107
```
```  1108 lemma polypoly_poly:
```
```  1109   assumes "isnpolyh p n0"
```
```  1110   shows "Ipoly (x # bs) p = poly (polypoly (x # bs) p) x"
```
```  1111   using assms
```
```  1112   by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def)
```
```  1113
```
```  1114 lemma polypoly'_poly:
```
```  1115   assumes "isnpolyh p n0"
```
```  1116   shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
```
```  1117   using polypoly_poly[OF assms, simplified polypoly_polypoly'[OF assms]] .
```
```  1118
```
```  1119
```
```  1120 lemma polypoly_poly_polybound0:
```
```  1121   assumes "isnpolyh p n0"
```
```  1122     and "polybound0 p"
```
```  1123   shows "polypoly bs p = [Ipoly bs p]"
```
```  1124   using assms
```
```  1125   unfolding polypoly_def
```
```  1126   apply (cases p)
```
```  1127   apply auto
```
```  1128   apply (rename_tac nat a, case_tac nat)
```
```  1129   apply auto
```
```  1130   done
```
```  1131
```
```  1132 lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
```
```  1133   by (induct p rule: head.induct) auto
```
```  1134
```
```  1135 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> headn p m = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
```
```  1136   by (cases p) auto
```
```  1137
```
```  1138 lemma head_eq_headn0: "head p = headn p 0"
```
```  1139   by (induct p rule: head.induct) simp_all
```
```  1140
```
```  1141 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> head p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
```
```  1142   by (simp add: head_eq_headn0)
```
```  1143
```
```  1144 lemma isnpolyh_zero_iff:
```
```  1145   assumes nq: "isnpolyh p n0"
```
```  1146     and eq :"\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a::{field_char_0,field, power})"
```
```  1147   shows "p = 0\<^sub>p"
```
```  1148   using nq eq
```
```  1149 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
```
```  1150   case less
```
```  1151   note np = \<open>isnpolyh p n0\<close> and zp = \<open>\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)\<close>
```
```  1152   {
```
```  1153     assume nz: "maxindex p = 0"
```
```  1154     then obtain c where "p = C c"
```
```  1155       using np by (cases p) auto
```
```  1156     with zp np have "p = 0\<^sub>p"
```
```  1157       unfolding wf_bs_def by simp
```
```  1158   }
```
```  1159   moreover
```
```  1160   {
```
```  1161     assume nz: "maxindex p \<noteq> 0"
```
```  1162     let ?h = "head p"
```
```  1163     let ?hd = "decrpoly ?h"
```
```  1164     let ?ihd = "maxindex ?hd"
```
```  1165     from head_isnpolyh[OF np] head_polybound0[OF np]
```
```  1166     have h: "isnpolyh ?h n0" "polybound0 ?h"
```
```  1167       by simp_all
```
```  1168     then have nhd: "isnpolyh ?hd (n0 - 1)"
```
```  1169       using decrpoly_normh by blast
```
```  1170
```
```  1171     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
```
```  1172     have mihn: "maxindex ?h \<le> maxindex p"
```
```  1173       by auto
```
```  1174     with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < maxindex p"
```
```  1175       by auto
```
```  1176     {
```
```  1177       fix bs :: "'a list"
```
```  1178       assume bs: "wf_bs bs ?hd"
```
```  1179       let ?ts = "take ?ihd bs"
```
```  1180       let ?rs = "drop ?ihd bs"
```
```  1181       have ts: "wf_bs ?ts ?hd"
```
```  1182         using bs unfolding wf_bs_def by simp
```
```  1183       have bs_ts_eq: "?ts @ ?rs = bs"
```
```  1184         by simp
```
```  1185       from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x # ?ts) ?h"
```
```  1186         by simp
```
```  1187       from ihd_lt_n have "\<forall>x. length (x # ?ts) \<le> maxindex p"
```
```  1188         by simp
```
```  1189       with length_le_list_ex obtain xs where xs: "length ((x # ?ts) @ xs) = maxindex p"
```
```  1190         by blast
```
```  1191       then have "\<forall>x. wf_bs ((x # ?ts) @ xs) p"
```
```  1192         unfolding wf_bs_def by simp
```
```  1193       with zp have "\<forall>x. Ipoly ((x # ?ts) @ xs) p = 0"
```
```  1194         by blast
```
```  1195       then have "\<forall>x. Ipoly (x # (?ts @ xs)) p = 0"
```
```  1196         by simp
```
```  1197       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
```
```  1198       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"
```
```  1199         by simp
```
```  1200       then have "poly (polypoly' (?ts @ xs) p) = poly []"
```
```  1201         by auto
```
```  1202       then have "\<forall>c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
```
```  1203         using poly_zero[where ?'a='a] by (simp add: polypoly'_def)
```
```  1204       with coefficients_head[of p, symmetric]
```
```  1205       have th0: "Ipoly (?ts @ xs) ?hd = 0"
```
```  1206         by simp
```
```  1207       from bs have wf'': "wf_bs ?ts ?hd"
```
```  1208         unfolding wf_bs_def by simp
```
```  1209       with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0"
```
```  1210         by simp
```
```  1211       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
```
```  1212         by simp
```
```  1213     }
```
```  1214     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
```
```  1215       by blast
```
```  1216     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p"
```
```  1217       by blast
```
```  1218     then have "?h = 0\<^sub>p" by simp
```
```  1219     with head_nz[OF np] have "p = 0\<^sub>p" by simp
```
```  1220   }
```
```  1221   ultimately show "p = 0\<^sub>p"
```
```  1222     by blast
```
```  1223 qed
```
```  1224
```
```  1225 lemma isnpolyh_unique:
```
```  1226   assumes np: "isnpolyh p n0"
```
```  1227     and nq: "isnpolyh q n1"
```
```  1228   shows "(\<forall>bs. \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (\<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup> :: 'a::{field_char_0,field,power})) \<longleftrightarrow> p = q"
```
```  1229 proof auto
```
```  1230   assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a) = \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
```
