src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
author wenzelm
Wed Mar 12 17:25:28 2014 +0100 (2014-03-12)
changeset 56066 cce36efe32eb
parent 56043 0b25c3d34b77
child 56073 29e308b56d23
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@56066
  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@56066
  1066
  unfolding wf_bs_def by (induct p q rule: polyadd.induct) (auto simp add: Let_def)
chaieb@33154
  1067
chaieb@33154
  1068
lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
wenzelm@52803
  1069
  unfolding wf_bs_def
wenzelm@52803
  1070
  apply (induct p q arbitrary: bs rule: polymul.induct)
chaieb@33154
  1071
  apply (simp_all add: wf_bs_polyadd)
chaieb@33154
  1072
  apply clarsimp
chaieb@33154
  1073
  apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
chaieb@33154
  1074
  apply auto
chaieb@33154
  1075
  done
chaieb@33154
  1076
chaieb@33154
  1077
lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
wenzelm@52658
  1078
  unfolding wf_bs_def by (induct p rule: polyneg.induct) auto
chaieb@33154
  1079
chaieb@33154
  1080
lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
wenzelm@56043
  1081
  unfolding polysub_def split_def fst_conv snd_conv
wenzelm@56043
  1082
  using wf_bs_polyadd wf_bs_polyneg by blast
chaieb@33154
  1083
wenzelm@52658
  1084
wenzelm@56043
  1085
subsection {* Canonicity of polynomial representation, see lemma isnpolyh_unique *}
chaieb@33154
  1086
chaieb@33154
  1087
definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
wenzelm@56043
  1088
definition "polypoly' bs p = map (Ipoly bs \<circ> decrpoly) (coefficients p)"
wenzelm@56043
  1089
definition "poly_nate bs p = map (Ipoly bs \<circ> decrpoly) (coefficients (polynate p))"
chaieb@33154
  1090
wenzelm@56043
  1091
lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall>q \<in> set (coefficients p). isnpolyh q n0"
chaieb@33154
  1092
proof (induct p arbitrary: n0 rule: coefficients.induct)
chaieb@33154
  1093
  case (1 c p n0)
wenzelm@56009
  1094
  have cp: "isnpolyh (CN c 0 p) n0"
wenzelm@56009
  1095
    by fact
wenzelm@56009
  1096
  then have norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
chaieb@33154
  1097
    by (auto simp add: isnpolyh_mono[where n'=0])
wenzelm@56009
  1098
  from "1.hyps"[OF norm(2)] norm(1) norm(4) show ?case
wenzelm@56009
  1099
    by simp
chaieb@33154
  1100
qed auto
chaieb@33154
  1101
wenzelm@56043
  1102
lemma coefficients_isconst: "isnpolyh p n \<Longrightarrow> \<forall>q \<in> set (coefficients p). isconstant q"
wenzelm@56043
  1103
  by (induct p arbitrary: n rule: coefficients.induct) (auto simp add: isnpolyh_Suc_const)
chaieb@33154
  1104
chaieb@33154
  1105
lemma polypoly_polypoly':
chaieb@33154
  1106
  assumes np: "isnpolyh p n0"
wenzelm@56043
  1107
  shows "polypoly (x # bs) p = polypoly' bs p"
wenzelm@56043
  1108
proof -
chaieb@33154
  1109
  let ?cf = "set (coefficients p)"
chaieb@33154
  1110
  from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
wenzelm@56043
  1111
  {
wenzelm@56043
  1112
    fix q
wenzelm@56043
  1113
    assume q: "q \<in> ?cf"
wenzelm@56043
  1114
    from q cn_norm have th: "isnpolyh q n0"
wenzelm@56043
  1115
      by blast
wenzelm@56043
  1116
    from coefficients_isconst[OF np] q have "isconstant q"
wenzelm@56043
  1117
      by blast
wenzelm@56043
  1118
    with isconstant_polybound0[OF th] have "polybound0 q"
wenzelm@56043
  1119
      by blast
wenzelm@56043
  1120
  }
wenzelm@56009
  1121
  then have "\<forall>q \<in> ?cf. polybound0 q" ..
wenzelm@56043
  1122
  then have "\<forall>q \<in> ?cf. Ipoly (x # bs) q = Ipoly bs (decrpoly q)"
chaieb@33154
  1123
    using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
chaieb@33154
  1124
    by auto
wenzelm@56043
  1125
  then show ?thesis
wenzelm@56043
  1126
    unfolding polypoly_def polypoly'_def by simp
chaieb@33154
  1127
qed
chaieb@33154
  1128
chaieb@33154
  1129
lemma polypoly_poly:
wenzelm@56043
  1130
  assumes "isnpolyh p n0"
wenzelm@56043
  1131
  shows "Ipoly (x # bs) p = poly (polypoly (x # bs) p) x"
wenzelm@56043
  1132
  using assms
wenzelm@52658
  1133
  by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def)
chaieb@33154
  1134
wenzelm@52803
  1135
lemma polypoly'_poly:
wenzelm@56043
  1136
  assumes "isnpolyh p n0"
wenzelm@52658
  1137
  shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
wenzelm@56043
  1138
  using polypoly_poly[OF assms, simplified polypoly_polypoly'[OF assms]] .
chaieb@33154
  1139
chaieb@33154
  1140
chaieb@33154
  1141
lemma polypoly_poly_polybound0:
wenzelm@56043
  1142
  assumes "isnpolyh p n0"
wenzelm@56043
  1143
    and "polybound0 p"
chaieb@33154
  1144
  shows "polypoly bs p = [Ipoly bs p]"
wenzelm@56043
  1145
  using assms
wenzelm@56043
  1146
  unfolding polypoly_def
wenzelm@52658
  1147
  apply (cases p)
wenzelm@52658
  1148
  apply auto
wenzelm@52658
  1149
  apply (case_tac nat)
wenzelm@52658
  1150
  apply auto
wenzelm@52658
  1151
  done
chaieb@33154
  1152
wenzelm@52803
  1153
lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
wenzelm@52658
  1154
  by (induct p rule: head.induct) auto
chaieb@33154
  1155
wenzelm@56043
  1156
lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> headn p m = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
  1157
  by (cases p) auto
chaieb@33154
  1158
chaieb@33154
  1159
lemma head_eq_headn0: "head p = headn p 0"
wenzelm@52658
  1160
  by (induct p rule: head.induct) simp_all
chaieb@33154
  1161
wenzelm@56043
  1162
lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> head p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
chaieb@33154
  1163
  by (simp add: head_eq_headn0)
chaieb@33154
  1164
wenzelm@52803
  1165
lemma isnpolyh_zero_iff:
wenzelm@52658
  1166
  assumes nq: "isnpolyh p n0"
wenzelm@56000
  1167
    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
  1168
  shows "p = 0\<^sub>p"
wenzelm@52658
  1169
  using nq eq
berghofe@34915
  1170
proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
berghofe@34915
  1171
  case less
berghofe@34915
  1172
  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
  1173
  {
wenzelm@56000
  1174
    assume nz: "maxindex p = 0"
wenzelm@56000
  1175
    then obtain c where "p = C c"
wenzelm@56000
  1176
      using np by (cases p) auto
wenzelm@56000
  1177
    with zp np have "p = 0\<^sub>p"
wenzelm@56000
  1178
      unfolding wf_bs_def by simp
wenzelm@56000
  1179
  }
chaieb@33154
  1180
  moreover
wenzelm@56000
  1181
  {
wenzelm@56000
  1182
    assume nz: "maxindex p \<noteq> 0"
chaieb@33154
  1183
    let ?h = "head p"
chaieb@33154
  1184
    let ?hd = "decrpoly ?h"
chaieb@33154
  1185
    let ?ihd = "maxindex ?hd"
wenzelm@56000
  1186
    from head_isnpolyh[OF np] head_polybound0[OF np]
wenzelm@56000
  1187
    have h: "isnpolyh ?h n0" "polybound0 ?h"
chaieb@33154
  1188
      by simp_all
wenzelm@56000
  1189
    then have nhd: "isnpolyh ?hd (n0 - 1)"
wenzelm@56000
  1190
      using decrpoly_normh by blast
wenzelm@52803
  1191
chaieb@33154
  1192
    from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
wenzelm@56000
  1193
    have mihn: "maxindex ?h \<le> maxindex p"
wenzelm@56000
  1194
      by auto
wenzelm@56000
  1195
    with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < maxindex p"
wenzelm@56000
  1196
      by auto
