src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
author chaieb
Sun Oct 25 08:57:36 2009 +0100 (2009-10-25)
changeset 33154 daa6ddece9f0
child 33268 02de0317f66f
permissions -rw-r--r--
Multivariate polynomials library over fields
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(*  Title:      HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
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    Author:     Amine Chaieb
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*)
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header {* Implementation and verification of mutivariate polynomials Library *}
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theory Reflected_Multivariate_Polynomial
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imports Parity Abstract_Rat Efficient_Nat List Polynomial_List
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begin
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  (* Impelementation *)
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subsection{* Datatype of polynomial expressions *} 
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datatype poly = C Num| Bound nat| Add poly poly|Sub poly poly
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  | Mul poly poly| Neg poly| Pw poly nat| CN poly nat poly
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ML{* @{term "Add"}*}
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syntax "_poly0" :: "poly" ("0\<^sub>p")
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translations "0\<^sub>p" \<rightleftharpoons> "C (0\<^sub>N)"
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syntax "_poly" :: "int \<Rightarrow> poly" ("_\<^sub>p")
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translations "i\<^sub>p" \<rightleftharpoons> "C (i\<^sub>N)"
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subsection{* Boundedness, substitution and all that *}
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consts polysize:: "poly \<Rightarrow> nat"
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primrec
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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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consts 
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  polybound0:: "poly \<Rightarrow> bool" (* a poly is INDEPENDENT of Bound 0 *)
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  polysubst0:: "poly \<Rightarrow> poly \<Rightarrow> poly" (* substitute a poly into a poly for Bound 0 *)
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primrec
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  "polybound0 (C c) = True"
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  "polybound0 (Bound n) = (n>0)"
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  "polybound0 (Neg a) = polybound0 a"
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  "polybound0 (Add a b) = (polybound0 a \<and> polybound0 b)"
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  "polybound0 (Sub a b) = (polybound0 a \<and> polybound0 b)" 
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  "polybound0 (Mul a b) = (polybound0 a \<and> polybound0 b)"
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  "polybound0 (Pw p n) = (polybound0 p)"
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  "polybound0 (CN c n p) = (n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p)"
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primrec
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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) = (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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consts 
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  decrpoly:: "poly \<Rightarrow> poly" 
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recdef decrpoly "measure polysize"
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  "decrpoly (Bound n) = Bound (n - 1)"
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  "decrpoly (Neg a) = Neg (decrpoly a)"
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  "decrpoly (Add a b) = Add (decrpoly a) (decrpoly b)"
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  "decrpoly (Sub a b) = Sub (decrpoly a) (decrpoly b)"
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  "decrpoly (Mul a b) = Mul (decrpoly a) (decrpoly b)"
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  "decrpoly (Pw p n) = Pw (decrpoly p) n"
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  "decrpoly (CN c n p) = CN (decrpoly c) (n - 1) (decrpoly p)"
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  "decrpoly a = a"
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subsection{* Degrees and heads and coefficients *}
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consts degree:: "poly \<Rightarrow> nat"
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recdef degree "measure size"
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  "degree (CN c 0 p) = 1 + degree p"
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  "degree p = 0"
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consts head:: "poly \<Rightarrow> poly"
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recdef head "measure size"
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  "head (CN c 0 p) = head p"
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  "head p = p"
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  (* More general notions of degree and head *)
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consts degreen:: "poly \<Rightarrow> nat \<Rightarrow> nat"
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recdef degreen "measure size"
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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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consts headn:: "poly \<Rightarrow> nat \<Rightarrow> poly"
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recdef headn "measure size"
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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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consts coefficients:: "poly \<Rightarrow> poly list"
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recdef coefficients "measure size"
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  "coefficients (CN c 0 p) = c#(coefficients p)"
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  "coefficients p = [p]"
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consts isconstant:: "poly \<Rightarrow> bool"
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recdef isconstant "measure size"
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  "isconstant (CN c 0 p) = False"
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  "isconstant p = True"
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consts behead:: "poly \<Rightarrow> poly"
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recdef behead "measure size"
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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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consts headconst:: "poly \<Rightarrow> Num"
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recdef headconst "measure size"
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  "headconst (CN c n p) = headconst p"
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  "headconst (C n) = n"
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subsection{* Operations for normalization *}
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consts 
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  polyadd :: "poly\<times>poly \<Rightarrow> poly"
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  polyneg :: "poly \<Rightarrow> poly" ("~\<^sub>p")
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  polysub :: "poly\<times>poly \<Rightarrow> poly"
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  polymul :: "poly\<times>poly \<Rightarrow> poly"
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  polypow :: "nat \<Rightarrow> poly \<Rightarrow> poly"
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syntax "_polyadd" :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "+\<^sub>p" 60)
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translations "a +\<^sub>p b" \<rightleftharpoons> "polyadd (a,b)"  
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syntax "_polymul" :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "*\<^sub>p" 60)
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translations "a *\<^sub>p b" \<rightleftharpoons> "polymul (a,b)"  
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syntax "_polysub" :: "poly \<Rightarrow> poly \<Rightarrow> poly" (infixl "-\<^sub>p" 60)
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translations "a -\<^sub>p b" \<rightleftharpoons> "polysub (a,b)"  
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syntax "_polypow" :: "nat \<Rightarrow> poly \<Rightarrow> poly" (infixl "^\<^sub>p" 60)
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translations "a ^\<^sub>p k" \<rightleftharpoons> "polypow k a" 
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recdef polyadd "measure (\<lambda> (a,b). polysize a + polysize b)"
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  "polyadd (C c, C c') = C (c+\<^sub>Nc')"
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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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stupid:  "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 (let 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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(hints recdef_simp add: Let_def measure_def split_def inv_image_def)
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(*
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declare stupid [simp del, code del]
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lemma [simp,code]: "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 (let 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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  by (simp add: Let_def stupid)
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*)
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recdef polyneg "measure size"
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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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defs polysub_def[code]: "polysub \<equiv> \<lambda> (p,q). polyadd (p,polyneg q)"
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recdef polymul "measure (\<lambda>(a,b). size a + size b)"
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  "polymul(C c, C c') = C (c*\<^sub>Nc')"
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  "polymul(C c, CN c' n' p') = 
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      (if c = 0\<^sub>N then 0\<^sub>p else CN (polymul(C c,c')) n' (polymul(C c, p')))"
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  "polymul(CN c n p, C c') = 
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      (if c' = 0\<^sub>N  then 0\<^sub>p else CN (polymul(c,C c')) n (polymul(p, C c')))"
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  "polymul(CN c n p, CN c' n' p') = 
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  (if n<n' then CN (polymul(c,CN c' n' p')) n (polymul(p,CN c' n' p'))
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  else if n' < n 
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  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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recdef polypow "measure id"
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  "polypow 0 = (\<lambda>p. 1\<^sub>p)"
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  "polypow n = (\<lambda>p. let q = polypow (n div 2) p ; d = polymul(q,q) in 
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                    if even n then d else polymul(p,d))"
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consts polynate :: "poly \<Rightarrow> poly"
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recdef polynate "measure polysize"
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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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fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly" 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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constdefs monic:: "poly \<Rightarrow> (poly \<times> bool)"
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  "monic p \<equiv> (let h = headconst p in if h = 0\<^sub>N then (p,False) else ((C (Ninv h)) *\<^sub>p p, 0>\<^sub>N h))"
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subsection{* Pseudo-division *}
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constdefs shift1:: "poly \<Rightarrow> poly"
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  "shift1 p \<equiv> CN 0\<^sub>p 0 p"
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consts funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
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primrec
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  "funpow 0 f x = x"
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  "funpow (Suc n) f x = funpow n f (f x)"
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function (tailrec) polydivide_aux :: "(poly \<times> nat \<times> poly \<times> nat \<times> 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 (let b = head s; m = degree s in
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  (if m < n then (k,s) else 
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  (let p'= funpow (m - n) shift1 p 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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  by pat_completeness auto
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constdefs polydivide:: "poly \<Rightarrow> poly \<Rightarrow> (nat \<times> poly)"
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  "polydivide s p \<equiv> polydivide_aux (head p,degree p,p,0, s)"
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fun poly_deriv_aux :: "nat \<Rightarrow> poly \<Rightarrow> poly" 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" 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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  (* Verification *)
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lemma nth_pos2[simp]: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
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using Nat.gr0_conv_Suc
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by clarsimp
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subsection{* Semantics of the polynomial representation *}
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consts Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{ring_char_0,power,division_by_zero,field}"
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primrec
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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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syntax "_Ipoly" :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{ring_char_0,power,division_by_zero,field}" ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
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translations "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup>" \<rightleftharpoons> "Ipoly bs p"  
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lemma Ipoly_CInt: "Ipoly bs (C (i,1)) = of_int i" 
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  by (simp add: INum_def)
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lemma Ipoly_CRat: "Ipoly bs (C (i, j)) = of_int i / of_int j" 
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  by (simp  add: INum_def)
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lemmas RIpoly_eqs = Ipoly.simps(2-7) Ipoly_CInt Ipoly_CRat
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subsection {* Normal form and normalization *}
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consts isnpolyh:: "poly \<Rightarrow> nat \<Rightarrow> bool"
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recdef isnpolyh "measure size"
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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: "\<lbrakk>n' \<le> n ; isnpolyh p n\<rbrakk> \<Longrightarrow> isnpolyh p n'"
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by (induct p rule: isnpolyh.induct, auto)
