src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
author krauss
Mon Feb 21 23:14:36 2011 +0100 (2011-02-21)
changeset 41811 7e338ccabff0
parent 41810 588c95c4b53e
child 41812 d46c2908a838
permissions -rw-r--r--
strengthened polymul.induct
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(*  Title:      HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
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    Author:     Amine Chaieb
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*)
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header {* Implementation and verification of multivariate polynomials *}
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theory Reflected_Multivariate_Polynomial
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imports Complex_Main "~~/src/HOL/Library/Abstract_Rat" Polynomial_List
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begin
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  (* Implementation *)
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subsection{* Datatype of polynomial expressions *} 
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datatype poly = C Num| Bound nat| Add poly poly|Sub poly poly
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  | Mul poly poly| Neg poly| Pw poly nat| CN poly nat poly
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abbreviation poly_0 :: "poly" ("0\<^sub>p") where "0\<^sub>p \<equiv> C (0\<^sub>N)"
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abbreviation poly_p :: "int \<Rightarrow> poly" ("_\<^sub>p") where "i\<^sub>p \<equiv> C (i\<^sub>N)"
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subsection{* Boundedness, substitution and all that *}
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primrec polysize:: "poly \<Rightarrow> nat" where
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  "polysize (C c) = 1"
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| "polysize (Bound n) = 1"
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| "polysize (Neg p) = 1 + polysize p"
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| "polysize (Add p q) = 1 + polysize p + polysize q"
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| "polysize (Sub p q) = 1 + polysize p + polysize q"
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| "polysize (Mul p q) = 1 + polysize p + polysize q"
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| "polysize (Pw p n) = 1 + polysize p"
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| "polysize (CN c n p) = 4 + polysize c + polysize p"
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primrec polybound0:: "poly \<Rightarrow> bool" (* a poly is INDEPENDENT of Bound 0 *) where
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  "polybound0 (C c) = True"
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| "polybound0 (Bound n) = (n>0)"
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| "polybound0 (Neg a) = polybound0 a"
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| "polybound0 (Add a b) = (polybound0 a \<and> polybound0 b)"
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| "polybound0 (Sub a b) = (polybound0 a \<and> polybound0 b)" 
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| "polybound0 (Mul a b) = (polybound0 a \<and> polybound0 b)"
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| "polybound0 (Pw p n) = (polybound0 p)"
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| "polybound0 (CN c n p) = (n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p)"
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primrec polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" (* substitute a poly into a poly for Bound 0 *) where
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  "polysubst0 t (C c) = (C c)"
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| "polysubst0 t (Bound n) = (if n=0 then t else Bound n)"
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| "polysubst0 t (Neg a) = Neg (polysubst0 t a)"
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| "polysubst0 t (Add a b) = Add (polysubst0 t a) (polysubst0 t b)"
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| "polysubst0 t (Sub a b) = Sub (polysubst0 t a) (polysubst0 t b)" 
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| "polysubst0 t (Mul a b) = Mul (polysubst0 t a) (polysubst0 t b)"
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| "polysubst0 t (Pw p n) = Pw (polysubst0 t p) n"
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| "polysubst0 t (CN c n p) = (if n=0 then Add (polysubst0 t c) (Mul t (polysubst0 t p))
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                             else CN (polysubst0 t c) n (polysubst0 t p))"
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fun decrpoly:: "poly \<Rightarrow> poly" 
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where
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  "decrpoly (Bound n) = Bound (n - 1)"
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| "decrpoly (Neg a) = Neg (decrpoly a)"
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| "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"
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| "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
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| "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
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| "decrpoly (Pw p n) = Pw (decrpoly p) n"
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| "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
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| "decrpoly a = a"
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subsection{* Degrees and heads and coefficients *}
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fun degree:: "poly \<Rightarrow> nat"
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where
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  "degree (CN c 0 p) = 1 + degree p"
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| "degree p = 0"
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fun head:: "poly \<Rightarrow> poly"
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where
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  "head (CN c 0 p) = head p"
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| "head p = p"
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(* More general notions of degree and head *)
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fun degreen:: "poly \<Rightarrow> nat \<Rightarrow> nat"
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where
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  "degreen (CN c n p) = (\<lambda>m. if n=m then 1 + degreen p n else 0)"
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 |"degreen p = (\<lambda>m. 0)"
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fun headn:: "poly \<Rightarrow> nat \<Rightarrow> poly"
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where
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  "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
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| "headn p = (\<lambda>m. p)"
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fun coefficients:: "poly \<Rightarrow> poly list"
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where
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  "coefficients (CN c 0 p) = c#(coefficients p)"
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| "coefficients p = [p]"
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fun isconstant:: "poly \<Rightarrow> bool"
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where
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  "isconstant (CN c 0 p) = False"
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| "isconstant p = True"
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fun behead:: "poly \<Rightarrow> poly"
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where
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  "behead (CN c 0 p) = (let p' = behead p in if p' = 0\<^sub>p then c else CN c 0 p')"
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| "behead p = 0\<^sub>p"
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fun headconst:: "poly \<Rightarrow> Num"
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where
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  "headconst (CN c n p) = headconst p"
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| "headconst (C n) = n"
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subsection{* Operations for normalization *}
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consts 
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  polyadd :: "poly\<times>poly \<Rightarrow> poly"
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  polysub :: "poly\<times>poly \<Rightarrow> poly"
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  polymul :: "poly\<times>poly \<Rightarrow> poly"
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abbreviation poly_add :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
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  where "a +\<^sub>p b \<equiv> polyadd (a,b)"
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abbreviation poly_mul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
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  where "a *\<^sub>p b \<equiv> polymul (a,b)"
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abbreviation poly_sub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
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  where "a -\<^sub>p b \<equiv> polysub (a,b)"
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recdef polyadd "measure (\<lambda> (a,b). polysize a + polysize b)"
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  "polyadd (C c, C c') = C (c+\<^sub>Nc')"
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  "polyadd (C c, CN c' n' p') = CN (polyadd (C c, c')) n' p'"
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  "polyadd (CN c n p, C c') = CN (polyadd (c, C c')) n p"
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  "polyadd (CN c n p, CN c' n' p') =
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    (if n < n' then CN (polyadd(c,CN c' n' p')) n p
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     else if n'<n then CN (polyadd(CN c n p, c')) n' p'
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     else (let cc' = polyadd (c,c') ; 
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               pp' = polyadd (p,p')
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           in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
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  "polyadd (a, b) = Add a b"
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(hints recdef_simp add: Let_def measure_def split_def inv_image_def recdef_cong del: if_cong)
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fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
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where
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  "polyneg (C c) = C (~\<^sub>N c)"
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| "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
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| "polyneg a = Neg a"
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defs polysub_def[code]: "polysub \<equiv> \<lambda> (p,q). polyadd (p,polyneg q)"
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recdef polymul "measure (\<lambda>(a,b). size a + size b)"
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  "polymul(C c, C c') = C (c*\<^sub>Nc')"
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  "polymul(C c, CN c' n' p') = 
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      (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul(C c,c')) n' (polymul(C c, p')))"
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  "polymul(CN c n p, C c') = 
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      (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul(c,C c')) n (polymul(p, C c')))"
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  "polymul(CN c n p, CN c' n' p') = 
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  (if n<n' then CN (polymul(c,CN c' n' p')) n (polymul(p,CN c' n' p'))
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  else if n' < n 
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  then CN (polymul(CN c n p,c')) n' (polymul(CN c n p,p'))
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  else polyadd(polymul(CN c n p, c'),CN 0\<^sub>p n' (polymul(CN c n p, p'))))"
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  "polymul (a,b) = Mul a b"
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(hints recdef_cong del: if_cong)
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fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
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where
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  "polypow 0 = (\<lambda>p. 1\<^sub>p)"
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| "polypow n = (\<lambda>p. let q = polypow (n div 2) p ; d = polymul(q,q) in 
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                    if even n then d else polymul(p,d))"
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abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
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  where "a ^\<^sub>p k \<equiv> polypow k a"
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function polynate :: "poly \<Rightarrow> poly"
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where
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  "polynate (Bound n) = CN 0\<^sub>p n 1\<^sub>p"
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| "polynate (Add p q) = (polynate p +\<^sub>p polynate q)"
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| "polynate (Sub p q) = (polynate p -\<^sub>p polynate q)"
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| "polynate (Mul p q) = (polynate p *\<^sub>p polynate q)"
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| "polynate (Neg p) = (~\<^sub>p (polynate p))"
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| "polynate (Pw p n) = ((polynate p) ^\<^sub>p n)"
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| "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
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| "polynate (C c) = C (normNum c)"
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by pat_completeness auto
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termination by (relation "measure polysize") auto
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fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly" where
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  "poly_cmul y (C x) = C (y *\<^sub>N x)"
