(*  Title:      HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy

     Author:     Amine Chaieb

 *)

 header {* Implementation and verification of multivariate polynomials *}

 theory Reflected_Multivariate_Polynomial

 imports Complex_Main "~~/src/HOL/Library/Abstract_Rat" Polynomial_List

 begin

   (* Implementation *)

 subsection{* Datatype of polynomial expressions *} 

 datatype poly = C Num| Bound nat| Add poly poly|Sub poly poly

   | Mul poly poly| Neg poly| Pw poly nat| CN poly nat poly

 abbreviation poly_0 :: "poly" ("0\<^sub>p") where "0\<^sub>p \<equiv> C (0\<^sub>N)"

 abbreviation poly_p :: "int \<Rightarrow> poly" ("_\<^sub>p") where "i\<^sub>p \<equiv> C (i\<^sub>N)"

 subsection{* Boundedness, substitution and all that *}

 primrec polysize:: "poly \<Rightarrow> nat" where

   "polysize (C c) = 1"

 | "polysize (Bound n) = 1"

 | "polysize (Neg p) = 1 + polysize p"

 | "polysize (Add p q) = 1 + polysize p + polysize q"

 | "polysize (Sub p q) = 1 + polysize p + polysize q"

 | "polysize (Mul p q) = 1 + polysize p + polysize q"

 | "polysize (Pw p n) = 1 + polysize p"

 | "polysize (CN c n p) = 4 + polysize c + polysize p"

 primrec polybound0:: "poly \<Rightarrow> bool" (* a poly is INDEPENDENT of Bound 0 *) where

   "polybound0 (C c) = True"

 | "polybound0 (Bound n) = (n>0)"

 | "polybound0 (Neg a) = polybound0 a"

 | "polybound0 (Add a b) = (polybound0 a \<and> polybound0 b)"

 | "polybound0 (Sub a b) = (polybound0 a \<and> polybound0 b)" 

 | "polybound0 (Mul a b) = (polybound0 a \<and> polybound0 b)"

 | "polybound0 (Pw p n) = (polybound0 p)"

 | "polybound0 (CN c n p) = (n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p)"

 primrec polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" (* substitute a poly into a poly for Bound 0 *) where

   "polysubst0 t (C c) = (C c)"

 | "polysubst0 t (Bound n) = (if n=0 then t else Bound n)"

 | "polysubst0 t (Neg a) = Neg (polysubst0 t a)"

 | "polysubst0 t (Add a b) = Add (polysubst0 t a) (polysubst0 t b)"

 | "polysubst0 t (Sub a b) = Sub (polysubst0 t a) (polysubst0 t b)" 

 | "polysubst0 t (Mul a b) = Mul (polysubst0 t a) (polysubst0 t b)"

 | "polysubst0 t (Pw p n) = Pw (polysubst0 t p) n"

 | "polysubst0 t (CN c n p) = (if n=0 then Add (polysubst0 t c) (Mul t (polysubst0 t p))

                              else CN (polysubst0 t c) n (polysubst0 t p))"

 fun decrpoly:: "poly \<Rightarrow> poly" 

 where

   "decrpoly (Bound n) = Bound (n - 1)"

 | "decrpoly (Neg a) = Neg (decrpoly a)"

 | "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"

 | "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"

 | "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"

 | "decrpoly (Pw p n) = Pw (decrpoly p) n"

 | "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"

 | "decrpoly a = a"

 subsection{* Degrees and heads and coefficients *}

 fun degree:: "poly \<Rightarrow> nat"

 where

   "degree (CN c 0 p) = 1 + degree p"

 | "degree p = 0"

 fun head:: "poly \<Rightarrow> poly"

 where

   "head (CN c 0 p) = head p"

 | "head p = p"

 (* More general notions of degree and head *)

 fun degreen:: "poly \<Rightarrow> nat \<Rightarrow> nat"

 where

   "degreen (CN c n p) = (\<lambda>m. if n=m then 1 + degreen p n else 0)"

  |"degreen p = (\<lambda>m. 0)"

 fun headn:: "poly \<Rightarrow> nat \<Rightarrow> poly"

 where

   "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"

 | "headn p = (\<lambda>m. p)"

 fun coefficients:: "poly \<Rightarrow> poly list"

 where

   "coefficients (CN c 0 p) = c#(coefficients p)"

 | "coefficients p = [p]"

 fun isconstant:: "poly \<Rightarrow> bool"

 where

   "isconstant (CN c 0 p) = False"

 | "isconstant p = True"

 fun behead:: "poly \<Rightarrow> poly"

 where

   "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"

 | "behead p = 0\<^sub>p"

 fun headconst:: "poly \<Rightarrow> Num"

 where

   "headconst (CN c n p) = headconst p"

 | "headconst (C n) = n"

 subsection{* Operations for normalization *}

 declare if_cong[fundef_cong del]

 declare let_cong[fundef_cong del]

 fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)

 where

   "polyadd (C c) (C c') = C (c+\<^sub>Nc')"

 |  "polyadd (C c) (CN c' n' p') = CN (polyadd (C c) c') n' p'"

 | "polyadd (CN c n p) (C c') = CN (polyadd c (C c')) n p"

 | "polyadd (CN c n p) (CN c' n' p') =

     (if n < n' then CN (polyadd c (CN c' n' p')) n p

      else if n'<n then CN (polyadd (CN c n p) c') n' p'

      else (let cc' = polyadd c c' ; 

                pp' = polyadd p p'

            in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"

 | "polyadd a b = Add a b"

 fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")

 where

   "polyneg (C c) = C (~\<^sub>N c)"

 | "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"

 | "polyneg a = Neg a"

 definition polysub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)

 where

   "p -\<^sub>p q = polyadd p (polyneg q)"

 fun polymul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)

 where

   "polymul (C c) (C c') = C (c*\<^sub>Nc')"

 | "polymul (C c) (CN c' n' p') = 

       (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul (C c) c') n' (polymul (C c) p'))"

 | "polymul (CN c n p) (C c') = 

       (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul c (C c')) n (polymul p (C c')))"

 | "polymul (CN c n p) (CN c' n' p') = 

   (if n<n' then CN (polymul c (CN c' n' p')) n (polymul p (CN c' n' p'))

   else if n' < n 

   then CN (polymul (CN c n p) c') n' (polymul (CN c n p) p')

   else polyadd (polymul (CN c n p) c') (CN 0\<^sub>p n' (polymul (CN c n p) p')))"

 | "polymul a b = Mul a b"

 declare if_cong[fundef_cong]

 declare let_cong[fundef_cong]

 fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"

 where

   "polypow 0 = (\<lambda>p. 1\<^sub>p)"

 | "polypow n = (\<lambda>p. let q = polypow (n div 2) p ; d = polymul q q in 

                     if even n then d else polymul p d)"

 abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)

   where "a ^\<^sub>p k \<equiv> polypow k a"

 function polynate :: "poly \<Rightarrow> poly"

 where

   "polynate (Bound n) = CN 0\<^sub>p n 1\<^sub>p"

