src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
 author wenzelm Mon Jul 15 11:29:19 2013 +0200 (2013-07-15) changeset 52658 1e7896c7f781 parent 50282 fe4d4bb9f4c2 child 52803 bcaa5bbf7e6b permissions -rw-r--r--
tuned specifications and proofs;
```     1 (*  Title:      HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
```
```     2     Author:     Amine Chaieb
```
```     3 *)
```
```     4
```
```     5 header {* Implementation and verification of multivariate polynomials *}
```
```     6
```
```     7 theory Reflected_Multivariate_Polynomial
```
```     8 imports Complex_Main "~~/src/HOL/Library/Abstract_Rat" Polynomial_List
```
```     9 begin
```
```    10
```
```    11 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) = True"
```
```    36 | "polybound0 (Bound n) = (n>0)"
```
```    37 | "polybound0 (Neg a) = polybound0 a"
```
```    38 | "polybound0 (Add a b) = (polybound0 a \<and> polybound0 b)"
```
```    39 | "polybound0 (Sub a b) = (polybound0 a \<and> polybound0 b)"
```
```    40 | "polybound0 (Mul a b) = (polybound0 a \<and> polybound0 b)"
```
```    41 | "polybound0 (Pw p n) = (polybound0 p)"
```
```    42 | "polybound0 (CN c n p) = (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) = (if n=0 then Add (polysubst0 t c) (Mul t (polysubst0 t p))
```
```    54                              else CN (polysubst0 t c) n (polysubst0 t p))"
```
```    55
```
```    56 fun decrpoly:: "poly \<Rightarrow> poly"
```
```    57 where
```
```    58   "decrpoly (Bound n) = Bound (n - 1)"
```
```    59 | "decrpoly (Neg a) = Neg (decrpoly a)"
```
```    60 | "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"
```
```    61 | "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
```
```    62 | "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
```
```    63 | "decrpoly (Pw p n) = Pw (decrpoly p) n"
```
```    64 | "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
```
```    65 | "decrpoly a = a"
```
```    66
```
```    67
```
```    68 subsection{* Degrees and heads and coefficients *}
```
```    69
```
```    70 fun degree:: "poly \<Rightarrow> nat"
```
```    71 where
```
```    72   "degree (CN c 0 p) = 1 + degree p"
```
```    73 | "degree p = 0"
```
```    74
```
```    75 fun head:: "poly \<Rightarrow> poly"
```
```    76 where
```
```    77   "head (CN c 0 p) = head p"
```
```    78 | "head p = p"
```
```    79
```
```    80 (* More general notions of degree and head *)
```
```    81 fun degreen:: "poly \<Rightarrow> nat \<Rightarrow> nat"
```
```    82 where
```
```    83   "degreen (CN c n p) = (\<lambda>m. if n=m then 1 + degreen p n else 0)"
```
```    84  |"degreen p = (\<lambda>m. 0)"
```
```    85
```
```    86 fun headn:: "poly \<Rightarrow> nat \<Rightarrow> poly"
```
```    87 where
```
```    88   "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
```
```    89 | "headn p = (\<lambda>m. p)"
```
```    90
```
```    91 fun coefficients:: "poly \<Rightarrow> poly list"
```
```    92 where
```
```    93   "coefficients (CN c 0 p) = c#(coefficients p)"
```
```    94 | "coefficients p = [p]"
```
```    95
```
```    96 fun isconstant:: "poly \<Rightarrow> bool"
```
```    97 where
```
```    98   "isconstant (CN c 0 p) = False"
```
```    99 | "isconstant p = True"
```
```   100
```
```   101 fun behead:: "poly \<Rightarrow> poly"
```
```   102 where
```
```   103   "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
```
```   104 | "behead p = 0\<^sub>p"
```
```   105
```
```   106 fun headconst:: "poly \<Rightarrow> Num"
```
```   107 where
```
```   108   "headconst (CN c n p) = headconst p"
```
```   109 | "headconst (C n) = n"
```
```   110
```
```   111
```
```   112 subsection{* Operations for normalization *}
```
```   113
```
```   114 declare if_cong[fundef_cong del]
```
```   115 declare let_cong[fundef_cong del]
```
```   116
```
```   117 fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
```
```   118 where
```
```   119   "polyadd (C c) (C c') = C (c+\<^sub>Nc')"
```
```   120 |  "polyadd (C c) (CN c' n' p') = CN (polyadd (C c) c') n' p'"
```
```   121 | "polyadd (CN c n p) (C c') = CN (polyadd c (C c')) n p"
```
```   122 | "polyadd (CN c n p) (CN c' n' p') =
```
```   123     (if n < n' then CN (polyadd c (CN c' n' p')) n p
```
```   124      else if n'<n then CN (polyadd (CN c n p) c') n' p'
```
```   125      else (let cc' = polyadd c c' ;
```
```   126                pp' = polyadd p p'
```
```   127            in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
```
```   128 | "polyadd a b = Add a b"
```
```   129
```
```   130
```
```   131 fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
```
```   132 where
```
```   133   "polyneg (C c) = C (~\<^sub>N c)"
```
```   134 | "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
```
```   135 | "polyneg a = Neg a"
```
```   136
```
```   137 definition polysub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
```
```   138   where "p -\<^sub>p q = polyadd p (polyneg q)"
```
```   139
```
```   140 fun polymul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
```
```   141 where
```
```   142   "polymul (C c) (C c') = C (c*\<^sub>Nc')"
```
```   143 | "polymul (C c) (CN c' n' p') =
```
```   144       (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul (C c) c') n' (polymul (C c) p'))"
```
```   145 | "polymul (CN c n p) (C c') =
```
```   146       (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul c (C c')) n (polymul p (C c')))"
```
```   147 | "polymul (CN c n p) (CN c' n' p') =
```
```   148   (if n<n' then CN (polymul c (CN c' n' p')) n (polymul p (CN c' n' p'))
```
```   149   else if n' < n
```
```   150   then CN (polymul (CN c n p) c') n' (polymul (CN c n p) p')
```
```   151   else polyadd (polymul (CN c n p) c') (CN 0\<^sub>p n' (polymul (CN c n p) p')))"
```
```   152 | "polymul a b = Mul a b"
```
```   153
```
```   154 declare if_cong[fundef_cong]
```
```   155 declare let_cong[fundef_cong]
```
```   156
```
```   157 fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
```
```   158 where
```
```   159   "polypow 0 = (\<lambda>p. (1)\<^sub>p)"
```
```   160 | "polypow n = (\<lambda>p. let q = polypow (n div 2) p ; d = polymul q q in
```
```   161                     if even n then d else polymul p d)"
```
```   162
```
```   163 abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
```
```   164   where "a ^\<^sub>p k \<equiv> polypow k a"
```
```   165
```
```   166 function polynate :: "poly \<Rightarrow> poly"
```
```   167 where
```
```   168   "polynate (Bound n) = CN 0\<^sub>p n (1)\<^sub>p"
```
```   169 | "polynate (Add p q) = (polynate p +\<^sub>p polynate q)"
```
```   170 | "polynate (Sub p q) = (polynate p -\<^sub>p polynate q)"
```
```   171 | "polynate (Mul p q) = (polynate p *\<^sub>p polynate q)"
```
```   172 | "polynate (Neg p) = (~\<^sub>p (polynate p))"
```
```   173 | "polynate (Pw p n) = ((polynate p) ^\<^sub>p n)"
```
```   174 | "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
```
```   175 | "polynate (C c) = C (normNum c)"
```
```   176 by pat_completeness auto
```
```   177 termination by (relation "measure polysize") auto
```
```   178
```
```   179 fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly"
```
```   180 where
```
```   181   "poly_cmul y (C x) = C (y *\<^sub>N x)"
```
```   182 | "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
```
```   183 | "poly_cmul y p = C y *\<^sub>p p"
```
```   184
```
```   185 definition monic :: "poly \<Rightarrow> (poly \<times> bool)" where
```
```   186   "monic p \<equiv> (let h = headconst p in if h = 0\<^sub>N then (p,False) else ((C (Ninv h)) *\<^sub>p p, 0>\<^sub>N h))"
```
```   187
```
```   188
```
```   189 subsection{* Pseudo-division *}
```
```   190
```
```   191 definition shift1 :: "poly \<Rightarrow> poly"
```
```   192   where "shift1 p \<equiv> CN 0\<^sub>p 0 p"
```
```   193
```
```   194 abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
```
```   195   where "funpow \<equiv> compow"
```
```   196
```
```   197 partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
```
```   198 where
```
```   199   "polydivide_aux a n p k s =
```
```   200     (if s = 0\<^sub>p then (k,s)
```
```   201     else (let b = head s; m = degree s in
```
```   202     (if m < n then (k,s) else
```
```   203     (let p'= funpow (m - n) shift1 p in
```
```   204     (if a = b then polydivide_aux a n p k (s -\<^sub>p p')
```
```   205     else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))))))"
```
```   206
```
```   207 definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)"
```
```   208   where "polydivide s p \<equiv> polydivide_aux (head p) (degree p) p 0 s"
```
```   209
```
```   210 fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly"
```
```   211 where
```
```   212   "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
```
```   213 | "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
```
```   214
```
```   215 fun poly_deriv :: "poly \<Rightarrow> poly"
```
```   216 where
```
```   217   "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
```
```   218 | "poly_deriv p = 0\<^sub>p"
```
```   219
```
```   220
```
```   221 subsection{* Semantics of the polynomial representation *}
```
```   222
```
```   223 primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0, field_inverse_zero, power}" where
```
```   224   "Ipoly bs (C c) = INum c"
```
```   225 | "Ipoly bs (Bound n) = bs!n"
```
```   226 | "Ipoly bs (Neg a) = - Ipoly bs a"
```
```   227 | "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
```
```   228 | "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
```
```   229 | "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
```
```   230 | "Ipoly bs (Pw t n) = (Ipoly bs t) ^ n"
```
```   231 | "Ipoly bs (CN c n p) = (Ipoly bs c) + (bs!n)*(Ipoly bs p)"
```
```   232
```
```   233 abbreviation
```
```   234   Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0, field_inverse_zero, power}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
```
```   235   where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"
```
```   236
```
```   237 lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i"
```
```   238   by (simp add: INum_def)
```
```   239 lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"
```
```   240   by (simp  add: INum_def)
```
```   241
```
```   242 lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
```
```   243
```
```   244
```
```   245 subsection {* Normal form and normalization *}
```
```   246
```
```   247 fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
```
```   248 where
```
```   249   "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
```
```   250 | "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))"
```
```   251 | "isnpolyh p = (\<lambda>k. False)"
```
```   252
```
```   253 lemma isnpolyh_mono: "\<lbrakk>n' \<le> n ; isnpolyh p n\<rbrakk> \<Longrightarrow> isnpolyh p n'"
```
```   254   by (induct p rule: isnpolyh.induct) auto
```
```   255
```
```   256 definition isnpoly :: "poly \<Rightarrow> bool"
```
```   257   where "isnpoly p \<equiv> isnpolyh p 0"
```
```   258
```
```   259 text{* polyadd preserves normal forms *}
```
```   260
```
```   261 lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk>
```
```   262       \<Longrightarrow> isnpolyh (polyadd p q) (min n0 n1)"
```
```   263 proof(induct p q arbitrary: n0 n1 rule: polyadd.induct)
```
```   264   case (2 ab c' n' p' n0 n1)
```
```   265   from 2 have  th1: "isnpolyh (C ab) (Suc n')" by simp
```
