src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
author wenzelm
Sat Mar 08 23:03:15 2014 +0100 (2014-03-08)
changeset 56000 899ad5a3ad00
parent 54489 03ff4d1e6784
child 56009 dda076a32aea
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>Nc')"
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| "polymul (C c) (CN c' n' p') =
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    (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul (C c) c') n' (polymul (C c) p'))"
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| "polymul (CN c n p) (C c') =
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    (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul c (C c')) n (polymul p (C c')))"
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| "polymul (CN c n p) (CN c' n' p') =
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    (if n < n' then CN (polymul c (CN c' n' p')) n (polymul p (CN c' n' p'))
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     else if n' < n 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')" by simp
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  from 2(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
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  with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
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  with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^sub>p c') (Suc n')" by simp
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  from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
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  thus ?case using 2 th3 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')" by simp
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  from 3(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
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  with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
krauss@41815
   292
  with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (Suc n')" by simp
wenzelm@52803
   293
  from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp
wenzelm@41807
   294
  thus ?case using 3 th3 by simp
chaieb@33154
   295
next
chaieb@33154
   296
  case (4 c n p c' n' p' n0 n1)
chaieb@33154
   297
  hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
wenzelm@52803
   298
  from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all
wenzelm@41807
   299
  from 4 have ngen0: "n \<ge> n0" by simp
wenzelm@52803
   300
  from 4 have n'gen1: "n' \<ge> n1" by simp
chaieb@33154
   301
  have "n < n' \<or> n' < n \<or> n = n'" by auto
wenzelm@52803
   302
  moreover {
wenzelm@52803
   303
    assume eq: "n = n'"
wenzelm@52803
   304
    with "4.hyps"(3)[OF nc nc']
chaieb@33154
   305
    have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
chaieb@33154
   306
    hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
chaieb@33154
   307
      using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
krauss@41812
   308
    from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
chaieb@33154
   309
    have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
wenzelm@52803
   310
    from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' have ?case by (simp add: Let_def) }
wenzelm@52803
   311
  moreover {
wenzelm@52803
   312
    assume lt: "n < n'"
chaieb@33154
   313
    have "min n0 n1 \<le> n0" by simp
wenzelm@52803
   314
    with 4 lt have th1:"min n0 n1 \<le> n" by auto
wenzelm@41807
   315
    from 4 have th21: "isnpolyh c (Suc n)" by simp
wenzelm@41807
   316
    from 4 have th22: "isnpolyh (CN c' n' p') n'" by simp
chaieb@33154
   317
    from lt have th23: "min (Suc n) n' = Suc n" by arith
krauss@41812
   318
    from "4.hyps"(1)[OF th21 th22]
krauss@41812
   319
    have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)" using th23 by simp
wenzelm@52803
   320
    with 4 lt th1 have ?case by simp }
wenzelm@52803
   321
  moreover {
wenzelm@52803
   322
    assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
chaieb@33154
   323
    have "min n0 n1 \<le> n1"  by simp
krauss@41815
   324
    with 4 gt have th1:"min n0 n1 \<le> n'" by auto
wenzelm@41807
   325
    from 4 have th21: "isnpolyh c' (Suc n')" by simp_all
wenzelm@41807
   326
    from 4 have th22: "isnpolyh (CN c n p) n" by simp
chaieb@33154
   327
    from gt have th23: "min n (Suc n') = Suc n'" by arith
krauss@41812
   328
    from "4.hyps"(2)[OF th22 th21]
krauss@41812
   329
    have "isnpolyh (polyadd (CN c n p) c') (Suc n')" using th23 by simp
wenzelm@52803
   330
    with 4 gt th1 have ?case by simp }
wenzelm@52803
   331
  ultimately show ?case by blast
chaieb@33154
   332
qed auto
chaieb@33154
   333
krauss@41812
   334
lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
wenzelm@52658
   335
  by (induct p q rule: polyadd.induct)
wenzelm@52658
   336
    (auto simp add: Let_def field_simps distrib_left[symmetric] simp del: distrib_left)
chaieb@33154
   337
krauss@41812
   338
lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd p q)"
chaieb@33154
   339
  using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
chaieb@33154
   340
krauss@41404
   341
text{* The degree of addition and other general lemmas needed for the normal form of polymul *}
chaieb@33154
   342
wenzelm@52803
   343
lemma polyadd_different_degreen:
wenzelm@52803
   344
  "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow>
krauss@41812
   345
  degreen (polyadd p q) m = max (degreen p m) (degreen q m)"
chaieb@33154
   346
proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
chaieb@33154
   347
  case (4 c n p c' n' p' m n0 n1)
krauss@41763
   348
  have "n' = n \<or> n < n' \<or> n' < n" by arith
krauss@41763
   349
  thus ?case
krauss@41763
   350
  proof (elim disjE)
krauss@41763
   351
    assume [simp]: "n' = n"
krauss@41812
   352
    from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
krauss@41763
   353
    show ?thesis by (auto simp: Let_def)
krauss@41763
   354
  next
krauss@41763
   355
    assume "n < n'"
krauss@41763
   356
    with 4 show ?thesis by auto
krauss@41763
   357
  next
krauss@41763
   358
    assume "n' < n"
krauss@41763
   359
    with 4 show ?thesis by auto
krauss@41763
   360
  qed
krauss@41763
   361
qed auto
chaieb@33154
   362
chaieb@33154
   363
lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
wenzelm@52658
   364
  by (induct p arbitrary: n rule: headn.induct) auto
chaieb@33154
   365
lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
wenzelm@52658
   366
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
   367
lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
wenzelm@52658
   368
  by (induct p arbitrary: n rule: degreen.induct) auto
chaieb@33154
   369
chaieb@33154
   370
lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
wenzelm@52658
   371
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
   372
chaieb@33154
   373
lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
chaieb@33154
   374
  using degree_isnpolyh_Suc by auto
chaieb@33154
   375
lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
chaieb@33154
   376
  using degreen_0 by auto
chaieb@33154
   377
chaieb@33154
   378
chaieb@33154
   379
lemma degreen_polyadd:
chaieb@33154
   380
  assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1"
chaieb@33154
   381
  shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
chaieb@33154
   382
  using np nq m
chaieb@33154
   383
proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
wenzelm@52803
   384
  case (2 c c' n' p' n0 n1)
wenzelm@52803
   385
  thus ?case  by (cases n') simp_all
chaieb@33154
   386
next
wenzelm@52803
   387
  case (3 c n p c' n0 n1)
wenzelm@52803
   388
  thus ?case by (cases n) auto
chaieb@33154
   389
next
wenzelm@52803
   390
  case (4 c n p c' n' p' n0 n1 m)
krauss@41763
   391
  have "n' = n \<or> n < n' \<or> n' < n" by arith
krauss@41763
   392
  thus ?case
krauss@41763
   393
  proof (elim disjE)
krauss@41763
   394
    assume [simp]: "n' = n"
krauss@41812
   395
    from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
krauss@41763
   396
    show ?thesis by (auto simp: Let_def)
krauss@41763
   397
  qed simp_all
chaieb@33154
   398
qed auto
chaieb@33154
   399
wenzelm@52803
   400
lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd p q = C c\<rbrakk>
chaieb@33154
   401
  \<Longrightarrow> degreen p m = degreen q m"
chaieb@33154
   402
proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
wenzelm@52803
   403
  case (4 c n p c' n' p' m n0 n1 x)
wenzelm@52803
   404
  { assume nn': "n' < n" hence ?case using 4 by simp }
wenzelm@52803
   405
  moreover
wenzelm@52803
   406
  { assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
wenzelm@52803
   407
    moreover { assume "n < n'" with 4 have ?case by simp }
wenzelm@52803
   408
    moreover { assume eq: "n = n'" hence ?case using 4
krauss@41763
   409
        apply (cases "p +\<^sub>p p' = 0\<^sub>p")
krauss@41763
   410
        apply (auto simp add: Let_def)
wenzelm@52658
   411
        apply blast
wenzelm@52658
   412
        done
wenzelm@52803
   413
    }
wenzelm@52803
   414
    ultimately have ?case by blast }
chaieb@33154
   415
  ultimately show ?case by blast
chaieb@33154
   416
qed simp_all
chaieb@33154
   417
chaieb@33154
   418
lemma polymul_properties:
wenzelm@56000
   419
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
   420
    and np: "isnpolyh p n0"
wenzelm@52658
   421
    and nq: "isnpolyh q n1"
wenzelm@52658
   422
    and m: "m \<le> min n0 n1"
wenzelm@52803
   423
  shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
wenzelm@52803
   424
    and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)"
wenzelm@52658
   425
    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
   426
  using np nq m
wenzelm@52658
   427
proof (induct p q arbitrary: n0 n1 m rule: polymul.induct)
wenzelm@52803
   428
  case (2 c c' n' p')
wenzelm@52803
   429
  { case (1 n0 n1)
krauss@41813
   430
    with "2.hyps"(4-6)[of n' n' n']
krauss@41813
   431
      and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']
krauss@41811
   432
    show ?case by (auto simp add: min_def)
chaieb@33154
   433
  next
wenzelm@52803
   434
    case (2 n0 n1) thus ?case by auto
chaieb@33154
   435
  next
wenzelm@52803
   436
    case (3 n0 n1) thus ?case  using "2.hyps" by auto }
chaieb@33154
   437
next
krauss@41813
   438
  case (3 c n p c')
wenzelm@52803
   439
  { case (1 n0 n1)
krauss@41813
   440
    with "3.hyps"(4-6)[of n n n]
krauss@41813
   441
      "3.hyps"(1-3)[of "Suc n" "Suc n" n]
krauss@41811
   442
    show ?case by (auto simp add: min_def)
chaieb@33154
   443
  next
krauss@41811
   444
    case (2 n0 n1) thus ?case by auto
chaieb@33154
   445
  next
wenzelm@52803
   446
    case (3 n0 n1) thus ?case  using "3.hyps" by auto }
chaieb@33154
   447
next
chaieb@33154
   448
  case (4 c n p c' n' p')
chaieb@33154
   449
  let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
krauss@41811
   450
    {
krauss@41811
   451
      case (1 n0 n1)
chaieb@33154
   452
      hence cnp: "isnpolyh ?cnp n" and cnp': "isnpolyh ?cnp' n'"
wenzelm@52803
   453
        and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)"
wenzelm@33268
   454
        and np': "isnpolyh p' n'" and nc': "isnpolyh c' (Suc n')"
wenzelm@33268
   455
        and nn0: "n \<ge> n0" and nn1:"n' \<ge> n1"
wenzelm@33268
   456
        by simp_all
krauss@41811
   457
      { assume "n < n'"
krauss@41813
   458
        with "4.hyps"(4-5)[OF np cnp', of n]
krauss@41813
   459
          "4.hyps"(1)[OF nc cnp', of n] nn0 cnp
krauss@41811
   460
        have ?case by (simp add: min_def)
krauss@41811
   461
      } moreover {
krauss@41811
   462
