src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
author nipkow
Thu Jun 14 15:45:53 2018 +0200 (12 months ago)
changeset 68442 477b3f7067c9
parent 67123 3fe40ff1b921
permissions -rw-r--r--
tuned
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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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section \<open>Implementation and verification of multivariate polynomials\<close>
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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 \<open>Datatype of polynomial expressions\<close>
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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\<open>Boundedness, substitution and all that\<close>
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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" \<comment> \<open>a poly is INDEPENDENT of Bound 0\<close>
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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" \<comment> \<open>substitute a poly into a poly for Bound 0\<close>
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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 \<open>Degrees and heads and coefficients\<close>
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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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text \<open>More general notions of degree and head.\<close>
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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 \<open>Operations for normalization\<close>
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declare if_cong[fundef_cong del]
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declare let_cong[fundef_cong del]
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fun polyadd :: "poly \<Rightarrow> poly \<Rightarrow> poly"  (infixl "+\<^sub>p" 60)
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  where
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    "polyadd (C c) (C c') = C (c +\<^sub>N c')"
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  | "polyadd (C c) (CN c' n' p') = CN (polyadd (C c) c') n' p'"
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  | "polyadd (CN c n p) (C c') = CN (polyadd c (C c')) n p"
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  | "polyadd (CN c n p) (CN c' n' p') =
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      (if n < n' then CN (polyadd c (CN c' n' p')) n p
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       else if n' < n then CN (polyadd (CN c n p) c') n' p'
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       else
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        let
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          cc' = polyadd c c';
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          pp' = polyadd p p'
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        in if pp' = 0\<^sub>p then cc' else CN cc' n pp')"
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  | "polyadd a b = Add a b"
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fun polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
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  where
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    "polyneg (C c) = C (~\<^sub>N c)"
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  | "polyneg (CN c n p) = CN (polyneg c) n (polyneg p)"
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  | "polyneg a = Neg a"
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definition polysub :: "poly \<Rightarrow> poly \<Rightarrow> poly"  (infixl "-\<^sub>p" 60)
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  where "p -\<^sub>p q = polyadd p (polyneg q)"
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fun polymul :: "poly \<Rightarrow> poly \<Rightarrow> poly"  (infixl "*\<^sub>p" 60)
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  where
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    "polymul (C c) (C c') = C (c *\<^sub>N c')"
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  | "polymul (C c) (CN c' n' p') =
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      (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul (C c) c') n' (polymul (C c) p'))"
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  | "polymul (CN c n p) (C c') =
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      (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul c (C c')) n (polymul p (C c')))"
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  | "polymul (CN c n p) (CN c' n' p') =
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      (if n < n' then CN (polymul c (CN c' n' p')) n (polymul p (CN c' n' p'))
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       else if n' < n then CN (polymul (CN c n p) c') n' (polymul (CN c n p) p')
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       else polyadd (polymul (CN c n p) c') (CN 0\<^sub>p n' (polymul (CN c n p) p')))"
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  | "polymul a b = Mul a b"
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declare if_cong[fundef_cong]
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declare let_cong[fundef_cong]
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fun polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
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  where
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    "polypow 0 = (\<lambda>p. (1)\<^sub>p)"
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  | "polypow n =
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      (\<lambda>p.
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        let
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          q = polypow (n div 2) p;
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          d = polymul q q
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        in if even n then d else polymul p d)"
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abbreviation poly_pow :: "poly \<Rightarrow> nat \<Rightarrow> poly"  (infixl "^\<^sub>p" 60)
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  where "a ^\<^sub>p k \<equiv> polypow k a"
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function polynate :: "poly \<Rightarrow> poly"
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  where
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    "polynate (Bound n) = CN 0\<^sub>p n (1)\<^sub>p"
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  | "polynate (Add p q) = polynate p +\<^sub>p polynate q"
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  | "polynate (Sub p q) = polynate p -\<^sub>p polynate q"
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  | "polynate (Mul p q) = polynate p *\<^sub>p polynate q"
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  | "polynate (Neg p) = ~\<^sub>p (polynate p)"
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  | "polynate (Pw p n) = polynate p ^\<^sub>p n"
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  | "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
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  | "polynate (C c) = C (normNum c)"
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  by pat_completeness auto
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termination by (relation "measure polysize") auto
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fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly"
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where
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  "poly_cmul y (C x) = C (y *\<^sub>N x)"
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| "poly_cmul y (CN c n p) = CN (poly_cmul y c) n (poly_cmul y p)"
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| "poly_cmul y p = C y *\<^sub>p p"
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definition monic :: "poly \<Rightarrow> poly \<times> bool"
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  where "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 \<open>Pseudo-division\<close>
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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 \<open>Semantics of the polynomial representation\<close>
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primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0,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,power}"  ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
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  where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"
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lemma Ipoly_CInt: "Ipoly bs (C (i, 1)) = of_int i"
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  by (simp add: INum_def)
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lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j"
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  by (simp  add: INum_def)
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lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
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subsection \<open>Normal form and normalization\<close>
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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 \<open>polyadd preserves normal forms\<close>
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lemma polyadd_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polyadd p q) (min n0 n1)"
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proof (induct p q arbitrary: n0 n1 rule: polyadd.induct)
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  case (2 ab c' n' p' n0 n1)
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  from 2 have  th1: "isnpolyh (C ab) (Suc n')"
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    by simp
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  from 2(3) have th2: "isnpolyh c' (Suc n')" and nplen1: "n' \<ge> n1"
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    by simp_all
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  with isnpolyh_mono have cp: "isnpolyh c' (Suc n')"
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    by simp
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  with 2(1)[OF th1 th2] have th3:"isnpolyh (C ab +\<^sub>p c') (Suc n')"
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    by simp
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  from nplen1 have n01len1: "min n0 n1 \<le> n'"
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    by simp
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  then show ?case using 2 th3
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    by simp
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next
krauss@41812
   292
  case (3 c' n' p' ab n1 n0)
wenzelm@56009
   293
  from 3 have  th1: "isnpolyh (C ab) (Suc n')"
wenzelm@56009
   294
    by simp
wenzelm@56009
   295
  from 3(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1"
wenzelm@56009
   296
    by simp_all
wenzelm@56009
   297
  with isnpolyh_mono have cp: "isnpolyh c' (Suc n')"
wenzelm@56009
   298
    by simp
wenzelm@56009
   299
  with 3(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C ab) (Suc n')"
wenzelm@56009
   300
    by simp
wenzelm@56009
   301
  from nplen1 have n01len1: "min n0 n1 \<le> n'"
wenzelm@56009
   302
    by simp
wenzelm@56009
   303
  then show ?case using 3 th3
wenzelm@56009
   304
    by simp
chaieb@33154
   305
next
chaieb@33154
   306
  case (4 c n p c' n' p' n0 n1)
wenzelm@56009
   307
  then have nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n"
wenzelm@56009
   308
    by simp_all
wenzelm@56009
   309
  from 4 have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'"
wenzelm@56009
   310
    by simp_all
wenzelm@56009
   311
  from 4 have ngen0: "n \<ge> n0"
wenzelm@56009
   312
    by simp
wenzelm@56009
   313
  from 4 have n'gen1: "n' \<ge> n1"
wenzelm@56009
   314
    by simp
wenzelm@60698
   315
  consider (eq) "n = n'" | (lt) "n < n'" | (gt) "n > n'"
wenzelm@60698
   316
    by arith
wenzelm@60698
   317
  then show ?case
wenzelm@60698
   318
  proof cases
wenzelm@60698
   319
    case eq
wenzelm@52803
   320
    with "4.hyps"(3)[OF nc nc']
wenzelm@56009
   321
    have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)"
wenzelm@56009
   322
      by auto
wenzelm@56009
   323
    then have ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
wenzelm@56009
   324
      using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1
wenzelm@56009
   325
      by auto
wenzelm@56009
   326
    from eq "4.hyps"(4)[OF np np'] have npp': "isnpolyh (p +\<^sub>p p') n"
wenzelm@56009
   327
      by simp
wenzelm@56009
   328
    have minle: "min n0 n1 \<le> n'"
wenzelm@56009
   329
      using ngen0 n'gen1 eq by simp
wenzelm@60698
   330
    from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' show ?thesis
wenzelm@56009
   331
      by (simp add: Let_def)
wenzelm@60698
   332
  next
wenzelm@60698
   333
    case lt
wenzelm@56009
   334
    have "min n0 n1 \<le> n0"
wenzelm@56009
   335
      by simp
wenzelm@56009
   336
    with 4 lt have th1:"min n0 n1 \<le> n"
wenzelm@56009
   337
      by auto
wenzelm@56009
   338
    from 4 have th21: "isnpolyh c (Suc n)"
wenzelm@56009
   339
      by simp
wenzelm@56009
   340
    from 4 have th22: "isnpolyh (CN c' n' p') n'"
wenzelm@56009
   341
      by simp
wenzelm@56009
   342
    from lt have th23: "min (Suc n) n' = Suc n"
wenzelm@56009
   343
      by arith
wenzelm@56009
   344
    from "4.hyps"(1)[OF th21 th22] have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)"
wenzelm@56009
   345
      using th23 by simp
wenzelm@60698
   346
    with 4 lt th1 show ?thesis
wenzelm@56009
   347
      by simp
wenzelm@60698
   348
  next
wenzelm@60698
   349
    case gt
wenzelm@56009
   350
    then have gt': "n' < n \<and> \<not> n < n'"
wenzelm@56009
   351
      by simp
wenzelm@56009