```   266   from 2(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
```
```   267   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
```
```   268   with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^sub>p c') (Suc n')" by simp
```
```   269   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
```
```   270   thus ?case using 2 th3 by simp
```
```   271 next
```
```   272   case (3 c' n' p' ab n1 n0)
```
```   273   from 3 have  th1: "isnpolyh (C ab) (Suc n')" by simp
```
```   274   from 3(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
```
```   275   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
```
```   276   with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (Suc n')" by simp
```
```   277   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
```
```   278   thus ?case using 3 th3 by simp
```
```   279 next
```
```   280   case (4 c n p c' n' p' n0 n1)
```
```   281   hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
```
```   282   from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all
```
```   283   from 4 have ngen0: "n \<ge> n0" by simp
```
```   284   from 4 have n'gen1: "n' \<ge> n1" by simp
```
```   285   have "n < n' \<or> n' < n \<or> n = n'" by auto
```
```   286   moreover {assume eq: "n = n'"
```
```   287     with "4.hyps"(3)[OF nc nc']
```
```   288     have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
```
```   289     hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
```
```   290       using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
```
```   291     from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
```
```   292     have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
```
```   293     from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}
```
```   294   moreover {assume lt: "n < n'"
```
```   295     have "min n0 n1 \<le> n0" by simp
```
```   296     with 4 lt have th1:"min n0 n1 \<le> n" by auto
```
```   297     from 4 have th21: "isnpolyh c (Suc n)" by simp
```
```   298     from 4 have th22: "isnpolyh (CN c' n' p') n'" by simp
```
```   299     from lt have th23: "min (Suc n) n' = Suc n" by arith
```
```   300     from "4.hyps"(1)[OF th21 th22]
```
```   301     have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)" using th23 by simp
```
```   302     with 4 lt th1 have ?case by simp }
```
```   303   moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
```
```   304     have "min n0 n1 \<le> n1"  by simp
```
```   305     with 4 gt have th1:"min n0 n1 \<le> n'" by auto
```
```   306     from 4 have th21: "isnpolyh c' (Suc n')" by simp_all
```
```   307     from 4 have th22: "isnpolyh (CN c n p) n" by simp
```
```   308     from gt have th23: "min n (Suc n') = Suc n'" by arith
```
```   309     from "4.hyps"(2)[OF th22 th21]
```
```   310     have "isnpolyh (polyadd (CN c n p) c') (Suc n')" using th23 by simp
```
```   311     with 4 gt th1 have ?case by simp}
```
```   312       ultimately show ?case by blast
```
```   313 qed auto
```
```   314
```
```   315 lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
```
```   316   by (induct p q rule: polyadd.induct)
```
```   317     (auto simp add: Let_def field_simps distrib_left[symmetric] simp del: distrib_left)
```
```   318
```
```   319 lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd p q)"
```
```   320   using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   321
```
```   322 text{* The degree of addition and other general lemmas needed for the normal form of polymul *}
```
```   323
```
```   324 lemma polyadd_different_degreen:
```
```   325   "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow>
```
```   326   degreen (polyadd p q) m = max (degreen p m) (degreen q m)"
```
```   327 proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
```
```   328   case (4 c n p c' n' p' m n0 n1)
```
```   329   have "n' = n \<or> n < n' \<or> n' < n" by arith
```
```   330   thus ?case
```
```   331   proof (elim disjE)
```
```   332     assume [simp]: "n' = n"
```
```   333     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
```
```   334     show ?thesis by (auto simp: Let_def)
```
```   335   next
```
```   336     assume "n < n'"
```
```   337     with 4 show ?thesis by auto
```
```   338   next
```
```   339     assume "n' < n"
```
```   340     with 4 show ?thesis by auto
```
```   341   qed
```
```   342 qed auto
```
```   343
```
```   344 lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
```
```   345   by (induct p arbitrary: n rule: headn.induct) auto
```
```   346 lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
```
```   347   by (induct p arbitrary: n rule: degree.induct) auto
```
```   348 lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
```
```   349   by (induct p arbitrary: n rule: degreen.induct) auto
```
```   350
```
```   351 lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
```
```   352   by (induct p arbitrary: n rule: degree.induct) auto
```
```   353
```
```   354 lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
```
```   355   using degree_isnpolyh_Suc by auto
```
```   356 lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
```
```   357   using degreen_0 by auto
```
```   358
```
```   359
```
```   360 lemma degreen_polyadd:
```
```   361   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1"
```
```   362   shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
```
```   363   using np nq m
```
```   364 proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
```
```   365   case (2 c c' n' p' n0 n1) thus ?case  by (cases n') simp_all
```
```   366 next
```
```   367   case (3 c n p c' n0 n1) thus ?case by (cases n) auto
```
```   368 next
```
```   369   case (4 c n p c' n' p' n0 n1 m)
```
```   370   have "n' = n \<or> n < n' \<or> n' < n" by arith
```
```   371   thus ?case
```
```   372   proof (elim disjE)
```
```   373     assume [simp]: "n' = n"
```
```   374     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
```
```   375     show ?thesis by (auto simp: Let_def)
```
```   376   qed simp_all
```
```   377 qed auto
```
```   378
```
```   379 lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd p q = C c\<rbrakk>
```
```   380   \<Longrightarrow> degreen p m = degreen q m"
```
```   381 proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
```
```   382   case (4 c n p c' n' p' m n0 n1 x)
```
```   383   {assume nn': "n' < n" hence ?case using 4 by simp}
```
```   384   moreover
```
```   385   {assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
```
```   386     moreover {assume "n < n'" with 4 have ?case by simp }
```
```   387     moreover {assume eq: "n = n'" hence ?case using 4
```
```   388         apply (cases "p +\<^sub>p p' = 0\<^sub>p")
```
```   389         apply (auto simp add: Let_def)
```
```   390         apply blast
```
```   391         done
```
```   392       }
```
```   393     ultimately have ?case by blast}
```
```   394   ultimately show ?case by blast
```
```   395 qed simp_all
```
```   396
```
```   397 lemma polymul_properties:
```
```   398   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   399     and np: "isnpolyh p n0"
```
```   400     and nq: "isnpolyh q n1"
```
```   401     and m: "m \<le> min n0 n1"
```
```   402   shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   403     and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)"
```
```   404     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)"
```
```   405   using np nq m
```
```   406 proof (induct p q arbitrary: n0 n1 m rule: polymul.induct)
```
```   407   case (2 c c' n' p')
```
```   408   { case (1 n0 n1)
```
```   409     with "2.hyps"(4-6)[of n' n' n']
```
```   410       and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']
```
```   411     show ?case by (auto simp add: min_def)
```
```   412   next
```
```   413     case (2 n0 n1) thus ?case by auto
```
```   414   next
```
```   415     case (3 n0 n1) thus ?case  using "2.hyps" by auto }
```
```   416 next
```
```   417   case (3 c n p c')
```
```   418   { case (1 n0 n1)
```
```   419     with "3.hyps"(4-6)[of n n n]
```
```   420       "3.hyps"(1-3)[of "Suc n" "Suc n" n]
```
```   421     show ?case by (auto simp add: min_def)
```
```   422   next
```
```   423     case (2 n0 n1) thus ?case by auto
```
```   424   next
```
```   425     case (3 n0 n1) thus ?case  using "3.hyps" by auto }
```
```   426 next
```
```   427   case (4 c n p c' n' p')
```
```   428   let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
```
```   429     {
```
```   430       case (1 n0 n1)
```
```   431       hence cnp: "isnpolyh ?cnp n" and cnp': "isnpolyh ?cnp' n'"
```
```   432         and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)"
```
```   433         and np': "isnpolyh p' n'" and nc': "isnpolyh c' (Suc n')"
```
```   434         and nn0: "n \<ge> n0" and nn1:"n' \<ge> n1"
```
```   435         by simp_all
```
```   436       { assume "n < n'"
```
```   437         with "4.hyps"(4-5)[OF np cnp', of n]
```
```   438           "4.hyps"(1)[OF nc cnp', of n] nn0 cnp
```
```   439         have ?case by (simp add: min_def)
```
```   440       } moreover {
```
```   441         assume "n' < n"
```
```   442         with "4.hyps"(16-17)[OF cnp np', of "n'"]
```
```   443           "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
```
```   444         have ?case
```
```   445           by (cases "Suc n' = n") (simp_all add: min_def)
```
```   446       } moreover {
```
```   447         assume "n' = n"
```
```   448         with "4.hyps"(16-17)[OF cnp np', of n]
```
```   449           "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0
```
```   450         have ?case
```
```   451           apply (auto intro!: polyadd_normh)
```
```   452           apply (simp_all add: min_def isnpolyh_mono[OF nn0])
```
```   453           done
```
```   454       }
```
```   455       ultimately show ?case by arith
```
```   456     next
```
```   457       fix n0 n1 m
```
```   458       assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1"
```
```   459       and m: "m \<le> min n0 n1"
```
```   460       let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
```
```   461       let ?d1 = "degreen ?cnp m"
```
```   462       let ?d2 = "degreen ?cnp' m"
```
```   463       let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
```
```   464       have "n'<n \<or> n < n' \<or> n' = n" by auto
```
```   465       moreover
```
```   466       {assume "n' < n \<or> n < n'"
```
```   467         with "4.hyps"(3,6,18) np np' m
```
```   468         have ?eq by auto }
```
```   469       moreover
```
```   470       {assume nn': "n' = n" hence nn:"\<not> n' < n \<and> \<not> n < n'" by arith
```
```   471         from "4.hyps"(16,18)[of n n' n]
```
```   472           "4.hyps"(13,14)[of n "Suc n'" n]
```
```   473           np np' nn'
```
```   474         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
```
```   475           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   476           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
```
```   477           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
```
```   478         {assume mn: "m = n"
```
```   479           from "4.hyps"(17,18)[OF norm(1,4), of n]
```
```   480             "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
```
```   481           have degs:  "degreen (?cnp *\<^sub>p c') n =
```
```   482             (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"
```
```   483             "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n" by (simp_all add: min_def)