```  1231   then have "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)"
```
```  1232     by simp
```
```  1233   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)"
```
```  1234     using wf_bs_polysub[where p=p and q=q] by auto
```
```  1235   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq] show "p = q"
```
```  1236     by blast
```
```  1237 qed
```
```  1238
```
```  1239
```
```  1240 text \<open>consequences of unicity on the algorithms for polynomial normalization\<close>
```
```  1241
```
```  1242 lemma polyadd_commute:
```
```  1243   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```  1244     and np: "isnpolyh p n0"
```
```  1245     and nq: "isnpolyh q n1"
```
```  1246   shows "p +\<^sub>p q = q +\<^sub>p p"
```
```  1247   using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]]
```
```  1248   by simp
```
```  1249
```
```  1250 lemma zero_normh: "isnpolyh 0\<^sub>p n"
```
```  1251   by simp
```
```  1252
```
```  1253 lemma one_normh: "isnpolyh (1)\<^sub>p n"
```
```  1254   by simp
```
```  1255
```
```  1256 lemma polyadd_0[simp]:
```
```  1257   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```  1258     and np: "isnpolyh p n0"
```
```  1259   shows "p +\<^sub>p 0\<^sub>p = p"
```
```  1260     and "0\<^sub>p +\<^sub>p p = p"
```
```  1261   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
```
```  1262     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
```
```  1263
```
```  1264 lemma polymul_1[simp]:
```
```  1265   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```  1266     and np: "isnpolyh p n0"
```
```  1267   shows "p *\<^sub>p (1)\<^sub>p = p"
```
```  1268     and "(1)\<^sub>p *\<^sub>p p = p"
```
```  1269   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
```
```  1270     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
```
```  1271
```
```  1272 lemma polymul_0[simp]:
```
```  1273   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```  1274     and np: "isnpolyh p n0"
```
```  1275   shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p"
```
```  1276     and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
```
```  1277   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
```
```  1278     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
```
```  1279
```
```  1280 lemma polymul_commute:
```
```  1281   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```  1282     and np: "isnpolyh p n0"
```
```  1283     and nq: "isnpolyh q n1"
```
```  1284   shows "p *\<^sub>p q = q *\<^sub>p p"
```
```  1285   using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np],
```
```  1286     where ?'a = "'a::{field_char_0,field, power}"]
```
```  1287   by simp
```
```  1288
```
```  1289 declare polyneg_polyneg [simp]
```
```  1290
```
```  1291 lemma isnpolyh_polynate_id [simp]:
```
```  1292   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```  1293     and np: "isnpolyh p n0"
```
```  1294   shows "polynate p = p"
```
```  1295   using isnpolyh_unique[where ?'a= "'a::{field_char_0,field}",
```
```  1296       OF polynate_norm[of p, unfolded isnpoly_def] np]
```
```  1297     polynate[where ?'a = "'a::{field_char_0,field}"]
```
```  1298   by simp
```
```  1299
```
```  1300 lemma polynate_idempotent[simp]:
```
```  1301   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```  1302   shows "polynate (polynate p) = polynate p"
```
```  1303   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
```
```  1304
```
```  1305 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
```
```  1306   unfolding poly_nate_def polypoly'_def ..
```
```  1307
```
```  1308 lemma poly_nate_poly:
```
```  1309   "poly (poly_nate bs p) = (\<lambda>x:: 'a ::{field_char_0,field}. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
```
```  1310   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
```
```  1311   unfolding poly_nate_polypoly' by auto
```
```  1312
```
```  1313
```
```  1314 subsection \<open>heads, degrees and all that\<close>
```
```  1315
```
```  1316 lemma degree_eq_degreen0: "degree p = degreen p 0"
```
```  1317   by (induct p rule: degree.induct) simp_all
```
```  1318
```
```  1319 lemma degree_polyneg:
```
```  1320   assumes "isnpolyh p n"
```
```  1321   shows "degree (polyneg p) = degree p"
```
```  1322   apply (induct p rule: polyneg.induct)
```
```  1323   using assms
```
```  1324   apply simp_all
```
```  1325   apply (case_tac na)
```
```  1326   apply auto
```
```  1327   done
```
```  1328
```
```  1329 lemma degree_polyadd:
```
```  1330   assumes np: "isnpolyh p n0"
```
```  1331     and nq: "isnpolyh q n1"
```
```  1332   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1333   using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
```
```  1334
```
```  1335
```
```  1336 lemma degree_polysub:
```
```  1337   assumes np: "isnpolyh p n0"
```
```  1338     and nq: "isnpolyh q n1"
```
```  1339   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1340 proof-
```
```  1341   from nq have nq': "isnpolyh (~\<^sub>p q) n1"
```
```  1342     using polyneg_normh by simp
```
```  1343   from degree_polyadd[OF np nq'] show ?thesis
```
```  1344     by (simp add: polysub_def degree_polyneg[OF nq])
```
```  1345 qed
```
```  1346
```
```  1347 lemma degree_polysub_samehead:
```
```  1348   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```  1349     and np: "isnpolyh p n0"
```
```  1350     and nq: "isnpolyh q n1"
```
```  1351     and h: "head p = head q"
```
```  1352     and d: "degree p = degree q"
```
```  1353   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
```
```  1354   unfolding polysub_def split_def fst_conv snd_conv
```
```  1355   using np nq h d
```
```  1356 proof (induct p q rule: polyadd.induct)
```
```  1357   case (1 c c')
```
```  1358   then show ?case
```
```  1359     by (simp add: Nsub_def Nsub0[simplified Nsub_def])
```
```  1360 next
```
```  1361   case (2 c c' n' p')
```
```  1362   from 2 have "degree (C c) = degree (CN c' n' p')"
```
```  1363     by simp
```
```  1364   then have nz: "n' > 0"
```
```  1365     by (cases n') auto