wenzelm@56000
  1197
    {
wenzelm@56000
  1198
      fix bs :: "'a list"
wenzelm@56000
  1199
      assume bs: "wf_bs bs ?hd"
chaieb@33154
  1200
      let ?ts = "take ?ihd bs"
chaieb@33154
  1201
      let ?rs = "drop ?ihd bs"
wenzelm@56000
  1202
      have ts: "wf_bs ?ts ?hd"
wenzelm@56000
  1203
        using bs unfolding wf_bs_def by simp
wenzelm@56000
  1204
      have bs_ts_eq: "?ts @ ?rs = bs"
wenzelm@56000
  1205
        by simp
wenzelm@56000
  1206
      from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x # ?ts) ?h"
wenzelm@56000
  1207
        by simp
wenzelm@56000
  1208
      from ihd_lt_n have "\<forall>x. length (x # ?ts) \<le> maxindex p"
wenzelm@56000
  1209
        by simp
wenzelm@56000
  1210
      with length_le_list_ex obtain xs where xs: "length ((x # ?ts) @ xs) = maxindex p"
wenzelm@56000
  1211
        by blast
wenzelm@56000
  1212
      then have "\<forall>x. wf_bs ((x # ?ts) @ xs) p"
wenzelm@56000
  1213
        unfolding wf_bs_def by simp
wenzelm@56000
  1214
      with zp have "\<forall>x. Ipoly ((x # ?ts) @ xs) p = 0"
wenzelm@56000
  1215
        by blast
wenzelm@56000
  1216
      then have "\<forall>x. Ipoly (x # (?ts @ xs)) p = 0"
wenzelm@56000
  1217
        by simp
chaieb@33154
  1218
      with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
wenzelm@56000
  1219
      have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"
wenzelm@56000
  1220
        by simp
wenzelm@56000
  1221
      then have "poly (polypoly' (?ts @ xs) p) = poly []"
wenzelm@56000
  1222
        by auto
wenzelm@56000
  1223
      then have "\<forall>c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
wenzelm@33268
  1224
        using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
chaieb@33154
  1225
      with coefficients_head[of p, symmetric]
wenzelm@56000
  1226
      have th0: "Ipoly (?ts @ xs) ?hd = 0"
wenzelm@56000
  1227
        by simp
wenzelm@56000
  1228
      from bs have wf'': "wf_bs ?ts ?hd"
wenzelm@56000
  1229
        unfolding wf_bs_def by simp
wenzelm@56000
  1230
      with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0"
wenzelm@56000
  1231
        by simp
wenzelm@56000
  1232
      with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
wenzelm@56000
  1233
        by simp
wenzelm@56000
  1234
    }
wenzelm@56000
  1235
    then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
wenzelm@56000
  1236
      by blast
wenzelm@56000
  1237
    from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p"
wenzelm@56000
  1238
      by blast
wenzelm@56000
  1239
    then have "?h = 0\<^sub>p" by simp
wenzelm@56000
  1240
    with head_nz[OF np] have "p = 0\<^sub>p" by simp
wenzelm@56000
  1241
  }
wenzelm@56000
  1242
  ultimately show "p = 0\<^sub>p"
wenzelm@56000
  1243
    by blast
chaieb@33154
  1244
qed
chaieb@33154
  1245
wenzelm@52803
  1246
lemma isnpolyh_unique:
wenzelm@56000
  1247
  assumes np: "isnpolyh p n0"
wenzelm@52658
  1248
    and nq: "isnpolyh q n1"
wenzelm@56000
  1249
  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
  1250
proof auto
wenzelm@56000
  1251
  assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a) = \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
wenzelm@56000
  1252
  then have "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)"
wenzelm@56000
  1253
    by simp
wenzelm@56000
  1254
  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
  1255
    using wf_bs_polysub[where p=p and q=q] by auto
wenzelm@56000
  1256
  with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq] show "p = q"
wenzelm@56000
  1257
    by blast
chaieb@33154
  1258
qed
chaieb@33154
  1259
chaieb@33154
  1260
krauss@41404
  1261
text{* consequences of unicity on the algorithms for polynomial normalization *}
chaieb@33154
  1262
wenzelm@52658
  1263
lemma polyadd_commute:
wenzelm@56000
  1264
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1265
    and np: "isnpolyh p n0"
wenzelm@52658
  1266
    and nq: "isnpolyh q n1"
wenzelm@52658
  1267
  shows "p +\<^sub>p q = q +\<^sub>p p"
wenzelm@56000
  1268
  using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]]
wenzelm@56000
  1269
  by simp
chaieb@33154
  1270
wenzelm@56000
  1271
lemma zero_normh: "isnpolyh 0\<^sub>p n"
wenzelm@56000
  1272
  by simp
wenzelm@56000
  1273
wenzelm@56000
  1274
lemma one_normh: "isnpolyh (1)\<^sub>p n"
wenzelm@56000
  1275
  by simp
wenzelm@52658
  1276
wenzelm@52803
  1277
lemma polyadd_0[simp]:
wenzelm@56000
  1278
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1279
    and np: "isnpolyh p n0"
wenzelm@56000
  1280
  shows "p +\<^sub>p 0\<^sub>p = p"
wenzelm@56000
  1281
    and "0\<^sub>p +\<^sub>p p = p"
wenzelm@52803
  1282
  using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
chaieb@33154
  1283
    isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
chaieb@33154
  1284
wenzelm@52803
  1285
lemma polymul_1[simp]:
wenzelm@56000
  1286
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1287
    and np: "isnpolyh p n0"
wenzelm@56000
  1288
  shows "p *\<^sub>p (1)\<^sub>p = p"
wenzelm@56000
  1289
    and "(1)\<^sub>p *\<^sub>p p = p"
wenzelm@52803
  1290
  using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
chaieb@33154
  1291
    isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
wenzelm@52658
  1292
wenzelm@52803
  1293
lemma polymul_0[simp]:
wenzelm@56000
  1294
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1295
    and np: "isnpolyh p n0"
wenzelm@56000
  1296
  shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p"
wenzelm@56000
  1297
    and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
wenzelm@52803
  1298
  using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
chaieb@33154
  1299
    isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
chaieb@33154
  1300
wenzelm@52803
  1301
lemma polymul_commute:
wenzelm@56000
  1302
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56000
  1303
    and np: "isnpolyh p n0"
wenzelm@52658
  1304
    and nq: "isnpolyh q n1"
chaieb@33154
  1305
  shows "p *\<^sub>p q = q *\<^sub>p p"
wenzelm@56043
  1306
  using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np],
wenzelm@56043
  1307
    where ?'a = "'a::{field_char_0,field_inverse_zero, power}"]
wenzelm@52658
  1308
  by simp
chaieb@33154
  1309
wenzelm@52658
  1310
declare polyneg_polyneg [simp]
wenzelm@52803
  1311
wenzelm@52803
  1312
lemma isnpolyh_polynate_id [simp]:
wenzelm@56000
  1313
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56000
  1314
    and np: "isnpolyh p n0"
wenzelm@52658
  1315
  shows "polynate p = p"
wenzelm@56043
  1316
  using isnpolyh_unique[where ?'a= "'a::{field_char_0,field_inverse_zero}",
wenzelm@56043
  1317
      OF polynate_norm[of p, unfolded isnpoly_def] np]
wenzelm@56043
  1318
    polynate[where ?'a = "'a::{field_char_0,field_inverse_zero}"]
wenzelm@52658
  1319
  by simp
chaieb@33154
  1320
wenzelm@52803
  1321
lemma polynate_idempotent[simp]:
wenzelm@56000
  1322
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
  1323
  shows "polynate (polynate p) = polynate p"
chaieb@33154
  1324
  using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
chaieb@33154
  1325
chaieb@33154
  1326
lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
chaieb@33154
  1327
  unfolding poly_nate_def polypoly'_def ..
