src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
author wenzelm
Mon Mar 10 23:03:15 2014 +0100 (2014-03-10)
changeset 56043 0b25c3d34b77
parent 56009 dda076a32aea
child 56066 cce36efe32eb
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
        apply blast
wenzelm@52658
   477
        done
wenzelm@52803
   478
    }
wenzelm@56009
   479
    ultimately have ?case by blast
wenzelm@56009
   480
  }
chaieb@33154
   481
  ultimately show ?case by blast
chaieb@33154
   482
qed simp_all
chaieb@33154
   483
chaieb@33154
   484
lemma polymul_properties:
wenzelm@56000
   485
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
   486
    and np: "isnpolyh p n0"
wenzelm@52658
   487
    and nq: "isnpolyh q n1"
wenzelm@52658
   488
    and m: "m \<le> min n0 n1"
wenzelm@52803
   489
  shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
wenzelm@56009
   490
    and "p *\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p \<or> q = 0\<^sub>p"
wenzelm@56009
   491
    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
   492
  using np nq m
wenzelm@52658
   493
proof (induct p q arbitrary: n0 n1 m rule: polymul.induct)
wenzelm@52803
   494
  case (2 c c' n' p')
wenzelm@56009
   495
  {
wenzelm@56009
   496
    case (1 n0 n1)
wenzelm@56009
   497
    with "2.hyps"(4-6)[of n' n' n'] and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']
krauss@41811
   498
    show ?case by (auto simp add: min_def)
chaieb@33154
   499
  next
wenzelm@56009
   500
    case (2 n0 n1)
wenzelm@56009
   501
    then show ?case by auto
chaieb@33154
   502
  next
wenzelm@56009
   503
    case (3 n0 n1)
wenzelm@56009
   504
    then show ?case  using "2.hyps" by auto
wenzelm@56009
   505
  }
chaieb@33154
   506
next
krauss@41813
   507
  case (3 c n p c')
wenzelm@56009
   508
  {
wenzelm@56009
   509
    case (1 n0 n1)
wenzelm@56009
   510
    with "3.hyps"(4-6)[of n n n] and "3.hyps"(1-3)[of "Suc n" "Suc n" n]
krauss@41811
   511
    show ?case by (auto simp add: min_def)
chaieb@33154
   512
  next
wenzelm@56009
   513
    case (2 n0 n1)
wenzelm@56009
   514
    then show ?case by auto
chaieb@33154
   515
  next
wenzelm@56009
   516
    case (3 n0 n1)
wenzelm@56009
   517
    then show ?case  using "3.hyps" by auto
wenzelm@56009
   518
  }
chaieb@33154
   519
next
chaieb@33154
   520
  case (4 c n p c' n' p')
chaieb@33154
   521
  let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
wenzelm@56009
   522
  {
wenzelm@56009
   523
    case (1 n0 n1)
wenzelm@56009
   524
    then have cnp: "isnpolyh ?cnp n"
wenzelm@56009
   525
      and cnp': "isnpolyh ?cnp' n'"
wenzelm@56009
   526
      and np: "isnpolyh p n"
wenzelm@56009
   527
      and nc: "isnpolyh c (Suc n)"
wenzelm@56009
   528
      and np': "isnpolyh p' n'"
wenzelm@56009
   529
      and nc': "isnpolyh c' (Suc n')"
wenzelm@56009
   530
      and nn0: "n \<ge> n0"
wenzelm@56009
   531
      and nn1: "n' \<ge> n1"
wenzelm@56009
   532
      by simp_all
krauss@41811
   533
    {
wenzelm@56009
   534
      assume "n < n'"
wenzelm@56009
   535
      with "4.hyps"(4-5)[OF np cnp', of n] and "4.hyps"(1)[OF nc cnp', of n] nn0 cnp
wenzelm@56009
   536
      have ?case by (simp add: min_def)
wenzelm@56009
   537
    } moreover {
wenzelm@56009
   538
      assume "n' < n"
wenzelm@56009
   539
      with "4.hyps"(16-17)[OF cnp np', of "n'"] and "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
wenzelm@56009
   540
      have ?case by (cases "Suc n' = n") (simp_all add: min_def)
wenzelm@56009
   541
    } moreover {
wenzelm@56009
   542
      assume "n' = n"
wenzelm@56009
   543
      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
   544
      have ?case
wenzelm@56009
   545
        apply (auto intro!: polyadd_normh)
wenzelm@56009
   546
        apply (simp_all add: min_def isnpolyh_mono[OF nn0])
wenzelm@56009
   547
        done
wenzelm@56009
   548
    }
wenzelm@56009
   549
    ultimately show ?case by arith
wenzelm@56009
   550
  next
wenzelm@56009
   551
    fix n0 n1 m
wenzelm@56009
   552
    assume np: "isnpolyh ?cnp n0"
wenzelm@56009
   553
    assume np':"isnpolyh ?cnp' n1"
wenzelm@56009
   554
    assume m: "m \<le> min n0 n1"
wenzelm@56009
   555
    let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
wenzelm@56009
   556
    let ?d1 = "degreen ?cnp m"
wenzelm@56009
   557
    let ?d2 = "degreen ?cnp' m"
wenzelm@56009
   558
    let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
wenzelm@56043
   559
    have "n' < n \<or> n < n' \<or> n' = n" by auto
wenzelm@56009
   560
    moreover
wenzelm@56009
   561
    {
wenzelm@56009
   562
      assume "n' < n \<or> n < n'"
wenzelm@56009
   563
      with "4.hyps"(3,6,18) np np' m have ?eq
wenzelm@56009
   564
        by auto
wenzelm@56009
   565
    }
wenzelm@56009
   566
    moreover
wenzelm@56009
   567
    {
wenzelm@56009
   568
      assume nn': "n' = n"
wenzelm@56009
   569
      then have nn: "\<not> n' < n \<and> \<not> n < n'" by arith
wenzelm@56009
   570
      from "4.hyps"(16,18)[of n n' n]
wenzelm@56009
   571
        "4.hyps"(13,14)[of n "Suc n'" n]
wenzelm@56009
   572
        np np' nn'
wenzelm@56043
   573
      have norm:
wenzelm@56043
   574
        "isnpolyh ?cnp n"
wenzelm@56043
   575
        "isnpolyh c' (Suc n)"
wenzelm@56043
   576
        "isnpolyh (?cnp *\<^sub>p c') n"
wenzelm@56043
   577
        "isnpolyh p' n"
wenzelm@56043
   578
        "isnpolyh (?cnp *\<^sub>p p') n"
wenzelm@56043
   579
        "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@56043
   580
        "?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p"
wenzelm@56043
   581
        "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
wenzelm@56043
   582
        by (auto simp add: min_def)
wenzelm@56009
   583
      {
wenzelm@56009
   584
        assume mn: "m = n"
wenzelm@56009
   585
        from "4.hyps"(17,18)[OF norm(1,4), of n]
wenzelm@56009
   586
          "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
wenzelm@56009
   587
        have degs:
wenzelm@56009
   588
          "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else ?d1 + degreen c' n)"
wenzelm@56009
   589
          "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n"
wenzelm@56009
   590
          by (simp_all add: min_def)
wenzelm@56009
   591
        from degs norm have th1: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
wenzelm@56009
   592
          by simp
wenzelm@56009
   593
        then have neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
wenzelm@56009
   594
          by simp
wenzelm@56009
   595
        have nmin: "n \<le> min n n"
wenzelm@56009
   596
          by (simp add: min_def)
wenzelm@56009
   597
        from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
wenzelm@56009
   598
        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
   599
            degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@56009
   600
          by simp
wenzelm@56009
   601
        from "4.hyps"(16-18)[OF norm(1,4), of n]
wenzelm@56009
   602
          "4.hyps"(13-15)[OF norm(1,2), of n]
wenzelm@56009
   603
          mn norm m nn' deg
wenzelm@56009
   604
        have ?eq by simp
krauss@41811
   605
      }
chaieb@33154
   606
      moreover
wenzelm@56009
   607
      {
wenzelm@56009
   608
        assume mn: "m \<noteq> n"
wenzelm@56009
   609
        then have mn': "m < n"
wenzelm@56009
   610
          using m np by auto
wenzelm@56009
   611
        from nn' m np have max1: "m \<le> max n n"
wenzelm@56009
   612
          by simp
wenzelm@56009
   613
        then have min1: "m \<le> min n n"
wenzelm@56009
   614
          by simp
wenzelm@56009
   615
        then have min2: "m \<le> min n (Suc n)"
wenzelm@56009
   616
          by simp
wenzelm@56009
   617
        from "4.hyps"(16-18)[OF norm(1,4) min1]
wenzelm@56009
   618
          "4.hyps"(13-15)[OF norm(1,2) min2]
wenzelm@56009
   619
          degreen_polyadd[OF norm(3,6) max1]
wenzelm@56009
   620
        have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m \<le>
wenzelm@56009
   621
            max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
wenzelm@56009
   622
          using mn nn' np np' by simp
wenzelm@56009
   623
        with "4.hyps"(16-18)[OF norm(1,4) min1]
wenzelm@56009
   624
          "4.hyps"(13-15)[OF norm(1,2) min2]
wenzelm@56009
   625
          degreen_0[OF norm(3) mn']
wenzelm@56009
   626
        have ?eq using nn' mn np np' by clarsimp
wenzelm@56009
   627
      }
wenzelm@56009
   628
      ultimately have ?eq by blast
wenzelm@56009
   629
    }
wenzelm@56009
   630
    ultimately show ?eq by blast
wenzelm@56009
   631
  }
wenzelm@56009
   632
  {
wenzelm@56009
   633
    case (2 n0 n1)
wenzelm@56009
   634
    then have np: "isnpolyh ?cnp n0"
wenzelm@56009
   635
      and np': "isnpolyh ?cnp' n1"
wenzelm@56043
   636
      and m: "m \<le> min n0 n1"
wenzelm@56043
   637
      by simp_all
wenzelm@56009
   638
    then have mn: "m \<le> n" by simp
wenzelm@56009
   639
    let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
wenzelm@56009
   640
    {
wenzelm@56009
   641
      assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
wenzelm@56009
   642
      then have nn: "\<not> n' < n \<and> \<not> n < n'"
wenzelm@56009
   643
        by simp
wenzelm@56009
   644
      from "4.hyps"(16-18) [of n n n]
wenzelm@56009
   645
        "4.hyps"(13-15)[of n "Suc n" n]
wenzelm@56009
   646
        np np' C(2) mn
wenzelm@56009
   647
      have norm:
wenzelm@56009
   648
        "isnpolyh ?cnp n"
wenzelm@56009
   649
        "isnpolyh c' (Suc n)"
wenzelm@56009
   650
        "isnpolyh (?cnp *\<^sub>p c') n"
wenzelm@56009
   651
        "isnpolyh p' n"
wenzelm@56009
   652
        "isnpolyh (?cnp *\<^sub>p p') n"
wenzelm@56009
   653
        "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@56009
   654
        "?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p"
wenzelm@56009
   655
        "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
wenzelm@56009
   656
        "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
wenzelm@56009
   657
        "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
wenzelm@56009
   658
        by (simp_all add: min_def)
wenzelm@56009
   659
      from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@56009
   660
        by simp
wenzelm@56009
   661
      have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
wenzelm@56009
   662
        using norm by simp
wenzelm@56009
   663
      from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"] degneq
wenzelm@56009
   664
      have False by simp
wenzelm@56009
   665
    }
wenzelm@56009
   666
    then show ?case using "4.hyps" by clarsimp
wenzelm@56009
   667
  }
chaieb@33154
   668
qed auto
chaieb@33154
   669
wenzelm@56009
   670
lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = Ipoly bs p * Ipoly bs q"
wenzelm@52658
   671
  by (induct p q rule: polymul.induct) (auto simp add: field_simps)
chaieb@33154
   672
wenzelm@52803
   673
lemma polymul_normh:
wenzelm@56000
   674
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56009
   675
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
wenzelm@52803
   676
  using polymul_properties(1) by blast
wenzelm@52658
   677
wenzelm@52803
   678
lemma polymul_eq0_iff:
wenzelm@56000
   679
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56009
   680
  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
   681
  using polymul_properties(2) by blast
wenzelm@52658
   682
wenzelm@52658
   683
lemma polymul_degreen:  (* FIXME duplicate? *)
wenzelm@56000
   684
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56009
   685
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> m \<le> min n0 n1 \<Longrightarrow>
wenzelm@56009
   686
    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