```
```  1366   then have "head (CN c' n' p') = CN c' n' p'"
```
```  1367     by (cases n') auto
```
```  1368   with 2 show ?case
```
```  1369     by simp
```
```  1370 next
```
```  1371   case (3 c n p c')
```
```  1372   then have "degree (C c') = degree (CN c n p)"
```
```  1373     by simp
```
```  1374   then have nz: "n > 0"
```
```  1375     by (cases n) auto
```
```  1376   then have "head (CN c n p) = CN c n p"
```
```  1377     by (cases n) auto
```
```  1378   with 3 show ?case by simp
```
```  1379 next
```
```  1380   case (4 c n p c' n' p')
```
```  1381   then have H:
```
```  1382     "isnpolyh (CN c n p) n0"
```
```  1383     "isnpolyh (CN c' n' p') n1"
```
```  1384     "head (CN c n p) = head (CN c' n' p')"
```
```  1385     "degree (CN c n p) = degree (CN c' n' p')"
```
```  1386     by simp_all
```
```  1387   then have degc: "degree c = 0" and degc': "degree c' = 0"
```
```  1388     by simp_all
```
```  1389   then have degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
```
```  1390     using H(1-2) degree_polyneg by auto
```
```  1391   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"
```
```  1392     by simp_all
```
```  1393   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc'
```
```  1394   have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"
```
```  1395     by simp
```
```  1396   from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'"
```
```  1397     by auto
```
```  1398   have "n = n' \<or> n < n' \<or> n > n'"
```
```  1399     by arith
```
```  1400   moreover
```
```  1401   {
```
```  1402     assume nn': "n = n'"
```
```  1403     have "n = 0 \<or> n > 0" by arith
```
```  1404     moreover
```
```  1405     {
```
```  1406       assume nz: "n = 0"
```
```  1407       then have ?case using 4 nn'
```
```  1408         by (auto simp add: Let_def degcmc')
```
```  1409     }
```
```  1410     moreover
```
```  1411     {
```
```  1412       assume nz: "n > 0"
```
```  1413       with nn' H(3) have  cc': "c = c'" and pp': "p = p'"
```
```  1414         by (cases n, auto)+
```
```  1415       then have ?case
```
```  1416         using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv]
```
```  1417         using polysub_same_0[OF c'nh, simplified polysub_def]
```
```  1418         using nn' 4 by (simp add: Let_def)
```
```  1419     }
```
```  1420     ultimately have ?case by blast
```
```  1421   }
```
```  1422   moreover
```
```  1423   {
```
```  1424     assume nn': "n < n'"
```
```  1425     then have n'p: "n' > 0"
```
```  1426       by simp
```
```  1427     then have headcnp':"head (CN c' n' p') = CN c' n' p'"
```
```  1428       by (cases n') simp_all
```
```  1429     have degcnp': "degree (CN c' n' p') = 0"
```
```  1430       and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
```
```  1431       using 4 nn' by (cases n', simp_all)
```
```  1432     then have "n > 0"
```
```  1433       by (cases n) simp_all
```
```  1434     then have headcnp: "head (CN c n p) = CN c n p"
```
```  1435       by (cases n) auto
```
```  1436     from H(3) headcnp headcnp' nn' have ?case
```
```  1437       by auto
```
```  1438   }
```
```  1439   moreover
```
```  1440   {
```
```  1441     assume nn': "n > n'"
```
```  1442     then have np: "n > 0" by simp
```
```  1443     then have headcnp:"head (CN c n p) = CN c n p"
```
```  1444       by (cases n) simp_all
```
```  1445     from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)"
```
```  1446       by simp
```
```  1447     from np have degcnp: "degree (CN c n p) = 0"
```
```  1448       by (cases n) simp_all
```
```  1449     with degcnpeq have "n' > 0"
```
```  1450       by (cases n') simp_all
```
```  1451     then have headcnp': "head (CN c' n' p') = CN c' n' p'"
```
```  1452       by (cases n') auto
```
```  1453     from H(3) headcnp headcnp' nn' have ?case by auto
```
```  1454   }
```
```  1455   ultimately show ?case by blast
```
```  1456 qed auto
```
```  1457
```
```  1458 lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
```
```  1459   by (induct p arbitrary: n0 rule: head.induct) (simp_all add: shift1_def)
```
```  1460
```
```  1461 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
```
```  1462 proof (induct k arbitrary: n0 p)
```
```  1463   case 0
```
```  1464   then show ?case
```
```  1465     by auto
```
```  1466 next
```
```  1467   case (Suc k n0 p)
```
```  1468   then have "isnpolyh (shift1 p) 0"
```
```  1469     by (simp add: shift1_isnpolyh)
```
```  1470   with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
```
```  1471     and "head (shift1 p) = head p"
```
```  1472     by (simp_all add: shift1_head)
```
```  1473   then show ?case
```
```  1474     by (simp add: funpow_swap1)
```
```  1475 qed
```
```  1476
```
```  1477 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
```
```  1478   by (simp add: shift1_def)
```
```  1479
```
```  1480 lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
```
```  1481   by (induct k arbitrary: p) (auto simp add: shift1_degree)
```
```  1482
```
```  1483 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
```
```  1484   by (induct n arbitrary: p) simp_all
```
```  1485
```
```  1486 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
```
```  1487   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1488 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
```
```  1489   by (induct p arbitrary: n rule: degreen.induct) auto
```
```  1490 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
```
```  1491   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1492 lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
```
```  1493   by (induct p rule: head.induct) auto
```
```  1494
```
```  1495 lemma polyadd_eq_const_degree:
```
```  1496   "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> polyadd p q = C c \<Longrightarrow> degree p = degree q"
```
```  1497   using polyadd_eq_const_degreen degree_eq_degreen0 by simp
```
```  1498
```
```  1499 lemma polyadd_head:
```
```  1500   assumes np: "isnpolyh p n0"
```
```  1501     and nq: "isnpolyh q n1"
```