```
```   484           from degs norm
```
```   485           have th1: "degreen(?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" by simp
```
```   486           hence neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   487             by simp
```
```   488           have nmin: "n \<le> min n n" by (simp add: min_def)
```
```   489           from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
```
```   490           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
```
```   491           from "4.hyps"(16-18)[OF norm(1,4), of n]
```
```   492             "4.hyps"(13-15)[OF norm(1,2), of n]
```
```   493             mn norm m nn' deg
```
```   494           have ?eq by simp}
```
```   495         moreover
```
```   496         {assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
```
```   497           from nn' m np have max1: "m \<le> max n n"  by simp
```
```   498           hence min1: "m \<le> min n n" by simp
```
```   499           hence min2: "m \<le> min n (Suc n)" by simp
```
```   500           from "4.hyps"(16-18)[OF norm(1,4) min1]
```
```   501             "4.hyps"(13-15)[OF norm(1,2) min2]
```
```   502             degreen_polyadd[OF norm(3,6) max1]
```
```   503
```
```   504           have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m
```
```   505             \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
```
```   506             using mn nn' np np' by simp
```
```   507           with "4.hyps"(16-18)[OF norm(1,4) min1]
```
```   508             "4.hyps"(13-15)[OF norm(1,2) min2]
```
```   509             degreen_0[OF norm(3) mn']
```
```   510           have ?eq using nn' mn np np' by clarsimp}
```
```   511         ultimately have ?eq by blast}
```
```   512       ultimately show ?eq by blast}
```
```   513     { case (2 n0 n1)
```
```   514       hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1"
```
```   515         and m: "m \<le> min n0 n1" by simp_all
```
```   516       hence mn: "m \<le> n" by simp
```
```   517       let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
```
```   518       {assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
```
```   519         hence nn: "\<not>n' < n \<and> \<not> n<n'" by simp
```
```   520         from "4.hyps"(16-18) [of n n n]
```
```   521           "4.hyps"(13-15)[of n "Suc n" n]
```
```   522           np np' C(2) mn
```
```   523         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
```
```   524           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
```
```   525           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
```
```   526           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
```
```   527           "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
```
```   528             "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
```
```   529           by (simp_all add: min_def)
```
```   530
```
```   531           from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```   532           have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
```
```   533             using norm by simp
```
```   534         from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"]  degneq
```
```   535         have "False" by simp }
```
```   536       thus ?case using "4.hyps" by clarsimp}
```
```   537 qed auto
```
```   538
```
```   539 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"
```
```   540   by (induct p q rule: polymul.induct) (auto simp add: field_simps)
```
```   541
```
```   542 lemma polymul_normh:
```
```   543   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   544   shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
```
```   545   using polymul_properties(1)  by blast
```
```   546
```
```   547 lemma polymul_eq0_iff:
```
```   548   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   549   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) "
```
```   550   using polymul_properties(2)  by blast
```
```   551
```
```   552 lemma polymul_degreen:  (* FIXME duplicate? *)
```
```   553   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   554   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow>
```
```   555     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)"
```
```   556   using polymul_properties(3) by blast
```
```   557
```
```   558 lemma polymul_norm:
```
```   559   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   560   shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul p q)"
```
```   561   using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
```
```   562
```
```   563 lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
```
```   564   by (induct p arbitrary: n0 rule: headconst.induct) auto
```
```   565
```
```   566 lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
```
```   567   by (induct p arbitrary: n0) auto
```
```   568
```
```   569 lemma monic_eqI:
```
```   570   assumes np: "isnpolyh p n0"
```
```   571   shows "INum (headconst p) * Ipoly bs (fst (monic p)) =
```
```   572     (Ipoly bs p ::'a::{field_char_0, field_inverse_zero, power})"
```
```   573   unfolding monic_def Let_def
```
```   574 proof (cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
```
```   575   let ?h = "headconst p"
```
```   576   assume pz: "p \<noteq> 0\<^sub>p"
```
```   577   {assume hz: "INum ?h = (0::'a)"
```
```   578     from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
```
```   579     from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
```
```   580     with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
```
```   581   thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast
```
```   582 qed
```
```   583
```
```   584
```
```   585 text{* polyneg is a negation and preserves normal forms *}
```
```   586
```
```   587 lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
```
```   588   by (induct p rule: polyneg.induct) auto
```
```   589
```
```   590 lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   591   by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: Nneg_def)
```
```   592 lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
```
```   593   by (induct p arbitrary: n0 rule: polyneg.induct) auto
```
```   594 lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "
```
```   595   by (induct p rule: polyneg.induct) (auto simp add: polyneg0)
```
```   596
```
```   597 lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
```
```   598   using isnpoly_def polyneg_normh by simp
```
```   599
```
```   600
```
```   601 text{* polysub is a substraction and preserves normal forms *}
```
```   602
```
```   603 lemma polysub[simp]: "Ipoly bs (polysub p q) = (Ipoly bs p) - (Ipoly bs q)"
```
```   604   by (simp add: polysub_def)
```
```   605 lemma polysub_normh: "\<And> n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
```
```   606   by (simp add: polysub_def polyneg_normh polyadd_normh)
```
```   607
```
```   608 lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub p q)"
```
```   609   using polyadd_norm polyneg_norm by (simp add: polysub_def)
```
```   610 lemma polysub_same_0[simp]:
```
```   611   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   612   shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p"
```
```   613   unfolding polysub_def split_def fst_conv snd_conv
```
```   614   by (induct p arbitrary: n0) (auto simp add: Let_def Nsub0[simplified Nsub_def])
```
```   615
```
```   616 lemma polysub_0:
```
```   617   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   618   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"
```
```   619   unfolding polysub_def split_def fst_conv snd_conv
```
```   620   by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
```
```   621     (auto simp: Nsub0[simplified Nsub_def] Let_def)
```
```   622
```
```   623 text{* polypow is a power function and preserves normal forms *}
```
```   624
```
```   625 lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{field_char_0, field_inverse_zero})) ^ n"
```
```   626 proof (induct n rule: polypow.induct)
```
```   627   case 1
```
```   628   thus ?case by simp
```
```   629 next
```
```   630   case (2 n)
```
```   631   let ?q = "polypow ((Suc n) div 2) p"
```
```   632   let ?d = "polymul ?q ?q"
```
```   633   have "odd (Suc n) \<or> even (Suc n)" by simp
```
```   634   moreover
```
```   635   {assume odd: "odd (Suc n)"
```
```   636     have th: "(Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat))))) = Suc n div 2 + Suc n div 2 + 1"
```
```   637       by arith
```
```   638     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)" by (simp add: Let_def)
```
```   639     also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"
```
```   640       using "2.hyps" by simp
```
```   641     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
```
```   642       by (simp only: power_add power_one_right) simp
```
```   643     also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat)))))"
```
```   644       by (simp only: th)
```
```   645     finally have ?case
```
```   646     using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
```
```   647   moreover
```
```   648   {assume even: "even (Suc n)"
```
```   649     have th: "(Suc (Suc (0\<Colon>nat))) * (Suc n div Suc (Suc (0\<Colon>nat))) = Suc n div 2 + Suc n div 2"
```
```   650       by arith
```
```   651     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
```
```   652     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
```
```   653       using "2.hyps" apply (simp only: power_add) by simp
```
```   654     finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}
```
```   655   ultimately show ?case by blast
```
```   656 qed
```
```   657
```
```   658 lemma polypow_normh:
```
```   659   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   660   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
```
```   661 proof (induct k arbitrary: n rule: polypow.induct)
```
```   662   case (2 k n)
```
```   663   let ?q = "polypow (Suc k div 2) p"
```
```   664   let ?d = "polymul ?q ?q"
```
```   665   from 2 have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
```
```   666   from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
```
```   667   from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n" by simp
```
```   668   from dn on show ?case by (simp add: Let_def)
```
```   669 qed auto
```
```   670
```
```   671 lemma polypow_norm:
```
```   672   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   673   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
```
```   674   by (simp add: polypow_normh isnpoly_def)
```
```   675
```
```   676 text{* Finally the whole normalization *}
```
```   677
```
```   678 lemma polynate [simp]:
```
```   679   "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0, field_inverse_zero})"
```
```   680   by (induct p rule:polynate.induct) auto
```
```   681
```
```   682 lemma polynate_norm[simp]:
```
```   683   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   684   shows "isnpoly (polynate p)"
```
```   685   by (induct p rule: polynate.induct)
```
```   686      (simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm,
```
```   687       simp_all add: isnpoly_def)
```
```   688
```
```   689 text{* shift1 *}
```
```   690
```
```   691
```
```   692 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
```
```   693   by (simp add: shift1_def)
```
```   694
```
```   695 lemma shift1_isnpoly:
```