   687
  using polymul_properties(3) by blast
wenzelm@52658
   688
wenzelm@52803
   689
lemma polymul_norm:
wenzelm@56000
   690
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56009
   691
  shows "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polymul p q)"
chaieb@33154
   692
  using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
chaieb@33154
   693
chaieb@33154
   694
lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
   695
  by (induct p arbitrary: n0 rule: headconst.induct) auto
chaieb@33154
   696
chaieb@33154
   697
lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
wenzelm@52658
   698
  by (induct p arbitrary: n0) auto
chaieb@33154
   699
wenzelm@52658
   700
lemma monic_eqI:
wenzelm@52803
   701
  assumes np: "isnpolyh p n0"
wenzelm@52658
   702
  shows "INum (headconst p) * Ipoly bs (fst (monic p)) =
wenzelm@56000
   703
    (Ipoly bs p ::'a::{field_char_0,field_inverse_zero, power})"
chaieb@33154
   704
  unfolding monic_def Let_def
wenzelm@52658
   705
proof (cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
chaieb@33154
   706
  let ?h = "headconst p"
chaieb@33154
   707
  assume pz: "p \<noteq> 0\<^sub>p"
wenzelm@56000
   708
  {
wenzelm@56000
   709
    assume hz: "INum ?h = (0::'a)"
wenzelm@56043
   710
    from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N"
wenzelm@56043
   711
      by simp_all
wenzelm@56043
   712
    from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N"
wenzelm@56043
   713
      by simp
wenzelm@56043
   714
    with headconst_zero[OF np] have "p = 0\<^sub>p"
wenzelm@56043
   715
      by blast
wenzelm@56043
   716
    with pz have False
wenzelm@56043
   717
      by blast
wenzelm@56043
   718
  }
wenzelm@56043
   719
  then show "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
wenzelm@56043
   720
    by blast
chaieb@33154
   721
qed
chaieb@33154
   722
chaieb@33154
   723
krauss@41404
   724
text{* polyneg is a negation and preserves normal forms *}
chaieb@33154
   725
chaieb@33154
   726
lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
wenzelm@52658
   727
  by (induct p rule: polyneg.induct) auto
chaieb@33154
   728
wenzelm@56009
   729
lemma polyneg0: "isnpolyh p n \<Longrightarrow> (~\<^sub>p p) = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
   730
  by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: Nneg_def)
wenzelm@56009
   731
chaieb@33154
   732
lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
wenzelm@52658
   733
  by (induct p arbitrary: n0 rule: polyneg.induct) auto
wenzelm@56009
   734
wenzelm@56009
   735
lemma polyneg_normh: "isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n"
wenzelm@56009
   736
  by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: polyneg0)
chaieb@33154
   737
chaieb@33154
   738
lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
chaieb@33154
   739
  using isnpoly_def polyneg_normh by simp
chaieb@33154
   740
chaieb@33154
   741
krauss@41404
   742
text{* polysub is a substraction and preserves normal forms *}
krauss@41404
   743
wenzelm@56009
   744
lemma polysub[simp]: "Ipoly bs (polysub p q) = Ipoly bs p - Ipoly bs q"
wenzelm@52658
   745
  by (simp add: polysub_def)
wenzelm@56009
   746
wenzelm@56009
   747
lemma polysub_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
wenzelm@52658
   748
  by (simp add: polysub_def polyneg_normh polyadd_normh)
chaieb@33154
   749
wenzelm@56009
   750
lemma polysub_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polysub p q)"
wenzelm@52803
   751
  using polyadd_norm polyneg_norm by (simp add: polysub_def)
wenzelm@56009
   752
wenzelm@52658
   753
lemma polysub_same_0[simp]:
wenzelm@56000
   754
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
krauss@41814
   755
  shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p"
wenzelm@52658
   756
  unfolding polysub_def split_def fst_conv snd_conv
wenzelm@52658
   757
  by (induct p arbitrary: n0) (auto simp add: Let_def Nsub0[simplified Nsub_def])
chaieb@33154
   758
wenzelm@52803
   759
lemma polysub_0:
wenzelm@56000
   760
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56009
   761
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p -\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = q"
chaieb@33154
   762
  unfolding polysub_def split_def fst_conv snd_conv
krauss@41763
   763
  by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
wenzelm@52658
   764
    (auto simp: Nsub0[simplified Nsub_def] Let_def)
chaieb@33154
   765
chaieb@33154
   766
text{* polypow is a power function and preserves normal forms *}
krauss@41404
   767
wenzelm@56009
   768
lemma polypow[simp]:
wenzelm@56009
   769
  "Ipoly bs (polypow n p) = (Ipoly bs p :: 'a::{field_char_0,field_inverse_zero}) ^ n"
wenzelm@52658
   770
proof (induct n rule: polypow.induct)
wenzelm@52658
   771
  case 1
wenzelm@56043
   772
  then show ?case
wenzelm@56043
   773
    by simp
chaieb@33154
   774
next
chaieb@33154
   775
  case (2 n)
chaieb@33154
   776
  let ?q = "polypow ((Suc n) div 2) p"
krauss@41813
   777
  let ?d = "polymul ?q ?q"
wenzelm@56043
   778
  have "odd (Suc n) \<or> even (Suc n)"
wenzelm@56043
   779
    by simp
wenzelm@52803
   780
  moreover
wenzelm@56043
   781
  {
wenzelm@56043
   782
    assume odd: "odd (Suc n)"
wenzelm@56000
   783
    have th: "(Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0)))) = Suc n div 2 + Suc n div 2 + 1"
wenzelm@52658
   784
      by arith
wenzelm@56043
   785
    from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)"
wenzelm@56043
   786
      by (simp add: Let_def)
wenzelm@56043
   787
    also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2) * (Ipoly bs p)^(Suc n div 2)"
chaieb@33154
   788
      using "2.hyps" by simp
chaieb@33154
   789
    also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
wenzelm@52658
   790
      by (simp only: power_add power_one_right) simp
wenzelm@56000
   791
    also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0))))"
chaieb@33154
   792
      by (simp only: th)
wenzelm@52803
   793
    finally have ?case
wenzelm@56043
   794
    using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp
wenzelm@56043
   795
  }
wenzelm@52803
   796
  moreover
wenzelm@56043
   797
  {
wenzelm@56043
   798
    assume even: "even (Suc n)"
wenzelm@56000
   799
    have th: "(Suc (Suc 0)) * (Suc n div Suc (Suc 0)) = Suc n div 2 + Suc n div 2"
wenzelm@52658
   800
      by arith
wenzelm@56043
   801
    from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d"
wenzelm@56043
   802
      by (simp add: Let_def)
chaieb@33154
   803
    also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
wenzelm@56043
   804
      using "2.hyps" by (simp only: power_add) simp
wenzelm@56043
   805
    finally have ?case using even_nat_div_two_times_two[OF even]
wenzelm@56043
   806
      by (simp only: th)
wenzelm@56043
   807
  }
chaieb@33154
   808
  ultimately show ?case by blast
chaieb@33154
   809
qed
chaieb@33154
   810
wenzelm@52803
   811
lemma polypow_normh:
wenzelm@56000
   812
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
   813
  shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
chaieb@33154
   814
proof (induct k arbitrary: n rule: polypow.induct)
wenzelm@56043
   815
  case 1
wenzelm@56043
   816
  then show ?case by auto
wenzelm@56043
   817
next
chaieb@33154
   818
  case (2 k n)
chaieb@33154
   819
  let ?q = "polypow (Suc k div 2) p"
krauss@41813
   820
  let ?d = "polymul ?q ?q"
wenzelm@56043
   821
  from 2 have th1: "isnpolyh ?q n" and th2: "isnpolyh p n"
wenzelm@56043
   822
    by blast+
wenzelm@56043
   823
  from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n"
wenzelm@56043
   824
    by simp
wenzelm@56043
   825
  from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n"
wenzelm@56043
   826
    by simp
wenzelm@56043
   827
  from dn on show ?case
wenzelm@56043
   828
    by (simp add: Let_def)
wenzelm@56043
   829
qed
chaieb@33154
   830
wenzelm@52803
   831
lemma polypow_norm:
wenzelm@56000
   832
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
   833
  shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
chaieb@33154
   834
  by (simp add: polypow_normh isnpoly_def)
chaieb@33154
   835
krauss@41404
   836
text{* Finally the whole normalization *}
chaieb@33154
   837
wenzelm@52658
   838
lemma polynate [simp]:
wenzelm@56000
   839
  "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0,field_inverse_zero})"
wenzelm@52658
   840
  by (induct p rule:polynate.induct) auto
chaieb@33154
   841
wenzelm@52803
   842
lemma polynate_norm[simp]:
wenzelm@56000
   843
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
   844
  shows "isnpoly (polynate p)"
wenzelm@52658
   845
  by (induct p rule: polynate.induct)
wenzelm@52658
   846
     (simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm,
wenzelm@52658
   847
      simp_all add: isnpoly_def)
chaieb@33154
   848
chaieb@33154
   849
text{* shift1 *}
chaieb@33154
   850
chaieb@33154
   851
chaieb@33154
   852
lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
wenzelm@52658
   853
  by (simp add: shift1_def)
chaieb@33154
   854
wenzelm@52803
   855
lemma shift1_isnpoly:
wenzelm@52658
   856
  assumes pn: "isnpoly p"
wenzelm@52658
   857
    and pnz: "p \<noteq> 0\<^sub>p"
wenzelm@52658
   858
  shows "isnpoly (shift1 p) "
wenzelm@52658
   859
  using pn pnz by (simp add: shift1_def isnpoly_def)
chaieb@33154
   860
chaieb@33154
   861
lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
chaieb@33154
   862
  by (simp add: shift1_def)
wenzelm@56043
   863
wenzelm@56043
   864
lemma funpow_shift1_isnpoly: "isnpoly p \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpoly (funpow n shift1 p)"
haftmann@39246
   865
  by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
chaieb@33154
   866
wenzelm@52803
   867
lemma funpow_isnpolyh:
wenzelm@56043
   868
  assumes f: "\<And>p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n"
wenzelm@52658
   869
    and np: "isnpolyh p n"
chaieb@33154
   870
  shows "isnpolyh (funpow k f p) n"
wenzelm@52658
   871
  using f np by (induct k arbitrary: p) auto
chaieb@33154
   872
wenzelm@52658
   873
lemma funpow_shift1:
wenzelm@56000
   874
  "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field_inverse_zero}) =
wenzelm@52658
   875
    Ipoly bs (Mul (Pw (Bound 0) n) p)"
wenzelm@52658
   876
  by (induct n arbitrary: p) (simp_all add: shift1_isnpoly shift1)
chaieb@33154
   877
wenzelm@56043
   878
lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
chaieb@33154
   879
  using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
chaieb@33154
   880
wenzelm@52803
   881
lemma funpow_shift1_1:
wenzelm@56000
   882
  "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field_inverse_zero}) =
wenzelm@52658
   883
    Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)"
chaieb@33154
   884
  by (simp add: funpow_shift1)
chaieb@33154
   885
chaieb@33154
   886
lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
wenzelm@45129
   887
  by (induct p rule: poly_cmul.induct) (auto simp add: field_simps)
chaieb@33154
   888
chaieb@33154
   889
lemma behead:
chaieb@33154
   890
  assumes np: "isnpolyh p n"
wenzelm@52658
   891
  shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) =
wenzelm@56000
   892
    (Ipoly bs p :: 'a :: {field_char_0,field_inverse_zero})"
chaieb@33154
   893
  using np
chaieb@33154
   894
proof (induct p arbitrary: n rule: behead.induct)
wenzelm@56009
   895
  case (1 c p n)
wenzelm@56009
   896
  then have pn: "isnpolyh p n" by simp
wenzelm@52803
   897
  from 1(1)[OF pn]
wenzelm@52803
   898
  have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
wenzelm@52658
   899
  then show ?case using "1.hyps"
wenzelm@52658
   900
    apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