 | "polynate (Add p q) = (polynate p +\<^sub>p polynate q)"

 | "polynate (Sub p q) = (polynate p -\<^sub>p polynate q)"

 | "polynate (Mul p q) = (polynate p *\<^sub>p polynate q)"

 | "polynate (Neg p) = (~\<^sub>p (polynate p))"

 | "polynate (Pw p n) = ((polynate p) ^\<^sub>p n)"

 | "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"

 | "polynate (C c) = C (normNum c)"

 by pat_completeness auto

 termination by (relation "measure polysize") auto

 fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly" where

   "poly_cmul y (C x) = C (y *\<^sub>N x)"

 | "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"

 | "poly_cmul y p = C y *\<^sub>p p"

 definition monic :: "poly \<Rightarrow> (poly \<times> bool)" where

   "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))"

 subsection{* Pseudo-division *}

 definition shift1 :: "poly \<Rightarrow> poly" where

   "shift1 p \<equiv> CN 0\<^sub>p 0 p"

 abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)" where

   "funpow \<equiv> compow"

 partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"

   where

   "polydivide_aux a n p k s = 

   (if s = 0\<^sub>p then (k,s)

   else (let b = head s; m = degree s in

   (if m < n then (k,s) else 

   (let p'= funpow (m - n) shift1 p in 

   (if a = b then polydivide_aux a n p k (s -\<^sub>p p') 

   else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))))))"

 definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)" where

   "polydivide s p \<equiv> polydivide_aux (head p) (degree p) p 0 s"

 fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly" where

   "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"

 | "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"

 fun poly_deriv :: "poly \<Rightarrow> poly" where

   "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"

 | "poly_deriv p = 0\<^sub>p"

 subsection{* Semantics of the polynomial representation *}

 primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0, field_inverse_zero, power}" where

   "Ipoly bs (C c) = INum c"

 | "Ipoly bs (Bound n) = bs!n"

 | "Ipoly bs (Neg a) = - Ipoly bs a"

 | "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"

 | "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"

 | "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"

 | "Ipoly bs (Pw t n) = (Ipoly bs t) ^ n"

 | "Ipoly bs (CN c n p) = (Ipoly bs c) + (bs!n)*(Ipoly bs p)"

 abbreviation

   Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0, field_inverse_zero, power}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")

   where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"

 lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i" 

   by (simp add: INum_def)

 lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j" 

   by (simp  add: INum_def)

 lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat

 subsection {* Normal form and normalization *}

 fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"

 where

   "isnpolyh (C c) = (\<lambda>k. isnormNum c)"

 | "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))"

 | "isnpolyh p = (\<lambda>k. False)"

 lemma isnpolyh_mono: "\<lbrakk>n' \<le> n ; isnpolyh p n\<rbrakk> \<Longrightarrow> isnpolyh p n'"

 by (induct p rule: isnpolyh.induct, auto)

 definition isnpoly :: "poly \<Rightarrow> bool" where

   "isnpoly p \<equiv> isnpolyh p 0"

 text{* polyadd preserves normal forms *}

 lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> 

       \<Longrightarrow> isnpolyh (polyadd p q) (min n0 n1)"

 proof(induct p q arbitrary: n0 n1 rule: polyadd.induct)

   case (2 ab c' n' p' n0 n1)

   from 2 have  th1: "isnpolyh (C ab) (Suc n')" by simp 

   from 2(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all

   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp

   with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^sub>p c') (Suc n')" by simp

   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp 

   thus ?case using 2 th3 by simp

 next

   case (3 c' n' p' ab n1 n0)

   from 3 have  th1: "isnpolyh (C ab) (Suc n')" by simp 

   from 3(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all

   with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp

   with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (Suc n')" by simp

   from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp 

   thus ?case using 3 th3 by simp

 next

   case (4 c n p c' n' p' n0 n1)

   hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all

   from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all 

   from 4 have ngen0: "n \<ge> n0" by simp

   from 4 have n'gen1: "n' \<ge> n1" by simp 

   have "n < n' \<or> n' < n \<or> n = n'" by auto

   moreover {assume eq: "n = n'"

     with "4.hyps"(3)[OF nc nc'] 

     have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto

     hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"

       using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto

     from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp

     have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp

     from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}

   moreover {assume lt: "n < n'"

     have "min n0 n1 \<le> n0" by simp

     with 4 lt have th1:"min n0 n1 \<le> n" by auto 

     from 4 have th21: "isnpolyh c (Suc n)" by simp

     from 4 have th22: "isnpolyh (CN c' n' p') n'" by simp

     from lt have th23: "min (Suc n) n' = Suc n" by arith

     from "4.hyps"(1)[OF th21 th22]

     have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)" using th23 by simp

     with 4 lt th1 have ?case by simp } 

   moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp

     have "min n0 n1 \<le> n1"  by simp

     with 4 gt have th1:"min n0 n1 \<le> n'" by auto

     from 4 have th21: "isnpolyh c' (Suc n')" by simp_all

     from 4 have th22: "isnpolyh (CN c n p) n" by simp

     from gt have th23: "min n (Suc n') = Suc n'" by arith

     from "4.hyps"(2)[OF th22 th21]

     have "isnpolyh (polyadd (CN c n p) c') (Suc n')" using th23 by simp

     with 4 gt th1 have ?case by simp}

       ultimately show ?case by blast

 qed auto

 lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"

 by (induct p q rule: polyadd.induct, auto simp add: Let_def field_simps right_distrib[symmetric] simp del: right_distrib)

 lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd p q)"

   using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp

 text{* The degree of addition and other general lemmas needed for the normal form of polymul *}

 lemma polyadd_different_degreen: 

   "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow> 

   degreen (polyadd p q) m = max (degreen p m) (degreen q m)"

 proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)

   case (4 c n p c' n' p' m n0 n1)

   have "n' = n \<or> n < n' \<or> n' < n" by arith

   thus ?case

   proof (elim disjE)

     assume [simp]: "n' = n"

     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)

     show ?thesis by (auto simp: Let_def)

   next

     assume "n < n'"

     with 4 show ?thesis by auto

   next

     assume "n' < n"

     with 4 show ?thesis by auto

   qed

 qed auto

 lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"

   by (induct p arbitrary: n rule: headn.induct, auto)

 lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"

   by (induct p arbitrary: n rule: degree.induct, auto)

 lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"

   by (induct p arbitrary: n rule: degreen.induct, auto)

 lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"

   by (induct p arbitrary: n rule: degree.induct, auto)

 lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"

   using degree_isnpolyh_Suc by auto

 lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"

   using degreen_0 by auto

 lemma degreen_polyadd:

   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1"

   shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"

   using np nq m

 proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)

   case (2 c c' n' p' n0 n1) thus ?case  by (cases n', simp_all)

 next

   case (3 c n p c' n0 n1) thus ?case by (cases n, auto)

 next

   case (4 c n p c' n' p' n0 n1 m) 

   have "n' = n \<or> n < n' \<or> n' < n" by arith

   thus ?case

   proof (elim disjE)

     assume [simp]: "n' = n"

     from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)

     show ?thesis by (auto simp: Let_def)

   qed simp_all

 qed auto

 lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd p q = C c\<rbrakk> 

   \<Longrightarrow> degreen p m = degreen q m"

 proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)

   case (4 c n p c' n' p' m n0 n1 x) 