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| "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
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| "poly_cmul y p = C y *\<^sub>p p"
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definition monic :: "poly \<Rightarrow> (poly \<times> bool)" where
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  "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))"
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subsection{* Pseudo-division *}
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definition shift1 :: "poly \<Rightarrow> poly" where
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  "shift1 p \<equiv> CN 0\<^sub>p 0 p"
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abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)" where
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  "funpow \<equiv> compow"
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partial_function (tailrec) polydivide_aux :: "poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> nat \<times> poly"
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  where
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  "polydivide_aux a n p k s = 
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  (if s = 0\<^sub>p then (k,s)
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  else (let b = head s; m = degree s in
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  (if m < n then (k,s) else 
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  (let p'= funpow (m - n) shift1 p in 
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  (if a = b then polydivide_aux a n p k (s -\<^sub>p p') 
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  else polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (b *\<^sub>p p')))))))"
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definition polydivide :: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)" where
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  "polydivide s p \<equiv> polydivide_aux (head p) (degree p) p 0 s"
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fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly" where
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  "poly_deriv_aux n (CN c 0 p) = CN (poly_cmul ((int n)\<^sub>N) c) 0 (poly_deriv_aux (n + 1) p)"
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| "poly_deriv_aux n p = poly_cmul ((int n)\<^sub>N) p"
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fun poly_deriv :: "poly \<Rightarrow> poly" where
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  "poly_deriv (CN c 0 p) = poly_deriv_aux 1 p"
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| "poly_deriv p = 0\<^sub>p"
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  (* Verification *)
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lemma nth_pos2[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
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using Nat.gr0_conv_Suc
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by clarsimp
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subsection{* Semantics of the polynomial representation *}
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primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0, field_inverse_zero, power}" where
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  "Ipoly bs (C c) = INum c"
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| "Ipoly bs (Bound n) = bs!n"
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| "Ipoly bs (Neg a) = - Ipoly bs a"
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| "Ipoly bs (Add a b) = Ipoly bs a + Ipoly bs b"
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| "Ipoly bs (Sub a b) = Ipoly bs a - Ipoly bs b"
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| "Ipoly bs (Mul a b) = Ipoly bs a * Ipoly bs b"
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| "Ipoly bs (Pw t n) = (Ipoly bs t) ^ n"
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| "Ipoly bs (CN c n p) = (Ipoly bs c) + (bs!n)*(Ipoly bs p)"
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abbreviation
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  Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0, field_inverse_zero, power}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
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  where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"
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lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i" 
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  by (simp add: INum_def)
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lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j" 
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  by (simp  add: INum_def)
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lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
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subsection {* Normal form and normalization *}
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fun isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
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where
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  "isnpolyh (C c) = (\<lambda>k. isnormNum c)"
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| "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))"
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| "isnpolyh p = (\<lambda>k. False)"
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lemma isnpolyh_mono: "\<lbrakk>n' \<le> n ; isnpolyh p n\<rbrakk> \<Longrightarrow> isnpolyh p n'"
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by (induct p rule: isnpolyh.induct, auto)
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definition isnpoly :: "poly \<Rightarrow> bool" where
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  "isnpoly p \<equiv> isnpolyh p 0"
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text{* polyadd preserves normal forms *}
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lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> 
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      \<Longrightarrow> isnpolyh (polyadd(p,q)) (min n0 n1)"
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proof(induct p q arbitrary: n0 n1 rule: polyadd.induct)
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  case (2 a b c' n' p' n0 n1)
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  from prems have  th1: "isnpolyh (C (a,b)) (Suc n')" by simp 
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  from prems(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
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  with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
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  with prems(1)[OF th1 th2] have th3:"isnpolyh (C (a,b) +\<^sub>p c') (Suc n')" by simp
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  from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp 
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  thus ?case using prems th3 by simp
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next
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  case (3 c' n' p' a b n1 n0)
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  from prems have  th1: "isnpolyh (C (a,b)) (Suc n')" by simp 
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  from prems(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
chaieb@33154
   272
  with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
chaieb@33154
   273
  with prems(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C (a,b)) (Suc n')" by simp
chaieb@33154
   274
  from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp 
chaieb@33154
   275
  thus ?case using prems th3 by simp
chaieb@33154
   276
next
chaieb@33154
   277
  case (4 c n p c' n' p' n0 n1)
chaieb@33154
   278
  hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
chaieb@33154
   279
  from prems have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all 
chaieb@33154
   280
  from prems have ngen0: "n \<ge> n0" by simp
chaieb@33154
   281
  from prems have n'gen1: "n' \<ge> n1" by simp 
chaieb@33154
   282
  have "n < n' \<or> n' < n \<or> n = n'" by auto
krauss@41763
   283
  moreover {assume eq: "n = n'"
krauss@41763
   284
    with prems(2)[OF nc nc'] 
chaieb@33154
   285
    have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
chaieb@33154
   286
    hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
chaieb@33154
   287
      using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
krauss@41763
   288
    from eq prems(1)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
chaieb@33154
   289
    have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
chaieb@33154
   290
    from minle npp' ncc'n01 prems ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}
chaieb@33154
   291
  moreover {assume lt: "n < n'"
chaieb@33154
   292
    have "min n0 n1 \<le> n0" by simp
chaieb@33154
   293
    with prems have th1:"min n0 n1 \<le> n" by auto 
chaieb@33154
   294
    from prems have th21: "isnpolyh c (Suc n)" by simp
chaieb@33154
   295
    from prems have th22: "isnpolyh (CN c' n' p') n'" by simp
chaieb@33154
   296
    from lt have th23: "min (Suc n) n' = Suc n" by arith
krauss@41763
   297
    from prems(4)[OF th21 th22]
chaieb@33154
   298
    have "isnpolyh (polyadd (c, CN c' n' p')) (Suc n)" using th23 by simp
chaieb@33154
   299
    with prems th1 have ?case by simp } 
chaieb@33154
   300
  moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
chaieb@33154
   301
    have "min n0 n1 \<le> n1"  by simp
chaieb@33154
   302
    with prems have th1:"min n0 n1 \<le> n'" by auto
chaieb@33154
   303
    from prems have th21: "isnpolyh c' (Suc n')" by simp_all
chaieb@33154
   304
    from prems have th22: "isnpolyh (CN c n p) n" by simp
chaieb@33154
   305
    from gt have th23: "min n (Suc n') = Suc n'" by arith
krauss@41763
   306
    from prems(3)[OF th22 th21]
chaieb@33154
   307
    have "isnpolyh (polyadd (CN c n p,c')) (Suc n')" using th23 by simp
chaieb@33154
   308
    with prems th1 have ?case by simp}
chaieb@33154
   309
      ultimately show ?case by blast
chaieb@33154
   310
qed auto
chaieb@33154
   311
chaieb@33154
   312
lemma polyadd[simp]: "Ipoly bs (polyadd (p,q)) = (Ipoly bs p) + (Ipoly bs q)"
haftmann@36349
   313
by (induct p q rule: polyadd.induct, auto simp add: Let_def field_simps right_distrib[symmetric] simp del: right_distrib)
chaieb@33154
   314
chaieb@33154
   315
lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd(p,q))"
chaieb@33154
   316
  using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
chaieb@33154
   317
krauss@41404
   318
text{* The degree of addition and other general lemmas needed for the normal form of polymul *}
chaieb@33154
   319
chaieb@33154
   320
lemma polyadd_different_degreen: 
chaieb@33154
   321
  "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow> 
chaieb@33154
   322
  degreen (polyadd(p,q)) m = max (degreen p m) (degreen q m)"
chaieb@33154
   323
proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
chaieb@33154
   324
  case (4 c n p c' n' p' m n0 n1)
krauss@41763
   325
  have "n' = n \<or> n < n' \<or> n' < n" by arith
krauss@41763
   326
  thus ?case
krauss@41763
   327
  proof (elim disjE)
krauss@41763
   328
    assume [simp]: "n' = n"
krauss@41763
   329
    from 4(1)[of n n m] 4(2)[of "Suc n" "Suc n" m] 4(5-7)
krauss@41763
   330
    show ?thesis by (auto simp: Let_def)
krauss@41763
   331
  next
krauss@41763
   332
    assume "n < n'"
krauss@41763
   333
    with 4 show ?thesis by auto
krauss@41763
   334
  next
krauss@41763
   335
    assume "n' < n"
krauss@41763
   336
    with 4 show ?thesis by auto
krauss@41763
   337
  qed
krauss@41763
   338
qed auto
chaieb@33154
   339
chaieb@33154
   340
lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
chaieb@33154
   341
  by (induct p arbitrary: n rule: headn.induct, auto)
chaieb@33154
   342
lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
chaieb@33154
   343
  by (induct p arbitrary: n rule: degree.induct, auto)
chaieb@33154
   344
lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
chaieb@33154
   345
  by (induct p arbitrary: n rule: degreen.induct, auto)
chaieb@33154
   346
chaieb@33154
   347
lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
chaieb@33154
   348
  by (induct p arbitrary: n rule: degree.induct, auto)
chaieb@33154
   349
chaieb@33154
   350
lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
chaieb@33154
   351
  using degree_isnpolyh_Suc by auto
chaieb@33154
   352
lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
chaieb@33154
   353
  using degreen_0 by auto
chaieb@33154
   354
chaieb@33154
   355
chaieb@33154
   356
lemma degreen_polyadd:
chaieb@33154
   357
  assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1"
chaieb@33154
   358
  shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
chaieb@33154
   359
  using np nq m
chaieb@33154
   360
proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
chaieb@33154
   361
  case (2 c c' n' p' n0 n1) thus ?case  by (cases n', simp_all)
chaieb@33154
   362
next
chaieb@33154
   363
  case (3 c n p c' n0 n1) thus ?case by (cases n, auto)
chaieb@33154
   364
next
chaieb@33154
   365
  case (4 c n p c' n' p' n0 n1 m) 
krauss@41763
   366
  have "n' = n \<or> n < n' \<or> n' < n" by arith
krauss@41763
   367
  thus ?case
krauss@41763
   368
  proof (elim disjE)
krauss@41763
   369
    assume [simp]: "n' = n"
krauss@41763
   370
    from 4(1)[of n n m] 4(2)[of "Suc n" "Suc n" m] 4(5-7)
krauss@41763
   371
    show ?thesis by (auto simp: Let_def)
krauss@41763
   372
  qed simp_all
chaieb@33154
   373
qed auto
chaieb@33154
   374
chaieb@33154
   375
lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk> 
chaieb@33154
   376
  \<Longrightarrow> degreen p m = degreen q m"
chaieb@33154
   377
proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
chaieb@33154
   378
  case (4 c n p c' n' p' m n0 n1 x) 
chaieb@33154
   379
  {assume nn': "n' < n" hence ?case using prems by simp}
chaieb@33154
   380
  moreover 
chaieb@33154
   381
  {assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
chaieb@33154
   382
    moreover {assume "n < n'" with prems have ?case by simp }
chaieb@33154
   383
    moreover {assume eq: "n = n'" hence ?case using prems 
krauss@41763
   384
        apply (cases "p +\<^sub>p p' = 0\<^sub>p")
krauss@41763
   385
        apply (auto simp add: Let_def)
krauss@41763
   386
        by blast
krauss@41763
   387
      }
chaieb@33154
   388
    ultimately have ?case by blast}
chaieb@33154
   389
  ultimately show ?case by blast
chaieb@33154
   390
qed simp_all
chaieb@33154
   391
chaieb@33154
   392
lemma polymul_properties:
haftmann@36409
   393
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
   394