   352
    have "min n0 n1 \<le> n1"
wenzelm@56009
   353
      by simp
wenzelm@56009
   354
    with 4 gt have th1: "min n0 n1 \<le> n'"
wenzelm@56009
   355
      by auto
wenzelm@56009
   356
    from 4 have th21: "isnpolyh c' (Suc n')"
wenzelm@56009
   357
      by simp_all
wenzelm@56009
   358
    from 4 have th22: "isnpolyh (CN c n p) n"
wenzelm@56009
   359
      by simp
wenzelm@56009
   360
    from gt have th23: "min n (Suc n') = Suc n'"
wenzelm@56009
   361
      by arith
wenzelm@56009
   362
    from "4.hyps"(2)[OF th22 th21] have "isnpolyh (polyadd (CN c n p) c') (Suc n')"
wenzelm@56009
   363
      using th23 by simp
wenzelm@60698
   364
    with 4 gt th1 show ?thesis
wenzelm@56009
   365
      by simp
wenzelm@60698
   366
  qed
chaieb@33154
   367
qed auto
chaieb@33154
   368
krauss@41812
   369
lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
wenzelm@52658
   370
  by (induct p q rule: polyadd.induct)
hoelzl@58776
   371
     (auto simp add: Let_def field_simps distrib_left[symmetric] simp del: distrib_left_NO_MATCH)
chaieb@33154
   372
wenzelm@56009
   373
lemma polyadd_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polyadd p q)"
wenzelm@60698
   374
  using polyadd_normh[of p 0 q 0] isnpoly_def by simp
chaieb@33154
   375
wenzelm@60698
   376
text \<open>The degree of addition and other general lemmas needed for the normal form of polymul.\<close>
chaieb@33154
   377
wenzelm@52803
   378
lemma polyadd_different_degreen:
wenzelm@56009
   379
  assumes "isnpolyh p n0"
wenzelm@56009
   380
    and "isnpolyh q n1"
wenzelm@56009
   381
    and "degreen p m \<noteq> degreen q m"
wenzelm@56009
   382
    and "m \<le> min n0 n1"
wenzelm@56009
   383
  shows "degreen (polyadd p q) m = max (degreen p m) (degreen q m)"
wenzelm@56009
   384
  using assms
chaieb@33154
   385
proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
chaieb@33154
   386
  case (4 c n p c' n' p' m n0 n1)
wenzelm@60698
   387
  show ?case
wenzelm@60698
   388
  proof (cases "n = n'")
wenzelm@60698
   389
    case True
wenzelm@60698
   390
    with 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
krauss@41763
   391
    show ?thesis by (auto simp: Let_def)
krauss@41763
   392
  next
wenzelm@60698
   393
    case False
krauss@41763
   394
    with 4 show ?thesis by auto
krauss@41763
   395
  qed
krauss@41763
   396
qed auto
chaieb@33154
   397
wenzelm@56009
   398
lemma headnz[simp]: "isnpolyh p n \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
wenzelm@52658
   399
  by (induct p arbitrary: n rule: headn.induct) auto
wenzelm@56009
   400
chaieb@33154
   401
lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
wenzelm@52658
   402
  by (induct p arbitrary: n rule: degree.induct) auto
wenzelm@56009
   403
chaieb@33154
   404
lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
wenzelm@52658
   405
  by (induct p arbitrary: n rule: degreen.induct) auto
chaieb@33154
   406
chaieb@33154
   407
lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
wenzelm@52658
   408
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
   409
chaieb@33154
   410
lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
chaieb@33154
   411
  using degree_isnpolyh_Suc by auto
wenzelm@56009
   412
chaieb@33154
   413
lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
chaieb@33154
   414
  using degreen_0 by auto
chaieb@33154
   415
chaieb@33154
   416
chaieb@33154
   417
lemma degreen_polyadd:
wenzelm@56009
   418
  assumes np: "isnpolyh p n0"
wenzelm@56009
   419
    and nq: "isnpolyh q n1"
wenzelm@56009
   420
    and m: "m \<le> max n0 n1"
chaieb@33154
   421
  shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
chaieb@33154
   422
  using np nq m
chaieb@33154
   423
proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
wenzelm@52803
   424
  case (2 c c' n' p' n0 n1)
wenzelm@56009
   425
  then show ?case
wenzelm@56009
   426
    by (cases n') simp_all
chaieb@33154
   427
next
wenzelm@52803
   428
  case (3 c n p c' n0 n1)
wenzelm@56009
   429
  then show ?case
wenzelm@56009
   430
    by (cases n) auto
chaieb@33154
   431
next
wenzelm@52803
   432
  case (4 c n p c' n' p' n0 n1 m)
wenzelm@60698
   433
  show ?case
wenzelm@60698
   434
  proof (cases "n = n'")
wenzelm@60698
   435
    case True
wenzelm@60698
   436
    with 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
krauss@41763
   437
    show ?thesis by (auto simp: Let_def)
wenzelm@60698
   438
  next
wenzelm@60698
   439
    case False
wenzelm@60698
   440
    then show ?thesis by simp
wenzelm@60698
   441
  qed
chaieb@33154
   442
qed auto
chaieb@33154
   443
wenzelm@56009
   444
lemma polyadd_eq_const_degreen:
wenzelm@56009
   445
  assumes "isnpolyh p n0"
wenzelm@56009
   446
    and "isnpolyh q n1"
wenzelm@56009
   447
    and "polyadd p q = C c"
wenzelm@56009
   448
  shows "degreen p m = degreen q m"
wenzelm@56009
   449
  using assms
chaieb@33154
   450
proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
wenzelm@52803
   451
  case (4 c n p c' n' p' m n0 n1 x)
wenzelm@60698
   452
  consider "n = n'" | "n > n' \<or> n < n'" by arith
wenzelm@60698
   453
  then show ?case
wenzelm@60698
   454
  proof cases
wenzelm@60698
   455
    case 1
wenzelm@60698
   456
    with 4 show ?thesis
wenzelm@60698
   457
      by (cases "p +\<^sub>p p' = 0\<^sub>p") (auto simp add: Let_def)
wenzelm@60698
   458
  next
wenzelm@60698
   459
    case 2
wenzelm@60698
   460
    with 4 show ?thesis by auto
wenzelm@60698
   461
  qed
chaieb@33154
   462
qed simp_all
chaieb@33154
   463
chaieb@33154
   464
lemma polymul_properties:
nipkow@68442
   465
  assumes "SORT_CONSTRAINT('a::field_char_0)"
wenzelm@52658
   466
    and np: "isnpolyh p n0"
wenzelm@52658
   467
    and nq: "isnpolyh q n1"
wenzelm@52658
   468
    and m: "m \<le> min n0 n1"
wenzelm@52803
   469
  shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
wenzelm@56009
   470
    and "p *\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p \<or> q = 0\<^sub>p"
wenzelm@56009
   471
    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
   472
  using np nq m
wenzelm@52658
   473
proof (induct p q arbitrary: n0 n1 m rule: polymul.induct)
wenzelm@52803
   474
  case (2 c c' n' p')
wenzelm@56009
   475
  {
wenzelm@56009
   476
    case (1 n0 n1)
wenzelm@56009
   477
    with "2.hyps"(4-6)[of n' n' n'] and "2.hyps"(1-3)[of "Suc n'" "Suc n'" n']
krauss@41811
   478
    show ?case by (auto simp add: min_def)
chaieb@33154
   479
  next
wenzelm@56009
   480
    case (2 n0 n1)
wenzelm@56009
   481
    then show ?case by auto
chaieb@33154
   482
  next
wenzelm@56009
   483
    case (3 n0 n1)
wenzelm@60698
   484
    then show ?case using "2.hyps" by auto
wenzelm@56009
   485
  }
chaieb@33154
   486
next
krauss@41813
   487
  case (3 c n p c')
wenzelm@56009
   488
  {
wenzelm@56009
   489
    case (1 n0 n1)
wenzelm@56009
   490
    with "3.hyps"(4-6)[of n n n] and "3.hyps"(1-3)[of "Suc n" "Suc n" n]
krauss@41811
   491
    show ?case by (auto simp add: min_def)
chaieb@33154
   492
  next
wenzelm@56009
   493
    case (2 n0 n1)
wenzelm@56009
   494
    then show ?case by auto
chaieb@33154
   495
  next
wenzelm@56009
   496
    case (3 n0 n1)
wenzelm@56009
   497
    then show ?case  using "3.hyps" by auto
wenzelm@56009
   498
  }
chaieb@33154
   499
next
chaieb@33154
   500
  case (4 c n p c' n' p')
chaieb@33154
   501
  let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
wenzelm@56009
   502
  {
wenzelm@56009
   503
    case (1 n0 n1)
wenzelm@56009
   504
    then have cnp: "isnpolyh ?cnp n"
wenzelm@56009
   505
      and cnp': "isnpolyh ?cnp' n'"
wenzelm@56009
   506
      and np: "isnpolyh p n"
wenzelm@56009
   507
      and nc: "isnpolyh c (Suc n)"
wenzelm@56009
   508
      and np': "isnpolyh p' n'"
wenzelm@56009
   509
      and nc': "isnpolyh c' (Suc n')"
wenzelm@56009
   510
      and nn0: "n \<ge> n0"
wenzelm@56009
   511
      and nn1: "n' \<ge> n1"
wenzelm@56009
   512
      by simp_all
wenzelm@67123
   513
    consider "n < n'" | "n' < n" | "n' = n" by arith
wenzelm@67123
   514
    then show ?case
wenzelm@67123
   515
    proof cases
wenzelm@67123
   516
      case 1
wenzelm@56009
   517
      with "4.hyps"(4-5)[OF np cnp', of n] and "4.hyps"(1)[OF nc cnp', of n] nn0 cnp
wenzelm@67123
   518
      show ?thesis by (simp add: min_def)
wenzelm@67123
   519
    next
wenzelm@67123
   520
      case 2
wenzelm@56009
   521
      with "4.hyps"(16-17)[OF cnp np', of "n'"] and "4.hyps"(13)[OF cnp nc', of "Suc n'"] nn1 cnp'
wenzelm@67123
   522
      show ?thesis by (cases "Suc n' = n") (simp_all add: min_def)
wenzelm@67123
   523
    next
wenzelm@67123
   524
      case 3
wenzelm@56009
   525
      with "4.hyps"(16-17)[OF cnp np', of n] and "4.hyps"(13)[OF cnp nc', of n] cnp cnp' nn1 nn0
wenzelm@67123
   526
      show ?thesis
wenzelm@67123
   527
        by (auto intro!: polyadd_normh) (simp_all add: min_def isnpolyh_mono[OF nn0])
wenzelm@67123
   528
    qed
wenzelm@56009
   529
  next
wenzelm@56009
   530
    fix n0 n1 m
wenzelm@56009
   531
    assume np: "isnpolyh ?cnp n0"
wenzelm@56009
   532
    assume np':"isnpolyh ?cnp' n1"
wenzelm@56009
   533
    assume m: "m \<le> min n0 n1"
wenzelm@56009
   534
    let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
wenzelm@56009
   535
    let ?d1 = "degreen ?cnp m"
wenzelm@56009
   536
    let ?d2 = "degreen ?cnp' m"
wenzelm@56009
   537
    let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
wenzelm@67123
   538
    consider "n' < n \<or> n < n'" | "n' = n" by linarith
wenzelm@67123
   539
    then show ?eq
wenzelm@67123
   540
    proof cases
wenzelm@67123
   541
      case 1
wenzelm@67123
   542
      with "4.hyps"(3,6,18) np np' m show ?thesis by auto
wenzelm@67123
   543
    next
wenzelm@67123
   544
      case 2
wenzelm@67123
   545
      have nn': "n' = n" by fact
wenzelm@56009
   546
      then have nn: "\<not> n' < n \<and> \<not> n < n'" by arith
wenzelm@56009
   547
      from "4.hyps"(16,18)[of n n' n]
wenzelm@56009
   548
        "4.hyps"(13,14)[of n "Suc n'" n]
wenzelm@56009
   549
        np np' nn'
wenzelm@56043
   550
      have norm:
wenzelm@56043
   551
        "isnpolyh ?cnp n"
wenzelm@56043
   552
        "isnpolyh c' (Suc n)"
wenzelm@56043
   553
        "isnpolyh (?cnp *\<^sub>p c') n"
wenzelm@56043
   554
        "isnpolyh p' n"
wenzelm@56043
   555
        "isnpolyh (?cnp *\<^sub>p p') n"
wenzelm@56043
   556
        "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@56043
   557
        "?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p"
wenzelm@56043
   558
        "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
wenzelm@56043
   559
        by (auto simp add: min_def)
wenzelm@67123
   560
      show ?thesis
wenzelm@67123
   561
      proof (cases "m = n")
wenzelm@67123
   562
        case mn: True
wenzelm@56009
   563
        from "4.hyps"(17,18)[OF norm(1,4), of n]
wenzelm@56009
   564
          "4.hyps"(13,15)[OF norm(1,2), of n] norm nn' mn
wenzelm@56009
   565
        have degs:
wenzelm@56009
   566
          "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else ?d1 + degreen c' n)"
wenzelm@56009
   567
          "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n"
wenzelm@56009
   568
          by (simp_all add: min_def)
wenzelm@56009
   569
        from degs norm have th1: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
wenzelm@56009
   570
          by simp
wenzelm@56009
   571
        then have neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
wenzelm@56009
   572
          by simp
wenzelm@56009
   573
        have nmin: "n \<le> min n n"
wenzelm@56009
   574
          by (simp add: min_def)
wenzelm@56009
   575
        from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
wenzelm@56009
   576
        have deg: "degreen (CN c n p *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n =
wenzelm@56009
   577
            degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@56009
   578
          by simp
wenzelm@56009
   579
        from "4.hyps"(16-18)[OF norm(1,4), of n]
wenzelm@56009
   580
          "4.hyps"(13-15)[OF norm(1,2), of n]
wenzelm@56009
   581
          mn norm m nn' deg
wenzelm@67123
   582
        show ?thesis by simp
wenzelm@67123
   583
      next
wenzelm@67123
   584
        case mn: False
wenzelm@56009
   585
        then have mn': "m < n"
wenzelm@56009
   586
          using m np by auto
wenzelm@56009
   587
        from nn' m np have max1: "m \<le> max n n"
wenzelm@56009
   588
          by simp
wenzelm@56009
   589
        then have min1: "m \<le> min n n"
wenzelm@56009
   590
          by simp
wenzelm@56009
   591
        then have min2: "m \<le> min n (Suc n)"
wenzelm@56009
   592
          by simp
wenzelm@56009
   593
        from "4.hyps"(16-18)[OF norm(1,4) min1]
wenzelm@56009
   594
          "4.hyps"(13-15)[OF norm(1,2) min2]
wenzelm@56009
   595
          degreen_polyadd[OF norm(3,6) max1]
wenzelm@56009
   596
        have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m \<le>
wenzelm@56009
   597
            max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
wenzelm@56009
   598
          using mn nn' np np' by simp
wenzelm@56009
   599
        with "4.hyps"(16-18)[OF norm(1,4) min1]
wenzelm@56009
   600
          "4.hyps"(13-15)[OF norm(1,2) min2]
wenzelm@56009
   601
          degreen_0[OF norm(3) mn']
wenzelm@67123
   602
          nn' mn np np'
wenzelm@67123
   603
        show ?thesis by clarsimp
wenzelm@67123
   604
      qed
wenzelm@67123
   605
    qed
wenzelm@56009
   606
  }
wenzelm@56009
   607
  {
wenzelm@56009
   608
    case (2 n0 n1)
wenzelm@56009
   609
    then have np: "isnpolyh ?cnp n0"
wenzelm@56009
   610
      and np': "isnpolyh ?cnp' n1"
wenzelm@56043
   611
      and m: "m \<le> min n0 n1"
wenzelm@56043
   612
      by simp_all
wenzelm@56009
   613
    then have mn: "m \<le> n" by simp
wenzelm@56009
   614
    let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
wenzelm@67123
   615
    have False if C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
wenzelm@67123
   616
    proof -
wenzelm@67123
   617
      from C have nn: "\<not> n' < n \<and> \<not> n < n'"