   {assume nn': "n' < n" hence ?case using 4 by simp}

   moreover 

   {assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith

     moreover {assume "n < n'" with 4 have ?case by simp }

     moreover {assume eq: "n = n'" hence ?case using 4 

         apply (cases "p +\<^sub>p p' = 0\<^sub>p")

         apply (auto simp add: Let_def)

         by blast

       }

     ultimately have ?case by blast}

   ultimately show ?case by blast

 qed simp_all

 lemma polymul_properties:

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> min n0 n1"

   shows "isnpolyh (p *\<^sub>p q) (min n0 n1)" 

   and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)" 

   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)"

   using np nq m

 proof(induct p q arbitrary: n0 n1 m rule: polymul.induct)

   case (2 c c' n' p') 

   { case (1 n0 n1) 

     with "2.hyps"(4-6)[of n' n' n']

       and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']

     show ?case by (auto simp add: min_def)

   next

     case (2 n0 n1) thus ?case by auto 

   next

     case (3 n0 n1) thus ?case  using "2.hyps" by auto } 

 next

   case (3 c n p c')

   { case (1 n0 n1) 

     with "3.hyps"(4-6)[of n n n]

       "3.hyps"(1-3)[of "Suc n" "Suc n" n]

     show ?case by (auto simp add: min_def)

   next

     case (2 n0 n1) thus ?case by auto

   next

     case (3 n0 n1) thus ?case  using "3.hyps" by auto } 

 next

   case (4 c n p c' n' p')

   let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"

     {

       case (1 n0 n1)

       hence cnp: "isnpolyh ?cnp n" and cnp': "isnpolyh ?cnp' n'"

         and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)" 

         and np': "isnpolyh p' n'" and nc': "isnpolyh c' (Suc n')"

         and nn0: "n \<ge> n0" and nn1:"n' \<ge> n1"

         by simp_all

       { assume "n < n'"

         with "4.hyps"(4-5)[OF np cnp', of n]

           "4.hyps"(1)[OF nc cnp', of n] nn0 cnp

         have ?case by (simp add: min_def)

       } moreover {

         assume "n' < n"

         with "4.hyps"(16-17)[OF cnp np', of "n'"]

           "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'

         have ?case

           by (cases "Suc n' = n", simp_all add: min_def)

       } moreover {

         assume "n' = n"

         with "4.hyps"(16-17)[OF cnp np', of n]

           "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0

         have ?case

           apply (auto intro!: polyadd_normh)

           apply (simp_all add: min_def isnpolyh_mono[OF nn0])

           done

       }

       ultimately show ?case by arith

     next

       fix n0 n1 m

       assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1"

       and m: "m \<le> min n0 n1"

       let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"

       let ?d1 = "degreen ?cnp m"

       let ?d2 = "degreen ?cnp' m"

       let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"

       have "n'<n \<or> n < n' \<or> n' = n" by auto

       moreover 

       {assume "n' < n \<or> n < n'"

         with "4.hyps"(3,6,18) np np' m 

         have ?eq by auto }

       moreover

       {assume nn': "n' = n" hence nn:"\<not> n' < n \<and> \<not> n < n'" by arith

         from "4.hyps"(16,18)[of n n' n]

           "4.hyps"(13,14)[of n "Suc n'" n]

           np np' nn'

         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"

           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"

           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)" 

           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)

         {assume mn: "m = n" 

           from "4.hyps"(17,18)[OF norm(1,4), of n]

             "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn

           have degs:  "degreen (?cnp *\<^sub>p c') n = 

             (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"

             "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n" by (simp_all add: min_def)

           from degs norm

           have th1: "degreen(?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" by simp

           hence neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"

             by simp

           have nmin: "n \<le> min n n" by (simp add: min_def)

           from polyadd_different_degreen[OF norm(3,6) neq nmin] th1

           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 

           from "4.hyps"(16-18)[OF norm(1,4), of n]

             "4.hyps"(13-15)[OF norm(1,2), of n]

             mn norm m nn' deg

           have ?eq by simp}

         moreover

         {assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto

           from nn' m np have max1: "m \<le> max n n"  by simp 

           hence min1: "m \<le> min n n" by simp     

           hence min2: "m \<le> min n (Suc n)" by simp

           from "4.hyps"(16-18)[OF norm(1,4) min1]

             "4.hyps"(13-15)[OF norm(1,2) min2]

             degreen_polyadd[OF norm(3,6) max1]

           have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m 

             \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"

             using mn nn' np np' by simp

           with "4.hyps"(16-18)[OF norm(1,4) min1]

             "4.hyps"(13-15)[OF norm(1,2) min2]

             degreen_0[OF norm(3) mn']

           have ?eq using nn' mn np np' by clarsimp}

         ultimately have ?eq by blast}

       ultimately show ?eq by blast}

     { case (2 n0 n1)

       hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1" 

         and m: "m \<le> min n0 n1" by simp_all

       hence mn: "m \<le> n" by simp

       let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"

       {assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"

         hence nn: "\<not>n' < n \<and> \<not> n<n'" by simp

         from "4.hyps"(16-18) [of n n n]

           "4.hyps"(13-15)[of n "Suc n" n]

           np np' C(2) mn

         have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"

           "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"

           "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)" 

           "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" 

           "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"

             "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"

           by (simp_all add: min_def)

           from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp

           have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" 

             using norm by simp

         from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"]  degneq

         have "False" by simp }

       thus ?case using "4.hyps" by clarsimp}

 qed auto

 lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"

 by(induct p q rule: polymul.induct, auto simp add: field_simps)

 lemma polymul_normh: 

     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"

   using polymul_properties(1)  by blast

 lemma polymul_eq0_iff: 

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   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) "

   using polymul_properties(2)  by blast

 lemma polymul_degreen:  

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   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)"

   using polymul_properties(3) by blast

 lemma polymul_norm:   

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul p q)"

   using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp

 lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"

   by (induct p arbitrary: n0 rule: headconst.induct, auto)

 lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"

   by (induct p arbitrary: n0, auto)

 lemma monic_eqI: assumes np: "isnpolyh p n0" 

   shows "INum (headconst p) * Ipoly bs (fst (monic p)) = (Ipoly bs p ::'a::{field_char_0, field_inverse_zero, power})"

   unfolding monic_def Let_def

 proof(cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])

   let ?h = "headconst p"

   assume pz: "p \<noteq> 0\<^sub>p"

