src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
author wenzelm
Mon Feb 21 23:54:53 2011 +0100 (2011-02-21)
changeset 41816 7a55699805dc
parent 41815 9a0cacbcd825
parent 41807 ab5d2d81f9fb
child 41842 d8f76db6a207
permissions -rw-r--r--
merged, resolving spurious conflicts and giving up Reflected_Multivariate_Polynomial.thy from ab5d2d81f9fb;
     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   (* Implementation *)
    12 
    13 subsection{* Datatype of polynomial expressions *} 
    14 
    15 datatype poly = C Num| Bound nat| Add poly poly|Sub poly poly
    16   | Mul poly poly| Neg poly| Pw poly nat| CN poly nat poly
    17 
    18 abbreviation poly_0 :: "poly" ("0\<^sub>p") where "0\<^sub>p \<equiv> C (0\<^sub>N)"
    19 abbreviation poly_p :: "int \<Rightarrow> poly" ("_\<^sub>p") where "i\<^sub>p \<equiv> C (i\<^sub>N)"
    20 
    21 subsection{* Boundedness, substitution and all that *}
    22 primrec polysize:: "poly \<Rightarrow> nat" where
    23   "polysize (C c) = 1"
    24 | "polysize (Bound n) = 1"
    25 | "polysize (Neg p) = 1 + polysize p"
    26 | "polysize (Add p q) = 1 + polysize p + polysize q"
    27 | "polysize (Sub p q) = 1 + polysize p + polysize q"
    28 | "polysize (Mul p q) = 1 + polysize p + polysize q"
    29 | "polysize (Pw p n) = 1 + polysize p"
    30 | "polysize (CN c n p) = 4 + polysize c + polysize p"
    31 
    32 primrec polybound0:: "poly \<Rightarrow> bool" (* a poly is INDEPENDENT of Bound 0 *) where
    33   "polybound0 (C c) = True"
    34 | "polybound0 (Bound n) = (n>0)"
    35 | "polybound0 (Neg a) = polybound0 a"
    36 | "polybound0 (Add a b) = (polybound0 a \<and> polybound0 b)"
    37 | "polybound0 (Sub a b) = (polybound0 a \<and> polybound0 b)" 
    38 | "polybound0 (Mul a b) = (polybound0 a \<and> polybound0 b)"
    39 | "polybound0 (Pw p n) = (polybound0 p)"
    40 | "polybound0 (CN c n p) = (n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p)"
    41 
    42 primrec polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" (* substitute a poly into a poly for Bound 0 *) where
    43   "polysubst0 t (C c) = (C c)"
    44 | "polysubst0 t (Bound n) = (if n=0 then t else Bound n)"
    45 | "polysubst0 t (Neg a) = Neg (polysubst0 t a)"
    46 | "polysubst0 t (Add a b) = Add (polysubst0 t a) (polysubst0 t b)"
    47 | "polysubst0 t (Sub a b) = Sub (polysubst0 t a) (polysubst0 t b)" 
    48 | "polysubst0 t (Mul a b) = Mul (polysubst0 t a) (polysubst0 t b)"
    49 | "polysubst0 t (Pw p n) = Pw (polysubst0 t p) n"
    50 | "polysubst0 t (CN c n p) = (if n=0 then Add (polysubst0 t c) (Mul t (polysubst0 t p))
    51                              else CN (polysubst0 t c) n (polysubst0 t p))"
    52 
    53 fun decrpoly:: "poly \<Rightarrow> poly" 
    54 where
    55   "decrpoly (Bound n) = Bound (n - 1)"
    56 | "decrpoly (Neg a) = Neg (decrpoly a)"
    57 | "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"
    58 | "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
    59 | "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
    60 | "decrpoly (Pw p n) = Pw (decrpoly p) n"
    61 | "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
    62 | "decrpoly a = a"
    63 
    64 subsection{* Degrees and heads and coefficients *}
    65 
    66 fun degree:: "poly \<Rightarrow> nat"
    67 where
    68   "degree (CN c 0 p) = 1 + degree p"
    69 | "degree p = 0"
    70 
    71 fun head:: "poly \<Rightarrow> poly"
    72 where
    73   "head (CN c 0 p) = head p"
    74 | "head p = p"
    75 
    76 (* More general notions of degree and head *)
    77 fun degreen:: "poly \<Rightarrow> nat \<Rightarrow> nat"
    78 where
    79   "degreen (CN c n p) = (\<lambda>m. if n=m then 1 + degreen p n else 0)"
    80  |"degreen p = (\<lambda>m. 0)"
    81 
    82 fun headn:: "poly \<Rightarrow> nat \<Rightarrow> poly"
    83 where
    84   "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
    85 | "headn p = (\<lambda>m. p)"
    86 
    87 fun coefficients:: "poly \<Rightarrow> poly list"
    88 where
    89   "coefficients (CN c 0 p) = c#(coefficients p)"
    90 | "coefficients p = [p]"
    91 
    92 fun isconstant:: "poly \<Rightarrow> bool"
    93 where
    94   "isconstant (CN c 0 p) = False"
    95 | "isconstant p = True"
    96 
    97 fun behead:: "poly \<Rightarrow> poly"
    98 where
    99   "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
   100 | "behead p = 0\<^sub>p"
   101 
   102 fun headconst:: "poly \<Rightarrow> Num"
   103 where
   104   "headconst (CN c n p) = headconst p"
   105 | "headconst (C n) = n"
   106 
   107 subsection{* Operations for normalization *}
   108 
   109 
   110 declare if_cong[fundef_cong del]
   111 declare let_cong[fundef_cong del]
   112 
   113 fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
   114 where
   115   "polyadd (C c) (C c') = C (c+\<^sub>Nc')"
   116 |  "polyadd (C c) (CN c' n' p') = CN (polyadd (C c) c') n' p'"
   117 | "polyadd (CN c n p) (C c') = CN (polyadd c (C c')) n p"
   118 | "polyadd (CN c n p) (CN c' n' p') =
   119     (if n < n' then CN (polyadd c (CN c' n' p')) n p
   120      else if n'<n then CN (polyadd (CN c n p) c') n' p'
   121      else (let cc' = polyadd c c' ; 
   122                pp' = polyadd p p'
   123            in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
   124 | "polyadd a b = Add a b"
   125 
   126 
   127 fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
   128 where
   129   "polyneg (C c) = C (~\<^sub>N c)"
   130 | "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
   131 | "polyneg a = Neg a"
   132 
   133 definition polysub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
   134 where
   135   "p -\<^sub>p q = polyadd p (polyneg q)"
   136 
   137 fun polymul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
   138 where
   139   "polymul (C c) (C c') = C (c*\<^sub>Nc')"
   140 | "polymul (C c) (CN c' n' p') = 
   141       (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul (C c) c') n' (polymul (C c) p'))"
   142 | "polymul (CN c n p) (C c') = 
   143       (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul c (C c')) n (polymul p (C c')))"
   144 | "polymul (CN c n p) (CN c' n' p') = 
   145   (if n<n' then CN (polymul c (CN c' n' p')) n (polymul p (CN c' n' p'))
   146   else if n' < n 
   147   then CN (polymul (CN c n p) c') n' (polymul (CN c n p) p')
   148   else polyadd (polymul (CN c n p) c') (CN 0\<^sub>p n' (polymul (CN c n p) p')))"
   149 | "polymul a b = Mul a b"
   150 
   151 declare if_cong[fundef_cong]
   152 declare let_cong[fundef_cong]
   153 
   154 fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
   155 where
   156   "polypow 0 = (\<lambda>p. 1\<^sub>p)"
   157 | "polypow n = (\<lambda>p. let q = polypow (n div 2) p ; d = polymul q q in 
   158                     if even n then d else polymul p d)"
   159 
   160 abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
   161   where "a ^\<^sub>p k \<equiv> polypow k a"
   162 
   163 function polynate :: "poly \<Rightarrow> poly"
   164 where
   165   "polynate (Bound n) = CN 0\<^sub>p n 1\<^sub>p"
   166 | "polynate (Add p q) = (polynate p +\<^sub>p polynate q)"
   167 | "polynate (Sub p q) = (polynate p -\<^sub>p polynate q)"
   168 | "polynate (Mul p q) = (polynate p *\<^sub>p polynate q)"
   169 | "polynate (Neg p) = (~\<^sub>p (polynate p))"
   170 | "polynate (Pw p n) = ((polynate p) ^\<^sub>p n)"
   171 | "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
   172 | "polynate (C c) = C (normNum c)"
   173 by pat_completeness auto
   174 termination by (relation "measure polysize") auto
   175 
   176 fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly" where
   177   "poly_cmul y (C x) = C (y *\<^sub>N x)"
   178 | "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
   179 | "poly_cmul y p = C y *\<^sub>p p"
   180 
   181 definition monic :: "poly \<Rightarrow> (poly \<times> bool)" where
   182   "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))"
   183 
   184 subsection{* Pseudo-division *}
   185 
   186 definition shift1 :: "poly \<Rightarrow> poly" where
   187   "shift1 p \<equiv> CN 0\<^sub>p 0 p"
   188 
   189 abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)" where
   190   "funpow \<equiv> compow"
   191 
   192 partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
   193   where
   194   "polydivide_aux a n p k s = 
   195   (if s = 0\<^sub>p then (k,s)
   196   else (let b = head s; m = degree s in
   197   (if m < n then (k,s) else 
   198   (let p'= funpow (m - n) shift1 p in 
   199   (if a = b then polydivide_aux a n p k (s -\<^sub>p p') 
   200   else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))))))"
   201 
   202 definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)" where
   203   "polydivide s p \<equiv> polydivide_aux (head p) (degree p) p 0 s"
   204 
   205 fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly" where
   206   "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
   207 | "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
   208 
   209 fun poly_deriv :: "poly \<Rightarrow> poly" where
   210   "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
   211 | "poly_deriv p = 0\<^sub>p"
   212 
   213   (* Verification *)
   214 lemma nth_pos2[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
   215 using Nat.gr0_conv_Suc
   216 by clarsimp
   217 
   218 subsection{* Semantics of the polynomial representation *}
   219 
   220 primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0, field_inverse_zero, power}" where
   221   "Ipoly bs (C c) = INum c"
   222 | "Ipoly bs (Bound n) = bs!n"
   223 | "Ipoly bs (Neg a) = - Ipoly bs a"
