src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
author wenzelm
Tue Mar 18 16:45:14 2014 +0100 (2014-03-18)
changeset 56207 52d5067ca06a
parent 56198 21dd034523e5
child 58249 180f1b3508ed
permissions -rw-r--r--
tuned proofs;
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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 Rat_Pair Polynomial_List
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begin
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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"
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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 *}
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where
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  "polybound0 (C c) \<longleftrightarrow> True"
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| "polybound0 (Bound n) \<longleftrightarrow> n > 0"
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| "polybound0 (Neg a) \<longleftrightarrow> polybound0 a"
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| "polybound0 (Add a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
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| "polybound0 (Sub a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
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| "polybound0 (Mul a b) \<longleftrightarrow> polybound0 a \<and> polybound0 b"
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| "polybound0 (Pw p n) \<longleftrightarrow> polybound0 p"
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| "polybound0 (CN c n p) \<longleftrightarrow> 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 *}
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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) =
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    (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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declare if_cong[fundef_cong del]
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declare let_cong[fundef_cong del]
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fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
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where
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  "polyadd (C c) (C c') = C (c +\<^sub>N c')"
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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
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      let
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        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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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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definition polysub :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
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  where "p -\<^sub>p q = polyadd p (polyneg q)"
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fun polymul :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
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where
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  "polymul (C c) (C c') = C (c *\<^sub>N c')"
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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 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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declare if_cong[fundef_cong]
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declare let_cong[fundef_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 =
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    (\<lambda>p.
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      let
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        q = polypow (n div 2) p;
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        d = polymul q q
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      in 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"
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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"
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where
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  "monic p =
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    (let h = headconst p
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     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"
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  where "shift1 p = CN 0\<^sub>p 0 p"
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abbreviation funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
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  where "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
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      let
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        b = head s;
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        m = degree s
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      in
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        if m < n then (k,s)
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        else
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          let p' = funpow (m - n) shift1 p
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          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"
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  where "polydivide s p = polydivide_aux (head p) (degree p) p 0 s"
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fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly"
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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"
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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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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}"
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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 Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0,field_inverse_zero,power}"
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    ("\<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: "n' \<le> n \<Longrightarrow> isnpolyh p n \<Longrightarrow> isnpolyh p n'"
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  by (induct p rule: isnpolyh.induct) auto
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definition isnpoly :: "poly \<Rightarrow> bool"
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  where "isnpoly p = isnpolyh p 0"
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text{* polyadd preserves normal forms *}
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lemma polyadd_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<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 ab c' n' p' n0 n1)
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  from 2 have  th1: "isnpolyh (C ab) (Suc n')"
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    by simp
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  from 2(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1"
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    by simp_all
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  with isnpolyh_mono have cp: "isnpolyh c' (Suc n')"
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    by simp
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  with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^sub>p c') (Suc n')"
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    by simp
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  from nplen1 have n01len1: "min n0 n1 \<le> n'"
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    by simp
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  then show ?case using 2 th3
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    by simp
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next
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  case (3 c' n' p' ab n1 n0)
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  from 3 have  th1: "isnpolyh (C ab) (Suc n')"
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   296
    by simp
wenzelm@56009
   297
  from 3(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1"
wenzelm@56009
   298
    by simp_all
wenzelm@56009
   299
  with isnpolyh_mono have cp: "isnpolyh c' (Suc n')"
wenzelm@56009
   300
    by simp
wenzelm@56009
   301
  with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (Suc n')"
wenzelm@56009
   302
    by simp
wenzelm@56009
   303
  from nplen1 have n01len1: "min n0 n1 \<le> n'"
wenzelm@56009
   304
    by simp
wenzelm@56009
   305
  then show ?case using 3 th3
wenzelm@56009
   306
    by simp
chaieb@33154
   307
next
chaieb@33154
   308
  case (4 c n p c' n' p' n0 n1)
wenzelm@56009
   309
  then have nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n"
wenzelm@56009
   310
    by simp_all
wenzelm@56009
   311
  from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'"
wenzelm@56009
   312
    by simp_all
wenzelm@56009
   313
  from 4 have ngen0: "n \<ge> n0"
wenzelm@56009
   314
    by simp
wenzelm@56009
   315
  from 4 have n'gen1: "n' \<ge> n1"
wenzelm@56009
   316
    by simp
wenzelm@56009
   317
  have "n < n' \<or> n' < n \<or> n = n'"
wenzelm@56009
   318
    by auto
wenzelm@56009
   319
  moreover
wenzelm@56009
   320
  {
wenzelm@52803
   321
    assume eq: "n = n'"
wenzelm@52803
   322
    with "4.hyps"(3)[OF nc nc']
wenzelm@56009
   323
    have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)"
wenzelm@56009
   324
      by auto
wenzelm@56009
   325
    then have ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
wenzelm@56009
   326
      using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1
wenzelm@56009
   327
      by auto
wenzelm@56009
   328
    from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n"
wenzelm@56009
   329
      by simp
wenzelm@56009
   330
    have minle: "min n0 n1 \<le> n'"
wenzelm@56009
   331
      using ngen0 n'gen1 eq by simp
wenzelm@56009
   332
    from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' have ?case
wenzelm@56009
   333
      by (simp add: Let_def)
wenzelm@56009
   334
  }
wenzelm@56009
   335
  moreover
wenzelm@56009
   336
  {
wenzelm@52803
   337
    assume lt: "n < n'"
wenzelm@56009
   338
    have "min n0 n1 \<le> n0"
wenzelm@56009
   339
      by simp
wenzelm@56009
   340
    with 4 lt have th1:"min n0 n1 \<le> n"
wenzelm@56009
   341
      by auto
wenzelm@56009
   342
    from 4 have th21: "isnpolyh c (Suc n)"
wenzelm@56009
   343
      by simp
wenzelm@56009
   344
    from 4 have th22: "isnpolyh (CN c' n' p') n'"
wenzelm@56009
   345
      by simp
wenzelm@56009
   346
    from lt have th23: "min (Suc n) n' = Suc n"
wenzelm@56009
   347
      by arith
wenzelm@56009
   348
    from "4.hyps"(1)[OF th21 th22] have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)"
wenzelm@56009
   349
      using th23 by simp
wenzelm@56009
   350
    with 4 lt th1 have ?case
wenzelm@56009
   351
      by simp
wenzelm@56009
   352
  }
wenzelm@56009
   353
  moreover
wenzelm@56009
   354
  {
wenzelm@56009
   355
    assume gt: "n' < n"
wenzelm@56009
   356
    then have gt': "n' < n \<and> \<not> n < n'"
wenzelm@56009
   357
      by simp
wenzelm@56009
   358
    have "min n0 n1 \<le> n1"
wenzelm@56009
   359
      by simp
wenzelm@56009
   360
    with 4 gt have th1: "min n0 n1 \<le> n'"
wenzelm@56009
   361
      by auto
wenzelm@56009
   362
    from 4 have th21: "isnpolyh c' (Suc n')"
wenzelm@56009
   363
      by simp_all
wenzelm@56009
   364
    from 4 have th22: "isnpolyh (CN c n p) n"
wenzelm@56009
   365
      by simp
wenzelm@56009
   366
    from gt have th23: "min n (Suc n') = Suc n'"
wenzelm@56009
   367
      by arith
wenzelm@56009
   368
    from "4.hyps"(2)[OF th22 th21] have "isnpolyh (polyadd (CN c n p) c') (Suc n')"
wenzelm@56009
   369
      using th23 by simp
wenzelm@56009
   370
    with 4 gt th1 have ?case
wenzelm@56009
   371
      by simp
wenzelm@56009
   372
  }
wenzelm@52803
   373
  ultimately show ?case by blast
chaieb@33154
   374
qed auto
chaieb@33154
   375
krauss@41812
   376
lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
wenzelm@52658
   377
  by (induct p q rule: polyadd.induct)
wenzelm@52658
   378
    (auto simp add: Let_def field_simps distrib_left[symmetric] simp del: distrib_left)
chaieb@33154
   379
wenzelm@56009
   380
lemma polyadd_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polyadd p q)"
chaieb@33154
   381
  using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
chaieb@33154
   382
krauss@41404
   383
text{* The degree of addition and other general lemmas needed for the normal form of polymul *}
chaieb@33154
   384
wenzelm@52803
   385
lemma polyadd_different_degreen:
wenzelm@56009
   386
  assumes "isnpolyh p n0"
wenzelm@56009
   387
    and "isnpolyh q n1"
wenzelm@56009
   388
    and "degreen p m \<noteq> degreen q m"
wenzelm@56009
   389
    and "m \<le> min n0 n1"
wenzelm@56009
   390
  shows "degreen (polyadd p q) m = max (degreen p m) (degreen q m)"
wenzelm@56009
   391
  using assms
chaieb@33154
   392
proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
chaieb@33154
   393
  case (4 c n p c' n' p' m n0 n1)
krauss@41763
   394
