src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
 author wenzelm Sat Mar 08 23:03:15 2014 +0100 (2014-03-08) changeset 56000 899ad5a3ad00 parent 54489 03ff4d1e6784 child 56009 dda076a32aea permissions -rw-r--r--
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')" by simp
```
```   282   from 2(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
```
```   283   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
```
```   284   with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^sub>p c') (Suc n')" by simp
```
```   285   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
```
```   286   thus ?case using 2 th3 by simp
```
```   287 next
```
```   288   case (3 c' n' p' ab n1 n0)
```
```   289   from 3 have  th1: "isnpolyh (C ab) (Suc n')" by simp
```
```   290   from 3(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
```
```   291   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
```
```   292   with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (Suc n')" by simp
```
```   293   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
```
```   294   thus ?case using 3 th3 by simp
```
```   295 next
```
```   296   case (4 c n p c' n' p' n0 n1)
```
```   297   hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
```
```   298   from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all
```
```   299   from 4 have ngen0: "n \<ge> n0" by simp
```
```   300   from 4 have n'gen1: "n' \<ge> n1" by simp
```
```   301   have "n < n' \<or> n' < n \<or> n = n'" by auto
```
```   302   moreover {
```
```   303     assume eq: "n = n'"
```
```   304     with "4.hyps"(3)[OF nc nc']
```
```   305     have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
```
```   306     hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
```
```   307       using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
```
```   308     from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
```
```   309     have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
```
```   310     from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' have ?case by (simp add: Let_def) }
```
```   311   moreover {
```
```   312     assume lt: "n < n'"
```
```   313     have "min n0 n1 \<le> n0" by simp
```
```   314     with 4 lt have th1:"min n0 n1 \<le> n" by auto
```
```   315     from 4 have th21: "isnpolyh c (Suc n)" by simp
```
```   316     from 4 have th22: "isnpolyh (CN c' n' p') n'" by simp
```
```   317     from lt have th23: "min (Suc n) n' = Suc n" by arith
```
```   318     from "4.hyps"(1)[OF th21 th22]
```
```   319     have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)" using th23 by simp
```
```   320     with 4 lt th1 have ?case by simp }
```
```   321   moreover {
```
```   322     assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
```
```   323     have "min n0 n1 \<le> n1"  by simp
```
```   324     with 4 gt have th1:"min n0 n1 \<le> n'" by auto
```
```   325     from 4 have th21: "isnpolyh c' (Suc n')" by simp_all
```
```   326     from 4 have th22: "isnpolyh (CN c n p) n" by simp
```
```   327     from gt have th23: "min n (Suc n') = Suc n'" by arith
```
```   328     from "4.hyps"(2)[OF th22 th21]
```
```   329     have "isnpolyh (polyadd (CN c n p) c') (Suc n')" using th23 by simp
```
```   330     with 4 gt th1 have ?case by simp }
```
```   331   ultimately show ?case by blast
```
```   332 qed auto
```
```   333
```
```   334 lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
```
```   335   by (induct p q rule: polyadd.induct)
```
```   336     (auto simp add: Let_def field_simps distrib_left[symmetric] simp del: distrib_left)
```
```   337
```
```   338 lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd p q)"
```
```   339   using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   340
```
```   341 text{* The degree of addition and other general lemmas needed for the normal form of polymul *}
```
```   342
```
```   343 lemma polyadd_different_degreen:
```
```   344   "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow>
```
```   345   degreen (polyadd p q) m = max (degreen p m) (degreen q m)"
```
```   346 proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
```
```   347   case (4 c n p c' n' p' m n0 n1)
```
```   348   have "n' = n \<or> n < n' \<or> n' < n" by arith
```
```   349   thus ?case
```
```   350   proof (elim disjE)
```
```   351     assume [simp]: "n' = n"
```
```   352     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
```
```   353     show ?thesis by (auto simp: Let_def)
```
```   354   next
```
```   355     assume "n < n'"
```
```   356     with 4 show ?thesis by auto
```
```   357   next
```
```   358     assume "n' < n"
```
```   359     with 4 show ?thesis by auto
```
```   360   qed
```
```   361 qed auto
```
```   362
```
```   363 lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
```
```   364   by (induct p arbitrary: n rule: headn.induct) auto
```
```   365 lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
```
```   366   by (induct p arbitrary: n rule: degree.induct) auto
```
```   367 lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
```
```   368   by (induct p arbitrary: n rule: degreen.induct) auto
```
```   369
```
```   370 lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
```
```   371   by (induct p arbitrary: n rule: degree.induct) auto
```
```   372
```
```   373 lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
```
```   374   using degree_isnpolyh_Suc by auto
```
```   375 lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
```
```   376   using degreen_0 by auto
```
```   377
```
```   378
```
```   379 lemma degreen_polyadd:
```
```   380   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1"
```
```   381   shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
```
```   382   using np nq m
```
```   383 proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
```
```   384   case (2 c c' n' p' n0 n1)
```
```   385   thus ?case  by (cases n') simp_all
```
```   386 next
```
```   387   case (3 c n p c' n0 n1)
```
```   388   thus ?case by (cases n) auto
```
```   389 next
```
```   390   case (4 c n p c' n' p' n0 n1 m)
```
```   391   have "n' = n \<or> n < n' \<or> n' < n" by arith
```
```   392   thus ?case
```
```   393   proof (elim disjE)
```
```   394     assume [simp]: "n' = n"
```
```   395     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
```
```   396     show ?thesis by (auto simp: Let_def)
```
```   397   qed simp_all
```
```   398 qed auto
```
```   399
```
```   400 lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd p q = C c\<rbrakk>
```
```   401   \<Longrightarrow> degreen p m = degreen q m"
```
```   402 proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
```
```   403   case (4 c n p c' n' p' m n0 n1 x)
```
```   404   { assume nn': "n' < n" hence ?case using 4 by simp }
```
```   405   moreover
```
```   406   { assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
```
```   407     moreover { assume "n < n'" with 4 have ?case by simp }
```
```   408     moreover { assume eq: "n = n'" hence ?case using 4
```
```   409         apply (cases "p +\<^sub>p p' = 0\<^sub>p")
```
```   410         apply (auto simp add: Let_def)
```
```   411         apply blast
```
```   412         done
```
```   413     }
```
```   414     ultimately have ?case by blast }
```
```   415   ultimately show ?case by blast
```
```   416 qed simp_all
```
```   417
```
```   418 lemma polymul_properties:
```
```   419   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   420     and np: "isnpolyh p n0"
```
```   421     and nq: "isnpolyh q n1"
```
```   422     and m: "m \<le> min n0 n1"
```
```   423   shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   424     and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)"
```
```   425     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)"
```
```   426   using np nq m
```
```   427 proof (induct p q arbitrary: n0 n1 m rule: polymul.induct)
```
```   428   case (2 c c' n' p')
```
```   429   { case (1 n0 n1)
```
```   430     with "2.hyps"(4-6)[of n' n' n']
```
```   431       and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']
```
```   432     show ?case by (auto simp add: min_def)
```
```   433   next
```
```   434     case (2 n0 n1) thus ?case by auto
```
```   435   next
```
```   436     case (3 n0 n1) thus ?case  using "2.hyps" by auto }
```
```   437 next
```
```   438   case (3 c n p c')
```
```   439   { case (1 n0 n1)
```
```   440     with "3.hyps"(4-6)[of n n n]
```
```   441       "3.hyps"(1-3)[of "Suc n" "Suc n" n]
```
```   442     show ?case by (auto simp add: min_def)
```
```   443   next
```
```   444     case (2 n0 n1) thus ?case by auto
```
```   445   next
```
```   446     case (3 n0 n1) thus ?case  using "3.hyps" by auto }
```
```   447 next
```
```   448   case (4 c n p c' n' p')
```
```   449   let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
```
```   450     {
```
```   451       case (1 n0 n1)
```
```   452       hence cnp: "isnpolyh ?cnp n" and cnp': "isnpolyh ?cnp' n'"
```
```   453         and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)"
```
```   454         and np': "isnpolyh p' n'" and nc': "isnpolyh c' (Suc n')"
```
```   455         and nn0: "n \<ge> n0" and nn1:"n' \<ge> n1"
```
```   456         by simp_all
```
```   457       { assume "n < n'"
```
```   458         with "4.hyps"(4-5)[OF np cnp', of n]
```
```   459           "4.hyps"(1)[OF nc cnp', of n] nn0 cnp
```
```   460         have ?case by (simp add: min_def)
```
```   461       } moreover {
```
```   462         assume "n' < n"
```
```   463         with "4.hyps"(16-17)[OF cnp np', of "n'"]
```
```   464           "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
```
```   465         have ?case
```
```   466           by (cases "Suc n' = n") (simp_all add: min_def)
```
```   467       } moreover {
```
```   468         assume "n' = n"
```
```   469         with "4.hyps"(16-17)[OF cnp np', of n]
```
```   470           "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0
```
```   471         have ?case
```
```   472           apply (auto intro!: polyadd_normh)
```