```   696   assumes pn: "isnpoly p"
```
```   697     and pnz: "p \<noteq> 0\<^sub>p"
```
```   698   shows "isnpoly (shift1 p) "
```
```   699   using pn pnz by (simp add: shift1_def isnpoly_def)
```
```   700
```
```   701 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
```
```   702   by (simp add: shift1_def)
```
```   703 lemma funpow_shift1_isnpoly:
```
```   704   "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
```
```   705   by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
```
```   706
```
```   707 lemma funpow_isnpolyh:
```
```   708   assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n"
```
```   709     and np: "isnpolyh p n"
```
```   710   shows "isnpolyh (funpow k f p) n"
```
```   711   using f np by (induct k arbitrary: p) auto
```
```   712
```
```   713 lemma funpow_shift1:
```
```   714   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) =
```
```   715     Ipoly bs (Mul (Pw (Bound 0) n) p)"
```
```   716   by (induct n arbitrary: p) (simp_all add: shift1_isnpoly shift1)
```
```   717
```
```   718 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
```
```   719   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
```
```   720
```
```   721 lemma funpow_shift1_1:
```
```   722   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) =
```
```   723     Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)"
```
```   724   by (simp add: funpow_shift1)
```
```   725
```
```   726 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
```
```   727   by (induct p rule: poly_cmul.induct) (auto simp add: field_simps)
```
```   728
```
```   729 lemma behead:
```
```   730   assumes np: "isnpolyh p n"
```
```   731   shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) =
```
```   732     (Ipoly bs p :: 'a :: {field_char_0, field_inverse_zero})"
```
```   733   using np
```
```   734 proof (induct p arbitrary: n rule: behead.induct)
```
```   735   case (1 c p n) hence pn: "isnpolyh p n" by simp
```
```   736   from 1(1)[OF pn]
```
```   737   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
```
```   738   then show ?case using "1.hyps"
```
```   739     apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
```
```   740     apply (simp_all add: th[symmetric] field_simps)
```
```   741     done
```
```   742 qed (auto simp add: Let_def)
```
```   743
```
```   744 lemma behead_isnpolyh:
```
```   745   assumes np: "isnpolyh p n"
```
```   746   shows "isnpolyh (behead p) n"
```
```   747   using np by (induct p rule: behead.induct) (auto simp add: Let_def isnpolyh_mono)
```
```   748
```
```   749
```
```   750 subsection{* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
```
```   751
```
```   752 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
```
```   753 proof (induct p arbitrary: n rule: poly.induct, auto)
```
```   754   case (goal1 c n p n')
```
```   755   hence "n = Suc (n - 1)" by simp
```
```   756   hence "isnpolyh p (Suc (n - 1))"  using `isnpolyh p n` by simp
```
```   757   with goal1(2) show ?case by simp
```
```   758 qed
```
```   759
```
```   760 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
```
```   761   by (induct p arbitrary: n0 rule: isconstant.induct) (auto simp add: isnpolyh_polybound0)
```
```   762
```
```   763 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
```
```   764   by (induct p) auto
```
```   765
```
```   766 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
```
```   767   apply (induct p arbitrary: n0)
```
```   768   apply auto
```
```   769   apply (atomize)
```
```   770   apply (erule_tac x = "Suc nat" in allE)
```
```   771   apply auto
```
```   772   done
```
```   773
```
```   774 lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
```
```   775   by (induct p  arbitrary: n0 rule: head.induct) (auto intro: isnpolyh_polybound0)
```
```   776
```
```   777 lemma polybound0_I:
```
```   778   assumes nb: "polybound0 a"
```
```   779   shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
```
```   780   using nb
```
```   781   by (induct a rule: poly.induct) auto
```
```   782
```
```   783 lemma polysubst0_I: "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
```
```   784   by (induct t) simp_all
```
```   785
```
```   786 lemma polysubst0_I':
```
```   787   assumes nb: "polybound0 a"
```
```   788   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t"
```
```   789   by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
```
```   790
```
```   791 lemma decrpoly:
```
```   792   assumes nb: "polybound0 t"
```
```   793   shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
```
```   794   using nb by (induct t rule: decrpoly.induct) simp_all
```
```   795
```
```   796 lemma polysubst0_polybound0:
```
```   797   assumes nb: "polybound0 t"
```
```   798   shows "polybound0 (polysubst0 t a)"
```
```   799   using nb by (induct a rule: poly.induct) auto
```
```   800
```
```   801 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
```
```   802   by (induct p arbitrary: n rule: degree.induct) (auto simp add: isnpolyh_polybound0)
```
```   803
```
```   804 primrec maxindex :: "poly \<Rightarrow> nat" where
```
```   805   "maxindex (Bound n) = n + 1"
```
```   806 | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
```
```   807 | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
```
```   808 | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
```
```   809 | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
```
```   810 | "maxindex (Neg p) = maxindex p"
```
```   811 | "maxindex (Pw p n) = maxindex p"
```
```   812 | "maxindex (C x) = 0"
```
```   813
```
```   814 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool"
```
```   815   where "wf_bs bs p = (length bs \<ge> maxindex p)"
```
```   816
```
```   817 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
```
```   818 proof (induct p rule: coefficients.induct)
```
```   819   case (1 c p)
```
```   820   show ?case
```
```   821   proof
```
```   822     fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"
```
```   823     hence "x = c \<or> x \<in> set (coefficients p)" by simp
```
```   824     moreover
```
```   825     {assume "x = c" hence "wf_bs bs x" using "1.prems"  unfolding wf_bs_def by simp}
```
```   826     moreover
```
```   827     {assume H: "x \<in> set (coefficients p)"
```
```   828       from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp
```
```   829       with "1.hyps" H have "wf_bs bs x" by blast }
```
```   830     ultimately  show "wf_bs bs x" by blast
```
```   831   qed
```
```   832 qed simp_all
```
```   833
```
```   834 lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
```
```   835   by (induct p rule: coefficients.induct) auto
```
```   836
```
```   837 lemma wf_bs_I: "wf_bs bs p ==> Ipoly (bs@bs') p = Ipoly bs p"
```
```   838   unfolding wf_bs_def by (induct p) (auto simp add: nth_append)
```
```   839
```
```   840 lemma take_maxindex_wf:
```
```   841   assumes wf: "wf_bs bs p"
```
```   842   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
```
```   843 proof-
```
```   844   let ?ip = "maxindex p"
```
```   845   let ?tbs = "take ?ip bs"
```
```   846   from wf have "length ?tbs = ?ip" unfolding wf_bs_def by simp
```
```   847   hence wf': "wf_bs ?tbs p" unfolding wf_bs_def by  simp
```
```   848   have eq: "bs = ?tbs @ (drop ?ip bs)" by simp
```
```   849   from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis using eq by simp
```
```   850 qed
```
```   851
```
```   852 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
```
```   853   by (induct p) auto
```
```   854
```
```   855 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
```
```   856   unfolding wf_bs_def by simp
```
```   857
```
```   858 lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
```
```   859   unfolding wf_bs_def by simp
```
```   860
```
```   861
```
```   862
```
```   863 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
```
```   864   by (induct p rule: coefficients.induct) (auto simp add: wf_bs_def)
```
```   865 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
```
```   866   by (induct p rule: coefficients.induct) simp_all
```
```   867
```
```   868
```
```   869 lemma coefficients_head: "last (coefficients p) = head p"
```
```   870   by (induct p rule: coefficients.induct) auto
```
```   871
```
```   872 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
```
```   873   unfolding wf_bs_def by (induct p rule: decrpoly.induct) auto
```
```   874
```
```   875 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
```
```   876   apply (rule exI[where x="replicate (n - length xs) z"])
```
```   877   apply simp
```
```   878   done
```
```   879
```
```   880 lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
```
```   881   apply (cases p)
```
```   882   apply auto
```
```   883   apply (case_tac "nat")
```
```   884   apply simp_all
```
```   885   done
```
```   886
```
```   887 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
```
```   888   unfolding wf_bs_def
```
```   889   apply (induct p q rule: polyadd.induct)
```
```   890   apply (auto simp add: Let_def)
```
```   891   done
```
```   892
```
```   893 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
```
```   894   unfolding wf_bs_def
```
```   895   apply (induct p q arbitrary: bs rule: polymul.induct)
```
```   896   apply (simp_all add: wf_bs_polyadd)
```
```   897   apply clarsimp
```
```   898   apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
```
```   899   apply auto
```
```   900   done
```
```   901
```
```   902 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
```
```   903   unfolding wf_bs_def by (induct p rule: polyneg.induct) auto
```
```   904
```
```   905 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
```
```   906   unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
```
```   907
```
```   908
```
```   909 subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}
```
```   910
```
```   911 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
```
```   912 definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)"
```
```   913 definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"
```
```   914
```
```   915 lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0"
```
```   916 proof (induct p arbitrary: n0 rule: coefficients.induct)
```
```   917   case (1 c p n0)
```
```   918   have cp: "isnpolyh (CN c 0 p) n0" by fact
```
```   919   hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
```
```   920     by (auto simp add: isnpolyh_mono[where n'=0])
```
```   921   from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp
```
```   922 qed auto
```
```   923
```
```   924 lemma coefficients_isconst:
```
```   925   "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
```
```   926   by (induct p arbitrary: n rule: coefficients.induct)
```
```   927     (auto simp add: isnpolyh_Suc_const)
```
```   928
```
```   929 lemma polypoly_polypoly':
```
```   930   assumes np: "isnpolyh p n0"
```
```   931   shows "polypoly (x#bs) p = polypoly' bs p"
```
```   932 proof-
```
```   933   let ?cf = "set (coefficients p)"
```
```   934   from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
```
```   935   {fix q assume q: "q \<in> ?cf"
```
```   936     from q cn_norm have th: "isnpolyh q n0" by blast
```
```   937     from coefficients_isconst[OF np] q have "isconstant q" by blast
```
```   938     with isconstant_polybound0[OF th] have "polybound0 q" by blast}
```
```   939   hence "\<forall>q \<in> ?cf. polybound0 q" ..