wenzelm@52658
   901
    apply (simp_all add: th[symmetric] field_simps)
wenzelm@52658
   902
    done
chaieb@33154
   903
qed (auto simp add: Let_def)
chaieb@33154
   904
chaieb@33154
   905
lemma behead_isnpolyh:
wenzelm@52658
   906
  assumes np: "isnpolyh p n"
wenzelm@52658
   907
  shows "isnpolyh (behead p) n"
wenzelm@52658
   908
  using np by (induct p rule: behead.induct) (auto simp add: Let_def isnpolyh_mono)
wenzelm@52658
   909
chaieb@33154
   910
krauss@41404
   911
subsection{* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
wenzelm@52658
   912
chaieb@33154
   913
lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
wenzelm@52658
   914
proof (induct p arbitrary: n rule: poly.induct, auto)
chaieb@33154
   915
  case (goal1 c n p n')
wenzelm@56009
   916
  then have "n = Suc (n - 1)"
wenzelm@56009
   917
    by simp
wenzelm@56009
   918
  then have "isnpolyh p (Suc (n - 1))"
wenzelm@56009
   919
    using `isnpolyh p n` by simp
wenzelm@56009
   920
  with goal1(2) show ?case
wenzelm@56009
   921
    by simp
chaieb@33154
   922
qed
chaieb@33154
   923
chaieb@33154
   924
lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
wenzelm@52658
   925
  by (induct p arbitrary: n0 rule: isconstant.induct) (auto simp add: isnpolyh_polybound0)
chaieb@33154
   926
wenzelm@52658
   927
lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
   928
  by (induct p) auto
chaieb@33154
   929
chaieb@33154
   930
lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
wenzelm@52658
   931
  apply (induct p arbitrary: n0)
wenzelm@52658
   932
  apply auto
wenzelm@56043
   933
  apply atomize
chaieb@33154
   934
  apply (erule_tac x = "Suc nat" in allE)
chaieb@33154
   935
  apply auto
chaieb@33154
   936
  done
chaieb@33154
   937
chaieb@33154
   938
lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
wenzelm@52658
   939
  by (induct p  arbitrary: n0 rule: head.induct) (auto intro: isnpolyh_polybound0)
chaieb@33154
   940
chaieb@33154
   941
lemma polybound0_I:
chaieb@33154
   942
  assumes nb: "polybound0 a"
wenzelm@56009
   943
  shows "Ipoly (b # bs) a = Ipoly (b' # bs) a"
wenzelm@52658
   944
  using nb
wenzelm@52803
   945
  by (induct a rule: poly.induct) auto
wenzelm@52658
   946
wenzelm@56009
   947
lemma polysubst0_I: "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b # bs) a) # bs) t"
chaieb@33154
   948
  by (induct t) simp_all
chaieb@33154
   949
chaieb@33154
   950
lemma polysubst0_I':
chaieb@33154
   951
  assumes nb: "polybound0 a"
wenzelm@56009
   952
  shows "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b' # bs) a) # bs) t"
chaieb@33154
   953
  by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
chaieb@33154
   954
wenzelm@52658
   955
lemma decrpoly:
wenzelm@52658
   956
  assumes nb: "polybound0 t"
wenzelm@56043
   957
  shows "Ipoly (x # bs) t = Ipoly bs (decrpoly t)"
wenzelm@52658
   958
  using nb by (induct t rule: decrpoly.induct) simp_all
chaieb@33154
   959
wenzelm@52658
   960
lemma polysubst0_polybound0:
wenzelm@52658
   961
  assumes nb: "polybound0 t"
chaieb@33154
   962
  shows "polybound0 (polysubst0 t a)"
wenzelm@52658
   963
  using nb by (induct a rule: poly.induct) auto
chaieb@33154
   964
chaieb@33154
   965
lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
wenzelm@52658
   966
  by (induct p arbitrary: n rule: degree.induct) (auto simp add: isnpolyh_polybound0)
chaieb@33154
   967
wenzelm@56043
   968
primrec maxindex :: "poly \<Rightarrow> nat"
wenzelm@56043
   969
where
chaieb@33154
   970
  "maxindex (Bound n) = n + 1"
chaieb@33154
   971
| "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
chaieb@33154
   972
| "maxindex (Add p q) = max (maxindex p) (maxindex q)"
chaieb@33154
   973
| "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
chaieb@33154
   974
| "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
chaieb@33154
   975
| "maxindex (Neg p) = maxindex p"
chaieb@33154
   976
| "maxindex (Pw p n) = maxindex p"
chaieb@33154
   977
| "maxindex (C x) = 0"
chaieb@33154
   978
wenzelm@52658
   979
definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool"
wenzelm@56000
   980
  where "wf_bs bs p \<longleftrightarrow> length bs \<ge> maxindex p"
chaieb@33154
   981
wenzelm@56043
   982
lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall>c \<in> set (coefficients p). wf_bs bs c"
wenzelm@52658
   983
proof (induct p rule: coefficients.induct)
wenzelm@52803
   984
  case (1 c p)
wenzelm@52803
   985
  show ?case
chaieb@33154
   986
  proof
wenzelm@56009
   987
    fix x
wenzelm@56009
   988
    assume xc: "x \<in> set (coefficients (CN c 0 p))"
wenzelm@56009
   989
    then have "x = c \<or> x \<in> set (coefficients p)"
wenzelm@56009
   990
      by simp
wenzelm@52803
   991
    moreover
wenzelm@56009
   992
    {
wenzelm@56009
   993
      assume "x = c"
wenzelm@56009
   994
      then have "wf_bs bs x"
wenzelm@56043
   995
        using "1.prems" unfolding wf_bs_def by simp
wenzelm@56009
   996
    }
wenzelm@56009
   997
    moreover
wenzelm@56009
   998
    {
wenzelm@56009
   999
      assume H: "x \<in> set (coefficients p)"
wenzelm@56009
  1000
      from "1.prems" have "wf_bs bs p"
wenzelm@56009
  1001
        unfolding wf_bs_def by simp
wenzelm@56009
  1002
      with "1.hyps" H have "wf_bs bs x"
wenzelm@56009
  1003
        by blast
wenzelm@56009
  1004
    }
wenzelm@56009
  1005
    ultimately  show "wf_bs bs x"
wenzelm@56009
  1006
      by blast
chaieb@33154
  1007
  qed
chaieb@33154
  1008
qed simp_all
chaieb@33154
  1009
wenzelm@56043
  1010
lemma maxindex_coefficients: "\<forall>c \<in> set (coefficients p). maxindex c \<le> maxindex p"
wenzelm@52658
  1011
  by (induct p rule: coefficients.induct) auto
chaieb@33154
  1012
wenzelm@56000
  1013
lemma wf_bs_I: "wf_bs bs p \<Longrightarrow> Ipoly (bs @ bs') p = Ipoly bs p"
wenzelm@52658
  1014
  unfolding wf_bs_def by (induct p) (auto simp add: nth_append)
chaieb@33154
  1015
wenzelm@52658
  1016
lemma take_maxindex_wf:
wenzelm@52803
  1017
  assumes wf: "wf_bs bs p"
chaieb@33154
  1018
  shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
wenzelm@56009
  1019
proof -
chaieb@33154
  1020
  let ?ip = "maxindex p"
chaieb@33154
  1021
  let ?tbs = "take ?ip bs"
wenzelm@56009
  1022
  from wf have "length ?tbs = ?ip"
wenzelm@56009
  1023
    unfolding wf_bs_def by simp
wenzelm@56009
  1024
  then have wf': "wf_bs ?tbs p"
wenzelm@56009
  1025
    unfolding wf_bs_def by  simp
wenzelm@56043
  1026
  have eq: "bs = ?tbs @ drop ?ip bs"
wenzelm@56009
  1027
    by simp
wenzelm@56009
  1028
  from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis
wenzelm@56009
  1029
    using eq by simp
chaieb@33154
  1030
qed
chaieb@33154
  1031
chaieb@33154
  1032
lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
wenzelm@52658
  1033
  by (induct p) auto
chaieb@33154
  1034
chaieb@33154
  1035
lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
chaieb@33154
  1036
  unfolding wf_bs_def by simp
chaieb@33154
  1037
chaieb@33154
  1038
lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
chaieb@33154
  1039
  unfolding wf_bs_def by simp
chaieb@33154
  1040
chaieb@33154
  1041
lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
wenzelm@52658
  1042
  by (induct p rule: coefficients.induct) (auto simp add: wf_bs_def)
wenzelm@56043
  1043
chaieb@33154
  1044
lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
wenzelm@52658
  1045
  by (induct p rule: coefficients.induct) simp_all
chaieb@33154
  1046
chaieb@33154
  1047
lemma coefficients_head: "last (coefficients p) = head p"
wenzelm@52658
  1048
  by (induct p rule: coefficients.induct) auto
chaieb@33154
  1049
chaieb@33154
  1050
lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
wenzelm@52658
  1051
  unfolding wf_bs_def by (induct p rule: decrpoly.induct) auto
chaieb@33154
  1052
wenzelm@56043
  1053
lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists>ys. length (xs @ ys) = n"
chaieb@33154
  1054
  apply (rule exI[where x="replicate (n - length xs) z"])
wenzelm@52658
  1055
  apply simp
wenzelm@52658
  1056
  done
wenzelm@52658
  1057
wenzelm@56043
  1058
lemma isnpolyh_Suc_const: "isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
wenzelm@52658
  1059
  apply (cases p)
wenzelm@52658
  1060
  apply auto
wenzelm@52658
  1061
  apply (case_tac "nat")
wenzelm@52658
  1062
  apply simp_all
wenzelm@52658
  1063
  done
chaieb@33154
  1064
chaieb@33154
  1065
lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
wenzelm@52803
  1066
  unfolding wf_bs_def
chaieb@33154
  1067
  apply (induct p q rule: polyadd.induct)
chaieb@33154
  1068
  apply (auto simp add: Let_def)
chaieb@33154
  1069
  done
chaieb@33154
  1070
chaieb@33154
  1071
lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
wenzelm@52803
  1072
  unfolding wf_bs_def
wenzelm@52803
  1073
  apply (induct p q arbitrary: bs rule: polymul.induct)
chaieb@33154
  1074
  apply (simp_all add: wf_bs_polyadd)
chaieb@33154
  1075
  apply clarsimp
chaieb@33154
  1076
  apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
chaieb@33154
  1077
  apply auto
chaieb@33154
  1078
  done
chaieb@33154
  1079
chaieb@33154
  1080
lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
wenzelm@52658
  1081
  unfolding wf_bs_def by (induct p rule: polyneg.induct) auto
chaieb@33154
  1082
chaieb@33154
  1083
lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
wenzelm@56043
  1084
  unfolding polysub_def split_def fst_conv snd_conv
wenzelm@56043
  1085
  using wf_bs_polyadd wf_bs_polyneg by blast
chaieb@33154
  1086
wenzelm@52658
  1087
wenzelm@56043
  1088
subsection {* Canonicity of polynomial representation, see lemma isnpolyh_unique *}
chaieb@33154
  1089
chaieb@33154
  1090
definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
wenzelm@56043
  1091
definition "polypoly' bs p = map (Ipoly bs \<circ> decrpoly) (coefficients p)"
wenzelm@56043
  1092
definition "poly_nate bs p = map (Ipoly bs \<circ> decrpoly) (coefficients (polynate p))"
chaieb@33154
  1093
wenzelm@56043
  1094
lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall>q \<in> set (coefficients p). isnpolyh q n0"
chaieb@33154
  1095
proof (induct p arbitrary: n0 rule: coefficients.induct)
chaieb@33154
  1096
  case (1 c p n0)
wenzelm@56009
  1097
  have cp: "isnpolyh (CN c 0 p) n0"
wenzelm@56009
  1098
    by fact
wenzelm@56009
  1099
  then have norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
chaieb@33154
  1100
    by (auto simp add: isnpolyh_mono[where n'=0])
wenzelm@56009
  1101
  from "1.hyps"[OF norm(2)] norm(1) norm(4) show ?case
wenzelm@56009
  1102
    by simp
chaieb@33154
  1103
qed auto
chaieb@33154
  1104
wenzelm@56043
  1105
lemma coefficients_isconst: "isnpolyh p n \<Longrightarrow> \<forall>q \<in> set (coefficients p). isconstant q"
wenzelm@56043
  1106
  by (induct p arbitrary: n rule: coefficients.induct) (auto simp add: isnpolyh_Suc_const)
chaieb@33154
  1107
chaieb@33154
  1108
lemma polypoly_polypoly':
chaieb@33154
  1109
  assumes np: "isnpolyh p n0"
wenzelm@56043
  1110
  shows "polypoly (x # bs) p = polypoly' bs p"
wenzelm@56043
  1111
proof -
chaieb@33154
  1112
  let ?cf = "set (coefficients p)"
chaieb@33154
  1113
  from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
wenzelm@56043
  1114
  {
wenzelm@56043
  1115
    fix q
wenzelm@56043
  1116
    assume q: "q \<in> ?cf"
wenzelm@56043
  1117
    from q cn_norm have th: "isnpolyh q n0"
wenzelm@56043
  1118
      by blast
wenzelm@56043
  1119
    from coefficients_isconst[OF np] q have "isconstant q"
wenzelm@56043
  1120
      by blast
wenzelm@56043
  1121
    with isconstant_polybound0[OF th] have "polybound0 q"
wenzelm@56043
  1122
      by blast
wenzelm@56043
  1123
  }
wenzelm@56009
  1124
  then have "\<forall>q \<in> ?cf. polybound0 q" ..