```   473           apply (simp_all add: min_def isnpolyh_mono[OF nn0])
```
```   474           done
```
```   475       }
```
```   476       ultimately show ?case by arith
```
```   477     next
```
```   478       fix n0 n1 m
```
```   479       assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1"
```
```   480       and m: "m \<le> min n0 n1"
```
```   481       let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
```
```   482       let ?d1 = "degreen ?cnp m"
```
```   483       let ?d2 = "degreen ?cnp' m"
```
```   484       let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
```
```   485       have "n'<n \<or> n < n' \<or> n' = n" by auto
```
```   486       moreover
```
```   487       {assume "n' < n \<or> n < n'"
```
```   488         with "4.hyps"(3,6,18) np np' m
```
```   489         have ?eq by auto }
```
```   490       moreover
```
```   491       { assume nn': "n' = n"
```
```   492         hence nn: "\<not> n' < n \<and> \<not> n < n'" by arith
```
```   493         from "4.hyps"(16,18)[of n n' n]
```
```   494           "4.hyps"(13,14)[of n "Suc n'" n]
```
```   495           np np' nn'
```
```   496         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
```
```   497           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   498           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
```
```   499           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
```
```   500         { assume mn: "m = n"
```
```   501           from "4.hyps"(17,18)[OF norm(1,4), of n]
```
```   502             "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
```
```   503           have degs:  "degreen (?cnp *\<^sub>p c') n =
```
```   504             (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"
```
```   505             "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n" by (simp_all add: min_def)
```
```   506           from degs norm
```
```   507           have th1: "degreen(?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" by simp
```
```   508           hence neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   509             by simp
```
```   510           have nmin: "n \<le> min n n" by (simp add: min_def)
```
```   511           from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
```
```   512           have deg: "degreen (CN c n p *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n = degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```   513           from "4.hyps"(16-18)[OF norm(1,4), of n]
```
```   514             "4.hyps"(13-15)[OF norm(1,2), of n]
```
```   515             mn norm m nn' deg
```
```   516           have ?eq by simp }
```
```   517         moreover
```
```   518         { assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
```
```   519           from nn' m np have max1: "m \<le> max n n"  by simp
```
```   520           hence min1: "m \<le> min n n" by simp
```
```   521           hence min2: "m \<le> min n (Suc n)" by simp
```
```   522           from "4.hyps"(16-18)[OF norm(1,4) min1]
```
```   523             "4.hyps"(13-15)[OF norm(1,2) min2]
```
```   524             degreen_polyadd[OF norm(3,6) max1]
```
```   525
```
```   526           have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m
```
```   527             \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
```
```   528             using mn nn' np np' by simp
```
```   529           with "4.hyps"(16-18)[OF norm(1,4) min1]
```
```   530             "4.hyps"(13-15)[OF norm(1,2) min2]
```
```   531             degreen_0[OF norm(3) mn']
```
```   532           have ?eq using nn' mn np np' by clarsimp }
```
```   533         ultimately have ?eq by blast }
```
```   534       ultimately show ?eq by blast }
```
```   535     { case (2 n0 n1)
```
```   536       hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1"
```
```   537         and m: "m \<le> min n0 n1" by simp_all
```
```   538       hence mn: "m \<le> n" by simp
```
```   539       let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
```
```   540       {assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
```
```   541         hence nn: "\<not>n' < n \<and> \<not> n<n'" by simp
```
```   542         from "4.hyps"(16-18) [of n n n]
```
```   543           "4.hyps"(13-15)[of n "Suc n" n]
```
```   544           np np' C(2) mn
```
```   545         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
```
```   546           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   547           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
```
```   548           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
```
```   549           "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
```
```   550             "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
```
```   551           by (simp_all add: min_def)
```
```   552
```
```   553           from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```   554           have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   555             using norm by simp
```
```   556         from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"]  degneq
```
```   557         have "False" by simp }
```
```   558       thus ?case using "4.hyps" by clarsimp }
```
```   559 qed auto
```
```   560
```
```   561 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"
```
```   562   by (induct p q rule: polymul.induct) (auto simp add: field_simps)
```
```   563
```
```   564 lemma polymul_normh:
```
```   565   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   566   shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   567   using polymul_properties(1) by blast
```
```   568
```
```   569 lemma polymul_eq0_iff:
```
```   570   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   571   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p) "
```
```   572   using polymul_properties(2) by blast
```
```   573
```
```   574 lemma polymul_degreen:  (* FIXME duplicate? *)
```
```   575   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   576   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow>
```
```   577     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)"
```
```   578   using polymul_properties(3) by blast
```
```   579
```
```   580 lemma polymul_norm:
```
```   581   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   582   shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul p q)"
```
```   583   using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   584
```
```   585 lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
```
```   586   by (induct p arbitrary: n0 rule: headconst.induct) auto
```
```   587
```
```   588 lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
```
```   589   by (induct p arbitrary: n0) auto
```
```   590
```
```   591 lemma monic_eqI:
```
```   592   assumes np: "isnpolyh p n0"
```
```   593   shows "INum (headconst p) * Ipoly bs (fst (monic p)) =
```
```   594     (Ipoly bs p ::'a::{field_char_0,field_inverse_zero, power})"
```
```   595   unfolding monic_def Let_def
```
```   596 proof (cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
```
```   597   let ?h = "headconst p"
```
```   598   assume pz: "p \<noteq> 0\<^sub>p"
```
```   599   {
```
```   600     assume hz: "INum ?h = (0::'a)"
```
```   601     from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
```
```   602     from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
```
```   603     with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
```
```   604   thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast
```
```   605 qed
```
```   606
```
```   607
```
```   608 text{* polyneg is a negation and preserves normal forms *}
```
```   609
```
```   610 lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
```
```   611   by (induct p rule: polyneg.induct) auto
```
```   612
```
```   613 lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   614   by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: Nneg_def)
```
```   615 lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
```
```   616   by (induct p arbitrary: n0 rule: polyneg.induct) auto
```
```   617 lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "
```
```   618   by (induct p rule: polyneg.induct) (auto simp add: polyneg0)
```
```   619
```
```   620 lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
```
```   621   using isnpoly_def polyneg_normh by simp
```
```   622
```
```   623
```
```   624 text{* polysub is a substraction and preserves normal forms *}
```
```   625
```
```   626 lemma polysub[simp]: "Ipoly bs (polysub p q) = (Ipoly bs p) - (Ipoly bs q)"
```
```   627   by (simp add: polysub_def)
```
```   628 lemma polysub_normh:
```
```   629   "\<And>n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
```
```   630   by (simp add: polysub_def polyneg_normh polyadd_normh)
```
```   631
```
```   632 lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub p q)"
```
```   633   using polyadd_norm polyneg_norm by (simp add: polysub_def)
```
```   634 lemma polysub_same_0[simp]:
```
```   635   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   636   shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p"
```
```   637   unfolding polysub_def split_def fst_conv snd_conv
```
```   638   by (induct p arbitrary: n0) (auto simp add: Let_def Nsub0[simplified Nsub_def])
```
```   639
```
```   640 lemma polysub_0:
```
```   641   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   642   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"
```
```   643   unfolding polysub_def split_def fst_conv snd_conv
```
```   644   by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
```
```   645     (auto simp: Nsub0[simplified Nsub_def] Let_def)
```
```   646
```
```   647 text{* polypow is a power function and preserves normal forms *}
```
```   648
```
```   649 lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{field_char_0,field_inverse_zero})) ^ n"
```
```   650 proof (induct n rule: polypow.induct)
```
```   651   case 1
```
```   652   thus ?case by simp
```
```   653 next
```
```   654   case (2 n)
```
```   655   let ?q = "polypow ((Suc n) div 2) p"
```