  have "n' = n \<or> n < n' \<or> n' < n" by arith
wenzelm@56009
   395
  then show ?case
krauss@41763
   396
  proof (elim disjE)
krauss@41763
   397
    assume [simp]: "n' = n"
krauss@41812
   398
    from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
krauss@41763
   399
    show ?thesis by (auto simp: Let_def)
krauss@41763
   400
  next
krauss@41763
   401
    assume "n < n'"
krauss@41763
   402
    with 4 show ?thesis by auto
krauss@41763
   403
  next
krauss@41763
   404
    assume "n' < n"
krauss@41763
   405
    with 4 show ?thesis by auto
krauss@41763
   406
  qed
krauss@41763
   407
qed auto
chaieb@33154
   408
wenzelm@56009
   409
lemma headnz[simp]: "isnpolyh p n \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
wenzelm@52658
   410
  by (induct p arbitrary: n rule: headn.induct) auto
wenzelm@56009
   411
chaieb@33154
   412
lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
wenzelm@52658
   413
  by (induct p arbitrary: n rule: degree.induct) auto
wenzelm@56009
   414
chaieb@33154
   415
lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
wenzelm@52658
   416
  by (induct p arbitrary: n rule: degreen.induct) auto
chaieb@33154
   417
chaieb@33154
   418
lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
wenzelm@52658
   419
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
   420
chaieb@33154
   421
lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
chaieb@33154
   422
  using degree_isnpolyh_Suc by auto
wenzelm@56009
   423
chaieb@33154
   424
lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
chaieb@33154
   425
  using degreen_0 by auto
chaieb@33154
   426
chaieb@33154
   427
chaieb@33154
   428
lemma degreen_polyadd:
wenzelm@56009
   429
  assumes np: "isnpolyh p n0"
wenzelm@56009
   430
    and nq: "isnpolyh q n1"
wenzelm@56009
   431
    and m: "m \<le> max n0 n1"
chaieb@33154
   432
  shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
chaieb@33154
   433
  using np nq m
chaieb@33154
   434
proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
wenzelm@52803
   435
  case (2 c c' n' p' n0 n1)
wenzelm@56009
   436
  then show ?case
wenzelm@56009
   437
    by (cases n') simp_all
chaieb@33154
   438
next
wenzelm@52803
   439
  case (3 c n p c' n0 n1)
wenzelm@56009
   440
  then show ?case
wenzelm@56009
   441
    by (cases n) auto
chaieb@33154
   442
next
wenzelm@52803
   443
  case (4 c n p c' n' p' n0 n1 m)
krauss@41763
   444
  have "n' = n \<or> n < n' \<or> n' < n" by arith
wenzelm@56009
   445
  then show ?case
krauss@41763
   446
  proof (elim disjE)
krauss@41763
   447
    assume [simp]: "n' = n"
krauss@41812
   448
    from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
krauss@41763
   449
    show ?thesis by (auto simp: Let_def)
krauss@41763
   450
  qed simp_all
chaieb@33154
   451
qed auto
chaieb@33154
   452
wenzelm@56009
   453
lemma polyadd_eq_const_degreen:
wenzelm@56009
   454
  assumes "isnpolyh p n0"
wenzelm@56009
   455
    and "isnpolyh q n1"
wenzelm@56009
   456
    and "polyadd p q = C c"
wenzelm@56009
   457
  shows "degreen p m = degreen q m"
wenzelm@56009
   458
  using assms
chaieb@33154
   459
proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
wenzelm@52803
   460
  case (4 c n p c' n' p' m n0 n1 x)
wenzelm@56009
   461
  {
wenzelm@56009
   462
    assume nn': "n' < n"
wenzelm@56009
   463
    then have ?case using 4 by simp
wenzelm@56009
   464
  }
wenzelm@52803
   465
  moreover
wenzelm@56009
   466
  {
wenzelm@56009
   467
    assume nn': "\<not> n' < n"
wenzelm@56009
   468
    then have "n < n' \<or> n = n'" by arith
wenzelm@52803
   469
    moreover { assume "n < n'" with 4 have ?case by simp }
wenzelm@56009
   470
    moreover
wenzelm@56009
   471
    {
wenzelm@56009
   472
      assume eq: "n = n'"
wenzelm@56009
   473
      then have ?case using 4
krauss@41763
   474
        apply (cases "p +\<^sub>p p' = 0\<^sub>p")
krauss@41763
   475
        apply (auto simp add: Let_def)
wenzelm@52658
   476
        done
wenzelm@52803
   477
    }
wenzelm@56009
   478
    ultimately have ?case by blast
wenzelm@56009
   479
  }
chaieb@33154
   480
  ultimately show ?case by blast
chaieb@33154
   481
qed simp_all
chaieb@33154
   482
chaieb@33154
   483
lemma polymul_properties:
wenzelm@56000
   484
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
   485
    and np: "isnpolyh p n0"
wenzelm@52658
   486
    and nq: "isnpolyh q n1"
wenzelm@52658
   487
    and m: "m \<le> min n0 n1"
wenzelm@52803
   488
  shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
wenzelm@56009
   489
    and "p *\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p \<or> q = 0\<^sub>p"
wenzelm@56009
   490
    and "degreen (p *\<^sub>p q) m = (if p = 0\<^sub>p \<or> q = 0\<^sub>p then 0 else degreen p m + degreen q m)"
chaieb@33154
   491
  using np nq m
wenzelm@52658
   492
proof (induct p q arbitrary: n0 n1 m rule: polymul.induct)
wenzelm@52803
   493
  case (2 c c' n' p')
wenzelm@56009
   494
  {
wenzelm@56009
   495
    case (1 n0 n1)
wenzelm@56009
   496
    with "2.hyps"(4-6)[of n' n' n'] and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']
krauss@41811
   497
    show ?case by (auto simp add: min_def)
chaieb@33154
   498
  next
wenzelm@56009
   499
    case (2 n0 n1)
wenzelm@56009
   500
    then show ?case by auto
chaieb@33154
   501
  next
wenzelm@56009
   502
    case (3 n0 n1)
wenzelm@56009
   503
    then show ?case  using "2.hyps" by auto
wenzelm@56009
   504
  }
chaieb@33154
   505
next
krauss@41813
   506
  case (3 c n p c')
wenzelm@56009
   507
  {
wenzelm@56009
   508
    case (1 n0 n1)
wenzelm@56009
   509
    with "3.hyps"(4-6)[of n n n] and "3.hyps"(1-3)[of "Suc n" "Suc n" n]
krauss@41811
   510
    show ?case by (auto simp add: min_def)
chaieb@33154
   511
  next
wenzelm@56009
   512
    case (2 n0 n1)
wenzelm@56009
   513
    then show ?case by auto
chaieb@33154
   514
  next
wenzelm@56009
   515
    case (3 n0 n1)
wenzelm@56009
   516
    then show ?case  using "3.hyps" by auto
wenzelm@56009
   517
  }
chaieb@33154
   518
next
chaieb@33154
   519
  case (4 c n p c' n' p')
chaieb@33154
   520
  let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
wenzelm@56009
   521
  {
wenzelm@56009
   522
    case (1 n0 n1)
wenzelm@56009
   523
    then have cnp: "isnpolyh ?cnp n"
wenzelm@56009
   524
      and cnp': "isnpolyh ?cnp' n'"
wenzelm@56009
   525
      and np: "isnpolyh p n"
wenzelm@56009
   526
      and nc: "isnpolyh c (Suc n)"
wenzelm@56009
   527
      and np': "isnpolyh p' n'"
wenzelm@56009
   528
      and nc': "isnpolyh c' (Suc n')"
wenzelm@56009
   529
      and nn0: "n \<ge> n0"
wenzelm@56009
   530
      and nn1: "n' \<ge> n1"
wenzelm@56009
   531
      by simp_all
krauss@41811
   532
    {
wenzelm@56009
   533
      assume "n < n'"
wenzelm@56009
   534
      with "4.hyps"(4-5)[OF np cnp', of n] and "4.hyps"(1)[OF nc cnp', of n] nn0 cnp
wenzelm@56009
   535
      have ?case by (simp add: min_def)
wenzelm@56009
   536
    } moreover {
wenzelm@56009
   537
      assume "n' < n"
wenzelm@56009
   538
      with "4.hyps"(16-17)[OF cnp np', of "n'"] and "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
wenzelm@56009
   539
      have ?case by (cases "Suc n' = n") (simp_all add: min_def)
wenzelm@56009
   540
    } moreover {
wenzelm@56009
   541
      assume "n' = n"
wenzelm@56009
   542
      with "4.hyps"(16-17)[OF cnp np', of n] and "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0
wenzelm@56009
   543
      have ?case
wenzelm@56009
   544
        apply (auto intro!: polyadd_normh)
wenzelm@56009
   545
        apply (simp_all add: min_def isnpolyh_mono[OF nn0])
wenzelm@56009
   546
        done
wenzelm@56009
   547
    }
wenzelm@56009
   548
    ultimately show ?case by arith
wenzelm@56009
   549
  next
wenzelm@56009
   550
    fix n0 n1 m
wenzelm@56009
   551
    assume np: "isnpolyh ?cnp n0"
wenzelm@56009
   552
    assume np':"isnpolyh ?cnp' n1"
wenzelm@56009
   553
    assume m: "m \<le> min n0 n1"
wenzelm@56009
   554
    let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
wenzelm@56009
   555
    let ?d1 = "degreen ?cnp m"
wenzelm@56009
   556
    let ?d2 = "degreen ?cnp' m"
wenzelm@56009
   557
    let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
wenzelm@56043
   558
    have "n' < n \<or> n < n' \<or> n' = n" by auto
wenzelm@56009
   559
    moreover
wenzelm@56009
   560
    {
wenzelm@56009
   561
      assume "n' < n \<or> n < n'"
wenzelm@56009
   562
      with "4.hyps"(3,6,18) np np' m have ?eq
wenzelm@56009
   563
        by auto
wenzelm@56009
   564
    }
wenzelm@56009
   565
    moreover
wenzelm@56009
   566
    {
wenzelm@56009
   567
      assume nn': "n' = n"
wenzelm@56009
   568
      then have nn: "\<not> n' < n \<and> \<not> n < n'" by arith
wenzelm@56009
   569
      from "4.hyps"(16,18)[of n n' n]
wenzelm@56009
   570
        "4.hyps"(13,14)[of n "Suc n'" n]
wenzelm@56009
   571
        np np' nn'
wenzelm@56043
   572
      have norm:
wenzelm@56043
   573
        "isnpolyh ?cnp n"
wenzelm@56043
   574
        "isnpolyh c' (Suc n)"
wenzelm@56043
   575
        "isnpolyh (?cnp *\<^sub>p c') n"
wenzelm@56043
   576
        "isnpolyh p' n"
wenzelm@56043
   577
        "isnpolyh (?cnp *\<^sub>p p') n"
wenzelm@56043
   578
        "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@56043
   579
        "?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p"
wenzelm@56043
   580
        "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
wenzelm@56043
   581
        by (auto simp add: min_def)
wenzelm@56009
   582
      {
wenzelm@56009
   583
        assume mn: "m = n"
wenzelm@56009
   584
        from "4.hyps"(17,18)[OF norm(1,4), of n]
wenzelm@56009
   585
          "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
wenzelm@56009
   586
        have degs:
wenzelm@56009
   587
          "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else ?d1 + degreen c' n)"
wenzelm@56009
   588
          "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n"
wenzelm@56009
   589
          by (simp_all add: min_def)
wenzelm@56009
   590
        from degs norm have th1: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
wenzelm@56009
   591
          by simp
wenzelm@56009
   592
        then have neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
wenzelm@56009
   593
          by simp
wenzelm@56009
   594
        have nmin: "n \<le> min n n"
wenzelm@56009
   595
          by (simp add: min_def)
wenzelm@56009
   596
        from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
wenzelm@56009
   597
        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 =
wenzelm@56009
   598
            degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@56009
   599
          by simp
wenzelm@56009
   600
        from "4.hyps"(16-18)[OF norm(1,4), of n]
wenzelm@56009
   601
          "4.hyps"(13-15)[OF norm(1,2), of n]
wenzelm@56009
   602
          mn norm m nn' deg
wenzelm@56009
   603
        have ?eq by simp
krauss@41811
   604
      }
chaieb@33154
   605
      moreover
wenzelm@56009
   606
      {
wenzelm@56009
   607
        assume mn: "m \<noteq> n"
wenzelm@56009
   608
        then have mn': "m < n"
wenzelm@56009
   609
          using m np by auto
wenzelm@56009
   610
        from nn' m np have max1: "m \<le> max n n"
wenzelm@56009
   611
          by simp
wenzelm@56009
   612
        then have min1: "m \<le> min n n"
wenzelm@56009
   613
          by simp
wenzelm@56009
   614
        then have min2: "m \<le> min n (Suc n)"
wenzelm@56009
   615
          by simp
wenzelm@56009
   616
        from "4.hyps"(16-18)[OF norm(1,4) min1]
wenzelm@56009
   617
          "4.hyps"(13-15)[OF norm(1,2) min2]
wenzelm@56009
   618
          degreen_polyadd[OF norm(3,6) max1]
wenzelm@56009
   619
        have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m \<le>
wenzelm@56009
   620
            max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
wenzelm@56009
   621
          using mn nn' np np' by simp
wenzelm@56009
   622
        with "4.hyps"(16-18)[OF norm(1,4) min1]
wenzelm@56009
   623
          "4.hyps"(13-15)[OF norm(1,2) min2]
wenzelm@56009
   624
          degreen_0[OF norm(3) mn']
wenzelm@56009
   625
        have ?eq using nn' mn np np' by clarsimp
wenzelm@56009
   626
      }
wenzelm@56009
   627
      ultimately have ?eq by blast
wenzelm@56009
   628
    }
wenzelm@56009
   629
    ultimately show ?eq by blast
wenzelm@56009
   630
  }
wenzelm@56009
   631
  {
wenzelm@56009
   632
    case (2 n0 n1)
wenzelm@56009
   633
    then have np: "isnpolyh ?cnp n0"
wenzelm@56009
   634
      and np': "isnpolyh ?cnp' n1"
wenzelm@56043
   635
      and m: "m \<le> min n0 n1"
wenzelm@56043
   636
      by simp_all
wenzelm@56009
   637
    then have mn: "m \<le> n" by simp
wenzelm@56009
   638
    let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
wenzelm@56009
   639
    {
wenzelm@56009
   640
      assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
wenzelm@56009
   641
      then have nn: "\<not> n' < n \<and> \<not> n < n'"
wenzelm@56009
   642
        by simp
wenzelm@56009
   643
      from "4.hyps"(16-18) [of n n n]
wenzelm@56009
   644
        "4.hyps"(13-15)[of n "Suc n" n]
wenzelm@56009
   645
        np np' C(2) mn
wenzelm@56009
   646
      have norm:
wenzelm@56009
   647
        "isnpolyh ?cnp n"
wenzelm@56009
   648
        "isnpolyh c' (Suc n)"
wenzelm@56009
