src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
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tuned proofs;
```     1 (*  Title:      HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
```
```     2     Author:     Amine Chaieb
```
```     3 *)
```
```     4
```
```     5 header {* Implementation and verification of multivariate polynomials *}
```
```     6
```
```     7 theory Reflected_Multivariate_Polynomial
```
```     8 imports Complex_Main Rat_Pair Polynomial_List
```
```     9 begin
```
```    10
```
```    11 subsection{* Datatype of polynomial expressions *}
```
```    12
```
```    13 datatype poly = C Num| Bound nat| Add poly poly|Sub poly poly
```
```    14   | Mul poly poly| Neg poly| Pw poly nat| CN poly nat poly
```
```    15
```
```    16 abbreviation poly_0 :: "poly" ("0\<^sub>p") where "0\<^sub>p \<equiv> C (0\<^sub>N)"
```
```    17 abbreviation poly_p :: "int \<Rightarrow> poly" ("'((_)')\<^sub>p") where "(i)\<^sub>p \<equiv> C (i)\<^sub>N"
```
```    18
```
```    19
```
```    20 subsection{* Boundedness, substitution and all that *}
```
```    21
```
```    22 primrec polysize:: "poly \<Rightarrow> nat"
```
```    23 where
```
```    24   "polysize (C c) = 1"
```
```    25 | "polysize (Bound n) = 1"
```
```    26 | "polysize (Neg p) = 1 + polysize p"
```
```    27 | "polysize (Add p q) = 1 + polysize p + polysize q"
```
```    28 | "polysize (Sub p q) = 1 + polysize p + polysize q"
```
```    29 | "polysize (Mul p q) = 1 + polysize p + polysize q"
```
```    30 | "polysize (Pw p n) = 1 + polysize p"
```
```    31 | "polysize (CN c n p) = 4 + polysize c + polysize p"
```
```    32
```
```    33 primrec polybound0:: "poly \<Rightarrow> bool" -- {* a poly is INDEPENDENT of Bound 0 *}
```
```    34 where
```
```    35   "polybound0 (C c) \<longleftrightarrow> True"
```
```    36 | "polybound0 (Bound n) \<longleftrightarrow> n > 0"
```
```    37 | "polybound0 (Neg a) \<longleftrightarrow> polybound0 a"
```
```    38 | "polybound0 (Add a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
```
```    39 | "polybound0 (Sub a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
```
```    40 | "polybound0 (Mul a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
```
```    41 | "polybound0 (Pw p n) \<longleftrightarrow> polybound0 p"
```
```    42 | "polybound0 (CN c n p) \<longleftrightarrow> n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p"
```
```    43
```
```    44 primrec polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" -- {* substitute a poly into a poly for Bound 0 *}
```
```    45 where
```
```    46   "polysubst0 t (C c) = C c"
```
```    47 | "polysubst0 t (Bound n) = (if n = 0 then t else Bound n)"
```
```    48 | "polysubst0 t (Neg a) = Neg (polysubst0 t a)"
```
```    49 | "polysubst0 t (Add a b) = Add (polysubst0 t a) (polysubst0 t b)"
```
```    50 | "polysubst0 t (Sub a b) = Sub (polysubst0 t a) (polysubst0 t b)"
```
```    51 | "polysubst0 t (Mul a b) = Mul (polysubst0 t a) (polysubst0 t b)"
```
```    52 | "polysubst0 t (Pw p n) = Pw (polysubst0 t p) n"
```
```    53 | "polysubst0 t (CN c n p) =
```
```    54     (if n = 0 then Add (polysubst0 t c) (Mul t (polysubst0 t p))
```
```    55      else CN (polysubst0 t c) n (polysubst0 t p))"
```
```    56
```
```    57 fun decrpoly:: "poly \<Rightarrow> poly"
```
```    58 where
```
```    59   "decrpoly (Bound n) = Bound (n - 1)"
```
```    60 | "decrpoly (Neg a) = Neg (decrpoly a)"
```
```    61 | "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"
```
```    62 | "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
```
```    63 | "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
```
```    64 | "decrpoly (Pw p n) = Pw (decrpoly p) n"
```
```    65 | "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
```
```    66 | "decrpoly a = a"
```
```    67
```
```    68
```
```    69 subsection{* Degrees and heads and coefficients *}
```
```    70
```
```    71 fun degree:: "poly \<Rightarrow> nat"
```
```    72 where
```
```    73   "degree (CN c 0 p) = 1 + degree p"
```
```    74 | "degree p = 0"
```
```    75
```
```    76 fun head:: "poly \<Rightarrow> poly"
```
```    77 where
```
```    78   "head (CN c 0 p) = head p"
```
```    79 | "head p = p"
```
```    80
```
```    81 (* More general notions of degree and head *)
```
```    82 fun degreen:: "poly \<Rightarrow> nat \<Rightarrow> nat"
```
```    83 where
```
```    84   "degreen (CN c n p) = (\<lambda>m. if n = m then 1 + degreen p n else 0)"
```
```    85 | "degreen p = (\<lambda>m. 0)"
```
```    86
```
```    87 fun headn:: "poly \<Rightarrow> nat \<Rightarrow> poly"
```
```    88 where
```
```    89   "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
```
```    90 | "headn p = (\<lambda>m. p)"
```
```    91
```
```    92 fun coefficients:: "poly \<Rightarrow> poly list"
```
```    93 where
```
```    94   "coefficients (CN c 0 p) = c # coefficients p"
```
```    95 | "coefficients p = [p]"
```
```    96
```
```    97 fun isconstant:: "poly \<Rightarrow> bool"
```
```    98 where
```
```    99   "isconstant (CN c 0 p) = False"
```
```   100 | "isconstant p = True"
```
```   101
```
```   102 fun behead:: "poly \<Rightarrow> poly"
```
```   103 where
```
```   104   "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
```
```   105 | "behead p = 0\<^sub>p"
```
```   106
```
```   107 fun headconst:: "poly \<Rightarrow> Num"
```
```   108 where
```
```   109   "headconst (CN c n p) = headconst p"
```
```   110 | "headconst (C n) = n"
```
```   111
```
```   112
```
```   113 subsection{* Operations for normalization *}
```
```   114
```
```   115 declare if_cong[fundef_cong del]
```
```   116 declare let_cong[fundef_cong del]
```
```   117
```
```   118 fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
```
```   119 where
```
```   120   "polyadd (C c) (C c') = C (c +\<^sub>N c')"
```
```   121 | "polyadd (C c) (CN c' n' p') = CN (polyadd (C c) c') n' p'"
```
```   122 | "polyadd (CN c n p) (C c') = CN (polyadd c (C c')) n p"
```
```   123 | "polyadd (CN c n p) (CN c' n' p') =
```
```   124     (if n < n' then CN (polyadd c (CN c' n' p')) n p
```
```   125      else if n' < n then CN (polyadd (CN c n p) c') n' p'
```
```   126      else
```
```   127       let
```
```   128         cc' = polyadd c c';
```
```   129         pp' = polyadd p p'
```
```   130       in if pp' = 0\<^sub>p then cc' else CN cc' n pp')"
```
```   131 | "polyadd a b = Add a b"
```
```   132
```
```   133
```
```   134 fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
```
```   135 where
```
```   136   "polyneg (C c) = C (~\<^sub>N c)"
```
```   137 | "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
```
```   138 | "polyneg a = Neg a"
```
```   139
```
```   140 definition polysub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
```
```   141   where "p -\<^sub>p q = polyadd p (polyneg q)"
```
```   142
```
```   143 fun polymul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
```
```   144 where
```
```   145   "polymul (C c) (C c') = C (c*\<^sub>Nc')"
```
```   146 | "polymul (C c) (CN c' n' p') =
```
```   147     (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul (C c) c') n' (polymul (C c) p'))"
```
```   148 | "polymul (CN c n p) (C c') =
```
```   149     (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul c (C c')) n (polymul p (C c')))"
```
```   150 | "polymul (CN c n p) (CN c' n' p') =
```
```   151     (if n < n' then CN (polymul c (CN c' n' p')) n (polymul p (CN c' n' p'))
```
```   152      else if n' < n then CN (polymul (CN c n p) c') n' (polymul (CN c n p) p')
```
```   153      else polyadd (polymul (CN c n p) c') (CN 0\<^sub>p n' (polymul (CN c n p) p')))"
```
```   154 | "polymul a b = Mul a b"
```
```   155
```
```   156 declare if_cong[fundef_cong]
```
```   157 declare let_cong[fundef_cong]
```
```   158
```
```   159 fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
```
```   160 where
```
```   161   "polypow 0 = (\<lambda>p. (1)\<^sub>p)"
```
```   162 | "polypow n =
```
```   163     (\<lambda>p.