   224 | "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
   225 | "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
   226 | "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
   227 | "Ipoly bs (Pw t n) = (Ipoly bs t) ^ n"
   228 | "Ipoly bs (CN c n p) = (Ipoly bs c) + (bs!n)*(Ipoly bs p)"
   229 
   230 abbreviation
   231   Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0, field_inverse_zero, power}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
   232   where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"
   233 
   234 lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i" 
   235   by (simp add: INum_def)
   236 lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j" 
   237   by (simp  add: INum_def)
   238 
   239 lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
   240 
   241 subsection {* Normal form and normalization *}
   242 
   243 fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
   244 where
   245   "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
   246 | "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))"
   247 | "isnpolyh p = (\<lambda>k. False)"
   248 
   249 lemma isnpolyh_mono: "\<lbrakk>n' \<le> n ; isnpolyh p n\<rbrakk> \<Longrightarrow> isnpolyh p n'"
   250 by (induct p rule: isnpolyh.induct, auto)
   251 
   252 definition isnpoly :: "poly \<Rightarrow> bool" where
   253   "isnpoly p \<equiv> isnpolyh p 0"
   254 
   255 text{* polyadd preserves normal forms *}
   256 
   257 lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> 
   258       \<Longrightarrow> isnpolyh (polyadd p q) (min n0 n1)"
   259 proof(induct p q arbitrary: n0 n1 rule: polyadd.induct)
   260   case (2 ab c' n' p' n0 n1)
   261   from 2 have  th1: "isnpolyh (C ab) (Suc n')" by simp 
   262   from 2(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
   263   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
   264   with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^sub>p c') (Suc n')" by simp
   265   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp 
   266   thus ?case using 2 th3 by simp
   267 next
   268   case (3 c' n' p' ab n1 n0)
   269   from 3 have  th1: "isnpolyh (C ab) (Suc n')" by simp 
   270   from 3(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
   271   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
   272   with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (Suc n')" by simp
   273   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp 
   274   thus ?case using 3 th3 by simp
   275 next
   276   case (4 c n p c' n' p' n0 n1)
   277   hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
   278   from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all 
   279   from 4 have ngen0: "n \<ge> n0" by simp
   280   from 4 have n'gen1: "n' \<ge> n1" by simp 
   281   have "n < n' \<or> n' < n \<or> n = n'" by auto
   282   moreover {assume eq: "n = n'"
   283     with "4.hyps"(3)[OF nc nc'] 
   284     have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
   285     hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
   286       using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
   287     from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
   288     have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
   289     from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}
   290   moreover {assume lt: "n < n'"
   291     have "min n0 n1 \<le> n0" by simp
   292     with 4 lt have th1:"min n0 n1 \<le> n" by auto 
   293     from 4 have th21: "isnpolyh c (Suc n)" by simp
   294     from 4 have th22: "isnpolyh (CN c' n' p') n'" by simp
   295     from lt have th23: "min (Suc n) n' = Suc n" by arith
   296     from "4.hyps"(1)[OF th21 th22]
   297     have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)" using th23 by simp
   298     with 4 lt th1 have ?case by simp } 
   299   moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
   300     have "min n0 n1 \<le> n1"  by simp
   301     with 4 gt have th1:"min n0 n1 \<le> n'" by auto
   302     from 4 have th21: "isnpolyh c' (Suc n')" by simp_all
   303     from 4 have th22: "isnpolyh (CN c n p) n" by simp
   304     from gt have th23: "min n (Suc n') = Suc n'" by arith
   305     from "4.hyps"(2)[OF th22 th21]
   306     have "isnpolyh (polyadd (CN c n p) c') (Suc n')" using th23 by simp
   307     with 4 gt th1 have ?case by simp}
   308       ultimately show ?case by blast
   309 qed auto
   310 
   311 lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
   312 by (induct p q rule: polyadd.induct, auto simp add: Let_def field_simps right_distrib[symmetric] simp del: right_distrib)
   313 
   314 lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd p q)"
   315   using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
   316 
   317 text{* The degree of addition and other general lemmas needed for the normal form of polymul *}
   318 
   319 lemma polyadd_different_degreen: 
   320   "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow> 
   321   degreen (polyadd p q) m = max (degreen p m) (degreen q m)"
   322 proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
   323   case (4 c n p c' n' p' m n0 n1)
   324   have "n' = n \<or> n < n' \<or> n' < n" by arith
   325   thus ?case
   326   proof (elim disjE)
   327     assume [simp]: "n' = n"
   328     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
   329     show ?thesis by (auto simp: Let_def)
   330   next
   331     assume "n < n'"
   332     with 4 show ?thesis by auto
   333   next
   334     assume "n' < n"
   335     with 4 show ?thesis by auto
   336   qed
   337 qed auto
   338 
   339 lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
   340   by (induct p arbitrary: n rule: headn.induct, auto)
   341 lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
   342   by (induct p arbitrary: n rule: degree.induct, auto)
   343 lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
   344   by (induct p arbitrary: n rule: degreen.induct, auto)
   345 
   346 lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
   347   by (induct p arbitrary: n rule: degree.induct, auto)
   348 
   349 lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
   350   using degree_isnpolyh_Suc by auto
   351 lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
   352   using degreen_0 by auto
   353 
   354 
   355 lemma degreen_polyadd:
   356   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1"
   357   shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
   358   using np nq m
   359 proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
   360   case (2 c c' n' p' n0 n1) thus ?case  by (cases n', simp_all)
   361 next
   362   case (3 c n p c' n0 n1) thus ?case by (cases n, auto)
   363 next
   364   case (4 c n p c' n' p' n0 n1 m) 
   365   have "n' = n \<or> n < n' \<or> n' < n" by arith
   366   thus ?case
   367   proof (elim disjE)
   368     assume [simp]: "n' = n"
   369     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
   370     show ?thesis by (auto simp: Let_def)
   371   qed simp_all
   372 qed auto
   373 
   374 lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd p q = C c\<rbrakk> 
   375   \<Longrightarrow> degreen p m = degreen q m"
   376 proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
   377   case (4 c n p c' n' p' m n0 n1 x) 
   378   {assume nn': "n' < n" hence ?case using 4 by simp}
   379   moreover 
   380   {assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
   381     moreover {assume "n < n'" with 4 have ?case by simp }
   382     moreover {assume eq: "n = n'" hence ?case using 4 
   383         apply (cases "p +\<^sub>p p' = 0\<^sub>p")
   384         apply (auto simp add: Let_def)
   385         by blast
   386       }
   387     ultimately have ?case by blast}
   388   ultimately show ?case by blast
   389 qed simp_all
   390 
   391 lemma polymul_properties:
   392   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   393   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> min n0 n1"
   394   shows "isnpolyh (p *\<^sub>p q) (min n0 n1)" 
   395   and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)" 
   396   and "degreen (p *\<^sub>p q) m = (if (p = 0\<^sub>p \<or> q = 0\<^sub>p) then 0 
   397                              else degreen p m + degreen q m)"
   398   using np nq m
   399 proof(induct p q arbitrary: n0 n1 m rule: polymul.induct)
   400   case (2 c c' n' p') 
   401   { case (1 n0 n1) 
   402     with "2.hyps"(4-6)[of n' n' n']
   403       and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']
   404     show ?case by (auto simp add: min_def)
   405   next
   406     case (2 n0 n1) thus ?case by auto 
   407   next
   408     case (3 n0 n1) thus ?case  using "2.hyps" by auto } 
   409 next
   410   case (3 c n p c')
   411   { case (1 n0 n1) 
   412     with "3.hyps"(4-6)[of n n n]
   413       "3.hyps"(1-3)[of "Suc n" "Suc n" n]
   414     show ?case by (auto simp add: min_def)
   415   next
   416     case (2 n0 n1) thus ?case by auto
   417   next
   418     case (3 n0 n1) thus ?case  using "3.hyps" by auto } 
   419 next
   420   case (4 c n p c' n' p')
   421   let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
   422     {
   423       case (1 n0 n1)
   424       hence cnp: "isnpolyh ?cnp n" and cnp': "isnpolyh ?cnp' n'"
   425         and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)" 
   426         and np': "isnpolyh p' n'" and nc': "isnpolyh c' (Suc n')"
   427         and nn0: "n \<ge> n0" and nn1:"n' \<ge> n1"
   428         by simp_all
   429       { assume "n < n'"
   430         with "4.hyps"(4-5)[OF np cnp', of n]
   431           "4.hyps"(1)[OF nc cnp', of n] nn0 cnp