   649
        "isnpolyh (?cnp *\<^sub>p c') n"
wenzelm@56009
   650
        "isnpolyh p' n"
wenzelm@56009
   651
        "isnpolyh (?cnp *\<^sub>p p') n"
wenzelm@56009
   652
        "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@56009
   653
        "?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p"
wenzelm@56009
   654
        "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
wenzelm@56009
   655
        "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
wenzelm@56009
   656
        "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
wenzelm@56009
   657
        by (simp_all add: min_def)
wenzelm@56009
   658
      from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@56009
   659
        by simp
wenzelm@56009
   660
      have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
wenzelm@56009
   661
        using norm by simp
wenzelm@56009
   662
      from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"] degneq
wenzelm@56009
   663
      have False by simp
wenzelm@56009
   664
    }
wenzelm@56009
   665
    then show ?case using "4.hyps" by clarsimp
wenzelm@56009
   666
  }
chaieb@33154
   667
qed auto
chaieb@33154
   668
wenzelm@56009
   669
lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = Ipoly bs p * Ipoly bs q"
wenzelm@52658
   670
  by (induct p q rule: polymul.induct) (auto simp add: field_simps)
chaieb@33154
   671
wenzelm@52803
   672
lemma polymul_normh:
wenzelm@56000
   673
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56009
   674
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
wenzelm@52803
   675
  using polymul_properties(1) by blast
wenzelm@52658
   676
wenzelm@52803
   677
lemma polymul_eq0_iff:
wenzelm@56000
   678
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56009
   679
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p *\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p \<or> q = 0\<^sub>p"
wenzelm@52803
   680
  using polymul_properties(2) by blast
wenzelm@52658
   681
wenzelm@56207
   682
lemma polymul_degreen:
wenzelm@56000
   683
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56009
   684
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> m \<le> min n0 n1 \<Longrightarrow>
wenzelm@56009
   685
    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)"
wenzelm@56207
   686
  by (fact polymul_properties(3))
wenzelm@52658
   687
wenzelm@52803
   688
lemma polymul_norm:
wenzelm@56000
   689
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56009
   690
  shows "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polymul p q)"
chaieb@33154
   691
  using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
chaieb@33154
   692
chaieb@33154
   693
lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
   694
  by (induct p arbitrary: n0 rule: headconst.induct) auto
chaieb@33154
   695
chaieb@33154
   696
lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
wenzelm@52658
   697
  by (induct p arbitrary: n0) auto
chaieb@33154
   698
wenzelm@52658
   699
lemma monic_eqI:
wenzelm@52803
   700
  assumes np: "isnpolyh p n0"
wenzelm@52658
   701
  shows "INum (headconst p) * Ipoly bs (fst (monic p)) =
wenzelm@56000
   702
    (Ipoly bs p ::'a::{field_char_0,field_inverse_zero, power})"
chaieb@33154
   703
  unfolding monic_def Let_def
wenzelm@52658
   704
proof (cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
chaieb@33154
   705
  let ?h = "headconst p"
chaieb@33154
   706
  assume pz: "p \<noteq> 0\<^sub>p"
wenzelm@56000
   707
  {
wenzelm@56000
   708
    assume hz: "INum ?h = (0::'a)"
wenzelm@56043
   709
    from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N"
wenzelm@56043
   710
      by simp_all
wenzelm@56043
   711
    from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N"
wenzelm@56043
   712
      by simp
wenzelm@56043
   713
    with headconst_zero[OF np] have "p = 0\<^sub>p"
wenzelm@56043
   714
      by blast
wenzelm@56043
   715
    with pz have False
wenzelm@56043
   716
      by blast
wenzelm@56043
   717
  }
wenzelm@56043
   718
  then show "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
wenzelm@56043
   719
    by blast
chaieb@33154
   720
qed
chaieb@33154
   721
chaieb@33154
   722
krauss@41404
   723
text{* polyneg is a negation and preserves normal forms *}
chaieb@33154
   724
chaieb@33154
   725
lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
wenzelm@52658
   726
  by (induct p rule: polyneg.induct) auto
chaieb@33154
   727
wenzelm@56009
   728
lemma polyneg0: "isnpolyh p n \<Longrightarrow> (~\<^sub>p p) = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
   729
  by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: Nneg_def)
wenzelm@56009
   730
chaieb@33154
   731
lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
wenzelm@52658
   732
  by (induct p arbitrary: n0 rule: polyneg.induct) auto
wenzelm@56009
   733
wenzelm@56009
   734
lemma polyneg_normh: "isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n"
wenzelm@56009
   735
  by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: polyneg0)
chaieb@33154
   736
chaieb@33154
   737
lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
chaieb@33154
   738
  using isnpoly_def polyneg_normh by simp
chaieb@33154
   739
chaieb@33154
   740
krauss@41404
   741
text{* polysub is a substraction and preserves normal forms *}
krauss@41404
   742
wenzelm@56009
   743
lemma polysub[simp]: "Ipoly bs (polysub p q) = Ipoly bs p - Ipoly bs q"
wenzelm@52658
   744
  by (simp add: polysub_def)
wenzelm@56009
   745
wenzelm@56009
   746
lemma polysub_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
wenzelm@52658
   747
  by (simp add: polysub_def polyneg_normh polyadd_normh)
chaieb@33154
   748
wenzelm@56009
   749
lemma polysub_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polysub p q)"
wenzelm@52803
   750
  using polyadd_norm polyneg_norm by (simp add: polysub_def)
wenzelm@56009
   751
wenzelm@52658
   752
lemma polysub_same_0[simp]:
wenzelm@56000
   753
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
krauss@41814
   754
  shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p"
wenzelm@52658
   755
  unfolding polysub_def split_def fst_conv snd_conv
wenzelm@52658
   756
  by (induct p arbitrary: n0) (auto simp add: Let_def Nsub0[simplified Nsub_def])
chaieb@33154
   757
wenzelm@52803
   758
lemma polysub_0:
wenzelm@56000
   759
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56009
   760
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p -\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = q"
chaieb@33154
   761
  unfolding polysub_def split_def fst_conv snd_conv
krauss@41763
   762
  by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
wenzelm@52658
   763
    (auto simp: Nsub0[simplified Nsub_def] Let_def)
chaieb@33154
   764
chaieb@33154
   765
text{* polypow is a power function and preserves normal forms *}
krauss@41404
   766
wenzelm@56009
   767
lemma polypow[simp]:
wenzelm@56009
   768
  "Ipoly bs (polypow n p) = (Ipoly bs p :: 'a::{field_char_0,field_inverse_zero}) ^ n"
wenzelm@52658
   769
proof (induct n rule: polypow.induct)
wenzelm@52658
   770
  case 1
wenzelm@56043
   771
  then show ?case
wenzelm@56043
   772
    by simp
chaieb@33154
   773
next
chaieb@33154
   774
  case (2 n)
chaieb@33154
   775
  let ?q = "polypow ((Suc n) div 2) p"
krauss@41813
   776
  let ?d = "polymul ?q ?q"
wenzelm@56043
   777
  have "odd (Suc n) \<or> even (Suc n)"
wenzelm@56043
   778
    by simp
wenzelm@52803
   779
  moreover
wenzelm@56043
   780
  {
wenzelm@56043
   781
    assume odd: "odd (Suc n)"
wenzelm@56000
   782
    have th: "(Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0)))) = Suc n div 2 + Suc n div 2 + 1"
wenzelm@52658
   783
      by arith
wenzelm@56043
   784
    from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)"
wenzelm@56043
   785
      by (simp add: Let_def)
wenzelm@56043
   786
    also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2) * (Ipoly bs p)^(Suc n div 2)"
chaieb@33154
   787
      using "2.hyps" by simp
chaieb@33154
   788
    also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
wenzelm@52658
   789
      by (simp only: power_add power_one_right) simp
wenzelm@56000
   790
    also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0))))"
chaieb@33154
   791
      by (simp only: th)
wenzelm@52803
   792
    finally have ?case
wenzelm@56043
   793
    using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp
wenzelm@56043
   794
  }
wenzelm@52803
   795
  moreover
wenzelm@56043
   796
  {
wenzelm@56043
   797
    assume even: "even (Suc n)"
wenzelm@56000
   798
    have th: "(Suc (Suc 0)) * (Suc n div Suc (Suc 0)) = Suc n div 2 + Suc n div 2"
wenzelm@52658
   799
      by arith
wenzelm@56043
   800
    from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d"
wenzelm@56043
   801
      by (simp add: Let_def)
chaieb@33154
   802
    also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
wenzelm@56043
   803
      using "2.hyps" by (simp only: power_add) simp
wenzelm@56043
   804
    finally have ?case using even_nat_div_two_times_two[OF even]
wenzelm@56043
   805
      by (simp only: th)
wenzelm@56043
   806
  }
chaieb@33154
   807
  ultimately show ?case by blast
chaieb@33154
   808
qed
chaieb@33154
   809
wenzelm@52803
   810
lemma polypow_normh:
wenzelm@56000
   811
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
   812
  shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
chaieb@33154
   813
proof (induct k arbitrary: n rule: polypow.induct)
wenzelm@56043
   814
  case 1
wenzelm@56043
   815
  then show ?case by auto
wenzelm@56043
   816
next
chaieb@33154
   817
  case (2 k n)
chaieb@33154
   818
  let ?q = "polypow (Suc k div 2) p"
krauss@41813
   819
  let ?d = "polymul ?q ?q"
wenzelm@56043
   820
  from 2 have th1: "isnpolyh ?q n" and th2: "isnpolyh p n"
wenzelm@56043
   821
    by blast+
wenzelm@56043
   822
  from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n"
wenzelm@56043
   823
    by simp
wenzelm@56043
   824
  from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n"
wenzelm@56043
   825
    by simp
wenzelm@56043
   826
  from dn on show ?case
wenzelm@56043
   827
    by (simp add: Let_def)
wenzelm@56043
   828
qed
chaieb@33154
   829
wenzelm@52803
   830
lemma polypow_norm:
wenzelm@56000
   831
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
   832
  shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
chaieb@33154
   833
  by (simp add: polypow_normh isnpoly_def)
chaieb@33154
   834
krauss@41404
   835
text{* Finally the whole normalization *}
chaieb@33154
   836
wenzelm@52658
   837
lemma polynate [simp]:
wenzelm@56000
   838
  "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0,field_inverse_zero})"
wenzelm@52658
   839
  by (induct p rule:polynate.induct) auto
chaieb@33154
   840
wenzelm@52803
   841
lemma polynate_norm[simp]:
wenzelm@56000
   842
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
   843
  shows "isnpoly (polynate p)"
wenzelm@52658
   844
  by (induct p rule: polynate.induct)
wenzelm@52658
   845
     (simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm,
wenzelm@52658
   846
      simp_all add: isnpoly_def)
chaieb@33154
   847
chaieb@33154
   848
text{* shift1 *}
chaieb@33154
   849
chaieb@33154
   850
chaieb@33154
   851
lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
wenzelm@52658
   852
  by (simp add: shift1_def)
chaieb@33154
   853
wenzelm@52803
   854
lemma shift1_isnpoly:
wenzelm@56207
   855
  assumes "isnpoly p"
wenzelm@56207
   856
    and "p \<noteq> 0\<^sub>p"
wenzelm@52658
   857
  shows "isnpoly (shift1 p) "
wenzelm@56207
   858
  using assms by (simp add: shift1_def isnpoly_def)
chaieb@33154
   859
chaieb@33154
   860
lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
chaieb@33154
   861
  by (simp add: shift1_def)
wenzelm@56043
   862
wenzelm@56043
   863
lemma funpow_shift1_isnpoly: "isnpoly p \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpoly (funpow n shift1 p)"
haftmann@39246
   864
  by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
chaieb@33154
   865
wenzelm@52803
   866
lemma funpow_isnpolyh:
wenzelm@56207
   867
  assumes "\<And>p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n"
wenzelm@56207
   868
    and "isnpolyh p n"
chaieb@33154
   869
  shows "isnpolyh (funpow k f p) n"
wenzelm@56207
   870
  using assms by (induct k arbitrary: p) auto
chaieb@33154
   871
wenzelm@52658
   872
lemma funpow_shift1:
wenzelm@56000
   873
  "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field_inverse_zero}) =
wenzelm@52658
   874
    Ipoly bs (Mul (Pw (Bound 0) n) p)"
wenzelm@52658
   875
  by (induct n arbitrary: p) (simp_all add: shift1_isnpoly shift1)
chaieb@33154
   876
wenzelm@56043
   877
lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
chaieb@33154
   878
  using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
chaieb@33154
   879
wenzelm@52803
   880
lemma funpow_shift1_1:
wenzelm@56000
   881
  "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field_inverse_zero}) =
wenzelm@52658
   882
    Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)"
chaieb@33154
   883
  by (simp add: funpow_shift1)
chaieb@33154
   884
chaieb@33154
   885
lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
wenzelm@45129
   886
  by (induct p rule: poly_cmul.induct) (auto simp add: field_simps)