```
```   940   hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"
```
```   941     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
```
```   942     by auto
```
```   943
```
```   944   thus ?thesis unfolding polypoly_def polypoly'_def by simp
```
```   945 qed
```
```   946
```
```   947 lemma polypoly_poly:
```
```   948   assumes np: "isnpolyh p n0"
```
```   949   shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
```
```   950   using np
```
```   951   by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def)
```
```   952
```
```   953 lemma polypoly'_poly:
```
```   954   assumes np: "isnpolyh p n0"
```
```   955   shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
```
```   956   using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
```
```   957
```
```   958
```
```   959 lemma polypoly_poly_polybound0:
```
```   960   assumes np: "isnpolyh p n0" and nb: "polybound0 p"
```
```   961   shows "polypoly bs p = [Ipoly bs p]"
```
```   962   using np nb unfolding polypoly_def
```
```   963   apply (cases p)
```
```   964   apply auto
```
```   965   apply (case_tac nat)
```
```   966   apply auto
```
```   967   done
```
```   968
```
```   969 lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
```
```   970   by (induct p rule: head.induct) auto
```
```   971
```
```   972 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   973   by (cases p) auto
```
```   974
```
```   975 lemma head_eq_headn0: "head p = headn p 0"
```
```   976   by (induct p rule: head.induct) simp_all
```
```   977
```
```   978 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```   979   by (simp add: head_eq_headn0)
```
```   980
```
```   981 lemma isnpolyh_zero_iff:
```
```   982   assumes nq: "isnpolyh p n0"
```
```   983     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})"
```
```   984   shows "p = 0\<^sub>p"
```
```   985   using nq eq
```
```   986 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
```
```   987   case less
```
```   988   note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
```
```   989   {assume nz: "maxindex p = 0"
```
```   990     then obtain c where "p = C c" using np by (cases p) auto
```
```   991     with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}
```
```   992   moreover
```
```   993   {assume nz: "maxindex p \<noteq> 0"
```
```   994     let ?h = "head p"
```
```   995     let ?hd = "decrpoly ?h"
```
```   996     let ?ihd = "maxindex ?hd"
```
```   997     from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h"
```
```   998       by simp_all
```
```   999     hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast
```
```  1000
```
```  1001     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
```
```  1002     have mihn: "maxindex ?h \<le> maxindex p" by auto
```
```  1003     with decr_maxindex[OF h(2)] nz  have ihd_lt_n: "?ihd < maxindex p" by auto
```
```  1004     {fix bs:: "'a list"  assume bs: "wf_bs bs ?hd"
```
```  1005       let ?ts = "take ?ihd bs"
```
```  1006       let ?rs = "drop ?ihd bs"
```
```  1007       have ts: "wf_bs ?ts ?hd" using bs unfolding wf_bs_def by simp
```
```  1008       have bs_ts_eq: "?ts@ ?rs = bs" by simp
```
```  1009       from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x#?ts) ?h" by simp
```
```  1010       from ihd_lt_n have "ALL x. length (x#?ts) \<le> maxindex p" by simp
```
```  1011       with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = maxindex p" by blast
```
```  1012       hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" unfolding wf_bs_def by simp
```
```  1013       with zp have "\<forall> x. Ipoly ((x#?ts) @ xs) p = 0" by blast
```
```  1014       hence "\<forall> x. Ipoly (x#(?ts @ xs)) p = 0" by simp
```
```  1015       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
```
```  1016       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"  by simp
```
```  1017       hence "poly (polypoly' (?ts @ xs) p) = poly []" by auto
```
```  1018       hence "\<forall> c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
```
```  1019         using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
```
```  1020       with coefficients_head[of p, symmetric]
```
```  1021       have th0: "Ipoly (?ts @ xs) ?hd = 0" by simp
```
```  1022       from bs have wf'': "wf_bs ?ts ?hd" unfolding wf_bs_def by simp
```
```  1023       with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0" by simp
```
```  1024       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by simp }
```
```  1025     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast
```
```  1026
```
```  1027     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast
```
```  1028     hence "?h = 0\<^sub>p" by simp
```
```  1029     with head_nz[OF np] have "p = 0\<^sub>p" by simp}
```
```  1030   ultimately show "p = 0\<^sub>p" by blast
```
```  1031 qed
```
```  1032
```
```  1033 lemma isnpolyh_unique:
```
```  1034   assumes np:"isnpolyh p n0"
```
```  1035     and nq: "isnpolyh q n1"
```
```  1036   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"
```
```  1037 proof(auto)
```
```  1038   assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a)= \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
```
```  1039   hence "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" by simp
```
```  1040   hence "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
```
```  1041     using wf_bs_polysub[where p=p and q=q] by auto
```
```  1042   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq]
```
```  1043   show "p = q" by blast
```
```  1044 qed
```
```  1045
```
```  1046
```
```  1047 text{* consequences of unicity on the algorithms for polynomial normalization *}
```
```  1048
```
```  1049 lemma polyadd_commute:
```
```  1050   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1051     and np: "isnpolyh p n0"
```
```  1052     and nq: "isnpolyh q n1"
```
```  1053   shows "p +\<^sub>p q = q +\<^sub>p p"
```
```  1054   using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]] by simp
```
```  1055
```
```  1056 lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp
```
```  1057 lemma one_normh: "isnpolyh (1)\<^sub>p n" by simp
```
```  1058
```
```  1059 lemma polyadd_0[simp]:
```
```  1060   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1061     and np: "isnpolyh p n0"
```
```  1062   shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>p p = p"
```
```  1063   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
```
```  1064     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
```
```  1065
```
```  1066 lemma polymul_1[simp]:
```
```  1067   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1068     and np: "isnpolyh p n0"
```
```  1069   shows "p *\<^sub>p (1)\<^sub>p = p" and "(1)\<^sub>p *\<^sub>p p = p"
```
```  1070   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
```
```  1071     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
```
```  1072
```
```  1073 lemma polymul_0[simp]:
```
```  1074   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1075     and np: "isnpolyh p n0"
```
```  1076   shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p" and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
```
```  1077   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
```
```  1078     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
```
```  1079
```
```  1080 lemma polymul_commute:
```
```  1081   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1082     and np:"isnpolyh p n0"
```
```  1083     and nq: "isnpolyh q n1"
```
```  1084   shows "p *\<^sub>p q = q *\<^sub>p p"
```
```  1085   using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np], where ?'a = "'a\<Colon>{field_char_0, field_inverse_zero, power}"]
```
```  1086   by simp
```
```  1087
```
```  1088 declare polyneg_polyneg [simp]
```
```  1089
```
```  1090 lemma isnpolyh_polynate_id [simp]:
```
```  1091   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1092     and np:"isnpolyh p n0"
```
```  1093   shows "polynate p = p"
```
```  1094   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}"]
```
```  1095   by simp
```
```  1096
```
```  1097 lemma polynate_idempotent[simp]:
```
```  1098   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1099   shows "polynate (polynate p) = polynate p"
```
```  1100   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
```
```  1101
```
```  1102 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
```
```  1103   unfolding poly_nate_def polypoly'_def ..