   432         have ?case by (simp add: min_def)
   433       } moreover {
   434         assume "n' < n"
   435         with "4.hyps"(16-17)[OF cnp np', of "n'"]
   436           "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
   437         have ?case
   438           by (cases "Suc n' = n", simp_all add: min_def)
   439       } moreover {
   440         assume "n' = n"
   441         with "4.hyps"(16-17)[OF cnp np', of n]
   442           "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0
   443         have ?case
   444           apply (auto intro!: polyadd_normh)
   445           apply (simp_all add: min_def isnpolyh_mono[OF nn0])
   446           done
   447       }
   448       ultimately show ?case by arith
   449     next
   450       fix n0 n1 m
   451       assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1"
   452       and m: "m \<le> min n0 n1"
   453       let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
   454       let ?d1 = "degreen ?cnp m"
   455       let ?d2 = "degreen ?cnp' m"
   456       let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
   457       have "n'<n \<or> n < n' \<or> n' = n" by auto
   458       moreover 
   459       {assume "n' < n \<or> n < n'"
   460         with "4.hyps"(3,6,18) np np' m 
   461         have ?eq by auto }
   462       moreover
   463       {assume nn': "n' = n" hence nn:"\<not> n' < n \<and> \<not> n < n'" by arith
   464         from "4.hyps"(16,18)[of n n' n]
   465           "4.hyps"(13,14)[of n "Suc n'" n]
   466           np np' nn'
   467         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
   468           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
   469           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)" 
   470           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
   471         {assume mn: "m = n" 
   472           from "4.hyps"(17,18)[OF norm(1,4), of n]
   473             "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
   474           have degs:  "degreen (?cnp *\<^sub>p c') n = 
   475             (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"
   476             "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n" by (simp_all add: min_def)
   477           from degs norm
   478           have th1: "degreen(?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" by simp
   479           hence neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
   480             by simp
   481           have nmin: "n \<le> min n n" by (simp add: min_def)
   482           from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
   483           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 
   484           from "4.hyps"(16-18)[OF norm(1,4), of n]
   485             "4.hyps"(13-15)[OF norm(1,2), of n]
   486             mn norm m nn' deg
   487           have ?eq by simp}
   488         moreover
   489         {assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
   490           from nn' m np have max1: "m \<le> max n n"  by simp 
   491           hence min1: "m \<le> min n n" by simp     
   492           hence min2: "m \<le> min n (Suc n)" by simp
   493           from "4.hyps"(16-18)[OF norm(1,4) min1]
   494             "4.hyps"(13-15)[OF norm(1,2) min2]
   495             degreen_polyadd[OF norm(3,6) max1]
   496 
   497           have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m 
   498             \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
   499             using mn nn' np np' by simp
   500           with "4.hyps"(16-18)[OF norm(1,4) min1]
   501             "4.hyps"(13-15)[OF norm(1,2) min2]
   502             degreen_0[OF norm(3) mn']
   503           have ?eq using nn' mn np np' by clarsimp}
   504         ultimately have ?eq by blast}
   505       ultimately show ?eq by blast}
   506     { case (2 n0 n1)
   507       hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1" 
   508         and m: "m \<le> min n0 n1" by simp_all
   509       hence mn: "m \<le> n" by simp
   510       let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
   511       {assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
   512         hence nn: "\<not>n' < n \<and> \<not> n<n'" by simp
   513         from "4.hyps"(16-18) [of n n n]
   514           "4.hyps"(13-15)[of n "Suc n" n]
   515           np np' C(2) mn
   516         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
   517           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
   518           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)" 
   519           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" 
   520           "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
   521             "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
   522           by (simp_all add: min_def)
   523             
   524           from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
   525           have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" 
   526             using norm by simp
   527         from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"]  degneq
   528         have "False" by simp }
   529       thus ?case using "4.hyps" by clarsimp}
   530 qed auto
   531 
   532 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"
   533 by(induct p q rule: polymul.induct, auto simp add: field_simps)
   534 
   535 lemma polymul_normh: 
   536     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   537   shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
   538   using polymul_properties(1)  by blast
   539 lemma polymul_eq0_iff: 
   540   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   541   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) "
   542   using polymul_properties(2)  by blast
   543 lemma polymul_degreen:  
   544   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   545   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow> 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)"
   546   using polymul_properties(3) by blast
   547 lemma polymul_norm:   
   548   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   549   shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul p q)"
   550   using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
   551 
   552 lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
   553   by (induct p arbitrary: n0 rule: headconst.induct, auto)
   554 
   555 lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
   556   by (induct p arbitrary: n0, auto)
   557 
   558 lemma monic_eqI: assumes np: "isnpolyh p n0" 
   559   shows "INum (headconst p) * Ipoly bs (fst (monic p)) = (Ipoly bs p ::'a::{field_char_0, field_inverse_zero, power})"
   560   unfolding monic_def Let_def
   561 proof(cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
   562   let ?h = "headconst p"
   563   assume pz: "p \<noteq> 0\<^sub>p"
   564   {assume hz: "INum ?h = (0::'a)"
   565     from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
   566     from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
   567     with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
   568   thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast
   569 qed
   570 
   571 
   572 text{* polyneg is a negation and preserves normal forms *}
   573 
   574 lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
   575 by (induct p rule: polyneg.induct, auto)
   576 
   577 lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"
   578   by (induct p arbitrary: n rule: polyneg.induct, auto simp add: Nneg_def)
   579 lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
   580   by (induct p arbitrary: n0 rule: polyneg.induct, auto)
   581 lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "
   582 by (induct p rule: polyneg.induct, auto simp add: polyneg0)
   583 
   584 lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
   585   using isnpoly_def polyneg_normh by simp
   586 
   587 
   588 text{* polysub is a substraction and preserves normal forms *}
   589 
   590 lemma polysub[simp]: "Ipoly bs (polysub p q) = (Ipoly bs p) - (Ipoly bs q)"
   591 by (simp add: polysub_def polyneg polyadd)
   592 lemma polysub_normh: "\<And> n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
   593 by (simp add: polysub_def polyneg_normh polyadd_normh)
   594 
   595 lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub p q)"
   596   using polyadd_norm polyneg_norm by (simp add: polysub_def) 
   597 lemma polysub_same_0[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   598   shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p"
   599 unfolding polysub_def split_def fst_conv snd_conv
   600 by (induct p arbitrary: n0,auto simp add: Let_def Nsub0[simplified Nsub_def])
   601 
   602 lemma polysub_0: 
   603   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   604   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"
   605   unfolding polysub_def split_def fst_conv snd_conv
   606   by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
   607   (auto simp: Nsub0[simplified Nsub_def] Let_def)
   608 
   609 text{* polypow is a power function and preserves normal forms *}
   610 
   611 lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{field_char_0, field_inverse_zero})) ^ n"
   612 proof(induct n rule: polypow.induct)
   613   case 1 thus ?case by simp
   614 next
   615   case (2 n)
   616   let ?q = "polypow ((Suc n) div 2) p"
   617   let ?d = "polymul ?q ?q"
   618   have "odd (Suc n) \<or> even (Suc n)" by simp
   619   moreover 
   620   {assume odd: "odd (Suc n)"
   621     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" by arith