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constdefs isnpoly:: "poly \<Rightarrow> bool"
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  "isnpoly p \<equiv> isnpolyh p 0"
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text{* polyadd preserves normal forms *}
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lemma polyadd_normh: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> 
chaieb@33154
   274
      \<Longrightarrow> isnpolyh (polyadd(p,q)) (min n0 n1)"
chaieb@33154
   275
proof(induct p q arbitrary: n0 n1 rule: polyadd.induct)
chaieb@33154
   276
  case (2 a b c' n' p' n0 n1)
chaieb@33154
   277
  from prems have  th1: "isnpolyh (C (a,b)) (Suc n')" by simp 
chaieb@33154
   278
  from prems(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
chaieb@33154
   279
  with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
chaieb@33154
   280
  with prems(1)[OF th1 th2] have th3:"isnpolyh (C (a,b) +\<^sub>p c') (Suc n')" by simp
chaieb@33154
   281
  from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp 
chaieb@33154
   282
  thus ?case using prems th3 by simp
chaieb@33154
   283
next
chaieb@33154
   284
  case (3 c' n' p' a b n1 n0)
chaieb@33154
   285
  from prems have  th1: "isnpolyh (C (a,b)) (Suc n')" by simp 
chaieb@33154
   286
  from prems(2) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1" by simp_all
chaieb@33154
   287
  with isnpolyh_mono have cp: "isnpolyh c' (Suc n')" by simp
chaieb@33154
   288
  with prems(1)[OF th2 th1] have th3:"isnpolyh (c' +\<^sub>p C (a,b)) (Suc n')" by simp
chaieb@33154
   289
  from nplen1 have n01len1: "min n0 n1 \<le> n'" by simp 
chaieb@33154
   290
  thus ?case using prems th3 by simp
chaieb@33154
   291
next
chaieb@33154
   292
  case (4 c n p c' n' p' n0 n1)
chaieb@33154
   293
  hence nc: "isnpolyh c (Suc n)" and np: "isnpolyh p n" by simp_all
chaieb@33154
   294
  from prems have nc': "isnpolyh c' (Suc n')" and np': "isnpolyh p' n'" by simp_all 
chaieb@33154
   295
  from prems have ngen0: "n \<ge> n0" by simp
chaieb@33154
   296
  from prems have n'gen1: "n' \<ge> n1" by simp 
chaieb@33154
   297
  have "n < n' \<or> n' < n \<or> n = n'" by auto
chaieb@33154
   298
  moreover {assume eq: "n = n'" hence eq': "\<not> n' < n \<and> \<not> n < n'" by simp
chaieb@33154
   299
    with prems(2)[rule_format, OF eq' nc nc'] 
chaieb@33154
   300
    have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)" by auto
chaieb@33154
   301
    hence ncc'n01: "isnpolyh (c +\<^sub>p c') (min n0 n1)"
chaieb@33154
   302
      using isnpolyh_mono[where n'="min n0 n1" and n="Suc n"] ngen0 n'gen1 by auto
chaieb@33154
   303
    from eq prems(1)[rule_format, OF eq' np np'] have npp': "isnpolyh (p +\<^sub>p p') n" by simp
chaieb@33154
   304
    have minle: "min n0 n1 \<le> n'" using ngen0 n'gen1 eq by simp
chaieb@33154
   305
    from minle npp' ncc'n01 prems ngen0 n'gen1 ncc' have ?case by (simp add: Let_def)}
chaieb@33154
   306
  moreover {assume lt: "n < n'"
chaieb@33154
   307
    have "min n0 n1 \<le> n0" by simp
chaieb@33154
   308
    with prems have th1:"min n0 n1 \<le> n" by auto 
chaieb@33154
   309
    from prems have th21: "isnpolyh c (Suc n)" by simp
chaieb@33154
   310
    from prems have th22: "isnpolyh (CN c' n' p') n'" by simp
chaieb@33154
   311
    from lt have th23: "min (Suc n) n' = Suc n" by arith
chaieb@33154
   312
    from prems(4)[rule_format, OF lt th21 th22]
chaieb@33154
   313
    have "isnpolyh (polyadd (c, CN c' n' p')) (Suc n)" using th23 by simp
chaieb@33154
   314
    with prems th1 have ?case by simp } 
chaieb@33154
   315
  moreover {assume gt: "n' < n" hence gt': "n' < n \<and> \<not> n < n'" by simp
chaieb@33154
   316
    have "min n0 n1 \<le> n1"  by simp
chaieb@33154
   317
    with prems have th1:"min n0 n1 \<le> n'" by auto
chaieb@33154
   318
    from prems have th21: "isnpolyh c' (Suc n')" by simp_all
chaieb@33154
   319
    from prems have th22: "isnpolyh (CN c n p) n" by simp
chaieb@33154
   320
    from gt have th23: "min n (Suc n') = Suc n'" by arith
chaieb@33154
   321
    from prems(3)[rule_format, OF  gt' th22 th21]
chaieb@33154
   322
    have "isnpolyh (polyadd (CN c n p,c')) (Suc n')" using th23 by simp
chaieb@33154
   323
    with prems th1 have ?case by simp}
chaieb@33154
   324
      ultimately show ?case by blast
chaieb@33154
   325
qed auto
chaieb@33154
   326
chaieb@33154
   327
lemma polyadd[simp]: "Ipoly bs (polyadd (p,q)) = (Ipoly bs p) + (Ipoly bs q)"
chaieb@33154
   328
by (induct p q rule: polyadd.induct, auto simp add: Let_def ring_simps right_distrib[symmetric] simp del: right_distrib)
chaieb@33154
   329
chaieb@33154
   330
lemma polyadd_norm: "\<lbrakk> isnpoly p ; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polyadd(p,q))"
chaieb@33154
   331
  using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
chaieb@33154
   332
chaieb@33154
   333
text{* The degree of addition and other general lemmas needed for the normal form of polymul*}
chaieb@33154
   334
chaieb@33154
   335
lemma polyadd_different_degreen: 
chaieb@33154
   336
  "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degreen p m \<noteq> degreen q m ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow> 
chaieb@33154
   337
  degreen (polyadd(p,q)) m = max (degreen p m) (degreen q m)"
chaieb@33154
   338
proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
chaieb@33154
   339
  case (4 c n p c' n' p' m n0 n1)
chaieb@33154
   340
  thus ?case 
chaieb@33154
   341
    apply (cases "n' < n", simp_all add: Let_def)
chaieb@33154
   342
    apply (cases "n = n'", simp_all)
chaieb@33154
   343
    apply (cases "n' = m", simp_all add: Let_def)
chaieb@33154
   344
    by (erule allE[where x="m"], erule allE[where x="Suc m"], 
chaieb@33154
   345
           erule allE[where x="m"], erule allE[where x="Suc m"], 
chaieb@33154
   346
           clarsimp,erule allE[where x="m"],erule allE[where x="Suc m"], simp)
chaieb@33154
   347
qed simp_all 
chaieb@33154
   348
chaieb@33154
   349
lemma headnz[simp]: "\<lbrakk>isnpolyh p n ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> headn p m \<noteq> 0\<^sub>p"
chaieb@33154
   350
  by (induct p arbitrary: n rule: headn.induct, auto)
chaieb@33154
   351
lemma degree_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> degree p = 0"
chaieb@33154
   352
  by (induct p arbitrary: n rule: degree.induct, auto)
chaieb@33154
   353
lemma degreen_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> degreen p m = 0"
chaieb@33154
   354
  by (induct p arbitrary: n rule: degreen.induct, auto)
chaieb@33154
   355
chaieb@33154
   356
lemma degree_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> degree p = 0"
chaieb@33154
   357
  by (induct p arbitrary: n rule: degree.induct, auto)
chaieb@33154
   358
chaieb@33154
   359
lemma degree_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degree c = 0"
chaieb@33154
   360
  using degree_isnpolyh_Suc by auto
chaieb@33154
   361
lemma degreen_npolyhCN[simp]: "isnpolyh (CN c n p) n0 \<Longrightarrow> degreen c n = 0"
chaieb@33154
   362
  using degreen_0 by auto
chaieb@33154
   363
chaieb@33154
   364
chaieb@33154
   365
lemma degreen_polyadd:
chaieb@33154
   366
  assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> max n0 n1"
chaieb@33154
   367
  shows "degreen (p +\<^sub>p q) m \<le> max (degreen p m) (degreen q m)"
chaieb@33154
   368
  using np nq m
chaieb@33154
   369
proof (induct p q arbitrary: n0 n1 m rule: polyadd.induct)
chaieb@33154
   370
  case (2 c c' n' p' n0 n1) thus ?case  by (cases n', simp_all)
chaieb@33154
   371
next
chaieb@33154
   372
  case (3 c n p c' n0 n1) thus ?case by (cases n, auto)
chaieb@33154
   373
next
chaieb@33154
   374
  case (4 c n p c' n' p' n0 n1 m) 
chaieb@33154
   375
  thus ?case 
chaieb@33154
   376
    apply (cases "n < n'", simp_all add: Let_def)
chaieb@33154
   377
    apply (cases "n' < n", simp_all)
chaieb@33154
   378
    apply (erule allE[where x="n"],erule allE[where x="Suc n"],clarify)
chaieb@33154
   379
    apply (erule allE[where x="n'"],erule allE[where x="Suc n'"],clarify)
chaieb@33154
   380
    by (erule allE[where x="m"],erule allE[where x="m"], auto)
chaieb@33154
   381
qed auto
chaieb@33154
   382
chaieb@33154
   383
chaieb@33154
   384
lemma polyadd_eq_const_degreen: "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk> 
chaieb@33154
   385
  \<Longrightarrow> degreen p m = degreen q m"
chaieb@33154
   386
proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
chaieb@33154
   387
  case (4 c n p c' n' p' m n0 n1 x) 
chaieb@33154
   388
  hence z: "CN c n p +\<^sub>p CN c' n' p' = C x" by simp
chaieb@33154
   389
  {assume nn': "n' < n" hence ?case using prems by simp}
chaieb@33154
   390
  moreover 
chaieb@33154
   391
  {assume nn':"\<not> n' < n" hence "n < n' \<or> n = n'" by arith
chaieb@33154
   392
    moreover {assume "n < n'" with prems have ?case by simp }
chaieb@33154
   393
    moreover {assume eq: "n = n'" hence ?case using prems 
chaieb@33154
   394
	by (cases "p +\<^sub>p p' = 0\<^sub>p", auto simp add: Let_def) }
chaieb@33154
   395
    ultimately have ?case by blast}
chaieb@33154
   396
  ultimately show ?case by blast
chaieb@33154
   397
qed simp_all
chaieb@33154
   398
chaieb@33154
   399
lemma polymul_properties:
chaieb@33154
   400
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
   401
  and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and m: "m \<le> min n0 n1"
chaieb@33154
   402
  shows "isnpolyh (p *\<^sub>p q) (min n0 n1)" 
chaieb@33154
   403
  and "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)" 
chaieb@33154
   404
  and "degreen (p *\<^sub>p q) m = (if (p = 0\<^sub>p \<or> q = 0\<^sub>p) then 0 
chaieb@33154
   405
                             else degreen p m + degreen q m)"
chaieb@33154
   406
  using np nq m
chaieb@33154
   407
proof(induct p q arbitrary: n0 n1 m rule: polymul.induct)
chaieb@33154
   408
  case (2 a b c' n' p') 
chaieb@33154
   409
  let ?c = "(a,b)"
chaieb@33154
   410
  { case (1 n0 n1) 
chaieb@33154
   411
    hence n: "isnpolyh (C ?c) n'" "isnpolyh c' (Suc n')" "isnpolyh p' n'" "isnormNum ?c" 
chaieb@33154
   412
      "isnpolyh (CN c' n' p') n1"
chaieb@33154
   413
      by simp_all
chaieb@33154
   414
    {assume "?c = 0\<^sub>N" hence ?case by auto}
chaieb@33154
   415
      moreover {assume cnz: "?c \<noteq> 0\<^sub>N" 
chaieb@33154
   416
	from "2.hyps"(1)[rule_format,where xb="n'",  OF cnz n(1) n(3)] 
chaieb@33154
   417
	  "2.hyps"(2)[rule_format, where x="Suc n'" 
chaieb@33154
   418
	  and xa="Suc n'" and xb = "n'", OF cnz ] cnz n have ?case
chaieb@33154
   419
	  by (auto simp add: min_def)}
chaieb@33154
   420
      ultimately show ?case by blast
chaieb@33154
   421
  next
chaieb@33154
   422
    case (2 n0 n1) thus ?case by auto 
chaieb@33154
   423
  next
chaieb@33154
   424
    case (3 n0 n1) thus ?case  using "2.hyps" by auto } 
chaieb@33154
   425
next
chaieb@33154
   426
  case (3 c n p a b){
chaieb@33154
   427
    let ?c' = "(a,b)"
chaieb@33154
   428
    case (1 n0 n1) 
chaieb@33154
   429
    hence n: "isnpolyh (C ?c') n" "isnpolyh c (Suc n)" "isnpolyh p n" "isnormNum ?c'" 
chaieb@33154
   430
      "isnpolyh (CN c n p) n0"
chaieb@33154
   431
      by simp_all
chaieb@33154
   432
    {assume "?c' = 0\<^sub>N" hence ?case by auto}
chaieb@33154
   433
      moreover {assume cnz: "?c' \<noteq> 0\<^sub>N"
chaieb@33154
   434
	from "3.hyps"(1)[rule_format,where xb="n",  OF cnz n(3) n(1)] 
chaieb@33154
   435
	  "3.hyps"(2)[rule_format, where x="Suc n" 
chaieb@33154
   436
	  and xa="Suc n" and xb = "n", OF cnz ] cnz n have ?case
chaieb@33154
   437
	  by (auto simp add: min_def)}
chaieb@33154
   438
      ultimately show ?case by blast
chaieb@33154
   439
  next
chaieb@33154
   440
    case (2 n0 n1) thus ?case apply auto done
chaieb@33154
   441
  next
chaieb@33154
   442
    case (3 n0 n1) thus ?case  using "3.hyps" by auto } 
chaieb@33154
   443
next
chaieb@33154
   444
  case (4 c n p c' n' p')
chaieb@33154
   445
  let ?cnp = "CN c n p" let ?cnp' = "CN c' n' p'"
chaieb@33154
   446
    {fix n0 n1
chaieb@33154
   447
      assume "isnpolyh ?cnp n0" and "isnpolyh ?cnp' n1"
chaieb@33154
   448
      hence cnp: "isnpolyh ?cnp n" and cnp': "isnpolyh ?cnp' n'"
chaieb@33154
   449
	and np: "isnpolyh p n" and nc: "isnpolyh c (Suc n)" 
chaieb@33154
   450
	and np': "isnpolyh p' n'" and nc': "isnpolyh c' (Suc n')"
chaieb@33154
   451
	and nn0: "n \<ge> n0" and nn1:"n' \<ge> n1"
chaieb@33154
   452
	by simp_all
chaieb@33154
   453
      have "n < n' \<or> n' < n \<or> n' = n" by auto
chaieb@33154
   454
      moreover
chaieb@33154
   455
      {assume nn': "n < n'"
chaieb@33154
   456
	with "4.hyps"(5)[rule_format, OF nn' np cnp', where xb ="n"] 
chaieb@33154
   457
	  "4.hyps"(6)[rule_format, OF nn' nc cnp', where xb="n"] nn' nn0 nn1 cnp
chaieb@33154
   458
	have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
chaieb@33154
   459
	  by (simp add: min_def) }
chaieb@33154
   460
      moreover
chaieb@33154
   461
chaieb@33154
   462
      {assume nn': "n > n'" hence stupid: "n' < n \<and> \<not> n < n'" by arith
chaieb@33154
   463
	with "4.hyps"(3)[rule_format, OF stupid cnp np', where xb="n'"]
chaieb@33154
   464
	  "4.hyps"(4)[rule_format, OF stupid cnp nc', where xb="Suc n'"] 
chaieb@33154
   465
	  nn' nn0 nn1 cnp'
chaieb@33154
   466
	have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
chaieb@33154
   467
	  by (cases "Suc n' = n", simp_all add: min_def)}
chaieb@33154
   468
      moreover
chaieb@33154
   469
      {assume nn': "n' = n" hence stupid: "\<not> n' < n \<and> \<not> n < n'" by arith
chaieb@33154
   470
	from "4.hyps"(1)[rule_format, OF stupid cnp np', where xb="n"]
chaieb@33154
   471
	  "4.hyps"(2)[rule_format, OF stupid cnp nc', where xb="n"] nn' cnp cnp' nn1
chaieb@33154
   472
	
chaieb@33154
   473
	have "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)"
chaieb@33154
   474
	  by simp (rule polyadd_normh,simp_all add: min_def isnpolyh_mono[OF nn0]) }
chaieb@33154
   475
      ultimately show "isnpolyh (?cnp *\<^sub>p ?cnp') (min n0 n1)" by blast }
chaieb@33154
   476
    note th = this
chaieb@33154
   477
    {fix n0 n1 m
chaieb@33154
   478
      assume np: "isnpolyh ?cnp n0" and np':"isnpolyh ?cnp' n1"
chaieb@33154
   479
      and m: "m \<le> min n0 n1"
chaieb@33154
   480
      let ?d = "degreen (?cnp *\<^sub>p ?cnp') m"
chaieb@33154
   481
      let ?d1 = "degreen ?cnp m"
chaieb@33154
   482
      let ?d2 = "degreen ?cnp' m"
chaieb@33154
   483
      let ?eq = "?d = (if ?cnp = 0\<^sub>p \<or> ?cnp' = 0\<^sub>p then 0  else ?d1 + ?d2)"
chaieb@33154
   484
      have "n'<n \<or> n < n' \<or> n' = n" by auto
chaieb@33154
   485
      moreover 
chaieb@33154
   486
      {assume "n' < n \<or> n < n'"
chaieb@33154
   487
	with "4.hyps" np np' m 
chaieb@33154
   488
	have ?eq apply (cases "n' < n", simp_all)
chaieb@33154
   489
	apply (erule allE[where x="n"],erule allE[where x="n"],auto) 
chaieb@33154
   490
	done }
chaieb@33154
   491
      moreover
chaieb@33154
   492
      {assume nn': "n' = n"  hence nn:"\<not> n' < n \<and> \<not> n < n'" by arith
chaieb@33154
   493
 	from "4.hyps"(1)[rule_format, OF nn, where x="n" and xa ="n'" and xb="n"]
chaieb@33154
   494
	  "4.hyps"(2)[rule_format, OF nn, where x="n" and xa ="Suc n'" and xb="n"] 
chaieb@33154
   495
	  np np' nn'
chaieb@33154
   496
	have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
chaieb@33154
   497
	  "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
chaieb@33154
   498
	  "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)" 
chaieb@33154
   499
	  "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" by (auto simp add: min_def)
chaieb@33154
   500
	{assume mn: "m = n" 
chaieb@33154
   501
	  from "4.hyps"(1)[rule_format, OF nn norm(1,4), where xb="n"]
chaieb@33154
   502
	    "4.hyps"(2)[rule_format, OF nn norm(1,2), where xb="n"] norm nn' mn
chaieb@33154
   503
	  have degs:  "degreen (?cnp *\<^sub>p c') n = 
chaieb@33154
   504
	    (if c'=0\<^sub>p then 0 else ?d1 + degreen c' n)"
chaieb@33154
   505
	    "degreen (?cnp *\<^sub>p p') n = ?d1  + degreen p' n" by (simp_all add: min_def)
chaieb@33154
   506
	  from degs norm
chaieb@33154
   507
	  have th1: "degreen(?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" by simp
chaieb@33154
   508
	  hence neq: "degreen (?cnp *\<^sub>p c') n \<noteq> degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n"
chaieb@33154
   509
	    by simp
chaieb@33154
   510
	  have nmin: "n \<le> min n n" by (simp add: min_def)
chaieb@33154
   511
	  from polyadd_different_degreen[OF norm(3,6) neq nmin] th1
chaieb@33154
   512
	  have deg: "degreen (CN c n p *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n = degreen (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp 