        assume "n' < n"
krauss@41813
   463
        with "4.hyps"(16-17)[OF cnp np', of "n'"]
krauss@41813
   464
          "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
krauss@41811
   465
        have ?case
wenzelm@52658
   466
          by (cases "Suc n' = n") (simp_all add: min_def)
krauss@41811
   467
      } moreover {
krauss@41811
   468
        assume "n' = n"
krauss@41813
   469
        with "4.hyps"(16-17)[OF cnp np', of n]
krauss@41813
   470
          "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0
krauss@41811
   471
        have ?case
krauss@41811
   472
          apply (auto intro!: polyadd_normh)
krauss@41811
   473
          apply (simp_all add: min_def isnpolyh_mono[OF nn0])
krauss@41811
   474
          done
krauss@41811
   475
      }
krauss@41811
   476
      ultimately show ?case by arith
krauss@41811
   477
    next
krauss@41811
   478
      fix n0 n1 m
chaieb@33154
   479
      assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1"
chaieb@33154
   480
      and m: "m \<le> min n0 n1"
chaieb@33154
   481
      let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
chaieb@33154
   482
      let ?d1 = "degreen ?cnp m"
chaieb@33154
   483
      let ?d2 = "degreen ?cnp' m"
chaieb@33154
   484
      let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
chaieb@33154
   485
      have "n'<n \<or> n < n' \<or> n' = n" by auto
wenzelm@52803
   486
      moreover
chaieb@33154
   487
      {assume "n' < n \<or> n < n'"
wenzelm@52803
   488
        with "4.hyps"(3,6,18) np np' m
krauss@41811
   489
        have ?eq by auto }
chaieb@33154
   490
      moreover
wenzelm@52803
   491
      { assume nn': "n' = n"
wenzelm@52803
   492
        hence nn: "\<not> n' < n \<and> \<not> n < n'" by arith
krauss@41813
   493
        from "4.hyps"(16,18)[of n n' n]
krauss@41813
   494
          "4.hyps"(13,14)[of n "Suc n'" n]
wenzelm@33268
   495
          np np' nn'
wenzelm@33268
   496
        have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
wenzelm@33268
   497
          "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@52803
   498
          "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
wenzelm@33268
   499
          "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
wenzelm@52803
   500
        { assume mn: "m = n"
krauss@41813
   501
          from "4.hyps"(17,18)[OF norm(1,4), of n]
krauss@41813
   502
            "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
wenzelm@52803
   503
          have degs:  "degreen (?cnp *\<^sub>p c') n =
wenzelm@33268
   504
            (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"
wenzelm@33268
   505
            "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n" by (simp_all add: min_def)
wenzelm@33268
   506
          from degs norm
wenzelm@33268
   507
          have th1: "degreen(?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" by simp
wenzelm@33268
   508
          hence neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
wenzelm@33268
   509
            by simp
wenzelm@33268
   510
          have nmin: "n \<le> min n n" by (simp add: min_def)
wenzelm@33268
   511
          from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
wenzelm@52803
   512
          have deg: "degreen (CN c n p *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n = degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
krauss@41813
   513
          from "4.hyps"(16-18)[OF norm(1,4), of n]
krauss@41813
   514
            "4.hyps"(13-15)[OF norm(1,2), of n]
wenzelm@33268
   515
            mn norm m nn' deg
wenzelm@52803
   516
          have ?eq by simp }
wenzelm@33268
   517
        moreover
wenzelm@52803
   518
        { assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
wenzelm@52803
   519
          from nn' m np have max1: "m \<le> max n n"  by simp
wenzelm@52803
   520
          hence min1: "m \<le> min n n" by simp
wenzelm@33268
   521
          hence min2: "m \<le> min n (Suc n)" by simp
krauss@41813
   522
          from "4.hyps"(16-18)[OF norm(1,4) min1]
krauss@41813
   523
            "4.hyps"(13-15)[OF norm(1,2) min2]
krauss@41811
   524
            degreen_polyadd[OF norm(3,6) max1]
chaieb@33154
   525
wenzelm@52803
   526
          have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m
krauss@41811
   527
            \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
krauss@41811
   528
            using mn nn' np np' by simp
krauss@41813
   529
          with "4.hyps"(16-18)[OF norm(1,4) min1]
krauss@41813
   530
            "4.hyps"(13-15)[OF norm(1,2) min2]
krauss@41811
   531
            degreen_0[OF norm(3) mn']
wenzelm@52803
   532
          have ?eq using nn' mn np np' by clarsimp }
wenzelm@52803
   533
        ultimately have ?eq by blast }
wenzelm@52803
   534
      ultimately show ?eq by blast }
chaieb@33154
   535
    { case (2 n0 n1)
wenzelm@52803
   536
      hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1"
wenzelm@33268
   537
        and m: "m \<le> min n0 n1" by simp_all
chaieb@33154
   538
      hence mn: "m \<le> n" by simp
chaieb@33154
   539
      let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
chaieb@33154
   540
      {assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
wenzelm@33268
   541
        hence nn: "\<not>n' < n \<and> \<not> n<n'" by simp
krauss@41813
   542
        from "4.hyps"(16-18) [of n n n]
krauss@41813
   543
          "4.hyps"(13-15)[of n "Suc n" n]
wenzelm@33268
   544
          np np' C(2) mn
wenzelm@33268
   545
        have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
wenzelm@33268
   546
          "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@52803
   547
          "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)"
wenzelm@52803
   548
          "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
wenzelm@33268
   549
          "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
wenzelm@33268
   550
            "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
wenzelm@33268
   551
          by (simp_all add: min_def)
wenzelm@52803
   552
wenzelm@33268
   553
          from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
wenzelm@52803
   554
          have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
wenzelm@33268
   555
            using norm by simp
wenzelm@33268
   556
        from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"]  degneq
wenzelm@33268
   557
        have "False" by simp }
wenzelm@52803
   558
      thus ?case using "4.hyps" by clarsimp }
chaieb@33154
   559
qed auto
chaieb@33154
   560
chaieb@33154
   561
lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"
wenzelm@52658
   562
  by (induct p q rule: polymul.induct) (auto simp add: field_simps)
chaieb@33154
   563
wenzelm@52803
   564
lemma polymul_normh:
wenzelm@56000
   565
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
   566
  shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
wenzelm@52803
   567
  using polymul_properties(1) by blast
wenzelm@52658
   568
wenzelm@52803
   569
lemma polymul_eq0_iff:
wenzelm@56000
   570
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
   571
  shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p) "
wenzelm@52803
   572
  using polymul_properties(2) by blast
wenzelm@52658
   573
wenzelm@52658
   574
lemma polymul_degreen:  (* FIXME duplicate? *)
wenzelm@56000
   575
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
   576
  shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow>
wenzelm@52658
   577
    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
   578
  using polymul_properties(3) by blast
wenzelm@52658
   579
wenzelm@52803
   580
lemma polymul_norm:
wenzelm@56000
   581
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
krauss@41813
   582
  shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul p q)"
chaieb@33154
   583
  using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
chaieb@33154
   584
chaieb@33154
   585
lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
   586
  by (induct p arbitrary: n0 rule: headconst.induct) auto
chaieb@33154
   587
chaieb@33154
   588
lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
wenzelm@52658
   589
  by (induct p arbitrary: n0) auto
chaieb@33154
   590
wenzelm@52658
   591
lemma monic_eqI:
wenzelm@52803
   592
  assumes np: "isnpolyh p n0"
wenzelm@52658
   593
  shows "INum (headconst p) * Ipoly bs (fst (monic p)) =
wenzelm@56000
   594
    (Ipoly bs p ::'a::{field_char_0,field_inverse_zero, power})"
chaieb@33154
   595
  unfolding monic_def Let_def
wenzelm@52658
   596
proof (cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
chaieb@33154
   597
  let ?h = "headconst p"
chaieb@33154
   598
  assume pz: "p \<noteq> 0\<^sub>p"
wenzelm@56000
   599
  {
wenzelm@56000
   600
    assume hz: "INum ?h = (0::'a)"
chaieb@33154
   601
    from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
chaieb@33154
   602
    from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
chaieb@33154
   603
    with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
chaieb@33154
   604
  thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast
chaieb@33154
   605
qed
chaieb@33154
   606
chaieb@33154
   607
krauss@41404
   608
text{* polyneg is a negation and preserves normal forms *}
chaieb@33154
   609
chaieb@33154
   610
lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
wenzelm@52658
   611
  by (induct p rule: polyneg.induct) auto
chaieb@33154
   612
chaieb@33154
   613
lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"
wenzelm@52658
   614
  by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: Nneg_def)
chaieb@33154
   615
lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
wenzelm@52658
   616
  by (induct p arbitrary: n0 rule: polyneg.induct) auto
chaieb@33154
   617
lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "
wenzelm@52658
   618
  by (induct p rule: polyneg.induct) (auto simp add: polyneg0)
chaieb@33154
   619
chaieb@33154
   620
lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
chaieb@33154
   621
  using isnpoly_def polyneg_normh by simp
chaieb@33154
   622
chaieb@33154
   623
krauss@41404
   624
text{* polysub is a substraction and preserves normal forms *}
krauss@41404
   625
krauss@41814
   626
lemma polysub[simp]: "Ipoly bs (polysub p q) = (Ipoly bs p) - (Ipoly bs q)"
wenzelm@52658
   627
  by (simp add: polysub_def)
wenzelm@52803
   628
lemma polysub_normh:
wenzelm@52803
   629
  "\<And>n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
wenzelm@52658
   630
  by (simp add: polysub_def polyneg_normh polyadd_normh)
chaieb@33154
   631
krauss@41814
   632
lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub p q)"
wenzelm@52803
   633
  using polyadd_norm polyneg_norm by (simp add: polysub_def)
wenzelm@52658
   634
lemma polysub_same_0[simp]:
wenzelm@56000
   635
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
krauss@41814
   636
  shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p"
wenzelm@52658
   637
  unfolding polysub_def split_def fst_conv snd_conv
wenzelm@52658
   638
  by (induct p arbitrary: n0) (auto simp add: Let_def Nsub0[simplified Nsub_def])
chaieb@33154
   639
wenzelm@52803
   640
lemma polysub_0:
wenzelm@56000
   641
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
   642
  shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"
chaieb@33154
   643
  unfolding polysub_def split_def fst_conv snd_conv
krauss@41763
   644
  by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
wenzelm@52658
   645
    (auto simp: Nsub0[simplified Nsub_def] Let_def)
chaieb@33154
   646
chaieb@33154
   647
text{* polypow is a power function and preserves normal forms *}