```
```  1104
```
```  1105 lemma poly_nate_poly:
```
```  1106   "poly (poly_nate bs p) = (\<lambda>x:: 'a ::{field_char_0, field_inverse_zero}. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
```
```  1107   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
```
```  1108   unfolding poly_nate_polypoly' by auto
```
```  1109
```
```  1110
```
```  1111 subsection{* heads, degrees and all that *}
```
```  1112
```
```  1113 lemma degree_eq_degreen0: "degree p = degreen p 0"
```
```  1114   by (induct p rule: degree.induct) simp_all
```
```  1115
```
```  1116 lemma degree_polyneg:
```
```  1117   assumes n: "isnpolyh p n"
```
```  1118   shows "degree (polyneg p) = degree p"
```
```  1119   apply (induct p arbitrary: n rule: polyneg.induct)
```
```  1120   using n apply simp_all
```
```  1121   apply (case_tac na)
```
```  1122   apply auto
```
```  1123   done
```
```  1124
```
```  1125 lemma degree_polyadd:
```
```  1126   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
```
```  1127   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1128   using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
```
```  1129
```
```  1130
```
```  1131 lemma degree_polysub:
```
```  1132   assumes np: "isnpolyh p n0"
```
```  1133     and nq: "isnpolyh q n1"
```
```  1134   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
```
```  1135 proof-
```
```  1136   from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp
```
```  1137   from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])
```
```  1138 qed
```
```  1139
```
```  1140 lemma degree_polysub_samehead:
```
```  1141   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1142     and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q"
```
```  1143     and d: "degree p = degree q"
```
```  1144   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
```
```  1145   unfolding polysub_def split_def fst_conv snd_conv
```
```  1146   using np nq h d
```
```  1147 proof (induct p q rule: polyadd.induct)
```
```  1148   case (1 c c')
```
```  1149   thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def])
```
```  1150 next
```
```  1151   case (2 c c' n' p')
```
```  1152   from 2 have "degree (C c) = degree (CN c' n' p')" by simp
```
```  1153   hence nz:"n' > 0" by (cases n') auto
```
```  1154   hence "head (CN c' n' p') = CN c' n' p'" by (cases n') auto
```
```  1155   with 2 show ?case by simp
```
```  1156 next
```
```  1157   case (3 c n p c')
```
```  1158   hence "degree (C c') = degree (CN c n p)" by simp
```
```  1159   hence nz:"n > 0" by (cases n) auto
```
```  1160   hence "head (CN c n p) = CN c n p" by (cases n) auto
```
```  1161   with 3 show ?case by simp
```
```  1162 next
```
```  1163   case (4 c n p c' n' p')
```
```  1164   hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1"
```
```  1165     "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+
```
```  1166   hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all
```
```  1167   hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
```
```  1168     using H(1-2) degree_polyneg by auto
```
```  1169   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"  by simp+
```
```  1170   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"  by simp
```
```  1171   from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto
```
```  1172   have "n = n' \<or> n < n' \<or> n > n'" by arith
```
```  1173   moreover
```
```  1174   {assume nn': "n = n'"
```
```  1175     have "n = 0 \<or> n >0" by arith
```
```  1176     moreover {assume nz: "n = 0" hence ?case using 4 nn' by (auto simp add: Let_def degcmc')}
```
```  1177     moreover {assume nz: "n > 0"
```
```  1178       with nn' H(3) have  cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
```
```  1179       hence ?case
```
```  1180         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]
```
```  1181         using nn' 4 by (simp add: Let_def)}
```
```  1182     ultimately have ?case by blast}
```
```  1183   moreover
```
```  1184   {assume nn': "n < n'" hence n'p: "n' > 0" by simp
```
```  1185     hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n') simp_all
```
```  1186     have degcnp': "degree (CN c' n' p') = 0" and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
```
```  1187       using 4 nn' by (cases n', simp_all)
```
```  1188     hence "n > 0" by (cases n) simp_all
```
```  1189     hence headcnp: "head (CN c n p) = CN c n p" by (cases n) auto
```
```  1190     from H(3) headcnp headcnp' nn' have ?case by auto}
```
```  1191   moreover
```
```  1192   {assume nn': "n > n'"  hence np: "n > 0" by simp
```
```  1193     hence headcnp:"head (CN c n p) = CN c n p"  by (cases n) simp_all
```
```  1194     from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
```
```  1195     from np have degcnp: "degree (CN c n p) = 0" by (cases n) simp_all
```
```  1196     with degcnpeq have "n' > 0" by (cases n') simp_all
```
```  1197     hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n') auto
```
```  1198     from H(3) headcnp headcnp' nn' have ?case by auto}
```
```  1199   ultimately show ?case  by blast
```
```  1200 qed auto
```
```  1201
```
```  1202 lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
```
```  1203   by (induct p arbitrary: n0 rule: head.induct) (simp_all add: shift1_def)
```
```  1204
```
```  1205 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
```
```  1206 proof (induct k arbitrary: n0 p)
```
```  1207   case 0
```
```  1208   thus ?case by auto
```
```  1209 next
```
```  1210   case (Suc k n0 p)
```
```  1211   hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
```
```  1212   with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
```
```  1213     and "head (shift1 p) = head p" by (simp_all add: shift1_head)
```
```  1214   thus ?case by (simp add: funpow_swap1)
```
```  1215 qed
```
```  1216
```
```  1217 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
```
```  1218   by (simp add: shift1_def)
```
```  1219 lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
```
```  1220   by (induct k arbitrary: p) (auto simp add: shift1_degree)
```
```  1221
```
```  1222 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
```
```  1223   by (induct n arbitrary: p) simp_all
```
```  1224
```
```  1225 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
```
```  1226   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1227 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
```
```  1228   by (induct p arbitrary: n rule: degreen.induct) auto
```
```  1229 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
```
```  1230   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1231 lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
```
```  1232   by (induct p rule: head.induct) auto
```
```  1233
```
```  1234 lemma polyadd_eq_const_degree:
```
```  1235   "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> polyadd p q = C c \<Longrightarrow> degree p = degree q"
```
```  1236   using polyadd_eq_const_degreen degree_eq_degreen0 by simp
```
```  1237
```
```  1238 lemma polyadd_head:
```
```  1239   assumes np: "isnpolyh p n0"
```
```  1240     and nq: "isnpolyh q n1"
```
```  1241     and deg: "degree p \<noteq> degree q"
```
```  1242   shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
```
```  1243   using np nq deg
```
```  1244   apply (induct p q arbitrary: n0 n1 rule: polyadd.induct)
```
```  1245   using np
```
```  1246   apply simp_all
```
```  1247   apply (case_tac n', simp, simp)
```
```  1248   apply (case_tac n, simp, simp)
```
```  1249   apply (case_tac n, case_tac n', simp add: Let_def)
```
```  1250   apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p")
```
```  1251   apply (auto simp add: polyadd_eq_const_degree)
```
```  1252   apply (metis head_nz)
```
```  1253   apply (metis head_nz)
```
```  1254   apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
```
```  1255   apply (metis degree.simps(9) gr0_conv_Suc nat_less_le order_le_less_trans)
```
```  1256   done
```
```  1257
```
```  1258 lemma polymul_head_polyeq:
```
```  1259   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1260   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"
```
```  1261 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
```
```  1262   case (2 c c' n' p' n0 n1)
```
```  1263   hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"  by (simp_all add: head_isnpolyh)
```
```  1264   thus ?case using 2 by (cases n') auto
```
```  1265 next
```
```  1266   case (3 c n p c' n0 n1)
```
```  1267   hence "isnpolyh (head (CN c n p)) n0" "isnormNum c'"  by (simp_all add: head_isnpolyh)
```
```  1268   thus ?case using 3 by (cases n) auto
```
```  1269 next
```
```  1270   case (4 c n p c' n' p' n0 n1)
```
```  1271   hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
```
```  1272     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
```
```  1273     by simp_all
```
```  1274   have "n < n' \<or> n' < n \<or> n = n'" by arith
```
```  1275   moreover
```
```  1276   {assume nn': "n < n'" hence ?case
```
```  1277       using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)]
```
```  1278       apply simp
```
```  1279       apply (cases n)
```
```  1280       apply simp
```
```  1281       apply (cases n')
```
```  1282       apply simp_all
```
```  1283       done }
```
```  1284   moreover {assume nn': "n'< n"
```
```  1285     hence ?case
```
```  1286       using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)]
```
```  1287       apply simp
```
```  1288       apply (cases n')
```
```  1289       apply simp
```
```  1290       apply (cases n)
```
```  1291       apply auto
```
```  1292       done }
```
```  1293   moreover {assume nn': "n' = n"
```
```  1294     from nn' polymul_normh[OF norm(5,4)]
```
```  1295     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
```
```  1296     from nn' polymul_normh[OF norm(5,3)] norm
```
```  1297     have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
```
```  1298     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
```
```  1299     have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
```
```  1300     from polyadd_normh[OF ncnpc' ncnpp0']
```
```  1301     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"
```
```  1302       by (simp add: min_def)
```
```  1303     {assume np: "n > 0"
```
```  1304       with nn' head_isnpolyh_Suc'[OF np nth]
```
```  1305         head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
```
```  1306       have ?case by simp}
```
```  1307     moreover
```
```  1308     {moreover assume nz: "n = 0"
```
```  1309       from polymul_degreen[OF norm(5,4), where m="0"]
```
```  1310         polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
```
```  1311       norm(5,6) degree_npolyhCN[OF norm(6)]
```
```  1312     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
```
```  1313     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
```
```  1314     from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
```
```  1315     have ?case   using norm "4.hyps"(6)[OF norm(5,3)]
```
```  1316         "4.hyps"(5)[OF norm(5,4)] nn' nz by simp }
```
```  1317     ultimately have ?case by (cases n) auto}
```
```  1318   ultimately show ?case by blast
```
```  1319 qed simp_all
```
```  1320
```
```  1321 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
```
```  1322   by (induct p rule: degree.induct) auto
```
```  1323
```
```  1324 lemma degree_head[simp]: "degree (head p) = 0"
```
```  1325   by (induct p rule: head.induct) auto
```
```  1326
```
```  1327 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
```
```  1328   by (cases n) simp_all
```
```  1329 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
```
```  1330   by (cases n) simp_all
```
```  1331
```
```  1332 lemma polyadd_different_degree:
```
```  1333   "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degree p \<noteq> degree q\<rbrakk> \<Longrightarrow>
```
```  1334     degree (polyadd p q) = max (degree p) (degree q)"