chaieb@33154
   513
	  from "4.hyps"(1)[rule_format, OF nn norm(1,4), where xb="n"]
chaieb@33154
   514
	    "4.hyps"(2)[rule_format, OF nn norm(1,2), where xb="n"]
chaieb@33154
   515
	    mn norm m nn' deg
chaieb@33154
   516
	  have ?eq by simp}
chaieb@33154
   517
	moreover
chaieb@33154
   518
	{assume mn: "m \<noteq> n" hence mn': "m < n" using m np by auto
chaieb@33154
   519
	  from nn' m np have max1: "m \<le> max n n"  by simp 
chaieb@33154
   520
	  hence min1: "m \<le> min n n" by simp	
chaieb@33154
   521
	  hence min2: "m \<le> min n (Suc n)" by simp
chaieb@33154
   522
	  {assume "c' = 0\<^sub>p"
chaieb@33154
   523
	    from `c' = 0\<^sub>p` have ?eq
chaieb@33154
   524
	      using "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
chaieb@33154
   525
	    "4.hyps"(2)[rule_format, OF nn norm(1,2) min2] mn nn'
chaieb@33154
   526
	      apply simp
chaieb@33154
   527
	      done}
chaieb@33154
   528
	  moreover
chaieb@33154
   529
	  {assume cnz: "c' \<noteq> 0\<^sub>p"
chaieb@33154
   530
	    from "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
chaieb@33154
   531
	      "4.hyps"(2)[rule_format, OF nn norm(1,2) min2]
chaieb@33154
   532
	      degreen_polyadd[OF norm(3,6) max1]
chaieb@33154
   533
chaieb@33154
   534
	    have "degreen (?cnp *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n (?cnp *\<^sub>p p')) m 
chaieb@33154
   535
	      \<le> max (degreen (?cnp *\<^sub>p c') m) (degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) m)"
chaieb@33154
   536
	      using mn nn' cnz np np' by simp
chaieb@33154
   537
	    with "4.hyps"(1)[rule_format, OF nn norm(1,4) min1]
chaieb@33154
   538
	      "4.hyps"(2)[rule_format, OF nn norm(1,2) min2]
chaieb@33154
   539
	      degreen_0[OF norm(3) mn'] have ?eq using nn' mn cnz np np' by clarsimp}
chaieb@33154
   540
	  ultimately have ?eq by blast }
chaieb@33154
   541
	ultimately have ?eq by blast}
chaieb@33154
   542
      ultimately show ?eq by blast}
chaieb@33154
   543
    note degth = this
chaieb@33154
   544
    { case (2 n0 n1)
chaieb@33154
   545
      hence np: "isnpolyh ?cnp n0" and np': "isnpolyh ?cnp' n1" 
chaieb@33154
   546
	and m: "m \<le> min n0 n1" by simp_all
chaieb@33154
   547
      hence mn: "m \<le> n" by simp
chaieb@33154
   548
      let ?c0p = "CN 0\<^sub>p n (?cnp *\<^sub>p p')"
chaieb@33154
   549
      {assume C: "?cnp *\<^sub>p c' +\<^sub>p ?c0p = 0\<^sub>p" "n' = n"
chaieb@33154
   550
	hence nn: "\<not>n' < n \<and> \<not> n<n'" by simp
chaieb@33154
   551
	from "4.hyps"(1) [rule_format, OF nn, where x="n" and xa = "n" and xb="n"] 
chaieb@33154
   552
	  "4.hyps"(2) [rule_format, OF nn, where x="n" and xa = "Suc n" and xb="n"] 
chaieb@33154
   553
	  np np' C(2) mn
chaieb@33154
   554
	have norm: "isnpolyh ?cnp n" "isnpolyh c' (Suc n)" "isnpolyh (?cnp *\<^sub>p c') n"
chaieb@33154
   555
	  "isnpolyh p' n" "isnpolyh (?cnp *\<^sub>p p') n" "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n"
chaieb@33154
   556
	  "(?cnp *\<^sub>p c' = 0\<^sub>p) = (c' = 0\<^sub>p)" 
chaieb@33154
   557
	  "?cnp *\<^sub>p p' \<noteq> 0\<^sub>p" 
chaieb@33154
   558
	  "degreen (?cnp *\<^sub>p c') n = (if c'=0\<^sub>p then 0 else degreen ?cnp n + degreen c' n)"
chaieb@33154
   559
	    "degreen (?cnp *\<^sub>p p') n = degreen ?cnp n + degreen p' n"
chaieb@33154
   560
	  by (simp_all add: min_def)
chaieb@33154
   561
	    
chaieb@33154
   562
	  from norm have cn: "isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp
chaieb@33154
   563
	  have degneq: "degreen (?cnp *\<^sub>p c') n < degreen (CN 0\<^sub>p n (?cnp *\<^sub>p p')) n" 
chaieb@33154
   564
	    using norm by simp
chaieb@33154
   565
	from polyadd_eq_const_degreen[OF norm(3) cn C(1), where m="n"]  degneq
chaieb@33154
   566
	have "False" by simp }
chaieb@33154
   567
      thus ?case using "4.hyps" by clarsimp}
chaieb@33154
   568
qed auto
chaieb@33154
   569
chaieb@33154
   570
lemma polymul[simp]: "Ipoly bs (p *\<^sub>p q) = (Ipoly bs p) * (Ipoly bs q)"
chaieb@33154
   571
by(induct p q rule: polymul.induct, auto simp add: ring_simps)
chaieb@33154
   572
chaieb@33154
   573
lemma polymul_normh: 
chaieb@33154
   574
    assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
   575
  shows "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (p *\<^sub>p q) (min n0 n1)"
chaieb@33154
   576
  using polymul_properties(1)  by blast
chaieb@33154
   577
lemma polymul_eq0_iff: 
chaieb@33154
   578
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
   579
  shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p) "
chaieb@33154
   580
  using polymul_properties(2)  by blast
chaieb@33154
   581
lemma polymul_degreen:  
chaieb@33154
   582
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
   583
  shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; m \<le> min n0 n1\<rbrakk> \<Longrightarrow> 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
   584
  using polymul_properties(3) by blast
chaieb@33154
   585
lemma polymul_norm:   
chaieb@33154
   586
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
   587
  shows "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polymul (p,q))"
chaieb@33154
   588
  using polymul_normh[of "p" "0" "q" "0"] isnpoly_def by simp
chaieb@33154
   589
chaieb@33154
   590
lemma headconst_zero: "isnpolyh p n0 \<Longrightarrow> headconst p = 0\<^sub>N \<longleftrightarrow> p = 0\<^sub>p"
chaieb@33154
   591
  by (induct p arbitrary: n0 rule: headconst.induct, auto)
chaieb@33154
   592
chaieb@33154
   593
lemma headconst_isnormNum: "isnpolyh p n0 \<Longrightarrow> isnormNum (headconst p)"
chaieb@33154
   594
  by (induct p arbitrary: n0, auto)
chaieb@33154
   595
chaieb@33154
   596
lemma monic_eqI: assumes np: "isnpolyh p n0" 
chaieb@33154
   597
  shows "INum (headconst p) * Ipoly bs (fst (monic p)) = (Ipoly bs p ::'a::{ring_char_0,power,division_by_zero,field})"
chaieb@33154
   598
  unfolding monic_def Let_def
chaieb@33154
   599
proof(cases "headconst p = 0\<^sub>N", simp_all add: headconst_zero[OF np])
chaieb@33154
   600
  let ?h = "headconst p"
chaieb@33154
   601
  assume pz: "p \<noteq> 0\<^sub>p"
chaieb@33154
   602
  {assume hz: "INum ?h = (0::'a)"
chaieb@33154
   603
    from headconst_isnormNum[OF np] have norm: "isnormNum ?h" "isnormNum 0\<^sub>N" by simp_all
chaieb@33154
   604
    from isnormNum_unique[where ?'a = 'a, OF norm] hz have "?h = 0\<^sub>N" by simp
chaieb@33154
   605
    with headconst_zero[OF np] have "p =0\<^sub>p" by blast with pz have "False" by blast}
chaieb@33154
   606
  thus "INum (headconst p) = (0::'a) \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by blast
chaieb@33154
   607
qed
chaieb@33154
   608
chaieb@33154
   609
chaieb@33154
   610
 
chaieb@33154
   611
chaieb@33154
   612
text{* polyneg is a negation and preserves normal form *}
chaieb@33154
   613
lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
chaieb@33154
   614
by (induct p rule: polyneg.induct, auto)
chaieb@33154
   615
chaieb@33154
   616
lemma polyneg0: "isnpolyh p n \<Longrightarrow> ((~\<^sub>p p) = 0\<^sub>p) = (p = 0\<^sub>p)"
chaieb@33154
   617
  by (induct p arbitrary: n rule: polyneg.induct, auto simp add: Nneg_def)
chaieb@33154
   618
lemma polyneg_polyneg: "isnpolyh p n0 \<Longrightarrow> ~\<^sub>p (~\<^sub>p p) = p"
chaieb@33154
   619
  by (induct p arbitrary: n0 rule: polyneg.induct, auto)
chaieb@33154
   620
lemma polyneg_normh: "\<And>n. isnpolyh p n \<Longrightarrow> isnpolyh (polyneg p) n "
chaieb@33154
   621
by (induct p rule: polyneg.induct, auto simp add: polyneg0)
chaieb@33154
   622
chaieb@33154
   623
lemma polyneg_norm: "isnpoly p \<Longrightarrow> isnpoly (polyneg p)"
chaieb@33154
   624
  using isnpoly_def polyneg_normh by simp
chaieb@33154
   625
chaieb@33154
   626
chaieb@33154
   627
text{* polysub is a substraction and preserves normalform *}
chaieb@33154
   628
lemma polysub[simp]: "Ipoly bs (polysub (p,q)) = (Ipoly bs p) - (Ipoly bs q)"
chaieb@33154
   629
by (simp add: polysub_def polyneg polyadd)
chaieb@33154
   630
lemma polysub_normh: "\<And> n0 n1. \<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> isnpolyh (polysub(p,q)) (min n0 n1)"
chaieb@33154
   631
by (simp add: polysub_def polyneg_normh polyadd_normh)
chaieb@33154
   632
chaieb@33154
   633
lemma polysub_norm: "\<lbrakk> isnpoly p; isnpoly q\<rbrakk> \<Longrightarrow> isnpoly (polysub(p,q))"
chaieb@33154
   634
  using polyadd_norm polyneg_norm by (simp add: polysub_def) 
chaieb@33154
   635
lemma polysub_same_0[simp]:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
   636
  shows "isnpolyh p n0 \<Longrightarrow> polysub (p, p) = 0\<^sub>p"
chaieb@33154
   637
unfolding polysub_def split_def fst_conv snd_conv
chaieb@33154
   638
by (induct p arbitrary: n0,auto simp add: Let_def Nsub0[simplified Nsub_def])
chaieb@33154
   639
chaieb@33154
   640
lemma polysub_0: 
chaieb@33154
   641
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
   642
  shows "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1\<rbrakk> \<Longrightarrow> (p -\<^sub>p q = 0\<^sub>p) = (p = q)"
chaieb@33154
   643
  unfolding polysub_def split_def fst_conv snd_conv
chaieb@33154
   644
  apply (induct p q arbitrary: n0 n1 rule:polyadd.induct, simp_all add: Nsub0[simplified Nsub_def])
chaieb@33154
   645
  apply (clarsimp simp add: Let_def)
chaieb@33154
   646
  apply (case_tac "n < n'", simp_all)
chaieb@33154
   647
  apply (case_tac "n' < n", simp_all)
chaieb@33154
   648
  apply (erule impE)+
chaieb@33154
   649
  apply (rule_tac x="Suc n" in exI, simp)
chaieb@33154
   650
  apply (rule_tac x="n" in exI, simp)
chaieb@33154
   651
  apply (erule impE)+
chaieb@33154
   652
  apply (rule_tac x="n" in exI, simp)
chaieb@33154
   653
  apply (rule_tac x="Suc n" in exI, simp)
chaieb@33154
   654
  apply (erule impE)+
chaieb@33154
   655
  apply (rule_tac x="Suc n" in exI, simp)
chaieb@33154
   656
  apply (rule_tac x="n" in exI, simp)
chaieb@33154
   657
  apply (erule impE)+
chaieb@33154
   658
  apply (rule_tac x="Suc n" in exI, simp)
chaieb@33154
   659
  apply clarsimp
chaieb@33154
   660
  done
chaieb@33154
   661
chaieb@33154
   662
text{* polypow is a power function and preserves normal forms *}
chaieb@33154
   663
lemma polypow[simp]: "Ipoly bs (polypow n p) = ((Ipoly bs p :: 'a::{ring_char_0,division_by_zero,field})) ^ n"
chaieb@33154
   664
proof(induct n rule: polypow.induct)
chaieb@33154
   665
  case 1 thus ?case by simp
chaieb@33154
   666
next
chaieb@33154
   667
  case (2 n)
chaieb@33154
   668
  let ?q = "polypow ((Suc n) div 2) p"
chaieb@33154
   669
  let ?d = "polymul(?q,?q)"
chaieb@33154
   670
  have "odd (Suc n) \<or> even (Suc n)" by simp
chaieb@33154
   671
  moreover 
chaieb@33154
   672
  {assume odd: "odd (Suc n)"
chaieb@33154
   673
    have th: "(Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat))))) = Suc n div 2 + Suc n div 2 + 1" by arith
chaieb@33154
   674
    from odd have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs (polymul(p, ?d))" by (simp add: Let_def)
chaieb@33154
   675
    also have "\<dots> = (Ipoly bs p) * (Ipoly bs p)^(Suc n div 2)*(Ipoly bs p)^(Suc n div 2)"
chaieb@33154
   676
      using "2.hyps" by simp
chaieb@33154
   677
    also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2 + 1)"
chaieb@33154
   678
      apply (simp only: power_add power_one_right) by simp
chaieb@33154
   679
    also have "\<dots> = (Ipoly bs p) ^ (Suc (Suc (Suc (0\<Colon>nat)) * (Suc n div Suc (Suc (0\<Colon>nat)))))"
chaieb@33154
   680
      by (simp only: th)
chaieb@33154
   681
    finally have ?case 
chaieb@33154
   682
    using odd_nat_div_two_times_two_plus_one[OF odd, symmetric] by simp  }
chaieb@33154
   683
  moreover 
chaieb@33154
   684
  {assume even: "even (Suc n)"
chaieb@33154
   685
    have th: "(Suc (Suc (0\<Colon>nat))) * (Suc n div Suc (Suc (0\<Colon>nat))) = Suc n div 2 + Suc n div 2" by arith
chaieb@33154
   686
    from even have "Ipoly bs (p ^\<^sub>p Suc n) = Ipoly bs ?d" by (simp add: Let_def)
chaieb@33154
   687
    also have "\<dots> = (Ipoly bs p) ^ (Suc n div 2 + Suc n div 2)"
chaieb@33154
   688
      using "2.hyps" apply (simp only: power_add) by simp
chaieb@33154
   689
    finally have ?case using even_nat_div_two_times_two[OF even] by (simp only: th)}
chaieb@33154
   690
  ultimately show ?case by blast
chaieb@33154
   691
qed
chaieb@33154
   692
chaieb@33154
   693
lemma polypow_normh: 
chaieb@33154
   694
    assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
   695
  shows "isnpolyh p n \<Longrightarrow> isnpolyh (polypow k p) n"
chaieb@33154
   696
proof (induct k arbitrary: n rule: polypow.induct)
chaieb@33154
   697
  case (2 k n)
chaieb@33154
   698
  let ?q = "polypow (Suc k div 2) p"
chaieb@33154
   699
  let ?d = "polymul (?q,?q)"
chaieb@33154
   700
  from prems have th1:"isnpolyh ?q n" and th2: "isnpolyh p n" by blast+
chaieb@33154
   701
  from polymul_normh[OF th1 th1] have dn: "isnpolyh ?d n" by simp
chaieb@33154
   702
  from polymul_normh[OF th2 dn] have on: "isnpolyh (polymul(p,?d)) n" by simp
chaieb@33154
   703
  from dn on show ?case by (simp add: Let_def)
chaieb@33154
   704
qed auto 
chaieb@33154
   705
chaieb@33154
   706
lemma polypow_norm:   
chaieb@33154
   707
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
   708
  shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"
chaieb@33154
   709
  by (simp add: polypow_normh isnpoly_def)
chaieb@33154
   710
chaieb@33154
   711
text{* Finally the whole normalization*}
chaieb@33154
   712
chaieb@33154
   713
lemma polynate[simp]: "Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{ring_char_0,division_by_zero,field})"
chaieb@33154
   714
by (induct p rule:polynate.induct, auto)
chaieb@33154
   715
chaieb@33154
   716
lemma polynate_norm[simp]: 
chaieb@33154
   717
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
   718
  shows "isnpoly (polynate p)"
chaieb@33154
   719
  by (induct p rule: polynate.induct, simp_all add: polyadd_norm polymul_norm polysub_norm polyneg_norm polypow_norm) (simp_all add: isnpoly_def)
chaieb@33154
   720
chaieb@33154
   721
text{* shift1 *}
chaieb@33154
   722
chaieb@33154
   723
chaieb@33154
   724
lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
chaieb@33154
   725
by (simp add: shift1_def polymul)
chaieb@33154
   726
chaieb@33154
   727
lemma shift1_isnpoly: 
chaieb@33154
   728
  assumes pn: "isnpoly p" and pnz: "p \<noteq> 0\<^sub>p" shows "isnpoly (shift1 p) "
chaieb@33154
   729
  using pn pnz by (simp add: shift1_def isnpoly_def )
chaieb@33154
   730
chaieb@33154
   731
lemma shift1_nz[simp]:"shift1 p \<noteq> 0\<^sub>p"
chaieb@33154
   732
  by (simp add: shift1_def)
chaieb@33154
   733
lemma funpow_shift1_isnpoly: 
chaieb@33154
   734
  "\<lbrakk> isnpoly p ; p \<noteq> 0\<^sub>p\<rbrakk> \<Longrightarrow> isnpoly (funpow n shift1 p)"
chaieb@33154
   735
  by (induct n arbitrary: p, auto simp add: shift1_isnpoly)
chaieb@33154
   736
chaieb@33154
   737
lemma funpow_isnpolyh: 
chaieb@33154
   738
  assumes f: "\<And> p. isnpolyh p n \<Longrightarrow> isnpolyh (f p) n "and np: "isnpolyh p n"
chaieb@33154
   739
  shows "isnpolyh (funpow k f p) n"
chaieb@33154
   740
  using f np by (induct k arbitrary: p, auto)
chaieb@33154
   741
chaieb@33154
   742
lemma funpow_shift1: "(Ipoly bs (funpow n shift1 p) :: 'a :: {ring_char_0,division_by_zero,field}) = Ipoly bs (Mul (Pw (Bound 0) n) p)"
chaieb@33154
   743
  by (induct n arbitrary: p, simp_all add: shift1_isnpoly shift1 power_Suc )
chaieb@33154
   744
chaieb@33154
   745
lemma shift1_isnpolyh: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> isnpolyh (shift1 p) 0"
chaieb@33154
   746
  using isnpolyh_mono[where n="n0" and n'="0" and p="p"] by (simp add: shift1_def)
chaieb@33154
   747
chaieb@33154
   748
lemma funpow_shift1_1: 
chaieb@33154
   749
  "(Ipoly bs (funpow n shift1 p) :: 'a :: {ring_char_0,division_by_zero,field}) = Ipoly bs (funpow n shift1 1\<^sub>p *\<^sub>p p)"
chaieb@33154
   750
  by (simp add: funpow_shift1)
chaieb@33154
   751
chaieb@33154
   752
lemma poly_cmul[simp]: "Ipoly bs (poly_cmul c p) = Ipoly bs (Mul (C c) p)"
chaieb@33154
   753
by (induct p  arbitrary: n0 rule: poly_cmul.induct, auto simp add: ring_simps)
chaieb@33154
   754
chaieb@33154
   755
lemma behead:
chaieb@33154
   756
  assumes np: "isnpolyh p n"
chaieb@33154
   757
  shows "Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = (Ipoly bs p :: 'a :: {ring_char_0,division_by_zero,field})"
chaieb@33154
   758
  using np
chaieb@33154
   759
proof (induct p arbitrary: n rule: behead.induct)
chaieb@33154
   760
  case (1 c p n) hence pn: "isnpolyh p n" by simp
chaieb@33154
   761
  from prems(2)[OF pn] 
chaieb@33154
   762
  have th:"Ipoly bs (Add (Mul (head p) (Pw (Bound 0) (degree p))) (behead p)) = Ipoly bs p" . 