wenzelm@56009
   618
        by simp
wenzelm@56009
   619
      from "4.hyps"(16-18) [of n n n]
wenzelm@56009
   620
        "4.hyps"(13-15)[of n "Suc n" n]
wenzelm@56009
   621
        np np' C(2) mn
wenzelm@56009
   622
      have norm:
wenzelm@56009
   623
        "isnpolyh ?cnp n"
wenzelm@56009
   624
        "isnpolyh c' (Suc n)"
wenzelm@56009
   625
        "isnpolyh (?cnp *\<^sub>p c') n"
wenzelm@56009
   626
        "isnpolyh p' n"
wenzelm@56009
   627
        "isnpolyh (?cnp *\<^sub>p p') n"
wenzelm@56009
   628
        "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@56009
   629
        "?cnp *\<^sub>p c' = 0\<^sub>p \<longleftrightarrow> c' = 0\<^sub>p"
wenzelm@56009
   630
        "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p"
wenzelm@56009
   631
        "degreen (?cnp *\<^sub>p c') n = (if c' = 0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
wenzelm@56009
   632
        "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
wenzelm@56009
   633
        by (simp_all add: min_def)
wenzelm@56009
   634
      from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
wenzelm@56009
   635
        by simp
wenzelm@56009
   636
      have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
wenzelm@56009
   637
        using norm by simp
wenzelm@56009
   638
      from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"] degneq
wenzelm@67123
   639
      show ?thesis by simp
wenzelm@67123
   640
    qed
wenzelm@56009
   641
    then show ?case using "4.hyps" by clarsimp
wenzelm@56009
   642
  }
chaieb@33154
   643
qed auto
chaieb@33154
   644
wenzelm@56009
   645
lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = Ipoly bs p * Ipoly bs q"
wenzelm@52658
   646
  by (induct p q rule: polymul.induct) (auto simp add: field_simps)
chaieb@33154
   647
wenzelm@52803
   648
lemma polymul_normh:
nipkow@68442
   649
  assumes "SORT_CONSTRAINT('a::field_char_0)"
wenzelm@56009
   650
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
wenzelm@52803
   651
  using polymul_properties(1) by blast
wenzelm@52658
   652
wenzelm@52803
   653
lemma polymul_eq0_iff:
nipkow@68442
   654
  assumes "SORT_CONSTRAINT('a::field_char_0)"
wenzelm@56009
   655
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p *\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p \<or> q = 0\<^sub>p"
wenzelm@52803
   656
  using polymul_properties(2) by blast
wenzelm@52658
   657
wenzelm@56207
   658
lemma polymul_degreen:
nipkow@68442
   659
  assumes "SORT_CONSTRAINT('a::field_char_0)"
wenzelm@56009
   660
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> m \<le> min n0 n1 \<Longrightarrow>
wenzelm@56009
   661
    degreen (p *\<^sub>p q) m = (if p = 0\<^sub>p \<or> q = 0\<^sub>p then 0 else degreen p m + degreen q m)"
wenzelm@56207
   662
  by (fact polymul_properties(3))
wenzelm@52658
   663
wenzelm@52803
   664
lemma polymul_norm:
nipkow@68442
   665
  assumes "SORT_CONSTRAINT('a::field_char_0)"
wenzelm@56009
   666
  shows "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polymul p q)"
chaieb@33154
   667
  using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
chaieb@33154
   668
chaieb@33154
   669
lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
   670
  by (induct p arbitrary: n0 rule: headconst.induct) auto
chaieb@33154
   671
chaieb@33154
   672
lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
wenzelm@52658
   673
  by (induct p arbitrary: n0) auto
chaieb@33154
   674
wenzelm@52658
   675
lemma monic_eqI:
wenzelm@52803
   676
  assumes np: "isnpolyh p n0"
wenzelm@52658
   677
  shows "INum (headconst p) * Ipoly bs (fst (monic p)) =
nipkow@68442
   678
    (Ipoly bs p ::'a::{field_char_0, power})"
chaieb@33154
   679
  unfolding monic_def Let_def
wenzelm@52658
   680
proof (cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
chaieb@33154
   681
  let ?h = "headconst p"
chaieb@33154
   682
  assume pz: "p \<noteq> 0\<^sub>p"
wenzelm@56000
   683
  {
wenzelm@56000
   684
    assume hz: "INum ?h = (0::'a)"
wenzelm@56043
   685
    from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N"
wenzelm@56043
   686
      by simp_all
wenzelm@56043
   687
    from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N"
wenzelm@56043
   688
      by simp
wenzelm@56043
   689
    with headconst_zero[OF np] have "p = 0\<^sub>p"
wenzelm@56043
   690
      by blast
wenzelm@56043
   691
    with pz have False
wenzelm@56043
   692
      by blast
wenzelm@56043
   693
  }
wenzelm@56043
   694
  then show "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0"
wenzelm@56043
   695
    by blast
chaieb@33154
   696
qed
chaieb@33154
   697
chaieb@33154
   698
wenzelm@60698
   699
text \<open>polyneg is a negation and preserves normal forms\<close>
chaieb@33154
   700
chaieb@33154
   701
lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
wenzelm@52658
   702
  by (induct p rule: polyneg.induct) auto
chaieb@33154
   703
wenzelm@56009
   704
lemma polyneg0: "isnpolyh p n \<Longrightarrow> (~\<^sub>p p) = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
   705
  by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: Nneg_def)
wenzelm@56009
   706
chaieb@33154
   707
lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
wenzelm@52658
   708
  by (induct p arbitrary: n0 rule: polyneg.induct) auto
wenzelm@56009
   709
wenzelm@56009
   710
lemma polyneg_normh: "isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n"
wenzelm@56009
   711
  by (induct p arbitrary: n rule: polyneg.induct) (auto simp add: polyneg0)
chaieb@33154
   712
chaieb@33154
   713
lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
chaieb@33154
   714
  using isnpoly_def polyneg_normh by simp
chaieb@33154
   715
chaieb@33154
   716
wenzelm@60698
   717
text \<open>polysub is a substraction and preserves normal forms\<close>
krauss@41404
   718
wenzelm@56009
   719
lemma polysub[simp]: "Ipoly bs (polysub p q) = Ipoly bs p - Ipoly bs q"
wenzelm@52658
   720
  by (simp add: polysub_def)
wenzelm@56009
   721
wenzelm@56009
   722
lemma polysub_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polysub p q) (min n0 n1)"
wenzelm@52658
   723
  by (simp add: polysub_def polyneg_normh polyadd_normh)
chaieb@33154
   724
wenzelm@56009
   725
lemma polysub_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polysub p q)"
wenzelm@52803
   726
  using polyadd_norm polyneg_norm by (simp add: polysub_def)
wenzelm@56009
   727
wenzelm@52658
   728
lemma polysub_same_0[simp]:
nipkow@68442
   729
  assumes "SORT_CONSTRAINT('a::field_char_0)"
krauss@41814
   730
  shows "isnpolyh p n0 \<Longrightarrow> polysub p p = 0\<^sub>p"
wenzelm@52658
   731
  unfolding polysub_def split_def fst_conv snd_conv
wenzelm@52658
   732
  by (induct p arbitrary: n0) (auto simp add: Let_def Nsub0[simplified Nsub_def])
chaieb@33154
   733
wenzelm@52803
   734
lemma polysub_0:
nipkow@68442
   735
  assumes "SORT_CONSTRAINT('a::field_char_0)"
wenzelm@56009
   736
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p -\<^sub>p q = 0\<^sub>p \<longleftrightarrow> p = q"
chaieb@33154
   737
  unfolding polysub_def split_def fst_conv snd_conv
krauss@41763
   738
  by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
wenzelm@52658
   739
    (auto simp: Nsub0[simplified Nsub_def] Let_def)
chaieb@33154
   740
wenzelm@60698
   741
text \<open>polypow is a power function and preserves normal forms\<close>
krauss@41404
   742
nipkow@68442
   743
lemma polypow[simp]: "Ipoly bs (polypow n p) = (Ipoly bs p :: 'a::field_char_0) ^ n"
wenzelm@52658
   744
proof (induct n rule: polypow.induct)
wenzelm@52658
   745
  case 1
wenzelm@67123
   746
  then show ?case by simp
chaieb@33154
   747
next
chaieb@33154
   748
  case (2 n)
chaieb@33154
   749
  let ?q = "polypow ((Suc n) div 2) p"
krauss@41813
   750
  let ?d = "polymul ?q ?q"
wenzelm@67123
   751
  consider "odd (Suc n)" | "even (Suc n)" by auto
wenzelm@67123
   752
  then show ?case
wenzelm@67123
   753
  proof cases
wenzelm@67123
   754
    case odd: 1
wenzelm@67123
   755
    have *: "(Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0)))) = Suc n div 2 + Suc n div 2 + 1"
wenzelm@52658
   756
      by arith
wenzelm@56043
   757
    from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul p ?d)"
wenzelm@56043
   758
      by (simp add: Let_def)
wenzelm@56043
   759
    also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2) * (Ipoly bs p)^(Suc n div 2)"
chaieb@33154
   760
      using "2.hyps" by simp
chaieb@33154
   761
    also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
wenzelm@52658
   762
      by (simp only: power_add power_one_right) simp
wenzelm@56000
   763
    also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc 0) * (Suc n div Suc (Suc 0))))"
wenzelm@67123
   764
      by (simp only: *)
wenzelm@67123
   765
    finally show ?thesis
wenzelm@67123
   766
      unfolding numeral_2_eq_2 [symmetric]
wenzelm@67123
   767
      using odd_two_times_div_two_nat [OF odd] by simp
wenzelm@67123
   768
  next
wenzelm@67123
   769
    case even: 2
wenzelm@56043
   770
    from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d"
wenzelm@56043
   771
      by (simp add: Let_def)
haftmann@58710
   772
    also have "\<dots> = (Ipoly bs p) ^ (2 * (Suc n div 2))"
haftmann@58710
   773
      using "2.hyps" by (simp only: mult_2 power_add) simp
wenzelm@67123
   774
    finally show ?thesis
wenzelm@67123
   775
      using even_two_times_div_two [OF even] by simp
wenzelm@67123
   776
  qed
chaieb@33154
   777
qed
chaieb@33154
   778
wenzelm@52803
   779
lemma polypow_normh:
nipkow@68442
   780
  assumes "SORT_CONSTRAINT('a::field_char_0)"
chaieb@33154
   781
  shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
chaieb@33154
   782
proof (induct k arbitrary: n rule: polypow.induct)
wenzelm@56043
   783
  case 1
wenzelm@56043
   784
  then show ?case by auto
wenzelm@56043
   785
next
chaieb@33154
   786
  case (2 k n)
chaieb@33154
   787
  let ?q = "polypow (Suc k div 2) p"
krauss@41813
   788
  let ?d = "polymul ?q ?q"
wenzelm@67123
   789
  from 2 have *: "isnpolyh ?q n" and **: "isnpolyh p n"
wenzelm@56043
   790
    by blast+
wenzelm@67123
   791
  from polymul_normh[OF * *] have dn: "isnpolyh ?d n"
wenzelm@56043
   792
    by simp
wenzelm@67123
   793
  from polymul_normh[OF ** dn] have on: "isnpolyh (polymul p ?d) n"
wenzelm@56043
   794
    by simp
haftmann@58710
   795
  from dn on show ?case by (simp, unfold Let_def) auto
wenzelm@56043
   796
qed
chaieb@33154
   797
wenzelm@52803
   798
lemma polypow_norm:
nipkow@68442
   799
  assumes "SORT_CONSTRAINT('a::field_char_0)"
chaieb@33154
   800
  shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
chaieb@33154
   801
  by (simp add: polypow_normh isnpoly_def)
chaieb@33154
   802
wenzelm@60698
   803
text \<open>Finally the whole normalization\<close>
chaieb@33154
   804
nipkow@68442
   805
lemma polynate [simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::field_char_0)"
wenzelm@52658
   806
  by (induct p rule:polynate.induct) auto
chaieb@33154
   807
wenzelm@52803
   808
lemma polynate_norm[simp]:
nipkow@68442
   809
  assumes "SORT_CONSTRAINT('a::field_char_0)"
chaieb@33154
   810
  shows "isnpoly (polynate p)"
wenzelm@52658
   811
  by (induct p rule: polynate.induct)
wenzelm@52658
   812
     (simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm,
wenzelm@52658
   813
      simp_all add: isnpoly_def)
chaieb@33154
   814
wenzelm@60698
   815
text \<open>shift1\<close>
chaieb@33154
   816
chaieb@33154
   817
lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
wenzelm@52658
   818
  by (simp add: shift1_def)
chaieb@33154
   819
wenzelm@52803
   820
lemma shift1_isnpoly:
wenzelm@56207
   821
  assumes "isnpoly p"
wenzelm@56207
   822
    and "p \<noteq> 0\<^sub>p"
wenzelm@52658
   823
  shows "isnpoly (shift1 p) "
wenzelm@56207
   824
  using assms by (simp add: shift1_def isnpoly_def)
chaieb@33154
   825
chaieb@33154
   826
lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
chaieb@33154
   827
  by (simp add: shift1_def)
wenzelm@56043
   828
wenzelm@56043
   829
lemma funpow_shift1_isnpoly: "isnpoly p \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpoly (funpow n shift1 p)"
haftmann@39246
   830
  by (induct n arbitrary: p) (auto simp add: shift1_isnpoly funpow_swap1)
chaieb@33154
   831
wenzelm@52803
   832
lemma funpow_isnpolyh:
wenzelm@56207
   833
  assumes "\<And>p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n"
wenzelm@56207
   834
    and "isnpolyh p n"
chaieb@33154
   835
  shows "isnpolyh (funpow k f p) n"
wenzelm@56207
   836
  using assms by (induct k arbitrary: p) auto
chaieb@33154
   837
wenzelm@52658
   838
lemma funpow_shift1:
nipkow@68442
   839
  "(Ipoly bs (funpow n shift1 p) :: 'a :: field_char_0) =
wenzelm@52658
   840
    Ipoly bs (Mul (Pw (Bound 0) n) p)"
wenzelm@52658
   841
  by (induct n arbitrary: p) (simp_all add: shift1_isnpoly shift1)
chaieb@33154
   842
wenzelm@56043
   843
lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
chaieb@33154
   844
  using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
chaieb@33154
   845
wenzelm@52803
   846
lemma funpow_shift1_1:
nipkow@68442
   847
  "(Ipoly bs (funpow n shift1 p) :: 'a :: field_char_0) =
wenzelm@52658
   848
    Ipoly bs (funpow n shift1 (1)\<^sub>p *\<^sub>p p)"
chaieb@33154
   849
  by (simp add: funpow_shift1)
chaieb@33154
   850
chaieb@33154
   851
lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
wenzelm@45129
   852
  by (induct p rule: poly_cmul.induct) (auto simp add: field_simps)
chaieb@33154
   853
chaieb@33154
   854
lemma behead:
wenzelm@56207
   855
  assumes "isnpolyh p n"
wenzelm@52658
   856
  shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) =
nipkow@68442
   857
    (Ipoly bs p :: 'a :: field_char_0)"
wenzelm@56207
   858
  using assms
chaieb@33154
   859
proof (induct p arbitrary: n rule: behead.induct)
wenzelm@56009
   860
  case (1 c p n)
wenzelm@56009
   861
  then have pn: "isnpolyh p n" by simp
wenzelm@52803
   862
  from 1(1)[OF pn]
wenzelm@52803
   863
  have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" .