```  1502     and deg: "degree p \<noteq> degree q"
```
```  1503   shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
```
```  1504   using np nq deg
```
```  1505   apply (induct p q arbitrary: n0 n1 rule: polyadd.induct)
```
```  1506   apply simp_all
```
```  1507   apply (case_tac n', simp, simp)
```
```  1508   apply (case_tac n, simp, simp)
```
```  1509   apply (case_tac n, case_tac n', simp add: Let_def)
```
```  1510   apply (auto simp add: polyadd_eq_const_degree)
```
```  1511   apply (metis head_nz)
```
```  1512   apply (metis head_nz)
```
```  1513   apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
```
```  1514   done
```
```  1515
```
```  1516 lemma polymul_head_polyeq:
```
```  1517   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```  1518   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"
```
```  1519 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
```
```  1520   case (2 c c' n' p' n0 n1)
```
```  1521   then have "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"
```
```  1522     by (simp_all add: head_isnpolyh)
```
```  1523   then show ?case
```
```  1524     using 2 by (cases n') auto
```
```  1525 next
```
```  1526   case (3 c n p c' n0 n1)
```
```  1527   then have "isnpolyh (head (CN c n p)) n0" "isnormNum c'"
```
```  1528     by (simp_all add: head_isnpolyh)
```
```  1529   then show ?case
```
```  1530     using 3 by (cases n) auto
```
```  1531 next
```
```  1532   case (4 c n p c' n' p' n0 n1)
```
```  1533   then have norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
```
```  1534     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
```
```  1535     by simp_all
```
```  1536   have "n < n' \<or> n' < n \<or> n = n'" by arith
```
```  1537   moreover
```
```  1538   {
```
```  1539     assume nn': "n < n'"
```
```  1540     then have ?case
```
```  1541       using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)]
```
```  1542       apply simp
```
```  1543       apply (cases n)
```
```  1544       apply simp
```
```  1545       apply (cases n')
```
```  1546       apply simp_all
```
```  1547       done
```
```  1548   }
```
```  1549   moreover {
```
```  1550     assume nn': "n'< n"
```
```  1551     then have ?case
```
```  1552       using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)]
```
```  1553       apply simp
```
```  1554       apply (cases n')
```
```  1555       apply simp
```
```  1556       apply (cases n)
```
```  1557       apply auto
```
```  1558       done
```
```  1559   }
```
```  1560   moreover
```
```  1561   {
```
```  1562     assume nn': "n' = n"
```
```  1563     from nn' polymul_normh[OF norm(5,4)]
```
```  1564     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
```
```  1565     from nn' polymul_normh[OF norm(5,3)] norm
```
```  1566     have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
```
```  1567     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
```
```  1568     have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```  1569     from polyadd_normh[OF ncnpc' ncnpp0']
```
```  1570     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"
```
```  1571       by (simp add: min_def)
```
```  1572     {
```
```  1573       assume np: "n > 0"
```
```  1574       with nn' head_isnpolyh_Suc'[OF np nth]
```
```  1575         head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
```
```  1576       have ?case by simp
```
```  1577     }
```
```  1578     moreover
```
```  1579     {
```
```  1580       assume nz: "n = 0"
```
```  1581       from polymul_degreen[OF norm(5,4), where m="0"]
```
```  1582         polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
```
```  1583       norm(5,6) degree_npolyhCN[OF norm(6)]
```
```  1584     have dth: "degree (CN c 0 p *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))"
```
```  1585       by simp
```
```  1586     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'))"
```
```  1587       by simp
```
```  1588     from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
```
```  1589     have ?case
```
```  1590       using norm "4.hyps"(6)[OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)] nn' nz
```
```  1591       by simp
```
```  1592     }
```
```  1593     ultimately have ?case
```
```  1594       by (cases n) auto
```
```  1595   }
```
```  1596   ultimately show ?case by blast
```
```  1597 qed simp_all
```
```  1598
```
```  1599 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
```
```  1600   by (induct p rule: degree.induct) auto
```
```  1601
```
```  1602 lemma degree_head[simp]: "degree (head p) = 0"
```
```  1603   by (induct p rule: head.induct) auto
```
```  1604
```
```  1605 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
```
```  1606   by (cases n) simp_all
```
```  1607
```
```  1608 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
```
```  1609   by (cases n) simp_all
```
```  1610
```
```  1611 lemma polyadd_different_degree:
```
```  1612   "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> degree p \<noteq> degree q \<Longrightarrow>
```
```  1613     degree (polyadd p q) = max (degree p) (degree q)"
```
```  1614   using polyadd_different_degreen degree_eq_degreen0 by simp
```
```  1615
```
```  1616 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
```
```  1617   by (induct p arbitrary: n0 rule: polyneg.induct) auto
```
```  1618
```
```  1619 lemma degree_polymul:
```
```  1620   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```  1621     and np: "isnpolyh p n0"
```
```  1622     and nq: "isnpolyh q n1"
```
```  1623   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
```
```  1624   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
```
```  1625
```
```  1626 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
```
```  1627   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1628
```
```  1629 lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head (polyneg p) = polyneg (head p)"
```
```  1630   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1631
```
```  1632
```