wenzelm@56043
  1125
  then have "\<forall>q \<in> ?cf. Ipoly (x # bs) q = Ipoly bs (decrpoly q)"
chaieb@33154
  1126
    using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
chaieb@33154
  1127
    by auto
wenzelm@56043
  1128
  then show ?thesis
wenzelm@56043
  1129
    unfolding polypoly_def polypoly'_def by simp
chaieb@33154
  1130
qed
chaieb@33154
  1131
chaieb@33154
  1132
lemma polypoly_poly:
wenzelm@56043
  1133
  assumes "isnpolyh p n0"
wenzelm@56043
  1134
  shows "Ipoly (x # bs) p = poly (polypoly (x # bs) p) x"
wenzelm@56043
  1135
  using assms
wenzelm@52658
  1136
  by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def)
chaieb@33154
  1137
wenzelm@52803
  1138
lemma polypoly'_poly:
wenzelm@56043
  1139
  assumes "isnpolyh p n0"
wenzelm@52658
  1140
  shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
wenzelm@56043
  1141
  using polypoly_poly[OF assms, simplified polypoly_polypoly'[OF assms]] .
chaieb@33154
  1142
chaieb@33154
  1143
chaieb@33154
  1144
lemma polypoly_poly_polybound0:
wenzelm@56043
  1145
  assumes "isnpolyh p n0"
wenzelm@56043
  1146
    and "polybound0 p"
chaieb@33154
  1147
  shows "polypoly bs p = [Ipoly bs p]"
wenzelm@56043
  1148
  using assms
wenzelm@56043
  1149
  unfolding polypoly_def
wenzelm@52658
  1150
  apply (cases p)
wenzelm@52658
  1151
  apply auto
wenzelm@52658
  1152
  apply (case_tac nat)
wenzelm@52658
  1153
  apply auto
wenzelm@52658
  1154
  done
chaieb@33154
  1155
wenzelm@52803
  1156
lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
wenzelm@52658
  1157
  by (induct p rule: head.induct) auto
chaieb@33154
  1158
wenzelm@56043
  1159
lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> headn p m = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
  1160
  by (cases p) auto
chaieb@33154
  1161
chaieb@33154
  1162
lemma head_eq_headn0: "head p = headn p 0"
wenzelm@52658
  1163
  by (induct p rule: head.induct) simp_all
chaieb@33154
  1164
wenzelm@56043
  1165
lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> head p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
chaieb@33154
  1166
  by (simp add: head_eq_headn0)
chaieb@33154
  1167
wenzelm@52803
  1168
lemma isnpolyh_zero_iff:
wenzelm@52658
  1169
  assumes nq: "isnpolyh p n0"
wenzelm@56000
  1170
    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
  1171
  shows "p = 0\<^sub>p"
wenzelm@52658
  1172
  using nq eq
berghofe@34915
  1173
proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
berghofe@34915
  1174
  case less
berghofe@34915
  1175
  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
  1176
  {
wenzelm@56000
  1177
    assume nz: "maxindex p = 0"
wenzelm@56000
  1178
    then obtain c where "p = C c"
wenzelm@56000
  1179
      using np by (cases p) auto
wenzelm@56000
  1180
    with zp np have "p = 0\<^sub>p"
wenzelm@56000
  1181
      unfolding wf_bs_def by simp
wenzelm@56000
  1182
  }
chaieb@33154
  1183
  moreover
wenzelm@56000
  1184
  {
wenzelm@56000
  1185
    assume nz: "maxindex p \<noteq> 0"
chaieb@33154
  1186
    let ?h = "head p"
chaieb@33154
  1187
    let ?hd = "decrpoly ?h"
chaieb@33154
  1188
    let ?ihd = "maxindex ?hd"
wenzelm@56000
  1189
    from head_isnpolyh[OF np] head_polybound0[OF np]
wenzelm@56000
  1190
    have h: "isnpolyh ?h n0" "polybound0 ?h"
chaieb@33154
  1191
      by simp_all
wenzelm@56000
  1192
    then have nhd: "isnpolyh ?hd (n0 - 1)"
wenzelm@56000
  1193
      using decrpoly_normh by blast
wenzelm@52803
  1194
chaieb@33154
  1195
    from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
wenzelm@56000
  1196
    have mihn: "maxindex ?h \<le> maxindex p"
wenzelm@56000
  1197
      by auto
wenzelm@56000
  1198
    with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < maxindex p"
wenzelm@56000
  1199
      by auto
wenzelm@56000
  1200
    {
wenzelm@56000
  1201
      fix bs :: "'a list"
wenzelm@56000
  1202
      assume bs: "wf_bs bs ?hd"
chaieb@33154
  1203
      let ?ts = "take ?ihd bs"
chaieb@33154
  1204
      let ?rs = "drop ?ihd bs"
wenzelm@56000
  1205
      have ts: "wf_bs ?ts ?hd"
wenzelm@56000
  1206
        using bs unfolding wf_bs_def by simp
wenzelm@56000
  1207
      have bs_ts_eq: "?ts @ ?rs = bs"
wenzelm@56000
  1208
        by simp
wenzelm@56000
  1209
      from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x # ?ts) ?h"
wenzelm@56000
  1210
        by simp
wenzelm@56000
  1211
      from ihd_lt_n have "\<forall>x. length (x # ?ts) \<le> maxindex p"
wenzelm@56000
  1212
        by simp
wenzelm@56000
  1213
      with length_le_list_ex obtain xs where xs: "length ((x # ?ts) @ xs) = maxindex p"
wenzelm@56000
  1214
        by blast
wenzelm@56000
  1215
      then have "\<forall>x. wf_bs ((x # ?ts) @ xs) p"
wenzelm@56000
  1216
        unfolding wf_bs_def by simp
wenzelm@56000
  1217
      with zp have "\<forall>x. Ipoly ((x # ?ts) @ xs) p = 0"
wenzelm@56000
  1218
        by blast
wenzelm@56000
  1219
      then have "\<forall>x. Ipoly (x # (?ts @ xs)) p = 0"
wenzelm@56000
  1220
        by simp
chaieb@33154
  1221
      with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
wenzelm@56000
  1222
      have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"
wenzelm@56000
  1223
        by simp
wenzelm@56000
  1224
      then have "poly (polypoly' (?ts @ xs) p) = poly []"
wenzelm@56000
  1225
        by auto
wenzelm@56000
  1226
      then have "\<forall>c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
wenzelm@33268
  1227
        using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
chaieb@33154
  1228
      with coefficients_head[of p, symmetric]
wenzelm@56000
  1229
      have th0: "Ipoly (?ts @ xs) ?hd = 0"
wenzelm@56000
  1230
        by simp
wenzelm@56000
  1231
      from bs have wf'': "wf_bs ?ts ?hd"
wenzelm@56000
  1232
        unfolding wf_bs_def by simp
wenzelm@56000
  1233
      with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0"
wenzelm@56000
  1234
        by simp
wenzelm@56000
  1235
      with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
wenzelm@56000
  1236
        by simp
wenzelm@56000
  1237
    }
wenzelm@56000
  1238
    then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
wenzelm@56000
  1239
      by blast
wenzelm@56000
  1240
    from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p"
wenzelm@56000
  1241
      by blast
wenzelm@56000
  1242
    then have "?h = 0\<^sub>p" by simp
wenzelm@56000
  1243
    with head_nz[OF np] have "p = 0\<^sub>p" by simp
wenzelm@56000
  1244
  }
wenzelm@56000
  1245
  ultimately show "p = 0\<^sub>p"
wenzelm@56000
  1246
    by blast
chaieb@33154
  1247
qed
chaieb@33154
  1248
wenzelm@52803
  1249
lemma isnpolyh_unique:
wenzelm@56000
  1250
  assumes np: "isnpolyh p n0"
wenzelm@52658
  1251
    and nq: "isnpolyh q n1"
wenzelm@56000
  1252
  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
  1253
proof auto
wenzelm@56000
  1254
  assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a) = \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
wenzelm@56000
  1255
  then have "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)"
wenzelm@56000
  1256
    by simp
wenzelm@56000
  1257
  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
  1258
    using wf_bs_polysub[where p=p and q=q] by auto
wenzelm@56000
  1259
  with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq] show "p = q"
wenzelm@56000
  1260
    by blast
chaieb@33154
  1261
qed
chaieb@33154
  1262
chaieb@33154
  1263
krauss@41404
  1264
text{* consequences of unicity on the algorithms for polynomial normalization *}
chaieb@33154
  1265
wenzelm@52658
  1266
lemma polyadd_commute:
wenzelm@56000
  1267
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1268
    and np: "isnpolyh p n0"
wenzelm@52658
  1269
    and nq: "isnpolyh q n1"
wenzelm@52658
  1270
  shows "p +\<^sub>p q = q +\<^sub>p p"
wenzelm@56000
  1271
  using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]]
wenzelm@56000
  1272
  by simp
chaieb@33154
  1273
wenzelm@56000
  1274
lemma zero_normh: "isnpolyh 0\<^sub>p n"
wenzelm@56000
  1275
  by simp
wenzelm@56000
  1276
wenzelm@56000
  1277
lemma one_normh: "isnpolyh (1)\<^sub>p n"
wenzelm@56000
  1278
  by simp
wenzelm@52658
  1279
wenzelm@52803
  1280
lemma polyadd_0[simp]:
wenzelm@56000
  1281
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1282
    and np: "isnpolyh p n0"
wenzelm@56000
  1283
  shows "p +\<^sub>p 0\<^sub>p = p"
wenzelm@56000
  1284
    and "0\<^sub>p +\<^sub>p p = p"
wenzelm@52803
  1285
  using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
chaieb@33154
  1286
    isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
chaieb@33154
  1287
wenzelm@52803
  1288
lemma polymul_1[simp]:
wenzelm@56000
  1289
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1290
    and np: "isnpolyh p n0"
wenzelm@56000
  1291
  shows "p *\<^sub>p (1)\<^sub>p = p"
wenzelm@56000
  1292
    and "(1)\<^sub>p *\<^sub>p p = p"
wenzelm@52803
  1293
  using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
chaieb@33154
  1294
    isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
wenzelm@52658
  1295
wenzelm@52803
  1296
lemma polymul_0[simp]:
wenzelm@56000
  1297
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1298
    and np: "isnpolyh p n0"
wenzelm@56000
  1299
  shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p"
wenzelm@56000
  1300
    and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
wenzelm@52803
  1301
  using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
chaieb@33154
  1302
    isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
chaieb@33154
  1303
wenzelm@52803
  1304
lemma polymul_commute:
wenzelm@56000
  1305
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56000
  1306
    and np: "isnpolyh p n0"
wenzelm@52658
  1307
    and nq: "isnpolyh q n1"
chaieb@33154
  1308
  shows "p *\<^sub>p q = q *\<^sub>p p"
wenzelm@56043
  1309
  using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np],
wenzelm@56043
  1310
    where ?'a = "'a::{field_char_0,field_inverse_zero, power}"]
wenzelm@52658
  1311
  by simp
chaieb@33154
  1312
wenzelm@52658
  1313
declare polyneg_polyneg [simp]
wenzelm@52803
  1314
wenzelm@52803
  1315
lemma isnpolyh_polynate_id [simp]:
wenzelm@56000
  1316
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56000
  1317
    and np: "isnpolyh p n0"
wenzelm@52658
  1318
  shows "polynate p = p"
wenzelm@56043
  1319
  using isnpolyh_unique[where ?'a= "'a::{field_char_0,field_inverse_zero}",
wenzelm@56043
  1320
      OF polynate_norm[of p, unfolded isnpoly_def] np]
wenzelm@56043
  1321
    polynate[where ?'a = "'a::{field_char_0,field_inverse_zero}"]