krauss@41404
   648
wenzelm@56000
   649
lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{field_char_0,field_inverse_zero})) ^ n"
wenzelm@52658
   650
proof (induct n rule: polypow.induct)
wenzelm@52658
   651
  case 1
wenzelm@52658
   652
  thus ?case by simp
chaieb@33154
   653
next
chaieb@33154
   654
  case (2 n)
chaieb@33154
   655
  let ?q = "polypow ((Suc n) div 2) p"
krauss@41813
   656
  let ?d = "polymul ?q ?q"
chaieb@33154
   657
  have "odd (Suc n) \<or> even (Suc n)" by simp
wenzelm@52803
   658
  moreover
wenzelm@52803
   659
  { assume odd: "odd (Suc n)"
wenzelm@56000
   660
    have th: "(Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0)))) = Suc n div 2 + Suc n div 2 + 1"
wenzelm@52658
   661
      by arith
krauss@41813
   662
    from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)" by (simp add: Let_def)
chaieb@33154
   663
    also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"
chaieb@33154
   664
      using "2.hyps" by simp
chaieb@33154
   665
    also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
wenzelm@52658
   666
      by (simp only: power_add power_one_right) simp
wenzelm@56000
   667
    also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0))))"
chaieb@33154
   668
      by (simp only: th)
wenzelm@52803
   669
    finally have ?case
chaieb@33154
   670
    using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
wenzelm@52803
   671
  moreover
wenzelm@52803
   672
  { assume even: "even (Suc n)"
wenzelm@56000
   673
    have th: "(Suc (Suc 0)) * (Suc n div Suc (Suc 0)) = Suc n div 2 + Suc n div 2"
wenzelm@52658
   674
      by arith
chaieb@33154
   675
    from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
chaieb@33154
   676
    also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
chaieb@33154
   677
      using "2.hyps" apply (simp only: power_add) by simp
chaieb@33154
   678
    finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}
chaieb@33154
   679
  ultimately show ?case by blast
chaieb@33154
   680
qed
chaieb@33154
   681
wenzelm@52803
   682
lemma polypow_normh:
wenzelm@56000
   683
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
   684
  shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
chaieb@33154
   685
proof (induct k arbitrary: n rule: polypow.induct)
chaieb@33154
   686
  case (2 k n)
chaieb@33154
   687
  let ?q = "polypow (Suc k div 2) p"
krauss@41813
   688
  let ?d = "polymul ?q ?q"
wenzelm@41807
   689
  from 2 have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
chaieb@33154
   690
  from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
krauss@41813
   691
  from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul p ?d) n" by simp
chaieb@33154
   692
  from dn on show ?case by (simp add: Let_def)
wenzelm@52803
   693
qed auto
chaieb@33154
   694
wenzelm@52803
   695
lemma polypow_norm:
wenzelm@56000
   696
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
   697
  shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
chaieb@33154
   698
  by (simp add: polypow_normh isnpoly_def)
chaieb@33154
   699
krauss@41404
   700
text{* Finally the whole normalization *}
chaieb@33154
   701
wenzelm@52658
   702
lemma polynate [simp]:
wenzelm@56000
   703
  "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0,field_inverse_zero})"
wenzelm@52658
   704
  by (induct p rule:polynate.induct) auto
chaieb@33154
   705
wenzelm@52803
   706
lemma polynate_norm[simp]:
wenzelm@56000
   707
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
   708
  shows "isnpoly (polynate p)"
wenzelm@52658
   709
  by (induct p rule: polynate.induct)
wenzelm@52658
   710
     (simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm,
wenzelm@52658
   711
      simp_all add: isnpoly_def)
chaieb@33154
   712
chaieb@33154
   713
text{* shift1 *}
chaieb@33154
   714
chaieb@33154
   715
chaieb@33154
   716
lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
wenzelm@52658
   717
  by (simp add: shift1_def)
chaieb@33154
   718
wenzelm@52803
   719
lemma shift1_isnpoly:
wenzelm@52658
   720
  assumes pn: "isnpoly p"
wenzelm@52658
   721
    and pnz: "p \<noteq> 0\<^sub>p"
wenzelm@52658
   722
  shows "isnpoly (shift1 p) "
wenzelm@52658
   723
  using pn pnz by (simp add: shift1_def isnpoly_def)
chaieb@33154
   724
chaieb@33154
   725
lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
chaieb@33154
   726
  by (simp add: shift1_def)
wenzelm@52803
   727
lemma funpow_shift1_isnpoly:
chaieb@33154
   728
  "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
haftmann@39246
   729
  by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
chaieb@33154
   730
wenzelm@52803
   731
lemma funpow_isnpolyh:
wenzelm@52658
   732
  assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n"
wenzelm@52658
   733
    and np: "isnpolyh p n"
chaieb@33154
   734
  shows "isnpolyh (funpow k f p) n"
wenzelm@52658
   735
  using f np by (induct k arbitrary: p) auto
chaieb@33154
   736
wenzelm@52658
   737
lemma funpow_shift1:
wenzelm@56000
   738
  "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field_inverse_zero}) =
wenzelm@52658
   739
    Ipoly bs (Mul (Pw (Bound 0) n) p)"
wenzelm@52658
   740
  by (induct n arbitrary: p) (simp_all add: shift1_isnpoly shift1)
chaieb@33154
   741
chaieb@33154
   742
lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
chaieb@33154
   743
  using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
chaieb@33154
   744
wenzelm@52803
   745
lemma funpow_shift1_1:
wenzelm@56000
   746
  "(Ipoly bs (funpow n shift1 p) :: 'a :: {field_char_0,field_inverse_zero}) =
wenzelm@52658
   747
    Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)"
chaieb@33154
   748
  by (simp add: funpow_shift1)
chaieb@33154
   749
chaieb@33154
   750
lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
wenzelm@45129
   751
  by (induct p rule: poly_cmul.induct) (auto simp add: field_simps)
chaieb@33154
   752
chaieb@33154
   753
lemma behead:
chaieb@33154
   754
  assumes np: "isnpolyh p n"
wenzelm@52658
   755
  shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) =
wenzelm@56000
   756
    (Ipoly bs p :: 'a :: {field_char_0,field_inverse_zero})"
chaieb@33154
   757
  using np
chaieb@33154
   758
proof (induct p arbitrary: n rule: behead.induct)
chaieb@33154
   759
  case (1 c p n) hence pn: "isnpolyh p n" by simp
wenzelm@52803
   760
  from 1(1)[OF pn]
wenzelm@52803
   761
  have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
wenzelm@52658
   762
  then show ?case using "1.hyps"
wenzelm@52658
   763
    apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
wenzelm@52658
   764
    apply (simp_all add: th[symmetric] field_simps)
wenzelm@52658
   765
    done
chaieb@33154
   766
qed (auto simp add: Let_def)
chaieb@33154
   767
chaieb@33154
   768
lemma behead_isnpolyh:
wenzelm@52658
   769
  assumes np: "isnpolyh p n"
wenzelm@52658
   770
  shows "isnpolyh (behead p) n"
wenzelm@52658
   771
  using np by (induct p rule: behead.induct) (auto simp add: Let_def isnpolyh_mono)
wenzelm@52658
   772
chaieb@33154
   773
krauss@41404
   774
subsection{* Miscellaneous lemmas about indexes, decrementation, substitution  etc ... *}
wenzelm@52658
   775
chaieb@33154
   776
lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
wenzelm@52658
   777
proof (induct p arbitrary: n rule: poly.induct, auto)
chaieb@33154
   778
  case (goal1 c n p n')
chaieb@33154
   779
  hence "n = Suc (n - 1)" by simp
chaieb@33154
   780
  hence "isnpolyh p (Suc (n - 1))"  using `isnpolyh p n` by simp
wenzelm@41807
   781
  with goal1(2) show ?case by simp
chaieb@33154
   782
qed
chaieb@33154
   783
chaieb@33154
   784
lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
wenzelm@52658
   785
  by (induct p arbitrary: n0 rule: isconstant.induct) (auto simp add: isnpolyh_polybound0)
chaieb@33154
   786
wenzelm@52658
   787
lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
   788
  by (induct p) auto
chaieb@33154
   789
chaieb@33154
   790
lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
wenzelm@52658
   791
  apply (induct p arbitrary: n0)
wenzelm@52658
   792
  apply auto
chaieb@33154
   793
  apply (atomize)
chaieb@33154
   794
  apply (erule_tac x = "Suc nat" in allE)
chaieb@33154
   795
  apply auto
chaieb@33154
   796
  done
chaieb@33154
   797
chaieb@33154
   798
lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
wenzelm@52658
   799
  by (induct p  arbitrary: n0 rule: head.induct) (auto intro: isnpolyh_polybound0)
chaieb@33154
   800
chaieb@33154
   801
lemma polybound0_I:
chaieb@33154
   802
  assumes nb: "polybound0 a"
chaieb@33154
   803
  shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
wenzelm@52658
   804
  using nb
wenzelm@52803
   805
  by (induct a rule: poly.induct) auto
wenzelm@52658
   806
wenzelm@52658
   807
lemma polysubst0_I: "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
chaieb@33154
   808
  by (induct t) simp_all
chaieb@33154
   809
chaieb@33154
   810
lemma polysubst0_I':
chaieb@33154
   811
  assumes nb: "polybound0 a"
chaieb@33154
   812
  shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t"
chaieb@33154
   813
  by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
chaieb@33154
   814
wenzelm@52658
   815
lemma decrpoly:
wenzelm@52658
   816
  assumes nb: "polybound0 t"
chaieb@33154
   817
  shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
wenzelm@52658
   818
  using nb by (induct t rule: decrpoly.induct) simp_all
chaieb@33154
   819
wenzelm@52658
   820
lemma polysubst0_polybound0:
wenzelm@52658
   821
  assumes nb: "polybound0 t"
chaieb@33154
   822
  shows "polybound0 (polysubst0 t a)"
wenzelm@52658
   823
  using nb by (induct a rule: poly.induct) auto
chaieb@33154
   824
chaieb@33154
   825
lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
wenzelm@52658
   826
  by (induct p arbitrary: n rule: degree.induct) (auto simp add: isnpolyh_polybound0)
chaieb@33154
   827
haftmann@39246
   828
primrec maxindex :: "poly \<Rightarrow> nat" where
chaieb@33154
   829
  "maxindex (Bound n) = n + 1"
chaieb@33154
   830
| "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
chaieb@33154
   831
| "maxindex (Add p q) = max (maxindex p) (maxindex q)"
chaieb@33154
   832
| "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
chaieb@33154
   833
| "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
chaieb@33154
   834
| "maxindex (Neg p) = maxindex p"
chaieb@33154
   835
| "maxindex (Pw p n) = maxindex p"
chaieb@33154
   836
| "maxindex (C x) = 0"
chaieb@33154
   837
wenzelm@52658
   838
definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool"
wenzelm@56000
   839
  where "wf_bs bs p \<longleftrightarrow> length bs \<ge> maxindex p"
chaieb@33154
   840
chaieb@33154
   841
lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
wenzelm@52658
   842
proof (induct p rule: coefficients.induct)
wenzelm@52803
   843
  case (1 c p)
wenzelm@52803
   844
  show ?case
chaieb@33154
   845
  proof
chaieb@33154
   846
    fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"
chaieb@33154
   847
    hence "x = c \<or> x \<in> set (coefficients p)" by simp
wenzelm@52803
   848
    moreover
chaieb@33154
   849
    {assume "x = c" hence "wf_bs bs x" using "1.prems"  unfolding wf_bs_def by simp}
wenzelm@52803
   850
    moreover
wenzelm@52803
   851
    {assume H: "x \<in> set (coefficients p)"
chaieb@33154
   852
      from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp
chaieb@33154
   853
      with "1.hyps" H have "wf_bs bs x" by blast }
chaieb@33154
   854
    ultimately  show "wf_bs bs x" by blast
chaieb@33154
   855
  qed
chaieb@33154
   856
qed simp_all
chaieb@33154
   857
chaieb@33154
   858
lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