```
```   164       let
```
```   165         q = polypow (n div 2) p;
```
```   166         d = polymul q q
```
```   167       in if even n then d else polymul p d)"
```
```   168
```
```   169 abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
```
```   170   where "a ^\<^sub>p k \<equiv> polypow k a"
```
```   171
```
```   172 function polynate :: "poly \<Rightarrow> poly"
```
```   173 where
```
```   174   "polynate (Bound n) = CN 0\<^sub>p n (1)\<^sub>p"
```
```   175 | "polynate (Add p q) = polynate p +\<^sub>p polynate q"
```
```   176 | "polynate (Sub p q) = polynate p -\<^sub>p polynate q"
```
```   177 | "polynate (Mul p q) = polynate p *\<^sub>p polynate q"
```
```   178 | "polynate (Neg p) = ~\<^sub>p (polynate p)"
```
```   179 | "polynate (Pw p n) = polynate p ^\<^sub>p n"
```
```   180 | "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
```
```   181 | "polynate (C c) = C (normNum c)"
```
```   182 by pat_completeness auto
```
```   183 termination by (relation "measure polysize") auto
```
```   184
```
```   185 fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly"
```
```   186 where
```
```   187   "poly_cmul y (C x) = C (y *\<^sub>N x)"
```
```   188 | "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
```
```   189 | "poly_cmul y p = C y *\<^sub>p p"
```
```   190
```
```   191 definition monic :: "poly \<Rightarrow> poly \<times> bool"
```
```   192 where
```
```   193   "monic p =
```
```   194     (let h = headconst p
```
```   195      in if h = 0\<^sub>N then (p, False) else (C (Ninv h) *\<^sub>p p, 0>\<^sub>N h))"
```
```   196
```
```   197
```
```   198 subsection {* Pseudo-division *}
```
```   199
```
```   200 definition shift1 :: "poly \<Rightarrow> poly"
```
```   201   where "shift1 p = CN 0\<^sub>p 0 p"
```
```   202
```
```   203 abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   204   where "funpow \<equiv> compow"
```
```   205
```
```   206 partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
```
```   207 where
```
```   208   "polydivide_aux a n p k s =
```
```   209     (if s = 0\<^sub>p then (k, s)
```
```   210      else
```
```   211       let
```
```   212         b = head s;
```
```   213         m = degree s
```
```   214       in
```
```   215         if m < n then (k,s)
```
```   216         else
```
```   217           let p' = funpow (m - n) shift1 p
```
```   218           in
```
```   219             if a = b then polydivide_aux a n p k (s -\<^sub>p p')
```
```   220             else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))"
```
```   221
```
```   222 definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
```
```   223   where "polydivide s p = polydivide_aux (head p) (degree p) p 0 s"
```
```   224
```
```   225 fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly"
```
```   226 where
```
```   227   "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
```
```   228 | "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
```
```   229
```
```   230 fun poly_deriv :: "poly \<Rightarrow> poly"
```
```   231 where
```
```   232   "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
```
```   233 | "poly_deriv p = 0\<^sub>p"
```
```   234
```
```   235
```
```   236 subsection{* Semantics of the polynomial representation *}
```
```   237
```
```   238 primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0,field_inverse_zero,power}"
```
```   239 where
```
```   240   "Ipoly bs (C c) = INum c"
```
```   241 | "Ipoly bs (Bound n) = bs!n"
```
```   242 | "Ipoly bs (Neg a) = - Ipoly bs a"
```
```   243 | "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
```
```   244 | "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
```
```   245 | "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
```
```   246 | "Ipoly bs (Pw t n) = Ipoly bs t ^ n"
```
```   247 | "Ipoly bs (CN c n p) = Ipoly bs c + (bs!n) * Ipoly bs p"
```
```   248
```
```   249 abbreviation Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0,field_inverse_zero,power}"
```
```   250     ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
```
```   251   where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"
```
```   252
```
```   253 lemma Ipoly_CInt: "Ipoly bs (C (i, 1)) = of_int i"
```
```   254   by (simp add: INum_def)
```
```   255
```
```   256 lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"
```
```   257   by (simp  add: INum_def)
```
```   258
```
```   259 lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
```
```   260
```
```   261
```
```   262 subsection {* Normal form and normalization *}
```
```   263
```
```   264 fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
```
```   265 where
```
```   266   "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
```
```   267 | "isnpolyh (CN c n p) = (\<lambda>k. n \<ge> k \<and> isnpolyh c (Suc n) \<and> isnpolyh p n \<and> p \<noteq> 0\<^sub>p)"
```
```   268 | "isnpolyh p = (\<lambda>k. False)"
```
```   269
```
```   270 lemma isnpolyh_mono: "n' \<le> n \<Longrightarrow> isnpolyh p n \<Longrightarrow> isnpolyh p n'"
```
```   271   by (induct p rule: isnpolyh.induct) auto
```
```   272
```
```   273 definition isnpoly :: "poly \<Rightarrow> bool"
```
```   274   where "isnpoly p = isnpolyh p 0"
```
```   275
```
```   276 text{* polyadd preserves normal forms *}
```
```   277
```
```   278 lemma polyadd_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polyadd p q) (min n0 n1)"
```
```   279 proof (induct p q arbitrary: n0 n1 rule: polyadd.induct)
```
```   280   case (2 ab c' n' p' n0 n1)
```
```   281   from 2 have  th1: "isnpolyh (C ab) (Suc n')"
```
```   282     by simp
```
```   283   from 2(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1"
```
```   284     by simp_all
```
```   285   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')"
```
```   286     by simp
```
```   287   with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^sub>p c') (Suc n')"
```
```   288     by simp
```
```   289   from nplen1 have n01len1: "min n0 n1 \<le> n'"
```
```   290     by simp
```
```   291   then show ?case using 2 th3
```
```   292     by simp
```
```   293 next
```
```   294   case (3 c' n' p' ab n1 n0)
```
```   295   from 3 have  th1: "isnpolyh (C ab) (Suc n')"
```
```   296     by simp
```
```   297   from 3(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1"
```
```   298     by simp_all
```
```   299   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')"
```
```   300     by simp
```
```   301   with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (Suc n')"
```
```   302     by simp
```
```   303   from nplen1 have n01len1: "min n0 n1 \<le> n'"
```
```   304     by simp
```
```   305   then show ?case using 3 th3
```
```   306     by simp
```
```   307 next
```
```   308   case (4 c n p c' n' p' n0 n1)
```
```   309   then have nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n"
```
```   310     by simp_all
```
```   311   from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'"
```
```   312     by simp_all
```
```   313   from 4 have ngen0: "n \<ge> n0"
```
```   314     by simp
```
```   315   from 4 have n'gen1: "n' \<ge> n1"
```
```   316     by simp
```
```   317   have "n < n' \<or> n' < n \<or> n = n'"
```
```   318     by auto
```
```   319   moreover
```
```   320   {
```
```   321     assume eq: "n = n'"
```
```   322     with "4.hyps"(3)[OF nc nc']
```
```   323     have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)"
```
```   324       by auto
```
```   325     then have ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
```
```   326       using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1
```
```   327       by auto
```
```   328     from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n"
```
```   329       by simp
```
```   330     have minle: "min n0 n1 \<le> n'"
```
```   331       using ngen0 n'gen1 eq by simp
```
```   332     from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' have ?case
```
```   333       by (simp add: Let_def)
```
```   334   }
```
```   335   moreover
```
```   336   {
```
```   337     assume lt: "n < n'"
```
```   338     have "min n0 n1 \<le> n0"
```
```   339       by simp
```
```   340     with 4 lt have th1:"min n0 n1 \<le> n"
```
```   341       by auto
```
```   342     from 4 have th21: "isnpolyh c (Suc n)"
```
```   343       by simp
```
```   344     from 4 have th22: "isnpolyh (CN c' n' p') n'"
```
```   345       by simp
```
```   346     from lt have th23: "min (Suc n) n' = Suc n"
```
```   347       by arith
```
```   348     from "4.hyps"(1)[OF th21 th22] have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)"
```
```   349       using th23 by simp
```
```   350     with 4 lt th1 have ?case
```
```   351       by simp
```
```   352   }
```
```   353   moreover
```
```   354   {
```
```   355     assume gt: "n' < n"
```
```   356     then have gt': "n' < n \<and> \<not> n < n'"
```
```   357       by simp
```
```   358     have "min n0 n1 \<le> n1"
```
```   359       by simp
```
```   360     with 4 gt have th1: "min n0 n1 \<le> n'"
```
```   361       by auto
```
```   362     from 4 have th21: "isnpolyh c' (Suc n')"
```
```   363       by simp_all
```
```   364     from 4 have th22: "isnpolyh (CN c n p) n"
```
```   365       by simp
```
```   366     from gt have th23: "min n (Suc n') = Suc n'"
```
```   367       by arith
```
```   368     from "4.hyps"(2)[OF th22 th21] have "isnpolyh (polyadd (CN c n p) c') (Suc n')"
```
```   369       using th23 by simp
```
```   370     with 4 gt th1 have ?case
```
```   371       by simp
```
```   372   }
```
```   373   ultimately show ?case by blast
```
```   374 qed auto
```
```   375
```
```   376 lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
```
```   377   by (induct p q rule: polyadd.induct)
```
```   378     (auto simp add: Let_def field_simps distrib_left[symmetric] simp del: distrib_left)
```
```   379
```
```   380 lemma polyadd_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polyadd p q)"
```
```   381   using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   382
```
```   383 text{* The degree of addition and other general lemmas needed for the normal form of polymul *}
```
```   384
```
```   385 lemma polyadd_different_degreen:
```
```   386   assumes "isnpolyh p n0"
```
```   387     and "isnpolyh q n1"
```
```   388     and "degreen p m \<noteq> degreen q m"
```
```   389     and "m \<le> min n0 n1"
```
```   390   shows "degreen (polyadd p q) m = max (degreen p m) (degreen q m)"
```
```   391   using assms
```
```   392 proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
```
```   393   case (4 c n p c' n' p' m n0 n1)
```
```   394   have "n' = n \<or> n < n' \<or> n' < n" by arith
```
```   395   then show ?case
```
```   396   proof (elim disjE)
```
```   397     assume [simp]: "n' = n"
```
```   398     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
```
```   399     show ?thesis by (auto simp: Let_def)
```
```   400   next
```
```   401     assume "n < n'"
```
```   402     with 4 show ?thesis by auto
```
```   403   next
```
```   404     assume "n' < n"
```
```   405     with 4 show ?thesis by auto
```
```   406   qed
```
```   407 qed auto
```
```   408
```
```   409 lemma headnz[simp]: "isnpolyh p n \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
```
```   410   by (induct p arbitrary: n rule: headn.induct) auto
```
```   411
```
```   412 lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
```
```   413   by (induct p arbitrary: n rule: degree.induct) auto
```
```   414
```
```   415 lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
```
```   416   by (induct p arbitrary: n rule: degreen.induct) auto
```
```   417
```
```   418 lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
```
```   419   by (induct p arbitrary: n rule: degree.induct) auto
```
```   420
```
```   421 lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
```
```   422   using degree_isnpolyh_Suc by auto
```
```   423
```
```   424 lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
```
```   425   using degreen_0 by auto
```
```   426
```
```   427
```
```   428 lemma degreen_polyadd:
```
```   429   assumes np: "isnpolyh p n0"
```
```   430     and nq: "isnpolyh q n1"
```
```   431     and m: "m \<le> max n0 n1"
```
```   432   shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
```
```   433   using np nq m
```
```   434 proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
```
```   435   case (2 c c' n' p' n0 n1)
```
```   436   then show ?case
```
```   437     by (cases n') simp_all
```
```   438 next
```
```   439   case (3 c n p c' n0 n1)
```
```   440   then show ?case
```
```   441     by (cases n) auto
```
```   442 next
```
```   443   case (4 c n p c' n' p' n0 n1 m)
```
```   444   have "n' = n \<or> n < n' \<or> n' < n" by arith
```
```   445   then show ?case
```
```   446   proof (elim disjE)
```
```   447     assume [simp]: "n' = n"
```
```   448     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
```
```   449     show ?thesis by (auto simp: Let_def)
```
```   450   qed simp_all
```
```   451 qed auto
```
```   452
```
```   453 lemma polyadd_eq_const_degreen:
```
```   454   assumes "isnpolyh p n0"
```
```   455     and "isnpolyh q n1"
```
```   456     and "polyadd p q = C c"
```
```   457   shows "degreen p m = degreen q m"
```
```   458   using assms
```
```   459 proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
```
```   460   case (4 c n p c' n' p' m n0 n1 x)
```
```   461   {
```
```   462     assume nn': "n' < n"
```
```   463     then have ?case using 4 by simp
```
```   464   }
```
```   465   moreover
```
```   466   {
```
```   467     assume nn': "\<not> n' < n"
```
```   468     then have "n < n' \<or> n = n'" by arith
```
```   469     moreover { assume "n < n'" with 4 have ?case by simp }
```
```   470     moreover
```
```   471     {
```
```   472       assume eq: "n = n'"
```
```   473       then have ?case using 4
```
```   474         apply (cases "p +\<^sub>p p' = 0\<^sub>p")
```
```   475         apply (auto simp add: Let_def)
```
```   476         apply blast
```
```   477         done
```
```   478     }
```
```   479     ultimately have ?case by blast
```
```   480   }
```
```   481   ultimately show ?case by blast
```
```   482 qed simp_all
```
```   483
```
```   484 lemma polymul_properties:
```