  and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> min n0 n1"
chaieb@33154
   395
  shows "isnpolyh (p *\<^sub>p q) (min n0 n1)" 
chaieb@33154
   396
  and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)" 
chaieb@33154
   397
  and "degreen (p *\<^sub>p q) m = (if (p = 0\<^sub>p \<or> q = 0\<^sub>p) then 0 
chaieb@33154
   398
                             else degreen p m + degreen q m)"
chaieb@33154
   399
  using np nq m
chaieb@33154
   400
proof(induct p q arbitrary: n0 n1 m rule: polymul.induct)
chaieb@33154
   401
  case (2 a b c' n' p') 
chaieb@33154
   402
  let ?c = "(a,b)"
chaieb@33154
   403
  { case (1 n0 n1) 
krauss@41811
   404
    with "2.hyps"(1-3)[of n' n' n']
krauss@41811
   405
      and "2.hyps"(4-6)[of "Suc n'" "Suc n'" n']
krauss@41811
   406
    show ?case by (auto simp add: min_def)
chaieb@33154
   407
  next
chaieb@33154
   408
    case (2 n0 n1) thus ?case by auto 
chaieb@33154
   409
  next
chaieb@33154
   410
    case (3 n0 n1) thus ?case  using "2.hyps" by auto } 
chaieb@33154
   411
next
krauss@41811
   412
  case (3 c n p a b)
krauss@41811
   413
  let ?c' = "(a,b)"
krauss@41811
   414
  { case (1 n0 n1) 
krauss@41811
   415
    with "3.hyps"(1-3)[of n n n]
krauss@41811
   416
      "3.hyps"(4-6)[of "Suc n" "Suc n" n]
krauss@41811
   417
    show ?case by (auto simp add: min_def)
chaieb@33154
   418
  next
krauss@41811
   419
    case (2 n0 n1) thus ?case by auto
chaieb@33154
   420
  next
chaieb@33154
   421
    case (3 n0 n1) thus ?case  using "3.hyps" by auto } 
chaieb@33154
   422
next
chaieb@33154
   423
  case (4 c n p c' n' p')
chaieb@33154
   424
  let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
krauss@41811
   425
    {
krauss@41811
   426
      case (1 n0 n1)
chaieb@33154
   427
      hence cnp: "isnpolyh ?cnp n" and cnp': "isnpolyh ?cnp' n'"
wenzelm@33268
   428
        and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)" 
wenzelm@33268
   429
        and np': "isnpolyh p' n'" and nc': "isnpolyh c' (Suc n')"
wenzelm@33268
   430
        and nn0: "n \<ge> n0" and nn1:"n' \<ge> n1"
wenzelm@33268
   431
        by simp_all
krauss@41811
   432
      { assume "n < n'"
krauss@41811
   433
        with "4.hyps"(13-14)[OF np cnp', of n]
krauss@41811
   434
          "4.hyps"(16)[OF nc cnp', of n] nn0 cnp
krauss@41811
   435
        have ?case by (simp add: min_def)
krauss@41811
   436
      } moreover {
krauss@41811
   437
        assume "n' < n"
krauss@41811
   438
        with "4.hyps"(1-2)[OF cnp np', of "n'"]
krauss@41811
   439
          "4.hyps"(4)[OF cnp nc', of "Suc n'"] nn1 cnp'
krauss@41811
   440
        have ?case
krauss@41811
   441
          by (cases "Suc n' = n", simp_all add: min_def)
krauss@41811
   442
      } moreover {
krauss@41811
   443
        assume "n' = n"
krauss@41811
   444
        with "4.hyps"(1-2)[OF cnp np', of n]
krauss@41811
   445
          "4.hyps"(4)[OF cnp nc', of n] cnp cnp' nn1 nn0
krauss@41811
   446
        have ?case
krauss@41811
   447
          apply (auto intro!: polyadd_normh)
krauss@41811
   448
          apply (simp_all add: min_def isnpolyh_mono[OF nn0])
krauss@41811
   449
          done
krauss@41811
   450
      }
krauss@41811
   451
      ultimately show ?case by arith
krauss@41811
   452
    next
krauss@41811
   453
      fix n0 n1 m
chaieb@33154
   454
      assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1"
chaieb@33154
   455
      and m: "m \<le> min n0 n1"
chaieb@33154
   456
      let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
chaieb@33154
   457
      let ?d1 = "degreen ?cnp m"
chaieb@33154
   458
      let ?d2 = "degreen ?cnp' m"
chaieb@33154
   459
      let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
chaieb@33154
   460
      have "n'<n \<or> n < n' \<or> n' = n" by auto
chaieb@33154
   461
      moreover 
chaieb@33154
   462
      {assume "n' < n \<or> n < n'"
krauss@41811
   463
        with "4.hyps"(3,15,18) np np' m 
krauss@41811
   464
        have ?eq by auto }
chaieb@33154
   465
      moreover
krauss@41811
   466
      {assume nn': "n' = n" hence nn:"\<not> n' < n \<and> \<not> n < n'" by arith
krauss@41811
   467
        from "4.hyps"(1,3)[of n n' n]
krauss@41811
   468
          "4.hyps"(4,5)[of n "Suc n'" n]
wenzelm@33268
   469
          np np' nn'
wenzelm@33268
   470
        have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
wenzelm@33268
   471
          "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@33268
   472
          "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)" 
wenzelm@33268
   473
          "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
wenzelm@33268
   474
        {assume mn: "m = n" 
krauss@41811
   475
          from "4.hyps"(2,3)[OF norm(1,4), of n]
krauss@41811
   476
            "4.hyps"(4,6)[OF norm(1,2), of n] norm nn' mn
wenzelm@33268
   477
          have degs:  "degreen (?cnp *\<^sub>p c') n = 
wenzelm@33268
   478
            (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"
wenzelm@33268
   479
            "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n" by (simp_all add: min_def)
wenzelm@33268
   480
          from degs norm
wenzelm@33268
   481
          have th1: "degreen(?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" by simp
wenzelm@33268
   482
          hence neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
wenzelm@33268
   483
            by simp
wenzelm@33268
   484
          have nmin: "n \<le> min n n" by (simp add: min_def)
wenzelm@33268
   485
          from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
wenzelm@33268
   486
          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 
krauss@41811
   487
          from "4.hyps"(1-3)[OF norm(1,4), of n]
krauss@41811
   488
            "4.hyps"(4-6)[OF norm(1,2), of n]
wenzelm@33268
   489
            mn norm m nn' deg
wenzelm@33268
   490
          have ?eq by simp}
wenzelm@33268
   491
        moreover
wenzelm@33268
   492
        {assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
wenzelm@33268
   493
          from nn' m np have max1: "m \<le> max n n"  by simp 
wenzelm@33268
   494
          hence min1: "m \<le> min n n" by simp     
wenzelm@33268
   495
          hence min2: "m \<le> min n (Suc n)" by simp
krauss@41811
   496
          from "4.hyps"(1-3)[OF norm(1,4) min1]
krauss@41811
   497
            "4.hyps"(4-6)[OF norm(1,2) min2]
krauss@41811
   498
            degreen_polyadd[OF norm(3,6) max1]
chaieb@33154
   499
krauss@41811
   500
          have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m 
krauss@41811
   501
            \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
krauss@41811
   502
            using mn nn' np np' by simp
krauss@41811
   503
          with "4.hyps"(1-3)[OF norm(1,4) min1]
krauss@41811
   504
            "4.hyps"(4-6)[OF norm(1,2) min2]
krauss@41811
   505
            degreen_0[OF norm(3) mn']
krauss@41811
   506
          have ?eq using nn' mn np np' by clarsimp}
wenzelm@33268
   507
        ultimately have ?eq by blast}
chaieb@33154
   508
      ultimately show ?eq by blast}
chaieb@33154
   509
    { case (2 n0 n1)
chaieb@33154
   510
      hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1" 
wenzelm@33268
   511
        and m: "m \<le> min n0 n1" by simp_all
chaieb@33154
   512
      hence mn: "m \<le> n" by simp
chaieb@33154
   513
      let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
chaieb@33154
   514
      {assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
wenzelm@33268
   515
        hence nn: "\<not>n' < n \<and> \<not> n<n'" by simp
krauss@41811
   516
        from "4.hyps"(1-3) [of n n n]
krauss@41811
   517
          "4.hyps"(4-6)[of n "Suc n" n]
wenzelm@33268
   518
          np np' C(2) mn
wenzelm@33268
   519
        have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
wenzelm@33268
   520
          "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@33268
   521
          "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)" 
wenzelm@33268
   522
          "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" 
wenzelm@33268
   523
          "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
wenzelm@33268
   524
            "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
wenzelm@33268
   525
          by (simp_all add: min_def)
wenzelm@33268
   526
            
wenzelm@33268
   527
          from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
wenzelm@33268
   528
          have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" 
wenzelm@33268
   529
            using norm by simp
wenzelm@33268
   530
        from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"]  degneq
wenzelm@33268
   531
        have "False" by simp }
chaieb@33154
   532
      thus ?case using "4.hyps" by clarsimp}
chaieb@33154
   533
qed auto
chaieb@33154
   534
chaieb@33154
   535
lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"
haftmann@36349
   536
by(induct p q rule: polymul.induct, auto simp add: field_simps)
chaieb@33154
   537
chaieb@33154
   538
lemma polymul_normh: 
haftmann@36409
   539
    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
   540
  shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
chaieb@33154
   541
  using polymul_properties(1)  by blast
chaieb@33154
   542
lemma polymul_eq0_iff: 
haftmann@36409
   543
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
   544
  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) "
chaieb@33154
   545
  using polymul_properties(2)  by blast
chaieb@33154
   546
lemma polymul_degreen:  
haftmann@36409
   547
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
   548
  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)"
chaieb@33154
   549
  using polymul_properties(3) by blast
chaieb@33154
   550
lemma polymul_norm:   
haftmann@36409
   551
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
   552
  shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul (p,q))"
chaieb@33154
   553
  using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
chaieb@33154
   554
chaieb@33154
   555
lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
chaieb@33154
   556
  by (induct p arbitrary: n0 rule: headconst.induct, auto)
chaieb@33154
   557
chaieb@33154
   558
lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
chaieb@33154
   559
  by (induct p arbitrary: n0, auto)
chaieb@33154
   560
chaieb@33154
   561
lemma monic_eqI: assumes np: "isnpolyh p n0" 
haftmann@36409
   562
  shows "INum (headconst p) * Ipoly bs (fst (monic p)) = (Ipoly bs p ::'a::{field_char_0, field_inverse_zero, power})"
chaieb@33154
   563
  unfolding monic_def Let_def
chaieb@33154
   564
proof(cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
chaieb@33154
   565
  let ?h = "headconst p"
chaieb@33154
   566
  assume pz: "p \<noteq> 0\<^sub>p"
chaieb@33154
   567
  {assume hz: "INum ?h = (0::'a)"
chaieb@33154
   568
    from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
chaieb@33154
   569
    from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
chaieb@33154
   570
    with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
chaieb@33154
   571
  thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast
chaieb@33154
   572
qed
chaieb@33154
   573
chaieb@33154
   574
krauss@41404
   575
text{* polyneg is a negation and preserves normal forms *}
chaieb@33154
   576
chaieb@33154
   577
lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
chaieb@33154
   578
by (induct p rule: polyneg.induct, auto)
chaieb@33154
   579
chaieb@33154
   580
lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"
chaieb@33154
   581
  by (induct p arbitrary: n rule: polyneg.induct, auto simp add: Nneg_def)
chaieb@33154
   582
lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
chaieb@33154
   583
  by (induct p arbitrary: n0 rule: polyneg.induct, auto)
chaieb@33154
   584
lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "
chaieb@33154
   585
by (induct p rule: polyneg.induct, auto simp add: polyneg0)
chaieb@33154
   586
chaieb@33154
   587
lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
chaieb@33154
   588
  using isnpoly_def polyneg_normh by simp
chaieb@33154
   589
chaieb@33154
   590
krauss@41404
   591
text{* polysub is a substraction and preserves normal forms *}
krauss@41404
   592
chaieb@33154
   593
lemma polysub[simp]: "Ipoly bs (polysub (p,q)) = (Ipoly bs p) - (Ipoly bs q)"
chaieb@33154
   594
by (simp add: polysub_def polyneg polyadd)
chaieb@33154
   595
lemma polysub_normh: "\<And> n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub(p,q)) (min n0 n1)"
chaieb@33154
   596
by (simp add: polysub_def polyneg_normh polyadd_normh)
chaieb@33154
   597
chaieb@33154
   598
lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub(p,q))"
chaieb@33154
   599
  using polyadd_norm polyneg_norm by (simp add: polysub_def) 
haftmann@36409
   600
lemma polysub_same_0[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
   601
  shows "isnpolyh p n0 \<Longrightarrow> polysub (p, p) = 0\<^sub>p"
chaieb@33154
   602
unfolding polysub_def split_def fst_conv snd_conv
chaieb@33154
   603
by (induct p arbitrary: n0,auto simp add: Let_def Nsub0[simplified Nsub_def])
chaieb@33154
   604
chaieb@33154
   605
lemma polysub_0: 
haftmann@36409
   606
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
   607
  shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"
chaieb@33154
   608
  unfolding polysub_def split_def fst_conv snd_conv
krauss@41763
   609
  by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
krauss@41763
   610
  (auto simp: Nsub0[simplified Nsub_def] Let_def)
chaieb@33154
   611
chaieb@33154
   612
text{* polypow is a power function and preserves normal forms *}
krauss@41404
   613
haftmann@36409
   614
lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{field_char_0, field_inverse_zero})) ^ n"
chaieb@33154
   615
proof(induct n rule: polypow.induct)
chaieb@33154
   616
  case 1 thus ?case by simp
chaieb@33154
   617
next
chaieb@33154
   618
  case (2 n)
chaieb@33154
   619
  let ?q = "polypow ((Suc n) div 2) p"
chaieb@33154
   620
  let ?d = "polymul(?q,?q)"
chaieb@33154
   621
  have "odd (Suc n) \<or> even (Suc n)" by simp
chaieb@33154
   622
  moreover 
chaieb@33154
   623
  {assume odd: "odd (Suc n)"
chaieb@33154
   624
    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
chaieb@33154
   625
    from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul(p, ?d))" by (simp add: Let_def)
chaieb@33154
   626
    also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"
chaieb@33154
   627
      using "2.hyps" by simp
chaieb@33154
   628
    also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
chaieb@33154
   629
      apply (simp only: power_add power_one_right) by simp
chaieb@33154
   630
    also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat)))))"
chaieb@33154
   631
      by (simp only: th)
chaieb@33154
   632
    finally have ?case 
chaieb@33154
   633
    using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
chaieb@33154
   634
  moreover 
chaieb@33154
   635