   {assume hz: "INum ?h = (0::'a)"

     from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all

     from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp

     with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}

   thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast

 qed

 text{* polyneg is a negation and preserves normal forms *}

 lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"

 by (induct p rule: polyneg.induct, auto)

 lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"

   by (induct p arbitrary: n rule: polyneg.induct, auto simp add: Nneg_def)

 lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"

   by (induct p arbitrary: n0 rule: polyneg.induct, auto)

 lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "

 by (induct p rule: polyneg.induct, auto simp add: polyneg0)

 lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"

   using isnpoly_def polyneg_normh by simp

 text{* polysub is a substraction and preserves normal forms *}

 lemma polysub[simp]: "Ipoly bs (polysub p q) = (Ipoly bs p) - (Ipoly bs q)"

 by (simp add: polysub_def polyneg polyadd)

 lemma polysub_normh: "\<And> n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"

 by (simp add: polysub_def polyneg_normh polyadd_normh)

 lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub p q)"

   using polyadd_norm polyneg_norm by (simp add: polysub_def) 

 lemma polysub_same_0[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p"

 unfolding polysub_def split_def fst_conv snd_conv

 by (induct p arbitrary: n0,auto simp add: Let_def Nsub0[simplified Nsub_def])

 lemma polysub_0: 

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"

   unfolding polysub_def split_def fst_conv snd_conv

   by (induct p q arbitrary: n0 n1 rule:polyadd.induct)

   (auto simp: Nsub0[simplified Nsub_def] Let_def)

 text{* polypow is a power function and preserves normal forms *}

 lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{field_char_0, field_inverse_zero})) ^ n"

 proof(induct n rule: polypow.induct)

   case 1 thus ?case by simp

 next

   case (2 n)

   let ?q = "polypow ((Suc n) div 2) p"

   let ?d = "polymul ?q ?q"

   have "odd (Suc n) \<or> even (Suc n)" by simp

   moreover 

   {assume odd: "odd (Suc n)"

     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

     from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)" by (simp add: Let_def)

     also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"

       using "2.hyps" by simp

     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"

       apply (simp only: power_add power_one_right) by simp

     also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat)))))"

       by (simp only: th)

     finally have ?case 

     using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }

   moreover 

   {assume even: "even (Suc n)"

     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

     from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)

     also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"

       using "2.hyps" apply (simp only: power_add) by simp

     finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}

   ultimately show ?case by blast

 qed

 lemma polypow_normh: 

     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"

 proof (induct k arbitrary: n rule: polypow.induct)

   case (2 k n)

   let ?q = "polypow (Suc k div 2) p"

   let ?d = "polymul ?q ?q"

   from 2 have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+

   from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp

   from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n" by simp

   from dn on show ?case by (simp add: Let_def)

 qed auto 

 lemma polypow_norm:   

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"

   by (simp add: polypow_normh isnpoly_def)

 text{* Finally the whole normalization *}

 lemma polynate[simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0, field_inverse_zero})"

 by (induct p rule:polynate.induct, auto)

 lemma polynate_norm[simp]: 

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   shows "isnpoly (polynate p)"

   by (induct p rule: polynate.induct, simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm) (simp_all add: isnpoly_def)

 text{* shift1 *}

 lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"

 by (simp add: shift1_def polymul)

 lemma shift1_isnpoly: 

   assumes pn: "isnpoly p" and pnz: "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p) "

   using pn pnz by (simp add: shift1_def isnpoly_def )

 lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"

   by (simp add: shift1_def)

 lemma funpow_shift1_isnpoly: 

   "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"

   by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)

 lemma funpow_isnpolyh: 

   assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n "and np: "isnpolyh p n"

   shows "isnpolyh (funpow k f p) n"

   using f np by (induct k arbitrary: p, auto)

 lemma funpow_shift1: "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (Mul (Pw (Bound 0) n) p)"

   by (induct n arbitrary: p, simp_all add: shift1_isnpoly shift1 power_Suc )

 lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"

   using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)

 lemma funpow_shift1_1: 

   "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (funpow n shift1 1\<^sub>p *\<^sub>p p)"

   by (simp add: funpow_shift1)

 lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"

   by (induct p rule: poly_cmul.induct) (auto simp add: field_simps)

 lemma behead:

   assumes np: "isnpolyh p n"

   shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = (Ipoly bs p :: 'a :: {field_char_0, field_inverse_zero})"

   using np

 proof (induct p arbitrary: n rule: behead.induct)

   case (1 c p n) hence pn: "isnpolyh p n" by simp

   from 1(1)[OF pn] 

   have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" . 

   then show ?case using "1.hyps" apply (simp add: Let_def,cases "behead p = 0\<^sub>p")

     by (simp_all add: th[symmetric] field_simps power_Suc)

 qed (auto simp add: Let_def)

 lemma behead_isnpolyh:

   assumes np: "isnpolyh p n" shows "isnpolyh (behead p) n"

   using np by (induct p rule: behead.induct, auto simp add: Let_def isnpolyh_mono)

 subsection{* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}

 lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"

 proof(induct p arbitrary: n rule: poly.induct, auto)

   case (goal1 c n p n')

   hence "n = Suc (n - 1)" by simp

   hence "isnpolyh p (Suc (n - 1))"  using `isnpolyh p n` by simp

   with goal1(2) show ?case by simp

 qed

 lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"

 by (induct p arbitrary: n0 rule: isconstant.induct, auto simp add: isnpolyh_polybound0)

 lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" by (induct p, auto)

 lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"

   apply (induct p arbitrary: n0, auto)

   apply (atomize)

   apply (erule_tac x = "Suc nat" in allE)

   apply auto

   done

 lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"

  by (induct p  arbitrary: n0 rule: head.induct, auto intro: isnpolyh_polybound0)

 lemma polybound0_I:

   assumes nb: "polybound0 a"

   shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"

 using nb

 by (induct a rule: poly.induct) auto 

 lemma polysubst0_I:

   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"

   by (induct t) simp_all

 lemma polysubst0_I':

   assumes nb: "polybound0 a"

   shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t"

   by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])

 lemma decrpoly: assumes nb: "polybound0 t"

   shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"

   using nb by (induct t rule: decrpoly.induct, simp_all)

 lemma polysubst0_polybound0: assumes nb: "polybound0 t"

   shows "polybound0 (polysubst0 t a)"

 using nb by (induct a rule: poly.induct, auto)

 lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"

   by (induct p arbitrary: n rule: degree.induct, auto simp add: isnpolyh_polybound0)

 primrec maxindex :: "poly \<Rightarrow> nat" where

   "maxindex (Bound n) = n + 1"

 | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"

 | "maxindex (Add p q) = max (maxindex p) (maxindex q)"

 | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"

 | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"

 | "maxindex (Neg p) = maxindex p"

 | "maxindex (Pw p n) = maxindex p"

 | "maxindex (C x) = 0"

 definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool" where

   "wf_bs bs p = (length bs \<ge> maxindex p)"

 lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"

 proof(induct p rule: coefficients.induct)

   case (1 c p) 

   show ?case 

   proof

     fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"

     hence "x = c \<or> x \<in> set (coefficients p)" by simp

     moreover 

     {assume "x = c" hence "wf_bs bs x" using "1.prems"  unfolding wf_bs_def by simp}

     moreover 

     {assume H: "x \<in> set (coefficients p)" 