wenzelm@52658
   864
  then show ?case using "1.hyps"
wenzelm@52658
   865
    apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
wenzelm@52658
   866
    apply (simp_all add: th[symmetric] field_simps)
wenzelm@52658
   867
    done
chaieb@33154
   868
qed (auto simp add: Let_def)
chaieb@33154
   869
chaieb@33154
   870
lemma behead_isnpolyh:
wenzelm@56207
   871
  assumes "isnpolyh p n"
wenzelm@52658
   872
  shows "isnpolyh (behead p) n"
wenzelm@56207
   873
  using assms by (induct p rule: behead.induct) (auto simp add: Let_def isnpolyh_mono)
wenzelm@52658
   874
chaieb@33154
   875
wenzelm@60533
   876
subsection \<open>Miscellaneous lemmas about indexes, decrementation, substitution  etc ...\<close>
wenzelm@52658
   877
chaieb@33154
   878
lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
wenzelm@61166
   879
proof (induct p arbitrary: n rule: poly.induct, auto, goal_cases)
wenzelm@60580
   880
  case prems: (1 c n p n')
wenzelm@56009
   881
  then have "n = Suc (n - 1)"
wenzelm@56009
   882
    by simp
wenzelm@56009
   883
  then have "isnpolyh p (Suc (n - 1))"
wenzelm@60533
   884
    using \<open>isnpolyh p n\<close> by simp
wenzelm@60580
   885
  with prems(2) show ?case
wenzelm@56009
   886
    by simp
chaieb@33154
   887
qed
chaieb@33154
   888
chaieb@33154
   889
lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
wenzelm@52658
   890
  by (induct p arbitrary: n0 rule: isconstant.induct) (auto simp add: isnpolyh_polybound0)
chaieb@33154
   891
wenzelm@52658
   892
lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
   893
  by (induct p) auto
chaieb@33154
   894
chaieb@33154
   895
lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
wenzelm@52658
   896
  apply (induct p arbitrary: n0)
wenzelm@67123
   897
         apply auto
wenzelm@56043
   898
  apply atomize
blanchet@58259
   899
  apply (rename_tac nat a b, erule_tac x = "Suc nat" in allE)
chaieb@33154
   900
  apply auto
chaieb@33154
   901
  done
chaieb@33154
   902
chaieb@33154
   903
lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
wenzelm@52658
   904
  by (induct p  arbitrary: n0 rule: head.induct) (auto intro: isnpolyh_polybound0)
chaieb@33154
   905
chaieb@33154
   906
lemma polybound0_I:
wenzelm@56207
   907
  assumes "polybound0 a"
wenzelm@56009
   908
  shows "Ipoly (b # bs) a = Ipoly (b' # bs) a"
wenzelm@56207
   909
  using assms by (induct a rule: poly.induct) auto
wenzelm@52658
   910
wenzelm@56009
   911
lemma polysubst0_I: "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b # bs) a) # bs) t"
chaieb@33154
   912
  by (induct t) simp_all
chaieb@33154
   913
chaieb@33154
   914
lemma polysubst0_I':
wenzelm@56207
   915
  assumes "polybound0 a"
wenzelm@56009
   916
  shows "Ipoly (b # bs) (polysubst0 a t) = Ipoly ((Ipoly (b' # bs) a) # bs) t"
wenzelm@56207
   917
  by (induct t) (simp_all add: polybound0_I[OF assms, where b="b" and b'="b'"])
chaieb@33154
   918
wenzelm@52658
   919
lemma decrpoly:
wenzelm@56207
   920
  assumes "polybound0 t"
wenzelm@56043
   921
  shows "Ipoly (x # bs) t = Ipoly bs (decrpoly t)"
wenzelm@56207
   922
  using assms by (induct t rule: decrpoly.induct) simp_all
chaieb@33154
   923
wenzelm@52658
   924
lemma polysubst0_polybound0:
wenzelm@56207
   925
  assumes "polybound0 t"
chaieb@33154
   926
  shows "polybound0 (polysubst0 t a)"
wenzelm@56207
   927
  using assms by (induct a rule: poly.induct) auto
chaieb@33154
   928
chaieb@33154
   929
lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
wenzelm@52658
   930
  by (induct p arbitrary: n rule: degree.induct) (auto simp add: isnpolyh_polybound0)
chaieb@33154
   931
wenzelm@56043
   932
primrec maxindex :: "poly \<Rightarrow> nat"
wenzelm@67123
   933
  where
wenzelm@67123
   934
    "maxindex (Bound n) = n + 1"
wenzelm@67123
   935
  | "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
wenzelm@67123
   936
  | "maxindex (Add p q) = max (maxindex p) (maxindex q)"
wenzelm@67123
   937
  | "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
wenzelm@67123
   938
  | "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
wenzelm@67123
   939
  | "maxindex (Neg p) = maxindex p"
wenzelm@67123
   940
  | "maxindex (Pw p n) = maxindex p"
wenzelm@67123
   941
  | "maxindex (C x) = 0"
chaieb@33154
   942
wenzelm@52658
   943
definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool"
wenzelm@56000
   944
  where "wf_bs bs p \<longleftrightarrow> length bs \<ge> maxindex p"
chaieb@33154
   945
wenzelm@56043
   946
lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall>c \<in> set (coefficients p). wf_bs bs c"
wenzelm@52658
   947
proof (induct p rule: coefficients.induct)
wenzelm@52803
   948
  case (1 c p)
wenzelm@52803
   949
  show ?case
chaieb@33154
   950
  proof
wenzelm@56009
   951
    fix x
wenzelm@67123
   952
    assume "x \<in> set (coefficients (CN c 0 p))"
wenzelm@67123
   953
    then consider "x = c" | "x \<in> set (coefficients p)"
wenzelm@67123
   954
      by auto
wenzelm@67123
   955
    then show "wf_bs bs x"
wenzelm@67123
   956
    proof cases
wenzelm@67123
   957
      case prems: 1
wenzelm@67123
   958
      then show ?thesis
wenzelm@67123
   959
        using "1.prems" by (simp add: wf_bs_def)
wenzelm@67123
   960
    next
wenzelm@67123
   961
      case prems: 2
wenzelm@56009
   962
      from "1.prems" have "wf_bs bs p"
wenzelm@67123
   963
        by (simp add: wf_bs_def)
wenzelm@67123
   964
      with "1.hyps" prems show ?thesis
wenzelm@56009
   965
        by blast
wenzelm@67123
   966
    qed
chaieb@33154
   967
  qed
chaieb@33154
   968
qed simp_all
chaieb@33154
   969
wenzelm@56043
   970
lemma maxindex_coefficients: "\<forall>c \<in> set (coefficients p). maxindex c \<le> maxindex p"
wenzelm@52658
   971
  by (induct p rule: coefficients.induct) auto
chaieb@33154
   972
wenzelm@56000
   973
lemma wf_bs_I: "wf_bs bs p \<Longrightarrow> Ipoly (bs @ bs') p = Ipoly bs p"
wenzelm@67123
   974
  by (induct p) (auto simp add: nth_append wf_bs_def)
chaieb@33154
   975
wenzelm@52658
   976
lemma take_maxindex_wf:
wenzelm@52803
   977
  assumes wf: "wf_bs bs p"
chaieb@33154
   978
  shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
wenzelm@56009
   979
proof -
chaieb@33154
   980
  let ?ip = "maxindex p"
chaieb@33154
   981
  let ?tbs = "take ?ip bs"
wenzelm@56009
   982
  from wf have "length ?tbs = ?ip"
wenzelm@56009
   983
    unfolding wf_bs_def by simp
wenzelm@56009
   984
  then have wf': "wf_bs ?tbs p"
wenzelm@56009
   985
    unfolding wf_bs_def by  simp
wenzelm@56043
   986
  have eq: "bs = ?tbs @ drop ?ip bs"
wenzelm@56009
   987
    by simp
wenzelm@56009
   988
  from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis
wenzelm@56009
   989
    using eq by simp
chaieb@33154
   990
qed
chaieb@33154
   991
chaieb@33154
   992
lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
wenzelm@52658
   993
  by (induct p) auto
chaieb@33154
   994
chaieb@33154
   995
lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
wenzelm@67123
   996
  by (simp add: wf_bs_def)
chaieb@33154
   997
wenzelm@56207
   998
lemma wf_bs_insensitive': "wf_bs (x # bs) p = wf_bs (y # bs) p"
wenzelm@67123
   999
  by (simp add: wf_bs_def)
chaieb@33154
  1000
wenzelm@56207
  1001
lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x # bs) p"
wenzelm@52658
  1002
  by (induct p rule: coefficients.induct) (auto simp add: wf_bs_def)
wenzelm@56043
  1003
chaieb@33154
  1004
lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
wenzelm@52658
  1005
  by (induct p rule: coefficients.induct) simp_all
chaieb@33154
  1006
chaieb@33154
  1007
lemma coefficients_head: "last (coefficients p) = head p"
wenzelm@52658
  1008
  by (induct p rule: coefficients.induct) auto
chaieb@33154
  1009
wenzelm@56207
  1010
lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x # bs) p"
wenzelm@52658
  1011
  unfolding wf_bs_def by (induct p rule: decrpoly.induct) auto
chaieb@33154
  1012
wenzelm@56043
  1013
lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists>ys. length (xs @ ys) = n"
wenzelm@67123
  1014
  by (rule exI[where x="replicate (n - length xs) z" for z]) simp
wenzelm@52658
  1015
wenzelm@56043
  1016
lemma isnpolyh_Suc_const: "isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
wenzelm@52658
  1017
  apply (cases p)
wenzelm@67123
  1018
         apply auto
blanchet@58259
  1019
  apply (rename_tac nat a, case_tac "nat")
wenzelm@67123
  1020
   apply simp_all
wenzelm@52658
  1021
  done
chaieb@33154
  1022
chaieb@33154
  1023
lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
wenzelm@67123
  1024
  by (induct p q rule: polyadd.induct) (auto simp add: Let_def wf_bs_def)
chaieb@33154
  1025
chaieb@33154
  1026
lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
wenzelm@52803
  1027
  apply (induct p q arbitrary: bs rule: polymul.induct)
wenzelm@67123
  1028
                      apply (simp_all add: wf_bs_polyadd wf_bs_def)
chaieb@33154
  1029
  apply clarsimp
chaieb@33154
  1030
  apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
chaieb@33154
  1031
  apply auto
chaieb@33154
  1032
  done
chaieb@33154
  1033
chaieb@33154
  1034
lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
wenzelm@67123
  1035
  by (induct p rule: polyneg.induct) (auto simp: wf_bs_def)
chaieb@33154
  1036
chaieb@33154
  1037
lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
wenzelm@56043
  1038
  unfolding polysub_def split_def fst_conv snd_conv
wenzelm@56043
  1039
  using wf_bs_polyadd wf_bs_polyneg by blast
chaieb@33154
  1040
wenzelm@52658
  1041
wenzelm@60533
  1042
subsection \<open>Canonicity of polynomial representation, see lemma isnpolyh_unique\<close>
chaieb@33154
  1043
chaieb@33154
  1044
definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
wenzelm@56043
  1045
definition "polypoly' bs p = map (Ipoly bs \<circ> decrpoly) (coefficients p)"
wenzelm@56043
  1046
definition "poly_nate bs p = map (Ipoly bs \<circ> decrpoly) (coefficients (polynate p))"
chaieb@33154
  1047
wenzelm@56043
  1048
lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall>q \<in> set (coefficients p). isnpolyh q n0"
chaieb@33154
  1049
proof (induct p arbitrary: n0 rule: coefficients.induct)
chaieb@33154
  1050
  case (1 c p n0)
wenzelm@56009
  1051
  have cp: "isnpolyh (CN c 0 p) n0"
wenzelm@56009
  1052
    by fact
wenzelm@56009
  1053
  then have norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
chaieb@33154
  1054
    by (auto simp add: isnpolyh_mono[where n'=0])
wenzelm@56009
  1055
  from "1.hyps"[OF norm(2)] norm(1) norm(4) show ?case
wenzelm@56009
  1056
    by simp
chaieb@33154
  1057
qed auto
chaieb@33154
  1058
wenzelm@56043
  1059
lemma coefficients_isconst: "isnpolyh p n \<Longrightarrow> \<forall>q \<in> set (coefficients p). isconstant q"
wenzelm@56043
  1060
  by (induct p arbitrary: n rule: coefficients.induct) (auto simp add: isnpolyh_Suc_const)
chaieb@33154
  1061
chaieb@33154
  1062
lemma polypoly_polypoly':
chaieb@33154
  1063
  assumes np: "isnpolyh p n0"
wenzelm@56043
  1064
  shows "polypoly (x # bs) p = polypoly' bs p"
wenzelm@56043
  1065
proof -
chaieb@33154
  1066
  let ?cf = "set (coefficients p)"
chaieb@33154
  1067
  from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
wenzelm@67123
  1068
  have "polybound0 q" if "q \<in> ?cf" for q
wenzelm@67123
  1069
  proof -
wenzelm@67123
  1070
    from that cn_norm have *: "isnpolyh q n0"
wenzelm@56043
  1071
      by blast
wenzelm@67123
  1072
    from coefficients_isconst[OF np] that have "isconstant q"
wenzelm@56043
  1073
      by blast
wenzelm@67123
  1074
    with isconstant_polybound0[OF *] show ?thesis
wenzelm@56043
  1075
      by blast
wenzelm@67123
  1076
  qed
wenzelm@56009
  1077
  then have "\<forall>q \<in> ?cf. polybound0 q" ..
wenzelm@56043
  1078
  then have "\<forall>q \<in> ?cf. Ipoly (x # bs) q = Ipoly bs (decrpoly q)"
chaieb@33154
  1079
    using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
chaieb@33154
  1080
    by auto
wenzelm@56043
  1081
  then show ?thesis
wenzelm@56043
  1082
    unfolding polypoly_def polypoly'_def by simp
chaieb@33154
  1083
qed
chaieb@33154
  1084
chaieb@33154
  1085
lemma polypoly_poly:
wenzelm@56043
  1086
  assumes "isnpolyh p n0"
wenzelm@56043
  1087
  shows "Ipoly (x # bs) p = poly (polypoly (x # bs) p) x"
wenzelm@56043
  1088
  using assms
wenzelm@52658
  1089
  by (induct p arbitrary: n0 bs rule: coefficients.induct) (auto simp add: polypoly_def)
chaieb@33154
  1090
wenzelm@52803
  1091
lemma polypoly'_poly:
wenzelm@56043
  1092
  assumes "isnpolyh p n0"
wenzelm@52658
  1093
  shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
wenzelm@56043
  1094
  using polypoly_poly[OF assms, simplified polypoly_polypoly'[OF assms]] .