  {assume even: "even (Suc n)"
chaieb@33154
   636
    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
chaieb@33154
   637
    from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
chaieb@33154
   638
    also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
chaieb@33154
   639
      using "2.hyps" apply (simp only: power_add) by simp
chaieb@33154
   640
    finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}
chaieb@33154
   641
  ultimately show ?case by blast
chaieb@33154
   642
qed
chaieb@33154
   643
chaieb@33154
   644
lemma polypow_normh: 
haftmann@36409
   645
    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
   646
  shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
chaieb@33154
   647
proof (induct k arbitrary: n rule: polypow.induct)
chaieb@33154
   648
  case (2 k n)
chaieb@33154
   649
  let ?q = "polypow (Suc k div 2) p"
chaieb@33154
   650
  let ?d = "polymul (?q,?q)"
chaieb@33154
   651
  from prems have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
chaieb@33154
   652
  from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
chaieb@33154
   653
  from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul(p,?d)) n" by simp
chaieb@33154
   654
  from dn on show ?case by (simp add: Let_def)
chaieb@33154
   655
qed auto 
chaieb@33154
   656
chaieb@33154
   657
lemma polypow_norm:   
haftmann@36409
   658
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
   659
  shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
chaieb@33154
   660
  by (simp add: polypow_normh isnpoly_def)
chaieb@33154
   661
krauss@41404
   662
text{* Finally the whole normalization *}
chaieb@33154
   663
haftmann@36409
   664
lemma polynate[simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0, field_inverse_zero})"
chaieb@33154
   665
by (induct p rule:polynate.induct, auto)
chaieb@33154
   666
chaieb@33154
   667
lemma polynate_norm[simp]: 
haftmann@36409
   668
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
   669
  shows "isnpoly (polynate p)"
chaieb@33154
   670
  by (induct p rule: polynate.induct, simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm) (simp_all add: isnpoly_def)
chaieb@33154
   671
chaieb@33154
   672
text{* shift1 *}
chaieb@33154
   673
chaieb@33154
   674
chaieb@33154
   675
lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
chaieb@33154
   676
by (simp add: shift1_def polymul)
chaieb@33154
   677
chaieb@33154
   678
lemma shift1_isnpoly: 
chaieb@33154
   679
  assumes pn: "isnpoly p" and pnz: "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p) "
chaieb@33154
   680
  using pn pnz by (simp add: shift1_def isnpoly_def )
chaieb@33154
   681
chaieb@33154
   682
lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
chaieb@33154
   683
  by (simp add: shift1_def)
chaieb@33154
   684
lemma funpow_shift1_isnpoly: 
chaieb@33154
   685
  "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
haftmann@39246
   686
  by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
chaieb@33154
   687
chaieb@33154
   688
lemma funpow_isnpolyh: 
chaieb@33154
   689
  assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n "and np: "isnpolyh p n"
chaieb@33154
   690
  shows "isnpolyh (funpow k f p) n"
chaieb@33154
   691
  using f np by (induct k arbitrary: p, auto)
chaieb@33154
   692
haftmann@36409
   693
lemma funpow_shift1: "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (Mul (Pw (Bound 0) n) p)"
chaieb@33154
   694
  by (induct n arbitrary: p, simp_all add: shift1_isnpoly shift1 power_Suc )
chaieb@33154
   695
chaieb@33154
   696
lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
chaieb@33154
   697
  using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
chaieb@33154
   698
chaieb@33154
   699
lemma funpow_shift1_1: 
haftmann@36409
   700
  "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0, field_inverse_zero}) = Ipoly bs (funpow n shift1 1\<^sub>p *\<^sub>p p)"
chaieb@33154
   701
  by (simp add: funpow_shift1)
chaieb@33154
   702
chaieb@33154
   703
lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
haftmann@36349
   704
by (induct p  arbitrary: n0 rule: poly_cmul.induct, auto simp add: field_simps)
chaieb@33154
   705
chaieb@33154
   706
lemma behead:
chaieb@33154
   707
  assumes np: "isnpolyh p n"
haftmann@36409
   708
  shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = (Ipoly bs p :: 'a :: {field_char_0, field_inverse_zero})"
chaieb@33154
   709
  using np
chaieb@33154
   710
proof (induct p arbitrary: n rule: behead.induct)
chaieb@33154
   711
  case (1 c p n) hence pn: "isnpolyh p n" by simp
chaieb@33154
   712
  from prems(2)[OF pn] 
chaieb@33154
   713
  have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" . 
chaieb@33154
   714
  then show ?case using "1.hyps" apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
haftmann@36349
   715
    by (simp_all add: th[symmetric] field_simps power_Suc)
chaieb@33154
   716
qed (auto simp add: Let_def)
chaieb@33154
   717
chaieb@33154
   718
lemma behead_isnpolyh:
chaieb@33154
   719
  assumes np: "isnpolyh p n" shows "isnpolyh (behead p) n"
chaieb@33154
   720
  using np by (induct p rule: behead.induct, auto simp add: Let_def isnpolyh_mono)
chaieb@33154
   721
krauss@41404
   722
subsection{* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
chaieb@33154
   723
lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
haftmann@39246
   724
proof(induct p arbitrary: n rule: poly.induct, auto)
chaieb@33154
   725
  case (goal1 c n p n')
chaieb@33154
   726
  hence "n = Suc (n - 1)" by simp
chaieb@33154
   727
  hence "isnpolyh p (Suc (n - 1))"  using `isnpolyh p n` by simp
chaieb@33154
   728
  with prems(2) show ?case by simp
chaieb@33154
   729
qed
chaieb@33154
   730
chaieb@33154
   731
lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
chaieb@33154
   732
by (induct p arbitrary: n0 rule: isconstant.induct, auto simp add: isnpolyh_polybound0)
chaieb@33154
   733
chaieb@33154
   734
lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" by (induct p, auto)
chaieb@33154
   735
chaieb@33154
   736
lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
chaieb@33154
   737
  apply (induct p arbitrary: n0, auto)
chaieb@33154
   738
  apply (atomize)
chaieb@33154
   739
  apply (erule_tac x = "Suc nat" in allE)
chaieb@33154
   740
  apply auto
chaieb@33154
   741
  done
chaieb@33154
   742
chaieb@33154
   743
lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
chaieb@33154
   744
 by (induct p  arbitrary: n0 rule: head.induct, auto intro: isnpolyh_polybound0)
chaieb@33154
   745
chaieb@33154
   746
lemma polybound0_I:
chaieb@33154
   747
  assumes nb: "polybound0 a"
chaieb@33154
   748
  shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
chaieb@33154
   749
using nb
haftmann@39246
   750
by (induct a rule: poly.induct) auto 
chaieb@33154
   751
lemma polysubst0_I:
chaieb@33154
   752
  shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
chaieb@33154
   753
  by (induct t) simp_all
chaieb@33154
   754
chaieb@33154
   755
lemma polysubst0_I':
chaieb@33154
   756
  assumes nb: "polybound0 a"
chaieb@33154
   757
  shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t"
chaieb@33154
   758
  by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
chaieb@33154
   759
chaieb@33154
   760
lemma decrpoly: assumes nb: "polybound0 t"
chaieb@33154
   761
  shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
chaieb@33154
   762
  using nb by (induct t rule: decrpoly.induct, simp_all)
chaieb@33154
   763
chaieb@33154
   764
lemma polysubst0_polybound0: assumes nb: "polybound0 t"
chaieb@33154
   765
  shows "polybound0 (polysubst0 t a)"
haftmann@39246
   766
using nb by (induct a rule: poly.induct, auto)
chaieb@33154
   767
chaieb@33154
   768
lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
chaieb@33154
   769
  by (induct p arbitrary: n rule: degree.induct, auto simp add: isnpolyh_polybound0)
chaieb@33154
   770
haftmann@39246
   771
primrec maxindex :: "poly \<Rightarrow> nat" where
chaieb@33154
   772
  "maxindex (Bound n) = n + 1"
chaieb@33154
   773
| "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
chaieb@33154
   774
| "maxindex (Add p q) = max (maxindex p) (maxindex q)"
chaieb@33154
   775
| "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
chaieb@33154
   776
| "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
chaieb@33154
   777
| "maxindex (Neg p) = maxindex p"
chaieb@33154
   778
| "maxindex (Pw p n) = maxindex p"
chaieb@33154
   779
| "maxindex (C x) = 0"
chaieb@33154
   780
chaieb@33154
   781
definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool" where
chaieb@33154
   782
  "wf_bs bs p = (length bs \<ge> maxindex p)"
chaieb@33154
   783
chaieb@33154
   784
lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
chaieb@33154
   785
proof(induct p rule: coefficients.induct)
chaieb@33154
   786
  case (1 c p) 
chaieb@33154
   787
  show ?case 
chaieb@33154
   788
  proof
chaieb@33154
   789
    fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"
chaieb@33154
   790
    hence "x = c \<or> x \<in> set (coefficients p)" by simp
chaieb@33154
   791
    moreover 
chaieb@33154
   792
    {assume "x = c" hence "wf_bs bs x" using "1.prems"  unfolding wf_bs_def by simp}
chaieb@33154
   793
    moreover 
chaieb@33154
   794
    {assume H: "x \<in> set (coefficients p)" 
chaieb@33154
   795
      from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp
chaieb@33154
   796
      with "1.hyps" H have "wf_bs bs x" by blast }
chaieb@33154
   797
    ultimately  show "wf_bs bs x" by blast
chaieb@33154
   798
  qed
chaieb@33154
   799
qed simp_all
chaieb@33154
   800
chaieb@33154
   801
lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
chaieb@33154
   802
by (induct p rule: coefficients.induct, auto)
chaieb@33154
   803
chaieb@33154
   804
lemma wf_bs_I: "wf_bs bs p ==> Ipoly (bs@bs') p = Ipoly bs p"
chaieb@33154
   805
  unfolding wf_bs_def by (induct p, auto simp add: nth_append)
chaieb@33154
   806
chaieb@33154
   807
lemma take_maxindex_wf: assumes wf: "wf_bs bs p" 
chaieb@33154
   808
  shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
chaieb@33154
   809
proof-
chaieb@33154
   810
  let ?ip = "maxindex p"
chaieb@33154
   811
  let ?tbs = "take ?ip bs"
chaieb@33154
   812
  from wf have "length ?tbs = ?ip" unfolding wf_bs_def by simp
chaieb@33154
   813
  hence wf': "wf_bs ?tbs p" unfolding wf_bs_def by  simp
chaieb@33154
   814
  have eq: "bs = ?tbs @ (drop ?ip bs)" by simp
chaieb@33154
   815
  from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis using eq by simp
chaieb@33154
   816
qed
chaieb@33154
   817
chaieb@33154
   818
lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
chaieb@33154
   819
  by (induct p, auto)
chaieb@33154
   820
chaieb@33154
   821
lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
chaieb@33154
   822
  unfolding wf_bs_def by simp
chaieb@33154
   823
chaieb@33154
   824
lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
chaieb@33154
   825
  unfolding wf_bs_def by simp
chaieb@33154
   826
chaieb@33154
   827
chaieb@33154
   828
chaieb@33154
   829
lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
chaieb@33154
   830
by(induct p rule: coefficients.induct, auto simp add: wf_bs_def)
chaieb@33154
   831
lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
chaieb@33154
   832
  by (induct p rule: coefficients.induct, simp_all)
chaieb@33154
   833
chaieb@33154
   834
chaieb@33154
   835
lemma coefficients_head: "last (coefficients p) = head p"
chaieb@33154
   836
  by (induct p rule: coefficients.induct, auto)
chaieb@33154
   837
chaieb@33154
   838
lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
chaieb@33154
   839
  unfolding wf_bs_def by (induct p rule: decrpoly.induct, auto)
chaieb@33154
   840
chaieb@33154
   841
lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
chaieb@33154
   842
  apply (rule exI[where x="replicate (n - length xs) z"])
chaieb@33154
   843
  by simp
chaieb@33154
   844
lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
chaieb@33154
   845
by (cases p, auto) (case_tac "nat", simp_all)
chaieb@33154
   846
chaieb@33154
   847
lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
chaieb@33154
   848
  unfolding wf_bs_def 
chaieb@33154
   849
  apply (induct p q rule: polyadd.induct)
chaieb@33154
   850
  apply (auto simp add: Let_def)
chaieb@33154
   851
  done
chaieb@33154
   852
chaieb@33154
   853
lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
krauss@41811
   854
  unfolding wf_bs_def 
chaieb@33154
   855
  apply (induct p q arbitrary: bs rule: polymul.induct) 
chaieb@33154
   856
  apply (simp_all add: wf_bs_polyadd)
chaieb@33154
   857
  apply clarsimp
chaieb@33154
   858
  apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
chaieb@33154
   859
  apply auto
chaieb@33154
   860
  done
chaieb@33154
   861
chaieb@33154
   862
lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
chaieb@33154
   863
  unfolding wf_bs_def by (induct p rule: polyneg.induct, auto)
chaieb@33154
   864
chaieb@33154
   865
lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
chaieb@33154
   866
  unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
chaieb@33154
   867
chaieb@33154
   868
subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}
chaieb@33154
   869
chaieb@33154
   870
definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
chaieb@33154
   871
definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)"
chaieb@33154
   872
definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"
chaieb@33154
   873
chaieb@33154
   874
lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0"
chaieb@33154
   875
proof (induct p arbitrary: n0 rule: coefficients.induct)
chaieb@33154
   876
  case (1 c p n0)
chaieb@33154
   877
  have cp: "isnpolyh (CN c 0 p) n0" by fact
chaieb@33154
   878
  hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
chaieb@33154
   879
    by (auto simp add: isnpolyh_mono[where n'=0])
chaieb@33154
   880
  from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp 
chaieb@33154
   881
qed auto
chaieb@33154
   882
chaieb@33154
   883
lemma coefficients_isconst:
chaieb@33154
   884
  "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
chaieb@33154
   885
  by (induct p arbitrary: n rule: coefficients.induct, 
chaieb@33154
   886
    auto simp add: isnpolyh_Suc_const)
chaieb@33154
   887
chaieb@33154
   888
lemma polypoly_polypoly':
chaieb@33154
   889
  assumes np: "isnpolyh p n0"
chaieb@33154
   890
  shows "polypoly (x#bs) p = polypoly' bs p"
chaieb@33154
   891
proof-
chaieb@33154
   892
  let ?cf = "set (coefficients p)"
chaieb@33154
   893
  from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
chaieb@33154
   894
  {fix q assume q: "q \<in> ?cf"
chaieb@33154
   895
    from q cn_norm have th: "isnpolyh q n0" by blast
chaieb@33154
   896
    from coefficients_isconst[OF np] q have "isconstant q" by blast
chaieb@33154
   897
    with isconstant_polybound0[OF th] have "polybound0 q" by blast}
chaieb@33154
   898
  hence "\<forall>q \<in> ?cf. polybound0 q" ..