   622     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)" by (simp add: Let_def)
   623     also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"
   624       using "2.hyps" by simp
   625     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
   626       apply (simp only: power_add power_one_right) by simp
   627     also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat)))))"
   628       by (simp only: th)
   629     finally have ?case 
   630     using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
   631   moreover 
   632   {assume even: "even (Suc n)"
   633     have th: "(Suc (Suc (0\<Colon>nat))) * (Suc n div Suc (Suc (0\<Colon>nat))) = Suc n div 2 + Suc n div 2" by arith
   634     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
   635     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
   636       using "2.hyps" apply (simp only: power_add) by simp
   637     finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}
   638   ultimately show ?case by blast
   639 qed
   640 
   641 lemma polypow_normh: 
   642     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   643   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
   644 proof (induct k arbitrary: n rule: polypow.induct)
   645   case (2 k n)
   646   let ?q = "polypow (Suc k div 2) p"
   647   let ?d = "polymul ?q ?q"
   648   from 2 have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
   649   from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
   650   from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n" by simp
   651   from dn on show ?case by (simp add: Let_def)
   652 qed auto 
   653 
   654 lemma polypow_norm:   
   655   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   656   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
   657   by (simp add: polypow_normh isnpoly_def)
   658 
   659 text{* Finally the whole normalization *}
   660 
   661 lemma polynate[simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0, field_inverse_zero})"
   662 by (induct p rule:polynate.induct, auto)
   663 
   664 lemma polynate_norm[simp]: 
   665   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   666   shows "isnpoly (polynate p)"
   667   by (induct p rule: polynate.induct, simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm) (simp_all add: isnpoly_def)
   668 
   669 text{* shift1 *}
   670 
   671 
   672 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
   673 by (simp add: shift1_def polymul)
   674 
   675 lemma shift1_isnpoly: 
   676   assumes pn: "isnpoly p" and pnz: "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p) "
   677   using pn pnz by (simp add: shift1_def isnpoly_def )
   678 
   679 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
   680   by (simp add: shift1_def)
   681 lemma funpow_shift1_isnpoly: 
   682   "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
   683   by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
   684 
   685 lemma funpow_isnpolyh: 
   686   assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n "and np: "isnpolyh p n"
   687   shows "isnpolyh (funpow k f p) n"
   688   using f np by (induct k arbitrary: p, auto)
   689 
   690 lemma funpow_shift1: "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (Mul (Pw (Bound 0) n) p)"
   691   by (induct n arbitrary: p, simp_all add: shift1_isnpoly shift1 power_Suc )
   692 
   693 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
   694   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
   695 
   696 lemma funpow_shift1_1: 
   697   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (funpow n shift1 1\<^sub>p *\<^sub>p p)"
   698   by (simp add: funpow_shift1)
   699 
   700 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
   701 by (induct p  arbitrary: n0 rule: poly_cmul.induct, auto simp add: field_simps)
   702 
   703 lemma behead:
   704   assumes np: "isnpolyh p n"
   705   shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = (Ipoly bs p :: 'a :: {field_char_0, field_inverse_zero})"
   706   using np
   707 proof (induct p arbitrary: n rule: behead.induct)
   708   case (1 c p n) hence pn: "isnpolyh p n" by simp
   709   from 1(1)[OF pn] 
   710   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" . 
   711   then show ?case using "1.hyps" apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
   712     by (simp_all add: th[symmetric] field_simps power_Suc)
   713 qed (auto simp add: Let_def)
   714 
   715 lemma behead_isnpolyh:
   716   assumes np: "isnpolyh p n" shows "isnpolyh (behead p) n"
   717   using np by (induct p rule: behead.induct, auto simp add: Let_def isnpolyh_mono)
   718 
   719 subsection{* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
   720 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
   721 proof(induct p arbitrary: n rule: poly.induct, auto)
   722   case (goal1 c n p n')
   723   hence "n = Suc (n - 1)" by simp
   724   hence "isnpolyh p (Suc (n - 1))"  using `isnpolyh p n` by simp
   725   with goal1(2) show ?case by simp
   726 qed
   727 
   728 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
   729 by (induct p arbitrary: n0 rule: isconstant.induct, auto simp add: isnpolyh_polybound0)
   730 
   731 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" by (induct p, auto)
   732 
   733 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
   734   apply (induct p arbitrary: n0, auto)
   735   apply (atomize)
   736   apply (erule_tac x = "Suc nat" in allE)
   737   apply auto
   738   done
   739 
   740 lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
   741  by (induct p  arbitrary: n0 rule: head.induct, auto intro: isnpolyh_polybound0)
   742 
   743 lemma polybound0_I:
   744   assumes nb: "polybound0 a"
   745   shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
   746 using nb
   747 by (induct a rule: poly.induct) auto 
   748 lemma polysubst0_I:
   749   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
   750   by (induct t) simp_all
   751 
   752 lemma polysubst0_I':
   753   assumes nb: "polybound0 a"
   754   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t"
   755   by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
   756 
   757 lemma decrpoly: assumes nb: "polybound0 t"
   758   shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
   759   using nb by (induct t rule: decrpoly.induct, simp_all)
   760 
   761 lemma polysubst0_polybound0: assumes nb: "polybound0 t"
   762   shows "polybound0 (polysubst0 t a)"
   763 using nb by (induct a rule: poly.induct, auto)
   764 
   765 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
   766   by (induct p arbitrary: n rule: degree.induct, auto simp add: isnpolyh_polybound0)
   767 
   768 primrec maxindex :: "poly \<Rightarrow> nat" where
   769   "maxindex (Bound n) = n + 1"
   770 | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
   771 | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
   772 | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
   773 | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
   774 | "maxindex (Neg p) = maxindex p"
   775 | "maxindex (Pw p n) = maxindex p"
   776 | "maxindex (C x) = 0"
   777 
   778 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool" where
   779   "wf_bs bs p = (length bs \<ge> maxindex p)"
   780 
   781 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
   782 proof(induct p rule: coefficients.induct)
   783   case (1 c p) 
   784   show ?case 
   785   proof
   786     fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"
   787     hence "x = c \<or> x \<in> set (coefficients p)" by simp
   788     moreover 
   789     {assume "x = c" hence "wf_bs bs x" using "1.prems"  unfolding wf_bs_def by simp}
   790     moreover 
   791     {assume H: "x \<in> set (coefficients p)" 
   792       from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp
   793       with "1.hyps" H have "wf_bs bs x" by blast }
   794     ultimately  show "wf_bs bs x" by blast
   795   qed
   796 qed simp_all
   797 
   798 lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
   799 by (induct p rule: coefficients.induct, auto)
   800 
   801 lemma wf_bs_I: "wf_bs bs p ==> Ipoly (bs@bs') p = Ipoly bs p"
   802   unfolding wf_bs_def by (induct p, auto simp add: nth_append)
   803 
   804 lemma take_maxindex_wf: assumes wf: "wf_bs bs p" 
   805   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
   806 proof-
   807   let ?ip = "maxindex p"
   808   let ?tbs = "take ?ip bs"
   809   from wf have "length ?tbs = ?ip" unfolding wf_bs_def by simp
   810   hence wf': "wf_bs ?tbs p" unfolding wf_bs_def by  simp
   811   have eq: "bs = ?tbs @ (drop ?ip bs)" by simp
   812   from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis using eq by simp
   813 qed
   814 
   815 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
   816   by (induct p, auto)
   817 
   818 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
   819   unfolding wf_bs_def by simp
   820 
   821 lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
   822   unfolding wf_bs_def by simp
   823 
   824 
   825 
   826 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
   827 by(induct p rule: coefficients.induct, auto simp add: wf_bs_def)
   828 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
   829   by (induct p rule: coefficients.induct, simp_all)
   830 
   831 
   832 lemma coefficients_head: "last (coefficients p) = head p"
   833   by (induct p rule: coefficients.induct, auto)
   834 
   835 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
   836   unfolding wf_bs_def by (induct p rule: decrpoly.induct, auto)
   837 
   838 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