chaieb@33154
  1095
chaieb@33154
  1096
chaieb@33154
  1097
lemma polypoly_poly_polybound0:
wenzelm@56043
  1098
  assumes "isnpolyh p n0"
wenzelm@56043
  1099
    and "polybound0 p"
chaieb@33154
  1100
  shows "polypoly bs p = [Ipoly bs p]"
wenzelm@56043
  1101
  using assms
wenzelm@56043
  1102
  unfolding polypoly_def
wenzelm@52658
  1103
  apply (cases p)
wenzelm@67123
  1104
         apply auto
blanchet@58259
  1105
  apply (rename_tac nat a, case_tac nat)
wenzelm@67123
  1106
   apply auto
wenzelm@52658
  1107
  done
chaieb@33154
  1108
wenzelm@52803
  1109
lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0"
wenzelm@52658
  1110
  by (induct p rule: head.induct) auto
chaieb@33154
  1111
wenzelm@56043
  1112
lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> headn p m = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
  1113
  by (cases p) auto
chaieb@33154
  1114
chaieb@33154
  1115
lemma head_eq_headn0: "head p = headn p 0"
wenzelm@52658
  1116
  by (induct p rule: head.induct) simp_all
chaieb@33154
  1117
wenzelm@56043
  1118
lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> head p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
chaieb@33154
  1119
  by (simp add: head_eq_headn0)
chaieb@33154
  1120
wenzelm@52803
  1121
lemma isnpolyh_zero_iff:
wenzelm@52658
  1122
  assumes nq: "isnpolyh p n0"
nipkow@68442
  1123
    and eq :"\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a::{field_char_0, power})"
chaieb@33154
  1124
  shows "p = 0\<^sub>p"
wenzelm@52658
  1125
  using nq eq
berghofe@34915
  1126
proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
berghofe@34915
  1127
  case less
wenzelm@60533
  1128
  note np = \<open>isnpolyh p n0\<close> and zp = \<open>\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)\<close>
wenzelm@67123
  1129
  show "p = 0\<^sub>p"
wenzelm@67123
  1130
  proof (cases "maxindex p = 0")
wenzelm@67123
  1131
    case True
wenzelm@67123
  1132
    with np obtain c where "p = C c" by (cases p) auto
wenzelm@67123
  1133
    with zp np show ?thesis by (simp add: wf_bs_def)
wenzelm@67123
  1134
  next
wenzelm@67123
  1135
    case nz: False
chaieb@33154
  1136
    let ?h = "head p"
chaieb@33154
  1137
    let ?hd = "decrpoly ?h"
chaieb@33154
  1138
    let ?ihd = "maxindex ?hd"
wenzelm@56000
  1139
    from head_isnpolyh[OF np] head_polybound0[OF np]
wenzelm@56000
  1140
    have h: "isnpolyh ?h n0" "polybound0 ?h"
chaieb@33154
  1141
      by simp_all
wenzelm@56000
  1142
    then have nhd: "isnpolyh ?hd (n0 - 1)"
wenzelm@56000
  1143
      using decrpoly_normh by blast
wenzelm@52803
  1144
chaieb@33154
  1145
    from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
wenzelm@56000
  1146
    have mihn: "maxindex ?h \<le> maxindex p"
wenzelm@56000
  1147
      by auto
wenzelm@56000
  1148
    with decr_maxindex[OF h(2)] nz have ihd_lt_n: "?ihd < maxindex p"
wenzelm@56000
  1149
      by auto
wenzelm@67123
  1150
wenzelm@67123
  1151
    have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" if bs: "wf_bs bs ?hd" for bs :: "'a list"
wenzelm@67123
  1152
    proof -
chaieb@33154
  1153
      let ?ts = "take ?ihd bs"
chaieb@33154
  1154
      let ?rs = "drop ?ihd bs"
wenzelm@67123
  1155
      from bs have ts: "wf_bs ?ts ?hd"
wenzelm@67123
  1156
        by (simp add: wf_bs_def)
wenzelm@56000
  1157
      have bs_ts_eq: "?ts @ ?rs = bs"
wenzelm@56000
  1158
        by simp
wenzelm@56000
  1159
      from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x # ?ts) ?h"
wenzelm@56000
  1160
        by simp
wenzelm@56000
  1161
      from ihd_lt_n have "\<forall>x. length (x # ?ts) \<le> maxindex p"
wenzelm@56000
  1162
        by simp
wenzelm@56000
  1163
      with length_le_list_ex obtain xs where xs: "length ((x # ?ts) @ xs) = maxindex p"
wenzelm@56000
  1164
        by blast
wenzelm@56000
  1165
      then have "\<forall>x. wf_bs ((x # ?ts) @ xs) p"
wenzelm@67123
  1166
        by (simp add: wf_bs_def)
wenzelm@56000
  1167
      with zp have "\<forall>x. Ipoly ((x # ?ts) @ xs) p = 0"
wenzelm@56000
  1168
        by blast
wenzelm@56000
  1169
      then have "\<forall>x. Ipoly (x # (?ts @ xs)) p = 0"
wenzelm@56000
  1170
        by simp
chaieb@33154
  1171
      with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
wenzelm@56000
  1172
      have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"
wenzelm@56000
  1173
        by simp
wenzelm@56000
  1174
      then have "poly (polypoly' (?ts @ xs) p) = poly []"
wenzelm@56000
  1175
        by auto
wenzelm@56000
  1176
      then have "\<forall>c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
wenzelm@60537
  1177
        using poly_zero[where ?'a='a] by (simp add: polypoly'_def)
chaieb@33154
  1178
      with coefficients_head[of p, symmetric]
wenzelm@67123
  1179
      have *: "Ipoly (?ts @ xs) ?hd = 0"
wenzelm@56000
  1180
        by simp
wenzelm@56000
  1181
      from bs have wf'': "wf_bs ?ts ?hd"
wenzelm@67123
  1182
        by (simp add: wf_bs_def)
wenzelm@67123
  1183
      with * wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0"
wenzelm@56000
  1184
        by simp
wenzelm@67123
  1185
      with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq show ?thesis
wenzelm@56000
  1186
        by simp
wenzelm@67123
  1187
    qed
wenzelm@56000
  1188
    then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
wenzelm@56000
  1189
      by blast
wenzelm@56000
  1190
    from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p"
wenzelm@56000
  1191
      by blast
wenzelm@56000
  1192
    then have "?h = 0\<^sub>p" by simp
wenzelm@67123
  1193
    with head_nz[OF np] show ?thesis by simp
wenzelm@67123
  1194
  qed
chaieb@33154
  1195
qed
chaieb@33154
  1196
wenzelm@52803
  1197
lemma isnpolyh_unique:
wenzelm@56000
  1198
  assumes np: "isnpolyh p n0"
wenzelm@52658
  1199
    and nq: "isnpolyh q n1"
nipkow@68442
  1200
  shows "(\<forall>bs. \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (\<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup> :: 'a::{field_char_0,power})) \<longleftrightarrow> p = q"
wenzelm@56000
  1201
proof auto
wenzelm@67123
  1202
  assume "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a) = \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
wenzelm@56000
  1203
  then have "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)"
wenzelm@56000
  1204
    by simp
wenzelm@56000
  1205
  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
  1206
    using wf_bs_polysub[where p=p and q=q] by auto
wenzelm@56000
  1207
  with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq] show "p = q"
wenzelm@56000
  1208
    by blast
chaieb@33154
  1209
qed
chaieb@33154
  1210
chaieb@33154
  1211
wenzelm@67123
  1212
text \<open>Consequences of unicity on the algorithms for polynomial normalization.\<close>
chaieb@33154
  1213
wenzelm@52658
  1214
lemma polyadd_commute:
nipkow@68442
  1215
  assumes "SORT_CONSTRAINT('a::field_char_0)"
wenzelm@52658
  1216
    and np: "isnpolyh p n0"
wenzelm@52658
  1217
    and nq: "isnpolyh q n1"
wenzelm@52658
  1218
  shows "p +\<^sub>p q = q +\<^sub>p p"
wenzelm@56000
  1219
  using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]]
wenzelm@56000
  1220
  by simp
chaieb@33154
  1221
wenzelm@56000
  1222
lemma zero_normh: "isnpolyh 0\<^sub>p n"
wenzelm@56000
  1223
  by simp
wenzelm@56000
  1224
wenzelm@56000
  1225
lemma one_normh: "isnpolyh (1)\<^sub>p n"
wenzelm@56000
  1226
  by simp
wenzelm@52658
  1227
wenzelm@52803
  1228
lemma polyadd_0[simp]:
nipkow@68442
  1229
  assumes "SORT_CONSTRAINT('a::field_char_0)"
wenzelm@52658
  1230
    and np: "isnpolyh p n0"
wenzelm@56000
  1231
  shows "p +\<^sub>p 0\<^sub>p = p"
wenzelm@56000
  1232
    and "0\<^sub>p +\<^sub>p p = p"
wenzelm@52803
  1233
  using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np]
chaieb@33154
  1234
    isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
chaieb@33154
  1235
wenzelm@52803
  1236
lemma polymul_1[simp]:
nipkow@68442
  1237
  assumes "SORT_CONSTRAINT('a::field_char_0)"
wenzelm@52658
  1238
    and np: "isnpolyh p n0"
wenzelm@56000
  1239
  shows "p *\<^sub>p (1)\<^sub>p = p"
wenzelm@56000
  1240
    and "(1)\<^sub>p *\<^sub>p p = p"
wenzelm@52803
  1241
  using isnpolyh_unique[OF polymul_normh[OF np one_normh] np]
chaieb@33154
  1242
    isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
wenzelm@52658
  1243
wenzelm@52803
  1244
lemma polymul_0[simp]:
nipkow@68442
  1245
  assumes "SORT_CONSTRAINT('a::field_char_0)"
wenzelm@52658
  1246
    and np: "isnpolyh p n0"
wenzelm@56000
  1247
  shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p"
wenzelm@56000
  1248
    and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
wenzelm@52803
  1249
  using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh]
chaieb@33154
  1250
    isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
chaieb@33154
  1251
wenzelm@52803
  1252
lemma polymul_commute:
nipkow@68442
  1253
  assumes "SORT_CONSTRAINT('a::field_char_0)"
wenzelm@56000
  1254
    and np: "isnpolyh p n0"
wenzelm@52658
  1255
    and nq: "isnpolyh q n1"
chaieb@33154
  1256
  shows "p *\<^sub>p q = q *\<^sub>p p"
wenzelm@56043
  1257
  using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np],
nipkow@68442
  1258
    where ?'a = "'a::{field_char_0, power}"]
wenzelm@52658
  1259
  by simp
chaieb@33154
  1260
wenzelm@52658
  1261
declare polyneg_polyneg [simp]
wenzelm@52803
  1262
wenzelm@52803
  1263
lemma isnpolyh_polynate_id [simp]:
nipkow@68442
  1264
  assumes "SORT_CONSTRAINT('a::field_char_0)"
wenzelm@56000
  1265
    and np: "isnpolyh p n0"
wenzelm@52658
  1266
  shows "polynate p = p"
nipkow@68442
  1267
  using isnpolyh_unique[where ?'a= "'a::field_char_0",
wenzelm@56043
  1268
      OF polynate_norm[of p, unfolded isnpoly_def] np]
nipkow@68442
  1269
    polynate[where ?'a = "'a::field_char_0"]
wenzelm@52658
  1270
  by simp
chaieb@33154
  1271
wenzelm@52803
  1272
lemma polynate_idempotent[simp]:
nipkow@68442
  1273
  assumes "SORT_CONSTRAINT('a::field_char_0)"
chaieb@33154
  1274
  shows "polynate (polynate p) = polynate p"
chaieb@33154
  1275
  using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
chaieb@33154
  1276
chaieb@33154
  1277
lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
chaieb@33154
  1278
  unfolding poly_nate_def polypoly'_def ..
wenzelm@52658
  1279
wenzelm@52658
  1280
lemma poly_nate_poly:
nipkow@68442
  1281
  "poly (poly_nate bs p) = (\<lambda>x:: 'a ::field_char_0. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
chaieb@33154
  1282
  using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
wenzelm@52658
  1283
  unfolding poly_nate_polypoly' by auto
wenzelm@52658
  1284
chaieb@33154
  1285
wenzelm@67123
  1286
subsection \<open>Heads, degrees and all that\<close>
wenzelm@52658
  1287
chaieb@33154
  1288
lemma degree_eq_degreen0: "degree p = degreen p 0"
wenzelm@52658
  1289
  by (induct p rule: degree.induct) simp_all
chaieb@33154
  1290
wenzelm@52658
  1291
lemma degree_polyneg:
wenzelm@56043
  1292
  assumes "isnpolyh p n"
chaieb@33154
  1293
  shows "degree (polyneg p) = degree p"
wenzelm@56043
  1294
  apply (induct p rule: polyneg.induct)
wenzelm@56043
  1295
  using assms
wenzelm@67123
  1296
         apply simp_all
wenzelm@52658
  1297
  apply (case_tac na)
wenzelm@67123
  1298
   apply auto
wenzelm@52658
  1299
  done
chaieb@33154
  1300
chaieb@33154
  1301
lemma degree_polyadd:
wenzelm@56043
  1302
  assumes np: "isnpolyh p n0"
wenzelm@56043
  1303
    and nq: "isnpolyh q n1"
chaieb@33154
  1304
  shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
wenzelm@52658
  1305
  using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
chaieb@33154
  1306
chaieb@33154
  1307
wenzelm@52658
  1308
lemma degree_polysub:
wenzelm@52658
  1309
  assumes np: "isnpolyh p n0"
wenzelm@52658
  1310
    and nq: "isnpolyh q n1"
chaieb@33154
  1311
  shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
chaieb@33154
  1312
proof-
wenzelm@56043
  1313
  from nq have nq': "isnpolyh (~\<^sub>p q) n1"
wenzelm@56043
  1314
    using polyneg_normh by simp
wenzelm@56043
  1315
  from degree_polyadd[OF np nq'] show ?thesis
wenzelm@56043
  1316
    by (simp add: polysub_def degree_polyneg[OF nq])
chaieb@33154
  1317
qed
chaieb@33154
  1318
wenzelm@52803
  1319
lemma degree_polysub_samehead:
nipkow@68442
  1320
  assumes "SORT_CONSTRAINT('a::field_char_0)"
wenzelm@56043
  1321
    and np: "isnpolyh p n0"
wenzelm@56043
  1322
    and nq: "isnpolyh q n1"
wenzelm@56043
  1323
    and h: "head p = head q"
wenzelm@52658
  1324
    and d: "degree p = degree q"
chaieb@33154
  1325
  shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
wenzelm@52658
  1326
  unfolding polysub_def split_def fst_conv snd_conv
wenzelm@52658
  1327
  using np nq h d
wenzelm@52658
  1328
proof (induct p q rule: polyadd.induct)
wenzelm@52658
  1329
  case (1 c c')
wenzelm@56009
  1330
  then show ?case
wenzelm@56009
  1331
    by (simp add: Nsub_def Nsub0[simplified Nsub_def])
chaieb@33154
  1332
next
wenzelm@52803
  1333
  case (2 c c' n' p')
wenzelm@56009
  1334
  from 2 have "degree (C c) = degree (CN c' n' p')"
wenzelm@56009
  1335
    by simp
wenzelm@56009
  1336
  then have nz: "n' > 0"
wenzelm@56009
  1337
    by (cases n') auto
wenzelm@56009
  1338
  then have "head (CN c' n' p') = CN c' n' p'"
wenzelm@56009
  1339
    by (cases n') auto
wenzelm@56009
  1340
  with 2 show ?case
wenzelm@56009
  1341
    by simp
chaieb@33154
  1342
next
wenzelm@52803
  1343
  case (3 c n p c')
wenzelm@56009
  1344
  then have "degree (C c') = degree (CN c n p)"
wenzelm@56009
  1345
    by simp
wenzelm@56009
  1346
  then have nz: "n > 0"
wenzelm@56009
  1347
    by (cases n) auto
wenzelm@56009
  1348
  then have "head (CN c n p) = CN c n p"
wenzelm@56009
  1349
    by (cases n) auto
wenzelm@41807
  1350
  with 3 show ?case by simp
chaieb@33154
  1351
next
chaieb@33154
  1352
  case (4 c n p c' n' p')
wenzelm@56009
  1353
  then have H:
wenzelm@56009
  1354
    "isnpolyh (CN c n p) n0"
wenzelm@56009
  1355
    "isnpolyh (CN c' n' p') n1"
wenzelm@56009
  1356
    "head (CN c n p) = head (CN c' n' p')"
wenzelm@56009
  1357
    "degree (CN c n p) = degree (CN c' n' p')"
wenzelm@56009
  1358
    by simp_all
wenzelm@56009
  1359
  then have degc: "degree c = 0" and degc': "degree c' = 0"
wenzelm@56009
  1360
    by simp_all
wenzelm@56009
  1361
  then have degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0"
chaieb@33154
  1362
    using H(1-2) degree_polyneg by auto
wenzelm@56009
  1363
  from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"
wenzelm@56009
  1364
    by simp_all
wenzelm@56009