```
```  1335   using polyadd_different_degreen degree_eq_degreen0 by simp
```
```  1336
```
```  1337 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
```
```  1338   by (induct p arbitrary: n0 rule: polyneg.induct) auto
```
```  1339
```
```  1340 lemma degree_polymul:
```
```  1341   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1342     and np: "isnpolyh p n0"
```
```  1343     and nq: "isnpolyh q n1"
```
```  1344   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
```
```  1345   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
```
```  1346
```
```  1347 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
```
```  1348   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1349
```
```  1350 lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"
```
```  1351   by (induct p arbitrary: n rule: degree.induct) auto
```
```  1352
```
```  1353
```
```  1354 subsection {* Correctness of polynomial pseudo division *}
```
```  1355
```
```  1356 lemma polydivide_aux_properties:
```
```  1357   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1358     and np: "isnpolyh p n0"
```
```  1359     and ns: "isnpolyh s n1"
```
```  1360     and ap: "head p = a"
```
```  1361     and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
```
```  1362   shows "(polydivide_aux a n p k s = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1363           \<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)))"
```
```  1364   using ns
```
```  1365 proof (induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
```
```  1366   case less
```
```  1367   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
```
```  1368   let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1369     \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
```
```  1370   let ?b = "head s"
```
```  1371   let ?p' = "funpow (degree s - n) shift1 p"
```
```  1372   let ?xdn = "funpow (degree s - n) shift1 (1)\<^sub>p"
```
```  1373   let ?akk' = "a ^\<^sub>p (k' - k)"
```
```  1374   note ns = `isnpolyh s n1`
```
```  1375   from np have np0: "isnpolyh p 0"
```
```  1376     using isnpolyh_mono[where n="n0" and n'="0" and p="p"]  by simp
```
```  1377   have np': "isnpolyh ?p' 0" using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def by simp
```
```  1378   have headp': "head ?p' = head p" using funpow_shift1_head[OF np pnz] by simp
```
```  1379   from funpow_shift1_isnpoly[where p="(1)\<^sub>p"] have nxdn: "isnpolyh ?xdn 0" by (simp add: isnpoly_def)
```
```  1380   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
```
```  1381   have nakk':"isnpolyh ?akk' 0" by blast
```
```  1382   { assume sz: "s = 0\<^sub>p"
```
```  1383     hence ?ths using np polydivide_aux.simps
```
```  1384       apply clarsimp
```
```  1385       apply (rule exI[where x="0\<^sub>p"])
```
```  1386       apply simp
```
```  1387       done }
```
```  1388   moreover
```
```  1389   {assume sz: "s \<noteq> 0\<^sub>p"
```
```  1390     {assume dn: "degree s < n"
```
```  1391       hence "?ths" using ns ndp np polydivide_aux.simps
```
```  1392         apply auto
```
```  1393         apply (rule exI[where x="0\<^sub>p"])
```
```  1394         apply simp
```
```  1395         done }
```
```  1396     moreover
```
```  1397     {assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith
```
```  1398       have degsp': "degree s = degree ?p'"
```
```  1399         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp
```
```  1400       {assume ba: "?b = a"
```
```  1401         hence headsp': "head s = head ?p'" using ap headp' by simp
```
```  1402         have nr: "isnpolyh (s -\<^sub>p ?p') 0" using polysub_normh[OF ns np'] by simp
```
```  1403         from degree_polysub_samehead[OF ns np' headsp' degsp']
```
```  1404         have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
```
```  1405         moreover
```
```  1406         {assume deglt:"degree (s -\<^sub>p ?p') < degree s"
```
```  1407           from polydivide_aux.simps sz dn' ba
```
```  1408           have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
```
```  1409             by (simp add: Let_def)
```
```  1410           {assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1411             from less(1)[OF deglt nr, of k k' r]
```
```  1412               trans[OF eq[symmetric] h1]
```
```  1413             have kk': "k \<le> k'" and nr:"\<exists>nr. isnpolyh r nr" and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1414               and q1:"\<exists>q nq. isnpolyh q nq \<and> (a ^\<^sub>p k' - k *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r)" by auto
```
```  1415             from q1 obtain q n1 where nq: "isnpolyh q n1"
```
```  1416               and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"  by blast
```
```  1417             from nr obtain nr where nr': "isnpolyh r nr" by blast
```
```  1418             from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0" by simp
```
```  1419             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
```
```  1420             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
```
```  1421             from polyadd_normh[OF polymul_normh[OF np
```
```  1422               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
```
```  1423             have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0" by simp
```
```  1424             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')) =
```
```  1425               Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
```
```  1426             hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
```
```  1427               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1428               by (simp add: field_simps)
```
```  1429             hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1430               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p *\<^sub>p p)
```
```  1431               + Ipoly bs p * Ipoly bs q + Ipoly bs r"
```
```  1432               by (auto simp only: funpow_shift1_1)
```
```  1433             hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1434               Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p)
```
```  1435               + Ipoly bs q) + Ipoly bs r" by (simp add: field_simps)
```
```  1436             hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1437               Ipoly bs (p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r)" by simp
```
```  1438             with isnpolyh_unique[OF nakks' nqr']
```
```  1439             have "a ^\<^sub>p (k' - k) *\<^sub>p s =
```
```  1440               p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r" by blast
```
```  1441             hence ?qths using nq'
```
```  1442               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q" in exI)
```
```  1443               apply (rule_tac x="0" in exI) by simp
```
```  1444             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"
```
```  1445               by blast } hence ?ths by blast }
```
```  1446         moreover
```
```  1447         {assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
```
```  1448           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}"]
```
```  1449           have " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs s = Ipoly bs ?p'" by simp
```
```  1450           hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)"
```
```  1451             using np nxdn
```
```  1452             apply simp
```
```  1453             apply (simp only: funpow_shift1_1)
```
```  1454             apply simp
```
```  1455             done
```
```  1456           hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]]
```
```  1457             by blast
```
```  1458           {assume h1: "polydivide_aux a n p k s = (k',r)"
```
```  1459             from polydivide_aux.simps sz dn' ba
```
```  1460             have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
```
```  1461               by (simp add: Let_def)
```
```  1462             also have "\<dots> = (k,0\<^sub>p)" using polydivide_aux.simps spz by simp
```
```  1463             finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
```
```  1464             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
```
```  1465               polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
```
```  1466               apply auto
```
```  1467               apply (rule exI[where x="?xdn"])
```
```  1468               apply (auto simp add: polymul_commute[of p])
```
```  1469               done} }
```
```  1470         ultimately have ?ths by blast }
```
```  1471       moreover
```
```  1472       {assume ba: "?b \<noteq> a"
```
```  1473         from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1474           polymul_normh[OF head_isnpolyh[OF ns] np']]
```
```  1475         have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by(simp add: min_def)
```
```  1476         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
```
```  1477           using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
```
```  1478             polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
```
```  1479             funpow_shift1_nz[OF pnz] by simp_all
```
```  1480         from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
```
```  1481           polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1482         have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
```
```  1483           using head_head[OF ns] funpow_shift1_head[OF np pnz]
```
```  1484             polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
```
```  1485           by (simp add: ap)
```
```  1486         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
```
```  1487           head_nz[OF np] pnz sz ap[symmetric]
```
```  1488           funpow_shift1_nz[OF pnz, where n="degree s - n"]
```
```  1489           polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"]  head_nz[OF ns]
```
```  1490           ndp dn
```
```  1491         have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p') "
```
```  1492           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
```
```  1493         {assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
```
```  1494           from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns] polymul_normh[OF head_isnpolyh[OF ns]np']]
```
```  1495           ap have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by simp
```
```  1496           {assume h1:"polydivide_aux a n p k s = (k', r)"
```
```  1497             from h1 polydivide_aux.simps sz dn' ba
```
```  1498             have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
```
```  1499               by (simp add: Let_def)
```
```  1500             with less(1)[OF dth nasbp', of "Suc k" k' r]
```
```  1501             obtain q nq nr where kk': "Suc k \<le> k'" and nr: "isnpolyh r nr" and nq: "isnpolyh q nq"
```
```  1502               and dr: "degree r = 0 \<or> degree r < degree p"
```
```  1503               and qr: "a ^\<^sub>p (k' - Suc k) *\<^sub>p ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = p *\<^sub>p q +\<^sub>p r" by auto
```
```  1504             from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith
```
```  1505             {fix bs:: "'a::{field_char_0, field_inverse_zero} list"
```
```  1506
```
```  1507             from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
```
```  1508             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)"
```
```  1509               by simp
```
```  1510             hence "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s =
```
```  1511               Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
```
```  1512               by (simp add: field_simps)
```
```  1513             hence "Ipoly bs a ^ (k' - k)  * Ipoly bs s =
```
```  1514               Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
```
```  1515               by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1516             hence "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1517               Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