       from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp

       with "1.hyps" H have "wf_bs bs x" by blast }

     ultimately  show "wf_bs bs x" by blast

   qed

 qed simp_all

 lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"

 by (induct p rule: coefficients.induct, auto)

 lemma wf_bs_I: "wf_bs bs p ==> Ipoly (bs@bs') p = Ipoly bs p"

   unfolding wf_bs_def by (induct p, auto simp add: nth_append)

 lemma take_maxindex_wf: assumes wf: "wf_bs bs p" 

   shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"

 proof-

   let ?ip = "maxindex p"

   let ?tbs = "take ?ip bs"

   from wf have "length ?tbs = ?ip" unfolding wf_bs_def by simp

   hence wf': "wf_bs ?tbs p" unfolding wf_bs_def by  simp

   have eq: "bs = ?tbs @ (drop ?ip bs)" by simp

   from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis using eq by simp

 qed

 lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"

   by (induct p, auto)

 lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"

   unfolding wf_bs_def by simp

 lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"

   unfolding wf_bs_def by simp

 lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"

 by(induct p rule: coefficients.induct, auto simp add: wf_bs_def)

 lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"

   by (induct p rule: coefficients.induct, simp_all)

 lemma coefficients_head: "last (coefficients p) = head p"

   by (induct p rule: coefficients.induct, auto)

 lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"

   unfolding wf_bs_def by (induct p rule: decrpoly.induct, auto)

 lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"

   apply (rule exI[where x="replicate (n - length xs) z"])

   by simp

 lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"

 by (cases p, auto) (case_tac "nat", simp_all)

 lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"

   unfolding wf_bs_def 

   apply (induct p q rule: polyadd.induct)

   apply (auto simp add: Let_def)

   done

 lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"

   unfolding wf_bs_def 

   apply (induct p q arbitrary: bs rule: polymul.induct) 

   apply (simp_all add: wf_bs_polyadd)

   apply clarsimp

   apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])

   apply auto

   done

 lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"

   unfolding wf_bs_def by (induct p rule: polyneg.induct, auto)

 lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"

   unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast

 subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}

 definition "polypoly bs p = map (Ipoly bs) (coefficients p)"

 definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)"

 definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"

 lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0"

 proof (induct p arbitrary: n0 rule: coefficients.induct)

   case (1 c p n0)

   have cp: "isnpolyh (CN c 0 p) n0" by fact

   hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"

     by (auto simp add: isnpolyh_mono[where n'=0])

   from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp 

 qed auto

 lemma coefficients_isconst:

   "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"

   by (induct p arbitrary: n rule: coefficients.induct, 

     auto simp add: isnpolyh_Suc_const)

 lemma polypoly_polypoly':

   assumes np: "isnpolyh p n0"

   shows "polypoly (x#bs) p = polypoly' bs p"

 proof-

   let ?cf = "set (coefficients p)"

   from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .

   {fix q assume q: "q \<in> ?cf"

     from q cn_norm have th: "isnpolyh q n0" by blast

     from coefficients_isconst[OF np] q have "isconstant q" by blast

     with isconstant_polybound0[OF th] have "polybound0 q" by blast}

   hence "\<forall>q \<in> ?cf. polybound0 q" ..

   hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"

     using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]

     by auto

   thus ?thesis unfolding polypoly_def polypoly'_def by simp 

 qed

 lemma polypoly_poly:

   assumes np: "isnpolyh p n0" shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"

   using np 

 by (induct p arbitrary: n0 bs rule: coefficients.induct, auto simp add: polypoly_def)

 lemma polypoly'_poly: 

   assumes np: "isnpolyh p n0" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"

   using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .

 lemma polypoly_poly_polybound0:

   assumes np: "isnpolyh p n0" and nb: "polybound0 p"

   shows "polypoly bs p = [Ipoly bs p]"

   using np nb unfolding polypoly_def 

   by (cases p, auto, case_tac nat, auto)

 lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0" 

   by (induct p rule: head.induct, auto)

 lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"

   by (cases p,auto)

 lemma head_eq_headn0: "head p = headn p 0"

   by (induct p rule: head.induct, simp_all)

 lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"

   by (simp add: head_eq_headn0)

 lemma isnpolyh_zero_iff: 

   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})"

   shows "p = 0\<^sub>p"

 using nq eq

 proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)

   case less

   note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`

   {assume nz: "maxindex p = 0"

     then obtain c where "p = C c" using np by (cases p, auto)

     with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}

   moreover

   {assume nz: "maxindex p \<noteq> 0"

     let ?h = "head p"

     let ?hd = "decrpoly ?h"

     let ?ihd = "maxindex ?hd"

     from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h" 

       by simp_all

     hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast

     from maxindex_coefficients[of p] coefficients_head[of p, symmetric]

     have mihn: "maxindex ?h \<le> maxindex p" by auto

     with decr_maxindex[OF h(2)] nz  have ihd_lt_n: "?ihd < maxindex p" by auto

     {fix bs:: "'a list"  assume bs: "wf_bs bs ?hd"

       let ?ts = "take ?ihd bs"

       let ?rs = "drop ?ihd bs"

       have ts: "wf_bs ?ts ?hd" using bs unfolding wf_bs_def by simp

       have bs_ts_eq: "?ts@ ?rs = bs" by simp

       from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x#?ts) ?h" by simp

       from ihd_lt_n have "ALL x. length (x#?ts) \<le> maxindex p" by simp

       with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = maxindex p" by blast

       hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" unfolding wf_bs_def by simp

       with zp have "\<forall> x. Ipoly ((x#?ts) @ xs) p = 0" by blast

       hence "\<forall> x. Ipoly (x#(?ts @ xs)) p = 0" by simp

       with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]

       have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"  by simp

       hence "poly (polypoly' (?ts @ xs) p) = poly []" by (auto intro: ext) 

       hence "\<forall> c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"

         using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)

       with coefficients_head[of p, symmetric]

       have th0: "Ipoly (?ts @ xs) ?hd = 0" by simp

       from bs have wf'': "wf_bs ?ts ?hd" unfolding wf_bs_def by simp

       with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0" by simp

       with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by simp }

     then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast

     from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast

     hence "?h = 0\<^sub>p" by simp

     with head_nz[OF np] have "p = 0\<^sub>p" by simp}

   ultimately show "p = 0\<^sub>p" by blast

 qed

 lemma isnpolyh_unique:  

   assumes np:"isnpolyh p n0" and nq: "isnpolyh q n1"

   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"

 proof(auto)

   assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a)= \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"

   hence "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" by simp

   hence "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" 

     using wf_bs_polysub[where p=p and q=q] by auto

   with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq]

   show "p = q" by blast

 qed

 text{* consequences of unicity on the algorithms for polynomial normalization *}

 lemma polyadd_commute:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" shows "p +\<^sub>p q = q +\<^sub>p p"

   using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]] by simp

 lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp

 lemma one_normh: "isnpolyh 1\<^sub>p n" by simp

 lemma polyadd_0[simp]: 

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   and np: "isnpolyh p n0" shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>p p = p"

   using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np] 

     isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all

 lemma polymul_1[simp]: 

     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   and np: "isnpolyh p n0" shows "p *\<^sub>p 1\<^sub>p = p" and "1\<^sub>p *\<^sub>p p = p"

   using isnpolyh_unique[OF polymul_normh[OF np one_normh] np] 

     isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all

 lemma polymul_0[simp]: 

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   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"

   using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh] 

     isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all

 lemma polymul_commute: 

     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   and np:"isnpolyh p n0" and nq: "isnpolyh q n1"

   shows "p *\<^sub>p q = q *\<^sub>p p"

 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

 declare polyneg_polyneg[simp]

 lemma isnpolyh_polynate_id[simp]: 

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   and np:"isnpolyh p n0" shows "polynate p = p"

   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

 lemma polynate_idempotent[simp]: 

     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   shows "polynate (polynate p) = polynate p"

   using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .

 lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"

   unfolding poly_nate_def polypoly'_def ..