wenzelm@52658
  1328
wenzelm@52658
  1329
lemma poly_nate_poly:
wenzelm@56000
  1330
  "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
  1331
  using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
wenzelm@52658
  1332
  unfolding poly_nate_polypoly' by auto
wenzelm@52658
  1333
chaieb@33154
  1334
chaieb@33154
  1335
subsection{* heads, degrees and all that *}
wenzelm@52658
  1336
chaieb@33154
  1337
lemma degree_eq_degreen0: "degree p = degreen p 0"
wenzelm@52658
  1338
  by (induct p rule: degree.induct) simp_all
chaieb@33154
  1339
wenzelm@52658
  1340
lemma degree_polyneg:
wenzelm@56043
  1341
  assumes "isnpolyh p n"
chaieb@33154
  1342
  shows "degree (polyneg p) = degree p"
wenzelm@56043
  1343
  apply (induct p rule: polyneg.induct)
wenzelm@56043
  1344
  using assms
wenzelm@56043
  1345
  apply simp_all
wenzelm@52658
  1346
  apply (case_tac na)
wenzelm@52658
  1347
  apply auto
wenzelm@52658
  1348
  done
chaieb@33154
  1349
chaieb@33154
  1350
lemma degree_polyadd:
wenzelm@56043
  1351
  assumes np: "isnpolyh p n0"
wenzelm@56043
  1352
    and nq: "isnpolyh q n1"
chaieb@33154
  1353
  shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
wenzelm@52658
  1354
  using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
chaieb@33154
  1355
chaieb@33154
  1356
wenzelm@52658
  1357
lemma degree_polysub:
wenzelm@52658
  1358
  assumes np: "isnpolyh p n0"
wenzelm@52658
  1359
    and nq: "isnpolyh q n1"
chaieb@33154
  1360
  shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
chaieb@33154
  1361
proof-
wenzelm@56043
  1362
  from nq have nq': "isnpolyh (~\<^sub>p q) n1"
wenzelm@56043
  1363
    using polyneg_normh by simp
wenzelm@56043
  1364
  from degree_polyadd[OF np nq'] show ?thesis
wenzelm@56043
  1365
    by (simp add: polysub_def degree_polyneg[OF nq])
chaieb@33154
  1366
qed
chaieb@33154
  1367
wenzelm@52803
  1368
lemma degree_polysub_samehead:
wenzelm@56000
  1369
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56043
  1370
    and np: "isnpolyh p n0"
wenzelm@56043
  1371
    and nq: "isnpolyh q n1"
wenzelm@56043
  1372
    and h: "head p = head q"
wenzelm@52658
  1373
    and d: "degree p = degree q"
chaieb@33154
  1374
  shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
wenzelm@52658
  1375
  unfolding polysub_def split_def fst_conv snd_conv
wenzelm@52658
  1376
  using np nq h d
wenzelm@52658
  1377
proof (induct p q rule: polyadd.induct)
wenzelm@52658
  1378
  case (1 c c')
wenzelm@56009
  1379
  then show ?case
wenzelm@56009
  1380
    by (simp add: Nsub_def Nsub0[simplified Nsub_def])
chaieb@33154
  1381
next
wenzelm@52803
  1382
  case (2 c c' n' p')
wenzelm@56009
  1383
  from 2 have "degree (C c) = degree (CN c' n' p')"
wenzelm@56009
  1384
    by simp
wenzelm@56009
  1385
  then have nz: "n' > 0"
wenzelm@56009
  1386
    by (cases n') auto
wenzelm@56009
  1387
  then have "head (CN c' n' p') = CN c' n' p'"
wenzelm@56009
  1388
    by (cases n') auto
wenzelm@56009
  1389
  with 2 show ?case
wenzelm@56009
  1390
    by simp
chaieb@33154
  1391
next
wenzelm@52803
  1392
  case (3 c n p c')
wenzelm@56009
  1393
  then have "degree (C c') = degree (CN c n p)"
wenzelm@56009
  1394
    by simp
wenzelm@56009
  1395
  then have nz: "n > 0"
wenzelm@56009
  1396
    by (cases n) auto
wenzelm@56009
  1397
  then have "head (CN c n p) = CN c n p"
wenzelm@56009
  1398
    by (cases n) auto
wenzelm@41807
  1399
  with 3 show ?case by simp
chaieb@33154
  1400
next
chaieb@33154
  1401
  case (4 c n p c' n' p')
wenzelm@56009
  1402
  then have H:
wenzelm@56009
  1403
    "isnpolyh (CN c n p) n0"
wenzelm@56009
  1404
    "isnpolyh (CN c' n' p') n1"
wenzelm@56009
  1405
    "head (CN c n p) = head (CN c' n' p')"
wenzelm@56009
  1406
    "degree (CN c n p) = degree (CN c' n' p')"
wenzelm@56009
  1407
    by simp_all
wenzelm@56009
  1408
  then have degc: "degree c = 0" and degc': "degree c' = 0"
wenzelm@56009
  1409
    by simp_all
wenzelm@56009
  1410
  then have degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
chaieb@33154
  1411
    using H(1-2) degree_polyneg by auto
wenzelm@56009
  1412
  from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"
wenzelm@56009
  1413
    by simp_all
wenzelm@56009
  1414
  from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc'
wenzelm@56009
  1415
  have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"
wenzelm@56009
  1416
    by simp
wenzelm@56009
  1417
  from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'"
wenzelm@56009
  1418
    by auto
wenzelm@56009
  1419
  have "n = n' \<or> n < n' \<or> n > n'"
wenzelm@56009
  1420
    by arith
chaieb@33154
  1421
  moreover
wenzelm@56009
  1422
  {
wenzelm@56009
  1423
    assume nn': "n = n'"
wenzelm@56009
  1424
    have "n = 0 \<or> n > 0" by arith
wenzelm@56066
  1425
    moreover
wenzelm@56066
  1426
    {
wenzelm@56009
  1427
      assume nz: "n = 0"
wenzelm@56009
  1428
      then have ?case using 4 nn'
wenzelm@56009
  1429
        by (auto simp add: Let_def degcmc')
wenzelm@56009
  1430
    }
wenzelm@56066
  1431
    moreover
wenzelm@56066
  1432
    {
wenzelm@56009
  1433
      assume nz: "n > 0"
wenzelm@56009
  1434
      with nn' H(3) have  cc': "c = c'" and pp': "p = p'"
wenzelm@56009
  1435
        by (cases n, auto)+
wenzelm@56009
  1436
      then have ?case
wenzelm@56009
  1437
        using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv]
wenzelm@56009
  1438
        using polysub_same_0[OF c'nh, simplified polysub_def]
wenzelm@56009
  1439
        using nn' 4 by (simp add: Let_def)
wenzelm@56009
  1440
    }
wenzelm@56009
  1441
    ultimately have ?case by blast
wenzelm@56009
  1442
  }
chaieb@33154
  1443
  moreover
wenzelm@56009
  1444
  {
wenzelm@56009
  1445
    assume nn': "n < n'"
wenzelm@56009
  1446
    then have n'p: "n' > 0"
wenzelm@56009
  1447
      by simp
wenzelm@56009
  1448
    then have headcnp':"head (CN c' n' p') = CN c' n' p'"
wenzelm@56009
  1449
      by (cases n') simp_all
wenzelm@56009
  1450
    have degcnp': "degree (CN c' n' p') = 0"
wenzelm@56009
  1451
      and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
wenzelm@52658
  1452
      using 4 nn' by (cases n', simp_all)
wenzelm@56009
  1453
    then have "n > 0"
wenzelm@56009
  1454
      by (cases n) simp_all
wenzelm@56009
  1455
    then have headcnp: "head (CN c n p) = CN c n p"
wenzelm@56009
  1456
      by (cases n) auto
wenzelm@56009
  1457
    from H(3) headcnp headcnp' nn' have ?case
wenzelm@56009
  1458
      by auto
wenzelm@56009
  1459
  }
chaieb@33154
  1460
  moreover
wenzelm@56009
  1461
  {
wenzelm@56009
  1462
    assume nn': "n > n'"
wenzelm@56009
  1463
    then have np: "n > 0" by simp
wenzelm@56009
  1464
    then have headcnp:"head (CN c n p) = CN c n p"
wenzelm@56009
  1465
      by (cases n) simp_all
wenzelm@56009
  1466
    from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)"
wenzelm@56009
  1467
      by simp
wenzelm@56009
  1468
    from np have degcnp: "degree (CN c n p) = 0"
wenzelm@56009
  1469
      by (cases n) simp_all
wenzelm@56009
  1470
    with degcnpeq have "n' > 0"
wenzelm@56009
  1471
      by (cases n') simp_all
wenzelm@56009
  1472
    then have headcnp': "head (CN c' n' p') = CN c' n' p'"
wenzelm@56009
  1473
      by (cases n') auto
wenzelm@56009
  1474
    from H(3) headcnp headcnp' nn' have ?case by auto
wenzelm@56009
  1475
  }
wenzelm@56009
  1476
  ultimately show ?case by blast
krauss@41812
  1477
qed auto
wenzelm@52803
  1478
chaieb@33154
  1479
lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
wenzelm@52658
  1480
  by (induct p arbitrary: n0 rule: head.induct) (simp_all add: shift1_def)
chaieb@33154
  1481
chaieb@33154
  1482
lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
wenzelm@52658
  1483
proof (induct k arbitrary: n0 p)
wenzelm@52658
  1484
  case 0
wenzelm@56009
  1485
  then show ?case by auto
wenzelm@52658
  1486
next
wenzelm@52658
  1487
  case (Suc k n0 p)
wenzelm@56066
  1488