chaieb@33154
   763
  then show ?case using "1.hyps" apply (simp add: Let_def,cases "behead p = 0\<^sub>p")
chaieb@33154
   764
    by (simp_all add: th[symmetric] ring_simps power_Suc)
chaieb@33154
   765
qed (auto simp add: Let_def)
chaieb@33154
   766
chaieb@33154
   767
lemma behead_isnpolyh:
chaieb@33154
   768
  assumes np: "isnpolyh p n" shows "isnpolyh (behead p) n"
chaieb@33154
   769
  using np by (induct p rule: behead.induct, auto simp add: Let_def isnpolyh_mono)
chaieb@33154
   770
chaieb@33154
   771
subsection{* Miscilanious lemmas about indexes, decrementation, substitution  etc ... *}
chaieb@33154
   772
lemma isnpolyh_polybound0: "isnpolyh p (Suc n) \<Longrightarrow> polybound0 p"
chaieb@33154
   773
proof(induct p arbitrary: n rule: polybound0.induct, auto)
chaieb@33154
   774
  case (goal1 c n p n')
chaieb@33154
   775
  hence "n = Suc (n - 1)" by simp
chaieb@33154
   776
  hence "isnpolyh p (Suc (n - 1))"  using `isnpolyh p n` by simp
chaieb@33154
   777
  with prems(2) show ?case by simp
chaieb@33154
   778
qed
chaieb@33154
   779
chaieb@33154
   780
lemma isconstant_polybound0: "isnpolyh p n0 \<Longrightarrow> isconstant p \<longleftrightarrow> polybound0 p"
chaieb@33154
   781
by (induct p arbitrary: n0 rule: isconstant.induct, auto simp add: isnpolyh_polybound0)
chaieb@33154
   782
chaieb@33154
   783
lemma decrpoly_zero[simp]: "decrpoly p = 0\<^sub>p \<longleftrightarrow> p = 0\<^sub>p" by (induct p, auto)
chaieb@33154
   784
chaieb@33154
   785
lemma decrpoly_normh: "isnpolyh p n0 \<Longrightarrow> polybound0 p \<Longrightarrow> isnpolyh (decrpoly p) (n0 - 1)"
chaieb@33154
   786
  apply (induct p arbitrary: n0, auto)
chaieb@33154
   787
  apply (atomize)
chaieb@33154
   788
  apply (erule_tac x = "Suc nat" in allE)
chaieb@33154
   789
  apply auto
chaieb@33154
   790
  done
chaieb@33154
   791
chaieb@33154
   792
lemma head_polybound0: "isnpolyh p n0 \<Longrightarrow> polybound0 (head p)"
chaieb@33154
   793
 by (induct p  arbitrary: n0 rule: head.induct, auto intro: isnpolyh_polybound0)
chaieb@33154
   794
chaieb@33154
   795
lemma polybound0_I:
chaieb@33154
   796
  assumes nb: "polybound0 a"
chaieb@33154
   797
  shows "Ipoly (b#bs) a = Ipoly (b'#bs) a"
chaieb@33154
   798
using nb
chaieb@33154
   799
by (induct a rule: polybound0.induct) auto 
chaieb@33154
   800
lemma polysubst0_I:
chaieb@33154
   801
  shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b#bs) a)#bs) t"
chaieb@33154
   802
  by (induct t) simp_all
chaieb@33154
   803
chaieb@33154
   804
lemma polysubst0_I':
chaieb@33154
   805
  assumes nb: "polybound0 a"
chaieb@33154
   806
  shows "Ipoly (b#bs) (polysubst0 a t) = Ipoly ((Ipoly (b'#bs) a)#bs) t"
chaieb@33154
   807
  by (induct t) (simp_all add: polybound0_I[OF nb, where b="b" and b'="b'"])
chaieb@33154
   808
chaieb@33154
   809
lemma decrpoly: assumes nb: "polybound0 t"
chaieb@33154
   810
  shows "Ipoly (x#bs) t = Ipoly bs (decrpoly t)"
chaieb@33154
   811
  using nb by (induct t rule: decrpoly.induct, simp_all)
chaieb@33154
   812
chaieb@33154
   813
lemma polysubst0_polybound0: assumes nb: "polybound0 t"
chaieb@33154
   814
  shows "polybound0 (polysubst0 t a)"
chaieb@33154
   815
using nb by (induct a rule: polysubst0.induct, auto)
chaieb@33154
   816
chaieb@33154
   817
lemma degree0_polybound0: "isnpolyh p n \<Longrightarrow> degree p = 0 \<Longrightarrow> polybound0 p"
chaieb@33154
   818
  by (induct p arbitrary: n rule: degree.induct, auto simp add: isnpolyh_polybound0)
chaieb@33154
   819
chaieb@33154
   820
fun maxindex :: "poly \<Rightarrow> nat" where
chaieb@33154
   821
  "maxindex (Bound n) = n + 1"
chaieb@33154
   822
| "maxindex (CN c n p) = max  (n + 1) (max (maxindex c) (maxindex p))"
chaieb@33154
   823
| "maxindex (Add p q) = max (maxindex p) (maxindex q)"
chaieb@33154
   824
| "maxindex (Sub p q) = max (maxindex p) (maxindex q)"
chaieb@33154
   825
| "maxindex (Mul p q) = max (maxindex p) (maxindex q)"
chaieb@33154
   826
| "maxindex (Neg p) = maxindex p"
chaieb@33154
   827
| "maxindex (Pw p n) = maxindex p"
chaieb@33154
   828
| "maxindex (C x) = 0"
chaieb@33154
   829
chaieb@33154
   830
definition wf_bs :: "'a list \<Rightarrow> poly \<Rightarrow> bool" where
chaieb@33154
   831
  "wf_bs bs p = (length bs \<ge> maxindex p)"
chaieb@33154
   832
chaieb@33154
   833
lemma wf_bs_coefficients: "wf_bs bs p \<Longrightarrow> \<forall> c \<in> set (coefficients p). wf_bs bs c"
chaieb@33154
   834
proof(induct p rule: coefficients.induct)
chaieb@33154
   835
  case (1 c p) 
chaieb@33154
   836
  show ?case 
chaieb@33154
   837
  proof
chaieb@33154
   838
    fix x assume xc: "x \<in> set (coefficients (CN c 0 p))"
chaieb@33154
   839
    hence "x = c \<or> x \<in> set (coefficients p)" by simp
chaieb@33154
   840
    moreover 
chaieb@33154
   841
    {assume "x = c" hence "wf_bs bs x" using "1.prems"  unfolding wf_bs_def by simp}
chaieb@33154
   842
    moreover 
chaieb@33154
   843
    {assume H: "x \<in> set (coefficients p)" 
chaieb@33154
   844
      from "1.prems" have "wf_bs bs p" unfolding wf_bs_def by simp
chaieb@33154
   845
      with "1.hyps" H have "wf_bs bs x" by blast }
chaieb@33154
   846
    ultimately  show "wf_bs bs x" by blast
chaieb@33154
   847
  qed
chaieb@33154
   848
qed simp_all
chaieb@33154
   849
chaieb@33154
   850
lemma maxindex_coefficients: " \<forall>c\<in> set (coefficients p). maxindex c \<le> maxindex p"
chaieb@33154
   851
by (induct p rule: coefficients.induct, auto)
chaieb@33154
   852
chaieb@33154
   853
lemma length_exists: "\<exists>xs. length xs = n" by (rule exI[where x="replicate n x"], simp)
chaieb@33154
   854
chaieb@33154
   855
lemma wf_bs_I: "wf_bs bs p ==> Ipoly (bs@bs') p = Ipoly bs p"
chaieb@33154
   856
  unfolding wf_bs_def by (induct p, auto simp add: nth_append)
chaieb@33154
   857
chaieb@33154
   858
lemma take_maxindex_wf: assumes wf: "wf_bs bs p" 
chaieb@33154
   859
  shows "Ipoly (take (maxindex p) bs) p = Ipoly bs p"
chaieb@33154
   860
proof-
chaieb@33154
   861
  let ?ip = "maxindex p"
chaieb@33154
   862
  let ?tbs = "take ?ip bs"
chaieb@33154
   863
  from wf have "length ?tbs = ?ip" unfolding wf_bs_def by simp
chaieb@33154
   864
  hence wf': "wf_bs ?tbs p" unfolding wf_bs_def by  simp
chaieb@33154
   865
  have eq: "bs = ?tbs @ (drop ?ip bs)" by simp
chaieb@33154
   866
  from wf_bs_I[OF wf', of "drop ?ip bs"] show ?thesis using eq by simp
chaieb@33154
   867
qed
chaieb@33154
   868
chaieb@33154
   869
lemma decr_maxindex: "polybound0 p \<Longrightarrow> maxindex (decrpoly p) = maxindex p - 1"
chaieb@33154
   870
  by (induct p, auto)
chaieb@33154
   871
chaieb@33154
   872
lemma wf_bs_insensitive: "length bs = length bs' \<Longrightarrow> wf_bs bs p = wf_bs bs' p"
chaieb@33154
   873
  unfolding wf_bs_def by simp
chaieb@33154
   874
chaieb@33154
   875
lemma wf_bs_insensitive': "wf_bs (x#bs) p = wf_bs (y#bs) p"
chaieb@33154
   876
  unfolding wf_bs_def by simp
chaieb@33154
   877
chaieb@33154
   878
chaieb@33154
   879
chaieb@33154
   880
lemma wf_bs_coefficients': "\<forall>c \<in> set (coefficients p). wf_bs bs c \<Longrightarrow> wf_bs (x#bs) p"
chaieb@33154
   881
by(induct p rule: coefficients.induct, auto simp add: wf_bs_def)
chaieb@33154
   882
lemma coefficients_Nil[simp]: "coefficients p \<noteq> []"
chaieb@33154
   883
  by (induct p rule: coefficients.induct, simp_all)
chaieb@33154
   884
chaieb@33154
   885
chaieb@33154
   886
lemma coefficients_head: "last (coefficients p) = head p"
chaieb@33154
   887
  by (induct p rule: coefficients.induct, auto)
chaieb@33154
   888
chaieb@33154
   889
lemma wf_bs_decrpoly: "wf_bs bs (decrpoly p) \<Longrightarrow> wf_bs (x#bs) p"
chaieb@33154
   890
  unfolding wf_bs_def by (induct p rule: decrpoly.induct, auto)
chaieb@33154
   891
chaieb@33154
   892
lemma length_le_list_ex: "length xs \<le> n \<Longrightarrow> \<exists> ys. length (xs @ ys) = n"
chaieb@33154
   893
  apply (rule exI[where x="replicate (n - length xs) z"])
chaieb@33154
   894
  by simp
chaieb@33154
   895
lemma isnpolyh_Suc_const:"isnpolyh p (Suc n) \<Longrightarrow> isconstant p"
chaieb@33154
   896
by (cases p, auto) (case_tac "nat", simp_all)
chaieb@33154
   897
chaieb@33154
   898
lemma wf_bs_polyadd: "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
chaieb@33154
   899
  unfolding wf_bs_def 
chaieb@33154
   900
  apply (induct p q rule: polyadd.induct)
chaieb@33154
   901
  apply (auto simp add: Let_def)
chaieb@33154
   902
  done
chaieb@33154
   903
chaieb@33154
   904
lemma wf_bs_polyul: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p *\<^sub>p q)"
chaieb@33154
   905
chaieb@33154
   906
 unfolding wf_bs_def 
chaieb@33154
   907
  apply (induct p q arbitrary: bs rule: polymul.induct) 
chaieb@33154
   908
  apply (simp_all add: wf_bs_polyadd)
chaieb@33154
   909
  apply clarsimp
chaieb@33154
   910
  apply (rule wf_bs_polyadd[unfolded wf_bs_def, rule_format])
chaieb@33154
   911
  apply auto
chaieb@33154
   912
  done
chaieb@33154
   913
chaieb@33154
   914
lemma wf_bs_polyneg: "wf_bs bs p \<Longrightarrow> wf_bs bs (~\<^sub>p p)"
chaieb@33154
   915
  unfolding wf_bs_def by (induct p rule: polyneg.induct, auto)
chaieb@33154
   916
chaieb@33154
   917
lemma wf_bs_polysub: "wf_bs bs p \<Longrightarrow> wf_bs bs q \<Longrightarrow> wf_bs bs (p -\<^sub>p q)"
chaieb@33154
   918
  unfolding polysub_def split_def fst_conv snd_conv using wf_bs_polyadd wf_bs_polyneg by blast
chaieb@33154
   919
chaieb@33154
   920
subsection{* Canonicity of polynomial representation, see lemma isnpolyh_unique*}
chaieb@33154
   921
chaieb@33154
   922
definition "polypoly bs p = map (Ipoly bs) (coefficients p)"
chaieb@33154
   923
definition "polypoly' bs p = map ((Ipoly bs o decrpoly)) (coefficients p)"
chaieb@33154
   924
definition "poly_nate bs p = map ((Ipoly bs o decrpoly)) (coefficients (polynate p))"
chaieb@33154
   925
chaieb@33154
   926
lemma coefficients_normh: "isnpolyh p n0 \<Longrightarrow> \<forall> q \<in> set (coefficients p). isnpolyh q n0"
chaieb@33154
   927
proof (induct p arbitrary: n0 rule: coefficients.induct)
chaieb@33154
   928
  case (1 c p n0)
chaieb@33154
   929
  have cp: "isnpolyh (CN c 0 p) n0" by fact
chaieb@33154
   930
  hence norm: "isnpolyh c 0" "isnpolyh p 0" "p \<noteq> 0\<^sub>p" "n0 = 0"
chaieb@33154
   931
    by (auto simp add: isnpolyh_mono[where n'=0])
chaieb@33154
   932
  from "1.hyps"[OF norm(2)] norm(1) norm(4)  show ?case by simp 
chaieb@33154
   933
qed auto
chaieb@33154
   934
chaieb@33154
   935
lemma coefficients_isconst:
chaieb@33154
   936
  "isnpolyh p n \<Longrightarrow> \<forall>q\<in>set (coefficients p). isconstant q"
chaieb@33154
   937
  by (induct p arbitrary: n rule: coefficients.induct, 
chaieb@33154
   938
    auto simp add: isnpolyh_Suc_const)
chaieb@33154
   939
chaieb@33154
   940
lemma polypoly_polypoly':
chaieb@33154
   941
  assumes np: "isnpolyh p n0"
chaieb@33154
   942
  shows "polypoly (x#bs) p = polypoly' bs p"
chaieb@33154
   943
proof-
chaieb@33154
   944
  let ?cf = "set (coefficients p)"
chaieb@33154
   945
  from coefficients_normh[OF np] have cn_norm: "\<forall> q\<in> ?cf. isnpolyh q n0" .
chaieb@33154
   946
  {fix q assume q: "q \<in> ?cf"
chaieb@33154
   947
    from q cn_norm have th: "isnpolyh q n0" by blast
chaieb@33154
   948
    from coefficients_isconst[OF np] q have "isconstant q" by blast
chaieb@33154
   949
    with isconstant_polybound0[OF th] have "polybound0 q" by blast}
chaieb@33154
   950
  hence "\<forall>q \<in> ?cf. polybound0 q" ..
chaieb@33154
   951
  hence "\<forall>q \<in> ?cf. Ipoly (x#bs) q = Ipoly bs (decrpoly q)"
chaieb@33154
   952
    using polybound0_I[where b=x and bs=bs and b'=y] decrpoly[where x=x and bs=bs]
chaieb@33154
   953
    by auto
chaieb@33154
   954
  
chaieb@33154
   955
  thus ?thesis unfolding polypoly_def polypoly'_def by simp 
chaieb@33154
   956
qed
chaieb@33154
   957
chaieb@33154
   958
lemma polypoly_poly:
chaieb@33154
   959
  assumes np: "isnpolyh p n0" shows "Ipoly (x#bs) p = poly (polypoly (x#bs) p) x"
chaieb@33154
   960
  using np 
chaieb@33154
   961
by (induct p arbitrary: n0 bs rule: coefficients.induct, auto simp add: polypoly_def)
chaieb@33154
   962
chaieb@33154
   963
lemma polypoly'_poly: 
chaieb@33154
   964
  assumes np: "isnpolyh p n0" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
chaieb@33154
   965
  using polypoly_poly[OF np, simplified polypoly_polypoly'[OF np]] .