  1365
  from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc'
wenzelm@56009
  1366
  have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"
wenzelm@56009
  1367
    by simp
wenzelm@56009
  1368
  from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'"
wenzelm@56009
  1369
    by auto
wenzelm@67123
  1370
  consider "n = n'" | "n < n'" | "n > n'"
wenzelm@56009
  1371
    by arith
wenzelm@67123
  1372
  then show ?case
wenzelm@67123
  1373
  proof cases
wenzelm@67123
  1374
    case nn': 1
wenzelm@67123
  1375
    consider "n = 0" | "n > 0" by arith
wenzelm@67123
  1376
    then show ?thesis
wenzelm@67123
  1377
    proof cases
wenzelm@67123
  1378
      case 1
wenzelm@67123
  1379
      with 4 nn' show ?thesis
wenzelm@56009
  1380
        by (auto simp add: Let_def degcmc')
wenzelm@67123
  1381
    next
wenzelm@67123
  1382
      case 2
wenzelm@67123
  1383
      with nn' H(3) have "c = c'" and "p = p'"
wenzelm@67123
  1384
        by (cases n; auto)+
wenzelm@67123
  1385
      with nn' 4 show ?thesis
wenzelm@56009
  1386
        using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv]
wenzelm@56009
  1387
        using polysub_same_0[OF c'nh, simplified polysub_def]
wenzelm@67123
  1388
        by (simp add: Let_def)
wenzelm@67123
  1389
    qed
wenzelm@67123
  1390
  next
wenzelm@67123
  1391
    case nn': 2
wenzelm@56009
  1392
    then have n'p: "n' > 0"
wenzelm@56009
  1393
      by simp
wenzelm@56009
  1394
    then have headcnp':"head (CN c' n' p') = CN c' n' p'"
wenzelm@56009
  1395
      by (cases n') simp_all
wenzelm@67123
  1396
    with 4 nn' have degcnp': "degree (CN c' n' p') = 0"
wenzelm@56009
  1397
      and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')"
wenzelm@67123
  1398
      by (cases n', simp_all)
wenzelm@56009
  1399
    then have "n > 0"
wenzelm@56009
  1400
      by (cases n) simp_all
wenzelm@56009
  1401
    then have headcnp: "head (CN c n p) = CN c n p"
wenzelm@56009
  1402
      by (cases n) auto
wenzelm@67123
  1403
    from H(3) headcnp headcnp' nn' show ?thesis
wenzelm@56009
  1404
      by auto
wenzelm@67123
  1405
  next
wenzelm@67123
  1406
    case nn': 3
wenzelm@56009
  1407
    then have np: "n > 0" by simp
wenzelm@56009
  1408
    then have headcnp:"head (CN c n p) = CN c n p"
wenzelm@56009
  1409
      by (cases n) simp_all
wenzelm@56009
  1410
    from 4 have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)"
wenzelm@56009
  1411
      by simp
wenzelm@56009
  1412
    from np have degcnp: "degree (CN c n p) = 0"
wenzelm@56009
  1413
      by (cases n) simp_all
wenzelm@56009
  1414
    with degcnpeq have "n' > 0"
wenzelm@56009
  1415
      by (cases n') simp_all
wenzelm@56009
  1416
    then have headcnp': "head (CN c' n' p') = CN c' n' p'"
wenzelm@56009
  1417
      by (cases n') auto
wenzelm@67123
  1418
    from H(3) headcnp headcnp' nn' show ?thesis by auto
wenzelm@67123
  1419
  qed
krauss@41812
  1420
qed auto
wenzelm@52803
  1421
chaieb@33154
  1422
lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
wenzelm@52658
  1423
  by (induct p arbitrary: n0 rule: head.induct) (simp_all add: shift1_def)
chaieb@33154
  1424
wenzelm@67123
  1425
lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
wenzelm@52658
  1426
proof (induct k arbitrary: n0 p)
wenzelm@52658
  1427
  case 0
wenzelm@56198
  1428
  then show ?case
wenzelm@56198
  1429
    by auto
wenzelm@52658
  1430
next
wenzelm@52658
  1431
  case (Suc k n0 p)
wenzelm@56066
  1432
  then have "isnpolyh (shift1 p) 0"
wenzelm@56066
  1433
    by (simp add: shift1_isnpolyh)
wenzelm@41807
  1434
  with Suc have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
wenzelm@56066
  1435
    and "head (shift1 p) = head p"
wenzelm@56066
  1436
    by (simp_all add: shift1_head)
wenzelm@56066
  1437
  then show ?case
wenzelm@56066
  1438
    by (simp add: funpow_swap1)
wenzelm@52658
  1439
qed
chaieb@33154
  1440
chaieb@33154
  1441
lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
chaieb@33154
  1442
  by (simp add: shift1_def)
wenzelm@56009
  1443
chaieb@33154
  1444
lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
wenzelm@46991
  1445
  by (induct k arbitrary: p) (auto simp add: shift1_degree)
chaieb@33154
  1446
chaieb@33154
  1447
lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
wenzelm@52658
  1448
  by (induct n arbitrary: p) simp_all
chaieb@33154
  1449
chaieb@33154
  1450
lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
wenzelm@52658
  1451
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
  1452
lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
wenzelm@52658
  1453
  by (induct p arbitrary: n rule: degreen.induct) auto
chaieb@33154
  1454
lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
wenzelm@52658
  1455
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
  1456
lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
wenzelm@52658
  1457
  by (induct p rule: head.induct) auto
chaieb@33154
  1458
wenzelm@52803
  1459
lemma polyadd_eq_const_degree:
wenzelm@52658
  1460
  "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> polyadd p q = C c \<Longrightarrow> degree p = degree q"
chaieb@33154
  1461
  using polyadd_eq_const_degreen degree_eq_degreen0 by simp
chaieb@33154
  1462
wenzelm@52658
  1463
lemma polyadd_head:
wenzelm@52658
  1464
  assumes np: "isnpolyh p n0"
wenzelm@52658
  1465
    and nq: "isnpolyh q n1"
wenzelm@52658
  1466
    and deg: "degree p \<noteq> degree q"
chaieb@33154
  1467
  shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
wenzelm@52658
  1468
  using np nq deg
wenzelm@52658
  1469
  apply (induct p q arbitrary: n0 n1 rule: polyadd.induct)
wenzelm@67123
  1470
                      apply simp_all
wenzelm@67123
  1471
    apply (case_tac n', simp, simp)
wenzelm@67123
  1472
   apply (case_tac n, simp, simp)
wenzelm@52658
  1473
  apply (case_tac n, case_tac n', simp add: Let_def)
wenzelm@67123
  1474
    apply (auto simp add: polyadd_eq_const_degree)[2]
wenzelm@67123
  1475
    apply (metis head_nz)
wenzelm@67123
  1476
   apply (metis head_nz)
wenzelm@52658
  1477
  apply (metis degree.simps(9) gr0_conv_Suc head.simps(1) less_Suc0 not_less_eq)
wenzelm@52658
  1478
  done
chaieb@33154
  1479
wenzelm@52803
  1480
lemma polymul_head_polyeq:
nipkow@68442
  1481
  assumes "SORT_CONSTRAINT('a::field_char_0)"
wenzelm@56066
  1482
  shows "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> p \<noteq> 0\<^sub>p \<Longrightarrow> q \<noteq> 0\<^sub>p \<Longrightarrow> head (p *\<^sub>p q) = head p *\<^sub>p head q"
chaieb@33154
  1483
proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
krauss@41813
  1484
  case (2 c c' n' p' n0 n1)
wenzelm@56009
  1485
  then have "isnpolyh (head (CN c' n' p')) n1" "isnormNum c"
wenzelm@56009
  1486
    by (simp_all add: head_isnpolyh)
wenzelm@56009
  1487
  then show ?case
wenzelm@56009
  1488
    using 2 by (cases n') auto
wenzelm@52803
  1489
next
wenzelm@52803
  1490
  case (3 c n p c' n0 n1)
wenzelm@56009
  1491
  then have "isnpolyh (head (CN c n p)) n0" "isnormNum c'"
wenzelm@56009
  1492
    by (simp_all add: head_isnpolyh)
wenzelm@56066
  1493
  then show ?case
wenzelm@56066
  1494
    using 3 by (cases n) auto
chaieb@33154
  1495
next
chaieb@33154
  1496
  case (4 c n p c' n' p' n0 n1)
wenzelm@56066
  1497
  then have norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
chaieb@33154
  1498
    "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
chaieb@33154
  1499
    by simp_all
wenzelm@67123
  1500
  consider "n < n'" | "n' < n" | "n' = n" by arith
wenzelm@67123
  1501
  then show ?case
wenzelm@67123
  1502
  proof cases
wenzelm@67123
  1503
    case nn': 1
wenzelm@67123
  1504
    then show ?thesis
wenzelm@52658
  1505
      using norm "4.hyps"(2)[OF norm(1,6)] "4.hyps"(1)[OF norm(2,6)]
wenzelm@52658
  1506
      apply simp
wenzelm@52658
  1507
      apply (cases n)
wenzelm@67123
  1508
       apply simp
wenzelm@52658
  1509
      apply (cases n')
wenzelm@67123
  1510
       apply simp_all
wenzelm@56009
  1511
      done
wenzelm@67123
  1512
  next
wenzelm@67123
  1513
    case nn': 2
wenzelm@67123
  1514
    then show ?thesis
wenzelm@52803
  1515
      using norm "4.hyps"(6) [OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)]
wenzelm@52658
  1516
      apply simp
wenzelm@52658
  1517
      apply (cases n')
wenzelm@67123
  1518
       apply simp
wenzelm@52658
  1519
      apply (cases n)
wenzelm@67123
  1520
       apply auto
wenzelm@56009
  1521
      done
wenzelm@67123
  1522
  next
wenzelm@67123
  1523
    case nn': 3
wenzelm@52803
  1524
    from nn' polymul_normh[OF norm(5,4)]
wenzelm@67123
  1525
    have ncnpc': "isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
wenzelm@52803
  1526
    from nn' polymul_normh[OF norm(5,3)] norm
wenzelm@67123
  1527
    have ncnpp': "isnpolyh (CN c n p *\<^sub>p p') n" by simp
chaieb@33154
  1528
    from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
wenzelm@67123
  1529
    have ncnpp0': "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
wenzelm@52803
  1530
    from polyadd_normh[OF ncnpc' ncnpp0']
wenzelm@52803
  1531
    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
  1532
      by (simp add: min_def)
wenzelm@67123
  1533
    consider "n > 0" | "n = 0" by auto
wenzelm@67123
  1534
    then show ?thesis
wenzelm@67123
  1535
    proof cases
wenzelm@67123
  1536
      case np: 1
chaieb@33154
  1537
      with nn' head_isnpolyh_Suc'[OF np nth]
wenzelm@33268
  1538
        head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
wenzelm@67123
  1539
      show ?thesis by simp
wenzelm@67123
  1540
    next
wenzelm@67123
  1541
      case nz: 2
chaieb@33154
  1542
      from polymul_degreen[OF norm(5,4), where m="0"]
wenzelm@33268
  1543
        polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
wenzelm@67123
  1544
        norm(5,6) degree_npolyhCN[OF norm(6)]
wenzelm@67123
  1545
      have dth: "degree (CN c 0 p *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))"
wenzelm@67123
  1546
        by simp
wenzelm@67123
  1547
      then have dth': "degree (CN c 0 p *\<^sub>p c') \<noteq> degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))"
wenzelm@67123
  1548
        by simp
wenzelm@67123
  1549
      from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
wenzelm@67123
  1550
      show ?thesis
wenzelm@67123
  1551
        using norm "4.hyps"(6)[OF norm(5,3)] "4.hyps"(5)[OF norm(5,4)] nn' nz
wenzelm@67123
  1552
        by simp
wenzelm@67123
  1553
    qed
wenzelm@67123
  1554
  qed
chaieb@33154
  1555
qed simp_all
chaieb@33154
  1556
chaieb@33154
  1557
lemma degree_coefficients: "degree p = length (coefficients p) - 1"
wenzelm@52658
  1558
  by (induct p rule: degree.induct) auto
chaieb@33154
  1559
chaieb@33154
  1560
lemma degree_head[simp]: "degree (head p) = 0"
wenzelm@52658
  1561
  by (induct p rule: head.induct) auto
chaieb@33154
  1562
krauss@41812
  1563
lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1 + degree p"
wenzelm@52658
  1564
  by (cases n) simp_all
wenzelm@56066
  1565
chaieb@33154
  1566
lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
wenzelm@52658
  1567
  by (cases n) simp_all
chaieb@33154
  1568
wenzelm@52658
  1569
lemma polyadd_different_degree:
wenzelm@56066
  1570
  "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> degree p \<noteq> degree q \<Longrightarrow>
wenzelm@52658
  1571
    degree (polyadd p q) = max (degree p) (degree q)"
chaieb@33154
  1572
  using polyadd_different_degreen degree_eq_degreen0 by simp
chaieb@33154
  1573
chaieb@33154
  1574
lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
wenzelm@52658
  1575
  by (induct p arbitrary: n0 rule: polyneg.induct) auto
chaieb@33154
  1576
chaieb@33154
  1577
lemma degree_polymul:
nipkow@68442
  1578
  assumes "SORT_CONSTRAINT('a::field_char_0)"
wenzelm@52658
  1579
    and np: "isnpolyh p n0"
wenzelm@52658
  1580
    and nq: "isnpolyh q n1"
chaieb@33154
  1581
  shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
chaieb@33154
  1582
  using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
chaieb@33154
  1583
chaieb@33154
  1584
lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
wenzelm@52658
  1585
  by (induct p arbitrary: n rule: degree.induct) auto
chaieb@33154
  1586
wenzelm@56207
  1587
lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head (polyneg p) = polyneg (head p)"
wenzelm@52658
  1588
  by (induct p arbitrary: n rule: degree.induct) auto
wenzelm@52658
  1589
chaieb@33154
  1590
wenzelm@60533
  1591
subsection \<open>Correctness of polynomial pseudo division\<close>
chaieb@33154
  1592
chaieb@33154
  1593
lemma polydivide_aux_properties:
nipkow@68442
  1594
  assumes "SORT_CONSTRAINT('a::field_char_0)"
wenzelm@52658
  1595
    and np: "isnpolyh p n0"
wenzelm@52658
  1596
    and ns: "isnpolyh s n1"
wenzelm@52658
  1597
    and ap: "head p = a"
wenzelm@56198
  1598
    and ndp: "degree p = n"
wenzelm@56198
  1599
    and pnz: "p \<noteq> 0\<^sub>p"
wenzelm@56198
  1600
  shows "polydivide_aux a n p k s = (k', r) \<longrightarrow> k' \<ge> k \<and> (degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@56066
  1601
    (\<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
  1602
  using ns
wenzelm@52658
  1603
proof (induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
berghofe@34915
  1604
  case less
chaieb@33154
  1605
  let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
wenzelm@56066
  1606
  let ?ths = "polydivide_aux a n p k s = (k', r) \<longrightarrow>  k \<le> k' \<and>
wenzelm@56066
  1607
    (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
chaieb@33154
  1608
  let ?b = "head s"
berghofe@34915
  1609
  let ?p' = "funpow (degree s - n) shift1 p"
wenzelm@50282
  1610
  let ?xdn = "funpow (degree s - n) shift1 (1)\<^sub>p"
chaieb@33154
  1611
  let ?akk' = "a ^\<^sub>p (k' - k)"
wenzelm@60533
  1612
  note ns = \<open>isnpolyh s n1\<close>
wenzelm@52803
  1613
  from np have np0: "isnpolyh p 0"
wenzelm@52803
  1614
    using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by simp
wenzelm@52803
  1615
  have np': "isnpolyh ?p' 0"
wenzelm@52803
  1616
    using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def
wenzelm@52803
  1617
    by simp
wenzelm@52803
  1618
  have headp': "head ?p' = head p"
wenzelm@52803
  1619
    using funpow_shift1_head[OF np pnz] by simp
wenzelm@52803
  1620
  from funpow_shift1_isnpoly[where p="(1)\<^sub>p"] have nxdn: "isnpolyh ?xdn 0"
wenzelm@52803
  1621
    by (simp add: isnpoly_def)
wenzelm@52803
  1622
  from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
chaieb@33154
  1623
  have nakk':"isnpolyh ?akk' 0" by blast