```   485   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   486     and np: "isnpolyh p n0"
```
```   487     and nq: "isnpolyh q n1"
```
```   488     and m: "m \<le> min n0 n1"
```
```   489   shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   490     and "p *\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p \<or> q = 0\<^sub>p"
```
```   491     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)"
```
```   492   using np nq m
```
```   493 proof (induct p q arbitrary: n0 n1 m rule: polymul.induct)
```
```   494   case (2 c c' n' p')
```
```   495   {
```
```   496     case (1 n0 n1)
```
```   497     with "2.hyps"(4-6)[of n' n' n'] and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']
```
```   498     show ?case by (auto simp add: min_def)
```
```   499   next
```
```   500     case (2 n0 n1)
```
```   501     then show ?case by auto
```
```   502   next
```
```   503     case (3 n0 n1)
```
```   504     then show ?case  using "2.hyps" by auto
```
```   505   }
```
```   506 next
```
```   507   case (3 c n p c')
```
```   508   {
```
```   509     case (1 n0 n1)
```
```   510     with "3.hyps"(4-6)[of n n n] and "3.hyps"(1-3)[of "Suc n" "Suc n" n]
```
```   511     show ?case by (auto simp add: min_def)
```
```   512   next
```
```   513     case (2 n0 n1)
```
```   514     then show ?case by auto
```
```   515   next
```
```   516     case (3 n0 n1)
```
```   517     then show ?case  using "3.hyps" by auto
```
```   518   }
```
```   519 next
```
```   520   case (4 c n p c' n' p')
```
```   521   let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
```
```   522   {
```
```   523     case (1 n0 n1)
```
```   524     then have cnp: "isnpolyh ?cnp n"
```
```   525       and cnp': "isnpolyh ?cnp' n'"
```
```   526       and np: "isnpolyh p n"
```
```   527       and nc: "isnpolyh c (Suc n)"
```
```   528       and np': "isnpolyh p' n'"
```
```   529       and nc': "isnpolyh c' (Suc n')"
```
```   530       and nn0: "n \<ge> n0"
```
```   531       and nn1: "n' \<ge> n1"
```
```   532       by simp_all
```
```   533     {
```
```   534       assume "n < n'"
```
```   535       with "4.hyps"(4-5)[OF np cnp', of n] and "4.hyps"(1)[OF nc cnp', of n] nn0 cnp
```
```   536       have ?case by (simp add: min_def)
```
```   537     } moreover {
```
```   538       assume "n' < n"
```
```   539       with "4.hyps"(16-17)[OF cnp np', of "n'"] and "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
```
```   540       have ?case by (cases "Suc n' = n") (simp_all add: min_def)
```
```   541     } moreover {
```
```   542       assume "n' = n"
```
```   543       with "4.hyps"(16-17)[OF cnp np', of n] and "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0
```
```   544       have ?case
```
```   545         apply (auto intro!: polyadd_normh)
```
```   546         apply (simp_all add: min_def isnpolyh_mono[OF nn0])
```
```   547         done
```
```   548     }
```
```   549     ultimately show ?case by arith
```
```   550   next
```
```   551     fix n0 n1 m
```
```   552     assume np: "isnpolyh ?cnp n0"
```
```   553     assume np':"isnpolyh ?cnp' n1"
```
```   554     assume m: "m \<le> min n0 n1"
```
```   555     let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
```
```   556     let ?d1 = "degreen ?cnp m"
```
```   557     let ?d2 = "degreen ?cnp' m"
```
```   558     let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
```
```   559     have "n'<n \<or> n < n' \<or> n' = n" by auto
```
```   560     moreover
```
```   561     {
```
```   562       assume "n' < n \<or> n < n'"
```
```   563       with "4.hyps"(3,6,18) np np' m have ?eq
```
```   564         by auto
```
```   565     }
```
```   566     moreover
```
```   567     {
```
```   568       assume nn': "n' = n"
```
```   569       then have nn: "\<not> n' < n \<and> \<not> n < n'" by arith
```
```   570       from "4.hyps"(16,18)[of n n' n]
```
```   571         "4.hyps"(13,14)[of n "Suc n'" n]
```
```   572         np np' nn'
```
```   573       have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
```
```   574         "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   575         "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
```
```   576         "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
```
```   577       {
```
```   578         assume mn: "m = n"
```
```   579         from "4.hyps"(17,18)[OF norm(1,4), of n]
```
```   580           "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
```
```   581         have degs:
```
```   582           "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else ?d1 + degreen c' n)"
```
```   583           "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n"
```
```   584           by (simp_all add: min_def)
```
```   585         from degs norm have th1: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   586           by simp
```
```   587         then have neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   588           by simp
```
```   589         have nmin: "n \<le> min n n"
```
```   590           by (simp add: min_def)
```
```   591         from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
```
```   592         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 =
```
```   593             degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   594           by simp
```
```   595         from "4.hyps"(16-18)[OF norm(1,4), of n]
```
```   596           "4.hyps"(13-15)[OF norm(1,2), of n]
```
```   597           mn norm m nn' deg
```
```   598         have ?eq by simp
```
```   599       }
```
```   600       moreover
```
```   601       {
```
```   602         assume mn: "m \<noteq> n"
```
```   603         then have mn': "m < n"
```
```   604           using m np by auto
```
```   605         from nn' m np have max1: "m \<le> max n n"
```
```   606           by simp
```
```   607         then have min1: "m \<le> min n n"
```
```   608           by simp
```
```   609         then have min2: "m \<le> min n (Suc n)"
```
```   610           by simp
```
```   611         from "4.hyps"(16-18)[OF norm(1,4) min1]
```
```   612           "4.hyps"(13-15)[OF norm(1,2) min2]
```
```   613           degreen_polyadd[OF norm(3,6) max1]
```
```   614         have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m \<le>
```
```   615             max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
```
```   616           using mn nn' np np' by simp
```
```   617         with "4.hyps"(16-18)[OF norm(1,4) min1]
```
```   618           "4.hyps"(13-15)[OF norm(1,2) min2]
```
```   619           degreen_0[OF norm(3) mn']
```
```   620         have ?eq using nn' mn np np' by clarsimp
```
```   621       }
```
```   622       ultimately have ?eq by blast
```
```   623     }
```
```   624     ultimately show ?eq by blast
```
```   625   }
```
```   626   {
```
```   627     case (2 n0 n1)
```
```   628     then have np: "isnpolyh ?cnp n0"
```
```   629       and np': "isnpolyh ?cnp' n1"
```
```   630       and m: "m \<le> min n0 n1" by simp_all
```
```   631     then have mn: "m \<le> n" by simp
```
```   632     let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
```
```   633     {
```
```   634       assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
```
```   635       then have nn: "\<not> n' < n \<and> \<not> n < n'"
```
```   636         by simp
```
```   637       from "4.hyps"(16-18) [of n n n]
```
```   638         "4.hyps"(13-15)[of n "Suc n" n]
```
```   639         np np' C(2) mn
```
```   640       have norm:
```
```   641         "isnpolyh ?cnp n"
```
```   642         "isnpolyh c' (Suc n)"
```
```   643         "isnpolyh (?cnp *\<^sub>p c') n"
```
```   644         "isnpolyh p' n"
```
```   645         "isnpolyh (?cnp *\<^sub>p p') n"
```
```   646         "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   647         "?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p"
```
```   648         "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
```
```   649         "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
```
```   650         "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
```
```   651         by (simp_all add: min_def)
```
```   652       from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   653         by simp
```
```   654       have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   655         using norm by simp
```
```   656       from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"] degneq
```
```   657       have False by simp
```
```   658     }
```
```   659     then show ?case using "4.hyps" by clarsimp
```
```   660   }
```
```   661 qed auto
```
```   662
```
```   663 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = Ipoly bs p * Ipoly bs q"
```
```   664   by (induct p q rule: polymul.induct) (auto simp add: field_simps)
```
```   665
```
```   666 lemma polymul_normh:
```
```   667   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   668   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   669   using polymul_properties(1) by blast
```
```   670
```
```   671 lemma polymul_eq0_iff:
```
```   672   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   673   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"
```
```   674   using polymul_properties(2) by blast
```
```   675
```
```   676 lemma polymul_degreen:  (* FIXME duplicate? *)
```
```   677   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   678   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> m \<le> min n0 n1 \<Longrightarrow>
```
```   679     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)"
```
```   680   using polymul_properties(3) by blast
```
```   681
```
```   682 lemma polymul_norm:
```
```   683   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   684   shows "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polymul p q)"
```
```   685   using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   686
```
```   687 lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
```
```   688   by (induct p arbitrary: n0 rule: headconst.induct) auto
```
```   689
```
```   690 lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
```
```   691   by (induct p arbitrary: n0) auto
```
```   692
```
```   693 lemma monic_eqI:
```
```   694   assumes np: "isnpolyh p n0"
```
```   695   shows "INum (headconst p) * Ipoly bs (fst (monic p)) =
```
```   696     (Ipoly bs p ::'a::{field_char_0,field_inverse_zero, power})"
```
```   697   unfolding monic_def Let_def
```
```   698 proof (cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
```
```   699   let ?h = "headconst p"
```
```   700   assume pz: "p \<noteq> 0\<^sub>p"
```
```   701   {
```
```   702     assume hz: "INum ?h = (0::'a)"
```
```   703     from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
```
```   704     from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
```
```   705     with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
```
```   706   then show "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast
```
```   707 qed
```
```   708
```
```   709
```
```   710 text{* polyneg is a negation and preserves normal forms *}
```
```   711
```
```   712 lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
```
```   713   by (induct p rule: polyneg.induct) auto
```
```   714
```
```   715 lemma polyneg0: "isnpolyh p n \<Longrightarrow> (~\<^sub>p p) = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
```
```   716   by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: Nneg_def)
```
```   717
```
```   718 lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
```
```   719   by (induct p arbitrary: n0 rule: polyneg.induct) auto
```
```   720
```
```   721 lemma polyneg_normh: "isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n"
```
```   722   by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: polyneg0)
```
```   723
```
```   724 lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
```
```   725   using isnpoly_def polyneg_normh by simp
```
```   726
```
```   727
```
```   728 text{* polysub is a substraction and preserves normal forms *}
```
```   729
```
```   730 lemma polysub[simp]: "Ipoly bs (polysub p q) = Ipoly bs p - Ipoly bs q"
```
```   731   by (simp add: polysub_def)
```
```   732
```
```   733 lemma polysub_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
```
```   734   by (simp add: polysub_def polyneg_normh polyadd_normh)
```
```   735
```
```   736 lemma polysub_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polysub p q)"
```
```   737   using polyadd_norm polyneg_norm by (simp add: polysub_def)
```
```   738
```
```   739 lemma polysub_same_0[simp]:
```
```   740   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   741   shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p"
```
```   742   unfolding polysub_def split_def fst_conv snd_conv
```
```   743   by (induct p arbitrary: n0) (auto simp add: Let_def Nsub0[simplified Nsub_def])
```
```   744
```
```   745 lemma polysub_0:
```
```   746   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   747   shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p -\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = q"
```
```   748   unfolding polysub_def split_def fst_conv snd_conv
```
```   749   by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
```
```   750     (auto simp: Nsub0[simplified Nsub_def] Let_def)
```
```   751
```
```   752 text{* polypow is a power function and preserves normal forms *}
```
```   753
```
```   754 lemma polypow[simp]:
```
```   755   "Ipoly bs (polypow n p) = (Ipoly bs p :: 'a::{field_char_0,field_inverse_zero}) ^ n"
```
```   756 proof (induct n rule: polypow.induct)
```
```   757   case 1
```
```   758   then show ?case by simp
```
```   759 next
```
```   760   case (2 n)
```
```   761   let ?q = "polypow ((Suc n) div 2) p"
```
```   762   let ?d = "polymul ?q ?q"
```
```   763   have "odd (Suc n) \<or> even (Suc n)" by simp
```
```   764   moreover
```
```   765   { assume odd: "odd (Suc n)"
```
```   766     have th: "(Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0)))) = Suc n div 2 + Suc n div 2 + 1"
```
```   767       by arith
```
```   768     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)" 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
```
```   776     using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
```
```   777   moreover
```