chaieb@33154
   887
chaieb@33154
   888
lemma behead:
wenzelm@56207
   889
  assumes "isnpolyh p n"
wenzelm@52658
   890
  shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) =
wenzelm@56000
   891
    (Ipoly bs p :: 'a :: {field_char_0,field_inverse_zero})"
wenzelm@56207
   892
  using assms
chaieb@33154
   893
proof (induct p arbitrary: n rule: behead.induct)
wenzelm@56009
   894
  case (1 c p n)
wenzelm@56009
   895
  then have pn: "isnpolyh p n" by simp
wenzelm@52803
   896
  from 1(1)[OF pn]
wenzelm@52803
   897
  have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
wenzelm@52658
   898
  then show ?case using "1.hyps"
wenzelm@52658
   899
    apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
wenzelm@52658
   900
    apply (simp_all add: th[symmetric] field_simps)
wenzelm@52658
   901
    done
chaieb@33154
   902
qed (auto simp add: Let_def)
chaieb@33154
   903
chaieb@33154
   904
lemma behead_isnpolyh:
wenzelm@56207
   905
  assumes "isnpolyh p n"
wenzelm@52658
   906
  shows "isnpolyh (behead p) n"
wenzelm@56207
   907
  using assms by (induct p rule: behead.induct) (auto simp add: Let_def isnpolyh_mono)
wenzelm@52658
   908
chaieb@33154
   909
wenzelm@56207
   910
subsection {* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
wenzelm@52658
   911
chaieb@33154
   912
lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
wenzelm@52658
   913
proof (induct p arbitrary: n rule: poly.induct, auto)
chaieb@33154
   914
  case (goal1 c n p n')
wenzelm@56009
   915
  then have "n = Suc (n - 1)"
wenzelm@56009
   916
    by simp
wenzelm@56009
   917
  then have "isnpolyh p (Suc (n - 1))"
wenzelm@56009
   918
    using `isnpolyh p n` by simp
wenzelm@56009
   919
  with goal1(2) show ?case
wenzelm@56009
   920
    by simp
chaieb@33154
   921
qed
chaieb@33154
   922
chaieb@33154
   923
lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
wenzelm@52658
   924
  by (induct p arbitrary: n0 rule: isconstant.induct) (auto simp add: isnpolyh_polybound0)
chaieb@33154
   925
wenzelm@52658
   926
lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
   927
  by (induct p) auto
chaieb@33154
   928
chaieb@33154
   929
lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
wenzelm@52658
   930
  apply (induct p arbitrary: n0)
wenzelm@52658
   931
  apply auto
wenzelm@56043
   932
  apply atomize
chaieb@33154
   933
  apply (erule_tac x = "Suc nat" in allE)
chaieb@33154
   934
  apply auto
chaieb@33154
   935
  done
chaieb@33154
   936
chaieb@33154
   937
lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
wenzelm@52658
   938
  by (induct p  arbitrary: n0 rule: head.induct) (auto intro: isnpolyh_polybound0)
chaieb@33154
   939
chaieb@33154
   940
lemma polybound0_I:
wenzelm@56207
   941
  assumes "polybound0 a"
wenzelm@56009
   942
  shows "Ipoly (b # bs) a = Ipoly (b' # bs) a"
wenzelm@56207
   943
  using assms by (induct a rule: poly.induct) auto
wenzelm@52658
   944
wenzelm@56009
   945
lemma polysubst0_I: "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b # bs) a) # bs) t"
chaieb@33154
   946
  by (induct t) simp_all
chaieb@33154
   947
chaieb@33154
   948
lemma polysubst0_I':
wenzelm@56207
   949
  assumes "polybound0 a"
wenzelm@56009
   950
  shows "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b' # bs) a) # bs) t"
wenzelm@56207
   951
  by (induct t) (simp_all add: polybound0_I[OF assms, where b="b" and b'="b'"])
chaieb@33154
   952
wenzelm@52658
   953
lemma decrpoly:
wenzelm@56207
   954
  assumes "polybound0 t"
wenzelm@56043
   955
  shows "Ipoly (x # bs) t = Ipoly bs (decrpoly t)"
wenzelm@56207
   956
  using assms by (induct t rule: decrpoly.induct) simp_all
chaieb@33154
   957
wenzelm@52658
   958
lemma polysubst0_polybound0:
wenzelm@56207
   959
  assumes "polybound0 t"
chaieb@33154
   960
  shows "polybound0 (polysubst0 t a)"
wenzelm@56207
   961
  using assms by (induct a rule: poly.induct) auto
chaieb@33154
   962
chaieb@33154
   963
lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
wenzelm@52658
   964
  by (induct p arbitrary: n rule: degree.induct) (auto simp add: isnpolyh_polybound0)
chaieb@33154
   965
wenzelm@56043
   966
primrec maxindex :: "poly \<Rightarrow> nat"
wenzelm@56043
   967
where
chaieb@33154
   968
  "maxindex (Bound n) = n + 1"
chaieb@33154
   969
| "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
chaieb@33154
   970
| "maxindex (Add p q) = max (maxindex p) (maxindex q)"
chaieb@33154
   971
| "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
chaieb@33154
   972
| "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
chaieb@33154
   973
| "maxindex (Neg p) = maxindex p"
chaieb@33154
   974
| "maxindex (Pw p n) = maxindex p"
chaieb@33154
   975
| "maxindex (C x) = 0"
chaieb@33154
   976
wenzelm@52658
   977
definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool"
wenzelm@56000
   978
  where "wf_bs bs p \<longleftrightarrow> length bs \<ge> maxindex p"
chaieb@33154
   979
wenzelm@56043
   980
lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall>c \<in> set (coefficients p). wf_bs bs c"
wenzelm@52658
   981
proof (induct p rule: coefficients.induct)
wenzelm@52803
   982
  case (1 c p)
wenzelm@52803
   983
  show ?case
chaieb@33154
   984
  proof
wenzelm@56009
   985
    fix x
wenzelm@56009
   986
    assume xc: "x \<in> set (coefficients (CN c 0 p))"
wenzelm@56009
   987
    then have "x = c \<or> x \<in> set (coefficients p)"
wenzelm@56009
   988
      by simp
wenzelm@52803
   989
    moreover
wenzelm@56009
   990
    {
wenzelm@56009
   991
      assume "x = c"
wenzelm@56009
   992
      then have "wf_bs bs x"
wenzelm@56043
   993
        using "1.prems" unfolding wf_bs_def by simp
wenzelm@56009
   994
    }
wenzelm@56009
   995
    moreover
wenzelm@56009
   996
    {
wenzelm@56009
   997
      assume H: "x \<in> set (coefficients p)"
wenzelm@56009
   998
      from "1.prems" have "wf_bs bs p"
wenzelm@56009
   999
        unfolding wf_bs_def by simp
wenzelm@56009
  1000
      with "1.hyps" H have "wf_bs bs x"
wenzelm@56009
  1001
        by blast
wenzelm@56009
  1002
    }
wenzelm@56066
  1003
    ultimately show "wf_bs bs x"
wenzelm@56009
  1004
      by blast
chaieb@33154
  1005
  qed
chaieb@33154
  1006
qed simp_all
chaieb@33154
  1007
wenzelm@56043
  1008
lemma maxindex_coefficients: "\<forall>c \<in> set (coefficients p). maxindex c \<le> maxindex p"
wenzelm@52658
  1009
  by (induct p rule: coefficients.induct) auto
chaieb@33154
  1010
wenzelm@56000
  1011
lemma wf_bs_I: "wf_bs bs p \<Longrightarrow> Ipoly (bs @ bs') p = Ipoly bs p"
wenzelm@52658
  1012
  unfolding wf_bs_def by (induct p) (auto simp add: nth_append)
chaieb@33154
  1013
wenzelm@52658
  1014
lemma take_maxindex_wf:
wenzelm@52803
  1015
  assumes wf: "wf_bs bs p"
chaieb@33154
  1016
  shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
wenzelm@56009
  1017
proof -
chaieb@33154
  1018
  let ?ip = "maxindex p"
chaieb@33154
  1019
  let ?tbs = "take ?ip bs"
wenzelm@56009
  1020
  from wf have "length ?tbs = ?ip"
wenzelm@56009
  1021
    unfolding wf_bs_def by simp
wenzelm@56009
  1022
  then have wf': "wf_bs ?tbs p"
wenzelm@56009
  1023
    unfolding wf_bs_def by  simp
wenzelm@56043
  1024
  have eq: "bs = ?tbs @ drop ?ip bs"
wenzelm@56009
  1025
    by simp
wenzelm@56009
  1026
  from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis
wenzelm@56009
  1027
    using eq by simp
chaieb@33154
  1028
qed
chaieb@33154
  1029
chaieb@33154
  1030
lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
wenzelm@52658
  1031
  by (induct p) auto
chaieb@33154
  1032
chaieb@33154
  1033
lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
chaieb@33154
  1034
  unfolding wf_bs_def by simp
chaieb@33154
  1035
wenzelm@56207
  1036
lemma wf_bs_insensitive': "wf_bs (x # bs) p = wf_bs (y # bs) p"
chaieb@33154
  1037
  unfolding wf_bs_def by simp
chaieb@33154
  1038
wenzelm@56207
  1039
lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x # bs) p"
wenzelm@52658
  1040
  by (induct p rule: coefficients.induct) (auto simp add: wf_bs_def)
wenzelm@56043
  1041
chaieb@33154
  1042
lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
wenzelm@52658
  1043
  by (induct p rule: coefficients.induct) simp_all
chaieb@33154
  1044
chaieb@33154
  1045
lemma coefficients_head: "last (coefficients p) = head p"
wenzelm@52658
  1046
  by (induct p rule: coefficients.induct) auto
chaieb@33154
  1047
wenzelm@56207
  1048
lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x # bs) p"
wenzelm@52658
  1049
  unfolding wf_bs_def by (induct p rule: decrpoly.induct) auto
chaieb@33154
  1050
wenzelm@56043
  1051
lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists>ys. length (xs @ ys) = n"
wenzelm@56207
  1052
  apply (rule exI[where x="replicate (n - length xs) z" for z])
wenzelm@52658
  1053
  apply simp
wenzelm@52658
  1054
  done
wenzelm@52658
  1055
wenzelm@56043
  1056
lemma isnpolyh_Suc_const: "isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
wenzelm@52658
  1057
  apply (cases p)
wenzelm@52658
  1058
  apply auto
wenzelm@52658
  1059
  apply (case_tac "nat")
wenzelm@52658
  1060
  apply simp_all
wenzelm@52658
  1061
  done
chaieb@33154
  1062
chaieb@33154
  1063
lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
wenzelm@56066
  1064
  unfolding wf_bs_def by (induct p q rule: polyadd.induct) (auto simp add: Let_def)
chaieb@33154
  1065
chaieb@33154
  1066
lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
wenzelm@52803
  1067
  unfolding wf_bs_def
wenzelm@52803
  1068
  apply (induct p q arbitrary: bs rule: polymul.induct)
chaieb@33154
  1069
  apply (simp_all add: wf_bs_polyadd)
chaieb@33154
  1070
  apply clarsimp
chaieb@33154
  1071
  apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
chaieb@33154
  1072
  apply auto
chaieb@33154
  1073
  done
chaieb@33154
  1074
chaieb@33154
  1075
lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
wenzelm@52658
  1076
  unfolding wf_bs_def by (induct p rule: polyneg.induct) auto
chaieb@33154
  1077
chaieb@33154
  1078
lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
wenzelm@56043
  1079
  unfolding polysub_def split_def fst_conv snd_conv
wenzelm@56043
  1080
  using wf_bs_polyadd wf_bs_polyneg by blast
chaieb@33154
  1081
wenzelm@52658
  1082
wenzelm@56043
  1083
subsection {* Canonicity of polynomial representation, see lemma isnpolyh_unique *}
chaieb@33154
  1084
chaieb@33154
  1085
definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
wenzelm@56043
  1086
definition "polypoly' bs p = map (Ipoly bs \<circ> decrpoly) (coefficients p)"
wenzelm@56043
  1087
definition "poly_nate bs p = map (Ipoly bs \<circ> decrpoly) (coefficients (polynate p))"
chaieb@33154
  1088
wenzelm@56043
  1089
lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall>q \<in> set (coefficients p). isnpolyh q n0"
chaieb@33154
  1090
proof (induct p arbitrary: n0 rule: coefficients.induct)
chaieb@33154
  1091
  case (1 c p n0)
wenzelm@56009
  1092
  have cp: "isnpolyh (CN c 0 p) n0"
wenzelm@56009
  1093
    by fact
wenzelm@56009
  1094
  then have norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
chaieb@33154
  1095
    by (auto simp add: isnpolyh_mono[where n'=0])
wenzelm@56009
  1096
  from "1.hyps"[OF norm(2)] norm(1) norm(4) show ?case
wenzelm@56009
  1097
    by simp
chaieb@33154
  1098
qed auto
chaieb@33154
  1099
wenzelm@56043
  1100
lemma coefficients_isconst: "isnpolyh p n \<Longrightarrow> \<forall>q \<in> set (coefficients p). isconstant q"
wenzelm@56043
  1101
  by (induct p arbitrary: n rule: coefficients.induct) (auto simp add: isnpolyh_Suc_const)
chaieb@33154
  1102
chaieb@33154
  1103
lemma polypoly_polypoly':
chaieb@33154
  1104
  assumes np: "isnpolyh p n0"
wenzelm@56043
  1105
  shows "polypoly (x # bs) p = polypoly' bs p"
wenzelm@56043
  1106
proof -
chaieb@33154
  1107
  let ?cf = "set (coefficients p)"
chaieb@33154
  1108
  from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
wenzelm@56043
  1109
  {
wenzelm@56043
  1110
    fix q
wenzelm@56043
  1111
    assume q: "q \<in> ?cf"
wenzelm@56043
  1112
    from q cn_norm have th: "isnpolyh q n0"
wenzelm@56043
  1113
      by blast
wenzelm@56043
  1114
    from coefficients_isconst[OF np] q have "isconstant q"
wenzelm@56043
  1115
      by blast
wenzelm@56043
  1116
    with isconstant_polybound0[OF th] have "polybound0 q"
wenzelm@56043
  1117
      by blast
wenzelm@56043
  1118
  }
wenzelm@56009
  1119
  then have "\<forall>q \<in> ?cf. polybound0 q" ..
wenzelm@56043
  1120
  then have "\<forall>q \<in> ?cf. Ipoly (x # bs) q = Ipoly bs (decrpoly q)"
chaieb@33154
  1121
    using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
chaieb@33154
  1122
    by auto
wenzelm@56043
  1123
  then show ?thesis
wenzelm@56043
  1124
    unfolding polypoly_def polypoly'_def by simp
chaieb@33154
  1125
qed
chaieb@33154
  1126
chaieb@33154
  1127
lemma polypoly_poly:
wenzelm@56043
  1128
  assumes "isnpolyh p n0"
wenzelm@56043
  1129
  shows "Ipoly (x # bs) p = poly (polypoly (x # bs) p) x"
wenzelm@56043
  1130
  using assms
wenzelm@52658
  1131
  by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def)
chaieb@33154
  1132
wenzelm@52803
  1133
lemma polypoly'_poly:
wenzelm@56043
  1134
  assumes "isnpolyh p n0"
wenzelm@52658
  1135
  shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
wenzelm@56043
  1136
  using polypoly_poly[OF assms, simplified polypoly_polypoly'[OF assms]] .