```   656   let ?d = "polymul ?q ?q"
```
```   657   have "odd (Suc n) \<or> even (Suc n)" by simp
```
```   658   moreover
```
```   659   { assume odd: "odd (Suc n)"
```
```   660     have th: "(Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0)))) = Suc n div 2 + Suc n div 2 + 1"
```
```   661       by arith
```
```   662     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)" by (simp add: Let_def)
```
```   663     also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"
```
```   664       using "2.hyps" by simp
```
```   665     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
```
```   666       by (simp only: power_add power_one_right) simp
```
```   667     also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0))))"
```
```   668       by (simp only: th)
```
```   669     finally have ?case
```
```   670     using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
```
```   671   moreover
```
```   672   { assume even: "even (Suc n)"
```
```   673     have th: "(Suc (Suc 0)) * (Suc n div Suc (Suc 0)) = Suc n div 2 + Suc n div 2"
```
```   674       by arith
```
```   675     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
```
```   676     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
```
```   677       using "2.hyps" apply (simp only: power_add) by simp
```
```   678     finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}
```
```   679   ultimately show ?case by blast
```
```   680 qed
```
```   681
```
```   682 lemma polypow_normh:
```
```   683   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   684   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
```
```   685 proof (induct k arbitrary: n rule: polypow.induct)
```
```   686   case (2 k n)
```
```   687   let ?q = "polypow (Suc k div 2) p"
```
```   688   let ?d = "polymul ?q ?q"
```
```   689   from 2 have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
```
```   690   from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
```
```   691   from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n" by simp
```
```   692   from dn on show ?case by (simp add: Let_def)
```
```   693 qed auto
```
```   694
```
```   695 lemma polypow_norm:
```
```   696   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   697   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
```
```   698   by (simp add: polypow_normh isnpoly_def)
```
```   699
```
```   700 text{* Finally the whole normalization *}
```
```   701
```
```   702 lemma polynate [simp]:
```
```   703   "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0,field_inverse_zero})"
```
```   704   by (induct p rule:polynate.induct) auto
```
```   705
```
```   706 lemma polynate_norm[simp]:
```
```   707   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```   708   shows "isnpoly (polynate p)"
```
```   709   by (induct p rule: polynate.induct)
```
```   710      (simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm,
```
```   711       simp_all add: isnpoly_def)
```
```   712
```
```   713 text{* shift1 *}
```
```   714
```
```   715
```
```   716 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
```
```   717   by (simp add: shift1_def)
```
```   718
```
```   719 lemma shift1_isnpoly:
```
```   720   assumes pn: "isnpoly p"
```
```   721     and pnz: "p \<noteq> 0\<^sub>p"
```
```   722   shows "isnpoly (shift1 p) "
```
```   723   using pn pnz by (simp add: shift1_def isnpoly_def)
```
```   724
```
```   725 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
```
```   726   by (simp add: shift1_def)
```
```   727 lemma funpow_shift1_isnpoly:
```
```   728   "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
```
```   729   by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
```
```   730
```
```   731 lemma funpow_isnpolyh:
```
```   732   assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n"
```
```   733     and np: "isnpolyh p n"
```
```   734   shows "isnpolyh (funpow k f p) n"
```
```   735   using f np by (induct k arbitrary: p) auto
```
```   736
```
```   737 lemma funpow_shift1:
```
```   738   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field_inverse_zero}) =
```
```   739     Ipoly bs (Mul (Pw (Bound 0) n) p)"
```
```   740   by (induct n arbitrary: p) (simp_all add: shift1_isnpoly shift1)
```
```   741
```
```   742 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
```
```   743   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
```
```   744
```
```   745 lemma funpow_shift1_1:
```
```   746   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field_inverse_zero}) =
```
```   747     Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)"
```
```   748   by (simp add: funpow_shift1)
```
```   749
```
```   750 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
```
```   751   by (induct p rule: poly_cmul.induct) (auto simp add: field_simps)
```
```   752
```
```   753 lemma behead:
```
```   754   assumes np: "isnpolyh p n"
```
```   755   shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) =
```
```   756     (Ipoly bs p :: 'a :: {field_char_0,field_inverse_zero})"
```
```   757   using np
```
```   758 proof (induct p arbitrary: n rule: behead.induct)
```
```   759   case (1 c p n) hence pn: "isnpolyh p n" by simp
```
```   760   from 1(1)[OF pn]
```
```   761   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
```
```   762   then show ?case using "1.hyps"
```
```   763     apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
```
```   764     apply (simp_all add: th[symmetric] field_simps)
```
```   765     done
```
```   766 qed (auto simp add: Let_def)
```
```   767
```
```   768 lemma behead_isnpolyh:
```
```   769   assumes np: "isnpolyh p n"
```
```   770   shows "isnpolyh (behead p) n"
```
```   771   using np by (induct p rule: behead.induct) (auto simp add: Let_def isnpolyh_mono)
```
```   772
```
```   773
```
```   774 subsection{* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
```
```   775
```
```   776 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
```
```   777 proof (induct p arbitrary: n rule: poly.induct, auto)
```
```   778   case (goal1 c n p n')
```
```   779   hence "n = Suc (n - 1)" by simp
```
```   780   hence "isnpolyh p (Suc (n - 1))"  using `isnpolyh p n` by simp
```
```   781   with goal1(2) show ?case by simp
```
```   782 qed
```
```   783
```
```   784 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
```
```   785   by (induct p arbitrary: n0 rule: isconstant.induct) (auto simp add: isnpolyh_polybound0)
```
```   786
```
```   787 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
```
```   788   by (induct p) auto
```
```   789
```
```   790 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
```
```   791   apply (induct p arbitrary: n0)
```
```   792   apply auto
```
```   793   apply (atomize)
```
```   794   apply (erule_tac x = "Suc nat" in allE)
```
```   795   apply auto
```
```   796   done
```
```   797
```
```   798 lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
```
```   799   by (induct p  arbitrary: n0 rule: head.induct) (auto intro: isnpolyh_polybound0)
```
```   800
```
```   801 lemma polybound0_I:
```
```   802   assumes nb: "polybound0 a"
```
```   803   shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
```
```   804   using nb
```
```   805   by (induct a rule: poly.induct) auto
```
```   806
```
```   807 lemma polysubst0_I: "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
```
```   808   by (induct t) simp_all
```
```   809
```
```   810 lemma polysubst0_I':
```
```   811   assumes nb: "polybound0 a"
```
```   812   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t"
```
```   813   by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
```
```   814
```
```   815 lemma decrpoly:
```
```   816   assumes nb: "polybound0 t"
```
```   817   shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
```
```   818   using nb by (induct t rule: decrpoly.induct) simp_all
```
```   819
```
```   820 lemma polysubst0_polybound0:
```
```   821   assumes nb: "polybound0 t"
```
```   822   shows "polybound0 (polysubst0 t a)"
```
```   823   using nb by (induct a rule: poly.induct) auto
```
```   824
```
```   825 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
```
```   826   by (induct p arbitrary: n rule: degree.induct) (auto simp add: isnpolyh_polybound0)
```
```   827
```
```   828 primrec maxindex :: "poly \<Rightarrow> nat" where
```
```   829   "maxindex (Bound n) = n + 1"
```
```   830 | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
```
```   831 | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
```
```   832 | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
```
```   833 | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
```
```   834 | "maxindex (Neg p) = maxindex p"
```
```   835 | "maxindex (Pw p n) = maxindex p"
```
```   836 | "maxindex (C x) = 0"
```
```   837
```
```   838 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool"
```
```   839   where "wf_bs bs p \<longleftrightarrow> length bs \<ge> maxindex p"
```
```   840
```
```   841 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
```
```   842 proof (induct p rule: coefficients.induct)
```
```   843   case (1 c p)
```
```   844   show ?case
```
```   845   proof
```
```   846     fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"
```
```   847     hence "x = c \<or> x \<in> set (coefficients p)" by simp
```
```   848     moreover
```
```   849     {assume "x = c" hence "wf_bs bs x" using "1.prems"  unfolding wf_bs_def by simp}
```
```   850     moreover
```
```   851     {assume H: "x \<in> set (coefficients p)"
```
```   852       from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp
```
```   853       with "1.hyps" H have "wf_bs bs x" by blast }
```
```   854     ultimately  show "wf_bs bs x" by blast
```
```   855   qed
```
```   856 qed simp_all
```
```   857
```
```   858 lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
```
```   859   by (induct p rule: coefficients.induct) auto
```
```   860
```
```   861 lemma wf_bs_I: "wf_bs bs p \<Longrightarrow> Ipoly (bs @ bs') p = Ipoly bs p"
```
```   862   unfolding wf_bs_def by (induct p) (auto simp add: nth_append)
```
```   863
```