```   778   { assume even: "even (Suc n)"
```
```   779     have th: "(Suc (Suc 0)) * (Suc n div Suc (Suc 0)) = Suc n div 2 + Suc n div 2"
```
```   780       by arith
```
```   781     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
```
```   782     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
```
```   783       using "2.hyps" apply (simp only: power_add) by simp
```
```   784     finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}
```
```   785   ultimately show ?case by blast
```
```   786 qed
```
```   787
```
```   788 lemma polypow_normh:
```
```   789   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   790   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
```
```   791 proof (induct k arbitrary: n rule: polypow.induct)
```
```   792   case (2 k n)
```
```   793   let ?q = "polypow (Suc k div 2) p"
```
```   794   let ?d = "polymul ?q ?q"
```
```   795   from 2 have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
```
```   796   from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
```
```   797   from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n" by simp
```
```   798   from dn on show ?case by (simp add: Let_def)
```
```   799 qed auto
```
```   800
```
```   801 lemma polypow_norm:
```
```   802   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   803   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
```
```   804   by (simp add: polypow_normh isnpoly_def)
```
```   805
```
```   806 text{* Finally the whole normalization *}
```
```   807
```
```   808 lemma polynate [simp]:
```
```   809   "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0,field_inverse_zero})"
```
```   810   by (induct p rule:polynate.induct) auto
```
```   811
```
```   812 lemma polynate_norm[simp]:
```
```   813   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   814   shows "isnpoly (polynate p)"
```
```   815   by (induct p rule: polynate.induct)
```
```   816      (simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm,
```
```   817       simp_all add: isnpoly_def)
```
```   818
```
```   819 text{* shift1 *}
```
```   820
```
```   821
```
```   822 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
```
```   823   by (simp add: shift1_def)
```
```   824
```
```   825 lemma shift1_isnpoly:
```
```   826   assumes pn: "isnpoly p"
```
```   827     and pnz: "p \<noteq> 0\<^sub>p"
```
```   828   shows "isnpoly (shift1 p) "
```
```   829   using pn pnz by (simp add: shift1_def isnpoly_def)
```
```   830
```
```   831 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
```
```   832   by (simp add: shift1_def)
```
```   833 lemma funpow_shift1_isnpoly:
```
```   834   "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
```
```   835   by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
```
```   836
```
```   837 lemma funpow_isnpolyh:
```
```   838   assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n"
```
```   839     and np: "isnpolyh p n"
```
```   840   shows "isnpolyh (funpow k f p) n"
```
```   841   using f np by (induct k arbitrary: p) auto
```
```   842
```
```   843 lemma funpow_shift1:
```
```   844   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field_inverse_zero}) =
```
```   845     Ipoly bs (Mul (Pw (Bound 0) n) p)"
```
```   846   by (induct n arbitrary: p) (simp_all add: shift1_isnpoly shift1)
```
```   847
```
```   848 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
```
```   849   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
```
```   850
```
```   851 lemma funpow_shift1_1:
```
```   852   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field_inverse_zero}) =
```
```   853     Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)"
```
```   854   by (simp add: funpow_shift1)
```
```   855
```
```   856 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
```
```   857   by (induct p rule: poly_cmul.induct) (auto simp add: field_simps)
```
```   858
```
```   859 lemma behead:
```
```   860   assumes np: "isnpolyh p n"
```
```   861   shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) =
```
```   862     (Ipoly bs p :: 'a :: {field_char_0,field_inverse_zero})"
```
```   863   using np
```
```   864 proof (induct p arbitrary: n rule: behead.induct)
```
```   865   case (1 c p n)
```
```   866   then have pn: "isnpolyh p n" by simp
```
```   867   from 1(1)[OF pn]
```
```   868   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
```
```   869   then show ?case using "1.hyps"
```
```   870     apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
```
```   871     apply (simp_all add: th[symmetric] field_simps)
```
```   872     done
```
```   873 qed (auto simp add: Let_def)
```
```   874
```
```   875 lemma behead_isnpolyh:
```
```   876   assumes np: "isnpolyh p n"
```
```   877   shows "isnpolyh (behead p) n"
```
```   878   using np by (induct p rule: behead.induct) (auto simp add: Let_def isnpolyh_mono)
```
```   879
```
```   880
```
```   881 subsection{* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
```
```   882
```
```   883 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
```
```   884 proof (induct p arbitrary: n rule: poly.induct, auto)
```
```   885   case (goal1 c n p n')
```
```   886   then have "n = Suc (n - 1)"
```
```   887     by simp
```
```   888   then have "isnpolyh p (Suc (n - 1))"
```
```   889     using `isnpolyh p n` by simp
```
```   890   with goal1(2) show ?case
```
```   891     by simp
```
```   892 qed
```
```   893
```
```   894 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
```
```   895   by (induct p arbitrary: n0 rule: isconstant.induct) (auto simp add: isnpolyh_polybound0)
```
```   896
```
```   897 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
```
```   898   by (induct p) auto
```
```   899
```
```   900 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
```
```   901   apply (induct p arbitrary: n0)
```
```   902   apply auto
```
```   903   apply (atomize)
```
```   904   apply (erule_tac x = "Suc nat" in allE)
```
```   905   apply auto
```
```   906   done
```
```   907
```
```   908 lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
```
```   909   by (induct p  arbitrary: n0 rule: head.induct) (auto intro: isnpolyh_polybound0)
```
```   910
```
```   911 lemma polybound0_I:
```
```   912   assumes nb: "polybound0 a"
```
```   913   shows "Ipoly (b # bs) a = Ipoly (b' # bs) a"
```
```   914   using nb
```
```   915   by (induct a rule: poly.induct) auto
```
```   916
```
```   917 lemma polysubst0_I: "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b # bs) a) # bs) t"
```
```   918   by (induct t) simp_all
```
```   919
```
```   920 lemma polysubst0_I':
```
```   921   assumes nb: "polybound0 a"
```
```   922   shows "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b' # bs) a) # bs) t"
```
```   923   by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
```
```   924
```
```   925 lemma decrpoly:
```
```   926   assumes nb: "polybound0 t"
```
```   927   shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
```
```   928   using nb by (induct t rule: decrpoly.induct) simp_all
```
```   929
```
```   930 lemma polysubst0_polybound0:
```
```   931   assumes nb: "polybound0 t"
```
```   932   shows "polybound0 (polysubst0 t a)"
```
```   933   using nb by (induct a rule: poly.induct) auto
```
```   934
```
```   935 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
```
```   936   by (induct p arbitrary: n rule: degree.induct) (auto simp add: isnpolyh_polybound0)
```
```   937
```
```   938 primrec maxindex :: "poly \<Rightarrow> nat" where
```
```   939   "maxindex (Bound n) = n + 1"
```
```   940 | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
```
```   941 | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
```
```   942 | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
```
```   943 | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
```
```   944 | "maxindex (Neg p) = maxindex p"
```
```   945 | "maxindex (Pw p n) = maxindex p"
```
```   946 | "maxindex (C x) = 0"
```
```   947
```
```   948 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool"
```
```   949   where "wf_bs bs p \<longleftrightarrow> length bs \<ge> maxindex p"
```
```   950
```
```   951 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
```
```   952 proof (induct p rule: coefficients.induct)
```
```   953   case (1 c p)
```
```   954   show ?case
```
```   955   proof
```
```   956     fix x
```
```   957     assume xc: "x \<in> set (coefficients (CN c 0 p))"
```
```   958     then have "x = c \<or> x \<in> set (coefficients p)"
```
```   959       by simp
```
```   960     moreover
```
```   961     {
```
```   962       assume "x = c"
```
```   963       then have "wf_bs bs x"
```
```   964         using "1.prems"  unfolding wf_bs_def by simp
```
```   965     }
```
```   966     moreover
```
```   967     {
```
```   968       assume H: "x \<in> set (coefficients p)"
```
```   969       from "1.prems" have "wf_bs bs p"
```
```   970         unfolding wf_bs_def by simp
```
```   971       with "1.hyps" H have "wf_bs bs x"
```
```   972         by blast
```
```   973     }
```
```   974     ultimately  show "wf_bs bs x"
```
```   975       by blast
```
```   976   qed
```
```   977 qed simp_all
```
```   978
```
```   979 lemma maxindex_coefficients: "\<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
```
```   980   by (induct p rule: coefficients.induct) auto
```
```   981
```
```   982 lemma wf_bs_I: "wf_bs bs p \<Longrightarrow> Ipoly (bs @ bs') p = Ipoly bs p"
```
```   983   unfolding wf_bs_def by (induct p) (auto simp add: nth_append)
```
```   984
```
```   985 lemma take_maxindex_wf:
```
```   986   assumes wf: "wf_bs bs p"
```
```   987   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
```
```   988 proof -
```
```   989   let ?ip = "maxindex p"
```
```   990   let ?tbs = "take ?ip bs"
```
```   991   from wf have "length ?tbs = ?ip"
```
```   992     unfolding wf_bs_def by simp
```
```   993   then have wf': "wf_bs ?tbs p"
```
```   994     unfolding wf_bs_def by  simp
```
```   995   have eq: "bs = ?tbs @ (drop ?ip bs)"
```
```   996     by simp
```
```   997   from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis
```
```   998     using eq by simp
```
```   999 qed
```
```  1000
```
```  1001 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
```
```  1002   by (induct p) auto
```
```  1003
```
```  1004 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
```
```  1005   unfolding wf_bs_def by simp
```
```  1006
```
```  1007 lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
```
```  1008   unfolding wf_bs_def by simp
```
```  1009
```
```  1010
```
```  1011
```
```  1012 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
```
```  1013   by (induct p rule: coefficients.induct) (auto simp add: wf_bs_def)
```
```  1014 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
```
```  1015   by (induct p rule: coefficients.induct) simp_all
```
```  1016
```
```  1017
```
```  1018 lemma coefficients_head: "last (coefficients p) = head p"
```
```  1019   by (induct p rule: coefficients.induct) auto
```
```  1020
```
```  1021 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
```
```  1022   unfolding wf_bs_def by (induct p rule: decrpoly.induct) auto
```
```  1023
```
```  1024 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
```
```  1025   apply (rule exI[where x="replicate (n - length xs) z"])
```
```  1026   apply simp
```
```  1027   done
```
```  1028
```
```  1029 lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
```
```  1030   apply (cases p)
```
```  1031   apply auto
```
```  1032   apply (case_tac "nat")
```
```  1033   apply simp_all
```
```  1034   done
```
```  1035
```
```  1036 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
```
```  1037   unfolding wf_bs_def
```
```  1038   apply (induct p q rule: polyadd.induct)
```
```  1039   apply (auto simp add: Let_def)
```
```  1040   done
```
```  1041
```
```  1042 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
```
```  1043   unfolding wf_bs_def
```
```  1044   apply (induct p q arbitrary: bs rule: polymul.induct)
```
```  1045   apply (simp_all add: wf_bs_polyadd)
```
```  1046   apply clarsimp
```
```  1047   apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
```
```  1048   apply auto
```
```  1049   done
```
```  1050
```
```  1051 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
```
```  1052   unfolding wf_bs_def by (induct p rule: polyneg.induct) auto
```
```  1053
```
```  1054 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
```
```  1055   unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
```
```  1056
```
```  1057
```
```  1058 subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}
```
```  1059
```
```  1060 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
```
```  1061 definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)"
```
```  1062 definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"
```
```  1063
```
```  1064 lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0"
```
```  1065 proof (induct p arbitrary: n0 rule: coefficients.induct)
```
```  1066   case (1 c p n0)
```
```  1067   have cp: "isnpolyh (CN c 0 p) n0"
```
```  1068     by fact
```
```  1069   then have norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
```
```  1070     by (auto simp add: isnpolyh_mono[where n'=0])
```
```  1071   from "1.hyps"[OF norm(2)] norm(1) norm(4) show ?case
```
```  1072     by simp
```
```  1073 qed auto
```
```  1074
```
```  1075 lemma coefficients_isconst:
```
```  1076   "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
```
```  1077   by (induct p arbitrary: n rule: coefficients.induct)
```
```  1078     (auto simp add: isnpolyh_Suc_const)
```
```  1079
```
```  1080 lemma polypoly_polypoly':
```
```  1081   assumes np: "isnpolyh p n0"
```
```  1082   shows "polypoly (x#bs) p = polypoly' bs p"
```
```  1083 proof-
```
```  1084   let ?cf = "set (coefficients p)"
```
```  1085   from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
```
```  1086   {fix q assume q: "q \<in> ?cf"
```
```  1087     from q cn_norm have th: "isnpolyh q n0" by blast
```
```  1088     from coefficients_isconst[OF np] q have "isconstant q" by blast
```
```  1089     with isconstant_polybound0[OF th] have "polybound0 q" by blast}
```
```  1090   then have "\<forall>q \<in> ?cf. polybound0 q" ..