chaieb@33154
   966
chaieb@33154
   967
chaieb@33154
   968
lemma polypoly_poly_polybound0:
chaieb@33154
   969
  assumes np: "isnpolyh p n0" and nb: "polybound0 p"
chaieb@33154
   970
  shows "polypoly bs p = [Ipoly bs p]"
chaieb@33154
   971
  using np nb unfolding polypoly_def 
chaieb@33154
   972
  by (cases p, auto, case_tac nat, auto)
chaieb@33154
   973
chaieb@33154
   974
lemma head_isnpolyh: "isnpolyh p n0 \<Longrightarrow> isnpolyh (head p) n0" 
chaieb@33154
   975
  by (induct p rule: head.induct, auto)
chaieb@33154
   976
chaieb@33154
   977
lemma headn_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
chaieb@33154
   978
  by (cases p,auto)
chaieb@33154
   979
chaieb@33154
   980
lemma head_eq_headn0: "head p = headn p 0"
chaieb@33154
   981
  by (induct p rule: head.induct, simp_all)
chaieb@33154
   982
chaieb@33154
   983
lemma head_nz[simp]: "isnpolyh p n0 \<Longrightarrow> (head p = 0\<^sub>p) = (p = 0\<^sub>p)"
chaieb@33154
   984
  by (simp add: head_eq_headn0)
chaieb@33154
   985
chaieb@33154
   986
lemma isnpolyh_zero_iff: 
chaieb@33154
   987
  assumes nq: "isnpolyh p n0" and eq :"\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a::{ring_char_0,power,division_by_zero,field})"
chaieb@33154
   988
  shows "p = 0\<^sub>p"
chaieb@33154
   989
using nq eq
chaieb@33154
   990
proof (induct n\<equiv>"maxindex p" arbitrary: p n0 rule: nat_less_induct)
chaieb@33154
   991
  fix n p n0
chaieb@33154
   992
  assume H: "\<forall>m<n. \<forall>p n0. isnpolyh p n0 \<longrightarrow>
chaieb@33154
   993
    (\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)) \<longrightarrow> m = maxindex p \<longrightarrow> p = 0\<^sub>p"
chaieb@33154
   994
    and np: "isnpolyh p n0" and zp: "\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" and n: "n = maxindex p"
chaieb@33154
   995
  {assume nz: "n = 0"
chaieb@33154
   996
    then obtain c where "p = C c" using n np by (cases p, auto)
chaieb@33154
   997
    with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}
chaieb@33154
   998
  moreover
chaieb@33154
   999
  {assume nz: "n \<noteq> 0"
chaieb@33154
  1000
    let ?h = "head p"
chaieb@33154
  1001
    let ?hd = "decrpoly ?h"
chaieb@33154
  1002
    let ?ihd = "maxindex ?hd"
chaieb@33154
  1003
    from head_isnpolyh[OF np] head_polybound0[OF np] have h:"isnpolyh ?h n0" "polybound0 ?h" 
chaieb@33154
  1004
      by simp_all
chaieb@33154
  1005
    hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast
chaieb@33154
  1006
    
chaieb@33154
  1007
    from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
chaieb@33154
  1008
    have mihn: "maxindex ?h \<le> n" unfolding n by auto
chaieb@33154
  1009
    with decr_maxindex[OF h(2)] nz  have ihd_lt_n: "?ihd < n" by auto
chaieb@33154
  1010
    {fix bs:: "'a list"  assume bs: "wf_bs bs ?hd"
chaieb@33154
  1011
      let ?ts = "take ?ihd bs"
chaieb@33154
  1012
      let ?rs = "drop ?ihd bs"
chaieb@33154
  1013
      have ts: "wf_bs ?ts ?hd" using bs unfolding wf_bs_def by simp
chaieb@33154
  1014
      have bs_ts_eq: "?ts@ ?rs = bs" by simp
chaieb@33154
  1015
      from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x#?ts) ?h" by simp
chaieb@33154
  1016
      from ihd_lt_n have "ALL x. length (x#?ts) \<le> n" by simp
chaieb@33154
  1017
      with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = n" by blast
chaieb@33154
  1018
      hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" using n unfolding wf_bs_def by simp
chaieb@33154
  1019
      with zp have "\<forall> x. Ipoly ((x#?ts) @ xs) p = 0" by blast
chaieb@33154
  1020
      hence "\<forall> x. Ipoly (x#(?ts @ xs)) p = 0" by simp
chaieb@33154
  1021
      with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
chaieb@33154
  1022
      have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"  by simp
chaieb@33154
  1023
      hence "poly (polypoly' (?ts @ xs) p) = poly []" by (auto intro: ext) 
chaieb@33154
  1024
      hence "\<forall> c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
chaieb@33154
  1025
	thm poly_zero
chaieb@33154
  1026
	using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
chaieb@33154
  1027
      with coefficients_head[of p, symmetric]
chaieb@33154
  1028
      have th0: "Ipoly (?ts @ xs) ?hd = 0" by simp
chaieb@33154
  1029
      from bs have wf'': "wf_bs ?ts ?hd" unfolding wf_bs_def by simp
chaieb@33154
  1030
      with th0 wf_bs_I[of ?ts ?hd xs] have "Ipoly ?ts ?hd = 0" by simp
chaieb@33154
  1031
      with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by simp }
chaieb@33154
  1032
    then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast
chaieb@33154
  1033
    
chaieb@33154
  1034
    from H[rule_format, OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast
chaieb@33154
  1035
    hence "?h = 0\<^sub>p" by simp
chaieb@33154
  1036
    with head_nz[OF np] have "p = 0\<^sub>p" by simp}
chaieb@33154
  1037
  ultimately show "p = 0\<^sub>p" by blast
chaieb@33154
  1038
qed
chaieb@33154
  1039
chaieb@33154
  1040
lemma isnpolyh_unique:  
chaieb@33154
  1041
  assumes np:"isnpolyh p n0" and nq: "isnpolyh q n1"
chaieb@33154
  1042
  shows "(\<forall>bs. \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (\<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup> :: 'a::{ring_char_0,power,division_by_zero,field})) \<longleftrightarrow>  p = q"
chaieb@33154
  1043
proof(auto)
chaieb@33154
  1044
  assume H: "\<forall>bs. (\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> ::'a)= \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
chaieb@33154
  1045
  hence "\<forall>bs.\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup>= (0::'a)" by simp
chaieb@33154
  1046
  hence "\<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
  1047
    using wf_bs_polysub[where p=p and q=q] by auto
chaieb@33154
  1048
  with isnpolyh_zero_iff[OF polysub_normh[OF np nq]] polysub_0[OF np nq]
chaieb@33154
  1049
  show "p = q" by blast
chaieb@33154
  1050
qed
chaieb@33154
  1051
chaieb@33154
  1052
chaieb@33154
  1053
text{* consequenses of unicity on the algorithms for polynomial normalization *}
chaieb@33154
  1054
chaieb@33154
  1055
lemma polyadd_commute:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
  1056
  and np: "isnpolyh p n0" and nq: "isnpolyh q n1" shows "p +\<^sub>p q = q +\<^sub>p p"
chaieb@33154
  1057
  using isnpolyh_unique[OF polyadd_normh[OF np nq] polyadd_normh[OF nq np]] by simp
chaieb@33154
  1058
chaieb@33154
  1059
lemma zero_normh: "isnpolyh 0\<^sub>p n" by simp
chaieb@33154
  1060
lemma one_normh: "isnpolyh 1\<^sub>p n" by simp
chaieb@33154
  1061
lemma polyadd_0[simp]: 
chaieb@33154
  1062
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
  1063
  and np: "isnpolyh p n0" shows "p +\<^sub>p 0\<^sub>p = p" and "0\<^sub>p +\<^sub>p p = p"
chaieb@33154
  1064
  using isnpolyh_unique[OF polyadd_normh[OF np zero_normh] np] 
chaieb@33154
  1065
    isnpolyh_unique[OF polyadd_normh[OF zero_normh np] np] by simp_all
chaieb@33154
  1066
chaieb@33154
  1067
lemma polymul_1[simp]: 
chaieb@33154
  1068
    assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
  1069
  and np: "isnpolyh p n0" shows "p *\<^sub>p 1\<^sub>p = p" and "1\<^sub>p *\<^sub>p p = p"
chaieb@33154
  1070
  using isnpolyh_unique[OF polymul_normh[OF np one_normh] np] 
chaieb@33154
  1071
    isnpolyh_unique[OF polymul_normh[OF one_normh np] np] by simp_all
chaieb@33154
  1072
lemma polymul_0[simp]: 
chaieb@33154
  1073
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
  1074
  and np: "isnpolyh p n0" shows "p *\<^sub>p 0\<^sub>p = 0\<^sub>p" and "0\<^sub>p *\<^sub>p p = 0\<^sub>p"
chaieb@33154
  1075
  using isnpolyh_unique[OF polymul_normh[OF np zero_normh] zero_normh] 
chaieb@33154
  1076
    isnpolyh_unique[OF polymul_normh[OF zero_normh np] zero_normh] by simp_all
chaieb@33154
  1077
chaieb@33154
  1078
lemma polymul_commute: 
chaieb@33154
  1079
    assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
  1080
  and np:"isnpolyh p n0" and nq: "isnpolyh q n1"
chaieb@33154
  1081
  shows "p *\<^sub>p q = q *\<^sub>p p"
chaieb@33154
  1082
using isnpolyh_unique[OF polymul_normh[OF np nq] polymul_normh[OF nq np], where ?'a = "'a\<Colon>{ring_char_0,power,division_by_zero,field}"] by simp
chaieb@33154
  1083
chaieb@33154
  1084
declare polyneg_polyneg[simp]
chaieb@33154
  1085
  
chaieb@33154
  1086
lemma isnpolyh_polynate_id[simp]: 
chaieb@33154
  1087
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
  1088
  and np:"isnpolyh p n0" shows "polynate p = p"
chaieb@33154
  1089
  using isnpolyh_unique[where ?'a= "'a::{ring_char_0,division_by_zero,field}", OF polynate_norm[of p, unfolded isnpoly_def] np] polynate[where ?'a = "'a::{ring_char_0,division_by_zero,field}"] by simp
chaieb@33154
  1090
chaieb@33154
  1091
lemma polynate_idempotent[simp]: 
chaieb@33154
  1092
    assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
  1093
  shows "polynate (polynate p) = polynate p"
chaieb@33154
  1094
  using isnpolyh_polynate_id[OF polynate_norm[of p, unfolded isnpoly_def]] .
chaieb@33154
  1095
chaieb@33154
  1096
lemma poly_nate_polypoly': "poly_nate bs p = polypoly' bs (polynate p)"
chaieb@33154
  1097
  unfolding poly_nate_def polypoly'_def ..
chaieb@33154
  1098
lemma poly_nate_poly: shows "poly (poly_nate bs p) = (\<lambda>x:: 'a ::{ring_char_0,division_by_zero,field}. \<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup>)"
chaieb@33154
  1099
  using polypoly'_poly[OF polynate_norm[unfolded isnpoly_def], symmetric, of bs p]
chaieb@33154
  1100
  unfolding poly_nate_polypoly' by (auto intro: ext)
chaieb@33154
  1101
chaieb@33154
  1102
subsection{* heads, degrees and all that *}
chaieb@33154
  1103
lemma degree_eq_degreen0: "degree p = degreen p 0"
chaieb@33154
  1104
  by (induct p rule: degree.induct, simp_all)
chaieb@33154
  1105
chaieb@33154
  1106
lemma degree_polyneg: assumes n: "isnpolyh p n"
chaieb@33154
  1107
  shows "degree (polyneg p) = degree p"
chaieb@33154
  1108
  using n
chaieb@33154
  1109
  by (induct p arbitrary: n rule: polyneg.induct, simp_all) (case_tac na, auto)
chaieb@33154
  1110
chaieb@33154
  1111
lemma degree_polyadd:
chaieb@33154
  1112
  assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
chaieb@33154
  1113
  shows "degree (p +\<^sub>p q) \<le> max (degree p) (degree q)"
chaieb@33154
  1114
using degreen_polyadd[OF np nq, where m= "0"] degree_eq_degreen0 by simp
chaieb@33154
  1115
chaieb@33154
  1116
chaieb@33154
  1117
lemma degree_polysub: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
chaieb@33154
  1118
  shows "degree (p -\<^sub>p q) \<le> max (degree p) (degree q)"
chaieb@33154
  1119
proof-
chaieb@33154
  1120
  from nq have nq': "isnpolyh (~\<^sub>p q) n1" using polyneg_normh by simp
chaieb@33154
  1121
  from degree_polyadd[OF np nq'] show ?thesis by (simp add: polysub_def degree_polyneg[OF nq])
chaieb@33154
  1122
qed
chaieb@33154
  1123
chaieb@33154
  1124
lemma degree_polysub_samehead: 
chaieb@33154
  1125
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
  1126
  and np: "isnpolyh p n0" and nq: "isnpolyh q n1" and h: "head p = head q" 
chaieb@33154
  1127
  and d: "degree p = degree q"
chaieb@33154
  1128
  shows "degree (p -\<^sub>p q) < degree p \<or> (p -\<^sub>p q = 0\<^sub>p)"
chaieb@33154
  1129
unfolding polysub_def split_def fst_conv snd_conv
chaieb@33154
  1130
using np nq h d
chaieb@33154
  1131
proof(induct p q rule:polyadd.induct)
chaieb@33154
  1132
  case (1 a b a' b') thus ?case by (simp add: Nsub_def Nsub0[simplified Nsub_def]) 
chaieb@33154
  1133
next
chaieb@33154
  1134
  case (2 a b c' n' p') 
chaieb@33154
  1135
  let ?c = "(a,b)"
chaieb@33154
  1136
  from prems have "degree (C ?c) = degree (CN c' n' p')" by simp
chaieb@33154
  1137
  hence nz:"n' > 0" by (cases n', auto)
chaieb@33154
  1138
  hence "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
chaieb@33154
  1139
  with prems show ?case by simp
chaieb@33154
  1140
next
chaieb@33154
  1141
  case (3 c n p a' b') 
chaieb@33154
  1142
  let ?c' = "(a',b')"
chaieb@33154
  1143
  from prems have "degree (C ?c') = degree (CN c n p)" by simp
chaieb@33154
  1144
  hence nz:"n > 0" by (cases n, auto)
chaieb@33154
  1145
  hence "head (CN c n p) = CN c n p" by (cases n, auto)
chaieb@33154
  1146
  with prems show ?case by simp
chaieb@33154
  1147
next
chaieb@33154
  1148
  case (4 c n p c' n' p')
chaieb@33154
  1149
  hence H: "isnpolyh (CN c n p) n0" "isnpolyh (CN c' n' p') n1" 
chaieb@33154
  1150
    "head (CN c n p) = head (CN c' n' p')" "degree (CN c n p) = degree (CN c' n' p')" by simp+
chaieb@33154
  1151
  hence degc: "degree c = 0" and degc': "degree c' = 0" by simp_all  
chaieb@33154
  1152
  hence degnc: "degree (~\<^sub>p c) = 0" and degnc': "degree (~\<^sub>p c') = 0" 
chaieb@33154
  1153
    using H(1-2) degree_polyneg by auto
chaieb@33154
  1154
  from H have cnh: "isnpolyh c (Suc n)" and c'nh: "isnpolyh c' (Suc n')"  by simp+
chaieb@33154
  1155
  from degree_polysub[OF cnh c'nh, simplified polysub_def] degc degc' have degcmc': "degree (c +\<^sub>p  ~\<^sub>pc') = 0"  by simp
chaieb@33154
  1156
  from H have pnh: "isnpolyh p n" and p'nh: "isnpolyh p' n'" by auto
chaieb@33154
  1157
  have "n = n' \<or> n < n' \<or> n > n'" by arith
chaieb@33154
  1158
  moreover
chaieb@33154
  1159
  {assume nn': "n = n'"
chaieb@33154
  1160
    have "n = 0 \<or> n >0" by arith
chaieb@33154
  1161
    moreover {assume nz: "n = 0" hence ?case using prems by (auto simp add: Let_def degcmc')}
chaieb@33154
  1162
    moreover {assume nz: "n > 0"
chaieb@33154
  1163
      with nn' H(3) have  cc':"c = c'" and pp': "p = p'" by (cases n, auto)+
chaieb@33154
  1164
      hence ?case using polysub_same_0[OF p'nh, simplified polysub_def split_def fst_conv snd_conv] polysub_same_0[OF c'nh, simplified polysub_def split_def fst_conv snd_conv] using nn' prems by (simp add: Let_def)}
chaieb@33154
  1165
    ultimately have ?case by blast}
chaieb@33154
  1166
  moreover
chaieb@33154
  1167
  {assume nn': "n < n'" hence n'p: "n' > 0" by simp 
chaieb@33154
  1168
    hence headcnp':"head (CN c' n' p') = CN c' n' p'"  by (cases n', simp_all)
chaieb@33154
  1169
    have degcnp': "degree (CN c' n' p') = 0" and degcnpeq: "degree (CN c n p) = degree (CN c' n' p')" using prems by (cases n', simp_all)
chaieb@33154