```   864 lemma take_maxindex_wf:
```
```   865   assumes wf: "wf_bs bs p"
```
```   866   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
```
```   867 proof-
```
```   868   let ?ip = "maxindex p"
```
```   869   let ?tbs = "take ?ip bs"
```
```   870   from wf have "length ?tbs = ?ip" unfolding wf_bs_def by simp
```
```   871   hence wf': "wf_bs ?tbs p" unfolding wf_bs_def by  simp
```
```   872   have eq: "bs = ?tbs @ (drop ?ip bs)" by simp
```
```   873   from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis using eq by simp
```
```   874 qed
```
```   875
```
```   876 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
```
```   877   by (induct p) auto
```
```   878
```
```   879 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
```
```   880   unfolding wf_bs_def by simp
```
```   881
```
```   882 lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
```
```   883   unfolding wf_bs_def by simp
```
```   884
```
```   885
```
```   886
```
```   887 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
```
```   888   by (induct p rule: coefficients.induct) (auto simp add: wf_bs_def)
```
```   889 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
```
```   890   by (induct p rule: coefficients.induct) simp_all
```
```   891
```
```   892
```
```   893 lemma coefficients_head: "last (coefficients p) = head p"
```
```   894   by (induct p rule: coefficients.induct) auto
```
```   895
```
```   896 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
```
```   897   unfolding wf_bs_def by (induct p rule: decrpoly.induct) auto
```
```   898
```
```   899 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
```
```   900   apply (rule exI[where x="replicate (n - length xs) z"])
```
```   901   apply simp
```
```   902   done
```
```   903
```
```   904 lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
```
```   905   apply (cases p)
```
```   906   apply auto
```
```   907   apply (case_tac "nat")
```
```   908   apply simp_all
```
```   909   done
```
```   910
```
```   911 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
```
```   912   unfolding wf_bs_def
```
```   913   apply (induct p q rule: polyadd.induct)
```
```   914   apply (auto simp add: Let_def)
```
```   915   done
```
```   916
```
```   917 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
```
```   918   unfolding wf_bs_def
```
```   919   apply (induct p q arbitrary: bs rule: polymul.induct)
```
```   920   apply (simp_all add: wf_bs_polyadd)
```
```   921   apply clarsimp
```
```   922   apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
```
```   923   apply auto
```
```   924   done
```
```   925
```
```   926 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
```
```   927   unfolding wf_bs_def by (induct p rule: polyneg.induct) auto
```
```   928
```
```   929 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
```
```   930   unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
```
```   931
```
```   932
```
```   933 subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}
```
```   934
```
```   935 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
```
```   936 definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)"
```
```   937 definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"
```
```   938
```
```   939 lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0"
```
```   940 proof (induct p arbitrary: n0 rule: coefficients.induct)
```
```   941   case (1 c p n0)
```
```   942   have cp: "isnpolyh (CN c 0 p) n0" by fact
```
```   943   hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
```
```   944     by (auto simp add: isnpolyh_mono[where n'=0])
```
```   945   from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp
```
```   946 qed auto
```
```   947
```
```   948 lemma coefficients_isconst:
```
```   949   "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
```
```   950   by (induct p arbitrary: n rule: coefficients.induct)
```
```   951     (auto simp add: isnpolyh_Suc_const)
```
```   952
```
```   953 lemma polypoly_polypoly':
```
```   954   assumes np: "isnpolyh p n0"
```
```   955   shows "polypoly (x#bs) p = polypoly' bs p"
```
```   956 proof-
```
```   957   let ?cf = "set (coefficients p)"
```
```   958   from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
```
```   959   {fix q assume q: "q \<in> ?cf"
```
```   960     from q cn_norm have th: "isnpolyh q n0" by blast
```
```   961     from coefficients_isconst[OF np] q have "isconstant q" by blast
```
```   962     with isconstant_polybound0[OF th] have "polybound0 q" by blast}
```
```   963   hence "\<forall>q \<in> ?cf. polybound0 q" ..
```
```   964   hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"
```
```   965     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
```
```   966     by auto
```
```   967
```
```   968   thus ?thesis unfolding polypoly_def polypoly'_def by simp
```
```   969 qed
```
```   970
```
```   971 lemma polypoly_poly:
```
```   972   assumes np: "isnpolyh p n0"
```
```   973   shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
```
```   974   using np
```
```   975   by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def)
```
```   976
```
```   977 lemma polypoly'_poly:
```
```   978   assumes np: "isnpolyh p n0"
```
```   979   shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
```
```   980   using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
```
```   981
```
```   982
```
```   983 lemma polypoly_poly_polybound0:
```
```   984   assumes np: "isnpolyh p n0" and nb: "polybound0 p"
```
```   985   shows "polypoly bs p = [Ipoly bs p]"
```
```   986   using np nb unfolding polypoly_def
```
```   987   apply (cases p)
```
```   988   apply auto
```
```   989   apply (case_tac nat)
```
```   990   apply auto
```
```   991   done
```
```   992
```
```   993 lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
```
```   994   by (induct p rule: head.induct) auto
```
```   995
```
```   996 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   997   by (cases p) auto
```
```   998
```
```   999 lemma head_eq_headn0: "head p = headn p 0"
```
```  1000   by (induct p rule: head.induct) simp_all
```
```  1001
```
```  1002 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```  1003   by (simp add: head_eq_headn0)
```
```  1004
```
```  1005 lemma isnpolyh_zero_iff:
```
```  1006   assumes nq: "isnpolyh p n0"
```
```  1007     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})"
```
```  1008   shows "p = 0\<^sub>p"
```
```  1009   using nq eq
```
```  1010 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
```
```  1011   case less
```
```  1012   note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
```
```  1013   {
```
```  1014     assume nz: "maxindex p = 0"
```
```  1015     then obtain c where "p = C c"
```
```  1016       using np by (cases p) auto
```
```  1017     with zp np have "p = 0\<^sub>p"
```
```  1018       unfolding wf_bs_def by simp
```
```  1019   }
```
```  1020   moreover
```
```  1021   {
```
```  1022     assume nz: "maxindex p \<noteq> 0"
```
```  1023     let ?h = "head p"
```
```  1024     let ?hd = "decrpoly ?h"
```
```  1025     let ?ihd = "maxindex ?hd"
```
```  1026     from head_isnpolyh[OF np] head_polybound0[OF np]
```
```  1027     have h: "isnpolyh ?h n0" "polybound0 ?h"
```
```  1028       by simp_all
```
```  1029     then have nhd: "isnpolyh ?hd (n0 - 1)"
```
```  1030       using decrpoly_normh by blast
```
```  1031
```
```  1032     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
```
```  1033     have mihn: "maxindex ?h \<le> maxindex p"
```
```  1034       by auto
```
```  1035     with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < maxindex p"
```
```  1036       by auto
```
```  1037     {
```
```  1038       fix bs :: "'a list"
```
```  1039       assume bs: "wf_bs bs ?hd"
```
```  1040       let ?ts = "take ?ihd bs"
```
```  1041       let ?rs = "drop ?ihd bs"
```
```  1042       have ts: "wf_bs ?ts ?hd"
```
```  1043         using bs unfolding wf_bs_def by simp
```
```  1044       have bs_ts_eq: "?ts @ ?rs = bs"
```
```  1045         by simp
```
```  1046       from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x # ?ts) ?h"
```
```  1047         by simp
```
```  1048       from ihd_lt_n have "\<forall>x. length (x # ?ts) \<le> maxindex p"
```
```  1049         by simp
```
```  1050       with length_le_list_ex obtain xs where xs: "length ((x # ?ts) @ xs) = maxindex p"
```
```  1051         by blast
```
```  1052       then have "\<forall>x. wf_bs ((x # ?ts) @ xs) p"
```
```  1053         unfolding wf_bs_def by simp
```
```  1054       with zp have "\<forall>x. Ipoly ((x # ?ts) @ xs) p = 0"
```
```  1055         by blast
```
```  1056       then have "\<forall>x. Ipoly (x # (?ts @ xs)) p = 0"
```
```  1057         by simp
```
```  1058       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
```
```  1059       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"
```
```  1060         by simp
```
```  1061       then have "poly (polypoly' (?ts @ xs) p) = poly []"
```
```  1062         by auto
```
```  1063       then have "\<forall>c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
```
```  1064         using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
```
```  1065       with coefficients_head[of p, symmetric]
```
```  1066       have th0: "Ipoly (?ts @ xs) ?hd = 0"
```
```  1067         by simp
```
```  1068       from bs have wf'': "wf_bs ?ts ?hd"
```
```  1069         unfolding wf_bs_def by simp
```
```  1070       with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0"
```
```  1071         by simp
```
```  1072       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
```
```  1073         by simp
```
```  1074     }
```
```  1075     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
```
```  1076       by blast
```
```  1077     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p"
```
```  1078       by blast
```
```  1079     then have "?h = 0\<^sub>p" by simp
```
```  1080     with head_nz[OF np] have "p = 0\<^sub>p" by simp
```
```  1081   }
```
```  1082   ultimately show "p = 0\<^sub>p"
```
```  1083     by blast
```