```
```  1091   then have "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"
```
```  1092     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
```
```  1093     by auto
```
```  1094   then show ?thesis unfolding polypoly_def polypoly'_def by simp
```
```  1095 qed
```
```  1096
```
```  1097 lemma polypoly_poly:
```
```  1098   assumes np: "isnpolyh p n0"
```
```  1099   shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
```
```  1100   using np
```
```  1101   by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def)
```
```  1102
```
```  1103 lemma polypoly'_poly:
```
```  1104   assumes np: "isnpolyh p n0"
```
```  1105   shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
```
```  1106   using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
```
```  1107
```
```  1108
```
```  1109 lemma polypoly_poly_polybound0:
```
```  1110   assumes np: "isnpolyh p n0" and nb: "polybound0 p"
```
```  1111   shows "polypoly bs p = [Ipoly bs p]"
```
```  1112   using np nb unfolding polypoly_def
```
```  1113   apply (cases p)
```
```  1114   apply auto
```
```  1115   apply (case_tac nat)
```
```  1116   apply auto
```
```  1117   done
```
```  1118
```
```  1119 lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
```
```  1120   by (induct p rule: head.induct) auto
```
```  1121
```
```  1122 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```  1123   by (cases p) auto
```
```  1124
```
```  1125 lemma head_eq_headn0: "head p = headn p 0"
```
```  1126   by (induct p rule: head.induct) simp_all
```
```  1127
```
```  1128 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```  1129   by (simp add: head_eq_headn0)
```
```  1130
```
```  1131 lemma isnpolyh_zero_iff:
```
```  1132   assumes nq: "isnpolyh p n0"
```
```  1133     and eq :"\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a::{field_char_0,field_inverse_zero, power})"
```
```  1134   shows "p = 0\<^sub>p"
```
```  1135   using nq eq
```
```  1136 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
```
```  1137   case less
```
```  1138   note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
```
```  1139   {
```
```  1140     assume nz: "maxindex p = 0"
```
```  1141     then obtain c where "p = C c"
```
```  1142       using np by (cases p) auto
```
```  1143     with zp np have "p = 0\<^sub>p"
```
```  1144       unfolding wf_bs_def by simp
```
```  1145   }
```
```  1146   moreover
```
```  1147   {
```
```  1148     assume nz: "maxindex p \<noteq> 0"
```
```  1149     let ?h = "head p"
```
```  1150     let ?hd = "decrpoly ?h"
```
```  1151     let ?ihd = "maxindex ?hd"
```
```  1152     from head_isnpolyh[OF np] head_polybound0[OF np]
```
```  1153     have h: "isnpolyh ?h n0" "polybound0 ?h"
```
```  1154       by simp_all
```
```  1155     then have nhd: "isnpolyh ?hd (n0 - 1)"
```
```  1156       using decrpoly_normh by blast
```
```  1157
```
```  1158     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
```
```  1159     have mihn: "maxindex ?h \<le> maxindex p"
```
```  1160       by auto
```
```  1161     with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < maxindex p"
```
```  1162       by auto
```
```  1163     {
```
```  1164       fix bs :: "'a list"
```
```  1165       assume bs: "wf_bs bs ?hd"
```
```  1166       let ?ts = "take ?ihd bs"
```
```  1167       let ?rs = "drop ?ihd bs"
```
```  1168       have ts: "wf_bs ?ts ?hd"
```
```  1169         using bs unfolding wf_bs_def by simp
```
```  1170       have bs_ts_eq: "?ts @ ?rs = bs"
```
```  1171         by simp
```
```  1172       from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x # ?ts) ?h"
```
```  1173         by simp
```
```  1174       from ihd_lt_n have "\<forall>x. length (x # ?ts) \<le> maxindex p"
```
```  1175         by simp
```
```  1176       with length_le_list_ex obtain xs where xs: "length ((x # ?ts) @ xs) = maxindex p"
```
```  1177         by blast
```
```  1178       then have "\<forall>x. wf_bs ((x # ?ts) @ xs) p"
```
```  1179         unfolding wf_bs_def by simp
```
```  1180       with zp have "\<forall>x. Ipoly ((x # ?ts) @ xs) p = 0"
```
```  1181         by blast
```
```  1182       then have "\<forall>x. Ipoly (x # (?ts @ xs)) p = 0"
```
```  1183         by simp
```
```  1184       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
```
```  1185       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"
```
```  1186         by simp
```
```  1187       then have "poly (polypoly' (?ts @ xs) p) = poly []"
```
```  1188         by auto
```
```  1189       then have "\<forall>c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
```
```  1190         using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
```
```  1191       with coefficients_head[of p, symmetric]
```
```  1192       have th0: "Ipoly (?ts @ xs) ?hd = 0"
```
```  1193         by simp
```
```  1194       from bs have wf'': "wf_bs ?ts ?hd"
```
```  1195         unfolding wf_bs_def by simp
```
```  1196       with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0"
```
```  1197         by simp
```
```  1198       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
```
```  1199         by simp
```
```  1200     }
```
```  1201     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
```
```  1202       by blast
```
```  1203     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p"
```
```  1204       by blast
```
```  1205     then have "?h = 0\<^sub>p" by simp
```
```  1206     with head_nz[OF np] have "p = 0\<^sub>p" by simp
```
```  1207   }
```
```  1208   ultimately show "p = 0\<^sub>p"
```
```  1209     by blast
```
```  1210 qed
```
```  1211
```
```  1212 lemma isnpolyh_unique:
```
```  1213   assumes np: "isnpolyh p n0"
```
```  1214     and nq: "isnpolyh q n1"
```
```  1215   shows "(\<forall>bs. \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (\<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup> :: 'a::{field_char_0,field_inverse_zero,power})) \<longleftrightarrow> p = q"
```
```  1216 proof auto
```
```  1217   assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a) = \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
```
```  1218   then have "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)"
```
```  1219     by simp
```
```  1220   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)"
```
```  1221     using wf_bs_polysub[where p=p and q=q] by auto
```
```  1222   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq] show "p = q"
```
```  1223     by blast
```
```  1224 qed
```
```  1225
```
```  1226
```
```  1227 text{* consequences of unicity on the algorithms for polynomial normalization *}
```
```  1228
```
```  1229 lemma polyadd_commute:
```
```  1230   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1231     and np: "isnpolyh p n0"
```
```  1232     and nq: "isnpolyh q n1"
```
```  1233   shows "p +\<^sub>p q = q +\<^sub>p p"
```
```  1234   using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]]
```
```  1235   by simp
```
```  1236
```
```  1237 lemma zero_normh: "isnpolyh 0\<^sub>p n"
```
```  1238   by simp
```
```  1239
```
```  1240 lemma one_normh: "isnpolyh (1)\<^sub>p n"
```
```  1241   by simp
```
```  1242
```
```  1243 lemma polyadd_0[simp]:
```
```  1244   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1245     and np: "isnpolyh p n0"
```
```  1246   shows "p +\<^sub>p 0\<^sub>p = p"
```
```  1247     and "0\<^sub>p +\<^sub>p p = p"
```
```  1248   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
```
```  1249     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
```
```  1250
```
```  1251 lemma polymul_1[simp]:
```
```  1252   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1253     and np: "isnpolyh p n0"
```
```  1254   shows "p *\<^sub>p (1)\<^sub>p = p"
```
```  1255     and "(1)\<^sub>p *\<^sub>p p = p"
```
```  1256   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
```
```  1257     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
```
```  1258
```
```  1259 lemma polymul_0[simp]:
```
```  1260   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1261     and np: "isnpolyh p n0"
```
```  1262   shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p"
```
```  1263     and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
```
```  1264   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
```
```  1265     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
```
```  1266
```
```  1267 lemma polymul_commute:
```
```  1268   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1269     and np: "isnpolyh p n0"
```
```  1270     and nq: "isnpolyh q n1"
```
```  1271   shows "p *\<^sub>p q = q *\<^sub>p p"
```
```  1272   using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np], where ?'a = "'a::{field_char_0,field_inverse_zero, power}"]
```
```  1273   by simp
```
```  1274
```
```  1275 declare polyneg_polyneg [simp]
```
```  1276
```
```  1277 lemma isnpolyh_polynate_id [simp]:
```
```  1278   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1279     and np: "isnpolyh p n0"
```
```  1280   shows "polynate p = p"
```
```  1281   using isnpolyh_unique[where ?'a= "'a::{field_char_0,field_inverse_zero}", OF polynate_norm[of p, unfolded isnpoly_def] np] polynate[where ?'a = "'a::{field_char_0,field_inverse_zero}"]
```
```  1282   by simp
```
```  1283
```
```  1284 lemma polynate_idempotent[simp]:
```
```  1285   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1286   shows "polynate (polynate p) = polynate p"
```
```  1287   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
```
```  1288
```
```  1289 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
```
```  1290   unfolding poly_nate_def polypoly'_def ..