chaieb@33154
   899
  hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"
chaieb@33154
   900
    using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
chaieb@33154
   901
    by auto
chaieb@33154
   902
  
chaieb@33154
   903
  thus ?thesis unfolding polypoly_def polypoly'_def by simp 
chaieb@33154
   904
qed
chaieb@33154
   905
chaieb@33154
   906
lemma polypoly_poly:
chaieb@33154
   907
  assumes np: "isnpolyh p n0" shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
chaieb@33154
   908
  using np 
chaieb@33154
   909
by (induct p arbitrary: n0 bs rule: coefficients.induct, auto simp add: polypoly_def)
chaieb@33154
   910
chaieb@33154
   911
lemma polypoly'_poly: 
chaieb@33154
   912
  assumes np: "isnpolyh p n0" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
chaieb@33154
   913
  using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
chaieb@33154
   914
chaieb@33154
   915
chaieb@33154
   916
lemma polypoly_poly_polybound0:
chaieb@33154
   917
  assumes np: "isnpolyh p n0" and nb: "polybound0 p"
chaieb@33154
   918
  shows "polypoly bs p = [Ipoly bs p]"
chaieb@33154
   919
  using np nb unfolding polypoly_def 
chaieb@33154
   920
  by (cases p, auto, case_tac nat, auto)
chaieb@33154
   921
chaieb@33154
   922
lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0" 
chaieb@33154
   923
  by (induct p rule: head.induct, auto)
chaieb@33154
   924
chaieb@33154
   925
lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
chaieb@33154
   926
  by (cases p,auto)
chaieb@33154
   927
chaieb@33154
   928
lemma head_eq_headn0: "head p = headn p 0"
chaieb@33154
   929
  by (induct p rule: head.induct, simp_all)
chaieb@33154
   930
chaieb@33154
   931
lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
chaieb@33154
   932
  by (simp add: head_eq_headn0)
chaieb@33154
   933
chaieb@33154
   934
lemma isnpolyh_zero_iff: 
haftmann@36409
   935
  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})"
chaieb@33154
   936
  shows "p = 0\<^sub>p"
chaieb@33154
   937
using nq eq
berghofe@34915
   938
proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
berghofe@34915
   939
  case less
berghofe@34915
   940
  note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
berghofe@34915
   941
  {assume nz: "maxindex p = 0"
berghofe@34915
   942
    then obtain c where "p = C c" using np by (cases p, auto)
chaieb@33154
   943
    with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}
chaieb@33154
   944
  moreover
berghofe@34915
   945
  {assume nz: "maxindex p \<noteq> 0"
chaieb@33154
   946
    let ?h = "head p"
chaieb@33154
   947
    let ?hd = "decrpoly ?h"
chaieb@33154
   948
    let ?ihd = "maxindex ?hd"
chaieb@33154
   949
    from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h" 
chaieb@33154
   950
      by simp_all
chaieb@33154
   951
    hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast
chaieb@33154
   952
    
chaieb@33154
   953
    from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
berghofe@34915
   954
    have mihn: "maxindex ?h \<le> maxindex p" by auto
berghofe@34915
   955
    with decr_maxindex[OF h(2)] nz  have ihd_lt_n: "?ihd < maxindex p" by auto
chaieb@33154
   956
    {fix bs:: "'a list"  assume bs: "wf_bs bs ?hd"
chaieb@33154
   957
      let ?ts = "take ?ihd bs"
chaieb@33154
   958
      let ?rs = "drop ?ihd bs"
chaieb@33154
   959
      have ts: "wf_bs ?ts ?hd" using bs unfolding wf_bs_def by simp
chaieb@33154
   960
      have bs_ts_eq: "?ts@ ?rs = bs" by simp
chaieb@33154
   961
      from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x#?ts) ?h" by simp
berghofe@34915
   962
      from ihd_lt_n have "ALL x. length (x#?ts) \<le> maxindex p" by simp
berghofe@34915
   963
      with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = maxindex p" by blast
berghofe@34915
   964
      hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" unfolding wf_bs_def by simp
chaieb@33154
   965
      with zp have "\<forall> x. Ipoly ((x#?ts) @ xs) p = 0" by blast
chaieb@33154
   966
      hence "\<forall> x. Ipoly (x#(?ts @ xs)) p = 0" by simp
chaieb@33154
   967
      with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
chaieb@33154
   968
      have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"  by simp
chaieb@33154
   969
      hence "poly (polypoly' (?ts @ xs) p) = poly []" by (auto intro: ext) 
chaieb@33154
   970
      hence "\<forall> c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
wenzelm@33268
   971
        using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
chaieb@33154
   972
      with coefficients_head[of p, symmetric]
chaieb@33154
   973
      have th0: "Ipoly (?ts @ xs) ?hd = 0" by simp
chaieb@33154
   974
      from bs have wf'': "wf_bs ?ts ?hd" unfolding wf_bs_def by simp
chaieb@33154
   975
      with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0" by simp
chaieb@33154
   976
      with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by simp }
chaieb@33154
   977
    then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast
chaieb@33154
   978
    
berghofe@34915
   979
    from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast
chaieb@33154
   980
    hence "?h = 0\<^sub>p" by simp
chaieb@33154
   981
    with head_nz[OF np] have "p = 0\<^sub>p" by simp}
chaieb@33154
   982
  ultimately show "p = 0\<^sub>p" by blast
chaieb@33154
   983
qed
chaieb@33154
   984
chaieb@33154
   985
lemma isnpolyh_unique:  
chaieb@33154
   986
  assumes np:"isnpolyh p n0" and nq: "isnpolyh q n1"
haftmann@36409
   987
  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"
chaieb@33154
   988
proof(auto)
chaieb@33154
   989
  assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a)= \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
chaieb@33154
   990
  hence "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" by simp
chaieb@33154
   991
  hence "\<forall>bs. wf_bs bs (p -\<^sub>p q) \<longrightarrow> \<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" 
chaieb@33154
   992
    using wf_bs_polysub[where p=p and q=q] by auto
chaieb@33154
   993
  with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq]
chaieb@33154
   994
  show "p = q" by blast
chaieb@33154
   995
qed
chaieb@33154
   996
chaieb@33154
   997
krauss@41404
   998
text{* consequences of unicity on the algorithms for polynomial normalization *}
chaieb@33154
   999
haftmann@36409
  1000
lemma polyadd_commute:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
  1001
  and np: "isnpolyh p n0" and nq: "isnpolyh q n1" shows "p +\<^sub>p q = q +\<^sub>p p"
chaieb@33154
  1002
  using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]] by simp
chaieb@33154
  1003
chaieb@33154
  1004
lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp
chaieb@33154
  1005
lemma one_normh: "isnpolyh 1\<^sub>p n" by simp
chaieb@33154
  1006
lemma polyadd_0[simp]: 
haftmann@36409
  1007
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
  1008
  and np: "isnpolyh p n0" shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>p p = p"
chaieb@33154
  1009
  using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np] 
chaieb@33154
  1010
    isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
chaieb@33154
  1011
chaieb@33154
  1012
lemma polymul_1[simp]: 
haftmann@36409
  1013
    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
  1014
  and np: "isnpolyh p n0" shows "p *\<^sub>p 1\<^sub>p = p" and "1\<^sub>p *\<^sub>p p = p"
chaieb@33154
  1015
  using isnpolyh_unique[OF polymul_normh[OF np one_normh] np] 
chaieb@33154
  1016
    isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
chaieb@33154
  1017
lemma polymul_0[simp]: 
haftmann@36409
  1018
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
  1019
  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"
chaieb@33154
  1020
  using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh] 
chaieb@33154
  1021
    isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
chaieb@33154
  1022
chaieb@33154
  1023
lemma polymul_commute: 
haftmann@36409
  1024
    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
  1025
  and np:"isnpolyh p n0" and nq: "isnpolyh q n1"
chaieb@33154
  1026
  shows "p *\<^sub>p q = q *\<^sub>p p"
haftmann@36409
  1027
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
chaieb@33154
  1028
chaieb@33154
  1029
declare polyneg_polyneg[simp]
chaieb@33154
  1030
  
chaieb@33154
  1031
lemma isnpolyh_polynate_id[simp]: 
haftmann@36409
  1032
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
  1033
  and np:"isnpolyh p n0" shows "polynate p = p"
haftmann@36409
  1034
  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
chaieb@33154
  1035
chaieb@33154
  1036
lemma polynate_idempotent[simp]: 
haftmann@36409
  1037
    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
  1038
  shows "polynate (polynate p) = polynate p"
chaieb@33154
  1039
  using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
chaieb@33154
  1040
chaieb@33154
  1041
lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
chaieb@33154
  1042
  unfolding poly_nate_def polypoly'_def ..