```  1084 qed
```
```  1085
```
```  1086 lemma isnpolyh_unique:
```
```  1087   assumes np: "isnpolyh p n0"
```
```  1088     and nq: "isnpolyh q n1"
```
```  1089   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"
```
```  1090 proof auto
```
```  1091   assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a) = \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
```
```  1092   then have "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)"
```
```  1093     by simp
```
```  1094   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)"
```
```  1095     using wf_bs_polysub[where p=p and q=q] by auto
```
```  1096   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq] show "p = q"
```
```  1097     by blast
```
```  1098 qed
```
```  1099
```
```  1100
```
```  1101 text{* consequences of unicity on the algorithms for polynomial normalization *}
```
```  1102
```
```  1103 lemma polyadd_commute:
```
```  1104   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1105     and np: "isnpolyh p n0"
```
```  1106     and nq: "isnpolyh q n1"
```
```  1107   shows "p +\<^sub>p q = q +\<^sub>p p"
```
```  1108   using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]]
```
```  1109   by simp
```
```  1110
```
```  1111 lemma zero_normh: "isnpolyh 0\<^sub>p n"
```
```  1112   by simp
```
```  1113
```
```  1114 lemma one_normh: "isnpolyh (1)\<^sub>p n"
```
```  1115   by simp
```
```  1116
```
```  1117 lemma polyadd_0[simp]:
```
```  1118   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1119     and np: "isnpolyh p n0"
```
```  1120   shows "p +\<^sub>p 0\<^sub>p = p"
```
```  1121     and "0\<^sub>p +\<^sub>p p = p"
```
```  1122   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
```
```  1123     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
```
```  1124
```
```  1125 lemma polymul_1[simp]:
```
```  1126   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1127     and np: "isnpolyh p n0"
```
```  1128   shows "p *\<^sub>p (1)\<^sub>p = p"
```
```  1129     and "(1)\<^sub>p *\<^sub>p p = p"
```
```  1130   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
```
```  1131     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
```
```  1132
```
```  1133 lemma polymul_0[simp]:
```
```  1134   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1135     and np: "isnpolyh p n0"
```
```  1136   shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p"
```
```  1137     and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
```
```  1138   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
```
```  1139     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
```
```  1140
```
```  1141 lemma polymul_commute:
```
```  1142   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1143     and np: "isnpolyh p n0"
```
```  1144     and nq: "isnpolyh q n1"
```
```  1145   shows "p *\<^sub>p q = q *\<^sub>p p"
```
```  1146   using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np], where ?'a = "'a::{field_char_0,field_inverse_zero, power}"]
```
```  1147   by simp
```
```  1148
```
```  1149 declare polyneg_polyneg [simp]
```
```  1150
```
```  1151 lemma isnpolyh_polynate_id [simp]:
```
```  1152   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1153     and np: "isnpolyh p n0"
```
```  1154   shows "polynate p = p"
```
```  1155   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}"]
```
```  1156   by simp
```
```  1157
```
```  1158 lemma polynate_idempotent[simp]:
```
```  1159   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1160   shows "polynate (polynate p) = polynate p"
```
```  1161   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
```
```  1162
```
```  1163 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
```
```  1164   unfolding poly_nate_def polypoly'_def ..
```
```  1165
```
```  1166 lemma poly_nate_poly:
```
```  1167   "poly (poly_nate bs p) = (\<lambda>x:: 'a ::{field_char_0,field_inverse_zero}. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
```
```  1168   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
```
```  1169   unfolding poly_nate_polypoly' by auto
```
```  1170
```
```  1171
```
```  1172 subsection{* heads, degrees and all that *}
```
```  1173
```
```  1174 lemma degree_eq_degreen0: "degree p = degreen p 0"
```
```  1175   by (induct p rule: degree.induct) simp_all
```
```  1176
```
```  1177 lemma degree_polyneg:
```
```  1178   assumes n: "isnpolyh p n"
```
```  1179   shows "degree (polyneg p) = degree p"
```
```  1180   apply (induct p arbitrary: n rule: polyneg.induct)
```
```  1181   using n apply simp_all
```
```  1182   apply (case_tac na)
```
```  1183   apply auto
```
```  1184   done
```
```  1185
```
```  1186 lemma degree_polyadd:
```
```  1187   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1188   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1189   using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
```
```  1190
```
```  1191
```
```  1192 lemma degree_polysub:
```
```  1193   assumes np: "isnpolyh p n0"
```
```  1194     and nq: "isnpolyh q n1"
```
```  1195   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1196 proof-
```
```  1197   from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp
```
```  1198   from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])
```
```  1199 qed
```
```  1200
```
```  1201 lemma degree_polysub_samehead:
```
```  1202   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1203     and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q"
```
```  1204     and d: "degree p = degree q"
```
```  1205   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
```
```  1206   unfolding polysub_def split_def fst_conv snd_conv
```
```  1207   using np nq h d
```
```  1208 proof (induct p q rule: polyadd.induct)
```
```  1209   case (1 c c')
```
```  1210   thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def])
```
```  1211 next
```
```  1212   case (2 c c' n' p')
```
```  1213   from 2 have "degree (C c) = degree (CN c' n' p')" by simp
```
```  1214   hence nz:"n' > 0" by (cases n') auto
```
```  1215   hence "head (CN c' n' p') = CN c' n' p'" by (cases n') auto
```
```  1216   with 2 show ?case by simp
```
```  1217 next
```
```  1218   case (3 c n p c')
```
```  1219   hence "degree (C c') = degree (CN c n p)" by simp
```
```  1220   hence nz:"n > 0" by (cases n) auto
```
```  1221   hence "head (CN c n p) = CN c n p" by (cases n) auto
```
```  1222   with 3 show ?case by simp
```
```  1223 next
```
```  1224   case (4 c n p c' n' p')
```
```  1225   hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1"
```
```  1226     "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+
```
```  1227   hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all
```
```  1228   hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
```
```  1229     using H(1-2) degree_polyneg by auto
```
```  1230   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"  by simp+
```
```  1231   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"  by simp
```
```  1232   from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto
```
```  1233   have "n = n' \<or> n < n' \<or> n > n'" by arith
```
```  1234   moreover
```
```  1235   {assume nn': "n = n'"
```
```  1236     have "n = 0 \<or> n >0" by arith
```
```  1237     moreover {assume nz: "n = 0" hence ?case using 4 nn' by (auto simp add: Let_def degcmc')}
```
```  1238     moreover {assume nz: "n > 0"
```
```  1239       with nn' H(3) have  cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
```
```  1240       hence ?case
```
```  1241         using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv] polysub_same_0[OF c'nh, simplified polysub_def]
```
```  1242         using nn' 4 by (simp add: Let_def) }
```
```  1243     ultimately have ?case by blast}
```
```  1244   moreover
```
```  1245   {assume nn': "n < n'" hence n'p: "n' > 0" by simp
```
```  1246     hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n') simp_all
```
```  1247     have degcnp': "degree (CN c' n' p') = 0" and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
```
```  1248       using 4 nn' by (cases n', simp_all)
```
```  1249     hence "n > 0" by (cases n) simp_all
```
```  1250     hence headcnp: "head (CN c n p) = CN c n p" by (cases n) auto
```
```  1251     from H(3) headcnp headcnp' nn' have ?case by auto}
```
```  1252   moreover
```
```  1253   {assume nn': "n > n'"  hence np: "n > 0" by simp
```
```  1254     hence headcnp:"head (CN c n p) = CN c n p"  by (cases n) simp_all
```
```  1255     from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
```
```  1256     from np have degcnp: "degree (CN c n p) = 0" by (cases n) simp_all
```
```  1257     with degcnpeq have "n' > 0" by (cases n') simp_all
```
```  1258     hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n') auto
```
```  1259     from H(3) headcnp headcnp' nn' have ?case by auto}
```
```  1260   ultimately show ?case  by blast
```
```  1261 qed auto
```
```  1262
```
```  1263 lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
```
```  1264   by (induct p arbitrary: n0 rule: head.induct) (simp_all add: shift1_def)
```
```  1265
```
```  1266 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
```
```  1267 proof (induct k arbitrary: n0 p)
```
```  1268   case 0
```
```  1269   thus ?case by auto
```
```  1270 next
```
```  1271   case (Suc k n0 p)
```
```  1272   hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
```
```  1273   with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
```
```  1274     and "head (shift1 p) = head p" by (simp_all add: shift1_head)
```
```  1275   thus ?case by (simp add: funpow_swap1)
```
```  1276 qed
```
```  1277
```
```  1278 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
```
```  1279   by (simp add: shift1_def)
```
```  1280 lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
```
```  1281   by (induct k arbitrary: p) (auto simp add: shift1_degree)
```
```  1282
```
```  1283 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
```
```  1284   by (induct n arbitrary: p) simp_all
```
```  1285
```
```  1286 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
```