  then have "isnpolyh (shift1 p) 0"
wenzelm@56066
  1489
    by (simp add: shift1_isnpolyh)
wenzelm@41807
  1490
  with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
wenzelm@56066
  1491
    and "head (shift1 p) = head p"
wenzelm@56066
  1492
    by (simp_all add: shift1_head)
wenzelm@56066
  1493
  then show ?case
wenzelm@56066
  1494
    by (simp add: funpow_swap1)
wenzelm@52658
  1495
qed
chaieb@33154
  1496
chaieb@33154
  1497
lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
chaieb@33154
  1498
  by (simp add: shift1_def)
wenzelm@56009
  1499
chaieb@33154
  1500
lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
wenzelm@46991
  1501
  by (induct k arbitrary: p) (auto simp add: shift1_degree)
chaieb@33154
  1502
chaieb@33154
  1503
lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
wenzelm@52658
  1504
  by (induct n arbitrary: p) simp_all
chaieb@33154
  1505
chaieb@33154
  1506
lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
wenzelm@52658
  1507
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
  1508
lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
wenzelm@52658
  1509
  by (induct p arbitrary: n rule: degreen.induct) auto
chaieb@33154
  1510
lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
wenzelm@52658
  1511
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
  1512
lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
wenzelm@52658
  1513
  by (induct p rule: head.induct) auto
chaieb@33154
  1514
wenzelm@52803
  1515
lemma polyadd_eq_const_degree:
wenzelm@52658
  1516
  "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> polyadd p q = C c \<Longrightarrow> degree p = degree q"
chaieb@33154
  1517
  using polyadd_eq_const_degreen degree_eq_degreen0 by simp
chaieb@33154
  1518
wenzelm@52658
  1519
lemma polyadd_head:
wenzelm@52658
  1520
  assumes np: "isnpolyh p n0"
wenzelm@52658
  1521
    and nq: "isnpolyh q n1"
wenzelm@52658
  1522
    and deg: "degree p \<noteq> degree q"
chaieb@33154
  1523
  shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
wenzelm@52658
  1524
  using np nq deg
wenzelm@52658
  1525
  apply (induct p q arbitrary: n0 n1 rule: polyadd.induct)
wenzelm@52658
  1526
  using np
wenzelm@52658
  1527
  apply simp_all
wenzelm@52658
  1528
  apply (case_tac n', simp, simp)
wenzelm@52658
  1529
  apply (case_tac n, simp, simp)
wenzelm@52658
  1530
  apply (case_tac n, case_tac n', simp add: Let_def)
haftmann@54489
  1531
  apply (auto simp add: polyadd_eq_const_degree)[2]
wenzelm@52658
  1532
  apply (metis head_nz)
wenzelm@52658
  1533
  apply (metis head_nz)
wenzelm@52658
  1534
  apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
wenzelm@52658
  1535
  done
chaieb@33154
  1536
wenzelm@52803
  1537
lemma polymul_head_polyeq:
wenzelm@56000
  1538
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56066
  1539
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> q \<noteq> 0\<^sub>p \<Longrightarrow> head (p *\<^sub>p q) = head p *\<^sub>p head q"
chaieb@33154
  1540
proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
krauss@41813
  1541
  case (2 c c' n' p' n0 n1)
wenzelm@56009
  1542
  then have "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"
wenzelm@56009
  1543
    by (simp_all add: head_isnpolyh)
wenzelm@56009
  1544
  then show ?case
wenzelm@56009
  1545
    using 2 by (cases n') auto
wenzelm@52803
  1546
next
wenzelm@52803
  1547
  case (3 c n p c' n0 n1)
wenzelm@56009
  1548
  then have "isnpolyh (head (CN c n p)) n0" "isnormNum c'"
wenzelm@56009
  1549
    by (simp_all add: head_isnpolyh)
wenzelm@56066
  1550
  then show ?case
wenzelm@56066
  1551
    using 3 by (cases n) auto
chaieb@33154
  1552
next
chaieb@33154
  1553
  case (4 c n p c' n' p' n0 n1)
wenzelm@56066
  1554
  then have norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
chaieb@33154
  1555
    "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
chaieb@33154
  1556
    by simp_all
chaieb@33154
  1557
  have "n < n' \<or> n' < n \<or> n = n'" by arith
wenzelm@52803
  1558
  moreover
wenzelm@56009
  1559
  {
wenzelm@56009
  1560
    assume nn': "n < n'"
wenzelm@56009
  1561
    then have ?case
wenzelm@52658
  1562
      using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)]
wenzelm@52658
  1563
      apply simp
wenzelm@52658
  1564
      apply (cases n)
wenzelm@52658
  1565
      apply simp
wenzelm@52658
  1566
      apply (cases n')
wenzelm@52658
  1567
      apply simp_all
wenzelm@56009
  1568
      done
wenzelm@56009
  1569
  }
wenzelm@56009
  1570
  moreover {
wenzelm@56009
  1571
    assume nn': "n'< n"
wenzelm@56009
  1572
    then have ?case
wenzelm@52803
  1573
      using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)]
wenzelm@52658
  1574
      apply simp
wenzelm@52658
  1575
      apply (cases n')
wenzelm@52658
  1576
      apply simp
wenzelm@52658
  1577
      apply (cases n)
wenzelm@52658
  1578
      apply auto
wenzelm@56009
  1579
      done
wenzelm@56009
  1580
  }
wenzelm@56066
  1581
  moreover
wenzelm@56066
  1582
  {
wenzelm@56009
  1583
    assume nn': "n' = n"
wenzelm@52803
  1584
    from nn' polymul_normh[OF norm(5,4)]
chaieb@33154
  1585
    have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
wenzelm@52803
  1586
    from nn' polymul_normh[OF norm(5,3)] norm
chaieb@33154
  1587
    have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
chaieb@33154
  1588
    from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
wenzelm@52803
  1589
    have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
wenzelm@52803
  1590
    from polyadd_normh[OF ncnpc' ncnpp0']
wenzelm@52803
  1591
    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
  1592
      by (simp add: min_def)
wenzelm@56009
  1593
    {
wenzelm@56009
  1594
      assume np: "n > 0"
chaieb@33154
  1595
      with nn' head_isnpolyh_Suc'[OF np nth]
wenzelm@33268
  1596
        head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
wenzelm@56009
  1597
      have ?case by simp
wenzelm@56009
  1598
    }
chaieb@33154
  1599
    moreover
wenzelm@56009
  1600
    {
wenzelm@56009
  1601
      assume nz: "n = 0"
chaieb@33154
  1602
      from polymul_degreen[OF norm(5,4), where m="0"]
wenzelm@33268
  1603
        polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
chaieb@33154
  1604
      norm(5,6) degree_npolyhCN[OF norm(6)]
wenzelm@56066
  1605
    have dth: "degree (CN c 0 p *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))"
wenzelm@56066
  1606
      by simp
wenzelm@56066
  1607
    then have dth': "degree (CN c 0 p *\<^sub>p c') \<noteq> degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))"
wenzelm@56066
  1608
      by simp
chaieb@33154
  1609
    from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
wenzelm@56066
  1610
    have ?case
wenzelm@56066
  1611
      using norm "4.hyps"(6)[OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)] nn' nz
wenzelm@56066
  1612
      by simp
wenzelm@56009
  1613
    }
wenzelm@56066
  1614
    ultimately have ?case
wenzelm@56066
  1615
      by (cases n) auto
wenzelm@56009
  1616
  }
chaieb@33154
  1617
  ultimately show ?case by blast
chaieb@33154
  1618
qed simp_all
chaieb@33154
  1619
chaieb@33154
  1620
lemma degree_coefficients: "degree p = length (coefficients p) - 1"
wenzelm@52658
  1621
  by (induct p rule: degree.induct) auto
chaieb@33154
  1622
chaieb@33154
  1623
lemma degree_head[simp]: "degree (head p) = 0"
wenzelm@52658
  1624
  by (induct p rule: head.induct) auto
chaieb@33154
  1625
krauss@41812
  1626
lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
wenzelm@52658
  1627
  by (cases n) simp_all
wenzelm@56066
  1628
chaieb@33154
  1629
lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
wenzelm@52658
  1630
  by (cases n) simp_all
chaieb@33154
  1631
wenzelm@52658
  1632
lemma polyadd_different_degree:
wenzelm@56066
  1633
  "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> degree p \<noteq> degree q \<Longrightarrow>
wenzelm@52658
  1634