  1170
    hence "n > 0" by (cases n, simp_all)
chaieb@33154
  1171
    hence headcnp: "head (CN c n p) = CN c n p" by (cases n, auto)
chaieb@33154
  1172
    from H(3) headcnp headcnp' nn' have ?case by auto}
chaieb@33154
  1173
  moreover
chaieb@33154
  1174
  {assume nn': "n > n'"  hence np: "n > 0" by simp 
chaieb@33154
  1175
    hence headcnp:"head (CN c n p) = CN c n p"  by (cases n, simp_all)
chaieb@33154
  1176
    from prems have degcnpeq: "degree (CN c' n' p') = degree (CN c n p)" by simp
chaieb@33154
  1177
    from np have degcnp: "degree (CN c n p) = 0" by (cases n, simp_all)
chaieb@33154
  1178
    with degcnpeq have "n' > 0" by (cases n', simp_all)
chaieb@33154
  1179
    hence headcnp': "head (CN c' n' p') = CN c' n' p'" by (cases n', auto)
chaieb@33154
  1180
    from H(3) headcnp headcnp' nn' have ?case by auto}
chaieb@33154
  1181
  ultimately show ?case  by blast
chaieb@33154
  1182
qed auto 
chaieb@33154
  1183
 
chaieb@33154
  1184
lemma shift1_head : "isnpolyh p n0 \<Longrightarrow> head (shift1 p) = head p"
chaieb@33154
  1185
by (induct p arbitrary: n0 rule: head.induct, simp_all add: shift1_def)
chaieb@33154
  1186
chaieb@33154
  1187
lemma funpow_shift1_head: "isnpolyh p n0 \<Longrightarrow> p\<noteq> 0\<^sub>p \<Longrightarrow> head (funpow k shift1 p) = head p"
chaieb@33154
  1188
proof(induct k arbitrary: n0 p)
chaieb@33154
  1189
  case (Suc k n0 p) hence "isnpolyh (shift1 p) 0" by (simp add: shift1_isnpolyh)
chaieb@33154
  1190
  with prems have "head (funpow k shift1 (shift1 p)) = head (shift1 p)"
chaieb@33154
  1191
    and "head (shift1 p) = head p" by (simp_all add: shift1_head) 
chaieb@33154
  1192
  thus ?case by simp
chaieb@33154
  1193
qed auto  
chaieb@33154
  1194
chaieb@33154
  1195
lemma shift1_degree: "degree (shift1 p) = 1 + degree p"
chaieb@33154
  1196
  by (simp add: shift1_def)
chaieb@33154
  1197
lemma funpow_shift1_degree: "degree (funpow k shift1 p) = k + degree p "
chaieb@33154
  1198
  by (induct k arbitrary: p, auto simp add: shift1_degree)
chaieb@33154
  1199
chaieb@33154
  1200
lemma funpow_shift1_nz: "p \<noteq> 0\<^sub>p \<Longrightarrow> funpow n shift1 p \<noteq> 0\<^sub>p"
chaieb@33154
  1201
  by (induct n arbitrary: p, simp_all add: funpow_def)
chaieb@33154
  1202
chaieb@33154
  1203
lemma head_isnpolyh_Suc[simp]: "isnpolyh p (Suc n) \<Longrightarrow> head p = p"
chaieb@33154
  1204
  by (induct p arbitrary: n rule: degree.induct, auto)
chaieb@33154
  1205
lemma headn_0[simp]: "isnpolyh p n \<Longrightarrow> m < n \<Longrightarrow> headn p m = p"
chaieb@33154
  1206
  by (induct p arbitrary: n rule: degreen.induct, auto)
chaieb@33154
  1207
lemma head_isnpolyh_Suc': "n > 0 \<Longrightarrow> isnpolyh p n \<Longrightarrow> head p = p"
chaieb@33154
  1208
  by (induct p arbitrary: n rule: degree.induct, auto)
chaieb@33154
  1209
lemma head_head[simp]: "isnpolyh p n0 \<Longrightarrow> head (head p) = head p"
chaieb@33154
  1210
  by (induct p rule: head.induct, auto)
chaieb@33154
  1211
chaieb@33154
  1212
lemma polyadd_eq_const_degree: 
chaieb@33154
  1213
  "\<lbrakk> isnpolyh p n0 ; isnpolyh q n1 ; polyadd (p,q) = C c\<rbrakk> \<Longrightarrow> degree p = degree q" 
chaieb@33154
  1214
  using polyadd_eq_const_degreen degree_eq_degreen0 by simp
chaieb@33154
  1215
chaieb@33154
  1216
lemma polyadd_head: assumes np: "isnpolyh p n0" and nq: "isnpolyh q n1"
chaieb@33154
  1217
  and deg: "degree p \<noteq> degree q"
chaieb@33154
  1218
  shows "head (p +\<^sub>p q) = (if degree p < degree q then head q else head p)"
chaieb@33154
  1219
using np nq deg
chaieb@33154
  1220
apply(induct p q arbitrary: n0 n1 rule: polyadd.induct,simp_all)
chaieb@33154
  1221
apply (case_tac n', simp, simp)
chaieb@33154
  1222
apply (case_tac n, simp, simp)
chaieb@33154
  1223
apply (case_tac n, case_tac n', simp add: Let_def)
chaieb@33154
  1224
apply (case_tac "pa +\<^sub>p p' = 0\<^sub>p")
chaieb@33154
  1225
apply (clarsimp simp add: polyadd_eq_const_degree)
chaieb@33154
  1226
apply clarsimp
chaieb@33154
  1227
apply (erule_tac impE,blast)
chaieb@33154
  1228
apply (erule_tac impE,blast)
chaieb@33154
  1229
apply clarsimp
chaieb@33154
  1230
apply simp
chaieb@33154
  1231
apply (case_tac n', simp_all)
chaieb@33154
  1232
done
chaieb@33154
  1233
chaieb@33154
  1234
lemma polymul_head_polyeq: 
chaieb@33154
  1235
   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
  1236
  shows "\<lbrakk>isnpolyh p n0; isnpolyh q n1 ; p \<noteq> 0\<^sub>p ; q \<noteq> 0\<^sub>p \<rbrakk> \<Longrightarrow> head (p *\<^sub>p q) = head p *\<^sub>p head q"
chaieb@33154
  1237
proof (induct p q arbitrary: n0 n1 rule: polymul.induct)
chaieb@33154
  1238
  case (2 a b c' n' p' n0 n1)
chaieb@33154
  1239
  hence "isnpolyh (head (CN c' n' p')) n1" "isnormNum (a,b)"  by (simp_all add: head_isnpolyh)
chaieb@33154
  1240
  thus ?case using prems by (cases n', auto) 
chaieb@33154
  1241
next 
chaieb@33154
  1242
  case (3 c n p a' b' n0 n1) 
chaieb@33154
  1243
  hence "isnpolyh (head (CN c n p)) n0" "isnormNum (a',b')"  by (simp_all add: head_isnpolyh)
chaieb@33154
  1244
  thus ?case using prems by (cases n, auto)
chaieb@33154
  1245
next
chaieb@33154
  1246
  case (4 c n p c' n' p' n0 n1)
chaieb@33154
  1247
  hence norm: "isnpolyh p n" "isnpolyh c (Suc n)" "isnpolyh p' n'" "isnpolyh c' (Suc n')"
chaieb@33154
  1248
    "isnpolyh (CN c n p) n" "isnpolyh (CN c' n' p') n'"
chaieb@33154
  1249
    by simp_all
chaieb@33154
  1250
  have "n < n' \<or> n' < n \<or> n = n'" by arith
chaieb@33154
  1251
  moreover 
chaieb@33154
  1252
  {assume nn': "n < n'" hence ?case 
chaieb@33154
  1253
      thm prems
chaieb@33154
  1254
      using norm 
chaieb@33154
  1255
    prems(6)[rule_format, OF nn' norm(1,6)]
chaieb@33154
  1256
    prems(7)[rule_format, OF nn' norm(2,6)] by (simp, cases n, simp,cases n', simp_all)}
chaieb@33154
  1257
  moreover {assume nn': "n'< n"
chaieb@33154
  1258
    hence stupid: "n' < n \<and> \<not> n < n'" by simp
chaieb@33154
  1259
    hence ?case using norm prems(4) [rule_format, OF stupid norm(5,3)]
chaieb@33154
  1260
      prems(5)[rule_format, OF stupid norm(5,4)] 
chaieb@33154
  1261
      by (simp,cases n',simp,cases n,auto)}
chaieb@33154
  1262
  moreover {assume nn': "n' = n"
chaieb@33154
  1263
    hence stupid: "\<not> n' < n \<and> \<not> n < n'" by simp
chaieb@33154
  1264
    from nn' polymul_normh[OF norm(5,4)] 
chaieb@33154
  1265
    have ncnpc':"isnpolyh (CN c n p *\<^sub>p c') n" by (simp add: min_def)
chaieb@33154
  1266
    from nn' polymul_normh[OF norm(5,3)] norm 
chaieb@33154
  1267
    have ncnpp':"isnpolyh (CN c n p *\<^sub>p p') n" by simp
chaieb@33154
  1268
    from nn' ncnpp' polymul_eq0_iff[OF norm(5,3)] norm(6)
chaieb@33154
  1269
    have ncnpp0':"isnpolyh (CN 0\<^sub>p n (CN c n p *\<^sub>p p')) n" by simp 
chaieb@33154
  1270
    from polyadd_normh[OF ncnpc' ncnpp0'] 
chaieb@33154
  1271
    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
  1272
      by (simp add: min_def)
chaieb@33154
  1273
    {assume np: "n > 0"
chaieb@33154
  1274
      with nn' head_isnpolyh_Suc'[OF np nth]
chaieb@33154
  1275
	head_isnpolyh_Suc'[OF np norm(5)] head_isnpolyh_Suc'[OF np norm(6)[simplified nn']]
chaieb@33154
  1276
      have ?case by simp}
chaieb@33154
  1277
    moreover
chaieb@33154
  1278
    {moreover assume nz: "n = 0"
chaieb@33154
  1279
      from polymul_degreen[OF norm(5,4), where m="0"]
chaieb@33154
  1280
	polymul_degreen[OF norm(5,3), where m="0"] nn' nz degree_eq_degreen0
chaieb@33154
  1281
      norm(5,6) degree_npolyhCN[OF norm(6)]
chaieb@33154
  1282
    have dth:"degree (CN c 0 p *\<^sub>p c') < degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))" by simp
chaieb@33154
  1283
    hence dth':"degree (CN c 0 p *\<^sub>p c') \<noteq> degree (CN 0\<^sub>p 0 (CN c 0 p *\<^sub>p p'))" by simp
chaieb@33154
  1284
    from polyadd_head[OF ncnpc'[simplified nz] ncnpp0'[simplified nz] dth'] dth
chaieb@33154
  1285
    have ?case   using norm prems(2)[rule_format, OF stupid norm(5,3)]
chaieb@33154
  1286
	prems(3)[rule_format, OF stupid norm(5,4)] nn' nz by simp }
chaieb@33154
  1287
    ultimately have ?case by (cases n) auto} 
chaieb@33154
  1288
  ultimately show ?case by blast
chaieb@33154
  1289
qed simp_all
chaieb@33154
  1290
chaieb@33154
  1291
lemma degree_coefficients: "degree p = length (coefficients p) - 1"
chaieb@33154
  1292
  by(induct p rule: degree.induct, auto)
chaieb@33154
  1293
chaieb@33154
  1294
lemma degree_head[simp]: "degree (head p) = 0"
chaieb@33154
  1295
  by (induct p rule: head.induct, auto)
chaieb@33154
  1296
chaieb@33154
  1297
lemma degree_CN: "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<le> 1+ degree p"
chaieb@33154
  1298
  by (cases n, simp_all)
chaieb@33154
  1299
lemma degree_CN': "isnpolyh p n \<Longrightarrow> degree (CN c n p) \<ge>  degree p"
chaieb@33154
  1300
  by (cases n, simp_all)
chaieb@33154
  1301
chaieb@33154
  1302
lemma polyadd_different_degree: "\<lbrakk>isnpolyh p n0 ; isnpolyh q n1; degree p \<noteq> degree q\<rbrakk> \<Longrightarrow> degree (polyadd(p,q)) = max (degree p) (degree q)"
chaieb@33154
  1303
  using polyadd_different_degreen degree_eq_degreen0 by simp
chaieb@33154
  1304
chaieb@33154
  1305
lemma degreen_polyneg: "isnpolyh p n0 \<Longrightarrow> degreen (~\<^sub>p p) m = degreen p m"
chaieb@33154
  1306
  by (induct p arbitrary: n0 rule: polyneg.induct, auto)
chaieb@33154
  1307
chaieb@33154
  1308
lemma degree_polymul:
chaieb@33154
  1309
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
  1310
  and np: "isnpolyh p n0" and nq: "isnpolyh q n1"
chaieb@33154
  1311
  shows "degree (p *\<^sub>p q) \<le> degree p + degree q"
chaieb@33154
  1312
  using polymul_degreen[OF np nq, where m="0"]  degree_eq_degreen0 by simp
chaieb@33154
  1313
chaieb@33154
  1314
lemma polyneg_degree: "isnpolyh p n \<Longrightarrow> degree (polyneg p) = degree p"
chaieb@33154
  1315
  by (induct p arbitrary: n rule: degree.induct, auto)
chaieb@33154
  1316
chaieb@33154
  1317
lemma polyneg_head: "isnpolyh p n \<Longrightarrow> head(polyneg p) = polyneg (head p)"
chaieb@33154
  1318
  by (induct p arbitrary: n rule: degree.induct, auto)
chaieb@33154
  1319
chaieb@33154
  1320
subsection {* Correctness of polynomial pseudo division *}
chaieb@33154
  1321
chaieb@33154
  1322
lemma polydivide_aux_real_domintros:
chaieb@33154
  1323
  assumes call1: "\<lbrakk>s \<noteq> 0\<^sub>p; \<not> degree s < n; a = head s\<rbrakk> 
chaieb@33154
  1324
  \<Longrightarrow> polydivide_aux_dom (a, n, p, k, s -\<^sub>p funpow (degree s - n) shift1 p)"
chaieb@33154
  1325
  and call2 : "\<lbrakk>s \<noteq> 0\<^sub>p; \<not> degree s < n; a \<noteq> head s\<rbrakk> 
chaieb@33154
  1326
  \<Longrightarrow> polydivide_aux_dom(a, n, p,Suc k, a *\<^sub>p s -\<^sub>p (head s *\<^sub>p funpow (degree s - n) shift1 p))"
chaieb@33154
  1327
  shows "polydivide_aux_dom (a, n, p, k, s)"
chaieb@33154
  1328
proof (rule accpI, erule polydivide_aux_rel.cases)
chaieb@33154
  1329
  fix y aa ka na pa sa x xa xb
chaieb@33154
  1330
  assume eqs: "y = (aa, na, pa, ka, sa -\<^sub>p xb)" "(a, n, p, k, s) = (aa, na, pa, ka, sa)"
chaieb@33154
  1331
     and \<Gamma>1': "sa \<noteq> 0\<^sub>p" "x = head sa" "xa = degree sa" "\<not> xa < na" 
chaieb@33154
  1332
    "xb = funpow (xa - na) shift1 pa" "aa = x"
chaieb@33154
  1333
chaieb@33154
  1334
  hence \<Gamma>1: "s \<noteq> 0\<^sub>p" "a = head s" "xa = degree s" "\<not> degree s < n" "\<not> xa < na" 
chaieb@33154
  1335
    "xb = funpow (xa - na) shift1 pa" "aa = x" by auto
chaieb@33154
  1336
chaieb@33154
  1337
  with call1 have "polydivide_aux_dom (a, n, p, k, s -\<^sub>p funpow (degree s - n) shift1 p)"
chaieb@33154
  1338
    by auto
chaieb@33154
  1339
  with eqs \<Gamma>1 show "polydivide_aux_dom y" by auto
chaieb@33154
  1340
next
chaieb@33154
  1341
  fix y aa ka na pa sa x xa xb
chaieb@33154
  1342
  assume eqs: "y = (aa, na, pa, Suc ka, aa *\<^sub>p sa -\<^sub>p (x *\<^sub>p xb))" 
chaieb@33154
  1343
    "(a, n, p, k, s) =(aa, na, pa, ka, sa)"
chaieb@33154
  1344
    and \<Gamma>2': "sa \<noteq> 0\<^sub>p" "x = head sa" "xa = degree sa" "\<not> xa < na"
chaieb@33154
  1345
    "xb = funpow (xa - na) shift1 pa" "aa \<noteq> x"
chaieb@33154
  1346
  hence \<Gamma>2: "s \<noteq> 0\<^sub>p" "a \<noteq> head s" "xa = degree s" "\<not> degree s < n"
chaieb@33154
  1347
    "xb = funpow (xa - na) shift1 pa" "aa \<noteq> x" by auto
chaieb@33154
  1348
  with call2 have "polydivide_aux_dom (a, n, p, Suc k, a *\<^sub>p s -\<^sub>p (head s *\<^sub>p funpow (degree s - n) shift1 p))" by auto
chaieb@33154
  1349
  with eqs \<Gamma>2'  show "polydivide_aux_dom y" by auto
chaieb@33154
  1350
qed
chaieb@33154
  1351
chaieb@33154
  1352
lemma polydivide_aux_properties:
chaieb@33154
  1353
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
  1354
  and np: "isnpolyh p n0" and ns: "isnpolyh s n1"
chaieb@33154
  1355
  and ap: "head p = a" and ndp: "degree p = n" and pnz: "p \<noteq> 0\<^sub>p"
chaieb@33154
  1356
  shows "polydivide_aux_dom (a,n,p,k,s) \<and> 
chaieb@33154
  1357
  (polydivide_aux (a,n,p,k,s) = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p) 
chaieb@33154
  1358
          \<and> (\<exists>nr. isnpolyh r nr) \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow (k' - k) a) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
chaieb@33154
  1359
  using ns
chaieb@33154
  1360
proof(induct d\<equiv>"degree s" arbitrary: s k k' r n1 rule: nat_less_induct)
chaieb@33154
  1361
  fix d s k k' r n1
chaieb@33154
  1362
  let ?D = "polydivide_aux_dom"
chaieb@33154
  1363
  let ?dths = "?D (a, n, p, k, s)"
chaieb@33154
  1364
  let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
chaieb@33154
  1365
  let ?qrths = "polydivide_aux (a, n, p, k, s) = (k', r) \<longrightarrow>  k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) 
chaieb@33154
  1366
    \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
chaieb@33154
  1367
  let ?ths = "?dths \<and> ?qrths"
chaieb@33154
  1368
  let ?b = "head s"
chaieb@33154
  1369
  let ?p' = "funpow (d - n) shift1 p"
chaieb@33154
  1370
  let ?xdn = "funpow (d - n) shift1 1\<^sub>p"
chaieb@33154
  1371
  let ?akk' = "a ^\<^sub>p (k' - k)"
chaieb@33154
  1372
  assume H: "\<forall>m<d. \<forall>x xa xb xc xd.