haftmann@36409
  1043
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>)"
chaieb@33154
  1044
  using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
chaieb@33154
  1045
  unfolding poly_nate_polypoly' by (auto intro: ext)
chaieb@33154
  1046
chaieb@33154
  1047
subsection{* heads, degrees and all that *}
chaieb@33154
  1048
lemma degree_eq_degreen0: "degree p = degreen p 0"
chaieb@33154
  1049
  by (induct p rule: degree.induct, simp_all)
chaieb@33154
  1050
chaieb@33154
  1051
lemma degree_polyneg: assumes n: "isnpolyh p n"
chaieb@33154
  1052
  shows "degree (polyneg p) = degree p"
chaieb@33154
  1053
  using n
chaieb@33154
  1054
  by (induct p arbitrary: n rule: polyneg.induct, simp_all) (case_tac na, auto)
chaieb@33154
  1055
chaieb@33154
  1056
lemma degree_polyadd:
chaieb@33154
  1057
  assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
chaieb@33154
  1058
  shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
chaieb@33154
  1059
using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
chaieb@33154
  1060
chaieb@33154
  1061
chaieb@33154
  1062
lemma degree_polysub: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
chaieb@33154
  1063
  shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
chaieb@33154
  1064
proof-
chaieb@33154
  1065
  from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp
chaieb@33154
  1066
  from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])
chaieb@33154
  1067
qed
chaieb@33154
  1068
chaieb@33154
  1069
lemma degree_polysub_samehead: 
haftmann@36409
  1070
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
  1071
  and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q" 
chaieb@33154
  1072
  and d: "degree p = degree q"
chaieb@33154
  1073
  shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
chaieb@33154
  1074
unfolding polysub_def split_def fst_conv snd_conv
chaieb@33154
  1075
using np nq h d
chaieb@33154
  1076
proof(induct p q rule:polyadd.induct)
chaieb@33154
  1077
  case (1 a b a' b') thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def]) 
chaieb@33154
  1078
next
chaieb@33154
  1079
  case (2 a b c' n' p') 
chaieb@33154
  1080
  let ?c = "(a,b)"
chaieb@33154
  1081
  from prems have "degree (C ?c) = degree (CN c' n' p')" by simp
chaieb@33154
  1082
  hence nz:"n' > 0" by (cases n', auto)
chaieb@33154
  1083
  hence "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
chaieb@33154
  1084
  with prems show ?case by simp
chaieb@33154
  1085
next
chaieb@33154
  1086
  case (3 c n p a' b') 
chaieb@33154
  1087
  let ?c' = "(a',b')"
chaieb@33154
  1088
  from prems have "degree (C ?c') = degree (CN c n p)" by simp
chaieb@33154
  1089
  hence nz:"n > 0" by (cases n, auto)
chaieb@33154
  1090
  hence "head (CN c n p) = CN c n p" by (cases n, auto)
chaieb@33154
  1091
  with prems show ?case by simp
chaieb@33154
  1092
next
chaieb@33154
  1093
  case (4 c n p c' n' p')
chaieb@33154
  1094
  hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1" 
chaieb@33154
  1095
    "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+
chaieb@33154
  1096
  hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all  
chaieb@33154
  1097
  hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0" 
chaieb@33154
  1098
    using H(1-2) degree_polyneg by auto
chaieb@33154
  1099
  from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"  by simp+
chaieb@33154
  1100
  from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"  by simp
chaieb@33154
  1101
  from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto
chaieb@33154
  1102
  have "n = n' \<or> n < n' \<or> n > n'" by arith
chaieb@33154
  1103
  moreover
chaieb@33154
  1104
  {assume nn': "n = n'"
chaieb@33154
  1105
    have "n = 0 \<or> n >0" by arith
chaieb@33154
  1106
    moreover {assume nz: "n = 0" hence ?case using prems by (auto simp add: Let_def degcmc')}
chaieb@33154
  1107
    moreover {assume nz: "n > 0"
chaieb@33154
  1108
      with nn' H(3) have  cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
chaieb@33154
  1109
      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 split_def fst_conv snd_conv] using nn' prems by (simp add: Let_def)}
chaieb@33154
  1110
    ultimately have ?case by blast}
chaieb@33154
  1111
  moreover
chaieb@33154
  1112
  {assume nn': "n < n'" hence n'p: "n' > 0" by simp 
chaieb@33154
  1113
    hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n', simp_all)
chaieb@33154
  1114
    have degcnp': "degree (CN c' n' p') = 0" and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')" using prems by (cases n', simp_all)
chaieb@33154
  1115
    hence "n > 0" by (cases n, simp_all)
chaieb@33154
  1116
    hence headcnp: "head (CN c n p) = CN c n p" by (cases n, auto)
chaieb@33154
  1117
    from H(3) headcnp headcnp' nn' have ?case by auto}
chaieb@33154
  1118
  moreover
chaieb@33154
  1119
  {assume nn': "n > n'"  hence np: "n > 0" by simp 
chaieb@33154
  1120
    hence headcnp:"head (CN c n p) = CN c n p"  by (cases n, simp_all)
chaieb@33154
  1121
    from prems have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
chaieb@33154
  1122
    from np have degcnp: "degree (CN c n p) = 0" by (cases n, simp_all)
chaieb@33154
  1123
    with degcnpeq have "n' > 0" by (cases n', simp_all)
chaieb@33154
  1124
    hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
chaieb@33154
  1125
    from H(3) headcnp headcnp' nn' have ?case by auto}
chaieb@33154
  1126
  ultimately show ?case  by blast
chaieb@33154
  1127
qed auto 
chaieb@33154
  1128
 
chaieb@33154
  1129
lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
chaieb@33154
  1130
by (induct p arbitrary: n0 rule: head.induct, simp_all add: shift1_def)
chaieb@33154
  1131
chaieb@33154
  1132
lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
chaieb@33154
  1133
proof(induct k arbitrary: n0 p)
chaieb@33154
  1134
  case (Suc k n0 p) hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
chaieb@33154
  1135
  with prems have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
chaieb@33154
  1136
    and "head (shift1 p) = head p" by (simp_all add: shift1_head) 
haftmann@39246
  1137
  thus ?case by (simp add: funpow_swap1)
chaieb@33154
  1138
qed auto  
chaieb@33154
  1139
chaieb@33154
  1140
lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
chaieb@33154
  1141
  by (simp add: shift1_def)
chaieb@33154
  1142
lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
chaieb@33154
  1143
  by (induct k arbitrary: p, auto simp add: shift1_degree)
chaieb@33154
  1144
chaieb@33154
  1145
lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
chaieb@33154
  1146
  by (induct n arbitrary: p, simp_all add: funpow_def)
chaieb@33154
  1147
chaieb@33154
  1148
lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
chaieb@33154
  1149
  by (induct p arbitrary: n rule: degree.induct, auto)
chaieb@33154
  1150
lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
chaieb@33154
  1151
  by (induct p arbitrary: n rule: degreen.induct, auto)
chaieb@33154
  1152
lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
chaieb@33154
  1153
  by (induct p arbitrary: n rule: degree.induct, auto)
chaieb@33154
  1154
lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
chaieb@33154
  1155
  by (induct p rule: head.induct, auto)
chaieb@33154
  1156
chaieb@33154
  1157
lemma polyadd_eq_const_degree: 
chaieb@33154
  1158
  "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk> \<Longrightarrow> degree p = degree q" 
chaieb@33154
  1159
  using polyadd_eq_const_degreen degree_eq_degreen0 by simp
chaieb@33154
  1160
chaieb@33154
  1161
lemma polyadd_head: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
chaieb@33154
  1162
  and deg: "degree p \<noteq> degree q"
chaieb@33154
  1163
  shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
chaieb@33154
  1164
using np nq deg
chaieb@33154
  1165
apply(induct p q arbitrary: n0 n1 rule: polyadd.induct,simp_all)
chaieb@33154
  1166
apply (case_tac n', simp, simp)
chaieb@33154
  1167
apply (case_tac n, simp, simp)
chaieb@33154
  1168
apply (case_tac n, case_tac n', simp add: Let_def)
chaieb@33154
  1169
apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p")
krauss@41763
  1170
apply (auto simp add: polyadd_eq_const_degree)
krauss@41763
  1171
apply (metis head_nz)
krauss@41763
  1172
apply (metis head_nz)
krauss@41763
  1173
apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
krauss@41763
  1174
by (metis degree.simps(9) gr0_conv_Suc nat_less_le order_le_less_trans)
chaieb@33154
  1175
chaieb@33154
  1176
lemma polymul_head_polyeq: 
haftmann@36409
  1177
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
  1178
  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"
chaieb@33154
  1179
proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
chaieb@33154
  1180
  case (2 a b c' n' p' n0 n1)
chaieb@33154
  1181
  hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum (a,b)"  by (simp_all add: head_isnpolyh)
chaieb@33154
  1182
  thus ?case using prems by (cases n', auto) 
chaieb@33154
  1183
next 
chaieb@33154
  1184
  case (3 c n p a' b' n0 n1) 
chaieb@33154
  1185
  hence "isnpolyh (head (CN c n p)) n0" "isnormNum (a',b')"  by (simp_all add: head_isnpolyh)
chaieb@33154
  1186
  thus ?case using prems by (cases n, auto)
chaieb@33154
  1187
next
chaieb@33154
  1188
  case (4 c n p c' n' p' n0 n1)
chaieb@33154
  1189
  hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
chaieb@33154
  1190
    "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
chaieb@33154
  1191
    by simp_all
chaieb@33154
  1192
  have "n < n' \<or> n' < n \<or> n = n'" by arith
chaieb@33154
  1193
  moreover 
chaieb@33154
  1194
  {assume nn': "n < n'" hence ?case 
chaieb@33154
  1195
      using norm 
krauss@41811
  1196
    "4.hyps"(5)[OF norm(1,6)]
krauss@41811
  1197
    "4.hyps"(6)[OF norm(2,6)] by (simp, cases n, simp,cases n', simp_all)}
chaieb@33154
  1198
  moreover {assume nn': "n'< n"
krauss@41811
  1199
    hence ?case using norm "4.hyps"(1) [OF norm(5,3)]
krauss@41811
  1200
      "4.hyps"(2)[OF norm(5,4)] 
chaieb@33154
  1201
      by (simp,cases n',simp,cases n,auto)}
chaieb@33154
  1202
  moreover {assume nn': "n' = n"
chaieb@33154
  1203
    from nn' polymul_normh[OF norm(5,4)] 
chaieb@33154
  1204
    have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
chaieb@33154
  1205
    from nn' polymul_normh[OF norm(5,3)] norm 
chaieb@33154
  1206
    have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
chaieb@33154
  1207
    from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
chaieb@33154
  1208
    have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp 
chaieb@33154
  1209
    from polyadd_normh[OF ncnpc' ncnpp0'] 
chaieb@33154
  1210
    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" 
chaieb@33154
  1211
      by (simp add: min_def)
chaieb@33154
  1212
    {assume np: "n > 0"
chaieb@33154
  1213
      with nn' head_isnpolyh_Suc'[OF np nth]
wenzelm@33268
  1214
        head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
chaieb@33154
  1215
      have ?case by simp}
chaieb@33154
  1216
    moreover
chaieb@33154
  1217
    {moreover assume nz: "n = 0"
chaieb@33154
  1218
      from polymul_degreen[OF norm(5,4), where m="0"]
wenzelm@33268
  1219
        polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
chaieb@33154
  1220
      norm(5,6) degree_npolyhCN[OF norm(6)]
chaieb@33154
  1221
    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
chaieb@33154
  1222
    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
chaieb@33154
  1223
    from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
krauss@41811
  1224
    have ?case   using norm "4.hyps"(1)[OF norm(5,3)]
krauss@41811
  1225
        "4.hyps"(2)[OF norm(5,4)] nn' nz by simp }
chaieb@33154
  1226
    ultimately have ?case by (cases n) auto} 
chaieb@33154
  1227
  ultimately show ?case by blast
chaieb@33154
  1228
qed simp_all
chaieb@33154
  1229
chaieb@33154
  1230
lemma degree_coefficients: "degree p = length (coefficients p) - 1"
chaieb@33154
  1231
  by(induct p rule: degree.induct, auto)
chaieb@33154
  1232
chaieb@33154
  1233
lemma degree_head[simp]: "degree (head p) = 0"
chaieb@33154
  1234
  by (induct p rule: head.induct, auto)
chaieb@33154
  1235
chaieb@33154
  1236
lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1+ degree p"
chaieb@33154
  1237
  by (cases n, simp_all)
chaieb@33154
  1238
lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
chaieb@33154
  1239
  by (cases n, simp_all)
chaieb@33154
  1240
chaieb@33154
  1241
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)"
chaieb@33154
  1242
  using polyadd_different_degreen degree_eq_degreen0 by simp
chaieb@33154
  1243
chaieb@33154
  1244
lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
chaieb@33154
  1245
  by (induct p arbitrary: n0 rule: polyneg.induct, auto)
chaieb@33154
  1246
chaieb@33154
  1247
lemma degree_polymul:
haftmann@36409
  1248
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
  1249
  and np: "isnpolyh p n0" and nq: "isnpolyh q n1"
chaieb@33154
  1250
  shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
chaieb@33154
  1251
  using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
chaieb@33154
  1252
chaieb@33154
  1253
lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
chaieb@33154
  1254
  by (induct p arbitrary: n rule: degree.induct, auto)
chaieb@33154
  1255
chaieb@33154
  1256
lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"
chaieb@33154
  1257
  by (induct p arbitrary: n rule: degree.induct, auto)
chaieb@33154
  1258
chaieb@33154
  1259