```  1633 subsection \<open>Correctness of polynomial pseudo division\<close>
```
```  1634
```
```  1635 lemma polydivide_aux_properties:
```
```  1636   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```  1637     and np: "isnpolyh p n0"
```
```  1638     and ns: "isnpolyh s n1"
```
```  1639     and ap: "head p = a"
```
```  1640     and ndp: "degree p = n"
```
```  1641     and pnz: "p \<noteq> 0\<^sub>p"
```
```  1642   shows "polydivide_aux a n p k s = (k', r) \<longrightarrow> k' \<ge> k \<and> (degree r = 0 \<or> degree r < degree p) \<and>
```
```  1643     (\<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)"
```
```  1644   using ns
```
```  1645 proof (induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
```
```  1646   case less
```
```  1647   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
```
```  1648   let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and>
```
```  1649     (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
```
```  1650   let ?b = "head s"
```
```  1651   let ?p' = "funpow (degree s - n) shift1 p"
```
```  1652   let ?xdn = "funpow (degree s - n) shift1 (1)\<^sub>p"
```
```  1653   let ?akk' = "a ^\<^sub>p (k' - k)"
```
```  1654   note ns = \<open>isnpolyh s n1\<close>
```
```  1655   from np have np0: "isnpolyh p 0"
```
```  1656     using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by simp
```
```  1657   have np': "isnpolyh ?p' 0"
```
```  1658     using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def
```
```  1659     by simp
```
```  1660   have headp': "head ?p' = head p"
```
```  1661     using funpow_shift1_head[OF np pnz] by simp
```
```  1662   from funpow_shift1_isnpoly[where p="(1)\<^sub>p"] have nxdn: "isnpolyh ?xdn 0"
```
```  1663     by (simp add: isnpoly_def)
```
```  1664   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
```
```  1665   have nakk':"isnpolyh ?akk' 0" by blast
```
```  1666   {
```
```  1667     assume sz: "s = 0\<^sub>p"
```
```  1668     then have ?ths
```
```  1669       using np polydivide_aux.simps
```
```  1670       apply clarsimp
```
```  1671       apply (rule exI[where x="0\<^sub>p"])
```
```  1672       apply simp
```
```  1673       done
```
```  1674   }
```
```  1675   moreover
```
```  1676   {
```
```  1677     assume sz: "s \<noteq> 0\<^sub>p"
```
```  1678     {
```
```  1679       assume dn: "degree s < n"
```
```  1680       then have "?ths"
```
```  1681         using ns ndp np polydivide_aux.simps
```
```  1682         apply auto
```
```  1683         apply (rule exI[where x="0\<^sub>p"])
```
```  1684         apply simp
```
```  1685         done
```
```  1686     }
```
```  1687     moreover
```
```  1688     {
```
```  1689       assume dn': "\<not> degree s < n"
```
```  1690       then have dn: "degree s \<ge> n"
```
```  1691         by arith
```
```  1692       have degsp': "degree s = degree ?p'"
```
```  1693         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"]
```
```  1694         by simp
```
```  1695       {
```
```  1696         assume ba: "?b = a"
```
```  1697         then have headsp': "head s = head ?p'"
```
```  1698           using ap headp' by simp
```
```  1699         have nr: "isnpolyh (s -\<^sub>p ?p') 0"
```
```  1700           using polysub_normh[OF ns np'] by simp
```
```  1701         from degree_polysub_samehead[OF ns np' headsp' degsp']
```
```  1702         have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p"
```
```  1703           by simp
```
```  1704         moreover
```
```  1705         {
```
```  1706           assume deglt:"degree (s -\<^sub>p ?p') < degree s"
```
```  1707           from polydivide_aux.simps sz dn' ba
```
```  1708           have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
```
```  1709             by (simp add: Let_def)
```
```  1710           {
```
```  1711             assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1712             from less(1)[OF deglt nr, of k k' r] trans[OF eq[symmetric] h1]
```
```  1713             have kk': "k \<le> k'"
```
```  1714               and nr: "\<exists>nr. isnpolyh r nr"
```
```  1715               and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1716               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"
```
```  1717               by auto
```
```  1718             from q1 obtain q n1 where nq: "isnpolyh q n1"
```
```  1719               and asp: "a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"
```
```  1720               by blast
```
```  1721             from nr obtain nr where nr': "isnpolyh r nr"
```
```  1722               by blast
```
```  1723             from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0"
```
```  1724               by simp
```
```  1725             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
```
```  1726             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
```
```  1727             from polyadd_normh[OF polymul_normh[OF np
```
```  1728               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
```
```  1729             have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0"
```
```  1730               by simp
```
```  1731             from asp have "\<forall>bs :: 'a::{field_char_0,field} list.
```
```  1732               Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)"
```
```  1733               by simp
```
```  1734             then have "\<forall>bs :: 'a::{field_char_0,field} list.
```
```  1735               Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
```
```  1736               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1737               by (simp add: field_simps)
```
```  1738             then have "\<forall>bs :: 'a::{field_char_0,field} list.
```
```  1739               Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1740               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p *\<^sub>p p) +
```
```  1741               Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1742               by (auto simp only: funpow_shift1_1)
```
```  1743             then have "\<forall>bs:: 'a::{field_char_0,field} list.