chaieb@33154
  1137
chaieb@33154
  1138
chaieb@33154
  1139
lemma polypoly_poly_polybound0:
wenzelm@56043
  1140
  assumes "isnpolyh p n0"
wenzelm@56043
  1141
    and "polybound0 p"
chaieb@33154
  1142
  shows "polypoly bs p = [Ipoly bs p]"
wenzelm@56043
  1143
  using assms
wenzelm@56043
  1144
  unfolding polypoly_def
wenzelm@52658
  1145
  apply (cases p)
wenzelm@52658
  1146
  apply auto
wenzelm@52658
  1147
  apply (case_tac nat)
wenzelm@52658
  1148
  apply auto
wenzelm@52658
  1149
  done
chaieb@33154
  1150
wenzelm@52803
  1151
lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
wenzelm@52658
  1152
  by (induct p rule: head.induct) auto
chaieb@33154
  1153
wenzelm@56043
  1154
lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> headn p m = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
  1155
  by (cases p) auto
chaieb@33154
  1156
chaieb@33154
  1157
lemma head_eq_headn0: "head p = headn p 0"
wenzelm@52658
  1158
  by (induct p rule: head.induct) simp_all
chaieb@33154
  1159
wenzelm@56043
  1160
lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> head p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
chaieb@33154
  1161
  by (simp add: head_eq_headn0)
chaieb@33154
  1162
wenzelm@52803
  1163
lemma isnpolyh_zero_iff:
wenzelm@52658
  1164
  assumes nq: "isnpolyh p n0"
wenzelm@56000
  1165
    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
  1166
  shows "p = 0\<^sub>p"
wenzelm@52658
  1167
  using nq eq
berghofe@34915
  1168
proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
berghofe@34915
  1169
  case less
berghofe@34915
  1170
  note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
wenzelm@56000
  1171
  {
wenzelm@56000
  1172
    assume nz: "maxindex p = 0"
wenzelm@56000
  1173
    then obtain c where "p = C c"
wenzelm@56000
  1174
      using np by (cases p) auto
wenzelm@56000
  1175
    with zp np have "p = 0\<^sub>p"
wenzelm@56000
  1176
      unfolding wf_bs_def by simp
wenzelm@56000
  1177
  }
chaieb@33154
  1178
  moreover
wenzelm@56000
  1179
  {
wenzelm@56000
  1180
    assume nz: "maxindex p \<noteq> 0"
chaieb@33154
  1181
    let ?h = "head p"
chaieb@33154
  1182
    let ?hd = "decrpoly ?h"
chaieb@33154
  1183
    let ?ihd = "maxindex ?hd"
wenzelm@56000
  1184
    from head_isnpolyh[OF np] head_polybound0[OF np]
wenzelm@56000
  1185
    have h: "isnpolyh ?h n0" "polybound0 ?h"
chaieb@33154
  1186
      by simp_all
wenzelm@56000
  1187
    then have nhd: "isnpolyh ?hd (n0 - 1)"
wenzelm@56000
  1188
      using decrpoly_normh by blast
wenzelm@52803
  1189
chaieb@33154
  1190
    from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
wenzelm@56000
  1191
    have mihn: "maxindex ?h \<le> maxindex p"
wenzelm@56000
  1192
      by auto
wenzelm@56000
  1193
    with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < maxindex p"
wenzelm@56000
  1194
      by auto
wenzelm@56000
  1195
    {
wenzelm@56000
  1196
      fix bs :: "'a list"
wenzelm@56000
  1197
      assume bs: "wf_bs bs ?hd"
chaieb@33154
  1198
      let ?ts = "take ?ihd bs"
chaieb@33154
  1199
      let ?rs = "drop ?ihd bs"
wenzelm@56000
  1200
      have ts: "wf_bs ?ts ?hd"
wenzelm@56000
  1201
        using bs unfolding wf_bs_def by simp
wenzelm@56000
  1202
      have bs_ts_eq: "?ts @ ?rs = bs"
wenzelm@56000
  1203
        by simp
wenzelm@56000
  1204
      from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x # ?ts) ?h"
wenzelm@56000
  1205
        by simp
wenzelm@56000
  1206
      from ihd_lt_n have "\<forall>x. length (x # ?ts) \<le> maxindex p"
wenzelm@56000
  1207
        by simp
wenzelm@56000
  1208
      with length_le_list_ex obtain xs where xs: "length ((x # ?ts) @ xs) = maxindex p"
wenzelm@56000
  1209
        by blast
wenzelm@56000
  1210
      then have "\<forall>x. wf_bs ((x # ?ts) @ xs) p"
wenzelm@56000
  1211
        unfolding wf_bs_def by simp
wenzelm@56000
  1212
      with zp have "\<forall>x. Ipoly ((x # ?ts) @ xs) p = 0"
wenzelm@56000
  1213
        by blast
wenzelm@56000
  1214
      then have "\<forall>x. Ipoly (x # (?ts @ xs)) p = 0"
wenzelm@56000
  1215
        by simp
chaieb@33154
  1216
      with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
wenzelm@56000
  1217
      have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"
wenzelm@56000
  1218
        by simp
wenzelm@56000
  1219
      then have "poly (polypoly' (?ts @ xs) p) = poly []"
wenzelm@56000
  1220
        by auto
wenzelm@56000
  1221
      then have "\<forall>c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
wenzelm@33268
  1222
        using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
chaieb@33154
  1223
      with coefficients_head[of p, symmetric]
wenzelm@56000
  1224
      have th0: "Ipoly (?ts @ xs) ?hd = 0"
wenzelm@56000
  1225
        by simp
wenzelm@56000
  1226
      from bs have wf'': "wf_bs ?ts ?hd"
wenzelm@56000
  1227
        unfolding wf_bs_def by simp
wenzelm@56000
  1228
      with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0"
wenzelm@56000
  1229
        by simp
wenzelm@56000
  1230
      with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
wenzelm@56000
  1231
        by simp
wenzelm@56000
  1232
    }
wenzelm@56000
  1233
    then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
wenzelm@56000
  1234
      by blast
wenzelm@56000
  1235
    from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p"
wenzelm@56000
  1236
      by blast
wenzelm@56000
  1237
    then have "?h = 0\<^sub>p" by simp
wenzelm@56000
  1238
    with head_nz[OF np] have "p = 0\<^sub>p" by simp
wenzelm@56000
  1239
  }
wenzelm@56000
  1240
  ultimately show "p = 0\<^sub>p"
wenzelm@56000
  1241
    by blast
chaieb@33154
  1242
qed
chaieb@33154
  1243
wenzelm@52803
  1244
lemma isnpolyh_unique:
wenzelm@56000
  1245
  assumes np: "isnpolyh p n0"
wenzelm@52658
  1246
    and nq: "isnpolyh q n1"
wenzelm@56000
  1247
  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"
wenzelm@56000
  1248
proof auto
wenzelm@56000
  1249
  assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a) = \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
wenzelm@56000
  1250
  then have "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)"
wenzelm@56000
  1251
    by simp
wenzelm@56000
  1252
  then have "\<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
  1253
    using wf_bs_polysub[where p=p and q=q] by auto
wenzelm@56000
  1254
  with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq] show "p = q"
wenzelm@56000
  1255
    by blast
chaieb@33154
  1256
qed
chaieb@33154
  1257
chaieb@33154
  1258
krauss@41404
  1259
text{* consequences of unicity on the algorithms for polynomial normalization *}
chaieb@33154
  1260
wenzelm@52658
  1261
lemma polyadd_commute:
wenzelm@56000
  1262
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1263
    and np: "isnpolyh p n0"
wenzelm@52658
  1264
    and nq: "isnpolyh q n1"
wenzelm@52658
  1265
  shows "p +\<^sub>p q = q +\<^sub>p p"
wenzelm@56000
  1266
  using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]]
wenzelm@56000
  1267
  by simp
chaieb@33154
  1268
wenzelm@56000
  1269
lemma zero_normh: "isnpolyh 0\<^sub>p n"
wenzelm@56000
  1270
  by simp
wenzelm@56000
  1271
wenzelm@56000
  1272
lemma one_normh: "isnpolyh (1)\<^sub>p n"
wenzelm@56000
  1273
  by simp
wenzelm@52658
  1274
wenzelm@52803
  1275
lemma polyadd_0[simp]:
wenzelm@56000
  1276
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1277
    and np: "isnpolyh p n0"
wenzelm@56000
  1278
  shows "p +\<^sub>p 0\<^sub>p = p"
wenzelm@56000
  1279
    and "0\<^sub>p +\<^sub>p p = p"
wenzelm@52803
  1280
  using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
chaieb@33154
  1281
    isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
chaieb@33154
  1282
wenzelm@52803
  1283
lemma polymul_1[simp]:
wenzelm@56000
  1284
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1285
    and np: "isnpolyh p n0"
wenzelm@56000
  1286
  shows "p *\<^sub>p (1)\<^sub>p = p"
wenzelm@56000
  1287
    and "(1)\<^sub>p *\<^sub>p p = p"
wenzelm@52803
  1288
  using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
chaieb@33154
  1289
    isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
wenzelm@52658
  1290
wenzelm@52803
  1291
lemma polymul_0[simp]:
wenzelm@56000
  1292
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1293
    and np: "isnpolyh p n0"
wenzelm@56000
  1294
  shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p"
wenzelm@56000
  1295
    and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
wenzelm@52803
  1296
  using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
chaieb@33154
  1297
    isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
chaieb@33154
  1298
wenzelm@52803
  1299
lemma polymul_commute:
wenzelm@56000
  1300
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56000
  1301
    and np: "isnpolyh p n0"
wenzelm@52658
  1302
    and nq: "isnpolyh q n1"
chaieb@33154
  1303
  shows "p *\<^sub>p q = q *\<^sub>p p"
wenzelm@56043
  1304
  using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np],
wenzelm@56043
  1305
    where ?'a = "'a::{field_char_0,field_inverse_zero, power}"]
wenzelm@52658
  1306
  by simp
chaieb@33154
  1307
wenzelm@52658
  1308
declare polyneg_polyneg [simp]
wenzelm@52803
  1309
wenzelm@52803
  1310
lemma isnpolyh_polynate_id [simp]:
wenzelm@56000
  1311
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56000
  1312
    and np: "isnpolyh p n0"
wenzelm@52658
  1313
  shows "polynate p = p"
wenzelm@56043
  1314
  using isnpolyh_unique[where ?'a= "'a::{field_char_0,field_inverse_zero}",
wenzelm@56043
  1315
      OF polynate_norm[of p, unfolded isnpoly_def] np]
wenzelm@56043
  1316
    polynate[where ?'a = "'a::{field_char_0,field_inverse_zero}"]
wenzelm@52658
  1317
  by simp
chaieb@33154
  1318
wenzelm@52803
  1319
lemma polynate_idempotent[simp]:
wenzelm@56000
  1320
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
  1321
  shows "polynate (polynate p) = polynate p"
chaieb@33154
  1322
  using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
chaieb@33154
  1323
chaieb@33154
  1324
lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
chaieb@33154
  1325
  unfolding poly_nate_def polypoly'_def ..
wenzelm@52658
  1326
wenzelm@52658
  1327
lemma poly_nate_poly:
wenzelm@56000
  1328
  "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
  1329
  using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
wenzelm@52658
  1330
  unfolding poly_nate_polypoly' by auto
wenzelm@52658
  1331
chaieb@33154
  1332
chaieb@33154
  1333
subsection{* heads, degrees and all that *}
wenzelm@52658
  1334
chaieb@33154
  1335
lemma degree_eq_degreen0: "degree p = degreen p 0"
wenzelm@52658
  1336
  by (induct p rule: degree.induct) simp_all
chaieb@33154
  1337
wenzelm@52658
  1338
lemma degree_polyneg:
wenzelm@56043
  1339
  assumes "isnpolyh p n"
chaieb@33154
  1340
  shows "degree (polyneg p) = degree p"
wenzelm@56043
  1341
  apply (induct p rule: polyneg.induct)
wenzelm@56043
  1342
  using assms
wenzelm@56043
  1343
  apply simp_all
wenzelm@52658
  1344
  apply (case_tac na)
wenzelm@52658
  1345
  apply auto
wenzelm@52658
  1346
  done
chaieb@33154
  1347
chaieb@33154
  1348
lemma degree_polyadd:
wenzelm@56043
  1349
  assumes np: "isnpolyh p n0"
wenzelm@56043
  1350
    and nq: "isnpolyh q n1"
chaieb@33154
  1351
  shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
wenzelm@52658
  1352
  using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
chaieb@33154
  1353
chaieb@33154
  1354
wenzelm@52658
  1355
lemma degree_polysub:
wenzelm@52658
  1356
  assumes np: "isnpolyh p n0"
wenzelm@52658
  1357
    and nq: "isnpolyh q n1"
chaieb@33154
  1358
  shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
chaieb@33154
  1359
proof-
wenzelm@56043
  1360
  from nq have nq': "isnpolyh (~\<^sub>p q) n1"
wenzelm@56043
  1361
    using polyneg_normh by simp
wenzelm@56043
  1362
  from degree_polyadd[OF np nq'] show ?thesis
wenzelm@56043
  1363
    by (simp add: polysub_def degree_polyneg[OF nq])
chaieb@33154
  1364
qed
chaieb@33154
  1365
wenzelm@52803
  1366
lemma degree_polysub_samehead:
wenzelm@56000
  1367
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56043
  1368
    and np: "isnpolyh p n0"
wenzelm@56043
  1369
    and nq: "isnpolyh q n1"
wenzelm@56043
  1370
    and h: "head p = head q"
wenzelm@52658
  1371
    and d: "degree p = degree q"
chaieb@33154
  1372
  shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
wenzelm@52658
  1373
  unfolding polysub_def split_def fst_conv snd_conv
wenzelm@52658
  1374
  using np nq h d
wenzelm@52658
  1375
proof (induct p q rule: polyadd.induct)
wenzelm@52658
  1376
  case (1 c c')
wenzelm@56009
  1377
  then show ?case
wenzelm@56009
  1378
    by (simp add: Nsub_def Nsub0[simplified Nsub_def])
chaieb@33154
  1379
next
wenzelm@52803
  1380
  case (2 c c' n' p')
wenzelm@56009
  1381
  from 2 have "degree (C c) = degree (CN c' n' p')"
wenzelm@56009
  1382
    by simp
wenzelm@56009
  1383
  then have nz: "n' > 0"
wenzelm@56009
  1384
    by (cases n') auto
wenzelm@56009
  1385
  then have "head (CN c' n' p') = CN c' n' p'"
wenzelm@56009
  1386
    by (cases n') auto
wenzelm@56009
  1387
  with 2 show ?case
wenzelm@56009
  1388
    by simp
chaieb@33154
  1389
next
wenzelm@52803
  1390
  case (3 c n p c')
wenzelm@56009
  1391
  then have "degree (C c') = degree (CN c n p)"
wenzelm@56009
  1392
    by simp
wenzelm@56009
  1393
  then have nz: "n > 0"
wenzelm@56009
  1394
    by (cases n) auto
wenzelm@56009
  1395
  then have "head (CN c n p) = CN c n p"
wenzelm@56009
  1396
    by (cases n) auto
wenzelm@41807
  1397
  with 3 show ?case by simp
chaieb@33154
  1398
next
chaieb@33154
  1399
  case (4 c n p c' n' p')
wenzelm@56009
  1400
  then have H:
wenzelm@56009
  1401
    "isnpolyh (CN c n p) n0"
wenzelm@56009
  1402
    "isnpolyh (CN c' n' p') n1"
wenzelm@56009
  1403
    "head (CN c n p) = head (CN c' n' p')"
wenzelm@56009
  1404
    "degree (CN c n p) = degree (CN c' n' p')"
wenzelm@56009
  1405
    by simp_all
wenzelm@56009
  1406
  then have degc: "degree c = 0" and degc': "degree c' = 0"
wenzelm@56009
  1407
    by simp_all
wenzelm@56009
  1408
  then have degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
chaieb@33154
  1409