wenzelm@67123
  1624
  show ?ths
wenzelm@67123
  1625
  proof (cases "s = 0\<^sub>p")
wenzelm@67123
  1626
    case True
wenzelm@67123
  1627
    with np show ?thesis
wenzelm@67123
  1628
      apply (clarsimp simp: polydivide_aux.simps)
wenzelm@52658
  1629
      apply (rule exI[where x="0\<^sub>p"])
wenzelm@52658
  1630
      apply simp
wenzelm@56066
  1631
      done
wenzelm@67123
  1632
  next
wenzelm@67123
  1633
    case sz: False
wenzelm@67123
  1634
    show ?thesis
wenzelm@67123
  1635
    proof (cases "degree s < n")
wenzelm@67123
  1636
      case True
wenzelm@67123
  1637
      then show ?thesis
wenzelm@56066
  1638
        using ns ndp np polydivide_aux.simps
wenzelm@52658
  1639
        apply auto
wenzelm@52658
  1640
        apply (rule exI[where x="0\<^sub>p"])
wenzelm@52658
  1641
        apply simp
wenzelm@56066
  1642
        done
wenzelm@67123
  1643
    next
wenzelm@67123
  1644
      case dn': False
wenzelm@56066
  1645
      then have dn: "degree s \<ge> n"
wenzelm@56066
  1646
        by arith
wenzelm@52803
  1647
      have degsp': "degree s = degree ?p'"
wenzelm@56066
  1648
        using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"]
wenzelm@56066
  1649
        by simp
wenzelm@67123
  1650
      show ?thesis
wenzelm@67123
  1651
      proof (cases "?b = a")
wenzelm@67123
  1652
        case ba: True
wenzelm@56066
  1653
        then have headsp': "head s = head ?p'"
wenzelm@52803
  1654
          using ap headp' by simp
wenzelm@52803
  1655
        have nr: "isnpolyh (s -\<^sub>p ?p') 0"
wenzelm@52803
  1656
          using polysub_normh[OF ns np'] by simp
berghofe@34915
  1657
        from degree_polysub_samehead[OF ns np' headsp' degsp']
wenzelm@67123
  1658
        consider "degree (s -\<^sub>p ?p') < degree s" | "s -\<^sub>p ?p' = 0\<^sub>p" by auto
wenzelm@67123
  1659
        then show ?thesis
wenzelm@67123
  1660
        proof cases
wenzelm@67123
  1661
          case deglt: 1
krauss@41403
  1662
          from polydivide_aux.simps sz dn' ba
krauss@41403
  1663
          have eq: "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
wenzelm@33268
  1664
            by (simp add: Let_def)
wenzelm@67123
  1665
          have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
wenzelm@67123
  1666
            if h1: "polydivide_aux a n p k s = (k', r)"
wenzelm@67123
  1667
          proof -
wenzelm@52803
  1668
            from less(1)[OF deglt nr, of k k' r] trans[OF eq[symmetric] h1]
wenzelm@52803
  1669
            have kk': "k \<le> k'"
wenzelm@56066
  1670
              and nr: "\<exists>nr. isnpolyh r nr"
wenzelm@52803
  1671
              and dr: "degree r = 0 \<or> degree r < degree p"
wenzelm@56066
  1672
              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
  1673
              by auto
wenzelm@52803
  1674
            from q1 obtain q n1 where nq: "isnpolyh q n1"
wenzelm@56066
  1675
              and asp: "a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"
wenzelm@56066
  1676
              by blast
wenzelm@56066
  1677
            from nr obtain nr where nr': "isnpolyh r nr"
wenzelm@56066
  1678
              by blast
wenzelm@52803
  1679
            from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0"
wenzelm@52803
  1680
              by simp
wenzelm@33268
  1681
            from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
wenzelm@33268
  1682
            have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
wenzelm@52803
  1683
            from polyadd_normh[OF polymul_normh[OF np
wenzelm@33268
  1684
              polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
wenzelm@52803
  1685
            have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0"
wenzelm@52803
  1686
              by simp
nipkow@68442
  1687
            from asp have "\<forall>bs :: 'a::field_char_0 list.
wenzelm@56066
  1688
              Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) = Ipoly bs (p *\<^sub>p q +\<^sub>p r)"
wenzelm@56066
  1689
              by simp
nipkow@68442
  1690
            then have "\<forall>bs :: 'a::field_char_0 list.
wenzelm@56066
  1691
              Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) =
wenzelm@52803
  1692
              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
haftmann@36349
  1693
              by (simp add: field_simps)
nipkow@68442
  1694
            then have "\<forall>bs :: 'a::field_char_0 list.
wenzelm@56066
  1695
              Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1696
              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p *\<^sub>p p) +
wenzelm@52803
  1697
              Ipoly bs p * Ipoly bs q + Ipoly bs r"
wenzelm@52803
  1698
              by (auto simp only: funpow_shift1_1)
nipkow@68442
  1699
            then have "\<forall>bs:: 'a::field_char_0 list.
wenzelm@56066
  1700
              Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1701
              Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 (1)\<^sub>p) +
wenzelm@52803
  1702
              Ipoly bs q) + Ipoly bs r"
wenzelm@52803
  1703
              by (simp add: field_simps)
nipkow@68442
  1704
            then have "\<forall>bs:: 'a::field_char_0 list.
wenzelm@56066
  1705
              Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@52803
  1706
              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
  1707
              by simp
wenzelm@33268
  1708
            with isnpolyh_unique[OF nakks' nqr']
wenzelm@52803
  1709
            have "a ^\<^sub>p (k' - k) *\<^sub>p s =
wenzelm@52803
  1710
              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
  1711
              by blast
wenzelm@67123
  1712
            with nq' have ?qths
wenzelm@50282
  1713
              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
  1714
              apply (rule_tac x="0" in exI)
wenzelm@52803
  1715
              apply simp
wenzelm@52803
  1716
              done
wenzelm@67123
  1717
            with kk' nr dr show ?thesis
wenzelm@52803
  1718
              by blast
wenzelm@67123
  1719
          qed
wenzelm@67123
  1720
          then show ?thesis by blast
wenzelm@67123
  1721
        next
wenzelm@67123
  1722
          case spz: 2
nipkow@68442
  1723
          from spz isnpolyh_unique[OF polysub_normh[OF ns np'], where q="0\<^sub>p", symmetric, where ?'a = "'a::field_char_0"]
nipkow@68442
  1724
          have "\<forall>bs:: 'a::field_char_0 list. Ipoly bs s = Ipoly bs ?p'"
wenzelm@52803
  1725
            by simp
nipkow@68442
  1726
          with np nxdn have "\<forall>bs:: 'a::field_char_0 list. Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)"
wenzelm@67123
  1727
            by (simp only: funpow_shift1_1) simp
wenzelm@56066
  1728
          then have sp': "s = ?xdn *\<^sub>p p"
wenzelm@56066
  1729
            using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]]
wenzelm@52658
  1730
            by blast
wenzelm@67123
  1731
          have ?thesis if h1: "polydivide_aux a n p k s = (k', r)"
wenzelm@67123
  1732
          proof -
wenzelm@67123
  1733
            from sz dn' ba
wenzelm@67123
  1734
            have "polydivide_aux a n p k s = polydivide_aux a n p k (s -\<^sub>p ?p')"
wenzelm@67123
  1735
              by (simp add: Let_def polydivide_aux.simps)
wenzelm@52803
  1736
            also have "\<dots> = (k,0\<^sub>p)"
wenzelm@67123
  1737
              using spz by (simp add: polydivide_aux.simps)
wenzelm@56066
  1738
            finally have "(k', r) = (k, 0\<^sub>p)"
wenzelm@67123
  1739
              by (simp add: h1)
berghofe@34915
  1740
            with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
wenzelm@67123
  1741
              polyadd_0(2)[OF polymul_normh[OF np nxdn]] show ?thesis
wenzelm@33268
  1742
              apply auto
wenzelm@52803
  1743
              apply (rule exI[where x="?xdn"])
berghofe@34915
  1744
              apply (auto simp add: polymul_commute[of p])
wenzelm@52803
  1745
              done
wenzelm@67123
  1746
          qed
wenzelm@67123
  1747
          then show ?thesis by blast
wenzelm@67123
  1748
        qed
wenzelm@67123
  1749
      next
wenzelm@67123
  1750
        case ba: False
wenzelm@52803
  1751
        from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
wenzelm@33268
  1752
          polymul_normh[OF head_isnpolyh[OF ns] np']]
wenzelm@52803
  1753
        have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
wenzelm@52803
  1754
          by (simp add: min_def)
wenzelm@33268
  1755
        have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
wenzelm@52803
  1756
          using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns]
wenzelm@33268
  1757
            polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
wenzelm@52803
  1758
            funpow_shift1_nz[OF pnz]
wenzelm@52803
  1759
          by simp_all
wenzelm@33268
  1760
        from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
wenzelm@67123
  1761
          polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz
wenzelm@67123
  1762
          funpow_shift1_nz[OF pnz, where n="degree s - n"]
wenzelm@52803
  1763
        have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')"
wenzelm@33268
  1764
          using head_head[OF ns] funpow_shift1_head[OF np pnz]
wenzelm@33268
  1765
            polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
wenzelm@33268
  1766
          by (simp add: ap)
wenzelm@33268
  1767
        from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
wenzelm@33268
  1768
          head_nz[OF np] pnz sz ap[symmetric]
berghofe@34915
  1769
          funpow_shift1_nz[OF pnz, where n="degree s - n"]
wenzelm@52803
  1770
          polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"] head_nz[OF ns]
berghofe@34915
  1771
          ndp dn
wenzelm@52803
  1772
        have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p')"
wenzelm@33268
  1773
          by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
wenzelm@67123
  1774
wenzelm@67123
  1775
        consider "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s" | "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
wenzelm@67123
  1776
          using degree_polysub_samehead[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns]
wenzelm@67123
  1777
            polymul_normh[OF head_isnpolyh[OF ns] np'] hdth degth]
wenzelm@67123
  1778
            polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
wenzelm@67123
  1779
            head_nz[OF np] pnz sz ap[symmetric]
wenzelm@67123
  1780
          by (auto simp add: degree_eq_degreen0[symmetric])
wenzelm@67123
  1781
        then show ?thesis
wenzelm@67123
  1782
        proof cases
wenzelm@67123
  1783
          case dth: 1
wenzelm@52803
  1784
          from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns]
wenzelm@52803
  1785
            polymul_normh[OF head_isnpolyh[OF ns]np']] ap
wenzelm@52803
  1786
          have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0"
wenzelm@52803
  1787
            by simp
wenzelm@67123
  1788
          have ?thesis if h1: "polydivide_aux a n p k s = (k', r)"
wenzelm@67123
  1789
          proof -
krauss@41403
  1790
            from h1 polydivide_aux.simps sz dn' ba
krauss@41403
  1791
            have eq:"polydivide_aux a n p (Suc k) ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
wenzelm@33268
  1792
              by (simp add: Let_def)
berghofe@34915
  1793
            with less(1)[OF dth nasbp', of "Suc k" k' r]
wenzelm@52803
  1794
            obtain q nq nr where kk': "Suc k \<le> k'"
wenzelm@52803
  1795
              and nr: "isnpolyh r nr"
wenzelm@52803
  1796
              and nq: "isnpolyh q nq"
wenzelm@33268
  1797
              and dr: "degree r = 0 \<or> degree r < degree p"
wenzelm@52803
  1798
              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
  1799
              by auto
wenzelm@56066
  1800
            from kk' have kk'': "Suc (k' - Suc k) = k' - k"
wenzelm@56066
  1801
              by arith
wenzelm@67123
  1802
            have "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@67123
  1803
                Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
nipkow@68442
  1804
              for bs :: "'a::field_char_0 list"
wenzelm@67123
  1805
            proof -
wenzelm@52803
  1806
              from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
wenzelm@52803
  1807
              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
  1808
                by simp
wenzelm@56066
  1809
              then have "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s =
wenzelm@52803
  1810
                Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
wenzelm@52803
  1811
                by (simp add: field_simps)
wenzelm@56066
  1812
              then have "Ipoly bs a ^ (k' - k)  * Ipoly bs s =
wenzelm@52803
  1813
                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
  1814
                by (simp add: kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
wenzelm@67123
  1815
              then show ?thesis
wenzelm@52803
  1816
                by (simp add: field_simps)
wenzelm@67123
  1817
            qed
nipkow@68442
  1818
            then have ieq: "\<forall>bs :: 'a::field_char_0 list.
wenzelm@56207
  1819
                Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
wenzelm@56207
  1820
                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
  1821
              by auto
wenzelm@33268
  1822
            let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
wenzelm@56207
  1823
            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
  1824
            have nqw: "isnpolyh ?q 0"
wenzelm@52803
  1825
              by simp
wenzelm@33268
  1826
            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
  1827
            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
  1828
              by blast
wenzelm@67123
  1829
            from dr kk' nr h1 asth nqw show ?thesis
wenzelm@52803
  1830
              apply simp
wenzelm@33268
  1831
              apply (rule conjI)
wenzelm@33268
  1832
              apply (rule exI[where x="nr"], simp)
wenzelm@33268
  1833
              apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
wenzelm@33268
  1834
              apply (rule exI[where x="0"], simp)
wenzelm@52803
  1835
              done
wenzelm@67123
  1836
          qed
wenzelm@67123
  1837
          then show ?thesis by blast
wenzelm@67123
  1838
        next
wenzelm@67123
  1839
          case spz: 2
nipkow@68442
  1840
          have hth: "\<forall>bs :: 'a::field_char_0 list.