```
```  1744               Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1745               Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p) +
```
```  1746               Ipoly bs q) + Ipoly bs r"
```
```  1747               by (simp add: field_simps)
```
```  1748             then have "\<forall>bs:: 'a::{field_char_0,field} list.
```
```  1749               Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1750               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)"
```
```  1751               by simp
```
```  1752             with isnpolyh_unique[OF nakks' nqr']
```
```  1753             have "a ^\<^sub>p (k' - k) *\<^sub>p s =
```
```  1754               p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r"
```
```  1755               by blast
```
```  1756             then have ?qths using nq'
```
```  1757               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q" in exI)
```
```  1758               apply (rule_tac x="0" in exI)
```
```  1759               apply simp
```
```  1760               done
```
```  1761             with kk' nr dr have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and>
```
```  1762               (\<exists>nr. isnpolyh r nr) \<and> ?qths"
```
```  1763               by blast
```
```  1764           }
```
```  1765           then have ?ths by blast
```
```  1766         }
```
```  1767         moreover
```
```  1768         {
```
```  1769           assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
```
```  1770           from spz isnpolyh_unique[OF polysub_normh[OF ns np'], where q="0\<^sub>p", symmetric, where ?'a = "'a::{field_char_0,field}"]
```
```  1771           have "\<forall>bs:: 'a::{field_char_0,field} list. Ipoly bs s = Ipoly bs ?p'"
```
```  1772             by simp
```
```  1773           then have "\<forall>bs:: 'a::{field_char_0,field} list. Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)"
```
```  1774             using np nxdn
```
```  1775             apply simp
```
```  1776             apply (simp only: funpow_shift1_1)
```
```  1777             apply simp
```
```  1778             done
```
```  1779           then have sp': "s = ?xdn *\<^sub>p p"
```
```  1780             using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]]
```
```  1781             by blast
```
```  1782           {
```
```  1783             assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1784             from polydivide_aux.simps sz dn' ba
```
```  1785             have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
```
```  1786               by (simp add: Let_def)
```
```  1787             also have "\<dots> = (k,0\<^sub>p)"
```
```  1788               using polydivide_aux.simps spz by simp
```
```  1789             finally have "(k', r) = (k, 0\<^sub>p)"
```
```  1790               using h1 by simp
```
```  1791             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
```
```  1792               polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
```
```  1793               apply auto
```
```  1794               apply (rule exI[where x="?xdn"])
```
```  1795               apply (auto simp add: polymul_commute[of p])
```
```  1796               done
```
```  1797           }
```
```  1798         }
```
```  1799         ultimately have ?ths by blast
```
```  1800       }
```
```  1801       moreover
```
```  1802       {
```
```  1803         assume ba: "?b \<noteq> a"
```
```  1804         from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1805           polymul_normh[OF head_isnpolyh[OF ns] np']]
```
```  1806         have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
```
```  1807           by (simp add: min_def)
```
```  1808         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
```
```  1809           using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1810             polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
```
```  1811             funpow_shift1_nz[OF pnz]
```
```  1812           by simp_all
```
```  1813         from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
```
```  1814           polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1815         have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
```
```  1816           using head_head[OF ns] funpow_shift1_head[OF np pnz]
```
```  1817             polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
```
```  1818           by (simp add: ap)
```
```  1819         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
```
```  1820           head_nz[OF np] pnz sz ap[symmetric]
```
```  1821           funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1822           polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"] head_nz[OF ns]
```
```  1823           ndp dn
```
```  1824         have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p')"
```
```  1825           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
```
```  1826         {
```
```  1827           assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
```
```  1828           from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns]
```
```  1829             polymul_normh[OF head_isnpolyh[OF ns]np']] ap
```
```  1830           have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
```
```  1831             by simp
```
```  1832           {
```
```  1833             assume h1:"polydivide_aux a n p k s = (k', r)"
```
```  1834             from h1 polydivide_aux.simps sz dn' ba
```
```  1835             have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
```
```  1836               by (simp add: Let_def)
```
```  1837             with less(1)[OF dth nasbp', of "Suc k" k' r]
```
```  1838             obtain q nq nr where kk': "Suc k \<le> k'"
```
```  1839               and nr: "isnpolyh r nr"
```
```  1840               and nq: "isnpolyh q nq"
```
```  1841               and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1842               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"
```
```  1843               by auto
```
```  1844             from kk' have kk'': "Suc (k' - Suc k) = k' - k"
```
```  1845               by arith
```
```  1846             {
```
```  1847               fix bs :: "'a::{field_char_0,field} list"
```
```  1848               from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
```
```  1849               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)"
```
```  1850                 by simp
```
```  1851               then have "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s =
```
```  1852                 Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
```
```  1853                 by (simp add: field_simps)
```
```  1854               then have "Ipoly bs a ^ (k' - k)  * Ipoly bs s =
```
```  1855                 Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
```
```  1856                 by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1857               then have "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1858                 Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
```
```  1859                 by (simp add: field_simps)
```
```  1860             }
```
```  1861             then have ieq:"\<forall>bs :: 'a::{field_char_0,field} list.