wenzelm@52658
   859
  by (induct p rule: coefficients.induct) auto
chaieb@33154
   860
wenzelm@56000
   861
lemma wf_bs_I: "wf_bs bs p \<Longrightarrow> Ipoly (bs @ bs') p = Ipoly bs p"
wenzelm@52658
   862
  unfolding wf_bs_def by (induct p) (auto simp add: nth_append)
chaieb@33154
   863
wenzelm@52658
   864
lemma take_maxindex_wf:
wenzelm@52803
   865
  assumes wf: "wf_bs bs p"
chaieb@33154
   866
  shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
chaieb@33154
   867
proof-
chaieb@33154
   868
  let ?ip = "maxindex p"
chaieb@33154
   869
  let ?tbs = "take ?ip bs"
chaieb@33154
   870
  from wf have "length ?tbs = ?ip" unfolding wf_bs_def by simp
chaieb@33154
   871
  hence wf': "wf_bs ?tbs p" unfolding wf_bs_def by  simp
chaieb@33154
   872
  have eq: "bs = ?tbs @ (drop ?ip bs)" by simp
chaieb@33154
   873
  from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis using eq by simp
chaieb@33154
   874
qed
chaieb@33154
   875
chaieb@33154
   876
lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
wenzelm@52658
   877
  by (induct p) auto
chaieb@33154
   878
chaieb@33154
   879
lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
chaieb@33154
   880
  unfolding wf_bs_def by simp
chaieb@33154
   881
chaieb@33154
   882
lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
chaieb@33154
   883
  unfolding wf_bs_def by simp
chaieb@33154
   884
chaieb@33154
   885
chaieb@33154
   886
chaieb@33154
   887
lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
wenzelm@52658
   888
  by (induct p rule: coefficients.induct) (auto simp add: wf_bs_def)
chaieb@33154
   889
lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
wenzelm@52658
   890
  by (induct p rule: coefficients.induct) simp_all
chaieb@33154
   891
chaieb@33154
   892
chaieb@33154
   893
lemma coefficients_head: "last (coefficients p) = head p"
wenzelm@52658
   894
  by (induct p rule: coefficients.induct) auto
chaieb@33154
   895
chaieb@33154
   896
lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
wenzelm@52658
   897
  unfolding wf_bs_def by (induct p rule: decrpoly.induct) auto
chaieb@33154
   898
chaieb@33154
   899
lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
chaieb@33154
   900
  apply (rule exI[where x="replicate (n - length xs) z"])
wenzelm@52658
   901
  apply simp
wenzelm@52658
   902
  done
wenzelm@52658
   903
chaieb@33154
   904
lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
wenzelm@52658
   905
  apply (cases p)
wenzelm@52658
   906
  apply auto
wenzelm@52658
   907
  apply (case_tac "nat")
wenzelm@52658
   908
  apply simp_all
wenzelm@52658
   909
  done
chaieb@33154
   910
chaieb@33154
   911
lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
wenzelm@52803
   912
  unfolding wf_bs_def
chaieb@33154
   913
  apply (induct p q rule: polyadd.induct)
chaieb@33154
   914
  apply (auto simp add: Let_def)
chaieb@33154
   915
  done
chaieb@33154
   916
chaieb@33154
   917
lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
wenzelm@52803
   918
  unfolding wf_bs_def
wenzelm@52803
   919
  apply (induct p q arbitrary: bs rule: polymul.induct)
chaieb@33154
   920
  apply (simp_all add: wf_bs_polyadd)
chaieb@33154
   921
  apply clarsimp
chaieb@33154
   922
  apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
chaieb@33154
   923
  apply auto
chaieb@33154
   924
  done
chaieb@33154
   925
chaieb@33154
   926
lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
wenzelm@52658
   927
  unfolding wf_bs_def by (induct p rule: polyneg.induct) auto
chaieb@33154
   928
chaieb@33154
   929
lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
chaieb@33154
   930
  unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
chaieb@33154
   931
wenzelm@52658
   932
chaieb@33154
   933
subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}
chaieb@33154
   934
chaieb@33154
   935
definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
chaieb@33154
   936
definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)"
chaieb@33154
   937
definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"
chaieb@33154
   938
chaieb@33154
   939
lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0"
chaieb@33154
   940
proof (induct p arbitrary: n0 rule: coefficients.induct)
chaieb@33154
   941
  case (1 c p n0)
chaieb@33154
   942
  have cp: "isnpolyh (CN c 0 p) n0" by fact
chaieb@33154
   943
  hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
chaieb@33154
   944
    by (auto simp add: isnpolyh_mono[where n'=0])
wenzelm@52803
   945
  from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp
chaieb@33154
   946
qed auto
chaieb@33154
   947
chaieb@33154
   948
lemma coefficients_isconst:
chaieb@33154
   949
  "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
wenzelm@52803
   950
  by (induct p arbitrary: n rule: coefficients.induct)
wenzelm@52658
   951
    (auto simp add: isnpolyh_Suc_const)
chaieb@33154
   952
chaieb@33154
   953
lemma polypoly_polypoly':
chaieb@33154
   954
  assumes np: "isnpolyh p n0"
chaieb@33154
   955
  shows "polypoly (x#bs) p = polypoly' bs p"
chaieb@33154
   956
proof-
chaieb@33154
   957
  let ?cf = "set (coefficients p)"
chaieb@33154
   958
  from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
chaieb@33154
   959
  {fix q assume q: "q \<in> ?cf"
chaieb@33154
   960
    from q cn_norm have th: "isnpolyh q n0" by blast
chaieb@33154
   961
    from coefficients_isconst[OF np] q have "isconstant q" by blast
chaieb@33154
   962
    with isconstant_polybound0[OF th] have "polybound0 q" by blast}
chaieb@33154
   963
  hence "\<forall>q \<in> ?cf. polybound0 q" ..
chaieb@33154
   964
  hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"
chaieb@33154
   965
    using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
chaieb@33154
   966
    by auto
wenzelm@52803
   967
wenzelm@52803
   968
  thus ?thesis unfolding polypoly_def polypoly'_def by simp
chaieb@33154
   969
qed
chaieb@33154
   970
chaieb@33154
   971
lemma polypoly_poly:
wenzelm@52658
   972
  assumes np: "isnpolyh p n0"
wenzelm@52658
   973
  shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
wenzelm@52803
   974
  using np
wenzelm@52658
   975
  by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def)
chaieb@33154
   976
wenzelm@52803
   977
lemma polypoly'_poly:
wenzelm@52658
   978
  assumes np: "isnpolyh p n0"
wenzelm@52658
   979
  shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
chaieb@33154
   980
  using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
chaieb@33154
   981
chaieb@33154
   982
chaieb@33154
   983
lemma polypoly_poly_polybound0:
chaieb@33154
   984
  assumes np: "isnpolyh p n0" and nb: "polybound0 p"
chaieb@33154
   985
  shows "polypoly bs p = [Ipoly bs p]"
wenzelm@52803
   986
  using np nb unfolding polypoly_def
wenzelm@52658
   987
  apply (cases p)
wenzelm@52658
   988
  apply auto
wenzelm@52658
   989
  apply (case_tac nat)
wenzelm@52658
   990
  apply auto
wenzelm@52658
   991
  done
chaieb@33154
   992
wenzelm@52803
   993
lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
wenzelm@52658
   994
  by (induct p rule: head.induct) auto
chaieb@33154
   995
chaieb@33154
   996
lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
wenzelm@52658
   997
  by (cases p) auto
chaieb@33154
   998
chaieb@33154
   999
lemma head_eq_headn0: "head p = headn p 0"
wenzelm@52658
  1000
  by (induct p rule: head.induct) simp_all
chaieb@33154
  1001
chaieb@33154
  1002
lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
chaieb@33154
  1003
  by (simp add: head_eq_headn0)
chaieb@33154
  1004
wenzelm@52803
  1005
lemma isnpolyh_zero_iff:
wenzelm@52658
  1006
  assumes nq: "isnpolyh p n0"
wenzelm@56000
  1007
    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
  1008
  shows "p = 0\<^sub>p"
wenzelm@52658
  1009
  using nq eq
berghofe@34915
  1010
proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
berghofe@34915
  1011
  case less
berghofe@34915
  1012
  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
  1013
  {
wenzelm@56000
  1014
    assume nz: "maxindex p = 0"
wenzelm@56000
  1015
    then obtain c where "p = C c"
wenzelm@56000
  1016
      using np by (cases p) auto
wenzelm@56000
  1017
    with zp np have "p = 0\<^sub>p"
wenzelm@56000
  1018
      unfolding wf_bs_def by simp
wenzelm@56000
  1019
  }
chaieb@33154
  1020
  moreover
wenzelm@56000
  1021
  {
wenzelm@56000
  1022
    assume nz: "maxindex p \<noteq> 0"
chaieb@33154
  1023
    let ?h = "head p"
chaieb@33154
  1024
    let ?hd = "decrpoly ?h"
chaieb@33154
  1025
    let ?ihd = "maxindex ?hd"
wenzelm@56000
  1026
    from head_isnpolyh[OF np] head_polybound0[OF np]
wenzelm@56000
  1027
    have h: "isnpolyh ?h n0" "polybound0 ?h"
chaieb@33154
  1028
      by simp_all
wenzelm@56000
  1029
    then have nhd: "isnpolyh ?hd (n0 - 1)"
wenzelm@56000
  1030
      using decrpoly_normh by blast
wenzelm@52803
  1031
chaieb@33154
  1032
    from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
wenzelm@56000
  1033
    have mihn: "maxindex ?h \<le> maxindex p"
wenzelm@56000
  1034
      by auto
wenzelm@56000
  1035
    with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < maxindex p"
wenzelm@56000
  1036
      by auto
wenzelm@56000
  1037
    {
wenzelm@56000
  1038
      fix bs :: "'a list"
wenzelm@56000
  1039
      assume bs: "wf_bs bs ?hd"
chaieb@33154
  1040
      let ?ts = "take ?ihd bs"
chaieb@33154
  1041
      let ?rs = "drop ?ihd bs"
wenzelm@56000
  1042
      have ts: "wf_bs ?ts ?hd"
wenzelm@56000
  1043
        using bs unfolding wf_bs_def by simp
wenzelm@56000
  1044
      have bs_ts_eq: "?ts @ ?rs = bs"
wenzelm@56000
  1045
        by simp
wenzelm@56000
  1046
      from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x # ?ts) ?h"
wenzelm@56000
  1047
        by simp
wenzelm@56000
  1048
      from ihd_lt_n have "\<forall>x. length (x # ?ts) \<le> maxindex p"
wenzelm@56000
  1049
        by simp
wenzelm@56000
  1050
      with length_le_list_ex obtain xs where xs: "length ((x # ?ts) @ xs) = maxindex p"
wenzelm@56000
  1051
        by blast
wenzelm@56000
  1052
      then have "\<forall>x. wf_bs ((x # ?ts) @ xs) p"
wenzelm@56000
  1053
        unfolding wf_bs_def by simp
wenzelm@56000
  1054
      with zp have "\<forall>x. Ipoly ((x # ?ts) @ xs) p = 0"
wenzelm@56000
  1055
        by blast
wenzelm@56000
  1056
      then have "\<forall>x. Ipoly (x # (?ts @ xs)) p = 0"
wenzelm@56000
  1057
        by simp
chaieb@33154
  1058
      with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
wenzelm@56000
  1059
      have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"
wenzelm@56000
  1060
        by simp
wenzelm@56000
  1061
      then have "poly (polypoly' (?ts @ xs) p) = poly []"
wenzelm@56000
  1062
        by auto
wenzelm@56000
  1063
      then have "\<forall>c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
wenzelm@33268
  1064
        using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
chaieb@33154
  1065
      with coefficients_head[of p, symmetric]
wenzelm@56000
  1066
      have th0: "Ipoly (?ts @ xs) ?hd = 0"
wenzelm@56000
  1067
        by simp
wenzelm@56000
  1068
      from bs have wf'': "wf_bs ?ts ?hd"
wenzelm@56000
  1069
        unfolding wf_bs_def by simp
wenzelm@56000
  1070
      with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0"
wenzelm@56000
  1071
        by simp
wenzelm@56000
  1072
      with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
wenzelm@56000
  1073
        by simp
wenzelm@56000
  1074
    }
wenzelm@56000
  1075
    then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
wenzelm@56000
  1076
      by blast
wenzelm@56000
  1077
    from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p"
wenzelm@56000
