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