    degree (polyadd p q) = max (degree p) (degree q)"
chaieb@33154
  1635
  using polyadd_different_degreen degree_eq_degreen0 by simp
chaieb@33154
  1636
chaieb@33154
  1637
lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
wenzelm@52658
  1638
  by (induct p arbitrary: n0 rule: polyneg.induct) auto
chaieb@33154
  1639
chaieb@33154
  1640
lemma degree_polymul:
wenzelm@56000
  1641
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1642
    and np: "isnpolyh p n0"
wenzelm@52658
  1643
    and nq: "isnpolyh q n1"
chaieb@33154
  1644
  shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
chaieb@33154
  1645
  using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
chaieb@33154
  1646
chaieb@33154
  1647
lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
wenzelm@52658
  1648
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
  1649
chaieb@33154
  1650
lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"
wenzelm@52658
  1651
  by (induct p arbitrary: n rule: degree.induct) auto
wenzelm@52658
  1652
chaieb@33154
  1653
chaieb@33154
  1654
subsection {* Correctness of polynomial pseudo division *}
chaieb@33154
  1655
chaieb@33154
  1656
lemma polydivide_aux_properties:
wenzelm@56000
  1657
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1658
    and np: "isnpolyh p n0"
wenzelm@52658
  1659
    and ns: "isnpolyh s n1"
wenzelm@52658
  1660
    and ap: "head p = a"
wenzelm@52658
  1661
    and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
wenzelm@56066
  1662
  shows "polydivide_aux a n p k s = (k',r) \<longrightarrow> k' \<ge> k \<and> (degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@56066
  1663
    (\<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
  1664
  using ns
wenzelm@52658
  1665
proof (induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
berghofe@34915
  1666
  case less
chaieb@33154
  1667
  let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
wenzelm@56066
  1668
  let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and>
wenzelm@56066
  1669
    (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
chaieb@33154
  1670
  let ?b = "head s"
berghofe@34915
  1671
  let ?p' = "funpow (degree s - n) shift1 p"
wenzelm@50282
  1672
  let ?xdn = "funpow (degree s - n) shift1 (1)\<^sub>p"
chaieb@33154
  1673
  let ?akk' = "a ^\<^sub>p (k' - k)"
berghofe@34915
  1674
  note ns = `isnpolyh s n1`
wenzelm@52803
  1675
  from np have np0: "isnpolyh p 0"
wenzelm@52803
  1676
    using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by simp
wenzelm@52803
  1677
  have np': "isnpolyh ?p' 0"
wenzelm@52803
  1678
    using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def
wenzelm@52803
  1679
    by simp
wenzelm@52803
  1680
  have headp': "head ?p' = head p"
wenzelm@52803
  1681
    using funpow_shift1_head[OF np pnz] by simp
wenzelm@52803
  1682
  from funpow_shift1_isnpoly[where p="(1)\<^sub>p"] have nxdn: "isnpolyh ?xdn 0"
wenzelm@52803
  1683
    by (simp add: isnpoly_def)
wenzelm@52803
  1684
  from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
chaieb@33154
  1685
  have nakk':"isnpolyh ?akk' 0" by blast
wenzelm@56066
  1686
  {
wenzelm@56066
  1687
    assume sz: "s = 0\<^sub>p"
wenzelm@56066
  1688
    then have ?ths
wenzelm@56066
  1689
      using np polydivide_aux.simps
wenzelm@52658
  1690
      apply clarsimp
wenzelm@52658
  1691
      apply (rule exI[where x="0\<^sub>p"])
wenzelm@52658
  1692
      apply simp
wenzelm@56066
  1693
      done
wenzelm@56066
  1694
  }
chaieb@33154
  1695
  moreover
wenzelm@56066
  1696
  {
wenzelm@56066
  1697
    assume sz: "s \<noteq> 0\<^sub>p"
wenzelm@56066
  1698
    {
wenzelm@56066
  1699
      assume dn: "degree s < n"
wenzelm@56066
  1700
      then have "?ths"
wenzelm@56066
  1701
        using ns ndp np polydivide_aux.simps
wenzelm@52658
  1702
        apply auto
wenzelm@52658
  1703
        apply (rule exI[where x="0\<^sub>p"])
wenzelm@52658
  1704
        apply simp
wenzelm@56066
  1705
        done
wenzelm@56066
  1706
    }
wenzelm@52803
  1707
    moreover
wenzelm@56066
  1708
    {
wenzelm@56066
  1709
      assume dn': "\<not> degree s < n"
wenzelm@56066
  1710
      then have dn: "degree s \<ge> n"
wenzelm@56066
  1711
        by arith
wenzelm@52803
  1712
      have degsp': "degree s = degree ?p'"
wenzelm@56066
  1713
        using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"]
wenzelm@56066
  1714
        by simp
wenzelm@56066
  1715
      {
wenzelm@56066
  1716
        assume ba: "?b = a"
wenzelm@56066
  1717
        then have headsp': "head s = head ?p'"
wenzelm@52803
  1718
          using ap headp' by simp
wenzelm@52803
  1719
        have nr: "isnpolyh (s -\<^sub>p ?p') 0"
wenzelm@52803
  1720
          using polysub_normh[OF ns np'] by simp
berghofe@34915
  1721
        from degree_polysub_samehead[OF ns np' headsp' degsp']
berghofe@34915
  1722
        have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
wenzelm@52803
  1723
        moreover
wenzelm@56066
  1724
        {
wenzelm@56066
  1725
          assume deglt:"degree (s -\<^sub>p ?p') < degree s"
krauss@41403
  1726
          from polydivide_aux.simps sz dn' ba
krauss@41403
  1727
          have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
wenzelm@33268
  1728
            by (simp add: Let_def)
wenzelm@56066
  1729
          {
wenzelm@56066
  1730
            assume h1: "polydivide_aux a n p k s = (k', r)"
wenzelm@52803
  1731
            from less(1)[OF deglt nr, of k k' r] trans[OF eq[symmetric] h1]
wenzelm@52803
  1732
            have kk': "k \<le> k'"
wenzelm@56066
  1733
              and nr: "\<exists>nr. isnpolyh r nr"
wenzelm@52803
  1734
              and dr: "degree r = 0 \<or> degree r < degree p"
wenzelm@56066
  1735
              and q1: "\<exists>q nq. isnpolyh q nq \<and> a ^\<^sub>p k' - k *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"
wenzelm@52803
  1736
              by auto
wenzelm@52803
  1737
            from q1 obtain q n1 where nq: "isnpolyh q n1"
wenzelm@56066
  1738
              and asp: "a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"
wenzelm@56066
  1739
              by blast
wenzelm@56066
  1740
            from nr obtain nr where nr': "isnpolyh r nr"
wenzelm@56066
  1741
              by blast
wenzelm@52803
  1742
            from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0"
wenzelm@52803
  1743
              by simp
wenzelm@33268
  1744
            from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
wenzelm@33268
  1745
            have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
wenzelm@52803
  1746
            from polyadd_normh[OF polymul_normh[OF np
wenzelm@33268
  1747
              polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
wenzelm@52803
  1748
            have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0"
wenzelm@52803
  1749
              by simp
wenzelm@56066
  1750
            from asp have "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
wenzelm@56066
  1751
              Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)"
wenzelm@56066
  1752
              by simp
wenzelm@56066
  1753
            then have "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
wenzelm@56066
  1754
              Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
wenzelm@52803
  1755
              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
haftmann@36349
  1756
              by (simp add: field_simps)
wenzelm@56066
  1757
            then have "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
wenzelm@56066
  1758
              Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1759
              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p *\<^sub>p p) +
wenzelm@52803
  1760
              Ipoly bs p * Ipoly bs q + Ipoly bs r"
wenzelm@52803
  1761
              by (auto simp only: funpow_shift1_1)
wenzelm@56066
  1762
            then have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list.