```  1287   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1288 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
```
```  1289   by (induct p arbitrary: n rule: degreen.induct) auto
```
```  1290 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
```
```  1291   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1292 lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
```
```  1293   by (induct p rule: head.induct) auto
```
```  1294
```
```  1295 lemma polyadd_eq_const_degree:
```
```  1296   "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> polyadd p q = C c \<Longrightarrow> degree p = degree q"
```
```  1297   using polyadd_eq_const_degreen degree_eq_degreen0 by simp
```
```  1298
```
```  1299 lemma polyadd_head:
```
```  1300   assumes np: "isnpolyh p n0"
```
```  1301     and nq: "isnpolyh q n1"
```
```  1302     and deg: "degree p \<noteq> degree q"
```
```  1303   shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
```
```  1304   using np nq deg
```
```  1305   apply (induct p q arbitrary: n0 n1 rule: polyadd.induct)
```
```  1306   using np
```
```  1307   apply simp_all
```
```  1308   apply (case_tac n', simp, simp)
```
```  1309   apply (case_tac n, simp, simp)
```
```  1310   apply (case_tac n, case_tac n', simp add: Let_def)
```
```  1311   apply (auto simp add: polyadd_eq_const_degree)
```
```  1312   apply (metis head_nz)
```
```  1313   apply (metis head_nz)
```
```  1314   apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
```
```  1315   done
```
```  1316
```
```  1317 lemma polymul_head_polyeq:
```
```  1318   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1319   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"
```
```  1320 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
```
```  1321   case (2 c c' n' p' n0 n1)
```
```  1322   hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"  by (simp_all add: head_isnpolyh)
```
```  1323   thus ?case using 2 by (cases n') auto
```
```  1324 next
```
```  1325   case (3 c n p c' n0 n1)
```
```  1326   hence "isnpolyh (head (CN c n p)) n0" "isnormNum c'"  by (simp_all add: head_isnpolyh)
```
```  1327   thus ?case using 3 by (cases n) auto
```
```  1328 next
```
```  1329   case (4 c n p c' n' p' n0 n1)
```
```  1330   hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
```
```  1331     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
```
```  1332     by simp_all
```
```  1333   have "n < n' \<or> n' < n \<or> n = n'" by arith
```
```  1334   moreover
```
```  1335   {assume nn': "n < n'" hence ?case
```
```  1336       using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)]
```
```  1337       apply simp
```
```  1338       apply (cases n)
```
```  1339       apply simp
```
```  1340       apply (cases n')
```
```  1341       apply simp_all
```
```  1342       done }
```
```  1343   moreover {assume nn': "n'< n"
```
```  1344     hence ?case
```
```  1345       using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)]
```
```  1346       apply simp
```
```  1347       apply (cases n')
```
```  1348       apply simp
```
```  1349       apply (cases n)
```
```  1350       apply auto
```
```  1351       done }
```
```  1352   moreover {assume nn': "n' = n"
```
```  1353     from nn' polymul_normh[OF norm(5,4)]
```
```  1354     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
```
```  1355     from nn' polymul_normh[OF norm(5,3)] norm
```
```  1356     have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
```
```  1357     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
```
```  1358     have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```  1359     from polyadd_normh[OF ncnpc' ncnpp0']
```
```  1360     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"
```
```  1361       by (simp add: min_def)
```
```  1362     {assume np: "n > 0"
```
```  1363       with nn' head_isnpolyh_Suc'[OF np nth]
```
```  1364         head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
```
```  1365       have ?case by simp}
```
```  1366     moreover
```
```  1367     { assume nz: "n = 0"
```
```  1368       from polymul_degreen[OF norm(5,4), where m="0"]
```
```  1369         polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
```
```  1370       norm(5,6) degree_npolyhCN[OF norm(6)]
```
```  1371     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
```
```  1372     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
```
```  1373     from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
```
```  1374     have ?case   using norm "4.hyps"(6)[OF norm(5,3)]
```
```  1375         "4.hyps"(5)[OF norm(5,4)] nn' nz by simp }
```
```  1376     ultimately have ?case by (cases n) auto}
```
```  1377   ultimately show ?case by blast
```
```  1378 qed simp_all
```
```  1379
```
```  1380 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
```
```  1381   by (induct p rule: degree.induct) auto
```
```  1382
```
```  1383 lemma degree_head[simp]: "degree (head p) = 0"
```
```  1384   by (induct p rule: head.induct) auto
```
```  1385
```
```  1386 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
```
```  1387   by (cases n) simp_all
```
```  1388 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
```
```  1389   by (cases n) simp_all
```
```  1390
```
```  1391 lemma polyadd_different_degree:
```
```  1392   "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degree p \<noteq> degree q\<rbrakk> \<Longrightarrow>
```
```  1393     degree (polyadd p q) = max (degree p) (degree q)"
```
```  1394   using polyadd_different_degreen degree_eq_degreen0 by simp
```
```  1395
```
```  1396 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
```
```  1397   by (induct p arbitrary: n0 rule: polyneg.induct) auto
```
```  1398
```
```  1399 lemma degree_polymul:
```
```  1400   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1401     and np: "isnpolyh p n0"
```
```  1402     and nq: "isnpolyh q n1"
```
```  1403   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
```
```  1404   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
```
```  1405
```
```  1406 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
```
```  1407   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1408
```
```  1409 lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"
```
```  1410   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1411
```
```  1412
```
```  1413 subsection {* Correctness of polynomial pseudo division *}
```
```  1414
```
```  1415 lemma polydivide_aux_properties:
```
```  1416   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1417     and np: "isnpolyh p n0"
```
```  1418     and ns: "isnpolyh s n1"
```
```  1419     and ap: "head p = a"
```
```  1420     and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
```
```  1421   shows "(polydivide_aux a n p k s = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1422           \<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)))"
```
```  1423   using ns
```
```  1424 proof (induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
```
```  1425   case less
```
```  1426   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
```
```  1427   let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1428     \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
```
```  1429   let ?b = "head s"
```
```  1430   let ?p' = "funpow (degree s - n) shift1 p"
```
```  1431   let ?xdn = "funpow (degree s - n) shift1 (1)\<^sub>p"
```
```  1432   let ?akk' = "a ^\<^sub>p (k' - k)"
```
```  1433   note ns = `isnpolyh s n1`
```
```  1434   from np have np0: "isnpolyh p 0"
```
```  1435     using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by simp
```
```  1436   have np': "isnpolyh ?p' 0"
```
```  1437     using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def
```
```  1438     by simp
```
```  1439   have headp': "head ?p' = head p"
```
```  1440     using funpow_shift1_head[OF np pnz] by simp
```
```  1441   from funpow_shift1_isnpoly[where p="(1)\<^sub>p"] have nxdn: "isnpolyh ?xdn 0"
```
```  1442     by (simp add: isnpoly_def)
```
```  1443   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
```
```  1444   have nakk':"isnpolyh ?akk' 0" by blast
```
```  1445   { assume sz: "s = 0\<^sub>p"
```
```  1446     hence ?ths using np polydivide_aux.simps
```
```  1447       apply clarsimp
```
```  1448       apply (rule exI[where x="0\<^sub>p"])
```
```  1449       apply simp
```
```  1450       done }
```
```  1451   moreover
```
```  1452   { assume sz: "s \<noteq> 0\<^sub>p"
```
```  1453     { assume dn: "degree s < n"
```
```  1454       hence "?ths" using ns ndp np polydivide_aux.simps
```
```  1455         apply auto
```
```  1456         apply (rule exI[where x="0\<^sub>p"])
```
```  1457         apply simp
```
```  1458         done }
```
```  1459     moreover
```
```  1460     { assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith
```
```  1461       have degsp': "degree s = degree ?p'"
```
```  1462         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp
```
```  1463       { assume ba: "?b = a"
```
```  1464         hence headsp': "head s = head ?p'"
```
```  1465           using ap headp' by simp
```
```  1466         have nr: "isnpolyh (s -\<^sub>p ?p') 0"
```
```  1467           using polysub_normh[OF ns np'] by simp
```
```  1468         from degree_polysub_samehead[OF ns np' headsp' degsp']
```
```  1469         have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
```
```  1470         moreover
```
```  1471         { assume deglt:"degree (s -\<^sub>p ?p') < degree s"
```
```  1472           from polydivide_aux.simps sz dn' ba
```
```  1473           have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
```
```  1474             by (simp add: Let_def)
```
```  1475           { assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1476             from less(1)[OF deglt nr, of k k' r] trans[OF eq[symmetric] h1]
```
```  1477             have kk': "k \<le> k'"
```
```  1478               and nr:"\<exists>nr. isnpolyh r nr"
```
```  1479               and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1480               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)"
```