```
```  1862                 Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1863                 Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)"
```
```  1864               by auto
```
```  1865             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
```
```  1866             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]]
```
```  1867             have nqw: "isnpolyh ?q 0"
```
```  1868               by simp
```
```  1869             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]]
```
```  1870             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"
```
```  1871               by blast
```
```  1872             from dr kk' nr h1 asth nqw have ?ths
```
```  1873               apply simp
```
```  1874               apply (rule conjI)
```
```  1875               apply (rule exI[where x="nr"], simp)
```
```  1876               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
```
```  1877               apply (rule exI[where x="0"], simp)
```
```  1878               done
```
```  1879           }
```
```  1880           then have ?ths by blast
```
```  1881         }
```
```  1882         moreover
```
```  1883         {
```
```  1884           assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
```
```  1885           {
```
```  1886             fix bs :: "'a::{field_char_0,field} list"
```
```  1887             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
```
```  1888             have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'"
```
```  1889               by simp
```
```  1890             then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
```
```  1891               by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1892             then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))"
```
```  1893               by simp
```
```  1894           }
```
```  1895           then have hth: "\<forall>bs :: 'a::{field_char_0,field} list.
```
```  1896             Ipoly bs (a *\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
```
```  1897           from hth have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
```
```  1898             using isnpolyh_unique[where ?'a = "'a::{field_char_0,field}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
```
```  1899                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
```
```  1900               simplified ap]
```
```  1901             by simp
```
```  1902           {
```
```  1903             assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1904             from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
```
```  1905             have "(k', r) = (Suc k, 0\<^sub>p)"
```
```  1906               by (simp add: Let_def)
```
```  1907             with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
```
```  1908               polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
```
```  1909             have ?ths
```
```  1910               apply (clarsimp simp add: Let_def)
```
```  1911               apply (rule exI[where x="?b *\<^sub>p ?xdn"])
```
```  1912               apply simp
```
```  1913               apply (rule exI[where x="0"], simp)
```
```  1914               done
```
```  1915           }
```
```  1916           then have ?ths by blast
```
```  1917         }
```
```  1918         ultimately have ?ths
```
```  1919           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"]
```
```  1920             head_nz[OF np] pnz sz ap[symmetric]
```
```  1921           by (auto simp add: degree_eq_degreen0[symmetric])
```
```  1922       }
```
```  1923       ultimately have ?ths by blast
```
```  1924     }
```
```  1925     ultimately have ?ths by blast
```
```  1926   }
```
```  1927   ultimately show ?ths by blast
```
```  1928 qed
```
```  1929
```
```  1930 lemma polydivide_properties:
```
```  1931   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```  1932     and np: "isnpolyh p n0"
```
```  1933     and ns: "isnpolyh s n1"
```
```  1934     and pnz: "p \<noteq> 0\<^sub>p"
```
```  1935   shows "\<exists>k r. polydivide s p = (k, r) \<and>
```
```  1936     (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) \<and>
```
```  1937     (\<exists>q n1. isnpolyh q n1 \<and> polypow k (head p) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
```
```  1938 proof -
```
```  1939   have trv: "head p = head p" "degree p = degree p"
```
```  1940     by simp_all
```
```  1941   from polydivide_def[where s="s" and p="p"] have ex: "\<exists> k r. polydivide s p = (k,r)"
```
```  1942     by auto
```
```  1943   then obtain k r where kr: "polydivide s p = (k,r)"
```
```  1944     by blast
```
```  1945   from trans[OF polydivide_def[where s="s"and p="p", symmetric] kr]
```
```  1946     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
```
```  1947   have "(degree r = 0 \<or> degree r < degree p) \<and>
```
```  1948     (\<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)"
```
```  1949     by blast
```
```  1950   with kr show ?thesis
```
```  1951     apply -
```
```  1952     apply (rule exI[where x="k"])
```
```  1953     apply (rule exI[where x="r"])
```
```  1954     apply simp
```
```  1955     done
```
```  1956 qed
```
```  1957
```
```  1958
```
```  1959 subsection \<open>More about polypoly and pnormal etc\<close>
```
```  1960
```
```  1961 definition "isnonconstant p \<longleftrightarrow> \<not> isconstant p"
```
```  1962
```
```  1963 lemma isnonconstant_pnormal_iff:
```
```  1964   assumes "isnonconstant p"
```
```  1965   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1966 proof
```
```  1967   let ?p = "polypoly bs p"
```
```  1968   assume H: "pnormal ?p"
```
```  1969   have csz: "coefficients p \<noteq> []"
```
```  1970     using assms by (cases p) auto
```
```  1971   from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"] pnormal_last_nonzero[OF H]
```
```  1972   show "Ipoly bs (head p) \<noteq> 0"
```
```  1973     by (simp add: polypoly_def)
```
```  1974 next
```
```  1975   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1976   let ?p = "polypoly bs p"
```
```  1977   have csz: "coefficients p \<noteq> []"
```
```  1978     using assms by (cases p) auto
```
```  1979   then have pz: "?p \<noteq> []"
```
```  1980     by (simp add: polypoly_def)
```
```  1981   then have lg: "length ?p > 0"
```
```  1982     by simp
```
```  1983   from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1984   have lz: "last ?p \<noteq> 0"
```
```  1985     by (simp add: polypoly_def)
```
```  1986   from pnormal_last_length[OF lg lz] show "pnormal ?p" .