   839   apply (rule exI[where x="replicate (n - length xs) z"])
   840   by simp
   841 lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
   842 by (cases p, auto) (case_tac "nat", simp_all)
   843 
   844 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
   845   unfolding wf_bs_def 
   846   apply (induct p q rule: polyadd.induct)
   847   apply (auto simp add: Let_def)
   848   done
   849 
   850 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
   851   unfolding wf_bs_def 
   852   apply (induct p q arbitrary: bs rule: polymul.induct) 
   853   apply (simp_all add: wf_bs_polyadd)
   854   apply clarsimp
   855   apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
   856   apply auto
   857   done
   858 
   859 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
   860   unfolding wf_bs_def by (induct p rule: polyneg.induct, auto)
   861 
   862 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
   863   unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
   864 
   865 subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}
   866 
   867 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
   868 definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)"
   869 definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"
   870 
   871 lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0"
   872 proof (induct p arbitrary: n0 rule: coefficients.induct)
   873   case (1 c p n0)
   874   have cp: "isnpolyh (CN c 0 p) n0" by fact
   875   hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
   876     by (auto simp add: isnpolyh_mono[where n'=0])
   877   from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp 
   878 qed auto
   879 
   880 lemma coefficients_isconst:
   881   "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
   882   by (induct p arbitrary: n rule: coefficients.induct, 
   883     auto simp add: isnpolyh_Suc_const)
   884 
   885 lemma polypoly_polypoly':
   886   assumes np: "isnpolyh p n0"
   887   shows "polypoly (x#bs) p = polypoly' bs p"
   888 proof-
   889   let ?cf = "set (coefficients p)"
   890   from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
   891   {fix q assume q: "q \<in> ?cf"
   892     from q cn_norm have th: "isnpolyh q n0" by blast
   893     from coefficients_isconst[OF np] q have "isconstant q" by blast
   894     with isconstant_polybound0[OF th] have "polybound0 q" by blast}
   895   hence "\<forall>q \<in> ?cf. polybound0 q" ..
   896   hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"
   897     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
   898     by auto
   899   
   900   thus ?thesis unfolding polypoly_def polypoly'_def by simp 
   901 qed
   902 
   903 lemma polypoly_poly:
   904   assumes np: "isnpolyh p n0" shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
   905   using np 
   906 by (induct p arbitrary: n0 bs rule: coefficients.induct, auto simp add: polypoly_def)
   907 
   908 lemma polypoly'_poly: 
   909   assumes np: "isnpolyh p n0" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
   910   using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
   911 
   912 
   913 lemma polypoly_poly_polybound0:
   914   assumes np: "isnpolyh p n0" and nb: "polybound0 p"
   915   shows "polypoly bs p = [Ipoly bs p]"
   916   using np nb unfolding polypoly_def 
   917   by (cases p, auto, case_tac nat, auto)
   918 
   919 lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0" 
   920   by (induct p rule: head.induct, auto)
   921 
   922 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
   923   by (cases p,auto)
   924 
   925 lemma head_eq_headn0: "head p = headn p 0"
   926   by (induct p rule: head.induct, simp_all)
   927 
   928 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
   929   by (simp add: head_eq_headn0)
   930 
   931 lemma isnpolyh_zero_iff: 
   932   assumes nq: "isnpolyh p n0" 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})"
   933   shows "p = 0\<^sub>p"
   934 using nq eq
   935 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
   936   case less
   937   note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
   938   {assume nz: "maxindex p = 0"
   939     then obtain c where "p = C c" using np by (cases p, auto)
   940     with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}
   941   moreover
   942   {assume nz: "maxindex p \<noteq> 0"
   943     let ?h = "head p"
   944     let ?hd = "decrpoly ?h"
   945     let ?ihd = "maxindex ?hd"
   946     from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h" 
   947       by simp_all
   948     hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast
   949     
   950     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
   951     have mihn: "maxindex ?h \<le> maxindex p" by auto
   952     with decr_maxindex[OF h(2)] nz  have ihd_lt_n: "?ihd < maxindex p" by auto
   953     {fix bs:: "'a list"  assume bs: "wf_bs bs ?hd"
   954       let ?ts = "take ?ihd bs"
   955       let ?rs = "drop ?ihd bs"
   956       have ts: "wf_bs ?ts ?hd" using bs unfolding wf_bs_def by simp
   957       have bs_ts_eq: "?ts@ ?rs = bs" by simp
   958       from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x#?ts) ?h" by simp
   959       from ihd_lt_n have "ALL x. length (x#?ts) \<le> maxindex p" by simp
   960       with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = maxindex p" by blast
   961       hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" unfolding wf_bs_def by simp
   962       with zp have "\<forall> x. Ipoly ((x#?ts) @ xs) p = 0" by blast
   963       hence "\<forall> x. Ipoly (x#(?ts @ xs)) p = 0" by simp
   964       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
   965       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"  by simp
   966       hence "poly (polypoly' (?ts @ xs) p) = poly []" by (auto intro: ext) 
   967       hence "\<forall> c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
   968         using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
   969       with coefficients_head[of p, symmetric]
   970       have th0: "Ipoly (?ts @ xs) ?hd = 0" by simp
   971       from bs have wf'': "wf_bs ?ts ?hd" unfolding wf_bs_def by simp
   972       with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0" by simp
   973       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by simp }
   974     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast
   975     
   976     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast
   977     hence "?h = 0\<^sub>p" by simp
   978     with head_nz[OF np] have "p = 0\<^sub>p" by simp}
   979   ultimately show "p = 0\<^sub>p" by blast
   980 qed
   981 
   982 lemma isnpolyh_unique:  
   983   assumes np:"isnpolyh p n0" and nq: "isnpolyh q n1"
   984   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"
   985 proof(auto)
   986   assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a)= \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
   987   hence "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" by simp
   988   hence "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" 
   989     using wf_bs_polysub[where p=p and q=q] by auto
   990   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq]
   991   show "p = q" by blast
   992 qed
   993 
   994 
   995 text{* consequences of unicity on the algorithms for polynomial normalization *}
   996 
   997 lemma polyadd_commute:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   998   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" shows "p +\<^sub>p q = q +\<^sub>p p"
   999   using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]] by simp
  1000 
  1001 lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp
  1002 lemma one_normh: "isnpolyh 1\<^sub>p n" by simp
  1003 lemma polyadd_0[simp]: 
  1004   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1005   and np: "isnpolyh p n0" shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>p p = p"
  1006   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np] 
  1007     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
  1008 
  1009 lemma polymul_1[simp]: 
  1010     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1011   and np: "isnpolyh p n0" shows "p *\<^sub>p 1\<^sub>p = p" and "1\<^sub>p *\<^sub>p p = p"
  1012   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np] 
  1013     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
  1014 lemma polymul_0[simp]: 
  1015   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1016   and np: "isnpolyh p n0" shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p" and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
  1017   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh] 
  1018     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
  1019 
  1020 lemma polymul_commute: 
  1021     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1022   and np:"isnpolyh p n0" and nq: "isnpolyh q n1"
  1023   shows "p *\<^sub>p q = q *\<^sub>p p"
  1024 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}"] by simp
  1025 
  1026 declare polyneg_polyneg[simp]
  1027   
  1028 lemma isnpolyh_polynate_id[simp]: 
  1029   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1030   and np:"isnpolyh p n0" shows "polynate p = p"
  1031   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}"] by simp
  1032 
  1033 lemma polynate_idempotent[simp]: 
  1034     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1035   shows "polynate (polynate p) = polynate p"
  1036   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
  1037 
  1038 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
  1039   unfolding poly_nate_def polypoly'_def ..