chaieb@33154
  1373
    isnpolyh x xd \<longrightarrow>
chaieb@33154
  1374
    m = degree x \<longrightarrow> ?D (a, n, p, xa, x) \<and>
chaieb@33154
  1375
    (polydivide_aux (a, n, p, xa, x) = (xb, xc) \<longrightarrow>
chaieb@33154
  1376
    xa \<le> xb \<and> (degree xc = 0 \<or> degree xc < degree p) \<and> 
chaieb@33154
  1377
    (\<exists> nr. isnpolyh xc nr) \<and>
chaieb@33154
  1378
    (\<exists>q n1. isnpolyh q n1 \<and> a ^\<^sub>p xb - xa *\<^sub>p x = p *\<^sub>p q +\<^sub>p xc))"
chaieb@33154
  1379
    and ns: "isnpolyh s n1" and ds: "d = degree s"
chaieb@33154
  1380
  from np have np0: "isnpolyh p 0" 
chaieb@33154
  1381
    using isnpolyh_mono[where n="n0" and n'="0" and p="p"]  by simp
chaieb@33154
  1382
  have np': "isnpolyh ?p' 0" using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="d -n"] isnpoly_def by simp
chaieb@33154
  1383
  have headp': "head ?p' = head p" using funpow_shift1_head[OF np pnz] by simp
chaieb@33154
  1384
  from funpow_shift1_isnpoly[where p="1\<^sub>p"] have nxdn: "isnpolyh ?xdn 0" by (simp add: isnpoly_def)
chaieb@33154
  1385
  from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap 
chaieb@33154
  1386
  have nakk':"isnpolyh ?akk' 0" by blast
chaieb@33154
  1387
  {assume sz: "s = 0\<^sub>p"
chaieb@33154
  1388
    hence dom: ?dths apply - apply (rule polydivide_aux_real_domintros) apply simp_all done
chaieb@33154
  1389
    from polydivide_aux.psimps[OF dom] sz
chaieb@33154
  1390
    have ?qrths using np apply clarsimp by (rule exI[where x="0\<^sub>p"], simp)
chaieb@33154
  1391
    hence ?ths using dom by blast}
chaieb@33154
  1392
  moreover
chaieb@33154
  1393
  {assume sz: "s \<noteq> 0\<^sub>p"
chaieb@33154
  1394
    {assume dn: "d < n"
chaieb@33154
  1395
      with sz ds  have dom:"?dths" by - (rule polydivide_aux_real_domintros,simp_all) 
chaieb@33154
  1396
      from polydivide_aux.psimps[OF dom] sz dn ds
chaieb@33154
  1397
      have "?qrths" using ns ndp np by auto (rule exI[where x="0\<^sub>p"],simp)
chaieb@33154
  1398
      with dom have ?ths by blast}
chaieb@33154
  1399
    moreover 
chaieb@33154
  1400
    {assume dn': "\<not> d < n" hence dn: "d \<ge> n" by arith
chaieb@33154
  1401
      have degsp': "degree s = degree ?p'" 
chaieb@33154
  1402
	using ds dn ndp funpow_shift1_degree[where k = "d - n" and p="p"] by simp
chaieb@33154
  1403
      {assume ba: "?b = a"
chaieb@33154
  1404
	hence headsp': "head s = head ?p'" using ap headp' by simp
chaieb@33154
  1405
	have nr: "isnpolyh (s -\<^sub>p ?p') 0" using polysub_normh[OF ns np'] by simp
chaieb@33154
  1406
	from ds degree_polysub_samehead[OF ns np' headsp' degsp']
chaieb@33154
  1407
	have "degree (s -\<^sub>p ?p') < d \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
chaieb@33154
  1408
	moreover 
chaieb@33154
  1409
	{assume deglt:"degree (s -\<^sub>p ?p') < d"
chaieb@33154
  1410
	  from  H[rule_format, OF deglt nr,simplified] 
chaieb@33154
  1411
	  have domsp: "?D (a, n, p, k, s -\<^sub>p ?p')" by blast 
chaieb@33154
  1412
	  have dom: ?dths apply (rule polydivide_aux_real_domintros) 
chaieb@33154
  1413
	    using ba ds dn' domsp by simp_all
chaieb@33154
  1414
	  from polydivide_aux.psimps[OF dom] sz dn' ba ds
chaieb@33154
  1415
	  have eq: "polydivide_aux (a,n,p,k,s) = polydivide_aux (a,n,p,k, s -\<^sub>p ?p')"
chaieb@33154
  1416
	    by (simp add: Let_def)
chaieb@33154
  1417
	  {assume h1: "polydivide_aux (a, n, p, k, s) = (k', r)"
chaieb@33154
  1418
	    from H[rule_format, OF deglt nr, where xa = "k" and xb="k'" and xc="r", simplified]
chaieb@33154
  1419
	      trans[OF eq[symmetric] h1]
chaieb@33154
  1420
	    have kk': "k \<le> k'" and nr:"\<exists>nr. isnpolyh r nr" and dr: "degree r = 0 \<or> degree r < degree p"
chaieb@33154
  1421
	      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)" by auto
chaieb@33154
  1422
	    from q1 obtain q n1 where nq: "isnpolyh q n1" 
chaieb@33154
  1423
	      and asp:"a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r"  by blast
chaieb@33154
  1424
	    from nr obtain nr where nr': "isnpolyh r nr" by blast
chaieb@33154
  1425
	    from polymul_normh[OF nakk' ns] have nakks': "isnpolyh (a ^\<^sub>p (k' - k) *\<^sub>p s) 0" by simp
chaieb@33154
  1426
	    from polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]
chaieb@33154
  1427
	    have nq': "isnpolyh (?akk' *\<^sub>p ?xdn +\<^sub>p q) 0" by simp
chaieb@33154
  1428
	    from polyadd_normh[OF polymul_normh[OF np 
chaieb@33154
  1429
	      polyadd_normh[OF polymul_normh[OF nakk' nxdn] nq]] nr']
chaieb@33154
  1430
	    have nqr': "isnpolyh (p *\<^sub>p (?akk' *\<^sub>p ?xdn +\<^sub>p q) +\<^sub>p r) 0" by simp 
chaieb@33154
  1431
	    from asp have "\<forall> (bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a^\<^sub>p (k' - k) *\<^sub>p (s -\<^sub>p ?p')) = 
chaieb@33154
  1432
	      Ipoly bs (p *\<^sub>p q +\<^sub>p r)" by simp
chaieb@33154
  1433
	    hence " \<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a^\<^sub>p (k' - k)*\<^sub>p s) = 
chaieb@33154
  1434
	      Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r" 
chaieb@33154
  1435
	      by (simp add: ring_simps)
chaieb@33154
  1436
	    hence " \<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 
chaieb@33154
  1437
	      Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (d - n) shift1 1\<^sub>p *\<^sub>p p) 
chaieb@33154
  1438
	      + Ipoly bs p * Ipoly bs q + Ipoly bs r"
chaieb@33154
  1439
	      by (auto simp only: funpow_shift1_1) 
chaieb@33154
  1440
	    hence "\<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 
chaieb@33154
  1441
	      Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (d - n) shift1 1\<^sub>p) 
chaieb@33154
  1442
	      + Ipoly bs q) + Ipoly bs r" by (simp add: ring_simps)
chaieb@33154
  1443
	    hence "\<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 
chaieb@33154
  1444
	      Ipoly bs (p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (d - n) shift1 1\<^sub>p) +\<^sub>p q) +\<^sub>p r)" by simp
chaieb@33154
  1445
	    with isnpolyh_unique[OF nakks' nqr']
chaieb@33154
  1446
	    have "a ^\<^sub>p (k' - k) *\<^sub>p s = 
chaieb@33154
  1447
	      p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (d - n) shift1 1\<^sub>p) +\<^sub>p q) +\<^sub>p r" by blast
chaieb@33154
  1448
	    hence ?qths using nq'
chaieb@33154
  1449
	      apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (d - n) shift1 1\<^sub>p) +\<^sub>p q" in exI)
chaieb@33154
  1450
	      apply (rule_tac x="0" in exI) by simp
chaieb@33154
  1451
	    with kk' nr dr have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
chaieb@33154
  1452
	      by blast } hence ?qrths by blast
chaieb@33154
  1453
	  with dom have ?ths by blast} 
chaieb@33154
  1454
	moreover 
chaieb@33154
  1455
	{assume spz:"s -\<^sub>p ?p' = 0\<^sub>p"
chaieb@33154
  1456
	  hence domsp: "?D (a, n, p, k, s -\<^sub>p ?p')" 
chaieb@33154
  1457
	    apply (simp) by (rule polydivide_aux_real_domintros, simp_all)
chaieb@33154
  1458
	  have dom: ?dths apply (rule polydivide_aux_real_domintros) 
chaieb@33154
  1459
	    using ba ds dn' domsp by simp_all
chaieb@33154
  1460
	  from spz isnpolyh_unique[OF polysub_normh[OF ns np'], where q="0\<^sub>p", symmetric, where ?'a = "'a::{ring_char_0,division_by_zero,field}"]
chaieb@33154
  1461
	  have " \<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs s = Ipoly bs ?p'" by simp
chaieb@33154
  1462
	  hence "\<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)" using np nxdn apply simp
chaieb@33154
  1463
	    by (simp only: funpow_shift1_1) simp
chaieb@33154
  1464
	  hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]] by blast
chaieb@33154
  1465
	  {assume h1: "polydivide_aux (a,n,p,k,s) = (k',r)"
chaieb@33154
  1466
	    from polydivide_aux.psimps[OF dom] sz dn' ba ds
chaieb@33154
  1467
	    have eq: "polydivide_aux (a,n,p,k,s) = polydivide_aux (a,n,p,k, s -\<^sub>p ?p')"
chaieb@33154
  1468
	      by (simp add: Let_def)
chaieb@33154
  1469
	    also have "\<dots> = (k,0\<^sub>p)" using polydivide_aux.psimps[OF domsp] spz by simp
chaieb@33154
  1470
	    finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
chaieb@33154
  1471
	    with sp' ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
chaieb@33154
  1472
	      polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?qrths
chaieb@33154
  1473
	      apply auto
chaieb@33154
  1474
	      apply (rule exI[where x="?xdn"]) 	      
chaieb@33154
  1475
	      apply auto
chaieb@33154
  1476
	      apply (rule polymul_commute)
chaieb@33154
  1477
	      apply simp_all
chaieb@33154
  1478
	      done}
chaieb@33154
  1479
	  with dom have ?ths by blast}
chaieb@33154
  1480
	ultimately have ?ths by blast }
chaieb@33154
  1481
      moreover
chaieb@33154
  1482
      {assume ba: "?b \<noteq> a"
chaieb@33154
  1483
	from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns] 
chaieb@33154
  1484
	  polymul_normh[OF head_isnpolyh[OF ns] np']]
chaieb@33154
  1485
	have nth: "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by(simp add: min_def)
chaieb@33154
  1486
	have nzths: "a *\<^sub>p s \<noteq> 0\<^sub>p" "?b *\<^sub>p ?p' \<noteq> 0\<^sub>p"
chaieb@33154
  1487
	  using polymul_eq0_iff[OF head_isnpolyh[OF np0, simplified ap] ns] 
chaieb@33154
  1488
	    polymul_eq0_iff[OF head_isnpolyh[OF ns] np']head_nz[OF np0] ap pnz sz head_nz[OF ns]
chaieb@33154
  1489
	    funpow_shift1_nz[OF pnz] by simp_all
chaieb@33154
  1490
	from polymul_head_polyeq[OF head_isnpolyh[OF np] ns] head_nz[OF np] sz ap head_head[OF np] pnz
chaieb@33154
  1491
	  polymul_head_polyeq[OF head_isnpolyh[OF ns] np'] head_nz [OF ns] sz funpow_shift1_nz[OF pnz, where n="d - n"]
chaieb@33154
  1492
	have hdth: "head (a *\<^sub>p s) = head (?b *\<^sub>p ?p')" 
chaieb@33154
  1493
	  using head_head[OF ns] funpow_shift1_head[OF np pnz]
chaieb@33154
  1494
	    polymul_commute[OF head_isnpolyh[OF np] head_isnpolyh[OF ns]]
chaieb@33154
  1495
	  by (simp add: ap)
chaieb@33154
  1496
	from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
chaieb@33154
  1497
	  head_nz[OF np] pnz sz ap[symmetric]
chaieb@33154
  1498
	  funpow_shift1_nz[OF pnz, where n="d - n"]
chaieb@33154
  1499
	  polymul_degreen[OF head_isnpolyh[OF ns] np', where m="0"]  head_nz[OF ns]
chaieb@33154
  1500
	  ndp ds[symmetric] dn
chaieb@33154
  1501
	have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p') "
chaieb@33154
  1502
	  by (simp add: degree_eq_degreen0[symmetric] funpow_shift1_degree)
chaieb@33154
  1503
	{assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < d"
chaieb@33154
  1504
	  have th: "?D (a, n, p, Suc k, (a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p'))"
chaieb@33154
  1505
	    using H[rule_format, OF dth nth, simplified] by blast 
chaieb@33154
  1506
	  have dom: ?dths
chaieb@33154
  1507
	    using ba ds dn' th apply simp apply (rule polydivide_aux_real_domintros)  
chaieb@33154
  1508
	    using ba ds dn'  by simp_all
chaieb@33154
  1509
	  from polysub_normh[OF polymul_normh[OF head_isnpolyh[OF np] ns] polymul_normh[OF head_isnpolyh[OF ns]np']]
chaieb@33154
  1510
	  ap have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by simp
chaieb@33154
  1511
	  {assume h1:"polydivide_aux (a,n,p,k,s) = (k', r)"
chaieb@33154
  1512
	    from h1  polydivide_aux.psimps[OF dom] sz dn' ba ds
chaieb@33154
  1513
	    have eq:"polydivide_aux (a,n,p,Suc k,(a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
chaieb@33154
  1514
	      by (simp add: Let_def)
chaieb@33154
  1515
	    with H[rule_format, OF dth nasbp', simplified, where xa="Suc k" and xb="k'" and xc="r"]
chaieb@33154
  1516
	    obtain q nq nr where kk': "Suc k \<le> k'" and nr: "isnpolyh r nr" and nq: "isnpolyh q nq" 
chaieb@33154
  1517
	      and dr: "degree r = 0 \<or> degree r < degree p"
chaieb@33154
  1518
	      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" by auto
chaieb@33154
  1519
	    from kk' have kk'':"Suc (k' - Suc k) = k' - k" by arith
chaieb@33154
  1520
	    {fix bs:: "'a::{ring_char_0,division_by_zero,field} list"
chaieb@33154
  1521
	      
chaieb@33154
  1522
	    from qr isnpolyh_unique[OF polypow_normh[OF head_isnpolyh[OF np], where k="k' - Suc k", simplified ap] nasbp', symmetric]
chaieb@33154
  1523
	    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)" by simp
chaieb@33154
  1524
	    hence "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s = Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
chaieb@33154
  1525
	      by (simp add: ring_simps power_Suc)
chaieb@33154
  1526
	    hence "Ipoly bs a ^ (k' - k)  * Ipoly bs s = Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
chaieb@33154
  1527
	      by (simp add:kk'' funpow_shift1_1[where n="d - n" and p="p"])
chaieb@33154
  1528
	    hence "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
chaieb@33154
  1529
	      by (simp add: ring_simps)}
chaieb@33154
  1530
	    hence ieq:"\<forall>(bs :: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = 
chaieb@33154
  1531
	      Ipoly bs (p *\<^sub>p (q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)) +\<^sub>p r)" by auto 
chaieb@33154
  1532
	    let ?q = "q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn)"
chaieb@33154
  1533
	    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]]
chaieb@33154
  1534
	    have nqw: "isnpolyh ?q 0" by simp
chaieb@33154
  1535
	    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]]
chaieb@33154
  1536