wenzelm@67123
  1841
            Ipoly bs (a *\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))"
wenzelm@67123
  1842
          proof
nipkow@68442
  1843
            fix bs :: "'a::field_char_0 list"
wenzelm@33268
  1844
            from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
wenzelm@52803
  1845
            have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'"
wenzelm@52803
  1846
              by simp
wenzelm@56066
  1847
            then have "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
wenzelm@52803
  1848
              by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
wenzelm@67123
  1849
            then show "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))"
wenzelm@52803
  1850
              by simp
wenzelm@67123
  1851
          qed
wenzelm@52803
  1852
          from hth have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)"
nipkow@68442
  1853
            using isnpolyh_unique[where ?'a = "'a::field_char_0", OF polymul_normh[OF head_isnpolyh[OF np] ns]
chaieb@33154
  1854
                    polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
wenzelm@56066
  1855
              simplified ap]
wenzelm@56066
  1856
            by simp
wenzelm@67123
  1857
          have ?ths if h1: "polydivide_aux a n p k s = (k', r)"
wenzelm@67123
  1858
          proof -
wenzelm@52803
  1859
            from h1 sz ba dn' spz polydivide_aux.simps polydivide_aux.simps
wenzelm@56066
  1860
            have "(k', r) = (Suc k, 0\<^sub>p)"
wenzelm@56066
  1861
              by (simp add: Let_def)
wenzelm@52803
  1862
            with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
wenzelm@52803
  1863
              polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
wenzelm@67123
  1864
            show ?thesis
wenzelm@52803
  1865
              apply (clarsimp simp add: Let_def)
wenzelm@52803
  1866
              apply (rule exI[where x="?b *\<^sub>p ?xdn"])
wenzelm@52803
  1867
              apply simp
wenzelm@52803
  1868
              apply (rule exI[where x="0"], simp)
wenzelm@52803
  1869
              done
wenzelm@67123
  1870
          qed
wenzelm@67123
  1871
          then show ?thesis by blast
wenzelm@67123
  1872
        qed
wenzelm@67123
  1873
      qed
wenzelm@67123
  1874
    qed
wenzelm@67123
  1875
  qed
chaieb@33154
  1876
qed
chaieb@33154
  1877
wenzelm@52803
  1878
lemma polydivide_properties:
nipkow@68442
  1879
  assumes "SORT_CONSTRAINT('a::field_char_0)"
wenzelm@56066
  1880
    and np: "isnpolyh p n0"
wenzelm@56066
  1881
    and ns: "isnpolyh s n1"
wenzelm@56066
  1882
    and pnz: "p \<noteq> 0\<^sub>p"
wenzelm@56066
  1883
  shows "\<exists>k r. polydivide s p = (k, r) \<and>
wenzelm@52803
  1884
    (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@56066
  1885
    (\<exists>q n1. isnpolyh q n1 \<and> polypow k (head p) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
wenzelm@52803
  1886
proof -
wenzelm@52803
  1887
  have trv: "head p = head p" "degree p = degree p"
wenzelm@52803
  1888
    by simp_all
wenzelm@52803
  1889
  from polydivide_def[where s="s" and p="p"] have ex: "\<exists> k r. polydivide s p = (k,r)"
wenzelm@52803
  1890
    by auto
wenzelm@52803
  1891
  then obtain k r where kr: "polydivide s p = (k,r)"
wenzelm@52803
  1892
    by blast
wenzelm@56000
  1893
  from trans[OF polydivide_def[where s="s"and p="p", symmetric] kr]
chaieb@33154
  1894
    polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
chaieb@33154
  1895
  have "(degree r = 0 \<or> degree r < degree p) \<and>
wenzelm@52803
  1896
    (\<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
  1897
    by blast
wenzelm@52803
  1898
  with kr show ?thesis
chaieb@33154
  1899
    apply -
chaieb@33154
  1900
    apply (rule exI[where x="k"])
chaieb@33154
  1901
    apply (rule exI[where x="r"])
chaieb@33154
  1902
    apply simp
chaieb@33154
  1903
    done
chaieb@33154
  1904
qed
chaieb@33154
  1905
wenzelm@52658
  1906
wenzelm@60533
  1907
subsection \<open>More about polypoly and pnormal etc\<close>
chaieb@33154
  1908
wenzelm@56000
  1909
definition "isnonconstant p \<longleftrightarrow> \<not> isconstant p"
chaieb@33154
  1910
wenzelm@52658
  1911
lemma isnonconstant_pnormal_iff:
wenzelm@56198
  1912
  assumes "isnonconstant p"
wenzelm@52803
  1913
  shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
chaieb@33154
  1914
proof
wenzelm@52803
  1915
  let ?p = "polypoly bs p"
wenzelm@67123
  1916
  assume *: "pnormal ?p"
wenzelm@67123
  1917
  have "coefficients p \<noteq> []"
wenzelm@56198
  1918
    using assms by (cases p) auto
wenzelm@67123
  1919
  from coefficients_head[of p] last_map[OF this, of "Ipoly bs"] pnormal_last_nonzero[OF *]
wenzelm@56066
  1920
  show "Ipoly bs (head p) \<noteq> 0"
wenzelm@56066
  1921
    by (simp add: polypoly_def)
chaieb@33154
  1922
next
wenzelm@67123
  1923
  assume *: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  1924
  let ?p = "polypoly bs p"
wenzelm@56066
  1925
  have csz: "coefficients p \<noteq> []"
wenzelm@56198
  1926
    using assms by (cases p) auto
wenzelm@56066
  1927
  then have pz: "?p \<noteq> []"
wenzelm@56066
  1928
    by (simp add: polypoly_def)
wenzelm@56066
  1929
  then have lg: "length ?p > 0"
wenzelm@56066
  1930
    by simp
wenzelm@67123
  1931
  from * coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]
wenzelm@56066
  1932
  have lz: "last ?p \<noteq> 0"
wenzelm@56066
  1933
    by (simp add: polypoly_def)
chaieb@33154
  1934
  from pnormal_last_length[OF lg lz] show "pnormal ?p" .
chaieb@33154
  1935
qed
chaieb@33154
  1936
chaieb@33154
  1937
lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
chaieb@33154
  1938
  unfolding isnonconstant_def
wenzelm@52658
  1939
  apply (cases p)
wenzelm@52658
  1940
  apply simp_all
blanchet@58259
  1941
  apply (rename_tac nat a, case_tac nat)
wenzelm@52658
  1942
  apply auto
chaieb@33154
  1943
  done
wenzelm@52658
  1944
wenzelm@52658
  1945
lemma isnonconstant_nonconstant:
wenzelm@56198
  1946
  assumes "isnonconstant p"
chaieb@33154
  1947
  shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
chaieb@33154
  1948
proof
chaieb@33154
  1949
  let ?p = "polypoly bs p"
wenzelm@67123
  1950
  assume "nonconstant ?p"
wenzelm@67123
  1951
  with isnonconstant_pnormal_iff[OF assms, of bs] show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
wenzelm@56066
  1952
    unfolding nonconstant_def by blast
chaieb@33154
  1953
next
chaieb@33154
  1954
  let ?p = "polypoly bs p"
wenzelm@67123
  1955
  assume "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
wenzelm@67123
  1956
  with isnonconstant_pnormal_iff[OF assms, of bs] have pn: "pnormal ?p"
wenzelm@56066
  1957
    by blast
wenzelm@67123
  1958
  have False if H: "?p = [x]" for x
wenzelm@67123
  1959
  proof -
wenzelm@56009
  1960
    from H have "length (coefficients p) = 1"
wenzelm@67123
  1961
      by (auto simp: polypoly_def)
wenzelm@56198
  1962
    with isnonconstant_coefficients_length[OF assms]
wenzelm@67123
  1963
    show ?thesis by arith
wenzelm@67123
  1964
  qed
wenzelm@56009
  1965
  then show "nonconstant ?p"
wenzelm@56009
  1966
    using pn unfolding nonconstant_def by blast
chaieb@33154
  1967
qed
chaieb@33154
  1968
wenzelm@56066
  1969
lemma pnormal_length: "p \<noteq> [] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
wenzelm@52658
  1970
  apply (induct p)
wenzelm@67123
  1971
   apply (simp_all add: pnormal_def)
wenzelm@52658
  1972
  apply (case_tac "p = []")
wenzelm@67123
  1973
   apply simp_all
wenzelm@52658
  1974
  done
chaieb@33154
  1975
wenzelm@52658
  1976
lemma degree_degree:
wenzelm@56207
  1977
  assumes "isnonconstant p"
chaieb@33154
  1978
  shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
wenzelm@67123
  1979
    (is "?lhs \<longleftrightarrow> ?rhs")
chaieb@33154
  1980
proof
wenzelm@52803
  1981
  let ?p = "polypoly bs p"
wenzelm@67123
  1982
  {
wenzelm@67123
  1983
    assume ?lhs
wenzelm@67123
  1984
    from isnonconstant_coefficients_length[OF assms] have "?p \<noteq> []"
wenzelm@67123
  1985
      by (auto simp: polypoly_def)
wenzelm@67123
  1986
    from \<open>?lhs\<close> degree_coefficients[of p] isnonconstant_coefficients_length[OF assms]
wenzelm@67123
  1987
    have "length (pnormalize ?p) = length ?p"
wenzelm@67123
  1988
      by (simp add: Polynomial_List.degree_def polypoly_def)
wenzelm@67123
  1989
    with pnormal_length[OF \<open>?p \<noteq> []\<close>] have "pnormal ?p"
wenzelm@67123
  1990
      by blast
wenzelm@67123
  1991
    with isnonconstant_pnormal_iff[OF assms] show ?rhs
wenzelm@67123
  1992
      by blast
wenzelm@67123
  1993
  next
wenzelm@67123
  1994
    assume ?rhs
wenzelm@67123
  1995
    with isnonconstant_pnormal_iff[OF assms] have "pnormal ?p"
wenzelm@67123
  1996
      by blast
wenzelm@67123
  1997
    with degree_coefficients[of p] isnonconstant_coefficients_length[OF assms] show ?lhs
wenzelm@67123
  1998
      by (auto simp: polypoly_def pnormal_def Polynomial_List.degree_def)
wenzelm@67123
  1999
  }
chaieb@33154
  2000
qed
chaieb@33154
  2001
wenzelm@52658
  2002
wenzelm@67123
  2003
section \<open>Swaps -- division by a certain variable\<close>
wenzelm@52658
  2004
wenzelm@56066
  2005
primrec swap :: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly"
wenzelm@67123
  2006
  where
wenzelm@67123
  2007
    "swap n m (C x) = C x"
wenzelm@67123
  2008
  | "swap n m (Bound k) = Bound (if k = n then m else if k = m then n else k)"
wenzelm@67123
  2009
  | "swap n m (Neg t) = Neg (swap n m t)"
wenzelm@67123
  2010
  | "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
wenzelm@67123
  2011
  | "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
wenzelm@67123
  2012
  | "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
wenzelm@67123
  2013
  | "swap n m (Pw t k) = Pw (swap n m t) k"
wenzelm@67123
  2014
  | "swap n m (CN c k p) = CN (swap n m c) (if k = n then m else if k=m then n else k) (swap n m p)"
chaieb@33154
  2015
wenzelm@52658
  2016
lemma swap:
wenzelm@56066
  2017
  assumes "n < length bs"
wenzelm@56066
  2018
    and "m < length bs"
chaieb@33154
  2019
  shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
chaieb@33154
  2020
proof (induct t)
wenzelm@52658
  2021
  case (Bound k)
wenzelm@56066
  2022
  then show ?case
wenzelm@56066
  2023
    using assms by simp
chaieb@33154
  2024
next
wenzelm@52658
  2025
  case (CN c k p)
wenzelm@56066
  2026
  then show ?case
wenzelm@56066
  2027
    using assms by simp
chaieb@33154
  2028
qed simp_all
chaieb@33154
  2029
wenzelm@52658
  2030
lemma swap_swap_id [simp]: "swap n m (swap m n t) = t"
wenzelm@52658
  2031
  by (induct t) simp_all
wenzelm@52658
  2032
wenzelm@52658
  2033
lemma swap_commute: "swap n m p = swap m n p"
wenzelm@52658
  2034
  by (induct p) simp_all
chaieb@33154
  2035
chaieb@33154
  2036
lemma swap_same_id[simp]: "swap n n t = t"
wenzelm@52658
  2037
  by (induct t) simp_all
chaieb@33154
  2038
chaieb@33154
  2039
definition "swapnorm n m t = polynate (swap n m t)"
chaieb@33154
  2040
wenzelm@52658
  2041
lemma swapnorm:
wenzelm@52658
  2042
  assumes nbs: "n < length bs"
wenzelm@52658
  2043
    and mbs: "m < length bs"
nipkow@68442
  2044
  shows "((Ipoly bs (swapnorm n m t) :: 'a::field_char_0)) =
wenzelm@52658
  2045
    Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
wenzelm@41807
  2046
  using swap[OF assms] swapnorm_def by simp
chaieb@33154
  2047
wenzelm@52658
  2048
lemma swapnorm_isnpoly [simp]:
nipkow@68442
  2049
  assumes "SORT_CONSTRAINT('a::field_char_0)"
chaieb@33154
  2050
  shows "isnpoly (swapnorm n m p)"
chaieb@33154
  2051
  unfolding swapnorm_def by simp
chaieb@33154
  2052
wenzelm@52803
  2053
definition "polydivideby n s p =
wenzelm@56000
  2054
  (let
wenzelm@56000
  2055
    ss = swapnorm 0 n s;
wenzelm@56000
  2056
    sp = swapnorm 0 n p;
wenzelm@56000
  2057
    h = head sp;
wenzelm@56000
  2058
    (k, r) = polydivide ss sp
wenzelm@56000
  2059
   in (k, swapnorm 0 n h, swapnorm 0 n r))"
chaieb@33154
  2060
wenzelm@56000
  2061
lemma swap_nz [simp]: "swap n m p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p"
wenzelm@52658
  2062
  by (induct p) simp_all
chaieb@33154
  2063
krauss@41808
  2064
fun isweaknpoly :: "poly \<Rightarrow> bool"
wenzelm@67123
  2065
  where
wenzelm@67123
  2066
    "isweaknpoly (C c) = True"
wenzelm@67123
  2067
  | "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
wenzelm@67123
  2068
  | "isweaknpoly p = False"
chaieb@33154
  2069
wenzelm@52803
  2070
lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p"
wenzelm@52658
  2071
  by (induct p arbitrary: n0) auto
chaieb@33154
  2072
wenzelm@52803
  2073
lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)"
wenzelm@52658
  2074
  by (induct p) auto
chaieb@33154
  2075
nipkow@62390
  2076
end