wenzelm@56066
  1763
              Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1764
              Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p) +
wenzelm@52803
  1765
              Ipoly bs q) + Ipoly bs r"
wenzelm@52803
  1766
              by (simp add: field_simps)
wenzelm@56066
  1767
            then have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list.
wenzelm@56066
  1768
              Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1769
              Ipoly bs (p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r)"
wenzelm@52803
  1770
              by simp
wenzelm@33268
  1771
            with isnpolyh_unique[OF nakks' nqr']
wenzelm@52803
  1772
            have "a ^\<^sub>p (k' - k) *\<^sub>p s =
wenzelm@52803
  1773
              p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q) +\<^sub>p r"
wenzelm@52803
  1774
              by blast
wenzelm@56066
  1775
            then have ?qths using nq'
wenzelm@50282
  1776
              apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 (1)\<^sub>p) +\<^sub>p q" in exI)
wenzelm@52803
  1777
              apply (rule_tac x="0" in exI)
wenzelm@52803
  1778
              apply simp
wenzelm@52803
  1779
              done
wenzelm@56066
  1780
            with kk' nr dr have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@56066
  1781
              (\<exists>nr. isnpolyh r nr) \<and> ?qths"
wenzelm@52803
  1782
              by blast
wenzelm@52803
  1783
          }
wenzelm@56066
  1784
          then have ?ths by blast
wenzelm@52803
  1785
        }
wenzelm@52803
  1786
        moreover
wenzelm@56066
  1787
        {
wenzelm@56066
  1788
          assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
wenzelm@56000
  1789
          from spz isnpolyh_unique[OF polysub_normh[OF ns np'], where q="0\<^sub>p", symmetric, where ?'a = "'a::{field_char_0,field_inverse_zero}"]
wenzelm@56066
  1790
          have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list. Ipoly bs s = Ipoly bs ?p'"
wenzelm@52803
  1791
            by simp
wenzelm@56066
  1792
          then have "\<forall>bs:: 'a::{field_char_0,field_inverse_zero} list. Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)"
wenzelm@52658
  1793
            using np nxdn
wenzelm@52658
  1794
            apply simp
wenzelm@52658
  1795
            apply (simp only: funpow_shift1_1)
wenzelm@52658
  1796
            apply simp
wenzelm@52658
  1797
            done
wenzelm@56066
  1798
          then have sp': "s = ?xdn *\<^sub>p p"
wenzelm@56066
  1799
            using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]]
wenzelm@52658
  1800
            by blast
wenzelm@56066
  1801
          {
wenzelm@56066
  1802
            assume h1: "polydivide_aux a n p k s = (k',r)"
krauss@41403
  1803
            from polydivide_aux.simps sz dn' ba
krauss@41403
  1804
            have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
wenzelm@33268
  1805
              by (simp add: Let_def)
wenzelm@52803
  1806
            also have "\<dots> = (k,0\<^sub>p)"
wenzelm@52803
  1807
              using polydivide_aux.simps spz by simp
wenzelm@56066
  1808
            finally have "(k', r) = (k, 0\<^sub>p)"
wenzelm@56066
  1809
              using h1 by simp
berghofe@34915
  1810
            with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
krauss@41403
  1811
              polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
wenzelm@33268
  1812
              apply auto
wenzelm@52803
  1813
              apply (rule exI[where x="?xdn"])
berghofe@34915
  1814
              apply (auto simp add: polymul_commute[of p])
wenzelm@52803
  1815
              done
wenzelm@52803
  1816
          }
wenzelm@52803
  1817
        }
wenzelm@52803
  1818
        ultimately have ?ths by blast
wenzelm@52803
  1819
      }
chaieb@33154
  1820
      moreover
wenzelm@56066
  1821
      {
wenzelm@56066
  1822
        assume ba: "?b \<noteq> a"
wenzelm@52803
  1823
        from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
wenzelm@33268
  1824
          polymul_normh[OF head_isnpolyh[OF ns] np']]
wenzelm@52803
  1825
        have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
wenzelm@52803
  1826
          by (simp add: min_def)
wenzelm@33268
  1827
        have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
wenzelm@52803
  1828
          using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
wenzelm@33268
  1829
            polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
wenzelm@52803
  1830
            funpow_shift1_nz[OF pnz]
wenzelm@52803
  1831
          by simp_all
wenzelm@33268
  1832
        from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
berghofe@34915
  1833
          polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="degree s - n"]
wenzelm@52803
  1834
        have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
wenzelm@33268
  1835
          using head_head[OF ns] funpow_shift1_head[OF np pnz]
wenzelm@33268
  1836
            polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
wenzelm@33268
  1837
          by (simp add: ap)
wenzelm@33268
  1838
        from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
wenzelm@33268
  1839
          head_nz[OF np] pnz sz ap[symmetric]
berghofe@34915
  1840
          funpow_shift1_nz[OF pnz, where n="degree s - n"]
wenzelm@52803
  1841
          polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"] head_nz[OF ns]
berghofe@34915
  1842
          ndp dn
wenzelm@52803
  1843
        have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p')"
wenzelm@33268
  1844
          by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
wenzelm@56066
  1845
        {
wenzelm@56066
  1846
          assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
wenzelm@52803
  1847
          from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns]
wenzelm@52803
  1848
            polymul_normh[OF head_isnpolyh[OF ns]np']] ap
wenzelm@52803
  1849
          have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
wenzelm@52803
  1850
            by simp
wenzelm@56066
  1851
          {
wenzelm@56066
  1852
            assume h1:"polydivide_aux a n p k s = (k', r)"
krauss@41403
  1853
            from h1 polydivide_aux.simps sz dn' ba
krauss@41403
  1854
            have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
wenzelm@33268
  1855
              by (simp add: Let_def)
berghofe@34915
  1856
            with less(1)[OF dth nasbp', of "Suc k" k' r]
wenzelm@52803
  1857
            obtain q nq nr where kk': "Suc k \<le> k'"
wenzelm@52803
  1858
              and nr: "isnpolyh r nr"
wenzelm@52803
  1859
              and nq: "isnpolyh q nq"
wenzelm@33268
  1860
              and dr: "degree r = 0 \<or> degree r < degree p"
wenzelm@52803
  1861
              and qr: "a ^\<^sub>p (k' - Suc k) *\<^sub>p ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = p *\<^sub>p q +\<^sub>p r"
wenzelm@52803
  1862
              by auto
wenzelm@56066
  1863
            from kk' have kk'': "Suc (k' - Suc k) = k' - k"
wenzelm@56066
  1864
              by arith
wenzelm@52803
  1865
            {
wenzelm@56066
  1866
              fix bs :: "'a::{field_char_0,field_inverse_zero} list"
wenzelm@52803
  1867
              from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
wenzelm@52803
  1868
              have "Ipoly bs (a ^\<^sub>p (k' - Suc k) *\<^sub>p ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p'))) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)"
wenzelm@52803
  1869
                by simp
wenzelm@56066
  1870
              then have "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s =
wenzelm@52803
  1871
                Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
wenzelm@52803
  1872
                by (simp add: field_simps)
wenzelm@56066
  1873
              then have "Ipoly bs a ^ (k' - k)  * Ipoly bs s =
wenzelm@52803
  1874
                Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
wenzelm@52803
  1875
                by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
wenzelm@56066
  1876
              then have "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1877
                Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
wenzelm@52803
  1878
                by (simp add: field_simps)
wenzelm@52803
  1879
            }
wenzelm@56066
  1880
            then have ieq:"\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
wenzelm@56066
  1881
              Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1882
              Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)"
wenzelm@52803
  1883
              by auto
wenzelm@33268
  1884
            let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
wenzelm@33268
  1885
            from polyadd_normh[OF nq polymul_normh[OF polymul_normh[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k"] head_isnpolyh[OF ns], simplified ap ] nxdn]]
wenzelm@52803
  1886
            have nqw: "isnpolyh ?q 0"
wenzelm@52803
  1887
              by simp
wenzelm@33268
  1888
            from ieq isnpolyh_unique[OF polymul_normh[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - k"] ns, simplified ap] polyadd_normh[OF polymul_normh[OF np nqw] nr]]
wenzelm@52803
  1889
            have asth: "(a ^\<^sub>p (k' - k) *\<^sub>p s) = p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r"
wenzelm@52803
  1890
              by blast
wenzelm@52803
  1891
            from dr kk' nr h1 asth nqw have ?ths
wenzelm@52803
  1892
              apply simp
wenzelm@33268
  1893
              apply (rule conjI)
wenzelm@33268
  1894
              apply (rule exI[where x="nr"], simp)
wenzelm@33268
  1895
              apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
wenzelm@33268
  1896
              apply (rule exI[where x="0"], simp)
wenzelm@52803
  1897
              done
wenzelm@52803
  1898
          }
wenzelm@56066
  1899
          then have ?ths by blast
wenzelm@52803
  1900
        }
wenzelm@52803
  1901
        moreover
wenzelm@56066
  1902
        {
wenzelm@56066
  1903
          assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
wenzelm@52803
  1904
          {
wenzelm@56000
  1905
            fix bs :: "'a::{field_char_0,field_inverse_zero} list"
wenzelm@33268
  1906
            from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
wenzelm@52803
  1907
            have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'"
wenzelm@52803
  1908
              by simp
wenzelm@56066
  1909
            then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
wenzelm@52803
  1910
              by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
wenzelm@56066
  1911
            then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))"
wenzelm@52803
  1912
              by simp
wenzelm@52803
  1913
          }
wenzelm@56066
  1914
          then have hth: "\<forall>bs :: 'a::{field_char_0,field_inverse_zero} list.