wenzelm@52658
  1322
  by simp
chaieb@33154
  1323
wenzelm@52803
  1324
lemma polynate_idempotent[simp]:
wenzelm@56000
  1325
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
  1326
  shows "polynate (polynate p) = polynate p"
chaieb@33154
  1327
  using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
chaieb@33154
  1328
chaieb@33154
  1329
lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
chaieb@33154
  1330
  unfolding poly_nate_def polypoly'_def ..
wenzelm@52658
  1331
wenzelm@52658
  1332
lemma poly_nate_poly:
wenzelm@56000
  1333
  "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
  1334
  using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
wenzelm@52658
  1335
  unfolding poly_nate_polypoly' by auto
wenzelm@52658
  1336
chaieb@33154
  1337
chaieb@33154
  1338
subsection{* heads, degrees and all that *}
wenzelm@52658
  1339
chaieb@33154
  1340
lemma degree_eq_degreen0: "degree p = degreen p 0"
wenzelm@52658
  1341
  by (induct p rule: degree.induct) simp_all
chaieb@33154
  1342
wenzelm@52658
  1343
lemma degree_polyneg:
wenzelm@56043
  1344
  assumes "isnpolyh p n"
chaieb@33154
  1345
  shows "degree (polyneg p) = degree p"
wenzelm@56043
  1346
  apply (induct p rule: polyneg.induct)
wenzelm@56043
  1347
  using assms
wenzelm@56043
  1348
  apply simp_all
wenzelm@52658
  1349
  apply (case_tac na)
wenzelm@52658
  1350
  apply auto
wenzelm@52658
  1351
  done
chaieb@33154
  1352
chaieb@33154
  1353
lemma degree_polyadd:
wenzelm@56043
  1354
  assumes np: "isnpolyh p n0"
wenzelm@56043
  1355
    and nq: "isnpolyh q n1"
chaieb@33154
  1356
  shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
wenzelm@52658
  1357
  using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
chaieb@33154
  1358
chaieb@33154
  1359
wenzelm@52658
  1360
lemma degree_polysub:
wenzelm@52658
  1361
  assumes np: "isnpolyh p n0"
wenzelm@52658
  1362
    and nq: "isnpolyh q n1"
chaieb@33154
  1363
  shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
chaieb@33154
  1364
proof-
wenzelm@56043
  1365
  from nq have nq': "isnpolyh (~\<^sub>p q) n1"
wenzelm@56043
  1366
    using polyneg_normh by simp
wenzelm@56043
  1367
  from degree_polyadd[OF np nq'] show ?thesis
wenzelm@56043
  1368
    by (simp add: polysub_def degree_polyneg[OF nq])
chaieb@33154
  1369
qed
chaieb@33154
  1370
wenzelm@52803
  1371
lemma degree_polysub_samehead:
wenzelm@56000
  1372
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56043
  1373
    and np: "isnpolyh p n0"
wenzelm@56043
  1374
    and nq: "isnpolyh q n1"
wenzelm@56043
  1375
    and h: "head p = head q"
wenzelm@52658
  1376
    and d: "degree p = degree q"
chaieb@33154
  1377
  shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
wenzelm@52658
  1378
  unfolding polysub_def split_def fst_conv snd_conv
wenzelm@52658
  1379
  using np nq h d
wenzelm@52658
  1380
proof (induct p q rule: polyadd.induct)
wenzelm@52658
  1381
  case (1 c c')
wenzelm@56009
  1382
  then show ?case
wenzelm@56009
  1383
    by (simp add: Nsub_def Nsub0[simplified Nsub_def])
chaieb@33154
  1384
next
wenzelm@52803
  1385
  case (2 c c' n' p')
wenzelm@56009
  1386
  from 2 have "degree (C c) = degree (CN c' n' p')"
wenzelm@56009
  1387
    by simp
wenzelm@56009
  1388
  then have nz: "n' > 0"
wenzelm@56009
  1389
    by (cases n') auto
wenzelm@56009
  1390
  then have "head (CN c' n' p') = CN c' n' p'"
wenzelm@56009
  1391
    by (cases n') auto
wenzelm@56009
  1392
  with 2 show ?case
wenzelm@56009
  1393
    by simp
chaieb@33154
  1394
next
wenzelm@52803
  1395
  case (3 c n p c')
wenzelm@56009
  1396
  then have "degree (C c') = degree (CN c n p)"
wenzelm@56009
  1397
    by simp
wenzelm@56009
  1398
  then have nz: "n > 0"
wenzelm@56009
  1399
    by (cases n) auto
wenzelm@56009
  1400
  then have "head (CN c n p) = CN c n p"
wenzelm@56009
  1401
    by (cases n) auto
wenzelm@41807
  1402
  with 3 show ?case by simp
chaieb@33154
  1403
next
chaieb@33154
  1404
  case (4 c n p c' n' p')
wenzelm@56009
  1405
  then have H:
wenzelm@56009
  1406
    "isnpolyh (CN c n p) n0"
wenzelm@56009
  1407
    "isnpolyh (CN c' n' p') n1"
wenzelm@56009
  1408
    "head (CN c n p) = head (CN c' n' p')"
wenzelm@56009
  1409
    "degree (CN c n p) = degree (CN c' n' p')"
wenzelm@56009
  1410
    by simp_all
wenzelm@56009
  1411
  then have degc: "degree c = 0" and degc': "degree c' = 0"
wenzelm@56009
  1412
    by simp_all
wenzelm@56009
  1413
  then have degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
chaieb@33154
  1414
    using H(1-2) degree_polyneg by auto
wenzelm@56009
  1415
  from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"
wenzelm@56009
  1416
    by simp_all
wenzelm@56009
  1417
  from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc'
wenzelm@56009
  1418
  have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"
wenzelm@56009
  1419
    by simp
wenzelm@56009
  1420
  from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'"
wenzelm@56009
  1421
    by auto
wenzelm@56009
  1422
  have "n = n' \<or> n < n' \<or> n > n'"
wenzelm@56009
  1423
    by arith
chaieb@33154
  1424
  moreover
wenzelm@56009
  1425
  {
wenzelm@56009
  1426
    assume nn': "n = n'"
wenzelm@56009
  1427
    have "n = 0 \<or> n > 0" by arith
wenzelm@56009
  1428
    moreover {
wenzelm@56009
  1429
      assume nz: "n = 0"
wenzelm@56009
  1430
      then have ?case using 4 nn'
wenzelm@56009
  1431
        by (auto simp add: Let_def degcmc')
wenzelm@56009
  1432
    }
wenzelm@56009
  1433
    moreover {
wenzelm@56009
  1434
      assume nz: "n > 0"
wenzelm@56009
  1435
      with nn' H(3) have  cc': "c = c'" and pp': "p = p'"
wenzelm@56009
  1436
        by (cases n, auto)+
wenzelm@56009
  1437
      then have ?case
wenzelm@56009
  1438
        using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv]
wenzelm@56009
  1439
        using polysub_same_0[OF c'nh, simplified polysub_def]
wenzelm@56009
  1440
        using nn' 4 by (simp add: Let_def)
wenzelm@56009
  1441
    }
wenzelm@56009
  1442
    ultimately have ?case by blast
wenzelm@56009
  1443
  }
chaieb@33154
  1444
  moreover
wenzelm@56009
  1445
  {
wenzelm@56009
  1446
    assume nn': "n < n'"
wenzelm@56009
  1447
    then have n'p: "n' > 0"
wenzelm@56009
  1448
      by simp
wenzelm@56009
  1449
    then have headcnp':"head (CN c' n' p') = CN c' n' p'"
wenzelm@56009
  1450
      by (cases n') simp_all
wenzelm@56009
  1451
    have degcnp': "degree (CN c' n' p') = 0"
wenzelm@56009
  1452
      and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
wenzelm@52658
  1453
      using 4 nn' by (cases n', simp_all)
wenzelm@56009
  1454
    then have "n > 0"
wenzelm@56009
  1455
      by (cases n) simp_all
wenzelm@56009
  1456
    then have headcnp: "head (CN c n p) = CN c n p"
wenzelm@56009
  1457
      by (cases n) auto
wenzelm@56009
  1458
    from H(3) headcnp headcnp' nn' have ?case
wenzelm@56009
  1459
      by auto
wenzelm@56009
  1460
  }
chaieb@33154
  1461
  moreover
wenzelm@56009
  1462
  {
wenzelm@56009
  1463
    assume nn': "n > n'"
wenzelm@56009
  1464
    then have np: "n > 0" by simp
wenzelm@56009
  1465
    then have headcnp:"head (CN c n p) = CN c n p"
wenzelm@56009
  1466
      by (cases n) simp_all
wenzelm@56009
  1467
    from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)"
wenzelm@56009
  1468
      by simp
wenzelm@56009
  1469
    from np have degcnp: "degree (CN c n p) = 0"
wenzelm@56009
  1470
      by (cases n) simp_all
wenzelm@56009
  1471
    with degcnpeq have "n' > 0"
wenzelm@56009
  1472
      by (cases n') simp_all
wenzelm@56009
  1473
    then have headcnp': "head (CN c' n' p') = CN c' n' p'"
wenzelm@56009
  1474
      by (cases n') auto
wenzelm@56009
  1475
    from H(3) headcnp headcnp' nn' have ?case by auto
wenzelm@56009
  1476
  }
wenzelm@56009
  1477
  ultimately show ?case by blast
krauss@41812
  1478
qed auto
wenzelm@52803
  1479
chaieb@33154
  1480
lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
wenzelm@52658
  1481
  by (induct p arbitrary: n0 rule: head.induct) (simp_all add: shift1_def)
chaieb@33154
  1482
chaieb@33154
  1483
lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
wenzelm@52658
  1484
proof (induct k arbitrary: n0 p)
wenzelm@52658
  1485
  case 0
wenzelm@56009
  1486
  then show ?case by auto
wenzelm@52658
  1487
next
wenzelm@52658
  1488
  case (Suc k n0 p)
wenzelm@56009
  1489
  then have "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
wenzelm@41807
  1490
  with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
wenzelm@52803
  1491
    and "head (shift1 p) = head p" by (simp_all add: shift1_head)
wenzelm@56009
  1492
  then show ?case by (simp add: funpow_swap1)
wenzelm@52658
  1493
qed
chaieb@33154
  1494
chaieb@33154
  1495
lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
chaieb@33154
  1496
  by (simp add: shift1_def)
wenzelm@56009
  1497
chaieb@33154
  1498
lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
wenzelm@46991
  1499
  by (induct k arbitrary: p) (auto simp add: shift1_degree)
chaieb@33154
  1500
chaieb@33154
  1501
lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
wenzelm@52658
  1502
  by (induct n arbitrary: p) simp_all
chaieb@33154
  1503
chaieb@33154
  1504
lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
wenzelm@52658
  1505
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
  1506
lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
wenzelm@52658
  1507
  by (induct p arbitrary: n rule: degreen.induct) auto
chaieb@33154
  1508
lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
wenzelm@52658
  1509
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
  1510
lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
wenzelm@52658
  1511
  by (induct p rule: head.induct) auto
chaieb@33154
  1512
wenzelm@52803
  1513
lemma polyadd_eq_const_degree:
wenzelm@52658
  1514
  "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> polyadd p q = C c \<Longrightarrow> degree p = degree q"
chaieb@33154
  1515
  using polyadd_eq_const_degreen degree_eq_degreen0 by simp
chaieb@33154
  1516
wenzelm@52658
  1517
lemma polyadd_head:
wenzelm@52658
  1518
  assumes np: "isnpolyh p n0"
wenzelm@52658
  1519
    and nq: "isnpolyh q n1"
wenzelm@52658
  1520
    and deg: "degree p \<noteq> degree q"
chaieb@33154
  1521
  shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
wenzelm@52658
  1522
  using np nq deg
wenzelm@52658
  1523
  apply (induct p q arbitrary: n0 n1 rule: polyadd.induct)
wenzelm@52658
  1524
  using np
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})"
chaieb@33154
  1537