  1040 lemma poly_nate_poly: shows "poly (poly_nate bs p) = (\<lambda>x:: 'a ::{field_char_0, field_inverse_zero}. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
  1041   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
  1042   unfolding poly_nate_polypoly' by (auto intro: ext)
  1043 
  1044 subsection{* heads, degrees and all that *}
  1045 lemma degree_eq_degreen0: "degree p = degreen p 0"
  1046   by (induct p rule: degree.induct, simp_all)
  1047 
  1048 lemma degree_polyneg: assumes n: "isnpolyh p n"
  1049   shows "degree (polyneg p) = degree p"
  1050   using n
  1051   by (induct p arbitrary: n rule: polyneg.induct, simp_all) (case_tac na, auto)
  1052 
  1053 lemma degree_polyadd:
  1054   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
  1055   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
  1056 using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
  1057 
  1058 
  1059 lemma degree_polysub: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
  1060   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
  1061 proof-
  1062   from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp
  1063   from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])
  1064 qed
  1065 
  1066 lemma degree_polysub_samehead: 
  1067   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1068   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q" 
  1069   and d: "degree p = degree q"
  1070   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
  1071 unfolding polysub_def split_def fst_conv snd_conv
  1072 using np nq h d
  1073 proof(induct p q rule:polyadd.induct)
  1074   case (1 c c') thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def]) 
  1075 next
  1076   case (2 c c' n' p') 
  1077   from 2 have "degree (C c) = degree (CN c' n' p')" by simp
  1078   hence nz:"n' > 0" by (cases n', auto)
  1079   hence "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
  1080   with 2 show ?case by simp
  1081 next
  1082   case (3 c n p c') 
  1083   hence "degree (C c') = degree (CN c n p)" by simp
  1084   hence nz:"n > 0" by (cases n, auto)
  1085   hence "head (CN c n p) = CN c n p" by (cases n, auto)
  1086   with 3 show ?case by simp
  1087 next
  1088   case (4 c n p c' n' p')
  1089   hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1" 
  1090     "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+
  1091   hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all  
  1092   hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0" 
  1093     using H(1-2) degree_polyneg by auto
  1094   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"  by simp+
  1095   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"  by simp
  1096   from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto
  1097   have "n = n' \<or> n < n' \<or> n > n'" by arith
  1098   moreover
  1099   {assume nn': "n = n'"
  1100     have "n = 0 \<or> n >0" by arith
  1101     moreover {assume nz: "n = 0" hence ?case using 4 nn' by (auto simp add: Let_def degcmc')}
  1102     moreover {assume nz: "n > 0"
  1103       with nn' H(3) have  cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
  1104       hence ?case 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] using nn' 4 by (simp add: Let_def)}
  1105     ultimately have ?case by blast}
  1106   moreover
  1107   {assume nn': "n < n'" hence n'p: "n' > 0" by simp 
  1108     hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n', simp_all)
  1109     have degcnp': "degree (CN c' n' p') = 0" and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')" using 4 nn' by (cases n', simp_all)
  1110     hence "n > 0" by (cases n, simp_all)
  1111     hence headcnp: "head (CN c n p) = CN c n p" by (cases n, auto)
  1112     from H(3) headcnp headcnp' nn' have ?case by auto}
  1113   moreover
  1114   {assume nn': "n > n'"  hence np: "n > 0" by simp 
  1115     hence headcnp:"head (CN c n p) = CN c n p"  by (cases n, simp_all)
  1116     from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
  1117     from np have degcnp: "degree (CN c n p) = 0" by (cases n, simp_all)
  1118     with degcnpeq have "n' > 0" by (cases n', simp_all)
  1119     hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
  1120     from H(3) headcnp headcnp' nn' have ?case by auto}
  1121   ultimately show ?case  by blast
  1122 qed auto
  1123  
  1124 lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
  1125 by (induct p arbitrary: n0 rule: head.induct, simp_all add: shift1_def)
  1126 
  1127 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
  1128 proof(induct k arbitrary: n0 p)
  1129   case (Suc k n0 p) hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
  1130   with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
  1131     and "head (shift1 p) = head p" by (simp_all add: shift1_head) 
  1132   thus ?case by (simp add: funpow_swap1)
  1133 qed auto  
  1134 
  1135 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
  1136   by (simp add: shift1_def)
  1137 lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
  1138   by (induct k arbitrary: p, auto simp add: shift1_degree)
  1139 
  1140 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
  1141   by (induct n arbitrary: p, simp_all add: funpow_def)
  1142 
  1143 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
  1144   by (induct p arbitrary: n rule: degree.induct, auto)
  1145 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
  1146   by (induct p arbitrary: n rule: degreen.induct, auto)
  1147 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
  1148   by (induct p arbitrary: n rule: degree.induct, auto)
  1149 lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
  1150   by (induct p rule: head.induct, auto)
  1151 
  1152 lemma polyadd_eq_const_degree: 
  1153   "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd p q = C c\<rbrakk> \<Longrightarrow> degree p = degree q" 
  1154   using polyadd_eq_const_degreen degree_eq_degreen0 by simp
  1155 
  1156 lemma polyadd_head: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
  1157   and deg: "degree p \<noteq> degree q"
  1158   shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
  1159 using np nq deg
  1160 apply(induct p q arbitrary: n0 n1 rule: polyadd.induct,simp_all)
  1161 apply (case_tac n', simp, simp)
  1162 apply (case_tac n, simp, simp)
  1163 apply (case_tac n, case_tac n', simp add: Let_def)
  1164 apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p")
  1165 apply (auto simp add: polyadd_eq_const_degree)
  1166 apply (metis head_nz)
  1167 apply (metis head_nz)
  1168 apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
  1169 by (metis degree.simps(9) gr0_conv_Suc nat_less_le order_le_less_trans)
  1170 
  1171 lemma polymul_head_polyeq: 
  1172    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1173   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"
  1174 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
  1175   case (2 c c' n' p' n0 n1)
  1176   hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"  by (simp_all add: head_isnpolyh)
  1177   thus ?case using 2 by (cases n', auto) 
  1178 next 
  1179   case (3 c n p c' n0 n1) 
  1180   hence "isnpolyh (head (CN c n p)) n0" "isnormNum c'"  by (simp_all add: head_isnpolyh)
  1181   thus ?case using 3 by (cases n, auto)
  1182 next
  1183   case (4 c n p c' n' p' n0 n1)
  1184   hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
  1185     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
  1186     by simp_all
  1187   have "n < n' \<or> n' < n \<or> n = n'" by arith
  1188   moreover 
  1189   {assume nn': "n < n'" hence ?case 
  1190       using norm 
  1191     "4.hyps"(2)[OF norm(1,6)]
  1192     "4.hyps"(1)[OF norm(2,6)] by (simp, cases n, simp,cases n', simp_all)}
  1193   moreover {assume nn': "n'< n"
  1194     hence ?case using norm "4.hyps"(6) [OF norm(5,3)]
  1195       "4.hyps"(5)[OF norm(5,4)] 
  1196       by (simp,cases n',simp,cases n,auto)}
  1197   moreover {assume nn': "n' = n"
  1198     from nn' polymul_normh[OF norm(5,4)] 
  1199     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
  1200     from nn' polymul_normh[OF norm(5,3)] norm 
  1201     have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
  1202     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
  1203     have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp 
  1204     from polyadd_normh[OF ncnpc' ncnpp0'] 
  1205     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" 
  1206       by (simp add: min_def)
  1207     {assume np: "n > 0"
  1208       with nn' head_isnpolyh_Suc'[OF np nth]
  1209         head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
  1210       have ?case by simp}
  1211     moreover
  1212     {moreover assume nz: "n = 0"
  1213       from polymul_degreen[OF norm(5,4), where m="0"]
  1214         polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
  1215       norm(5,6) degree_npolyhCN[OF norm(6)]
  1216     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
  1217     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
  1218     from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
  1219     have ?case   using norm "4.hyps"(6)[OF norm(5,3)]
  1220         "4.hyps"(5)[OF norm(5,4)] nn' nz by simp }
  1221     ultimately have ?case by (cases n) auto} 
  1222   ultimately show ?case by blast
  1223 qed simp_all
  1224 
  1225 lemma degree_coefficients: "degree p = length (coefficients p) - 1"
  1226   by(induct p rule: degree.induct, auto)
  1227 
  1228 lemma degree_head[simp]: "degree (head p) = 0"
  1229   by (induct p rule: head.induct, auto)
  1230 
  1231 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
  1232   by (cases n, simp_all)
  1233 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
  1234   by (cases n, simp_all)
  1235 
  1236 lemma polyadd_different_degree: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degree p \<noteq> degree q\<rbrakk> \<Longrightarrow> degree (polyadd p q) = max (degree p) (degree q)"
  1237   using polyadd_different_degreen degree_eq_degreen0 by simp
  1238 