subsection {* Correctness of polynomial pseudo division *}
chaieb@33154
  1260
chaieb@33154
  1261
lemma polydivide_aux_properties:
haftmann@36409
  1262
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
  1263
  and np: "isnpolyh p n0" and ns: "isnpolyh s n1"
chaieb@33154
  1264
  and ap: "head p = a" and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
krauss@41403
  1265
  shows "(polydivide_aux a n p k s = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p) 
chaieb@33154
  1266
          \<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)))"
chaieb@33154
  1267
  using ns
berghofe@34915
  1268
proof(induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
berghofe@34915
  1269
  case less
chaieb@33154
  1270
  let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
krauss@41403
  1271
  let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) 
chaieb@33154
  1272
    \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
chaieb@33154
  1273
  let ?b = "head s"
berghofe@34915
  1274
  let ?p' = "funpow (degree s - n) shift1 p"
berghofe@34915
  1275
  let ?xdn = "funpow (degree s - n) shift1 1\<^sub>p"
chaieb@33154
  1276
  let ?akk' = "a ^\<^sub>p (k' - k)"
berghofe@34915
  1277
  note ns = `isnpolyh s n1`
chaieb@33154
  1278
  from np have np0: "isnpolyh p 0" 
chaieb@33154
  1279
    using isnpolyh_mono[where n="n0" and n'="0" and p="p"]  by simp
berghofe@34915
  1280
  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
chaieb@33154
  1281
  have headp': "head ?p' = head p" using funpow_shift1_head[OF np pnz] by simp
chaieb@33154
  1282
  from funpow_shift1_isnpoly[where p="1\<^sub>p"] have nxdn: "isnpolyh ?xdn 0" by (simp add: isnpoly_def)
chaieb@33154
  1283
  from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap 
chaieb@33154
  1284
  have nakk':"isnpolyh ?akk' 0" by blast
chaieb@33154
  1285
  {assume sz: "s = 0\<^sub>p"
krauss@41403
  1286
   hence ?ths using np polydivide_aux.simps apply clarsimp by (rule exI[where x="0\<^sub>p"], simp) }
chaieb@33154
  1287
  moreover
chaieb@33154
  1288
  {assume sz: "s \<noteq> 0\<^sub>p"
berghofe@34915
  1289
    {assume dn: "degree s < n"
krauss@41403
  1290
      hence "?ths" using ns ndp np polydivide_aux.simps by auto (rule exI[where x="0\<^sub>p"],simp) }
chaieb@33154
  1291
    moreover 
berghofe@34915
  1292
    {assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith
chaieb@33154
  1293
      have degsp': "degree s = degree ?p'" 
berghofe@34915
  1294
        using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp
chaieb@33154
  1295
      {assume ba: "?b = a"
wenzelm@33268
  1296
        hence headsp': "head s = head ?p'" using ap headp' by simp
wenzelm@33268
  1297
        have nr: "isnpolyh (s -\<^sub>p ?p') 0" using polysub_normh[OF ns np'] by simp
berghofe@34915
  1298
        from degree_polysub_samehead[OF ns np' headsp' degsp']
berghofe@34915
  1299
        have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
wenzelm@33268
  1300
        moreover 
berghofe@34915
  1301
        {assume deglt:"degree (s -\<^sub>p ?p') < degree s"
krauss@41403
  1302
          from polydivide_aux.simps sz dn' ba
krauss@41403
  1303
          have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
wenzelm@33268
  1304
            by (simp add: Let_def)
krauss@41403
  1305
          {assume h1: "polydivide_aux a n p k s = (k', r)"
berghofe@34915
  1306
            from less(1)[OF deglt nr, of k k' r]
wenzelm@33268
  1307
              trans[OF eq[symmetric] h1]
wenzelm@33268
  1308
            have kk': "k \<le> k'" and nr:"\<exists>nr. isnpolyh r nr" and dr: "degree r = 0 \<or> degree r < degree p"
wenzelm@33268
  1309
              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
wenzelm@33268
  1310
            from q1 obtain q n1 where nq: "isnpolyh q n1" 
wenzelm@33268
  1311
              and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"  by blast
wenzelm@33268
  1312
            from nr obtain nr where nr': "isnpolyh r nr" by blast
wenzelm@33268
  1313
            from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0" by simp
wenzelm@33268
  1314
            from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
wenzelm@33268
  1315
            have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
wenzelm@33268
  1316
            from polyadd_normh[OF polymul_normh[OF np 
wenzelm@33268
  1317
              polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
wenzelm@33268
  1318
            have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0" by simp 
haftmann@36409
  1319
            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')) = 
wenzelm@33268
  1320
              Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
haftmann@36409
  1321
            hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) = 
wenzelm@33268
  1322
              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r" 
haftmann@36349
  1323
              by (simp add: field_simps)
haftmann@36409
  1324
            hence " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 
berghofe@34915
  1325
              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p *\<^sub>p p) 
wenzelm@33268
  1326
              + Ipoly bs p * Ipoly bs q + Ipoly bs r"
wenzelm@33268
  1327
              by (auto simp only: funpow_shift1_1) 
haftmann@36409
  1328
            hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 
berghofe@34915
  1329
              Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p) 
haftmann@36349
  1330
              + Ipoly bs q) + Ipoly bs r" by (simp add: field_simps)
haftmann@36409
  1331
            hence "\<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 
berghofe@34915
  1332
              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
wenzelm@33268
  1333
            with isnpolyh_unique[OF nakks' nqr']
wenzelm@33268
  1334
            have "a ^\<^sub>p (k' - k) *\<^sub>p s = 
berghofe@34915
  1335
              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
wenzelm@33268
  1336
            hence ?qths using nq'
berghofe@34915
  1337
              apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 1\<^sub>p) +\<^sub>p q" in exI)
wenzelm@33268
  1338
              apply (rule_tac x="0" in exI) by simp
wenzelm@33268
  1339
            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"
krauss@41403
  1340
              by blast } hence ?ths by blast }
wenzelm@33268
  1341
        moreover 
wenzelm@33268
  1342
        {assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
haftmann@36409
  1343
          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}"]
haftmann@36409
  1344
          have " \<forall>(bs:: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs s = Ipoly bs ?p'" by simp
haftmann@36409
  1345
          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
wenzelm@33268
  1346
            by (simp only: funpow_shift1_1) simp
wenzelm@33268
  1347
          hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]] by blast
krauss@41403
  1348
          {assume h1: "polydivide_aux a n p k s = (k',r)"
krauss@41403
  1349
            from polydivide_aux.simps sz dn' ba
krauss@41403
  1350
            have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
wenzelm@33268
  1351
              by (simp add: Let_def)
krauss@41403
  1352
            also have "\<dots> = (k,0\<^sub>p)" using polydivide_aux.simps spz by simp
wenzelm@33268
  1353
            finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
berghofe@34915
  1354
            with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
krauss@41403
  1355
              polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
wenzelm@33268
  1356
              apply auto
wenzelm@33268
  1357
              apply (rule exI[where x="?xdn"])        
berghofe@34915
  1358
              apply (auto simp add: polymul_commute[of p])
krauss@41403
  1359
              done} }
wenzelm@33268
  1360
        ultimately have ?ths by blast }
chaieb@33154
  1361
      moreover
chaieb@33154
  1362
      {assume ba: "?b \<noteq> a"
wenzelm@33268
  1363
        from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns] 
wenzelm@33268
  1364
          polymul_normh[OF head_isnpolyh[OF ns] np']]
wenzelm@33268
  1365
        have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by(simp add: min_def)
wenzelm@33268
  1366
        have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
wenzelm@33268
  1367
          using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns] 
wenzelm@33268
  1368
            polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
wenzelm@33268
  1369
            funpow_shift1_nz[OF pnz] by simp_all
wenzelm@33268
  1370
        from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
berghofe@34915
  1371
          polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
wenzelm@33268
  1372
        have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')" 
wenzelm@33268
  1373
          using head_head[OF ns] funpow_shift1_head[OF np pnz]
wenzelm@33268
  1374
            polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
wenzelm@33268
  1375
          by (simp add: ap)
wenzelm@33268
  1376
        from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
wenzelm@33268
  1377
          head_nz[OF np] pnz sz ap[symmetric]
berghofe@34915
  1378
          funpow_shift1_nz[OF pnz, where n="degree s - n"]
wenzelm@33268
  1379
          polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"]  head_nz[OF ns]
berghofe@34915
  1380
          ndp dn
wenzelm@33268
  1381
        have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p') "
wenzelm@33268
  1382
          by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
berghofe@34915
  1383
        {assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
wenzelm@33268
  1384
          from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns] polymul_normh[OF head_isnpolyh[OF ns]np']]
wenzelm@33268
  1385
          ap have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by simp
krauss@41403
  1386
          {assume h1:"polydivide_aux a n p k s = (k', r)"
krauss@41403
  1387
            from h1 polydivide_aux.simps sz dn' ba
krauss@41403
  1388
            have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
wenzelm@33268
  1389
              by (simp add: Let_def)
berghofe@34915
  1390
            with less(1)[OF dth nasbp', of "Suc k" k' r]
wenzelm@33268
  1391
            obtain q nq nr where kk': "Suc k \<le> k'" and nr: "isnpolyh r nr" and nq: "isnpolyh q nq" 
wenzelm@33268
  1392
              and dr: "degree r = 0 \<or> degree r < degree p"
wenzelm@33268
  1393
              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
wenzelm@33268
  1394
            from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith
haftmann@36409
  1395
            {fix bs:: "'a::{field_char_0, field_inverse_zero} list"
wenzelm@33268
  1396
              
wenzelm@33268
  1397
            from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
wenzelm@33268
  1398
            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
wenzelm@33268
  1399
            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"
haftmann@36349
  1400
              by (simp add: field_simps power_Suc)
wenzelm@33268
  1401
            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"
berghofe@34915
  1402
              by (simp add:kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
wenzelm@33268
  1403
            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"
haftmann@36349
  1404
              by (simp add: field_simps)}
haftmann@36409
  1405
            hence ieq:"\<forall>(bs :: 'a::{field_char_0, field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 
wenzelm@33268
  1406
              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 
wenzelm@33268
  1407
            let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
wenzelm@33268
  1408
            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]]
wenzelm@33268
  1409
            have nqw: "isnpolyh ?q 0" by simp
wenzelm@33268
  1410
            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]]
wenzelm@33268
  1411
            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
krauss@41403
  1412
            from dr kk' nr h1 asth nqw have ?ths apply simp
wenzelm@33268
  1413
              apply (rule conjI)
wenzelm@33268
  1414
              apply (rule exI[where x="nr"], simp)
wenzelm@33268
  1415
              apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
wenzelm@33268
  1416
              apply (rule exI[where x="0"], simp)
wenzelm@33268
  1417
              done}
krauss@41403
  1418
          hence ?ths by blast }
wenzelm@33268
  1419
        moreover 
wenzelm@33268
  1420
        {assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
haftmann@36409
  1421
          {fix bs :: "'a::{field_char_0, field_inverse_zero} list"
wenzelm@33268
  1422
            from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
wenzelm@33268
  1423
          have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'" by simp
wenzelm@33268
  1424
          hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p" 
berghofe@34915
  1425
            by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
wenzelm@33268
  1426
          hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" by simp
wenzelm@33268
  1427
        }
haftmann@36409
  1428
        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))" ..