  shows "\<lbrakk>isnpolyh p n0; isnpolyh q n1 ; p \<noteq> 0\<^sub>p ; q \<noteq> 0\<^sub>p \<rbrakk> \<Longrightarrow> head (p *\<^sub>p q) = head p *\<^sub>p head q"
chaieb@33154
  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@56009
  1548
  then show ?case using 3
wenzelm@56009
  1549
    by (cases n) auto
chaieb@33154
  1550
next
chaieb@33154
  1551
  case (4 c n p c' n' p' n0 n1)
chaieb@33154
  1552
  hence 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@56009
  1579
  moreover {
wenzelm@56009
  1580
    assume nn': "n' = n"
wenzelm@52803
  1581
    from nn' polymul_normh[OF norm(5,4)]
chaieb@33154
  1582
    have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
wenzelm@52803
  1583
    from nn' polymul_normh[OF norm(5,3)] norm
chaieb@33154
  1584
    have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
chaieb@33154
  1585
    from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
wenzelm@52803
  1586
    have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
wenzelm@52803
  1587
    from polyadd_normh[OF ncnpc' ncnpp0']
wenzelm@52803
  1588
    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
  1589
      by (simp add: min_def)
wenzelm@56009
  1590
    {
wenzelm@56009
  1591
      assume np: "n > 0"
chaieb@33154
  1592
      with nn' head_isnpolyh_Suc'[OF np nth]
wenzelm@33268
  1593
        head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
wenzelm@56009
  1594
      have ?case by simp
wenzelm@56009
  1595
    }
chaieb@33154
  1596
    moreover
wenzelm@56009
  1597
    {
wenzelm@56009
  1598
      assume nz: "n = 0"
chaieb@33154
  1599
      from polymul_degreen[OF norm(5,4), where m="0"]
wenzelm@33268
  1600
        polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
chaieb@33154
  1601
      norm(5,6) degree_npolyhCN[OF norm(6)]
chaieb@33154
  1602
    have dth:"degree (CN c 0 p *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))" by simp
chaieb@33154
  1603
    hence dth':"degree (CN c 0 p *\<^sub>p c') \<noteq> degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))" by simp
chaieb@33154
  1604
    from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
krauss@41813
  1605
    have ?case   using norm "4.hyps"(6)[OF norm(5,3)]
wenzelm@56009
  1606
        "4.hyps"(5)[OF norm(5,4)] nn' nz by simp
wenzelm@56009
  1607
    }
wenzelm@56009
  1608
    ultimately have ?case by (cases n) auto
wenzelm@56009
  1609
  }
chaieb@33154
  1610
  ultimately show ?case by blast
chaieb@33154
  1611
qed simp_all
chaieb@33154
  1612
chaieb@33154
  1613
lemma degree_coefficients: "degree p = length (coefficients p) - 1"
wenzelm@52658
  1614
  by (induct p rule: degree.induct) auto
chaieb@33154
  1615
chaieb@33154
  1616
lemma degree_head[simp]: "degree (head p) = 0"
wenzelm@52658
  1617
  by (induct p rule: head.induct) auto
chaieb@33154
  1618
krauss@41812
  1619
lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
wenzelm@52658
  1620
  by (cases n) simp_all
chaieb@33154
  1621
lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
wenzelm@52658
  1622
  by (cases n) simp_all
chaieb@33154
  1623
wenzelm@52658
  1624
lemma polyadd_different_degree:
wenzelm@52658
  1625
  "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degree p \<noteq> degree q\<rbrakk> \<Longrightarrow>
wenzelm@52658
  1626
    degree (polyadd p q) = max (degree p) (degree q)"
chaieb@33154
  1627
  using polyadd_different_degreen degree_eq_degreen0 by simp
chaieb@33154
  1628
chaieb@33154
  1629
lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
wenzelm@52658
  1630
  by (induct p arbitrary: n0 rule: polyneg.induct) auto
chaieb@33154
  1631
chaieb@33154
  1632
lemma degree_polymul:
wenzelm@56000
  1633
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1634
    and np: "isnpolyh p n0"
wenzelm@52658
  1635
    and nq: "isnpolyh q n1"
chaieb@33154
  1636
  shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
chaieb@33154
  1637
  using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
chaieb@33154
  1638
chaieb@33154
  1639
lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
wenzelm@52658
  1640
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
  1641
chaieb@33154
  1642
lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"
wenzelm@52658
  1643
  by (induct p arbitrary: n rule: degree.induct) auto
wenzelm@52658
  1644
chaieb@33154
  1645
chaieb@33154
  1646
subsection {* Correctness of polynomial pseudo division *}
chaieb@33154
  1647
chaieb@33154
  1648
lemma polydivide_aux_properties:
wenzelm@56000
  1649
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1650
    and np: "isnpolyh p n0"
wenzelm@52658
  1651
    and ns: "isnpolyh s n1"
wenzelm@52658
  1652
    and ap: "head p = a"
wenzelm@52658
  1653
    and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
wenzelm@52803
  1654
  shows "(polydivide_aux a n p k s = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p)
chaieb@33154
  1655
          \<and> (\<exists>nr. isnpolyh r nr) \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow (k' - k) a) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
chaieb@33154
  1656
  using ns
wenzelm@52658
  1657
proof (induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
berghofe@34915
  1658
  case less
chaieb@33154
  1659
  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@52803
  1660
  let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p)
chaieb@33154
  1661
    \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
chaieb@33154
  1662
  let ?b = "head s"
berghofe@34915
  1663
  let ?p' = "funpow (degree s - n) shift1 p"
wenzelm@50282
  1664
  let ?xdn = "funpow (degree s - n) shift1 (1)\<^sub>p"
chaieb@33154
  1665
  let ?akk' = "a ^\<^sub>p (k' - k)"
berghofe@34915
  1666
  note ns = `isnpolyh s n1`
wenzelm@52803
  1667
  from np have np0: "isnpolyh p 0"
wenzelm@52803
  1668
    using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by simp
wenzelm@52803
  1669
  have np': "isnpolyh ?p' 0"
wenzelm@52803
  1670
    using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def
wenzelm@52803
  1671
    by simp
wenzelm@52803
  1672
  have headp': "head ?p' = head p"
wenzelm@52803
  1673
    using funpow_shift1_head[OF np pnz] by simp
wenzelm@52803
  1674
  from funpow_shift1_isnpoly[where p="(1)\<^sub>p"] have nxdn: "isnpolyh ?xdn 0"
wenzelm@52803
  1675
    by (simp add: isnpoly_def)
wenzelm@52803
  1676
  from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
chaieb@33154
  1677
  have nakk':"isnpolyh ?akk' 0" by blast
wenzelm@52658
  1678
  { assume sz: "s = 0\<^sub>p"
wenzelm@52658
  1679
    hence ?ths using np polydivide_aux.simps
wenzelm@52658
  1680
      apply clarsimp
wenzelm@52658
  1681
      apply (rule exI[where x="0\<^sub>p"])
wenzelm@52658
  1682
      apply simp
wenzelm@52658
  1683
      done }
chaieb@33154
  1684
  moreover
wenzelm@52803
  1685
  { assume sz: "s \<noteq> 0\<^sub>p"
wenzelm@52803
  1686
    { assume dn: "degree s < n"
wenzelm@52658
  1687
      hence "?ths" using ns ndp np polydivide_aux.simps
wenzelm@52658
  1688
        apply auto
wenzelm@52658
  1689
        apply (rule exI[where x="0\<^sub>p"])
wenzelm@52658
  1690
        apply simp
wenzelm@52658
  1691
        done }
wenzelm@52803
  1692
    moreover
wenzelm@52803
  1693
    { assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith
wenzelm@52803
  1694
      have degsp': "degree s = degree ?p'"
berghofe@34915
  1695
        using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp
wenzelm@52803
  1696
      { assume ba: "?b = a"
wenzelm@52803
  1697
        hence headsp': "head s = head ?p'"
wenzelm@52803
  1698
          using ap headp' by simp
wenzelm@52803
  1699
        have nr: "isnpolyh (s -\<^sub>p ?p') 0"
wenzelm@52803
  1700
          using polysub_normh[OF ns np'] by simp
berghofe@34915
  1701
        from degree_polysub_samehead[OF ns np' headsp' degsp']
berghofe@34915
  1702
        have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
wenzelm@52803
  1703
        moreover
wenzelm@52803
  1704
        { assume deglt:"degree (s -\<^sub>p ?p') < degree s"
krauss@41403
  1705
          from polydivide_aux.simps sz dn' ba
krauss@41403
  1706
          have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
wenzelm@33268
  1707
            by (simp add: Let_def)
wenzelm@52803
  1708
          { assume h1: "polydivide_aux a n p k s = (k', r)"
wenzelm@52803
  1709
            from less(1)[OF deglt nr, of k k' r] trans[OF eq[symmetric] h1]
wenzelm@52803
  1710
            have kk': "k \<le> k'"
wenzelm@52803
  1711
              and nr:"\<exists>nr. isnpolyh r nr"
wenzelm@52803
  1712
              and dr: "degree r = 0 \<or> degree r < degree p"
wenzelm@52803
  1713
              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
  1714
              by auto
wenzelm@52803
  1715
            from q1 obtain q n1 where nq: "isnpolyh q n1"
wenzelm@52803
  1716
              and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r" by blast
wenzelm@33268
  1717
            from nr obtain nr where nr': "isnpolyh r nr" by blast
wenzelm@52803
  1718
            from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0"
wenzelm@52803
  1719
              by simp
wenzelm@33268
  1720
            from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
wenzelm@33268
  1721
            have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
wenzelm@52803
  1722
            from polyadd_normh[OF polymul_normh[OF np
wenzelm@33268
  1723
              polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
wenzelm@52803
  1724
            have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0"
wenzelm@52803
  1725
              by simp
wenzelm@56000
  1726
            from asp have "\<forall> (bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) =
wenzelm@33268
  1727
              Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
wenzelm@56000
  1728
            hence " \<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
wenzelm@52803
  1729
              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
haftmann@36349
  1730
              by (simp add: field_simps)
wenzelm@56000
  1731
            hence " \<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1732
              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p *\<^sub>p p) +
wenzelm@52803
  1733
              Ipoly bs p * Ipoly bs q + Ipoly bs r"
wenzelm@52803
  1734
              by (auto simp only: funpow_shift1_1)
wenzelm@56000
  1735
            hence "\<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1736
              Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p) +