 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>)"

   using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]

   unfolding poly_nate_polypoly' by (auto intro: ext)

 subsection{* heads, degrees and all that *}

 lemma degree_eq_degreen0: "degree p = degreen p 0"

   by (induct p rule: degree.induct, simp_all)

 lemma degree_polyneg: assumes n: "isnpolyh p n"

   shows "degree (polyneg p) = degree p"

   using n

   by (induct p arbitrary: n rule: polyneg.induct, simp_all) (case_tac na, auto)

 lemma degree_polyadd:

   assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"

   shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"

 using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp

 lemma degree_polysub: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"

   shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"

 proof-

   from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp

   from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])

 qed

 lemma degree_polysub_samehead: 

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q" 

   and d: "degree p = degree q"

   shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"

 unfolding polysub_def split_def fst_conv snd_conv

 using np nq h d

 proof(induct p q rule:polyadd.induct)

   case (1 c c') thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def]) 

 next

   case (2 c c' n' p') 

   from 2 have "degree (C c) = degree (CN c' n' p')" by simp

   hence nz:"n' > 0" by (cases n', auto)

   hence "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)

   with 2 show ?case by simp

 next

   case (3 c n p c') 

   hence "degree (C c') = degree (CN c n p)" by simp

   hence nz:"n > 0" by (cases n, auto)

   hence "head (CN c n p) = CN c n p" by (cases n, auto)

   with 3 show ?case by simp

 next

   case (4 c n p c' n' p')

   hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1" 

     "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+

   hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all  

   hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0" 

     using H(1-2) degree_polyneg by auto

   from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"  by simp+

   from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"  by simp

   from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto

   have "n = n' \<or> n < n' \<or> n > n'" by arith

   moreover

   {assume nn': "n = n'"

     have "n = 0 \<or> n >0" by arith

     moreover {assume nz: "n = 0" hence ?case using 4 nn' by (auto simp add: Let_def degcmc')}

     moreover {assume nz: "n > 0"

       with nn' H(3) have  cc':"c = c'" and pp': "p = p'" by (cases n, auto)+

       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)}

     ultimately have ?case by blast}

   moreover

   {assume nn': "n < n'" hence n'p: "n' > 0" by simp 

     hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n', simp_all)

     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)

     hence "n > 0" by (cases n, simp_all)

     hence headcnp: "head (CN c n p) = CN c n p" by (cases n, auto)

     from H(3) headcnp headcnp' nn' have ?case by auto}

   moreover

   {assume nn': "n > n'"  hence np: "n > 0" by simp 

     hence headcnp:"head (CN c n p) = CN c n p"  by (cases n, simp_all)

     from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp

     from np have degcnp: "degree (CN c n p) = 0" by (cases n, simp_all)

     with degcnpeq have "n' > 0" by (cases n', simp_all)

     hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)

     from H(3) headcnp headcnp' nn' have ?case by auto}

   ultimately show ?case  by blast

 qed auto

 lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"

 by (induct p arbitrary: n0 rule: head.induct, simp_all add: shift1_def)

 lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"

 proof(induct k arbitrary: n0 p)

   case (Suc k n0 p) hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)

   with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"

     and "head (shift1 p) = head p" by (simp_all add: shift1_head) 

   thus ?case by (simp add: funpow_swap1)

 qed auto  

 lemma shift1_degree: "degree (shift1 p) = 1 + degree p"

   by (simp add: shift1_def)

 lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "

   by (induct k arbitrary: p) (auto simp add: shift1_degree)

 lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"

   by (induct n arbitrary: p) (simp_all add: funpow.simps)

 lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"

   by (induct p arbitrary: n rule: degree.induct, auto)

 lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"

   by (induct p arbitrary: n rule: degreen.induct, auto)

 lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"

   by (induct p arbitrary: n rule: degree.induct, auto)

 lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"

   by (induct p rule: head.induct, auto)

 lemma polyadd_eq_const_degree: 

   "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd p q = C c\<rbrakk> \<Longrightarrow> degree p = degree q" 

   using polyadd_eq_const_degreen degree_eq_degreen0 by simp

 lemma polyadd_head: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"

   and deg: "degree p \<noteq> degree q"

   shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"

 using np nq deg

 apply(induct p q arbitrary: n0 n1 rule: polyadd.induct,simp_all)

 apply (case_tac n', simp, simp)

 apply (case_tac n, simp, simp)

 apply (case_tac n, case_tac n', simp add: Let_def)

 apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p")

 apply (auto simp add: polyadd_eq_const_degree)

 apply (metis head_nz)

 apply (metis head_nz)

 apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)

 by (metis degree.simps(9) gr0_conv_Suc nat_less_le order_le_less_trans)

 lemma polymul_head_polyeq: 

    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   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"

 proof (induct p q arbitrary: n0 n1 rule: polymul.induct)

   case (2 c c' n' p' n0 n1)

   hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"  by (simp_all add: head_isnpolyh)

   thus ?case using 2 by (cases n', auto) 

 next 

   case (3 c n p c' n0 n1) 

   hence "isnpolyh (head (CN c n p)) n0" "isnormNum c'"  by (simp_all add: head_isnpolyh)

   thus ?case using 3 by (cases n, auto)

 next

   case (4 c n p c' n' p' n0 n1)

   hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"

     "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"

     by simp_all

   have "n < n' \<or> n' < n \<or> n = n'" by arith

   moreover 

   {assume nn': "n < n'" hence ?case 

       using norm 

     "4.hyps"(2)[OF norm(1,6)]

     "4.hyps"(1)[OF norm(2,6)] by (simp, cases n, simp,cases n', simp_all)}

   moreover {assume nn': "n'< n"

     hence ?case using norm "4.hyps"(6) [OF norm(5,3)]

       "4.hyps"(5)[OF norm(5,4)] 

       by (simp,cases n',simp,cases n,auto)}

   moreover {assume nn': "n' = n"

     from nn' polymul_normh[OF norm(5,4)] 

     have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)

     from nn' polymul_normh[OF norm(5,3)] norm 

     have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp

     from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)

     have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp 

     from polyadd_normh[OF ncnpc' ncnpp0'] 