  1078
      by blast
wenzelm@56000
  1079
    then have "?h = 0\<^sub>p" by simp
wenzelm@56000
  1080
    with head_nz[OF np] have "p = 0\<^sub>p" by simp
wenzelm@56000
  1081
  }
wenzelm@56000
  1082
  ultimately show "p = 0\<^sub>p"
wenzelm@56000
  1083
    by blast
chaieb@33154
  1084
qed
chaieb@33154
  1085
wenzelm@52803
  1086
lemma isnpolyh_unique:
wenzelm@56000
  1087
  assumes np: "isnpolyh p n0"
wenzelm@52658
  1088
    and nq: "isnpolyh q n1"
wenzelm@56000
  1089
  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
  1090
proof auto
wenzelm@56000
  1091
  assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a) = \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
wenzelm@56000
  1092
  then have "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)"
wenzelm@56000
  1093
    by simp
wenzelm@56000
  1094
  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
  1095
    using wf_bs_polysub[where p=p and q=q] by auto
wenzelm@56000
  1096
  with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq] show "p = q"
wenzelm@56000
  1097
    by blast
chaieb@33154
  1098
qed
chaieb@33154
  1099
chaieb@33154
  1100
krauss@41404
  1101
text{* consequences of unicity on the algorithms for polynomial normalization *}
chaieb@33154
  1102
wenzelm@52658
  1103
lemma polyadd_commute:
wenzelm@56000
  1104
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1105
    and np: "isnpolyh p n0"
wenzelm@52658
  1106
    and nq: "isnpolyh q n1"
wenzelm@52658
  1107
  shows "p +\<^sub>p q = q +\<^sub>p p"
wenzelm@56000
  1108
  using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]]
wenzelm@56000
  1109
  by simp
chaieb@33154
  1110
wenzelm@56000
  1111
lemma zero_normh: "isnpolyh 0\<^sub>p n"
wenzelm@56000
  1112
  by simp
wenzelm@56000
  1113
wenzelm@56000
  1114
lemma one_normh: "isnpolyh (1)\<^sub>p n"
wenzelm@56000
  1115
  by simp
wenzelm@52658
  1116
wenzelm@52803
  1117
lemma polyadd_0[simp]:
wenzelm@56000
  1118
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1119
    and np: "isnpolyh p n0"
wenzelm@56000
  1120
  shows "p +\<^sub>p 0\<^sub>p = p"
wenzelm@56000
  1121
    and "0\<^sub>p +\<^sub>p p = p"
wenzelm@52803
  1122
  using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
chaieb@33154
  1123
    isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
chaieb@33154
  1124
wenzelm@52803
  1125
lemma polymul_1[simp]:
wenzelm@56000
  1126
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1127
    and np: "isnpolyh p n0"
wenzelm@56000
  1128
  shows "p *\<^sub>p (1)\<^sub>p = p"
wenzelm@56000
  1129
    and "(1)\<^sub>p *\<^sub>p p = p"
wenzelm@52803
  1130
  using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
chaieb@33154
  1131
    isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
wenzelm@52658
  1132
wenzelm@52803
  1133
lemma polymul_0[simp]:
wenzelm@56000
  1134
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1135
    and np: "isnpolyh p n0"
wenzelm@56000
  1136
  shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p"
wenzelm@56000
  1137
    and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
wenzelm@52803
  1138
  using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
chaieb@33154
  1139
    isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
chaieb@33154
  1140
wenzelm@52803
  1141
lemma polymul_commute:
wenzelm@56000
  1142
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56000
  1143
    and np: "isnpolyh p n0"
wenzelm@52658
  1144
    and nq: "isnpolyh q n1"
chaieb@33154
  1145
  shows "p *\<^sub>p q = q *\<^sub>p p"
wenzelm@56000
  1146
  using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np], where ?'a = "'a::{field_char_0,field_inverse_zero, power}"]
wenzelm@52658
  1147
  by simp
chaieb@33154
  1148
wenzelm@52658
  1149
declare polyneg_polyneg [simp]
wenzelm@52803
  1150
wenzelm@52803
  1151
lemma isnpolyh_polynate_id [simp]:
wenzelm@56000
  1152
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@56000
  1153
    and np: "isnpolyh p n0"
wenzelm@52658
  1154
  shows "polynate p = p"
wenzelm@56000
  1155
  using isnpolyh_unique[where ?'a= "'a::{field_char_0,field_inverse_zero}", OF polynate_norm[of p, unfolded isnpoly_def] np] polynate[where ?'a = "'a::{field_char_0,field_inverse_zero}"]
wenzelm@52658
  1156
  by simp
chaieb@33154
  1157
wenzelm@52803
  1158
lemma polynate_idempotent[simp]:
wenzelm@56000
  1159
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
  1160
  shows "polynate (polynate p) = polynate p"
chaieb@33154
  1161
  using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
chaieb@33154
  1162
chaieb@33154
  1163
lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
chaieb@33154
  1164
  unfolding poly_nate_def polypoly'_def ..
wenzelm@52658
  1165
wenzelm@52658
  1166
lemma poly_nate_poly:
wenzelm@56000
  1167
  "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
  1168
  using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
wenzelm@52658
  1169
  unfolding poly_nate_polypoly' by auto
wenzelm@52658
  1170
chaieb@33154
  1171
chaieb@33154
  1172
subsection{* heads, degrees and all that *}
wenzelm@52658
  1173
chaieb@33154
  1174
lemma degree_eq_degreen0: "degree p = degreen p 0"
wenzelm@52658
  1175
  by (induct p rule: degree.induct) simp_all
chaieb@33154
  1176
wenzelm@52658
  1177
lemma degree_polyneg:
wenzelm@52658
  1178
  assumes n: "isnpolyh p n"
chaieb@33154
  1179
  shows "degree (polyneg p) = degree p"
wenzelm@52658
  1180
  apply (induct p arbitrary: n rule: polyneg.induct)
wenzelm@52658
  1181
  using n apply simp_all
wenzelm@52658
  1182
  apply (case_tac na)
wenzelm@52658
  1183
  apply auto
wenzelm@52658
  1184
  done
chaieb@33154
  1185
chaieb@33154
  1186
lemma degree_polyadd:
chaieb@33154
  1187
  assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
chaieb@33154
  1188
  shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
wenzelm@52658
  1189
  using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
chaieb@33154
  1190
chaieb@33154
  1191
wenzelm@52658
  1192
lemma degree_polysub:
wenzelm@52658
  1193
  assumes np: "isnpolyh p n0"
wenzelm@52658
  1194
    and nq: "isnpolyh q n1"
chaieb@33154
  1195
  shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
chaieb@33154
  1196
proof-
chaieb@33154
  1197
  from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp
chaieb@33154
  1198
  from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])
chaieb@33154
  1199
qed
chaieb@33154
  1200
wenzelm@52803
  1201
lemma degree_polysub_samehead:
wenzelm@56000
  1202
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52803
  1203
    and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q"
wenzelm@52658
  1204
    and d: "degree p = degree q"
chaieb@33154
  1205
  shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
wenzelm@52658
  1206
  unfolding polysub_def split_def fst_conv snd_conv
wenzelm@52658
  1207
  using np nq h d
wenzelm@52658
  1208
proof (induct p q rule: polyadd.induct)
wenzelm@52658
  1209
  case (1 c c')
wenzelm@52803
  1210
  thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def])
chaieb@33154
  1211
next
wenzelm@52803
  1212
  case (2 c c' n' p')
krauss@41815
  1213
  from 2 have "degree (C c) = degree (CN c' n' p')" by simp
wenzelm@52658
  1214
  hence nz:"n' > 0" by (cases n') auto
wenzelm@52658
  1215
  hence "head (CN c' n' p') = CN c' n' p'" by (cases n') auto
wenzelm@41807
  1216
  with 2 show ?case by simp
chaieb@33154
  1217
next
wenzelm@52803
  1218
  case (3 c n p c')
krauss@41815
  1219
  hence "degree (C c') = degree (CN c n p)" by simp
wenzelm@52658
  1220
  hence nz:"n > 0" by (cases n) auto
wenzelm@52658
  1221
  hence "head (CN c n p) = CN c n p" by (cases n) auto
wenzelm@41807
  1222
  with 3 show ?case by simp
chaieb@33154
  1223
next
chaieb@33154
  1224
  case (4 c n p c' n' p')
wenzelm@52803
  1225
  hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1"
chaieb@33154
  1226
    "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+
wenzelm@52803
  1227
  hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all
wenzelm@52803
  1228
  hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
chaieb@33154
  1229
    using H(1-2) degree_polyneg by auto
chaieb@33154
  1230
  from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"  by simp+
chaieb@33154
  1231
  from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"  by simp
chaieb@33154
  1232
  from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto
chaieb@33154
  1233
  have "n = n' \<or> n < n' \<or> n > n'" by arith
chaieb@33154
  1234
  moreover
chaieb@33154
  1235
  {assume nn': "n = n'"
chaieb@33154
  1236
    have "n = 0 \<or> n >0" by arith
wenzelm@41807
  1237
    moreover {assume nz: "n = 0" hence ?case using 4 nn' by (auto simp add: Let_def degcmc')}
chaieb@33154
  1238
    moreover {assume nz: "n > 0"
chaieb@33154
  1239
      with nn' H(3) have  cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
wenzelm@52658
  1240
      hence ?case
wenzelm@52658
  1241
        using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv] polysub_same_0[OF c'nh, simplified polysub_def]
wenzelm@52803
  1242
        using nn' 4 by (simp add: Let_def) }
chaieb@33154
  1243
    ultimately have ?case by blast}
chaieb@33154
  1244
  moreover
wenzelm@52803
  1245
  {assume nn': "n < n'" hence n'p: "n' > 0" by simp
wenzelm@52658
  1246
    hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n') simp_all
wenzelm@52658
  1247
    have degcnp': "degree (CN c' n' p') = 0" and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
wenzelm@52658
  1248
      using 4 nn' by (cases n', simp_all)
wenzelm@52658
  1249
    hence "n > 0" by (cases n) simp_all
wenzelm@52658
  1250
    hence headcnp: "head (CN c n p) = CN c n p" by (cases n) auto
chaieb@33154
  1251
    from H(3) headcnp headcnp' nn' have ?case by auto}
chaieb@33154
  1252
  moreover
wenzelm@52803
  1253
  {assume nn': "n > n'"  hence np: "n > 0" by simp
wenzelm@52658
  1254
    hence headcnp:"head (CN c n p) = CN c n p"  by (cases n) simp_all
wenzelm@41807
  1255
    from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
wenzelm@52658
  1256
    from np have degcnp: "degree (CN c n p) = 0" by (cases n) simp_all
wenzelm@52658
  1257
    with degcnpeq have "n' > 0" by (cases n') simp_all
wenzelm@52658
  1258
    hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n') auto
chaieb@33154
  1259
    from H(3) headcnp headcnp' nn' have ?case by auto}
chaieb@33154
  1260
  ultimately show ?case  by blast
krauss@41812
  1261
qed auto
wenzelm@52803
  1262
chaieb@33154
  1263
lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
wenzelm@52658
  1264
  by (induct p arbitrary: n0 rule: head.induct) (simp_all add: shift1_def)
chaieb@33154
  1265
chaieb@33154
  1266
lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
wenzelm@52658
  1267
proof (induct k arbitrary: n0 p)
wenzelm@52658
  1268
  case 0
wenzelm@52658
  1269
  thus ?case by auto
wenzelm@52658
  1270
next
wenzelm@52658
  1271
  case (Suc k n0 p)
wenzelm@52658
  1272
  hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
wenzelm@41807
  1273
  with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
wenzelm@52803
  1274
    and "head (shift1 p) = head p" by (simp_all add: shift1_head)
haftmann@39246
  1275
  thus ?case by (simp add: funpow_swap1)
wenzelm@52658
  1276
qed
chaieb@33154
  1277
chaieb@33154
  1278
lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
chaieb@33154
  1279
  by (simp add: shift1_def)
chaieb@33154
  1280
lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
wenzelm@46991
  1281
  by (induct k arbitrary: p) (auto simp add: shift1_degree)
chaieb@33154
  1282
chaieb@33154
  1283
lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