wenzelm@52803
  1737
              Ipoly bs q) + Ipoly bs r"
wenzelm@52803
  1738
              by (simp add: field_simps)
wenzelm@56000
  1739
            hence "\<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1740
              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
  1741
              by simp
wenzelm@33268
  1742
            with isnpolyh_unique[OF nakks' nqr']
wenzelm@52803
  1743
            have "a ^\<^sub>p (k' - k) *\<^sub>p s =
wenzelm@52803
  1744
              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
  1745
              by blast
wenzelm@33268
  1746
            hence ?qths using nq'
wenzelm@50282
  1747
              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
  1748
              apply (rule_tac x="0" in exI)
wenzelm@52803
  1749
              apply simp
wenzelm@52803
  1750
              done
wenzelm@33268
  1751
            with kk' nr dr have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
wenzelm@52803
  1752
              by blast
wenzelm@52803
  1753
          }
wenzelm@52803
  1754
          hence ?ths by blast
wenzelm@52803
  1755
        }
wenzelm@52803
  1756
        moreover
wenzelm@52803
  1757
        { assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
wenzelm@56000
  1758
          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@56000
  1759
          have " \<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs s = Ipoly bs ?p'"
wenzelm@52803
  1760
            by simp
wenzelm@56000
  1761
          hence "\<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)"
wenzelm@52658
  1762
            using np nxdn
wenzelm@52658
  1763
            apply simp
wenzelm@52658
  1764
            apply (simp only: funpow_shift1_1)
wenzelm@52658
  1765
            apply simp
wenzelm@52658
  1766
            done
wenzelm@52658
  1767
          hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]]
wenzelm@52658
  1768
            by blast
wenzelm@52803
  1769
          { assume h1: "polydivide_aux a n p k s = (k',r)"
krauss@41403
  1770
            from polydivide_aux.simps sz dn' ba
krauss@41403
  1771
            have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
wenzelm@33268
  1772
              by (simp add: Let_def)
wenzelm@52803
  1773
            also have "\<dots> = (k,0\<^sub>p)"
wenzelm@52803
  1774
              using polydivide_aux.simps spz by simp
wenzelm@33268
  1775
            finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
berghofe@34915
  1776
            with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
krauss@41403
  1777
              polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
wenzelm@33268
  1778
              apply auto
wenzelm@52803
  1779
              apply (rule exI[where x="?xdn"])
berghofe@34915
  1780
              apply (auto simp add: polymul_commute[of p])
wenzelm@52803
  1781
              done
wenzelm@52803
  1782
          }
wenzelm@52803
  1783
        }
wenzelm@52803
  1784
        ultimately have ?ths by blast
wenzelm@52803
  1785
      }
chaieb@33154
  1786
      moreover
wenzelm@52803
  1787
      { assume ba: "?b \<noteq> a"
wenzelm@52803
  1788
        from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
wenzelm@33268
  1789
          polymul_normh[OF head_isnpolyh[OF ns] np']]
wenzelm@52803
  1790
        have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
wenzelm@52803
  1791
          by (simp add: min_def)
wenzelm@33268
  1792
        have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
wenzelm@52803
  1793
          using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
wenzelm@33268
  1794
            polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
wenzelm@52803
  1795
            funpow_shift1_nz[OF pnz]
wenzelm@52803
  1796
          by simp_all
wenzelm@33268
  1797
        from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
berghofe@34915
  1798
          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
  1799
        have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
wenzelm@33268
  1800
          using head_head[OF ns] funpow_shift1_head[OF np pnz]
wenzelm@33268
  1801
            polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
wenzelm@33268
  1802
          by (simp add: ap)
wenzelm@33268
  1803
        from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
wenzelm@33268
  1804
          head_nz[OF np] pnz sz ap[symmetric]
berghofe@34915
  1805
          funpow_shift1_nz[OF pnz, where n="degree s - n"]
wenzelm@52803
  1806
          polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"] head_nz[OF ns]
berghofe@34915
  1807
          ndp dn
wenzelm@52803
  1808
        have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p')"
wenzelm@33268
  1809
          by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
wenzelm@52803
  1810
        { assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
wenzelm@52803
  1811
          from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns]
wenzelm@52803
  1812
            polymul_normh[OF head_isnpolyh[OF ns]np']] ap
wenzelm@52803
  1813
          have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
wenzelm@52803
  1814
            by simp
wenzelm@52803
  1815
          { assume h1:"polydivide_aux a n p k s = (k', r)"
krauss@41403
  1816
            from h1 polydivide_aux.simps sz dn' ba
krauss@41403
  1817
            have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
wenzelm@33268
  1818
              by (simp add: Let_def)
berghofe@34915
  1819
            with less(1)[OF dth nasbp', of "Suc k" k' r]
wenzelm@52803
  1820
            obtain q nq nr where kk': "Suc k \<le> k'"
wenzelm@52803
  1821
              and nr: "isnpolyh r nr"
wenzelm@52803
  1822
              and nq: "isnpolyh q nq"
wenzelm@33268
  1823
              and dr: "degree r = 0 \<or> degree r < degree p"
wenzelm@52803
  1824
              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
  1825
              by auto
wenzelm@33268
  1826
            from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith
wenzelm@52803
  1827
            {
wenzelm@56000
  1828
              fix bs:: "'a::{field_char_0,field_inverse_zero} list"
wenzelm@52803
  1829
              from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
wenzelm@52803
  1830
              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
  1831
                by simp
wenzelm@52803
  1832
              hence "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s =
wenzelm@52803
  1833
                Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
wenzelm@52803
  1834
                by (simp add: field_simps)
wenzelm@52803
  1835
              hence "Ipoly bs a ^ (k' - k)  * Ipoly bs s =
wenzelm@52803
  1836
                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
  1837
                by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
wenzelm@52803
  1838
              hence "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1839
                Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
wenzelm@52803
  1840
                by (simp add: field_simps)
wenzelm@52803
  1841
            }
wenzelm@56000
  1842
            hence ieq:"\<forall>(bs :: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1843
              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
  1844
              by auto
wenzelm@33268
  1845
            let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
wenzelm@33268
  1846
            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
  1847
            have nqw: "isnpolyh ?q 0"
wenzelm@52803
  1848
              by simp
wenzelm@33268
  1849
            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
  1850
            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
  1851
              by blast
wenzelm@52803
  1852
            from dr kk' nr h1 asth nqw have ?ths
wenzelm@52803
  1853
              apply simp
wenzelm@33268
  1854
              apply (rule conjI)
wenzelm@33268
  1855
              apply (rule exI[where x="nr"], simp)
wenzelm@33268
  1856
              apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
wenzelm@33268
  1857
              apply (rule exI[where x="0"], simp)
wenzelm@52803
  1858
              done
wenzelm@52803
  1859
          }
wenzelm@52803
  1860
          hence ?ths by blast
wenzelm@52803
  1861
        }
wenzelm@52803
  1862
        moreover
wenzelm@52803
  1863
        { assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
wenzelm@52803
  1864
          {
wenzelm@56000
  1865
            fix bs :: "'a::{field_char_0,field_inverse_zero} list"
wenzelm@33268
  1866
            from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
wenzelm@52803
  1867
            have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'"
wenzelm@52803
  1868
              by simp
wenzelm@52803
  1869
            hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
wenzelm@52803
  1870
              by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
wenzelm@52803
  1871
            hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))"
wenzelm@52803
  1872
              by simp
wenzelm@52803
  1873
          }
wenzelm@56000
  1874
          hence hth: "\<forall> (bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a*\<^sub>p s) =
wenzelm@52658
  1875
            Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
wenzelm@52803
  1876
          from hth have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
wenzelm@56000
  1877
            using isnpolyh_unique[where ?'a = "'a::{field_char_0,field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
chaieb@33154
  1878
                    polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
wenzelm@33268
  1879
              simplified ap] by simp
wenzelm@52803
  1880
          { assume h1: "polydivide_aux a n p k s = (k', r)"
wenzelm@52803
  1881
            from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
wenzelm@52803
  1882
            have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
wenzelm@52803
  1883
            with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
wenzelm@52803
  1884
              polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
wenzelm@52803
  1885
            have ?ths
wenzelm@52803
  1886
              apply (clarsimp simp add: Let_def)
wenzelm@52803
  1887
              apply (rule exI[where x="?b *\<^sub>p ?xdn"])
wenzelm@52803
  1888
              apply simp
wenzelm@52803
  1889
              apply (rule exI[where x="0"], simp)
