(*  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"

  (* Verification *)

lemma nth_pos2[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"

using Nat.gr0_conv_Suc

by clarsimp

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

  from prems(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 prems(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 prems th3 by simp

next

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

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

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

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

  from prems 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 prems 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 prems have th1:"min n0 n1 \<le> n" by auto 

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

    from prems 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 prems 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 prems have th1:"min n0 n1 \<le> n'" by auto

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

    from prems 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 prems 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 prems by simp}

  moreover 

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

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

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

        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 prems 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  arbitrary: n0 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 prems(2)[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 prems(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