```
```  1291
```
```  1292 lemma poly_nate_poly:
```
```  1293   "poly (poly_nate bs p) = (\<lambda>x:: 'a ::{field_char_0,field_inverse_zero}. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
```
```  1294   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
```
```  1295   unfolding poly_nate_polypoly' by auto
```
```  1296
```
```  1297
```
```  1298 subsection{* heads, degrees and all that *}
```
```  1299
```
```  1300 lemma degree_eq_degreen0: "degree p = degreen p 0"
```
```  1301   by (induct p rule: degree.induct) simp_all
```
```  1302
```
```  1303 lemma degree_polyneg:
```
```  1304   assumes n: "isnpolyh p n"
```
```  1305   shows "degree (polyneg p) = degree p"
```
```  1306   apply (induct p arbitrary: n rule: polyneg.induct)
```
```  1307   using n apply simp_all
```
```  1308   apply (case_tac na)
```
```  1309   apply auto
```
```  1310   done
```
```  1311
```
```  1312 lemma degree_polyadd:
```
```  1313   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1314   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1315   using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
```
```  1316
```
```  1317
```
```  1318 lemma degree_polysub:
```
```  1319   assumes np: "isnpolyh p n0"
```
```  1320     and nq: "isnpolyh q n1"
```
```  1321   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1322 proof-
```
```  1323   from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp
```
```  1324   from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])
```
```  1325 qed
```
```  1326
```
```  1327 lemma degree_polysub_samehead:
```
```  1328   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1329     and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q"
```
```  1330     and d: "degree p = degree q"
```
```  1331   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
```
```  1332   unfolding polysub_def split_def fst_conv snd_conv
```
```  1333   using np nq h d
```
```  1334 proof (induct p q rule: polyadd.induct)
```
```  1335   case (1 c c')
```
```  1336   then show ?case
```
```  1337     by (simp add: Nsub_def Nsub0[simplified Nsub_def])
```
```  1338 next
```
```  1339   case (2 c c' n' p')
```
```  1340   from 2 have "degree (C c) = degree (CN c' n' p')"
```
```  1341     by simp
```
```  1342   then have nz: "n' > 0"
```
```  1343     by (cases n') auto
```
```  1344   then have "head (CN c' n' p') = CN c' n' p'"
```
```  1345     by (cases n') auto
```
```  1346   with 2 show ?case
```
```  1347     by simp
```
```  1348 next
```
```  1349   case (3 c n p c')
```
```  1350   then have "degree (C c') = degree (CN c n p)"
```
```  1351     by simp
```
```  1352   then have nz: "n > 0"
```
```  1353     by (cases n) auto
```
```  1354   then have "head (CN c n p) = CN c n p"
```
```  1355     by (cases n) auto
```
```  1356   with 3 show ?case by simp
```
```  1357 next
```
```  1358   case (4 c n p c' n' p')
```
```  1359   then have H:
```
```  1360     "isnpolyh (CN c n p) n0"
```
```  1361     "isnpolyh (CN c' n' p') n1"
```
```  1362     "head (CN c n p) = head (CN c' n' p')"
```
```  1363     "degree (CN c n p) = degree (CN c' n' p')"
```
```  1364     by simp_all
```
```  1365   then have degc: "degree c = 0" and degc': "degree c' = 0"
```
```  1366     by simp_all
```
```  1367   then have degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
```
```  1368     using H(1-2) degree_polyneg by auto
```
```  1369   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"
```
```  1370     by simp_all
```
```  1371   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc'
```
```  1372   have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"
```
```  1373     by simp
```
```  1374   from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'"
```
```  1375     by auto
```
```  1376   have "n = n' \<or> n < n' \<or> n > n'"
```
```  1377     by arith
```
```  1378   moreover
```
```  1379   {
```
```  1380     assume nn': "n = n'"
```
```  1381     have "n = 0 \<or> n > 0" by arith
```
```  1382     moreover {
```
```  1383       assume nz: "n = 0"
```
```  1384       then have ?case using 4 nn'
```
```  1385         by (auto simp add: Let_def degcmc')
```
```  1386     }
```
```  1387     moreover {
```
```  1388       assume nz: "n > 0"
```
```  1389       with nn' H(3) have  cc': "c = c'" and pp': "p = p'"
```
```  1390         by (cases n, auto)+
```
```  1391       then have ?case
```
```  1392         using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv]
```
```  1393         using polysub_same_0[OF c'nh, simplified polysub_def]
```
```  1394         using nn' 4 by (simp add: Let_def)
```
```  1395     }
```
```  1396     ultimately have ?case by blast
```
```  1397   }
```
```  1398   moreover
```
```  1399   {
```
```  1400     assume nn': "n < n'"
```
```  1401     then have n'p: "n' > 0"
```
```  1402       by simp
```
```  1403     then have headcnp':"head (CN c' n' p') = CN c' n' p'"
```
```  1404       by (cases n') simp_all
```
```  1405     have degcnp': "degree (CN c' n' p') = 0"
```
```  1406       and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
```
```  1407       using 4 nn' by (cases n', simp_all)
```
```  1408     then have "n > 0"
```
```  1409       by (cases n) simp_all
```
```  1410     then have headcnp: "head (CN c n p) = CN c n p"
```
```  1411       by (cases n) auto
```
```  1412     from H(3) headcnp headcnp' nn' have ?case
```
```  1413       by auto
```
```  1414   }
```
```  1415   moreover
```
```  1416   {
```
```  1417     assume nn': "n > n'"
```
```  1418     then have np: "n > 0" by simp
```
```  1419     then have headcnp:"head (CN c n p) = CN c n p"
```
```  1420       by (cases n) simp_all
```
```  1421     from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)"
```
```  1422       by simp
```
```  1423     from np have degcnp: "degree (CN c n p) = 0"
```
```  1424       by (cases n) simp_all
```
```  1425     with degcnpeq have "n' > 0"
```
```  1426       by (cases n') simp_all
```
```  1427     then have headcnp': "head (CN c' n' p') = CN c' n' p'"
```
```  1428       by (cases n') auto
```
```  1429     from H(3) headcnp headcnp' nn' have ?case by auto
```
```  1430   }
```
```  1431   ultimately show ?case by blast
```
```  1432 qed auto
```
```  1433
```
```  1434 lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
```
```  1435   by (induct p arbitrary: n0 rule: head.induct) (simp_all add: shift1_def)
```
```  1436
```
```  1437 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
```
```  1438 proof (induct k arbitrary: n0 p)
```
```  1439   case 0
```
```  1440   then show ?case by auto
```
```  1441 next
```
```  1442   case (Suc k n0 p)
```
```  1443   then have "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
```
```  1444   with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
```
```  1445     and "head (shift1 p) = head p" by (simp_all add: shift1_head)
```
```  1446   then show ?case by (simp add: funpow_swap1)
```
```  1447 qed
```
```  1448
```
```  1449 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
```
```  1450   by (simp add: shift1_def)
```
```  1451
```
```  1452 lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
```
```  1453   by (induct k arbitrary: p) (auto simp add: shift1_degree)
```
```  1454
```
```  1455 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
```
```  1456   by (induct n arbitrary: p) simp_all
```
```  1457
```
```  1458 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
```
```  1459   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1460 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
```
```  1461   by (induct p arbitrary: n rule: degreen.induct) auto
```
```  1462 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
```
```  1463   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1464 lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
```
```  1465   by (induct p rule: head.induct) auto
```
```  1466
```
```  1467 lemma polyadd_eq_const_degree:
```
```  1468   "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> polyadd p q = C c \<Longrightarrow> degree p = degree q"
```
```  1469   using polyadd_eq_const_degreen degree_eq_degreen0 by simp
```
```  1470
```
```  1471 lemma polyadd_head:
```
```  1472   assumes np: "isnpolyh p n0"
```
```  1473     and nq: "isnpolyh q n1"
```
```  1474     and deg: "degree p \<noteq> degree q"
```
```  1475   shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
```
```  1476   using np nq deg
```
```  1477   apply (induct p q arbitrary: n0 n1 rule: polyadd.induct)
```
```  1478   using np
```
```  1479   apply simp_all
```
```  1480   apply (case_tac n', simp, simp)
```
```  1481   apply (case_tac n, simp, simp)
```
```  1482   apply (case_tac n, case_tac n', simp add: Let_def)
```
```  1483   apply (auto simp add: polyadd_eq_const_degree)
```
```  1484   apply (metis head_nz)
```
```  1485   apply (metis head_nz)
```
```  1486   apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
```
```  1487   done
```
```  1488
```
```  1489 lemma polymul_head_polyeq:
```
```  1490   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1491   shows "\<lbrakk>isnpolyh p n0; isnpolyh q n1 ; p \<noteq> 0\<^sub>p ; q \<noteq> 0\<^sub>p \<rbrakk> \<Longrightarrow> head (p *\<^sub>p q) = head p *\<^sub>p head q"
```
```  1492 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
```
```  1493   case (2 c c' n' p' n0 n1)
```
```  1494   then have "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"
```
```  1495     by (simp_all add: head_isnpolyh)
```
```  1496   then show ?case
```
```  1497     using 2 by (cases n') auto
```
```  1498 next
```
```  1499   case (3 c n p c' n0 n1)
```
```  1500   then have "isnpolyh (head (CN c n p)) n0" "isnormNum c'"
```
```  1501     by (simp_all add: head_isnpolyh)
```
```  1502   then show ?case using 3
```
```  1503     by (cases n) auto
```
```  1504 next
```
```  1505   case (4 c n p c' n' p' n0 n1)
```
```  1506   hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
```
```  1507     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
```
```  1508     by simp_all
```
```  1509   have "n < n' \<or> n' < n \<or> n = n'" by arith
```
```  1510   moreover
```
```  1511   {
```
```  1512     assume nn': "n < n'"
```
```  1513     then have ?case
```
```  1514       using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)]
```
```  1515       apply simp
```
```  1516       apply (cases n)
```
```  1517       apply simp
```
```  1518       apply (cases n')
```
```  1519       apply simp_all
```
```  1520       done
```
```  1521   }
```
```  1522   moreover {
```
```  1523     assume nn': "n'< n"
```
```  1524     then have ?case
```
```  1525       using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)]
```
```  1526       apply simp
```
```  1527       apply (cases n')
```
```  1528       apply simp
```
```  1529       apply (cases n)
```
```  1530       apply auto
```
```  1531       done
```
```  1532   }
```
```  1533   moreover {
```
```  1534     assume nn': "n' = n"
```
```  1535     from nn' polymul_normh[OF norm(5,4)]
```
```  1536     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
```
```  1537     from nn' polymul_normh[OF norm(5,3)] norm
```
```  1538     have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
```
```  1539     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
```
```  1540     have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```  1541     from polyadd_normh[OF ncnpc' ncnpp0']
```
```  1542     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"
```
```  1543       by (simp add: min_def)
```
```  1544     {
```
```  1545       assume np: "n > 0"
```
```  1546       with nn' head_isnpolyh_Suc'[OF np nth]
```
```  1547         head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
```
```  1548       have ?case by simp
```
```  1549     }
```
```  1550     moreover
```
```  1551     {
```
```  1552       assume nz: "n = 0"
```
```  1553       from polymul_degreen[OF norm(5,4), where m="0"]
```
```  1554         polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
```
```  1555       norm(5,6) degree_npolyhCN[OF norm(6)]
```
```  1556     have dth:"degree (CN c 0 p *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))" by simp
```
```  1557     hence dth':"degree (CN c 0 p *\<^sub>p c') \<noteq> degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))" by simp
```
```  1558     from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
```
```  1559     have ?case   using norm "4.hyps"(6)[OF norm(5,3)]
```
```  1560         "4.hyps"(5)[OF norm(5,4)] nn' nz by simp
```
```  1561     }
```
```  1562     ultimately have ?case by (cases n) auto
```
```  1563   }
```
```  1564   ultimately show ?case by blast
```
```  1565 qed simp_all
```
```  1566
```
```  1567 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
```
```  1568   by (induct p rule: degree.induct) auto
```
```  1569
```
```  1570 lemma degree_head[simp]: "degree (head p) = 0"