     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" 

       by (simp add: min_def)

     {assume np: "n > 0"

       with nn' head_isnpolyh_Suc'[OF np nth]

         head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]

       have ?case by simp}

     moreover

     {moreover assume nz: "n = 0"

       from polymul_degreen[OF norm(5,4), where m="0"]

         polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0

       norm(5,6) degree_npolyhCN[OF norm(6)]

     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

     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

     from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth

     have ?case   using norm "4.hyps"(6)[OF norm(5,3)]

         "4.hyps"(5)[OF norm(5,4)] nn' nz by simp }

     ultimately have ?case by (cases n) auto} 

   ultimately show ?case by blast

 qed simp_all

 lemma degree_coefficients: "degree p = length (coefficients p) - 1"

   by(induct p rule: degree.induct, auto)

 lemma degree_head[simp]: "degree (head p) = 0"

   by (induct p rule: head.induct, auto)

 lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"

   by (cases n, simp_all)

 lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"

   by (cases n, simp_all)

 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)"

   using polyadd_different_degreen degree_eq_degreen0 by simp

 lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"

   by (induct p arbitrary: n0 rule: polyneg.induct, auto)

 lemma degree_polymul:

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   and np: "isnpolyh p n0" and nq: "isnpolyh q n1"

   shows "degree (p *\<^sub>p q) \<le> degree p + degree q"

   using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp

 lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"

   by (induct p arbitrary: n rule: degree.induct, auto)

 lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"

   by (induct p arbitrary: n rule: degree.induct, auto)

 subsection {* Correctness of polynomial pseudo division *}

 lemma polydivide_aux_properties:

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   and np: "isnpolyh p n0" and ns: "isnpolyh s n1"

   and ap: "head p = a" and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"

   shows "(polydivide_aux a n p k s = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p) 

           \<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)))"

   using ns

 proof(induct "degree s" arbitrary: s k k' r n1 rule: less_induct)

   case less

   let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"

   let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) 

     \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"

   let ?b = "head s"

   let ?p' = "funpow (degree s - n) shift1 p"

   let ?xdn = "funpow (degree s - n) shift1 1\<^sub>p"

   let ?akk' = "a ^\<^sub>p (k' - k)"

   note ns = `isnpolyh s n1`

   from np have np0: "isnpolyh p 0" 

     using isnpolyh_mono[where n="n0" and n'="0" and p="p"]  by simp

   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

   have headp': "head ?p' = head p" using funpow_shift1_head[OF np pnz] by simp

   from funpow_shift1_isnpoly[where p="1\<^sub>p"] have nxdn: "isnpolyh ?xdn 0" by (simp add: isnpoly_def)

   from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap 

   have nakk':"isnpolyh ?akk' 0" by blast

   {assume sz: "s = 0\<^sub>p"

    hence ?ths using np polydivide_aux.simps apply clarsimp by (rule exI[where x="0\<^sub>p"], simp) }

   moreover

   {assume sz: "s \<noteq> 0\<^sub>p"

     {assume dn: "degree s < n"

       hence "?ths" using ns ndp np polydivide_aux.simps by auto (rule exI[where x="0\<^sub>p"],simp) }

     moreover 

     {assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith

       have degsp': "degree s = degree ?p'" 

         using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp

       {assume ba: "?b = a"

         hence headsp': "head s = head ?p'" using ap headp' by simp

         have nr: "isnpolyh (s -\<^sub>p ?p') 0" using polysub_normh[OF ns np'] by simp

         from degree_polysub_samehead[OF ns np' headsp' degsp']

         have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp

         moreover 

         {assume deglt:"degree (s -\<^sub>p ?p') < degree s"

           from polydivide_aux.simps sz dn' ba

           have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"

             by (simp add: Let_def)

           {assume h1: "polydivide_aux a n p k s = (k', r)"

             from less(1)[OF deglt nr, of k k' r]

               trans[OF eq[symmetric] h1]

             have kk': "k \<le> k'" and nr:"\<exists>nr. isnpolyh r nr" and dr: "degree r = 0 \<or> degree r < degree p"

               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

             from q1 obtain q n1 where nq: "isnpolyh q n1" 

               and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"  by blast

             from nr obtain nr where nr': "isnpolyh r nr" by blast

             from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0" by simp

             from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]

             have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp

             from polyadd_normh[OF polymul_normh[OF np 

               polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']

             have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0" by simp 

             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')) = 

               Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp

             hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) = 

               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r" 

               by (simp add: field_simps)

             hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 

               Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p *\<^sub>p p) 

               + Ipoly bs p * Ipoly bs q + Ipoly bs r"

               by (auto simp only: funpow_shift1_1) 

             hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 

               Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p) 

               + Ipoly bs q) + Ipoly bs r" by (simp add: field_simps)

             hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 

               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

             with isnpolyh_unique[OF nakks' nqr']

             have "a ^\<^sub>p (k' - k) *\<^sub>p s = 

               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

             hence ?qths using nq'

               apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 1\<^sub>p) +\<^sub>p q" in exI)

               apply (rule_tac x="0" in exI) by simp

             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"

               by blast } hence ?ths by blast }

         moreover 

         {assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"

           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}"]

           have " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs s = Ipoly bs ?p'" by simp

           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

             by (simp only: funpow_shift1_1) simp

           hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]] by blast

           {assume h1: "polydivide_aux a n p k s = (k',r)"

             from polydivide_aux.simps sz dn' ba

             have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"

               by (simp add: Let_def)

             also have "\<dots> = (k,0\<^sub>p)" using polydivide_aux.simps spz by simp

             finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp

             with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]

               polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths

               apply auto

               apply (rule exI[where x="?xdn"])        

               apply (auto simp add: polymul_commute[of p])

               done} }

         ultimately have ?ths by blast }

       moreover

       {assume ba: "?b \<noteq> a"

         from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns] 

           polymul_normh[OF head_isnpolyh[OF ns] np']]

         have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by(simp add: min_def)

         have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"

           using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns] 

             polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]

             funpow_shift1_nz[OF pnz] by simp_all

         from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz

           polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]

         have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')" 

           using head_head[OF ns] funpow_shift1_head[OF np pnz]

             polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]

           by (simp add: ap)

         from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]

           head_nz[OF np] pnz sz ap[symmetric]

           funpow_shift1_nz[OF pnz, where n="degree s - n"]

           polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"]  head_nz[OF ns]

           ndp dn

         have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p') "

           by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)

         {assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"

           from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns] polymul_normh[OF head_isnpolyh[OF ns]np']]

           ap have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by simp

           {assume h1:"polydivide_aux a n p k s = (k', r)"

             from h1 polydivide_aux.simps sz dn' ba

             have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"

               by (simp add: Let_def)

             with less(1)[OF dth nasbp', of "Suc k" k' r]

             obtain q nq nr where kk': "Suc k \<le> k'" and nr: "isnpolyh r nr" and nq: "isnpolyh q nq" 

               and dr: "degree r = 0 \<or> degree r < degree p"