wenzelm@33268
  1429
          from hth
wenzelm@33268
  1430
          have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)" 
haftmann@36409
  1431
            using isnpolyh_unique[where ?'a = "'a::{field_char_0, field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns] 
chaieb@33154
  1432
                    polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
wenzelm@33268
  1433
              simplified ap] by simp
krauss@41403
  1434
          {assume h1: "polydivide_aux a n p k s = (k', r)"
krauss@41403
  1435
          from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
wenzelm@33268
  1436
          have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
wenzelm@33268
  1437
          with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
wenzelm@33268
  1438
            polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
krauss@41403
  1439
          have ?ths apply (clarsimp simp add: Let_def)
wenzelm@33268
  1440
            apply (rule exI[where x="?b *\<^sub>p ?xdn"]) apply simp
wenzelm@33268
  1441
            apply (rule exI[where x="0"], simp)
wenzelm@33268
  1442
            done}
krauss@41403
  1443
        hence ?ths by blast}
wenzelm@33268
  1444
        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"]
berghofe@34915
  1445
          head_nz[OF np] pnz sz ap[symmetric]
wenzelm@33268
  1446
          by (simp add: degree_eq_degreen0[symmetric]) blast }
chaieb@33154
  1447
      ultimately have ?ths by blast
chaieb@33154
  1448
    }
chaieb@33154
  1449
    ultimately have ?ths by blast}
chaieb@33154
  1450
  ultimately show ?ths by blast
chaieb@33154
  1451
qed
chaieb@33154
  1452
chaieb@33154
  1453
lemma polydivide_properties: 
haftmann@36409
  1454
  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
  1455
  and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
chaieb@33154
  1456
  shows "(\<exists> k r. polydivide s p = (k,r) \<and> (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) 
chaieb@33154
  1457
  \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
chaieb@33154
  1458
proof-
chaieb@33154
  1459
  have trv: "head p = head p" "degree p = degree p" by simp_all
krauss@41403
  1460
  from polydivide_def[where s="s" and p="p"] 
chaieb@33154
  1461
  have ex: "\<exists> k r. polydivide s p = (k,r)" by auto
chaieb@33154
  1462
  then obtain k r where kr: "polydivide s p = (k,r)" by blast
krauss@41403
  1463
  from trans[OF meta_eq_to_obj_eq[OF polydivide_def[where s="s"and p="p"], symmetric] kr]
chaieb@33154
  1464
    polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
chaieb@33154
  1465
  have "(degree r = 0 \<or> degree r < degree p) \<and>
chaieb@33154
  1466
   (\<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
chaieb@33154
  1467
  with kr show ?thesis 
chaieb@33154
  1468
    apply -
chaieb@33154
  1469
    apply (rule exI[where x="k"])
chaieb@33154
  1470
    apply (rule exI[where x="r"])
chaieb@33154
  1471
    apply simp
chaieb@33154
  1472
    done
chaieb@33154
  1473
qed
chaieb@33154
  1474
chaieb@33154
  1475
subsection{* More about polypoly and pnormal etc *}
chaieb@33154
  1476
chaieb@33154
  1477
definition "isnonconstant p = (\<not> isconstant p)"
chaieb@33154
  1478
chaieb@33154
  1479
lemma isnonconstant_pnormal_iff: assumes nc: "isnonconstant p" 
chaieb@33154
  1480
  shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0" 
chaieb@33154
  1481
proof
chaieb@33154
  1482
  let ?p = "polypoly bs p"  
chaieb@33154
  1483
  assume H: "pnormal ?p"
chaieb@33154
  1484
  have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
chaieb@33154
  1485
  
chaieb@33154
  1486
  from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]  
chaieb@33154
  1487
    pnormal_last_nonzero[OF H]
chaieb@33154
  1488
  show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)
chaieb@33154
  1489
next
chaieb@33154
  1490
  assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  1491
  let ?p = "polypoly bs p"
chaieb@33154
  1492
  have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
chaieb@33154
  1493
  hence pz: "?p \<noteq> []" by (simp add: polypoly_def) 
chaieb@33154
  1494
  hence lg: "length ?p > 0" by simp
chaieb@33154
  1495
  from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"] 
chaieb@33154
  1496
  have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)
chaieb@33154
  1497
  from pnormal_last_length[OF lg lz] show "pnormal ?p" .
chaieb@33154
  1498
qed
chaieb@33154
  1499
chaieb@33154
  1500
lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
chaieb@33154
  1501
  unfolding isnonconstant_def
chaieb@33154
  1502
  apply (cases p, simp_all)
chaieb@33154
  1503
  apply (case_tac nat, auto)
chaieb@33154
  1504
  done
chaieb@33154
  1505
lemma isnonconstant_nonconstant: assumes inc: "isnonconstant p"
chaieb@33154
  1506
  shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
chaieb@33154
  1507
proof
chaieb@33154
  1508
  let ?p = "polypoly bs p"
chaieb@33154
  1509
  assume nc: "nonconstant ?p"
chaieb@33154
  1510
  from isnonconstant_pnormal_iff[OF inc, of bs] nc
chaieb@33154
  1511
  show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast
chaieb@33154
  1512
next
chaieb@33154
  1513
  let ?p = "polypoly bs p"
chaieb@33154
  1514
  assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  1515
  from isnonconstant_pnormal_iff[OF inc, of bs] h
chaieb@33154
  1516
  have pn: "pnormal ?p" by blast
chaieb@33154
  1517
  {fix x assume H: "?p = [x]" 
chaieb@33154
  1518
    from H have "length (coefficients p) = 1" unfolding polypoly_def by auto
chaieb@33154
  1519
    with isnonconstant_coefficients_length[OF inc] have False by arith}
chaieb@33154
  1520
  thus "nonconstant ?p" using pn unfolding nonconstant_def by blast  
chaieb@33154
  1521
qed
chaieb@33154
  1522
chaieb@33154
  1523
lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
chaieb@33154
  1524
  unfolding pnormal_def
haftmann@39246
  1525
 apply (induct p)
haftmann@39246
  1526
 apply (simp_all, case_tac "p=[]", simp_all)
chaieb@33154
  1527
 done
chaieb@33154
  1528
chaieb@33154
  1529
lemma degree_degree: assumes inc: "isnonconstant p"
chaieb@33154
  1530
  shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  1531
proof
chaieb@33154
  1532
  let  ?p = "polypoly bs p"
chaieb@33154
  1533
  assume H: "degree p = Polynomial_List.degree ?p"
chaieb@33154
  1534
  from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
chaieb@33154
  1535
    unfolding polypoly_def by auto
chaieb@33154
  1536
  from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
chaieb@33154
  1537
  have lg:"length (pnormalize ?p) = length ?p"
chaieb@33154
  1538
    unfolding Polynomial_List.degree_def polypoly_def by simp
chaieb@33154
  1539
  hence "pnormal ?p" using pnormal_length[OF pz] by blast 
chaieb@33154
  1540
  with isnonconstant_pnormal_iff[OF inc]  
chaieb@33154
  1541
  show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast
chaieb@33154
  1542
next
chaieb@33154
  1543
  let  ?p = "polypoly bs p"  
chaieb@33154
  1544
  assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  1545
  with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast
chaieb@33154
  1546
  with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
chaieb@33154
  1547
  show "degree p = Polynomial_List.degree ?p" 
chaieb@33154
  1548
    unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
chaieb@33154
  1549
qed
chaieb@33154
  1550
chaieb@33154
  1551
section{* Swaps ; Division by a certain variable *}
haftmann@39246
  1552
primrec swap:: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly" where
chaieb@33154
  1553
  "swap n m (C x) = C x"
haftmann@39246
  1554
| "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"
haftmann@39246
  1555
| "swap n m (Neg t) = Neg (swap n m t)"
haftmann@39246
  1556
| "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
haftmann@39246
  1557
| "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
haftmann@39246
  1558
| "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
haftmann@39246
  1559
| "swap n m (Pw t k) = Pw (swap n m t) k"
haftmann@39246
  1560
| "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)
chaieb@33154
  1561
  (swap n m p)"
chaieb@33154
  1562
chaieb@33154
  1563
lemma swap: assumes nbs: "n < length bs" and mbs: "m < length bs"
chaieb@33154
  1564
  shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
chaieb@33154
  1565
proof (induct t)
chaieb@33154
  1566
  case (Bound k) thus ?case using nbs mbs by simp 
chaieb@33154
  1567
next
chaieb@33154
  1568
  case (CN c k p) thus ?case using nbs mbs by simp 
chaieb@33154
  1569
qed simp_all
chaieb@33154
  1570
lemma swap_swap_id[simp]: "swap n m (swap m n t) = t"
chaieb@33154
  1571
  by (induct t,simp_all)
chaieb@33154
  1572
chaieb@33154
  1573
lemma swap_commute: "swap n m p = swap m n p" by (induct p, simp_all)
chaieb@33154
  1574
chaieb@33154
  1575
lemma swap_same_id[simp]: "swap n n t = t"
chaieb@33154
  1576
  by (induct t, simp_all)
chaieb@33154
  1577
chaieb@33154
  1578
definition "swapnorm n m t = polynate (swap n m t)"
chaieb@33154
  1579
chaieb@33154
  1580
lemma swapnorm: assumes nbs: "n < length bs" and mbs: "m < length bs"
haftmann@36409
  1581
  shows "((Ipoly bs (swapnorm n m t) :: 'a\<Colon>{field_char_0, field_inverse_zero})) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
chaieb@33154
  1582
  using swap[OF prems] swapnorm_def by simp
chaieb@33154
  1583
chaieb@33154
  1584
lemma swapnorm_isnpoly[simp]: 
haftmann@36409
  1585
    assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
chaieb@33154
  1586
  shows "isnpoly (swapnorm n m p)"
chaieb@33154
  1587
  unfolding swapnorm_def by simp
chaieb@33154
  1588
chaieb@33154
  1589
definition "polydivideby n s p = 
chaieb@33154
  1590
    (let ss = swapnorm 0 n s ; sp = swapnorm 0 n p ; h = head sp; (k,r) = polydivide ss sp
chaieb@33154
  1591
     in (k,swapnorm 0 n h,swapnorm 0 n r))"
chaieb@33154
  1592
chaieb@33154
  1593
lemma swap_nz [simp]: " (swap n m p = 0\<^sub>p) = (p = 0\<^sub>p)" by (induct p, simp_all)
chaieb@33154
  1594
krauss@41808
  1595
fun isweaknpoly :: "poly \<Rightarrow> bool"
krauss@41808
  1596
where
chaieb@33154
  1597
  "isweaknpoly (C c) = True"
krauss@41808
  1598
| "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
krauss@41808
  1599
| "isweaknpoly p = False"
chaieb@33154
  1600
chaieb@33154
  1601
lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p" 
chaieb@33154
  1602
  by (induct p arbitrary: n0, auto)
chaieb@33154
  1603
chaieb@33154
  1604
lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)" 
chaieb@33154
  1605
  by (induct p, auto)
chaieb@33154
  1606
chaieb@33154
  1607
end