```
```  1571   by (induct p rule: head.induct) auto
```
```  1572
```
```  1573 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
```
```  1574   by (cases n) simp_all
```
```  1575 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
```
```  1576   by (cases n) simp_all
```
```  1577
```
```  1578 lemma polyadd_different_degree:
```
```  1579   "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degree p \<noteq> degree q\<rbrakk> \<Longrightarrow>
```
```  1580     degree (polyadd p q) = max (degree p) (degree q)"
```
```  1581   using polyadd_different_degreen degree_eq_degreen0 by simp
```
```  1582
```
```  1583 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
```
```  1584   by (induct p arbitrary: n0 rule: polyneg.induct) auto
```
```  1585
```
```  1586 lemma degree_polymul:
```
```  1587   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1588     and np: "isnpolyh p n0"
```
```  1589     and nq: "isnpolyh q n1"
```
```  1590   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
```
```  1591   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
```
```  1592
```
```  1593 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
```
```  1594   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1595
```
```  1596 lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"
```
```  1597   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1598
```
```  1599
```
```  1600 subsection {* Correctness of polynomial pseudo division *}
```
```  1601
```
```  1602 lemma polydivide_aux_properties:
```
```  1603   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1604     and np: "isnpolyh p n0"
```
```  1605     and ns: "isnpolyh s n1"
```
```  1606     and ap: "head p = a"
```
```  1607     and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
```
```  1608   shows "(polydivide_aux a n p k s = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1609           \<and> (\<exists>nr. isnpolyh r nr) \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow (k' - k) a) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
```
```  1610   using ns
```
```  1611 proof (induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
```
```  1612   case less
```
```  1613   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
```
```  1614   let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1615     \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
```
```  1616   let ?b = "head s"
```
```  1617   let ?p' = "funpow (degree s - n) shift1 p"
```
```  1618   let ?xdn = "funpow (degree s - n) shift1 (1)\<^sub>p"
```
```  1619   let ?akk' = "a ^\<^sub>p (k' - k)"
```
```  1620   note ns = `isnpolyh s n1`
```
```  1621   from np have np0: "isnpolyh p 0"
```
```  1622     using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by simp
```
```  1623   have np': "isnpolyh ?p' 0"
```
```  1624     using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def
```
```  1625     by simp
```
```  1626   have headp': "head ?p' = head p"
```
```  1627     using funpow_shift1_head[OF np pnz] by simp
```
```  1628   from funpow_shift1_isnpoly[where p="(1)\<^sub>p"] have nxdn: "isnpolyh ?xdn 0"
```
```  1629     by (simp add: isnpoly_def)
```
```  1630   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
```
```  1631   have nakk':"isnpolyh ?akk' 0" by blast
```
```  1632   { assume sz: "s = 0\<^sub>p"
```
```  1633     hence ?ths using np polydivide_aux.simps
```
```  1634       apply clarsimp
```
```  1635       apply (rule exI[where x="0\<^sub>p"])
```
```  1636       apply simp
```
```  1637       done }
```
```  1638   moreover
```
```  1639   { assume sz: "s \<noteq> 0\<^sub>p"
```
```  1640     { assume dn: "degree s < n"
```
```  1641       hence "?ths" using ns ndp np polydivide_aux.simps
```
```  1642         apply auto
```
```  1643         apply (rule exI[where x="0\<^sub>p"])
```
```  1644         apply simp
```
```  1645         done }
```
```  1646     moreover
```
```  1647     { assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith
```
```  1648       have degsp': "degree s = degree ?p'"
```
```  1649         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp
```
```  1650       { assume ba: "?b = a"
```
```  1651         hence headsp': "head s = head ?p'"
```
```  1652           using ap headp' by simp
```
```  1653         have nr: "isnpolyh (s -\<^sub>p ?p') 0"
```
```  1654           using polysub_normh[OF ns np'] by simp
```
```  1655         from degree_polysub_samehead[OF ns np' headsp' degsp']
```
```  1656         have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
```
```  1657         moreover
```
```  1658         { assume deglt:"degree (s -\<^sub>p ?p') < degree s"
```
```  1659           from polydivide_aux.simps sz dn' ba
```
```  1660           have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
```
```  1661             by (simp add: Let_def)
```
```  1662           { assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1663             from less(1)[OF deglt nr, of k k' r] trans[OF eq[symmetric] h1]
```
```  1664             have kk': "k \<le> k'"
```
```  1665               and nr:"\<exists>nr. isnpolyh r nr"
```
```  1666               and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1667               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)"
```
```  1668               by auto
```
```  1669             from q1 obtain q n1 where nq: "isnpolyh q n1"
```
```  1670               and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r" by blast
```
```  1671             from nr obtain nr where nr': "isnpolyh r nr" by blast
```
```  1672             from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0"
```
```  1673               by simp
```
```  1674             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
```
```  1675             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
```
```  1676             from polyadd_normh[OF polymul_normh[OF np
```
```  1677               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
```
```  1678             have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0"
```
```  1679               by simp
```
```  1680             from asp have "\<forall> (bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) =
```
```  1681               Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
```
```  1682             hence " \<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
```
```  1683               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1684               by (simp add: field_simps)
```
```  1685             hence " \<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1686               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p *\<^sub>p p) +
```
```  1687               Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1688               by (auto simp only: funpow_shift1_1)
```
```  1689             hence "\<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1690               Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p) +
```
```  1691               Ipoly bs q) + Ipoly bs r"
```
```  1692               by (simp add: field_simps)
```
```  1693             hence "\<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1694               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)"
```
```  1695               by simp
```
```  1696             with isnpolyh_unique[OF nakks' nqr']
```
```  1697             have "a ^\<^sub>p (k' - k) *\<^sub>p s =
```
```  1698               p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r"
```
```  1699               by blast
```
```  1700             hence ?qths using nq'
```
```  1701               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q" in exI)
```
```  1702               apply (rule_tac x="0" in exI)
```
```  1703               apply simp
```
```  1704               done
```
```  1705             with kk' nr dr have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
```
```  1706               by blast
```
```  1707           }
```
```  1708           hence ?ths by blast
```
```  1709         }
```
```  1710         moreover
```
```  1711         { assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
```
```  1712           from spz isnpolyh_unique[OF polysub_normh[OF ns np'], where q="0\<^sub>p", symmetric, where ?'a = "'a::{field_char_0,field_inverse_zero}"]
```
```  1713           have " \<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs s = Ipoly bs ?p'"
```
```  1714             by simp
```
```  1715           hence "\<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)"
```
```  1716             using np nxdn
```
```  1717             apply simp
```
```  1718             apply (simp only: funpow_shift1_1)
```
```  1719             apply simp
```
```  1720             done
```
```  1721           hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]]
```
```  1722             by blast
```
```  1723           { assume h1: "polydivide_aux a n p k s = (k',r)"
```
```  1724             from polydivide_aux.simps sz dn' ba
```
```  1725             have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
```
```  1726               by (simp add: Let_def)
```
```  1727             also have "\<dots> = (k,0\<^sub>p)"
```
```  1728               using polydivide_aux.simps spz by simp
```
```  1729             finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
```
```  1730             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
```
```  1731               polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
```
```  1732               apply auto
```
```  1733               apply (rule exI[where x="?xdn"])
```
```  1734               apply (auto simp add: polymul_commute[of p])
```
```  1735               done
```
```  1736           }
```
```  1737         }
```
```  1738         ultimately have ?ths by blast
```
```  1739       }
```
```  1740       moreover
```
```  1741       { assume ba: "?b \<noteq> a"
```
```  1742         from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1743           polymul_normh[OF head_isnpolyh[OF ns] np']]
```
```  1744         have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
```
```  1745           by (simp add: min_def)
```
```  1746         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
```
```  1747           using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1748             polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
```
```  1749             funpow_shift1_nz[OF pnz]
```
```  1750           by simp_all
```
```  1751         from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
```
```  1752           polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1753         have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
```
```  1754           using head_head[OF ns] funpow_shift1_head[OF np pnz]
```
```  1755             polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
```
```  1756           by (simp add: ap)
```
```  1757         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
```
```  1758           head_nz[OF np] pnz sz ap[symmetric]
```
```  1759           funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1760           polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"] head_nz[OF ns]
```
```  1761           ndp dn
```
```  1762         have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p')"
```
```  1763           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
```
```  1764         { assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
```
```  1765           from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns]
```
```  1766             polymul_normh[OF head_isnpolyh[OF ns]np']] ap
```
```  1767           have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
```
```  1768             by simp
```
```  1769           { assume h1:"polydivide_aux a n p k s = (k', r)"
```
```  1770             from h1 polydivide_aux.simps sz dn' ba
```
```  1771             have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
```
```  1772               by (simp add: Let_def)
```
```  1773             with less(1)[OF dth nasbp', of "Suc k" k' r]
```
```  1774             obtain q nq nr where kk': "Suc k \<le> k'"
```
```  1775               and nr: "isnpolyh r nr"
```
```  1776               and nq: "isnpolyh q nq"
```
```  1777               and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1778               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"
```
```  1779               by auto
```
```  1780             from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith
```
```  1781             {
```
```  1782               fix bs:: "'a::{field_char_0,field_inverse_zero} list"
```
```  1783               from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
```
```  1784               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)"
```
```  1785                 by simp
```