    using H(1-2) degree_polyneg by auto
wenzelm@56009
  1410
  from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"
wenzelm@56009
  1411
    by simp_all
wenzelm@56009
  1412
  from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc'
wenzelm@56009
  1413
  have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"
wenzelm@56009
  1414
    by simp
wenzelm@56009
  1415
  from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'"
wenzelm@56009
  1416
    by auto
wenzelm@56009
  1417
  have "n = n' \<or> n < n' \<or> n > n'"
wenzelm@56009
  1418
    by arith
chaieb@33154
  1419
  moreover
wenzelm@56009
  1420
  {
wenzelm@56009
  1421
    assume nn': "n = n'"
wenzelm@56009
  1422
    have "n = 0 \<or> n > 0" by arith
wenzelm@56066
  1423
    moreover
wenzelm@56066
  1424
    {
wenzelm@56009
  1425
      assume nz: "n = 0"
wenzelm@56009
  1426
      then have ?case using 4 nn'
wenzelm@56009
  1427
        by (auto simp add: Let_def degcmc')
wenzelm@56009
  1428
    }
wenzelm@56066
  1429
    moreover
wenzelm@56066
  1430
    {
wenzelm@56009
  1431
      assume nz: "n > 0"
wenzelm@56009
  1432
      with nn' H(3) have  cc': "c = c'" and pp': "p = p'"
wenzelm@56009
  1433
        by (cases n, auto)+
wenzelm@56009
  1434
      then have ?case
wenzelm@56009
  1435
        using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv]
wenzelm@56009
  1436
        using polysub_same_0[OF c'nh, simplified polysub_def]
wenzelm@56009
  1437
        using nn' 4 by (simp add: Let_def)
wenzelm@56009
  1438
    }
wenzelm@56009
  1439
    ultimately have ?case by blast
wenzelm@56009
  1440
  }
chaieb@33154
  1441
  moreover
wenzelm@56009
  1442
  {
wenzelm@56009
  1443
    assume nn': "n < n'"
wenzelm@56009
  1444
    then have n'p: "n' > 0"
wenzelm@56009
  1445
      by simp
wenzelm@56009
  1446
    then have headcnp':"head (CN c' n' p') = CN c' n' p'"
wenzelm@56009
  1447
      by (cases n') simp_all
wenzelm@56009
  1448
    have degcnp': "degree (CN c' n' p') = 0"
wenzelm@56009
  1449
      and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
wenzelm@52658
  1450
      using 4 nn' by (cases n', simp_all)
wenzelm@56009
  1451
    then have "n > 0"
wenzelm@56009
  1452
      by (cases n) simp_all
wenzelm@56009
  1453
    then have headcnp: "head (CN c n p) = CN c n p"
wenzelm@56009
  1454
      by (cases n) auto
wenzelm@56009
  1455
    from H(3) headcnp headcnp' nn' have ?case
wenzelm@56009
  1456
      by auto
wenzelm@56009
  1457
  }
chaieb@33154
  1458
  moreover
wenzelm@56009
  1459
  {
wenzelm@56009
  1460
    assume nn': "n > n'"
wenzelm@56009
  1461
    then have np: "n > 0" by simp
wenzelm@56009
  1462
    then have headcnp:"head (CN c n p) = CN c n p"
wenzelm@56009
  1463
      by (cases n) simp_all
wenzelm@56009
  1464
    from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)"
wenzelm@56009
  1465
      by simp
wenzelm@56009
  1466
    from np have degcnp: "degree (CN c n p) = 0"
wenzelm@56009
  1467
      by (cases n) simp_all
wenzelm@56009
  1468
    with degcnpeq have "n' > 0"
wenzelm@56009
  1469
      by (cases n') simp_all
wenzelm@56009
  1470
    then have headcnp': "head (CN c' n' p') = CN c' n' p'"
wenzelm@56009
  1471
      by (cases n') auto
wenzelm@56009
  1472
    from H(3) headcnp headcnp' nn' have ?case by auto
wenzelm@56009
  1473
  }
wenzelm@56009
  1474
  ultimately show ?case by blast
krauss@41812
  1475
qed auto
wenzelm@52803
  1476
chaieb@33154
  1477
lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
wenzelm@52658
  1478
  by (induct p arbitrary: n0 rule: head.induct) (simp_all add: shift1_def)
chaieb@33154
  1479
chaieb@33154
  1480
lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
wenzelm@52658
  1481
proof (induct k arbitrary: n0 p)
wenzelm@52658
  1482
  case 0
wenzelm@56198
  1483
  then show ?case
wenzelm@56198
  1484
    by auto
wenzelm@52658
  1485
next
wenzelm@52658
  1486
  case (Suc k n0 p)
wenzelm@56066
  1487
  then have "isnpolyh (shift1 p) 0"
wenzelm@56066
  1488
    by (simp add: shift1_isnpolyh)
wenzelm@41807
  1489
  with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
wenzelm@56066
  1490
    and "head (shift1 p) = head p"
wenzelm@56066
  1491
    by (simp_all add: shift1_head)
wenzelm@56066
  1492
  then show ?case
wenzelm@56066
  1493
    by (simp add: funpow_swap1)
wenzelm@52658
  1494
qed
chaieb@33154
  1495
chaieb@33154
  1496
lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
chaieb@33154
  1497
  by (simp add: shift1_def)
wenzelm@56009
  1498
chaieb@33154
  1499
lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
wenzelm@46991
  1500
  by (induct k arbitrary: p) (auto simp add: shift1_degree)
chaieb@33154
  1501
chaieb@33154
  1502
lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
wenzelm@52658
  1503
  by (induct n arbitrary: p) simp_all
chaieb@33154
  1504
chaieb@33154
  1505
lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
wenzelm@52658
  1506
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
  1507
lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
wenzelm@52658
  1508
  by (induct p arbitrary: n rule: degreen.induct) auto
chaieb@33154
  1509
lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
wenzelm@52658
  1510
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
  1511
lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
wenzelm@52658
  1512
  by (induct p rule: head.induct) auto
chaieb@33154
  1513
wenzelm@52803
  1514
lemma polyadd_eq_const_degree:
wenzelm@52658
  1515
  "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> polyadd p q = C c \<Longrightarrow> degree p = degree q"
chaieb@33154
  1516
  using polyadd_eq_const_degreen degree_eq_degreen0 by simp
chaieb@33154
  1517
wenzelm@52658
  1518
lemma polyadd_head:
wenzelm@52658
  1519
  assumes np: "isnpolyh p n0"
wenzelm@52658
  1520
    and nq: "isnpolyh q n1"
wenzelm@52658
  1521
    and deg: "degree p \<noteq> degree q"
chaieb@33154
  1522
  shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
wenzelm@52658
  1523
  using np nq deg
wenzelm@52658
  1524
  apply (induct p q arbitrary: n0 n1 rule: polyadd.induct)
wenzelm@52658
  1525
  apply simp_all
wenzelm@52658
  1526
  apply (case_tac n', simp, simp)
wenzelm@52658
  1527
  apply (case_tac n, simp, simp)
wenzelm@52658
  1528
  apply (case_tac n, case_tac n', simp add: Let_def)
haftmann@54489
  1529
  apply (auto simp add: polyadd_eq_const_degree)[2]
wenzelm@52658
  1530
  apply (metis head_nz)
wenzelm@52658
  1531
  apply (metis head_nz)
wenzelm@52658
  1532
  apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
wenzelm@52658
  1533
  done
chaieb@33154
  1534
wenzelm@52803
  1535
lemma polymul_head_polyeq:
wenzelm@56000
  1536
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56066
  1537
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> q \<noteq> 0\<^sub>p \<Longrightarrow> head (p *\<^sub>p q) = head p *\<^sub>p head q"
chaieb@33154
  1538
proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
krauss@41813
  1539
  case (2 c c' n' p' n0 n1)
wenzelm@56009
  1540
  then have "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"
wenzelm@56009
  1541
    by (simp_all add: head_isnpolyh)
wenzelm@56009
  1542
  then show ?case
wenzelm@56009
  1543
    using 2 by (cases n') auto
wenzelm@52803
  1544
next
wenzelm@52803
  1545
  case (3 c n p c' n0 n1)
wenzelm@56009
  1546
  then have "isnpolyh (head (CN c n p)) n0" "isnormNum c'"
wenzelm@56009
  1547
    by (simp_all add: head_isnpolyh)
wenzelm@56066
  1548
  then show ?case
wenzelm@56066
  1549
    using 3 by (cases n) auto
chaieb@33154
  1550
next
chaieb@33154
  1551
  case (4 c n p c' n' p' n0 n1)
wenzelm@56066
  1552
  then have norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
chaieb@33154
  1553
    "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
chaieb@33154
  1554
    by simp_all
chaieb@33154
  1555
  have "n < n' \<or> n' < n \<or> n = n'" by arith
wenzelm@52803
  1556
  moreover
wenzelm@56009
  1557
  {
wenzelm@56009
  1558
    assume nn': "n < n'"
wenzelm@56009
  1559
    then have ?case
wenzelm@52658
  1560
      using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)]
wenzelm@52658
  1561
      apply simp
wenzelm@52658
  1562
      apply (cases n)
wenzelm@52658
  1563
      apply simp
wenzelm@52658
  1564
      apply (cases n')
wenzelm@52658
  1565
      apply simp_all
wenzelm@56009
  1566
      done
wenzelm@56009
  1567
  }
wenzelm@56009
  1568
  moreover {
wenzelm@56009
  1569
    assume nn': "n'< n"
wenzelm@56009
  1570
    then have ?case
wenzelm@52803
  1571
      using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)]
wenzelm@52658
  1572
      apply simp
wenzelm@52658
  1573
      apply (cases n')
wenzelm@52658
  1574
      apply simp
wenzelm@52658
  1575
      apply (cases n)
wenzelm@52658
  1576
      apply auto
wenzelm@56009
  1577
      done
wenzelm@56009
  1578
  }
wenzelm@56066
  1579
  moreover
wenzelm@56066
  1580
  {
wenzelm@56009
  1581
    assume nn': "n' = n"
wenzelm@52803
  1582
    from nn' polymul_normh[OF norm(5,4)]
chaieb@33154
  1583
    have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
wenzelm@52803
  1584
    from nn' polymul_normh[OF norm(5,3)] norm
chaieb@33154
  1585
    have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
chaieb@33154
  1586
    from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
wenzelm@52803
  1587
    have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
wenzelm@52803
  1588
    from polyadd_normh[OF ncnpc' ncnpp0']
wenzelm@52803
  1589
    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
  1590
      by (simp add: min_def)
wenzelm@56009
  1591
    {
wenzelm@56009
  1592
      assume np: "n > 0"
chaieb@33154
  1593
      with nn' head_isnpolyh_Suc'[OF np nth]
wenzelm@33268
  1594
        head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
wenzelm@56009
  1595
      have ?case by simp
wenzelm@56009
  1596
    }
chaieb@33154
  1597
    moreover
wenzelm@56009
  1598
    {
wenzelm@56009
  1599
      assume nz: "n = 0"
chaieb@33154
  1600
      from polymul_degreen[OF norm(5,4), where m="0"]
wenzelm@33268
  1601
        polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
chaieb@33154
  1602
      norm(5,6) degree_npolyhCN[OF norm(6)]
wenzelm@56066
  1603
    have dth: "degree (CN c 0 p *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))"
wenzelm@56066
  1604
      by simp
wenzelm@56066
  1605
    then have dth': "degree (CN c 0 p *\<^sub>p c') \<noteq> degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))"
wenzelm@56066
  1606
      by simp
chaieb@33154
  1607
    from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
wenzelm@56066
  1608
    have ?case
wenzelm@56066
  1609
      using norm "4.hyps"(6)[OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)] nn' nz
wenzelm@56066
  1610
      by simp
wenzelm@56009
  1611
    }
wenzelm@56066
  1612
    ultimately have ?case
wenzelm@56066
  1613
      by (cases n) auto
wenzelm@56009
  1614
  }
chaieb@33154
  1615
  ultimately show ?case by blast
chaieb@33154
  1616
qed simp_all
chaieb@33154
  1617
chaieb@33154
  1618
lemma degree_coefficients: "degree p = length (coefficients p) - 1"
wenzelm@52658
  1619
  by (induct p rule: degree.induct) auto
chaieb@33154
  1620
chaieb@33154
  1621
lemma degree_head[simp]: "degree (head p) = 0"
wenzelm@52658
  1622
  by (induct p rule: head.induct) auto
chaieb@33154
  1623
krauss@41812
  1624
lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
wenzelm@52658
  1625
  by (cases n) simp_all
wenzelm@56066
  1626
chaieb@33154
  1627
lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
wenzelm@52658
  1628
  by (cases n) simp_all
chaieb@33154
  1629
wenzelm@52658
  1630
lemma polyadd_different_degree:
wenzelm@56066
  1631
  "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> degree p \<noteq> degree q \<Longrightarrow>
wenzelm@52658
  1632
    degree (polyadd p q) = max (degree p) (degree q)"
chaieb@33154
  1633
  using polyadd_different_degreen degree_eq_degreen0 by simp
chaieb@33154
  1634
chaieb@33154
  1635
lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
wenzelm@52658
  1636
  by (induct p arbitrary: n0 rule: polyneg.induct) auto
chaieb@33154
  1637
chaieb@33154
  1638
lemma degree_polymul:
wenzelm@56000
  1639
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1640
    and np: "isnpolyh p n0"
wenzelm@52658
  1641
    and nq: "isnpolyh q n1"
chaieb@33154
  1642
  shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
chaieb@33154
  1643
  using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
chaieb@33154
  1644
chaieb@33154
  1645
lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
wenzelm@52658
  1646
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
  1647
wenzelm@56207
  1648
lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head (polyneg p) = polyneg (head p)"
wenzelm@52658
  1649
  by (induct p arbitrary: n rule: degree.induct) auto
wenzelm@52658
  1650
chaieb@33154
  1651
chaieb@33154
  1652
subsection {* Correctness of polynomial pseudo division *}
chaieb@33154
  1653
chaieb@33154
  1654
lemma polydivide_aux_properties:
wenzelm@56000
  1655
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1656
    and np: "isnpolyh p n0"
wenzelm@52658
  1657
    and ns: "isnpolyh s n1"
wenzelm@52658
  1658
    and ap: "head p = a"
wenzelm@56198
  1659
    and ndp: "degree p = n"
wenzelm@56198
  1660
    and pnz: "p \<noteq> 0\<^sub>p"
wenzelm@56198
  1661
  shows "polydivide_aux a n p k s = (k', r) \<longrightarrow> k' \<ge> k \<and> (degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@56066
  1662
    (\<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
  1663
  using ns
wenzelm@52658
  1664
proof (induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
berghofe@34915
  1665
  case less
chaieb@33154
  1666
  let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
wenzelm@56066
  1667
  let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and>
wenzelm@56066
  1668
    (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
chaieb@33154
  1669
  let ?b = "head s"
berghofe@34915
  1670
  let ?p' = "funpow (degree s - n) shift1 p"
wenzelm@50282
  1671
  let ?xdn = "funpow (degree s - n) shift1 (1)\<^sub>p"
chaieb@33154
  1672
  let ?akk' = "a ^\<^sub>p (k' - k)"