	    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" by blast
chaieb@33154
  1537
	    from dr kk' nr h1 asth nqw have ?qrths apply simp
chaieb@33154
  1538
	      apply (rule conjI)
chaieb@33154
  1539
	      apply (rule exI[where x="nr"], simp)
chaieb@33154
  1540
	      apply (rule exI[where x="(q +\<^sub>p (a ^\<^sub>p (k' - Suc k) *\<^sub>p ?b *\<^sub>p ?xdn))"], simp)
chaieb@33154
  1541
	      apply (rule exI[where x="0"], simp)
chaieb@33154
  1542
	      done}
chaieb@33154
  1543
	  hence ?qrths by blast
chaieb@33154
  1544
	  with dom have ?ths by blast}
chaieb@33154
  1545
	moreover 
chaieb@33154
  1546
	{assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
chaieb@33154
  1547
	  hence domsp: "?D (a, n, p, Suc k, a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p'))" 
chaieb@33154
  1548
	    apply (simp) by (rule polydivide_aux_real_domintros, simp_all)
chaieb@33154
  1549
	  have dom: ?dths using sz ba dn' ds domsp 
chaieb@33154
  1550
	    by - (rule polydivide_aux_real_domintros, simp_all)
chaieb@33154
  1551
	  {fix bs :: "'a::{ring_char_0,division_by_zero,field} list"
chaieb@33154
  1552
	    from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
chaieb@33154
  1553
	  have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'" by simp
chaieb@33154
  1554
	  hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p" 
chaieb@33154
  1555
	    by (simp add: funpow_shift1_1[where n="d - n" and p="p"])
chaieb@33154
  1556
	  hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" by simp
chaieb@33154
  1557
	}
chaieb@33154
  1558
	hence hth: "\<forall> (bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
chaieb@33154
  1559
	  from hth
chaieb@33154
  1560
	  have asq: "a *\<^sub>p s = p *\<^sub>p (?b *\<^sub>p ?xdn)" 
chaieb@33154
  1561
	    using isnpolyh_unique[where ?'a = "'a::{ring_char_0,division_by_zero,field}", OF polymul_normh[OF head_isnpolyh[OF np] ns] 
chaieb@33154
  1562
                    polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
chaieb@33154
  1563
	      simplified ap] by simp
chaieb@33154
  1564
	  {assume h1: "polydivide_aux (a,n,p,k,s) = (k', r)"
chaieb@33154
  1565
	  from h1 sz ds ba dn' spz polydivide_aux.psimps[OF dom] polydivide_aux.psimps[OF domsp] 
chaieb@33154
  1566
	  have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
chaieb@33154
  1567
	  with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
chaieb@33154
  1568
	    polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
chaieb@33154
  1569
	  have ?qrths apply (clarsimp simp add: Let_def)
chaieb@33154
  1570
	    apply (rule exI[where x="?b *\<^sub>p ?xdn"]) apply simp
chaieb@33154
  1571
	    apply (rule exI[where x="0"], simp)
chaieb@33154
  1572
	    done}
chaieb@33154
  1573
	hence ?qrths by blast
chaieb@33154
  1574
	with dom have ?ths by blast}
chaieb@33154
  1575
	ultimately have ?ths using  degree_polysub_samehead[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns] polymul_normh[OF head_isnpolyh[OF ns] np'] hdth degth] polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
chaieb@33154
  1576
	  head_nz[OF np] pnz sz ap[symmetric] ds[symmetric] 
chaieb@33154
  1577
	  by (simp add: degree_eq_degreen0[symmetric]) blast }
chaieb@33154
  1578
      ultimately have ?ths by blast
chaieb@33154
  1579
    }
chaieb@33154
  1580
    ultimately have ?ths by blast}
chaieb@33154
  1581
  ultimately show ?ths by blast
chaieb@33154
  1582
qed
chaieb@33154
  1583
chaieb@33154
  1584
lemma polydivide_properties: 
chaieb@33154
  1585
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
  1586
  and np: "isnpolyh p n0" and ns: "isnpolyh s n1" and pnz: "p \<noteq> 0\<^sub>p"
chaieb@33154
  1587
  shows "(\<exists> k r. polydivide s p = (k,r) \<and> (\<exists>nr. isnpolyh r nr) \<and> (degree r = 0 \<or> degree r < degree p) 
chaieb@33154
  1588
  \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow k (head p)) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
chaieb@33154
  1589
proof-
chaieb@33154
  1590
  have trv: "head p = head p" "degree p = degree p" by simp_all
chaieb@33154
  1591
  from polydivide_aux_properties[OF np ns trv pnz, where k="0"] 
chaieb@33154
  1592
  have d: "polydivide_aux_dom (head p, degree p, p, 0, s)" by blast
chaieb@33154
  1593
  from polydivide_def[where s="s" and p="p"] polydivide_aux.psimps[OF d]
chaieb@33154
  1594
  have ex: "\<exists> k r. polydivide s p = (k,r)" by auto
chaieb@33154
  1595
  then obtain k r where kr: "polydivide s p = (k,r)" by blast
chaieb@33154
  1596
  from trans[OF meta_eq_to_obj_eq[OF polydivide_def[where s="s" and p="p"], symmetric] kr]
chaieb@33154
  1597
    polydivide_aux_properties[OF np ns trv pnz, where k="0" and k'="k" and r="r"]
chaieb@33154
  1598
  have "(degree r = 0 \<or> degree r < degree p) \<and>
chaieb@33154
  1599
   (\<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)" by blast
chaieb@33154
  1600
  with kr show ?thesis 
chaieb@33154
  1601
    apply -
chaieb@33154
  1602
    apply (rule exI[where x="k"])
chaieb@33154
  1603
    apply (rule exI[where x="r"])
chaieb@33154
  1604
    apply simp
chaieb@33154
  1605
    done
chaieb@33154
  1606
qed
chaieb@33154
  1607
chaieb@33154
  1608
subsection{* More about polypoly and pnormal etc *}
chaieb@33154
  1609
chaieb@33154
  1610
definition "isnonconstant p = (\<not> isconstant p)"
chaieb@33154
  1611
chaieb@33154
  1612
lemma last_map: "xs \<noteq> [] ==> last (map f xs) = f (last xs)" by (induct xs, auto)
chaieb@33154
  1613
chaieb@33154
  1614
lemma isnonconstant_pnormal_iff: assumes nc: "isnonconstant p" 
chaieb@33154
  1615
  shows "pnormal (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0" 
chaieb@33154
  1616
proof
chaieb@33154
  1617
  let ?p = "polypoly bs p"  
chaieb@33154
  1618
  assume H: "pnormal ?p"
chaieb@33154
  1619
  have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
chaieb@33154
  1620
  
chaieb@33154
  1621
  from coefficients_head[of p] last_map[OF csz, of "Ipoly bs"]  
chaieb@33154
  1622
    pnormal_last_nonzero[OF H]
chaieb@33154
  1623
  show "Ipoly bs (head p) \<noteq> 0" by (simp add: polypoly_def)
chaieb@33154
  1624
next
chaieb@33154
  1625
  assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  1626
  let ?p = "polypoly bs p"
chaieb@33154
  1627
  have csz: "coefficients p \<noteq> []" using nc by (cases p, auto)
chaieb@33154
  1628
  hence pz: "?p \<noteq> []" by (simp add: polypoly_def) 
chaieb@33154
  1629
  hence lg: "length ?p > 0" by simp
chaieb@33154
  1630
  from h coefficients_head[of p] last_map[OF csz, of "Ipoly bs"] 
chaieb@33154
  1631
  have lz: "last ?p \<noteq> 0" by (simp add: polypoly_def)
chaieb@33154
  1632
  from pnormal_last_length[OF lg lz] show "pnormal ?p" .
chaieb@33154
  1633
qed
chaieb@33154
  1634
chaieb@33154
  1635
lemma isnonconstant_coefficients_length: "isnonconstant p \<Longrightarrow> length (coefficients p) > 1"
chaieb@33154
  1636
  unfolding isnonconstant_def
chaieb@33154
  1637
  apply (cases p, simp_all)
chaieb@33154
  1638
  apply (case_tac nat, auto)
chaieb@33154
  1639
  done
chaieb@33154
  1640
lemma isnonconstant_nonconstant: assumes inc: "isnonconstant p"
chaieb@33154
  1641
  shows "nonconstant (polypoly bs p) \<longleftrightarrow> Ipoly bs (head p) \<noteq> 0"
chaieb@33154
  1642
proof
chaieb@33154
  1643
  let ?p = "polypoly bs p"
chaieb@33154
  1644
  assume nc: "nonconstant ?p"
chaieb@33154
  1645
  from isnonconstant_pnormal_iff[OF inc, of bs] nc
chaieb@33154
  1646
  show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" unfolding nonconstant_def by blast
chaieb@33154
  1647
next
chaieb@33154
  1648
  let ?p = "polypoly bs p"
chaieb@33154
  1649
  assume h: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  1650
  from isnonconstant_pnormal_iff[OF inc, of bs] h
chaieb@33154
  1651
  have pn: "pnormal ?p" by blast
chaieb@33154
  1652
  {fix x assume H: "?p = [x]" 
chaieb@33154
  1653
    from H have "length (coefficients p) = 1" unfolding polypoly_def by auto
chaieb@33154
  1654
    with isnonconstant_coefficients_length[OF inc] have False by arith}
chaieb@33154
  1655
  thus "nonconstant ?p" using pn unfolding nonconstant_def by blast  
chaieb@33154
  1656
qed
chaieb@33154
  1657
chaieb@33154
  1658
lemma pnormal_length: "p\<noteq>[] \<Longrightarrow> pnormal p \<longleftrightarrow> length (pnormalize p) = length p"
chaieb@33154
  1659
  unfolding pnormal_def
chaieb@33154
  1660
 apply (induct p rule: pnormalize.induct, simp_all)
chaieb@33154
  1661
 apply (case_tac "p=[]", simp_all)
chaieb@33154
  1662
 done
chaieb@33154
  1663
chaieb@33154
  1664
lemma degree_degree: assumes inc: "isnonconstant p"
chaieb@33154
  1665
  shows "degree p = Polynomial_List.degree (polypoly bs p) \<longleftrightarrow> \<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  1666
proof
chaieb@33154
  1667
  let  ?p = "polypoly bs p"
chaieb@33154
  1668
  assume H: "degree p = Polynomial_List.degree ?p"
chaieb@33154
  1669
  from isnonconstant_coefficients_length[OF inc] have pz: "?p \<noteq> []"
chaieb@33154
  1670
    unfolding polypoly_def by auto
chaieb@33154
  1671
  from H degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
chaieb@33154
  1672
  have lg:"length (pnormalize ?p) = length ?p"
chaieb@33154
  1673
    unfolding Polynomial_List.degree_def polypoly_def by simp
chaieb@33154
  1674
  hence "pnormal ?p" using pnormal_length[OF pz] by blast 
chaieb@33154
  1675
  with isnonconstant_pnormal_iff[OF inc]  
chaieb@33154
  1676
  show "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0" by blast
chaieb@33154
  1677
next
chaieb@33154
  1678
  let  ?p = "polypoly bs p"  
chaieb@33154
  1679
  assume H: "\<lparr>head p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<noteq> 0"
chaieb@33154
  1680
  with isnonconstant_pnormal_iff[OF inc] have "pnormal ?p" by blast
chaieb@33154
  1681
  with degree_coefficients[of p] isnonconstant_coefficients_length[OF inc]
chaieb@33154
  1682
  show "degree p = Polynomial_List.degree ?p" 
chaieb@33154
  1683
    unfolding polypoly_def pnormal_def Polynomial_List.degree_def by auto
chaieb@33154
  1684
qed
chaieb@33154
  1685
chaieb@33154
  1686
section{* Swaps ; Division by a certain variable *}
chaieb@33154
  1687
consts swap:: "nat \<Rightarrow> nat \<Rightarrow> poly \<Rightarrow> poly"
chaieb@33154
  1688
primrec
chaieb@33154
  1689
  "swap n m (C x) = C x"
chaieb@33154
  1690
  "swap n m (Bound k) = Bound (if k = n then m else if k=m then n else k)"
chaieb@33154
  1691
  "swap n m (Neg t) = Neg (swap n m t)"
chaieb@33154
  1692
  "swap n m (Add s t) = Add (swap n m s) (swap n m t)"
chaieb@33154
  1693
  "swap n m (Sub s t) = Sub (swap n m s) (swap n m t)"
chaieb@33154
  1694
  "swap n m (Mul s t) = Mul (swap n m s) (swap n m t)"
chaieb@33154
  1695
  "swap n m (Pw t k) = Pw (swap n m t) k"
chaieb@33154
  1696
  "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)
chaieb@33154
  1697
  (swap n m p)"
chaieb@33154
  1698
chaieb@33154
  1699
lemma swap: assumes nbs: "n < length bs" and mbs: "m < length bs"
chaieb@33154
  1700
  shows "Ipoly bs (swap n m t) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
chaieb@33154
  1701
proof (induct t)
chaieb@33154
  1702
  case (Bound k) thus ?case using nbs mbs by simp 
chaieb@33154
  1703
next
chaieb@33154
  1704
  case (CN c k p) thus ?case using nbs mbs by simp 
chaieb@33154
  1705
qed simp_all
chaieb@33154
  1706
lemma swap_swap_id[simp]: "swap n m (swap m n t) = t"
chaieb@33154
  1707
  by (induct t,simp_all)
chaieb@33154
  1708
chaieb@33154
  1709
lemma swap_commute: "swap n m p = swap m n p" by (induct p, simp_all)
chaieb@33154
  1710
chaieb@33154
  1711
lemma swap_same_id[simp]: "swap n n t = t"
chaieb@33154
  1712
  by (induct t, simp_all)
chaieb@33154
  1713
chaieb@33154
  1714
definition "swapnorm n m t = polynate (swap n m t)"
chaieb@33154
  1715
chaieb@33154
  1716
lemma swapnorm: assumes nbs: "n < length bs" and mbs: "m < length bs"
chaieb@33154
  1717
  shows "((Ipoly bs (swapnorm n m t) :: 'a\<Colon>{ring_char_0,division_by_zero,field})) = Ipoly ((bs[n:= bs!m])[m:= bs!n]) t"
chaieb@33154
  1718
  using swap[OF prems] swapnorm_def by simp
chaieb@33154
  1719
chaieb@33154
  1720
lemma swapnorm_isnpoly[simp]: 
chaieb@33154
  1721
    assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
chaieb@33154
  1722
  shows "isnpoly (swapnorm n m p)"
chaieb@33154
  1723
  unfolding swapnorm_def by simp
chaieb@33154
  1724
chaieb@33154
  1725
definition "polydivideby n s p = 
chaieb@33154
  1726
    (let ss = swapnorm 0 n s ; sp = swapnorm 0 n p ; h = head sp; (k,r) = polydivide ss sp
chaieb@33154
  1727
     in (k,swapnorm 0 n h,swapnorm 0 n r))"
chaieb@33154
  1728
chaieb@33154
  1729
lemma swap_nz [simp]: " (swap n m p = 0\<^sub>p) = (p = 0\<^sub>p)" by (induct p, simp_all)
chaieb@33154
  1730
chaieb@33154
  1731
consts isweaknpoly :: "poly \<Rightarrow> bool"
chaieb@33154
  1732
recdef isweaknpoly "measure size"
chaieb@33154
  1733
  "isweaknpoly (C c) = True"
chaieb@33154
  1734
  "isweaknpoly (CN c n p) \<longleftrightarrow> isweaknpoly c \<and> isweaknpoly p"
chaieb@33154
  1735
  "isweaknpoly p = False"	
chaieb@33154
  1736
chaieb@33154
  1737
lemma isnpolyh_isweaknpoly: "isnpolyh p n0 \<Longrightarrow> isweaknpoly p" 
chaieb@33154
  1738
  by (induct p arbitrary: n0, auto)
chaieb@33154
  1739
chaieb@33154
  1740
lemma swap_isweanpoly: "isweaknpoly p \<Longrightarrow> isweaknpoly (swap n m p)" 
chaieb@33154
  1741
  by (induct p, auto)
chaieb@33154
  1742
chaieb@33154
  1743
end