```  1786               hence "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s =
```
```  1787                 Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
```
```  1788                 by (simp add: field_simps)
```
```  1789               hence "Ipoly bs a ^ (k' - k)  * Ipoly bs s =
```
```  1790                 Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
```
```  1791                 by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1792               hence "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1793                 Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
```
```  1794                 by (simp add: field_simps)
```
```  1795             }
```
```  1796             hence ieq:"\<forall>(bs :: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1797               Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)"
```
```  1798               by auto
```
```  1799             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
```
```  1800             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]]
```
```  1801             have nqw: "isnpolyh ?q 0"
```
```  1802               by simp
```
```  1803             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]]
```
```  1804             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"
```
```  1805               by blast
```
```  1806             from dr kk' nr h1 asth nqw have ?ths
```
```  1807               apply simp
```
```  1808               apply (rule conjI)
```
```  1809               apply (rule exI[where x="nr"], simp)
```
```  1810               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
```
```  1811               apply (rule exI[where x="0"], simp)
```
```  1812               done
```
```  1813           }
```
```  1814           hence ?ths by blast
```
```  1815         }
```
```  1816         moreover
```
```  1817         { assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
```
```  1818           {
```
```  1819             fix bs :: "'a::{field_char_0,field_inverse_zero} list"
```
```  1820             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
```
```  1821             have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'"
```
```  1822               by simp
```
```  1823             hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
```
```  1824               by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1825             hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))"
```
```  1826               by simp
```
```  1827           }
```
```  1828           hence hth: "\<forall> (bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a*\<^sub>p s) =
```
```  1829             Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
```
```  1830           from hth have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
```
```  1831             using isnpolyh_unique[where ?'a = "'a::{field_char_0,field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
```
```  1832                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
```
```  1833               simplified ap] by simp
```
```  1834           { assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1835             from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
```
```  1836             have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
```
```  1837             with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
```
```  1838               polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
```
```  1839             have ?ths
```
```  1840               apply (clarsimp simp add: Let_def)
```
```  1841               apply (rule exI[where x="?b *\<^sub>p ?xdn"])
```
```  1842               apply simp
```
```  1843               apply (rule exI[where x="0"], simp)
```
```  1844               done
```
```  1845           }
```
```  1846           hence ?ths by blast
```
```  1847         }
```
```  1848         ultimately have ?ths
```
```  1849           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"]
```
```  1850             head_nz[OF np] pnz sz ap[symmetric]
```
```  1851           by (simp add: degree_eq_degreen0[symmetric]) blast
```
```  1852       }
```
```  1853       ultimately have ?ths by blast
```
```  1854     }
```
```  1855     ultimately have ?ths by blast
```
```  1856   }
```
```  1857   ultimately show ?ths by blast
```
```  1858 qed
```
```  1859
```
```  1860 lemma polydivide_properties:
```
```  1861   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1862     and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
```
```  1863   shows "\<exists>k r. polydivide s p = (k,r) \<and>
```
```  1864     (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) \<and>
```
```  1865     (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r))"
```
```  1866 proof -
```
```  1867   have trv: "head p = head p" "degree p = degree p"
```
```  1868     by simp_all
```
```  1869   from polydivide_def[where s="s" and p="p"] have ex: "\<exists> k r. polydivide s p = (k,r)"
```
```  1870     by auto
```
```  1871   then obtain k r where kr: "polydivide s p = (k,r)"
```
```  1872     by blast
```
```  1873   from trans[OF polydivide_def[where s="s"and p="p", symmetric] kr]
```
```  1874     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
```
```  1875   have "(degree r = 0 \<or> degree r < degree p) \<and>
```
```  1876     (\<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)"
```
```  1877     by blast
```
```  1878   with kr show ?thesis
```
```  1879     apply -
```
```  1880     apply (rule exI[where x="k"])
```
```  1881     apply (rule exI[where x="r"])
```
```  1882     apply simp
```
```  1883     done
```
```  1884 qed
```
```  1885
```
```  1886
```
```  1887 subsection{* More about polypoly and pnormal etc *}
```
```  1888
```
```  1889 definition "isnonconstant p \<longleftrightarrow> \<not> isconstant p"
```
```  1890
```
```  1891 lemma isnonconstant_pnormal_iff:
```
```  1892   assumes nc: "isnonconstant p"
```
```  1893   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1894 proof
```
```  1895   let ?p = "polypoly bs p"
```
```  1896   assume H: "pnormal ?p"
```
```  1897   have csz: "coefficients p \<noteq> []" using nc by (cases p) auto
```
```  1898
```
```  1899   from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1900     pnormal_last_nonzero[OF H]
```
```  1901   show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)
```
```  1902 next
```
```  1903   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1904   let ?p = "polypoly bs p"
```
```  1905   have csz: "coefficients p \<noteq> []" using nc by (cases p) auto
```
```  1906   hence pz: "?p \<noteq> []" by (simp add: polypoly_def)
```
```  1907   hence lg: "length ?p > 0" by simp
```
```  1908   from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1909   have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)
```
```  1910   from pnormal_last_length[OF lg lz] show "pnormal ?p" .
```
```  1911 qed
```
```  1912
```
```  1913 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
```
```  1914   unfolding isnonconstant_def
```
```  1915   apply (cases p)
```
```  1916   apply simp_all
```
```  1917   apply (case_tac nat)
```
```  1918   apply auto
```
```  1919   done
```
```  1920
```
```  1921 lemma isnonconstant_nonconstant:
```
```  1922   assumes inc: "isnonconstant p"
```
```  1923   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1924 proof
```
```  1925   let ?p = "polypoly bs p"
```
```  1926   assume nc: "nonconstant ?p"
```
```  1927   from isnonconstant_pnormal_iff[OF inc, of bs] nc
```
```  1928   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast
```
```  1929 next
```
```  1930   let ?p = "polypoly bs p"
```
```  1931   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1932   from isnonconstant_pnormal_iff[OF inc, of bs] h
```
```  1933   have pn: "pnormal ?p" by blast
```
```  1934   {
```
```  1935     fix x
```
```  1936     assume H: "?p = [x]"
```
```  1937     from H have "length (coefficients p) = 1"
```
```  1938       unfolding polypoly_def by auto
```
```  1939     with isnonconstant_coefficients_length[OF inc]
```
```  1940       have False by arith
```
```  1941   }
```
```  1942   then show "nonconstant ?p"
```
```  1943     using pn unfolding nonconstant_def by blast
```
```  1944 qed
```
```  1945
```
```  1946 lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
```
```  1947   apply (induct p)
```
```  1948   apply (simp_all add: pnormal_def)
```
```  1949   apply (case_tac "p = []")
```
```  1950   apply simp_all
```
```  1951   done
```
```  1952
```
```  1953 lemma degree_degree:
```
```  1954   assumes inc: "isnonconstant p"
```
```  1955   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1956 proof
```
```  1957   let ?p = "polypoly bs p"
```
```  1958   assume H: "degree p = Polynomial_List.degree ?p"
```
```  1959   from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
```
```  1960     unfolding polypoly_def by auto
```
```  1961   from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
```
```  1962   have lg:"length (pnormalize ?p) = length ?p"
```
```  1963     unfolding Polynomial_List.degree_def polypoly_def by simp
```
```  1964   hence "pnormal ?p" using pnormal_length[OF pz] by blast
```
```  1965   with isnonconstant_pnormal_iff[OF inc]
```
```  1966   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast
```
```  1967 next
```
```  1968   let  ?p = "polypoly bs p"
```
```  1969   assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1970   with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast
```
```  1971   with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
```
```  1972   show "degree p = Polynomial_List.degree ?p"
```
```  1973     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
```
```  1974 qed
```
```  1975
```
```  1976
```
```  1977 section {* Swaps ; Division by a certain variable *}
```
```  1978
```
```  1979 primrec swap :: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly" where
```
```  1980   "swap n m (C x) = C x"
```
```  1981 | "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"
```
```  1982 | "swap n m (Neg t) = Neg (swap n m t)"
```
```  1983 | "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
```
```  1984 | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
```
```  1985 | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
```
```  1986 | "swap n m (Pw t k) = Pw (swap n m t) k"
```
```  1987 | "swap n m (CN c k p) =
```
```  1988     CN (swap n m c) (if k = n then m else if k=m then n else k) (swap n m p)"
```
```  1989
```
```  1990 lemma swap:
```
```  1991   assumes nbs: "n < length bs"
```
```  1992     and mbs: "m < length bs"
```
```  1993   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  1994 proof (induct t)
```
```  1995   case (Bound k)
```
```  1996   then show ?case using nbs mbs by simp
```
```  1997 next
```
```  1998   case (CN c k p)
```
```  1999   then show ?case using nbs mbs by simp
```
```  2000 qed simp_all
```
```  2001
```
```  2002 lemma swap_swap_id [simp]: "swap n m (swap m n t) = t"
```
```  2003   by (induct t) simp_all
```
```  2004
```
```  2005 lemma swap_commute: "swap n m p = swap m n p"
```
```  2006   by (induct p) simp_all
```
```  2007
```
```  2008 lemma swap_same_id[simp]: "swap n n t = t"
```
```  2009   by (induct t) simp_all
```
```  2010
```
```  2011 definition "swapnorm n m t = polynate (swap n m t)"
```
```  2012
```
```  2013 lemma swapnorm:
```
```  2014   assumes nbs: "n < length bs"
```
```  2015     and mbs: "m < length bs"
```
```  2016   shows "((Ipoly bs (swapnorm n m t) :: 'a::{field_char_0,field_inverse_zero})) =
```
```  2017     Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  2018   using swap[OF assms] swapnorm_def by simp
```
```  2019
```
```  2020 lemma swapnorm_isnpoly [simp]:
```
```  2021   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  2022   shows "isnpoly (swapnorm n m p)"
```
```  2023   unfolding swapnorm_def by simp
```
```  2024
```
```  2025 definition "polydivideby n s p =
```
```  2026   (let
```
```  2027     ss = swapnorm 0 n s;
```
```  2028     sp = swapnorm 0 n p;
```
```  2029     h = head sp;
```
```  2030     (k, r) = polydivide ss sp
```
```  2031    in (k, swapnorm 0 n h, swapnorm 0 n r))"
```
```  2032
```
```  2033 lemma swap_nz [simp]: "swap n m p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
```
```  2034   by (induct p) simp_all
```
```  2035
```
```  2036 fun isweaknpoly :: "poly \<Rightarrow> bool"
```
```  2037 where
```
```  2038   "isweaknpoly (C c) = True"
```
```  2039 | "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
```
```  2040 | "isweaknpoly p = False"
```
```  2041
```
```  2042 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
```
```  2043   by (induct p arbitrary: n0) auto
```
```  2044
```
```  2045 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
```
```  2046   by (induct p) auto
```
```  2047
```
`  2048 end`