wenzelm@52803
  1890
              done
wenzelm@52803
  1891
          }
wenzelm@52803
  1892
          hence ?ths by blast
wenzelm@52803
  1893
        }
wenzelm@52658
  1894
        ultimately have ?ths
wenzelm@52658
  1895
          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
  1896
            head_nz[OF np] pnz sz ap[symmetric]
wenzelm@52803
  1897
          by (simp add: degree_eq_degreen0[symmetric]) blast
wenzelm@52803
  1898
      }
chaieb@33154
  1899
      ultimately have ?ths by blast
chaieb@33154
  1900
    }
wenzelm@52803
  1901
    ultimately have ?ths by blast
wenzelm@52803
  1902
  }
chaieb@33154
  1903
  ultimately show ?ths by blast
chaieb@33154
  1904
qed
chaieb@33154
  1905
wenzelm@52803
  1906
lemma polydivide_properties:
wenzelm@56000
  1907
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52803
  1908
    and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
wenzelm@52803
  1909
  shows "\<exists>k r. polydivide s p = (k,r) \<and>
wenzelm@52803
  1910
    (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@52803
  1911
    (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r))"
wenzelm@52803
  1912
proof -
wenzelm@52803
  1913
  have trv: "head p = head p" "degree p = degree p"
wenzelm@52803
  1914
    by simp_all
wenzelm@52803
  1915
  from polydivide_def[where s="s" and p="p"] have ex: "\<exists> k r. polydivide s p = (k,r)"
wenzelm@52803
  1916
    by auto
wenzelm@52803
  1917
  then obtain k r where kr: "polydivide s p = (k,r)"
wenzelm@52803
  1918
    by blast
wenzelm@56000
  1919
  from trans[OF polydivide_def[where s="s"and p="p", symmetric] kr]
chaieb@33154
  1920
    polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
chaieb@33154
  1921
  have "(degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@52803
  1922
    (\<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
  1923
    by blast
wenzelm@52803
  1924
  with kr show ?thesis
chaieb@33154
  1925
    apply -
chaieb@33154
  1926
    apply (rule exI[where x="k"])
chaieb@33154
  1927
    apply (rule exI[where x="r"])
chaieb@33154
  1928
    apply simp
chaieb@33154
  1929
    done
chaieb@33154
  1930
qed
chaieb@33154
  1931
wenzelm@52658
  1932
chaieb@33154
  1933
subsection{* More about polypoly and pnormal etc *}
chaieb@33154
  1934
wenzelm@56000
  1935
definition "isnonconstant p \<longleftrightarrow> \<not> isconstant p"
chaieb@33154
  1936
wenzelm@52658
  1937
lemma isnonconstant_pnormal_iff:
wenzelm@52803
  1938
  assumes nc: "isnonconstant p"
wenzelm@52803
  1939
  shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
chaieb@33154
  1940
proof
wenzelm@52803
  1941
  let ?p = "polypoly bs p"
chaieb@33154
  1942
  assume H: "pnormal ?p"
wenzelm@52658
  1943
  have csz: "coefficients p \<noteq> []" using nc by (cases p) auto
wenzelm@52803
  1944
wenzelm@52803
  1945
  from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
chaieb@33154
  1946
    pnormal_last_nonzero[OF H]
chaieb@33154
  1947
  show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)
chaieb@33154
  1948
next
chaieb@33154
  1949
  assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  1950
  let ?p = "polypoly bs p"
wenzelm@52658
  1951
  have csz: "coefficients p \<noteq> []" using nc by (cases p) auto
wenzelm@52803
  1952
  hence pz: "?p \<noteq> []" by (simp add: polypoly_def)
chaieb@33154
  1953
  hence lg: "length ?p > 0" by simp
wenzelm@52803
  1954
  from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
chaieb@33154
  1955
  have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)
chaieb@33154
  1956
  from pnormal_last_length[OF lg lz] show "pnormal ?p" .
chaieb@33154
  1957
qed
chaieb@33154
  1958
chaieb@33154
  1959
lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
chaieb@33154
  1960
  unfolding isnonconstant_def
wenzelm@52658
  1961
  apply (cases p)
wenzelm@52658
  1962
  apply simp_all
wenzelm@52658
  1963
  apply (case_tac nat)
wenzelm@52658
  1964
  apply auto
chaieb@33154
  1965
  done
wenzelm@52658
  1966
wenzelm@52658
  1967
lemma isnonconstant_nonconstant:
wenzelm@52658
  1968
  assumes inc: "isnonconstant p"
chaieb@33154
  1969
  shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
chaieb@33154
  1970
proof
chaieb@33154
  1971
  let ?p = "polypoly bs p"
chaieb@33154
  1972
  assume nc: "nonconstant ?p"
chaieb@33154
  1973
  from isnonconstant_pnormal_iff[OF inc, of bs] nc
chaieb@33154
  1974
  show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast
chaieb@33154
  1975
next
chaieb@33154
  1976
  let ?p = "polypoly bs p"
chaieb@33154
  1977
  assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  1978
  from isnonconstant_pnormal_iff[OF inc, of bs] h
chaieb@33154
  1979
  have pn: "pnormal ?p" by blast
wenzelm@56009
  1980
  {
wenzelm@56009
  1981
    fix x
wenzelm@56009
  1982
    assume H: "?p = [x]"
wenzelm@56009
  1983
    from H have "length (coefficients p) = 1"
wenzelm@56009
  1984
      unfolding polypoly_def by auto
wenzelm@56009
  1985
    with isnonconstant_coefficients_length[OF inc]
wenzelm@56009
  1986
      have False by arith
wenzelm@56009
  1987
  }
wenzelm@56009
  1988
  then show "nonconstant ?p"
wenzelm@56009
  1989
    using pn unfolding nonconstant_def by blast
chaieb@33154
  1990
qed
chaieb@33154
  1991
chaieb@33154
  1992
lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
wenzelm@52658
  1993
  apply (induct p)
wenzelm@52658
  1994
  apply (simp_all add: pnormal_def)
wenzelm@52658
  1995
  apply (case_tac "p = []")
wenzelm@52658
  1996
  apply simp_all
wenzelm@52658
  1997
  done
chaieb@33154
  1998
wenzelm@52658
  1999
lemma degree_degree:
wenzelm@52658
  2000
  assumes inc: "isnonconstant p"
chaieb@33154
  2001
  shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  2002
proof
wenzelm@52803
  2003
  let ?p = "polypoly bs p"
chaieb@33154
  2004
  assume H: "degree p = Polynomial_List.degree ?p"
chaieb@33154
  2005
  from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
chaieb@33154
  2006
    unfolding polypoly_def by auto
chaieb@33154
  2007
  from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
chaieb@33154
  2008
  have lg:"length (pnormalize ?p) = length ?p"
chaieb@33154
  2009
    unfolding Polynomial_List.degree_def polypoly_def by simp
wenzelm@52803
  2010
  hence "pnormal ?p" using pnormal_length[OF pz] by blast
wenzelm@52803
  2011
  with isnonconstant_pnormal_iff[OF inc]
chaieb@33154
  2012
  show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast
chaieb@33154
  2013
next
wenzelm@52803
  2014
  let  ?p = "polypoly bs p"
chaieb@33154
  2015
  assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  2016
  with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast
chaieb@33154
  2017
  with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
wenzelm@52803
  2018
  show "degree p = Polynomial_List.degree ?p"
chaieb@33154
  2019
    unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
chaieb@33154
  2020
qed
chaieb@33154
  2021
wenzelm@52658
  2022
wenzelm@52803
  2023
section {* Swaps ; Division by a certain variable *}
wenzelm@52658
  2024
wenzelm@52803
  2025
primrec swap :: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly" where
chaieb@33154
  2026
  "swap n m (C x) = C x"
haftmann@39246
  2027
| "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"
haftmann@39246
  2028
| "swap n m (Neg t) = Neg (swap n m t)"
haftmann@39246
  2029
| "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
haftmann@39246
  2030
| "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
haftmann@39246
  2031
| "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
haftmann@39246
  2032
| "swap n m (Pw t k) = Pw (swap n m t) k"
wenzelm@52803
  2033
| "swap n m (CN c k p) =
wenzelm@52803
  2034
    CN (swap n m c) (if k = n then m else if k=m then n else k) (swap n m p)"
chaieb@33154
  2035
wenzelm@52658
  2036
lemma swap:
wenzelm@52658
  2037
  assumes nbs: "n < length bs"
wenzelm@52658
  2038
    and mbs: "m < length bs"
chaieb@33154
  2039
  shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
chaieb@33154
  2040
proof (induct t)
wenzelm@52658
  2041
  case (Bound k)
wenzelm@56009
  2042
  then show ?case using nbs mbs by simp
chaieb@33154
  2043
next
wenzelm@52658
  2044
  case (CN c k p)
wenzelm@56009
  2045
  then show ?case using nbs mbs by simp
chaieb@33154
  2046
qed simp_all
chaieb@33154
  2047
wenzelm@52658
  2048
lemma swap_swap_id [simp]: "swap n m (swap m n t) = t"
wenzelm@52658
  2049
  by (induct t) simp_all
wenzelm@52658
  2050
wenzelm@52658
  2051
lemma swap_commute: "swap n m p = swap m n p"
wenzelm@52658
  2052
  by (induct p) simp_all
chaieb@33154
  2053
chaieb@33154
  2054
lemma swap_same_id[simp]: "swap n n t = t"
wenzelm@52658
  2055
  by (induct t) simp_all
chaieb@33154
  2056
chaieb@33154
  2057
definition "swapnorm n m t = polynate (swap n m t)"
chaieb@33154
  2058
wenzelm@52658
  2059
lemma swapnorm:
wenzelm@52658
  2060
  assumes nbs: "n < length bs"
wenzelm@52658
  2061
    and mbs: "m < length bs"
wenzelm@56000
  2062
  shows "((Ipoly bs (swapnorm n m t) :: 'a::{field_char_0,field_inverse_zero})) =
wenzelm@52658
  2063
    Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
wenzelm@41807
  2064
  using swap[OF assms] swapnorm_def by simp
chaieb@33154
  2065
wenzelm@52658
  2066
lemma swapnorm_isnpoly [simp]:
wenzelm@56000
  2067
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
  2068
  shows "isnpoly (swapnorm n m p)"
chaieb@33154
  2069
  unfolding swapnorm_def by simp
chaieb@33154
  2070
wenzelm@52803
  2071
definition "polydivideby n s p =
wenzelm@56000
  2072
  (let
wenzelm@56000
  2073
    ss = swapnorm 0 n s;
wenzelm@56000
  2074
    sp = swapnorm 0 n p;
wenzelm@56000
  2075
    h = head sp;
wenzelm@56000
  2076
    (k, r) = polydivide ss sp
wenzelm@56000
  2077
   in (k, swapnorm 0 n h, swapnorm 0 n r))"
chaieb@33154
  2078
wenzelm@56000
  2079
lemma swap_nz [simp]: "swap n m p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
  2080
  by (induct p) simp_all
chaieb@33154
  2081
krauss@41808
  2082
fun isweaknpoly :: "poly \<Rightarrow> bool"
krauss@41808
  2083
where
chaieb@33154
  2084
  "isweaknpoly (C c) = True"
krauss@41808
  2085
| "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
krauss@41808
  2086
| "isweaknpoly p = False"
chaieb@33154
  2087
wenzelm@52803
  2088
lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
wenzelm@52658
  2089
  by (induct p arbitrary: n0) auto
chaieb@33154
  2090
wenzelm@52803
  2091
lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
wenzelm@52658
  2092
  by (induct p) auto
chaieb@33154
  2093
chaieb@33154
  2094
end