wenzelm@56066
  1915
            Ipoly bs (a *\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
wenzelm@52803
  1916
          from hth have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
wenzelm@56000
  1917
            using isnpolyh_unique[where ?'a = "'a::{field_char_0,field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
chaieb@33154
  1918
                    polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
wenzelm@56066
  1919
              simplified ap]
wenzelm@56066
  1920
            by simp
wenzelm@56066
  1921
          {
wenzelm@56066
  1922
            assume h1: "polydivide_aux a n p k s = (k', r)"
wenzelm@52803
  1923
            from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
wenzelm@56066
  1924
            have "(k', r) = (Suc k, 0\<^sub>p)"
wenzelm@56066
  1925
              by (simp add: Let_def)
wenzelm@52803
  1926
            with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
wenzelm@52803
  1927
              polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
wenzelm@52803
  1928
            have ?ths
wenzelm@52803
  1929
              apply (clarsimp simp add: Let_def)
wenzelm@52803
  1930
              apply (rule exI[where x="?b *\<^sub>p ?xdn"])
wenzelm@52803
  1931
              apply simp
wenzelm@52803
  1932
              apply (rule exI[where x="0"], simp)
wenzelm@52803
  1933
              done
wenzelm@52803
  1934
          }
wenzelm@56066
  1935
          then have ?ths by blast
wenzelm@52803
  1936
        }
wenzelm@52658
  1937
        ultimately have ?ths
wenzelm@56066
  1938
          using degree_polysub_samehead[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns] polymul_normh[OF head_isnpolyh[OF ns] np'] hdth degth] polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
wenzelm@52658
  1939
            head_nz[OF np] pnz sz ap[symmetric]
wenzelm@56066
  1940
          by (auto simp add: degree_eq_degreen0[symmetric])
wenzelm@52803
  1941
      }
chaieb@33154
  1942
      ultimately have ?ths by blast
chaieb@33154
  1943
    }
wenzelm@52803
  1944
    ultimately have ?ths by blast
wenzelm@52803
  1945
  }
chaieb@33154
  1946
  ultimately show ?ths by blast
chaieb@33154
  1947
qed
chaieb@33154
  1948
wenzelm@52803
  1949
lemma polydivide_properties:
wenzelm@56000
  1950
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56066
  1951
    and np: "isnpolyh p n0"
wenzelm@56066
  1952
    and ns: "isnpolyh s n1"
wenzelm@56066
  1953
    and pnz: "p \<noteq> 0\<^sub>p"
wenzelm@56066
  1954
  shows "\<exists>k r. polydivide s p = (k, r) \<and>
wenzelm@52803
  1955
    (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@56066
  1956
    (\<exists>q n1. isnpolyh q n1 \<and> polypow k (head p) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
wenzelm@52803
  1957
proof -
wenzelm@52803
  1958
  have trv: "head p = head p" "degree p = degree p"
wenzelm@52803
  1959
    by simp_all
wenzelm@52803
  1960
  from polydivide_def[where s="s" and p="p"] have ex: "\<exists> k r. polydivide s p = (k,r)"
wenzelm@52803
  1961
    by auto
wenzelm@52803
  1962
  then obtain k r where kr: "polydivide s p = (k,r)"
wenzelm@52803
  1963
    by blast
wenzelm@56000
  1964
  from trans[OF polydivide_def[where s="s"and p="p", symmetric] kr]
chaieb@33154
  1965
    polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
chaieb@33154
  1966
  have "(degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@52803
  1967
    (\<exists>nr. isnpolyh r nr) \<and> (\<exists>q n1. isnpolyh q n1 \<and> head p ^\<^sub>p k - 0 *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
wenzelm@52803
  1968
    by blast
wenzelm@52803
  1969
  with kr show ?thesis
chaieb@33154
  1970
    apply -
chaieb@33154
  1971
    apply (rule exI[where x="k"])
chaieb@33154
  1972
    apply (rule exI[where x="r"])
chaieb@33154
  1973
    apply simp
chaieb@33154
  1974
    done
chaieb@33154
  1975
qed
chaieb@33154
  1976
wenzelm@52658
  1977
wenzelm@56066
  1978
subsection {* More about polypoly and pnormal etc *}
chaieb@33154
  1979
wenzelm@56000
  1980
definition "isnonconstant p \<longleftrightarrow> \<not> isconstant p"
chaieb@33154
  1981
wenzelm@52658
  1982
lemma isnonconstant_pnormal_iff:
wenzelm@52803
  1983
  assumes nc: "isnonconstant p"
wenzelm@52803
  1984
  shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
chaieb@33154
  1985
proof
wenzelm@52803
  1986
  let ?p = "polypoly bs p"
chaieb@33154
  1987
  assume H: "pnormal ?p"
wenzelm@56066
  1988
  have csz: "coefficients p \<noteq> []"
wenzelm@56066
  1989
    using nc by (cases p) auto
wenzelm@56066
  1990
  from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"] pnormal_last_nonzero[OF H]
wenzelm@56066
  1991
  show "Ipoly bs (head p) \<noteq> 0"
wenzelm@56066
  1992
    by (simp add: polypoly_def)
chaieb@33154
  1993
next
chaieb@33154
  1994
  assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  1995
  let ?p = "polypoly bs p"
wenzelm@56066
  1996
  have csz: "coefficients p \<noteq> []"
wenzelm@56066
  1997
    using nc by (cases p) auto
wenzelm@56066
  1998
  then have pz: "?p \<noteq> []"
wenzelm@56066
  1999
    by (simp add: polypoly_def)
wenzelm@56066
  2000
  then have lg: "length ?p > 0"
wenzelm@56066
  2001
    by simp
wenzelm@52803
  2002
  from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
wenzelm@56066
  2003
  have lz: "last ?p \<noteq> 0"
wenzelm@56066
  2004
    by (simp add: polypoly_def)
chaieb@33154
  2005
  from pnormal_last_length[OF lg lz] show "pnormal ?p" .