berghofe@34915
  1673
  note ns = `isnpolyh s n1`
wenzelm@52803
  1674
  from np have np0: "isnpolyh p 0"
wenzelm@52803
  1675
    using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by simp
wenzelm@52803
  1676
  have np': "isnpolyh ?p' 0"
wenzelm@52803
  1677
    using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def
wenzelm@52803
  1678
    by simp
wenzelm@52803
  1679
  have headp': "head ?p' = head p"
wenzelm@52803
  1680
    using funpow_shift1_head[OF np pnz] by simp
wenzelm@52803
  1681
  from funpow_shift1_isnpoly[where p="(1)\<^sub>p"] have nxdn: "isnpolyh ?xdn 0"
wenzelm@52803
  1682
    by (simp add: isnpoly_def)
wenzelm@52803
  1683
  from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
chaieb@33154
  1684
  have nakk':"isnpolyh ?akk' 0" by blast
wenzelm@56066
  1685
  {
wenzelm@56066
  1686
    assume sz: "s = 0\<^sub>p"
wenzelm@56066
  1687
    then have ?ths
wenzelm@56066
  1688
      using np polydivide_aux.simps
wenzelm@52658
  1689
      apply clarsimp
wenzelm@52658
  1690
      apply (rule exI[where x="0\<^sub>p"])
wenzelm@52658
  1691
      apply simp
wenzelm@56066
  1692
      done
wenzelm@56066
  1693
  }
chaieb@33154
  1694
  moreover
wenzelm@56066
  1695
  {
wenzelm@56066
  1696
    assume sz: "s \<noteq> 0\<^sub>p"
wenzelm@56066
  1697
    {
wenzelm@56066
  1698
      assume dn: "degree s < n"
wenzelm@56066
  1699
      then have "?ths"
wenzelm@56066
  1700
        using ns ndp np polydivide_aux.simps
wenzelm@52658
  1701
        apply auto
wenzelm@52658
  1702
        apply (rule exI[where x="0\<^sub>p"])
wenzelm@52658
  1703
        apply simp
wenzelm@56066
  1704
        done
wenzelm@56066
  1705
    }
wenzelm@52803
  1706
    moreover
wenzelm@56066
  1707
    {
wenzelm@56066
  1708
      assume dn': "\<not> degree s < n"
wenzelm@56066
  1709
      then have dn: "degree s \<ge> n"
wenzelm@56066
  1710
        by arith
wenzelm@52803
  1711
      have degsp': "degree s = degree ?p'"
wenzelm@56066
  1712
        using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"]
wenzelm@56066
  1713
        by simp
wenzelm@56066
  1714
      {
wenzelm@56066
  1715
        assume ba: "?b = a"
wenzelm@56066
  1716
        then have headsp': "head s = head ?p'"
wenzelm@52803
  1717
          using ap headp' by simp
wenzelm@52803
  1718
        have nr: "isnpolyh (s -\<^sub>p ?p') 0"
wenzelm@52803
  1719
          using polysub_normh[OF ns np'] by simp
berghofe@34915
  1720
        from degree_polysub_samehead[OF ns np' headsp' degsp']
wenzelm@56207
  1721
        have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p"
wenzelm@56207
  1722
          by simp
wenzelm@52803
  1723
        moreover
wenzelm@56066
  1724
        {
wenzelm@56066
  1725
          assume deglt:"degree (s -\<^sub>p ?p') < degree s"
krauss@41403
  1726
          from polydivide_aux.simps sz dn' ba
krauss@41403
  1727
          have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
wenzelm@33268
  1728
            by (simp add: Let_def)
wenzelm@56066
  1729
          {
wenzelm@56066
  1730
            assume h1: "polydivide_aux a n p k s = (k', r)"
wenzelm@52803
  1731
            from less(1)[OF deglt nr, of k k' r] trans[OF eq[symmetric] h1]
wenzelm@52803
  1732
            have kk': "k \<le> k'"
wenzelm@56066
  1733
              and nr: "\<exists>nr. isnpolyh r nr"
wenzelm@52803
  1734
              and dr: "degree r = 0 \<or> degree r < degree p"
wenzelm@56066
  1735
              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"
wenzelm@52803
  1736
              by auto
wenzelm@52803
  1737
            from q1 obtain q n1 where nq: "isnpolyh q n1"
wenzelm@56066
  1738
              and asp: "a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"
wenzelm@56066
  1739
              by blast
wenzelm@56066
  1740
            from nr obtain nr where nr': "isnpolyh r nr"
wenzelm@56066
  1741
              by blast
wenzelm@52803
  1742
            from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0"
wenzelm@52803
  1743
              by simp
wenzelm@33268
  1744
            from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
wenzelm@33268
  1745
            have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
wenzelm@52803
  1746
            from polyadd_normh[OF polymul_normh[OF np
wenzelm@33268
  1747
              polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
wenzelm@52803
  1748
            have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0"
wenzelm@52803
  1749
              by simp
wenzelm@56066
  1750
            from asp have "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
wenzelm@56066
  1751
              Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)"
wenzelm@56066
  1752
              by simp
wenzelm@56066
  1753
            then have "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
wenzelm@56066
  1754
              Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
wenzelm@52803
  1755
              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
haftmann@36349
  1756
              by (simp add: field_simps)
wenzelm@56066
  1757
            then have "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
wenzelm@56066
  1758
              Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1759
              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p *\<^sub>p p) +
wenzelm@52803
  1760
              Ipoly bs p * Ipoly bs q + Ipoly bs r"
wenzelm@52803
  1761
              by (auto simp only: funpow_shift1_1)
wenzelm@56066
  1762
            then have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list.
wenzelm@56066
  1763
              Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1764
              Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p) +
wenzelm@52803
  1765
              Ipoly bs q) + Ipoly bs r"
wenzelm@52803
  1766
              by (simp add: field_simps)
wenzelm@56066
  1767
            then have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list.
wenzelm@56066
  1768
              Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1769
              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)"
wenzelm@52803
  1770
              by simp
wenzelm@33268
  1771
            with isnpolyh_unique[OF nakks' nqr']
wenzelm@52803
  1772
            have "a ^\<^sub>p (k' - k) *\<^sub>p s =
wenzelm@52803
  1773
              p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r"
wenzelm@52803
  1774
              by blast
wenzelm@56066
  1775
            then have ?qths using nq'
wenzelm@50282
  1776
              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@52803
  1777
              apply (rule_tac x="0" in exI)
wenzelm@52803
  1778
              apply simp
wenzelm@52803
  1779
              done
wenzelm@56066
  1780
            with kk' nr dr have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@56066
  1781
              (\<exists>nr. isnpolyh r nr) \<and> ?qths"
wenzelm@52803
  1782
              by blast
wenzelm@52803
  1783
          }
wenzelm@56066
  1784
          then have ?ths by blast
wenzelm@52803
  1785
        }
wenzelm@52803
  1786
        moreover
wenzelm@56066
  1787
        {
wenzelm@56066
  1788
          assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
wenzelm@56000
  1789
          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}"]
wenzelm@56066
  1790
          have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list. Ipoly bs s = Ipoly bs ?p'"
wenzelm@52803
  1791
            by simp
wenzelm@56066
  1792
          then have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list. Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)"
wenzelm@52658
  1793
            using np nxdn
wenzelm@52658
  1794
            apply simp
wenzelm@52658
  1795
            apply (simp only: funpow_shift1_1)
wenzelm@52658
  1796
            apply simp
wenzelm@52658
  1797
            done
wenzelm@56066
  1798
          then have sp': "s = ?xdn *\<^sub>p p"
wenzelm@56066
  1799
            using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]]
wenzelm@52658
  1800
            by blast
wenzelm@56066
  1801
          {
wenzelm@56207
  1802
            assume h1: "polydivide_aux a n p k s = (k', r)"
krauss@41403
  1803
            from polydivide_aux.simps sz dn' ba
krauss@41403
  1804
            have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
wenzelm@33268
  1805
              by (simp add: Let_def)
wenzelm@52803
  1806
            also have "\<dots> = (k,0\<^sub>p)"
wenzelm@52803
  1807
              using polydivide_aux.simps spz by simp
wenzelm@56066
  1808
            finally have "(k', r) = (k, 0\<^sub>p)"
wenzelm@56066
  1809
              using h1 by simp
berghofe@34915
  1810
            with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
krauss@41403
  1811
              polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
wenzelm@33268
  1812
              apply auto
wenzelm@52803
  1813
              apply (rule exI[where x="?xdn"])
berghofe@34915
  1814
              apply (auto simp add: polymul_commute[of p])
wenzelm@52803
  1815
              done
wenzelm@52803
  1816
          }
wenzelm@52803
  1817
        }
wenzelm@52803
  1818
        ultimately have ?ths by blast
wenzelm@52803
  1819
      }
chaieb@33154
  1820
      moreover
wenzelm@56066
  1821
      {
wenzelm@56066
  1822
        assume ba: "?b \<noteq> a"
wenzelm@52803
  1823
        from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
wenzelm@33268
  1824
          polymul_normh[OF head_isnpolyh[OF ns] np']]
wenzelm@52803
  1825
        have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
wenzelm@52803
  1826
          by (simp add: min_def)
wenzelm@33268
  1827
        have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
wenzelm@52803
  1828
          using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
wenzelm@33268
  1829
            polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
wenzelm@52803
  1830
            funpow_shift1_nz[OF pnz]
wenzelm@52803
  1831
          by simp_all
wenzelm@33268
  1832
        from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
berghofe@34915
  1833
          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@52803
  1834
        have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
wenzelm@33268
  1835
          using head_head[OF ns] funpow_shift1_head[OF np pnz]
wenzelm@33268
  1836
            polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
wenzelm@33268
  1837
          by (simp add: ap)
wenzelm@33268
  1838
        from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
wenzelm@33268
  1839
          head_nz[OF np] pnz sz ap[symmetric]
berghofe@34915
  1840
          funpow_shift1_nz[OF pnz, where n="degree s - n"]
wenzelm@52803
  1841
          polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"] head_nz[OF ns]
berghofe@34915
  1842
          ndp dn
wenzelm@52803
  1843
        have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p')"
wenzelm@33268
  1844
          by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
wenzelm@56066
  1845
        {
wenzelm@56066
  1846
          assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
wenzelm@52803
  1847
          from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns]
wenzelm@52803
  1848
            polymul_normh[OF head_isnpolyh[OF ns]np']] ap
wenzelm@52803
  1849
          have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
wenzelm@52803
  1850
            by simp
wenzelm@56066
  1851
          {
wenzelm@56066
  1852
            assume h1:"polydivide_aux a n p k s = (k', r)"
krauss@41403
  1853
            from h1 polydivide_aux.simps sz dn' ba
krauss@41403
  1854
            have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
wenzelm@33268
  1855
              by (simp add: Let_def)
berghofe@34915
  1856
            with less(1)[OF dth nasbp', of "Suc k" k' r]
wenzelm@52803
  1857
            obtain q nq nr where kk': "Suc k \<le> k'"
wenzelm@52803
  1858
              and nr: "isnpolyh r nr"
wenzelm@52803
  1859
              and nq: "isnpolyh q nq"
wenzelm@33268
  1860
              and dr: "degree r = 0 \<or> degree r < degree p"
wenzelm@52803
  1861
              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"
wenzelm@52803
  1862
              by auto
wenzelm@56066
  1863
            from kk' have kk'': "Suc (k' - Suc k) = k' - k"
wenzelm@56066
  1864
              by arith
wenzelm@52803
  1865
            {
wenzelm@56066
  1866
              fix bs :: "'a::{field_char_0,field_inverse_zero} list"
wenzelm@52803
  1867
              from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
wenzelm@52803
  1868
              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)"
wenzelm@52803
  1869
                by simp
wenzelm@56066
  1870
              then have "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s =
wenzelm@52803
  1871
                Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
wenzelm@52803
  1872
                by (simp add: field_simps)
wenzelm@56066
  1873
              then have "Ipoly bs a ^ (k' - k)  * Ipoly bs s =
wenzelm@52803
  1874
                Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
wenzelm@52803
  1875
                by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
wenzelm@56066
  1876
              then have "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1877
                Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
wenzelm@52803
  1878
                by (simp add: field_simps)
wenzelm@52803
  1879
            }
wenzelm@56066
  1880
            then have ieq:"\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
wenzelm@56207
  1881
                Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@56207
  1882
                Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)"
wenzelm@52803
  1883
              by auto
wenzelm@33268
  1884
            let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
wenzelm@56207
  1885
            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@52803
  1886
            have nqw: "isnpolyh ?q 0"
wenzelm@52803
  1887
              by simp
wenzelm@33268
  1888