               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

             from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith

             {fix bs:: "'a::{field_char_0, field_inverse_zero} list"

             from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]

             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

             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"

               by (simp add: field_simps power_Suc)

             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"

               by (simp add:kk'' funpow_shift1_1[where n="degree s - n" and p="p"])

             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"

               by (simp add: field_simps)}

             hence ieq:"\<forall>(bs :: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 

               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 

             let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"

             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]]

             have nqw: "isnpolyh ?q 0" by simp

             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]]

             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

             from dr kk' nr h1 asth nqw have ?ths apply simp

               apply (rule conjI)

               apply (rule exI[where x="nr"], simp)

               apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)

               apply (rule exI[where x="0"], simp)

               done}

           hence ?ths by blast }

         moreover 

         {assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"

           {fix bs :: "'a::{field_char_0, field_inverse_zero} list"

             from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz

           have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'" by simp

           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p" 

             by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])

           hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" by simp

         }

         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))" ..

           from hth

           have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)" 

             using isnpolyh_unique[where ?'a = "'a::{field_char_0, field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns] 

                     polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],

               simplified ap] by simp

           {assume h1: "polydivide_aux a n p k s = (k', r)"

           from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps

           have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)

           with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]

             polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq

           have ?ths apply (clarsimp simp add: Let_def)

             apply (rule exI[where x="?b *\<^sub>p ?xdn"]) apply simp

             apply (rule exI[where x="0"], simp)

             done}

         hence ?ths by blast}

         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"]

           head_nz[OF np] pnz sz ap[symmetric]

           by (simp add: degree_eq_degreen0[symmetric]) blast }

       ultimately have ?ths by blast

     }

     ultimately have ?ths by blast}

   ultimately show ?ths by blast

 qed

 lemma polydivide_properties: 

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"

   shows "(\<exists> k r. polydivide s p = (k,r) \<and> (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) 

   \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"

 proof-

   have trv: "head p = head p" "degree p = degree p" by simp_all

   from polydivide_def[where s="s" and p="p"] 

   have ex: "\<exists> k r. polydivide s p = (k,r)" by auto

   then obtain k r where kr: "polydivide s p = (k,r)" by blast

   from trans[OF meta_eq_to_obj_eq[OF polydivide_def[where s="s"and p="p"], symmetric] kr]

     polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]

   have "(degree r = 0 \<or> degree r < degree p) \<and>

    (\<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

   with kr show ?thesis 

     apply -

     apply (rule exI[where x="k"])

     apply (rule exI[where x="r"])

     apply simp

     done

 qed

 subsection{* More about polypoly and pnormal etc *}

 definition "isnonconstant p = (\<not> isconstant p)"

 lemma isnonconstant_pnormal_iff: assumes nc: "isnonconstant p" 

   shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0" 

 proof

   let ?p = "polypoly bs p"  

   assume H: "pnormal ?p"

   have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)

   from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]  

     pnormal_last_nonzero[OF H]

   show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)

 next

   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"

   let ?p = "polypoly bs p"

   have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)

   hence pz: "?p \<noteq> []" by (simp add: polypoly_def) 

   hence lg: "length ?p > 0" by simp

   from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"] 

   have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)

   from pnormal_last_length[OF lg lz] show "pnormal ?p" .

 qed

 lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"

   unfolding isnonconstant_def

   apply (cases p, simp_all)

   apply (case_tac nat, auto)

   done

 lemma isnonconstant_nonconstant: assumes inc: "isnonconstant p"

   shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"

 proof

   let ?p = "polypoly bs p"

   assume nc: "nonconstant ?p"

   from isnonconstant_pnormal_iff[OF inc, of bs] nc

   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast

 next

   let ?p = "polypoly bs p"

   assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"

   from isnonconstant_pnormal_iff[OF inc, of bs] h

   have pn: "pnormal ?p" by blast

   {fix x assume H: "?p = [x]" 

     from H have "length (coefficients p) = 1" unfolding polypoly_def by auto

     with isnonconstant_coefficients_length[OF inc] have False by arith}

   thus "nonconstant ?p" using pn unfolding nonconstant_def by blast  

 qed

 lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"

   unfolding pnormal_def

  apply (induct p)

  apply (simp_all, case_tac "p=[]", simp_all)

  done

 lemma degree_degree: assumes inc: "isnonconstant p"

   shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"

 proof

   let  ?p = "polypoly bs p"

   assume H: "degree p = Polynomial_List.degree ?p"

   from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"

     unfolding polypoly_def by auto

   from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]

   have lg:"length (pnormalize ?p) = length ?p"

     unfolding Polynomial_List.degree_def polypoly_def by simp

   hence "pnormal ?p" using pnormal_length[OF pz] by blast 

   with isnonconstant_pnormal_iff[OF inc]  

   show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast

 next

   let  ?p = "polypoly bs p"  

   assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"

   with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast

   with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]

   show "degree p = Polynomial_List.degree ?p" 

     unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto

 qed

 section{* Swaps ; Division by a certain variable *}

 primrec swap:: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly" where

   "swap n m (C x) = C x"

 | "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"

 | "swap n m (Neg t) = Neg (swap n m t)"

 | "swap n m (Add s t) = Add (swap n m s) (swap n m t)"

 | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"

 | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"

 | "swap n m (Pw t k) = Pw (swap n m t) k"

 | "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)

   (swap n m p)"

 lemma swap: assumes nbs: "n < length bs" and mbs: "m < length bs"

   shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"

 proof (induct t)

   case (Bound k) thus ?case using nbs mbs by simp 

 next

   case (CN c k p) thus ?case using nbs mbs by simp 

 qed simp_all

 lemma swap_swap_id[simp]: "swap n m (swap m n t) = t"

   by (induct t,simp_all)

 lemma swap_commute: "swap n m p = swap m n p" by (induct p, simp_all)

 lemma swap_same_id[simp]: "swap n n t = t"

   by (induct t, simp_all)

 definition "swapnorm n m t = polynate (swap n m t)"

 lemma swapnorm: assumes nbs: "n < length bs" and mbs: "m < length bs"

   shows "((Ipoly bs (swapnorm n m t) :: 'a\<Colon>{field_char_0, field_inverse_zero})) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"

   using swap[OF assms] swapnorm_def by simp

 lemma swapnorm_isnpoly[simp]: 

     assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   shows "isnpoly (swapnorm n m p)"

   unfolding swapnorm_def by simp

 definition "polydivideby n s p = 

     (let ss = swapnorm 0 n s ; sp = swapnorm 0 n p ; h = head sp; (k,r) = polydivide ss sp

      in (k,swapnorm 0 n h,swapnorm 0 n r))"

 lemma swap_nz [simp]: " (swap n m p = 0\<^sub>p) = (p = 0\<^sub>p)" by (induct p, simp_all)

 fun isweaknpoly :: "poly \<Rightarrow> bool"

 where

   "isweaknpoly (C c) = True"

 | "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"

 | "isweaknpoly p = False"

 lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p" 

   by (induct p arbitrary: n0, auto)

 lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)" 

   by (induct p, auto)

 end