```  1481               by auto
```
```  1482             from q1 obtain q n1 where nq: "isnpolyh q n1"
```
```  1483               and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r" by blast
```
```  1484             from nr obtain nr where nr': "isnpolyh r nr" by blast
```
```  1485             from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0"
```
```  1486               by simp
```
```  1487             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
```
```  1488             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
```
```  1489             from polyadd_normh[OF polymul_normh[OF np
```
```  1490               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
```
```  1491             have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0"
```
```  1492               by simp
```
```  1493             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')) =
```
```  1494               Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
```
```  1495             hence " \<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
```
```  1496               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1497               by (simp add: field_simps)
```
```  1498             hence " \<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1499               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p *\<^sub>p p) +
```
```  1500               Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1501               by (auto simp only: funpow_shift1_1)
```
```  1502             hence "\<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1503               Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p) +
```
```  1504               Ipoly bs q) + Ipoly bs r"
```
```  1505               by (simp add: field_simps)
```
```  1506             hence "\<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1507               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)"
```
```  1508               by simp
```
```  1509             with isnpolyh_unique[OF nakks' nqr']
```
```  1510             have "a ^\<^sub>p (k' - k) *\<^sub>p s =
```
```  1511               p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r"
```
```  1512               by blast
```
```  1513             hence ?qths using nq'
```
```  1514               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q" in exI)
```
```  1515               apply (rule_tac x="0" in exI)
```
```  1516               apply simp
```
```  1517               done
```
```  1518             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"
```
```  1519               by blast
```
```  1520           }
```
```  1521           hence ?ths by blast
```
```  1522         }
```
```  1523         moreover
```
```  1524         { assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
```
```  1525           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}"]
```
```  1526           have " \<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs s = Ipoly bs ?p'"
```
```  1527             by simp
```
```  1528           hence "\<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)"
```
```  1529             using np nxdn
```
```  1530             apply simp
```
```  1531             apply (simp only: funpow_shift1_1)
```
```  1532             apply simp
```
```  1533             done
```
```  1534           hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]]
```
```  1535             by blast
```
```  1536           { assume h1: "polydivide_aux a n p k s = (k',r)"
```
```  1537             from polydivide_aux.simps sz dn' ba
```
```  1538             have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
```
```  1539               by (simp add: Let_def)
```
```  1540             also have "\<dots> = (k,0\<^sub>p)"
```
```  1541               using polydivide_aux.simps spz by simp
```
```  1542             finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
```
```  1543             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
```
```  1544               polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
```
```  1545               apply auto
```
```  1546               apply (rule exI[where x="?xdn"])
```
```  1547               apply (auto simp add: polymul_commute[of p])
```
```  1548               done
```
```  1549           }
```
```  1550         }
```
```  1551         ultimately have ?ths by blast
```
```  1552       }
```
```  1553       moreover
```
```  1554       { assume ba: "?b \<noteq> a"
```
```  1555         from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1556           polymul_normh[OF head_isnpolyh[OF ns] np']]
```
```  1557         have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
```
```  1558           by (simp add: min_def)
```
```  1559         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
```
```  1560           using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1561             polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
```
```  1562             funpow_shift1_nz[OF pnz]
```
```  1563           by simp_all
```
```  1564         from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
```
```  1565           polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1566         have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
```
```  1567           using head_head[OF ns] funpow_shift1_head[OF np pnz]
```
```  1568             polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
```
```  1569           by (simp add: ap)
```
```  1570         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
```
```  1571           head_nz[OF np] pnz sz ap[symmetric]
```
```  1572           funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1573           polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"] head_nz[OF ns]
```
```  1574           ndp dn
```
```  1575         have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p')"
```
```  1576           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
```
```  1577         { assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
```
```  1578           from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns]
```
```  1579             polymul_normh[OF head_isnpolyh[OF ns]np']] ap
```
```  1580           have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
```
```  1581             by simp
```
```  1582           { assume h1:"polydivide_aux a n p k s = (k', r)"
```
```  1583             from h1 polydivide_aux.simps sz dn' ba
```
```  1584             have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
```
```  1585               by (simp add: Let_def)
```
```  1586             with less(1)[OF dth nasbp', of "Suc k" k' r]
```
```  1587             obtain q nq nr where kk': "Suc k \<le> k'"
```
```  1588               and nr: "isnpolyh r nr"
```
```  1589               and nq: "isnpolyh q nq"
```
```  1590               and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1591               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"
```
```  1592               by auto
```
```  1593             from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith
```
```  1594             {
```
```  1595               fix bs:: "'a::{field_char_0,field_inverse_zero} list"
```
```  1596               from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
```
```  1597               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)"
```
```  1598                 by simp
```
```  1599               hence "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s =
```
```  1600                 Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
```
```  1601                 by (simp add: field_simps)
```
```  1602               hence "Ipoly bs a ^ (k' - k)  * Ipoly bs s =
```
```  1603                 Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
```
```  1604                 by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1605               hence "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1606                 Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
```
```  1607                 by (simp add: field_simps)
```
```  1608             }
```
```  1609             hence ieq:"\<forall>(bs :: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1610               Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)"
```
```  1611               by auto
```
```  1612             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
```
```  1613             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]]
```
```  1614             have nqw: "isnpolyh ?q 0"
```
```  1615               by simp
```
```  1616             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]]
```
```  1617             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"
```
```  1618               by blast
```
```  1619             from dr kk' nr h1 asth nqw have ?ths
```
```  1620               apply simp
```
```  1621               apply (rule conjI)
```
```  1622               apply (rule exI[where x="nr"], simp)
```
```  1623               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
```
```  1624               apply (rule exI[where x="0"], simp)
```
```  1625               done
```
```  1626           }
```
```  1627           hence ?ths by blast
```
```  1628         }
```
```  1629         moreover
```
```  1630         { assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
```
```  1631           {
```
```  1632             fix bs :: "'a::{field_char_0,field_inverse_zero} list"
```
```  1633             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
```
```  1634             have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'"
```
```  1635               by simp
```
```  1636             hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
```
```  1637               by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1638             hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))"
```
```  1639               by simp
```
```  1640           }
```
```  1641           hence hth: "\<forall> (bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a*\<^sub>p s) =
```