            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@52803
  1889
            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"
wenzelm@52803
  1890
              by blast
wenzelm@52803
  1891
            from dr kk' nr h1 asth nqw have ?ths
wenzelm@52803
  1892
              apply simp
wenzelm@33268
  1893
              apply (rule conjI)
wenzelm@33268
  1894
              apply (rule exI[where x="nr"], simp)
wenzelm@33268
  1895
              apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
wenzelm@33268
  1896
              apply (rule exI[where x="0"], simp)
wenzelm@52803
  1897
              done
wenzelm@52803
  1898
          }
wenzelm@56066
  1899
          then have ?ths by blast
wenzelm@52803
  1900
        }
wenzelm@52803
  1901
        moreover
wenzelm@56066
  1902
        {
wenzelm@56066
  1903
          assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
wenzelm@52803
  1904
          {
wenzelm@56000
  1905
            fix bs :: "'a::{field_char_0,field_inverse_zero} list"
wenzelm@33268
  1906
            from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
wenzelm@52803
  1907
            have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'"
wenzelm@52803
  1908
              by simp
wenzelm@56066
  1909
            then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
wenzelm@52803
  1910
              by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
wenzelm@56066
  1911
            then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))"
wenzelm@52803
  1912
              by simp
wenzelm@52803
  1913
          }
wenzelm@56066
  1914
          then have hth: "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
wenzelm@56066
  1915
            Ipoly bs (a *\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
wenzelm@52803
  1916
          from hth have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
wenzelm@56000
  1917
            using isnpolyh_unique[where ?'a = "'a::{field_char_0,field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
chaieb@33154
  1918
                    polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
wenzelm@56066
  1919
              simplified ap]
wenzelm@56066
  1920
            by simp
wenzelm@56066
  1921
          {
wenzelm@56066
  1922
            assume h1: "polydivide_aux a n p k s = (k', r)"
wenzelm@52803
  1923
            from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
wenzelm@56066
  1924
            have "(k', r) = (Suc k, 0\<^sub>p)"
wenzelm@56066
  1925
              by (simp add: Let_def)
wenzelm@52803
  1926
            with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
wenzelm@52803
  1927
              polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
wenzelm@52803
  1928
            have ?ths
wenzelm@52803
  1929
              apply (clarsimp simp add: Let_def)
wenzelm@52803
  1930
              apply (rule exI[where x="?b *\<^sub>p ?xdn"])
wenzelm@52803
  1931
              apply simp
wenzelm@52803
  1932
              apply (rule exI[where x="0"], simp)
wenzelm@52803
  1933
              done
wenzelm@52803
  1934
          }
wenzelm@56066
  1935
          then have ?ths by blast
wenzelm@52803
  1936
        }
wenzelm@52658
  1937
        ultimately have ?ths
wenzelm@56066
  1938
          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"]
wenzelm@52658
  1939
            head_nz[OF np] pnz sz ap[symmetric]
wenzelm@56066
  1940
          by (auto simp add: degree_eq_degreen0[symmetric])
wenzelm@52803
  1941
      }
chaieb@33154
  1942
      ultimately have ?ths by blast
chaieb@33154
  1943
    }
wenzelm@52803
  1944
    ultimately have ?ths by blast
wenzelm@52803
  1945
  }
chaieb@33154
  1946
  ultimately show ?ths by blast
chaieb@33154
  1947
qed
chaieb@33154
  1948
wenzelm@52803
  1949
lemma polydivide_properties:
wenzelm@56000
  1950
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56066
  1951
    and np: "isnpolyh p n0"
wenzelm@56066
  1952
    and ns: "isnpolyh s n1"
wenzelm@56066
  1953
    and pnz: "p \<noteq> 0\<^sub>p"
wenzelm@56066
  1954
  shows "\<exists>k r. polydivide s p = (k, r) \<and>
wenzelm@52803
  1955
    (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@56066
  1956
    (\<exists>q n1. isnpolyh q n1 \<and> polypow k (head p) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
wenzelm@52803
  1957
proof -
wenzelm@52803
  1958
  have trv: "head p = head p" "degree p = degree p"
wenzelm@52803
  1959
    by simp_all
wenzelm@52803
  1960
  from polydivide_def[where s="s" and p="p"] have ex: "\<exists> k r. polydivide s p = (k,r)"
wenzelm@52803
  1961
    by auto
wenzelm@52803
  1962
  then obtain k r where kr: "polydivide s p = (k,r)"
wenzelm@52803
  1963
    by blast
wenzelm@56000
  1964
  from trans[OF polydivide_def[where s="s"and p="p", symmetric] kr]
chaieb@33154
  1965
    polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
chaieb@33154
  1966
  have "(degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@52803
  1967
    (\<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)"
wenzelm@52803
  1968
    by blast
wenzelm@52803
  1969
  with kr show ?thesis
chaieb@33154
  1970
    apply -
chaieb@33154
  1971
    apply (rule exI[where x="k"])
chaieb@33154
  1972
    apply (rule exI[where x="r"])
chaieb@33154
  1973
    apply simp
chaieb@33154
  1974
    done
chaieb@33154
  1975
qed
chaieb@33154
  1976
wenzelm@52658
  1977
wenzelm@56066
  1978
subsection {* More about polypoly and pnormal etc *}
chaieb@33154
  1979
wenzelm@56000
  1980
definition "isnonconstant p \<longleftrightarrow> \<not> isconstant p"
chaieb@33154
  1981
wenzelm@52658
  1982
lemma isnonconstant_pnormal_iff:
wenzelm@56198
  1983
  assumes "isnonconstant p"
wenzelm@52803
  1984
  shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
chaieb@33154
  1985
proof
wenzelm@52803
  1986
  let ?p = "polypoly bs p"
chaieb@33154
  1987
  assume H: "pnormal ?p"
wenzelm@56066
  1988
  have csz: "coefficients p \<noteq> []"
wenzelm@56198
  1989
    using assms by (cases p) auto
wenzelm@56066
  1990
  from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"] pnormal_last_nonzero[OF H]
wenzelm@56066
  1991
  show "Ipoly bs (head p) \<noteq> 0"
wenzelm@56066
  1992
    by (simp add: polypoly_def)
chaieb@33154
  1993
next
chaieb@33154
  1994
  assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  1995
  let ?p = "polypoly bs p"
wenzelm@56066
  1996
  have csz: "coefficients p \<noteq> []"
wenzelm@56198
  1997
    using assms by (cases p) auto
wenzelm@56066
  1998
  then have pz: "?p \<noteq> []"
wenzelm@56066
  1999
    by (simp add: polypoly_def)
wenzelm@56066
  2000
  then have lg: "length ?p > 0"
wenzelm@56066
  2001
    by simp
wenzelm@52803
  2002
  from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
wenzelm@56066
  2003
  have lz: "last ?p \<noteq> 0"
wenzelm@56066
  2004
    by (simp add: polypoly_def)
chaieb@33154
  2005
  from pnormal_last_length[OF lg lz] show "pnormal ?p" .
chaieb@33154
  2006
qed
chaieb@33154
  2007
chaieb@33154
  2008
lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
chaieb@33154
  2009
  unfolding isnonconstant_def
wenzelm@52658
  2010
  apply (cases p)
wenzelm@52658
  2011
  apply simp_all
wenzelm@52658
  2012
  apply (case_tac nat)
wenzelm@52658
  2013
  apply auto
chaieb@33154
  2014
  done
wenzelm@52658
  2015
wenzelm@52658
  2016
lemma isnonconstant_nonconstant:
wenzelm@56198
  2017
  assumes "isnonconstant p"
chaieb@33154
  2018
  shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
chaieb@33154
  2019
proof
chaieb@33154
  2020
  let ?p = "polypoly bs p"
chaieb@33154
  2021
  assume nc: "nonconstant ?p"
wenzelm@56198
  2022
  from isnonconstant_pnormal_iff[OF assms, of bs] nc
wenzelm@56066
  2023
  show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
wenzelm@56066
  2024
    unfolding nonconstant_def by blast
chaieb@33154
  2025
next
chaieb@33154
  2026
  let ?p = "polypoly bs p"
chaieb@33154
  2027
  assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
wenzelm@56198
  2028
  from isnonconstant_pnormal_iff[OF assms, of bs] h
wenzelm@56066
  2029
  have pn: "pnormal ?p"
wenzelm@56066
  2030
    by blast
wenzelm@56009
  2031
  {
wenzelm@56009
  2032
    fix x
wenzelm@56009
  2033
    assume H: "?p = [x]"
wenzelm@56009
  2034
    from H have "length (coefficients p) = 1"
wenzelm@56009
  2035
      unfolding polypoly_def by auto
wenzelm@56198
  2036
    with isnonconstant_coefficients_length[OF assms]
wenzelm@56198
  2037
    have False by arith
wenzelm@56009
  2038
  }
wenzelm@56009
  2039
  then show "nonconstant ?p"
wenzelm@56009
  2040
    using pn unfolding nonconstant_def by blast
chaieb@33154
  2041
qed
chaieb@33154
  2042
wenzelm@56066
  2043
lemma pnormal_length: "p \<noteq> [] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
wenzelm@52658
  2044
  apply (induct p)
wenzelm@52658
  2045
  apply (simp_all add: pnormal_def)
wenzelm@52658
  2046
  apply (case_tac "p = []")
wenzelm@52658
  2047
  apply simp_all
wenzelm@52658
  2048
  done
chaieb@33154
  2049
wenzelm@52658
  2050
lemma degree_degree:
wenzelm@56207
  2051
  assumes "isnonconstant p"
chaieb@33154
  2052
  shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  2053
proof
wenzelm@52803
  2054
  let ?p = "polypoly bs p"
chaieb@33154
  2055
  assume H: "degree p = Polynomial_List.degree ?p"
wenzelm@56207
  2056
  from isnonconstant_coefficients_length[OF assms] have pz: "?p \<noteq> []"
chaieb@33154
  2057
    unfolding polypoly_def by auto
wenzelm@56207
  2058
  from H degree_coefficients[of p] isnonconstant_coefficients_length[OF assms]
wenzelm@56066
  2059
  have lg: "length (pnormalize ?p) = length ?p"
chaieb@33154
  2060
    unfolding Polynomial_List.degree_def polypoly_def by simp
wenzelm@56066
  2061
  then have "pnormal ?p"
wenzelm@56066
  2062
    using pnormal_length[OF pz] by blast
wenzelm@56207
  2063
  with isnonconstant_pnormal_iff[OF assms] show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
wenzelm@56066
  2064
    by blast
chaieb@33154
  2065
next
wenzelm@56066
  2066
  let ?p = "polypoly bs p"
chaieb@33154
  2067
  assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
wenzelm@56207
  2068
  with isnonconstant_pnormal_iff[OF assms] have "pnormal ?p"
wenzelm@56066
  2069
    by blast
wenzelm@56207
  2070
  with degree_coefficients[of p] isnonconstant_coefficients_length[OF assms]
wenzelm@52803
  2071
  show "degree p = Polynomial_List.degree ?p"
chaieb@33154
  2072
    unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
chaieb@33154
  2073
qed
chaieb@33154
  2074
wenzelm@52658
  2075
wenzelm@52803
  2076
section {* Swaps ; Division by a certain variable *}
wenzelm@52658
  2077
wenzelm@56066
  2078
primrec swap :: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly"
wenzelm@56066
  2079
where
chaieb@33154
  2080
  "swap n m (C x) = C x"
wenzelm@56198
  2081
| "swap n m (Bound k) = Bound (if k = n then m else if k = m then n else k)"
haftmann@39246
  2082
| "swap n m (Neg t) = Neg (swap n m t)"
haftmann@39246
  2083
| "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
haftmann@39246
  2084
| "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
haftmann@39246
  2085
| "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
haftmann@39246
  2086
| "swap n m (Pw t k) = Pw (swap n m t) k"
wenzelm@56066
  2087
| "swap n m (CN c k p) = CN (swap n m c) (if k = n then m else if k=m then n else k) (swap n m p)"
chaieb@33154
  2088
wenzelm@52658
  2089
lemma swap:
wenzelm@56066
  2090
  assumes "n < length bs"
wenzelm@56066
  2091
    and "m < length bs"
chaieb@33154
  2092
  shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
chaieb@33154
  2093
proof (induct t)
wenzelm@52658
  2094
  case (Bound k)
wenzelm@56066
  2095
  then show ?case
wenzelm@56066
  2096
    using assms by simp
chaieb@33154
  2097
next
wenzelm@52658
  2098
  case (CN c k p)
wenzelm@56066
  2099
  then show ?case
wenzelm@56066
  2100
    using assms by simp
chaieb@33154
  2101
qed simp_all
chaieb@33154
  2102
wenzelm@52658
  2103
lemma swap_swap_id [simp]: "swap n m (swap m n t) = t"
wenzelm@52658
  2104
  by (induct t) simp_all
wenzelm@52658
  2105
wenzelm@52658
  2106
lemma swap_commute: "swap n m p = swap m n p"
wenzelm@52658
  2107
  by (induct p) simp_all
chaieb@33154
  2108
chaieb@33154
  2109
lemma swap_same_id[simp]: "swap n n t = t"
wenzelm@52658
  2110
  by (induct t) simp_all
chaieb@33154
  2111
chaieb@33154
  2112
definition "swapnorm n m t = polynate (swap n m t)"
chaieb@33154
  2113
wenzelm@52658
  2114
lemma swapnorm:
wenzelm@52658
  2115
  assumes nbs: "n < length bs"
wenzelm@52658
  2116
    and mbs: "m < length bs"
wenzelm@56000
  2117
  shows "((Ipoly bs (swapnorm n m t) :: 'a::{field_char_0,field_inverse_zero})) =
wenzelm@52658
  2118
    Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
wenzelm@41807
  2119
  using swap[OF assms] swapnorm_def by simp
chaieb@33154
  2120
wenzelm@52658
  2121
lemma swapnorm_isnpoly [simp]:
wenzelm@56000
  2122
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
  2123
  shows "isnpoly (swapnorm n m p)"
chaieb@33154
  2124
  unfolding swapnorm_def by simp
chaieb@33154
  2125
wenzelm@52803
  2126
definition "polydivideby n s p =
wenzelm@56000
  2127
  (let
wenzelm@56000
  2128
    ss = swapnorm 0 n s;
wenzelm@56000
  2129
    sp = swapnorm 0 n p;
wenzelm@56000
  2130
    h = head sp;
wenzelm@56000
  2131
    (k, r) = polydivide ss sp
wenzelm@56000
  2132
   in (k, swapnorm 0 n h, swapnorm 0 n r))"
chaieb@33154
  2133
wenzelm@56000
  2134
lemma swap_nz [simp]: "swap n m p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
  2135
  by (induct p) simp_all
chaieb@33154
  2136
krauss@41808
  2137
fun isweaknpoly :: "poly \<Rightarrow> bool"
krauss@41808
  2138
where
chaieb@33154
  2139
  "isweaknpoly (C c) = True"
krauss@41808
  2140
| "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
krauss@41808
  2141
| "isweaknpoly p = False"
chaieb@33154
  2142
wenzelm@52803
  2143
lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
wenzelm@52658
  2144
  by (induct p arbitrary: n0) auto
chaieb@33154
  2145
wenzelm@52803
  2146
lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
wenzelm@52658
  2147
  by (induct p) auto
chaieb@33154
  2148
chaieb@33154
  2149
end