wenzelm@52658
  1284
  by (induct n arbitrary: p) simp_all
chaieb@33154
  1285
chaieb@33154
  1286
lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
wenzelm@52658
  1287
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
  1288
lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
wenzelm@52658
  1289
  by (induct p arbitrary: n rule: degreen.induct) auto
chaieb@33154
  1290
lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
wenzelm@52658
  1291
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
  1292
lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
wenzelm@52658
  1293
  by (induct p rule: head.induct) auto
chaieb@33154
  1294
wenzelm@52803
  1295
lemma polyadd_eq_const_degree:
wenzelm@52658
  1296
  "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> polyadd p q = C c \<Longrightarrow> degree p = degree q"
chaieb@33154
  1297
  using polyadd_eq_const_degreen degree_eq_degreen0 by simp
chaieb@33154
  1298
wenzelm@52658
  1299
lemma polyadd_head:
wenzelm@52658
  1300
  assumes np: "isnpolyh p n0"
wenzelm@52658
  1301
    and nq: "isnpolyh q n1"
wenzelm@52658
  1302
    and deg: "degree p \<noteq> degree q"
chaieb@33154
  1303
  shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
wenzelm@52658
  1304
  using np nq deg
wenzelm@52658
  1305
  apply (induct p q arbitrary: n0 n1 rule: polyadd.induct)
wenzelm@52658
  1306
  using np
wenzelm@52658
  1307
  apply simp_all
wenzelm@52658
  1308
  apply (case_tac n', simp, simp)
wenzelm@52658
  1309
  apply (case_tac n, simp, simp)
wenzelm@52658
  1310
  apply (case_tac n, case_tac n', simp add: Let_def)
haftmann@54489
  1311
  apply (auto simp add: polyadd_eq_const_degree)[2]
wenzelm@52658
  1312
  apply (metis head_nz)
wenzelm@52658
  1313
  apply (metis head_nz)
wenzelm@52658
  1314
  apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
wenzelm@52658
  1315
  done
chaieb@33154
  1316
wenzelm@52803
  1317
lemma polymul_head_polyeq:
wenzelm@56000
  1318
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
  1319
  shows "\<lbrakk>isnpolyh p n0; isnpolyh q n1 ; p \<noteq> 0\<^sub>p ; q \<noteq> 0\<^sub>p \<rbrakk> \<Longrightarrow> head (p *\<^sub>p q) = head p *\<^sub>p head q"
chaieb@33154
  1320
proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
krauss@41813
  1321
  case (2 c c' n' p' n0 n1)
krauss@41813
  1322
  hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"  by (simp_all add: head_isnpolyh)
wenzelm@52658
  1323
  thus ?case using 2 by (cases n') auto
wenzelm@52803
  1324
next
wenzelm@52803
  1325
  case (3 c n p c' n0 n1)
krauss@41813
  1326
  hence "isnpolyh (head (CN c n p)) n0" "isnormNum c'"  by (simp_all add: head_isnpolyh)
wenzelm@52658
  1327
  thus ?case using 3 by (cases n) auto
chaieb@33154
  1328
next
chaieb@33154
  1329
  case (4 c n p c' n' p' n0 n1)
chaieb@33154
  1330
  hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
chaieb@33154
  1331
    "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
chaieb@33154
  1332
    by simp_all
chaieb@33154
  1333
  have "n < n' \<or> n' < n \<or> n = n'" by arith
wenzelm@52803
  1334
  moreover
wenzelm@52803
  1335
  {assume nn': "n < n'" hence ?case
wenzelm@52658
  1336
      using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)]
wenzelm@52658
  1337
      apply simp
wenzelm@52658
  1338
      apply (cases n)
wenzelm@52658
  1339
      apply simp
wenzelm@52658
  1340
      apply (cases n')
wenzelm@52658
  1341
      apply simp_all
wenzelm@52658
  1342
      done }
chaieb@33154
  1343
  moreover {assume nn': "n'< n"
wenzelm@52658
  1344
    hence ?case
wenzelm@52803
  1345
      using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)]
wenzelm@52658
  1346
      apply simp
wenzelm@52658
  1347
      apply (cases n')
wenzelm@52658
  1348
      apply simp
wenzelm@52658
  1349
      apply (cases n)
wenzelm@52658
  1350
      apply auto
wenzelm@52658
  1351
      done }
chaieb@33154
  1352
  moreover {assume nn': "n' = n"
wenzelm@52803
  1353
    from nn' polymul_normh[OF norm(5,4)]
chaieb@33154
  1354
    have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
wenzelm@52803
  1355
    from nn' polymul_normh[OF norm(5,3)] norm
chaieb@33154
  1356
    have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
chaieb@33154
  1357
    from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
wenzelm@52803
  1358
    have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
wenzelm@52803
  1359
    from polyadd_normh[OF ncnpc' ncnpp0']
wenzelm@52803
  1360
    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
  1361
      by (simp add: min_def)
chaieb@33154
  1362
    {assume np: "n > 0"
chaieb@33154
  1363
      with nn' head_isnpolyh_Suc'[OF np nth]
wenzelm@33268
  1364
        head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
chaieb@33154
  1365
      have ?case by simp}
chaieb@33154
  1366
    moreover
wenzelm@53374
  1367
    { assume nz: "n = 0"
chaieb@33154
  1368
      from polymul_degreen[OF norm(5,4), where m="0"]
wenzelm@33268
  1369
        polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
chaieb@33154
  1370
      norm(5,6) degree_npolyhCN[OF norm(6)]
chaieb@33154
  1371
    have dth:"degree (CN c 0 p *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))" by simp
chaieb@33154
  1372
    hence dth':"degree (CN c 0 p *\<^sub>p c') \<noteq> degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))" by simp
chaieb@33154
  1373
    from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
krauss@41813
  1374
    have ?case   using norm "4.hyps"(6)[OF norm(5,3)]
krauss@41813
  1375
        "4.hyps"(5)[OF norm(5,4)] nn' nz by simp }
wenzelm@52803
  1376
    ultimately have ?case by (cases n) auto}
chaieb@33154
  1377
  ultimately show ?case by blast
chaieb@33154
  1378
qed simp_all
chaieb@33154
  1379
chaieb@33154
  1380
lemma degree_coefficients: "degree p = length (coefficients p) - 1"
wenzelm@52658
  1381
  by (induct p rule: degree.induct) auto
chaieb@33154
  1382
chaieb@33154
  1383
lemma degree_head[simp]: "degree (head p) = 0"
wenzelm@52658
  1384
  by (induct p rule: head.induct) auto
chaieb@33154
  1385
krauss@41812
  1386
lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
wenzelm@52658
  1387
  by (cases n) simp_all
chaieb@33154
  1388
lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
wenzelm@52658
  1389
  by (cases n) simp_all
chaieb@33154
  1390
wenzelm@52658
  1391
lemma polyadd_different_degree:
wenzelm@52658
  1392
  "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degree p \<noteq> degree q\<rbrakk> \<Longrightarrow>
wenzelm@52658
  1393
    degree (polyadd p q) = max (degree p) (degree q)"
chaieb@33154
  1394
  using polyadd_different_degreen degree_eq_degreen0 by simp
chaieb@33154
  1395
chaieb@33154
  1396
lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
wenzelm@52658
  1397
  by (induct p arbitrary: n0 rule: polyneg.induct) auto
chaieb@33154
  1398
chaieb@33154
  1399
lemma degree_polymul:
wenzelm@56000
  1400
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1401
    and np: "isnpolyh p n0"
wenzelm@52658
  1402
    and nq: "isnpolyh q n1"
chaieb@33154
  1403
  shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
chaieb@33154
  1404
  using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
chaieb@33154
  1405
chaieb@33154
  1406
lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
wenzelm@52658
  1407
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
  1408
chaieb@33154
  1409
lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"
wenzelm@52658
  1410
  by (induct p arbitrary: n rule: degree.induct) auto
wenzelm@52658
  1411
chaieb@33154
  1412
chaieb@33154
  1413
subsection {* Correctness of polynomial pseudo division *}
chaieb@33154
  1414
chaieb@33154
  1415
lemma polydivide_aux_properties:
wenzelm@56000
  1416
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52658
  1417
    and np: "isnpolyh p n0"
wenzelm@52658
  1418
    and ns: "isnpolyh s n1"
wenzelm@52658
  1419
    and ap: "head p = a"
wenzelm@52658
  1420
    and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
wenzelm@52803
  1421
  shows "(polydivide_aux a n p k s = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p)
chaieb@33154
  1422
          \<and> (\<exists>nr. isnpolyh r nr) \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow (k' - k) a) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
chaieb@33154
  1423
  using ns
wenzelm@52658
  1424
proof (induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
berghofe@34915
  1425
  case less
chaieb@33154
  1426
  let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
wenzelm@52803
  1427
  let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p)
chaieb@33154
  1428
    \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
chaieb@33154
  1429
  let ?b = "head s"
berghofe@34915
  1430
  let ?p' = "funpow (degree s - n) shift1 p"
wenzelm@50282
  1431
  let ?xdn = "funpow (degree s - n) shift1 (1)\<^sub>p"
chaieb@33154
  1432
  let ?akk' = "a ^\<^sub>p (k' - k)"
berghofe@34915
  1433
  note ns = `isnpolyh s n1`
wenzelm@52803
  1434
  from np have np0: "isnpolyh p 0"
wenzelm@52803
  1435
    using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by simp
wenzelm@52803
  1436
  have np': "isnpolyh ?p' 0"
wenzelm@52803
  1437
    using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def
wenzelm@52803
  1438
    by simp
wenzelm@52803
  1439
  have headp': "head ?p' = head p"
wenzelm@52803
  1440
    using funpow_shift1_head[OF np pnz] by simp
wenzelm@52803
  1441
  from funpow_shift1_isnpoly[where p="(1)\<^sub>p"] have nxdn: "isnpolyh ?xdn 0"
wenzelm@52803
  1442
    by (simp add: isnpoly_def)
wenzelm@52803
  1443
  from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
chaieb@33154
  1444
  have nakk':"isnpolyh ?akk' 0" by blast
wenzelm@52658
  1445
  { assume sz: "s = 0\<^sub>p"
wenzelm@52658
  1446
    hence ?ths using np polydivide_aux.simps
wenzelm@52658
  1447
      apply clarsimp
wenzelm@52658
  1448
      apply (rule exI[where x="0\<^sub>p"])
wenzelm@52658
  1449
      apply simp
wenzelm@52658
  1450
      done }
chaieb@33154
  1451
  moreover
wenzelm@52803
  1452
  { assume sz: "s \<noteq> 0\<^sub>p"
wenzelm@52803
  1453
    { assume dn: "degree s < n"
wenzelm@52658
  1454
      hence "?ths" using ns ndp np polydivide_aux.simps
wenzelm@52658
  1455
        apply auto
wenzelm@52658
  1456
        apply (rule exI[where x="0\<^sub>p"])
wenzelm@52658
  1457
        apply simp
wenzelm@52658
  1458
        done }
wenzelm@52803
  1459
    moreover
wenzelm@52803
  1460
    { assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith
wenzelm@52803
  1461
      have degsp': "degree s = degree ?p'"
berghofe@34915
  1462
        using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp
wenzelm@52803
  1463
      { assume ba: "?b = a"
wenzelm@52803
  1464
        hence headsp': "head s = head ?p'"
wenzelm@52803
  1465
          using ap headp' by simp
wenzelm@52803
  1466
        have nr: "isnpolyh (s -\<^sub>p ?p') 0"
wenzelm@52803
  1467
          using polysub_normh[OF ns np'] by simp
berghofe@34915
  1468
        from degree_polysub_samehead[OF ns np' headsp' degsp']
berghofe@34915
  1469
        have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
wenzelm@52803
  1470
        moreover
wenzelm@52803
  1471
        { assume deglt:"degree (s -\<^sub>p ?p') < degree s"
krauss@41403
  1472
          from polydivide_aux.simps sz dn' ba
krauss@41403
  1473
          have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
wenzelm@33268
  1474
            by (simp add: Let_def)
wenzelm@52803
  1475
          { assume h1: "polydivide_aux a n p k s = (k', r)"
wenzelm@52803
  1476