```
```  1987 qed
```
```  1988
```
```  1989 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
```
```  1990   unfolding isnonconstant_def
```
```  1991   apply (cases p)
```
```  1992   apply simp_all
```
```  1993   apply (rename_tac nat a, case_tac nat)
```
```  1994   apply auto
```
```  1995   done
```
```  1996
```
```  1997 lemma isnonconstant_nonconstant:
```
```  1998   assumes "isnonconstant p"
```
```  1999   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  2000 proof
```
```  2001   let ?p = "polypoly bs p"
```
```  2002   assume nc: "nonconstant ?p"
```
```  2003   from isnonconstant_pnormal_iff[OF assms, of bs] nc
```
```  2004   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  2005     unfolding nonconstant_def by blast
```
```  2006 next
```
```  2007   let ?p = "polypoly bs p"
```
```  2008   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  2009   from isnonconstant_pnormal_iff[OF assms, of bs] h
```
```  2010   have pn: "pnormal ?p"
```
```  2011     by blast
```
```  2012   {
```
```  2013     fix x
```
```  2014     assume H: "?p = [x]"
```
```  2015     from H have "length (coefficients p) = 1"
```
```  2016       unfolding polypoly_def by auto
```
```  2017     with isnonconstant_coefficients_length[OF assms]
```
```  2018     have False by arith
```
```  2019   }
```
```  2020   then show "nonconstant ?p"
```
```  2021     using pn unfolding nonconstant_def by blast
```
```  2022 qed
```
```  2023
```
```  2024 lemma pnormal_length: "p \<noteq> [] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
```
```  2025   apply (induct p)
```
```  2026   apply (simp_all add: pnormal_def)
```
```  2027   apply (case_tac "p = []")
```
```  2028   apply simp_all
```
```  2029   done
```
```  2030
```
```  2031 lemma degree_degree:
```
```  2032   assumes "isnonconstant p"
```
```  2033   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  2034 proof
```
```  2035   let ?p = "polypoly bs p"
```
```  2036   assume H: "degree p = Polynomial_List.degree ?p"
```
```  2037   from isnonconstant_coefficients_length[OF assms] have pz: "?p \<noteq> []"
```
```  2038     unfolding polypoly_def by auto
```
```  2039   from H degree_coefficients[of p] isnonconstant_coefficients_length[OF assms]
```
```  2040   have lg: "length (pnormalize ?p) = length ?p"
```
```  2041     unfolding Polynomial_List.degree_def polypoly_def by simp
```
```  2042   then have "pnormal ?p"
```
```  2043     using pnormal_length[OF pz] by blast
```
```  2044   with isnonconstant_pnormal_iff[OF assms] show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  2045     by blast
```
```  2046 next
```
```  2047   let ?p = "polypoly bs p"
```
```  2048   assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  2049   with isnonconstant_pnormal_iff[OF assms] have "pnormal ?p"
```
```  2050     by blast
```
```  2051   with degree_coefficients[of p] isnonconstant_coefficients_length[OF assms]
```
```  2052   show "degree p = Polynomial_List.degree ?p"
```
```  2053     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
```
```  2054 qed
```
```  2055
```
```  2056
```
```  2057 section \<open>Swaps ; Division by a certain variable\<close>
```
```  2058
```
```  2059 primrec swap :: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly"
```
```  2060 where
```
```  2061   "swap n m (C x) = C x"
```
```  2062 | "swap n m (Bound k) = Bound (if k = n then m else if k = m then n else k)"
```
```  2063 | "swap n m (Neg t) = Neg (swap n m t)"
```
```  2064 | "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
```
```  2065 | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
```
```  2066 | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
```
```  2067 | "swap n m (Pw t k) = Pw (swap n m t) k"
```
```  2068 | "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)"
```
```  2069
```
```  2070 lemma swap:
```
```  2071   assumes "n < length bs"
```
```  2072     and "m < length bs"
```
```  2073   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  2074 proof (induct t)
```
```  2075   case (Bound k)
```
```  2076   then show ?case
```
```  2077     using assms by simp
```
```  2078 next
```
```  2079   case (CN c k p)
```
```  2080   then show ?case
```
```  2081     using assms by simp
```
```  2082 qed simp_all
```
```  2083
```
```  2084 lemma swap_swap_id [simp]: "swap n m (swap m n t) = t"
```
```  2085   by (induct t) simp_all
```
```  2086
```
```  2087 lemma swap_commute: "swap n m p = swap m n p"
```
```  2088   by (induct p) simp_all
```
```  2089
```
```  2090 lemma swap_same_id[simp]: "swap n n t = t"
```
```  2091   by (induct t) simp_all
```
```  2092
```
```  2093 definition "swapnorm n m t = polynate (swap n m t)"
```
```  2094
```
```  2095 lemma swapnorm:
```
```  2096   assumes nbs: "n < length bs"
```
```  2097     and mbs: "m < length bs"
```
```  2098   shows "((Ipoly bs (swapnorm n m t) :: 'a::{field_char_0,field})) =
```
```  2099     Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  2100   using swap[OF assms] swapnorm_def by simp
```
```  2101
```
```  2102 lemma swapnorm_isnpoly [simp]:
```
```  2103   assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
```
```  2104   shows "isnpoly (swapnorm n m p)"
```
```  2105   unfolding swapnorm_def by simp
```
```  2106
```
```  2107 definition "polydivideby n s p =
```
```  2108   (let
```
```  2109     ss = swapnorm 0 n s;
```
```  2110     sp = swapnorm 0 n p;
```
```  2111     h = head sp;
```
```  2112     (k, r) = polydivide ss sp
```
```  2113    in (k, swapnorm 0 n h, swapnorm 0 n r))"
```
```  2114
```
```  2115 lemma swap_nz [simp]: "swap n m p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
```
```  2116   by (induct p) simp_all
```
```  2117
```
```  2118 fun isweaknpoly :: "poly \<Rightarrow> bool"
```
```  2119 where
```
```  2120   "isweaknpoly (C c) = True"
```
```  2121 | "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
```
```  2122 | "isweaknpoly p = False"
```
```  2123
```
```  2124 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
```
```  2125   by (induct p arbitrary: n0) auto
```
```  2126
```
```  2127 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
```
```  2128   by (induct p) auto
```
```  2129
```
`  2130 end`