  1239 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
  1240   by (induct p arbitrary: n0 rule: polyneg.induct, auto)
  1241 
  1242 lemma degree_polymul:
  1243   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1244   and np: "isnpolyh p n0" and nq: "isnpolyh q n1"
  1245   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
  1246   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
  1247 
  1248 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
  1249   by (induct p arbitrary: n rule: degree.induct, auto)
  1250 
  1251 lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"
  1252   by (induct p arbitrary: n rule: degree.induct, auto)
  1253 
  1254 subsection {* Correctness of polynomial pseudo division *}
  1255 
  1256 lemma polydivide_aux_properties:
  1257   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1258   and np: "isnpolyh p n0" and ns: "isnpolyh s n1"
  1259   and ap: "head p = a" and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
  1260   shows "(polydivide_aux a n p k s = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p) 
  1261           \<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)))"
  1262   using ns
  1263 proof(induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
  1264   case less
  1265   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
  1266   let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) 
  1267     \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
  1268   let ?b = "head s"
  1269   let ?p' = "funpow (degree s - n) shift1 p"
  1270   let ?xdn = "funpow (degree s - n) shift1 1\<^sub>p"
  1271   let ?akk' = "a ^\<^sub>p (k' - k)"
  1272   note ns = `isnpolyh s n1`
  1273   from np have np0: "isnpolyh p 0" 
  1274     using isnpolyh_mono[where n="n0" and n'="0" and p="p"]  by simp
  1275   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
  1276   have headp': "head ?p' = head p" using funpow_shift1_head[OF np pnz] by simp
  1277   from funpow_shift1_isnpoly[where p="1\<^sub>p"] have nxdn: "isnpolyh ?xdn 0" by (simp add: isnpoly_def)
  1278   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap 
  1279   have nakk':"isnpolyh ?akk' 0" by blast
  1280   {assume sz: "s = 0\<^sub>p"
  1281    hence ?ths using np polydivide_aux.simps apply clarsimp by (rule exI[where x="0\<^sub>p"], simp) }
  1282   moreover
  1283   {assume sz: "s \<noteq> 0\<^sub>p"
  1284     {assume dn: "degree s < n"
  1285       hence "?ths" using ns ndp np polydivide_aux.simps by auto (rule exI[where x="0\<^sub>p"],simp) }
  1286     moreover 
  1287     {assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith
  1288       have degsp': "degree s = degree ?p'" 
  1289         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp
  1290       {assume ba: "?b = a"
  1291         hence headsp': "head s = head ?p'" using ap headp' by simp
  1292         have nr: "isnpolyh (s -\<^sub>p ?p') 0" using polysub_normh[OF ns np'] by simp
  1293         from degree_polysub_samehead[OF ns np' headsp' degsp']
  1294         have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
  1295         moreover 
  1296         {assume deglt:"degree (s -\<^sub>p ?p') < degree s"
  1297           from polydivide_aux.simps sz dn' ba
  1298           have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
  1299             by (simp add: Let_def)
  1300           {assume h1: "polydivide_aux a n p k s = (k', r)"
  1301             from less(1)[OF deglt nr, of k k' r]
  1302               trans[OF eq[symmetric] h1]
  1303             have kk': "k \<le> k'" and nr:"\<exists>nr. isnpolyh r nr" and dr: "degree r = 0 \<or> degree r < degree p"
  1304               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
  1305             from q1 obtain q n1 where nq: "isnpolyh q n1" 
  1306               and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"  by blast
  1307             from nr obtain nr where nr': "isnpolyh r nr" by blast
  1308             from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0" by simp
  1309             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
  1310             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
  1311             from polyadd_normh[OF polymul_normh[OF np 
  1312               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
  1313             have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0" by simp 
  1314             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')) = 
  1315               Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
  1316             hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) = 
  1317               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r" 
  1318               by (simp add: field_simps)
  1319             hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 
  1320               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p *\<^sub>p p) 
  1321               + Ipoly bs p * Ipoly bs q + Ipoly bs r"
  1322               by (auto simp only: funpow_shift1_1) 
  1323             hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 
  1324               Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p) 
  1325               + Ipoly bs q) + Ipoly bs r" by (simp add: field_simps)
  1326             hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 
  1327               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
  1328             with isnpolyh_unique[OF nakks' nqr']
  1329             have "a ^\<^sub>p (k' - k) *\<^sub>p s = 
  1330               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
  1331             hence ?qths using nq'
  1332               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 1\<^sub>p) +\<^sub>p q" in exI)
  1333               apply (rule_tac x="0" in exI) by simp
  1334             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"
  1335               by blast } hence ?ths by blast }
  1336         moreover 
  1337         {assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
  1338           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}"]
  1339           have " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs s = Ipoly bs ?p'" by simp
  1340           hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)" using np nxdn apply simp
  1341             by (simp only: funpow_shift1_1) simp
  1342           hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]] by blast
  1343           {assume h1: "polydivide_aux a n p k s = (k',r)"
  1344             from polydivide_aux.simps sz dn' ba
  1345             have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
  1346               by (simp add: Let_def)
  1347             also have "\<dots> = (k,0\<^sub>p)" using polydivide_aux.simps spz by simp
  1348             finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
  1349             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
  1350               polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
  1351               apply auto
  1352               apply (rule exI[where x="?xdn"])        
  1353               apply (auto simp add: polymul_commute[of p])
  1354               done} }
  1355         ultimately have ?ths by blast }
  1356       moreover
  1357       {assume ba: "?b \<noteq> a"
  1358         from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns] 
  1359           polymul_normh[OF head_isnpolyh[OF ns] np']]
  1360         have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by(simp add: min_def)
  1361         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
  1362           using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns] 
  1363             polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
  1364             funpow_shift1_nz[OF pnz] by simp_all
  1365         from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
  1366           polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
  1367         have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')" 
  1368           using head_head[OF ns] funpow_shift1_head[OF np pnz]
  1369             polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
  1370           by (simp add: ap)
  1371         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
  1372           head_nz[OF np] pnz sz ap[symmetric]
  1373           funpow_shift1_nz[OF pnz, where n="degree s - n"]
  1374           polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"]  head_nz[OF ns]
  1375           ndp dn
  1376         have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p') "
  1377           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
  1378         {assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
  1379           from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns] polymul_normh[OF head_isnpolyh[OF ns]np']]
  1380           ap have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by simp
  1381           {assume h1:"polydivide_aux a n p k s = (k', r)"
  1382             from h1 polydivide_aux.simps sz dn' ba
  1383             have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
  1384               by (simp add: Let_def)
  1385             with less(1)[OF dth nasbp', of "Suc k" k' r]
  1386             obtain q nq nr where kk': "Suc k \<le> k'" and nr: "isnpolyh r nr" and nq: "isnpolyh q nq" 
  1387               and dr: "degree r = 0 \<or> degree r < degree p"
  1388               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
  1389             from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith
  1390             {fix bs:: "'a::{field_char_0, field_inverse_zero} list"
  1391               
  1392             from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
  1393             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)" by simp
  1394             hence "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s = Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
  1395               by (simp add: field_simps power_Suc)
  1396             hence "Ipoly bs a ^ (k' - k)  * Ipoly bs s = Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