            from less(1)[OF deglt nr, of k k' r] trans[OF eq[symmetric] h1]
wenzelm@52803
  1477
            have kk': "k \<le> k'"
wenzelm@52803
  1478
              and nr:"\<exists>nr. isnpolyh r nr"
wenzelm@52803
  1479
              and dr: "degree r = 0 \<or> degree r < degree p"
wenzelm@52803
  1480
              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
  1481
              by auto
wenzelm@52803
  1482
            from q1 obtain q n1 where nq: "isnpolyh q n1"
wenzelm@52803
  1483
              and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r" by blast
wenzelm@33268
  1484
            from nr obtain nr where nr': "isnpolyh r nr" by blast
wenzelm@52803
  1485
            from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0"
wenzelm@52803
  1486
              by simp
wenzelm@33268
  1487
            from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
wenzelm@33268
  1488
            have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
wenzelm@52803
  1489
            from polyadd_normh[OF polymul_normh[OF np
wenzelm@33268
  1490
              polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
wenzelm@52803
  1491
            have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0"
wenzelm@52803
  1492
              by simp
wenzelm@56000
  1493
            from asp have "\<forall> (bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) =
wenzelm@33268
  1494
              Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
wenzelm@56000
  1495
            hence " \<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
wenzelm@52803
  1496
              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
haftmann@36349
  1497
              by (simp add: field_simps)
wenzelm@56000
  1498
            hence " \<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1499
              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p *\<^sub>p p) +
wenzelm@52803
  1500
              Ipoly bs p * Ipoly bs q + Ipoly bs r"
wenzelm@52803
  1501
              by (auto simp only: funpow_shift1_1)
wenzelm@56000
  1502
            hence "\<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1503
              Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p) +
wenzelm@52803
  1504
              Ipoly bs q) + Ipoly bs r"
wenzelm@52803
  1505
              by (simp add: field_simps)
wenzelm@56000
  1506
            hence "\<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1507
              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
  1508
              by simp
wenzelm@33268
  1509
            with isnpolyh_unique[OF nakks' nqr']
wenzelm@52803
  1510
            have "a ^\<^sub>p (k' - k) *\<^sub>p s =
wenzelm@52803
  1511
              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
  1512
              by blast
wenzelm@33268
  1513
            hence ?qths using nq'
wenzelm@50282
  1514
              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
  1515
              apply (rule_tac x="0" in exI)
wenzelm@52803
  1516
              apply simp
wenzelm@52803
  1517
              done
wenzelm@33268
  1518
            with kk' nr dr have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
wenzelm@52803
  1519
              by blast
wenzelm@52803
  1520
          }
wenzelm@52803
  1521
          hence ?ths by blast
wenzelm@52803
  1522
        }
wenzelm@52803
  1523
        moreover
wenzelm@52803
  1524
        { assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
wenzelm@56000
  1525
          from spz isnpolyh_unique[OF polysub_normh[OF ns np'], where q="0\<^sub>p", symmetric, where ?'a = "'a::{field_char_0,field_inverse_zero}"]
wenzelm@56000
  1526
          have " \<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs s = Ipoly bs ?p'"
wenzelm@52803
  1527
            by simp
wenzelm@56000
  1528
          hence "\<forall>(bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)"
wenzelm@52658
  1529
            using np nxdn
wenzelm@52658
  1530
            apply simp
wenzelm@52658
  1531
            apply (simp only: funpow_shift1_1)
wenzelm@52658
  1532
            apply simp
wenzelm@52658
  1533
            done
wenzelm@52658
  1534
          hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]]
wenzelm@52658
  1535
            by blast
wenzelm@52803
  1536
          { assume h1: "polydivide_aux a n p k s = (k',r)"
krauss@41403
  1537
            from polydivide_aux.simps sz dn' ba
krauss@41403
  1538
            have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
wenzelm@33268
  1539
              by (simp add: Let_def)
wenzelm@52803
  1540
            also have "\<dots> = (k,0\<^sub>p)"
wenzelm@52803
  1541
              using polydivide_aux.simps spz by simp
wenzelm@33268
  1542
            finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
berghofe@34915
  1543
            with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
krauss@41403
  1544
              polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?ths
wenzelm@33268
  1545
              apply auto
wenzelm@52803
  1546
              apply (rule exI[where x="?xdn"])
berghofe@34915
  1547
              apply (auto simp add: polymul_commute[of p])
wenzelm@52803
  1548
              done
wenzelm@52803
  1549
          }
wenzelm@52803
  1550
        }
wenzelm@52803
  1551
        ultimately have ?ths by blast
wenzelm@52803
  1552
      }
chaieb@33154
  1553
      moreover
wenzelm@52803
  1554
      { assume ba: "?b \<noteq> a"
wenzelm@52803
  1555
        from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
wenzelm@33268
  1556
          polymul_normh[OF head_isnpolyh[OF ns] np']]
wenzelm@52803
  1557
        have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
wenzelm@52803
  1558
          by (simp add: min_def)
wenzelm@33268
  1559
        have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
wenzelm@52803
  1560
          using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
wenzelm@33268
  1561
            polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
wenzelm@52803
  1562
            funpow_shift1_nz[OF pnz]
wenzelm@52803
  1563
          by simp_all
wenzelm@33268
  1564
        from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
berghofe@34915
  1565
          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
  1566
        have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
wenzelm@33268
  1567
          using head_head[OF ns] funpow_shift1_head[OF np pnz]
wenzelm@33268
  1568
            polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
wenzelm@33268
  1569
          by (simp add: ap)
wenzelm@33268
  1570
        from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
wenzelm@33268
  1571
          head_nz[OF np] pnz sz ap[symmetric]
berghofe@34915
  1572
          funpow_shift1_nz[OF pnz, where n="degree s - n"]
wenzelm@52803
  1573
          polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"] head_nz[OF ns]
berghofe@34915
  1574
          ndp dn
wenzelm@52803
  1575
        have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p')"
wenzelm@33268
  1576
          by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
wenzelm@52803
  1577
        { assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
wenzelm@52803
  1578
          from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns]
wenzelm@52803
  1579
            polymul_normh[OF head_isnpolyh[OF ns]np']] ap
wenzelm@52803
  1580
          have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
wenzelm@52803
  1581
            by simp
wenzelm@52803
  1582
          { assume h1:"polydivide_aux a n p k s = (k', r)"
krauss@41403
  1583
            from h1 polydivide_aux.simps sz dn' ba
krauss@41403
  1584
            have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
wenzelm@33268
  1585
              by (simp add: Let_def)
berghofe@34915
  1586
            with less(1)[OF dth nasbp', of "Suc k" k' r]
wenzelm@52803
  1587
            obtain q nq nr where kk': "Suc k \<le> k'"
wenzelm@52803
  1588
              and nr: "isnpolyh r nr"
wenzelm@52803
  1589
              and nq: "isnpolyh q nq"
wenzelm@33268
  1590
              and dr: "degree r = 0 \<or> degree r < degree p"
wenzelm@52803
  1591
              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
  1592
              by auto
wenzelm@33268
  1593
            from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith
wenzelm@52803
  1594
            {
wenzelm@56000
  1595
              fix bs:: "'a::{field_char_0,field_inverse_zero} list"
wenzelm@52803
  1596
              from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
wenzelm@52803
  1597
              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
  1598
                by simp
wenzelm@52803
  1599
              hence "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s =
wenzelm@52803
  1600
                Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
wenzelm@52803
  1601
                by (simp add: field_simps)
wenzelm@52803
  1602
              hence "Ipoly bs a ^ (k' - k)  * Ipoly bs s =
wenzelm@52803
  1603
                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
  1604
                by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
wenzelm@52803
  1605
              hence "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1606
                Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
wenzelm@52803
  1607
                by (simp add: field_simps)
wenzelm@52803
  1608
            }
wenzelm@56000
  1609
            hence ieq:"\<forall>(bs :: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1610
              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
  1611
              by auto
wenzelm@33268
  1612
            let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
wenzelm@33268
  1613
            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
  1614
            have nqw: "isnpolyh ?q 0"
wenzelm@52803
  1615
              by simp
wenzelm@33268
  1616
            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
  1617
            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
  1618
              by blast
wenzelm@52803
  1619
            from dr kk' nr h1 asth nqw have ?ths
wenzelm@52803
  1620
              apply simp
wenzelm@33268
  1621
              apply (rule conjI)
wenzelm@33268
  1622
              apply (rule exI[where x="nr"], simp)
wenzelm@33268
  1623
              apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
wenzelm@33268
  1624
              apply (rule exI[where x="0"], simp)
wenzelm@52803
  1625
              done
wenzelm@52803
  1626
          }
wenzelm@52803
  1627
          hence ?ths by blast
wenzelm@52803
  1628
        }
wenzelm@52803
  1629
        moreover
wenzelm@52803
  1630
        { assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
wenzelm@52803
  1631
          {
wenzelm@56000
  1632
            fix bs :: "'a::{field_char_0,field_inverse_zero} list"
wenzelm@33268
  1633
            from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
wenzelm@52803
  1634
            have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'"
wenzelm@52803
  1635
              by simp
wenzelm@52803
  1636
            hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
wenzelm@52803
  1637
              by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
wenzelm@52803
  1638
            hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))"
wenzelm@52803
  1639
              by simp
wenzelm@52803
  1640
          }
wenzelm@56000
  1641
          hence hth: "\<forall> (bs:: 'a::{field_char_0,field_inverse_zero} list). Ipoly bs (a*\<^sub>p s) =