```  1642             Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
```
```  1643           from hth have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
```
```  1644             using isnpolyh_unique[where ?'a = "'a::{field_char_0,field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
```
```  1645                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
```
```  1646               simplified ap] by simp
```
```  1647           { assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1648             from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
```
```  1649             have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
```
```  1650             with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
```
```  1651               polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
```
```  1652             have ?ths
```
```  1653               apply (clarsimp simp add: Let_def)
```
```  1654               apply (rule exI[where x="?b *\<^sub>p ?xdn"])
```
```  1655               apply simp
```
```  1656               apply (rule exI[where x="0"], simp)
```
```  1657               done
```
```  1658           }
```
```  1659           hence ?ths by blast
```
```  1660         }
```
```  1661         ultimately have ?ths
```
```  1662           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"]
```
```  1663             head_nz[OF np] pnz sz ap[symmetric]
```
```  1664           by (simp add: degree_eq_degreen0[symmetric]) blast
```
```  1665       }
```
```  1666       ultimately have ?ths by blast
```
```  1667     }
```
```  1668     ultimately have ?ths by blast
```
```  1669   }
```
```  1670   ultimately show ?ths by blast
```
```  1671 qed
```
```  1672
```
```  1673 lemma polydivide_properties:
```
```  1674   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1675     and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
```
```  1676   shows "\<exists>k r. polydivide s p = (k,r) \<and>
```
```  1677     (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) \<and>
```
```  1678     (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r))"
```
```  1679 proof -
```
```  1680   have trv: "head p = head p" "degree p = degree p"
```
```  1681     by simp_all
```
```  1682   from polydivide_def[where s="s" and p="p"] have ex: "\<exists> k r. polydivide s p = (k,r)"
```
```  1683     by auto
```
```  1684   then obtain k r where kr: "polydivide s p = (k,r)"
```
```  1685     by blast
```
```  1686   from trans[OF polydivide_def[where s="s"and p="p", symmetric] kr]
```
```  1687     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
```
```  1688   have "(degree r = 0 \<or> degree r < degree p) \<and>
```
```  1689     (\<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)"
```
```  1690     by blast
```
```  1691   with kr show ?thesis
```
```  1692     apply -
```
```  1693     apply (rule exI[where x="k"])
```
```  1694     apply (rule exI[where x="r"])
```
```  1695     apply simp
```
```  1696     done
```
```  1697 qed
```
```  1698
```
```  1699
```
```  1700 subsection{* More about polypoly and pnormal etc *}
```
```  1701
```
```  1702 definition "isnonconstant p \<longleftrightarrow> \<not> isconstant p"
```
```  1703
```
```  1704 lemma isnonconstant_pnormal_iff:
```
```  1705   assumes nc: "isnonconstant p"
```
```  1706   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1707 proof
```
```  1708   let ?p = "polypoly bs p"
```
```  1709   assume H: "pnormal ?p"
```
```  1710   have csz: "coefficients p \<noteq> []" using nc by (cases p) auto
```
```  1711
```
```  1712   from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1713     pnormal_last_nonzero[OF H]
```
```  1714   show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)
```
```  1715 next
```
```  1716   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1717   let ?p = "polypoly bs p"
```
```  1718   have csz: "coefficients p \<noteq> []" using nc by (cases p) auto
```
```  1719   hence pz: "?p \<noteq> []" by (simp add: polypoly_def)
```
```  1720   hence lg: "length ?p > 0" by simp
```
```  1721   from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1722   have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)
```
```  1723   from pnormal_last_length[OF lg lz] show "pnormal ?p" .
```
```  1724 qed
```
```  1725
```
```  1726 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
```
```  1727   unfolding isnonconstant_def
```
```  1728   apply (cases p)
```
```  1729   apply simp_all
```
```  1730   apply (case_tac nat)
```
```  1731   apply auto
```
```  1732   done
```
```  1733
```
```  1734 lemma isnonconstant_nonconstant:
```
```  1735   assumes inc: "isnonconstant p"
```
```  1736   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1737 proof
```
```  1738   let ?p = "polypoly bs p"
```
```  1739   assume nc: "nonconstant ?p"
```
```  1740   from isnonconstant_pnormal_iff[OF inc, of bs] nc
```
```  1741   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast
```
```  1742 next
```
```  1743   let ?p = "polypoly bs p"
```
```  1744   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1745   from isnonconstant_pnormal_iff[OF inc, of bs] h
```
```  1746   have pn: "pnormal ?p" by blast
```
```  1747   { fix x assume H: "?p = [x]"
```
```  1748     from H have "length (coefficients p) = 1" unfolding polypoly_def by auto
```
```  1749     with isnonconstant_coefficients_length[OF inc] have False by arith }
```
```  1750   thus "nonconstant ?p" using pn unfolding nonconstant_def by blast
```
```  1751 qed
```
```  1752
```
```  1753 lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
```
```  1754   apply (induct p)
```
```  1755   apply (simp_all add: pnormal_def)
```
```  1756   apply (case_tac "p = []")
```
```  1757   apply simp_all
```
```  1758   done
```
```  1759
```
```  1760 lemma degree_degree:
```
```  1761   assumes inc: "isnonconstant p"
```
```  1762   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1763 proof
```
```  1764   let ?p = "polypoly bs p"
```
```  1765   assume H: "degree p = Polynomial_List.degree ?p"
```
```  1766   from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
```
```  1767     unfolding polypoly_def by auto
```
```  1768   from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
```
```  1769   have lg:"length (pnormalize ?p) = length ?p"
```
```  1770     unfolding Polynomial_List.degree_def polypoly_def by simp
```
```  1771   hence "pnormal ?p" using pnormal_length[OF pz] by blast
```
```  1772   with isnonconstant_pnormal_iff[OF inc]
```
```  1773   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast
```
```  1774 next
```
```  1775   let  ?p = "polypoly bs p"
```
```  1776   assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1777   with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast
```
```  1778   with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
```
```  1779   show "degree p = Polynomial_List.degree ?p"
```
```  1780     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
```
```  1781 qed
```
```  1782
```
```  1783
```
```  1784 section {* Swaps ; Division by a certain variable *}
```
```  1785
```
```  1786 primrec swap :: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly" where
```
```  1787   "swap n m (C x) = C x"
```
```  1788 | "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"
```
```  1789 | "swap n m (Neg t) = Neg (swap n m t)"
```
```  1790 | "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
```
```  1791 | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
```
```  1792 | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
```
```  1793 | "swap n m (Pw t k) = Pw (swap n m t) k"
```
```  1794 | "swap n m (CN c k p) =
```
```  1795     CN (swap n m c) (if k = n then m else if k=m then n else k) (swap n m p)"
```
```  1796
```
```  1797 lemma swap:
```
```  1798   assumes nbs: "n < length bs"
```
```  1799     and mbs: "m < length bs"
```
```  1800   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  1801 proof (induct t)
```
```  1802   case (Bound k)
```
```  1803   thus ?case using nbs mbs by simp
```
```  1804 next
```
```  1805   case (CN c k p)
```
```  1806   thus ?case using nbs mbs by simp
```
```  1807 qed simp_all
```
```  1808
```
```  1809 lemma swap_swap_id [simp]: "swap n m (swap m n t) = t"
```
```  1810   by (induct t) simp_all
```
```  1811
```
```  1812 lemma swap_commute: "swap n m p = swap m n p"
```
```  1813   by (induct p) simp_all
```
```  1814
```
```  1815 lemma swap_same_id[simp]: "swap n n t = t"
```
```  1816   by (induct t) simp_all
```
```  1817
```
```  1818 definition "swapnorm n m t = polynate (swap n m t)"
```
```  1819
```
```  1820 lemma swapnorm:
```
```  1821   assumes nbs: "n < length bs"
```
```  1822     and mbs: "m < length bs"
```
```  1823   shows "((Ipoly bs (swapnorm n m t) :: 'a::{field_char_0,field_inverse_zero})) =
```
```  1824     Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  1825   using swap[OF assms] swapnorm_def by simp
```
```  1826
```
```  1827 lemma swapnorm_isnpoly [simp]:
```
```  1828   assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
```
```  1829   shows "isnpoly (swapnorm n m p)"
```
```  1830   unfolding swapnorm_def by simp
```
```  1831
```
```  1832 definition "polydivideby n s p =
```
```  1833   (let
```
```  1834     ss = swapnorm 0 n s;
```
```  1835     sp = swapnorm 0 n p;
```
```  1836     h = head sp;
```
```  1837     (k, r) = polydivide ss sp
```
```  1838    in (k, swapnorm 0 n h, swapnorm 0 n r))"
```
```  1839
```
```  1840 lemma swap_nz [simp]: "swap n m p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
```
```  1841   by (induct p) simp_all
```
```  1842
```
```  1843 fun isweaknpoly :: "poly \<Rightarrow> bool"
```
```  1844 where
```
```  1845   "isweaknpoly (C c) = True"
```
```  1846 | "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
```
```  1847 | "isweaknpoly p = False"
```
```  1848
```
```  1849 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
```
```  1850   by (induct p arbitrary: n0) auto
```
```  1851
```
```  1852 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
```
```  1853   by (induct p) auto
```
```  1854
```
`  1855 end`