  1397               by (simp add:kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
  1398             hence "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
  1399               by (simp add: field_simps)}
  1400             hence ieq:"\<forall>(bs :: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 
  1401               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 
  1402             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
  1403             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]]
  1404             have nqw: "isnpolyh ?q 0" by simp
  1405             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]]
  1406             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
  1407             from dr kk' nr h1 asth nqw have ?ths apply simp
  1408               apply (rule conjI)
  1409               apply (rule exI[where x="nr"], simp)
  1410               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
  1411               apply (rule exI[where x="0"], simp)
  1412               done}
  1413           hence ?ths by blast }
  1414         moreover 
  1415         {assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
  1416           {fix bs :: "'a::{field_char_0, field_inverse_zero} list"
  1417             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
  1418           have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'" by simp
  1419           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p" 
  1420             by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
  1421           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" by simp
  1422         }
  1423         hence hth: "\<forall> (bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
  1424           from hth
  1425           have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)" 
  1426             using isnpolyh_unique[where ?'a = "'a::{field_char_0, field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns] 
  1427                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
  1428               simplified ap] by simp
  1429           {assume h1: "polydivide_aux a n p k s = (k', r)"
  1430           from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
  1431           have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
  1432           with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
  1433             polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
  1434           have ?ths apply (clarsimp simp add: Let_def)
  1435             apply (rule exI[where x="?b *\<^sub>p ?xdn"]) apply simp
  1436             apply (rule exI[where x="0"], simp)
  1437             done}
  1438         hence ?ths by blast}
  1439         ultimately have ?ths 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"]
  1440           head_nz[OF np] pnz sz ap[symmetric]
  1441           by (simp add: degree_eq_degreen0[symmetric]) blast }
  1442       ultimately have ?ths by blast
  1443     }
  1444     ultimately have ?ths by blast}
  1445   ultimately show ?ths by blast
  1446 qed
  1447 
  1448 lemma polydivide_properties: 
  1449   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1450   and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
  1451   shows "(\<exists> k r. polydivide s p = (k,r) \<and> (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) 
  1452   \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
  1453 proof-
  1454   have trv: "head p = head p" "degree p = degree p" by simp_all
  1455   from polydivide_def[where s="s" and p="p"] 
  1456   have ex: "\<exists> k r. polydivide s p = (k,r)" by auto
  1457   then obtain k r where kr: "polydivide s p = (k,r)" by blast
  1458   from trans[OF meta_eq_to_obj_eq[OF polydivide_def[where s="s"and p="p"], symmetric] kr]
  1459     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
  1460   have "(degree r = 0 \<or> degree r < degree p) \<and>
  1461    (\<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
  1462   with kr show ?thesis 
  1463     apply -
  1464     apply (rule exI[where x="k"])
  1465     apply (rule exI[where x="r"])
  1466     apply simp
  1467     done
  1468 qed
  1469 
  1470 subsection{* More about polypoly and pnormal etc *}
  1471 
  1472 definition "isnonconstant p = (\<not> isconstant p)"
  1473 
  1474 lemma isnonconstant_pnormal_iff: assumes nc: "isnonconstant p" 
  1475   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0" 
  1476 proof
  1477   let ?p = "polypoly bs p"  
  1478   assume H: "pnormal ?p"
  1479   have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
  1480   
  1481   from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]  
  1482     pnormal_last_nonzero[OF H]
  1483   show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)
  1484 next
  1485   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
  1486   let ?p = "polypoly bs p"
  1487   have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
  1488   hence pz: "?p \<noteq> []" by (simp add: polypoly_def) 
  1489   hence lg: "length ?p > 0" by simp
  1490   from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"] 
  1491   have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)
  1492   from pnormal_last_length[OF lg lz] show "pnormal ?p" .
  1493 qed
  1494 
  1495 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
  1496   unfolding isnonconstant_def
  1497   apply (cases p, simp_all)
  1498   apply (case_tac nat, auto)
  1499   done
  1500 lemma isnonconstant_nonconstant: assumes inc: "isnonconstant p"
  1501   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
  1502 proof
  1503   let ?p = "polypoly bs p"
  1504   assume nc: "nonconstant ?p"
  1505   from isnonconstant_pnormal_iff[OF inc, of bs] nc
  1506   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast
  1507 next
  1508   let ?p = "polypoly bs p"
  1509   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
  1510   from isnonconstant_pnormal_iff[OF inc, of bs] h
  1511   have pn: "pnormal ?p" by blast
  1512   {fix x assume H: "?p = [x]" 
  1513     from H have "length (coefficients p) = 1" unfolding polypoly_def by auto
  1514     with isnonconstant_coefficients_length[OF inc] have False by arith}
  1515   thus "nonconstant ?p" using pn unfolding nonconstant_def by blast  
  1516 qed
  1517 
  1518 lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
  1519   unfolding pnormal_def
  1520  apply (induct p)
  1521  apply (simp_all, case_tac "p=[]", simp_all)
  1522  done
  1523 
  1524 lemma degree_degree: assumes inc: "isnonconstant p"
  1525   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
  1526 proof
  1527   let  ?p = "polypoly bs p"
  1528   assume H: "degree p = Polynomial_List.degree ?p"
  1529   from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
  1530     unfolding polypoly_def by auto
  1531   from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
  1532   have lg:"length (pnormalize ?p) = length ?p"
  1533     unfolding Polynomial_List.degree_def polypoly_def by simp
  1534   hence "pnormal ?p" using pnormal_length[OF pz] by blast 
  1535   with isnonconstant_pnormal_iff[OF inc]  
  1536   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast
  1537 next
  1538   let  ?p = "polypoly bs p"  
  1539   assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
  1540   with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast
  1541   with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
  1542   show "degree p = Polynomial_List.degree ?p" 
  1543     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
  1544 qed
  1545 
  1546 section{* Swaps ; Division by a certain variable *}
  1547 primrec swap:: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly" where
  1548   "swap n m (C x) = C x"
  1549 | "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"
  1550 | "swap n m (Neg t) = Neg (swap n m t)"
  1551 | "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
  1552 | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
  1553 | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
  1554 | "swap n m (Pw t k) = Pw (swap n m t) k"
  1555 | "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)
  1556   (swap n m p)"
  1557 
  1558 lemma swap: assumes nbs: "n < length bs" and mbs: "m < length bs"
  1559   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
  1560 proof (induct t)
  1561   case (Bound k) thus ?case using nbs mbs by simp 
  1562 next
  1563   case (CN c k p) thus ?case using nbs mbs by simp 
  1564 qed simp_all
  1565 lemma swap_swap_id[simp]: "swap n m (swap m n t) = t"
  1566   by (induct t,simp_all)
  1567 
  1568 lemma swap_commute: "swap n m p = swap m n p" by (induct p, simp_all)
  1569 
  1570 lemma swap_same_id[simp]: "swap n n t = t"
  1571   by (induct t, simp_all)
  1572 
  1573 definition "swapnorm n m t = polynate (swap n m t)"
  1574 
  1575 lemma swapnorm: assumes nbs: "n < length bs" and mbs: "m < length bs"
  1576   shows "((Ipoly bs (swapnorm n m t) :: 'a\<Colon>{field_char_0, field_inverse_zero})) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
  1577   using swap[OF assms] swapnorm_def by simp
  1578 
  1579 lemma swapnorm_isnpoly[simp]: 
  1580     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1581   shows "isnpoly (swapnorm n m p)"
  1582   unfolding swapnorm_def by simp
  1583 
  1584 definition "polydivideby n s p = 
  1585     (let ss = swapnorm 0 n s ; sp = swapnorm 0 n p ; h = head sp; (k,r) = polydivide ss sp
  1586      in (k,swapnorm 0 n h,swapnorm 0 n r))"
  1587 
  1588 lemma swap_nz [simp]: " (swap n m p = 0\<^sub>p) = (p = 0\<^sub>p)" by (induct p, simp_all)
  1589 
  1590 fun isweaknpoly :: "poly \<Rightarrow> bool"
  1591 where
  1592   "isweaknpoly (C c) = True"
  1593 | "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
  1594 | "isweaknpoly p = False"
  1595 
  1596 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p" 
  1597   by (induct p arbitrary: n0, auto)
  1598 
  1599 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)" 
  1600   by (induct p, auto)
  1601 
  1602 end