wenzelm@52658
  1642
            Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
wenzelm@52803
  1643
          from hth have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
wenzelm@56000
  1644
            using isnpolyh_unique[where ?'a = "'a::{field_char_0,field_inverse_zero}", OF polymul_normh[OF head_isnpolyh[OF np] ns]
chaieb@33154
  1645
                    polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
wenzelm@33268
  1646
              simplified ap] by simp
wenzelm@52803
  1647
          { assume h1: "polydivide_aux a n p k s = (k', r)"
wenzelm@52803
  1648
            from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
wenzelm@52803
  1649
            have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
wenzelm@52803
  1650
            with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
wenzelm@52803
  1651
              polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
wenzelm@52803
  1652
            have ?ths
wenzelm@52803
  1653
              apply (clarsimp simp add: Let_def)
wenzelm@52803
  1654
              apply (rule exI[where x="?b *\<^sub>p ?xdn"])
wenzelm@52803
  1655
              apply simp
wenzelm@52803
  1656
              apply (rule exI[where x="0"], simp)
wenzelm@52803
  1657
              done
wenzelm@52803
  1658
          }
wenzelm@52803
  1659
          hence ?ths by blast
wenzelm@52803
  1660
        }
wenzelm@52658
  1661
        ultimately have ?ths
wenzelm@52658
  1662
          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
  1663
            head_nz[OF np] pnz sz ap[symmetric]
wenzelm@52803
  1664
          by (simp add: degree_eq_degreen0[symmetric]) blast
wenzelm@52803
  1665
      }
chaieb@33154
  1666
      ultimately have ?ths by blast
chaieb@33154
  1667
    }
wenzelm@52803
  1668
    ultimately have ?ths by blast
wenzelm@52803
  1669
  }
chaieb@33154
  1670
  ultimately show ?ths by blast
chaieb@33154
  1671
qed
chaieb@33154
  1672
wenzelm@52803
  1673
lemma polydivide_properties:
wenzelm@56000
  1674
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
wenzelm@52803
  1675
    and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
wenzelm@52803
  1676
  shows "\<exists>k r. polydivide s p = (k,r) \<and>
wenzelm@52803
  1677
    (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@52803
  1678
    (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r))"
wenzelm@52803
  1679
proof -
wenzelm@52803
  1680
  have trv: "head p = head p" "degree p = degree p"
wenzelm@52803
  1681
    by simp_all
wenzelm@52803
  1682
  from polydivide_def[where s="s" and p="p"] have ex: "\<exists> k r. polydivide s p = (k,r)"
wenzelm@52803
  1683
    by auto
wenzelm@52803
  1684
  then obtain k r where kr: "polydivide s p = (k,r)"
wenzelm@52803
  1685
    by blast
wenzelm@56000
  1686
  from trans[OF polydivide_def[where s="s"and p="p", symmetric] kr]
chaieb@33154
  1687
    polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
chaieb@33154
  1688
  have "(degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@52803
  1689
    (\<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
  1690
    by blast
wenzelm@52803
  1691
  with kr show ?thesis
chaieb@33154
  1692
    apply -
chaieb@33154
  1693
    apply (rule exI[where x="k"])
chaieb@33154
  1694
    apply (rule exI[where x="r"])
chaieb@33154
  1695
    apply simp
chaieb@33154
  1696
    done
chaieb@33154
  1697
qed
chaieb@33154
  1698
wenzelm@52658
  1699
chaieb@33154
  1700
subsection{* More about polypoly and pnormal etc *}
chaieb@33154
  1701
wenzelm@56000
  1702
definition "isnonconstant p \<longleftrightarrow> \<not> isconstant p"
chaieb@33154
  1703
wenzelm@52658
  1704
lemma isnonconstant_pnormal_iff:
wenzelm@52803
  1705
  assumes nc: "isnonconstant p"
wenzelm@52803
  1706
  shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
chaieb@33154
  1707
proof
wenzelm@52803
  1708
  let ?p = "polypoly bs p"
chaieb@33154
  1709
  assume H: "pnormal ?p"
wenzelm@52658
  1710
  have csz: "coefficients p \<noteq> []" using nc by (cases p) auto
wenzelm@52803
  1711
wenzelm@52803
  1712
  from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
chaieb@33154
  1713
    pnormal_last_nonzero[OF H]
chaieb@33154
  1714
  show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)
chaieb@33154
  1715
next
chaieb@33154
  1716
  assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  1717
  let ?p = "polypoly bs p"
wenzelm@52658
  1718
  have csz: "coefficients p \<noteq> []" using nc by (cases p) auto
wenzelm@52803
  1719
  hence pz: "?p \<noteq> []" by (simp add: polypoly_def)
chaieb@33154
  1720
  hence lg: "length ?p > 0" by simp
wenzelm@52803
  1721
  from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
chaieb@33154
  1722
  have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)
chaieb@33154
  1723
  from pnormal_last_length[OF lg lz] show "pnormal ?p" .
chaieb@33154
  1724
qed
chaieb@33154
  1725
chaieb@33154
  1726
lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
chaieb@33154
  1727
  unfolding isnonconstant_def
wenzelm@52658
  1728
  apply (cases p)
wenzelm@52658
  1729
  apply simp_all
wenzelm@52658
  1730
  apply (case_tac nat)
wenzelm@52658
  1731
  apply auto
chaieb@33154
  1732
  done
wenzelm@52658
  1733
wenzelm@52658
  1734
lemma isnonconstant_nonconstant:
wenzelm@52658
  1735
  assumes inc: "isnonconstant p"
chaieb@33154
  1736
  shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
chaieb@33154
  1737
proof
chaieb@33154
  1738
  let ?p = "polypoly bs p"
chaieb@33154
  1739
  assume nc: "nonconstant ?p"
chaieb@33154
  1740
  from isnonconstant_pnormal_iff[OF inc, of bs] nc
chaieb@33154
  1741
  show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast
chaieb@33154
  1742
next
chaieb@33154
  1743
  let ?p = "polypoly bs p"
chaieb@33154
  1744
  assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  1745
  from isnonconstant_pnormal_iff[OF inc, of bs] h
chaieb@33154
  1746
  have pn: "pnormal ?p" by blast
wenzelm@52803
  1747
  { fix x assume H: "?p = [x]"
chaieb@33154
  1748
    from H have "length (coefficients p) = 1" unfolding polypoly_def by auto
wenzelm@52803
  1749
    with isnonconstant_coefficients_length[OF inc] have False by arith }
wenzelm@52803
  1750
  thus "nonconstant ?p" using pn unfolding nonconstant_def by blast
chaieb@33154
  1751
qed
chaieb@33154
  1752
chaieb@33154
  1753
lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
wenzelm@52658
  1754
  apply (induct p)
wenzelm@52658
  1755
  apply (simp_all add: pnormal_def)
wenzelm@52658
  1756
  apply (case_tac "p = []")
wenzelm@52658
  1757
  apply simp_all
wenzelm@52658
  1758
  done
chaieb@33154
  1759
wenzelm@52658
  1760
lemma degree_degree:
wenzelm@52658
  1761
  assumes inc: "isnonconstant p"
chaieb@33154
  1762
  shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  1763
proof
wenzelm@52803
  1764
  let ?p = "polypoly bs p"
chaieb@33154
  1765
  assume H: "degree p = Polynomial_List.degree ?p"
chaieb@33154
  1766
  from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
chaieb@33154
  1767
    unfolding polypoly_def by auto
chaieb@33154
  1768
  from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
chaieb@33154
  1769
  have lg:"length (pnormalize ?p) = length ?p"
chaieb@33154
  1770
    unfolding Polynomial_List.degree_def polypoly_def by simp
wenzelm@52803
  1771
  hence "pnormal ?p" using pnormal_length[OF pz] by blast
wenzelm@52803
  1772
  with isnonconstant_pnormal_iff[OF inc]
chaieb@33154
  1773
  show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast
chaieb@33154
  1774
next
wenzelm@52803
  1775
  let  ?p = "polypoly bs p"
chaieb@33154
  1776
  assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  1777
  with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast
chaieb@33154
  1778
  with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
wenzelm@52803
  1779
  show "degree p = Polynomial_List.degree ?p"
chaieb@33154
  1780
    unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
chaieb@33154
  1781
qed
chaieb@33154
  1782
wenzelm@52658
  1783
wenzelm@52803
  1784
section {* Swaps ; Division by a certain variable *}
wenzelm@52658
  1785
wenzelm@52803
  1786
primrec swap :: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly" where
chaieb@33154
  1787
  "swap n m (C x) = C x"
haftmann@39246
  1788
| "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"
haftmann@39246
  1789
| "swap n m (Neg t) = Neg (swap n m t)"
haftmann@39246
  1790
| "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
haftmann@39246
  1791
| "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
haftmann@39246
  1792
| "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
haftmann@39246
  1793
| "swap n m (Pw t k) = Pw (swap n m t) k"
wenzelm@52803
  1794
| "swap n m (CN c k p) =
wenzelm@52803
  1795
    CN (swap n m c) (if k = n then m else if k=m then n else k) (swap n m p)"
chaieb@33154
  1796
wenzelm@52658
  1797
lemma swap:
wenzelm@52658
  1798
  assumes nbs: "n < length bs"
wenzelm@52658
  1799
    and mbs: "m < length bs"
chaieb@33154
  1800
  shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
chaieb@33154
  1801
proof (induct t)
wenzelm@52658
  1802
  case (Bound k)
wenzelm@52803
  1803
  thus ?case using nbs mbs by simp
chaieb@33154
  1804
next
wenzelm@52658
  1805
  case (CN c k p)
wenzelm@52803
  1806
  thus ?case using nbs mbs by simp
chaieb@33154
  1807
qed simp_all
chaieb@33154
  1808
wenzelm@52658
  1809
lemma swap_swap_id [simp]: "swap n m (swap m n t) = t"
wenzelm@52658
  1810
  by (induct t) simp_all
wenzelm@52658
  1811
wenzelm@52658
  1812
lemma swap_commute: "swap n m p = swap m n p"
wenzelm@52658
  1813
  by (induct p) simp_all
chaieb@33154
  1814
chaieb@33154
  1815
lemma swap_same_id[simp]: "swap n n t = t"
wenzelm@52658
  1816
  by (induct t) simp_all
chaieb@33154
  1817
chaieb@33154
  1818
definition "swapnorm n m t = polynate (swap n m t)"
chaieb@33154
  1819
wenzelm@52658
  1820
lemma swapnorm:
wenzelm@52658
  1821
  assumes nbs: "n < length bs"
wenzelm@52658
  1822
    and mbs: "m < length bs"
wenzelm@56000
  1823
  shows "((Ipoly bs (swapnorm n m t) :: 'a::{field_char_0,field_inverse_zero})) =
wenzelm@52658
  1824
    Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
wenzelm@41807
  1825
  using swap[OF assms] swapnorm_def by simp
chaieb@33154
  1826
wenzelm@52658
  1827
lemma swapnorm_isnpoly [simp]:
wenzelm@56000
  1828
  assumes "SORT_CONSTRAINT('a::{field_char_0,field_inverse_zero})"
chaieb@33154
  1829
  shows "isnpoly (swapnorm n m p)"
chaieb@33154
  1830
  unfolding swapnorm_def by simp
chaieb@33154
  1831
wenzelm@52803
  1832
definition "polydivideby n s p =
wenzelm@56000
  1833
  (let
wenzelm@56000
  1834
    ss = swapnorm 0 n s;
wenzelm@56000
  1835
    sp = swapnorm 0 n p;
wenzelm@56000
  1836
    h = head sp;
wenzelm@56000
  1837
    (k, r) = polydivide ss sp
wenzelm@56000
  1838
   in (k, swapnorm 0 n h, swapnorm 0 n r))"
chaieb@33154
  1839
wenzelm@56000
  1840
lemma swap_nz [simp]: "swap n m p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
  1841
  by (induct p) simp_all
chaieb@33154
  1842
krauss@41808
  1843
fun isweaknpoly :: "poly \<Rightarrow> bool"
krauss@41808
  1844
where
chaieb@33154
  1845
  "isweaknpoly (C c) = True"
krauss@41808
  1846
| "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
krauss@41808
  1847
| "isweaknpoly p = False"
chaieb@33154
  1848
wenzelm@52803
  1849
lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
wenzelm@52658
  1850
  by (induct p arbitrary: n0) auto
chaieb@33154
  1851
wenzelm@52803
  1852
lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
wenzelm@52658
  1853
  by (induct p) auto
chaieb@33154
  1854
chaieb@33154
  1855
end