chaieb@33154
  2006
qed
chaieb@33154
  2007
chaieb@33154
  2008
lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
chaieb@33154
  2009
  unfolding isnonconstant_def
wenzelm@52658
  2010
  apply (cases p)
wenzelm@52658
  2011
  apply simp_all
wenzelm@52658
  2012
  apply (case_tac nat)
wenzelm@52658
  2013
  apply auto
chaieb@33154
  2014
  done
wenzelm@52658
  2015
wenzelm@52658
  2016
lemma isnonconstant_nonconstant:
wenzelm@52658
  2017
  assumes inc: "isnonconstant p"
chaieb@33154
  2018
  shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
chaieb@33154
  2019
proof
chaieb@33154
  2020
  let ?p = "polypoly bs p"
chaieb@33154
  2021
  assume nc: "nonconstant ?p"
chaieb@33154
  2022
  from isnonconstant_pnormal_iff[OF inc, of bs] nc
wenzelm@56066
  2023
  show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
wenzelm@56066
  2024
    unfolding nonconstant_def by blast
chaieb@33154
  2025
next
chaieb@33154
  2026
  let ?p = "polypoly bs p"
chaieb@33154
  2027
  assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  2028
  from isnonconstant_pnormal_iff[OF inc, of bs] h
wenzelm@56066
  2029
  have pn: "pnormal ?p"
wenzelm@56066
  2030
    by blast
wenzelm@56009
  2031
  {
wenzelm@56009
  2032
    fix x
wenzelm@56009
  2033
    assume H: "?p = [x]"
wenzelm@56009
  2034
    from H have "length (coefficients p) = 1"
wenzelm@56009
  2035
      unfolding polypoly_def by auto
wenzelm@56009
  2036
    with isnonconstant_coefficients_length[OF inc]
wenzelm@56009
  2037
      have False by arith
wenzelm@56009
  2038
  }
wenzelm@56009
  2039
  then show "nonconstant ?p"
wenzelm@56009
  2040
    using pn unfolding nonconstant_def by blast
chaieb@33154
  2041
qed
chaieb@33154
  2042
wenzelm@56066
  2043
lemma pnormal_length: "p \<noteq> [] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
wenzelm@52658
  2044
  apply (induct p)
wenzelm@52658
  2045
  apply (simp_all add: pnormal_def)
wenzelm@52658
  2046
  apply (case_tac "p = []")
wenzelm@52658
  2047
  apply simp_all
wenzelm@52658
  2048
  done
chaieb@33154
  2049
wenzelm@52658
  2050
lemma degree_degree:
wenzelm@52658
  2051
  assumes inc: "isnonconstant p"
chaieb@33154
  2052
  shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  2053
proof
wenzelm@52803
  2054
  let ?p = "polypoly bs p"
chaieb@33154
  2055
  assume H: "degree p = Polynomial_List.degree ?p"
chaieb@33154
  2056
  from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
chaieb@33154
  2057
    unfolding polypoly_def by auto
chaieb@33154
  2058
  from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
wenzelm@56066
  2059
  have lg: "length (pnormalize ?p) = length ?p"
chaieb@33154
  2060
    unfolding Polynomial_List.degree_def polypoly_def by simp
wenzelm@56066
  2061
  then have "pnormal ?p"
wenzelm@56066
  2062
    using pnormal_length[OF pz] by blast
wenzelm@56066
  2063
  with isnonconstant_pnormal_iff[OF inc] show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
wenzelm@56066
  2064
    by blast
chaieb@33154
  2065
next
wenzelm@56066
  2066
  let ?p = "polypoly bs p"
chaieb@33154
  2067
  assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
wenzelm@56066
  2068
  with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p"
wenzelm@56066
  2069
    by blast
chaieb@33154
  2070
  with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
wenzelm@52803
  2071
  show "degree p = Polynomial_List.degree ?p"
chaieb@33154
  2072
    unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
chaieb@33154
  2073
qed
chaieb@33154
  2074
wenzelm@52658
  2075
wenzelm@52803
  2076
section {* Swaps ; Division by a certain variable *}
wenzelm@52658
  2077
wenzelm@56066
  2078
primrec swap :: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly"
wenzelm@56066
  2079
where
chaieb@33154
  2080
  "swap n m (C x) = C x"
haftmann@39246
  2081
| "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"
haftmann@39246
  2082
| "swap n m (Neg t) = Neg (swap n m t)"
haftmann@39246
  2083
| "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
haftmann@39246
  2084
| "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
haftmann@39246
  2085
| "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
haftmann@39246
  2086
| "swap n m (Pw t k) = Pw (swap n m t) k"
wenzelm@56066
  2087
| "swap n m (CN c k p) = CN (swap n m c) (if k = n then m else if k=m then n else k) (swap n m p)"
chaieb@33154
  2088
wenzelm@52658
  2089
lemma swap:
wenzelm@56066
  2090
  assumes "n < length bs"
wenzelm@56066
  2091
    and "m < length bs"
chaieb@33154
  2092
  shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
chaieb@33154
  2093
proof (induct t)
wenzelm@52658
  2094
  case (Bound k)
wenzelm@56066
  2095
  then show ?case
wenzelm@56066
  2096
    using assms by simp
chaieb@33154
  2097
next
wenzelm@52658
  2098
  case (CN c k p)
wenzelm@56066
  2099
  then show ?case
wenzelm@56066
  2100
    using assms by simp
chaieb@33154
  2101
qed simp_all
chaieb@33154
  2102
wenzelm@52658
  2103
lemma swap_swap_id [simp]: "swap n m (swap m n t) = t"
wenzelm@52658
  2104
  by (induct t) simp_all
wenzelm@52658
  2105
wenzelm@52658
  2106
lemma swap_commute: "swap n m p = swap m n p"
wenzelm@52658
  2107
  by (induct p) simp_all
chaieb@33154
  2108
chaieb@33154
  2109
lemma swap_same_id[simp]: "swap n n t = t"
wenzelm@52658
  2110
  by (induct t) simp_all
chaieb@33154
  2111
chaieb@33154
  2112
definition "swapnorm n m t = polynate (swap n m t)"
chaieb@33154
  2113
wenzelm@52658
  2114
lemma swapnorm:
wenzelm@52658
  2115
  assumes nbs: "n < length bs"
wenzelm@52658
  2116
    and mbs: "m < length bs"
wenzelm@56000
  2117
  shows "((Ipoly bs (swapnorm n m t) :: 'a::{field_char_0,field_inverse_zero})) =
wenzelm@52658
  2118
    Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
wenzelm@41807
  2119
  using swap[OF assms] swapnorm_def by simp
chaieb@33154
  2120
wenzelm@52658
  2121
lemma swapnorm_isnpoly [simp]:
wenzelm@56000
  2122
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
  2123
  shows "isnpoly (swapnorm n m p)"
chaieb@33154
  2124
  unfolding swapnorm_def by simp
chaieb@33154
  2125
wenzelm@52803
  2126
definition "polydivideby n s p =
wenzelm@56000
  2127
  (let
wenzelm@56000
  2128
    ss = swapnorm 0 n s;
wenzelm@56000
  2129
    sp = swapnorm 0 n p;
wenzelm@56000
  2130
    h = head sp;
wenzelm@56000
  2131
    (k, r) = polydivide ss sp
wenzelm@56000
  2132
   in (k, swapnorm 0 n h, swapnorm 0 n r))"
chaieb@33154
  2133
wenzelm@56000
  2134
lemma swap_nz [simp]: "swap n m p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
  2135
  by (induct p) simp_all
chaieb@33154
  2136
krauss@41808
  2137
fun isweaknpoly :: "poly \<Rightarrow> bool"
krauss@41808
  2138
where
chaieb@33154
  2139
  "isweaknpoly (C c) = True"
krauss@41808
  2140
| "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
krauss@41808
  2141
| "isweaknpoly p = False"
chaieb@33154
  2142
wenzelm@52803
  2143
lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
wenzelm@52658
  2144
  by (induct p arbitrary: n0) auto
chaieb@33154
  2145
wenzelm@52803
  2146
lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
wenzelm@52658
  2147
  by (induct p) auto
chaieb@33154
  2148
chaieb@33154
  2149
end