```
```  1518               by (simp add: field_simps)}
```
```  1519             hence ieq:"\<forall>(bs :: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
```
```  1520               Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)" by auto
```
```  1521             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
```
```  1522             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]]
```
```  1523             have nqw: "isnpolyh ?q 0" by simp
```
```  1524             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]]
```
```  1525             have asth: "(a ^\<^sub>p (k' - k) *\<^sub>p s) = p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r" by blast
```
```  1526             from dr kk' nr h1 asth nqw have ?ths apply simp
```
```  1527               apply (rule conjI)
```
```  1528               apply (rule exI[where x="nr"], simp)
```
```  1529               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
```
```  1530               apply (rule exI[where x="0"], simp)
```
```  1531               done}
```
```  1532           hence ?ths by blast }
```
```  1533         moreover
```
```  1534         {assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
```
```  1535           {fix bs :: "'a::{field_char_0, field_inverse_zero} list"
```
```  1536             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
```
```  1537           have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'" by simp
```
```  1538           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
```
```  1539             by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
```
```  1540           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" by simp
```
```  1541         }
```
```  1542         hence hth: "\<forall> (bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a*\<^sub>p s) =
```
```  1543             Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
```
```  1544           from hth
```
```  1545           have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
```
```  1546             using isnpolyh_unique[where ?'a = "'a::{field_char_0, field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
```
```  1547                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
```
```  1548               simplified ap] by simp
```
```  1549           {assume h1: "polydivide_aux a n p k s = (k', r)"
```
```  1550           from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
```
```  1551           have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
```
```  1552           with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
```
```  1553             polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
```
```  1554           have ?ths
```
```  1555             apply (clarsimp simp add: Let_def)
```
```  1556             apply (rule exI[where x="?b *\<^sub>p ?xdn"])
```
```  1557             apply simp
```
```  1558             apply (rule exI[where x="0"], simp)
```
```  1559             done }
```
```  1560         hence ?ths by blast }
```
```  1561         ultimately have ?ths
```
```  1562           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"]
```
```  1563             head_nz[OF np] pnz sz ap[symmetric]
```
```  1564           by (simp add: degree_eq_degreen0[symmetric]) blast }
```
```  1565       ultimately have ?ths by blast
```
```  1566     }
```
```  1567     ultimately have ?ths by blast }
```
```  1568   ultimately show ?ths by blast
```
```  1569 qed
```
```  1570
```
```  1571 lemma polydivide_properties:
```
```  1572   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1573   and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
```
```  1574   shows "(\<exists> k r. polydivide s p = (k,r) \<and> (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p)
```
```  1575   \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
```
```  1576 proof-
```
```  1577   have trv: "head p = head p" "degree p = degree p" by simp_all
```
```  1578   from polydivide_def[where s="s" and p="p"]
```
```  1579   have ex: "\<exists> k r. polydivide s p = (k,r)" by auto
```
```  1580   then obtain k r where kr: "polydivide s p = (k,r)" by blast
```
```  1581   from trans[OF meta_eq_to_obj_eq[OF polydivide_def[where s="s"and p="p"], symmetric] kr]
```
```  1582     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
```
```  1583   have "(degree r = 0 \<or> degree r < degree p) \<and>
```
```  1584    (\<exists>nr. isnpolyh r nr) \<and> (\<exists>q n1. isnpolyh q n1 \<and> head p ^\<^sub>p k - 0 *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)" by blast
```
```  1585   with kr show ?thesis
```
```  1586     apply -
```
```  1587     apply (rule exI[where x="k"])
```
```  1588     apply (rule exI[where x="r"])
```
```  1589     apply simp
```
```  1590     done
```
```  1591 qed
```
```  1592
```
```  1593
```
```  1594 subsection{* More about polypoly and pnormal etc *}
```
```  1595
```
```  1596 definition "isnonconstant p = (\<not> isconstant p)"
```
```  1597
```
```  1598 lemma isnonconstant_pnormal_iff:
```
```  1599   assumes nc: "isnonconstant p"
```
```  1600   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1601 proof
```
```  1602   let ?p = "polypoly bs p"
```
```  1603   assume H: "pnormal ?p"
```
```  1604   have csz: "coefficients p \<noteq> []" using nc by (cases p) auto
```
```  1605
```
```  1606   from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1607     pnormal_last_nonzero[OF H]
```
```  1608   show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)
```
```  1609 next
```
```  1610   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1611   let ?p = "polypoly bs p"
```
```  1612   have csz: "coefficients p \<noteq> []" using nc by (cases p) auto
```
```  1613   hence pz: "?p \<noteq> []" by (simp add: polypoly_def)
```
```  1614   hence lg: "length ?p > 0" by simp
```
```  1615   from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
```
```  1616   have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)
```
```  1617   from pnormal_last_length[OF lg lz] show "pnormal ?p" .
```
```  1618 qed
```
```  1619
```
```  1620 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
```
```  1621   unfolding isnonconstant_def
```
```  1622   apply (cases p)
```
```  1623   apply simp_all
```
```  1624   apply (case_tac nat)
```
```  1625   apply auto
```
```  1626   done
```
```  1627
```
```  1628 lemma isnonconstant_nonconstant:
```
```  1629   assumes inc: "isnonconstant p"
```
```  1630   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
```
```  1631 proof
```
```  1632   let ?p = "polypoly bs p"
```
```  1633   assume nc: "nonconstant ?p"
```
```  1634   from isnonconstant_pnormal_iff[OF inc, of bs] nc
```
```  1635   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast
```
```  1636 next
```
```  1637   let ?p = "polypoly bs p"
```
```  1638   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1639   from isnonconstant_pnormal_iff[OF inc, of bs] h
```
```  1640   have pn: "pnormal ?p" by blast
```
```  1641   {fix x assume H: "?p = [x]"
```
```  1642     from H have "length (coefficients p) = 1" unfolding polypoly_def by auto
```
```  1643     with isnonconstant_coefficients_length[OF inc] have False by arith}
```
```  1644   thus "nonconstant ?p" using pn unfolding nonconstant_def by blast
```
```  1645 qed
```
```  1646
```
```  1647 lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
```
```  1648   apply (induct p)
```
```  1649   apply (simp_all add: pnormal_def)
```
```  1650   apply (case_tac "p = []")
```
```  1651   apply simp_all
```
```  1652   done
```
```  1653
```
```  1654 lemma degree_degree:
```
```  1655   assumes inc: "isnonconstant p"
```
```  1656   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1657 proof
```
```  1658   let  ?p = "polypoly bs p"
```
```  1659   assume H: "degree p = Polynomial_List.degree ?p"
```
```  1660   from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
```
```  1661     unfolding polypoly_def by auto
```
```  1662   from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
```
```  1663   have lg:"length (pnormalize ?p) = length ?p"
```
```  1664     unfolding Polynomial_List.degree_def polypoly_def by simp
```
```  1665   hence "pnormal ?p" using pnormal_length[OF pz] by blast
```
```  1666   with isnonconstant_pnormal_iff[OF inc]
```
```  1667   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast
```
```  1668 next
```
```  1669   let  ?p = "polypoly bs p"
```
```  1670   assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
```
```  1671   with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast
```
```  1672   with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
```
```  1673   show "degree p = Polynomial_List.degree ?p"
```
```  1674     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
```
```  1675 qed
```
```  1676
```
```  1677
```
```  1678 section{* Swaps ; Division by a certain variable *}
```
```  1679
```
```  1680 primrec swap:: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly" where
```
```  1681   "swap n m (C x) = C x"
```
```  1682 | "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"
```
```  1683 | "swap n m (Neg t) = Neg (swap n m t)"
```
```  1684 | "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
```
```  1685 | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
```
```  1686 | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
```
```  1687 | "swap n m (Pw t k) = Pw (swap n m t) k"
```
```  1688 | "swap n m (CN c k p) = CN (swap n m c) (if k = n then m else if k=m then n else k)
```
```  1689   (swap n m p)"
```
```  1690
```
```  1691 lemma swap:
```
```  1692   assumes nbs: "n < length bs"
```
```  1693     and mbs: "m < length bs"
```
```  1694   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  1695 proof (induct t)
```
```  1696   case (Bound k)
```
```  1697   thus ?case using nbs mbs by simp
```
```  1698 next
```
```  1699   case (CN c k p)
```
```  1700   thus ?case using nbs mbs by simp
```
```  1701 qed simp_all
```
```  1702
```
```  1703 lemma swap_swap_id [simp]: "swap n m (swap m n t) = t"
```
```  1704   by (induct t) simp_all
```
```  1705
```
```  1706 lemma swap_commute: "swap n m p = swap m n p"
```
```  1707   by (induct p) simp_all
```
```  1708
```
```  1709 lemma swap_same_id[simp]: "swap n n t = t"
```
```  1710   by (induct t) simp_all
```
```  1711
```
```  1712 definition "swapnorm n m t = polynate (swap n m t)"
```
```  1713
```
```  1714 lemma swapnorm:
```
```  1715   assumes nbs: "n < length bs"
```
```  1716     and mbs: "m < length bs"
```
```  1717   shows "((Ipoly bs (swapnorm n m t) :: 'a\<Colon>{field_char_0, field_inverse_zero})) =
```
```  1718     Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
```
```  1719   using swap[OF assms] swapnorm_def by simp
```
```  1720
```
```  1721 lemma swapnorm_isnpoly [simp]:
```
```  1722   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1723   shows "isnpoly (swapnorm n m p)"
```
```  1724   unfolding swapnorm_def by simp
```
```  1725
```
```  1726 definition "polydivideby n s p =
```
```  1727   (let ss = swapnorm 0 n s ; sp = swapnorm 0 n p ; h = head sp; (k,r) = polydivide ss sp
```
```  1728    in (k,swapnorm 0 n h,swapnorm 0 n r))"
```
```  1729
```
```  1730 lemma swap_nz [simp]: " (swap n m p = 0\<^sub>p) = (p = 0\<^sub>p)"
```
```  1731   by (induct p) simp_all
```
```  1732
```
```  1733 fun isweaknpoly :: "poly \<Rightarrow> bool"
```
```  1734 where
```
```  1735   "isweaknpoly (C c) = True"
```
```  1736 | "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
```
```  1737 | "isweaknpoly p = False"
```
```  1738
```
```  1739 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
```
```  1740   by (induct p arbitrary: n0) auto
```
```  1741
```
```  1742 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
```
```  1743   by (induct p) auto
```
```  1744
```
`  1745 end`